Proceedings SOR th Proceedings of the 11 International Symposium on OPERATIONAL RESEARCH Rupnik V. and L. Bogataj (Editors): The 1st Symposium on Operational Research, SOR'93. Proceedings. Ljubljana: Slovenian Society Informatika, Section for Operational Research, 1993, 310 pp. SOR '11 Rupnik V. and M. Bogataj (Editors): The 2nd International Symposium on Operational Research in Slovenia, SOR'94. Proceedings. Ljubljana: Slovenian Society Informatika, Section for Operational Research, 1994, 275 pp. Rupnik V., L. Zadnik Stirn and S. Drobne (Editors.): The 4th International Symposium on Operational Research in Slovenia, SOR'97. Proceedings. Ljubljana: Slovenian Society Informatika, Section for Operational Research, 1997, 366 pp. ISBN 961-616505-4. Rupnik V., L. Zadnik Stirn and S. Drobne (Editors.): The 5th International Symposium on Operational Research SOR '99, Proceedings. Ljubljana: Slovenian Society Informatika, Section for Operational Research, 1999, 300 pp. ISBN 961-6165-08-9. Lenart L., L. Zadnik Stirn and S. Drobne (Editors.): The 6th International Symposium on Operational Research SOR '01, Proceedings. Ljubljana: Slovenian Society Informatika, Section for Operational Research, 2001, 403 pp. ISBN 961-6165-12-7. Zadnik Stirn L., M. Bastiè and S. Drobne (Editors): The 7th International Symposium on Operational Research SOR’03, Proceedings. Ljubljana: Slovenian Society Informatika, Section for Operational Research, 2003, 424 pp. ISBN 961-6165-15-1. Zadnik Stirn L. and S. Drobne (Editors): The 8th International Symposium on Operational Research SOR’05, Proceedings. Ljubljana: Slovenian Society Informatika, Section for Operational Research, 2005, 426 pp. ISBN 961-6165-20-8. Zadnik Stirn L. and S. Drobne (Editors): The 9th International Symposium on Operational Research SOR’07, Proceedings. Ljubljana: Slovenian Society Informatika, Section for Operational Research, 2007, 460 pp. ISBN 978-961-6165-25-9. Proceedings SOR'11 Rupnik V. and M. Bogataj (Editors): The 3rd International Symposium on Operational Research in Slovenia, SOR'95. Proceedings. Ljubljana: Slovenian Society Informatika, Section for Operational Research, 1995, 175 pp. Dolenjske Toplice, Slovenia September 28-30, 2011 Zadnik Stirn L., J. Žerovnik, S. Drobne and A. Lisec (Editors): The 10th International Symposium on Operational Research SOR’09, Proceedings. Ljubljana: Slovenian Society Informatika, Section for Operational Research, 2009, 604 pp. ISBN 978-9616165-30-3. Edited by: L. Zadnik Stirn • J. Žerovnik • J. Povh • S. Drobne • A. Lisec Pantone 3115 CV Pantone Yellow Black SOR ’11 Proceedings The 11th International Symposium on Operational Research in Slovenia Dolenjske Toplice, SLOVENIA, September 28 - 30, 2011 Edited by: L. Zadnik Stirn, J. Žerovnik, J. Povh, S. Drobne and A. Lisec Slovenian Society Informatika (SDI) Section for Operational Research (SOR)  2011 Lidija Zadnik Stirn – Janez Žerovnik – Janez Povh – Samo Drobne – Anka Lisec Proceedings of the 11th International Symposium on Operational Research SOR'11 in Slovenia, Dolenjske Toplice, September 28 - 30, 2011. Organiser : Slovenian Society Informatika – Section for Operational Research, SI 1000 Ljubljana, Vožarski pot 12, Slovenia (www.drustvo-informatika.si/sekcije/sor/) Under the auspices of the Slovenian Research Agency First published in Slovenia in 2011 by Slovenian Society Informatika – Section for Operational Research, SI 1000 Ljubljana, Vožarski pot 12, Slovenia (www.drustvo-informatika.si/sekcije/sor/) CIP - Kataložni zapis o publikaciji Narodna in univerzitetna knjižnica, Ljubljana 519.8(082) 519.8:005.745(082) 519.81:519.233.3/.5(082) INTERNATIONAL Symposium on Operational Research in Slovenia (11 ; 2011 ; Dolenjske Toplice) SOR '11 proceedings / The 11th International Symposium on Operational Research in Slovenia, Dolenjske Toplice, Slovenia, September 28-30, 2011 ; [organiser] Slovenian Society Informatika (SDI), Section for Operational Research (SOR). - Ljubljana : Slovenian Society Informatika, Section for Operational Research, 2011 ISBN 978-961-6165-35-8 1. Slovensko društvo Informatika. Sekcija za operacijske raziskave 257839872 All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted by any other means without the prior written permission of the copyright holder. Proceedings of the 11th International Symposium on Operational Research in Slovenia (SOR'11) is cited in: ISI (Index to Scientific & Technical Proceedings on CD-ROM and ISI/ISTP&B online database), Current Mathematical Publications, Mathematical Review, MathSci, Zentralblatt für Mathematic / Mathematics Abstracts, MATH on STN International, CompactMath, INSPEC, Journal of Economic Literature Technical editor : Samo Drobne Designed by : Studio LUMINA and Samo Drobne Printed by : Birografika BORI, Ljubljana, Slovenia The 11th International Symposium on Operational Research in Slovenia - SOR ’11 Dolenjske Toplice, SLOVENIA, September 28 - 30, 2011 Program Committee: L. Zadnik Stirn, University of Ljubljana, Biotechnical Faculty, Ljubljana, Slovenia, Chair J. Žerovnik, University of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, Slovenia, Chair Z. Babić, University of Split, Faculty of Economics, Department for Quantitative Methods, Split, Croatia M. Bastič, University of Maribor, Faculty of Business and Economics, Maribor, Slovenia M. Bogataj, University of Ljubljana, Faculty of Maritime Studies and Transport, Portorož, Slovenia K. Cechlarova, P.J. Šafarik University, Faculty of Science, Košice, Slovakia T. Csendes, University of Szeged, Department of Applied Informatics, Szeged, Hungary V. Čančer, University of Maribor, Faculty of Business and Economics, Maribor, Slovenia S. Drobne, University of Ljubljana, Faculty of Civil Engineering and Geodesy, Ljubljana, Slovenia L. Ferbar, University of Ljubljana, Faculty of Economics, Ljubljana, Slovenia M. Gavalec, University of Hradec Králové, Faculty of Informatics and Management, Hradec Králové, Czech Republic R. W. Grubbström, Linköping University, Linköping Institute of Technology, Linköping, Sweden J. Jablonsky, University of Economics, Faculty of Informatics and Statistics, Praha, Czech Republic P. Köchel, Chemnitz University of Technology, Faculty of Informatics, Chemnitz, Germany J. Kušar, University of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, Slovenia L. Lenart, Institute Jožef Stefan, Ljubljana, Slovenia A. Lisec, University of Ljubljana, Faculty of Civil Engineering and Geodesy, Ljubljana, Slovenia L. Neralić, University of Zagreb, Faculty of Economics & Business, Zagreb, Croatia I. Pesek, University of Maribor, Faculty of Natural Sciences and Mathematics, Maribor, Slovenia J. Povh, Faculty of Information Studies, Novo mesto, Slovenia M. S. Rauner, University of Vienna, Department of Innovation and Technology Management, Vienna, Austria A. Schaerf, University of Udine, Department of Electrical, Management and Mechanical Engineering, Udine, Italy M. Sniedovich, University of Melbourne, Department of Mathematics and Statistics, Melbourne, Australia K. Šorić, University of Zagreb, Faculty of Economics & Business, Zagreb, Croatia D. Škulj, University of Ljubljana, Faculty of Social Sciences, Ljubljana, Slovenia P. Šparl, University of Maribor, Faculty of Organizational Sciences, Kranj, Slovenia T. Trzaskalik, Karol Adamiecki University of Economics, Department of Operational Research Katowice, Poland B. Zmazek, University of Maribor, Faculty of Natural Sciences and Mathematics, Maribor, Slovenia D. Yuan, Linköping University, Department of Science, Linköping, Sweden Organizing Committee: J. Povh, Faculty of Information Studies, Novo mesto, Slovenia, Chair S. Drobne, University of Ljubljana, Faculty of Civil Engineering and Geodesy, Ljubljana, Slovenia J. Gabrič, Faculty of Information Studies, Novo mesto, Slovenia A. Lisec, University of Ljubljana, Faculty of Civil Engineering and Geodesy, Ljubljana, Slovenia K. Macedoni, Faculty of Information Studies, Novo mesto, Slovenia L. Zadnik Stirn, University of Ljubljana, Biotechnical Faculty, Ljubljana, Slovenia J. Žerovnik, University of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, Slovenia The 11th International Symposium on Operational Research in Slovenia - SOR ’11 Dolenjske Toplice, SLOVENIA, September 28 - 30, 2011 Chairs: Z. Babić, University of Split, Faculty of Economics, Split, Croatia M. Bogataj, MEDIFAS, Nova Gorica, Slovenia A. Brezavšček, University of Maribor, Faculty of Organizational Sciences, Kranj, Slovenia V. Čančer, University of Maribor, Faculty of Economics and Business Maribor, Maribor, Slovenia Arie M. C. A. Koster, RWTH Aachen University, Lehrstuhl II für Mathematik, Aachen, Germany J. Kušar, University of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, Slovenia A. Lisec, University of Ljubljana, Faculty of Civil Engineering and Geodesy, Ljubljana, Slovenia Z. Lukać, University of Zagreb, Faculty Economics and Business, Zagreb, Croatia U. Pferschy, Department of Statistics and Operational Research, University of Graz, Graz, Austria S. Pivac, University of Split, Faculty of Economics, Split, Croatia J. Povh, Faculty of Information Studies in Novo mesto, Novo mesto, Slovenia P. Šparl, University of Maribor, Faculty of Organizational Sciences, Kranj, Slovenia L. Zadnik Stirn, University of Ljubljana, Biotechical Faculty, Ljubljana, Slovenia J. Žerovnik, University of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, Slovenia Preface The publication, Proceedings of The 11th International Symposium on Operations Research, called SOR’11, contains papers presented at SOR’11(http://sor11.fis.unm.si/) which was organized by Slovenian Society INFORMATIKA, Section for Operations Research (SSISOR) and Faculty of Information Studies (FIS), Novo mesto, Slovenia, and held in Dolenjske Toplice, Slovenia, from September 28 through September 30, 2011. In the Proceedings are published only the articles blindly reviewed and accepted by two independent reviewers. So, with the Proceedings SOR’11, the scientific activities of the symposium SOR’11 are available to all who participated in the symposium and to all who are interested in the contents and are considering to participate in the future SOR symposia. SOR’11 stands under the auspices of the Slovenian Research Agency and PASCAL2, European Network of Excellence. The opening address was given by Prof. Dr. L. Zadnik Stirn, the President of the Slovenian Section of Operations Research, Mr. Niko Schlamberger, the President of Slovenian Society INFORMATIKA, Prof. Dr. Janez Povh, the dean of Faculty of Information Studies, Novo mesto, and presidents/representatives of Operations Research Societies from neighboring countries. SOR’11 is the scientific event in the area of operations research, one of the traditional series of the biannual international OR conferences, organized in Slovenia by SSI-SOR. It is a continuity of ten previous symposia. The main objective of SOR’11 is to advance knowledge, interest and education in OR in Slovenia and worldwide in order to build the intellectual and social capital that are essential in maintaining the identity of OR, especially at a time when interdisciplinary collaboration is proclaimed as significantly important in resolving problems facing the current challenging times. Further, when joining IFORS and EURO, SSI-SOR agreed to work together with diverse disciplines, i.e. to balance the depth of theoretical knowledge in OR and the understanding of theory, methods and problems in other areas within and beyond OR. We are sure that SOR’11 creates the advantage of these objectives, contributes to the quality and reputation of OR with presenting and exchanging new developments, opinions, experiences in the OR theory and practice. SOR’11 was highlighted by a distinguished set of six keynote speakers. Thus, the first part of the Proceedings SOR’11 comprises invited papers, presented by outstanding scientists: Professor Dr. Erling D. Andersen, MOSEK ApS, Denmark, Professor Dr. Walter Gutjahr, University of Vienna, Department of Statistics and Decision Support Systems, Austria, Professor Dr. Horst W. Hamacher, University of Kaiserslautern, Department of Mathematics, Germany, Professor Dr. Arie M.C.A. Koster, Lehrstuhl II für Mathematik, RWTH Aachen, Germany, Professor Dr. Zrinka Lukač, University of Zagreb, Faculty of Economics & Business, Croatia and Professor Dr. Ulrich Pferschy, University of Graz, Department of Statistics and Operations Research, Austria. The second part of the Proceedings includes 47 papers written by 100 authors and co-authors. Most of the authors of the contributed papers came from Slovenia (37), then from Croatia (29), Serbia (13), Bosnia and Herzegovina (4), Germany (5), Spain (3), Austria (2), Hungary (2), Iran (2), Uruguay (2) and Denmark (1). The papers published in the Proceedings are divided into sections: (the number of papers in each section is given in parentheses): Plenary Lectures (6), Graphs and their Applications (3), Production and Inventory (12), OR Applications in Telecommunication and Navigation Systems (3), Econometric Models and Statistics(6), Finance and Investments (6), Multiple Criteria Decision Making (6), Pascal2 session (3), Mathematical Programming and Optimization (3), Location and Transport (5). The Proceedings of the previous ten International Symposia on Operations Research organized by Slovenian Section of Operations Research are indexed in the following secondary and tertiary publications: Current Mathematical Publications, Mathematical Review and MathScinet, Zentralblatt fuer Mathematik/Mathematics Abstracts, MATH on STN International and CompactMath, INSPEC. Also the present Proceedings SOR’11 will be submitted and is supposed to be indexed in the same basis. On behalf of the organizers we would like to express our sincere thanks to all who have supported us in preparing this event - SOR’11. We would not have succeeded in attracting so many distinguished speakers from all over the world without the engagement and the advice of active members of Slovenian Section of Operations Research. Many thanks to them. Further, we would like to express our deepest gratitude to prominent keynote speakers, to the members of the Program and Organizing Committees, to the referees who raised the quality of the SOR’11 by their useful suggestions, section’s chairs, and to all the numerous people - far too many to be listed here individually - who helped in carrying out The 11th International Symposium on Operations Research SOR’11 and in putting together these Proceedings. At last, we appreciate the authors’ efforts in preparing and presenting the papers, which made The 11th Symposium on Operational Research SOR’11 successful. The success of the scientific events at SOR’11 and the present proceedings should be seen as a result of joint effort. Dolenjske Toplice, September 28, 2011 Lidija Zadnik Stirn Janez Žerovnik Janez Povh Samo Drobne Anka Lisec (Editors) Sponsors of the 11th International Symposium on Operational Research in Slovenia (SOR’11) The following institution and network are gratefully acknowledged for their financial support of the symposium:  Slovenian Research Agency, The Republic of Slovenia  PASCAL 2, European Network of Excellence Contents Plenary Lectures 1 Erling D. Andersen Ten Years of Experience with Conic Quadratic Optimization 3 Katharina Gerhardt, Horst W. Hamacher and Stefan Ruzika Operations Research Methods in the Planning, Control and Adaptation of Evacuation Plans 5 Walter J. Gutjahr A Note on the Utility-Theoretic Justification of the Pareto Approach in Stochastic Multi-Objective Combinatorial Optimization 7 Arie M. C. A. Koster Robust Optimization of Telecommunication Networks 13 Zrinka Lukać Metaheuristic Optimization 17 Ulrich Pferschy Mathematical Models and Solutions for Network Design Problems 23 Section I: Pascal 2 session 29 Petra Šparl A Survey on Known Upper Bounds for Multichromatic Number for Hexagonal Graphs 31 Janez Žerovnik A Generalization of the Ball-Packing Problem 37 Janez Žerovnik and Bojan Kuzma Ball Packing with two Ball Sizes: Some Bounds Based on Geometry 45 Section II: Graphs and Their Applications 51 Marjan Čeh, Frank Gielsdorf and Anka Lisec Homogenization of Digital Cadastre Index Map Improving Geometrical Quality 53 Dušan Hvalica Solving Job Shop Problems in the Context of Hypergraphs 61 Pablo Sartor Del Giudice and Franco Robledo Amoza An Heuristic for the Edge-survivable Generalized Steiner Problem 67 Section III: OR Applications in Telecommunication and Navigation Systems Minja Marinović, Milan Stanojević and Dragana Makajić-Nikolić LP Model for Day-ahead Planning in Energy Trading 73 75 Polona Pavlovčič Prešeren and Bojan Stopar GNSS Orbit Re-Construction Using Wavelet Neural Networks 81 Željko V. Račić Application of Data Mining in Telecommunications Companies 87 Section IV: Mathematical Programming and Optimization Kristijan Cafuta, Igor Klep and Janez Povh Trace Optimization Using Semidefinite Programming 93 95 Igor Đukanović, Jelena Govorčin, Nebojša Gvozdenović and Janez Povh On Semidefinite Programming Based Heuristics for the Graph Colouring Problem 103 Maja Remic, Gašper Žerovnik and Janez Žerovnik Experimental Comparison of Basic and Cardinality Constrained Bin Packing Problem Algorithms 109 Section V: Multiple Criteria Decision Making 115 Andrej Bregar Investigation of the Aggregation-Disaggregation Approach to Multi-Criteria Negotiations: Consolidation of Simulation and Case Studies 117 Vesna Čančer How to Use the 5WS & H Technique to Determine the Weights of Interacting Criteria 125 Samo Drobne and Marija Bogataj Economic Criteria in Decision-Making on Number of Functional Regions: the Case of Slovenia 131 Milena Đurović, Gordana Savić, Marija Kuzmanović and Milan Martić Towards Criteria Selection in DEA by Conjoint Analysis 137 Petra Grošelj and Lidija Zadnik Stirn Interval Comparison Matrices in Group AHP 143 Višnja Vojvodić Rosenzweig, Hrvoje Volarević and Mario Varović A Goal Programming Approach to Ranking Banks 149 Section VI: Econometric Models and Statistics 157 Martina Basarac Sectoral Growth Drivers of Wood Processing and Furniture Manufacturing in Croatia 159 Mirjana Čižmešija and Jelena Knezović ARIMA Models and the Box – Jenkins Approach in Analaysing and Forecasting Variables in Field of Sustainable Development – The Case of Croatia 165 Samo Drobne, Marija Bogataj, Danijela Tuljak Suban and Urška Železnik Regression-Neuro-Fuzzy Approach to Analyse Distance Function in Internal Inter-Regional Migrations in EU Countries 171 Ksenija Dumićić, Anita Čeh Časni and Irena Palić Multivariate Regression Analysis of Personal Consumption in Croatia 177 Ksenija Dumičić and Berislav Zmuk Impact of Applied Acceptance Sampling Plan on Decisions in Quality Management 183 Amir Jamak, Alen Savatić and Mehmet Can Principal Component Analysis for Authorship Attribution 189 Section VII: Production and Inventory 197 Zoran Babić and Tunjo Perić Volume Discounts in Multiproduct Supplier Selection Problem 199 Tomaž Berlec, Primož Potočnik, Edvard Govekar and Marko Starbek Predicting Manufacturing Due Date 205 Alenka Brezavšček A Simple Discrete Approximation for the Renewal Function 213 Mirjana Čižmešija, Maja Copak and Nataša Kurnoga Živadinović Industrial Confidence Indicator and Manufacturing Industry in the EURO Area 221 Stanislav Gorenc, Mitja Hrast, Tomaž Berlec and Marko Starbek Layout of Company's Functional Units 227 Mitja Hrast, Stanislav Gorenc, Tomaž Berlec and Marko Starbek Virtual Factory – the Reality of the Present 233 Danijel Kovačić and Marija Bogataj Reverse Logistics Facility Location in Extended MRP Theory 239 Marija Kuzmanović, Biljana Panić, Mirko Vujošević and Slobodan Vujić Risk Assessment and Management in the Mining Industry 245 Lidija Rihar, Stanislav Gorenc, Janez Kušar and Marko Starbek Teamwork in Simultaneous Product Realization Process 251 Viljem Rupnik An Axiomatic Approach to the Generalised Evaluation Procedure 257 Slavko Šimundić, Siniša Franjić and Danijel Barbarić Computer Manipulation Methods, Theirs Detecting, Reporting and Sanctioning 261 Ilko Vrankić, Mira Oraić and Zrinka Lukač Allocative Efficiency and Optimal Adjustments of a Producer 269 Section VIII: Finance and Investments 277 Draženka Čizmić Price and Volume Measures in National Accounts 279 Alan Domić The Open Economy New Keynesian Phillips Curve with Adjusted Measures of Real Marginal Cost: Estimates for Croatia 285 Margareta Gardijan Option Leverage 291 Alenka Kavkler and Mejra Festic Modelling Stock Exchange Index Returns in 12 New Member States with a Tree-Based Approach 297 Snježana Pivac, Blanka Škrabić Perić and Anita Udiljak Analysis of Croatian Tax Revenue Indicators and Comparison with Selectes Countries 303 Josip Arnerić, Nada Pleli and Tihana Škrinjarić Impact of Social Security on Fertility – a Panel Data Analysis 309 Section IX: Location and Transport 315 Karlo Bala, Dejan Brcanov and Nebojša Gvozdenović Where to Place Cross Docking Points? 317 Samo Drobne, Marija Bogataj, Mateja Zupan and Anka Lisec Dynamics and Local Policy in Commuting: Attractiveness and Stickness of Slovenian Municipalities 323 Eloy Hontoria, M.Victoria de-la-Fuente Aragon, Lorenzo Ros-McDonnell and Marija Bogataj Monte Carlo Simulation Applied to the Logistics of Ceramics 329 Arash Motaghedi-Larijani and Mohamad-Saied Jabalameli Proposing Mixed non Linear Programming Model for Network Design Problem under Perspective of third Party Logistics 335 Attila Tóth and Miklós Krész A Flexible Optimization Framework for Driver Scheduling 341 APPENDIX Authors' addresses Author index A Andersen Erling D. ............................3 Arnerić Josip .................................309 B Babić Zoran ..................................199 Bala Karlo .....................................317 Barbarić Danijel ...........................261 Basarac Martina ............................159 Berlec Tomaž ...............205, 227, 233 Bogataj Marija .................................... .......................131, 171, 239, 323, 329 Brcanov Dejan ..............................317 Bregar Andrej ...............................117 Brezavšček Alenka .......................213 C Cafuta Kristijan ..............................95 Can Mehmet .................................189 Copak Maja ..................................221 Č Čančer Vesna ................................125 Čeh Časni Anita ............................177 Čeh Marjan .....................................53 Čizmić Draženka ..........................279 Čižmešija Mirjana ................165, 221 G Gardijan Margareta ....................... 291 Gerhardt Katarina ............................ 5 Gielsdorf Frank .............................. 53 Gorenc Stanislav .......... 227, 233, 251 Govekar Edvard ........................... 205 Govorčin Jelena ........................... 103 Grošelj Petra ................................ 143 Gutjahr Walter J. .............................. 7 Gvozdenović Nebojša .......... 103, 317 H Hamacher Horst W. ......................... 5 Hontoria Eloy ............................... 329 Hrast Mitja ........................... 227, 233 Hvalica Dušan ................................ 61 J Jabalameli Mohamad-Saied ......... 335 Jamak Amir .................................. 189 D Domić Alan ..................................285 Drobne Samo ................131, 171, 323 Dumičić Ksenija ...................177, 183 K Kavkler Alenka ............................ 297 Klep Igor ........................................ 95 Knezović Jelena ........................... 165 Koster Arie M. C. A. ...................... 13 Kovačić Danijel ........................... 239 Krész Miklós ................................ 341 Kurnoga Živadinović Nataša ....... 221 Kušar Janez .................................. 251 Kuzma Bojan ................................. 45 Kuzmanović Marija ............. 137, 245 Đ Đukanović Igor .............................103 Đurović Milena .............................137 L Lisec Anka ............................. 53, 323 Lukać Zrinka .......................... 17, 269 F Festic Mejra ..................................297 Franjić Siniša .................................261 de la Fuente Aragon M. Victoria ..329 M Makajić-Nikolić Dragana .............. 75 Marinović Minja ............................ 75 Martić Milan ................................ 137 Motaghedi-Larijani Arash ............ 335 O Oraić Mira .....................................269 P Palić Irena .....................................177 Panić Biljana .................................245 Pavlovčič Prešeren Polona .............81 Perić Tunjo ...................................199 Pferschy Ulrich ...............................23 Pivac Snježana ..............................303 Pleli Nada .....................................309 Potočnik Primož ............................205 Povh Janez ..............................95, 103 R Račić Željko V. ...............................87 Remic Maja ..................................109 Rihar Lidija ...................................251 Robledo Amoza Franco ...................67 Ros-McDonnell Lorenzo ..............329 Rupnik Viljem ..............................257 Ruzika Stefan ...................................5 S Sartor Del Giudice Pablo ................67 Savatić Alen ..................................189 Savić Gordana ..............................137 Starbek Marko ......205, 227, 233, 251 Stanojević Milan .............................75 Stopar Bojan ...................................81 Š Šimundić Slavko ........................... 261 Škrabić Perić Blanka .................... 303 Škrinjarić Tihana .......................... 309 Šparl Petra ...................................... 31 T Tóth Attila .................................... 341 Tuljak Suban Danijela ................. 171 U Udiljak Anita ................................ 303 V Varović Mario ............................... 149 Vojvodić Rosenzweig Višnja ....... 149 Volarević Hrvoje .......................... 149 Vrankić Ilko .................................. 269 Vujić Slobodan ............................ 245 Vujošević Mirko .......................... 245 Z Zadnik Stirn Lidija ....................... 143 Zmuk Berislav .............................. 183 Zupan Mateja ................................ 323 Ž Železnik Urška ............................. 171 Žerovnik Gašper .......................... 109 Žerovnik Janez ................. 37, 45, 109 The 11th International Symposium on Operational Research in Slovenia SOR ’11 Dolenjske Toplice, SLOVENIA September 28 - 30, 2011 Plenary Lectures 1 2 TEN YEARS OF EXPERIENCE WITH CONIC QUADRATIC OPTIMIZATION Erling D. Andersen MOSEK ApS, Fruebjergvej 3, Box 16 2100 Copenhagen O, Denmark Email: e.d.andersen@mosek.com Abstract: For about 10 years the software package MOSEK has been capable of solving large-scale sparse conic quadratic optimization (CQO) problems. Based on the experience gained with CQO during those 10 years we will present a few important applications of conic quadratic optimization, discuss properties of the CQO problem and give an overview of how MOSEK solve a CQO problem. We will also present numerical results demonstrating the performance of MOSEK on CQO problems and discuss future developments. 3 4 OPERATIONS RESEARCH METHODS IN THE PLANNING, CONTROL, AND ADAPTATION OF EVACUATION PLANS Katharina Gerhardt, Horst W. Hamacher and Stefan Ruzika Department of Mathematics, University of Kaiserslautern (Germany), Email: {Gerhardt, Hamacher, Ruzika}@mathematik.uni-kl.de Extended Abstract: One of the basic emergency measures in the case of (bomb) threats, attacks, large-scale accidents, and natural disasters is the evacuation of the affected buildings and regions. Here, the most important goal is to evacuate occupants, i.e. to take them away from the risk area and bring them to safety, as fast and reliable as possible. Unfortunately, several tragic scenarios are known from the past, for instance, the love parade disaster in Duisburg, Germany, of July 2010, where 19 people were killed and more than 300 injured while trying to leave the location in which the parade was to take place. It is not claimed that the application of operations research methods can avoid such tragic incidences, but it is reasonable to assume that they will yield insight to make them less likely to occur or help to mitigate the consequences. The research project REPKA which is co-ordinated at the University of Kaiserslautern is primarily concerned with regional evacuation. In particular, the situation is considered where a large crowd has already left a building and must then be brought further away to safety. In order to reliably predict evacuation data like evacuation times or number of evacuees per given time period we propose an approach in which real-world data is sandwiched between a lower bound computed by optimization methods and upper bounds delivered by simulation. In our presentation we will focus on the following operations research issues.  Representation of pedestrian movements in evacuation scenarios by dynamic network flows.  Solution of maximal, earliest arrival, and quickest dynamic flows.  Integration of location decisions with regard to the placement of emergency and commercial units.  Control and adaptation of evacuation plans using the visualization tool of REPKAOptimizer.  Practical experience with the sandwich method applied to the evacuation of the Betzenberg, home of the Fritz-Walter football stadium in Kaiserslautern (Germany). Reference: REPKA webpage http://www.repka-evakuierung.de and further information therein Acknowledgement This paper is supported in part by the Federal Ministry for Education and Research (Bundesministerium für Bildung und Forschung, BMBF) as project REPKA under FKZ 13N9961 (TU Kaiserslautern). 5 6 A Note on the Utility-Theoretic Justification of the Pareto Approach in Stochastic Multi-Objective Combinatorial Optimization Walter J. Gutjahr Department of Statistics and Operations Research University of Vienna 1 Introduction For a precise mathematical formulation of stochastic multi-objective optimization (SMOO) problems, the multi-objective and the stochastic approach have been proposed [1, 2, 3]. The former reduces the problem in a first step to a deterministic multi-objective problem, whereas the latter starts by reducing it to a stochastic single-objective problem. In the case of a finite decision set, i.e., in combinatorial SMOO, the multi-objective approach is especially attractive, because it allows the computation of a list of Pareto-efficient solutions from which the decision maker (DM) can select her/his preferred solution without being forced to describe her/his preference structure or utility function in formal terms. However, some authors have questioned the appropriateness of the multi-objective approach for the general case of a SMOO problem, see [3]. The aim of this note is to show that there are conditions under which the multi-objective approach (using the Pareto concept for solving the resulting deterministic multi-objective problem) can be proven to be well-founded in decision theory in the sense that it is consistent with expected-utility maximization. Two examples of such conditions will be given. 2 Problem Definition We consider stochastic multi-objective combinatorial optimization (SMOCO) problems of the following form: max (f1 (x, ω), . . . , fm (x, ω)) s.t. x ∈ S. (1) Therein, S is a finite set of decisions, and ω denotes a random influence. More formally, for each x and j, the function fj (x, ·) is a random variable on a probability space (Ω, A, P ), and ω is an element of the sample space Ω. Each ω determines a specific realization of the random variables fj (x, ·), i.e., a random scenario. We shall 1 7 identify ω with the scenario determined by it. Throughout the paper, we assume that the joint distribution of the random variables fj (x, ·) is known.1 Mathematically, the considered problem is still under-specified by (1). First, the formulation contains more than one objective, such that it is not yet clear what is meant by “maximize”. Secondly, since each objective also depends on ω, it still needs to be defined what is done to obtain a unique evaluation of the objective(s). In the literature, two ways have been described to re-formulate (1) in a precise sense (see [1, 2, 3]): • The multi-objective approach applies operators as expectation, variance etc. to each component of the vector (f1 (x, ω), . . . , fm (x, ω)). In this way, it reduces the stochastic multi-objective problem to a deterministic multi-objective problem. After that, some method of treating a multi-objective problem is applied. We focus here on the case where the expectation operator is applied to each fj (x, ω). This gives the multi-objective problem max (F1 (x), . . . , Fm (x)) s.t. x ∈ S, (2) where Fj (x) = E(fj (x, ω)) (j = 1, . . . , m). • The stochastic approach aggregates the random objectives f1 (x, ω), . . . , fm (x, ω) by some aggregation function A and applies then an operator such as the expectation operator. E.g., it solves max E(A(f1 (x, ω), . . . , fm (x, ω))) s.t. x ∈ S. (3) We see that now the order has been reversed: First, the original problem is reduced to a single-objective stochastic optimization problem, and then some method of stochastic optimization is applied to solve this resulting problem. By the Pareto approach, we understand the multi-objective approach in the version given by (2) in the special case where as the solution concept for multi-objective optimization, the computation of the set of Pareto-efficient (or: non-dominated) solutions2 is chosen. Thus, the Pareto approach reformulates problem (1) as the problem of determining all non-dominated vectors of expected objective function values. Caballero et al. [3] argue that in general, the stochastic approach is more adequate than the multi-objective approach, because it takes dependencies between the objectives into account (cf. also Ben Abdelaziz [2]). However, the stochastic approach has the disadvantage that it forces the DM to unveil her preferences in advance (i.e., before computational solution) by specifying the aggregation function A. 1 A simple special case occurs if Ω = {ω1 , . . . , ωn } is finite. Then the joint distribution is defined n by a vector of probabilities pi assigned to the scenarios ωi (i = 1, . . . , n) with i=1 pi = 1. 2 A solution x ∈ S is called Pareto-efficient with respect to objectives F1 , . . . , Fm , if there is no other solution y ∈ S with Fj (y) ≥ Fj (x) ∀j and ∃j : Fj (y) > Fj (x). We also extend this notion from the solution space to the objective space by calling the vector (F1 (x), . . . , Fm (x)) Pareto-efficient, if solution x is Pareto-efficient. 2 8 In many practical situations, the DM is unable or unwilling to do that. Therefore, it makes sense to look for conditions under which the multi-objective approach in general and the Pareto approach in particular are justifiable by decision-theoretic considerations. 3 Utility-Theoretic Analysis In deterministic multi-objective optimization, the attractiveness of the Pareto concept stems from the following observation, which is easy to prove: For each each vector (F1 , . . . , Fm ) of objective functions on S and each utility function u : Rm → R that is increasing in each component, it holds that if solution x∗ ∈ S is optimal w.r.t. u in the sense that x∗ maximizes u(F1 (x∗ ), . . . , Fm (x∗ )) on S, then (F1 (x∗ ), . . . , Fm (x∗ )) must be Pareto-efficient. In other words, it is safe to omit dominated solutions (for this purpose, the utility function needs not to be known!) since after this omission, the optimal solution w.r.t. the implicit utility function u can still be found in the Pareto set (the set of Pareto-optimal solutions), so the DM can inspect this set and choose the best solution according to her or his true preferences. Unfortunately, the generalization of this assumption to problem (2) does not hold anymore in the general case: Simple examples show that the solution of the problem max E(u(f1 (x), . . . , fm (x))) s.t. x ∈ S (4) needs not to be a Pareto-efficient solution of (2). The essential reason is that in general, the expectation operator cannot be interchanged with the utility function u. Thus, it can happen that the solution with maximum expected utility does not occur in the Pareto set. This does not necessarily imply that in these circumstances, the Pareto approach is inadequate3 ; it only means that it is not consistent with the principle of expected-utility maximization (sometimes called the Bernoulli principle). In the following, we give two examples of situations where there is no conflict between the Pareto approach and expected-utility maximization. For this purpose, let us start with a definition. Definition 1. Let U denote a set of increasing utility functions u : Rm → R. We say that for an m-dimensional SMOCO problem (f1 , . . . , fm ) on S, the Pareto approach is U-consistent with expected-utility maximization (short: U-consistent), if for each u ∈ U and each x∗ ∈ S, the validity of E(u(f1 (x∗ , ω), . . . , fm (x∗ , ω))) = max E(u(f1 (x, ω), . . . , fm (x, ω))) x∈S implies that (E(f1 (x∗ , ω)), . . . , E(fm (x∗ , ω))) is Pareto-efficient. 3 The Pareto approach could still be justified then by considering utilities of expectations instead of expected utilities. 3 9 Theorem 1. Let fj (x, ω) be objective functions with fj (x, ω) = fj (x) deterministic for j = 1, . . . , k, and fj (x, ω) ≥ 0 ∀x ∈ S for j = k + 1, . . . , m (0 ≤ k ≤ m). Furthermore, let the class U consist of all utility functions u of the form u(y1 , . . . , ym ) = ϕ(y1 , . . . , yk ) + m  ψj (y1 , . . . , yk ) yj , j=k+1 where ϕ(y1 , . . . , yk ) and ψj (y1 , . . . , yk ) (j = k + 1, . . . , m) are increasing in y1 , . . . , yk . Then for (f1 , . . . , fm ), the Pareto approach is U-consistent. Proof. With f¯j (x) = E(fj (x, ω)), we have E(u(f1 (x, ω), . . . , fm (x, ω))) = ϕ(f1 (x), . . . , fk (x)) m  + ψj (f1 (x), . . . , fk (x)) E(fj (x, ω)) j=k+1 = ϕ(f¯1 (x), . . . , f¯k (x)) + m  ψj (f¯1 (x), . . . , f¯k (x)) f¯j (x) = u(f¯1 (x), . . . , f¯m (x)). (5) j=k+1 The function u is increasing. Therefore, by the observation that in the deterministic context, the optimality of x∗ ∈ S w.r.t. u implies the Pareto-efficiency of the vector of objective function values, for a maximizer x∗ of (5), the vector (f¯1 (x∗ ), . . . , f¯m (x∗ )) must be Pareto-efficient. 2 It can be seen that in the sufficient condition given in Theorem 1 for U-consistency, the set U needs not to be restricted to utility functions exhibiting utility independence in the sense mof Keeney and Raiffa [5], i.e., utility functions u of the form u(y1 , . . . , ym ) = j=1 uj (yj ). Evidently, also the case of linear utility functions u(y1 , . . . , ym ) = a1 y1 + . . . + am ym is covered by Theorem 1. As already outlined in [3], the multi-objective approach does not fall back behind the stochastic approach in this situation. However, it should be noted that in this case, only so-called supported solutions can be optimal, i.e., solutions that are optimal under a weighted sum of objectives with suitable weights. The conditions of Theorem 1, on the other hand, also extend to situations where under certain utility functions, the optimal solution is unsupported. Example. Consider the case where the decision x determines a work plan for a project P. Each work plan x consumes a certain amount r1 (x) of a resource (say, working time) and yields a (stochastic) profit f2 (x, ω). In total, B units of the resource are available. The remaining f1 (x) = B − r1 (x) units of the resource can be used in a project Q, yielding there a (deterministic) profit ϕ(f1 (x)) that is increasing in f1 (x). Note that ϕ needs not to be linear, and that the DM may shy away from providing an explicit mathematical representation of ϕ. Then, we can consider the bi-objective maximization problem with objective functions f1 (x) and f2 (x, ω), where the first objective function is deterministic, the second is stochastic. The overall utility function will be of the form u(f1 , f2 ) = u1 (f1 ) + u2 (f2 ) with 4 10 u1 (f1 ) = ϕ(f1 ) denoting the profit from project Q and u2 (f2 ) = f2 denoting the profit from project P. Evidently, we are within the conditions of Theorem 1. As a consequence, the Pareto approach will produce a set of solutions among which the solution with maximal expected utility occurs. An approach aggregating expectations by a weighted sum, on the other hand, may miss the solution with maximal expected utility even if all possible weights are considered.  Utility functions can frequently be approximated by quadratic functions. The following theorem gives a second sufficient condition for consistency if all utilities are quadratic: Theorem 2. Let fj have values in some interval [αj , βj ] (j = 1, . . . , m), and let the m class U consist of all increasing utility functions of the form u(y1 , . . . , ym ) = j=1 uj (yj ) with uj (yj ) = aj + bj yj + cj yj2 (j = 1, . . . , m), where uj is increasing on [αj , βj ]. Furthermore, let the variance var(fj (x, ω)) be independent of x, i.e., var(fj (x, ω)) = σj2 ∀x ∈ S. Then for (f1 , . . . , fm ), the Pareto approach is U-consistent. Proof. E(u(f1 (x, ω), . . . , fm (x, ω))) = m    aj + bj E(fj (x, ω)) + cj E((fj (x, ω)2 ) j=1 = m   m    aj + bj E(fj (x, ω)) + cj (E(fj (x, ω))) + cj E((fj (x, ω)2 ) − (E(fj (x, ω)))2 2 j=1 j=1 = u(E(f1 (x, ω)), . . . , E(fm (x, ω))) + m  cj σj2 , j=1 where the last term is constant. Again by reference to the deterministic special case (as in the proof of Theorem 1), we conclude that for a maximizer x∗ of u, the vector (E(f1 (x, ω), . . . , E(fm (x, ω))) must be Pareto-efficient. 2 4 Conclusions Two examples of sufficient conditions for the consistency of the Pareto approach with expected-utility maximization have been given. Although the conditions may look rather restrictive, a large class of practically relevant problems in multi-objective decision analysis is covered by them. Also other sufficient conditions may be provided, as well as relaxed versions guaranteeing approximate solutions to utility optimization problems on a certain approximation level. Results of this type are of interest, since during the last years, advances in the computational solution of SMOCO problems based on the Pareto approach by suitable meta-heuristic techniques have been made (cf. [4]). 5 11 References [1] Ben Abdelaziz, F. L’efficacité en Programmation Multi-objectifs Stochastique. Thesis, Université de Laval, Québec. [2] Ben Abdelaziz F. Solution approaches for the multiobjective stochastic programming. European Journal of Operations Research, in press. [3] Caballero R, Cerda E, Muños MM, Rey L. Stochastic approach versus multiobjective approach for obtaining efficient solutions in stochastic multiobjective programming problems. European Journal of Operations Research 2004; 158; 633-648. [4] Gutjahr WJ. Recent trends in metaheuristics for stochastic combinatorial optimization. Central European Journal of Computer Science 2011; 1; 58-66. [5] Keeney R, Raiffa H. Decisions with Multiple Objectives: Preferences and Value Trade Offs. John Wiley and Sons, New York, 1976. 6 12 Robust Optimization of Telecommunication Networks Arie M.C.A. Koster∗ 9th September 2011 Abstract Robust Optimization is an emerging field of Operations Research, focussing on dealing with data uncertainty in optimization problems. In this talk, we discuss a variety of robust optimization approaches and their application to both fixed and wireless telecommunication network design problems. In particular, we show how demand uncertainty can be incorporated in classical network design by LP duality. The robust optimization approach is evaluated with real-life Internet traffic data. Keywords: network design, robust optimization, price of robustness, integer linear programming Extended Abstract Incorporating uncertainty within the mathematical analysis of operational research has been an effort since its very first beginnings. In the 1950s, Dantzig [13] introduced Stochastic Programming using probabilities for the possible realizations of the uncertain data. The main limitation of such probabilistic approaches is that the distribution of the uncertain data must be known a priori which is often not the case for “real-world” problems. Stochastic programming may also result in extremely hard to solve optimization problems. A promising alternative to handle data uncertainties is the usage of so-called chance-constraints which were introduced in [24]. Chance-constrained programming is a one-stage concept for which the probability distribution of the uncertain data has to be known completely. The aim of this concept is to find the best solution remaining feasible for a given infeasibility probability tolerance. In [14] chance constraints for combinatorial optimization problems are studied. In 1973, Soyster [26] suggested another approach based on implicitely describing the uncertain data introducing so-called uncertainty sets and establishing the concept of Robust Optimization. Using this framework we do not need any information about the probabilistic distribution of the uncertainty. Instead a solution is said to be robust if it is feasible for all realizations of the data in the given uncertainty set. In Robust Optimization we aim at finding the cost-optimal robust solution. This approach has been further developed by Ben-Tal and Nemirovski [3, 4, 5], Bertsimas and Sim [7], and others using different convex and bounded uncertainty sets. They introduce the concept of robust counterparts for uncertain linear programs. In [3] it is shown that these can be solved by deterministic linear programs or deterministic conic quadratic programs if the uncertainty set is polyhedral or ellipsoidal, respectively. Bertsimas and Sim [7] introduced a polyhedral uncertainty set that easily allows to control the price of robustness by varying the number Γ of coefficients in a row of the given linear program that are allowed to deviate from its nominal values simultaneously. By changing this parameter Γ the practitioner is enabled to regulate the trade-off between the degree of uncertainty taking into account and the cost of this additional feature. Robust optimization is also a well known method in telecommunication network design. We distinguish between robust network design using static or dynamic routing which refers to the flexibility of flow to respond to the realization of the demand (while the capacity remains fixed). Static routing means that for every node pair the same paths are used with the same splitting independent of the realization of demand. Contrary, dynamic routing allows for full flexibility in rerouting the traffic if the demand changes. The concept of different routing schemes is strongly related to different levels of recourse in multi-stage ∗ RWTH Aachen University, Lehrstuhl {koster,kutschka}@math2.rwth-aachen.de II für Mathematik, 1 13 Wüllnerstr. 5b, D-52062 Aachen, Germany, stochastic and robust optimization [6]. We refer to [21] and [23] for a discussion on how to embed the two classical routing schemes in these more general frameworks. For general two-stage robust network design check [1]. We also note that recently there has been some progress in defining routing schemes in between static and dynamic, see for instance [2, 22, 23, 25]. In this presentation, we will discuss three recent applications of robust optimization to problems from telecommunications: • The design of a backbone network under demand uncertainty is described by – an undirected connected graph G = (V, E) representing a potential network topology, – on each of the links e ∈ E capacity can be installed in integral units and costs κe per unit, – a set of commodities K represents potential traffic demands with for each commodity k ∈ K a nodepair (sk , tk ) and a demand value dk ≥ 0 for traffic from source sk ∈ V to target tk ∈ V . The actual demand values are considered to be uncertain. The traffic for commodity k is realized by a splittable multi-path flow between sk and tk . Of course, the actual multi-commodity flow depends on the realization of the demand d. A routing template describes for every commodity the percental splitting of the traffic among the paths from sk to tk . Along the lines of the Γ-robustness approach of [7], we define for every commodity k ∈ K a nominal demand value d¯k and a deviation dˆk . It is assumed that dk ∈ [0, d¯k + dˆk ] and, given a value Γ ∈ Z, at most Γ commodities deviate from their nominal values simultaneously. The robust network design problem is to find a minimum-cost installation of integral capacities and a routing template for every commodity such that actual flow does not exceed the link capacities independent of the realization of demands. Theoretical and practical aspects of this problem are studied in a series of papers: [16, 17, 18, 19, 20]. In particular, the problem is formulated as an integer linear program, valid inenequalities are derived, and optimal network designs are evaluated on the basis of real-life traffic data. • The green design of a wireless network under demand uncertainty. In wireless network planning we have to install a number of base stations (BSs) from a set S of BS candidates (including candidate sites at the same location but with different configurations) such that a set T of traffic nodes (TNs) can be assigned to the BSs. Each BS s ∈ S has an available downlink (DL) bandwidth bs , a basic power consumption ps and a power consumption p̃s per TN served by s, whereas each TN t ∈ T requests a data rate wt . The spectral efficiency between any BS-TN pair (s, t) is denoted by est . This parameter gives the ratio between data rate and bandwidth. It incorporates, e. g., modulation and coding scheme that is supported by the associated signal-to-noise ratio (SNR). To establish a link from a BS to a TN the corresponding spectral efficiency must exceed the threshold emin . The amount of bandwidth that is allocated to TN t from BS s, if t is served by s, is ewstt . The objective is to minimize the total amount of energy consumed by the deployed BSs and the number of TNs which are not served. A scaling parameter λ is needed to combine the two objectives reasonably. A TN can be assigned to at most one BS due to hard handover in future wireless networks. The required data rate of a TN varies over time due to mobility of the users and their behavour (voice, web browsing, data transfer). Again, we assume that the data rate varies within an interval [0, w̄t +ŵt ] with w̄t the nominal data rate and w̄t + ŵt the peak data rate. In [12] the Γ-robustness approach is applied to this model, including valid inequalities to improve the performance of CPLEX. In a case study, we observed energy savings either by deploying less BSs or serving more TNs with the same number of BSs. A subproblem within this context is the robust knapsack problem [15]. In [8, 9] this classical problem is further generalized to include the possibility to exchange items in a second stage, the so-called recoverable robust knapsack problem. In [8], the case of discrete scenarios (i.e., a set of weight vectors) is studied, whereas in [9] the case of Γ-scenarios is studied. • The design of reliable fixed broadband wireless networks is considered in [10, 11]. Here, the problem is similar to the design of backbone networks, except that radio links are used instead of optical fibres. 2 14 Hence, we cannot install multiple capacity units on a single link. The capacity of a link is determined by the chosen channel bandwidth and the bandwidth efficiency (modulation scheme). The quality of the signal varies over time resulting in a fluctuation of the provided capacity. The design of such a network can be formulated as a chance-constrained optimization problem. Here, the probability that the provided capacity exceed the required bandwidth has to be close to 1. In case the probabilities of the links are independent, the problem can be reformulated as integer linear programming [10] and the performance of ILP solvers can again be improved by valid inequalities [11]. Acknowledgement This extended abstract is based on research carried out in collaboration with many people. In particular, I would like to thank Grit Claßen, Manuel Kutschka, Napoleao Nepomuceno, and Christian Raack for the fruitful teamwork. This work was partly supported by the Federal Ministry of Education and Research (BMBF grant 03MS616A, project ROBUKOM - Robust Communication Networks, www.robukom.de). References [1] A. Atamtürk and M. Zhang. Two-Stage Robust Network Flow and Design Under Demand Uncertainty. Operations Research, 55(4):662–673, 2007. [2] W. Ben-Ameur. Between fully dynamic routing and robust stable routing. In Design and Reliable Communication Networks, 2007. DRCN 2007. 6th International Workshop on, pages 1–6. IEEE, 2007. [3] A. Ben-Tal and A. Nemirovski. Robust solutions of uncertain linear programs. Operations Research Letters, 25(1):1–14, 1999. [4] A. Ben-Tal and A. Nemirovski. Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88:411–424, 2000. [5] A. Ben-Tal and A. Nemirovski. Robust optimization - methodology and application. Mathematical Programming, 92:453–480, 2002. [6] A. Ben-Tal, A. Goryashko, E. Guslitzer, and A. Nemirovski. Adjustable robust solutions of uncertain linear programs. Mathematical Programming, 99(2):351–376, 2004. [7] D. Bertsimas and M. Sim. The Price of Robustness. Operations Research, 52(1):35–53, 2004. [8] C. Büsing, A. M. C. A. Koster, and M. Kutschka. Recoverable Robust Knapsacks: the discrete scenario case. Optimization Letters, 5:379–392, 2011. ISSN 1862-4472. [9] C. Büsing, A. M. C. A. Koster, and M. Kutschka. Recoverable Robust Knapsacks: Γ-Scenarios. In J. Pahl, T. Reiners, and S. Voß, editors, Network Optimization, 5th International Conference, INOC 2011, Hamburg, Germany, June 2011, Proceedings, volume 6701. Springer, 2011. [10] G. Claßen, D. Coudert, A. M. C. A. Koster, and N. Nepomuceno. Bandwidth allocation for reliable fixed broadband wireless networks. In Proceedings of IEEE WoWMoM 2011, 2011. [11] G. Claßen, D. Coudert, A. M. C. A. Koster, and N. Nepomuceno. A chance-constrained model & cutting planes for fixed broadband wireless networks. In J. Pahl, T. Reiners, and S. Voß, editors, Network Optimization, 5th International Conference, INOC 2011, Hamburg, Germany, June 2011, Proceedings, volume 6701, pages 37–42. Springer, 2011. 3 15 [12] G. Claßen, A. M. C. A. Koster, and A. Schmeink. Robust Planning of Green Wireless Networks. In Proceedings of NetGCoop 2011, 2011. to appear. [13] G. B. Dantzig. Linear programming under uncertainty. Management Science, pages 197–206, 1955. [14] O. Klopfenstein. Solving chance-constrained combinatorial problems to optimality. Computational Optimization and Applications, 45(3):607–638, 2010. ISSN 0926-6003. [15] O. Klopfenstein and D. Nace. Valid inequalities for a robust knapsack polyhedron – Application to the robust bandwidth packing problem. In Proc. of International Network Optimization Conference INOC, 2009. [16] A. M.C.A. Koster, M. Kutschka, and C. Raack. Cutset inequalities for robust network design. In Proceedings of INOC 2011, pages 118–123. International Network Optimization Conference, 2011. [17] Arie M. C. A. Koster, Manuel Kutschka, and Christian Raack. Robust network design: Formulations, valid inequalities, and computations. ZIB Report 11-34, Zuse Institute Berlin, August 2011. [18] Arie M.C.A. Koster and Manuel Kutschka. Network design under demand uncertainties: A case study on the abilene and geant network data. In Tagungsband der 12. ITG-Fachtagung Photonische Netze, volume 228, pages 154–161, Leipzig, Germany, 2011. VDE Verlag GmbH. http://www.vdeverlag.de/proceedings-de/453346030.html. [19] Arie M.C.A. Koster, Manuel Kutschka, and C. Raack. Towards robust network design using integer linear programming techniques. In Proceedings Next Generation Internet, NGI 2010. IEEE Xplore, 2010. http://dx.doi.org/10.1109/NGI.2010.5534462. [20] Arie M.C.A. Koster, Manuel Kutschka, and C. Raack. On the robustness of optimal network designs. In Proceedings IEEE International Conference on Communications, ICC 2011, pages 1–5. IEEE Xplore, 2011. http://dx.doi.org/10.1109/icc.2011.5962479. [21] S. Mudchanatongsuk, F. Ordóñez, and J. Liu. Robust solutions for network design under transportation cost and demand uncertainty. Journal of the Operational Research Society, 59:652–662, 2008. [22] A. Ouorou and J.-P. Vial. A model for robust capacity planning for telecommunications networks under demand uncertainty. In 6th International Workshop on Design and Reliable Communication Networks, 2007. DRCN 2007, pages 1–4, 2007. [23] M. Poss and C. Raack. Affine recourse for the robust network design problem: between static and dynamic routing. ZIB Report 11-03, Zuse Institute Berlin, February 2011. [24] A. Prékopa. Stochastic Programming, volume 324 of Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht/Boston/London, 1995. [25] M. G. Scutellà. On improving optimal oblivious routing. Operations Research Letters, 37(3):197– 200, 2009. [26] A. L. Soyster. Convex programming with set-inclusive constraints and applications to inexact linear programming. Operations Research, 21:1154–1157, 1973. 4 16 METAHEURISTIC OPTIMIZATION Zrinka Lukač University of Zagreb, Faculty of Economics & Business Trg J. F. Kennedy 6, 10000 Zagreb, Croatia zlukac@efzg.hr Abstract. Many real life problems are so complex in their nature that they cannot be efficiently and reliably solved within a reasonable amount of time by using traditional methods which guaranty the optimality of the solution. However, in recent years metaheuristics have proved to offer a promising approach to tackle this kind of problems. We give a short survey of the most important singlesolution based metaheuristics for combinatorial optimization problems from the conceptual point of view and analyze their similarities and differences. The special attention is given to concepts of intensification and diversification, which are the two driving forces of any metaheuristics. For each metaheuristics presented, we study the mechanisms through which the intensification and diversification effects are achieved. Keywords: metaheuristics, single-solution based methods, intensification, diversification 1 INTRODUCTION Many real life optimization problems are so complex that traditional methods offer very little help in search of their solutions. Very often it is hard to tell how the solution looks like and where to look for the solution. At the same time the brute-force approach is out of the question because solution space is too wide. Metaheuristics offer a promising approach to tackle this kind of problems due to their ability to find acceptable solutions within a reasonable amount of computational time. The term metaheuristic was first introduced by F. Glover in [4] and it refers to master strategies that orchestrate the interaction between strategies which find locally optimal solutions and higher level strategies which make possible escaping from local optima and searching the whole solution space. We give a survey of the most important single-based metaheuristics for combinatorial optimization problems as of today from the conceptual point of view and analyze their similarities and differences. Due to space limitation and extensiveness of the field, we consider single-solution based methods only, such as local search, tabu search, simulated annealing, greedy randomized adaptive search procedure, variable neighborhood search, iterated local search and guided local search. Finding the right balance between intensification and diversification plays one of the key roles in design of any successful metaheuristic. Hereby diversification refers to the effective exploration of the whole search space, while the intensification refers to the thorough exploration of the promising regions which might contain good solutions. According to Glover and Laguna [6], the main difference between the intensification and diversification is that during an intensification stage the search focuses on examining neighbors of elite solutions, while the diversification stage encourages the search process to examine unvisited regions and to generate solutions that differ in various significant ways from those seen before. These two opposing principals are the driving forces of any metaheuristic. Therefore for each method presented we highlight the mechanisms through which the intensification and the diversification are achieved and controlled. 2 SINGLE-SOLUTION BASED METHODS Single-solution based methods (also called trajectory methods) conduct the search by transforming a single solution by means of some iterative procedure. In general, after 17 creating an initial solution, in each iteration they create a set of candidate solutions and replace the current solution by some candidate solution chosen according to a given evaluation criterion. Note that the new solution is not necessarily improving. The procedure stops once the predefined stopping criteria is met. Very often the effectiveness of the search procedure relies heavily on the definition of the neighborhood structure. In general, singlesolution based methods are more intensification oriented. 2.1 Basic Local Search Local search is very likely the oldest and the simplest metaheuristic method [9]. It starts from a given initial solution and in each iteration picks a neighbor which improves the objective function. The algorithm stops when all candidate neighbors are worse than the current solution. Different strategies may be applied when choosing a neighbor, such as best improvement (choose the best neighbor), first improvement (chose the first neighbor better than the current solution) and random selection (chose a new solution randomly from the set of neighbors who are improving the current solution). The main disadvantage of the algorithm is that it does not posses the ability to escape local optima. Therefore, the intensification component is the dominant one. 2.2 Simulated Annealing Simulated Annealing [1,3,8,9] was one of the first metaheuristics to employ the idea of accepting solutions of a worse quality than the current one as a means of escaping from local optimum. It is inspired by the annealing process in metallurgy which involves heating and controlled cooling of a material so that atoms place themselves in a pattern that corresponds to the global energy minimum, thus reducing the defects in a material. In the initialization phase, one has to set the initial temperature T as well as the temperature cooling schedule. They are of crucial importance for the performance of the algorithm. The search starts from some initial solution s. Next, at each iteration a new candidate solution s’ is picked randomly from the set of neighbors of the current solution, N(s). If s’ improves the objective, then it becomes a new current solution. If not, it may still be accepted as a new current solution, but with acceptance probability p which is a function of the temperature parameter T and the difference of the objective function value in s’ and s. The acceptance probability decreases as the temperature parameter decreases. The temperature parameter is updated at the end of each iteration according to the cooling schedule. The search stops once the stopping criterion is met. The diversification and intensification mechanisms are controlled by the cooling schedule and the acceptance probability function. The diversification component is dominant at the beginning of the search since the acceptance probability is higher at higher temperatures. As the temperature decreases, the acceptance probability decreases and the bias moves towards the intensification. Therefore, the decrease of the temperature parameter drives the system from diversification to intensification, which in turn leads to the convergence of the system. At a fixed temperature, the acceptance probability decreases as the difference between the objective function value in s’ and s increases. 2.3 Tabu Search Tabu Search [1,3,4,5,6,9] was originally proposed by Glover [4] and is one of the most used metaheuristic methods. The main idea behind the search is the use of memory which enables both escaping from local optimum and diversification of the search. While simulated 18 annealing may accept degrading solutions at any time, tabu search will do so only if it is trapped in local optima. A short-term memory is implemented in the form of tabu list which contains a constant number of the most recently visited solutions or their attributes (depending on the definition of the search space). The length of a tabu list is controlled by a parameter called tabu tenure. At each iteration, the set of candidate solutions is restricted to the set of neighbors which are not tabu. The best of them is selected as a new solution. However, if the tabu list contains only some of the solution attributes, tabu list may be too restrictive in a sense that it may reject good solutions which have not yet been visited. In order to overcome this problem, a tabu move may be accepted if a set of predefined conditions (so called aspiration criteria) is satisfied. The tabu list is updated at the end of each iteration. The procedure stops once the stopping criteria are met. Tabu Search behaves like a best improvement local search algorithm. The difference is that the use of tabu list enables avoiding cycles and escaping from local optima. The use of tabu list has both an intensification and diversification effect on the search. In every step it imposes restrictions on the set of possible solution, thus having a diversification effect. At the same time, such a restriction determines the set of neighbors and in that way influences the choice of the best neighbor, thus having an intensifying effect. The length of the tabu list determines the balance between these two effects. The shorter the tabu list, the lower the influence of the diversifying effect. The longer the tabu list, the higher the influence of the diversifying effect. Intensification may also be achieved by means of medium term memory, which is very often represented in the form of the recency-based memory. For each solution (or its attribute), the recency based memory remembers the most recent iteration (or the number of successive iterations) it was involved in. The idea is to extract common features of elite solutions and then intensify the search around solutions which contain them. On the other hand, diversification may be achieved by means of long-term memory, which is very often represented in the form of frequency-based memory. For each solution (or its attribute), it memorizes how many times it has been visited. In this way it is possible to identify the regions which have already been explored and guide the search towards the unvisited regions. 2.4 Greedy Randomized Search Procedure (GRASP) Greedy Randomized Search Procedure [1,2,5,9] is an iterative greedy heuristic for combinatorial optimization problems in which each iteration consists of a construction and a local search step. The construction step is applied first. It builds a feasible solution by adding one new component at a time. A new component is added to the partial solution by randomly choosing an element from the restricted candidate list, which consists of  best candidate elements whose addition to the existing partial solution does not destroy feasibility. Hereby, the elements are evaluated according to some greedy function, usually the cost of incorporating this element into the partial solution already being constructed. After a new component is added to the partial solution, the candidate list is updated and the incremental costs are reevaluated. Once a complete solution is built, a local search step is applied in order to improve the solution. Very often a simple local search is performed. The process iterates until a predefined number of iterations is performed. The output of the algorithm is the best solution found in the process. Local search step has an intensifying effect on the search. On the other hand, diversification is achieved by means of restricted candidate list. Here, the length of the restricted candidate list, , is a crucial parameter which determines the diversifying effect of 19 the search. For =1, the construction step is equivalent to a deterministic greedy heuristics, since the best element from the candidate list is always added to the solution. For  equal to the number of candidate elements, construction is completely random, thus achieving the maximum diversifying effect. 2.5 Variable Neighborhood Search (VNS) Variable Neighborhood Search [1,3,5,7,9] is a metaheuristics method which conducts the local search by systematically changing the neighborhood structures. In that way, it explores increasingly distant neighborhoods of the current solution. It jumps from this solution to a new one if and only if a new solution improves the current solution. VNS is based on three simple observations. The first one is that a local optimum with respect to one neighborhood structure is not necessary so with another. The second one is that a global optimum is a local optimum with respect to all possible neighborhood structures. The third one is that for many problems local optima with respect to one or several neighborhood structures are relatively close to each other [7]. The basic VNS scheme can be described as follows. In the initialization phase, an initial solution s is generated and a finite number, kmax, of neighborhood structures to be used in search is defined. At the beginning of each iteration, the neighborhood counter k is set to 1. Then a shaking step begins. It refers to picking a random solution s’ from the k-th neighborhood structure of the current solution s. Next, a local search step is applied with s’ as the starting point. It should be noted that the local search step is independent of the neighborhood structures defined in the initialization phase. Let s’’ denote the so obtained optimum. If s’’ improves the objective, it becomes a new current solution s. Otherwise, the neighborhood counter is incremented by 1 and the shaking and the local search step are conducted again. This time shaking is conducted with respect to the new neighborhood structure. Once all the neighborhood structures are exhausted, i.e. when k = kmax, the stopping criteria are checked. The stopping criteria might be a maximum number of iterations, maximum number of iteration without improvement, a maximum CPU time allowed, etc. If the maximum number of neighborhood structures is exhausted and the stopping criteria still do not hold, the counter k is set to 1, and the whole process of running shaking and local search steps starts all over again. The output of the algorithm is the best solution found. The key issue in design of VNS metaheuristic is the choice of neighborhood structures. Local search step contributes to the intensification of the search, while diversification is achieved through shaking stage and the process of changing neighborhoods in case of no improvements. 2.6 Iterated Local Search (ILS) Iterated local search [1,3,5,9] is the most general of all the methods presented so far. It is a framework for some other metaheuristics (like VNS), but it can also incorporate other metaheuristics as its subcomponents. The main idea is very simple. It starts from some initial solution and applies local search on it. Then, at each iteration, it perturbs the obtained local optimum and applies local search on the perturbed solution. If the so obtained solution satisfies an acceptance criterion, it becomes a new solution. The process iterates until the stopping criterion is met. The local search component of the ILS framework can be any of the single-solution based metaheuristics. Its contribution is towards the intensification of the search. The balance between the intensification and the diversification is achieved through choice of 20 perturbation method and acceptance criterion. The choice of perturbation method is crucial. If a perturbation is too small, it might not be able to escape from local optimum and prevent cycling. Therefore small perturbations contribute to the intensification of the search. If a perturbation is too large, it might loose the memory of good properties of local optima already found. Large perturbations contribute to the diversification of the search. The acceptance criterion, on the other hand, acts as a counterbalance to perturbation. The effect of the acceptance criterion on the intensification and the diversification of the search is highly dependent on its definition. If only improving solutions in terms of objective function values are accepted, then it has a strong intensification effect. If any solution is accepted, regardless of its quality, then it has a strong diversifying effect. Acceptance criterion can take a range of values in-between these two extreme cases. 2.7 Guided local search (GLS) Guided local search [1,3,5,9,10] is another general metaheuristics. It acts like an upper level strategy on top of other metaheuristics in order to improve their efficiency. Unlike all the other methods presented so far, its main approach to escaping local optima and guiding the search is by means of dynamically changing the objective function. For a given optimization problem, a set of solution features is defined first, where solution features are solution characteristics used to discriminate between the solutions. A cost and a penalty is associated to each feature. In order to escape from local optimum, the method modifies the cost function by penalizing the unfavorable solution features. A penalty associated with each feature measures the importance of the feature. The higher the penalty, the higher the importance of the feature and the associated cost of having that feature in the solution. The method takes into account the current penalty value of the feature by considering the utility of penalizing that feature. If a feature is not present in the local optimum, then the utility of penalizing is 0. Furthermore, the higher the cost of the feature, the greater the utility of penalizing it. The more times the feature has been penalized, the lower the utility of penalizing it again. The penalty of the selected feature is always increased by 1, where the scaling of the penalty is normalized by some parameter . The algorithm starts form some initial solution, with all the penalties initialized to 0. It applies some local search method until a local optimum with respect to the modified objective function (the original objective function plus the penalties) is achieved. Here the local search step can consist of some other metaheuristic. The feature utilities are computed next and the penalties of the features having the maximum utility are incremented. The process iterates until the stopping criteria are met. The output of the algorithm is the best solution found. The efficiency of the search is affected by the choice of the solution features, their costs and parameter . The search will intensify in regions defined by lower costs of the features. On the other hand, diversification is achieved by penalizing the features of the local optima found so far. In that way, the search is diverted from searching the regions around local optima and directed towards the unexplored regions of the search space. For majority of problems, GLS is not very sensitive to the choice of . However, large values of  will create greater diversification effect, while small values of  will result in intensification of the search around local optima. The more effective the metehauristics used in the local search is, the lesser the value of  and the lesser the values of penalties are needed. 21 3 CONCLUSION Over the last couple of decades metaheuristics have proved to be a promising approach to tackle complex real life optimization problems. In this paper we have presented some of the most important single-solution based metaheuristics as of today. Special attention has been given to the principles of intensification and diversification, the two driving principals whose balance is crucial for design of any effective metaheuristics. We have presented each method from the conceptual point of view and highlighted the control mechanisms through which the intensification and diversification effects are achieved. It should be noted that the very same mechanism often has both an intensification and diversification effect on the search. In general, the higher the influence of the objective function criterion, the higher the intensification effect. On the other hand, diversification is achieved by using criteria other than objective function value. Therefore, in order to design an efficient problem specific metaheuristic, special attention should be given to finding the right balance between intensification and diversification effects. References [1] Blum, C., Roli, A., 2003. Metaheuristics in Combinatorial Optimization: Overview and Conceptual Comparison. ACM Computing Surveys 35 (3), pp. 268-308 [2] Feo, T. A., Resende, M. G. C., 1995. Greedy randomized adaptive search procedures. Journal of Global Optimization 6, pp. 109–133 [3] Gendreau, M., Potvin, J. (Eds), 2010. Handbook of Metaheuristics, 2nd ed., Springer [4] Glover, F. 1986. Future paths for integer programming and links to artificial intelligence. Computers & Operations Research 13, pp. 533–549 [5] Glover, F., Kochenberg, G. A. (Eds), 2003. Handbook of Metaheuristics, Kluwer Academic Publishers [6] Glover, F., Laguna, M., 1997. Tabu Search. Kluwer Academic Publishers. [7] Hansen, P., Mladenović, N., 2001. Variable neighborhood search: Principles and applications. European Journal of Operations Research 130, pp. 449–467 [8] Kirkpatrick, S., Gelatt, C. D., Vecchi, M. P., 1983. Optimization by simulated annealing. Science 220 (4598), pp. 671-680 [9] Talbi, E. G., 2009. Metaheuristics: From design to implementation. John Wiley & Sons [10] Voudouris, C., Tsang, E. 1999. Guided local search. European Journal of Operations Research 113 (2), pp. 469–499 22 Mathematical Models and Solutions for Network Design Problems Ulrich Pferschy Department of Statistics and Operations Research, University of Graz, Universitaetsstrasse 15, A-8010 Graz, Austria. pferschy@uni-graz.at Abstract We discuss two real-world scenarios of network design problems arising in the design of district heating systems and in the design of fiber optic networks. In both cases we can choose which customers to connect to the network, depending on the resulting profit, while the necessary connections incur construction costs. Maximizing the total profit minus the total cost yields the prize collecting Steiner tree problem (PCST) with additional connectivity constraints. We present different mathematical solution methods, namely cut-based models, multi-commodity flow formulations and path models. The computational behavior of these models will be illustrated. Keywords: Network Design, Prize Collecting Steiner Tree, Branch-and-Cut 1 Introduction Classical network design problems consider the most cost efficient way to connect a given set of customers to a network, often with additional conditions such as network reliability, capacity or degree constraints. A more complex problem arises if the owner of the network (e.g. a provider of public utilities) is free to choose which customers to connect. Indeed, it may be unprofitable to connect customers which pay a low prize for the network service but cause a huge connection cost because of their geographic position. Hence, we are faced with two decisions: On one hand, a subset of profitable customers has to be selected form a given ground set, on the other hand, the selected customers have to be connected by a network with lowest possible cost. In this situation we are faced with a trade-off between maximizing the sum of profits over all selected customers and minimizing the construction cost of the network. This leads to a prize-collecting objective function, which we studied extensively in Ljubic et al. [18]. The motivation of that paper was the connection of houses to a district heating system. More formally, we formulate this problem as an optimization problem on an undirected graph G = (V, E, c, p), where the vertices V are associated with profits, p : V → R+ , and the edges E with costs, c : E → R+ . The graph corresponds to the topology of the customer vertices with edges representing street segments or potential cable connections. The cost c of each edge is the cost of establishing the connection, e.g., of laying a pipe or cable. The profit p of each vertex gives an estimate of the potential gain of revenue caused if the associated customer is connected to the network. Then we 1 23 can formulate the Linear Prize-Collecting Steiner Tree problem (PCST) as finding a connected subgraph T = (VT , ET ) of G, VT ⊆ V , ET ⊆ E that maximizes total profits minus construction costs, i.e.   profit(T ) = p(v) − c(e) . (1) v∈VT e∈ET A different objective function, where the ratio of profit over costs (resembling the return on investment) is maximized was considered by Klau et al. [14]. Obviously, every optimal solution T will be a tree. Note that vertices of the graph, which represent junctions of cables or streets but do not incur customer profits, can be easily represented as vertices with zero profit. This problem was introduced in the literature (in a slightly modified form) by Segev [20] in 1987. Approximation algorithms were presented by Bienstock et al. [2], Goemans and Williamson [10], Johnson et al. [12] and Feofiloff et al. [7] Preprocessing and reduction procedures were described by Duin and Volgenant [5] and Uchoa [21]. Metaheuristics are given by Canuto et al. [3] and Klau et al. [13]. Fischetti [8] and Goemans [9] studied the polyhedral structure of a closely related problem. Exact algorithms and lower bounds required for enumeration schemes were developed by Engevall et al. [6], Lucena and Resende [19] and more recently by Haouari et al. [11]. 2 A Branch-and-Cut Algorithm The currently best algorithmic framework for the exact solution of PCST was developed in Ljubic et al. [18], on which this article is partially based. Their approach starts with a number of preprocessing steps to reduce the size of the given graph. These are a Least-Cost Test where each edge cost cij is replaced by the cost of the shortest path from i to j. The Degree-l Test to replace a connected set of junction vertices by a minimum spanning tree (see Uchoa [21] for extension of this idea), and a Minimum Adjacency Test, which merges two adjacent vertices i and j if min{pi , pj } − cij > 0 and cij = min cit . it∈E The general approach of [18] is an integer linear programming formulation defined on a directed graph model and using connectivity inequalities corresponding to minimum weight cuts in the graph to guarantee connectivity of the solution. Related models were used by Wong [24] and Fischetti [8]. An undirected cut model was given by Aneja [1]. Assuming that E contains all arcs (i, j) and (j, i) we introduce binary variables x ∈ {0, 1}E and y ∈ {0, 1}V to represent the solution tree T = (VT , ET ) with the following interpretation:   1 (i, j) ∈ ET 1 i ∈ VT xij = ∀(i, j) ∈ E, yi = ∀i ∈ V, i = r . 0 otherwise 0 otherwise 2 24 In a slightly simplified way the cut model for PCST can be written as follows: (CU T ) max  pi yi − i∈V subject to   cij xij (2) ij∈E xji = yi ji∈E − x(δ (S)) ≥ yk  xri ≥ 1 ∀i ∈ V \ {r} (3) k ∈ S, r ∈ S, ∀S ⊂ V (4) (5) ri∈E xij , yi ∈ {0, 1} ∀(i, j) ∈ E, ∀i ∈ V \ {r} (6) A given root vertex of the network is denoted by r ∈ V . The objective function (2) corresponds directly to our goal as given in (1). Constraints (3) guarantee that an arc is incident to a vertex i if and only if i is included in the solution set VT , while (5) enforces the inclusion of the root r. Finally, the crucial cut constraints (4) guarantee the connectedness of the solution tree. As usual, δ − (S) = {(i, j) | i ∈ S̄, j ∈ S} denotes the set of edges reaching into set S and starting from its complement. Note that disconnectivity would imply the existence of a cut S separating r and v which would clearly violate the corresponding cut constraint. The classical branch-and-cut approach requires the separation of the cut constraints. This means that instead of enumerating all (exponentially many) cut sets S we compute only the most violated cut, i.e. the cut with smallest cut value. Such a min-cut computation is well-known to be equivalent to a max-flow problem. This can be done very efficiently by Cherkassky and Goldberg’s maximum flow algorithm [4]. It turned out in the computational experiments (see [18] for details) that this straightforward approach fails to solve large test instances to optimality within reasonable running time. Thus, several algorithmic improvements were introduced. First of all, it is beneficial to add more than one cut to the ILP model in each iteration. Such a nested cut approach leads to faster changes of the value of the LP-relaxation. Furthermore, instead of computing flows only from the root to each customer vertex, it is also convenient to compute backcuts arising from a hypothetical flow from the customer back to the root vertex. Extensive computational experiments showed that the resulting algorithmic framework performs better than previous algorithms on benchmark instances from the literature and is able to solve problems with up to 60.000 edges to optimality within 10 minutes. Also real-world instances provided by a German telecom provider could be solved to optimality in reasonable running time. 3 Redundant Connectivity A classical issue for network design problems is reliability. In particular, the failure of a single link of the network, i.e. the disruption of a cable by construction work or technical failure, should 3 25 not disconnect any customer from the network root. Clearly, this requirement is fulfilled by a biconnected network structure where each vertex has two disjoint paths to the root. However, the construction of a biconnected network incurs extremely high costs. Thus, in many applications the network provider strives for a relaxed version of reliability. It may be acceptable that the disruption of a single edge of the network disconnects a limited set of customers from the network, as long as the vast majority of customer vertices remains connected. Such a scenario occurred in the design of fiber optic networks taking into account the last mile connections to the customers. This problem is currently a major issue for many telecommunication providers and was studied in the research project NETQUEST led by the Carinthia University of Applied Sciences and supported by the Austrian Research Promotion Agency (FFG), together with partners from industry. A formal model of the above idea was described in Wagner et al. [22, 23] and Leitner et al. [16, 15] and works as follows: For each customer vertex k we are given a distance value bmax (k) and require that vertex k is connected by a single path of length at most bmax (k) to a biconnected vertex, i.e. to a vertex with two disjoint paths to the root. This means that any failure of an edge will disconnect only a limited area of a certain diameter from the network. For many application scenarios this model offers a reasonable compromise between construction cost and network reliability. Of course, very important customers might still be biconnected by setting their distance value to 0. A direct extension of the directed cut model sketched in Section 2 for this model was given by Wagner et al. [22]. However, modeling the single path of length bmax (k) turned out to be against the spirit of the cut idea. It requires to identify for each vertex whether it is single or biconnected and then establishes the length of the path connection to the nearest biconnected vertex. Computational experiments showed only moderate success of this approach. The multi-commodity flow formulation introduced in Wagner et al. [23] models the connection of each customer vertex to the root by a different commodity within a multi-commodity flow problem. There are two different flows from the root to each vertex. Flows to biconnected vertices are required to be disjoint while flows on the last segment of length bmax (k) will coincide. This model was better suited to capture the essence of the relaxed connectivity. Finally, a column generation approach was introduced by Leitner and Raidl [15] and further developed in Leitner et al. [17]. It is based on the implicit enumeration of the sets Pk for all customer vertices k consisting of all feasible connections of k to the root. Each element p ∈ Pk is a fork-like structure consisting of two disjoint paths from r to a certain junction vertex zk and a single path connecting zk and k with length at most bmax (k). Binary assignment variables fpk assign a unique connection p ∈ Pk to each customer vertex k. Instead of enumerating the exponentially many fork connections they are generated iteratively in a branch-and-price framework (see Leitner et al. [17] for more details). It turns out that the associated pricing problem asks for the computation of the cheapest fork connection with positive edge costs. While this could be done very efficiently in polynomial time for the special cases of bmax (k) = ∞ (single connection) and bmax (k) = 0 (strictly biconnected), a mixed-integer linear programming model has to be solved for the general case. Alternatively, it can also be shown that the resulting problem is equivalent to an elementary shortest path problem with resource constraints. 4 26 Computational experiments show that the standard column generation approach suffers from primal degeneracy which worsens its performance considerably. Thus we applied stabilization techniques based on the selection of dual variables in the LP-relaxation (see Leitner et al. [16]). Computational results show that the branch-and-price approach performs reasonably well on small and medium sized instances while large test instances still pose an interesting challenge. References [1] Y.P. Aneja. An integer linear programming approach to the Steiner problem in graphs. Networks, 10:167–178, 1980. [2] D. 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Pferschy, P. Mutzel, G. Raidl, and P. Bachhiesl. A multi-commodity flow approach for the design of the last mile in real-world fiber optic networks. In Operations Research Proceedings 2006, pages 197–202. Springer, 2007. [24] R.T. Wong. A dual ascent based approach for the Steiner tree problem in directed graphs. Mathematical Programming, 28:271–287, 1984. 6 28 The 11th International Symposium on Operational Research in Slovenia SOR ’11 Dolenjske Toplice, SLOVENIA September 28 - 30, 2011 Section I: Pascal 2 session 29 30 A SURVEY ON KNOWN UPPER BOUNDS FOR MULTICHROMATIC NUMBER FOR HEXAGONAL GRAPHS Petra Šparl University of Maribor, Faculty of Organizational Sciences Kidričeva cesta 55a, SI-4000 Kranj, Slovenia petra.sparl@fov.uni-mb.si and Institute of Mathematics, Physics and Mechanics Jadranska 19, SI-1000 Ljubljana, Slovenia Abstract An optimization problem in the design of cellular networks is to assign sets of frequencies to transmitters to avoid unacceptable interference. Frequency assignment problem can be abstracted as a multicoloring problem on a weighted subgraph of the infinite triangular lattice, called hexagonal graph. In this paper a survey of known upper bounds for multichromatic number χm (G) in terms of weighted clique number ωm (G) for hexagonal graphs and some open problems are presented. Keywords: graph algorithm, approximation algorithm, graph coloring, frequency planning, cellular networks, local distributed algorithm 1 INTRODUCTION A fundamental problem concerning cellular networks is to assign sets of frequencies (colors) to transmitters (vertices) in order to avoid unacceptable interferences [2]. The number of frequencies demanded at a transmitter may vary between transmitters. In a usual cellular model, transmitters are centers of hexagonal cells and the corresponding adjacency graph is a subgraph of the infinite triangular lattice. An integer p(v) is assigned to each vertex of the triangular lattice and is called the demand of vertex v. The vertex-weighted graph induced on the subset of the triangular lattice of vertices of positive demand is called a hexagonal graph, and is denoted G = 1 31 (V, E, p). Hexagonal graphs arise naturally in studies of cellular networks, such as known Philadelphia examples [10]. A proper n-[p]coloring of G (also called multicoloring) is a mapping f : V (G) → 2{1,...,n} such that |f (v)| ≥ p(v) for any vertex v ∈ V (G) and f (v) ∩ f (u) = ∅ for any two adjacent vertices u and v. The least integer n for which a proper n-[p]coloring exists, denoted by χm (G), is called the multichromatic number of G. Another invariant of interest in this context is the (weighted) clique number ωm (G), defined as follows: the weight of a clique of G is the sum of demands on its vertices and ωm (G) is the maximal clique weight on G. Clearly, χm (G) ≥ ωm (G). Even for graphs with regular structure, such as hexagonal graphs, the problem of determining χm (G) is not trivial. It has been showed [6] that it is NP-complete to decide whether χm (G) = ωm (G) for an arbitrary hexagonal graph G. Hence, it is unlikely to expect that a polynomial time algorithm for computing χm (G) for an arbitrary hexagonal graph G can be devised. Therefore, it is of interest to study approximation algorithms for the discussed problem. In the last decade several results, reporting upper bounds for χm (G) in terms of ωm (G) for hexagonal graphs, appeared in the literature. In the continuation we will talk about so called distributed and k-local algorithms. An algorithm is distributed if it uses a distributed computation, such that for solving one big problem a set of interconnected processors is used. In the case of algorithms for graph multicoloring it means that we put processors in several vertices of a given graph such that each processor covers only a particular part of the graph. An algorithm for a graph multicoloring is distributed if computation of a single processor does not depend on the size of the graph. An algorithm is a k-local algorithm if instead of a global information it needs only some local information of a given graph G. In the case of a graph multicoloring this means that every vertex v ∈ G, that has to be multicolored, uses only information about the demands of vertices whose graph distance from v is less than or equal to k. The paper is organized as follows. In the next two sections the known results for multicoloring arbitrary hexagonal graphs and triangle-free hexagonal graphs are stated, respectively. In the last section some open problems are mentioned. 2 32 2 KNOWN RESULTS FOR ARBITRARY HEXAGONAL GRAPHS In this section a survey of known upper bounds for weighted chromatic number in terms of weighted clique number for an arbitrary hexagonal graph G are given. The most of listed results are obtained by using rich structural properties of hexagonal graphs, especially the fact that there exists a natural 3-coloring of the vertices of the infinite triangular lattice, which gives rise to the partition of the vertex set of any hexagonal graph into three independent sets. For arbitrary hexagonal graphs the upper bound χm (G) ≤ (4/3)ωm (G)+ C, where C is an absolute constant, is the best known for both distributed and non-distributed model of computation. The bound was almost at the same time, but independently, proved by several authors [6, 7, 18]. All proofs are constructive, thus implying the existence of 4/3-approximation algorithms. But  algorithm presented in [7, 8] is distributed and  only the 4ωm (G) bound. Later a distributed algorithm which gives guarantees the 3   a proper multicoloring of a hexagonal graph with at most 4ωm (G)+1 colors, 3   which is less than or equal to 4ωm3(G) , is given in [12]. Figure 1 shows two simple hexagonal graphs H and C9 , where numbers present demands of vertices. Note that ωm (H) = 3 and χm (H) = 4, which means that the bound χm (G) ≤ (4/3)ωm (G) + C is optimal for the general case. 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 2 1 4 4 1 4 4 1 4 4 4 4 C9 H Figure 1: Two simple examples of hexagonal graphs. An obvious drawback of the mentioned distributed algorithms is that each vertex needs one piece of global information, more precisely ωm (G). Since so called k-local algorithms are more effective in the practice, several local algorithms for multicoloring hexagonal graphs appeared in the literature, where vertices can communicate to its neighbors to obtain some local information on graph G. A framework for studying distributed on3 33 line assignment in cellular networks was developed in [5]. In particular 3competitive 0-local, 3/2-competitive 1-local, 17/12-competitive 2-local and 4/3-competitive 4-local algorithm are presented. In [14] a 2-local algorithm with competitive ratio 4/3 is given. The best ratio for 1-local case for general hexagonal graphs was first improved to 13/9 in [1], later to 17/12 in [19], and finally to 7/5 in [16]. 3 KNOWN RESULTS FOR TRIANGLE-FREE HEXAGONAL GRAPHS It turned out that better bounds for weighted chromatic number in terms of weighted clique number can be obtained for triangle-free hexagonal graphs. The conjecture proposed by McDiarmid and Reed [6] is that χm (G) ≤ (9/8)ωm (G) + C, where C is an absolute constant, holds for triangle-free hexagonal graphs. It is not difficult to see that for the cycle C9 , depicted on the right picture of Figure 1, it holds ωm (C9 ) = 8 and χm (C9 ) = 9, which means that the bound 9/8ωm (G) is the best possible for an arbitrary triangle-free hexagonal graph G. Several algorithms reporting upper bounds for χm (G) for triangle-free hexagonal graphs can be found in the literature. The bound χm (G) ≤ (7/6)ωm (G) + C, where C is an absolute constant, is the best known for the triangle-free hexagonal graphs at the moment. The history of the results for triangle-free hexagonal graphs is the following. In [4] a distributed algorithm with competitive ratio 5/4 was presented. Later a 2-local distributed algorithm with the same ratio was given in [13], while an inductive proof for ratio 7/6 is reported in [3]. Better results are obtained for special sub-classes of triangle-free hexagonal graphs, where some particular configurations are forbidden. Namely, a 2-local 7/6-competitive algorithm for multicoloring triangle-free hexagonal graphs with no adjacent centers (i.e. vertices which has at least two neighbors in G, which are not on the same line) is given in [15] and a 1-local 4/3-competitive algorithm for multicoloring a similar sub-class of hexagonal graphs is presented in [20]. A special case of a proper multicoloring is when p is a constant function. For example, a 7-[3]coloring of a graph G is an assignment of three colors between 1 and 7 to each vertex v ∈ G. An elegant idea that implies the existence of a 14-[6]coloring is presented in [11]. Recently an algorithm to find a 7-[3]coloring of an arbitrary triangle-free hexagonal graph G was given in [9], which implies that χm (G) ≤ (7/6)ωm (G) + C. This provides a shorter alternative proof to the inductive proof of Havet [3] and improves the short 4 34 proof of [11] that implied the existence of a 14-[6]coloring. Note that the algorithm in [9] is not linear because it uses a 4-coloring of a planar graph. This was improved very recently in [17], where a linear time algorithm for 7-[3]coloring of triangle-free hexagonal graphs is presented. 4 OPEN PROBLEMS In the case of multicoloring problem for an arbitrary hexagonal graph the remaining question is whether one can find a 1-local algorithm with competitive ratio 4/3. The 7-[3]coloring algorithm for multicoloring triangle-free hexagonal graphs presented in [17] is not distributed since the construction in one Lemma in the paper does not work in constant time if long paths exist. Therefore, it is an interesting question whether there exists a distributed algorithm for 7-[3]coloring of an arbitrary triangle-free hexagonal graph, using its rich structural properties. At last, the conjecture of McDiarmid and Reed, that for triangle-free hexagonal graphs the inequality χm (G) ≤ (9/8)ωm (G) + C holds, still remains opened. References [1] F.Y.L. Chin, Y. Zhang, and H. Zhu, A 1-local 13/9-competitive Algorithm for Multicoloring Hexagonal Graphs, Proc. of the 13th Annual International Computing and Combinatorics Conference (COCOON), volume 4598 of LNCS, pp. 526-536, 2007. [2] W. K. Hale, Frequency Assignment: Theory and Applications, Proc. of the IEEE 68(12):1497-1514, 1980. [3] F. Havet, Channel Assignment and Multicoloring of the Induced Subgraphs of the Triangular Lattice, Discrete Mathematics 233(1-3):219-231, 2001. [4] F. Havet and J. Žerovnik, Finding a Five Bicolouring of a Triangle-free Subgraph of the Triangular Lattice, Discrete Mathematics 244(1-3):103-108, 2002. [5] J. Janssen, D. Krizanc, L. Narayanan, and S. Shende, Distributed Online Frequency Assignment in Cellular Networks, Journal of Algorithms 36(2):119151, 2000. [6] C. McDiarmid and B. Reed, Channel Assignment and Weighted Colouring, Networks 36(2):114-117, 2000. 5 35 [7] L. Narayanan and S. Shende, Static Frequency Assignment in Cellular Networks, Algorithmica 29(3):396-410, 2001. [8] L. Narayanan and S. Shende, Corrigendum to Static Frequency Assignment in Cellular Networks, Algorithmica 32(4):697, 2002. [9] I. Sau, P. Šparl and J. Žerovnik, Simpler Multicoloring of Triangle-free Hexagonal Graphs, Accepted in Discrete Mathematics. [10] D. H. Smith, S. Hurley and S. M. Allen, A new lower bound for the channel assignment problem, IEEE Transactions on Vehicular Technology 49(4):12651272, 2000. [11] K. S. Sudeep and S. Vishwanathan, A technique for multicoloring triangle-free hexagonal graphs, Discrete Mathematics 300(1-3):256-259, 2005. [12] P. Šparl, S. Ubeda, and J. Žerovnik, Upper bounds for the span of frequency plans in cellular networks, International Journal of Applied Mathematics 9(2):115-139, 2002. [13] P. Šparl and J. Žerovnik, 2-local 5/4-competitive algorithm for multicoloring triangle-free hexagonal graphs, Information Processing Letters 90(5):239-246, 2004. [14] P. Šparl and J. Žerovnik, 2-Local 4/3-Competitive Algorithm For Multicoloring Hexagonal Graphs, Journal of Algorithms 55(1):29-41, 2005. [15] P. Šparl and J. Žerovnik, 2-local 7/6-competitive algorithm for multicoloring a sub-class of hexagonal graphs, International Journal of Computer Mathematics 87(9):2003-2013, 2010. [16] P. Šparl, R. Witkowski and J. Žerovnik, A 1-local 7/5-competitive Algorithm for Multicoloring Hexagonal Graphs, Submitted for publication in Algoritmica. [17] P. Šparl, R. Witkowski and J. Žerovnik, A Linear Time Algorithm for 7[3]coloring triangle-free Hexagonal Graphs, Submitted for publication in Inf. Proc. Letters. [18] S.Ubeda and J.Žerovnik, Upper bounds for the span of triangular lattice graphs: application to frequency planing for cellular networks, Research report No. 9728, ENS Lyon, September 1997. [19] R. Witkowski, A 1-local 17/12-competitive Algorithm for Multicoloring Hexagonal Graphs, Proc. of Fundamentals of Computation Theory, volume 5699 of LNCS, pp. 346-356 (2009). [20] R.Witkowski, 1-local 4/3-competitive Algorithm for Multicoloring a subclass of Hexagonal Graphs, Submitted for publication in Discrete Applied Mathematics. 6 36 A generalization of the ball packing problem Janez Žerovnik FME, University of Ljubljana, Aškerčeva 6, SI-1000 Ljubljana, Slovenia and Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia janez.zerovnik@fs.uni-lj.si, janez.zerovnik@imfm.si May 28, 2011 1 Introduction The packing problems, and in particular ball (or sphere) packing problems, have applications in many areas of science and technology, for example in Nuclear technics, Chemistry, Physics, Computer science, Telecommunications, etc. (see [26], and many later papers, ommited due to space limitation) Different versions of the basic problem and different aspects are studied in available scientific literature. They are very interesting NP-hard combinatorial optimization problems; that is, no procedure is able to exactly solve them in deterministic polynomial time. Packing problems are encountered in a variety of real-world applications including production and packing for the textile, apparel, naval, automobile, aerospace, and food industries. They are bottleneck problems in computer aided design where design plans are to be generated for industrial plants, electronic modules, nuclear and thermal plants, etc. Packing problems consist of packing a set of geometric objects/items of fixed dimensions and shape into a region of predetermined shape while accounting for the design and technological considerations of the problem. The packing identifies the arrangement and positions of the geometric objects that determine the dimensions of the containing shape and reach the extremum of a specific objective function. However, the search for exact local extrema is time consuming without any guarantee of a sufficiently good convergence to optimum. The literature on the ball packing and applications is enormous, the research is extensive, and there is a great variety of specific aspects that are elaborated. Later we will mention some work on the original ball packing problem related to the well-known Kepler’s conjecture, whre the maximal density packing of uniform balls is considered. On the other hand, there is not so much known 1 37 Figure 1: One of the two optimal arrangements and a diagram from Johannes Kepler’s 1611 Strena Seu de Nive Sexangula. about packing of balls with different sizes. Based on the searches through some available literature databases, it is clear that there is a number of simulation tools developed, and among them some recently developed simulation tools allow different shapes and different sizes of objects [8, 15, 16] . In certain applications, it is important to achieve near maximal density with balls of different sizes, where in addition, size of the balls introduces a cost that may be a cost of production or depend on the availability. This seems to be a new optimization problem because we did not find any similar approach in literature search. We also provide some preliminary remarks on the maximal density bounds under some natural assumptions on the cost of balls. 2 History of the ball packing problem The Kepler conjecture, named after Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is slightly greater than 74%. It may be interesting to note that the solution of Kepler’s conjecture is included as a part of 18th problem in the famous list of Hilbert’s problem list back in 1900 [29]. Recently Thomas Hales, following an approach suggested by Fejes Toth, published a proof of the Kepler conjecture. Hales’ proof is a proof by exhaustion involving checking of many individual cases using complex computer calculations. For more details, see [11, 12, 7]. The basic problem of packing balls of equal size into infinite 3D space can be naturally generalized in many ways. Obviously, it is interesting to study maximal possible density when the region in 3D is bounded. In this case, given the shape (and size) of the region one asks how many balls of given size can be 2 38 packed into the region. The second generalization is to allow balls of different sizes, where again there are many possibilities: there can be two, three or more ball sizes given, but one can also assume that the ball sizes are from an interval (approximating the realistic situation), or, furthermore, that the ball sizes are taken randomly according to some probability distribution. This type of problems can be generaly regarded as optimization problems, and in most cases, they are studied as discrete optimization problems in the literature. As the problems are usually NP-hard, the optimal solutions are very difficult to find. In other words, there is no polynomial time algorithm that would guarantee optimality of the solution. Therefore many classical and modern metaheuristcs are applied for particular versions of the problem. The meta-heuristcs for integer and nonconvex programming include taboo search, simulated annealing, etc. [1]. The hardness of the problem is well illustrated by the fact that even two dimensional problems of packing circles are NP-hard, when restricted to circles of equal sizes and to simple rectangular or circular regions, see for example recent survey paper [13]. As usual, the best results are reported when general meta-heuristics are accompanied with clever hints that are based on properties of particular type of problem or instance (compare with [31]). 3 New optimization problem From the point of view of Discrete optimization, or more generally, in view of Operational research, a possible approach to the problem is as follows. A general problem in the very spirit of discrete optimization that may be of interest for reasons mentioned before is to allow different ball (or, circle) sizes, but at the same time control the distribution of sizes by introduction of cost, so that each circle (ball) has a cost c(r) depending on the radius r. The motivation is that very small circles (balls) are expensive to produce and/or handle. Of course, we also have the primary optimization objective, the maximization of density. This naturally leads to the variety of optimization goals that may either be expressed as multicriterial optimization, or a single cost optimization where the cost function is a combination of the two criteria. The general problem should apply to arbitrary region (volume), however already simple regions such as rectangle, circle or elipsis in 2D, and cube, polyhedra, ball, or ellipsoid in 3D may be both hard and interesting to consider. 3.1 FORMAL PROBLEM DEFINITION General problem: INPUT: An arbitrary region R in 3D, and a treasury of balls with cost c(r) of a ball of radius r. 3 39 TASK: Pack the balls in R so that the volume of packed balls is maximised at minimal cost. The general problem has to be defined more precisely, in particular one should define in more detail the following: • regions, • treasury of balls, • objective function. Objective function. There are obviously two conflicting goals in the task. This can be defined more precisely by introducing a goal function that would combine the cost of material (i.e. the balls) and the value (quality) of achieved density. Hence standard methods apply: • multicriterial optimization (search for (un)dominated solutions) • priorities: first objective, second objective • design a single objective function AIM = VALUE - COST Note that when single criteria optimization is considered, one also needs to evaluate the profit of densities in the same units as the ball cost. Alternatively, one can formulate optimization tasks as follows: • given target density X%, find minimal cost solution. • given X units of money, find a mixture of balls that gives maximal density. • provide maximal density with cost at most XX • etc. 3.2 Theoretical bounds When evaluating the quality of results (i.e. density) of the solutions provided by heuristics, it is very useful to have good bounds for the optimal solutions. For example, recall that Hales’ proof of Kepler’s conjecture assures that the optimal packing density of equal sized balls is 74%, provided the volume is unbounded. The above densitiy applies to unbounded volumes, and are therefore may be good estimates (upper bounds) in case when the regions are ”regular” in the sense that the surface (boundary) is not too complex and the ball raduis r is small in comparison to the diameter of the region. One indeed has to be very cautious Clearly, Kepler’s maximal density can be improved if balls of more than one size are allowed. Simple reasoning implies that densities arbitrary close to 100% can be obtained with a long enough sequence of allowed ball sizes. When 4 40 restricted to only two balls sizes, a straightforward upper bound is 0.74 + 0.26 * 0.74 ≈ 93%. Straightforward here means that it is intuitively ”clear”, however it is not at all clear that it can be easily proved formally. It seems natural that the cost of different balls will most likely enforce smallest possible difference between two ball sizes. This leads to a question, which seems to be tractable: find the densities with two ball sizes at fixed ratio between radii. Then, depending on the cost of balls, it may be easy to find the optimal solutions, in some cases. We elaborate this question in more detail elsewhere. References [1] Emile Aarts and Jan Karel Lenstra, Local Search in Combinatorial Optimization, Princeton University Press, Princeton 2003. [2] Azeddine Benabbou, Houman Borouchaki, Patrick Laug, and Jian Lu, Sphere Packing and Applications to Granular Structure Modeling pages 118 Proceedings of the 17th International Meshing Roundtable, Garimella, Rao V. (Ed.), 2008. [3] A. Benabbou, H. Borouchaki, P. Laug, and J. Lu, Geometrical modeling of granular structures in two and three dimensions. Application to nanostructures, International Journal for Numerical Methods in Engineering 80 (2009) 425454. [4] J. Brest, J. Žerovnik, A heuristic for the asymmetric traveling salesman problem. In: The 6th Metaheuristics International Conference, August 2226, 2005, Viena, Austria. MIC 2005. Wien: Universitaett Wien, [2005], 145150. (See also: An approximation algorithm for the asymmetric traveling salesman problem, Ricerca Operativa 28 (1999) 59-67.) [5] H.J. Brouwers, Particle-size distributions and packing fraction of geometric random packing, Physical Review E, Statistical, Nonlinear and Soft Matter Physics 74 (2006) 031309 (14 pages). [6] Aleksandar Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato and P. M. Chaikin, Improving the Density of Jammed Disordered Packings using Ellipsoids, Science 303 (2004) 990-993. [7] Alireza Entezari, Towards Computing on Non-Cartesian Lattices, IEEE Transactions on visualization and computer graphics (submitted). [8] M. Gan, N. Gopinathan, X. Jia and R.A. Williams Predicting Packing Characteristics of Particles of Arbitrary Shapes, KONA 22 (2004) 82-93. [9] T. Gensane, Dense Packings of Equal Spheres in a Cube, Electronic J. Combinatorics 11, No. 1, R33, 2004. 5 41 [10] T. C. Hales, P. Sarnak, and M. C. Pugh, Advances in random matrix theory, zeta functions, and sphere packing PNAS u November 21, 2000 u vol. 97 u no. 24 u 1296312964 The Proceedings of the National Academy of Sciences USA (PNAS) [11] T. Hales, Cannonballs and honeycombs, Notices of the American Mathematical Society 47 (2000) 440449. [12] T. Hales, A proof of the Kepler conjecture, Annals of Mathematics. Second Series 162 (2005) 10651185. [13] Mhand Hifi and Rym M’Hallah, A Literature Review on Circle and Sphere Packing Problems: Models and Methodologies, Advances in Operations Research, Volume 2009, Article ID 150624, 22 pages. [14] H.M. Jaeger and S. R. Nagel, Physics of Granular States, Science 255 (1992) 1523-1531. [15] X. Jia, M. Gan, R.A. Williams and D. Rhodes, Validation of a digital packing algorithm in predicting powder packing densities Powder Technology 174 (2007) 10-13. [16] X. Jia and R. A. Williams, A packing algorithm for particles of arbitrary shapes, Powder Technology 120 (2001) 175-186. [17] Jong Cheol Kim, David M. Martin and Chang Sung Lim, Effect of rearrangement on simulated particle packing, Powder Technology 126 (2002) 211-216 . [18] B. Korte, J. Vygen, Combinatorial Optimization: Theory and Algorithms, Algorithms and Combinatorics 21, Springer, Berlin 2000. [19] E.J.Maginn, Molecular simulation of ionic liquids: currecnt status and futute opportunities, J. Phys.: Condens. Matter 21 (2009) 373101 (17pp). [20] Weining Man, Aleksandar Donev, Frank H. Stillinger, Matthew T. Sullivan, William B. Russel, David Heeger, Souhil Inati, Salvatore Torquato and P. M. Chaikin, Experiments on Random Packings of Ellipsoids, Physical Review Letters 94 (2005) 198001 (4 pages). [21] V. V. Pavlov, Distinctive features of a crystal, rystal-like properties of a liquid and atomic quantum effects 13th International Conference on Liquid and Amorphous Metals IOP Publishing Journal of Physics: Conference Series 98 (2008) 052008 [22] Igor Pesek, Andrea Schaerf, Janez Žerovnik, Hybrid local search techniques for the resource-constrained project scheduling problem. Lect. notes comput. sci., vol. 4771, 57-68. 6 42 [23] Igor Pesek, Iztok Saje, Janez Žerovnik, Frequency assignment - case study. Part I, Problem definition. Frequency assignment - case study. Part II, Computational results, Proceedings of the 9th International Symposium on Operational Research in Slovenia, Nova Gorica, 2007, pp. 63-68 and 69-74. [24] Nadge Reboul, Eric Vincens, and Bernard Cambou, A statistical analysis of void size distribution in a simulated narrowly graded packing of spheres, Granular Matter (2008) 10:457468. [25] Roland Roth, Fundamental measure theory for hard-sphere mixtures: a review, J. Phys.: Condens. Matter 22 (2010) 063102 (18pp.). [26] G.D.Scott and D.M. Kilgour, The density of random close packing of spheres, Brit. J. Appl. Phys. (J. Phsy. D) 2 (1969) 863866. [27] Yu Shi and Yuwen Zhang, Simulation of random packing of spherical particles with different size distributions Appl Phys A (2008) 92: 621626. [28] A.Vesel and J.Žerovnik, Improved lower bound on the Shannon capacity of C7 , Information Processing Letters 81 (2002) 277-282. [29] Benjamin H. Yandell, The Honors Class. Hilbertov Problems and Their Solvers, A K Peters, 2002. [30] J. Q. Xu, R. P. Zou, and A. B. Yu, Analysis of the packing structure of wet spheres by Voronoi-Delaunay tessellation, Granular Matter (2007) 9:455463. [31] J. Žerovnik, Simulated annealing type metaheuristics - to cool or not to cool, SOR ’03 proceedings. Ljubljana: Slovenian Society Informatika, Section for Operational Research, 2003, (6 pp.). [32] Frederick S Hillier, Gerald J Lieberman, Introduction to Operations Research, McGraw-Hill New York 2005. 7 43 44 BALL PACKING WITH TWO DIFFERENT BALL SIZES: SOME BOUNDS BASED ON GEOMETRY Janez Žerovnik FME, University of Ljubljana, Aškerčeva 6, SI-1000 Ljubljana, Slovenia and IMFM, Jadranska 19, SI-1000 Ljubljana, Slovenia janez.zerovnik@fs.uni-lj.si, janez.zerovnik@imfm.si Bojan Kuzma FAMNIT, University of Primorska Glagoljaška 8, SI-6000 Koper, Slovenia and IMFM, Jadranska 19, SI-1000 Ljubljana, Slovenia bojan.kuzma@upr.si Abstract: Densities of ball packing with two different sizes of balls is studied. For several ratios between the radii estimates for densities on unbounded regions are given. Keywords: combinatorial optimization, ball packing, sphere packing, Kepler conjecture. 1 Introduction The packing problems, and in particular ball (or sphere) packing problems, have applications in many areas of science and technology. In another short paper (this proceedings) we have introduced an optimization problem in which the goals are both maximizing the density and minimizing the cost that depends on the size of the balls. To the best of our knowledge, the optimization problem in this form has not been studied before. Besides experimental studies of particular versions of the problem (to be worked out elsewhere), it may also be important to study lower and upper bounds for the optimal solutions. The general problem is namely NP-hard and the same is expected for most of the specific problems. The seemingly simple basic problem of packing balls of uniform size in unbounded 3D space has been a challenge for mathematicians for several centuries. The Kepler’s conjecture, says that no arrangement of equally sized spheres filling space has greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is slightly greater than 74%. It may be interesting to note that the solution of Kepler’s conjecture is included as a part of the 18th problem in the famous list of Hilbert’s problem list back in 1900 [4]. A proof, widely believed to be rigorous, appeared only recently [1]. A minor dispute over validity stems from the fact that the bulk of the proof consists of computer calculations and has spurred a new project Flyspeck to formally verify the calculations [2]. In this short note we report on some preliminary observations on densities that can be achieved with balls of two different sizes. The region is unbounded, and as we have no boundary, only the ratio between the radii of large and small balls is important. For 101 45 several ratios we provide constructions and compute the densities. These densities may be seen as lower bounds for the maximal density that can be obtained with the ratio. 2 Basic geometry First we give some constants (ratios) that are computed from the optimal packing of spheres of uniform size (see for example [3, 5]). Definitions: • R, size of the basic balls • v, altitute of the equilateral triangle (three kissing balls centered at the vertices) • h, distance between two layers in the optimal arrangement, height of a tetrahedron (four kissing balls centered at vertices) • rF , maximal radius of a ball that can be moved between three kissing balls • dT , distance from the centre of gravity of tetrahedron to a face • dO , distance from the centre of gravity of octahedron to a face • rT , maximum radius of a ball in tetrahedronal void • rO , maximum radius of a ball in octahedronal void Formulas: √ • v = 3R  1.732R √ 2 6 R  1.633R 3 √ = ( 2 3 3 − 1)R  0.155R √ = 66 R  0.408R √ = 36 R  0.816R √ = ( 26 − 1)R  0.225R • h= • rF • dT • dO • rT √ • rO = ( 2 − 1)R  0.414R In the optimal packing (both hexagonal close packing and cubic close packing) the number of octahedral voids is equal to the number of balls and the number of tetrahedral voids is twice this number. 102 46 3 Maximal density with two ball sizes Based on the above ratios, we estimate the densities of the packings that are obtained by first packing the big balls in one of the optimal ways, and then filling the voids with the balls of smaller radius. First we consider the radii rT , rO , and rF = 0.154R. (For simplicity, we will set R = 1, so for example rF = 0.154R = 0.154.) The last radius, rF , is small enough to fit into the narrowness between three kissing balls, and hence the balls of this size or smaller might flow through already rigid structure composed of the large balls. This may be an interesting feature when simulating random arrangements. First we count how many smaller balls can be put into the voids between larger balls. Recall that in general the problem of positioning the optimal number of smaller balls into a bounded region is a difficult optimization problem. Let kT denote the number of smaller balls that fit into the tetrahedral void, and let kO denote the number of smaller balls that fit into octahedral void. The exact numbers of kT and kO are very difficult to find, near optimal constructions improving the bounds may be found by heuristic search. The numbers in the Table 1 for the second and the third radius are a rough estimate obtained by a simple algorithm. Design of a more sophisticated algorithm is in progress. We wish to emphasize that while the optimization problems that arise in the cases from Table 1 are hard, there is some hope to obtain good constructions by hand because the number of balls in the optimal solutions are rather small. On the other hand, when smaller radii (or better, larger ratios between radii) is considered, the only realistic approach is to design and run suitable heuristic algorithms to obtain near optimal solutions. Table 1: Fitting the smaller balls into the tetrahedral and octahedral voids smaller ball radius r rO rT rF =0.154 tetrahedral void kT 0 1 3 octahedral void kO 1 4 20 The figures from Table 1, more precisely the constructions behind, give rise to infinite arrangements of balls with higher densities than 74%. They may be understood as obvious lower bounds on densities with given ratio between sizes of large and small balls. Note that all the above constructions are regular in the sense that we find certain solutions how to fit a number of balls in both types of voids, sometimes assuming that each face of the void meets a ball but there is always at least half of the ball inside the void. Furthermore, we found certain arrangements and have no proofs (yet) that they are best possible, even under the assumptions mentioned. We continue by a rough calculation of the densities obtained. These numbers are, as argued above, lower bounds on densities for each fixed ratio of ball sizes, but at the same time any of them gives a lower bound on the maximal density that may be achieved 103 47 when two ball sizes are allowed. Finally, we mention an obvious estimate of the maximal density that can be achieved with two ball sizes. Provided that the smaller balls are much smaller that the large balls, the voids are much larger than the smaller balls, and consequently one may believe that the density will be close to the optimal density on infinite region, hence the estimate 0.74 + 0.26 ∗ 0.74 = 0.9324 However we have to be careful, because it is known that for certain region boundaries, it is possible to have higher density than 74%, so we can not claim that the estimate is indeed a proven upper bound on the maximal possible density on infinite region with two ball sizes. In order to compute the lower bounds from our constructions we first compute the proportions of the number of larger and smaller balls. The number of small balls is computed as follows: roughly speaking, for each large ball we have two tetrahedral and one octahedral void (provided the region is infinite or at least large enough). Therefore for each large ball, we can use n2 = 2 ∗ kT + 1 ∗ kO smaller balls, assuming n1 = 1. The densities are computed as follows: if the volume of a ball with R = 1 is taken as unit then a smaller ball has volume of r3 units. The densities in the table are thus computed by the formula 0.74 + n2 ∗ r3 /0.74. Table 2: Proportions of number of balls in the constructions and respective densities smaller ball radius r rO rT 0.154 r << 1 4 large balls n1 1 1 1 1 small balls n2 1 6 26 >> 1 density 0,8357 0,8322 0,8681  0, 9324 Concluding remarks Constructions for ball packing of balls in unbounded regions with two allowed ball sizes were considered. It is clear that the densities higly depend on the ratio between ball sizes. However, even for each fixed ratio we have in general a difficult mathematical problem that in most cases does not allow analytical solution and it is therefore natural to consider the question as an optimization problem. These optmization problems are expected or known to be NP-hard, so it may be interesting to use heuristic methods to compute near optimal solutions. 104 48 Figure 1: Tetrahedral and octahedral void with attached four, respectively, six, touching balls. Black dots are their centra. 105 49 On the other hand, although the preliminary results given here may lead to conjecture that for relative small proportions of radii the density need not uniformly increase, it is much less doubt that the density will uniformly increase when the radii proportion will increase to infinity, i.e. when the smaller radius will be very small. Last but not least, the prelimiary analysis given here provides some ideas on the lower bounds on densities with certain assumptions mentioned in the text. Perhaps the most important is that we have assumed that in the optimal constructions the large balls are arranged according to one of the optimal arrangements and the smaller balls only cover the voids of this arrangements. If this assumption is dropped, there are many more possible arrangements. However, our lower bounds and the estimated upper bound remain valid. References [1] T. C. Hales, A proof of the Kepler conjecture. Ann. of Math. (2) 162 (2005), no. 3. [2] T. C. Hales, J. Harrison, S. McLaughlin, T. Nipkow, S. Obua, R. Zumkeller, A revision of the proof of the Kepler conjecture. Discrete Comput. Geom. 44 (2010), no. 1, 1-34. [3] P. Krishna and D. Padney, Close-Packed Structures, University college Cardiff, 1981. [4] Benjamin H. Yandell, The Honors Class. Hilbert’s Problems and Their Solvers, A K Peters, 2002. [5] http://en.wikipedia.org/wiki/Tetrahedron 106 50 The 11th International Symposium on Operational Research in Slovenia SOR ’11 Dolenjske Toplice, SLOVENIA September 28 - 30, 2011 Section II: Graphs and Their Applications 51 52 HOMOGENIZATION OF DIGITAL CADASTRE INDEX MAP IMPROVING GEOMETRICAL QUALITY Marjan Čeh*, Frank Gielsdorf** and Anka Lisec*** * University of Ljubljana, Faculty of Civil and Geodetic Engineering, Ljubljana, Slovenia, e-mail: mceh@fgg.uni-lj.si **Technet GmbH, Berlin, Germany, e-mail: frank.gielsdorf@technet-gmbh.com *** University of Ljubljana, Faculty of Civil and Geodetic Engineering, Ljubljana, Slovenia, e-mail: anka.lisec@fgg.uni-lj.si Abstract: In Slovenia, as well as in the comparable traditional cadastral societies, a digital land cadastre index map is a composition of digitized various analogue land cadastre maps and measurement data sets with different positional accuracy, depending on the scale of the map and on the methodology (quality) of data acquisition. The growing role of spatial data quality for GIS applications as well as the needs of advanced land administration systems, calls for improvement of the geometrical quality of land cadastre index maps, which is being associated also with the problem of heterogeneity of graphical data. For this purpose, the homogenisation process of digital land cadastre index maps is needed. In the article, we present the approach to the improvement of land cadastre index map geometrical quality, based on surveying measurements, where the basic principles of geodetic profession are to be respected (coordinate geometry, topology, adjustments, error propagation law etc.). Keywords: cadastre map, homogenization, membrane method, proximity fitting. 1 INTRODUCTION Efficiency and transparency of land administration systems are one of crucial importance in the society since land has always remained at the foundation of human life. Here, land administration system can be understood as conceptualization of rights, restrictions, and responsibilities related to people, policies and land (places), and it has been closely linked with the land evidences, land (cadastral) maps since ancient times (see [2] and [8]). Problems concerning effective land administration supporting suitable use of land resources are becoming more and more important all over the world. In recent time, there has been a revival of interest in the role and operation of land administration systems, which have been strongly influenced by development of information technology, in particular by development of geographical information technology. The core sub-system of the advanced land administration system has become a high quality digital data on land and real property organized in the multipurpose geographical (land) information systems GIS (LIS). Focusing on the “parcel based cadastral societies”, a common characteristic is the heterogeneous graphical subsystem of digital land cadastre data. Coordinates of land parcel border vertexes, are mainly derived from positional heterogeneous analogue maps that were digitized and geo-referenced. The analogue maps, which in Slovenia originate from the beginning of the 19th century, had been maintained throughout the centuries with manual technique of graphical adjustment. After digitalization of the heterogeneous analogue cadastral maps, integration of hundreds of analogue map sheets and mapping units (cadastral communities) has been executed that caused additional errors that were adjusted manually. Therefore improvement of geometrical accuracy of land cadastre index map has become a challenging issue in Slovenia. In addition, the integration of contemporary measurement data due to the permanent maintenance of the land cadastre index map is an important issue [1]. The resulting geometrical quality of digital land cadastral index map reflects the quality of underlying analogue cadastral maps and its integration, The problem of geometrical improvement of graphical subsystem of digital land cadastre is common for the 53 traditional cadastral societies – not only Slovenia, but also Austria, Germany etc., and has been proven as serious problem in the field of spatial data analysis and spatial decisions since land cadastre index map has became widely used in the framework of different land administration systems (agriculture, spatial planning, utilities, environmental) through overlay in geographical information systems (GIS). 2 RESEARCH PROBLEM The problem of measurement based spatial data is a wide problem in the field of GIS and spatial data infrastructure [9]. In general, GIS is designed to consider geometrical parameters (coordinates) as deterministic value. This is often misunderstanding of the nature of spatial data in GIS; coordinates are namely always calculated from the observations, which are redundant aleatory values. Consequently, coordinates are aleatory values as well as correlated. The accuracy of relative geometry is higher than the absolute accuracy – coordinates and relative measures are therefore not equivalent. As point positions remain unchanged in reality, new determined (measured) coordinates lead to a virtual displacement of the related point in GIS. Without any consideration of neighbourhood relationships to other points at all, the relative geometry between updated and unchanged points in GIS would be highly violated – such an integration of new coordinates are not to be accepted. Here, a virtual displacement of points the different accuracy of relative and absolute geometry has to be considered [4]. Spatial data integration has therefore become a widely used term, covering a variety of processes. In this paper it is used to describe methods that attempt to improve the geometrical accuracy of land cadastre vector dataset (index map) based on integration with more accurate measurement data at identical points. Land cadastre positional data are predominantly based on observations for direct mapping but the original measurements data are most often not stored. In some countries surveys for cadastral datasets, for example in the area of former Habsburg monarchy, started around 200 years ago. The quality of measurement equipment as well as methodology has improved dramatically since than. Therefore, the problem of integrating new measurements in the existing datasets appears. A transformation is necessary to fit the best results of the new measurements into the old dataset. Adding new measurements to the old datasets, by storing point coordinates only, does not improve the quality of the dataset. Surveying measurements result in more accurate point coordinates providing the basis for improvement of spatial data – and its homogenisation. So called data conflation (spatial data integration) can involve shifting one dataset (land cadastre vector index map) to align with other target datasets (measured identical points). Another approach is Positional Accuracy Improvement (PAI), which is the main topic of our paper. Localized pockets of higher accuracy data (contemporary measurements of higher positional accuracy) are used to improve the positional accuracy of surrounding (neighbourhood) low level positional accuracy datasets, for example land cadastre vector index map. Rather than considering this higher accuracy data to be a target dataset, methods developed for PAI determine the best fit positioning solutions using positional information of both dataset; these methods are based on surveying adjustment theory and are also able to preserve geometric properties such as straight or parallel lines (see also [4][5]). 3 APPROACH (PROBLEM SOLVING) The geometrical quality of the cadastral index map can be improved by integration of precise geodetic measurements, existing field book measurements as well as other sources of data 54 (photogrammetric measurements etc.). The new measurements data might be the result of the sporadic cadastral data maintenance activities or result of systematic projects of mass measurements. The integration of new, more accurate spatial data into existing data sets should result in more accurate point coordinates (location) and should improve and homogenise geometrical accuracy of neighbouring cadastral (spatial) data. Here it should be stressed, that measured coordinates are random variables and they are stochastically dependent. If the coordinates of land plot boundary in GIS would be stochastically independent we could just exchange the less accurate coordinates by more accurate ones. But such an approach would neglect geometrical neighbourhood relationships. Here, wider areas (not only the measured land parcel) should gain from this accuracy improvement without loosing its internal geometrical quality. The current system for maintaining of land cadastre index map in Slovenia does not support such strategies, and update transformations lead to inconsistency because proximity fitting principle is avoided. To keep neighbourhood relationships, proximity fitting methods should be applied. Spatial heterogeneity implies each location has intrinsic uniqueness, conditions vary from place to place. The lengths of the edges in network may be equal, but they have different relative interpretations within the clusters based on absolute distances. It is clear that relative proximity is more important than absolute proximity in geo-referenced settings and thus geospatial clustering [7]. Proximity fitting methods substitute a usual single system transformation. The result of a usual transformation can be seen as the first step of proximity fitting. At transformation phase an artificial coordinate differences between identical points, the connection points, and the new points are introduced into the adjustment. These „pseudo observations‟ are weighted dependent on the distances. There are no direct neighbourhood relationships between the interpolated points. Additionally, the result depends on the number of identity points which create residuals. The method is not suitable to model direct neighbourhood relationships. The resulting displacements of the new points are dependent of the density and distribution of the identical points. In a second step the mapping approach is extended by the introduction of relative geometry information. Advanced methods use the Delaunay triangulation to model neighbourhood relationships directly. The resulting displacements are here independent of the density and distribution of identity points. Suggested approach is called „membrane method‟. This method uses as functional model coordinate differences along the triangle sites, what leads to linear residual equations with a very stabile convergence behaviour. The stochastic model is derived from finite element methods, and it simulates the behaviour of a rubber membrane. The proximity fitting is run as an adjustment calculation [3][4]. 3.1 Geometric data homogenization with adjustment techniques The coordinates in GIS (cadastre index map) result from the evaluation of measured values. In a first step these measured values often were local coordinates of digitized analogue maps which were transformed into a global reference frame. These so determined global coordinates describe the geometry of the GIS objects unique whereby the coordinates have to be addressed as random variables. During the process of PAI new measured values with higher accuracy are introduced. The new measured values are redundant to the already existing coordinates. Therefore, the determination of new coordinates with improved positional accuracy is a typical adjustment problem. But measured values have two essential properties [4]: - They are random variables. Because it is impossible to measure a value with arbitrary accuracy, which leads to the fact that any measured value contains some uncertainty. 55 - They are redundant. Commonly there exist more measured values then necessary to be able to calculate unique point coordinates. A function of random variables results again in a random variable. Because of point coordinates are functions of measurement values they are like them random variables. The uncertainties contained in measured values lead necessarily to uncertainties in point coordinates. For the unique determination of a number of coordinates the exact same number of measured values is necessary. 3.1.1 Error Propagation for Linear Functions The law of error propagation describes the propagation of accuracies for linear functions of random variables. Applying this law to an adjustment it is possible to calculate the standard deviations of the unknown parameters and those of the residual errors. This case can be explained with a simple example. Fig. 1 shows points in one dimensional coordinate system. A 1 dd1 2 3 4 5 Ij-1 Ij Dd2 Dd3 Dd4 Dd5 Ddj Ij+1 Ddj+1 nn Ddn Nx Figure 1: Points in a 1-dimensional coordinate system. The coordinate of the control point A is known and fixed – A(xA); based on measured distances di, we want to calculate the coordinate xi of the new point i: j x j  x A   di (1) i 1 According to the law of error propagation (see [3]), the standard deviations of xi is: i  x2    x2 . i j 1 (2) j 3.1.2 Adjustment Considering the Correlations We want to interpret the standard deviations of the distances di and those of the coordinate xi. The standard deviation of the coordinate σxi represents the absolute accuracy of the coordinates in relation to the reference frame. On the other hand the standard deviations of the distances σdi represent the relative accuracy of the coordinates related to each other. If two calculated random values are functions of partial the same random variable arguments they are stochastically dependent. The degree of their stochastic dependency is quantified by their covariance. For one dimensional coordinate system (Fig. 1), the parameters xj-1 and xj are stochastically dependent because they are functions of partial the same random variables. The distances d1…dj are arguments of both functions (1). For this purpose generalisation of the law of error propagation is supposed. The general form of the law of error propagation can be represented in matrix notation. If there is a system of linear equations F describing the functional dependency of parameters xj on the arguments di X  F d (3) and the standard deviations of di are known then the variances and covariances of the parameters xj can be calculated by: 56 Cxx  F  Cdd  F T . (4) In this formula the functional matrix F contains the coefficients of the linear functions (1). The matrix Cdd is called the covariance matrix of observations and contains the variances of observations on its principal diagonal and their covariances on its secondary diagonals. In the most common case of stochastically independent observations Cdd is a diagonal matrix. Cxx is the covariance matrix of the unknown parameters and contains their variances and covariances. For the study case (Fig. 1) the Cdd matrix contains the variances of xj-1 and xj as well as their covariance: Cx j1x j   x2j1 cov( x j 1 , x j )  .   x2j cov( x j 1 , x j )  (5) If we solve the matrix equation for the general law of error propagation in a symbolic way then we get the expression:  x2 j 1  x j   x2j1   x2j  2  cov( x j 1 , x j ) , (6) where cov( x j 1 , x j )   x j1x j   x j1   x j  f (d j ) . (7) This formula (6) which does not neglect the covariance between dependent random variables yields the right result (see [3]). 3.1.3 The positional accuracy improvement of cadastral index map The problem of local geometrical distortions is illustrated with the following case. There is a GIS layer containing parcel boundaries of 4 boundary points. A surveyor may have determined coordinates (new high accurate measurements) of 4 boundary points. The positional standard deviations of the measurements are about 2 cm and substantially much more exact than the graphic coordinates in GIS layer. The introduction of the new coordinates would cause the following situation in the GIS (Fig. 2): Figure 2: (Left) cadastral boundaries (solid lines) and boundaries determined by higher accuracy measurements (dashed lines). (Middle) polygon xady is distorted if points abcd are replaced by ABCD. (Right) consideration of neighbourhood. In example the new determined points A, B, C and D are introduced in a map based cadastral data set. Neglecting the correlations leads to a massive distortion of the (left) parcel xady. The simplest way to use the exact (high accurate) coordinates would be just to exchange them with the existing graphic coordinates. Unfortunately this simple action would cause the distortion of the adjacent parcel geometry! An obviously much better approach would keep 57 the geometrical neighbourhood relations seen in the following picture (Fig. 2). The neighbourhood accuracy of two points which are descended from a digitized map is higher than their absolute positional accuracy. For this reason, the correlation is a function of the point distance. The smaller the point distance the greater the correlation. The distance-dependent correlations of the coordinates have two essential reasons: - The origin measurements were done with respect to the principle of neighborhood (e.g. tape distance measurements). - The manual mapping was done according to the principle of neighbourhood as well. Homogenization which models the distance-dependent correlations directly uses sophisticated adjustment algorithms for the calculation and analysis of coordinates. Firstly, topological neighbourhood information is determined. This is performed by a Delaunay triangulation over all GIS points (control points and new points) of the origin system. The triangle sides are used as carriers of neighbourhood information. All cadastral index map points are used for a Delaunay triangulation. Along the triangle sides, artificial coordinate difference observations are generated which are subsequently introduced in an adjustment calculation according to the least squares method. The observation values are derived from the coordinates in the origin system. The triangular net is acting like a homogeneous membrane; its elasticity is given by the weights following the map digitization accuracy. The remaining divergences of the control points are propagated thus on the new points (Fig. 4). Figure 3: Homogenization: Residuals of the identical points (gray dots) are transmitted by triangle sides (dashed lines), the neighbourhood is considered. 4 CONCLUSIONS The aim of this paper is to present method for homogenization of cadastre index maps. The method, suggested here, follows homogenization applying adjustment techniques, which uses simulated and real measurements. The method has been applied to analyse how heterogeneous cadastre map parts get fitted to the real positions by the use of point positions, relative measurements and geometrical constraints. The significance of adjustment techniques for transformation problems is recognized long ago. To apply the least squares method following C.F. Gauß is the usual geodesic practice for two dimensional transformations with redundant identities from one Cartesian system into another. This classical adjustment method can be expanded to a simultaneous 58 transformation of multiple systems, subsequently called „Interconnected Transformation‟ which is not the subject of these paper. If neighbourhood geometry is to be maintained through the inclusion of relative distances, as well as geometric and topological properties, there will be redundancy in the data integration process. It will not be possible to obtain a solution that perfectly meets every constraint. Instead, an adjustment problem is set up, for which an optimal solution can be determined using the method of weighted least squares. All of the available information is considered, and observations are weighted by their recorded accuracy values to determine the solution that best fits the datasets being integrated. Moreover, the method of least squares generates precision values for the calculated parameters, thereby enabling update of the positional accuracy of the upgraded dataset (see also [6]). It is known that the best consideration of neighborhood relationships is warranted using proximity fitting adjustment methods where artificial observations between points are integrated. Advanced adjustment programs use for that task finite element methods based on triangles. Nevertheless, the real observations can completely be introduced in these proximity fitting adjustment processes. References [1] Čeh, M., Lisec, A., Ferlan, M., Šumrada, R., 2011. Geodetsko podprta prenova grafičnega dela zemljiškega katastra (The renovation of the land cadastre‟s graphical part based on surveying principles). Geodetski vestnik 55(2): 257-268. [2] Dale, P., Mclaughlin, J.D., 1999. Land Administration. Oxford etc., UK: Oxford University Press. [3] Gielsdorf, F., 2010. Data Integration with Adjustment Techniques. Technet GmbH. (http://www.technet-gmbh.de/) [4] Gielsdorf, F., Gruendig, L., Aschoff, B., 2004. Positional Accuracy Improvement – A Necessary Tool for Updating and Integrating of GIS Data. FIG Working week, May 22-27, 2004, Athens. [5] Hope, S., Kealy, A., 2008. Using Topological Relationship to Inform a Data Integration Process. Transactions in GIS 12(2): 267-283. [6] Hope, S., Gielsdorf, F., 2007. Quality Considerations for Optimal Positioning When Integrating Spatial Data. Spatial Sciences Institute International Biennial Conference 2007, Hobart. [7] Ickjai, L., 2005. Geosaptial Clustering in data-rich environments: features and Issues. In: Khosla, R., Howlett, R.J., Jain, L.C. (eds.), Knowledge-Based Intelligent Information and engineering systems, 9th International Conference KES 2005 Proceedings, Part IV. [8] Larsson, G., 1997. Land management – public policy, control and participation. Stockholm: Ljungflöts Offset. [9] Navratil, G., Franz, M., Pontikakis, E., 2004, Measurement-Based GIS Revisited. 7th AGILE Conference on Geographic Information Science, April 29-May 1, 2004, Heraklion. (http://plone.itc.nl/agile_old/Conference/greece2004/papers/P-07_Navratil.pdf) 59 60 SOLVING JOB SHOP PROBLEMS IN THE CONTEXT OF HYPERGRAPHS Dušan Hvalica University of Ljubljana, Faculty of Economics dusan.hvalica@ef.uni-lj.si Abstract A class of hypergraphs, called B-tails, is introduced. It is shown that a certain hypegraph can be assigned to any job shop problem in such a way that minimal B-tails correspond to optimal schedules. Keywords: scheduling, job shop problem, hypergraph 1 Introduction In coping with ‘system nervousness’, i.e., the fact that a relatively small change of the environment can result in substantial change of the system [4], the use of sensitivity analysis (the determination of the bounds within which a given schedule remains optimal) has been proposed to resolve the problem, whether to reschedule or not [5]. The proposed approach is based on AND/OR graphs, and is applicable not only in ‘pure’ scheduling and rescheduling but also in alternative process scheduling problems, where the choice to be made is not only when to perform every activity, but also which of the available set of activities is to be used [1]. However, the drawback of this approach is that the AND/OR graphs produced in the process are quite large. Here we show that scheduling can be discussed also in the environment of hypergraphs and that the corresponding hypergraph is substantially smaller. 61 2 Hypergraphs An (oriented) hypergraph G is defined1 as G = (V, A), where V and A are the sets of nodes and hyperarcs, respectively. A hyperarc E is defined as E = (T (E), H(E)), where T (E), H(E) ⊂ V; the sets T (E) and H(E) are called the tail and head of E, respectively. A hyperarc, whose head has (only) one element, is called a B-arc (backward (hyper)arc), a hypergraph, the hyperarcs of which are all B-arcs, is a B-graph. A subhypergraph of a hypergraph G = (V, A) is such a hypergraph G1 = (V1 , A1 ) that V1 ⊂ V and A1 ⊂ A. When convenient, we shall denote V1 = V(G1 ) and A1 = A(G1 ). For any node u its backward star BS(u) is defined by BS(u) = {E ; u ∈ H(E)}, while its forward star F S(u) is F S(u) = {E ; u ∈ T (E)}. A node u for which BS(u) = ∅ or F S(u) = ∅ will be called a tip node. For any subhypergraph H ⊂ G the set of its nodes v such that BS(v)∩A(H) = ∅ will be denoted by B(H) while the set of its nodes v such that F S(v)∩A(H) = ∅ will be denoted by F (H). A path is a sequence u1 , E1 , u2 , E2 , . . . , Eq−1 , uq , such that ui ∈ H(Ei−1 ) for 1 < i 6 q and ui ∈ T (Ei ) for 1 6 i < q. If uq ∈ T (E1 ), such a path is called a cycle. If u1 , E1 , u2 , E2 , . . . , uq−1 , Eq−1 , uq is a path, then uq , Eq−1 , uq−1 , . . . u2 , E1 , u1 will be called a reversed path. B-tails 3 Let G be a hypergraph and x ∈ V(G). Definition A hypergraph D is a B-tail of x if 1. for every E ∈ A(D) (a) in D there exists a path . . . , E, . . . , x, (b) |H(E) \ F (D)| = 1, unless x ∈ H(E), when |H(E) \ F (D)| = 0, 2. for every t ∈ V(D) (a) if t ∈ / B(G), then |BS(t) ∩ A(D)| > 1, (b) if t ∈ / F (D), then |BS(t) ∩ A(D)| 6 1, (c) |BS(x) ∩ A(D)| 6 1, 3. there are no cycles in D. 1 For details on hypergraphs, cf. [2]. 62 Thus, to get a B-tail of a node x, one must select one of the hyperarcs from the back star of every node (starting at x), however, this must be done in such a way, that no cycles are completed and that for hyperarcs, which are not B-arcs, only one of the nodes from their head has successors in the B-tail. As we shall see, B-tails are suitable for handling job shop problems, specifically, to any job shop a hypergraph can be assigned such that specific B-tails in it correspond to feasible schedules. 4 The job shop problem The job shop problem consists of the following: n jobs and m machines are given. A job is a set of operations, which are totally ordered by the precedence relation; each operation can be performed only on one of the machines. Thus, operations that take part in different jobs but must be performed on the same machine can not be performed simultaneously. The problem is to find a schedule of all operations so that the total duration time is minimal. It is known that the job shop problem is N P-complete [3]. Thus, it is at least as hard as any other problem that belongs to N P and therefore hard to solve. With every job shop problem a disjunctive graph G = (O, A, E) can be associated in the following way: • the nodes (elements of O) represent operations; • an arc (oi , oj ) ∈ A exists if and only if oi precedes oj (in one of the jobs); • an edge {oi , oj } ∈ E exists if and only if oi and oj must be processed on the same machine. Thus, a disjunctive graph is partly directed and partly undirected. Finally, two nodes – usually called source and sink – are added to G, together with an arc (source, n) for each n ∈ O for which there are no incoming arcs in A and an arc (n,sink) for each n ∈ O for which there are no outgoing arcs in A. The graph is weighted by c(oi , oj ) = d(oi ), where (oi , oj ) ∈ A and d(oi ) is the duration of oi . The cost of a path P is defined by2 X c(P ) = c(x, y) (x,y)∈P 2 Alternatively one can weight the nodes: w(oi ) = d(oi ). 63 An orientation Ω is an assignment E → O × O such that Ω({oi , oj }) ∈ {(oi , oj ), (oj , oi )} Thus, every orientation turns a disjunctive graph G = (O, A, E) into a directed graph (O, A ∪ Ω(E)). If the latter is acyclic, it corresponds to a feasible schedule for the original job shop problem. Moreover, the one in which the maximal path (from source to sink) is minimal, corresponds to the optimal schedule, i.e., to the solution of the job shop problem. Example 1 Consider the following job shop problem: we have four machines m1 , m2 , m3 and m4 and two jobs J1 = {o1 , o2 , o3 } and J2 = {o4 , o5 , o6 }. Operation o1 must be performed on m1 , operations o2 and o5 on m2 , operations o3 and o4 on m3 , while o6 must be performed on m4 ; the processing times are d(o1 ) = 5, d(o2 ) = 4, d(o3 ) = 2, d(o4 ) = 6, d(o5 ) = 3 and d(o6 ) = 1. The corresponding disjunctive graph is in Fig. 2. 5 o1 4 o2 o3 2 0 o7 o0 o4 0 6 o5 3 o6 1 Figure 2 Orientation Ω1 : ½ {o2 , o5 } 7→ (o2 , o5 ) {o3 , o4 } 7→ (o3 , o4 ) yields an acyclic directed graph, so that it corresponds to a feasible schedule o1 , o2 , o3 , o4 , o5 , o6 with makespan 21. We shall show that to any job shop problem a hypergraph can be assigned so that specific B-tails in it correspond to feasible schedules. Starting with the disjunctive graph G = (O, A, E) the corresponding hypergraph Gh is obtained in the following way: • first form the hypergraph G0 for which — V(G0 ) = O, Of course, the cost of a path P is then c(P ) = X n∈P 64 w(n) — for every t ∈ V(G0 ) we have BS(t) = {(Tt , {t})}, where Tt = {u; (u, t) ∈ A} (so that every node in G0 has only one incoming hyperarc, which is a B-arc); • then for every t ∈ V(G0 ) such that t is incident to an edge in G, we — add to G0 a new node ut,t , — for every v such that {t, v} ∈ E add to G0 a new node ut,v , — add to G0 hyperarc ({t}, ∪v∈Et {ut,v }), where Et = {v; {t, v} ∈ E}; • then for every t ∈ V(G0 ) such that t is incident to an edge in G, we — for every E ∈ BS(t) do the following: ∗ for every v such that {t, v} ∈ E add to G0 the hyperarc (T (E) ∪ {uv,t }, {t}), ∗ replace E with (T (E) ∪ {ut,t }, {t}), • for every edge-connected component C, add to G0 the hyperarc ({source}, {ut,t ; t ∈ V(C)}). Clearly, the only nodes in Gh with more than one hyperarc in their back star are the nodes that are incident to an edge in G and the only hyperarcs which are not B-arcs are of the form ({t}, ∪v∈Et {ut,v }) or ({source}, {ut,t ; t ∈ V(C)}). Example 2 For instance, the hypergraph, corresponding to the disjunctive graph from Example 1, is depicted in Fig. 3. o7 1 o6 u3,4 o3 o2 o1 u3,3 u5,2 1 1 1 o5 u4,3 u2,5 o4 1 1 1 u2,2 u5,5 u4,4 o0 1 Figure 3 65 The following applies (due to space limitations we omit the proof): Proposition 3 B-tails of the node sink in hypergraph Gh correspond to feasible schedules for the job shop problem. In any particular schedule, operation oi precedes operation oj if and only if in the corresponding B-tail there is a path oi , . . . , oj . The cost of a B-tail Q is defined as the highest of the costs of paths, completely contained in Q. For the hypergraph Gh the cost of any path P is just the total duration of the operations lying on P . By Proposition 3 it is also the minimal time in which all these operations can be completed. Thus, if Q is a B-tail of sink, then its cost is equal to the makespan by the schedule, corresponding to Q. Of course, the minimal B-tail of sink corresponds to the solution of the job shop problem. An algorithm for searching for a minimal B-tail can be designed in the spirit of the A∗ algorithm so that an accurate heuristics can significantly improve effectiveness of the alghorithm (but this discussion must be omitted here). 5 Conclusion We have demonstrated that hypergraphs are suitable for handling the job shop problem, specifically, that solving the job shop problem translates into searching for a minimal B-tail. References [1] Beck, J. C., Fox, M. S., 2000. Constraint-directed techniques for scheduling alternative activities. Artificial Intelligence 121, 211-250. [2] C. Berge, C., 1989. Graphs and Hypergraphs. North-Holland,Amsterdam. [3] Garey, M. R., 1979. Computers and Intractability. W. H. Freeman and Company, San Francisco. [4] Grubbström, R. W., Tang, O., 2000. Modelling Rescheduling Activities in a Multi-Period Production-Inventory System. International Journal of Production Economics 68, No 2, 123-135. [5] Hvalica, D., Bogataj, L.,2005. Sensitivity results considering rescheduling by AND/OR graphs. Int. j. prod. econ. [Print ed.] 93/94, 455-464. 66 An heuristic for the edge-survivable General Steiner Problem Pablo Sartor Del Giudice and Franco Robledo Amoza Instituto de Computación, Facultad de Ingeniería, Universidad de la República Julio Herrera y Reissig 565 – Postal Code 11.300 – Montevideo, Uruguay psartor@um.edu.uy; frobledo@fing.edu.uy Abstract: The Generalized Steiner Problem with Edge-Connectivity constraints (GSP-EC) consists of computing the minimal cost subnetwork of a given feasible network where some pairs of nodes must satisfy edge-connectivity requirements and models the design of communications networks where connection lines can fail. In this paper we present an algorithm based on the GRASP metaheuristic to solve this version of the problem, known to be NP-Complete. Promising results are obtained when testing the algorithm over a set of heterogeneous network and optimal or near-optimal solutions are found. Keywords: network design; optimization; graph theory; survivability; Steiner problem; GRASP. 1. INTRODUCTION The design of communication networks often involves two antagonistic goals. One one hand the resulting design must bear the lowest possible cost; on the other hand, certain survability requirements must be met, i.e. the network must be capable to resist failures in some of its components. One way to do it is by specifying a connectivity level (a positive integer) and constraining the design process to only consider topologies that have at least that amount of disjoint paths (either edge or node disjoint) between each pair of nodes. In the most general case, the connectivity level can be fixed independently for each pair of nodes (heterogeneous connectivity requirements), some of them having even no requirement at all. This problem is known as Generalized Steiner Problem (GSP) [9] and is an NP-Complete problem [18]. Some references on the GSP and related problems are [1], [2], [3], [4], [6], [7], [10], [11] most of them using polyedral approaches and addressing particular cases (specific types of topology and or connectivity levels). Topologies verifying edge-disjoint path connectivity constraints ensure that the network can survive to failures in the connection lines; while node-disjoint path constraints ensure that the network can survive to failures both in switch sites as well as in connection lines. Finding a minimal cost subnetwork satisfying edgeconnectivity requirements is modeled as a GSP edge-connected (GSP-EC) problem. Due to the intrinsic complexity of the problem, heuristic approaches have to be used to cope with general real-sized instances; this work presents one inspired in the ideas of recent work [13], [14], [16]. The remainder of this paper is organized as follows. Notation, auxiliary definitions and formal definition of the GSP-EC are introduced in Section 2. The GRASP metaheuristic and the particular implementation that we propose for the GSP-EC are presented in section 3. Experimental results obtained when applying the algorithms on a test set of GSP-EC instances with up to one hundred nodes and four hundred edges are presented in Section 4. Finally conclusions are presented in Section 5. 2. PROBLEM FORMALIZATION AND DEFINITIONS We use the following notation to formalize the GSP-EC: G = (V, E, C): simple undirected graph with weighted edges; V: Nodes of G; E: Edges of G; C: E → ℜ+: edge weights; T ⊆ V: Terminal nodes (the ones for which connectivity requirements exist); R: R ∈ Z |T|×|T| : Symmetrical integer matrix of connectivity requirements with rii = 0 ∀i ∈ T. 67 The set 𝑉 models existing sites among which a certain set 𝐸 of feasible links could be deployed, being the cost of including a certain link in the solution given by the matrix 𝐶. The set 𝑇 models those sites for which at least one connectivity requirement involving other site has to be met; these requirements are specified using the matrix 𝑅. Nodes in the set 𝑉\𝑇 (named Steiner nodes) model sites that can potentially be used (because doing so reduces the total topology cost or because it is impossible to avoid using them when connecting a given pair of terminals) but for which no requirements exist. Using this notation the GSP-EC can be defined as follows: Definition 2.1: GSP-EC. Given the graph 𝐺 with edge weights 𝐶, the teminals set 𝑇 and the connectivity requirements matrix 𝑅, the objective is to find a minimum cost subgraph 𝐺𝑇 = 𝑉, 𝐸𝑇 , 𝐶 where every pair of terminals 𝑖, 𝑗 is connected by 𝑟𝑖𝑗 edge-disjoint paths. 3. THE “GRASP” METAHEURISTIC GRASP (Greedy Randomized Adaptive Search Procedure) is a metaheuristic that proved to perform very well for a variety of combinatorial optimization problems. A GRASP is an iterative “multistart local optimization” procedure which performs two consecutive phases during each iteration: Construction Phase (ConstPhase): it builds a feasible solution that chooses (following some randomized criteria) which elements to add from a list of candidates defined with some greedy approach; Local Search Phase (LocalSearchPhase): it explores the neigborhood of the feasible solution delivered by the Construction Phase, moving consecutively to lower cost solutions until a local optimum is reached. Typical parameters are the size of the list of candidates, the amount of iterations to run 𝑀𝑎𝑥𝐼𝑡𝑒𝑟 and a seed for random number generation. After having run 𝑀𝑎𝑥𝐼𝑡𝑒𝑟 iterations the procedure returns the best solution found. Details of this metaheuristic can be found in [12]. 3.1 Construction Phase Algorithm The algorithm, shown in Figure 1, is an adaptation of the one found in [13] to the edgeconnected case. It proceeds by building a graph which satisfies the requirements of the matrix 𝑅, starting with an edgeless graph and adding one new path in each iteration to the solution 𝐺𝑠𝑜𝑙 under construction. The matrix 𝑀 = (𝑚𝑖𝑗 ) records the amount of connection requirements not yet satisfied in 𝐺𝑠𝑜𝑙 between the terminal nodes 𝑖, 𝑗; the sets 𝑃𝑖𝑗 will record the 𝑟𝑖𝑗 disjoint paths found for connecting the nodes 𝑖, 𝑗. One improvement over the previous algorithm is to alter the costs of the matrix 𝐶 to introduce random and enable the chance to build an optimal solution no matter what the problem instance is. We have proven that a sufficient condition to ensure this is that all edges have their costs altered independently from the others and the altered costs take values in (0, +∞) with any probability distribution that assigns non-zero probabilities to any open subinterval of (0, +∞). Loop 3-15 is repeated until all terminal nodes have their connectivity requirements satisfied, or until for a certain pair of terminals i; j, the algorithm fails to find a path a certain number of times MAX ATTEMPT. In each iteration, one pending connection requirement is chosen and the shortest-path is computed considering a modified cost matrix 𝐶′ where edges already introduced in the solution under construction during previous iterations have cost zero, enabling edge-reusing among different pairs of terminals. Finally, the algorithm ends by returning the feasible solution 𝐺𝑠𝑜𝑙 together with the path set 𝑃 which “certifies” that the requirements specified by 𝑅 were met. 68 3.2 Local Search Phase Algorithms Any local search algorithm needs a precise definition of the neighbourhood concept; we propose two different ones, which we will chain inside our suggested LocalSearchPhase algorithm. They are defined in terms of the structural decomposition of graphs in “key-node” and “key-paths” [13] plus a new structural component that we define below. Definition 3.2.1 Key-star: Given a GSP-EC instance, a feasible solution 𝐺𝑠𝑜𝑙 and any of its nodes 𝑣, the key-star associated to 𝑣 is the subgraph of 𝐺𝑠𝑜𝑙 obtained through the union of all key-paths having 𝑣 as an endpoint. Definition 3.2.2 Path-Based Local Search Neighbourhood1: Our first neighbourhood is based on the replacement of any key-path 𝑘 by another key-path with the same endpoints, built with any edge from the feasible connections graph 𝐺 (even some of 𝐺𝑠𝑜𝑙 ), provided no connectivity levels are lost when reusing edges. Let 𝑘 be a key-path of a certain solution 𝐺𝑠𝑜𝑙 and 𝑃 a set of paths which “certificates” its feasibility (as the one returned by ConstPhase). We will denote by 𝐽𝑘 (𝐺𝑠𝑜𝑙 ) the set of paths {𝑝 ∈ 𝐺𝑠𝑜𝑙 : 𝑘 ⊆ 𝑝}. These are the paths which contain the key-path 𝑘. We will also denote by 𝜒𝑘 (𝐺𝑠𝑜𝑙 ) the edge set 𝑞=𝑖..𝑗 ∈𝐽 𝑘 (𝐺𝑠𝑜𝑙 ) 𝐸(𝑃𝑖𝑗 \ 𝑞). These are the edges that, if used to replace the key-path 𝑘 in 𝑃 (obtaining a path set 𝑃′) would turn to be shared by some paths from 𝐺𝑠𝑜𝑙 with the same endpoints, thus invalidating the resulting set 𝑃′ as a feasibility certificate. The algorithm LocalSearchPhase1 (shown in Figure 1) then considers the replacement of key-paths 𝑘 by other paths 𝑝 such that 𝑐𝑜𝑠𝑡 𝑝 < 𝑐𝑜𝑠𝑡(𝑘) and the edges of 𝑝 are chosen from the set (𝐸\𝜒𝑘 𝐺𝑠𝑜𝑙 ) 𝑘. Definition 3.2.3 Key-Star-Based Local Search Neighbourhood2: This is a second neighbourhood based on the replacement of key-stars, which frequently allows to improve feasible solutions that are locally optimal when only considering Neighbourhood1. In the case of the GSP-NC, as no node sharing is allowed among disjoint paths, all key-stars are trees (named key-trees); a key-tree replacement neighbourhood for the GSP-NC can be found in [13], [14]. Due to the possibilty of sharing nodes among edge-disjoint paths, when working with GSP-EC problems, we must work with key-stars. Unlike [13], [14] we will allow the root node to be a terminal node in order to get a broader neighbourhood. In the GSP-NC any key-tree can be replaced by any tree with the same leaves with no loss of connectivity levels. In the GSP-EC, if the replacing structure is also a keystar the same holds true; but it does not for other general structures (non-star trees included). We propose an algorithm that given a key-star 𝑘, deterministically seeks for the lowest cost replacing keystar 𝑘′ able to “repair” the paths from 𝑃 broken when removing the edges of 𝑘. For allowing as much reusing of edges as possible, we can extend our previous definition of 𝐽𝑘 (𝐺𝑠𝑜𝑙 ) and 𝜒𝑘 𝐺𝑠𝑜𝑙 to consider key-stars 𝑘 instead of key-paths. Figure 1 presents the LocalSearchPhase2 algorithm, making use of a BestKeyStar algorithm shown in Figure 2. Given a keystar 𝑘 we denote by 𝜃𝑘 its root node; by 𝜓𝑘 the set of its leaf nodes; and by 𝛿𝑘,𝑚 (being 𝑚 the root node of 𝑘 or one of its leaves) the highest amount of key-paths that join 𝑚 in 𝑘 with any other node that is root or leaf in 𝑘. Figure 2 also depicts the process of determining which the best key-star to replace a given one is. It illustrates (a) the feasible graph 𝐺 with a key-star that keep4s connected the leaf nodes 𝑡, 𝑢, 𝑣; (b) the graph 𝐺′ obtained after adding the virtual nodes 𝑤 linked to 𝑡, 𝑢, 𝑣 by the appropriate amount of edges, and a “candidate” root node 𝑧; (c) the shortest paths found to connect 𝑧 and 𝑤 (using a polynomial-time algorithm minimum-cost k-edge-disjoint paths algorithm like the one in [5]; and (d) the new key-star obtained after removing the virtual node 𝑤. 69 Figure 1: Algorithms for Local Search Phase (1 and 2) Figure 2: Algorithm for determination of the best key star to replace a given one 3.3 GRASP algorithm description Finally we can put together the pieces and build a GRASP algorithm to solve the GSP-EC using a compound local search phase that will operate by applying key-path movements until no further improvements are possible followed by a (single) key-star replacement, and so on. The cycle is repeated until no further improvements are found with none of both movements. 4. PERFORMANCE TESTS This section presents the results obtained after testing our algorithms with twenty-one test cases. The algorithms were implemented in C/C++ and tested on a 2 GB RAM, Intel Core 2 Duo, 2.0 GHz machine running Microsoft Windows Vista. For every instance we ran 100 70 GRASP iterations. To our best knowledge, no library containing benchmark instances related to the GSP-NC nor GSP-EC exists; we have built a set of twenty-one test cases that are based in cases found in the following public libraries: steinlib [8]: instances of the Steiner problem; in many cases the optimal solution is known, in others the best solution known is available; tsplib [15]: instances of diverse graph theory related problems, including a “Traveling Salesman Problem” section. The main characteristics of the twenty-one test cases are shown in Table 1 where we show the amount of nodes (V), feasible edges (E), terminal nodes (T), Steiner (non terminal) nodes (St), the level of edge-connectivity requirements (one, two, three or mixed) (Redund.) and the optimal costs when available (Opt). Source data of the twenty-one instances as well as the best solutions found are available in [17]. Computational results of the tests can be also seen in Table 1. Here follows the meaning of each column: (Reqs.) total amount of requirements satisfied by the best solution found; (t(ms)) the average running time in milliseconds per iteration; (Cost) the cost of the best solution found; (%LSI) “local search improvement” – the percentage of cost improvement achieved by the local search phase when compared to the cost of the solution delivered by the construction phase, for the best solution found. Case V E T St Redund. Opt Reqs. t(ms) Cost %LSI b01-r1 50 63 9 41 1-EC 82 36 77 82 3.0 b01-r2 50 63 9 41 2-EC NA 42 80 98 3.4 b03-r1 50 63 25 25 1-EC 138 300 2611 138 10.6 b03-r2 50 63 25 25 2-EC NA 378 3108 188 4.1 b05-r1 50 100 13 37 1-EC 61 78 298 61 9.2 b05-r2 50 100 13 37 2-EC NA 144 1389 120 5.2 b11-r1 75 150 19 56 1-EC 88 171 1477 88 13.8 b11-r2 75 150 19 56 2-EC NA 324 4901 180 3.4 b17-r1 100 200 25 75 1-EC 131 300 6214 131 10.2 b17-r2 100 200 25 75 2-EC NA 531 15143 244 3.0 cc3-4p-r1 64 288 8 56 1-EC 2338 28 388 2338 10.0 cc3-4p-r3 64 288 8 56 3-EC NA 84 2221 5991 4.6 cc6-2p-r1 64 192 12 52 1-EC 3271 66 2971 3271 2.4 cc6-2p-r2 64 192 12 52 2-EC NA 132 4801 5962 10.2 cc6-2p-r123 64 192 12 52 1,2,3-EC NA 140 6317 8422 9.8 hc-6p-r1 64 192 32 32 1-EC 4003 496 25314 4033 6.8 hc-6p-r2 64 192 32 32 2-EC NA 992 28442 6652 3.5 hc-6p-r123 64 192 32 32 1,2,3-EC NA 957 26551 7930 5.2 bayg29-r2 29 406 11 18 2-EC NA 110 975 6856.88 4.6 bayg29-r3 29 406 11 18 3-EC NA 165 2413 11722 4.2 att48-r2 48 300 10 38 2-EC NA 90 1313 23214 13.0 265 6524 - 6.7 Averages Table 1: Test cases and results 5. CONCLUSIONS In this work we overcame the problems introduced by edge-disjointess (when compared to node-disjointness on previously proposed algorithms). Our algorithm GRASP-GSP was 71 shown to find good quality solutions to the GSP-EC when applied to a series of heterogeneous test cases with up to 100 nodes and up to 406 edges. For all cases with known optimal cost the algorithm was able to find solutions with costs no more than 0.74% higher than the optimal cost. Significant cost reductions averaging 6.7% are achieved after applying the local search phase over the greedy solutions built by the construction phase. Execution times were comparable to the ones of previous similar works like [13], [14] for the nodeconnected version of the GSP. In all cases the maximum possible amount of connection requirements (allowed by the topology of the original graph 𝐺 and the requirements in 𝑅) was reached and always returning edge-minimal solutions. References [1] Agrawal, Klein, Ravi. When trees collide: An approximation algorithm for the generalized steiner problem on networks. SIAM Journal on Computing, 24(3):440–456, 1995. [2] Baïou. Le problème du sous-graphe Steiner 2-arête connexe: Approche polyédrale. PhD thesis, Université de Rennes I, Rennes, France, 1996. [3] Baïou, Mahjoub. Steiner 2-edge connected subgraph polytope on series-parallel graphs. SIAM Journal on Discrete Mathematics, 10(1):505 – 514, 1997. [4] Baïou. On the dominant of the Steiner 2-edge connected subgraph polytope. Discrete Applied Mathematics, 112(1-3):3 – 10, 2001. [5] Bhandari. Optimal physical diversity algorithms and survivable networks. In Computers and Communications, 1997. Proceedings, Second IEEE Symposium on, pages 433 –441, July 1997. [6] Coullard, Rais, Wagner, Rardin. Linear-time algorithms for the 2-Connected Steiner Subgraph Problem on Special Classes of Graphs. Networks, 23(1):195 – 206, 1993. [7] Kerivin, Mahjoub. Design of Survivable Networks: A survey. Networks, 46(1):1 – 21, 2005. [8] Koch. Konrad-Zuse-Zentrum für Informationstechnik Berlin. Steinlib test data library. Available at http://steinlib.zib.de/steinlib.php. [9] Krarup. The generalized steiner problem. Technical report, DIKU, University of Copenhagen, 1979. [10] Mahjoub, Pesneau. On the Steiner 2-edge connected subgraph polytope. RAIRO Operations Research, 42(1):259–283, 2008. [11] Monma, Munson, Pulleyblank. Minimum-weight two connected spanning networks. Mathematical Programming, 46(1):153 – 171, 1990. [12] Resende, Ribeiro. Greedy randomized adaptive search procedures. In F. Glover and G. Kochenberger, editors, Handbook of Metaheuristics, pages 219–249. Kluwer Academic Publishers, 2003. [13] Robledo, Canale. Designing backbone networks using the generalized steiner problem. In Design of Reliable Communication Networks, 2009. DRCN 2009. 7th International Workshop on, volume 1, pages 327 – 334, October 2009. [14] Robledo. GRASP heuristics for Wide Area Network design. PhD thesis, IRISA, Université de Rennes I, Rennes, France, february 2005. [15] Ruprecht-Karls-Universitat Heidelberg. Tsplib network optimization problems library. Available at http://comopt.ifi.uniheidelberg.de/software/TSPLIB95. [16] Sartor. Problema general de Steiner en grafos: resultados y algoritmos GRASP para la versión arista-disjunta. M.Sc. thesis, Universidad de la República, Montevideo, Uruguay, 2011. [17] Universidad de Montevideo. GSP-EC test set with best results found. Available at http://www2.um.edu.uy/psartor/grasp-gsp-ec.zip. [18] Winter. Steiner problem in networks: A survey. Networks., 17(2):129–167, 1987. 72 The 11th International Symposium on Operational Research in Slovenia SOR ’11 Dolenjske Toplice, SLOVENIA September 28 - 30, 2011 Section III: OR Applications in Telecommunication and Navigation Systems 73 74 LP MODEL FOR DAY-AHEAD PLANNING IN ENERGY TRADING Marinović Minja, Milan Stanojević and Dragana Makajić Nikolić University of Belgrade, Faculty of Organizational Sciences Jove Ilića 154, 11000 Belgrade, Serbia marinovic75709@fon.bg.ac.rs, milans@fon.rs, gis@fon.rs Abstract: In this paper, the problem of day-ahead planning in the trading section of energy trading companies has been considered. It has been assumed that the demand and supply are arranged and that the additional MW and transmission capacity can be purchased. The problem is observed as directed multiple-source and multiple-sink network and then represented by LP mathematical model of total daily profit maximization subject to flow constraints. A numerical example is presented to illustrate the application of the model. Keywords: energy trading, day-ahead planning, network flow, LP model. 1 INTRODUCTION During XX century, power production and transmission were carried out between monopolistic public power companies. In the last twenty years, electricity markets have been deregulated allowing customers to choose their provider and new producers. The concept of a single European electricity market foresees seamless competition throughout the electricity supply chain, within and between EU Member States and adjoining countries. Customers enjoy a choice between competing electricity retailers, who source their requirements in liquid and competitive wholesale markets, irrespective of national or control area boundaries. Market participants actively compete to meet the demand in their own countries and to supply electricity across borders into neighborhood markets. Cross-border trading and supply is an integral part of this competition, as market participants enjoyed nondiscriminatory access to interconnected transmission lines. Energy Trading Companies (ETCs) are buying transmission capacity from Transmission System Operators (TSOs) [1]. TSOs consistently release to the market a truly maximum amount of cross-border transmission capacity. ETCs are trying to manage the risks associated with fluctuating prices through buying and selling gas and electricity contracts. Both traders and end-users apply financial instruments such as futures, options and derivatives to protect their exposures to prices and to speculate on price fluctuations. There are few ways of trading electricity but two main ways are via the telephone in bilateral transactions (so called ―Over The Counter― or OTC, usually through the intermediation of a broker), or it is traded through futures markets such as Nordpool or the EEX. Some key factors influencing energy prices include geopolitical factors, global economic growth, short term weather impacting demand, supply disruptions from maintenance or unexpected outages, fuel price movements and product swapping in response to relative prices [2]. The literature concerning different issues in energy trading and transmission is extensive. Kristiansen in [5] analyses the auction prices at the cross-border annual, monthly and daily capacity auction in area of energy trading in case of Denmark and Germany. Triki et al. [10] consider the multiple interrelated markets for electricity and propose a multi-stage mixed-integer stochastic model for capacity allocation strategy in a multi-auction competitive market. A generalized network flow model of the national integrated energy system that incorporates the production, transportation of coal, natural gas, and electricity storage with respect to the entire electric energy sector of the U.S. economy is proposed in [9]. The authors have formulated the multi period generalized flow problem as an 75 optimization model in which the total costs are minimized subject to energy balance constraints. The problem of energy allocation between spot markets and bilateral contracts is formulated as a general portfolio optimization quadratic programming problem in [6]. The proposed methodology with risky assets can be applied to a market where pricing, either zonal or nodal, is adopted to mitigate transmission congestion. Purchala et al. in [8] propose a zonal network model, aggregating individual nodes within each zone into virtual equivalent nodes, and all cross-border lines into equivalent border links. Using flow-based modeling, the feasibility of the least granularity zonal model where the price zones are defined by the political borders, is analyzed. The authors in [3] consider multiple-source multiple-sink flow network systems such as electric and power systems. They observe the problem in which resources are transmitted from resource-supplying (source) node(s) to resource-demanding (sink) node(s) through unreliable flow networks. Nowak et al. [7] analyzed the simultaneous optimization of power production and day-ahead power trading and formulated it as a stochastic integer programming model. The rest of the paper is organized as follows. In section 2 are description of the main assumptions of the observed problem. The LP mathematical model for day-ahead planning is presented in Section 3. Section 4 is dedicated to the experimental results to illustrate the model, along with comments on the main results of its simulation. Conclusions along with perspectives regarding further work are finally presented in section 5,. 2 PROBLEM DESCRIPTION The focus of this paper is on electricity trading from the perspective of the ETC. The main task of the trading section is to optimize the portfolio of energy products, ensuring clients’ demands are met, whatever the circumstances. The trading section also enables companies to respond to the ever-changing state of the region’s transmission grid and production capacities. The organization of this section is traditional. The trading department deals with spot and longer term arrangements. The scheduling and portfolio Management department makes schedules, takes care about cross border capacities allocations and creates paths in order to optimize the whole portfolio, managing different energy sources, customers in different countries and cross border energy flows and costs. Finally, the settlements department is dealing with invoicing, preparing deal confirmations and statistics necessary for all local ETCs. Trading section of ETC is dealing with at least two optimization problems: first one is long term planning and consist determination of transmission capacity that will be used for next period, while the second one is day-ahead planning and represents finding the optimal routes that will satisfy the short term demand considering the available transmission capacity. Subject of this paper is day-ahead problem. The assumptions of the observed problem are:  Daily demand is known, and all arranged demands must be satisfied.  Energy selling prices are known for each country.  Daily supply is known and some of them are arranged.  Energy buying prices are known for each country.  If there is a surplus or shortage of arranged supply, it will be traded through futures markets.  Available daily capacity is purchased and presented in MW.  It is possible to buy extra daily transmission capacity if it is necessary.  Profit presents difference between selling price and cost of buying energy and transmission capacity. 76 Decision that should be made is: where and how much ETC should buy and on which ways that energy should be transferred so that they maximize total daily profit. 3 MODEL FORMULATION Described problem originally can be modeled as directed multiple-source and multiple-sink network. All ETCs from one country are represented by one node (Figure 1). Figure 1: Cross-border energy flow presented as network Obtained network then can be converted into single-source and single-sink problem by introducing supersource s and supersink t. [4] The notation used to define sets, parameters and decision variables is as follows.  Sets o V – set of all nodes o A - set of arcs presenting existance of purchased capacity between nodes, o B - set of arcs presenting existance of additional capacity between nodes,  Parameters o cij – daily capacity in MW of arc (i,j)A, i≠s, j≠t; o csj – daily offer in MW of supplier j, (s,j)A; o cit – daily demand in MW of buyer i, (i,t)A; o dij – flow price in €/MW on arc (i,j)A, i≠s, j≠t: o dsj – purchase price in €/MW from supplier j, (s,j)A; o dit – selling price in €/MW for buyer i, (i,t)A; o pij – daily capacity in MW which is possible to buy on arc (i,j) B, i≠s, j≠t; o qij – flow price for additional capacity in €/MW on arc (i,j)B, o qsj – purchase price for additional supply in €/MW from supplier j, (s,j)B; o r – daily total arranged demand r   cit ( i ,t )A  Variables: o xij – optimal flow in purchased arc (i,j)A o yij – optimal flow in additional arc (i,j)B Using given notation, the LP mathematical model for day-ahead planning can be stated as: 77 (max) f ( x, y)  d ( i ,t )A x  it it  ( i , j )A, j t dij xij   ( i , j )B qij yij (1) Subject to.  ( s , j )A   ( s , j )B ysj  r (2) xit  r ( i ,t )A  xsj  (3) ( i , j )A xij   ( i , j )B yij   ( j ,i )A x ji   ( j ,i )B y ji (4) 0  xij  cij , (i, j )  A (5) 0  yij  pij , (i, j )  B (6) Objective function (1) maximizes the total profit. Equations (2-4) represent standard constraints for value of the flow. Constraints (5-6) refer to the purchased and additional arcs capacities. The optimal values of variables xsj, (s,j)A and ysj, (s,j)B represent the amount of MW that should be transmitted from regular and additional supplier s. Variables, xit, (i,t)A represent the amount of MW which should be delivered to the buyer i. The rest of the variables, xij, (i,j)A, i≠s, j≠t and yij, (i,j)B, i≠s, j≠t are related to the optimal flow between purchased and additional capacities, respectively. 4 A NUMERICAL EXAMPLE A hypothetical example is as follows: Suppose that an ETC is in the middle of the day-ahead planning and that there are arranged trading with three buyers and three suppliers. Energy can be transmitted using 19 purchased capacities. Input data are given in Table 1. The main values in the table refer to energy flow while values in parentheses represent prices. The daily supplies and purchase prices from suppliers are given in the first row and the daily demands and selling prices for buyer are given in the last column of the table. The rest of the data represent available daily flow capacities and prices. Table 1: Demand, supply and purchased transmission capacities and prices s s 1 2 3 4 5 6 7 8 9 t 1 2 3 50(2) 4 30(16) 45(2.5) 5 50(14) 30(3) 6 8 60(15) 9 t 35(2) 30(4) 20(3) 30(1.5) 50(4) 7 25(2) 25(3.5) 15(4.5) 40(1) 20(5) 30(3.5) 20(20) 30(25) 25(2) 30(3) 50(4) 35(1.5) 15(2.5) 60(22) Since there is a surplus of arranged supply, ETC must sell that amount of energy through futures markets. Also, if it is necessary, ETC can purchase additional daily transmission capacity. 78 The corresponding data are given in Table 2. Table 2: Additional transmission capacities and prices s s 1 2 3 4 5 6 7 8 9 t 1 2 3 30(12) 4 5 6 7 8 9 t 15(6) 25(10) 40(9.5) 45(11) 25(12) 40(10) 20(7) 30(8) 25(14) 10(13) 30(8) 25(7) 25(11) 35(15) 15(7) 25(12) 40(13) 10(9) GNU Linear Programming Kit (GLPK) has been used for modeling and solving. GLPK is an open source software for solving linear and mixed integer mathematical programming problems. The optimal solution obtained for the described problem by using GLPK is given in Table 3 and represented graphically in Figure 2. Table 3: Optimal solution Arcs flow Optimal value x[s,4] 30 x[s,5] 45 x[s,8] 35 x[4,7] 30 x[5,6] 20 x[8,9] 35 x[6,t] 20 x[7,t] 30 x[9,t] 60 y[5,9] 25 1 4 Maximum value 30 50 60 40 30 35 20 30 60 45 30 7 30 30 35 8 20 s 45 5 25 60 20 2 3 6 t 35 9 Figure 2: Graph representation of the optimal solution Total daily profit is 447.5€, 5MW from the supplier 5 and 25MW from the supplier 8 will be sold through futures markets. In order to transmit arranged demand it is necessary to buy additional 25MW of transmission capacity from node 5 to node 9. 79 5 CONCLUSIONS The optimization problems that appear in the trading section of Energy Trading Companies have been considered in this paper and one of them, day-ahead energy planning, has been formulated as linear problem. Day-ahead energy planning implies finding the optimal routes that will satisfy the daily demand using the purchased and additional energy transmission capacity. A hypothetical example has been presented in this paper. Since developed model is linear, it can be used to solve real life problems of large dimensions. As a topic of further research, the trading through futures markets can be taken into consideration. Another topic of future research is modeling long term planning strategy of Energy Trading Companies. ACKNOWLEDGMENTS This research was partially supported by the Ministry of Education and Science, Republic of Serbia, Projects numbers TR32013 and TR35045. References [1] EFET, Electricity markets harmonisation - New edition, http://www.efet.org/Position_Papers/ Electricity_Market,_Position_Papers,_European_Level_5492.aspx?urlID2r=72 [2] HARCOURT, Harcourt strategy sheet – Energy & Power Trading, http://www.harcourt.ch/ cms/hauptseite/zeigeBereich/11/gibDatei/1088/26-energypowertrading.pdf [3] Hsieh, C. C., Chen, Y. T., 2005. Resource allocation decisions under various demands and cost requirements in an unreliable flow network Computers & Operations Research Vol. 32 pp. 2771–2784 [4] Hu, Т.C., Shing, M.T., 1982. ―Combinatorial Algorithms‖, Dover Publication, Inc., NY [5] Kristiansen, T., 2007. An assessment of the Danish–German cross-border auctions. Energy Policy Vol. 35, pp. 3369–3382 [6] Liu, M., Wu, F. F., 2007. Portfolio optimization in electricity markets, Electric Power Systems Research Vol. 77 pp. 1000–1009 [7] Nowak, M. P. Schultz, R. D., Westphalenm M., A., 2005. Stochastic Integer Programming Model for Incorporating Day-Ahead Trading of Electricity into Hydro-Thermal Unit Commitment, Optimization and Engineering,Vol. 6, pp. 163–176 [8] Purchala, K.; Haesen, E.; Meeus, L.; Belmans, R., 2005. Zonal network model of European interconnected electricity network, CIGRE/IEEE PES, International Symposium New Orleans, pp. 362 - 369 [9] Quelhas, A., Gil, E., McCalley, J. D., Ryan, S. M., 2007. A Multiperiod Generalized Network Flow Model of the U.S. Integrated Energy System: Part I—Model Description. IEEE Transactions on Power Systems Vol. 22 No. 2 pp. 829 – 836 [10] Triki, C., Beraldi, P., Gross, G., 2005. Optimal capacity allocation in multi-auction electricity markets under uncertainty. Computers & Operations Research Vol. 32 pp. 201–217 80 GNSS ORBIT RE-CONSTRUCTION USING WAVELET NEURAL NETWORKS Polona Pavlovčič Prešeren and Bojan Stopar University of Ljubljana, Slovenija polona.pavlovcic@fgg.uni-lj.si, bojan.stopar@fgg.uni-lj.si Abstract: This paper presents a slightly different approach of setting continuous GNSS (Global Navigation Satellite System) orbit from discrete precise ephemerides, as contained in GNSS processing software packages, where the orbit problem is solved using known mathematical tools such as numerical integration and polynomial interpolation. This time we present solution based on the use of wavelet neural networks (WNN), which is proved to be universal approximator. Numerical studies showed that WNN employement in GNSS orbit re-construction can be useful alternative to known solutions in GNSS orbit processing strategies. Keywords: GNSS orbit re-construction, ephemerides, numerical integration, polynomial inerpolation, wavelet neural networks 1 INTRODUCTION Processing of GNSS-observations is based on two essential input data: observations and ephemerides. The latter serve to determine the positions of GNSS-satellites at any time since in the GNSS-processing satellites are treated as points of known positions. Satellites are moving along the orbit and circle the Earth in a half of sidereal day. This means that satellite positions are time-dependent quantities with large amount of positional data. To reduce this amount of data satellite positions are presented in a compressed form and termed as ephemerides. GNSS satellites broadcast information in the form of broadcast ephemerides, which are pre-prepared in master control station and uploaded to the satellites. In case of NAVSTAR GPS (Global Positioning System) technology broadcast ephemerides are presented by Keplerian orbital elements, but in the case of GLONASS (rus. Globalnaja Navigacionnaja Sputnikovaja Sistema) technology by discrete satellite positions. In case of more precise ephemeris data, known as precise ephemerides, provided by IGS (International GPS Service) and IGLOS (International GLONASS Service) over the web, data include positions and clock corrections of all active satellites in a 15-minute tabular form. Using both broadcast and precise ephemerides requires numerical methods for orbit reconstruction. This means that in GNSS-processing orbit function has to be re-constructed from ephemeris data and only from the resulting continuous function user can calculate the positions of a satellite at specific time. 1.1 GNSS ephemeris data form Broadcast and precise ephemerides use different data and therefore each uses different reconstruction method. GNSS control segment derives GNSS broadcast ephemeris data by extrapolating the estimated orbit for 1 to 14 days into the future. Parameters in broadcast ephemerides are obtained from a curve fitting to the predicted satellite orbit over an interval of 4 to 6 hours. GPS broadcast ephemerides contain orbital parameters (a, e, i, Ω, ) and specific coefficients of perturbed orbit motion, which take into account orbit perturbation derived from Earth’s gravity field, solar pressure, attraction from Sun and Moon. Orbital parameters (Figure 1) are used to compute orbit function in an inertial frame, which means that is rotated to terrestrial frame (i.e. Earth fixed). Since Keplerian elements 81 in broadcast ephemerides only describe the satellite's orbit over the interval of applicability, they should't be treated as true Keplerian elements of satellite motion. A sample of GPSbroadcast orbit for one satellite is given in Figure 2. a e i Ω is semimajor axis of the orbit is numerical eccentricity, computed from b: √ inclination of the orbital plane is right ascension of the ascending node is argument of perigee argument of latitude a Figure 1: Keplerian orbital parameters [1] Figure 2: GPS navigation message with one set of broadcast ephemerides Since in geodesy many applications have required orbits of better accuracy than those from broadcast ephemerides, in 1992 IGS started to produce precise ephemerides. The precise ephemerides's problem of arrival time delay was partly solved in 1996 with rapid ephemerides (IGR) and eventually in 2001 with ultra-rapid ephemerides (IGU). IGU, IGR and IGS final ephemerides are distinguished according to criteria of time delay of access and accuracy of data. However, comparing to broadcast all sets of precise ephemerides have better accuracy that is constant for the whole interval, but these data are available in smaller discrete time intervals [2]. Precise ephemerides are packaged in daily (or one day and a half in case of IGU) SP3 files and contain discrete positions (x,y,z) and clock rate's for all satellites at 15-minute interval. 1.2 GNSS orbit re-construction GPS broadcast orbits are re-constructed from ephemerides using a method, proposed in ICDGPS-200C, Table 20-IV 1997 [3]. GNSS data processing softwares use the interpolation functions for orbit reconstruction using discrete tabular data (numerical integration is not used very often, since SP3 precise orbits include only positional data). However, those functions have restrictions since they can be used only for central interpolation, while the accuracy of the calculation is 82 not good enough at the beginning and end of the interval. The solution of the problem are presented by providing more functions for definition area, known as several successive interpolations [4]. However, the problem is still not resolved at the beginning and end of the computational area (Figure3). Interpolation by polynomials is done for each poisition component separately. Most used Lagrange's interpolation has a better performance at the boundaries (as trigonometric) which makes it more convinient for real time applications [5]. Here we present a different approach of satellite orbit construction from 15-minute tabular data, allowing identification of only one function throughout the interval. This approach is based on WNN learning. 2 WAVELET NEURAL NETWORKS Wavelet neural networks [6] are a special case of feedforward networks, that combine two traditionally separate theories into optimal functional tool for solving non-linear problems. The first wavelet theory is used for optimization of theorems of approximation and scaling, while feed-forward networks keep properties of universal approximation and effective learning. Figure 3: Wavelet neural network architecture Wavelet networks are three-layer networks, i.e. they consist from a single hidden layer in which activation functions are dilated ( ) and translated ( ) versions of a single function, called mother wavelet function. Daughter wavelets are formed by [7]: ( ) ⁄ ( ) (1) where x consists of known data in the input layer and vector y consists of known data in the output layer. And the WNN output f is given by: ∑ ( ) (2) can be any suitable basis function, such as Gaussian, Gaussian wavelet, Mexican hat or Morlet function. 83 Gaussian: ( ) Gaussian wavelet: ( ) ( Mexican hat: ( ) ( Morlet: ( ) Gaussian ( ) ) ( ) ) ( (3) (4) ( ) ), ( Gaussian wavelet ( ) (5) ) (6) Mexican hat Morlet Figure 4: Wavelet functions as types of activation functions WNN learning is done by two steps: first by fixing dilation and translation parameters, and next by finding weights solving the system of linear equations. The translation parameters can be chosen randomly from the input data, while the dilation parameters are usually fixed. The network output O for all the input data can be expresseed as: (7) where: [ ( ) ( ) ( ) ( ) ] (8) The weights matrix is solved by pseudoinversion: ( ) (9) For the evaluation of WNN learning mean square function is used: ( ) 3 ∑ ( )) ( (10) NUMERICAL SIMULATIONS The WNN, used for orbit approximation, consisted of one neuron in the input layer (time) and three neurons in the output layer, representing the three components of the precise Two experiments have been done to verify the performance of WNN: the first experiment is to determine the applicability of WNN for function approximation, and second the performance of WNN in prediction. WNN learning was evaluated using Eq. 10 and further compared to positions computed from other (polynomial inetrpolation) orbit computation methods. Approximation results were evaluated using root mean square function as the error criteria: √ where is the WNN output and ∑ ( ) known value. Smaller RMSE, better the accuracy. 84 (11) In Tables 1-3 experimental results of WNN in approximating with different types of activation functions in hidden layer (in our case wavelet functions) are presented. Minimal and maximal deviations are shown, as well as RMSE as final error criteria. As seen WNNs approximate function efficiently, which probably results due to the fast oscillating characteristics of the wavelet functions. Among all the wavelet basis functions used, Gaussian function yields the lowest accuracy. For the x-component best results gives the Gaussian wavelet, while for y and z component Morlet function is best choice in above presented situation. WNN can be used also for prediction. Simulations showed method performed well for 20 minutes after last known precise broadcast orbit correction used for training. Figure 4 shows different WNNs performances, i.e. deviations in radial componet. As seen from Figure 4 after deviations after 20 minutes of last known data dont' exceed 2 cm. This is an important ascpect especially in situations when the user cannot get new ephemeris data for example during the reception gaps. Figure 4: WNNs’ performance in prediction (radial component) 85 4 CONCLUSIONS In this paper WNN is addressed to estimate the GNSS-orbit function. This product could support GNSS software packages especially near the end of the precise ephemerides. The advantage of such orbit function construction is the ability to determine unique orbit function for the whole are of given points (i.e. satellite's positions), but also allows to use orbit function outside the area (when using interpolation polynomials extrapolation can be critical). References [1] Rothacher, M., Mervart, L., 1998. Bernese GPS Software Version 4.2. Astronomical Institute, University of Bern, Switzerland. [2] Spofford, P.R., Remondi, B.W., 1999. The National Geodetic Survey Standard GPS Format SP3. [3] Neta. B., Sagovac, C. P, Danielson, D. A., Clynch, J. R., 1996. Fast interpolation for Global Positioning System (GPS) Satellite Orbits. In: Proc. AIAA/AAS Astrodynamics Specialist Conference, San Diego, CA, Paper Number AIAA 96-3658. [4] Schenewerk, M., 2003. A brief of basic GPS orbit interpolation strategies. GPS Solutions 6(4), pp. 265–267. [5] Zhang, Q., Benveniste, A., 1992. Wavelet Networks. IEEE Transcastions on neural networks, Vol. 3, No.6. 1992 [6] Zhang, Q., 1997. Using wavelet network in nonparametric estimation. IEEE Trans. Neural Networks, 8(2):227–236. [7] ICD-GPS-200C. http://www.navcen.uscg.gov/gps/geninfo 86 APPLICATION OF DATA MINING IN TELECOMMUNICATIONS COMPANIES Željko V. Račić University of Banja Luka, Faculty of Economics Majke Jugovića 4, 78000 Banja Luka, Bosnia and Herzegovina&Republic of Srpska zeljko.racic@efbl.org Abstract: Application of Business Intelligence and Data Mining is necessary in process of obtaining competitive advantage and survival in the market. Application of tools and methods of data search results lead to a decrease of loss of income and indicates how a company should manage its business, ie. how to make the organization more "intelligent", and how to make its human resources to be knowledge society. In our work we will use the demographic group and REFII model, with use of a software solution Time Explorer, developed in Visual FoxPro tool. Keywords: Business Intelligence, Data Mining, Knowledge Discovery from Databases, model, Customer Relationship Management REFII 1 INTRODUCTION Telecommunications companies face a tough competition today, the demands of the Internet, the possibilities of broadband services (teleconferencing, videoconferencing, video on demand) and the ever increasing demands. By monitoring the evolution of the telecommunications industry, more and more is evident that many telecommunications companies are moving away from a business model based on the strategy of development of infrastructure / products / services and accept a business model based on the strategy focused on customer service. Business Intelligence (BI) means a category of funds data analysis, reporting, querying, that in a business process it can help to use out large amounts of data that are synthesized in valuable information which will be based on prudent business decisions. Business intelligence involves the following four factors: the decision support system (DSS-Decision Support System), Data Warehouse and / or Data Mart, Best Analytical Processing and data mining applications. For the successful operation of modern companies need to focus right information at the appropriate parts of the company at the right time. It is required to digitize all the processes in the organization and make the organization of "intelligent one", and its human resources to their staff knowledge. To get the large amount of information provided from data that have business significance and serve as decision support, it is necessary to perform data mining analysis, or conduct the process of discovering knowledge from data (KDD - Knowledge Discovery from Databases). Data Mining is the analysis (often very large) of data sets with the goal of finding unexpected relations and patterns in data sets or summary view of data, so that the owner or user of data provides new, understandable and useful information. Data mining plays a major role in every aspect of CRM. Only by applying data mining techniques, can be a great hope for the transformation of numerous company records in the user database into a kind of a comprehensive image of its users. An important prerequisite for successful analysis of DM's is"purity" of the data from the database, and great attention should be paid to the DM model building and selection of appropriate techniques. The choice of data mining techniques is just one of the phases in the generic data mining model.This paper presents the process of DM analysis on the example of Telekom Srpske segmentation. There are several DM techniques that can be used for the segmentation of the user. Segmentation of users is also known as clustering technique. Clustering is a research DM technique. 87 2 CUSTOMER SEGMENTATION ANALYSIS USING DATA MINING Identifying characteristics of users is the target's activity analysis and gives views of the users, which directly involve the business strategy for CRM (customer relationship management). CRM is a business model based on a strategy focused to customer service. Strategy focused on customer service lies at the basis of the available modern information technologies that enable rapid and efficient understanding of the needs and behavior of service users, so that understanding is to be as better as possible used to increase the competitiveness of the company. CRM in telecommunications companies as a result of the convergence of services, provides a view of the user - one call, one view, one bill. Necessary conditions of successful business strategy for telecommunications companies are understanding users’ behaviour and proper management of relationships with these customers. The reason for this is also illustrated by the following data: 1. 5% savings in operational costs, the company can raise profits in the range of 25-60%, 2. 35% of users using only one telecom service or have only one job with telecom, potentially are ready to go in the competition, 3. about 20% of customers make 80% profit to telecom, 4. five times the higher cost of acquiring a new customer than retain an existing. Using the DM model we can get a model for the segmentation of customers to identify groups with similar characteristics that will help better understanding of the user. To know who our customers are, is a good starting point for understanding the rapid changes in the market for telecommunications industry. Customer segmentation is the basis for taking targeted action for each user group. Thus, for example, we offer free minutes to a particular group of users such as users of telecom services are mostly used on weekends or those who have the most incoming calls. Customer segmentation can be done based on different criteria. It can be simple criteria such as age, gender, demographic characteristics or a combination of these. The DM deals with the segmentation even though it actually works on clustering. Clustering means grouping of users with similar characteristics while simultaneously trying to maximize the differences between these groups. The goal of segmentation in the DM is to discover groups of users that come from the data that the company has about them, instead of ordering a group on the basis of deciding which features are the most important. DM tells what business rules should be, instead of what is thought it should be. There are two DM techniques for segmentation of users: demographics and neuronal clustering. Neuronal clustering is mainly used when the majority of variables are numerical, while demographic clustering is mainly used when the most variables are categorical, because it can operate with categorical values not converting them into numeric. These techniques are complementary and can be used together. Neuronal clustering involves determining the maximum number of clusters. In our case, we can identify four clusters. The number of clusters is base for CRM marketing department of the company, and it is very difficult to use a large number of clusters and data arising from the same. The decision about a number of clusters depends on where and how to use the results of segmentation and should be discussed with the marketing perspective. There are software’s for this scaling of input data (IBM's Intelligent Miner for Data Mining). The main difference between the demographic and neuronal clustering is that the demographic clustering automatically determines the number of clusters, whereas for neural clustering is needed to decide how many clusters are to be performed. We will describe how a generic DM model can be used to define groups of users in the telecom industry and how this can be used in the CRM system. 88 3 CUSTOMER SEGMENTATION TELEKOM SRPSKE (CASE STUDY) From a large set of questions about users’ behaviours of telephone services from Telekom Srpske, and for which we shall give an answer using the DM analysis, we recommend the following: How are the users of telephone services grouped due to: the frequency of calls by days of a week, total call duration and total number of spent pulse? The data in CDR format are taken from the headquarters in Banja Luka for a period of six months of the 2011. In order to obtain answers to the above question, we used the DM technique - cluster analysis. In this case, the entire available data set (total traffic), the purpose of exploration, cluster analysis was performed using a neural network (Kohonen Feature Map - Kohonen selforganizing map). Cluster analysis was based on the variables: the frequency and duration of calls by call type (local, international ...), the days of the week and hours in the day when the calls were completed, a specified number of clusters was four. This is just one of many possible combinations. The clusters are shown in Figure 1: Figure 1: Cluster analysis of frequency and duration of calls by call type 89 The variable "total number of impulses" contains rough information about the profitability of customers, and it is easy to see that each cluster 1 and cluster 2 are very different in profitability. This confirms the distribution of the total number of impulses in these clusters, with Figure 2. Figure 2: The distribution of pulses from Cluster 1 and Cluster 2 Characteristics of Cluster 1 are (33% of users belong to this cluster): 1. Local calls are above average frequency and duration in certain periods of time (early in a day during the whole week), 2. low level of use of services, 3. low level of other types of calls and to other networks. From the description of Cluster 1, we can conclude that it is the largest group of users of low profitability. Behaviour based on the method of a call indicates that this is the home users (natural persons), mostly unemployed (students, pensioners), with low levels of use of services that are charged. Marketing efforts directed towards this group of users whose consumption is highly correlated with the prices of services, should concern the campaign to retain customers and prevent their leaving to the competition (it is five times more expensive to attract new users than to retain existing). These users need to be offered discounts during 90 the day when level of calling is below average (the introduction of service-type happy hour cheaper or free calls during certain periods of the day). Also they need to offer new services with discounts. Characteristics of Cluster 2 are (29% of users belong to this cluster): 1. Local calls are evenly spread throughout the time frame (except Saturday and Sunday) uniformed frequency and duration, 2. Calls to MOBI'S network of evenly spread throughout the time framework of uniformed frequency and duration (except Saturday and Sunday) 3. All calls are above average frequency and duration in the middle of all working days. The characteristics of this cluster proved the claim that about 20% of users made 80% profit of telecommunications company. This means that this is a very profitable segment of customers (business users) with high purchasing power. Marketing campaign for this segmentation group should concern keeping profitable customers and increase profits brought by these users (a campaign to increase the loyalty of profitable customers, and crossselling and up-selling through service packages and business solutions). 4 CONCLUSIONS Data mining and extraction of hidden knowledge from large databases is a powerful new technology with great potential to help companies to focus on the most important information in their databases. The most common areas of use of data mining, KDD ie. (Knowledge Data Discovery) are: telecommunications, customer retention and to prevent their transition to competition, profiling of users, the techniques of direct marketing, TV, business with credit cards, credit card fraud, investment analysis, risk management, banking, bioinformatics, the collection of loans, education, chemical industry, pharmaceutical industry, insurance, industrial processes - detecting defects, manufacturing, marketing, ecommerce. Application areas where the greatest benefits are achieved are: improving relationships with customers (CRM), customer retention and preventing their transition to competition, the detection of fraud (Fraud Detection) to detect and predict failures (Fault Detection and Prediction). CRM system can help to build customer loyalty through: better service with a view to the user (one call, one view, one bill); better marketing with analysis of the users (targeted campaigns using analytical CRM tools for in-depth analysis of customer data in order to find models and usage of indicators, rather than "spray and pray" campaigns) to identify potential churn-era and proactive action, linking behaviour, "user-in focus" with rewards for employees. Keeping existing customers is more valuable than attracting new ones, because efforts to customers’ retention avoid direct acquisition costs. Until recently, the market of the Republika Srpska, Telekom Srpske, was protected by the state and, generally, there was no need for technology to support decision making and improving relationships with customers because customers had no choice. However, by the emergence of more operators, it leads to rapid maturation of the market of Republika Srpska, and BiH. However, due to specific conditions in the environment, the market is still not mature in the true sense of the word. In order that Telekom Srpske remains the leading operator in Republika Srpska (or become in BiH), the company's strategy must be constantly improving the CRM function, quality improvement and expansion of services and providers. Since data mining plays a major role in every aspect of CRM, just by applying data mining techniques Telekom Srpske 91 can hope for a better understanding of the user. The primary goal of Telekom Srpske management must be the creation of information infrastructure (decision support systems) that will provide the necessary support to data meaning, such as data warehouse, database supplied from many operational systems and the inspection of the entire company with one point of view. Current practices show that the implementation of data mining in the business of the company brings a quick return on investment, actively applying the knowledge gained and the most important is advantage in a competitive environment. References [1] Amor, D. 2002. The E-Business Revolution, Prentice Hall, New York. [2] Atkinson, R.D. i dr., 2002. The 2002 New Economy Index, PPI, Washington D.C. [3] Blenkhorn, L. D. 2005. Fleisher, S. C. Outsource Competitive Intelligence? A Viable Option, Competitive Intelligence Magazine, 8/br. 6. [4] Inmon, B., William, H. 1992. Building the data warehouse, John Wiley & Sons, Inc. [5] Liautaud, B. 2001. E-Business Intelligence: Turning Information into Knowledge into Profit, McGraw-Hill, New York. [6] Linoff, G., Berry, J.A. 2000. Mastering Data Mining-The Art and Science of Customer Relationship Management, John Wiley & Sons, Inc. [7] Pyle, D. 1999. Data preparation for Data Mining, Morgan Kaufmann Publishers. [8] Pyle, D. 2003. Business Modelling and Data Mining, Morgan Kaufmann Publishers. [9] Rudd, P.O. 2001. Data Mining Cookbook - Modelling Data for Marketing, Risk, and Customer Relationship Management, Wiley. [10] Solomon, M.R., Rabolt N.J. 2004. Consumer Behaviour in Fashion, Prentice Hall. [11] Sachs, R.J., 1993. Macroeconomics, Prentice Hall, New York. [12] Turban, E., McLean, E., Wetherbe, J. 2004. Information Technology for Management, John Wiley & Sons, Inc. 92 The 11th International Symposium on Operational Research in Slovenia SOR ’11 Dolenjske Toplice, SLOVENIA September 28 - 30, 2011 Section IV: Mathematical Programming and Optimization 93 94 TRACE OPTIMIZATION USING SEMIDEFINITE PROGRAMMING Kristijan Cafuta University of Ljubljana, Faculty of Electrical Engineering Tržaška 25, 1000 Ljubljana, Slovenia, E-mail: kristijan.cafuta@fe.uni-lj.si Igor Klep University of Maribor, Faculty of Natural Sciences and Mathematics Koroška 160, 2000 Maribor, Slovenia University of Ljubljana, Faculty of Mathematics and Physics Jadranska 19, 1111 Ljubljana, Slovenia, E-mail: igor.klep@fmf.uni-lj.si Janez Povh Faculty of Information Studies in Novo Mesto Novi trg 5, 8000 Novo mesto, Slovenia, E-mail: janez.povh@fis.unm.si Abstract: In this paper we present the algorithm and its implementation in the software package NCSOStools for finding sums of Hermitian squares and commutators decompositions for polynomials in noncommuting variables. It is based on noncommutative analogs of the classical Gram matrix method and the Newton polytope method, which allow us to use semidefinite programming. Keywords: noncommutative polynomial, sum of squares, semidefinite programming, Matlab toolbox, free positivity. 1 INTRODUCTION The main question studied in this paper is whether a given real polynomial in noncommuting variables (nc polynomial) can be decomposed as a sum of Hermitian squares and commutators. Using semidefinite programming (SDP) one can obtain numeric evidence and may ask if an exact proof (or certificate) for the answer based on an algorithm can also be given. 1.1 Motivation The interest in finding decompositions of an nc polynomial as a sum of Hermitian squares and commutators is based on the following fact. If such a decomposition exists, the nc polynomial is necessarily trace-positive, i.e. all of its evaluations at tuples of symmetric matrices have nonnegative trace. Following Helton’s seminal paper [5], this can be considered as a specific case of free real algebraic geometry in the study of positivity of nc polynomials. Much of today’s interest in real algebraic geometry is due to its powerful applications. For instance, the use of sums of squares and the truncated moment problem for polynomial optimization on Rn established by Lasserre and Parrilo [9, 8, 12, 11] is nowadays a common fact with applications to control theory, mathematical finance or operations research. A nice survey on connections to control theory, systems engineering and optimization is given by Helton, McCullough, de Oliveira, Putinar [3]. Applications of the free case to quantum physics are explained e.g. by Pironio, Navascués, Acı́n [13] who also consider computational aspects related to sums of Hermitian squares (without commutators). As a consequence of this surge of interest in free real algebraic geometry and sums of (Hermitian) squares of nc polynomials we have developed NCSOStools [1] – an open source Matlab toolbox for solving such problems using semidefinite programming (SDP). As a side product our toolbox implements symbolic computation with noncommuting variables in Matlab. 95 2 PRELIMINARIES 2.1 Words, nc polynomials and involution Fix n ∈ N and let hXi be the set of words in the n noncommuting Pletters X1 , . . . , Xn (including the empty word denoted by 1). We consider linear combinations w aw w with aw ∈ R, w ∈ hXi of words in the n letters X which we call nc polynomials. The set of nc polynomials is actually a free algebra, which we denote by RhXi. An element of the form aw where a ∈ R \ {0} and w ∈ hXi is called a monomial and a its coefficient. The length of the longest word in an nc polynomial f ∈ RhXi is the degree of f and is denoted by deg f . The set of all nc polynomials of degree ≤ d will be denoted by RhXi≤d . If an nc polynomial f involves only two variables, we write f ∈ RhX, Y i instead of RhX1 , X2 i. We equip RhXi with the involution ∗ that fixes R ∪ {X} pointwise and thus reverses words, e.g. (X1 X22 X3 − 2X33 )∗ = X3 X22 X1 − 2X33 . Hence RhXi is the ∗-algebra freely generated by n symmetric letters. Let Sym RhXi denote the set of all symmetric elements, that is, Sym RhXi := {f ∈ RhXi | f = f ∗ }. 2.2 Sum of Hermitian squares and commutators An nc polynomial of the form g ∗ g is called a Hermitian square and the set of all sums of Hermitian squares will be denoted by Σ2 . Clearly, Σ2 ( Sym RhXi. The involution ∗ extends naturally to matrices (in particular, to vectors) over RhXi. For instance, if V = (vi ) is a (column) vector of nc polynomials vi ∈ RhXi, then V ∗ is the row vector with components vi∗ . We use V t to denote the row vector with components vi . Example 2.1 The polynomial f = X 2 − X 2 Y − Y X 2 + Y X 2 Y + XY 2 X is a sum of Hermitian squares, in fact, f = (X − XY )∗ (X − XY ) + (Y X)∗ (Y X). The next notation we introduce is cyclic equivalence which resembles the fact that we are interested in the trace of a given polynomial under matrix evaluations. Definition 2.2 An element of the form [p, q] := pq−qp, where p, q are polynomials from RhXi, cyc is called a commutator. Nc polynomials f, g ∈ RhXi are called cyclically equivalent (f ∼ g) if f − g is a sum of commutators: f −g = k X i=1 [pi , qi ] = k X (pi qi − qi pi ) for some k ∈ N and pi , qi ∈ RhXi. i=1 cyc It is clear that ∼ is an equivalence relation. The following remark shows how to test it. Remark 2.3 cyc (a) For v, w ∈ hXi, we have v ∼ w if and only if there are v1 , v2 ∈ hXi such that v = v1 v2 cyc and w = v2 v1 . That is, v ∼ w if and only if w is a cyclic permutation of v. P P (b) Nc polynomials f = w∈hXi aw w and g = w∈hXi bw w (aw , bw ∈ R) are cyclically equivalent if and only if for each v ∈ hXi, X X aw = bw . (1) w∈hXi cyc w ∼ v w∈hXi cyc w ∼ v 96 cyc Example 2.4 We have 2X 2 Y 2 X 3 + XY 2 X 2 + XY 2 X 4 ∼ 3Y X 5 Y + Y X 3 Y as 2X 2 Y 2 X 3 + XY 2 X 2 + XY 2 X 4 − (3Y X 5 Y + Y X 3 Y ) = = [2X 2 Y, Y X 3 ] + [XY, Y X 4 ] + [XY, Y X 2 ]. Let cyc Θ2 := {f ∈ RhXi | ∃g ∈ Σ2 : f ∼ g} denote the convex cone of all nc polynomials cyclically equivalent to a sum of Hermitian squares. By definition, the elements in Θ2 are exactly nc polynomials which can be written as sums of Hermitian squares and commutators. Example 2.5 Consider the nc polynomial f = X 2 Y 2 + XY 2 X + XY XY + Y X 2 Y + Y XY X + Y 2 X 2 . This f is of the form f = (XY XY + Y XY X + XY 2 X + Y X 2 Y ) + 2XY 2 X + (sum of commutators) = (XY + Y X)∗ (XY + Y X) + 2(Y X)∗ (Y X) + (sum of commutators), √ hence we have f ∈ Θ2 taking the polynomials g1 = (XY + Y X), g2 = 2Y X in the certificate. 2.3 Semidefinite programming Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function over the intersection of the cone of positive semidefinite matrices with an affine space. More precisely, given symmetric matrices C, A1 , . . . , Am ∈ SRs×s and a vector b ∈ Rm , we formulate a semidefinite program in standard primal form (in the sequel we refer to problems of this type by PSDP) as follows: inf hC, Gi s. t. hAi , Gi = bi , i = 1, . . . , m G  0. (PSDP) Here h , i stands for the standard scalar product of matrices: hA, Bi = tr(B t A). The dual problem to (PSDP) is the semidefinite program in the standard dual form sup hb, P yi (DSDP) s. t. i yi Ai  C. P Here y ∈ Rm , and the difference C − i yi Ai is usually denoted by Z. The mentioned matrix C is arbitrary. One can use C = I, a commonly used heuristic for matrix rank minimization. Often however, a solution of high-rank is desired. Then C = 0 is used, since under a strict feasibility assumption the interior point methods yield solutions in the relative interior of the optimal face, which is in our case the whole feasibility set. If strict complementarity is additionally provided, the interior point methods lead to the analytic center of the feasibility set [4]. Even though these assumptions do not always hold for the instances of SDP we construct, in our computational experiments the choice C = 0 in the objective function almost always gave a solution of higher rank than the choice C = I. The relevance of SDP increased with the ability to solve these problems efficiently in theory and in practice. Given an ε > 0 we can obtain by interior point methods [10] an ε-optimal solution with polynomially many iterations, where each iteration takes polynomially many real number operations (provided that both (PSDP) and (DSDP) have non-empty interiors of feasible sets and we have good initial points). 97 There exist several general purpose open source packages (cf. SeDuMi , SDPA , SDPT3 ) which can efficiently find ε-optimal solutions. If the problem is of medium size (i.e., s ≤ 1000 and m ≤ 10.000), these packages are based on interior point methods, while packages for larger semidefinite programs use some variant of the first order methods. Nevertheless, once s ≥ 3000 or m ≥ 250000, the problem must share some special property, otherwise state-of-the-art solvers will fail to solve it for complexity reasons. 3 COMPUTATIONAL ALGORITHM In this section we discuss an algorithm based on the Gram matrix method for testing the membership in Θ2 . The algorithm with the aid of semidefinite programming is presented in Section 3.2. 3.1 The tracial Gram matrix method Testing whether a given f ∈ RhXi is an element of Σ2 or Θ2 can be done efficiently by using semidefinite programming as first observed in [7, Section 3]. The method behind it is a variant of the Gram matrix method and is based on the following proposition, which is a natural extension of the results for sums of Hermitian squares (cf. [5, Section 2.2] or [6, Theorem 3.1 and Algorithm 1]), which are in turn variants of the classical result for polynomials in commuting variables due to Choi, Lam and Reznick ([2, Section 2]; see also [11]). Proposition 3.1 Let W be the vector of all words w ∈ hXi satisfying 2 deg(w) ≤ deg(f ), where f ∈ RhXi. Then (a) f ∈ Σ2 if and only if there exists a positive semidefinite matrix G such that f = W ∗ GW ; (2) (b) f ∈ Θ2 if and only if there exists a positive semidefinite matrix G such that cyc f ∼ W ∗ GW ; (3) Moreover, given a positive semidefinite matrix G of rank r satisfying (2) or (3), respectively, P cyc P one can construct nc polynomials g1 , . . . , gr ∈ RhXi such that f = ri=1 gi∗ gi or f ∼ ri=1 gi∗ gi , respectively. The matrix G satisfying (2) is called a Gram matrix for f , while matrix G satisfying (3) is called a (tracial) Gram matrix for f . For an nc polynomial f ∈ Σ2 or f ∈ RhXi the (tracial) Gram matrix is not unique, hence determining whether f ∈ Σ2 or f ∈ Θ2 amounts to finding a positive semidefinite Gram matrix from the affine set of all (tracial) Gram matrices for f . Problems like this can (in theory) be solved exactly using quantifier elimination. However, this only works for problems of small size, so a numerical approach is needed in practice. Thus we turn to semidefinite programming. 3.2 Sums of Hermitian squares and commutators and SDP In this subsection we P present a conceptual algorithm based on SDP for checking whether an nc polynomial f = w∈RhXi aw w of degree ≤ 2d is cyclically equivalent to a sum of Hermitian squares. Following Proposition 3.1 we must determine whether there exists a positive semidefcyc inite matrix G such that f ∼ W ∗ GW . This is a semidefinite feasibility problem in the matrix 98 variable G, where the constraints hAi , Gi = bi are essentially equations (1). Note that since cyc w∗ ∼ 6 w in general, these constraints need not be symmetric. As we restrict our attention to polynomials which are cyclically equivalent to symmetric polynomials (the others are clearly not in Θ2 ), we may always merge the equations corresponding to a particular word and its involution. We formalize this lesson as follows: Proposition 3.2 If f = P w aw w ∈ Θ2 then for every v ∈ hXi, X X aw = aw . cyc (4) cyc w ∼ v∗ w∼v Corollary 3.3 Given f ∈ RhXi we have: (1) if f does not satisfy (4), then f 6∈ Θ2 ; (2) if f satisfies (4), then we can determine whether f ∈ Θ2 by solving the following SDP with only symmetric constraints: inf s. t. X hC, Gi Gp,q = cyc p,q, p∗ q ∼ v cyc ∨ p∗ q ∼ v ∗ X (aw + aw∗ ), ∀v ∈ W (CSOHSSDP ) cyc w∼v G  0. The constraints in (CSOHSSDP ) are hAv , Gi = bv , where bv = Av = Av∗ is the symmetric matrix defined by  cyc cyc   2; if p∗ q ∼ v & p∗ q ∼ v ∗ , cyc cyc (Av )p,q = 1; if p∗ q ∼ v & p∗ q 6 ∼ v ∗ ,   0; otherwise. P cyc w∼v (aw + aw∗ ) and The conceptual algorithm to determine whether a given polynomial is cyclically equivalent to a sum of Hermitian squares (the Gram matrix method ) is now as follows: Input: f ∈ RhXi with f = P w∈hXi aw w, where aw ∈ R. Step 1: If f does not satisfy (4), then f 6∈ Θ2 . Stop. Step 2: Construct W . Step 3: Construct data Av , bv , C corresponding to (CSOHSSDP ). Step 4: Solve (CSOHSSDP ) to obtain G. If it is not feasible, then f 6∈ Θ2 . Stop. Step 5: Compute a decomposition G = Rt R. cyc Output: Sum of Hermitian squares cyclically equivalent to f : f ∼ P ∗ i gi gi , where gi denotes the i-th component of RW . Algorithm 1: The Gram matrix method for finding Θ2 -certificates. 99 Example 3.4 We conclude this presentation considering the polynomial f = S8,2 (X, Y ) = X 6 Y 2 + X 5 Y XY + X 5 Y 2 X + X 4 Y X 2 Y + X 4 Y XY X + X 4 Y 2 X 2 + + X 3 Y X 3 Y + X 3 Y X 2 Y X + X 3 Y XY X 2 + X 3 Y 2 X 3 + X 2 Y X 4 Y + X 2 Y X 3 Y X+ + X 2 Y X 2 Y X 2 + X 2 Y XY X 3 + X 2 Y 2 X 4 + XY X 5 Y + XY X 4 Y X + XY X 3 Y X 2 + + XY X 2 Y X 3 + XY XY X 4 + XY 2 X 5 + Y X 6 Y + Y X 5 Y X + Y X 4 Y X 2 + + Y X 3 Y X 3 + Y X 2 Y X 4 + Y XY X 5 + Y 2 X 6 . To prove that f ∈ Θ2 with the aid of NCSOStools, proceed as follows: (1) Define two noncommuting variables: >> NCvars x y (2) For a numerical test whether f ∈ Θ2 , run >> params.obj = 0; >> [IsCycEq,G,W,sohs,g,SDP_data] = NCcycSos(BMV(8,2), params); where our polynomial f is constructed using BMV(8,2). This yields a floating point Gram matrix G   3.9135 2.0912 −0.1590 0.9430  2.0912 4.4341 1.0570 −0.1298  G= −0.1590 1.0570 4.1435 1.9088  0.9430 −0.1298 1.9088 4.0865 for the word vector  t W = X 3 Y X 2 Y X XY X 2 Y X 3 . The rest of the output: IsCycEq = 1 since f is (numerically) an element of Θ2 ; sohs cyc P ∗ is a vector of nc polynomials gi with f ∼ i gi gi = g; SDP data is the SDP data for (CSOHSSDP ) constructed from f . (3) To round and project the obtained floating point solution G, feed G and SDP data into RprojRldlt: >> [Grat,L,D,P,err]=RprojRldlt(G,SDP_data,true) This produces a rational Gram matrix Grat for f with respect to W and its LDU decomposition P LDLt P t , where P is a permutation matrix, L lower unitriangular, and D a diagonal matrix with positive entries. Finally, the obtained rational sum of Hermitian squares certificate for f = S8,2 (X, Y ) is cyc f ∼ 4 X λi gi∗ gi i=1 for 1 1 g1 = X 3 Y + X 2 Y X + Y X 3 2 4 1 1 2 2 g2 = X Y X + XY X − Y X 3 3 6 13 g3 = XY X 2 + Y X 3 22 g4 = Y X 3 and λ1 = 4, λ2 = 3, λ3 = 100 11 , 3 λ4 = 105 . 44 4 Conclusions In this paper we consider polynomials in noncommuting variables which can be decomposed into a sum of Hermitian squares and commutators. Such nc polynomials are cyclically equivalent to a sum of Hermitian squares and are trace-positive. In the first part of the paper we present a systematic way to find such a decomposition of a given nc polynomial using computer algebra system. The main part of the method, which is a variant of the classical Gram matrix method, is a construction of semidefinite programming feasibility problem. We also present some examples illustrating our results. We conclude with a demonstration of how to use the proposed algorithm for practical problem solving implemented in the computer algebra system NCSOStools. References [1] K. Cafuta, I. Klep, and J. Povh. NCSOStools: a computer algebra system for symbolic and numerical computation with noncommutative polynomials. Optim. Methods Softw., 26(3):363–380, 2011. http://ncsostools.fis.unm.si/. [2] M. Choi, T. Lam, and B. Reznick. Sums of squares of real polynomials. In K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), volume 58 of Proc. Sympos. Pure Math., pages 103–126. Amer. Math. Soc., Providence, RI, 1995. [3] M. de Oliveira, J. Helton, S. McCullough, and M. Putinar. Engineering systems and free semi-algebraic geometry. In Emerging applications of algebraic geometry, volume 149 of IMA Vol. Math. Appl., pages 17–61. Springer, New York, 2008. [4] M. Halická, E. de Klerk, and C. Roos. On the convergence of the central path in semidefinite optimization. SIAM J. Optim., 12(4):1090–1099, 2002. [5] J. Helton. “Positive” noncommutative polynomials are sums of squares. Ann. of Math. (2), 156(2):675–694, 2002. [6] I. Klep and J. Povh. Semidefinite programming and sums of hermitian squares of noncommutative polynomials. J. Pure Appl. Algebra, 214:740–749, 2010. [7] I. Klep and M. Schweighofer. Sums of Hermitian squares and the BMV conjecture. J. Stat. Phys, 133(4):739–760, 2008. [8] J. B. Lassere. Moments, positive polynomials and their applications, volume 1 of Imperial College Press Optimization Series. Imperial College Press, London, 2009. [9] J. B. Lasserre. Global optimization with polynomials and the problem of moments. SIAM J. Optim., 11(3):796–817, 2000/01. [10] Y. Nesterov and A. Nemirovskii. Interior-point polynomial algorithms in convex programming, volume 13 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. [11] P. Parrilo. Semidefinite programming relaxations for semialgebraic problems. Math. Program., 96(2, Ser. B):293–320, 2003. [12] P. Parrilo and B. Sturmfels. Minimizing polynomial functions. In Algorithmic and quantitative real algebraic geometry (Piscataway, NJ, 2001), volume 60 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 83–99. Amer. Math. Soc., Providence, RI, 2003. [13] S. Pironio, M. Navascués, and A. Acı́n. Convergent relaxations of polynomial optimization problems with noncommuting variables. SIAM J. Optim., 20(5):2157–2180, 2010. 101 102 On semidefinite programming based heuristics for the graph coloring problem Igor Dukanović∗ , Jelena Govorčin† , Nebojša Gvozdenović‡1, Janez Povh†2 ∗ † Univerza v Mariboru, Ekonomsko-poslovna fakulteta, Razlagova 14, 2000 Maribor Fakulteta za informacijske študije v Novem mestu, Novi trg 5, 8000 Novo mesto, Slovenia ‡ Univerzitet u Novom Sadu, Ekonomski fakultet Subotica, Segedinski put 9-11, 24000 Subotica, Serbia email: igor.mat@uni-mb.si,jelena.govorcin@fis.unm.si, nebojsa.gvozdenovic@gmail.com,janez.povh@fis.unm.si Abstract: The Lovász theta number is a well-known lower bound on the chromatic number of a graph G, and ΨK (G) is its impressive strengthening. We apply semidefinite programming formulation of both functions to obtain suboptimal (matrix) solutions in a polynomial time. These matrices carry valuable information on how to color the graph. The resulting graph coloring heuristics utilizing these two functions are compared on medium sized graphs. Keywords: semidefinite programming, graph coloring problem, boundary point method Math. Subj. Class. (2010): Primary 90C22, 13J30; Secondary 14P10, 47A57, 08B20 1 Introduction Let G(V, E) be a simple (i.e. loopless and undirected) graph, where V denotes the set of vertices of G, and E the set of edges of G. Since we represent graphs by matrices, we assume that vertex set is V = {1, 2, ..., n}. Symmetric matrix X is positive semidefinite (denoted by X  0), if all its eigenvalues are non-negative. Map c : V → {1, ..., C} such that ij ∈ E ⇒ c(i) 6= c(j) (1) is a graph C-coloring. Subset S ⊆ V is stable, if no pair of vertices in S is connected by an edge, i.e. i, j ∈ S ⇒ ij ∈ / E. Vertices colored by the same color i therefore form a stable set −1 c (i). So graph C-coloring forms a partition of the vertex set into C stable sets, and the graph coloring problem is to find such a partition with the smallest cardinality χ(G). 2 Some formulations of χ(G) and their SDP relaxations Given a simple graph G(V, E), V = {1, ..., n}, and a C-coloring c we can assign to each stable set c−1 (i), 1 ≤ i ≤ C, its characteristic vector xi ∈ {0, 1}n , defined coordinate-wise by xij The coloring matrix X := 1Supported 2Supported P ix =  i (xi )⊤ 1, if c(j) = i, 0, otherwise. (2) has the following obvious properties by the Serbian Ministry of Education and Science (projects III 44006 and OI 174018) by the Slovenian Research Agency (bilateral project no. BI-SR/10-11-040). 1 103 xij ∈ {0, 1} for all i, j ∈ V, (3) xii = 1 for all i ∈ V, (4) xij (5) = 0 for all ij ∈ E, X̃  0, (6) and rank(X) = C. In fact any X  0 satisfying (3)-(5) represents a legal coloring[3]. Following standard homogenization procedure ([7, 5, 9] we rather add to the matrix X the zeroeth row and column (which do not represent any vertex), and study the matrices ⊤     C  X C e⊤ 1 1 (7) X̃ = := · e X xi xi i=1 where e denotes the vector of all ones. Obviously X̃  0. In fact, it is easy to see [3] that   t e⊤ C = rank(X̃) = min{t|  0, X feasible for (3) − (5)}. e X (8) Therefore χ(G) = min t  t e⊤ s. t. X̃ = 0 e X X̃ feasible for (3) − (5). (GCP) Removing binary constraint (3) yields the Lovász Θ function([7, 3, 4]) Θ(G) = min t  t e⊤ s. t. X̃ =  0, e X xij = 0 for all ij ∈ E, xii = 1 for all i ∈ V, (Θ) a well-known relaxation of the chromatic number. Let X̃ be an optimal solution of (Θ). Karger, Motwani and Sudan [6] defined a heuristic with the best theoretical upper bound on the number of colors needed to legally color a graph. A pair i, j ∈ V is likely to be colored by the same color by their heuristic only when xij is large. A recursive variant of their heuristic performs well on medium sized graphs [3], and benefits from strengthening SDP bound (Θ) toward χ(G). Therefore in the next section we introduce ΨK (G), an impressive strengthening of Θ(G) based on moment matrix idea[4]. 3 Function ΨK (G) Let K ⊆ V be a clique, i.e. a set of pairwise connected vertices. W.l.o.g. we may assume K = {1, . . . , k}, i.e. each pair of the first k vertices of V forms an edge, for some k < n. We define for the given coloring c the moment coloring vectors y i ∈ {0, 1}1+n+k(1+n) , 1 ≤ i ≤ C based on the clique K as follows y i := [1, xi1 , . . . , xin , xi1 , xi1 xi1 , xi1 xi2 , . . . , xi1 xin , . . . xik , xik xi1 , . . . , xik xin ]⊤ where xij are from (2). 104 (9) For easier referencing to elements of y i we label components of y i by monomials of order 2 in (commutative) variables z0 := 1, z1 , . . . , zn : z02 , z0 z1 , . . . , z0 zn , z1 z0 , z12 , z1 z2 , . . . , z1 zn , . . . , zk z0 , zk z1 , . . . , zk zn . | {z } | {z } | {z } k-th block 0-th block 1-st block For any n-tuple of values (xi1 , . . . , xin ) we have yzi p zq = yzi q zp , 0 ≤ p, q ≤ n. Vectors [xi1 , . . . , xin ] have actually only 0 − 1 values and since we fix z0 = 1 we also have the following equations for every y i : yzi p2 = yzi 0 zp These equations imply that several components from y i can be deleted. In numerical implementations they need to be deleted, but here we keep them as the larger matrix (10) has nicer structure making the arguments simpler. Resembling (7) we can construct the matrix Y := C X y i (y i )⊤  0. (10) i=1 It has a (k + 1) × (k + 1) block structure  00 Y Y 01 · · ·  Y 10 Y 11 · · ·  Y = . .. ..  .. . . Y k0 · · · · · · Y 0k Y 1k .. . Y kk      where blocks are actually indexed by z0 , . . . , zk and each block is a (n + 1) × (n + 1) matrix pq (indexed again by z0 , . . . , zn ). Hence yst denotes the zs zt -th element in zp zq -th block. Notice that Y 00 = X̃ from (7). The elements of the matrix Y satisfy the following obvious properties ij ypq ∈ {0, 1} ∀ i, j, p, q, i + j + p + q 6= 0 (11) ij ypq = 1, if all the vertices in {i, j, p, q} bear the same color (12) ij ypq (13) ij ypq = 0, if {i, j, p, q} contains an edge, = ′ ′ ypi ′jq′ if zi zj zp zq = zi′ zj ′ zp′ zq′ . (14) Since the index 0 does not make a difference, we also have Y 0i = Y ii , 1 ≤ i ≤ k. (15) Meanwhile the ”off-diagonal” blocks Y ij = Y ji , 1 ≤ i < j ≤ k, equal 0 as ij ∈ E since i, j belong to the clique K. So Y has the following block structure:  0 Y Y1 1  Y Y1   2 0 Y = Y  .. ..  . . Y k ··· Y2 0 Y2 .. . ··· ··· ··· .. . ··· ··· Yk 0 0       0  Yk where Y i := Y ii = Y 0i . Though Y is a large matrix the condition Y  0 is easy to check as (due to a simple manipulation by a certain unitary matrix [4]) 105 Y  0 ⇐⇒ Y − 0 k X Y i  0 and Y i  0 for each 1 ≤ i ≤ k. i=1 After relaxing the binary constraint (like from (GCP) to (Θ)) we define ΨK (G) = min t 0 = t, s. t. y00 yii0 = 1, ∀i ∈ {1, ..., n} i ypq = 0, whenever {i, p, q} contains an edge ′ i i ypq = yp′ q′ , whenever zi zp zq = zi′ zp′ zq′ Pk 0 i Y − i=1 Y  0, Y i  0∀i ∈ {1, ..., k}. (ΨK ) Now Θ(G) ≤ ΨK (G) ≤ χ(G). Let Ȳ be an optimal solution of (ΨK ). It turns out that from the principal block Ȳ 0 one can extract better information about graph coloring than from an optimal solution X̃ of (Θ). i , 1 ≤ i ≤ k, inspired by (12) is that all Yet another interpretation of a large element ȳpq three vertices i, p and q should be colored by the same color. 4 Heuristics For a given graph G(V, E) we numericaly solve semidefinite program (Θ). Let X̃ be the computed matrix solution, and xij its largest off-diagonal entry. Then with large probability there exists an optimal partition of V into χ(G) stable sets such that i and j belong to the same stable set (the same element of this partition). In fact typpically there exists many such optimal partitions. Therefore we want to color vertices i and j by the same color. But finding a coloring c of the graph G(V, E) such that c(i) = c(j) is equivalent task to finding a coloring c′ of a graph G′ = G/{i,j} with one vertex less. In the graph G/{i,j} vertices i and j are merged into new vertex i′ where the neighborhood of i′ s the union of the neighborhoods of vertices i and j. Let c′ be by this algorithm recursively obtained coloring of G′ . We define c(j) := c(i) := c′ (i′ ) while for all other vertices c := c′ . This recursive heuristic (with details and fine tuning explained in [3]) is very competitive especially in terms of quality on medium sized graphs. It has been observed already in [3] that it benefits from replacing (Θ) by its strengthening toward (GCP). So we replace Θ(G) by ΨK (G) and likewise merge the two vertices i and j corresponding 0 in the numerical optimal matrix solution Ȳ 0 of (Ψ ). The to the largest off-diagonal entry ȳij K later program though quite large can efficiently be solved by a very robust boundary point method [10, 8]. Additional advantage of solving (ΨK ) is that we can also merge three vertices l, p, q corl in the blocks Y l , l = 1, ..., k, (see (12)) an responding to the largest off-diagonal element ypq approach that we plan to investigate further. 5 Preliminary numerical results There is an extensive numerical evidence [4] that ΨK (G) strengthens Θ(G) a lot when K is a largest or an almost largest clique. It is relatively cheap to efficiently find a suboptimal clique [1], however computational complexity increases roughly |K| times when we switch from Θ(G) to ΨK (G). As in our recursive heuristic we need to compute such SDP bound on χ(G) almost n = |V | times we have investigated how the bound improves with and increase in the size of a clique. A rough demonstration is presented in the Table 1. Notice that all our results are obtained on eight random graphs G1 , ..., G8 with 20, 40, 60 or 80 vertices and with edge 106 density 0.4 or 0.6. These graphs are exactly the toughest for a competitive interior point method approach (see [2]). graph G1 G2 G3 G4 G5 G6 G7 n |E| Θ(G) Ψ2 (G) Ψ3 (G) Ψ4 (G) 20 81 4.9999 4.9999 4.9999 4.9999 20 98 5.9999 5.9998 5.9999 5.9999 40 304 6.0029 6.0057 6.0372 6.1217 40 489 9.2509 9.4696 9.5133 9.5882 60 694 6.9999 6.9880 7.0772 7.1620 60 1062 10.6542 10.9544 11.0745 11.1591 80 1221 7.4595 7.7838 7.9111 8.0381 Table 1. Comparison of Θ(G) with ΨK (G) for K a clique of size 2,3 and 4 These examples in Table 1 demonstrate that ΨK is indeed a significant strengthening of Θ. However, the improvements are consistent with general observations from discrete optimization: we have to put a lot of effort for a small strengthening of such bound. There are also time complexity issues. Computing approximate colorings by applying ΨK is very time consuming, e.g. applying Ψ4 on G7 takes approximately 10 hours on laptop with two 2GBh processors and 2 Gb Ram. The Table 2 compares colorings obtained by applying Θ and ΨK . Coloring small graphs by applying the recursive variant of Karger-Motwani-Sudan’s heuristics typically finds an optimal χ(G)-coloring and on medium graphs an almost optimal one [3] - a hard to bit result. By comparing the lower bound in Table 1 with the number of colors in Table 2 notice that the colorings of G1 , G2 and G3 are indeed optimal, while already on the graph G7 , a graph with just 80 vertices, we were able to improve the coloring impressively by using two colors less - a rare instance and hard to obtain result. On the other hand notice an unusual result in Table 1 for the graph G5 where for a stronger bound ΨK (G5 ) with K an edge, i.e. |K| = 2, we have computed its approximation which is smaller then Θ(G5 ). The problem is due to the fact that for boundary point method we do not have a simple duality-gap-like stopping criterium. Likewise the numerical results for graphs G5 and G6 in Table 2 where stronger bounds occasionaly produced worse colorings are just a bit unusual. This kind of numerical unstability is not very unusual for this heuristic approach, and it should be possible to avoid by more carefull choice of stopping criterium. 6 Conclusions Replacing Lovász theta function Θ(G) by its impressive strengthening ΨK (G) can improve colorings obtained by Karger-Motwani-Sudan-motivated heuristics (see [3]) impressively graph/bound Θ Ψ2 Ψ3 Ψ4 G1 5 5 5 5 G2 6 6 6 6 G3 7 7 7 7 G4 11 11 11 11 G5 9 10 10 9 G6 13 14 13 13 G7 12 11 11 10 Table 2. Number of colors used by heuristics depending on the bound Θ or ΨK , |K| = 2, 3, 4 107 already on quite small graphs. Tradeoff between computational complexity (size of the clique K, stopping criteria for computing ΨK by the boundary point method, number of merged vertices (2 or 3)) versus quality of the produced coloring need to be investigated further. As numerical evidence shows (see G5 , G6 , G7 in Table 2) ΨK can jump over an integer thus improving the Θ bound on the chromatic number by 1. Such improvement can to a much more serious extent be exploited in a branch and bound algorithm for finding an optimal coloring. Thus we plan to substantially improve the speed reported in [11] where Θ was applied in a branch and bound exact coloring algorithm. References [1] S. Burer, R.D.C. Monteiro, and Y. Zhang. Maximum stable set formulations and heuristics based on continuous optimization. Math. Program. Ser. A, 94:137—-166, 2002. [2] I. Dukanovic and F. Rendl. Semidefinite programming relaxations for graph coloring and maximal clique problems. Mathematical Programming, 109:345–365, 2007. [3] I. Dukanovic and F. Rendl. A semidefinite programming-based heuristic for graph coloring. Discrete Applied Mathematics, 156(2):180 – 189, 2008. [4] N. Gvozdenović and M. Laurent. Computing semidefinite programming lower bounds for the (fractional) chromatic number via block-diagonalization. SIAM J. Optim., 19(2):592–615, 2008. [5] N. Gvozdenović, M. Laurent, and F. Vallentin. Block-diagonal semidefinite programming hierarchies for 0/1 programming. Oper. Res. Lett., 37(1):27–31, 2009. [6] D. Karger, R. Motwani, and M. Sudan. Approximate graph coloring by semidefinite programming. Journal of the Association for Computing Machinery, 45:246–265, 1998. [7] L. Lovász. On the Shannon capacity of a graph. IEEE Transactions on Information Theory, 25:1–7, 1979. [8] J. Malick, J. Povh, F. Rendl, and A. Wiegele. Regularization methods for semidefinite programming. SIAM J. Optim., 20(1):336–356, 2009. [9] J. Povh and F. Rendl. Copositive and semidefinite relaxations of the quadratic assignment problem. Discrete Optim., 6(3):231–241, 2009. [10] J. Povh, F. Rendl, and A. Wiegele. A boundary point method to solve semidefinite programs. Computing, 78(3):277–286, 2006. [11] A. T. Wilson. Applying the Boundary Point Method to an SDP Relaxation of the Maximum Independent Set Problem For a Branch And Bound Algorithm. New Mexico Institute of Mining and Technology, Socorro, New Mexico, 2009. 108 EXPERIMENTAL COMPARISON OF BASIC AND CARDINALITY CONSTRAINED BIN PACKING PROBLEM ALGORITHMS Maja Remic University of Ljubljana, Faculty of Computer and Information Science Tržaška 25, SI-1000 Ljubljana, Slovenia maja.remic@siol.net Gašper Žerovnik Jožef Stefan Institute, Reactor Physics Division Jamova 39, SI-1000 Ljubljana, Slovenia gasper.zerovnik@ijs.si and Janez Žerovnik Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1111 Ljubljana, and University of Ljubljana, Faculty for Mechanical Engineering, Aškerčeva 6, SI-1000 Ljubljana, Slovenia janez.zerovnik@imfm.uni-lj.si Abstract: Bin packing is an NP­hard optimization problem of packing items of given sizes into minimum number of capacity­limited bins. Besides the basic problem, numerous other variants of bin packing exist. The cardinality constrained bin packing adds an additional constraint that the number of items in a bin must not exceed a given limit Nmax. The FFD, RFF and Zhang's bin packing problem algorithms are compared to the three cardinality constrained bin packing problem specific algorithms on random lists of items with 0%, 10%, 30% and 50% of large items. The behaviour of all algorithms when Nmax increases is also studied. Results show that all three specific algorithms outperform the general algorithms on lists with low percentage of big items. One of the specific algorithms performs better or equally even on lists with high percentage of big items and is therefore of significant interest. The behaviour when Nmax increases shows that all three specific algorithms can be used for solving the general bin packing problem as well. Keywords: cardinality constrained bin packing problem, approximation algorithms, comparison, FFD, RFF, Zhang's algorithm. 1 INTRODUCTION The cardinality constrained bin packing problem can be described as follows: A list I of n items with specified size (or weight/volume/etc.) xi has to be arranged into bins of limited capacity Cmax and maximum number of item per bin (cardinality constraint) Nmax to minimize the number of bins m used. The minimization problem has been shown to be NP-hard [8]. The cardinality constrained bin packing problem can, for example, be applied to optimization of the spent nuclear fuel deposition in deep repository [7]. Swedish concept of deep repository in hard rock [3] is currently seriously regarded in Slovenia as an option for nuclear power plant Krško decommissioning program [6]. Motivated by this application, several heuristics for the cardinality constrained bin packing problem with Nmax =4 were designed in [8]. The new heuristics were compared against an obvious adaptation of the first fit decreasing (FFD) algorithm and were proven to clearly outperform the FFD algorithm on the datasets of interest. It is well known that FFD algorithm is a good approximation algorithm for the general bin packing problem (more precisely, it is well known [2] that FFD always gives a solution with at most 11/9 OPT(I) + C bins, where OPT(I) stands for the value of optimal solution). Therefore, it is natural to ask how the generalized versions of the new algorithms behave on the bin packing problem with arbitrary cardinality constraints. A 109 preliminary experimental study is presented in this report which shows that the obvious adaptations of the new algorithms are competitive also on the general cardinality constrained bin packing problem. 2 ALGORITHM ADJUSTMENTS In this paper, three algorithms (referred to as Algorithm 1, 2, and 3) from Ref. [8] (see Appendix) are experimentally compared to First Fit Decreasing (FFD) [1], Refined First Fit (RFF) [4] and Zhang [5] algorithms. Since the latter three are designed for the original bin packing problem, the following modifications had to be performed in order to adapt to the additional cardinality constraint:  For any item, the FFD algorithm chooses the first bin with both enough space (capacity constraint) and at least 1 empty slot (cardinality constraint).  Similarly, the RFF algorithm places each item in the first possible bin of the suitable group.  Zhang algorithm closes the (active or additional) bin also when the cardinality constraint is reached. All input data for all algorithms have been sorted by size, even though for RFF it is not necessary. 3 EXPERIMENTAL COMPARISON OF ALGORITHMS First, the quality of solutions, obtained by different algorithms, was experimentally compared for different input data distributions. The main purpose of this investigation was to asses the performance of algorithms from [8], specifically designed for the cardinality constrained bin packing, relative to the (suitably adapted) algorithms for the general bin packing problem. Similar experiment was already presented in [8], with two significant differences. In this paper, RFF and Zhang algorithms were added for the comparison. Furthermore, in the present experiments additional instances with different fractions of large items were regarded. 3.1 Experiments with fixed Nmax = 4 As in [8], default values of Nmax = 4 items per bin and capacity Cmax = 1 were adopted. Four different types of input data of length n = 100 were used:  lists without large items (i.e. items, larger than Cmax/2),  lists with 10% large items,  lists with 30% large items, and  lists with 50% large items. All input lists were generated by default Octave random number generator, using uniform distribution with compositions given above. The input data distributions are shown on Fig. 1. The experiment has been repeated 10000 times with input data generated from different random seeds. 110 Figure 1: Probability density P(x) over item size x for different types of input data. The theoretical lower bound MIN(I) for the solutions of cardinality constrained bin packing problem can be expressed as:  n      xi   n   MIN ( I )  max   i 1  ,   ,   Cmax   N max        (1) which is a useful and conservative approximation for the optimal solution OPT ( I )  MIN ( I ) . The quality of the obtained solution A(I) is measured by A( I ) / OPT ( I ) and can be estimated by A( I ) / MIN ( I ) . The values for all algorithms, averaged over 10000 instances, are presented in Table 1. Table 1: Average values of the A(I) / MIN(I) ratio for selected algorithms at different lists. A(I) / MIN(I) 0% large items 10% large items 30% large items 50% large items FFD RFF Zhang Algorithm 1 Algorithm 2 Algorithm 3 1.1273 ± 0.0033 1.3318 ± 0.0041 1.3606 ± 0.0034 1.0262 ± 0.0026 1.0493 ± 0.0039 1.0256 ± 0.0024 1.0933 ± 0.0036 1.3199 ± 0.0045 1.3530 ± 0.0040 1.0698 ± 0.0024 1.0552 ± 0.0030 1.0571 ± 0.0022 1.0348 ± 0.0018 1.3251 ± 0.0038 1.2680 ± 0.0026 1.1421 ± 0.0034 1.0504 ± 0.0019 1.0992 ± 0.0022 1.0418 ± 0.0018 1.3323 ± 0.0034 1.2581 ± 0.0024 1.2059 ± 0.0038 1.0440 ± 0.0017 1.1434 ± 0.0024 The results clearly show the influence of the fraction of large items on the quality of the solution for individual algorithms. The Zhang and FFD algorithms work much better with significant fractions of large items, while the opposite can be stated for Algorithm 1 and 3. For RFF and Algorithm 2, the sensitivity of the quality of solutions on large item fractions is insignificant. Direct comparison between algorithms shows that Zhang and RFF in general give much worse solutions than FFD. For RFF that kind of behavior is expected since it is (in contrast to other five algorithms) basically an online algorithm. Worse performance of the Zhang algorithm is probably due to linear time complexity. All three algorithms from [8] perform 111 significantly better than others for small (up to 10%) fractions of large items, whereas for 50% of large items, only Algorithm 2 is comparable to FFD. This is expected since Algorithm 2 was designed for input data with significant fraction of large items [8] while Algorithms 1 and 3 were designed for input data with (almost) no large items. Let us mention at this point that Algorithm 3 is random number based [8], therefore its solution may be improved significantly when taking advantage of multi-start mode. 3.2 Increasing the cardinality constraint The purpose of the second experiment was to examine the behaviour of the algorithms when increasing the cardinality constraint Nmax. When Nmax is converged to the total number of items n, the cardinality constraint becomes irrelevant, consequently the cardinality constrained bin packing problem converges to the original bin packing problem in this limit. In this experiment, list length n = 100 and capacity Cmax = 1 was used throughout. Two different types of input data lists were used: lists without large items and with 50% large items (Fig. 1), which were generated in exactly the same way as in the first experiment. The cardinality constraint was changed between Nmax = 2 and Nmax = n/4 since the latter was experimentally observed to be enough to achieve the convergence. The experiment was repeated 1000 times with different input data, each generated with different random number sequence. The results are shown in Figs. 2 and 3. Figure 2: Average number of bins as a function of the cardinality constraint for lists without large items. Figure 3: Average number of bins as a function of the cardinality constraint for lists with 50% of large items. 112 Expectedly, the number of bins used in most cases decreases with the increase of Nmax, up to the point where Nmax no longer has impact and the capacity Cmax of the bin becomes the limiting factor in Eq. (1). The speed of convergence depends on the percentage of large items in the list. When the percentage is high, the impact of Nmax is smaller, meaning faster convergence. This is evident in lists with 50% large items (Fig. 3). For lists without large items, all three algorithms for solving the cardinality constrained bin packing problem perform best at Nmax = 4, which is the setting the algorithms were developed for, and deteriorate slightly as Nmax increases. Similarly, for lists with 50% large items, both Algorithm 1 and 3 experience unexpected swings at Nmax < 5. This behaviour can be explained by the functioning of those two algorithms, since at small Nmax, larger items are considered first. It is notable that the increasing of Nmax has little influence over the final number of bins used by each algorithm. Even at the non-limiting Nmax, the algorithms for solving the general bin packing problem find no better solutions than the specific algorithms, the FFD algorithm being the only exception and performing slightly better on lists without large items. 4 CONCLUSION As the experiments have shown, all three (cardinality constrained bin packing problem) specific algorithms perform better compared to the general (bin packing) algorithms on lists without large items or with a low percentage of large items, while having a similar time complexity. In such cases, the use of specific algorithms is recommended. In case of input data with significant fractions of large items, the general FFD algorithm perform best, closely followed by the specific Algorithm 2. All specific algorithms can be generalized to the basic bin packing problem by sufficiently increasing the cardinality constraint. Surprisingly or not, the quality of solutions is hardly compromised. The best performer is again Algorithm 2, therefore it can be regarded as a good alternative to the FFD algorithm even for the basic bin packing problem. References [1] Coffman, E. G., Garey, M. R., Johnson, D. S., 1997. Approximation algorithms for bin­packing: A survey. In: Approximation Algorithms for NP­Hard Problems, PWS, Boston, pp. 46­93. [2] Korte B, Vygen J., 2000. Combinatorial Optimization, Theory and Algorithms. Springer-Verlag: Berlin. [3] Milnes AG., 2002. Guide to the documentation of 25 year of geoscientific research (1976-2000). TR-02-18, VBB Anlaggning AB, Swedish Nuclear Fuel and Waste Management Co. [4] Yao, A. C., 1980. New algorithms for bin packing. Journal of ACM, vol. 2, issue 27, 207-227. [5] Zhang, G. Cai, X., Wong, C. K., 2000. Linear­time­approximation algorithms for bin packing. Operation Research Letters, vol. 5, issue 26, pp. 217­222. [6] Železnik N, Mele I, T. Jenko T, Lokner V, Levanat I, Rapić A., 2004. Program razgradnje NEK in odlaganja NSRAO in IJG (Program of NPP Krško Decommissioning and SF \& LILW Disposal). ARAO-T-1123-03, ARAO (Agency for Radwaste Management), Ljubljana, Slovenia, and APO (Agency for Hazardous Waste), Zagreb, Croatia. [7] Žerovnik, G., Snoj, L., Ravnik, M., 2009. Optimization of Spent Nuclear Fuel Filling in Canisters for Deep Repository. Nuclear Science and Engineering, vol. 163, issue 2, pp. 183-190. [8] Žerovnik, G., Žerovnik, J., 2010. Constructive heuristics for the canister filling problem. Central European Journal of Operations Research, DOI:10.1007/s10100­010­0164­5. 113 5 APPENDIX While FFD, RFD and Zhang’s algorithm are well known, the Algorithms 1, 2 and 3 first appear in a very recent publication [5], and therefore we give a brief outline here. Generic Algorithm quicksort X; m := 0; while X   do begin set aux := ; C := Cmax; for k: = K down to 1 do begin x := CHOOSE(X); (*) if x is defined then X := X \{x}; C := C - x ; aux := aux  {x} endif; endfor; m := m + 1; Rn = aux; endwhile; return R = {R1,R2, ... ,Rn}. The algorithms differ only in implementation of the function CHOOSE. CHOOSE in Algorithm1: CHOOSE(X) := x, the largest element x of X that satisfies x ≤ C/k; CHOOSE in Algorithm2: if k = K then CHOOSE(X) := x, the largest element of X that satisfies x ≤ C; if k < K then CHOOSE(X) := x, the largest element of X that satisfies x ≤ C/k; CHOOSE in Algorithm3: if k > Floor(K/2) then CHOOSE(X) := x, a random element of X that satisfies x ≤ C; if k ≤ Floor(K/2) then CHOOSE(X) := x, the largest element of X that satisfies x ≤ C/k; Remark: Clearly if CHOOSE is simply CHOOSE(X) := x, where x is largest element of X that satisfies x ≤ C; we have the algorithm which puts each element in the first bin in which there is enough room. If K is large enough, this variant is equivalent to the FFD Algorithm. 114 The 11th International Symposium on Operational Research in Slovenia SOR ’11 Dolenjske Toplice, SLOVENIA September 28 - 30, 2011 Section V: Multiple Criteria Decision Making 115 116 INVESTIGATION OF THE AGGREGATION-DISAGGREGATION APPROACH TO MULTI-CRITERIA NEGOTIATIONS: CONSOLIDATION OF SIMULATION AND CASE STUDIES Andrej Bregar Informatika d.d., Vetrinjska ulica 2, 2000 Maribor andrej.bregar@informatika.si Abstract: The characteristics of a state-of-the-art outranking relation and dichotomic sorting based aggregation-disaggregation procedure for multi-criteria negotiations are uniformly analysed with two research methods – a simulation study and a case study. Results are compared according to a standard holistic model for the evaluation of group decision-making methods and systems, which consists of several key quality factors, such as ability to direct the decision-making process, credibility of the decision, robustness, and cognitive complexity. Both studies consistently prove that the observed approach to automated negotiations is efficient. Keywords: Multi-criteria decision analysis, Decision support systems, Negotiations, Group decisionmaking, Consensus seeking, Preference aggregation and disaggregation, Outranking relation, Sorting, Simulation experiments, Case studies, Comparative studies 1 INTRODUCTION To aid conflicting and opposing decision-makers in reaching consensual or compromise solutions, many methods and decision support systems for negotiations and group decisionmaking have been introduced [10]. They are based on various preference models. Among the most widely used are the approaches that capture preferential information in the form of pseudo-criteria to construct outranking relations on pairs of alternatives. These methods and systems include group PROMETHEE [8], ELECTRE TRI for groups [6], ELECTRE-GD [11] and the collective preorder inference method [9]. On the other end, state-of-the-art approaches to negotiations and group decision-making aim at improving human-computer interaction and efficiency of outcomes by applying the aggregation-disaggregation analysis [13]. They are, however, mainly based on the utility function [12] and oriented graphs [7]. Recently, two methods that combine the principles of group decision-making, outranking and aggregation-disaggregation have been introduced. The interactive dichotomic sorting procedure for multi-agent consensus seeking is the result of our research work [4], while the group version of the IRIS system has been developed at the LAMSADE laboratory [5]. It is necessary to determine the characteristics and efficiency of the outranking relation based aggregation-disaggregation methods that are applied to multi-criteria negotiations and group decision-making processes, especially in the context of convergence, autonomous guidance, conflict resolution, robustness and cognitive complexity. For this purpose, our consensus seeking procedure has been evaluated with a simulation study [1] and tested with a case study [3]. However, the results of the latter have neither been assessed according to a formal evaluation model nor systematically compared with the results of the former. The goal of the paper is thus to: 1. uniformly analyse the results of the case study and the simulation study with regard to a holistic model for the evaluation of group decision-making methods/systems [2]; 2. compare the results of both studies, and consequently determine, if they expose any divergence in the characteristics of the consensus seeking procedure. The rest of the paper is organized as follows. Section 2 provides a brief description of the evaluated method. Section 3 presents the simulation model, while Section 4 summarizes the 117 case study. In Section 5, experimental and case based results are presented, analysed and compared. Finally, Section 6 gives a resume and some directions for further work. 2 DICHOTOMIC SORTING PROCEDURE FOR GROUP DECISION ANALYSIS The procedure is based on the ELECTRE TRI outranking method [14], which is slightly modified so that preferences are modelled in a symmetrically-asymmetrical manner in the neighbourhood of the reference profile b. The purpose of the profile is to divide the set of alternatives into two exclusive categories – all acceptable choices are sorted into the positive class C +, while unsatisfactory ones are the members of the negative class C –. The decisionmaker has to provide six preferential parameters for each criterion xj, including the value of the profile gj(b), the importance weight wj, and the thresholds of preference (pj), indifference (qj), discordance (uj) and veto (vj). Additionally, he can also specify the upper and lower allowed limits of these parameters, which constrain their automatic adjustment in the process of unification with the common opinion of the group. In order to reduce the cognitive load and enable a rational convergence of individual judgements towards the consensual solution, several mechanisms are applied:  Preferences may be specified with fuzzy variables or by holistic assessments.  The most discordant decision-maker who has to conform to the collective opinion is identified by computing the consensus and agreement degrees.  Several robustness metrics reveal if preferences of an individual are firmly stated.  The centralized agent negotiation architecture and protocol eliminate the need for a human moderator and minimize the activity of each decision-maker.  An optimization algorithm is implemented for the purpose of automatic preference unification, which is based on the inference of parameter values. 3 SIMULATION STUDY In order to evaluate the efficiency of the aggregation-disaggregation approach to group consensus seeking with a simulation study, several independent variables have been defined:  Number of criteria may be n  {4, 7, 10}. Number of observed alternatives is fixed to m = 8 because only the m : n ratio is significant. Thus, three fundamental situations are considered: n > m, n  m and n < m.  Number of decision-makers may also be o  {4, 7, 10}. Situations with less than four decision-makers are irrelevant because the experimental study focuses on multilateral rather than bilateral negotiations.  Preferential parameters of the decision model are obtained in the following way:  The referential profile is sampled from the normal distribution N(50, 15).  Thresholds are calculated relatively to the profile. The deviations are sampled from the set of real values {0.2, 0.4, 0.6, 0.8, 1} corresponding to linguistic modifiers {very weak, weak, moderate, strong, very strong}.  Criteria importance weights are sampled from the uniform distribution on the [0, 1] interval. Afterwards, they are normalized to sum to 1.  Criteria-wise values of alternatives are sampled from the uniform distribution on the [0, 100] interval.  The cut-level is λ = 0.5, which is the most common value.  The sensitivity threshold is set to ψ = 0.3. An experimental combination is hence determined by the pair. For each of nine possible combinations, 20 samples have been generated. This number of simulation cases is 118 sufficient to obtain statistically significant results, and is small enough to cope with a high complexity of applied nonlinear optimization algorithms. In the original simulation study, four dependent variables have been observed: (1.) ability to reach a compromise, (2.) ability of autonomous guidance and conflict resolution, (3.) convergence of opinions, and (4.) robustness of the consensual decision. For the purpose of consolidation with the case study, additional dependent variables have been introduced according to the standard evaluation model, as is thoroughly explained in Section 5. Consequently, post-processing and additional synthesis of experimental data have been performed. 4 CASE STUDY A case from the information and communications technology domain is studied. It deals with the selection of a software development subcontractor. Five quantitative criteria are considered: payment, time required for the realization of the project, number of successfully finished projects in the past, experience with similar applications, and availability of suitable technologies. There are eight alternatives (subcontractors) available. To hide their identity, symbols ai are used. Similarly, six non-autocratic decision-makers in the group are denoted with DMk. They have agreed on the values of two technical parameters. The cut-level, which determines the outranking relation between two alternatives, is set to  = 0.5. The robustness threshold, which disallows an alternative to be sorted into a different class, is set to  = 0.3. Four group members have directly specified the initial values and lower/upper limits of the profile, thresholds and weights. Two have reduced their cognitive load in the following way:  They have set the initial values of the reference profile only. By disregarding the allowed limits, they have forced the optimization algorithm to search the entire space of criteria values.  They have uniformly specified the initial values and limits of all thresholds.  DM2 has defined the limits of threshold intervals, while the initial values have been automatically calculated as centre points.  DM4 has specified all threshold values and limitations relatively to the profile with linguistic modifiers from the set {very weak, weak, moderate, strong, very strong}. Each iteration of the dichotomic sorting based aggregation-disaggregation procedure for consensus seeking consists of several steps: 1. At first, fuzzy and strict outranking relations are calculated. The former are expressed with values on the [0, 1] interval. The latter are derived with the -cut. 2. Robustness degrees rk(ai) are computed. Each shows to what extent the k-th decision-maker's preferential parameters must deviate for the category of the i-th alternative to change. The bigger adjustment that is required, the more robustly ai is evaluated from the perspective of DMk. If the robustness exceeds the  threshold, then the decision-maker or his agent should not be asked to conform to the majority opinion. 3. Alternatives are rank ordered according to received votes. For a single alternative this is the number of outranking relations stating that it is an acceptable choice. 4. Since alternatives can differ considerably with regard to compromise votes, the group gets directed on the basis of consensus and agreement degrees. The agreement degree  k shows to what extent DMk is in concordance with the collective opinion of the whole group. DMk with the lowest  k must generally reconsider his preferences in order to conform to the group. However, if some of his assessments are robust, which means that the rk(ai)   inequality holds true for at least one alternative, someone else is selected for conformation. The decision 119 support system thus verifies for each decision-maker, whether the required reassignment of alternatives is feasible. In the first iteration, each decision-maker is able to sort at least one alternative with low agreement and robustness degrees to a different category. Hence, a matrix of required strict outranking relations is induced from the initial outranking matrix by changing the relation either from 0 to 1 or vice versa for each alternative with i < 0.5 and r(ai) < . The most discordant decision-maker DM2 is able to reassign alternatives a1, a3, a4 and a7, but he is not allowed to change the category of a2 because of its high robustness r2(a2) = 0.399 > 0.3 = . DM2 Initial fuzzy relations Initial strict relations Robustness degrees Agreement degrees Required strict relations Reassignments a1 0.688 1 0.263 0.200 0 1 a2 0.725 1 0.399 0.400 1 0 a3 0.600 1 0.053 0.200 0 1 a4 0.750 1 0.194 0.000 0 1 a5 0.700 1 0.345 0.600 1 0 a6 0.267 0 0.541 1.000 0 0 a7 0.588 1 0.288 0.000 0 1 a8 0.900 1 0.724 0.600 1 0 The optimization algorithm is now applied to reassign alternatives and to simultaneously infer new values of preferential parameters. As these values do not violate specified interval constraints, a new negotiation iteration is started. The overall consensus degree rises from 0.458 to 0.625 in the second iteration, and in the final iteration, it reaches 0.917. Seven iterations are needed to find a consensual solution. All subsequent iterations, except the second, proceed identically as the first, which means that the negotiation procedure operates totally automatically. Only in the second iteration, the decision-maker's constraints are violated, so he must manually check newly inferred values of preferential parameters and confirm their acceptability. 5 COMPARATIVE ANALYSIS OF RESULTS The case study does not refer to the same cases as the simulation. As has been explained in Section 3, 180 different samples (cases) have been randomly generated for nine possible experimental combinations of the latter. On the contrary, exactly one specific case has been defined within the scope of the former. However, the results of both studies are directly comparable because they are analysed according to the same set of criteria that constitute a formal model for the evaluation of group decision-making methods and systems [2]. Since the model is holistic and complex, only a subset of its criteria and metrics is considered. It is structured on Figure 1. All metrics are quantitative. They can be directly applied to data of both studies. They hence enable a consistent comparison of simulation and case study results. For most metrics, results are computed as scalars (M1.1, M1.2, M1.3, M2.3, M3.2, M3.3, M4.1, M4.2, M5.1, M5.2, M6.1, M6.2, M7.1, M7.2, M7.3, M7.4). Other metrics produce vectors or matrices. Where appropriate, the comparability of results for various experimental combinations of the simulation study is determined with a one-way ANOVA analysis of variance. The M5.1 and M5.2 metrics are operationalized with the Kemeny-Snell distance between ordinal rank orders. Exact definitions of metrics and general criteria are omitted. The interested reader should refer to the related literature [1, 2]. The results of scalar metrics are listed in the table. They provide a direct comparison of simulation and case studies. It can be stated that the dichotomic sorting based aggregationdisaggregation procedure for multi-criteria group consensus seeking is efficient with regard to both research methods. The following conclusions can be drawn: 120 1. The best alternative is always unique. It has a large distance to the second best one, and especially to the set of all other suboptimal choices. The distance considerably increases when a compromise solution iteratively progresses to the final consensual solution. Group maintenance (ability to direct the process of group decision-making) Ability to reach a compromise M1.1: Percentage of cases in which the best alternative is not unique M1.2: Distance from the best to the second best alternative M1.3: Distance from the best alternative to the subset of all suboptimal alternatives M1.4: Average degree of compromise for the i-th best alternative M1.5: Average number of alternatives ranked in the k-th place M1.6: Average robustness of alternatives ranked in the k-th place Ability of autonomous guidance and conflict resolution M2.1: Progression of the compromise solution over negotiation iterations M2.2: Agreement degrees of individuals that are reached in each iteration M2.3: Absolute increase of the calculated agreement degrees Convergence of opinions M3.1: Monotonous iterative progression of the consensus degree M3.2: Improvement of the consensus degree compared to the initial compromise M3.3: Number of iterations Analysis Credibility of analysis with regard to robustness Improved richness of discriminating information (consensus versus compromise) M4.1: Difference in the distance from the best to the second best alternative M4.2: Difference in the distance from the best alternative to all other alternatives M4.3: Change of the average number of votes for the i-th best alternative M4.4: Change of the average number of alternatives ranked in the k-th place M4.5: Change of the average robustness of alternatives ranked in the k-th place Sensitivity to changes in the problem structure M5.1: Number of rank reversals when an existing alternative is discarded M5.2: Percentage of rank reversal cases when an existing alternative is discarded Sensitivity to changes in the decision-making group M6.1: Number of rank reversals when a decision-maker leaves the group M6.2: Percentage of rank reversal cases when a decision-maker leaves the group Complexity of analysis M7.1: Total number of preferential parameters M7.2: Quantity of inputs that are required for the first negotiation iteration M7.3: Number of manual adjustments of parameters in each subsequent iteration M7.4: Amount of data analysed in each iteration Figure 1: Applied criteria and metrics of the evaluation model 2. The overall agreement and consensus degrees increase drastically from the first to the last iteration of the problem solving process. This means that opinions of decision-makers are considerably more unified at the end than at the beginning of negotiation. Convergence is fast, since maximally seven iterations are required to reach unanimity with regard to the assessments of alternatives. Hence, it is unlikely that any decision-maker would drop out of the group because of personal time constraints, motivation or difficulties in communication. 3. Robustness of assessments of alternatives increases over subsequent iterations of the decision-making process. Although the best alternative is already sufficiently distant from suboptimal alternatives in the initial compromise solution, the distance is even larger in the case of the final consensual solution. 121 4. Decision analysis is unsensitive to changes in the problem structure. If an alternative is excluded from the set of available choices, this has no effect on the evaluation of remaining alternatives. Their rank order is thus preserved. 5. Decision analysis is not significantly sensitive to any changes in the decisionmaking group. If an individual stops participating in the consensus seeking process, this has a minor or even negligible influence on the evaluation of alternatives. Their rank order changes only slightly, in a way that rank reversals never occur. This means that it is impossible for the preference between two alternatives to revert ([a P b  b P a]). It can only happen that preference gets exchanged with indifference (a P b  b I a), or vice versa (a I b  b P a). This conclusion is not only verifiable by a formal proof, but also by observing the discrepancy between the intensity and frequency of rank order changes (M6.1 versus M6.2). 6. Cognitive load of participants of the negotiation process is relatively high only in the first iteration, when initial preferences have to be set. In all subsequent iterations, the negotiation support system operates almost automatically. In the simulation study, no human interaction was required, while in the case study, the decision-makers had to continuously observe just a small set of crucial parameters, and were asked to make a minimal amount of manual adjustments. Metric M1.1 M1.2 M1.3 M2.3 M3.2 M3.3 M4.1 M4.2 M5.1 M5.2 M6.1 M6.2 M7.1 M7.2 M7.3 M7.4 Case study 0% 0.333 0.905 0.295 0.459 7 0.333 0.548 0 0% 0.010 19.048 % 132 110 0.167 6.194 Simulation study 0% 0.276 0.646 0.386 0.399 (average, min, max) = (4.150, 2, 7) 0.165 0.131 0 0% (0.015, 0.008, 0.026) (26.081 %, 14.381 %, 31.833 %) maximally n · (m + 18) maximally n · (m + 18) 0 0 The results of the simulation study and the case study are consistent. A relatively high deviation can be observed only with regard to the M4.2 metric. However, even in this case it is evident that both studies consistently detect an improvement of the consensual solution over the initial compromise solution. Similarly, the comparability of results for various experimental combinations of the simulation study is also determined with regard to several metrics – M1.2, M1.3, M6.1 and M6.2. A one-way ANOVA analysis of variance returns p-values 0.611, 0.248, 0.108 and 0.292, respectively. No statistically significant difference among experimental combinations can hence be found for the usual threshold α = 0.05, nor even for the less common α = 0.1. Figure 2 shows a relatively fast monotonous increase of total consensus degrees towards the highest possible value of 1. Although the upper limit is not utterly reached, this is not a drawback, because rubust assignments of alternatives must prevent decision-makers to fully conform to the collective opinion of the group. Analogous to scalar metrics, M3.1 reveals that the observed method exhibits similar characteristics according to both research studies. Figure 3 demonstrates an iterative progression of the compromise solution towards the consensual solution in the case study. Decision-makers eventually agree on sorting six non- 122 optimal alternatives into the negative category C –, and assigning the optimal alternative a8 to the positive category C +. Seven out of eight alternatives thus converge towards the upper or lower limit of votes, respectively. Only a5 is not subjected to convergence because of its high robustness. The negotiation mechanism is consequently not allowed to propose its irrational reassignment. For the case study, metric M2.2 is depicted on Figure 4. It shows that agreement degrees of individuals generally increase, yet may instantaneously fall in certain cases. This happens when a group member adjusts his preferential parameters in order to unify with the collective opinion of the whole group. The discordance of one or more other decisionmakers can then consequently decrease. Figure 2: Iterative increase of the consensus degree Figure 3: Progression of the compromise solution Figure 4: Agreement degrees of decision-makers for each iteration Figures 5, 6 and 7 correspond to metrics M1.4, M1.5, M1.6, M4.3, M4.4 and M4.5. It can be seen that the weak ordinal rank order of alternatives is more efficient in the last than in the first iteration of the consensus seeking procedure. The implication is that the robustness of assessments, as well as the discrimination among acceptable and unsatisfactory alternatives improve. Similar results may be observed for the simulation study (above) and the case study (below). Figure 5: Number of votes for the i-th best alternative Metric M7.2 Figure 6: Number of alternatives ranked in the k-th place Figure 7: Robustness of the i-th best alternative Metric M7.4 (metric M7.3) Iter. 2 Iter. 3 Iter. 4 Iter. 5 Iter. 6 Iter. 7 Total Average DM1 n · (m + 18) + 2 2 2 2 2 2 2 12 2.000 DM2 n · (m + 1) + 13 26 2 2 26 2 2 60 10.000 DM3 n · (m + 4) + 14 2 2 2 2 2 2 12 2.000 DM4 n · (m + 18) + 2 2 57 (1) 2 2 2 2 67 (1) 11.167 DM5 n · (m + 18) + 2 2 2 26 2 2 2 36 6.000 DM6 n · (m + 18) + 2 2 2 2 2 26 2 36 6.000 123 Finally, the table above gives details about metrics M7.2, M7.3 and M7.4 for each decisionmaker and each iteration of the case study. All decision-makers have to provide m·n criteriawise evaluations of alternatives and n values of the referential profile in the first iteration. Other parameters vary according to the form in which preferences are expressed. In each subsequent iteration, only two values have to be mandatory observed by the decision-maker: the overall consensus degree and the total individual agreement degree. An active decisionmaker must additionally process m categories, robustness degrees and ranks for each choice. If an individual is active and has to manually adjust preferential parameters, newly derived values of parameters and violations of constraints are relevant for him as well. It should be noted that only one manual adjustment is required throughout the negotiation process. 6 CONCLUSION Both research methods – the simulation study and the case study – produce comparable results. Their complementary application clearly and consistently exposes the characteristics of the evaluated aggregation-disaggregation approach to negotiations, which is proved to be efficient. Within the scope of further research work, a questionnaire survey will also be performed to obtain and analyse the feedback of decision-makers on the usefulness of such negotiation and group decision support systems. References [1] Bregar, A. Efficiency of problem localization in group decision-making. Proceedings of 10th [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] International Symposium on Operational Research in Slovenia, 139–149, 2009. Bregar, A. Celovit model vrednotenja skupinskih odločitvenih metod in sistemov. Proceedings of 17th Symposium Dnevi Slovenske Informatike (DSI), 13 pp., 2010. Bregar, A. Metode na temelju prednostne relacije in njihova uporaba v postopkih sku-pinskega odločanja: študija primera. Proceedings of 18th DSI, 15 pp., 2011. Bregar, A., Györkös, J., Jurič, M. B. Interactive aggregation/disaggregation dichotomic sorting procedure for group decision analysis based on the threshold model. Informa-tica, 19 (2), 161–190, 2008. Damart, S., Dias, L. C., Mousseau, V. Supporting groups in sorting decisions: Metho-dology and use of a multi-criteria aggregation/disaggregation DSS. Decision Support Systems, 43 (4), 1464–1475, 2007. Dias, L. C., Clímaco, J. N. ELECTRE TRI for groups with imprecise information on parameter values. Group Decision and Negotiation (GDN), 9 (5), 355–377, 2000. El Fallah, S. A., Moraitis, P., Tsoukiàs, A. An aggregation-disaggregation approach for automated negotiation in multi-agent systems. Proceedings of ICSC Symposium on Multi-Agents and Mobile Agents, 7 pp., 2000. Espinasse, B., Picolet, G., Chouraqui, E. Negotiation support systems: A multi-criteria and multiagent approach. European Journal of Operational Research (EJOR), 103 (2), 389–409, 1997. Jabeur, K., Martel, J.-M., Ben Khélifa, S. A distance-based collective preorder inte-grating the relative importance of the group's members. GDN, 13 (4), 327–349, 2004. Kilgour, D. M., Colin, E. Handbook of Group Decision and Negotiation. Springer, Dordrecht, 2010. Leyva-López, J. C., Fernández-González, E. A new method for group decision support based on the ELECTRE III methodology. EJOR, 148 (1), 14–27, 2003. Matsatsinis, N. F., Grigoroudis, E., Samaras, A. P. Aggregation and disaggregation of preferences for collective decision-making. GDN, 14 (3), 217–232, 2005. Matsatsinis, N. F., Samaras, A. P. MCDA and preference disaggregation in group deci-sion support systems. EJOR, 130 (2), 414–429, 2001. Mousseau, V., Slowinski, R., Zielniewicz, P. A user-oriented implementation of the ELECTRE TRI method. Computers & Operations Research, 27 (7–8), 757–777, 2000. 124 HOW TO USE THE 5WS & H TECHNIQUE TO DETERMINE THE WEIGHTS OF INTERACTING CRITERIA Vesna Čančer University of Maribor, Faculty of Economics and Business Razlagova 14, SI-2000 Maribor, Slovenia vesna.cancer@uni-mb.si Abstract: This paper introduces the use of the 5Ws & H technique, for the establishing of the criteria weights. It adapts and completes the steps of this technique based on questions to establish the importance of criteria by the methods based on interval scale. It extends its use to the weighting step of MCDM where synergies and redundancies among criteria are considered by using the Choquet integral. The applicability of the proposed approach is also discussed and introduced via a practical case – the selection of the most appropriate blade. Keywords: 5Ws & H technique, Choquet integral, SMART, SWING, synthesis, weighting. 1 INTRODUCTION When solving problems with multi-criteria decision-making (MCDM) methods, decision makers are encouraged to follow one of the MCDM procedures. The frame procedure of MCDM for the group of methods based on assigning weights includes the following steps: problem definition, elimination of unacceptable alternatives, problem structuring, measuring local alternatives’ values, criteria weighting, synthesis and ranking, and sensitivity analysis [4]. The procedure was well-verified in practice [3], but lacked the support of the problem definition techniques. To eliminate this deficiency, in terms of prescriptive approach, we adapted and completed the steps of the 5Ws & H technique, which is based on questions [2], to establish the weights of criteria and of the subsets of them. The discrete Choquet integral [5, 10] was used to consider interactions among criteria. The applicability of the proposed approach is introduced via a practical case – the selection of the most appropriate blade. 2 WEIGHTING AND AGGREGATION TOOLS IN MCDM The most common aggregation tool that is used in MCDM is the weighted arithmetic mean. Under the assumption of independence among criteria, it requires the assignment of a weight to each criterion [9]. This step is usually carried out by decision makers and thus reflects their point of view on the multi-criteria decision problem [9]. Since, in practical applications, decision makers very often tell the relative importance of criteria directly with difficulty, the criteria’s importance can be expressed by using the methods based on ordinal (e.g. SMARTER), interval (e.g. SWING and SMART) and the ratio scale (i.e. AHP) [1]. In this paper, special attention is given to the use of the methods for establishing the judgements on criteria‘s importance, based on the interval scale. In SMART, a decision maker is first asked to assign 10 points to the least important attribute change from the worst criterion level to its best level, and then to give points ( 10, but  100) to reflect the importance of the attribute change from the worst criterion level to the best level relative to the least important attribute range [6]. In SWING, a decision maker is asked first to assign 100 points to the most important attribute change from the worst criterion level to the best level, and then to assign points ( 100, but  10) to reflect the importance of the attribute change from the worst criterion level to the best level relative to the most important attribute change [6]. In SMART and SWING, the weight of the jth criterion, wj, is obtained by [6]: 125 wj  tj , m t j 1 (1) j where tj corresponds to the points given to the jth criterion, and m is the number of criteria. When the criteria are structured in two levels, the weight of the sth attribute of the jth criterion, wjs, is obtained in SMART and SWING by: w js  t js pj , (2)  t js s 1 where tjs corresponds to the points given to the sth attribute of the jth criterion, and pj is the number of the jth criterion sub-criteria. The local values of alternatives with respect to each criterion on the lowest level (attribute) can be measured by value functions, pair-wise comparisons or directly [1]. During synthesis, the additive model is usually used [1]. When the criteria are structured in two levels, the aggregate alternatives’ values are obtained by [4]:  pj  v( X i )   w j   w js v js ( X i )  ,for each i = 1, 2, …, n   j 1  s 1  m (3) where v(Xi) is the value of the ith alternative, and vjs(Xi) is the local value of the ith alternative with respect to the sth attribute of the jth criterion. If there is interaction among the criteria, decision makers usually return to the hierarchy and redefine the criteria. They can also use other models to obtain the aggregated alternatives’ values; it has already been delineated how to complete the additive model into the multiplicative one [4]. Further, the concept of fuzzy measure has been introduced [10]: it is useful to substitute to the weight vector w a non-additive set function on K allowing to define a weight not only on each criterion, but also on each subset of criteria. A suitable aggregation operator, which generalizes the weighted arithmetic mean, is the discrete Choquet integral. Following [5, 10], this integral is viewed here as an m-variable aggregation function; let us adopt a function-like notation instead of the usual integral form, where the integrand is a set of m real values, denoted by v = (v1, …, vm) n. The (discrete) Choquet integral of v n with respect to w is defined by: m   C w (v)   v( j ) w( K ( j ) )  w( K ( j 1) ) , j 1 (4) where (.) is a permutation on K – the set of criteria, such that v(1)  …  v(m). Also, K(j) = (j), …, (m). 3 THE CRITERIA WEIGHTING BASED ON QUESTIONS In MCDM, by using the groups of methods based on assigning weights, it is assumed that decision makers are able to express their judgments about the criteria’s importance. However, very often decision makers are not aware of the relationships among different criteria. Marichal and Roubens [11] have already emphasized that it is important to ask the decision maker the kind of good questions to determine the weights of interacting criteria from a 126 reference set; since questions are not evident from [11], we propose the use of the problem definition method based on questions to determine the weights of interacting criteria. According to Cook [2], the 5Ws & H technique is a structured method that examines a problem from multiple viewpoints. It is based on who, what, when, where, why and how questions. When the technique is used for problem definition, the process may be summarized as follows [2]. We state the problem starting with 'In what ways might ... ?' and write down the questions that are relevant to the problem. Participants answer the questions, examine responses to each question and use them to stimulate new problem definition. Any redefinitions suggested are written down so that one redefinition that best captures the problem we are trying to resolve is selected. In this paper, we propose the following process of establishing the judgments about the criteria’s importance by the 5Ws & H technique: 1. In what ways might the criteria weights be determined? 2. The questions regarding the considered problem are put and written down. 3. The questions are answered and the weights are determined and re-determined. 4 A PRACTICAL CASE: BLADE SELECTION This section describes the process of establishing judgments about the criteria’s importance by using the 5Ws & H technique in decision-making about blade selection. The MCDM model for the selection of the most suitable blade was built together with an IT company with the aim of presenting possible solutions to their current and potential customers: mediumsized and large companies. The blades that can be offered are described as alternatives in Table 2. The criteria structure is presented in Table 1. The decision maker with appropriate knowledge for the problem definition techniques and for MCDM (in this case, the co-ordinator) asks (Q) and answers (A) the typical question of the first step of the 5Ws & H technique process: Q: In what ways might the criteria weights be determined? A: Directly, by using several methods based on the interval scale (e.g. SWING, SMART), ordinal (SMARTER) and ratio scale (AHP); individually, in groups; by assuming independence between two criteria, by considering interactions among multiple criteria. In the second step, the co-ordinator asked, and in the third step, answered the questions regarding the responsibility and competency regarding expressing judgments about the criteria’s importance. After the participants of the group for solving the problem were defined (project manager, seller, engineers in the considered IT company, responsible for pre-sales support, customers), they answered the questions, successively put by the co-ordinator. To determine the first level criteria’s importance, the SWING method was used: Q: Which criterion change from the worst to the best level is considered the most important? A: The change from worst to best costs. Q: With respect to this change importance, how many points less and how many points are given to other first-level criteria changes? A: 20 points less, i.e. 80 points are given to the change from the worst to the best vision, and 40 points less (i.e. 60 points) are given to the technology change. Similar questions were asked to determine the importance of the attributes of technology. To determine the weights of the vision attributes, the SMART method was used: Q: Which criterion change from the worst to the best level is considered the least important? 127 A: The change from worst to best innovativeness. Q: With respect to this change importance, how many points more and how many points are given to other vision attributes changes? A: 10 points more, i.e. 20 points are given to the change from the worst to the best product development, and 15 points more (i.e. 25) are given to the market strategy change. Similar questions were asked to determine the importance of the attributes of the costs. The weights in Table 1 were determined by considering the above-written answers, (1) for the weights of the first level criteria and (2) for the weights of the second level criteria. Table 1: The criteria structure and the weights for the selection of blade. First level criteria Weights of the first level criteria TECHNOLOGY w1 = 0.250 COSTS w2 = 0.417 VISION w3 = 0.333 Second level criteria Chassis Blade number Connectivity Deployment Features Energy efficiency Purchase price Management Market strategy Product development Innovativeness Weights of the second level criteria w11 = 0.138 w12 = 0.345 w13 = 0.207 w14 = 0.103 w15 = 0.207 w21 = 0.444 w22 = 0.333 w23 = 0.222 w31 = 0.455 w32 = 0.364 w33 = 0.182 Table 2: Alternatives’ data with respect to the attributes. Data type Chassis Blade number Connectivity Deployment Features Energy efficiency Purchase price Management Market strategy Product development Innovativeness Quantitative: number of choices Quantitative: number Qualitative, verbal evaluation Quantitative, MU: h Quantitative: number Quantitative: in 1000 kWH Quantitative, MU: 1000 € Quantitative, MU: 1000 € Qualitative, verbal evaluation Qualitative, verbal evaluation Qualitative, verbal evaluation Alternative 1 Alternative 2 Alternative 3 5 3 2 14 16 10 Flexible Limited Flexible 12 8 8 3 3 1 190 240 165 100 140 130 400 400 600 Good Very good Not enough Good Very good Good Medium Medium Low Measuring local alternatives' values Value function, LB: 1, UB: 5 Value function, LB: 6, UB: 16 Pair-wise comparisons Value function, LB: 4, UB: 16 Value function, LB: 1, UB: 4 Value function, LB: 60, UB: 240 Value function, LB: 60, UB: 140 Value function, LB:400, UB: 600 Pair-wise comparisons Pair-wise comparisons Pair-wise comparisons Symbols: MU – measurement unit, € – Euro, h – hour, kwH – kilo watt hour, LB – lower bound, UB – upper bound, Alternative 1 – IBM BladeCenter [8], Alternative 2 – HP BladeSystem c7000 Enclosure [7], Alternative 3 – Oracle’s Sun Blade 6000 [12]; Sources: [7, 8, 12]; own experience of the observed company’s pre-sales support engineers 128 Table 2 shows the alternatives’ data with respect to the criteria of the lowest hierarchy level, together with the methods that are used to measure the local alternatives’ values. The alternatives’ values with respect to the higher level criteria and the aggregate alternatives’ values obtained by the additive model (3) are presented in Table 3. They allow us to report that Alternative 1 is the most appropriate alternative. Table 3: The alternatives’ values, obtained with the additive model. Value with respect to ‘technology’ v1 Value with respect to ‘costs’ v2 Value with respect to ‘vision’ v3 Aggregate alternative’s value v Rank Alternative 1 0.675 0.512 0.244 0.463 1. Alternative 2 0.650 0.222 0.601 0.455 2. Alternative 3 0.330 0.227 0.156 0.229 3. This is a complex MCDM process, where interactions among criteria should be considered; the co-ordinator found that the Choquet integral as an aggregation function has proven quite useful and convenient in this direction. Since engineers in the considered IT company, who are responsible for pre-sales support, can evaluate the synergies and redundancies between factors on the bases of their professional experience, detailed data from the principal, and the project goals directly, he asked them the following questions: Q: Which synergies should be taken into consideration? A: The customers’ (especially top) managers that make the blade purchase decisions are interested in the interactions among higher level criteria, in this case among ‘technology’, ‘costs’ and ‘vision’. Q: Where are the synergies/redundancies in this model? A: Synergy: between ‘costs’ and ‘vision,’ redundancy: between ‘technology’ and ‘vision’. Q: Why is there synergy between ‘costs’ and ‘vision’ and what does it mean? A: Appropriate vision enables better cost controlling. In the concept of the Choquet integral this means: w2,3 > w2 + w3; w2 + w3 = 0.75 (Table 1), w2,3 = 0.85. Q: Why is there redundancy between ‘technology’ and ‘vision’ and what does it mean? A: Because the vision determines the technology. For the concept of the Choquet integral, this means that w1,3 < w1 + w3; note that w1 + w3 = 0.583 (see Table 1), w1,3 = 0.45. Considering the above-written answers, the weights were re-determined. Table 4 presents the Choquet integrals for the selection of the most suitable blade, obtained by (4). Table 4: The alternatives’ values, obtained by considering interactions among criteria with the Choquet integral. Choquet integral C Rank Alternative 1 0.463 1. Alternative 2 0.405 2. Alternative 3 0.229 3. Studying the results in Table 4 we can report that considering interactions among criteria did not change the final rank of alternatives. However, when comparing it with the values obtained by the additive model (Table 3), it can be concluded that redundancy between ‘technology’ and ‘vision’ decreased the value of the Choquet integral C of Alternative 2 (note that v2 < v3 < v1). Because v3 < v2 < v1 for alternatives 1 and 3, the above mentioned redundancy did not influence C of alternatives 1 and 3, and the synergy between ‘costs’ and ‘vision’ did not come into forefront. Although the aggregate values of alternatives 1 and 2, obtained by the additive model (3) (Table 3), are extremely sensitive to the changes of the 129 weights of ‘costs’ and ‘vision’, the redundancy between ‘technology’ and ‘vision’ considerably decreases C of Alternative 2 and thus strengthen the decision-making basis. 5 DISCUSSION AND CONCLUSIONS The problem definition techniques based on questions (W, 5Ws & H, Why, the 5 Whys) and visualisation (cognitive mapping, fishbone diagrams, mind mapping) are usually applied in problem definition – the first step of the frame procedure of MCDM, based on assigning weights, in order to find and describe a problem, relevant criteria and alternatives. However, we illustrated that in this paper adapted and completed steps of the 5Ws & H technique enable decision makers to consider several aspects regarding establishing the criteria weights. The described approach requires the co-ordinator that can be a member of the decisionmaking group, but has knowledge on both the considered problem definition (5Ws & H) technique and the computer supported MCDM methods based on assigning weights (e.g. SMART and SWING). It enables other participants to focus on professional aspects of the considered problem without knowing the particularities of weighting and aggregation procedures for interacting criteria. We recognize further application possibilities of the methods based on questions in measuring the local alternatives’ values. References 1 Belton, V., Stewart, T. J. (2002): Multiple Criteria Decision Analysis: An Integrated Approach. Boston, Dordrecht, London: Kluwer Academic Publishers. 2 Cook, P. (1998): Best Practice Creativity. Aldershot: Gower Publishing Limited. 3 Čančer, V. (2008): A Frame Procedure for Multi-criteria Decision-making: Formulation and Applications. In: V. Boljunčić, L. Neralić and K. Šorić (eds.): KOI 2008 Proceedings. Pula, Zagreb: Croatian Operational Research Society. 4 Čančer, V. (2009): Considering Interactions in Multi-criteria Decision-making. In: L. Zadnik Stirn, J. Žerovnik, S. Drobne and A. Lisec (eds.): SOR ’09 Proceedings. Ljubljana: Slovenian Society Informatika, Section for Operational Research. 5 Grabisch, M. (1995): Fuzzy Integral in Multicriteria Decision Making. Fuzzy Sets and Systems, 69(3): 279-298. 6 Helsinki University of Technology (2002): Value http://www.mcda.hut.fi/value_tree/theory, consulted in August 2011. Tree Analysis. 7 Hewlett-Packard Development Company (2011): HP BladeSystem c7000 Enclosure. http:// h18004.www1.hp.com/products/quickspecs/12810_div/12810_div.pdf, consulted in June 2011. 8 IBM Systems and Technology (2011): IBM BladeCenter – Build smarter IT. http://public.dhe.ibm.com/common/ssi/ecm/en/blb03002usen/BLB03002USEN.PDF, consulted in May 2011. 9 Kojadinovic, I. (2004): Estimation of the weights of interacting criteria from the set of profiles by means of information-theoretic functional. European Journal of Operational Research, 155(3): 741-751. 10 Marichal, J. L (2000): An Axiomatic Approach of the Discrete Choquet Integral as a Tool to Aggregate Interacting Criteria. IEEE Trans. Fuzzy Systems, 8: 800-807 [11] Marichal, J. L., Roubens, M. (2000): Determination of weights of interacting criteria from a reference set. European Journal of Operational Research, 124(3): 641-650. 12 Oracle (2010): Sun Blade 6000 Chassis. http://www.oracle.com/us/products/serversstorage/servers/blades/033613.pdf, consulted in May 2011. 130 ECONOMIC CRITERIA IN DECISION-MAKING ON NUMBER OF FUNCTIONAL REGIONS: THE CASE OF SLOVENIA Samo Drobne* and Marija Bogataj** * University of Ljubljana, Faculty of Civil and Geodetic Engineering, Ljubljana, Slovenia, e-mail: samo.drobne@fgg.uni-lj.si ** University of Ljubljana, Faculty of Maritime Studies and Transport, Portorož, & MEDIFAS, Nova Gorica, Slovenia, e-mail: marija.bogataj@guest.arnes.si Abstract: In this paper, we suggest the method for decision-making on number of functional regions. The method considers economic variables of the average monthly gross earnings in the functional region as well as the guidelines for the size of the regions. The method was tested in case study of Slovenia. There are also some recommendations to improve the method. Keywords: functional region, commuting, decision-making, number of functional regions, Slovenia. 1 INTRODUCTION The concept of regions is anchored deep in the history of Europe. Nowadays, the idea of regions is often connected with the integration of European Union (EU). However, different actors understand the very concept of a region quite differently. Administrative or statistic regions are defined by their borders and they are required to cover whole the respective territory and to be of comparable size. But, the functional regions of economy and/or society are product of interrelations, they are quite diverse in terms of their size and population, and they may overlap as well as not fully cover the territory [12]. The NUTS (Nomenclature of Territorial Units for Statistics) classification is a hierarchical system for dividing up the economic territory of the EU for the purpose of (a) the collection, development and harmonisation of EU regional statistics; (b) socio-economic analyses of the regions; (c) framing of EU regional policies. For the purpose of socioeconomic analyses, three levels of regions have been established inside each EU member: (b1) major socio-economic regions at NUTS 1 level, (b2) basic regions for the application of regional policies at NUTS 2 level, and (b3) small regions for specific diagnoses at NUTS 3 level. For the purpose of framing of EU regional policies areas eligible under the other priority objectives have mainly been classified at the NUTS 3 level. The current NUTS classification lists 1303 regions at NUTS 3 level [10]. In Slovenia, there are twelve “statistical regions” at NUTS 3 level also called “development regions”. The first version of statistical regions is from the middle of seventies of the previous century. At that time, statistical regions were established for the purpose of regional planning and cooperation in various fields. The first regionalization of statistical regions was supported by exhaustive gravity analysis of labour markets, education areas and supply markets in twelve regional and their sub-regional centres – that is the reason, why Slovenian regions at NUTS 3 level are very stable [17]. Regions on NUTS 3 level are normally functional regions. A functional region is characterised by its agglomeration of activities and by its intra-regional transport infrastructure, facilitating a large mobility of people, products, and inputs within its interaction borders. The basic characteristic of a functional region is the integrated labour market, in which intra-regional commuting as well as intra-regional job search and search for labour demand is much more intensive than the inter-regional counterparts [11]. Consequently, the border of a labour market region is a good approximation of the border of a functional region [1,3,4,5,9,11,15]. 131 There are several approaches and methods for delimitation of functional regions; for taxonomy and discussion about approaches see [11]. In this paper we analyse functional regions of Slovenia by commuting zone approach using Intramax method1 and suggest model for decision-making on number of functional regions in the country. 2 MATERIALS AND METHODOLOGY In 2009, there were a total of 390,500 labour commuters (persons in employment who commute) between 210 municipalities of total 812,315 labour commuters in Slovenia. Labour commuter is person in employment whose territorial unit of workplace is not the same as territorial unit of residence. The source of data was the Statistical Register of Employment (SRDAP), which was kept by Statistical Office of the Republic of Slovenia. SRDAP covers persons in paid employment and self-employed persons who are at least 15 years old and who have on the basis of the employment contract compulsory social insurance or are employed on the territory of the Republic of Slovenia. Employment can be permanent or temporary, full time or part time [18]. To analyse economic criteria in decision-making on number of regions, set of functional regions were modelled first using Intramax method. The method, which was introduced by Masser and Brown in [13] and improved in [14], carries out a regionalization of an interaction matrix. The objective of the Intramax procedure is to maximise the proportion within the group interaction at each stage of the grouping process, while taking account of the variations in the row and column totals of the matrix. In the grouping process, two areas (municipalities in our case) are grouped together for which the objective function T is maximised [2]: max T , Tij T ji , T  Oi  D j O j  Di (1) where Tij is the interaction between origin location i and destination location j , Oi   Tij is the total of interactions originating from origin i , D j   Tij is total of j i interactions coming to destination j , and Oi and D j  0 . The Intramax analysis is a stepwise analysis. In each step two areas are grouped together and the interaction between the two municipalities becomes internal interaction for the new resulting area. This new area takes the place of the two parent areas at the next step of the analyses. So with N areas after N  1 steps all areas are grouped together into one area (region) and all interaction become internal. In the analysis on number of regions, the Flowmap software [2], with implemented Intramax method, was used to delimitate functional regions of Slovenia. In Flowmap, the outcome of an Intramax analysis is a report in table form and a dendogram which municipalities are grouped and how. We modelled fourteen systems of two to fifteen functional regions based on commuting data between 210 municipalities of Slovenia in 2009. This set of functional regions was used to develop economic criteria on decisionmaking on number of regions in the country. Here we considered two economic criteria: (a) the EU guidelines for the size of region, and (b) economic homogeneity of regions. 1 Functional regions of Slovenia defined by labour market approach are in [6,7,8]. 132 There are EU guidelines for size of regions (population) on NUTS 3 level. The thresholds in the Tab. 1 are used as guidelines for establishing the regions, but they are not applied rigidly. Table 1: Guidelines for establishing the regions at NUTS 3 level [10]. Minimum population (EU) 150,000 Level NUTS 3 Maximum population (EU) 800,000 On the other side, the most useful and the most often used economic criterion in different regional development analysis is gross domestic product (GDP). Economic prosperity can be determined in three ways, all of which should, in principle, give the same result. There are the product (or output) approach, the income approach, and the expenditure approach. The income approach measures GDP by adding incomes that firms pay households for the factors of production they hire- wages for labour, interest for capital, rent for land and profits for entrepreneurship. Normally, GDP is measured only for regions at NUTS 3 level, or higher. There are no data for GDP at lower levels of regions. For that reason, we chose average monthly gross earnings per person in paid employment [19] in the municipality of destination as a measure of economic prosperity. The average monthly gross earnings were grouped according pre-modelled functional regions of Slovenia. This economic criterion was calculated per persons in employment as well as per capita in functional region. Model for decision-making on number of regions in the country is based on the variation of average monthly gross earnings per capita in the (functional) region and variation of population in the (functional) region regarding the EU guidelines: min f ( K , a) K f  a  CVGEAR ( SI )  (1  a)  CDPOP ( EU ) (2) where K is the number of regions in the country, CVGEAR (SI ) is coefficient of variation of average monthly gross earnings per capita in the Slovene functional region, CDPOP (EU ) is coefficient of deviation of population in the region regarding the EU guidelines, a is the weight for Slovenian criterion, respectively 1  a is weight for EU criterion, and CVGEAR ( SI )   GEAR GEAR CDPOP ( EU )  1 K 2  Di K i (3) ( POP( EU ) min  POP( EU ) max ) 2  POPi  POP( EU ) min  2 Di   POPi  POP( EU ) max  otherwise   ( POPi  POP( EU ) min ) 2  ( POPi  POP( EU ) max ) 2  0 (4) (5) where POP(EU ) min is the minimum population in region regarding the EU guidelines and POP(EU ) max is the maximum population in region regarding the EU guidelines. Model (2) allows decision-makers to include domestic and/or EU preferences; i.e. weights for domestic ( a ) respectively EU criteria ( 1  a ). 133 3 RESULTS Using data on commuting between 210 municipalities in Slovenia in 2009 and Intramax procedure in Flowmap software, fourteen systems of two to fifteen functional regions were modelled. According the EU guidelines for regions at NUTS 3 level (see Tab. 1) the lowest suggested system is system of five functional regions (minimum of population in functional region is 199,011 and maximum is 821,703). Besides EU recommendations on population in the regions, here suggested model for defining appropriate number of (functional) regions in the country considers in addition economic variable on average monthly gross earnings in the (functional) region. Results in Chart 1 show that the systems of three, four, nine and fourteen functional regions are the most interesting systems of functional regions considering solely analysed economic variable in the region (variation of economic variable has a local minimum). Chart 1: Standard deviation of average monthly gross earnings in fourteen systems of two to fifteen functional regions in Slovenia in 2009. But, using the model (2) – i.e. considering both (Slovenian and EU) requirements – where the weight for Slovenian criterion a is changing from 0.1 to 0.9, there are two suggestions for the number of functional regions in Slovenia: five functional regions for 0.1  a  0.8 and seven functional regions for a  0.9 . So, laying (great) stress on EU guidelines suggests us to consider the system of smaller number of (five) functional regions (Fig. 1 on the left), while forcing the most economically homogeneous regions (the more uniform distribution of average monthly gross earnings per capita in the functional regions) gives us the system of seven functional regions of Slovenia in 2009 (Fig. 1 on the right). Considering solely “local” economic variable and the fact that EU guidelines for regions are not applied rigidly for EU members the system of fourteen (functional) regions (Fig. 2) becomes also a candidate for the “right” number of functionally modelled regions. The system of fourteen functional regions is the most close to the current system of twelve statistical regions of Slovenia. 4 CONCLUSIONS In the paper, we suggested the method for decision-making on number of functional regions in the country. Software implementation of algorithms for delimitation of functional regions enables modelling of many different systems of functional regions. So, the question about the “right” number of functional regions should not be ignored. Suggested model for 134 decision-making on the number of functional regions considers the average monthly gross earnings per capita in the functional region as well as the guidelines for the size of the regions. The model for decision-making on number of functional regions could be improved by considering other more holistic economic parameters (i.e. GDP) or by including other functional criterion in decision-making (e.g. travelling costs in functional region as whole or to regional centre as future administrative centre). In our case study of Slovenia, functional regions were delimitated by Intramax method considering flows of labour commuters between 210 municipalities in Slovenia in 2009. The results show that there were two (conditionally three) systems of functional regions in 2009. But, for more adequate results, functional regions should be studied at longer time horizon. The complex territorial organization of most EU member’s political and administrative systems is rooted in history and tradition as well as in a strong political will. Most parts of the provincial structure (states) and of the district structure of administration have been already inherited from the past and reflect the administrative entities of different social systems. But, for various motivations, the creation of a middle layer of regional government or administration should be established in some new member states of the EU where no intermediate level, except state and municipality level, of territorial organisation is organised [6,8,16]. This is also the case in Slovenia. Here suggested model could help decision-makers to decide about new level of administrative regions. Figure 1: Five (left) and seven (right) functional regions of Slovenia in 2009. Figure 2: Fourteen functional regions of Slovenia in 2009. References [1] Andersen A. K. (2002). Are commuting areas relevant for the delimitation of administrative regions in Denmark?, Regional Studies, 36:833–844. 135 [2] Breukelman J., G. Brink, T. de Jong, H. Floor (2009). Manual Flowmap 7.3. Faculty of Geographical Sciences, Utrecht University, The Netherlands. (http://flowmap.geo.uu.nl; accessed: 15.8.2010). [3] Casado-Di´Az J. M. (2000). Local labour market areas in Spain: A case study. Regional Studies, 34:843–856. [4] Coombes M. G., Green A. E. and Openshaw S. (1986). An efficient algorithm to generate official statistical reporting areas: The case of the 1984 travel-to-work-areas revision in Britain. Journal of the Operational Research Society, 37:943–953. [5] Cörvers F., M. Hensen and D. Bongaerts (2009). Delimitation and coherence of functional and administrative regions. Regional Studies, 43(1):19-31. [6] Drobne, S., M. Konjar and A. Lisec (2009). Delimitation of Functional Regions Using Labour Market Approach. In: Zadnik Stirn L., J. Žerovnik, S. Drobne and A. Lisec (Eds.), Proceedings of SOR’09, 10th International Symposium on Operational Research in Slovenia. Ljubljana. Slovenian Society Informatika (SDI), Section for Operational Research (SOR): 417–425. [7] Drobne, S., M. Konjar and A. Lisec (2010). Razmejitev funkcionalnih regij Slovenije na podlagi analize trga dela = Delimitation of functional regions of Slovenia based on labour market analysis. Geod. vestn. 54(3):481-500. [8] Drobne, S., A. Lisec, M. Konjar, A. Zavodnik Lamovšek and A. Pogačnik (2009). Functional vs. Administrative regions: Case of Slovenia. In: Vujošević M. (Ed.), Thematic Conference Proceedings. Vol. 1. Belgrade. Institute of Architecture and Urban & Spatial Planning of Serbia: 395–416. [9] Eurostat (1992). Commission Regulation amending annexes I, II and III to Regulation (EC) No1059/2003 of the European Parliament and of the Council on the establishment of a common classification of territorial units for statistics (NUTS). CPS 2006/60/1/EN, Eurostat, Luxembourg. [10] Eurostat (2011). NUTS - Nomenclature of territorial units for statistics. (http://aesop2005.scix.net/cgi-bin/papers/Show?667; accessed: 1.5.2011). [11] Karlsson C. and M. Olsson (2006). The identification of functional regions: theory, methods, and applications. Ann Reg Sci, 40:1–18. [12] Maier K. (2005). New policy? New regions? New borders? AESOP 2005 Congress, 13-17 July 2005, Vienna, Austria, 9 pp. (http://aesop2005.scix.net/cgi-bin/papers/Show?667; accessed: 25.2.2011). [13] Masser, I. and P. J. B. Brown (1975). Hierarchical aggregation procedures for interaction data. Environment and Planning A, 7(5): 509-523. [14] Masser, I. and P.J.B. Brown (1977). Spatial representation and spatial Interaction. Papers of the Regional Science Association 38, 71-92. [15] OECD (2002). Redefining territories – The functional regions. Organisation for Economic Cooperation and Development. Paris, France. [16] Schrerrer W. (2006). Economic Aspects of a “Middle Layer” of Administration and Government: Some Experience from Austria, Uprava, 4(2-3):25-34. [17] SORS (2011a). Administrative-territorial division = “Upravno-teritorialna razdelitev”, Statistical Office of the Republic of Slovenia, Ljubljana. (http://www.stat.si › Splošno ›; accessed: 25.3.2011). [18] SORS (2011b). Persons in employment (excluding farmers) by municipalities of residence and municipalities of workplace by sex, municipalities, Slovenia, annually. Statistical Office of the Republic of Slovenia. Ljubljana. (http://pxweb.stat.si/pxweb/Database/Municipalities/Municipalities.asp; accessed: 1.6.2010) [19] SORS (2011c). Average monthly gross and net earnings per person in paid employment and persons in employment, 2010, municipalities, Slovenia. Statistical yearbook, Statistical Office of the Republic of Slovenia. Ljubljana. (http://www.stat.si/letopis/2010/31_10/31-08-10.htm; accessed: 25.2.2011) 136 TOWARDS CRITERIA SELECTION IN DEA BY CONJOINT ANALYSIS Milena Djurovic, Gordana Savic, Marija Kuzmanovic and Milan Martic University of Belgrade, Faculty of Organizational Sciences Jove Ilica 154, Belgrade, Serbia djurovicmilena86@gmail.com; goca@fon.rs; mari@fon.rs; milan@fon.rs Abstract: Data envelopment analysis (DEA) is a method for evaluating the relative efficiency of decision making units (DMUs) with multiple inputs and outputs. DEA results depend heavily on the criteria included into analysis. Therefore the selection of adequate criteria is one of the most important phases in DEA application. In this paper, statistical technique, Conjoint analysis is suggested as supporting method for the selection of the criteria. Conjoint analysis determines the relative importance of each criterion, based on stakeholders’ preference, which can be used as guide for selection of inputs and outputs for efficiency assessment of DMUs. Keywords: Data envelopment analysis, criteria selection, stakeholder preferences, Conjoint analysis. 1 INTRODUCTION Data envelopment analysis (DEA) is a linear programming-based procedure that measures the relative efficiencies of peer decision-making units (DMUs). This special mathematical technique was firstly introduced by Charnes, Cooper and Rhodes [4]. The DEA methodology has been widely applied in areas such as banking [12], energy, flexible manufacturing cells, highway maintenance, individual physician practice, and telecommunications. For a given set of inputs and outputs, DEA produces a single comprehensive measure of performance (efficiency score) for each DMU. Obtained results rely heavily on the set of inputs and outputs that are used in the analysis. Therefore, one of the most important DEA phases is criteria selection. The effort increases significantly when the available data are growing. In the literature relatively little attention has been paid to how, in a real-world situation, these inputs and outputs should be chosen. Many authors treat the inputs and outputs used in their studies as simply „givens“ and then go on to deal with the DEA methodology. On the other hand, statistical methods, such as a regression and a correlation analysis, have been used in order to decrease the number of the criteria. Finally, criteria selection, as well as proper DEA model selection, may differ depends on particular case, the objectives and the purposes of the analysis. The aim of this paper is to show possibilities of using Conjoint statistical techniques as support tool for DEA inputs and outputs selection. The paper is organized as follows: Section 2 describes DEA basics and implementation process, followed by literature survey concerning the criteria (inputs and outputs) selection. The Conjoint analysis, including the procedure for the determining the criteria importance is given in the Section 3. The proposed methodological framework of DEA criteria selection by Conjoint analysis is given in the Section 4. Finally, main conclusions are summarized in Section 5. 2 DATA ENVELOPMENT ANALYSIS The creators of DEA [4], introduced the basic DEA CCR model as a new way to measure efficiency of DMUs, and since then a lot of variations of DEA models have been developed: the BCC model [2] which assumes variable return to scale, the additive model [1]which is 137 non-radial, Banker and Morey [3] model which involves qualitative inputs and outputs, Golany and Roll [8] model in which input-output weights are restricted to certain ranges of values. DEA empirically identifies the efficient frontier of a set of DMUs based on the input and output variables. Assume that there are n DMUs, and the jth DMU, produces s outputs ( yij ,.., y sj ) by using m inputs ( xij ,.., xmj ). The efficiency score of the observed DMUk is given as ratio of the virtual outputs (sum of weighted outputs) to the virtual inputs (sum of weighted inputs). The basic CCR ratio model is as follows: s max  k   ur yrk r 1 m v x i 1 i ik s.t s m u y v x r 1 r rj i 1 ur  0, r  1, i ij (1)  1, j  1, , s, vi  0, i  1, ,n ,m where ur  0, are weights assigned to the rth outputs, r  1, , s , and vi  0, are weights assigned to the ith inputs, i  1, , m in order assess DMUk as efficient as possible. This basic CCR DEA ratio model, which can easily be linearized, should be solved n times, once for each DMUk. The index  k shows relative efficiency of DMUk, obtained as maximum possible achievement in comparison with the other DMUs under the evaluation. Emrouznejad and Witte [6] suggest a complete procedure of DEA efficiency assesment. It is a framework which can be further adapted and modified along the specific needs of the researcher. They suggested a COOPER-framework, which involves six interrelated phases: (1) Concepts and objectives, (2) On structuring data, (3) Operational models, (4) Performance comparison model, (5) Evaluation, and (6) Results and deployment. The first two phases of the COOPER-framework correspond to defining the problem and understanding how decision making units operate. The last two phases correspond to summarization of the results and documentation of the project for non-DEA experts. Obviously, the phases are interrelated and affect each other. The attention in this paper is paid to the phases 1, 2 and 3, since the objective definition in the model (1) obviously indicated that efficiency of DMUk is crucially related to the criteria selection. Jenkins and Anderson [10] claimed the greater the criteria, the less constrained are the model weights assigned to the criteria, and the less discerning are the DEA results. The number of criteria may be large, and it is not clear which one to choose. Even worse, different selections of criteria can lead to different efficiency evaluation results. It is obviously possible to consider all criteria for evaluation, but too many of them may lead to too many efficient units, and it gives rise to difficulties in distinguishing truly efficient units from inefficient ones. For this reason, the problem of selecting adequate criteria becomes an important issue for the improvement of discrimination power of DEA. While it is advantageous to limit the number of variables, there is no consensus on how best to do this. Banker, Charnes and Cooper [2] suggested that the number of DMUs should be at least three times larger than the number of criteria. As noted by Golany and Roll [8], 138 few studies give an overall view of DEA as an application procedure that must focus on the choice of data variables in addition to the methodology of DEA. Several methods have been proposed that involve the analysis of correlation among the criteria, with the goal of choosing a set of criteria that are not highly correlated with each other. Unfortunately, studies have shown that these approaches yield results which are often inconsistent in the sense that removing criteria that are highly correlated with others can still have a large effect on the DEA results [15]. Morita and Haba [14] select criteria so as to distinguish between two groups based on external information, where a 2-level orthogonal layout experiment is utilized and optimal variables can be found statistically. Ediridsinghe and Zhang [5] have proposed a generalized DEA approach to select criteria by maximizing the correlation between the DEA score and the external performance index. They utilize a two-step heuristic algorithm that combines random sampling and local search to find an optimal combination of inputs and outputs. Morita and Avkiran [13] considered the criteria selection method based on discriminant analysis using external evaluation. They used a 3-level orthogonal layout experiment to find an appropriate combination of inputs and outputs, where experiments are independent of each other. Lim [11] proposed a method for selection of better combinations of input-output factors. It is designed to select better combinations of input-output factors that are well suited for evaluating substantial performance of DMUs. Several selected DEA models with different combinations of input-output factors are evaluated, and the relationship between the computed efficiency scores and a single performance criterion of DMUs is investigated using decision tree. Based on the results of decision tree analysis, a relatively better DEA model can be chosen, which is expected to effectively assess the true performance of DMUs. 3 CONJOINT ANALYSIS Conjoint analysis is a multivariate technique that can be used to understand how individual’s preferences for products or services are developed [9]. Specifically, Conjoint analysis is used to gain insights into how customers value various product attributes based on their valuation of the complete product. Green and Krieger [7] pointed out the potential usefulness of Conjoint analysis to deal with some marketing problems, in particular to develop new multiattribute products with optimal utility levels over other competitive products, to estimate market shares in alternative competitive scenarios, to benefit segmentation, and to design promotion strategies, among other uses. Implementation of Conjoint analysis can be simply described as follows. Researchers at first develop a set of alternative products (real or hypothetical) in terms of bundles of quantitative and qualitative attributes through fractional factorial designs. These products, referred to as profiles, are then presented to the customers during the survey. The customers are asked to rank order or rate these alternatives, or choose the best one. Because the products are represented in terms of bundles of attributes at mixed “good” and “bad” levels, the customers have to evaluate the total utility from all of the attribute levels simultaneously to make their judgments. Based on these judgments, the researchers can estimate the part-worths for the attribute levels by assuming certain composition rules. The manner that respondents combine the part-worths of attribute levels in total utility of product can be explained by these roles. The simplest and most commonly used model is the linear additive model. This model assumes that the overall utility derived from any combination of attributes of a given good or service is obtained from the sum of the separate part-worths of the attributes. Thus, respondent i’s predicted utility for profile j can be specified as follows: 139 Lk K U ij   ikl x jkl   ij , i  1,..., I , j  1,..., J (2) k 1 l 1 where I is the number of respondents; J is the number of profiles; K is the number of attributes; Lk is the number of levels of attribute k. ikl is respondent i’s utility with respect to level l of attribute k. x jkl is such a (0,1) variable that it equals 1 if profile j has attribute k at level l, otherwise it equals 0.  ij is a stochastic error term. The parameters ikl , also known as part-worth utilities, can be used to establish a number of things. Firstly, the value of these coefficients indicates the amount of any effect that an attribute has on overall utility of the profiles– the larger the coefficient, the greater the impact. Secondly, part-worths can be used for preference-based segmentation. Namely, given that part worth utilities are calculated at the individual level, if preference heterogeneity is present, the researcher can find it. Respondents who place similar value to the various attribute levels will be grouped together into a segment. Thirdly, part-worths can be used to calculate the relative importance of each attribute, also known as an importance value. Importance values are calculated by taking the utility range for each attribute separately, and then dividing it by the sum of the utility ranges for all of the factors: FI ik  max{ik1 , ik 2 ,...ikLk }  min{ik1 , ik 2 ,...ikLk }   max{ K k 1 ik 1 , ik 2 ,...ikLk }  min{ik1 , ik 2 ,...ikLk }  , i  1,..., I , k  1,..., K (3) where FIik is the relative importance that ith respondent assigned to the factor k. The results are then averaged to include all of the respondents: I FI k   FI ik I , k  1,..., K . (4) i 1 If the market is characterized by heterogeneous customer preferences, it is possible to determine the importance of each attribute for each of isolated market segments. 4 DEA CRITERIA SELECTION USING CONJOINT ANALYSIS Here is a framework designed to help the selection of the criteria relevant to the analysis and to obtain results that best reflect the research objectives. The adequate criteria selection is especially significant in the case of non-profit sector as well as for the studies that include a large number of categorical (discrete, intangible) variables. The main phases of proposed framework are shown in Figure 1. In the first phase, the research objectives and stakeholders should be defined. In the second phase, Conjoint analysis should be conducted. Based on the objectives defined in the first stage, selection of the key attributes and their levels are performed. After the data are collected, Conjoint parameters estimation should be done, as described in Section 4. In the real-world applications, the parameters obtained from Conjoint analysis shows respondents' preferences to the particular criteria. One of the important objectives of Conjoint analysis is to determine what combination of a limited number of criteria is most influential on respondent choice or decision making. Particularly, FIk, k = 1,.., K, represents the importance of each criterion, which may be starting point for DEA criteria selection (see Fig. 1). 140 METHODOLOGICAL FRAMEWORK REAL WORLD APPLICATION OBJECTIVE DEFINING Evaluation of university teachers STOCKHOLDERS DETERMINING Students, Teachers, University representatives CONJOINT ANALYSIS CONJOINT ANALYSIS Attributes and attribute levels selection Experimental design construction Data collection Parameter estimation 9 Attributes, 1152 possible profiles Full factorial, orthogonal, design (16 profiles) Rating approach for data collection Parameter estimation using OLS Criteria (Attribute) Attributes (Criteria) importance FIk DATA ENVELOPMENT ANALYSIS Avg. Importance FI (%) Clear and understandable presentation 20.56% Methodical and systematic approach 16.96% Tempo of lectures 13.35% Preparedness 7.12% Punctuality 8.05% Encouraging the interaction 7.28% Informing students about their work 8.12% Concerning the students comments and answering to the questions 8.01% Availability (at the consultation or via e-mail) 10.56% DATA ENVELOPMENT ANALYSIS Criteria (Attribute) Criteria Selection & Division on the input and output sets DEA model selection Evaluation Results and Deployment Avg. Importance FI (%) Clear and understandable presentation 20.56% Methodical and systematic approach 16.96% Tempo of lectures 13.35% Availability (at the consultation or via e-mail) 10.56% DEA model selection Evaluation Results and Deployment Figure 1: Model of criteria selection in DEA using Conjoint data The illustrative example is given in the right part of the Fig. 1. The study is motivated by teachers’ evaluation on the University of Belgrade, Serbia. The survey, usually, carries out twice a year and results are meant to represent students’ standpoint. But the final estimation of each teacher is calculated as average of mean values of 9 criteria given as attributes set up by the University representatives. A survey, partially presented here, is conducted with 98 undergraduate students. The results obtained by Conjoint analysis shows that some criteria, such as Clear and understandable presentation is far more significant than the others. Four criteria, shown in DATA ENVELOPMENT ANALYSIS rectangle, are distinguished as the most important from the students’ point of view. These criteria are going to be used as DEA outputs, since marks are desirable to be as higher as possible. 5 CONCLUSION The aim of each entity is to provide the most reliable, useful and inexpensive business analysis. It can be DEA which can help the management to decrease cost, to make processes easier and to focusing on the key business competencies. DEA is an effective tool for evaluating and managing operational performance in a wide variety of settings. Since DEA gives different indexes of efficiencies with different combination of criteria, the selection of inputs and outputs is one of the most important steps in DEA. A large number of criteria require a great effort to obtain efficiency index of each DMU. DEA efficiency index is relative measure, depends on the number of DMUs and number and structure of criteria included into the analysis. The criteria reduction has been usually done by statistical methods, such as regression and correlation analysis. This paper 141 has suggested using Conjoint analysis as supporting tool for more realistic criteria selection. The stakeholders' preferences obtained by Conjoint analysis represents starting point for making the most suitable combination of criteria used in next phase of DEA efficiency measurement. The described framework provides better criteria selection which is well suited to the stakeholders and allows selection of different criteria combination suited to the different objectives and the number of DMUs. This paper also provides some interesting and promising lines for further research. References [1] Ahn, T., Charnes, A., Cooper, W. W. 1988. Efficiency Characterizations in Different DEA Models. Socio-Economic Planning Sciences, 22 (6), 253-257. [2] Banker, R., Charnes, A., Cooper, W. W. 1984. Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis. Management Science, 30 (9), 1078-1092. [3] Banker, R. D., Morey, R. C. 1986. The Use of Categorical Variables in Data Envelopment Analysis. Management Science, 32 (12), 1613-1627. [4] Charnes, A., Cooper, W.W., Rhodes, E., 1978. Measuring efficiency of decision making units. European Journal of Operations Research 2, 429-44. [5] Edirisinghe, N.C.P., Zhang, X. 2007. Generalized DEA model of fundamental analysis and its application to portfolio optimization. Journal of Banking & Finance, 31, 3311–3335. [6] Emrouznejad, A., Witte, K. 2010. COOPER-framework: A unified process for non-parametric projects. European Journal of Operational Research, 207, 1573–1586 [7] Green, P.E., Krieger, A.M. 1991. Segmenting markets with conjoint analysis. Journal of Marketing, 55, 20–31. [8] Golany, B., Roll, Y. 1989. An application procedure for DEA. OMEGA International Journal of Management Science 17 (3), 237–250. [9] Hair, J. F., Anderson, R. E., Tathan, R. L., and Black, W. C. 1995. Multivariate Data Analysis, Englewood Cliffs, NJ: Prentice Hall. [10] Jenkins, L., Anderson, M., 2003. A multivariate statistical approach to reducing the number of variables in data envelopment analysis. European Journal of Operational Research 147, 51–61. [11] Lim, S., 2008. A Decision Tree-Based Method for Selection of Input-Output Factors in DEA. Proceedings of the 2008 International Conference on Machine Learning; Models, Technologies and Applications, MLMTA'08, July 14-17, 2008, Las Vegas, Nevada, USA, 762-769. [12] Luo, X. 2003. Evaluating the Profitability and Marketability Efficiency of Large Banks - an Application of Data Envelopment Analysis. Journal of Business Research, 56 (8), 627-635. [13] Morita, H., Avkiran, K. N. 2009. Selecting inputs and outputs in data envelopment analysis by designing statistical experiments. Journal of the Operations Research, Society of Japan, Vol. 52, No.2, 163-173 [14] Morita, H., Haba, Y. 2005. Variable selection in data envelopment analysis based on external information. Proceedings of the eighth Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainty, 181–187. [15] Nunamaker, T.R., 1985. Using data envelopment analysis to measure the efficiency of non-profit organizations: A critical evaluation. Managerial and Decision Economics 6 (1), 50–58. 142 INTERVAL COMPARISON MATRICES IN GROUP AHP Petra Grošelj and Lidija Zadnik Stirn University of Ljubljana, Biotechnical Faculty Jamnikarjeva 101, 1000 Ljubljana, Slovenia petra.groselj@bf.uni-lj.si, lidija.zadnik@bf.uni-lj.si Abstract: Analytic hierarchy process is a well-known approach for handling multi-criteria decision making problems. In group decision making the main difficulty is aggregating individual preferences into joint weights, so that the weights satisfy all decision makers as much as possible. One possibility, when the group cannot reach a consensus on a single judgment is to express it with an interval. In the paper a new way of constructing the interval comparison matrix from individual judgments is proposed. The problem of generating weights from interval comparison matrices is discussed and a numerical example from the field of natural resources management is provided. Keywords: multiple criteria decision making; group decision making; analytic hierarchy process; interval judgments; management of natural resources 1 INTRODUCTION Analytic hierarchy process (AHP) is a well-known approach for handling multi-criteria decision making problems. It is based on pairwise comparisons. The 1-9 ratio scale is used for expressing the strength of preference between the compared objects. Since one decision maker is limited by his/her knowledge, experiences and perspective, group decision making is often applied inside the AHP model. In group decision making the main difficulty is aggregating individual preferences into joint weights, so that the weights satisfy all decision makers as much as possible. Many approaches have been applied, including aggregating individual judgments or priorities [1] and aggregating the base of data envelopment analysis concepts [2]. In one decision maker’s case intervals can be used instead of the crisp values employed in pairwise comparisons. The exact values sometimes cannot express the subjectivity and the lack of information of decision maker, or the complexity and uncertainty of the real world decision problems. Interval judgments are more natural in such cases. Another possibility for interval judgments arises in group decision making when the group cannot reach a consensus on a single judgment and expresses it with an interval [3]. Several analysts have examined the problem of generating weights from interval comparison matrices recently. A derived priority vector can be a vector of interval weights, the representative vector from the assurance region or a weakly efficient vector of crisp weights in the case when assurance region is empty [4]. In the paper we discuss the problem of combining individual judgments into group interval judgment and provide a new concept of aggregation. For deriving the interval weights from interval comparison matrix we employed the concept of Liu [5], which is based on constructing two crisp matrices from the interval matrix and deriving weights from them. 2 THE CONSTRUCTION OF GROUP INTERVAL MATRIX Let m be the number of decision makers included in the process of evaluating n criteria (or alternatives) according to the element on the next higher level. Let A( k )   aij( k )  , k=1,…,n nn be their comparison matrices. Group intervals can be constructed using minimum and maximum judgments for the endpoints of the intervals [6]. So the outstanding judgments 143 determine the group intervals. Despite eliminating outliers, the intermediate values do not impact the endpoint of the intervals. We suggest another approach to overcome this drawback. The lower bound of the interval should be influenced by all values that are lower or equal to the median. Since it is the lower bound the influence should not be equal for all values. The degree of influence should be greater for smaller values and smaller for the values that are closer to the median. Similarly, the upper bound of the interval should be influenced by all values that are greater than or equal to the median. The simplest way for mathematical record of such approach is employing the Ordered Weighted Geometric (OWG) operator [7]. Definition 1: An OWG operator of dimension m is a mapping F : m  , that has associated a weighting vector W   w1 ,..., wm  having the properties: wi  0,1 , m m i 1 i 1  wi  1 and such that: F  a1 ,..., am    ciwi (1) where ci is the ith largest value from the set a1 ,..., am  . We selected the OWG operator since it preserves reciprocity. Different vectors W assign different weights to the values a1 ,..., am . We assume that all decision makers are equally important. We define two vectors WL   w1L ,..., wmL  and WU   w1U ,..., wmU  for the lower and upper bounds of the intervals, respectively. The description is made separately for even or odd number of decision makers. If m is an odd number, then m2 1 is the median of numbers 1, 2,..., m and s m1  ( m1)(8 m3) is the sum of numbers from 1 to 2 m1 2 . Then we employ the judgments of decision makers that are smaller than or equal to the median to influence the lower bound of the group interval. The judgments of decision makers that are greater than or equal to the median influence the upper bound of the group interval:   m1 m1  m 1 m 1  1 2 2 1 WLodd   0,..., 0, , ,..., 2 , 2  and WUodd   2 , 2 ,..., , , 0,..., 0  . (2) s m1 s m1  s m1 s m1  m1 s m1 s m1  s m1 s m1  m1 2 2 2 2  2 2 2  2  2 2  If n is an even number, then median of numbers 1, 2,..., m is not an integer and s m  2 the sum of numbers from 1 to n 2 m ( m  2) 8 is , which are smaller than median. Then  m2 m 1 2 WLeven   0,..., 0, , ,..., 2 , 2 sm sm sm sm  m 2 2 2 2  2   m m2   and WUeven   2 , 2 ,..., 2 , 1 , 0,..., 0  . sm sm   sm sm  m 2 2   2 2 2  (3) Then the aggregated interval group matrix Agroup is defined as Agroup  1     m ( k ) wkL m ( k ) wUk     c  ,   c21      k 1 21 k 1      m ( k ) wkL m ( k ) wUk     cn1  ,   cn1   k 1    k 1  m ( k ) wkL m ( k ) wUk    c12  ,   c12   k 1  k 1  1  m ( k ) wkL m ( k ) wUk    cn 2  ,   cn 2   k 1  k 1  144  m ( k ) wkL m ( k ) wUk     c1n  ,   c1n    k 1  k 1  m m L U   ( k ) wk ( k ) wk    c2 n  ,   c2 n    k 1  k 1     1   (4) where cij( k ) is the kth largest value from the set aij1 ,..., aijm  . 3 DERIVING MATRIX INTERVAL WEIGHTS FROM INTERVAL COMPARISON For deriving interval weights from interval comparison matrix Agroup we use the approach of separating Agroup into two crisp comparison matrices ALgroup   aijL  and AUgroup   aijU  [5]. m wkL wU  m k  Let lij , uij  :   cij( k )  ,   cij( k )   for i,j=1,…,n. Then k 1  k 1   lij , i  j  aijL   1, i  j , u , i  j  ij uij , i  j  aijU   1, i  j . l , i  j  ij (5) Matrices ALgroup and AUgroup are reciprocal comparison matrices, since the OWG operator preserves reciprocity. The weights can be obtained from ALgroup and AUgroup in many ways, which are suitable for crisp comparison matrices. We employed the eigenvector method [8], where the priority vector, derived from a comparison matrix A, is the eigenvector belonging to the maximal eigenvalue of matrix A. This is the most commonly used method for deriving weights in AHP. The results are the vectors AL  1AL ,..., nAL  and AU  1AU ,..., nAU  for matrices ALgroup and AUgroup , respectively. The interval weights belonging to Agroup are defined as i  iL , iU   min iA , iA  , max iA , iA  . L U L U (6) For ranking interval weights the matrix of degrees of preference could be used:  p P   21    pn1 p12  pn 2 p1n  p2 n      (7) In recent years the possibility- degree formula for pij has been used several times [9-12]: pij  P i   j   max 0, iU   Lj   max 0, iL  Uj  (iU  iL )  (Uj   Lj ) , i,j=1,…,n, i  j (8) The preference ranking order is provided using row-column elimination method [10]. 4 CASE STUDY Natura 2000 is a European network of ecologically significant areas of nature, as specified on the basis of the EU Bird and Habitat Directives. The Slovenian Decree of special protected areas [13] directed that the Natura 2000 sites will be managed only throughout sectorial management plans. The agricultural priorities are outlined in the Rural 145 Development Programme of the Republic of Slovenia 2007 – 2013 [14] and the objectives are divided on four axes: 1. Axis 1 – improving the competitiveness of the agricultural and forestry sector The activities under the first axis should support modernization and innovations and raise the qualification and competitive position. They should contribute to improved employment possibilities, increased productivity, and added value in agriculture and forestry. 2. Axis 2 – improving the environment and rural areas The activities under the second axis should contribute to environmental and water resource protection, conservation of natural resources, and implementation of nature friendly technologies in agriculture and forestry. They should provide sustainable development of rural areas and ensure a favorable biodiversity status and the preservation of habitats in the Natura 2000 sites. 3. Axis 3 – the quality of life in the rural areas and diversification of the rural economy The activities under the third axis promote entrepreneurship and raise the quality of life in rural areas through enhanced employment opportunities, rural economic development, and natural and cultural heritage conservation. 4. Axis 4 – LEADER initiative LEADER is a bottom - up method of delivering support for rural development through implementing local development strategies. The activities under the fourth axis should stimulate the cooperation and connection of local action groups (LAS). To assure the best results for the Natura 2000 sites we should rank these four aims and seek a balance between them. The weighting depends on the view of the objectives which differ among stakeholders. With an objective of incorporating different perspectives, we identified three main stakeholders for the Natura 2000 sites: representatives of environmental protection, representatives of farmers, and the government. The pairwise comparisons of the four objectives and results are represented by matrices A, B and C for environmental, farmer, and government views, respectively. 1 14 13 2  1 4 1 2 3 1  , B  3 A  3 12 1 2  1 1 1 1   1  2 3 2 1 3 1 1 1 12 3 8  2 1 4 6 1 13 12    C  ,  13 14 1 3 3 1 2 1 1 1   2 12 1   8 6 3 1 The priorities of the three stakeholders, gained by the eigenvector method are presented in Table 1. Table 1: The priorities and the ranking of the four axes for three stakeholders. axis 1 axis 2 axis 3 axis 4 A environmantalist priorities ranks 0.1397 3 0.4647 1 0.2799 2 0.1156 4 B farmer priorities ranks 0.3015 2 0.1100 4 0.3584 1 0.2301 3 C government priorities ranks 0.3372 2 0.4832 1 0.1265 3 0.0531 4 The ranking differs between the stakeholders. The environmentalist prefers axis 2, which is the most favors nature protection. Farmers favor rural economic development, which is reflect by axis 3 and 1, which are the highest evaluated. The government weights indicate that it is focused on sharing funds for particular objectives. 146 The comparison matrices A, B and C are aggregated in the Agroup (4) according to the OWG operator (1). The associated lower and upper weighted vectors (2) are defined as WL   0, 13 , 23  and WU   23 , 13 , 0  . The intervals in the matrix Agroup (9) are presented on four decimals. Agroup  1  0.6057,3.1748   0.4807, 2.0801   0.1984, 0.7937  0.3150,1.6510 0.4807, 2.0801 1.2599,5.0397   1 0.6057,3.1748 0.9086, 4.7622 1 0.3150,1.6510  2.0000, 2.6207   1 0.2100,1.1006 0.3816, 0.5000  (9) The interval weights arising from the Agroup are composed from the eigenvectors belonging to the maximal eigenvalues of matrices ALgroup and AUgroup by the equation (6):  0.158, 0.414   0.281, 0.355      0.161, 0.368     0.071, 0.193  (10) Ranking of interval weights (10) has been done over the matrix of degrees of preference (7) 0.403 0.547 0.909   0.597 59.7% 54.7% 90.2%  0.692 1  1 3 4 . P , which presents the ranking 2  0.453 0.308  0.902    0 0.098    0.091 The final ranking sets axis 2 – the protection of nature - as the most important aim in managing Nature 2000 sites. This objective is expected to be the most vital. The second most significant aim is axis 1. Improving the competitiveness of the agricultural and forestry sectors will indirectly contribute to improved environmental, water, and air quality through new technologies and renewable energy sources [14]. Axis 3 is third in importance, but close to axis 2 and should upgrade, complement, and refine the effects of axes 1 and 2 [14]. Axis 4 is ranked last but should be included in implementing the other three axes. 5 CONCLUSIONS In the paper we discussed the problem of aggregating individuals’ judgments into group interval judgments, deriving interval weights from the interval comparison matrix, and the ranking of interval weights. The case study indicates that the group approach with interval matrices could be appropriate and contributes to managing group decision problems from different areas. Additional issues must be addressed however:  We assumed that all decision makers are equally important for our analysis. 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Journal of Computational Information Systems, 6(14), pp. 4811-4818. 148 11th International Conference on Operational Research, SOR 2011 A GOAL PROGRAMMING APPROACH TO RANKING BANKS Višnja Vojvodić Rosenzweig Ekonomski fakultet u Zagrebu Kennedyjev trg 6, 10000 Zagreb Phone: ++385 1 2383 333; E-mail: vvojvodic@efzg.hr Hrvoje Volarević Zagrebačka škola ekonomije i managementa Jordanovac 110, 10000 Zagreb Phone: ++385 1 2354 151; E-mail: hrvoje.volarevic@zsem.hr Mario Varović Zagrebačka škola ekonomije i managementa Jordanovac 110, 10000 Zagreb Phone: ++385 1 2354 151; E-mail: mario.varovic@zsem.hr Abstract: Ranking of commercial banks based on seven proposed criteria is performed by using goal programming, in which the goal of every bank is the best business performance (evaluated with multiple criteria), and which is represented by a Score. The Score is obtained by calculating weights as a solution of a goal programming problem. Profitability indicators are the most important indicators for the five observed Croatian banks. Other indicators, for credit risk and productivity, are far less important for the final ranking of the chosen banks. Key words: Commercial banks, Multi-criteria ranking, Goal programming, Business performance. 1. INTRODUCTION Banks play an extremely important role in each country's economy, particularly in countries with a rather less developed financial system, as is the case with the Republic of Croatia. The banking sector in the Republic of Croatia consists of thirty banks that are mostly owned by foreign proprietors, generally by Italian, Austrian, French, and Hungarian banks. The dominant position, based on their total assets and the size of equity, is occupied by two largest Croatian Banks, Zagrebačka banka d.d. and Privredna banka Zagreb d.d. In addition to these, the top ten Croatian banks also include Erste & Steiermarkische bank d.d., Raiffeisenbank Austria d.d., Hypo-Alpe-Adria-bank d.d., Societe Generale - Splitska banka d.d., Hrvatska poštanska banka d.d., OTP banka d.d., Volksbank d.d., and Podravska banka d.d. The scope of this study encompasses the following five banks: Erste & Steiermarkische bank (ERSTE), Raiffeisenbank Austria (RBA), Hypo-Alpe-Adria-bank (HYPO), Hrvatska poštanska banka (HPB) and Podravska banka (POBA). These banks were chosen primarily because of their comparability with regard to the criteria of total assets and size of equity, as well as for the online availability of their annual reports with financial statements for the year 2010. Moreover, because of the fact that only one out of the five - HPB bank has domestic (Croatian) ownership, these five banks represent a representative sample for the Croatian banking sector. The two largest banks that participate in over 50% of the Croatian banking sector are excluded from analysis since their results would not be comparable with the financial position of the other banks studied. Particular emphasis will be put on the interpretation of the results relating to the HPB Bank, since it is the only large bank in Croatia owned by domestic capital, i.e. mainly a state-owned bank. The results of the analyses will imply certain conclusions and recommendations for the purpose of repositioning the HPB bank, but also other banks covered in the study, on the Croatian banking market. 149 11th International Conference on Operational Research, SOR 2011 A mathematical multicriteria decision making model will be used, that will consist of seven individual criteria classified into three basic groups - profitability, credit risk, and productivity. Multicriteria business performance of each bank will be evaluated using a score calculated as the weighted sum of relative values of individual indicators. There is an assumption that each bank goal is the maximum score that they wish to obtain. The score is dependent on the weights assigned to individual indicators. The deviation from the goal will be measured using two distance functions. The formulated mathematical model uses goal programming to determine the weights and the score for each bank. This approach is used in paper [6]; however, in that paper the goal of each bank is the score closest to the performance of all indicators, which will not be the case here. The rest of this paper is presented in the following manner. All seven criteria are presented in the second section, followed by formulation of the multicriteria optimalisation model in the third. The approach to solving this kind of a model is illustrated in the fourth section on the basis of the examples that include five banks and seven selected attributes (criteria). The closing considerations are presented in the final section of this paper. 2. SELECTION OF CRITERIA Ranking of commercial banks is a classic problem of multicriteria decision-making. In the first place, it is necessary to select the criteria on the basis of the ranking of the banks in a descending order (from the best to the worst). In this paper seven individual criteria have been chosen, categorized in three fundamental groups (profitability, credit risk, and productivity) as follows: 1. Return on average assets – ROAA represents one of the most well-known indicators of profitability that is often used not only in the banking sector, but also in the real sector. The value of this indicator is obtained from the next relation: X1 = Return on average assets (ROAA) = profit before taxation / average assets of the bank (1) Profit before taxation can be found in the Income statement (P&L), while the average assets of the bank are calculated as the arithmetical mean of the balance sheet's positions on the asset side for two consecutive business years (in this case for the years 2009 and 2010). The obtained values are expressed as percentages, and are desirable to be as high as possible for each bank. 2. Return on average equity – ROAE also represents a well-known profitability indicator, as well as Return on average assets. The value of this indicator is obtained as follows: X2 = Return on average equity (ROAE) = profit after taxation / average equity of the bank (2) Profit after taxation is the final entry of the Income statement, while the average equity of the bank is calculated in the same way as the average assets of the bank (arithmetical mean of the balance sheet's positions of the equity for the two sequential business years). The obtained values are also expressed as percentages, and are desirable to be as high as possible for each bank. 150 11th International Conference on Operational Research, SOR 2011 3. Income from interest bearing assets and expenses on interest bearing liabilities represents a specific indicator of profitability that is solely applied to the banking sector. The value of this indicator is obtained as follows: X3 = Income from interest bearing assets and expenses on interest bearing liabilities = (interest income / average interest bearing assets) / (interest expenses / average interest bearing liabilities) (3) Interest income and interest expenses represent the initial positions in the Income statement of every business bank because they define the financial result that is derived from basic banking activity - receiving deposits and lending loans. Interest bearing assets are the total of all positions on the asset side of the balance sheet that represent the ground for calculating active interest, by which banks’ income is generated. On the other hand, interest bearing liabilities are the total of all positions on the liability side of the balance sheet as the ground for calculating passive interest that make banks’ expenditures. The obtained values are expressed as absolute values and it is desirable that the obtained results of this ratio be as high as possible in order to confirm the profitability of banks dealings. 4. Coverage represents the indicator commonly used in banks for credit risk evaluation. The value of this indicator is obtained as follows: X4 = Coverage = (total of investments impairment + provisions) / (total of investments + contingent liabilities) (4) The numerator of the ratio consists of the total of investments impairment and provisions, where the impairment stands for the cumulative of all recognized losses for bad and doubtful loans that are not expected to be repaid, that is reimbursed, while the term provisions refers to the balance sheet position on the liability side that is recognized in the banks expenses as future observed and estimated liabilities (for example provisions for legal actions, that is lawsuits filed against the bank). The denominator of the ratio consists of the total of investments comprised divided by the total of all balance sheet positions on the asset side of the bank that represent the basis for generating income, and the other part of the denominator relates to contingent liabilities that are, as a rule, booked on the off-balance sheet, and consist of given guarantees and open letters of credit as typical banking affairs. The obtained values are expressed as percentages, and it is desirable that the obtained results of this ratio should be as high as possible, which implies that the bank management is aware of possible credit risk in business activities and of the necessity for its anticipation. 5. Quality of investments represents an indicator that pertains to the credit risk assessment, as well as coverage, because it assesses the percentage of bank investments that can be reimbursed. The value of this indicator is obtained as follows: X5 = Quality of investments = (1 – (total of investments impairment / total of investments)) (5) The equation listed above puts in ratio two positions from the asset side of the bank’s balance sheet. The obtained values are expressed as percentages and their maximum value is 100%, which means that all the bank’s investments can be repaid and that 151 11th International Conference on Operational Research, SOR 2011 there is no need for investment impairment. Taking into account the existing risk when making credit investments, this situation should not be expected to be realistic. 6. Assets per employee is a typical banking indicator that belongs to the category of productivity indicators because it represents the ratio of the realized output (total of assets, i.e. total bank’s property) against actors in bank business operations (which means all bank's employees). The value of this indicator is obtained as follows: X6 = Assets per employee = total assets / total number of employees (6) The values in this equation are obtained from the balance sheet and the notes accompanying financial statements (information about the number of employees). The obtained values are expressed as absolute values, i.e. money units, and are desirable to be as high as possible. 7. Interest income per employee represents the banking indicator that also belongs to the category of productivity indicators. The value of this indicator is obtained as follows: X7 = Interest income per employee = Interest income / total number of employees (7) The numerator of the ratio is obtained from the Income statement, while the denominator consists of the number of employees that can be found in the notes accompanying financial statements. The obtained values are also expressed as absolute values, i.e. money units, and are desirable to be as high as possible, just as with all the previous indicators. Based on the former formulas, the calculated values of all seven individual criteria (X1,...,X7) for the five selected banks, and all the obtained results are presented in the following decision-making table ( Tab. 1): Table 1. The values of seven individual indicators (X1, X2, X3, X4, X5, X6 and X7), categorized into three basic groups (profitability, credit risk, and productivity) for the five selected banks (ERSTE, HPB, HYPO, POBA and RBA). BANK: X1 PROFITABILITY: X2 X3 CREDIT RISK: X4 X5 PRODUCTIVITY: X6 X7 1. ERSTE 1,52% 10,55% 2,26 4,37% 95,65% 26,17 1,51 2. HPB 0,40% 5,55% 2,05 5,44% 94,10% 14,61 0,81 3. HYPO 0,72% 3,56% 1,77 5,71% 93,90% 22,82 1,24 4. POBA 0,58% 3,52% 2,20 5,58% 94,53% 9,11 0,54 5. RBA 1,13% 6,77% 2,17 2,85% 96,97% 17,44 0,97 All obtained results of individual indicators are positively directed, but the benefit criteria are not displayed in the same measurement units. Therefore the next step is the transformation of the positively directed criteria values. The percentage transformation is used here as it leads to proportional changes in the results. The obtained results are presented in Table 2. 152 11th International Conference on Operational Research, SOR 2011 Table 2. The transformed values of seven individual criteria (X1, X2, X3, X4, X5, X6 and X7) as part of the three basic groups (profitability, credit risk, and productivity) for the five selected banks (ERSTE, HPB, HYPO, POBA and RBA). BANK: X1 PROFITABILITY: X2 X3 CREDIT RISK: X4 X5 PRODUCTIVITY: X6 X7 1. ERSTE 0,3508 0,3522 0,2163 0,1825 0,2013 0,2903 0,2979 2. HPB 0,0912 0,1854 0,1964 0,2271 0,1980 0,1621 0,1597 3. HYPO 0,1663 0,1189 0,1692 0,2383 0,1976 0,2531 0,2436 4. POBA 0,1326 0,1176 0,2108 0,2329 0,1989 0,1011 0,1071 5. RBA 0,2591 0,2259 0,2072 0,1191 0,2041 0,1934 0,1918 3. MULTICRITERIA PROBLEM AND GOAL PROGRAMMING The weighted sum model is the most frequently used approach for the estimation of multicriteria performance of specific alternatives that are also used in this paper. To each bank i we assign score Si based on the values of individual indicators (attributes) and weights assigned to them. The weights wj of indicators j determine the score and by varying different weight different scores can be obtained for the same bank. Since the score of the alternative is its multicriteria value, it is assumed here that the goal of each bank is the maximum value of the score. In that sense the goal programming problem will be formulated. The notations in the model are as follows: i - Bank, i = 1,…,n. j – Indicator (Attribute), j = 1,…,p. wj –Weight of Attribute j, j = 1,…,p. xij – Value of Indicator j of Alternative i. Si - Score Alternative i, Si = w1 xi1 + …+ wp xip. As it was mentioned earlier, the goal for every bank i is the highest score, and therefore it is valid to define: gi = max {Si (w): w1 +…+ wp = 1, w1,…,wp ≥ 0} (8) If d = (d1,…,dn) represents a vector whose components di are deviations from components gi of the goal g = (g1,…,gn), and S is vector S = S(w) = (S1,…,Sn), the problem (GP) that we are solving is as follows: (GP) Min ||g-S(w))||α With limitations: S(w) +d =g , d ≥ 0 (9) w1 +…+ wp = 1 w1,…,wp ≥ 0 153 11th International Conference on Operational Research, SOR 2011 The solution of the problem depends on the selection of the norm i.e. on the values of the weights (wj) of the goal programming problem (GP). 4. IMPLEMENTATION The problem is solved for the five selected banks and the seven individual indicators. In this paper, the norm suggested by Dinckelbach and Isermann is used, as the first one: || g-S(w) ||α =|| g-S(w) ||∞ + (1/α)|| g-S(w) ||1, α ≥ 1 (10) The problem is solved for α = 1, 10 and 100. For all mentioned values of parameter α, the same solution is obtained. The following weights for every individual criterion are obtained: w1 = 0.3951, w2 = 0.2235, w3 = 0, w4 = 0.3783, w5 = 0, w6 = 0 and w7 = 0.032. The banks scores are (S1 – ERSTE, S2 – HPB, S3 – HYPO, S4 – POBA, S5 – RBA): S1 = 0.2855, S2 = 0.1655, S3 = 0.1855, S4 = 0.1665, S5 = 0.2001. Apart from using the Dinckelbach and Isermann's norm, the problem is also solved using the Euclid's norm in which the sum of square deviations is the smallest. The following weights are obtained for every individual criterion: w1 = 0.24, w2 = 0.22, w3 = 0.19, w4 = 0.22, w5 = 0, w6 = 0 and w7 = 0.13. The banks’ scores are (S1 – ERSTE, S2 – HPB, S3 – HYPO, S4 – POBA, S5 – RBA): S1 = 0.28, S2 = 0.17, S3 = 0.19, S4 = 0.16, S5 = 0.20. The results are rounded up to two decimal points, unlike the previous problem, since this is a square programming problem. The final ranking list of the five selected banks for both norms we used is as follows: I. Dinckelbach and Isermann's norm: II. Euclid's norm: ERSTE (S1) RBA (S5) HYPO (S3) POBA (S4) HPB (S2) ERSTE (S1) RBA (S5) HYPO (S3) HPB (S2) POBA (S4) As one can see from the obtained results, the score (Si) of every bank is approximately the same regardless of the norm used in the model, and the ranking is approximately the same in both cases. The only difference in the ranking is between the two banks with the lowest rank (HPB and POBA); their rank changes according to the norm used. Furthermore, in both cases the largest weights are assigned to profitability indicators (over 60%) while the weight of the fifth indicator equals zero because all the banks have approximately the same values of that indicator (quality of investments). Moreover, the 154 11th International Conference on Operational Research, SOR 2011 weight of the sixth indicator (assets per employee) equals zero because its values are approximately the same as the values of the seventh indicator from the list of indicators (interest income per employee). The first place of the ranking list is taken by a bank with moderate risk in business activities (ERSTE), while the bank with the highest risk in business activities (RBA) sits in the second place On the other hand, HPB has small risk and small productivity, and therefore has small profitability, which puts the bank in the last or next to the last place in the total ranking (it changes places with POBA depending of the norm used). HYPO bank in both observed cases firmly holds the third position. 5. CONCLUSION The commercial bank ranking problem can be efficiently solved with goal programming. The first step is to determine the criteria in advance, as the basis for executing multicriteria ranking and find the best business performance of the selected banks accordingly. The second step consists of using a goal programming mathematical model, in which the decision maker has the choice of using different norms. Two norms (Dinckelbach and Isermann, and Euklid's norm) are used in this paper, and the obtained results, weights, and scores are approximately the same in both cases. The obtained results for the five proposed banks suggest that the most important indicators in the model are profitability indicators, whose weights prevail in relation to the remaining two groups of indicators – credit risk and productivity – that have far less importance for the final bank ranking. This conclusion exclusively applies to the banking sector in the Republic of Croatia, while results might be different for some other countries and their banking markets [6]. Having analyzed the obtained score values for every bank selected in the model, it is beyond question that the two banks with the best score (ERSTE and RBA) have the adequate ratio for accomplished profitability and productivity, related to embedded risk in the business process. On the other hand, the same cannot be said for HPB and POBA that achieve just the opposite results, while HYPO is somewhere in between, which means there is room for improvement. HPB bank needs to improve its productivity and increase embedded risk in the business process. In that way, the bank ought to strengthen its market share in the Croatian banking sector, which would eventually lead to its repositioning regarding other banks. An alternative solution for HPB bank, as the only large bank in Croatia owned by domestic capital, would be referring to the possible recapitalization from its strategic partner, which should lead to necessary restructuring of its current business activity. 155 11th International Conference on Operational Research, SOR 2011 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] Atrill, P., McLaney, E., 2006, Accounting and Finance for Non-Specialists, 5th edition, Prentice Hall, Harlow, England. Blocher, E., J., Chen, K., H., Lin, T., W., 2002, Cost Management: A Strategic Emphasis, McGraw-Hill/Irwin, New York, USA. Ehrgott, M., Klamroth, K., Schwehm, C., 2004, An MCDM approach to portfolio optimization, European Journal of Operational Research, Vol. 155, pp. 752-770. Feroz, E., H., Kim, S., Raab, R., L., 2003, Financial Statement Analysis: A Data Envelopment Analysis Approach, Journal of the Operational Research Society, Vol. 54, pp. 48-58. Gallizo, J., L., Jimenez, F., Salvador, M., 2003, Evaluating the effects of financial ratio adjustment in European financial statements, European Accounting Review, Vol. 12(2), pp. 357-377. Garcia, F., Guijarro F., Moya I., 2010, Ranking Spanish savings banks: A multicriteria approach, Mathematical and Computer Modeling, Vol.52, pp. 1058-106. Horngren, C., T., Oliver, M., S., 2010, Managerial Accounting, Upper Saddle River, New Jersey, Pearson Prentice Hall. Ng, W., L., 2007, An Efficient and simple model for multiple criteria supplier selection problem, European Journal of Operational Research, doi:10.1016/j.ejor.2007.01.018. Sawaragi, Y., Nakayama, H., Tanino, T., 1985, Theory of Multiobjective Optimization, Academic Press, Inc., Orlando, USA. Triantaphyyllou, E., 2000, Multi-Criteria Decision Making Methods: A Comparative Study, Kluwer Academic Publishers. 156 The 11th International Symposium on Operational Research in Slovenia SOR ’11 Dolenjske Toplice, SLOVENIA September 28 - 30, 2011 Section VI: Econometric Models and Statistics 157 158 SECTORIAL GROWTH DRIVERS OF WOOD PROCESSING AND FURNITURE MANUFACTURING IN CROATIA Martina Basarac Economic Research Division Croatian Academy of Sciences and Arts Strossmayerov trg 2, 10000 Zagreb, Croatia mbasarac@hazu.hr Abstract: Reducing the share of manufacturing industry in gross domestic product, even in the years of higher rates of economic growth, leads to a decrease in competitiveness of Croatian economy. Therefore, this paper investigates the trends in the wood processing industry and furniture manufacturing, as one of the oldest industries in Croatia. Using the econometric models, we estimate which macroeconomic indicators influence sectorial production. According to the multiple linear regression analysis results, one can conclude that production is affected by both real gross domestic product growth rates and unit labour costs. Real export values also play an important role. Keywords: wood processing, manufacture of furniture, regression analysis, Croatia. 1 INTRODUCTION As we slowly emerge from the first global recession since World War II, governments and businesses share an overarching aim – to steer their economies toward increasing competitiveness and growth [3]. In this effort, long-term economic growth and development of the country depend, among others, on the competitive performance of individual industries. The efficacy and competitiveness of these industries in turn depend on the relative wealth (and hence cost) of resources, as well as the actions and possibilities to use them in a seminal and sustainable way. Therefore, analysing specific sectors is the key to understanding competitiveness and growth. Within this context, the issue of manufacturing production, as an important source of attractive jobs and export revenue, has frequently been addressed in the analysis of the Croatian economy. Nonetheless, numerous questions relating to the forest-based industries have not been adequately explored empirically (with the exception of [4] and [5]). The issue of scarce empirical analysis is even more pronounced since reducing the share of gross value added of manufacturing industry in gross domestic product (GDP), even in the years of higher rates of economy growth, leads to a decrease in Croatian economy competitiveness (see Figure 1). 19 8 18 6 in % 17 4 16 2 15 2007 2006 2005 2004 2003 2002 2001 2000 1999 -2 1998 13 1997 0 1996 14 The share of manufacturing industry in GDP (right axis) Real GDP growth rate (left axis) Source: Croatian Bureau of Statistics – CBS (at the end of 2010 CBS released revised annual GDP data for period from 1995 to 2007). Figure 1: The share of gross value added of manufacturing industry in gross domestic product, in period 19962007. 159 In that sense, a detail analysis of wood processing industry and furniture manufacturing, rather than looking at the aggregate macroeconomic level, will reveal notable insights. The study highlights the patterns and trends of the observed Croatian industries and shows the importance of these sectors, their major characteristics and level of international competitiveness. Furthermore, in order to address its driving factors, the current study on sectorial growth drivers aims to identify the key determinants of Croatian wood-based industries. Namely, many determinants, e.g. macroeconomic stability, affect industrial sectors and vary considerably between them, defining the environment within which industries operate. Therefore, it is important to focus on industry performance of output, because an industry analysis can contribute to the understanding of forces underlying competitiveness. The following regression analysis fills the gap in literature on the determinants of industrial production and shows the importance of the forest-based industries in Croatian economy. Using the econometric models, we estimate which macroeconomic indicators (e.g. economic activity, unit labour costs and international trade) influence sectorial production and to what extent. The results show that these models explain the dependent variable quite well. The remainder of this paper is organized as follows. In the next section we briefly discuss relevant characteristics of selected industries, addressing some important issues and general aspects of the wood-based industries. Third section provides a structural overview of the sectors. Fourth section gives a description of all variables used in models, as well as the analysis and interpretation of the estimated models. Finally, section five concludes. 2 WOOD PROCESSING INDUSTRY AND FURNITURE MANUFACTURING IN CROATIA Almost 48% of Croatian territory is covered by woods and forests. As one of the oldest industry in Croatia, the wood processing is labour-intensive, low-technology sector dominated by small and medium-sized companies. Although it is a relatively small sector of the Croatian economy, it is a large employer. Furthermore, wood processing industry is the significant consumer of forestry products and supplies its products mainly to the furniture and the construction industry. On the other hand, as demonstrated by the advanced European Union countries, modern furniture industry is becoming more of a capital-intensive sector. In general, wood processing and manufacture of furniture have always been significant export-oriented parts of Croatian economy. These sectors have been developed on high quality of forest raw material, long wood-processing tradition and good quality of human resources. In spite of tradition and prerequisites such as available infrastructure, long-term principle of sustainable management, labour and raw potential, past years show negative economic trends in certain macroeconomic indicators. These sectors should be competitive, profitable and internationally important, with high degree of post-processing products (especially in furniture), and with high share of value added. However, although the crisis started in the financial sector, its impact on the ‘real economy’ has now materialised; it has spread throughout the whole economy as all sectors are interconnected. Yet, this industry has been showing signs of deteriorating competitiveness even before the crisis started. In this respect, many governments are tempted to focus on emerging, innovative sectors as the key to their economies’ future competitiveness. Boosting the competitiveness of such sectors is not sufficient to sustain economy-wide growth in large, diversified economies [3]. Therein is the potential for enhancing the competitiveness of forest-based industries. This study covers forest-based activities regarding the Manufacture of wood and of products of wood and cork, except furniture; manufacture of articles of straw and plaiting 160 materials (refer to Section C – Manufacturing, Division 16) and the Manufacture of furniture (refer to Section C – Manufacturing, Division 31). The activities are classified according to NACE Rev. 2, the statistical classification of economic activities which has been in force since 2007 in Croatia. 3 SECTORIAL PERFORMANCES The evolution of Croatian production indices for wood and wood products manufacturing on the one hand, and furniture on the other, were relatively similar in the ten years through until 2007. For both of these activities, output growth was relatively strong in the period between 2000 and 2007 (albeit with a temporary fall in the output of furniture in 2005). The growth in output over the ten year period through until 2010 averaged 1.9% per annum for wood and wood products and 3.0 % per annum for furniture. The workforce trends in wood processing register more than 19.000 workers in both industrial sectors, accompanied by increasing labour productivity and decreasing unit labour costs. However, in the nineties as well as before the global crisis, that number was about 28.000 workers (in 1997). Namely, the global financial crisis continued to significantly impact the Croatian forest sector in 2009 and 2010. Furthermore, global demand shrank tremendously, including the demand for Croatia’s forest products exports. As exports are important to its economy, Croatia’s forest sector faced great challenges. According to data, we had the highest coverage of import by export for wood products in 2000 (204.6%). But, in 2007, 2008 and 2009 exports of most forest products have been seriously affected by the economic downturn (133.6, 129.0 and 134.8%). Moreover, in the past few years, total exports exceeded imports owing to the increased export of raw wood and wood products, while the import of furniture still exceeds exports. In 2010, the most important export markets for wood products were Italy, Slovenia and Egypt to which more than 50% of product value has been exported. Although the Italy remained the largest export market for Croatian wood products (accounting for a 36% share of Croatian exports) in 2010, this share recorded a sharp decrease relative to the corresponding value for the year 1997 (i.e. 57%). On the other hand, the most important export markets for furniture products were Germany, Italy and Slovakia (in 2010). The indicators presented encompass key dimensions of industrial performance and the relevant characteristics of wood processing and manufacture of furniture in Croatia. 4 EMPIRICAL ANALYSES Since there has been conducted only a few research on macroeconomic drivers of sector growth, we adjust the list of variables to be considered to include those that will be relevant for a wood processing and manufacture of furniture growth study. The general idea is to concentrate on three measures – economic activity, unit labour costs and international trade. Furthermore, we expect to find that two variables (real GDP growth rate and export) boost industrial production in wood processing and manufacture of furniture, while an increase of unit labour costs (as a proxy for cost competitiveness) is expected to have negative impact on production. 4.1 Data and Methodology Data used in the analysis encompass the period from the first quarter 2000 to the second quarter 2010, providing altogether 42 observations. The source of the data and thus the 161 construction of the variables are based on official data as published by the Croatian Bureau of Statistics [1]. In a small open economy in transition, such as Croatia, performance in international trade plays an important role. The value of exports was deflated using the exchange rate. Data for unit labour costs were obtained by multiplying the average gross wage and the number of persons employed in observed industries, and then divided by industrial production. Economic activity is proxied by real GDP growth rate. All variables are expressed in indices, 2000=100 (except the real GDP growth rate). The described approach enables us to investigate the microeconomic dynamics behind growth in observed sectors and to analyze the impact of different macroeconomic variables on industrial production. All series are seasonally adjusted and expressed in logarithms (except real GDP growth rate). Before the regression was specified, in order to avoid spurious regression, all of the time series were tested for the presence of the unit root. If the mean and variance are constant over time, then the series is stationary. On the other hand, if the mean and variance change over time, the series is non-stationary and it should be transformed to stationary ones by taking the first difference. A series that has stationary first differences is I(1) or integrated of order 1. In order to analyze the observed data, an augmented Dickey-Fuller test [2] (ADF test) is applied. Table 1 presents the results of the ADF test on the presence of the unit root. The results of ADF test in levels and first differences, suggest that all series are I(1). This means that the transformation of the original series by using first differences in the model is sufficient to obtain stationary series. Table 1: Test Values for ADF test, in levels and in differences. Name of the variable LIND_WOOD DLIND_WOOD LIND_FUR DLIND_FUR GDP_GR DGDP_GR LULC_WOOD DLULC_WOOD LULC_FUR DULC_FUR LEX_WOOD DLEX_WOOD LEX_FUR DLEX_FUR In levels/in first differences in levels in first differences in levels in first differences in levels in first differences in levels in first differences in levels in first differences in levels in first differences in levels in first differences ADF test statistic Trend & Intercept intercept -1.631649(1) -3.427235(0) -9.290167(0)* -9.220420(0)* -2.465237(2) -3.751371(2) -4.421063(0)* -5.038612(7)* -1.208661(0) -1.898058(0) -6.119240(0)* -6.083650(0)* -3.480046(0) -3.844119(0) -7.006564(0)* -6.898132(0)* -3.328738(2) -3.479316(2) -6.059656(0)* -5.955493(0)* -1.093496(0) -1.441408(0) -6.069704(0)* -5.975776(0)* -1.811743(0) -0.890117(0) -5.244475(0)* -5.533666(0)* Note: IND_WOOD, IND_FUR – value of the industrial production of wood processing and manufacture of furniture; GDP_GR – value of real GDP growth rate; ULC_WOOD, ULC_FUR – unit labour costs of wood processing and manufacture of furniture; EX_WOOD, EX_FUR – value of export of wood processing and manufacture of furniture. L and D denote natural logarithm and first differences respectively. Numbers in the brackets are the lag length (automatic based on SIC, MAXLAG=9). *Null hypothesis on the existence of unit root rejected at the 1 percent significance level. Source: Author’s calculations. 162 4.2 Estimation Results The next step is a multiple linear regression analysis. We estimate two regressions, with the industrial production index for both industries as dependant variables. As explanatory variables we use real GDP growth rate, and industry specific data on unit labour costs and real exports. The results of the regression analysis are presented in Table 2 and Table 3. Table 2: Results of a time series regression for Division 16. Dependent Variable: DLIND_WOOD Included observations: 41 after adjustments Variable Coefficient Std. Error t-Statistic 0.011167 0.005156 2.165.966 DGDP_GR -0.772018 0.129237 -5.973.667 DLULC_WOOD 0.324378 0.144652 2.242.471 DLEX_WOOD 0.005394 0.008258 0.653138 C R-squared = 0.567393; Adjusted R-squared = 0.532317 F-statistic = 1.617598; Prob(F-statistic) = 0.000001 Breusch-Godfrey Serial Correlation LM Test: F-statistic = 1.684259; Prob. F(3,34) = 0.1888 Obs*R-squared = 5.304714; Prob. Chi-Square(2) = 0.1508 Heteroskedasticity Test: Breusch-Pagan-Godfrey F-statistic = 0.768821; Prob. F(1,38) = 0.5188 Obs*R-squared =2.405838; Prob. Chi-Square(1) = 0.4925 Jarque-Bera Test = 1.831706; Probability = 0.400175 Variance Inflation Factor = 2.31 Tolerance = 0.43 Prob. 0.0368** 0.0000* 0.0310** 0.5177 Note: *p-value less than 0.01; **p-value less than 0.05; ***p-value less than 0.1. Source: Author’s calculations. Table 3: Results of a time series regression for Division 31. Dependent Variable: DLIND_FUR Included observations: 41 after adjustments Variable Coefficient Std. Error t-Statistic 0.003711 0.002177 1.705.201 DGDP_GR -0.299460 0.056020 -5.345.625 DLULC_FUR 0.060623 0.039675 1.527.981 DLEX_FUR 0.007238 0.003618 2.000.849 C R-squared = 0.469493; Adjusted R-squared = 0.426479 F-statistic = 1.091486; Prob(F-statistic) = 0.000028 Breusch-Godfrey Serial Correlation LM Test: F-statistic = 0.988891; Prob. F(2,35) = 0.3821 Obs*R-squared = 2.192913; Prob. Chi-Square(2) = 0.3341 Heteroskedasticity Test: Breusch-Pagan-Godfrey F-statistic = 0.181084; Prob. F(1,38) = 0.9086 Obs*R-squared =0.593271; Prob. Chi-Square(1) = 0.8980 Jarque-Bera Test = 0.567908; Probability = 0.752801 Variance Inflation Factor = 1.88 Tolerance = 0.53 Note: *p-value less than 0.01; **p-value less than 0.05; ***p-value less than 0.1. Source: Author’s calculations. 163 Prob. 0.0965*** 0.0000* 0.1350 0.0528*** Both models satisfy all diagnostic tests (autocorrelation, heteroskedasticity, normality and, multicollinearity), which are shown below the tables. Ramsey’s RESET Test shows that there is no evidence of specification error in any of the models. Furthermore, the CUSUM and CUSUM of Squares tests clearly indicate stability in both equations during the observed period (because the cumulative sum of the recursive residuals and the cumulative sum of squares are within the 5% significance lines). The results of the regression analysis indicate that real GDP growth rate and unit labour costs are statistically significant in both models, and have the expected signs. More specifically, the coefficient -0.772 (in Table 2) means that: on average, holding DGDP_GR and DLEX_WOOD fixed, an increase of one index point of unit labour costs is predicted to decrease industrial production by 0.772 index points. Hence, an increase in the unit labour costs has a negative impact on production in both industries while an increase in real GDP growth rate has a positive and statistically significant impact on production in wood processing and manufacture of furniture. Furthermore, real exports has statistically significant and positive coefficient only in wood processing industry, suggesting that an increase in export leads to the increase in output. This is in line with basic characteristics of analysed industries. 5 CONCLUDING REMARKS Strong and healthy forest-based industries with high levels of export competitiveness are essential in order to fully exploit the Croatia's potential for growth and to enhance and sustain its overall economic development. The economic importance of the domestic wood industry will increase in the future, with its primary influence in both its export orientation and its existing raw material potential. As industry-specific results are of interest, the analysis is conducted for each industry separately. Regression on macroeconomic variables presents similar stories: for the wood processing and furniture manufacturing, real GDP growth rates and unit labour costs play an important role. For wood processing industry model, real export is also statistically significant and has theoretically plausible sign. The models proved their adequacy in terms of various diagnostic tests. This empirical investigation leads to the conclusion that the main possibilities for the Croatian wood industry to maintain and enhance its competitiveness lays in export-oriented production and lower unit labour costs. References [1] Central Bureau of Statistics, Republic of Croatia, First releases and Statistical Reports, http://www.dzs.hr. [2] Dickey, D. A., Fuller, W. A., 1979. Distributions of the estimators for autoregressive time series with unit root. Journal of the American Statistical Association, 74, pp. 427-431. [3] McKinsey Global Institute, 2010. How to compete and grow: A sector guide to policy. McKinsey and Company, 53 p. [4] Motik, D., Pirc, A., 2009. Comparison of wood products production and consumption in the Republic of Croatia. In: Bičanić, K. (ed.), Competitiveness of wood processing and furniture manufacturing, Šibenik, WoodEMA, International Association, pp. 27-32. [5] Tkalec, M., Vizek, M., 2010. The Impact of Macroeconomic Policies on Manufacturing Production in Croatia. Economic Trends and Economic Policy, Vol. 19, No. 121, pp. 61-92. 164 ARIMA MODELS AND THE BOX – JENKINS APPROACH IN ANALYSING AND FORECASTING VARIABLES IN FIELD OF SUSTAINABLE DEVELOPMENT – THE CASE OF CROATIA Mirjana Čižmešija and Jelena Knezović University of Zagreb, Faculty of Economics and Business Trg J. F. Kennedyja 6 10000 Zagreb, Croatia mcizmesija@efzg.hr jelenaknezovic@yahoo.com Abstract: The purpose of the paper is applying the Box-Jenkins approach in developing an appropriate ARIMA model with the aim to analyze and forecast the total energy consumption in Croatia as a most important variable in the field of sustainable development. The analysis was conducted for the period since 1992 up to 2008 and forecasts values were determined up to 2014. There are two ARIMA models selected as representative: ARIMA (1,1,0) and ARIMA (1,1,1). Keywords: sustainable development, total energy consumption, Box-Jenkins approach, ARIMA model, ADF test. 1 INTORODUCTION Macroeconomic forecasting is a vital element of the total economic policy in a certain country or a region. The variables of interest usually are: gross domestic product, unemployment rate, industrial production etc. In this paper, the justification and motivation for analysing and forecasting total energy consumption in Croatia is its relevance in recent years, its impact on economic development and its significant role in the formation of macroeconomic policies in the state. The importance of this impact was recognized even fifty years ago, because of increasing scarcity of energy. During this time a growing need has been detected of developing a suitable forecasting model that would be able to predict energy consumption trends in the country. 2 THE CONCEPT OF SUSTAINABLE DEVELOPMENT The concept of sustainable development becomes a relevant issue during the 1970s. At that time people began to realize that it is very difficult to have a healthy society and a growing economy in a world with so much poverty and environment degradation. In accordance with this direction, one should develop a model of economic development that will not be as detrimental for environment and social development. There exist different understandings of the sole concept of sustainable development; therefore it is very difficult to determine its unique definition. A large number of such definitions are based on the report of Burtland (1987), which says that sustainability means ''meeting the needs of present generations without compromising the ability of future generations to meet their own needs''. In the context of such an understanding, Goddland and Ledec (1984) comprehend the concept of sustainable development as a model of social and structural-economic transformation that displays economic and social benefits of now living people, without compromising the benefits of future generations. On the other hand, Ress (1988), as well as Robbinson and Tinker (1995), see sustainability in the context of merging economic, social and ecological systems, emphasizing the importance of limited ecological capacity. Despite the differences in perception of the concept of sustainable development, in general we can say that it is a process towards achieving a balance between economic, social and environmental 165 requirements to ensure ''meeting the needs of present generations without compromising the ability of future generations to meet their own needs''. Croatia has adopted the ''Resolution on Environmental Protection” already in 1972. In 1992, after the World conference in Rio de Janeiro, where the Declaration was adopted, Croatia elect for sustainable development. But only since 2000, the topic of sustainable development becomes a current issue of public and economic interest. In February 2009, pursuant to article 44 paragraph 4 of the Environmental Protection Act (Official Gazette, No. 110/07) the Strategy for Sustainable Development of the Republic of Croatia has been adopted in the Croatian Parliament. It is a document that focuses on the long-term Croatian economic and social development and environmental protection to ensure sustainable development. In order to raise awareness of sustainable development, in this paper the variable of interest is the total energy consumption in Croatia. In the past few decades, the role of energy in economic growth is becoming increasingly important and a very common scientific topic of many authors. It is very clear that the role of energy in economic development should be given more attention. Since October 2009, when the Croatian Parliament adopted the Strategy of Energy Development, Croatia has a strategic document that relates to energy development. In accordance with this, the Strategy is not only important in terms of energy but also in terms of political and socially important documents because the energy situation in one country ''spills over'' to other very importance areas. 3 BOX – JENKINS (ARIMA) METHODOLOGY AND DATASET The Box-Jenkins methodology is an iterative approach of identifying, fitting and checking ARIMA models with time series data (Hanke and Wichern, 2009). The chosen model can be used for forecasting. Forecasts follow directly from the form of the fitted model. In this paper the dataset of total energy consumption in Croatia is used. The yearly data cover the period from 1992 to 2008 and are taken from the Croatian Bureau of Statistics (http://www.dzs.hr)1. Phenomena in nature do not behave deterministically; there is indeed a wide variety of different influences acting on the observed variables. In this case we should use the appropriate analytical model that expresses the correlation of the time series with itself, lagged by 1,2, or more periods. In such model, values of the observed series with a shift in time take the role of independent variables. It is necessary to find an appropriate analytical expression (model) that can express the dependence of the current value of the phenomena of its lagged values (Šošić, 2006). Consequently, an economic phenomenon can be defined as a stochastic process Yt , t  0,1,2, .... Time series of economic and energy variables often have nonstationarity problem that can be resolved through appropriate procedures.2 One of the reasons why the time series of these variables have the characteristics of nonstationarity may be constant changes in legal and technical principles and rules which certainly affect economic relations that have implications for changes in the time series of variables from this area. ARIMA ( p, d , q) models are used to analyze processes with nonstationary components, in other words, they are used to model the processes that contain a periodic variation in time. The mentioned models are often used to describe the dynamics of a large number of economic variables. Precisely because of these reasons they are also suitable for the analysis of total energy consumption in Croatia. 1 2 The data used in paper are the newest available. Differentiation of time series of original value. 166 ARIMA ( p, d , q) can be expressed as follows: ( B)(1  B) d Yt  ( B)t , (1) where  (B) indicates an autoregressive polynomial of order p , and  (B) represents the moving average polynomial of order q , assuming that zero points of these polynomials all lie outside the unit circle and the polynomials have no common zero point. d is a positive number and indicates the order of differencing (Bahovec and Erjavec, 2009). In other words, d numerically represents how many times a time series is differentiated to eventually become stationary. 4 EMPIRICAL RESULTS This paper first presents the results of unit root tests, Dickey-Fuller and Augmented DickeyFuller tests which are used to identify the order of integration for the variable total energy consumption (TCE) in Croatia. After unit root tests result, further follows an ARIMA model in the analysis of mentioned variable using Box-Jenkins approach to model selection and forecasting selected model. The forecast follows directly from the form of the fitted model (Hanke and Wichern, 2009). Source: http://www.dzs.hr and authors´ calculation Figure 1: Total energy consumption (TCE) and first differences of total energy consumption (DTCE) in Croatia (expressed in petajoule) The time series shown in Figure 1 indicates the presence of an upward trend in total energy consumption in Croatia for the observed period. This means that the total energy consumption in Croatia (TCE) for the observed period has the characteristics of nonstationarity.3 If the series is nonstationary, it can often be converted to a stationary series by differencing. That is, the original series is replaced by a series of differences (Hanke and Wichern, 2009). To eliminate the nonstationarity of the time series, the first differences (DTCE) were calculated. A series of first differences is tested and it has been found to be 3 Same conclusion is made by analyzing the correlogram, and based on the results of Ljung-Box test. Results are available upon request to the authors. 167 stationary4. It is concluded that the time series of total energy consumption in Croatia is integrated order of 1, I (1) . Stationarity was also noticed from the Sample autocorrelation function (SACF) and Sample partial autocorrelation function (SPACF) for time series D(TCE)5. Both functions have a tendency to decrease. In the identification phase, the initial models were chosen: ARIMA(1,1,0) and ARIMA(1,1,1) with estimated parameters, as shown in tables 1 and 2. Table 1. Estimation of ARIMA(1,1,0) model Variable Coefficient Std. Error t-Statistic 452,4773 0,920802 77,73031 0,063305 5,821119 14,54558 C AR(1) R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0,937936 0,933503 9,743446 1329,086 -58,06028 2,398034 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) Prob. 0 0 370,2412 37,7843 7,507535 7,604109 211,5739 0 Source: Authors' calculations (in EViews 5.0) ARIMA(1,1,0) with the backward shift operator can be expressed as: (1  0,92B)(1  B)Yt  t (2) p-value for the AR parameter is 0,0000 which leads to the conclusion that the AR parameter is significant in the observed model. It is known that the AR ( p) process has the property of invertibility by definition; therefore it is not necessary to examine the conditions for satisfying this property. AR ( p) process is stationary if it has MA (q ) representation, or an AR (1) process is stationary if   1 . In table 1 the AR parameter is   0,92 which is less than 1. This means that the ARIMA model satisfies the property of stationarity. ARIMA(1,1,1) with the backward shift operator can be expressed as: (1  0,935B)(1  B)Yt  (1  0,940B) t 4 5 (3) p-value is less than the significance level   0,05 . ADF test results are available upon request to the authors. Results are available upon request to the authors. 168 Table 2. Estimation of ARIMA(1,1,1) model Variable Coefficient C AR(1) MA(1) 487,1824 0,935301 -0,940203 0,959462 0,953225 8,171812 868,1206 -54,65295 1,623505 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Std. Error t-Statistic 46,27584 10,52779 0,025053 37,33249 0,056029 -16,78054 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) Prob. 0.0000 0.0000 0.0000 370,2412 37,7843 7,206619 7,351479 153,842 0 Source: Authors' calculations (in EViews 5.0) Empirical level of significance (p-value) for all three parameters is equal to 0.0000 which is less than any conventional theoretical significance level. As values of AR and MA parameters are less than 1, this means that the ARIMA(1,1,1) model has the property of both stationarity and invertibility. At the phase of diagnostic checking the following criteria are used: adjusted R square ( R 2 ), residual sum of squares ( RSS ) and information criteria. Since the model is used for forecasting purposes, to select the most appropriate model, measures of predictive efficiency can also be used. Forecasting errors that are often compared are: Mean Square Error (MSE), Root Mean Square Error (RMSE), Mean Absolute Percent Error (MAPE) and Mean Absolute Error (MAE). Table 3. Comparison of criteria for ARIMA(1,1,0) and ARIMA(1,1,1) models Criteria R2 SR AIC RMSE MAPE MAE ARIMA(1,1,0) 0,933503 1329,086 7,507535 9,320695 2,242391 8,164889 ARIMA(1,1,1) 0,953225 868,1206 7,351479 9,598116 2,192870 7,836630 Source: Authors’ calculations (in EViews 5.0) In table 3 we can see a comparison of measures of representativeness that were used in the evaluation of selected models. Between those two models, ARIMA(1,1,0) and ARIMA(1,1,1), there are very small differences when we take into account the adjusted R square and Akaike information criterion ( R 2 ) is slightly larger and AIC is slightly smaller for ARIMA(1,1,1) than for ARIMA(1,1,0). If we compare the predictive efficiency of models, MAPE and MAE are smaller for ARIMA(1,1,1) model, while RMSE is slightly greater than the RMSE for the ARIMA(1,1,0) model6. After comparing the two potentially ''good'' models with the selected criteria, it is difficult to draw conclusions on the selection of the final model. Although, looking at the relations between values of selected criteria, more of them are in favour of the ARIMA(1,1,1) model. But differences are almost minimal. As the art of selecting the model 6 In the selected models, there is no residual autocorrelation up to the fourteenth lag. 169 depends on experience and the ability of analysts, we decided to employ both models in forecasting future values of total energy consumption in Croatia. 5 FORECASTING AND CONCLUDING REMARKS Using the selected models, prognostic values for the next 6 years were found (see figure 2.) 500 ARIMA(1,1,0) petajoule 450 400 ARIMA(1,1,1) 350 300 250 2014 2012 2010 2008 2006 2004 2002 2000 1998 1996 1994 1992 200 ACTUAL VALUES OF TOTAL ENERGY CONSUMPTION (TCE) Source: Authors' calculations Figure 2. Actual and prognostic values of time series variable total energy consumption in Croatia for the models ARIMA (1,1,0) and ARIMA (1,1,1) Although the two selected forecasting models are almost equally representative, planning and forecasting, besides the quantitative part, also contains the judgmental approach. So in conditions in which the Croatian economy came (high energy prices and insufficient orientation to renewable energy sources and prediction of the recession conditions in the next three years) it is expected for the ARIMA(1,1,0) to be more appropriate forecasting model than the model ARIMA(1,1,1) because it predicts a slightly lower level of total energy consumption in Croatia up to 2014. References [1] Bahovec, V., and Erjavec, N., (2009), Uvod u ekonometrijsku analizu, Zagreb, Element [2] Henke, J. E., Wichern and Dean W., (2009), Business forecasting, Pearson Prentice Hall [3] Ledec, G., and Goodland, R. J., A., (1984), The role of environmental management in sustainable development, New York [4] Robinson, J., and Tinker, J.,(1995), Reconciling ecological, economic and social imperatives: Toward an analytical framework, SDRI Discussion Papers Series 1995-1, Sustainable Development Institute, Vancuver, Canada [5] Rees, W., E., (1988), Sustainable development and how to achieve it, University of British Columbia [6] Šošić, I. (2006), Primijenjena statistika, Zagreb, Školska knjiga 170 REGRESSION-NEURO-FUZZY APPROACH TO ANALYSE DISTANCE FUNCTION IN INTERNAL INTER-REGIONAL MIGRATIONS IN EU COUNTRIES Samo Drobne*, Marija Bogataj**, Danijela Tuljak Suban*** and Urška Železnik**** * University of Ljubljana, Faculty of Civil and Geodetic Engineering, Ljubljana, Slovenia, e-mail: samo.drobne@fgg.uni-lj.si ** University of Ljubljana, Faculty of Maritime Studies and Transport, Portorož, & MEDIFAS, Nova Gorica, Slovenia, e-mail: marija.bogataj@guest.arnes.si *** University of Ljubljana, Faculty of Maritime Studies and Transport, Portorož, e-mail: danijela.tuljak@fpp.uni-lj.si **** Dolenja vas 30, 1410 Zagorje ob Savi, e-mail: u.zeleznik@gmail.com Abstract: In this paper, we analyse the distance regression and neuro-fuzzy distance functions in internal inter-regional migration flows of EU and candidate countries. The significant impact of time distance on internal migration flows is presented. To analyse the perception of distance by analysed countries, we introduced neuro-fuzzy modelling approach. Keywords: migration, distance function, regression analysis, fuzzy logic, neuro-fuzzy, NUTS 2, EU. 1 INTRODUCTION Migrations, and periodical commuting, are regarded as the most important factors influencing the demographic and socio-economic compositions of regions at different levels. For the general process of regional changes, an understanding of inter-regional migrations (and daily or weekly commuting) is vital. As Codwallader (1992) and many other researchers (e.g. Anjomani, (2002) and Chun (1996)) have pointed out, policy-makers have become increasingly aware of the role of migrations as migration of the human resources for any production or services and in the context of any other socio-economic issues, especially as regional growth. The growth of regions relates closely to population growth, which is mostly a result of migrations.1 In this paper, we analyse the distance function in interregional migration for twenty EU and candidate countries in 2006 using regression-neurofuzzy approach. From Tab. 1 it is evident that the most active population in migration were in Turkey and United Kingdom, but Poland and Slovakia were the less active countries. 2 MODEL OF INTERNAL INTER-REGIONAL MIGRATION IN EU COUNTRIES Data for internal inter-region migrations in analysed EU and candidate countries in 2006 were obtained from EUROSTAT (2010). For those countries which did not reported for internal inter-regional flows for analysed year of 2006, geometric mean of data for 2004, 2005 and 2007 was calculated. In the regression analysis of the parameters in the gravity model, time distance function (d (t ) ij ) was analysed where d (t ) ij describes time spending distance by car between regional centres on NUTS 2 level.2 Time distance function (d (t ) ij ) was later considered by neuro-fuzzy reasoning. Data on road network in 2005 were obtained from JRC-IPTS (2010). Using data on roads, we constructed network models, 1 The migration between regions can be slowed down by daily commuting, which is becoming a surrogate for migration, if the commuting is bringing higher social well-being than any migration (Nijkamp, 1987). 2 The NUTS (Nomenclature of Territorial Units for Statistics) classification is a hierarchical system for dividing up the economic territory of the EU. Regions at NUTS 2 level are used for the application of regional policies. 171 which were the basis for calculation of optimal (shortest) time-spending distances between regional centres on NUTS 2 level. Table 1: Population and number of internal inter-regional migrants on NUTS 2 level in EU countries in 2006 (source: (EUROSTAT, 2010) and own calculation; * Slovenia is analysed for NUTS 3 regions). ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2.1 Country code Country Austria Belgium Bulgaria Croatia Czech Republic Denmark Finland France Germany Hungary Italy The Netherlands Poland Romania Slovakia Slovenia* Spain Sweden Turkey United Kingdom AT BE BG HR CZ DK FI FR DE HU IT NL PL RO SK SI ES SE TR UK Number of NUTS 2 regions 9 11 6 3 7 5 5 26 39 7 21 12 16 8 4 12 19 8 26 37 Population (in 1000) 8282 10,548 7699 4442 10,269 5437 5266 63,438 82,376 10,071 58,942 16,346 38,141 21,588 5391 2007 44,116 9081 71,553 60,596 Migrants Number 89,677 147,469 49,999 17,684 73,985 107,518 75,573 731,312 1,437,174 166,713 335,643 265,057 113,348 89,454 16,110 8499 533,128 164,755 2,099,033 1,729,963 Percentage 1.08% 1.40% 0.65% 0.40% 0.72% 1.98% 1.44% 1.15% 1.74% 1.66% 0.57% 1.62% 0.30% 0.41% 0.03% 0.42% 1.21% 1.81% 2.93% 2.85% Regression model Gravity models, based on classical physics, posit that the flow of migrants between two nodes will be proportional to the population at both nodes, and inversely proportional to the distance between them (Sen and Smith, 1998). In our previous work for case studies of Slovenia (Bogataj, Drobne and Bogataj, 1995; Bogataj and Drobne, 2005; Drobne, Bogataj and Bogataj, 2008) we proved that there is correlation between gross migration and daily commuting. For this reason, we used time-spending distance between regional centres to analyse the impact of distance on migration flows.3 The gravity model that has been analysed for twenty countries is M ij    Pi 1  Pj2  (d (t ) ij )    Pi 1  Pj2  d (t )ij (1) where we denote with i the region of origin and with j the region of destination, M ij is the number of inter-regional migrants, Pi and Pj are the populations in the region of origin respectively destination, d (t )ij is travel time between the region of origin and region of destination,  is the intercept, and 1 ,  2 ,  are the powers. Intercept and the powers were estimated in the regression analysis. Tab. 2 shows the results of the inter-regional migrating gravity model (1) between regions on NUTS 2 level by analysed country for 2006. The most reliable gravity models were estimated for those countries where adjusted R2 is high and (intercept and) powers are significant; those countries are: Austria, Denmark, France, Germany, Hungary, Italy, The Netherlands, Poland, Spain, Sweden, Turkey and United Kingdom. We got estimators of less reliable models for Belgium, Bulgaria, Croatia, Czech 3 We assume that in internal inter-regional migrations barriers like institutional and language barriers do not exist or do not play significant role. 172 Republic, Finland, Romania and Slovakia. Note that regression analysis for Slovenia was done for NUTS 3 regions (while there were only two NUTS 2 regions in Slovenia) and that there are only three regions on NUTS 2 level in Croatia. Table 2: Powers (exponents) of the inter-regional migrating gravity model (1) between regions on NUTS 2 level by country in 2006 (parentheses denote intercept and/or powers which significance is bad, P-value>0.1; in the most of other cases the significance is very good, 0.01