JET 23 GENERALISED FUZZY LINEAR PROGRAMMING GENERALIZIRANO MEHKO LINEARNO PROGRAMIRANJE Janez Usenik R 3, Maja Žulj 14 Keywords: linear programming, fuzzy linear programming, generalised linear programming, generalised fuzzy linear programming Abstract Linear programming is one of the widely used methods for optimising business systems, which includes organisational, financial, logistic and control subsystems of energy systems in general. It is possible to express numerous real-world problems in a form of linear program and then solve by simplex method [1]. In the development of linear programming, we are facing a number of upgrades and generalisations, as well as replenishment. Particularly interesting in recent years is an option that decision variables and coefficients are fuzzy numbers. In this case we are dealing with fuzzy linear programming. If we also include in a fuzzy linear program a generalisation with respect to Wolfe’s modified simplex method [1], we obtain a generalised fuzzy linear program (GFLP). Usenik and Žulj introduced methods for solving those programs and proved the existence of the optimal solution in [2]. In the article, the simplex algorithm which enables the determining of an optimal solution for GFLP is described. There is a numerical example at the end of the article that illustrates the algorithm. Povzetek Linearno programiranje je najbolj uporabljena metoda optimizacije poslovnih sistemov, med katere štejemo tudi organizacijske, finančne, logistične in nasploh upravljalne podsisteme energetskega sistema. Veliko praktičnih problemov je mogoče izraziti v obliki linearnega programa, ki ga nato rešimo s simpleksno metodo [1]. Razvoj linearnega programiranja je doživel vrsto nadgradenj, posplošitev in dopolnitev. V zadnjih letih je še posebej zanimiva možnost, da so odločitvene spremenljivke in koeficienti mehka števila – v tem primeru gre za mehko linearno programiranje. Ko pa v ta program uvedemo še pojem generalizacije v Wolfejev pomenu [1], govorimo o generaliziranem mehkem linearnem programiranju (GMLP). Usenik in Žulj [2] sta razvila postopke reševanja takšnih programov in dokazala eksistenco optimalne rešitve. V članku opišemo algoritem simpleksnega postopka za GMLP , ki omogoča izračun optimalne rešitve, in na koncu dodamo numerični primer, ki ilustrira izvedeni algoritem. R 3 Corresponding author: Prof. Janez Usenik, PhD, Tel.: +38640 647 686, E-mail address: janez.usenik@guest.um.si 14 University of Maribor, Faculty of Energy Technology, Hočevarjev trg 1, 8270 Krško, Slovenia, E-mail address: maja.zulj@um.si JET Volume 16 (2023) p.p. 23-40 Issue 1, 2023 Type of article: 1.01 http://www.fe.um.si/si/jet.htm 24 JET Janez Usenik, Maja Žulj 1 INTRODUCTION Linear programming is one of the most frequently used techniques in operations research, introduced in 1939 by Russian mathematician Leonid Vitalijevič Kantorovič, who also proposed a method for solving it. Between 1946 and 1947, American mathematician Georg Bernard Dantzig defined a general formulation of linear programming. In 1947, he introduced the so-called simplex method, a method that enables the successful solving of any linear programming problem [1]. Linear programming can be used in economic science and in the management of business or organisational systems, as well as in actions like production planning, the optimisation of the technological process, an optimal logistics service, optimal outsourcing, etc. Linear programming proved to be of considerable applicative importance at the time computers became more capable. Nowadays, a variety of competent computer software programs exist that even enable problems of enormous dimensions to be easily solved. The theory of linear programming is developing in different ways. Let us point out two alternatives in this field. In the first alternative we use a dynamic approach, where we study the optimal behaviour of variables, which are functions of time in this approach. This problem is called continuous variable dynamic linear programming [3], [4]. In [5] we can find a generalisation of c/b/A-continuous variable dynamic linear programming in the sense of Wolfe generalisation [1]. This generalisation is based on the condition that elements of some columns of matrix ( ) At , at the time of formulating the problem, are unknown, but linked convexly in columns. In this case, we talk about generalised continuous variable dynamic linear programming. The second alternative, which has attracted a lot of interest, especially in the last 20 years, is linear programming in conditions represented by fuzzy logic. In this matter, we are dealing with fuzzy linear programming. It is a tool for modelling imprecise data and it is based on fuzzy sets [6]. In 1978, Zimmermann proposed the formulation of fuzzy linear programming problems in [7]. Since then, researchers have developed a relatively large number of different methods to solve such problems. It also turns out that there are no obstacles for generalisation in fuzzy linear programming. In this case, we talk about generalised fuzzy linear programming. Usenik and Žulj introduced methods for solving those programs and proved the existence of the optimal solution in [2]. JET Volume 16 (2023) Issue 1, 2023 JET 25 Generalised fuzzy linear programming (2.1) 2.2 Fuzzy linear programming 26 JET Janez Usenik, Maja Žulj 3 GENERALISED FUZZY LINEAR PROGRAMMING In the article we are dealing with the generalisation of fuzzy linear programming in accordance with the Wolfe approach [1], [3], [5]. Therefore, we tackle the problem concerning a linear program where technological coefficients and coefficients in an objective function are not precisely known. But we know that these coefficients are integrated in some known convex composition, i.e., the limited material, financial, logistic, energy, ecological or human resource that is required to produce some product, service, or similar. Consider the fully fuzzy linear program in standard and canonical form: JET Volume 16 (2023) Issue 1, 2023 (2.2) JET 27 Generalised fuzzy linear programming 28 JET Janez Usenik, Maja Žulj JET Volume 16 (2023) Issue 1, 2023 JET 29 Generalised fuzzy linear programming 3.2 Initial feasible solution Solving of the generalised fuzzy linear program is based on Dantzig simplex algorithm and Wolfe’s modified simplex method. Thus, we solve the problem in two phases. In the first phase we look for an optimal feasible basic solution to problem (3.03) without varying vectors 30 JET Janez Usenik, Maja Žulj JET Volume 16 (2023) Issue 1, 2023 JET 31 Generalised fuzzy linear programming 32 JET Janez Usenik, Maja Žulj JET Volume 16 (2023) Issue 1, 2023 3.4 Further feasible solutions JET 33 Generalised fuzzy linear programming (3,10) 34 JET Janez Usenik, Maja Žulj JET Volume 16 (2023) Issue 1, 2023 3.5 The existence of the optimal solution JET 35 Generalised fuzzy linear programming 3.6 Degenerate solution In the given algorithm, degeneracy is also possible. The approach for solving problems in such cases is briefly described in [2]. 4 NUMERICAL EXAMPLE We want to solve the problem: 3.6 Degenerate solution 36 JET Janez Usenik, Maja Žulj Initial main program without varying columns is of the form: JET Volume 16 (2023) Issue 1, 2023 JET 37 Generalised fuzzy linear programming Modified main program of the first degree: 38 JET Janez Usenik, Maja Žulj JET Volume 16 (2023) Issue 1, 2023 JET 39 Generalised fuzzy linear programming