Volume 11 Issue 4 Article 2 12-31-2009 Cutting as a continuous business process Peter Trkman Jure Erjavec Miro Gradišar Follow this and additional works at: https://www.ebrjournal.net/home Recommended Citation Trkman, P., Erjavec, J., & Gradišar, M. (2009). Cutting as a continuous business process. Economic and Business Review, 11(4). https://doi.org/10.15458/2335-4216.1271 This Original Article is brought to you for free and open access by Economic and Business Review. It has been accepted for inclusion in Economic and Business Review by an authorized editor of Economic and Business Review. 285 ECONOMIC AND BUSINESS REVIEW | VOL. 11 | No. 4 | 2009 | 285–300 CUTTING AS A CONTINUOUS BUSINESS PROCESS PETER TRKMAN* JURE ERJAVEC** MIRO GRADIŠAR*** ABSTRACT: A review of state-of-the-art methods for cutting stock problem optimisation shows that the current methods lead to near-optimum results for the instantaneous opti- misation of trim loss. Further optimisation of this activity would not bring a considerable improvement. Th erefore, the paper treats cutting stock as a continuous business process and not as an isolated activity. An exact method for a general one-dimensional cutting stock problem is presented and tested. Th e method is mainly suitable for smaller orders. It is then applied to continuous cutting and used to develop a method for assessing cutting costs in consecutive time periods. Th e modifi ed method fi nds a good solution over the whole time-span, rather than just local optima. Key words: Cutting stock problem; Continuous cutting; Supply chain management; Exact solution; Business proc- ess management UDK: 005.81:004 JEL classification codes: C61, D81, L23 1. INTRODUCTION Th e cutting stock problem (‘CSP’) was fi rst defi ned more than 50 years ago (Paull, 1956) and has since then attracted many researchers. Th ere has been a rise in research interest in this topic over the last decade when many diff erent versions of the model and solution approaches have been studied (e.g. Trkman, Gradišar, 2007). Since near optimum solutions with a trim loss of less than 0.1% have been found (Gradišar, Trk- man, 2005) the research interest has shift ed to lowering the overall costs of the cutting process. * Corresponding author: University of Ljubljana, Faculty of Economics, Kardeljeva ploščad 17, 1000 Ljubljana, Slovenia, Email: peter.trkman@ef.uni-lj.si ** University of Ljubljana, Faculty of Economics, Kardeljeva ploščad 17, 1000 Ljubljana, Slovenia, Email: jure.erjavec@ef.uni-lj.si *** University of Ljubljana, Faculty of Economics, Kardeljeva ploščad 17, 1000 Ljubljana, Slovenia, Email: miro.gradisar@ef.uni-lj.si ECONOMIC AND BUSINESS REVIEW | VOL. 11 | No. 4 | 2009286 As the research goals have diverted away from optimising only trim loss, the considera- tion of cutting stock as a process instead of just an isolated activity has become more important. With the advent of concepts such as lean manufacturing and supply chain integration cutting orders are becoming smaller and thus exact methods can easily be used. We therefore develop an exact method for solving the CSP and integrate it into a broader view of the cutting stock process. Th e importance of treating the cutting stock process as a continuous business process and its eff ects on a decrease in overall costs is also presented. Th e structure of this paper is as follows. In the second section the CSP is defi ned and diff erent approaches to solving it in the past are presented. Th e third section presents an exact solution method suitable for CSPs with smaller orders. Th e next section defi nes cutting as a business process and emphasises the importance of treating cutting as one of the processes in the company and in the supply chain. Th e fi ft h section broadens the view by adding in the time component while the last section summarises the paper and points to topics for further research. 2. THE CUTTING STOCK PROBLEM Th ere are many diff erent variations of the CSP in practice since it is common in a range of industries. Th e materials used for cutting can take many diff erent shapes, from rolls, scrolls, coils, plates, logs etc. Th e basic problem is how to cut the material in stock into the desired number of ordered pieces while at the same time minimising the trim loss which results from the cutting (Gass, 1985). Th e stock can consist of materials of dif- ferent dimensions. Th e cutting can then be done on several diff erent cutting machines. Th e knives of each cutting machine can be set to any combination of orders so that the entire length of orders does not exceed the length of the piece from stock. Th e orders are defi ned as the number of pieces of a certain dimension or the total length needed for one order. Th ere are diff erent types of cutting stock problem with regard to the number of dimen- sions. Th e most frequently researched one is the problem of one-dimensional cutting. In this paper we focus on the latter. However, the main point that CSP optimisation should include the whole process can also be applied to other types of cutting (two-, three- or even four-dimensional). With one-dimensional cutting only one dimension is signifi - cant as the other dimensions are fi xed, negligible or do not even exist (for example the »cutting« of time, money etc.). Approaches to solving the one-dimensional cutting stock problem can be divided into two main groups (Dyckhoff , 1990): – Th e pattern-oriented approach: cutting patterns and their frequencies are defi ned by diff erent methods. Most are based on the algorithm presented by Gilmore and Go- mory (1961, 1963). Th ese methods are only usable if all the items in stock are of the same length or if there are only a few standard lengths in stock. P. TRKMAN, J. ERJAVEC, M. GRADIŠAR | CUTTING AS A CONTINUOUS BUSINESS PROCESS 287 – Th e item-oriented approach: each piece from stock is treated individually when preparing the cutting plan whereas with the pattern-oriented approach only pat- terns are defi ned. Methods that utilise the item-oriented approach have a broader aspect of usability since they can be used with either standard or non-standard stock lengths. Pattern-oriented approaches are therefore more useful and fl exible with larger problems of standard lengths of stock pieces, while item-oriented approaches are applicable to dif- ferent lengths of stock pieces. Both approaches utilise two diff erent methods: exact and heuristic. Exact methods are based on algorithms which lead to the optimal solution of the prob- lem, which is their main advantage. However, the time needed for solutions increases exponentially, meaning they are limited to smaller problems. Th e most commonly used methods for exact solutions are linear programming, branch and bound method and dynamic programming. Heuristic methods lead to near-optimum solutions and are usually better at solving larger problems. Heuristic methods take diff erent approaches to problem-solving: state- space search, problem reduction, cut-off , aspiration level, repeated exhaustion reduction, sampling (Hinxman, 1980; Nilsson, 1971). Several methods were developed in the past using the diff erent approaches mentioned above. Some of them solve the basic cutting problem by minimising the trim loss while others expand the initial cutting stock problem to additional criteria. Th e more interesting ones are the integer programming model applying the branch & bound (‘B&B’) approach used by Degraeve and Vandebroek (1998), cutting non-standard lengths (Belov, Scheithauer, 2002), cutting materials of diff erent quality (Carnieri et al., 1993), minimising the number of patterns used for cutting (Umetani et al., 2003; Vanderbeck, 2000), minimising the costs of changing the patterns (Shilling, Georgadis, 2002), multiphase methods (Zak, 2002) etc. Also interesting are methods developed for specifi c industries such as the steel industry (Armbuster, 2002), wood industry (Čižman, Urh, 2006), metal industry (Chu, Antonio, 1999) and car industry (Dowsland et al., 2007). 3. EXACT SOLUTION Most standard problems related to one-dimensional stock cutting are known to be NP- complete and in general a solution can be found by using approximate methods and heuristics. However, the constantly growing processing power is pushing the complexity limit for exact methods up slightly. Th e most important factor in the usability of exact solutions is the size of the problem. More information (future orders, available supply, lead times etc.) is available by treating ECONOMIC AND BUSINESS REVIEW | VOL. 11 | No. 4 | 2009288 cutting as one of the business processes in the entire supply chain. Th is enables greater fl exibility when determining the size of orders and can lead to smaller orders (Muff ato, Payaro, 2004), thus making them more suitable for exact methods. Th erefore we present an exact solution of general one-dimensional cutting stock problem (G1D-CSP) where all stock lengths can be diff erent. Either the branch & bound (‘B&B’) method or some dynamic programming can be used. Th e B&B exact method was chosen. First, B&B is a standard method and, second, many OR packages with B&B exist. Some of them allow the use of B&B as a subroutine so it can be included in other computer applications. In our case, it is included in an application which collects data, checks whether there is an abundance or shortage of material, solves the appropriate model and displays the results. Th e problem is defi ned as follows: For every customer order a certain number of stock lengths is available. In general all stock lengths are diff erent. We consider the lengths as integers. If they are not originally integers we assume that it is always possible to multiply them by a factor and transform them into integers. It is necessary to cut a certain number of order lengths into the re- quired number of pieces. Th e following notation is used: si = order lengths; i = 1,...,n. bi = the required number of pieces of order length si. dj = stock lengths; j = 1,...,m. xij = the number of pieces of order length si having been cut from stock length j. UB = the upper bound for the trim loss tj = the extent of the trim loss relating to stock length dj) δj = the remainder of the stock length dj Two cases are possible: Case 1: the order can be fulfi lled as an abundance of material is in stock. ∑ = m j jt 1 min (minimise the trim loss which is smaller than UB) (1) s.t. jdxs n i jjiji ∀=+×∑ =1 )( δ (knapsack constraints) (2) ∑ = ∀= m j iij ibx 1 (demand constraints – the numbers of pieces are all fi xed) (3) ∑ = ∀≤−×+ n i j j j ij jys d x 1 0)1( min max (yj indicates whether stock length j is not used in the cutting plan) (4) P. TRKMAN, J. ERJAVEC, M. GRADIŠAR | CUTTING AS A CONTINUOUS BUSINESS PROCESS 289 UB – δ j + UB ju j ∀≤−× 0)1( (uj indicates whether the remainder of stock length j is greater than UB) (5) ∑ = ≤ m j ju 1 1 (the maximum number of residual lengths that can be larger than UB) (6) jdyut jjjjj ∀≤×+−− 0)(max)(δ (7) UB ≥ max s j (8) 0≥ijx , integer ji,∀ (9) jt j ∀≥ 0 jj ∀≥ 0δ }1,0{∈ju }1,0{∈jy (10) If 0 1 >∑ = n i ijx then according to condition (4) 0=jy . If 0 1 =∑ = n i ijx this allows either 0=jy or 1=jy . Since 1=jy is less costly than 0=jy , the optimal solution will choose 1=jy if 0 1 =∑ = n i ijx . In summary, we have shown that 0 1 >∑ = n i ijx will imply 0=jy and 0 1 =∑ = n i ijx will imply 1=jy . Condition (5) can be explained similarly. Case 2: the order cannot be fulfi lled entirely due to a shortage of material (the distribution of uncut order lengths is not important). ∑ = n i j 1 min δ (minimise the sum of trim losses) (1) s.t. same as in case 1 (2) ∑ = ∀≤ m j iij ibx 1 (demand constraints) (3) 0≥ijx ,integer ji,∀ (4) jj ∀≥ 0δ (5) Unutilised stock length that is larger than some UB could be used further and is not considered as waste. Th e question is how to determine UB. Th e answer depends on the quantity of stock lengths available. Let us consider case 1 fi rst. If suffi cient stock lengths are available there will be cutting plans with ”no trim loss” but ever growing stocks. To prevent this, an additional condi- tion (case 1, condition (6)) has to be set: only one residual length may be longer than the UB. UB can be set arbitrarily between 0 and max si. UB = min si is found in practice (Gradišar et al., 1997). ECONOMIC AND BUSINESS REVIEW | VOL. 11 | No. 4 | 2009290 However, in case 2 UB is not included in the model. If, for example, UB is reduced to min si this would lead to the following problem: as the aim of the algorithm is minimisation of the overall trim loss this could lead to unfulfi lled requirements for the longest order lengths, even if the overall trim loss is small and the aim is achieved according to the logic of the algorithm. Trim losses which would be longer than UB but shorter than the longest order lengths could remain unutilised. For that reason, UB should not be less than max si. On the other hand, if UB were set to max si any trim loss longer than max si can certainly be used to cut an additional order length so UB equal or longer than max si would not have any infl uence on the solution. Th e method was tested on 270 problem instances, namely, 150 with an abundance and 120 with a shortage of material. To test the correlation between the time limit and trim loss each problem instance in the experiments was solved within six diff erent time lim- its. All problems with an abundance of stock were solved twice – once with UB set to max si and once with UB set to min si. Th e results with UB=min si are shown in Table 1, while full results can be found in (Gradišar et al., 2002). Th e “opt” column shows the number of optimally solved problems within the given time limit (each problem class contained 10 problem instances). A comparison of the exact method with state-of-the-art heuristic methods is shown in Table 2. Th e results show the approximate limit of the usability of the exact method. Since the size of the problems in section 5 is below that limit the exact method is used. P. TRKMAN, J. ERJAVEC, M. GRADIŠAR | CUTTING AS A CONTINUOUS BUSINESS PROCESS 291 TA BL E 1: R es ul ts fo r U B= m in s i 2 se co nd s 10 s ec on ds 20 s ec on ds 30 s ec on ds 45 s ec on ds 60 s ec on ds Ca se n o. Tr im lo ss op t Tr im lo ss op t Tr im lo ss op t Tr im lo ss op t Tr im lo ss op t Tr im lo ss op t cm % cm % cm % cm % cm % cm % 1 1 0. 00 27 % 9 1 0. 00 27 % 10 1 0. 00 27 % 10 1 0. 00 27 % 10 1 0. 00 27 % 10 1 0. 00 27 % 10 2 0 0. 00 00 % 10 0 0. 00 00 % 10 0 0. 00 00 % 10 0 0. 00 00 % 10 0 0. 00 00 % 10 0 0. 00 00 % 10 3 3 0. 00 26 % 8 1 0. 00 09 % 9 0 0. 00 00 % 10 0 0. 00 00 % 10 0 0. 00 00 % 10 0 0. 00 00 % 10 4 13 01 1. 67 61 % 5 10 28 1. 34 45 % 9 10 28 1. 34 45 % 9 10 28 1. 34 45 % 9 10 28 1. 34 45 % 9 10 28 1. 34 45 % 9 5 83 1 0. 49 78 % 0 35 1 0. 21 19 % 1 29 5 0. 17 59 % 2 25 5 0. 14 95 % 2 22 8 0. 13 27 % 2 21 2 0. 12 29 % 2 6 85 2 0. 35 52 % 0 29 5 0. 12 40 % 0 83 0. 03 64 % 1 69 0. 03 01 % 1 62 0. 02 69 % 2 62 0. 02 69 % 2 7 17 02 2. 39 12 % 9 17 02 2. 39 12 % 10 17 02 2. 39 12 % 10 17 02 2. 39 12 % 10 17 02 2. 39 12 % 10 17 02 2. 39 12 % 10 8 28 84 1. 33 26 % 0 23 25 1. 07 91 % 0 17 55 0. 81 12 % 0 15 41 0. 71 16 % 0 15 19 0. 69 96 % 0 14 57 0. 67 06 % 0 9 41 03 1. 19 64 % 0 30 45 0. 88 37 % 0 25 55 0. 74 73 % 0 22 01 0. 64 67 % 0 19 11 0. 55 95 % 0 17 80 0. 51 96 % 0 10 15 7 0. 22 95 % 1 53 0. 07 30 % 2 27 0. 03 83 % 3 27 0. 03 83 % 3 22 0. 03 05 % 4 18 0. 02 46 % 5 11 96 0. 06 71 % 2 25 0. 01 64 % 4 10 0. 00 65 % 6 10 0. 00 65 % 6 8 0. 00 52 % 6 8 0. 00 52 % 6 12 26 9 0. 11 89 % 0 83 0. 03 72 % 7 64 0. 02 85 % 7 39 0. 01 75 % 8 27 0. 01 22 % 8 23 0. 01 03 % 8 13 99 0. 12 45 % 2 41 0. 05 15 % 4 33 0. 04 19 % 4 17 0. 02 19 % 5 9 0. 01 13 % 6 8 0. 01 00 % 6 14 30 22 1. 01 56 % 0 20 48 0. 68 39 % 0 17 03 0. 56 87 % 0 16 66 0. 55 72 % 0 16 66 0. 55 72 % 0 16 13 0. 53 93 % 0 15 66 52 1. 44 17 % 0 27 53 0. 59 13 % 0 25 65 0. 54 83 % 0 25 02 0. 53 34 % 0 23 24 0. 49 23 % 0 18 33 0. 38 66 % 0 16 26 6 0. 36 86 % 1 11 4 0. 15 69 % 5 49 0. 06 81 % 8 49 0. 06 81 % 9 49 0. 06 81 % 9 49 0. 06 81 % 9 17 40 11 1. 38 46 % 0 30 13 1. 04 66 % 0 26 30 0. 91 36 % 0 25 80 0. 89 53 % 0 22 72 0. 78 93 % 0 20 74 0. 72 36 % 0 18 95 99 1. 66 17 % 0 79 36 1. 22 51 % 0 70 41 1. 08 60 % 0 65 33 1. 00 99 % 0 65 33 1. 00 99 % 0 65 33 1. 00 99 % 0 19 29 0. 04 02 % 5 10 0. 01 42 % 9 5 0. 00 71 % 9 3 0. 00 43 % 9 0 0. 00 00 % 10 0 0. 00 00 % 10 20 11 79 0. 51 50 % 0 75 6 0. 32 03 % 0 63 8 0. 27 07 % 0 55 7 0. 23 61 % 0 43 4 0. 18 27 % 1 42 9 0. 18 03 % 1 21 77 53 2. 10 27 % 0 11 43 0. 34 01 % 0 60 0 0. 16 91 % 0 56 8 0. 15 99 % 0 44 4 0. 12 35 % 1 43 8 0. 12 17 % 1 22 14 1 0. 18 31 % 4 37 0. 04 77 % 5 21 0. 02 75 % 7 4 0. 00 51 % 8 1 0. 00 13 % 9 1 0. 00 13 % 9 23 25 80 0. 86 12 % 0 19 41 0. 64 56 % 0 15 51 0. 51 26 % 0 13 57 0. 45 17 % 0 10 71 0. 35 59 % 0 10 43 0. 34 68 % 0 24 16 25 0 3. 06 32 % 0 13 97 7 2. 35 67 % 0 76 39 1. 15 72 % 0 71 01 1. 07 36 % 0 68 56 1. 03 87 % 0 56 15 0. 85 31 % 0 25 42 7 0. 57 28 % 1 11 3 0. 14 57 % 2 93 0. 11 97 % 2 93 0. 11 97 % 2 48 0. 06 36 % 2 41 0. 05 49 % 3 26 20 37 0. 68 72 % 0 13 19 0. 44 46 % 0 10 54 0. 35 47 % 0 10 54 0. 35 47 % 0 99 3 0. 33 25 % 0 98 4 0. 32 94 % 0 27 11 15 0 1. 61 80 % 0 75 18 1. 09 54 % 0 59 84 0. 87 30 % 0 59 84 0. 87 30 % 0 58 59 0. 85 53 % 0 53 16 0. 77 42 % 0 To ta ls 77 39 4 0. 87 07 % 57 51 62 8 0. 56 78 % 87 39 12 6 0. 45 6% 98 36 94 1 0. 43 34 % 10 2 35 06 7 0. 41 06 % 10 9 32 26 8 0. 38 95 % 11 0 ECONOMIC AND BUSINESS REVIEW | VOL. 11 | No. 4 | 2009292 TABLE 2: Trim loss with an exact method aft er 60 seconds and with the CUT procedure Abundance of material Trim loss CUT Trim loss B&B Case no. cm % cm % 1 Y 8 0.0213% 1 0.0027% 2 Y 0 0.0000% 0 0.0000% 3 Y 0 0.0000% 0 0.0000% 4 Y/N 1182 1.5460% 1028 1.3445% 5 Y 28 0.0162% 212 0.1229% 6 Y 9 0.0039% 62 0.0269% 7 N 1940 2.7256% 1702 2.3912% 8 Y 213 0.0980% 1457 0.6706% 9 Y 285 0.0832% 1780 0.5196% 10 Y/N 59 0.0807% 18 0.0246% 11 Y 0 0.0000% 8 0.0052% 12 Y 2 0.0009% 23 0.0103% 13 N 88 0.1103% 8 0.0100% 14 Y/N 172 0.0575% 1613 0.5393% 15 Y 22 0.0046% 1833 0.3866% 16 N 227 0.3155% 49 0.0681% 17 N 272 0.0949% 2074 0.7236% 18 Y/N 541 0.0836% 6533 1.0099% 19 N 7 0.0095% 0 0.0000% 20 Y 10 0.0042% 429 0.1803% 21 Y 0 0.0000% 438 0.1217% 22 N 47 0.0618% 1 0.0013% 23 N 36 0.0120% 1043 0.3468% 24 Y 159 0.0242% 5615 0.8531% 25 N 81 0.1085% 41 0.0549% 26 N 93 0.0311% 984 0.3294% 27 N 112 0.0163% 5316 0.7742% Source: (Trkman, Gradišar, 2003a) 4. CUTTING AS A BUSINESS PROCESS However, most of the reviewed methods for cutting optimisation try to optimise only the traditional criteria for the suitability of the solutions which are: trim loss, overproduction, average inventory level, average number of diff erent order lengths and average number of diff erent stock lengths (Venkateswarlu, 2001). Contemporary cutting stock solutions already lead to near-optimal results for such problems (for example, Gradišar et al., 2002a) and therefore research attention has been diverted P. TRKMAN, J. ERJAVEC, M. GRADIŠAR | CUTTING AS A CONTINUOUS BUSINESS PROCESS 293 to solutions with other goals (Trkman, Gradišar, 2003; Yang et al., 2006), such as total cutting costs, cutting time, opportunity costs etc. Most criteria are limited to a single activity (cutting) in a process. Th is opens up several research topics since the relevance of multiple criteria for evaluating supply chain effi ciency has been em- phasised (Meixell, Gargeya, 2005). In addition, the value of information technology implementation and other changes should be measured at the process level (Dava- manirajan et al., 2006). Our paper therefore presents a novel approach to treating the CSP as a business process. Treating cutting as one of the processes in the company is in accordance with research fi ndings that companies with a higher level of business process maturity outperform companies with less business process maturity (Lockamy, McCormack, 2004; Škrinjar et al., 2007). Business process is defi ned as a sequence of logically connected activities which are nec- essary to achieve wanted business outcomes (Srivardhana, Pawlowski, 2007; Davenport, Short, 1990). It is also defi ned as a structure of logically connected executing and control- ling activities that produce a product or a service as an outcome (Kovačič et al., 2004). If an activity can be defi ned as a process of its own it is called a sub-process. A sub-process is therefore a collection of interrelated activities within a larger process. Activity is the basic unit of a process for which it is no longer reasonable to divide it into smaller parts. Depending on the industry the cutting process can be one of the core operational proc- esses or just a supporting process for one of the core processes. Cutting is the core proc- ess in companies whose main service is cutting (for example, saw mills). As a sub-process it is involved in many industrial companies where materials need to be cut from larger parts. Th e importance of the cutting process is thus related to the type of industry and company involved. Th e cutting process itself consists of several activities. Th e level of dividing the cutting process into smaller activities mainly depends on the purpose of modelling. For exam- ple, activities in the cutting (sub)process can be the following: acceptance of the order, moving materials from the warehouse to the place of cutting, preparation of the cutting plan, returning unused material to the warehouse etc. It is important to model processes to better understand them, including the cutting process. Th e modelling involves converting all activities and knowledge about the busi- ness system into models which describe the processes within the organisation (Scholz- Reiter, Stickel, 1996; Giaglis, 2001). Processes are usually modelled in order to be rede- signed. What is important to note here and can be seen in Figure 1 is that the cutting should be viewed as only one of the activities or a sub-process which is connected with other processes and activities in the company and the entire supply chain (Erjavec et al., 2009). Only then can optimisation of the cutting itself lead to lowering the costs of the process of making new products or creating added value. ECONOMIC AND BUSINESS REVIEW | VOL. 11 | No. 4 | 2009294 FIGURE 1: Th e cutting stock process in connection to other processes in the company Source: Erjavec et al., 2008 Th e successful co-ordination of business processes and supply chain management is im- portant for certain factors that are key to optimising the cutting, for example: – E-procurement can bring a decrease of 42%-65% in purchasing transaction costs and a similar decrease in lead times (Davila et al., 2003; Presutti, 2003). Th is means that a company can order and deliver smaller quantities of materials. Diff erent cutting opti- misation methods are therefore required (for example, the exact method described in the following section). – A successful information interchange between companies in the supply chain can lead to a decrease in the uncertainty of future demand which leads to a better basis for stock planning and less cutting waste. It also lowers the risk of order non-fulfi lment. – Successful cutting process harmonisation within the supply chain can ease mass cus- tomisation (see e.g. Trkman, Gradišar, 2002). Th e examples shown above are a consequence of better supply chain management in the entire supply chain. It has been proven that optimisation of the whole chain can namely bring about considerably better results than the optimisation of processes within a single company (Trkman et al., 2007; Kobayashi et al., 2003). 5. CONSECUTIVE CUTTING Most approaches treat cutting as a one-off activity while, in reality, it is more likely to be a continuous business process. Th is opens up a new research fi eld and a number of re- search questions for both research and practical use since several new factors connected to business processes have to be taken into consideration. Th erefore, the development of methods, similar to the one described in this section, is vital. Further important research questions are outlined in the last section. Th ere is a constant fl ow of incoming stocks and orders that need to be fulfi lled with the company oft en having the ability to aff ect the size of the ordered stocks while, on the other hand, being able to anticipate future orders. Consecutive cutting is thus the P. TRKMAN, J. ERJAVEC, M. GRADIŠAR | CUTTING AS A CONTINUOUS BUSINESS PROCESS 295 periodical cutting of materials, replenishing stocks and anticipating future orders. Th e main goal is to minimise the sum of costs in all time periods. It is therefore important for companies to treat cutting as a business process and evaluate its costs compared to the cutting itself (Erjavec et al., 2009). Not many papers deal with cutting in the way described above. One of the fi rst approach- es is presented in Trkman, Gradišar (2007) where the authors deal with nine consecutive periods of time in which cutting is commenced and stocks are replenished while the number of pieces returned to the stock is unlimited. Two diff erent approaches are used in the paper to minimise overall losses. Both are summarised later in this section. In order to assess the eff ectiveness of both approaches we compared them with the exact method described in section 3 herein and two other methods (Gradišar et al., 1999; Gradišar, Trkman, 2005), as shown in Table 3. TABLE 3: Basic strengths and weaknesses of each method Method Strength Weakness CUT Fast solving, suitable for large problems Solution is not always optimal C-CUT Relatively fast solving, suitable for large problems Solution is not always optimal, but usually better than CUT Exact Leads to the optimal solution for smaller problems (e.g. with consecutive cutting) Not usable for larger problems Consecutive cutting – model 1 Average solving speed, lower loss in the optimal solution because of looser restrictions Leads to a large amount of partially cut pieces and consequently higher costs of warehousing and problems with later cuttings. Not suitable for consecutive cutting. Consecutive cutting – model 2 Average solving speed No saturation of stock with partially used stock lengths over time periods – a smaller number of partially cut pieces returned to stock Suitable for consecutive cutting No benchmark testing is possible as this is the fi rst method proposed for such a problem Since the fi rst approach in Trkman, Gradišar (2007) (Consecutive cutting – model 1) is unsuitable because of the steep rise in the amount of pieces returned to stock1 the authors suggest another approach. Th e costs of returning the partially cut pieces are now consid- ered. Th is lowers the total amount of partially cut pieces returned to stock (Consecutive cutting – model 2). Th e results of the second approach can be seen in Table 4. 1 Th e number of pieces returned to stock is not limited. Th erefore, many shorter ones are returned to stock and their amount rises throughout periods of time. ECONOMIC AND BUSINESS REVIEW | VOL. 11 | No. 4 | 2009296 TABLE 4: Results for consecutive cutting – model 2 Period UB Per 1 Per 2 Per 3 Per 4 Per 5 Per 6 Per 7 Per 8 Per 9 Total 400 0/20/4/3 1/22/5/0 0/20/8/2 7/54/10/60 0/20/12/1 0/20/14/12 5/50/16/60 0/0/16/1 4/28/18/60 17/234/18/200 600 0/20/4/2 1/22/5/0 0/20/8/5 0/40/10/60 0/20/12/2 6/32/14/60 6/120/16/60 9/18/19/35 12/164/20/60 34/456/20/285 800 0/20/4/0 0/20/5/0 0/20/8/16 13/66/10/60 0/0/12/0 0/10/14/2 13/226/16/60 0/20/16/5 2/24/18/60 28/406/18/203 1000 0/20/4/0 0/20/5/0 0/20/8/8 4/88/10/60 0/0/13/1 0/20/14/10 7/74/16/60 0/20/17/25 9/38/19/60 20/300/19/224 1200 0/20/4/0 0/20/5/0 0/20/8/8 7/134/10/60 0/20/13/4 8/16/15/6 4/88/16/60 0/20/15/15 8/76/18/60 27/414/18/214 1400 0/20/4/0 0/20/5/0 3/26/8/20 6/92/9/60 0/20/12/5 0/20/14/15 0/20/16/60 0/20/16/38 8/56/18/60 17/294/18/258 1600 0/20/4/0 0/20/5/0 3/26/8/60 0/40/9/60 0/20/11/1 0/20/13/7 0/20/15/60 0/20/14/7 0/20/16/60 3/240/16/255 1800 0/20/4/0 0/20/5/0 3/26/8/60 0/80/10/60 0/20/13/2 0/20/15/11 3/66/17/60 0/20/16/28 0/40/18/60 6/312/18/282 2000 0/20/4/0 0/20/5/0 0/20/8/6 0/60/9/60 0/0/11/0 0/20/13/5 1/22/15/60 0/20/15/8 0/120/17/60 1/302/17/198 Source: Trkman, Gradišar (2007) Table 4 shows the trim loss as the fi rst value, the sum of the trim loss and return costs as the second value, the number of stock lengths at the end of period as the third value and computation time (in seconds) as the fourth value for diff erent values of UB in each of the nine time periods. All problems are solved within the time limit of one minute. Th e number of stock lengths does not increase signifi cantly and the costs of the trim loss and return costs stay around the same level over all nine periods. Th is deviation of costs from period to period is likely due to coincidence. In specifi c demand and supply patterns (e.g. if the ratio between total supply and total demand were to increase considerably) a decrease in later time periods would be possible (or vice versa). See (Trkman, Gradišar, 2003) for a detailed analysis of factors that aff ect the quality of solutions. Unfortunately, the number of examples solved in this paper does not allow a full analysis of changes in time; it would also be beyond the scope of the paper. As such, the method is suitable for cutting in consecutive time periods. With the de- scribed modifi cation of the model local “optimums” lead to good solutions over the whole timespan. Th e proposed approach can help a company make better decisions which result in lower costs and higher profi ts. Th e results in Table 5 are expanded in order to also consider the time value of money. We assume that each of the nine periods is one month and that the discount rate is 10% p.a. P. TRKMAN, J. ERJAVEC, M. GRADIŠAR | CUTTING AS A CONTINUOUS BUSINESS PROCESS 297 TABLE 5: Cutting costs discounted to the fi rst period Period UB Per 1 Per 2 Per 3 Per 4 Per 5 Per 6 Per 7 Per 8 Per 9 Total 400 20 22 22 53 19 19 48 0 26 227 600 20 22 22 39 19 31 114 17 154 436 800 20 20 20 64 0 10 215 19 23 391 1000 20 20 20 86 0 19 71 19 36 290 1200 20 20 20 131 19 15 84 19 71 399 1400 20 20 26 90 19 19 19 19 53 284 1600 20 20 26 39 19 19 19 19 19 200 1800 20 20 26 78 19 19 63 19 38 302 2000 20 20 20 59 0 19 21 19 113 290 Table 5 shows the discounted costs with regard to the size of UB in each of the nine time periods. Th e discounted costs do not signifi cantly increase during the additional periods of time which implies that the saturation of stock with partially used pieces is not prob- lematic. Th e total costs are the sum of discounted costs for all nine periods which are not statistically connected to the size of UB. Th e idea of Table 5 is not to compare the costs in diff erent periods since it depends on the discount rate. Th e approach is a very simple application of a well-known technique in fi nance. Interestingly, the time value of money is almost always completely ignored in cutting stock research and is rarely included in the model. 6. CONCLUSION AND FURTHER RESEARCH Trim loss minimisation has already yielded near-optimal results and they do not need to be optimised any further. Th e cutting process therefore needs to be treated as a busi- ness process which is incorporated into an entire supply chain. In addition, it needs to be treated as a continuous process which is being executed over consecutive time periods. Its optimisation needs to consider several inputs that come not only from the cutting company but also from its suppliers and buyers. Th e main contribution of the paper is hence its presentation of the challenges of the cutting stock process and the development and testing of a method that is suitable for cutting in consecutive time periods. Th e new approach opens up a new fi eld in CSP research, namely the optimisation of problems over successive time periods rather than just instantaneous optimisation. It is likely that most of the methods developed earlier will need to be tested and (if necessary) modifi ed appropriately to match new business needs. Treating cutting in a broader view as described above opens several other research ques- tions: ECONOMIC AND BUSINESS REVIEW | VOL. 11 | No. 4 | 2009298 – Th e inclusion of warehousing costs, which can be included in the objective function. Another possible approach is to develop a simulation model which is used to assess the optimal size of the warehouse while taking diff erent costs into consideration. One such approach has been suggested by Erjavec et al. (2009). – Th e optimal size of an order to replenish stocks. While this is a well-researched topic there have been no attempts to assess it in relation to cutting stock. What distinguish- es this particular problem from others is the importance of order size. – Th e appropriateness of mass customisation with cutting. 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