An Ant Colony Optimization Approach for the Multi-Level Unconstrained Lot-Sizing Problem

An Ant Colony Optimization Approach for the Multi-Level Unconstrained Lot-Sizing Problem Jörg Homberger Stuttgart University of Applied Sciences Schellingstr. 24, 70174 Stuttgart, Germany e-mail: joerg.homberger@hft-stuttgart.de Hermann Gehring University of Hagen Profilstr. 8, Postfach 940, 58084 Hagen, Germany email: hermann.gehring@fernuni-hagen.de Abstract An Ant Colony Optimization approach for the Multi- Level Unconstrained Lot-Sizing Problem (MLULSP) is described and evaluated using 176 benchmark problems from the literature, with problem sizes varying from 5 to 500 products and up to 52 periods. The approach consists of a binary encoding of production plans. The lot-sizing decisions are mapped on a routing graph to apply the metaheuristic concept of ant systems. The proposed approach is competitive with the best known solution methods. It was possible with the new method to calculate new best solutions for 11 of the benchmark problems. 1. Introduction and Literature Review Models of Material Requirements Planning (MRP) are essential components of Production Planning and Control Systems (PPS) and of Enterprise Resource Planning Systems (ERP) as well [17][27][20]. A comprehensive overview of the different MRP models is given by Tempelmeier [25]. This paper deals with the Multi-Level Unconstrained Lot-Sizing Problem (MLULSP) which belongs to the class of dynamic lotsizing problems with multi-level production of one or more end products. The MLULSP aims at the calculation of lot-sizes for end products, intermediate products, and components that minimize the sum of set-up and inventory-holding costs, while meeting the given demand for end products over several periods [29][28]. According to the basic idea of MRP, capacity constraints are not considered in MLULSP. Hence, the practical application of the MLULSP covers mainly three situations: (1) The MLULSP is applied to the planning of lot-sizes in cases where due to the practical conditions complete or current information on the availability of capacities of resources is not known [22][10][20]. (2) In the case of given capacity constraints, the model can be used as a component within successive planning approaches of PPS. For this purpose the MRP is followed by a capacity requirements planning aiming at the calculation of feasible solutions with respect to the available capacities [22][10][20]. (3) Finally, in the area of purchasing the MLULSP is suited to determine earliest possible requirement times for (raw) material entering the production process [27]. Apart from capacity constraints, additional issues coming from correlations over time, random events, etc. are often to be considered in practical planning situations [4]. In these cases, a direct application of the MLULSP is not possible. In spite of its limited applicability to practical problems, the MLULSP is used in the current scientific literature as a basis for the development and evaluation of new methods for the planning of lot-sizes [9][20][16]. The reasons are as follows: (1) Since the MLULSP is a well established standard-problem of lot-size planning, a multitude of benchmark problems and solutions are reported in the literature. This favours the meaningful evaluation of new lot-sizing planning approaches. (2) The MLULSP considers only a few constraints (e.g., predecessor and successor relations between products) and ignores special practical conditions. Within the development of solution methods the focus can therefore be on metaheuristic components of methods. The Ant Colony Optimization (ACO) metaheuristic was introduced by Colorni et al. [8]. The basic idea of ACO is to imitate real ants searching for food based on pheromones. In each iteration, m alternative solutions of the optimization problem at hand are built by artificial ants (search processes) using a randomized constructive procedure and a pheromone matrix that represents an adaptive memory of previously visited solutions. A solution is generated stepwise. In each step, a further component of the solution is selected in accordance with a probability distribution and then added to the partial solution constructed so far. The probability distribution is defined by artificial pheromones. For a given solution 978-0-7695-3450-3/09 $25.00 2009 IEEE 1

component, a pheromone value describes the solution quality obtained by solutions containing this component. At the end of each iteration all constructed solutions are evaluated and the pheromone values of the solution components of the best solution or ant are increased according to an adaption rule. In this way, the components of good solutions get higher pheromone values and future ants will use this information to generate new and better solutions. To avoid local optima, some kind of pheromone evaporation is used, i.e., pheromone values are decreased during the search. Three standard ACO algorithms are distinguished. The Ant Colony System (ACS) [12][14], the MAX-MIN Ant System (MMAS) proposed by Stützle and Hoos [24], and the rank-based Ant System (AS rank ) [6]. These describe essential different pheromone update mechanisms. Usually the ACS is fastest, the MMAS gives best solutions but is slower, and the AS rank is a good compromise. Until now, ACO approaches have been applied successively to combinatorial optimization problems, especially vehicle routing problems [6][24][18][19]. For solving the MLULSP, different heuristics, metaheuristics, and exact optimization procedures were developed. An overview can be found in Dellaert and Jeunet [10]. Best results for large instances could be achieved by the following metaheuristics: a hybrid genetic algorithm (HGA) [9], a MMAS [20], and a parallel genetic algorithm (PGA) [16]. On the one hand, these methods differ with respect to the given metaheuristic concept, and on the other hand, with respect to the problem representation used. The HGA is based on a binary encoding of production plans, where each bit represents a set-up decision. In MMAS, production plans are represented as a sequence of products. Finally, a redundant binary encoding of production plans, which is based on a pre-selection of production periods, is used in PGA. In this paper, a new ACO approach for solving the MLULSP, denoted as Simple Ant System (SAS) in the following, is described. The aim is to develop and to test a new pheromone matrix for solving multi-level lotsizing problems. For the adaption of pheromone values only some simple rules are applied (i.e., only some of the rules described in ACS are used).the SAS differs from MMAS as follows: The SAS is based on a direct encoding of solutions by means of binary production decisions. The MMAS of Pitakaso et al. [20], on the other hand, is based on a indirect encoding of solutions using the sequence of the different items of the product structure. The MMAS is used to calculate a good sequence of items; for each item a modified Wagner-Whitin algorithm is applied separately. Apart from the chosen representation of solutions, a new pheromone matrix, and therefore a different interpretation of pheromones and a different rule of their adaptation, are used in SAS. Finally, problem specific heuristics like, e.g., the Wagner-Whitin algorithm, are not used for the construction of solutions. In this way, it is tried to extend the part of the solution space which is accessible for the search process. The formulation of the MLULSP is given in Section 2. In Section 3 the proposed SAS is described. To analyze the performance of the new metaheuristic for solving the MLULSP, the method is compared with best known solution methods in Section 4. Section 5 contains some conclusions and objectives for further research. 2. The Multi-Level Unconstrained Lot- Sizing Problem Based on the model formulation by Steinberg and Napier [23], the MLULSP is represented as mixedinteger program in the following. The notation used is described in Table 1. N T d i,t Γ(i) Γ -1 (i) q i,j t i s i h i x i,t l i,t y i,t M Table 1. Notation of the MLULSP. number of items finite planning horizon total requirement for item i in period t all direct successors of item i all direct predecessors of item i quantity of item i required to produce one unit of item j lead time to assemble, to manufacture or to purchase item i set-up cost for item i inventory-holding cost for item i delivered quantity (lot-size) of item i at the beginning of period t inventory positions for item i at the end of period t binary (set-up) variable which indicates if an item i is produced in period t (y i,t = 1) or not (y i,t = 0) a large number By using the introduced notation, the MLULSP is modelled as follows [9]: 2

minimize N T Cost = i= 1 t = 1 s i y i,t + h i l i,t, (1) subject to l i,t = l i,t-1 + x i,t d i,t, i = 1,..., N, (2) d i,t = q i, j x j,t + t, i, i = 1,..., N Γ(i) Ø, (3) i j (i) x i,t M y i,t 0, i = 1,..., N, (4) l i,0 = 0, i = 1,..., N, (5) l i,t 0, i = 1,..., N, (6) x i,t 0, i = 1,..., N, (7) y i,t {0, 1}, i = 1,..., N. (8) The objective function (1) aims at the minimization of the sum Cost of set-up and inventory costs for all items over the entire planning horizon. Equation (2) is the inventory balance equation. The constraints (3) ensure that a lot of item j in period t+t i triggers a corresponding demand d i,t in each predecessor item i, i Γ -1 (j). The demand triggered by a lot is also designated the dependent demand. Constraint (4) captures the fact that a set-up cost is incurred whenever an item is produced. Constraints (5) - (7) express that backlog is not allowed and that production is either positive or zero. Finally, constraint (8) represents the binary character of decisions on set-ups. As is usual in the literature, q i,j = 1, for i, j = 1,..., N, will be assumed in this paper without loss of generality [13]. Arkin et al. [3] show that the MLULSP is NP-hard for general multi-level bill-of-material (BOM) structures, i.e., for product structures, where each item can have more than one successor and predecessor [21]. exactly T bits are reserved for each item (see [16]). The bits c i,t, i = 1,..., N, and t = 1,..., T, represent a preselection of periods that can be used when required for production, where c i,t = 1 when period t is pre-selected as a possible production period for item i, and c i,t = 0 otherwise. 3.2. Pheromone Matrix ACO approaches are well suited for solving problems which require the construction of an efficient path between two nodes of a graph. Pheromone values imposed on the arcs of the graph are used to construct efficient paths. In order to enable the construction of encoded solutions of the MLULSP by means of the pheromone-based approach, the binary decision variables c i,t, i = 1,..., N, t = 1,..., T, are represented in a graph. The subgraph G i consists of T+1 nodes and T(T+1)/2 arcs. The nodes k i,t, t = 1,..., T, represent the possible production periods and the node k i,t+1 (also denoted as node stop) represents the end of the planning for item i. From node k i,x, x = 1,..., T, arcs emanate to the nodes k i,z, x < z T+1. An arc (k i,x, k i,z ), z < T+1, represents the decisions c i,y = 0, x < y < z, and c i,z = 1. This means, after the production in period x a further production may not start earlier than in period z. An arc (k i,x, k i,t+1 ) represents the decisions c i,y = 0, x < y T. This means, after the production of period x, item i may not be produced in any of the further periods. For each arc (k i,x, k i,z ) a pheromone value τ i,x,z is now introduced and kept in a three-dimensional pheromone matrix. In Figure 1, an example of an encoded solution c for a simple MLULSP instance which consists of only two items (one end item and one component) is given. 3. Ant Colony Optimization Approach 3.1. Representation of Solutions A solution of the MLULSP can basically be understood as a N T matrix, in which the delivered quantity, x i,t, is given for each item and each period. As it is not easy to directly derive feasible solutions, an indirect problem representation is used in this paper. In conformity with Dellaert and Jeunet [9], a binary encoding is chosen to represent solutions. A solution is represented by a N T bit matrix, denoted as c, in which 3

encoded solution c period t: c 1,t : c 2,t : 1 2 3 4 1 0 1 0 1 0 0 0 heuristic information i,x, y, i.e., each ant k in its current position x decides to go to the period y with the probability i,x, k y (nr ) given in formula 9. graph G 1 k 1,1 k 1,2 k 1,3 k 1,4 stop τ 1,1,2 τ 1,1,3 τ 1,1,4 τ 1,1,stop τ 1,1,stop τ 1,2,3 τ 1,2,4 τ 1,2,stop τ 1,2,stop 1,3,4 τ τ 1,3,stop i,x,y( nr) i,x,y ( nr) = (9) k i, x, y ( i,x,z( nr) i,x,z ) z> x The local heuristic information i,x, y is calculated by the following rule: Starting with period x, the next period y is determined by the Groff criterion [15]. The heuristic information i,x, y is now defined as i,x, y = (T-x) for y = y, and i,x, y = 1 otherwise. graph G 2 k 2,1 k 2,2 k 2,3 k 2,4 stop τ 2,1,2 τ 2,1,3 τ 2,1,4 2,1,stop τ 2,1,stop τ τ 2,2,3 τ 2,2,4 τ 2,2,stop τ 2,2,stop τ 2,3,4 τ 2,3,stop Figure 1. Example of a Graph and of the introduced Pheromones. On the one hand, all possible paths leading from the production in period t = 1 to the end of planning (node stop) are shown in Figure 1. On the other hand, the encoded solution (i.e., possible set-up decisions for item 1 and 2) is given in graph G 1 and G 2, and the respective solution paths are highlighted using bold types. 3.3. Generation of Encoded Solutions In each iteration, m encoded solutions are generated. In the terminology of ACO, the construction of a solution correspondents to starting an artificial ant. In order to construct an encoded solution c, all subgraphs G i are traversed by initialized ants in the sequence i = 1,..., N. The traversion of subgraph G i starts at the node representing the first period of demand of item i. Hence, an according production for the first period of demand is planned and the calculation of a feasible solution is guaranteed [9]. In each construction step, each ant moves from the current period x to another period y > x. The next period y is selected on the basis of a probabilistic decision. The probabilistic selection in iteration nr is biased by the pheromone information i, x, y(nr) and some local 3.4. Decoding Rule The decoding rule, i.e., the decoding of a binary represented solution c, follows the ideas of Homberger [16]. The following two decoding steps per level are executed successively for each item i N (for a detailed description we refer to the mentioned publications): (1) For each period t = 1,..., T, calculate the demand d i,t subject to the BOM and the lots for Γ(i) which have been determined already. (2) For each period t = 1,..., T, calculate the set-ups y i,t, the quantity stored l i,t, and the delivery quantity x i,t subject to d i,t and c i,t. For this purpose, for each period t of the planning horizon a check is made to determine whether a demand d i,t, d i,t > 0, was calculated in the first decoding step. Two cases can be distinguished for selecting the set-up y i,t : (1) The period t in the encoded solution c is pre-selected (c i,t = 1). The demand d i,t is then produced in period t; i.e., y i,t is set to 1, and x i,t is actualized as follows: x i,t := x i,t + d i,t. (2) The period t is not pre-selected (c i,t = 0). Then a lot is run in period w (w < t, c i,w = 1), which is the last period before t and which is pre-selected for the production of i. The quantity d i,t produced in period w is stored until period t, i.e., y i,w is set to 1, y i,t is set to 0, and x i,w is actualized as follows: x i,w := x i,w + d i,t. The calculated delivery quantities represent the (decoded) solution con. 3.5. Initialization of Pheromones and Adaptation Rule The initialization of the pheromones pursues the goal to generate each possible solution with equal probability. To achieve this goal, pheromones are initialized in such a manner that each period is selected as a possible production period with a probability of 0.5; i.e., the variables c i,t should receive the value 1 with probability 4

0.5. This is achieved by an initialization according to formula 10. i,x,y 1 2 if 1< y < T + 1, i,x,y = init if y = 1, (10) i,x,y 1 if y = T + 1. In each iteration, the best decoded solution con of m generated decoded solutions is selected and the pheromone information is updated by the following updating rule. First, all current pheromone values are decreased by a constant factor ρ. This procedure is also called pheromone evaporation. Second, the selected solution con is used to update the pheromones. Therefore, the pheromone values on the arcs which represent the selected solution con are increased with respect to the cost value Cost con of the solution con. The pheromone update rule is given by i, x, y( nr + 1) = * i, x,y nr + Δ i,x, y ( ), (11) with Δ i,x, y = 1/Cost con, if the ant which constructs the selected solution con has visited arc (x, y), and * Δ i,x,y = 0 otherwise. * 4. Computational Results 4.1. Problem Instances To evaluate the SAS, three classes of benchmark problems were used. Class 1 consists of 96 problem instances which were developed by Coleman and McKnew [7] on the basis of a work by Veral and LaForge [26] and by Benton and Srivastava [5]. Optimum solutions are known for all instances. Class 2 covers 40 problem instances with problem sizes of N = 40 and N = 50 products and of T = 12 and T = 24 periods. The instances are based on the product structures published by Afentakis et al. [2], Afentakis and Gavish [1], and Dellaert and Jeunet [9]. All lead times were set to zero. Class 3 covers the 40 problem instances with a problem size of N = 500 products and T = 36 and T = 52 periods generated by Dellaert and Jeunet [9]. A lead time of one period is assumed for all products in this case. 4.2. Analysis of the Solution Quality The SAS was implemented in Java. Each calculation run was carried out on a Dual-Core CPU (2.4 GHz, 2 GB RAM) operating under Red Hat Enterprise Linux Version 4 Update 4. In the following, SAS is compared to other methods on the basis of results reported in the literature. For each class and each method, the average solution quality is provided in Table 2. Table 2. Summary of Total Cost of Various Solution Methods. Method class 1 class 2 class 3 HGA 810.74 263,931.8 40,817,600 MMAS 810.79 263,796.3 40,371,702 PGA 810.67 263,359.6 39,809,739 SAS 810.67 263,347.1 39,790,184 The SAS is competitive with the best known solution methods. It was possible to calculate new best solutions for 11 instances. 4.3. Analysis of the Performance The mean execution times in seconds for HGA, MMAS, PGA, and SAS (to reach the solution quality identified in Table 2) are provided in Table 3. Table 3. Executing Times of Various Solution Methods. Method class 1 class 2 class 3 HGA 5 60 3,810 MMAS 1 20 2,400 PGA 5 60 3,600 SAS 3 32 2,800 The executing times of SAS are competitive with the computing times reported for other approaches. The MMAS by Pitakaso et al. [20] is a bit faster than the approach at hand. It should be remarked that comparison of the computing times should be made with caution because of the different test environments. To make a fair comparison of the efficiency of SAS and MMAS, the computing times of SAS are converted according to the different machines by using the information provided by Dongarra [11]. On this basis, the average computing times per instance of SAS are 5 seconds for class 1, 258 seconds for class 2, and 5040 seconds for class 3. In comparison to the average computing times of MMAS as given in Table 3, the respective times of SAS are approximately five times higher for class 1, and two times higher for the classes 2 and 3. It should be considered, however, that SAS was developed and tested under Java, and MMAS, on the other hand, under C. The latter programming language is regarded as to be more efficient. 5

In order to examine the solution quality of SAS on dependence of the computing time, all problem instances were solved again with SAS. As termination criterion a maximum computing time per run was now used which is comparable to the computing time reported by Pitakaso et al. [20], i.e., shorter computing times were now used. The obtained average solution quality amounts for 810.74 for class 1, 263,798.2 for class 2, and 40,611,991 for class 3. These results may be interpreted as follows. For the classes 1 and 2, the reduction of the computing time leads hardly to worsenings, i.e., MMAS and SAS provide solutions of nearly same quality for these classes. In the case of class 3, SAS generates on average a little more worse results than MMAS for comparable computing times. In order to demonstrate the solution quality of SAS on dependence of the computing time, the convergence behaviour per run for each calculated benchmark instance of class 2 is exemplarily shown in Figure 2. In this diagram, the dependence of the solution quality of the best solution or ant on the number of iterations is considered. Cost [1000 monetary units] 1,800 1.800.000,00 176 benchmark problems from the literature, with problem sizes varying from 5 to 500 products and covering 12 to 52 periods. The SAS is competetive with the best known solution methods described in the literature. It was possible to calculate new best solutions for 11 of the benchmark problems. Further research on the MLULSP will follow two directions: (1) Problem-specific extension of the described ant system SAS. The MLULSP underlying this paper is based on the assumption that capacity constraints are not to be considered. In order to eliminate this in many cases rather restrictive issue, it is intended to extend SAS in such a manner that capacitated problems can be solved, too. This requires the replacement of the problem-specific components of SAS, namely the Groff heuristic (Section 3.3) which is used to generate encoded solutions and the decoding procedure (Section 3.4), by new components which consider the capacity constraints. (2) Metaheuristic extension of the ant system SAS. In SAS only simple rules are used for the adaption of pheromone values. It is intended to refine the adaption concept by using and evaluating the mentioned three standard-algorithms ACS, MMAS, and AS rank. 6. References 1.600.000,00 1.400.000,00 1.200.000,00 1,000 1.000.000,00 800.000,00 600.000,00 400.000,00 200 200.000,00 0,00 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 0 Iterations 50 Figure 2. Convergence Diagram of SAS for Class 2. From Figure 2 it can be depicted that beyond an iteration number of about thirty improvements could scarcely be obtained. 5. Conclusions and Further Research A new Ant Colony Optimization approach for the MLULSP, denoted as SAS, is presented. The proposed procedure is based on a new pheromone matrix for solving multi-level lot-sizing problems and on the redundant binary encoding for the MLULSP introduced by Homberger [16]. The method was evaluated using [1] P. Afentakis, B. Gavish, Optimal lot-sizing algorithms for complex product structures, Oper. Res. 34, 1986, pp. 237-249. [2] P. Afentakis, B. Gavish, and U.S. Karmarkar, Computationally efficient optimal solutions to the lot-sizing problem in multistage assembly systems, Management Sci. 30, 1984, pp. 222-239. [3] E. Arkin, D. Joneja, and R. Roundy, Computational complexity of uncapacitated multi-echelon production planning problems, Oper. Res. Lett. 8, 1989, pp. 61-66. [4] G. Belvaux, L. Wolsey, Modelling practical lot-sizing problems as mixed-integer programs, Management Sci. 47, 2001, pp. 993-1007. [5] W.C. Benton, R. Srivastava, Product structure complexity and multilevel lot sizing using alternative costing policies, Decision Sci. 16, 1985, pp. 357-369. [6] B. Bullnheimer, R.F. Hartl, and C. Strauss, A new rankbased version of the ant system: a computational study, Central European Journal of Operations Research 7(1), 1999, pp. 25-38. [7] B.J. Coleman, M.A. McKnew, An improved heuristic for multilevel lot sizing in material requirements planning, Decision Sci. 22, 1991, pp. 136-156. [8] A.M. Colorni, M. Dorigo, and V. Maniezzo, Distributed optimization by ant colonies, in: Varela, F.J., P. Bourgine (eds.), Proceedings of the First European Conference on 6

Artificial Life, Elsevier Publishing, New York, USA, 1991, pp. 134-142. [9] N.P. Dellaert, J. Jeunet, Solving large unconstrained multilevel lot-sizing problems using a hybrid genetic algorithm, Internat. J. of Production Res. 38, 2000, pp. 1083-1099. [10] N.P. Dellaert, J. Jeunet, Randomized cost-modification procedures for multilevel lot-sizing heuristics, Eur. J. of Oper. Res. 148, 2003, pp. 211-228. [11] J.J. Dongarra, Performance of various computers using standard linear equations software, University of Tennessee Computer Science Technical Report CS - 89 85, 1994. [12] M. Dorigo, L.M. Gambardella, Ant colony system: a cooperative learning approach to the traveling salesman problem, IEEE Transactions on Evolutionary Computation 1(1), 1997, pp. 53-66. [13] A. Fink, Untersuchungen zur effizienten Lösbarkeit dynamischer, unkapazitierter, mehrstufiger Mehrprodukt- Losgrößenprobleme, Research Report AP-Nr. 97/11, TU Braunschweig, Germany, 1997. [14] L.M. Gambardella, M. Dorigo, An ant colony system hybridized with a new local search for the sequential ordering problem, INFORMS Journal on Computing 12(3), 2000, pp. 237-255. [15] G.K. Groff, A lot-sizing rule for time-phased components demand, Production and Inventory Management 20(1), 1979, pp. 47-53. [16] J. Homberger, A parallel genetic algorithm for the multilevel unconstrained lot-sizing problem, INFORMS Journal on Computing 20(1), 2008, pp. 124-132. [17] M. Jäger, Kennliniengestützte Parametereinstellung von PPS-Systemen, PhD Thesis, University of Hannover, Germany, 2000. [18] D. Merkle, M. Middendorf, An ant algorithm with a new pheromone evaluation rule for total tardiness problems, in: Boers, E.J.W. et al. (eds), Proceedings of the EvoWorkshops, Volume 1803 of LNCS, 2000, pp. 287-296 [19] D. Merkle, M. Middendorf, and H. Schmeck, Ant colony optimization for resource constrained project scheduling, IEEE Transactions on Evolutionary Computation 6, 2002, pp. 333-346. [20] R. Pitakaso, C. Almeder, K.F. Doerner, and R.F. Hartl, A MAX-MIN ant system for unconstrained multi-level lot-sizing problems, Comput. and Oper. Res. 34, 2007, pp. 2533-2552. [21] M. Salomon, R. Kuik, Statistical search methods for lotsizing problems, Ann. of Oper. Res. 41, 1993, pp. 453-468. [22] A. Segerstedt, A capacity-constrained multilevel inventory and production control problem, International Journal of Production Economics 45, 1996, pp. 449-461. [23] E. Steinberg, H.A. Napier, Optimal multi-level lot sizing for requirements planning systems, Management Sci. 26, 1980, pp. 1258-1271. [24] T. Stützle, H. Hoos, MAX-MIN ant system, Future Generation Computer Systems 16, 2000, pp. 889-914. [25] H. Tempelmeier, Material-Logistik. Modelle und Algorithmen für die Produktionsplanung und -steuerung und das Supply Chain Management, 5 th ed., Springer, Berlin, 2003. [26] E.A. Veral, R.L. LaForge, The performance of a simple incremental lot-sizing rule in a multilevel inventory environment, Decision Sci. 16, 1985, pp. 57-72. [27] S. Voß, D.L. Woodruff, Introduction to computational optimization models for production planning in a supply chain. 2 nd ed., Springer, Berlin, Heidelberg, 2006. [28] L.E. Yelle, Materials requirements lot sizing: a multilevel approach, Internat. J. of Production Res. 17, 1979, pp. 223-232. [29] W. Zangwill, Minimum concave cost flows in certain networks, Management Sci. 14, 1968, pp. 429-450. 7