Hierarchical approach to production scheduling in make-to-order assembly

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1 International Journal of Production Research, Vol. 44, No. 4, 15 February 26, Hierarchical approach to production scheduling in make-to-order assembly TADEUSZ SAWIK* Department of Computer Integrated Manufacturing, AGH University of Science and Technology, Al. Mickiewicza 3, 3-59 Kraków, Poland (Revision received June 25) This paper presents a hierarchical framework and integer programming formulations for multi-objective production scheduling in a make-to-order manufacturing environment. Integer programming formulations are proposed for the long-term assignment of customer orders to planning periods and the short-term machine assignment and scheduling of production lots in a hybrid flow shop with multi-capacity machines and due-date-related performance measures. The integer programs have been enhanced by the addition of cutting constraints derived for both the long-term customer order assignment and the short-term machine scheduling. Numerical examples modelled on a real-world make-to-order assembly system in the electronics industry are provided and computational results are reported. Keywords: Multi-objective production scheduling; Flexible flow shop; Integer programming; Make-to-order assembly 1. Introduction Optimized production schedules are crucial in make-to-order discrete manufacturing environments. One of the basic goals of production scheduling is to maximize the customer service level, that is, to maximize the fraction of customer orders filled on or before their due-dates (Hopp and Spearman 1996). A typical customer due-date performance measure is the minimization of the number of tardy orders (Markland et al. 199, Hopp and Spearman 1996, Proud 1999, Silver et al. 1999, Shapiro 21). Simultaneously, increasing competition forces manufacturers to achieve low unit costs by utilizing renewable production resources (e.g. machines and people) and by minimizing the inventory (Leachman et al. 1996, Younis and Saad 1996, Neumann and Zimmermann 1999). Both the input inventory of purchased materials waiting for processing in the system and the output inventory of finished products waiting for delivery to the customer should be minimized. Minimizing the number of tardy orders simultaneously reduces the input inventory of purchased materials that should be available before the order due-date, whereas the output inventory can be reduced by minimizing the number of early orders. * ghsawik@cyf-kr.edu.pl International Journal of Production Research ISSN print/issn online # 26 Taylor & Francis DOI: 1.18/

2 82 T. Sawik Customer Orders Master Scheduling Long-Term Production Schedule Batching Production Lots Machine Assignment & Scheduling Short-Term Machine Schedule Figure 1. Production scheduling for make-to-order assembly. Figure 1 shows a hierarchical framework (Schneeweiss 1999, Sawik 25a) for production scheduling in a make-to-order assembly. First, at the top level, long-term (e.g. monthly) master scheduling (Hopp and Spearman 1996, Sawik 25b) allocates customer orders among planning periods (e.g. days) to optimize some due-date-related criterion. Then, at the medium level, short-term (e.g. daily) lot streaming (also called lot splitting or batching; Dauzere-Peres and Lasserre 1997) divides production orders into production lots, each to be processed as a separate job. Finally, at the base level, detailed machine assignment and scheduling translates subsets of production lots into short-term (e.g. daily) machine scheduling to maximize throughput or, equivalently, to minimize makespan (Sawik 22, Kaczmarczyk et al. 24). The purpose of this paper is to present integer programming formulations for multi-objective, long-term master production scheduling and for short-term machine assignment and scheduling in make-to-order assembly with customer due-date-related performance measures. In master production scheduling a set of customer orders for different products and with various due-dates is allocated among planning periods so that each order is fully completed in exactly one period, and machines are selected for assignment in every period. The master scheduling objective is to minimize, in decreasing order of importance, the number of unscheduled, tardy and early orders and to level machine assignments over a planning horizon. The basic integer programming formulation is strengthened by the addition of cutting constraints that are derived by relating the demand on required capacity to available capacity for each processing stage and each subset of orders with the same due-date. The customer orders assigned to one planning period are split into production lots, each to be processed as a separate job. The objective of short-term machine assignment and scheduling is to determine the assignment of production lots to machines to minimize the number of operators required to attend the machines as a primary optimality criterion and to minimize makespan

3 Hierarchical approach to production scheduling in make-to-order assembly 83 as a secondary criterion. The mixed integer program proposed for machine scheduling is also strengthened by the addition of cutting constraints derived for batch scheduling. The proposed formulations have been applied to production scheduling in a hybrid flow shop with multi-capacity machines. The formulations, however, are quite versatile and can be applied for production scheduling in various make-to-order manufacturing environments. The integer programming approach has been widely used in the literature on production planning and scheduling (Nemhauser and Wolsey 1988, Shapiro 1993, Silver et al. 1999). For example, an integer goal programming formulation for master scheduling with a due-date-related criterion in make-to-order manufacturing has been presented by Markland et al. (199), a mixed integer program for lot streaming in job-shop scheduling has been proposed by Dauzere-Peres and Lasserre (1997), a mixed integer program for the problem of coordination fabrication and assembly in make-to-order production has been proposed by Kolisch (2), and an integer programming formulation for production scheduling with various due-date-related criteria is given by Sawik (25a). In industrial practice, however, the application of integer programming for production scheduling is limited, in particular for scheduling in a make-to-order manufacturing environment. For example, a hierarchical approach and integer programs for production scheduling in a make-to-order company have been presented by Carravilla and Pinho de Sousa (1995); however, computational results are based on the developed heuristics. An integer goal programming formulation for production scheduling with a due-date-related criterion has also been presented by Markland et al. (199), and the focus is again on the application of heuristic approaches. Bradley and Arntzen (1999) apply mixed integer programming for the simultaneous production, capacity and inventory planning in seasonal demand environments. A major disadvantage of the integer programming approach for scheduling is the need to solve large mixed integer programs. However, recent advances in integer programming have resulted in commercial software that can handle large mixed integer programs and find a proven optimal solution within a reasonable computation time, in particular if a strong formulation is applied (Kaczmarczyk et al. 24, Sawik 25b). In this paper, such formulations are proposed and their applications are illustrated with a set of numerical examples modelled on a real-world problem of production scheduling in a distribution centre for high-tech products. From a literature review, one can see that the problem of the multi-objective scheduling of a hybrid flow shop in make-to-order assembly, where, at the top level, customer orders have to be assigned to planning periods subject to scarce processing capacity, and where machine assignment and scheduling have to be performed for every period at the base level such that the customer service level is maximized and production resources are best utilized, has not been treated thus far. The major contribution of this paper is that it proposes a hierarchical framework with integer programming formulations for the multi-objective production scheduling in a make-to-order manufacturing environment, where both maximization of the customer service level and utilization of the production resources are integrated in the objective function. In addition, new cutting constraints are derived for the integer programs and the approach is tested computationally on a set of instances modelled after a real-world production scheduling problem.

4 84 T. Sawik The paper is organized as follows. Section 2 provides a description of maketo-order production scheduling in a hybrid flow shop with multi-capacity machines. An integer programming formulation for long-term master scheduling is presented in section 3. Model enhancements and additional cutting constraints are described in section 4. Two-level machine assignment and scheduling and the corresponding integer programs are proposed in section 5. Numerical examples modelled after a real-world make-to-order assembly system and computational results are provided in section 6. Conclusions are presented in the last section. 2. Problem description The production system under study is a hybrid flow shop (Bla_zewicz et al. 1994) that consists of m processing stages in series (see figure 2) and an output buffer of limited capacity B to hold completed products before delivery to the customer. Typically, the capacity B of the output buffer is not large in order to maintain low inventory costs and to limit the early completion of customer orders before the customerrequired shipping dates (Hall et al. 1998). Each stage i 2 I ¼f1,..., mg is made up of m i 1 parallel, multi-capacity machines. Let J i ¼f1,..., m i g be the set of indices of the parallel machines in stage i and denote by s ij 1 the capacity (number of products that can be processed simultaneously) of machine j 2 J i in stage i 2 I. Assume that in every stage i the parallel machines j 2 J i are ordered according to non-increasing capacity s ij, i.e. s i1 s i2 s imi. There is no limitation on storage space between the machines, so that finite intermediate buffers need not be considered for short-term scheduling, unlike in many other hybrid flow shop environments (Kaczmarczyk et al. 24). The planning horizon consists of h planning periods (e.g. working days) of equal length L (e.g. hours or minutes). Let T ¼f1,..., hg be the set of planning periods. In the system, various types of products are produced in a make-to-order environment responding directly to customer orders. Let G be the set of customer orders, each to be fully completed in exactly one planning period. Each order g 2 G is described by a triple ðr g, d g, o g Þ, where r g is the order ready period (e.g. the earliest period of material availability), d g is the customer due-date (e.g. customer-required shipping date), and o g is the quantity of ordered products. Each order requires processing in various processing stages; however, some orders may bypass some stages. Let G i G be the subset of orders that must be s 11 s 21 s m1 s 11 s 22 s m2 B s 1m1 s 2m2 s mmm Stage 1 Stage 2 Stage m Figure 2. A flexible assembly line with multi-capacity machines and finite output buffer.

5 Hierarchical approach to production scheduling in make-to-order assembly 85 processed in stage i, and let p ig > be the processing time in stage i of each product in order g 2 G i. The orders are processed and transferred among the stages in lots of various size and each lot is to be processed as a separate job. Let b g be the size of the production and transportation lot for order g. The size of the production and transportation lots is a fixed parameter and not a decision variable to be optimized in the framework of a lot-sizing model (Tempelmeier and Derstroff 1996, Kimms 1997, Wolsey 1997). Set-ups and machine changeovers between different product types are negligible, and the lot size b g depends only on product type and the capacity of the containers used for product storage and transportation. For each type of product, parameter b g is determined so that the processing of each lot b g as a separate job is fully completed in all stages of the hybrid flow shop in exactly one planning period, i.e. P i2i p ikdb g =s imi e < L. Processing of products in some stages additionally requires human operators and the number of operators required per machine depends on the type of product. The objective of long-term master production scheduling is to assign customer orders to planning periods and to select machines for assignment in every period so as to minimize, in decreasing order of importance, the number of unscheduled, tardy and early orders and to level machine assignments over the planning horizon. Leveling machine assignments and, in turn, balancing the number of people required for machine attendance, results in lower operational costs (Neumann and Zimmermann 1999). For each planning period, given the set of production lots of all customer orders assigned to this period and the set of machines selected for assignment in this period, the objective of short-term machine assignment and scheduling is to determine an assignment of the production lots to machines and a detailed processing schedule to minimize the number of people as a primary optimality criterion and to minimize makespan as a secondary criterion. 3. Master production scheduling In this section the integer programming model is presented for master scheduling where each customer order must be fully completed in exactly one planning period. Due to the discrete nature of orders it is unlikely that any time period will be filled exactly to capacity. As a result, some orders may be left unscheduled during the planning horizon. The orders that cannot be scheduled in periods 1 through h due to insufficient capacity are assigned, at a significant penalty, to a dummy planning period h ¼ h þ 1 with infinite capacity. Let T ¼f1,..., h, h þ 1g be the enlarged set of planning periods with a dummy period h ¼ h þ 1 included. A weighting approach is applied, where the quad-objective scheduling problem is reduced to a single objective problem by introducing the weights 1, 2, 3 and 4 representing the relative importance of the four objective functions to be minimized: respectively the number of unscheduled orders, tardy orders, early orders, and the maximum machine assignments in a single planning period (to level machine assignments over the horizon). The integer program for master production scheduling is presented below (for a definition of the decision variables, see table 1).

6 86 T. Sawik Table 1. Master production scheduling: notation. Indices i j g t Input parameters r g, d g, o g b g p ig s ij B D G i Gðd Þ G i ðd Þ L L it T Processing stage, i 2 I ¼f1,..., mg Parallel machine in stage i, j 2 J i ¼f1,..., m i g Customer order, g 2 G Planning period, t 2 T ¼f1,..., hg Ready period, due-date, size of order g Production and transportation lot for order g Processing time in stage i of each product in order g Capacity of machine j in stage i (number of products that can be processed simultaneously) Output buffer capacity fd g : g 2 Gg, set of distinct due-dates of all customer orders fg 2 G : p ig > g, subset of customer orders to be processed in stage i fg 2 G : d g ¼ dg, subset of customer orders with identical due-date d G i \ Gðd Þ, subset of customer orders with due-date d to be processed in stage i Length of each planning period Length of time available in period t on each machine in stage i T [fhþ1g, h þ 1, a dummy period with infinite capacity Decision variables v gt 1, if order g is performed in period t; otherwise v gt ¼ (order assignment variable) w ijt 1, if machine j 2 J i in stage i 2 I is selected for assignment in period t; otherwise w ijt ¼ (machine selection variable) M max Maximum number of machines selected for assignment in a single planning period 3.1 Model MPS: master production scheduling Minimize 1 v ghþ1 þ 2 v gt þ 3 v gt þ 4 M max, g2g g2g:t>d g g2g:t<d g ð1þ subject to 1. Order assignment constraint: each customer order is assigned to exactly one planning period, v gt ¼ 1, g 2 G: ð2þ t2t 2. Machine assignment constraints: in every stage the machines are selected for assignment in the order of non-increasing capacity,

7 Hierarchical approach to production scheduling in make-to-order assembly 87 in every period the number of machines selected for assignment in each stage is not greater than the maximum number of assigned transfer lots, in every period the number of machines selected for assignment cannot exceed the maximum number of machine assignments to be minimized, in every period demand on capacity in each processing stage cannot be greater than the total capacity of the machines selected for assignment in this period, w ijþ1t w ijt, i 2 I, j 2 J i, t 2 T : j < m i, ð3þ w ijt do g =b g ev gt, j2j i g2g i i 2 I, t 2 T, ð4þ w ijt M max, t 2 T, ð5þ i2i, j2j i p ig o g v gt L it s ij w ijt, g2g i j2j i i 2 I, t 2 T: ð6þ 3. Output buffer capacity constraint: in every period the total number of products completed before their due-dates and waiting for shipping to customers cannot exceed the output buffer capacity, g2g, 2T:r g t<d g o g v g B, t 2 T: ð7þ 4. Variable integrality constraints: v gt 2f, 1g, g 2 G, t 2 T : t r g, ð8þ w ijt 2f, 1g, i 2 I, j 2 J i, t 2 T, ð9þ M max, integer, ð1þ where de is the least integer not less than (do g =b g e denotes the number of production lots of customer order g). The objective of the master scheduling problem is to minimize the weighted sum of unscheduled, tardy and early orders and to level machine assignments over the planning horizon. In objective function (1), unscheduled orders are penalized at a higher rate than tardy orders and tardy orders are penalized at a higher rate than early orders and maximum machine assignments, i.e. 1 > 2 > 3 4. The left-hand side of capacity constraint (6) denotes the total processing time required for completing, in stage i, all customer orders assigned to period t, whereas the right-hand side of (6) is the total available processing time on all machines selected for assignment in stage i in period t. The cutting constraints (3) and (4) reduce the computational effort required to find the proven optimal solution to the integer program MPS.

8 88 T. Sawik The amount L it of time available on each machine in stage i in period t accounts for the flow shop configuration of the production system and the transfer lot sizes; L it is bounded as follows: L max g2g i :r g td g L min db g =s l1 ep lg! max l2i:l<i g2g i :r g td g g2g i :r g td g db g =s l1 ep lg! min l2i:l<i! db g =s l1 ep lg L it l2i:l>i g2g i :r g td g db g =s l1 ep lg!: l2i:l>i A correct estimation of the L it used for master scheduling is crucial to ensure the feasibility of daily schedules at the detailed machine assignment and scheduling level. A too large value of L it may lead to unfeasible daily schedules, i.e. some orders assigned to the same planning period will not be completed during a single day scheduling horizon. ð11þ 4. Model enhancements A necessary condition for problem MPS to have a feasible solution with all customer orders completed during the planning horizon is that, for each processing stage i 2 I, the total demand on capacity does not exceed the total capacity available, that is P g2g p igo g P t2t L 1, 8i 2 I: ð12þ it Pj2J i s ij If the customer orders were arbitrarily divisible and could be completed during more than one planning period, all orders were ready for completion at the beginning of the horizon and the output buffer capacity was unlimited to allow for any earliness in completing the orders, then the feasibility condition (12) would be sufficient. Due to the discrete nature of customer orders, however, it is possible that some time periods will not be filled exactly to capacity. As a result the necessary condition (12) is not sufficient for all orders to be scheduled during the planning horizon. Nevertheless, the integer program MPS can be strengthened by the incorporation of additional cutting constraints on decision variables. It is possible to generate such constraints by relating for each due-date the local demand on required capacity to available capacity and the cumulative demand on required capacity to available cumulative capacity. For each due-date d 2 D the local demand on capacity in processing stage i required to complete all orders on their due-dates is p ig o g : g2g i ðd Þ If the local demand on capacity in stage i exceeds the maximum available capacity L id Pj2J i s ij, then some orders with due-date d must be re-allocated to earlier or later periods that have excess capacity in stage i. For each due-date d 2 D denote by lcr(d ) the local capacity ratio defined as lcrðd Þ¼maxðlcriði, d ÞÞ, d 2 D, ð13þ i2i

9 Hierarchical approach to production scheduling in make-to-order assembly 89 where lcriði, d Þ is the local capacity ratio for due-date d with respect to processing stage i, i.e. the ratio of the total processing time required to complete in stage i all the orders with due-date d to the available processing time: P g2g lcrði, d Þ¼ i ðd Þ p igo g P, d 2 D, i 2 I: ð14þ L id j2j i s ij Denote by D l and D l1, respectively, the subset of locally satisfiable, unsatisfied due-dates D l ¼fd2D : lcrðd Þ1g, ð15þ D l1 ¼fd2D : lcrðd Þ > 1g: ð16þ For each processing stage and each locally unsatisfied due-date, at least one order should be moved to another period to reduce local demand on capacity with respect to that stage, at least to the level of the available capacity. Hence, some orders with due-dates in periods with the local demand on capacity exceeding the available capacity are potentially subject to earliness or tardiness. 4.1 Cutting constraints on the assignment of orders with locally unsatisfied due-dates v gt 1, d 2 D l1, i 2 I : lcriði, d Þ > 1, ð17þ g2g i ðd Þ, t2t:t6¼d p ig o g v gt ðlcriði, d Þ 1ÞL id s ij, d 2 D l1, i 2 I : lcriði, d Þ > 1: ð18þ g2g i ðd Þ, t2t:t6¼d j2j i Constraint (17) ensures that, for each due-date d 2 D l1 and each processing stage i, at least one order locally unsatisfied with respect to that stage will not be completed on the due-date. In addition, (18) guarantees that, for each due-date d 2 D l1 and each processing stage i, the total work of orders locally unsatisfied with respect to that stage and moved to other periods is not less than the surplus of demand on capacity with respect to the available capacity. Furthermore, for each processing stage and each locally satisfied due-date the work inflow of orders with other due-dates less the work outflow of orders with this due-date cannot exceed the slack capacity. 4.2 Flow balance cutting constraint for locally satisfied due-dates p ig o g v gd p ig o g v gt ð1 lcriði, d ÞÞL id s ij, j2j i d 2D, g2g i ðd Þ:d 6¼d g2g i ðd Þ, t2t:t6¼d d 2 D l, i 2 I: Constraint (19) ensures that, for each locally satisfied due-date and each processing stage, the total processing time of the orders with other due-dates and assigned to this period is not greater than the slack capacity plus the total processing time of the orders moved early or late from this period. ð19þ

10 81 T. Sawik A due-date d 2 D can be satisfied if, for each processing stage i and any period t < d, the cumulative demand on capacity of all orders with due-dates not greater than d and ready periods not less than t does not exceed the cumulative capacity available in this stage in periods t through d. For each due-date d 2 D denote by ccr(d ) the cumulative capacity ratio defined as ccrðd Þ¼maxðccriði, d ÞÞ, d 2 D, ð2þ i2i where ccriði, d Þ is the cumulative capacity ratio for due-date d with respect to processing stage i P! ccrði, d Þ¼ max t2t:t<d f2d, g2gð f Þ:tr g fd p igo g P 2T:td L i P j2j i s ij, d 2 D, i 2 I: ð21þ Denote by D c and D c1, respectively, the subset of globally satisfiable, unsatisfied due dates D c ¼fd 2 D : ccrðd Þ1g, ð22þ D c1 ¼fd 2 D : ccrðd Þ > 1g: ð23þ If the subset D c1 is empty, i.e. for each due-date the cumulative demand on capacity is not greater than the available cumulative demand, then all orders can be completed by their due-dates. Otherwise, only the orders with due-dates for which both the local demand on capacity does not exceed the available local capacity and the cumulative demand on capacity does not exceed the available cumulative capacity can be completed by their due-dates. The due-dates both locally and globally satisfiable are referred to as satisfiable due-dates. The subset D of satisfiable due-dates is defined as D ¼ D l \ D c ¼fd2D : ccrðd Þ1 and lcrðd Þ1g: ð24þ The cutting constraints for orders with satisfiable due-dates are presented below. 4.3 Cutting constraints on the assignment of orders with satisfiable due-dates v gt ¼, g 2 Gðd Þ, ifd c1 ¼ 1 then d 2 D else d 2 D, t 2 T : t > d g, ð25þ d g v gt ¼ 1, g 2 Gðd Þ, if D c1 ¼ 1 then d 2 D else d 2 D: ð26þ t¼r g Constraints (25) and (26) guarantee non-delayed completion of each order with a satisfiable due-date. In contrast, the due-dates with local and cumulative demand on capacity exceeding the available capacity are referred to as potentially unsatisfied due-dates. The subset D1 of potentially unsatisfied due dates is defined as D1 ¼ D l1 \ D c1 ¼fd 2 D : ccrðd Þ > 1 and lcrðd Þ > 1g: ð27þ In order to meet a potentially unsatisfied due-date d 2 D1, some orders with earlier due-dates should be moved forward (delayed) to periods with slack capacity.

11 Hierarchical approach to production scheduling in make-to-order assembly 811 In particular, for each stage i at least one order with a locally unsatisfied due-date not greater than the earliest potentially unsatisfied due-date firstðd1þ ¼ minfd : d 2 D1g will be tardy and re-allocated to some period in the set of locally satisfiable due-dates after firstðd1þ to reduce the cumulative demand on capacity with respect to stage i in periods 1 through firstðd1þ. The cutting constraints on decision variables for orders with potentially unsatisfied due-dates are presented below. 4.4 Cutting constraints on the assignment of orders with potentially unsatisfied due-dates v gt 1, i 2 I : lcriði, d Þ > 1, g2g i ðd Þ, d2d l1, t2d l :dfirstðd1þ<t ccriði, firstðd1þþ > 1: Constraint (28) ensures that at least one order locally unsatisfied with respect to stage i such that ccriði, firstðd1þþ > 1 and with a due-date not greater than firstðd1þ will be tardy and moved to some period t > firstðd1þ in the set of locally satisfiable due-dates. In addition, the following bounds on the maximum machine assignments M max can be added to model MPS: M M max M: ð29þ The lower bound M and upper bound M on M max are defined as follows: M ¼ ( min!ðiþ :!ðiþ s ij ) p ig o g =hl, ð3þ i2i j¼1 g2g i ( ( ( ))) M ¼ max min m i, min ðid Þ : ðid Þ s ij p ig o g =L i : ð31þ d2d i2i j¼1 g2g i :g2gðd Þ The lower bound M is calculated assuming that all orders are evenly distributed over a monthly planning horizon with full processing time L available in each stage in each daily planning period. The upper bound M assumes that each order is completed exactly on its due-date with limited processing time L i available in every period in each stage i. The cutting constraints (17) (19), (25), (26), (28) and (29) should be added to model MPS to reduce the computational effort required to find an optimal solution. ð28þ 5. Machine assignment and scheduling In this section, two-level machine assignment and scheduling is proposed (Sawik 22) and the corresponding integer programming formulations are presented. The customer orders assigned to one planning period are split into production lots, each to be processed as a separate job. Each order g 2 G is divided into do g =b g e production lots: bo g =b g c lots, each of size b g and at most one lot of size o g bo g =b g cb g (bc is the greatest integer not greater than ).

12 812 T. Sawik Figure 3. A two-level machine assignment and scheduling. Let K ¼f1,..., ng be the set of indices of all lots, and let ~ b k be the size of lot k 2 K, where n ¼ P g2g:v gt ¼1 do g=b g e is the total number of production lots of all customer orders assigned to one planning period t 2 T. Each production lot must be processed in various stages, however some lots may bypass some stages. Let e k 2 I and f k 2 I, f k e k, be the number of first and last processing stages for lot k, respectively. Denote by K i K the subset of lots that must visit stage i, and let ~p ik > be the processing time in stage i of each product in lot k 2 K i. All products in a lot are processed consecutively on the same machine and transported together between the machines, and therefore each lot can be scheduled as a separate job. The actual processing time in stage i of lot k 2 K i depends on the capacity s ij of parallel machine j 2 J i selected for processing and is ~p ik d ~ b k =s ij e. Processing of production lot k on machine j in stage i requires q ijk operators to be assigned. For each planning period, given the set of production lots of all customer orders assigned to this period and the set of machines selected for assignment in this period, the objective of short-term machine assignment and scheduling is to determine the assignment of the production lots to machines and a detailed processing schedule to minimize the number of people as a primary optimality criterion and to minimize makespan as a secondary criterion. The two-level machine assignment and scheduling is shown in figure 3. First, at the top level, the machine assignment is solved to best allocate a daily subset of production lots among the selected machines and to minimize the number of people required to complete the daily production plan. Then, at the base level, the shortest assembly schedule is found for prefixed assignment of the production lots. Figure 3 also illustrates the linkage between the master scheduling and the machine assignment. 5.1 Machine assignment The integer program for machine assignment is presented below (for notation, see table 2).

13 Hierarchical approach to production scheduling in make-to-order assembly 813 Table 2. Machine assignment and scheduling: notation. Indices i j k Input parameters a ij Processing stage, i 2 I ¼f1,..., mg Parallel machine in stage i, j 2 J i ¼f1,..., m i g Production lot, k 2 K ¼f1,..., ng 1, if in stage i machine j has been selected for assignment at the master scheduling level; otherwise a ij ¼ ~b k Size of lot k e k, f k First and last processing stage for lot k ~p ik Processing time in stage i of each product in lot k q ijk Number of operators required for processing of lot k on machine j in stage i K i Subset of production lots to be processed in stage i L Length of the daily scheduling horizon L i Length of time available on each machine in stage i Decision variables C max Schedule length P sum Total number of people required for assignment c ik Completion time of production lot k in stage i x ijk 1, if production lot k is assigned to machine j 2 J i in stage i 2 I; otherwise x ijk ¼ (assignment variable) y kl 1, if lot k precedes lot l; otherwise y kl ¼ (sequencing variable) Model A: machine assignment Minimize P sum ¼ q ijk x ijk, ð32þ i2i, j2j i, k2k i subject to 1. Product assignment constraints: in every stage, each lot is assigned to exactly one selected machine, at least one lot is assigned to every selected machine, the total processing time of all lots assigned to each machine does not exceed the processing time available during the scheduling horizon, x ijk ¼ 1, i 2 I, k 2 K i, ð33þ j2j i x ijk a ij, i 2 I, j 2 J i, k 2 K i, ð34þ x ijk a ij, i 2 I, j 2 J i, ð35þ k2k i ~p ik d b ~ k =s ij ex ijk a ij L i, i 2 I, j 2 J i : ð36þ k2k i 2. Variable integrality constraint: x ijk 2f, 1g, i 2 I, j 2 J i, k 2 K i : ð37þ The objective (32) is to minimize the number of people assignments.

14 814 T. Sawik The duration L i of time available on each machine in stage i can be calculated as! L i ¼ L ð1 Þmax ~p hk d b ~ k =s h1 e k2k i h2i:h<i! ð1 Þmax ~p hk d b ~ k =s h1 e k2k i h2i:h>i!! min ~p hk d b ~ k =s h1 e min ~p hk d b ~ k =s h1 e, ð38þ k2k i k2k i h2i:h<i h2i:h>i where 1. The coefficient reflects the idle time of the machine waiting for the first production lot from upstream stages and the machine idle time during processing of the last production lot at downstream stages. 5.2 Machine scheduling The mixed integer program for machine scheduling with prefixed machine assignments is presented below (for notation, see table 2), where the formulation is a priori improved by the addition of cutting constraints Model SjA: scheduling with prefixed machine assignment Minimize C max, ð39þ subject to 1. Product non-interference constraints: no two different production lots that have been assigned to the same machine can be processed simultaneously, c ik þ Ly kl c il þ ~p ik d b ~ k =s ij e, i 2 I, j 2 J i, k, l 2 K i : k < l, x ijk x ijl ¼ 1, c il þ Lð1 y kl Þc ik þ ~p il d b ~ l =s ij e, i 2 I, j 2 J i, k, l 2 K i : k < l, x ijk x ijl ¼ 1: 2. Product completion constraints: each lot must be completed on the selected machines in the first stage, and in all downstream stages of its processing route, each production lot bypasses stages not in its processing route, the completion time of each lot in its first processing stage cannot be greater than the sum of the processing times of all lots assigned to the same machine, ð4þ ð41þ c ek k ~p ek kd b ~ k =s ek je, k 2 K, j 2 J ek : x ek jk ¼ 1, ð42þ c ik c i 1k ~p ik d b ~ k =s ij e, i 2 I, j 2 J i, k 2 K i : i > e k, x ijk ¼ 1, ð43þ c ik ¼ c i 1k, i 2 I, k 2 K : i > 1, ~p ik ¼, ð44þ c ek k ~p ek ld b ~ l =s ek je, j 2 J ek, k 2 K : x ek jk ¼ 1: ð45þ l2k ek :x ek jl¼1

15 Hierarchical approach to production scheduling in make-to-order assembly Makespan constraints: each lot must be completed sufficiently early to have all of its remaining tasks finished by the end of the schedule, the maximum completion time of the last task determines the minimized makespan, the makespan is not less than the maximum workload of a machine and must not exceed the scheduling horizon, c ik þ ~p hk d b ~ k =s hj ec max, i 2 I, k 2 K i : i < f k, ð46þ h2i, j2j h :h>i, x hjk ¼1 c fk k C max, k 2 K, ð47þ! ~p ik d ~ b k =s ij eþmin ~p hk d b ~ k =s h1 e k2k k2k i :x ijk ¼1 i h2i:h<i! þ min ~p hk d b ~ k =s h1 e C max L, i 2 I, j 2 J i : ð48þ k2k i h2i:h>i 4. Variable non-negativity and integrality constraints: c ik, i 2 I, k 2 K i, ð49þ C max, ð5þ y kl 2f, 1g, k, l 2 K : k < l: ð51þ The cutting constraints (45), (46) and (48) may help to reduce the computational effort required to find the proven optimal solution to the mixed integer program SjA. 6. Computational examples In this section numerical examples and computational results are presented to illustrate the possible applications of the proposed approach. The examples are modelled after a real-world distribution centre for high-tech products (figure 4), where finished products are assembled for shipping to customers. The distribution centre can be modelled as a hybrid flow shop consisting of six processing stages with parallel multi-capacity machines and a finite capacity output buffer. In the distribution centre, 1 product types of three product groups are assembled. The processing stages are the following: material preparation stage, where all materials required for the assembly of each product are prepared, postponement stage, where products for some orders are customized, three flashing/flexing stages, one for each group of products, where the required software is downloaded, and a packing stage, where products and required accessories are packed for shipping. Each flashing/flexing stage consists of many parallel machines of different capacity. After all processing stages a finite output buffer is introduced to hold completed products before delivery to the customer. The finite output buffer limits the possibility of order re-allocation to earlier periods with excess capacity, which can affect tardiness. Customer orders require processing in at most four stages: material preparation stage, postponement stage, one flashing/flexing stage, and packing

16 816 T. Sawik Flexing Group 1 2 machines Material Preparation Postponement Flexing Group 2 Packing 1 machines 1 machines 2 machines 1 machines Flexing Group 3 2 machines Figure 4. Distribution centre. stage (see figure 4). However, some orders do not need postponement. Each customer order must be completed in one day. Customer orders are split into production lots of fixed sizes, each to be processed as a separate job. The processing time of each production lot in the flashing/flexing stage depends on the capacity of the machine selected for the assignment. A daily production consists of various production lots from different customer orders and each lot requires a different processing route. In the computational experiments, four test problems are constructed with the following four regular patterns of demand (see figure 5), which best model the actual demand patterns that are encountered in the distribution centre: 1. Increasing, with demand skewed towards the end of the planning horizon. 2. Decreasing, with demand skewed towards the beginning of the planning horizon. 3. Unimodal, where demand peaks in the middle of the planning horizon and falls under available capacity in the first and last days of the horizon. 4. Bimodal, where demand peaks at the beginning and at the end of the planning horizon and slumps in mid-horizon. Pattern 1 may require some orders to be completed earlier to reduce demand on capacity at the end of the planning horizon, whereas for pattern 2 some orders are moved later in time if at the beginning of the planning horizon demand on capacity exceeds available capacity. Patterns 3 and 4 may require that orders be moved both early and late to reach feasibility, if demand on capacity exceeds available capacity in mid-horizon, at the beginning or at the end of the planning horizon. A brief description of the production system, production process, products and customer orders is given below. 1. Production system. Six processing stages 1 machines in material preparation stage 1, 1 machines in postponement stage 2, 2 parallel machines in flashing/flexing stage 3

17 Hierarchical approach to production scheduling in make-to-order assembly Increasing Demand Pattern Decreasing Demand Pattern Unimodal Demand Pattern Bimodal Demand Pattern Product 1 Product 9 Product 8 Product 7 Product 6 Product 5 Product 4 Product 3 Product 2 Product 1 Product 1 Product 9 Product 8 Product 7 Product 6 Product 5 Product 4 Product 3 Product 2 Product 1 Product 1 Product 9 Product 8 Product 7 Product 6 Product 5 Product 4 Product 3 Product 2 Product 1 Product 1 Product 9 Product 8 Product 7 Product 6 Product 5 Product 4 Product 3 Product 2 Product 1 Figure 5. Demand patterns.

18 818 T. Sawik for product group 1, 2 parallel machines in flashing/flexing stage 4 for product group 2, 2 parallel machines in flashing/flexing stage 5 for product group 3, and 1 parallel machines in packing stage 6 capacity of multi-capacity machines in stages 3, 4 and 5: s 3j ¼ 8, j ¼ 1,..., 2, s 4j ¼ 4, j ¼ 1,..., 2, s 5j ¼ 2, j ¼ 1,...,2 workforce requirements: in material preparation stage 1 and postponement stage 2 each machine requires one operator, each flashing/flexing stage 3, 4 and 5 requires 1 operators for all machines, in packing stage 6, the number of operators depends on the type of packed product and ranges from four to 1 operators per machine. output buffer capacity B ¼ 45 products 2. Products. 1 product types of three product groups, each to be processed on a separate group of flashing/flexing machines 1 customer orders, each consisting of several suborders (customerrequired shipping volumes). Every suborder has a different volume ranging from five to 52 products, an identical ready period and a different due-date, and each suborder is to be completed in a single day. The total number of suborders is 812, 812, 74 and 671, respectively, for the increasing, decreasing, unimodal and bimodal demand patterns production and transfer lot sizes: 5, 5, 75, 25, 25, 25, 5, 5, 75 and 25, respectively, for product types 1, 2, 3, 4, 5, 6, 7, 8, 9 and 1 3. The processing times (in seconds) for product types are shown in table Planning horizon: h ¼ 3 days, each of length L ¼ 18 min. Notice that the suborders in the computational examples play the role of orders in the mathematical formulation. Now, the master scheduling objective is to determine an assignment of customer suborders over the planning horizon. In the example presented below, all customer orders are assumed to be ready for completion at the beginning of the planning horizon, i.e. r g ¼ 1, g 2 G, and the order due-dates are distributed over the entire horizon according to the demand patterns Table 3. Processing times (in seconds) for product types. Stage Product type

19 Hierarchical approach to production scheduling in make-to-order assembly 819 shown in figure 5. In the objective function (1) of model MPS the weights used for unscheduled orders, tardy orders, early orders and maximum machine assignments were 1 ¼ 9, 2 ¼ 1, 3 ¼ 1 and 4 ¼ 1, respectively. Figure 6 presents the master schedules for the example with various demand patterns, where, for each planning period, detailed values of the decision variables have been replaced by aggregate values of the total production. The corresponding machine assignments are shown in figure 7. The results indicate that, for all demand patterns with the exception of increasing demand, the optimal production schedules are directly driven by the demand for products at the end of the planning horizon where the demand on capacity does not exceed available capacity. This, however, results in greater fluctuations of machine assignments at the end of the planning horizon for all demand patterns with the exception of increasing demand. In contrast, the machine assignments are leveled as long as the demand on capacity exceeds available capacity, i.e. at the beginning and end of the planning horizon, respectively, for the decreasing and increasing demand patterns, and in mid-horizon for the unimodal demand pattern. As an example of detailed machine assignment and scheduling, a daily machine schedule was found for the increasing demand pattern and day 15 of the master schedule. The customer orders (suborders) for various product types assigned to day 15 were split into 34 production lots of different sizes to be scheduled during a daily horizon (table 4). The Gantt chart in figure 8 shows the detailed machine schedule for day 15. In figure 8, MP, PT, F1, F2, F3 and PK indicate Material Processing, Postponement, three groups of Flashing/Flexing, and Packing, respectively. In the Gantt chart, different numbers indicate production lots of 1 different product types. The daily schedule length is C max ¼ 149 min. The characteristics of the integer programs for the example with increasing demand pattern and the solution results are summarized in table 5. The size of the mixed integer programming models for the example problem is represented by the total number of variables, Var., the number of binary variables, Bin., the number of constraints, Cons., and the number of non-zero elements in the constraint matrix, Nonz. The last two columns of table 5 present the optimal values of the objective functions and the CPU time in seconds required to prove optimality of the solution. The computational experiments were performed using AMPL programming language and the CPLE v. 9.1 solver on a laptop with a Pentium IV processor running at 1.8 GHz. 6.1 Material availability In the previous examples the materials required for completing all customer orders were assumed to be available at the beginning of the monthly planning horizon, i.e. the ready periods for all orders were identical and equal to the first day of the month. In this subsection the impact of material availability on master scheduling is analysed by reducing the length of the interval between the order due-date d g and the ready period r g. In the computational experiments the ready period is r g ¼ maxð1, d g 3Þ for each order g, i.e. the length of the interval between the order due-date and the ready period is not greater than 3 days. When order ready periods and due-dates are closer, the limited order earliness due to the limited

20 82 T. Sawik Number of Products Number of Products Increasing Demand Decreasing Demand Demand Production Demand Production Number of Products Number of Products Unimodal Demand Bimodal Demand Demand Production Demand Production Figure 6. Master schedules for various demand patterns.

21 Hierarchical approach to production scheduling in make-to-order assembly 821 Machines Machines Machines Machines Decreasing Demand Increasing Demand Unimodal Demand Bimodal Demand Figure 7. Machine assignments for various demand patterns.

22 822 T. Sawik Table 4. The customer orders (suborders) for various product types assigned to day 15 were split into 34 production lots of different sizes to be scheduled during a daily horizon. Order No Product type No Lot No Lot size material availability restricts the re-allocation of orders to the earlier periods with surplus capacity, which may result in a greater number of tardy orders. On the other hand, both the input inventory of raw materials waiting for processing in the system and the output inventory of finished products completed before the due-dates and waiting for delivery to the customers can be reduced when order ready periods and due-dates are closer. The characteristics of the integer program for the test problems and the solution results are summarized in table 6 and 7, respectively, for the basic and enhanced integer program MPS. The size of the integer programming model is represented by the total number of variables, Var., the number of binary variables, Bin., the number of constraints, Cons., and the number of non-zero coefficients, Nonz., in the constraint matrix. The last two (or three) columns of each table present the best solution values and the CPU time in seconds required to prove optimality of the solution (or % GAP after 36 s of CPU time, if optimality is not proven). Table 6 shows that, for the basic integer program, the CPLE solver was not always able to prove optimality within the allowed 36 s of CPU time; however, the best solutions were found much earlier than the time limit. On the other hand, table 7 indicates that when the integer program MPS was strengthened by the addition of the cutting constraints presented in section 4, the CPLE solver was capable of finding the proven optimal solutions for all test examples. For the example data the optimal master schedule (figure 9) is directly driven by the demand for products as long as the demand on capacity does not exceed

23 Hierarchical approach to production scheduling in make-to-order assembly 823 Figure 8. Machine schedule for day 15. available capacity. The results indicate that, for the increasing and decreasing demand patterns, the optimal production is very close to demand at the beginning and end, respectively, of the planning horizon, whereas for unimodal demand, the production level is very close to demand at the beginning and end of the horizon. The optimal machine and people assignments are leveled when demand on capacity exceeds available capacity and fluctuate when the production schedule is driven by the demand for products (figure 1).

24 824 T. Sawik Table 5. Computational results for the increasing demand pattern. Problem Var. Bin. Cons. Nonz. Solution values CPU a Master scheduling , 16, 41 b 47 Machine assignment P sum ¼ 53.2 Machine scheduling C max ¼ a CPU time in seconds required to prove optimality on a Pentium IV PC running at 1.8 GHz/CPLE v b Number of tardy suborders, early suborders and maximum machine assignments, respectively. Table 6. Computational results: basic integer programm MPS. Demand Ready period Var. Bin. Cons. Nonz. Solution values a CPU b GAP c Increasing r g ¼ , 16, Increasing r g ¼ maxð1, d g 3Þ , 31, Decreasing r g ¼ , 16, Decreasing r g ¼ maxð1, d g 3Þ , 22, Unimodal r g ¼ , 17, Unimodal r g ¼ maxð1, d g 3Þ , 43, Bimodal r g ¼ , 23, Bimodal r g ¼ maxð1, d g 3Þ , 4, a Number of tardy suborders, early suborders and maximum machine assignments, respectively. b CPU time in seconds to prove optimality on a Pentium IV PC running at 1.8 GHz/CPLE v c Percent GAP after 36 s of CPU time. Table 7. Computational results: enhanced integer program MPS. Demand Ready period Var. Bin. Cons. Nonz. Solution a values CPU b Increasing r g ¼ , 16, Increasing r g ¼ maxð1, d g 3Þ , 3, Decreasing r g ¼ , 15, Decreasing r g ¼ maxð1, d g 3Þ , 21, Unimodal r g ¼ , 17, Unimodal r g ¼ maxð1, d g 3Þ , 39, Bimodal r g ¼ , 23, Bimodal r g ¼ maxð1, d g 3Þ , 4, a Number of tardy suborders, early suborders and maximum machine assignments, respectively. b CPU time in seconds to prove optimality on a Pentium IV PC running at 1.8 GHz/CPLE v The results also indicate that both the input inventory of raw materials (figure 11) and the output inventory of finished products (figure 12) are smaller when materials are supplied only a few days before the customer-required shipping days. The length of the interval between material supply and due-date, however, must be sufficiently large to avoid additional tardy orders due to limited order earliness. In the example, the minimum length of the interval was found to be 3 days, for which the numbers of tardy orders were identical for both r g ¼ 1 and r g ¼ maxð1, d g 3Þ, g 2 G. The number of tardy orders increases as the interval length decreases and is less than 3 days.