Assembly line balancing: Two resource constrained cases

Transcription

1 Int. J. Production Economics 96 (2005) Assembly line balancing: Two resource constrained cases K.ur-sad A&gpak a, Hadi b, * a Department of Industrial Engineering, Faculty of Engineering, Gaziantep University, Gaziantep, Turkey b Department of Industrial Engineering, Faculty of Engineering and Architecture, Gazi University, Maltepe, Ankara, Turkey Received 15 April 2003; accepted 25 March 2004 Abstract In this paper, a new approach on traditional assembly line balancingproblem is presented. The goal of proposed approach is to establish balance of the assembly line with minimum number of station and resources. For this purpose, 0 1 integer-programming models are developed. These models are solved using GAMS-CPLEX mathematical programming software for a numerical example. r 2004 Elsevier B.V. All rights reserved. Keywords: Assembly line balancing; Resource constraint; 0 1 integer programming 1. Introduction The idea of line balancingwas first introduced by Bryton (1954) in his graduate thesis. The first published scientific study belonged to Salveson (1955). For more than 45 years, many studies were made on this subject. Duringthis period various new balancingproblem concepts such as U-type, two-sided, parallel, flexible assembly line, etc., and solution algorithms for those problems have been produced. The common thingfor all these problems is usingboth the operator and the machine in the most efficient way, at the same time providingflexibility in production. The purpose of the cases presented in this paper is to provide flexibility in production while increasingthe productivity. Nowadays assembly lines move towards cellular manufacturingin terms of variety of production. As a result of this, usage of special equipment and/or professional workers, which are able to perform more than one process, is increasing. In order to benefit from continuous productions advantages, these equipment and workers must be added to the line in a way by which high efficiency measures (maximum usage, minimum number of stations) can be achieved. *Correspondingauthor. Tel.: ; fax: address: hgokcen@gazi.edu.tr (H. G.ok@en) /$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi: /j.ijpe

2 130 K. A &gpak, H. Gök@en / Int. J. Production Economics 96 (2005) In this paper, assembly line balancingproblem, for which lots of variations has been examined till now, is studied with a different perspective. Efficient usage of resources that carry out assembly line operations has been targeted by balancing the line with the minimum number of stations. In fact, the issue of line balancingwith the minimum number of resources has always been a serious problem in industry. The paper is organized as follows: the second part will address the traditional assembly line balancing problem and an integer programming model developed for the resource constrained cases. In the third part, the solutions of traditional assembly line balancingand the resource constrained case on a numerical example are given, and discussed. In the last part, conclusion and suggestions for future studies are addressed. 2. Assembly line balancing problem 2.1. Traditional assembly line balancing problem and literature review Assembly lines consist of successive workstations at which products are processed. Workstations are defined as places where some tasks (operations) on products are performed. Products stay at each workstation for the cycle time (C), which corresponds to the time interval between successively completed units. While designing the line, the list of tasks to be done, task times required to perform each task and the precedence relations between them are analyzed. While the tasks are beinggrouped into stations based on this analysis, the following goals are regarded: 1. Minimization of the number of workstations for a given cycle time. 2. Minimization of cycle time for a given number of work stations (Baybars, 1986). A grouping which satisfies a determined goal, is called a balance. This problem is called the line balancing problem. Up to now, many optimal or heuristic techniques have been developed for the solution of this problem (Baybars, 1986; Ghosh and Gagnon, 1989; Erel and Sarin, 1998). The problem analyzed in this paper is similar to the studies which is called Assembly System Design Problems (ASDP) in the literature. ASDP has sought to optimize an economic criterion such as total cost with machine selection (Ghosh and Gagnon, 1989; Nicosia et al., 2002; Pinnoi and Wilhelm, 1998; Graves and Lamar, 1983; Yamada and Matsui, 2003). But in this study, workforce/machine assignment and line balancingproblem is analyzed with a different approach Resource constrained assembly line balancing (RCALB) problem This problem was met in a factory where some machinery is manufactured and assembled. In this factory, there are limited number of specific machines and limited number of workers that can use these machines. For example, there is a special cuttingtool that can cut metals in a specific width and shape. In this situation, the problem is assigning these tools/machines and workers to the stations. In assembly lines, where specific operation robots are used, the importance of simultaneously balancingof the resources and the assembly line can be understood better. At this point, RCALB problem should be defined for the line balancingliterature. This new problem deals with maximization of resource usage/minimization of number of resources used for a given C and maximum number of stations. The resources expressed in description may be workforce or machines.

3 K. A &gpak, H. Gök@en / Int. J. Production Economics 96 (2005) In practice, existence of limited amount of workforce/machine type, which can perform only some specific tasks in a system, is common (for example there may be only a specific number of workforce/ machine, which/who are only able to weld or only able to drill). The new model proposed for the assembly line balancingproblems foresees that while tasks are being assigned to the workstations, the tasks that can be performed by the same resource should be assigned to the same workstation. This model enables resource saving. During this classification, number of resources is minimized as much as possible. This resource constrained case also helps smoothingthe production flow. In part 3, traditional balance and the effect of proposed model are explained on a numerical example more clearly. In RCALB problem, we may come across with two cases dependingon resource type: Case 1: There is no task that can be assigned to different resources. In this case, the intersection of the sets of tasks that can be performed by resources is an empty set. For example let us assume an assembly line with 9 tasks; resource A can do tasks 1, 3, 5, 7, 9 and resource B can do tasks 2, 4, 6, 8. So there is no common task which can be performed by resource A and B. This case will be called as RCALB Type 1 problem. Case 2: There are some tasks that can be assigned to different resources, which means some tasks can be performed by alternative resources. For example, let us assume an assembly line with 9 tasks, resource A can perform tasks 1, 2, 5, 7, 9; resource B can perform tasks 2, 4, 5, 6, 8. As seen, tasks 2 and 5 can be performed by both resource A and B. This type will be called as RCALB Type 2. Traditional assembly line balancingproblem without resource constrained is given as follows (Patterson and Albracht, 1975; G.ok@en and Erel, 1998). Objective function: Minimum Xm max z j : Constraints: j¼1 ð1þ X L i ðx ij Þ¼1; j¼e i i ¼ 1; y; n; ð2þ X t i ðx ij ÞpC; iaw k j ¼ 1; y; m max ; ð3þ X L a X L b jðx aj Þ jðx bj Þp0; for 8ða; bþap; j¼e a j¼e b ð4þ X ðx ij Þ jjw j jjz j p0; iaw j j ¼ 1; y; m max ; ð5þ x ij ; z j Af0; 1g8i; j: Notations: C: cycle time. m max : maximum number of stations which can be estimated from a heuristic procedure. W j : subset of all tasks that can be assigned to station j: jjw j jj : number of tasks in set W j :

4 132 K. A &gpak, H. Gök@en / Int. J. Production Economics 96 (2005) E i : earliest station task i can be assigned to, given the precedence relations. L i : latest station task i can be assigned to, given the precedence relations. P: set of tasks that precedes from a task. 1; if task i in the precedence diagram is assigned to workstation j; x ij ¼ 0; otherwise; 1; if there is any task assigned to workstation j; z j ¼ 0; otherwise: Constraint (2) assures that all tasks are assigned to at most one station. Constraint (3) ensures that the sum of task times assigned to each station does not exceed the cycle time. Constraint (4) ensures the precedence relationships between the tasks are not violated. Lastly, the objective of the formulation is to minimize the number of work stations. Resource constraint which express encounter of first case (Type I RCALB problem), for proposed model is given below: X x ij jjk jr jjm jr p0; r ¼ 1; y; R; ð6þ iak jr where K jr is the set of tasks that can be performed in workstation j with resource r: M jr defines the resource r in workstation j: If there is resource r in workstation j; M jr value is equal to 1, in otherwise, this value is 0. jjk jr jj is number of elements in set K jr : Constraint (6) ensures that if at least one task is done in workstation j with resource r; then resource r is used in workstation j; and M jr value gets 1. Now, the objective of the proposed model can be defined as minimization of the number of resources that is assigned to workstations: Minimum XR Xm max M jr : ð7þ r¼1 j¼1 Complete 0 1 integer programming formulation of proposed model for Type I RCALB is given below: Minimum XR Xm max M jr : ð8þ Constraints: Eqs. (2) (5), X r¼1 j¼1 iak jr x ij jjk jr jjm jr p0; r ¼ 1; y; R; ð9þ Xm max j¼1 z j pm max ; ð10þ x ij ; z j Af0; 1g8i; j: If the problem is Type II RCALB, common tasks that can be performed by different resources, are shown as a separate set. So, the resource constraint in Type I RCALB model is modified to constraint (11): X x ij jjn jr jjm jr p0; r ¼ 1; y; R; ð11þ ian jr where N jr is the set of tasks that can be done in workstation j with resource r excludingall the common tasks that can be performed by different resources. Assigning the common tasks to stations is satisfied by

5 new constraint (12): x ij X M jr p0; rav i r ¼ 1; y; R; 8 iafk jr -N jr g; j ¼ 1; y; m max ; ð12þ where V i is the set of resources that can do task i: Constraint (12) ensures that if task i is assigned to workstation j; then it is sufficient that at least one of the resources that can perform task i has been or will be assigned to workstation j: Proposed model for Type II RCALB includingconstraints (11) and (12) is given below: Minimum XR Xm max M jr : ð13þ Constraints: Eqs. (2) (5), X r¼1 j¼1 K. A &gpak, H. Gök@en / Int. J. Production Economics 96 (2005) ian jr x ij jjn jr jjm jr r0; r ¼ 1; y; R; ð14þ x ij X rav i M jr p0; r ¼ 1; y; R; 8 iafk jr -N jr g; j ¼ 1; y; m max ; ð15þ Xm max j¼1 z j pm max ; ð16þ x ij ; z j Af0; 1g 8i; j: 3. Numerical example A precedence diagram with 11 tasks is given in Fig. 1. The performance times of the tasks and the resources are presented in Table 1. The problem is solved for both the traditional ALB model and the proposed model, and the results are compared. All models in this study have been solved usinggams- CPLEX mathematical programming software package. While solvingthe problem, cycle time was assumed as 9 and the maximum number of stations was estimated as 7 (note that the theoretical minimum number of stations for numerical example is calculated as 6). Table 2 shows the results obtained when the model given in Appendix A is solved Fig. 1. Precedence diagram with 11 tasks.

6 134 K. A &gpak, H. Gök@en / Int. J. Production Economics 96 (2005) Table 1 Task times and resources Task no. Task time Resource 1 6 A 2 2 B 3 5 A 4 7 B 5 1 A 6 2 B 7 3 A 8 6 B 9 5 A 10 5 B 11 4 A Table 2 Balance with proposed model (C ¼ 9; m max ¼ 7) No. of station Task Resource 1 1 A 2 2, 4 B 3 3,5,7 A 4 6, 8 B 5 10 B 6 9, 11 A Table 3 Balance with traditional model (C ¼ 9; m max ¼ 7) No. of station Task Resource 1 1,2,5 A,B 2 6, 8 B 3 3 A 4 4 B 5 7, 9 A 6 10, 11 A, B When the same problem is solved usingthe traditional assembly line balancingmodel, we get the results given in Table 3. As seen from the Tables 2 and 3, a total of 8 resources (4 units of resource A and 4 units of resource B) are beingused in the traditional line balancingmodel, while this number could be reduced to 6 (3 units of resource A and 3 units of resource B) with the proposed model. Thus, 2 units of resources (one unit of resource A and one unit of resource B) are saved. Besides, the required number of stations is calculated as 6 in the line balanced with the proposed model. This number is same with the theoretical minimum number of stations, which means that the solution is also optimal in terms of station assignment. When the number of station is optimal, the ever best situation for the number of resources and stations is satisfied if the number of resources is equal to the number of stations, and this situation is achieved in our solution. Consequently, the solution is the best also in terms of the number of resources, and while reachingthis best, the best number of stations is also achieved.

7 K. A &gpak, H. Gök@en / Int. J. Production Economics 96 (2005) Table 4 Balance with proposed model (C ¼ 9; m max ¼ 7) No. of station Task Resource 1 1, 2 A 2 4, 5 B 3 6, 8 B 4 10 B 5 3, 7 A 6 9, 11 A In the example, if we assume that both resources can perform the tasks of 2 and 5, the problem becomes RCALB Type 2. The model given in Appendix B is solved with same cycle time and maximum number of station value given before (the theoretical minimum number of stations is 6), and the obtained results are given in Table 4. As seen from Table 4, the balance is achieved with 6 stations and 6 resources. Task 5, which was previously performed by resource A, alongwith the task 4 are performed by resource B. Similarly, it is considerable that the task 2, which was previously performed by B, alongwith the task 1 are performed by resource A. When the problem is solved without any resource constrained case, the number of stations is again found as 6, while the number of resources needed being calculated as 8 (Table 3). So, it can be stated that the optimal number of resources has been found under the optimal number of stations. 4. Conclusion and future work Although, resource constrained cases are widely experienced in practice, there has not been sufficient interest in the literature. This paper presents a new approach on traditional assembly line balancing problems. The goal of proposed approach is to establish balance of assembly line with minimum number of stations and resources. For this purpose, 0 1 integer-programming models are developed. With this proposed model, an important need with regard to line balancingwhile maintainingthe flexibility of production is met. Because assembly line balancingproblems are NP-hard nature (Ghosh and Gagnon, 1989), large-scale problems are also quite hard to solve using this proposed model. In the future studies, the models explained here can also be modified as a goal programming model by addingthe deviational variables. Especially, when there is limited number of resources, the objective function (Eqs. (7) and (13)) can be defined as a goal constraint: P R P mmax r¼1 j¼1 M jr dm þ þ d M pm max; where M max is the maximum number of resources. The P station constraints (Eqs. (10) and (16)) can be expressed as a goal by adding deviational variables: mmax j¼1 z j dst þ þ d ST pm max; where dst þ ; d ST ; dþ M ; d M are deviational variables for number of stations and resources. Also, several problems such as minimization of number of the stations for a given number of resources, and minimization of cycle time for a given number of the stations and resources can be examined, and some heuristic algorithms can be developed for the large-scale problems. Acknowledgements This research was supported in part by the State PlanningOrganization (DPT) of Turkish Prime Ministry under Grant no. 2002K

8 136 K. A &gpak, H. Gök@en / Int. J. Production Economics 96 (2005) Appendix A E i and L i values: " E i ¼ t i þ X # þ t j!=c ; jap i " L i ¼ m max þ 1 t i þ X # þ t j!=c : jas i P i : The set of tasks which must proceed task i: S i : The set of tasks which must succeed task i: Task: E i : L i : Objective: Minimum Z ¼ MAKð1; 1Þ þmakð1; 2Þ þmakð1; 3Þ þmakð1; 4Þ þmakð1; 5Þ þ MAKð1; 6ÞþMAKð1; 7ÞþMAKð2; 1ÞþMAKð2; 2ÞþMAKð2; 3Þ þ MAKð2; 4ÞþMAKð2; 5ÞþMAKð2; 6ÞþMAKð2; 7Þ Constraints Assignment constraints: Xð1; 1ÞþXð1; 2Þ ¼1; Xð2; 1ÞþXð2; 2ÞþXð2; 3ÞþXð2; 4ÞþXð2; 5Þ ¼1; Xð3; 2ÞþXð3; 3ÞþXð3; 4ÞþXð3; 5ÞþXð3; 6Þ ¼1; Xð4; 2ÞþXð4; 3ÞþXð4; 4ÞþXð4; 5Þ ¼1; Xð5; 1ÞþXð5; 2ÞþXð5; 3ÞþXð5; 4ÞþXð5; 5ÞþXð5; 6Þ ¼1; Xð6; 2ÞþXð6; 3ÞþXð6; 4ÞþXð6; 5ÞþXð6; 6Þ ¼1; Xð7; 3ÞþXð7; 4ÞþXð7; 5ÞþXð7; 6Þ ¼1; Xð8; 2ÞþXð8; 3ÞþXð8; 4ÞþXð8; 5ÞþXð8; 6Þ ¼1; Xð9; 3ÞþXð9; 4ÞþXð9; 5ÞþXð9; 6ÞþXð9; 7Þ ¼1; Xð10; 3ÞþXð10; 4ÞþXð10; 5ÞþXð10; 6ÞþXð10; 7Þ ¼1; Xð11; 6ÞþXð11; 7Þ ¼1: Cycle time constraints: 6 Xð1; 1Þþ2 Xð2; 1ÞþXð5; 1Þp9; 6 Xð1; 2Þþ2 Xð2; 2Þþ5 Xð3; 2Þþ7 Xð4; 2ÞþXð5; 2Þþ2 Xð6; 2Þþ6 Xð8; 2Þp9;

9 K. A &gpak, H. Gök@en / Int. J. Production Economics 96 (2005) Xð2; 3Þ þ5 Xð3; 3Þþ7 Xð4; 3ÞþXð5; 3Þþ2 Xð6; 3Þþ3 Xð7; 3Þþ6 Xð8; 3Þþ5 Xð9; 3Þ þ 5 Xð10; 3Þp9; 2 Xð2; 4Þ þ5 Xð3; 4Þþ7 Xð4; 4ÞþXð5; 4Þþ2 Xð6; 4Þþ3 Xð7; 4Þþ6 Xð8; 4Þþ5 Xð9; 4Þ þ 5 Xð10; 4Þp9; 2 Xð2; 5Þ þ5 Xð3; 5Þþ7 Xð4; 5ÞþXð5; 5Þþ2 Xð6; 5Þþ3 Xð7; 5Þþ6 Xð8; 5Þþ5 Xð9; 5Þ þ 5 Xð10; 5Þp9; 5 Xð3; 6Þ þxð5; 6Þþ2 Xð6; 6Þþ3 Xð7; 6Þþ6 Xð8; 6Þþ5 Xð9; 6Þ þ 5 Xð10; 6Þþ4 Xð11; 6Þp9; 5 Xð9; 7Þþ5 Xð10; 7Þþ4 Xð11; 7Þp9: Resource constraints: Xð1; 1ÞþXð5; 1Þ 2 MAKð1; 1Þp0; Xð2; 1Þ MAKð2; 1Þp0; Xð1; 2ÞþXð3; 2ÞþXð5; 2Þ 3 MAKð1; 2Þp0; Xð2; 2ÞþXð4; 2ÞþXð6; 2ÞþXð8; 2Þ 4 MAKð2; 2Þp0; Xð3; 3ÞþXð5; 3ÞþXð7; 3ÞþXð9; 3Þ 4 MAKð1; 3Þp0; Xð2; 3ÞþXð4; 3ÞþXð6; 3ÞþXð8; 3ÞþXð10; 3Þ 5 MAKð2; 3Þp0; Xð3; 4ÞþXð5; 4ÞþXð7; 4ÞþXð9; 4Þ 4 MAKð1; 4Þp0; Xð2; 4ÞþXð4; 4ÞþXð6; 4ÞþXð8; 4ÞþXð10; 4Þ 5 MAKð2; 4Þp0; Xð3; 5ÞþXð5; 5ÞþXð7; 5ÞþXð9; 5Þ 4 MAKð1; 5Þp0; Xð2; 5ÞþXð4; 5ÞþXð6; 5ÞþXð8; 5ÞþXð10; 5Þ 5 MAKð2; 5Þp0; Xð3; 6ÞþXð5; 6ÞþXð7; 6ÞþXð9; 6ÞþXð11; 6Þ 5 MAKð1; 6Þp0; Xð6; 6ÞþXð8; 6ÞþXð10; 6Þ 3 MAKð2; 6Þp0; Xð9; 7ÞþXð11; 7Þ 2 MAKð1; 7Þp0; Xð10; 7Þ MAKð2; 7Þp0: Station constraints: Xð1; 1ÞþXð2; 1ÞþXð5; 1Þ 3 ISTð1Þp0; Xð1; 2ÞþXð2; 2ÞþXð3; 2ÞþXð4; 2ÞþXð5; 2ÞþXð6; 2ÞþXð8; 2Þ 7 ISTð2Þp0;

10 138 K. A &gpak, H. Gök@en / Int. J. Production Economics 96 (2005) Xð2; 3ÞþXð3; 3ÞþXð4; 3ÞþXð5; 3ÞþXð6; 3ÞþXð7; 3ÞþXð8; 3ÞþXð9; 3ÞþXð10; 3Þ 9 ISTð3Þp0; Xð2; 4ÞþXð3; 4ÞþXð4; 4ÞþXð5; 4ÞþXð6; 4ÞþXð7; 4ÞþXð8; 4ÞþXð9; 4ÞþXð10; 4Þ 9 ISTð4Þp0; Xð2; 5ÞþXð3; 5ÞþXð4; 5ÞþXð5; 5ÞþXð6; 5ÞþXð7; 5ÞþXð8; 5ÞþXð9; 5ÞþXð10; 5Þ 9 ISTð5Þp0; Xð3; 6ÞþXð5; 6ÞþXð6; 6ÞþXð7; 6ÞþXð8; 6ÞþXð9; 6ÞþXð10; 6ÞþXð11; 6Þ 8 ISTð6Þp0; Xð9; 7ÞþXð10; 7ÞþXð11; 7Þ 3 ISTð7Þp0; ISTð1ÞþISTð2ÞþISTð3ÞþISTð4ÞþISTð5ÞþISTð6ÞþISTð7Þp7: Precedence constraints: Xð1; 1Þþ2 Xð1; 2Þ Xð2; 1Þ 2 Xð2; 2Þ 3 Xð2; 3Þ 4 Xð2; 4Þ 5 Xð2; 5Þp0; Xð1; 1Þþ2 Xð1; 2Þ 2 Xð3; 2Þ 3 Xð3; 3Þ 4 Xð3; 4Þ 5 Xð3; 5Þ 6 Xð3; 6Þp0; Xð1; 1Þþ2 Xð1; 2Þ 2 Xð4; 2Þ 3 Xð4; 3Þ 4 Xð4; 4Þ 5 Xð4; 5Þp0; Xð1; 1Þþ2 Xð1; 2Þ Xð5; 1Þ 2 Xð5; 2Þ 3 Xð5; 3Þ 4 Xð5; 4Þ 5 Xð5; 5Þ 6 Xð5; 6Þp0; Xð2; 1Þ þ2 Xð2; 2Þþ3 Xð2; 3Þþ4 Xð2; 4Þþ5 Xð2; 5Þ 2 Xð6; 2Þ 3 Xð6; 3Þ 4 Xð6; 4Þ 5 Xð6; 5Þ 6 Xð6; 6Þp0; 2 Xð3; 2Þ þ3 Xð3; 3Þþ4 Xð3; 4Þþ5 Xð3; 5Þþ6 Xð3; 6Þ 3 Xð7; 3Þ 4 Xð7; 4Þ 5 Xð7; 5Þ 6 Xð7; 6Þp0; 2 Xð4; 2Þþ3 Xð4; 3Þþ4 Xð4; 4Þþ5 Xð4; 5Þ 3 Xð7; 3Þ 4 Xð7; 4Þ 5 Xð7; 5Þ 6 Xð7; 6Þp0; Xð5; 1Þ þ2 Xð5; 2Þþ3 Xð5; 3Þþ4 Xð5; 4Þþ5 Xð5; 5Þþ6 Xð5; 6Þ 3 Xð7; 3Þ 4 Xð7; 4Þ 5 Xð7; 5Þ 6 Xð7; 6Þp0; 2 Xð6; 2Þ þ3 Xð6; 3Þþ4 Xð6; 4Þþ5 Xð6; 5Þþ6 Xð6; 6Þ 2 Xð8; 2Þ 3 Xð8; 3Þ 4 Xð8; 4Þ 5 Xð8; 5Þ 6 Xð8; 6Þp0; 3 Xð7; 3Þ þ4 Xð7; 4Þþ5 Xð7; 5Þþ6 Xð7; 6Þ 3 Xð9; 3Þ 4 Xð9; 4Þ 5 Xð9; 5Þ 6 Xð9; 6Þ 7 Xð9; 7Þx0; 2 Xð8; 2Þ þ3 Xð8; 3Þþ4 Xð8; 4Þþ5 Xð8; 5Þþ6 Xð8; 6Þ 3 Xð10; 3Þ 4 Xð10; 4Þ 5 Xð10; 5Þ 6 Xð10; 6Þ 7 Xð10; 7Þp0; 3 Xð9; 3Þþ4 Xð9; 4Þþ5 Xð9; 5Þþ6 Xð9; 6Þþ7 Xð9; 7Þ 6 Xð11; 6Þ 7 Xð11; 7Þp0; 3 Xð10; 3Þþ4 Xð10; 4Þþ5 Xð10; 5Þþ6 Xð10; 6Þþ7 Xð10; 7Þ 6 Xð11; 6Þ 7 Xð11; 7Þp0:

11 K. A &gpak, H. Gök@en / Int. J. Production Economics 96 (2005) Appendix B Objective: Minimum Z ¼ MAKð1; 1Þ þmakð1; 2Þ þmakð1; 3Þ þmakð1; 4Þ þmakð1; 5Þ þmakð1; 6ÞþMAKð1; 7ÞþMAKð2; 1ÞþMAKð2; 2ÞþMAKð2; 3Þ þ MAKð2; 4Þ þmakð2; 5Þ þmakð2; 6ÞþMAKð2; 7Þ Constraints Assignment constraints, Precedence constraints, Station constraints and Cycle time constraints are the same as those given in the model at Appendix A. Resource constraints: Xð1; 1Þ MAKð1; 1Þp0; Xð1; 2ÞþXð3; 2Þ 2 MAKð1; 2Þp0; Xð4; 2ÞþXð6; 2ÞþXð8; 2Þ 3 MAKð2; 2Þp0; Xð3; 3ÞþXð7; 3ÞþXð9; 3Þ 3 MAKð1; 3Þp0; Xð4; 3ÞþXð6; 3ÞþXð8; 3ÞþXð10; 3Þ 4 MAKð2; 3Þp0; Xð3; 4ÞþXð7; 4ÞþXð9; 4Þ 3 MAKð1; 4Þp0; Xð4; 4ÞþXð6; 4ÞþXð8; 4ÞþXð10; 4Þ 4 MAKð2; 4Þp0; Xð3; 5ÞþXð7; 5ÞþXð9; 5Þ 3 MAKð1; 5Þp0; Xð4; 5ÞþXð6; 5ÞþXð8; 5ÞþXð10; 5Þ 4 MAKð2; 5Þp0; Xð3; 6ÞþXð7; 6ÞþXð9; 6ÞþXð11; 6Þ 4 MAKð1; 6Þp0; Xð6; 6ÞþXð8; 6ÞþXð10; 6Þ 3 MAKð2; 6Þp0; Xð9; 7ÞþXð11; 7Þ 2 MAKð1; 7Þp0; Xð10; 7Þ MAKð2; 7Þp0; Xð2; 1Þ MAKð1; 1Þ MAKð2; 1Þp0; Xð2; 2Þ MAKð1; 2Þ MAKð2; 2Þp0; Xð2; 3Þ MAKð1; 3Þ MAKð2; 3Þp0; Xð2; 4Þ MAKð1; 4Þ MAKð2; 4Þp0; Xð2; 5Þ MAKð1; 5Þ MAKð2; 5Þp0; Xð5; 1Þ MAKð1; 1Þ MAKð2; 1Þp0; Xð5; 2Þ MAKð1; 2Þ MAKð2; 2Þp0;

12 140 K. A &gpak, H. Gök@en / Int. J. Production Economics 96 (2005) Xð5; 3Þ MAKð1; 3Þ MAKð2; 3Þp0; Xð5; 4Þ MAKð1; 4Þ MAKð2; 4Þp0; Xð5; 5Þ MAKð1; 5Þ MAKð2; 5Þp0; Xð5; 6Þ MAKð1; 6Þ MAKð2; 6Þp0: References Baybars, I., A survey of exact algorithms for the simple assembly line balancing problem. Management Science 32 (8), Bryton, B., Balancingof a continuous production line. M.S. Thesis, Northwestern University, Evanston, IL. Erel, E., Sarin, S.C., A survey of the assembly line balancingprocedures. Production Planningand Control 9 (5), Ghosh, S., Gagnon, J., A comprehensive literature review and analysis of the design, balancing and scheduling of assembly systems. International Journal of Production Research 27 (4), G.ok@en, H., Erel, E., Binary integer formulation for mixed model assembly line balancing problem. Computers and Industrial Engineering 34 (2), Graves, S.C., Lamar, B.W., An integer programming procedure for assembly system design problems. Operations Research 31 (3), Nicosia, G., Pacciarelli, D., Pacifici, A., Optimally balancingassembly lines with different workstations. Discrete Applied Mathematics 118, Patterson, J.H., Albracht, J.J., Assembly line balancing: Zero one programming with fibonacci search. Operations Research 23, Pinnoi, A., Wilhelm, W.E., Assembly system design: A branch and cut approach. Management Science 44 (1), Salveson, M.E., The assembly line balancingproblem. Journal of Industrial Engineering6 (3), Yamada, T., Matsui, M., A management design approach to assembly line systems. International Journal of Production Economics 84,