A shortest route formulation of simple U-type assembly line balancing problem

Transcription

1 Applied Mathematical Modelling 29 (2005) A shortest route formulation of simple U-type assembly line balancing problem Hadi Gökçen a, *,Kürsßat Ağpak b, Cevriye Gencer a, Emel Kizilkaya c a Faculty of Engineering and Architecture, Department of Industrial Engineering, Gazi University, Maltepe, Ankara, Turkey b Faculty of Engineering, Department of Industrial Engineering, Gaziantep University, Gaziantep, Turkey c Faculty of Engineering, Department of Industrial Engineering, Erciyes University, Kayseri, Turkey Received 1 May 2003; received in revised form 1 October 2004; accepted 14 October 2004 Available online 13 December 2004 Abstract In this paper, a shortest route formulation of simple U-type assembly line balancing (SULB) problem is presented and illustrated on a numerical example. This model is based on the shortest route model developed in [Manage. Sci. 11 (2) (1964) 308.] for the traditional single model assembly line balancing problem. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Assembly line balancing; U-type lines; Network programming 1. Introduction Traditional line balancing is the process of allocating work (task) to the workstations in such a manner that all workstations have approximately the same amount of task assigned to them. In task assignments to the workstations, precedence relations among these tasks should not be violated. The assembly line balancing (ALB) problem has been studied extensively since Since * Corresponding author. Tel.: ; fax: address: hgokcen@gazi.edu.tr (H. Gökçen) X/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi: /j.apm

2 374 H. Gökçen et al. / Applied Mathematical Modelling 29 (2005) that time, several techniques have been proposed for the solution of ALB problem. (See the review paper of Baybars [2]; Ghosh and Gagnon [3]; Erel and Sarin [4]). Recently, U-type layouts have been utilized in many production lines in place of the traditional straight-line configuration due to the use of just-in-time production principles. In 1994, Miltenburg and Wijngaard [5] presented a new problem derived from the traditional ALB problem where production lines are arranged as U-type lines instead of straight lines. The U-type assembly line is an attractive alternative for assembly production systems since operators become multiskilled by performing tasks located on different parts of assembly line. Moreover, since the U-type line disposition allows for more possibilities on how to assign tasks to workstations, the number of workstations needed for a U-type line layout is nevermore than the number of workstations needed for the traditional straight line. The reason for this is that in the traditional ALB problem, for a given cycle time (maximum amount of time units that can be spent at each workstation, or the time interval between two successive outputs), the set of possible assignable tasks is conformed by those tasks whose predecessors have already been assigned to workstations, whereas in the U-type line balancing problem, the set of assignable tasks is determined by all those tasks whose predecessors and successors have already been assigned [6]. Precedence diagram, traditional and U-type line balancing results are given in Fig. 1. The problem has 7 nodes (tasks). The numbers within the nodes represent tasks and the arrow (or arcs) connecting the nodes specifies the precedence relations. The numbers next to the nodes represent (a) (b) Enter Exit (c) Fig. 1. Precedence diagram (a), traditional (b) and U-type (c) line balancing results.

3 H. Gökçen et al. / Applied Mathematical Modelling 29 (2005) task performance times. When the problem was solved with a cycle time of 10, it can be seen that all tasks are performed at 5 workstations in traditional assembly line, whereas all tasks are performed at 4 workstations in U-type assembly line. As seen in Fig. 1, U-line design, workstations can include tasks located on different parts of the production line. For example the first workstation consists of tasks 1 and 7, where 1 is located at the beginning of the line while 7 is located at the end of the line. There is a small and growing literature on SULB problem. SULB problem was first modeled by Miltenburg and Wijngaard [5]. Urban [7] proposed an integer programming formulation for determining the optimal balance. Scholl and Klein [8] proposed the branch and bound procedure ULINO (U Line Optimizer) for the problem. In addition, some other studies on U-type line balancing problem can be found in Miltenburg and Sparling [9], Ohno and Nakade [10], Nakade and Ohno [11], Ohno et al. [12], Miltenburg [13], Sparling and Miltenburg [14], Ajenblit and Wainwright [6], Nakade and Ohno [15], Miltenburg [16], Miltenburg [17], Guerriero and Miltenburg [18]. Like the traditional line balancing problem, U-line balancing problem is also NP hard nature (Miltenburg and Sparling [9]). 2. Shortest route formulation First shortest route formulation of the traditional single model assembly line balancing problem is presented by Klein [19]. The network had directed arcs representing possible assignments of tasks to workstations, and each path from source to sink represented a possible line design. Then, Gutjahr and Nemhauser [1] developed an algorithm to solve this problem based on finding the shortest route in a directed network. The model was considerably superior to the model of Klein [19] in the sense that only a portion of the feasible orderings was generated [20]. The model proposed for U-type line balancing problems in this paper is based on the Gutjahr and NemhauserÕs [1] algorithm developed for the traditional ALB problems. In this proposed model, we assumed that the task performance times are known constant, precedence relations of tasks are known, parallel workstations are not allowed and no work-in process inventory buffer is allowed between the workstations. The SULB problem defined by Miltenburg and Wijngaard [5] is given as follows: Miltenburg and WijngaardÕs [5] definition follows from that given by Gutjahr and Nemhauser [1] for the traditional line balancing problem. Given set of tasks H ={ij i = 1,2,3,...,N}, a set of precedence constraints P ={(x,y)j task x must be completed before task y}, a set of task times T ={t i j i = 1,2,3,...,N}, cycle time C and a number of workstation K, find a collection of subsets of H, (S 1,S 2,...,S N ) where S k ={ij task i is done at workstation k}, that satisfy the following conditions: [ K k¼1 S k ¼ H; \ S k S j ¼ [; k6¼j ð1þ ð2þ

4 376 H. Gökçen et al. / Applied Mathematical Modelling 29 (2005) X t i 6 C; i2s k k ¼ 1; 2;...; N: ð3þ For each task y, if ðx; yþ 2P; x 2 S k ; y 2 S j ; then k 6 j; for all x; or if ðy; zþ 2P; y 2 S j ; z 2 S i ; then i 6 j; for all z: ð4þ " # KC XK X k¼1 i2s k t i is minimized: Condition 1 ensures that all tasks are assigned to a workstation. As a result of condition 2, each task is assigned only once. Condition 3 ensures that the work content of any workstation does not exceed the cycle time. Condition 4 ensures that the precedence constraints are not violated on the U-line. As a result of the objective function, the number of workstations will be minimized [5]. ð5þ 2.1. Network model The network model consists of developing a finite directed network for which the arcs represent workstations in the assembly line and the nodes correspond to possible first workstation assignments of tasks. The arc lengths are the idle times of workstations. Thus the optimization procedure is to find the shortest path in the network or to find the minimum number of arcs. Node generation, arc construction and finding the shortest path are given below in detail Generation of nodes The node generation process used in this paper is similar to the node generation process developed by Gutjahr and Nemhauser [1] for traditional single model ALB problem. The basic difference between two processes arises from the nature of the traditional and U-type line balancing problems. In the traditional line-balancing problem, tasks from a set of assignable tasks (those tasks whose predecessors have already been assigned) are selected for a workstation. In the U-type line, the set of assignable tasks is the union of the set of tasks whose predecessors have already been assigned and the set of tasks whose successors have already been assigned. Tasks are selected from this set to for a workstation [5]. In state generation, the following properties should be satisfied. (i) No state elements can be generated as duplicate. (ii) All sets generated are states. (iii) Every state is generated [1]. The node generation process can be defined as follows: The empty set is considered as the first state generated. The tasks that are available for assignments (tasks without any predecessors according to the left side of the precedence diagram, and tasks without any successors according to the right side of the precedence diagram) are placed in stage 1 and are considered marked tasks. Then all sets of task combination related to marked tasks are generated. Each set is defined as a

5 H. Gökçen et al. / Applied Mathematical Modelling 29 (2005) state. The unmarked immediate followers, F(S), (according to left side of the diagram) and predecessors (according to right side of the diagram) of a state are augmented to the current stage to construct the states of the next stage. For any state S of stage s, the unmarked immediate followers and predecessors are placed in two lists called F(S) and P(S) respectively. For each subset R F(S) and B P(S), the R [ S and B [ S are states for stage s + 1. For each state of stage s, the unmarked immediate followers and predecessors are determined and placed as marked tasks for stage s + 1. When all the tasks are marked or F(S) and P(S) are empty for the current stage, the node generation process is completed. With this node generation process, all possible feasible states can be generated. The final node in the network consists of all tasks Constructing the arcs and finding the shortest path Each state, which is generated by previous process, corresponds to the nodes in the directed network. Let G i, i =1,...,r represent the set of tasks in node i where r is the total number of nodes. G 0 is P equal to zero and G r denotes the set of all tasks in the precedence diagram. Also let T ðg i Þ¼ j2g i t j, where T(G i ) represents the total task times in G i. The paths from node 0 to node r in the network can be constructed as follows: the goal is to find the shortest path of the network. Finding the shortest path from node 0 to node r can be achieved by finding any path from node 0 to node r with the least number of arcs. In other words, a path with the least number of arcs from node 0 to node r gives the optimal solution of the problem. In constructing the network, we begin the initial node (node 0). All arcs are connected from this node. If T(G i ) 6 C, there is an arc from node 0 to node i. These nodes are called the first nodes. If G i G j and T(G j ) T(G i ) 6 C for node i among the first nodes, an arc to node j is constructed. These nodes are also called the second nodes. Network construction is repeated until node r is reached. Each directed arc (ij) from node i to node j in the network is assigned a distance of [C T(G j )+T(G i )]. This length gives the idle time of each workstation. Note that, no arc enters node 0 and no arc leaves node r. After the network construction is completed, a shortest route (or path) from node 0 to node r is determined by considering the arc lengths (or idle time of workstation). Each arc in the network corresponds to the workstation in the assembly line. If the shortest route from node 0 to node r is (0,i,j,k,r), workstation assignments can be determined as [(G r G k ), (G k G j ), (G j G i ), (G i G 0 )]. Each set gives the workstation assignments. 3. Illustrative example Precedence diagram and processing (or task) times of example U-type line balancing problem with 7 tasks were given in Fig. 1(a). The cycle time of the line was 10. The state generation process is shown in Table 1. Initially, unmarked immediate follower and predecessor tasks are considered as marked and task 1 and task 7 are placed in stage 1. The unmarked immediate followers of task 1 are tasks 2 and 3, and the immediate predecessors of task 7 are tasks 4, 5 and 6. They are placed in list F(S) and P(S) respectively. FP(S) is union of the F(S) and P(S), that is, {2,3} [ {4,5,6} = {2,3,4,5,6}. These tasks of FP(S) are placed in stage 2 as marked tasks. Task 1 is augmented to all subsets of the list F(S) that include tasks 2 and 3, and task 7 is also augmented to all subsets of the list P(S) that include tasks 4, 5 and 6, and task 1 and 7 are augmented to all subsets

6 378 H. Gökçen et al. / Applied Mathematical Modelling 29 (2005) Table 1 State generation Stage Marked tasks State no. State elements State times Unmarked immediate Successors Predecessors , , ,5,6 3 1,7 10 2,3 4,5,6 2 2,3,4,5,6 4 1, , ,2, , , , , 4, , 4, , 5, ,4,5, , 7, , 7, , 7, , 7, , 7, ,7,2, ,7,2, ,7,2, ,7,2, ,7,3, ,7,3, ,7,3, ,7,4, ,7,4, ,7,5, ,7,2,3, ,7,2,3, ,7,2,3, ,7,3,4, ,7,3,4, ,7,4,5, ,7,2,3,4, ,7,2,3,4, ,7,3,4,5, ,7,2,3,4,5,6 37 of the list F(S) and P(S) that include tasks 2,3,4,5 and 6 to form the states of stage 2. For example, at stage 1 one of the states generated contains the task 1. Its unmarked immediate followers are 2 and 3. So, in stage 2, from the state element 1, the states generated are {1,2}, {1,3} and {1,2,3}. For each state in stage 2, F(S) and P(S) lists are determined. However for all states of

7 H. Gökçen et al. / Applied Mathematical Modelling 29 (2005) Fig. 2. Resulting network of the example problem. stage 2, F(S) and P(S) lists are empty. So, procedure is terminated in stage 2. Total number of states generated for this problem is determined as 38. The network constructed with the constraint of T(G j ) T(G i ) 6 C is shown in Fig. 2. Nodes 1,2,3,4 and 7 define the first nodes, because the constraint of T(G i ) 6 C is satisfied for these nodes. From node 1,1 new node; from node 2,8 new nodes; from node 3,5 new nodes and lastly from node 4,1 new node can be constructed with G 1 G j and T(G j ) T(G 1 ) 6 C. These new nodes are called as second nodes. All the tasks in the problem are represented by node 38. This node is first obtained with 4 arcs. As seen from the Fig. 2, optimal route (or path) is determined as There are 4 arcs in the path, that is, the optimal solution has 4 workstations in the U-type assembly line. Tasks in G 38 G 31 {4,5} constitute a workstation assignment. Similarly G 31 G 15 {2,6}, G 15 G 3 {3} and G 3 G 0 {1,7} tasks are the optimal workstation assignments. The numbers in Fig. 2 correspond to the ones given in Table 1. The numbers next to the arcs in Fig. 2 represent the station idle times. Workstation assignments of this problem were demonstrated as schematic in Fig. 1(c). 4. Concluding remarks Recently, U-type layouts have been utilized in many production lines in place of the traditional straight-line configuration due to the use of just-in-time principles. The shape of U-lines improves visibility and allows the construction of stations containing tasks on both sides of the line. This arrangement, combined with cross-trained operators, provides greater flexibility in station construction than is available on a comparable straight production line. The shortest route model developed here for U-type assembly line balancing problem is a new approach and provides a different perspective for interested U-type assembly line balancing researchers. Furthermore, model can also be used as a framework to develop effective heuristic procedures to solve the simple U-type line-balancing problem.

8 380 H. Gökçen et al. / Applied Mathematical Modelling 29 (2005) Acknowledgement This research was supported in part by the State Planning Organization (DPT) of Turkey Prime Ministry under grant no. 2002K References [1] A.L. Gutjahr, G.L. Nemhauser, An algorithm for the line balancing problem, Manage. Sci. 11 (2) (1964) [2] I. Baybars, A survey of exact algorithms for the simple assembly line balancing problem, Manage. Sci. 32 (1986) [3] S. Ghosh, J. Gagnon, A comprehensive literature review and analysis of the design, balancing and scheduling of assembly systems, Int. J. Prod. Res. 27 (4) (1989) [4] E. Erel, S.C. Sarin, A survey of the assembly line balancing procedures, Prod. Plann. Control 9 (1998) [5] J. Miltenburg, J. Wijngaard, The U-line balancing problem, Manage. Sci. 40 (10) (1994) [6] D.A. Ajenblit, R.L. Wainwright, Applying genetic algorithms to the U-shaped assembly line balancing problem, IEEE World Congr. Comput. Intell. (1998) [7] T.L. Urban, Note: optimal balancing of U-shaped assembly lines, Manage. Sci. 44 (5) (1998) [8] A. Scholl, R. Klein, ULINO: optimally balancing U-shaped JIT assembly lines, Int. J. Prod. Res. 37 (4) (1999) [9] J. Miltenburg, D. Sparling, Optimal solution algorithms for the simple U-line balancing problem, Working paper. (1994). [10] K. Ohno, K. Nakade, Analysis and optimization of U-shaped production line, J. Oper. Res. Soc. Jpn. 40 (1) (1997) [11] K. Nakade, K. Ohno, Stochastic analysis of a U-shaped production line with multiple workers, Comput. Indus. Eng. 33 (3 4) (1997) [12] K. Ohno, K. Nakade, J.G. Shantikumar, Bounds and approximations for cycle times of a U-shaped production line, Oper. Res. Lett. 21 (1997) [13] J. Miltenburg, Balancing U-lines in a multiple U-line facility, Eur. J. Oper. Res. 109 (1998) [14] D. Sparling, J. Miltenburg, The mixed-model U-line balancing problem, Int. J. Prod. Res. 36 (2) (1998) [15] K. Nakade, K. Ohno, An optimal worker allocation for a U-shaped production line, Int. J. Prod. Econom (1999) [16] J. Miltenburg, One-piece flow manufacturing on U-shaped production lines: a tutorial, IIE Trans. 33 (2001) [17] J. Miltenburg, U-shaped production lines: a review of theory and practice, Int. J. Prod. Econom. 70 (2001) [18] F. Guerriero, J. Miltenburg, The stochastic U-line balancing problem, Naval Res. Logis. 50 (2003) [19] M. Klein, On assembly line balancing, Oper. Res. 11 (1963) [20] E. Erel, H. Gokcen, Shortest-route formulation of mixed-model assembly line balancing problem, Eur. J. Oper. Res. 116 (1999)