# A Tabu Search Algorithm for the Parallel Assembly Line Balancing Problem

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1 G.U. Journal o Science (): (9) IN PRESS A Tabu Search Algorithm or the Parallel Assembly Line Balancing Problem Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU Selçuk University, Faculty o Engineering, Department o Industrial Engineering, 7 Konya, Turkey Gazi University, Faculty o Engineering and Architecture, Department o Industrial Engineering, 67 Ankara, Turkey Received: 8..9 Revised: 7..9 Accepted: ABSTRACT In a production acility there may be more than one straight assembly line located in parallel. Balancing o parallel assembly lines will provide the lexibility to minimize the total number o workstations due to common resource. This type o problem is called as parallel assembly line balancing (PALB) problem. In this paper, a tabu search based approach is proposed or PALB problem with aim o maximizing line eiciency (LE) (or minimizing number o stations) and minimizing variation o workloads (V). This study is based on the study o Gökçen et al. []. The proposed approach is illustrated on a numerical example and its perormance is tested on a set o well-known problems in the literature. This study is the irst multi objective parallel assembly line balancing study in the literature. Key Words: Assembly line balancing; Parallel assembly lines; Tabu search.. INTRODUCTION Assembly line balancing (ALB) is an attempt to allocate equal amounts o work to the various workstations along the line. The undamental line balancing problem is how to assign a set o tasks to an ordered set o workstations, such that the precedence relations and some perormance measures (minimizing the number o workstation, cycle time, idle time, etc.) are satisied []. The irst analytical study on ALB problem was done by Salveson [3], who presented a mathematical ormulation o ALB problem and suggested a solution procedure. Ater that time, many heuristics and optimal procedures have been proposed or the solution o ALB problem. Assembly lines can be classiied into two general groups: traditional assembly lines (with single and multi/mixed products) and U-type assembly lines (with single and multi/mixed products). In traditional assembly lines, the line coniguration is straight and the entrance and the exit o the assembly line are in dierent position. For the studies on traditional assembly line balancing problem, the review papers o Baybars [], Ghosh and Gagnon [], Erel and Sarin [], and Scholl and Becker [6] can be investigated. U-type assembly line coniguration which has improved the line balance eiciency has been widely used by manuacturers. Design o these lines is dier rom the others, that is, entrance and exit o the assembly line is in same position. U-type assembly line balancing problem was irst studied by Miltenburg and Wijngaard [7]. Then, Urban [8] has developed an integer programming ormulation o the U-type assembly line balancing problem. Several solution techniques have been developed or the U-type assembly line balancing problems to date. A detailed review o exact and heuristic procedures or solving this problem can be seen in Becker and Scholl [9]. Although the literature on traditional and U-type ALB problems is extensive, the studies on parallel lines are quite little. In designing the parallel lines, Süer and Dagli [] have suggested heuristic procedures and Corresponding author,

2 3 G.U. J. Sci., ():33-33 (9)/ Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU algorithms to dynamically determine the number o lines and the line coniguration. Also, Süer [] has studied alternative line design strategies or a single product. Other researches involving parallel workstation have ocused on the simple assembly line balancing problem [] and mixed-model production line balancing problem [3-]. These studies on parallel lines are logically dierent rom the approach o Gökçen et al. []. The new problem presented by Gökçen et al. [] which more than one assembly line is balanced simultaneously with common resources has been derived rom the traditional and U-type ALB problem. Aim o the PALB problem is to minimize the number o workstations by balancing o two or more assembly lines together. In the literature, only two studies have been published to solve the PALB problem. First study belongs to Gökçen et al. []. They have developed a mathematical model and heuristic procedure or PALB problem. Second one is the study o Benzer et al. [6]. They have proposed a network model or PALB problem. The presented study is directly related to the PALB problem o Gökçen et al. []. In this study a new approach or PALB problem is developed with aim o maximizing LE (or minimizing number o stations) and minimizing V. This study is the irst multi objective parallel assembly line balancing study in the literature. The reminder o this paper is organized as ollows: In the ollowing section; PALB problem is explained and Gökçen et al. [] s mathematical ormulation is presented. In Section 3, the proposed approach is described and clariied on an example problem. The perormance o the proposed approach is tested and discussed in Section. Finally, the conclusions are given in Section.. PALB PROBLEM The idea o common balancing o more than one line is deined irst by Gökçen et al. []. PALB problem aims to minimize the number o workstations by balancing o two or more assembly lines together. Task assignment to common workstations o the parallel assembly lines is realized by using the precedence diagrams o each product manuactured on each assembly line separately. The common assumptions o PALB problem are presented as ollows: Only one product is produced on each assembly line. Precedence diagrams or each product are known. Task perormance times o each product are known. Operators working in each workstation o the each line are multi-skilled (lexible workers). It can also be worked each side o any line. The advantages o the parallel assembly lines are: (i) it can provide to produce similar products or dierent models o the same product on adjacent lines, (ii) it can reduce the idle time and increase the eiciency o the assembly lines, (iii) it can provide to able to make production with dierent cycle time or each o the assembly lines, (iv) it can improve visibility and communication skills between operators, (v) it can reduce operator requirements. Gökçen et al. [] s mathematical ormulation and notations used or PALB problem are given below; C : cycle time h : line number, h =,, H. k : station number, k =,, K. M hk : total number o tasks (that can be) assigned to station k on line h. n h : number o tasks on line h. t hi : perormance time o task i on line h. K max : maximum number o stations. P h : set o precedence relationships in precedence diagram o line h. x hik = i task i in line h is assigned to station k; otherwise. U hk = i station k is utilized in line h; otherwise. z k = i station k is utilized; otherwise. Objective Function: Min Kmax z k () k= k min Constraints: Kmax xhik =, i=,...,nh, h=,..,h. k= () nh xhik M hk i= U hk, h=,...,h k =,...,K max. U hk + U( h+ a) k =, nh nh+ thixhik + t( h+ ) ix( h+ ) ik Czk, k =,...,Kmax, h=,...,h-. i= i= h =,...,H- a=...h-h k =,...,K max. Kmax ( K max k+ )( xhrk xhsk ), k= {, } x hik, zk, U hk or all h, i, k. (r,s) Ph. (3) () () (6)

4 36 G.U. J. Sci., ():33-33 (9)/ Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU (a) (b) Figure. The precedence diagram and the task times o (a) the 7-task problem and (b) the 9-task problem. (,,7) (,3) (,6) (,3) (,,7) (,8) (6,9) (a) (b) I (,) () (3) (,6) III II IV (,3) (,6,7) () (7) V VI (,8) (9) Assembly Line I Assembly Line II (c) Figure. The task assignments o (a) traditional assembly line or the 7-task problem, (b) traditional assembly line or the 9-task problem and (c) parallel assembly lines.

7 G.U. J. Sci., ():33-33 (9)/ Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU 39 (V) Solution space (, D ) B Maximum extreme point E (, ) F min A Minimum extreme point G C Ideal point max (, min ) max (LE) Figure 3. Geometric interpretation o the MDM. Task [Line] [] [] 3[] [] [] 6[] 7[] [] [] 3[] [] [] 6[] 7[] 8[] 9[] Assignment order (a) Assembly Line I (,) (7) () () (3) (6) () LE = 8.39 V =.63 () (3,,7) (,8) (b) (6) (9) k=, TL = {Ø}, x*=x, *=(x ), x =x. Assembly Line II Figure. (a) Initial assignment order and (b) the initial line balance. Table. Candidate moves and objective unction values or the irst iteration. m (Swap) LE V (m(x )) m (Swap) LE V (m(x )) [] [] [] 3[] [] 3[] [] [] [] [] [] [] [] [] [] 6[] [] 6[] [] 7[] [] 7[] [] 8[] [] [] * [] 9[] [] []

8 3 G.U. J. Sci., ():33-33 (9)/ Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU Task [Line] [] [] 3[] [] [] 6[] 7[] [] [] 3[] [] [] 6[] 7[] 8[] 9[] Assignment Order (a) (,) (,7) () (3) (6) Assembly Line I (,) (3,,7) (,8) (6) (9) Assembly Line II [] LE = 8.39 V =.7 (b) [] 3[] [] [] 6[] 7[] [] [] 3[] [] [] 6[] 7[] 8[] 9[] [] 3[] [] [] 6[] 7[] [] (c) [] 3[] [] [] 6[] 7[] 8[] Figure. (a) New assignment order, (b) new line balance and (c) tabu list ater the irst iteration. (,) (,) (7) (6) (3) Assembly Line I (,3) () (,6) (7) () (8,9) Assembly Line II LE* = 9.6 V*=.8 Figure 6. Best solution o the example problem.

9 G.U. J. Sci., ():33-33 (9)/ Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU 3 Table. Computational results or same cycle time case. Test problems Number o task (Line -Line ) Cycle time Gökçen et al. [] TS Number o Station (K) K LE% V Kilbridge ,96, ,9, 9 99,9,7 99,, ,9, ,9, Hahn 3-93,7, ,7, ,6, ,7, ,7, Wee-Mag ,7, ,33,8 3 78,9, ,79, ,9, ,7, 7 7 6,3, ,69, ,, ,6,6 Arcus ,, ,, ,, ,, ,, ,,3 68 9,8,8 77 9,, ,,3 86 9,,88 Lutz * 96,9, * 96,38, ,3, ,73, ,,36 Mukherje ,, ,, 9 96,,7 96,7, 96,9, ,, ,37, ,8, ,7, ,3, ,33, ,37, ,37,9 Arcus * 96,, * 97,86, ,8, ,6, * 97,9, 76 3* 97,9,

10 3 G.U. J. Sci., ():33-33 (9)/ Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU Problem (Line- Line) Table 3. Computational results or dierent cycle time case. No. o tasks (Line-Line) Cycle time (Line-Line) Optimal no. o station (Line-Line) Theoretical min. no. o station (K min ) TS K LE% V Mitchell-Heskiao ,,3 Mitchell-Heskiao ,7,8 Mitchell-Heskiao ,67, Heskiao-Sawyer ,, Heskiao-Sawyer ,6,9 Heskiao-Sawyer ,, Sawyer-Kilbridge ,3,39 Sawyer-Kilbridge ,73,3 Sawyer-Kilbridge ,,3 Sawyer-Kilbridge ,8,7 Kilbridge-Tonge ,8, Kilbridge-Tonge ,7,6 Kilbridge-Tonge ,83,3 Kilbridge-Tonge ,6,9 Tonge-Arcus ,3,3 Tonge-Arcus ,, Tonge-Arcus ,, Tonge-Arcus ,6,3 Tonge-Arcus ,6, Tonge-Arcus ,7, Arcus-Arcus ,96,6 Arcus-Arcus ,3,39 Arcus-Arcus ,,3 Arcus-Arcus ,7, Arcus-Arcus ,6, Arcus-Arcus ,,9 Arcus-Arcus ,8,8 6. CONCLUSION In this paper, a tabu search based approach is proposed or PALB problem with the aim o maximizing LE (or minimizing number o stations) and minimizing V. Perormance o the proposed approach is tested on 8 well known test problems in the literature. Obtained results rom the test problems with same cycle time are compared with results o Gökçen et al. [] s. Moreover, main objective o proposed approach is not only optimizing the single objective, but also optimize two objectives simultaneously. According to the comparison results, the results o the proposed approach that optimize the two perormance criteria, is better than Gökçen et al. [] s results. For problems with dierent cycle times, new problems are generated and eiciency o the approach is evaluated. The results o the computational study on various test problems indicate that the proposed approach is successul and it has suicient perormance. To the best knowledge o the authors, this study is irst multi objective parallel line balancing study in the literature. REFERENCES [] Gökçen, H., Ağpak, K., Benzer, R., Balancing o parallel assembly lines, Int. J. Prod. Econ., 3: 6-69 (6). [] Ghosh, S., Gagnon, J., A comprehensive literature review and analysis o the design, balancing and scheduling o assembly systems, Int. J. Prod. Res., 7: (989). [3] Salveson, M.E., The assembly line balancing problem, J. Ind. Eng., 6: 8- (9). [] Baybars, I., A survey o exact algorithms or the simple assembly line balancing problem, Manage. Sci., 3: (986). [] Erel, E., Sarin, S.C., A survey o the assembly line balancing procedures, Prod. Plan. Control., 9: -3 (998). [6] Scholl, A., Becker, C., State-o-the-art exact and heuristic solution procedures or simple assembly line balancing, Eur. J. Oper. Res., 68(3): (6).

11 G.U. J. Sci., ():33-33 (9)/ Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU 33 [7] Miltenburg, J., Wijngaard, J., The U-line line balancing problem, Manage. Sci., : (99). [8] Urban, T.L., Optimal balancing o U-shaped assembly lines, Manage. Sci., : (998). [9] Becker, C., Scholl, A., A survey on problems and methods in generalized assembly line balancing, Eur. J. Oper. Res., 68: 69-7 (6). [] Süer, G.A., Dagli, C., A knowledge-based system or selection o resource allocation rules and algorithms, In: A. Mital and S. Anand, eds. Handbook o expert system applications in manuacturing; Structures and rules, Chapman and hall, 8-7 (99). [] Chiang, W.C., The application o a tabu search metaheuristic to the assembly line balancing problem, Ann. Oper. Res., 9-7 (998). [] Lapierre, S.D., Ruiz, A., Soriano, P., Balancing assembly lines with tabu search, Eur. J. Oper. Res., 68 (3): (6). [3] Hwang, R.K., Katayama, H., Gen, M., U-shaped assembly line balancing problem with genetic algorithm, Int. J. Prod. Res., 6 (6): (8). [] Tabucanon, M.T., Multiple criteria decision making in industry (Studies in production and engineering economics; 8), Elsevier, Amsterdam- Oxord-New York-Tokyo (988). [] Süer, G.A., Designing parallel assembly lines, Comput. Ind. Eng., 3: 67-7 (998). [] Simaria, A.S., Vilarinho, P.M., The simple assembly line balancing problem with parallel workstations-a simulated annealing approach, Int. J. Ind. Eng., 8: 3- (). [3] Askin, R.G., Zhou, M., A parallel station heuristic or the mixed-model production line balancing problem, Int. J. Prod. Res., 3: 39-3 (997). [] McMullen, P.R., Frazier, G.V., A heuristic or solving mixed-model line balancing problems with stochastic task durations and parallel stations, Int. J. Prod. Econ., : 77-9 (997). [] Vilarinho, P.M., Simaria, A.S., A two-stage heuristic method or balancing mixed-model assembly lines with parallel workstations, Int. J. Prod. Res., : - (). [6] Benzer, R., Gökçen, H., Çetinyokuş, T., Çerçioğlu, H., A network model or parallel line balancing problem, Math. Probl. Eng., Art. No: 6 (7). [7] Gutjahr, A.L., Nemhauser, G.L., An algorithm or the line balancing problem, Manage. Sci., (): 38-3 (96). [8] Glover, F., Tabu search, part I, ORSA Journal o Computing, : 9-6 (989). [9] Glover, F., Tabu search, part II, ORSA Journal o Computing, : -3 (99). [] Scholl, A., Voß, S., Simple assembly line balancing-heuristic approaches, J. Heuristics, : 7- (996).

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