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 wellknown 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 Utype 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. Utype 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. Utype assembly line balancing problem was irst studied by Miltenburg and Wijngaard [7]. Then, Urban [8] has developed an integer programming ormulation o the Utype assembly line balancing problem. Several solution techniques have been developed or the Utype 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 Utype 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., ():3333 (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 mixedmodel 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 Utype 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 multiskilled (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=...hh 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)
3 G.U. J. Sci., ():3333 (9)/ Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU 3 The objective o the above ormulation is to minimize the total number o workstations utilized in the production acility. Constraint () ensures the assignment o all tasks to a station and also each task is assigned only once. Constraint (3) ensures that the work content o any station does not exceed the cycle time. Constraints () and () ensure that an operator working at station k and line h can perorm task(s) rom only one adjacent line (i.e. operator in line h can perorm tasks in line h+ or h). Constraint (6) ensures that the precedence constraints are not violated on the line h precedence diagrams. In order to explain the PALB concepts, we have used two classical problems (Merten s 7task problem and Jaeschke s 9task problem) which precedence diagrams or two dierent products (two assembly lines) are given In Figure. The numbers within the nodes represent tasks and the arrow (or arcs) connecting the nodes speciies the precedence relations. The numbers next to the nodes represent task perormance times. When each product in the problem is balanced with a cycle time o, it can be seen that all tasks are perormed at 7 workstations in the traditional assembly line (Figure a, b), whereas all tasks are perormed at 6 workstations (one less than the traditional line solutions) in parallel assembly line (Figure c). The operator o Station I in Figure c, irst completes tasks and on the line I and then completes task on the line II. As mentioned beore, operators in a parallel assembly line may work on two dierent items within the same cycle time. The workstations which are opened both Assembly Line I and Assembly Line II are called as common workstations, and the other workstations are called as separate workstations. In this case, Workstations I, II, V and VI are common workstations, and Workstations III and IV are separate workstations. 3. THE PROPOSED TABU SEARCH ALGORITHM 3.. Introduction to the proposed tabu search algorithm In the mathematical complexity, ALB problem is NPhard class o combinatorial optimization problems [7]. The combinatorial structure o this problem makes it diicult to obtain an optimal solution when the problem size increases. PALB problem is also NPhard class. In addition to the mathematical complexity o ALB problem, PALB problem has also an additional level o complexity, since two or more assembly lines are balanced simultaneously. A binary mathematical ormulation is presented in Section is NPhard. Since it is impossible or the mathematical ormulation presented in Section to reach the optimal or easible solutions in a reasonable computation time or largesized problems, the easible solutions which may be close optimal solutions can be obtained by heuristic or approximate solution methods. Tabu search (TS), deined and developed primarily by Glover [8, 9], is one o the most eective heuristic optimization methods using local search techniques to ind possible optimal or nearoptimal solutions o many combinatorial optimization problems. There are several applications o tabu search algorithm or ALB problems in the literature []. Tabu search algorithm consists o several elements called as move, neighborhood, initial solution, search strategy, memory structure, aspiration criterion and stopping rules. In this section, the speciic characteristics o the proposed TS algorithm to TALB problem are presented. The proposed TS algorithm starts with an initial solution (x ) and stores it as the current solution (x k ) and the best solution (x*). The cost o initial solution ((x )) becomes the current value o the objective unction ((x k )) and the best value o the objective unction ((x*)). The neighborhood solutions o x k are then generated by a move (m). These are candidate solutions. They are evaluated by the objective unction and a candidate solution ( x k ) which is the best not tabu or satisies the aspiration criterion is selected as the new x k. This selection is called a move and added to tabu list (TL), the oldest move is removed rom tabu list i it is overloaded. I the new x k is better than x*, it is stored as the new x*. Otherwise, x* remains unchanged. This searching process is repeated until the termination criterion is met. In the remainder o this section, a detailed description o the proposed TS algorithm is given. Initial solution: Initial solution can be obtained by using a constructive heuristic or by getting a easible solution generated randomly. The proposed approach uses an assignment order o tasks which is generated randomly. Move: In this study, in generating a new neighborhood solution rom current solution, swap has been used. This neighborhood contains all those permutations M(x k ) obtained rom x k by swapping the assignment order o tasks placed at the bth position and the randomly selected ath position, i.e., x k = (AO,., AO a,, AO b,, AO n ) m(x k ) = (AO,., AO b,, AO a,, AO n ) M(x k ) = (AO,., AO b,, AO a,, AO n ) ; b {,..., n}, a b. where, m(x k ): a neighborhood o x k which is obtained by swapping the assignment order o tasks in positions a and b. n: total number o tasks on parallel assembly lines H ( n= n h ) h= AO j : assignment order o task j
4 36 G.U. J. Sci., ():3333 (9)/ Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU (a) (b) Figure. The precedence diagram and the task times o (a) the 7task problem and (b) the 9task 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 7task problem, (b) traditional assembly line or the 9task problem and (c) parallel assembly lines.
5 G.U. J. Sci., ():3333 (9)/ Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU 37 Tabu list: In this study, tabu list which consists o twodimensional array has been used to check i a move rom a solution to its neighborhood is orbidden or allowed. When a pair o tasks is declared as tabu, TL[a][b] is determined as current iteration number + tabu size. I TL[a][b] is empty, then the assignment orders o task a and task b are not orbidden or swap move. Otherwise, i TL[a][b] is T, then the assignment orders o task a and task b cannot be moved until iteration T. Initially, TL is empty and tabu size is n. Size o neighborhood: Size o neighborhood determines how many neighborhood solutions will be searched beore choosing a move which leads to next step. In this study, neighborhood structure o swap is used and the size o neighborhood solutions is also (n). Aspiration criterion: In this study, i a move is in the tabu list and it gives a better solution than the best objective unction value obtained so ar, then this move is applicable. Termination rule: In this study, iteration number is used as termination rule and considered as n. 3.. Building a easible solution Cycle times o parallel assembly lines can be same or dierent. For both cases, the ollowing procedure can be used. a) Normalize operation times o tasks. tih = tih / ch i=,..., nh h=,..., H. c =. b) Deine the cycle time o system. In obtaining a easible solution, alized operation times should be used and cycle time condition (C=.) should be controlled or each assembly line. Feasible solution procedure is given below: a) Start with m(x iter ) k =, IY k =. b) Establish an assignable task set rom the tasks that have not assigned yet, by treating the precedence relations or all independent assembly lines. In this set, there may be several tasks rom both assembly lines. I all tasks are assigned, calculate the objective unction. c) Select the irst task i (on assembly line h) with highest priority assignment order rom the assignable task set that satisies the IY k + t ih. constraint. Assign the related task to station k, calculate the station workload, IY k = IY k + t ih and go to b. I the cycle time constraint is not satisied, then k = k +, IY k = and go to b Perormance criteria and objective unction Two perormance criteria (LE and V) have been taken into consideration. These perormance criteria have also been used by Hwang et al. [3]. Number o stations can be decreased by maximizing the LE. Also workload dierence between stations can also be reduced by the minimizing the V (that is, it is possible to distribute these workloads to stations as equal as possible). Calculations o (LE) and (V) or a given solution with K stations are as ollows; H nh t ih h= i= = K ( V ) = (7) H n h t t K ih ih i s h= i= s= K K max t ih y= i y K The objective unctions which are given in Equations (7) and (8) should be combined into a single objective unction. In order to combine these objective unctions into a single objective unction, we have used the minimum deviation method (MDM) which is applicable when the analyst has partial iation o the objectives. Aim is to ind the best compromise solution which minimizes the sum o individual objective s ractional deviations []. Let and ( V ) be the least desirable objective value o (LE) and (V), respectively, which are obtained rom initial solution. The objective unction used in this study is ormulated as ollows: (8) Minimize: = max min ( LE) ( V ) ( V ) + max min ( V ) ( V ) where, max and min ( V ) are the target values o LE and V, respectively. Since minimum level o V and maximum level o LE represent the perect balance, the target values o V and LE are and, respectively. Equation (9) guarantees maximizing the (LE) and minimizing the (V) simultaneously. Geometrical interpretation o minimization o is given in Figure 3. (9) Points A and B give the target values o objectives (LE) and (V), respectively. The denominator max min and ( V ) ( V )
6 38 G.U. J. Sci., ():3333 (9)/ Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU can be represented by the length o BD and AD, respectively. I E is any point in the solution space which is candidate or the best solution, then the length o EF and EG represent the magnitude o max min ( LE) and ( V ) ( V ), respectively. The objective unction given in Equation (9) is the minimization o ( EF / BD + EG / AD ).. NUMERICAL EXAMPLE In this section, an explanatory example given in Figure is used to describe the proposed approach clearly. Cycle times o the lines are assumed as 8 (or line ) and (or line ), respectively. Initial solution A random task assignment order list with size o (n +n ) is established rom the tasks o the each parallel line. Initial assignment order and the parallel assembly line balance with 9 stations are given in Figure a and b, respectively. New solution generation By using swap operator, all neighborhood solutions (M(x )) are generated and objective unction values ((m(x ))) are calculated. For this purpose, a random task o any line is selected and whole examination is applied between assignment orders. In the Table, all candidate moves (m) and objective unction values or the irst iteration are given. Randomly selected task or irst assembly line is task o and the assignment order o this task is 9. As it can be seen rom Table, neighborhood solutions are generated and the best objective unction value among them is belong to move o [][]. So, move o [][] is accepted as the new solution and this move is added to tabu list. New assignment order, new line balance and the tabu list ater the irst iteration are shown in Figure (a), (b) and (c), respectively. Final solution Total number o tasks on parallel assembly lines has been used as termination criterion. The best solution with 8 stations achieved ater 6 iterations is given in Figure 6.. COMPUTATIONAL RESULTS Perormance o the proposed approach is tested on 8 well known test problems ( test problems with same cycle time and 7 test problems with dierent cycle time) in the literature. The number o parallel assembly line is considered as two or all test problems. The proposed approach is coded by using the Visual Basic 6. programming language, and the set o test problems are solved on a Pentium IV 3. GHz PC with MB RAM. All parameters o the algorithm are obtained experimentally (iteration number, (n +n ) and tabu size, n + ). Each test problem is solved ive times n using by these parameters and only best solutions are reported here. Table represents the computational results or same cycle time case. Aim o the proposed approach is to minimize the number o stations while the station workload is being smoothed, that is, two perormance criteria are optimized simultaneously. Here, results o the proposed approach is compared with results o Gökçen et al. [] s study, to orm an idea about the quality o obtained results. From the comparison results, it is seen that proposed approach have optimized the two perormance criteria. In Table, in 6 o the test problems, the proposed approach has obtained better solution than Gökçen et al. [] s. Only one o them is worse than that solution and other solutions are the same with them. Clearly, the proposed approach results are better than the Gökçen et al. [] s results or this problem set. Table 3 represents the computational results or dierent cycle time case. 7 test problems are combined rom the literature problems. These problems are dierent rom the Gökçen et al. [] s. Columns o the Table 3 represent problem types, number o tasks, cycle times and optimal number o stations or each line (independently), theoretical minimum number o workstations and the results obtained by the proposed approach, respectively. The theoretical minimum number o workstation (K min ) is a lower bound or the solution and calculated by Equation. + t K = hi min () Ch h i where C h is the cycle time o line h, and [X] + denotes the smallest integer greater than or equal to X. We know that the optimal number o stations is not less than the K min. For the values o the number o stations, which is dierent rom the K min in Table 3, it is not possible to say anything about whether the values o the number o stations are optimal or not. This comparison may only give an idea about the perormance o the procedure. As seen rom Table 3, in 9 o 7 test problems, number o stations obtained rom the proposed approach is equal to the K min (i.e. optimal value). In 7 o the test problems, the proposed approach obtained station more than the K min. In o the results, the proposed approach produced stations more than the K min. Ideal LE value is % and the V value is. But, it is so diicult to reach these values except or perect balance. As seen rom Table 3, obtained LE and V values are quite near to % and. Moreover, main objective o proposed approach is not only optimizing the single objective, but only optimize two objectives simultaneously. For this reason, as a result, it can be seen that the perormance o the proposed approach is successul and it has suicient perormance.
7 G.U. J. Sci., ():3333 (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., ():3333 (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., ():3333 (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 393,7, ,7, ,6, ,7, ,7, WeeMag ,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., ():3333 (9)/ Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU Problem (Line Line) Table 3. Computational results or dierent cycle time case. No. o tasks (LineLine) Cycle time (LineLine) Optimal no. o station (LineLine) Theoretical min. no. o station (K min ) TS K LE% V MitchellHeskiao ,,3 MitchellHeskiao ,7,8 MitchellHeskiao ,67, HeskiaoSawyer ,, HeskiaoSawyer ,6,9 HeskiaoSawyer ,, SawyerKilbridge ,3,39 SawyerKilbridge ,73,3 SawyerKilbridge ,,3 SawyerKilbridge ,8,7 KilbridgeTonge ,8, KilbridgeTonge ,7,6 KilbridgeTonge ,83,3 KilbridgeTonge ,6,9 TongeArcus ,3,3 TongeArcus ,, TongeArcus ,, TongeArcus ,6,3 TongeArcus ,6, TongeArcus ,7, ArcusArcus ,96,6 ArcusArcus ,3,39 ArcusArcus ,,3 ArcusArcus ,7, ArcusArcus ,6, ArcusArcus ,,9 ArcusArcus ,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: 669 (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., Stateotheart exact and heuristic solution procedures or simple assembly line balancing, Eur. J. Oper. Res., 68(3): (6).
11 G.U. J. Sci., ():3333 (9)/ Uğur ÖZCAN, Hakan ÇERÇĐOĞLU, Hadi GÖKÇEN, Bilal TOKLU 33 [7] Miltenburg, J., Wijngaard, J., The Uline line balancing problem, Manage. Sci., : (99). [8] Urban, T.L., Optimal balancing o Ushaped 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: 697 (6). [] Süer, G.A., Dagli, C., A knowledgebased 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, 87 (99). [] Chiang, W.C., The application o a tabu search metaheuristic to the assembly line balancing problem, Ann. Oper. Res., 97 (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., Ushaped 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 OxordNew YorkTokyo (988). [] Süer, G.A., Designing parallel assembly lines, Comput. Ind. Eng., 3: 677 (998). [] Simaria, A.S., Vilarinho, P.M., The simple assembly line balancing problem with parallel workstationsa simulated annealing approach, Int. J. Ind. Eng., 8: 3 (). [3] Askin, R.G., Zhou, M., A parallel station heuristic or the mixedmodel production line balancing problem, Int. J. Prod. Res., 3: 393 (997). [] McMullen, P.R., Frazier, G.V., A heuristic or solving mixedmodel line balancing problems with stochastic task durations and parallel stations, Int. J. Prod. Econ., : 779 (997). [] Vilarinho, P.M., Simaria, A.S., A twostage heuristic method or balancing mixedmodel 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., (): 383 (96). [8] Glover, F., Tabu search, part I, ORSA Journal o Computing, : 96 (989). [9] Glover, F., Tabu search, part II, ORSA Journal o Computing, : 3 (99). [] Scholl, A., Voß, S., Simple assembly line balancingheuristic approaches, J. Heuristics, : 7 (996).
Online publication date: 23 April 2010
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