An efficient tabu search algorithm to the cell formation problem with
alternative routings and machine reliability considerations
q
Shu-Hsing Chung
a, Tai-Hsi Wu
b,⇑, Chin-Chih Chang
a aDepartment of Industrial Engineering and Management, National Chiao Tung University, 1001, Ta Hsueh Road, Hsinchu 300, Taiwan
b
Department of Business Administration, National Taipei University, 151, University Road, San Shia, Taipei 237, Taiwan
a r t i c l e
i n f o
Article history:
Received 13 November 2008
Received in revised form 25 August 2010 Accepted 25 August 2010
Available online 31 August 2010 Keywords:
Cell formation
Alternative process routings Machine reliability Tabu search Mutation operator
a b s t r a c t
Cell formation is the first step in the design of cellular manufacturing systems. In this study, an efficient tabu search algorithm based on a similarity coefficient is proposed to solve the cell formation problem with alternative process routings and machine reliability considerations. In the proposed algorithm, good initial solutions are first generated and later on improved by a tabu search algorithm combining the mutation operator and an effective neighborhood solution searching mechanism. Computational experi-ences from test problems show that the proposed approach is extremely effective and efficient. When compared with the mathematical programming approach which took three hours to solve problems, the proposed algorithm is able to produce optimal solutions in less than 2 s.
Ó 2010 Published by Elsevier Ltd.
1. Introduction
Cellular manufacturing is the implementation of Group Technology (GT), a manufacturing philosophy in which similar parts are identified and grouped into part families; meanwhile, machines are grouped into machine cells to take advantage of their similarities in manufacturing and design. GT was originally intro-duced byMitrovanov (1966)and was popularized in the west by
Burbidge (1975). The implementation of cellular manufacturing has been reported to result in significant benefits such as reduc-tions in set-up times, work-in-progress inventory, throughput times and material handling costs, simplified scheduling and im-proved quality (Wemmerlov & Hyer, 1987).
Although cellular manufacturing may provide great benefits, the design of cellular manufacturing systems (CMS) is complex for real life problems. It has been known that the cell formation problem (CFP) in CMS is one of the NP-hard combinational prob-lems (Ballakur & Steudel, 1987). Many models and solution ap-proaches have been developed to identify machine cells and part families, as it becomes difficult to obtain optimal solutions in an acceptable amount of time, especially for large-sized problems. These approaches can be classified into three main categories: mathematical programming (MP) models (e.g.,Albadawi, Bashir,
& Chen, 2005; Boctor, 1991; Kumar, Kusiak, & Vannelli, 1986; Lozano, Adenso-Diaz, & Onieva, 1999; Lozano, Guerrero, Eguia, & Onieva, 1999; Srinivasan, Narendran, & Mahadevan, 1990; Wang, 2003), heuristic/meta-heuristic solution algorithms (e.g., Diaz, Lozano, Racero, & Guerrero, 2001; Lei & Wu, 2005; Sofianopoulou, 1999; Sun, Lin, & Batta, 1995; Wu, Chang, & Chung, 2008; Wu, Chung, & Chang, 2008; Wu, Low, & Wu, 2004), and similarity coefficient methods (SCM) (e.g., Alhourani & Seifoddini, 2007; McAuley, 1972; Nair & Narendran, 1998; Yin & Yasuda, 2006).
Most of the above CF researches assume that each part has a un-ique process routing. However, it is well known that alternatives may exist in any level of a process plan. When each part has alter-native process routings (APR), the CFP becomes the generalized CFP (Kusiak, 1987). Explicit consideration of APR may result in additional flexibility in the CMS design.
Over the past four decades, the machine-part cell formation problem has been the subject of numerous studies. Many research-ers have applied various methodologies in an effort to determine the optimal clustering of machines and the optimal groupings of parts into families. However, only a limited amount of research in the context of the CFP has dealt with machine breakdowns or reliability issues (e.g., Das, Lashkari, & Sengupta, 2007; Diallo, Perreval, & Quillot, 2001; Jabal Ameli & Arkat, 2008; Jabal Ameli, Arkat, & Barzinpour, 2008; Logendran & Talkington, 1997; Savsar, 2000; Zakarian & Kusiak, 1997). Traditionally, CF and work alloca-tion are performed, assuming that all the machines are 100% reliable. However, this is not always the case. Machines are key elements in manufacturing systems and oftentimes it is not
0360-8352/$ - see front matter Ó 2010 Published by Elsevier Ltd. doi:10.1016/j.cie.2010.08.016
q
This manuscript was processed by Area Editor Gursel A. Suer.
⇑ Corresponding author. Tel.: +886 2 86746574; fax: +886 2 86715912. E-mail addresses:[email protected](S.-H. Chung),[email protected]. edu.tw(T.-H. Wu),[email protected](C.-C. Chang).
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possible to handle their breakdowns as quickly as the production requirements dictate. Their breakdowns can dramatically affect system performance measures and bring about detrimental effects on the due date performance. Machine failures should hence be ta-ken into account during the design of CMS to improve the overall performance of the system (Jeon, Broering, Leep, Parsaei, & Wong, 1998). The machine breakdown cost generally consists of machine repairing costs, production suspension costs, and capacity lost costs, etc. (Jabal Ameli & Arkat, 2008).
Jabal Ameli and Arkat (2008)formulated a binary integer pro-gramming model for the CFP accounting for APR and the machine reliability issue. They solved the model using a mathematical pro-gramming approach. However, since CFP is an NP-hard problem, it usually takes a large amount of computational efforts to solve the problem when classical optimization methods are used, especially for large-sized problems. Thus, there is a need to develop an efficient and effective solution approach capable of handling this problem.
Due to their excellent performances in solving combinatorial optimization problems, meta-heuristic algorithms such as genetic algorithm (GA), simulated annealing (SA), neural network (NN) and tabu search (TS) are grouped into another class of search meth-ods that have been adapted to solve the CF problem and its variants efficiently. Among the aforementioned meta-heuristic algorithms, TS has been successfully used to solve many problems appeared in manufacturing system including cell formation problems ( Loz-ano, Adenso-Diaz et al., 1999;Lozano, Guerrero et al., 1999). TS uses flexible memory structures to store information and attri-butes of solutions from the recent history of the search. TS gives some recently or frequently visited solutions (moves) a tabu restriction to keep the solution search process from being trapped at a local optimum.
The mutation operator of the genetic algorithm (GA) is another well-known technique, famous for its capability to escape from lo-cal solutions and prevent premature convergence. It is used mainly to increase the diversity of the population and to ensure that an extensive search will be performed.
This study anticipates the synergy effects between the TS and the GA by presenting an efficient algorithm using the TS, together with the mutation operator from the GA, to increase the quality and efficiency of solutions.
The remainder of this article is organized as follows: Section2
describes the problem definition including the CFP with alternative routings and the issues of machine reliability. The mathematical model presented byJabal Ameli and Arkat (2008)for solving the problem is given and reviewed in this section as well. Section3
details the proposed TS algorithm including the generation of
initial solutions and solution improvement procedures. Computa-tional results on test problems are reported in Section4. Section
5concludes the paper.
2. Problem definition
This section describes the problem definition of CFP accounting for APR and the machine reliability issue. A 0–1 integer program-ming model formulated byJabal Ameli and Arkat (2008)for solving this complicated problem is introduced as well.
2.1. Cell formation problem with alternative process routings In a simple CFP, cell formation in a given 0–1 machine-part inci-dence matrix involves the rearrangement of its rows and columns to create part families and machine cells. Researches usually at-tempt to determine a rearrangement in which the intercellular movement can be minimized and the utilization of the machines within a cell maximized. After the rearrangement, blocks can be observed along the diagonal of the matrix. In the matrix, any 1s outside the diagonal blocks are called ‘‘exceptional elements’’; any 0s inside the diagonal blocks are called ‘‘voids’’.
Cases in which each part may have more than one process rout-ings are more complicated than the simple CFP. A process routing for a given part is a set of machines that have passed by this part. It is assumed that the sequence of machines in each process routing is identical with the operation sequence of the corresponding part. When parts are allowed to have more than one process routing, such as the case shown inTable 1, the CFP becomes generalized, wherein cases are more complicated than the simple cell formation problem. Under this circumstance, the formation of part families, machine cells, and selection of routings for each part need to be determined to achieve the decision objectives, such as the minimi-zation of intercellular movement or the maximiminimi-zation of grouping efficacy.
2.2. Machine reliability
Machines are key elements in manufacturing systems. Thus, machine reliability should be taken into account during the design of the CMS. The reliability of a machine is defined as R ¼ expðktÞ, where k is the machine failure rate and t is the machine operating time. A common way of dealing with machines reliability concern in the design phase of a manufacturing system is by the evaluation of the quantities of the mean time between failures (MTBF). MTBF Nomenclature m number of machines M machines set p number of parts P parts set NC number of cells C cells set
Vi production volume for part i
Qi number of routings for part i
Um maximum number of machines in each cell
Lm minimum number of machines in each cell
Ai unit cost of intercellular movement for part i
Kij number of operations in routing j of part i the
opera-tions of part i along route j are processed on a machines’ set of uð1Þij ;uð2Þij ; . . . ;uijðkÞ;uðkþ1Þij ; . . . ;uðKij1Þ
ij ;u ðKijÞ
ij
n o
uðkÞij machine index for the kth operation of part i along route j
TðkÞij processing time for the kth operation of part i along route j
Bk breakdown cost for machine k
MTBFk mean time between failures for machine k
uðKijÞ
ij Machine’s index in routing j of part i
Ykl 1, if machine k locates in cell l; 0, otherwise
Zij 1, if routing j of part i selected; 0, otherwise
Xijklsl 1, if routing j of part i is selected; machine k locates in
can be obtained by taking the reciprocal of k. As long as the break-down cost for each machine is known in advance, the cost caused by machine unreliability can be acquired after simple calculation.
Jabal Ameli and Arkat (2008)have presented a mathematical approach to calculate the machine breakdown cost. This is achieved by dividing the production time by the MTBF and then multiplying this quantity by the unit machine breakdown cost. The decision objective of their research is to minimize the sum of total intercellular movement cost and the machine breakdown cost. The 0–1 integer programming model that they formulated is given below, and the notations are introduced first.
Intercellular movement cost:
Inter C ¼X p i¼1 XQi j¼1 X Kij1 k¼1 XNC l¼1 AiViX ij uðkÞ ij l uðkþ1Þ ij l ð1Þ
Machine breakdown cost:
TBC ¼X p i¼1 XQi j¼1 XKij k¼1 Zij ViT ðkÞ ij B uðkÞij MTBF uðkÞ ij ð2Þ
The 0–1 integer programming model is as follows:
Min TC ¼ Inter C þ TBC ð3Þ s.t. XQi j¼1 Zij¼ 1;
8
i 2 P ð4Þ Lm6 Xm k¼1 Ykl6Um;8
l 2 C ð5Þ XNC l¼1 Ykl¼ 1;8
k 2 M ð6Þ Xijklsl6Zij;8
i; j; k; l; s ð7Þ Xijklsl6Ykl;8
i; j; k; l; s ð8Þ Xijklsl6ð1 YslÞ;8
i; j; k; l; s ð9Þ Zijþ Yklþ ð1 YslÞ Xijklsl62;8
i; j; k; l; s ð10Þ Ykl; Zij; Xijklsl2 f0; 1g8
i; j; k; l; s ð11ÞIn the above model, Eqs.(1) and (2)show the calculation of the total intercellular movement cost and the machine breakdown cost,
respectively. Eq.(3)is the objective function, which seeks the min-imization of total cost of intercellular movement and machine breakdown. Eq.(4)indicates that only one process routing will be assigned to each part. Eq. (5)assigns the upper and lower limits of the cell size. Eq.(6)provides a restriction that each machine will be assigned to exactly one cell. Eqs.(7)–(9)ensure that if one of the primary binary variables takes has a zero value, then their corre-sponding new variables will take a zero value as well. Eq.(10) en-sures that if all of the primary variables take unit values, then their corresponding new variables take unit values as well. Eq.
(11)indicates that Ykl, Zijand Xijklslare 0–1 binary decision variables.
Although the original objective function has been transformed into linear form, which makes several linear programming soft-ware empowered to solve this model, the large number of binary variables and constraints, makes it difficult to obtain optimal solu-tions when the problem sizes increase. Developing good heuristic approaches is more appropriate than using the exact method in terms of solution efficiency, especially for large-sized problems. This paper, thus, presents an efficient tabu search algorithm possessing the features of both the tabu search and the genetic algorithm and is collaborated with the delicate design of neighbor-hood solution searching. The proposed algorithm is described and explained in detail in the next section.
3. Proposed tabu search (TS) algorithm
When designing a heuristic search algorithm, there are several important things to keep in mind. The first is to develop a mecha-nism for searching the neighborhood solutions for improvement. Since the neighborhood is what will be searched next, the choice of the neighborhood function will strongly influence the direction that the search takes. Another item that needs to be considered is the mechanism for allowing the technique to escape local optima and settle only on a global optimum.
TS is a meta-heuristic algorithm developed by Glover which has been successfully used to generate solutions for a wide variety of combinatorial problems. The main ideas of TS are to avoid recently visited area of the solution space and to guide the search towards new and promising areas. Non-improving moves are allowed to es-cape from the local optima, and attributes of recently performed moves are declared tabu or forbidden for a number of iterations to avoid cycling. For more details about the tabu search methodol-ogy, seeGlover (1989)andGlover (1990).
In GA, some initial solutions are selected to be parents to gener-ate offspring via the crossover operator. All the solutions are then evaluated and selected based on Darwin’s concept of survival of the fittest. The process of reproduction, evaluation, and selection is repeated until the stopping criterion is met. In GA, the mutation
Table 1
APR and processing times for the numerical example (Bhide, Bhandwale, & Kesavadas, 2005).
PV 75 130 110 145 110 105 140 115 PN P1 P2 P3 P4 P5 P6 P7 P8 RN R1 R2 R3 R1 R2 R3 R1 R2 R1 R2 R1 R2 R1 R2 R1 R2 R3 R4 R1 R2 M1 * 1(5) 1(4) 1(4) 1(4) 1(5) 1(5) 1(4) 1(4) 1(5) 1(5) 1(4) M2 1(5) 1(5) 1(5) 1(5) 1(3) 1(5) 1(3) 1(3) 2(3) 1(4) M3 2(3) 2(3) 2(3) 2(3) M4 2(3) 2(4) 3(3) 3(5) 3(5) 2(4) 2(4) 2(4) M5 3(4) 2(4) 3(3) 2(3) 2(3) 3(4) 4(4) 3(3) 3(3) M6 2(5) 2(4) 3(4) 2(3) 2(4) M7 2(5) 3(5) 2(5) 2(5) M8 3(4) 4(4) 3(4) 4(3) 4(3) 4(4) 5(4) 3(5) 3(5) 4(5) M9 4(5) 3(5) 3(3) 3(5) 4(5) 4(5) 3(5) 4(5)
PV: Production Volume; PN: Part Number; RN: Routing Number.
operator is usually applied to solutions at hand with a certain probability to escape from local solutions and/or to prevent prema-ture convergence. This special feaprema-ture of the mutation operator provides a higher degree of diversification in the solution searching process. It is expected that the synergy effects from both the TS and the GA can be appreciated through a proper collocation of both techniques.
The proposed TS procedure consists of two stages: the initial solu-tion construcsolu-tion and the improvements stage. The similarity coeffi-cient-based method (SCM) is adapted in the first stage to produce good initial solutions, while the TS continuously improves and gen-erates more effective solutions through the neighborhood moves in the second stage. The details of these procedures are given below.
3.1. Initial solution construction
When generating the initial solutions, the SCM-based procedure follows three steps: (1) formation of machine cells; (2) selection of routings for each part; and (3) formation of part families.
3.1.1. Formation of machine cells
The part-based SCM ofKusiak and Cho (1992)and the machine-based SCM ofWon and Kim (1997)are the two most widely used generalized similarity coefficient methods for considering alterna-tive process routings. They are an extension of the Single Linkage Clustering Algorithm (SLCA) ofMcAuley (1972). Compared to the machine-based, the part-based SCM suffers from a computational burden since the number of parts in a cell formation problem is usually much greater than the number of machines. The ma-chine-based SCM is adapted in this research.
According toSeifoddini and Djassemi (1995), incorporation of production volume into the similarity measures increases the chance of components with high production volumes being pro-cessed within a single cell. As a result, there will be fewer intercel-lular movements and lower material handling costs. Won and Kim’s machine-based SCM is thus modified to incorporate the pro-duction volume information. Consider a specific machine-part ma-trix with alternative routings and the information of production volume, the corresponding similarity matrix for machines can be obtained by using the following formula:
Sij¼
Nij
Niþ Nj Nij ð12Þ
where
Sij similarity coefficient between machines i and j
Ni Ppk¼1Vkaki,
Nj ¼Ppk¼1Vkakj,
Nij ¼Ppk¼1Vkakij
P number of parts
Vk production volume of part k
ak
i 1 if i 2 some routing of part k
0 otherwise
ak
j 1 if j 2 some routing of part k
0 otherwise
ak
ij 1 if i; j 2 some routing of part k synchronously
0 otherwise
After calculating the similarity matrix for each pair of machines, the initial machines assignment is generated by using the following rule: the higher similarity measure a pair of machines has, they should be placed in the same cell with higher priority. This process is repeated until all machines have been assigned to cells. Consider
the numerical example with alternative process routings inTable 1. The corresponding similarity matrix for machines can be obtained by using Eq.(12)and is listed inTable 2.
Suppose there are two cells to be formed. The largest coefficient in the similarity matrix ofTable 3is 0.72, indicating that machines 2 and 6 must be assigned to cell 1. The second largest coefficient in the matrix, 0.70, appears in pair (4, 3). Because machines 3 and 4 have not been assigned to any cell, they are assigned to cell 2. Pair (8, 5) is the next choice. When determining to which cell machine 5 should be assigned, the similarity coefficients of machine 5 with machines in each cell are examined respectively. For cell 1, the largest similarity coefficient with machine 5 appears in machine 2, which is equal to 0.34. For cell 2, the largest similarity coefficient with machine 5 appears in machine 4, which is equal to 0.48. Ma-chine 5 is thus assigned to cell 2 together with maMa-chines 3 and 4. By repeating the same logic, it can finally be determined that ma-chines 2, 6 and 7 should be assigned to cell 1; while mama-chines 1, 3, 4, 5, 8 and 9 are assigned to cell 2. The sample problem displayed in
Table 1is thus rearranged as shown inFig. 1.
3.1.2. Selection of routings for each part
The next task deals with assigning a routing for each part after the machine cells have been obtained. The part routings are
Table 2
Similarity matrix for machines in numerical example.
Machine 1 2 3 4 5 6 7 8 9 1 – 2 0.15 – 3 0.42 0.00 – 4 0.18 0.17 0.70 – 5 0.00 0.34 0.24 0.48 – 6 0.16 0.72 0.00 0.00 0.00 – 7 0.17 0.32 0.00 0.23 0.00 0.00 – 8 0.00 0.00 0.00 0.43 0.68 0.00 0.18 – 9 0.00 0.00 0.00 0.00 0.58 0.32 0.26 0.00 – Table 3
Machine reliability information for the numerical example.
Machine Breakdown cost MTBF (min)
M1 900 5400 M2 2000 3060 M3 2000 4380 M4 1600 3600 M5 1500 4560 M6 1800 3720 M7 1400 4260 M8 1700 3480 M9 1500 3900 PV 75 130 110 145 110 105 140 115 PN P1 P2 P3 P4 P5 P6 P7 P8 RN R1 R2 R3 R1 R2 R3 R1 R2 R1 R2 R1 R2 R1 R2 R1 R2 R3 R4 R1 R2 M2 1 1 1 1 1 1 1 1 2 1 M6 2 2 3 2 2 M7 2 3 2 2 M1 1 1 1 1 1 1 1 1 1 1 1 M3 2 2 2 2 M4 2 2 3 3 3 2 2 2 M5 3 2 3 2 2 3 4 3 3 M8 3 4 3 4 4 4 5 3 3 4 M9 4 3 3 3 4 4 3 4
assigned to machine cells that would result in the least cost of intercellular movement and the machine breakdown. The unit intercellular movement for each trip made is assumed to be five, and machine breakdown information such as shown in Table 3
has to be given in order to calculate the cost. The part routing assignment procedure is described below:
Step 1: Read the results of machine cells formed by the machine-based similarity matrix.
Step 2: For each part with alternative routings, find the routing that will result in the least sum of the intercellular movement cost (Inter_C) and the machine breakdown cost (TBC). If a tie happens, make a random selection.
Step 3: Repeat Step 2 until the process routing has been deter-mined for each part. Results of machines assignment shown inFig. 2are used to demonstrate the above procedure.
3.1.3. Formation of part families
Wu, Chang, et al. (2008) presented a simple procedure for assigning machines to manufacturing cells in which the number of voids and exceptional elements – major components comprising the formula of grouping efficacy – are explicitly considered. Instead of using this procedure for assigning machines to cells, their ap-proach is adapted for assigning parts to cells in this study. The pro-cedure is summarized as follows:
Step 1: Read the results of machines assignment and routing selection for each part.
Step 2: For each part, find the cell to which a part assignment will result in the least sum of number of exceptional elements and the number of voids. If a tie happens, assign the part to a cell with the least number of voids.
Step 3: Repeat Step 2 until all parts have been assigned to cells. Results of machines assignment and routings selection shown inFig. 2are used to demonstrate the above procedure. After calcu-lating the sum of numbers of voids and exceptional elements for each part-cell combination, it can be observed inFig. 3that parts 5, 6 and 8 are assigned to cell 1, while parts 1–4 and 7 are assigned to cell 2. The initial solution matrix for this CF problem has been
generated and shown inFig. 3with total intercellular movement cost 1075 and machine breakdown cost 5236.
3.2. Solution improvements
The initial solution generated in Section3.1is to be improved through the tabu search iteratively to produce more effective solu-tions. The elements comprising the proposed tabu search algo-rithm are described below.
3.2.1. Configuration
An easy way to represent a configuration of a feasible solution of the CF problem is a string, the size of which is equal to the num-ber of machines, as shown inFig. 4. In such a configuration, the jth bit of the string stores the identifier of the cell to which the ma-chine is assigned. From the string (2, 1, 2, 2, 2, 1, 1, 2, 2), it is known that machines 2, 6, 7 are assigned to cell 1, while machines 1, 3, 4, 5, 8 and 9 are assigned to cell 2.
PV 75 130 110 145 110 105 140 115 PN P1 P2 P3 P4 P5 P6 P7 P8 RN R1 R2 R3 R1 R2 R3 R1 R2 R1 R2 R1 R2 R1 R2 R1 R2 R3 R4 R1 R2 M2 1 1 1 1 1 1 1 1 2 1 M6 2 2 3 2 2 M7 2 3 2 2 M1 1 1 1 1 1 1 1 1 1 1 1 M3 2 2 2 2 M4 2 2 3 3 3 2 2 2 M5 3 2 3 2 2 3 4 3 3 M8 3 4 3 4 4 4 5 3 3 4 M9 4 3 3 3 4 4 3 4 TBC 405 573 488 700 703 807 530 532 925 1116 608 661 772 499 773 815 1004 931 244 523 Inter_C 0 375 375 0 650 650 0 0 0 0 550 1100 525 1050 0 700 700 1400 575 0 TC 405 948 863 700 1353 1457 530 532 925 1116 1158 1761 1297 1549 773 1515 1704 2331 819 523
Fig. 2. Selection of routings.
PV 110 105 115 75 130 110 145 140 PN P5 P6 P8 P1 P2 P3 P4 P7 RN R1 R1 R2 R1 R1 R1 R1 R1 M2 1 1 1 M6 2 M7 2 2 M1 1 1 1 1 1 M3 2 2 M4 2 2 3 2 M5 3 3 3 3 M8 3 4 4 4 M9 3 4 4 TBC 608 772 523 405 700 530 925 773 Inter_C 550 525 0 0 0 0 0 0 TC 1158 1297 523 405 700 530 925 773
3.2.2. Neighborhood solution searching
In this study, the neighborhood of a given solution is defined as the set of all feasible solutions reachable by an insertion-move. The insertion-move is an operation that moves a machine j from its current cell i (source cell) to a new cell i0 (destination cell). The
new move is denoted as (i0, j). For the insertion-move, a move that
results in the most improvement in the objective function value from the current solution is selected – that is:
Zði0;jÞ ¼ Maxfobjði0;jÞ objði;jÞ;
8
i; i02 NF andðR NT or 2 NAÞ; i0–i;
8
j 2 Mg ð13Þwhere NFis the set of solutions satisfying the upper and lower limits
of cell size; NTis the set of tabu list; NAis the set of solutions
satis-fying the aspiration criterion; M is the set for machines. The above formula implies that every possible move will be evaluated as long as it is not in tabu status and it satisfies the upper and lower limits of the cell size.
3.2.3. Tabu list
In the process of tabu search, certain moves are characterized as tabu for some iterations (tabu tenure/tabu list size) to avoid repe-tition of previously visited solutions. In this paper, a tabu list TL[m][NC][NC] with a three-dimensional array (m NC NC) is used to check if a move from a solution to its neighborhood is for-bidden or allowed, where m is the number of machines and NC is the number of cells. If machine j moves from its current cell i to a new cell i0, then moving machine j from cell i0to cell i will be
for-bidden for a certain number of iterations, which is equal to the tabu list size (e.g. TL[j][i0][i] = tls). Previous studies have shown that the best tabu list size is between 5 and 12 in many applications, with 7 being the most recommended one (Glover, 1990). This sug-gestion is followed in this study.
3.2.4. Aspiration criterion
The tabu restriction may be overridden if the move will result in a solution that is better than the best solution found thus far. This aspiration criterion is applied in the proposed algorithm.
3.2.5. Machine mutation strategy
The mechanism of mutation aims at maintaining diversity in the population so that the large areas of the space are searched. In this study, when the number of moves has not been improved within a certain number of iterations, mut_check, the machine mutation strategy is implemented by reassigning a machine to any cells other than the current one based on a prescribed proba-bility
c
. That is, all machines are probable to change cell when ma-chine mutation is applied. For each mama-chine in the incumbent solution, a random number from (0, 1) is first drawn. If the value is greater thanc
, then the machine is assigned to another randomly determined cell; otherwise, it stays in the current cell. Note that when machines change cell, the resulting solution would not be ac-cepted unless it satisfies the upper and lower limits of cell size. Through this strategy, the search is able to explore a large solution space, thereby enhancing the possibility of finding the optimum solution in a very short time. The procedure of the machine muta-tion strategy in the pseudo-code format is shown inFig. 5. The va-lue of mut_check is determined by the formula, m(NC 1)/2; whilec
is set to 0.8 in this study.3.2.6. Stopping criterion
The proposed solution procedure will be terminated if a maxi-mum number of iterations Nmaxhas been reached, or the solution
has not been improved within a certain number of iterations stag_ check. After intensive testing, the values of Nmaxand stag_check are
set at 9000, 3000 (one third of Nmax), respectively.
3.3. Proposed algorithm TSM
This section describes the proposed TS algorithm with mutation (TSM) in detail. It is evident that the number of cells to be formed will affect the grouping solutions obtained in the CF problem. Un-like many researches in literature where the number of cells to be formed is prescribed beforehand, the number of cells resulting in the best objective values will be automatically calculated and used in the proposed TSM. However, to preserve flexibility, users are permitted to specify the preferred number of cells when imple-menting the algorithm. Before explaining the solution procedure, notations in addition to those in Section2.2are introduced.
Nmax maximum number of iterations
counter_stag number of times the incumbent solution did not improve
counter_mut number of times the mutation strategy has been implemented
NT set of tabu list
NA set of solutions satisfying aspiration criterion
NF set of solutions satisfying the upper and lower
limits of the cell size
m0 initial solution of machines assignment
mc current solution of machines assignment
m0 neighborhood solution of machines assignment m* incumbent solution of machines assignment of
current cell size
m** best solution of machines assignment so far
p0 initial solution of parts assignment
pc current solution of parts assignment
p0 neighborhood solution of parts assignment p* incumbent solution of parts assignment of
current cell size
p** best solution of parts assignment so far
r0 initial solution of routings selection
rc current solution of routings selection
r0 neighborhood solution of routings selection r* incumbent solution of routings selection of
current cell size
r** best solution of routings selection so far
Machine # 1 2 3 4 5 6 7 8 9
Cell # 2 1 2 2 2 1 1 2 2
Fig. 4. Configuration of a feasible solution to the CF problem.
S0 objective function value of initial cell configuration
(m0, p0, r0)
Sc objective function value of current cell configuration
(mc, pc, rc)
S0 objective function value of neighborhood cell configuration (m0, p0, r0)
S* objective function value of incumbent cell configuration
(m*, p*, r*)
S** objective function value of best cell configuration
(m**, p**, r**)so far
The proposed algorithm TSM is described as follows:
Step 1: Set NC ¼ dm=Ume.
Step 2: Generate initial cell configuration (m0, p0, r0) using the
initial solution construction procedure in Section3.1. Calculate the objective function value S0.
Step 3: Initialization: Let counter_iter = 0, counter_stag = 0, mc
m0, pc p0, rc r0, Sc S0, S* S0, NT= £.
Step 4: If counter_iter 6 Nmax and counter_stag 6 stag_check,
repeat Steps 5–10; otherwise, go to Step 11.
Step 5: If counter_mut P mut_check, then apply the mutation operator, as mentioned in Section3.2.5, to generate a new cell configuration (mc, pc, rc) and let counter_mut = 0.
Step 6: Search for a best neighborhood cell configuration {(m0, p0, r0)|m02 NF and m0
RNT or m0
2 NA} by performing the
insertion-move. Calculate the objective function value S0. Step 7: Update tabu list NT.
Step 8: If S0< S*then S* S0
, m* m0
, p* p0
, r* r0
, counter_ stag = 0, counter_mut = 0; otherwise, counter_stag = counter_ stag + 1, counter_mut = counter_mut + 1.
Step 9: Let Sc S 0 , mc m 0 , pc p 0 , rc r 0 . Step 10: counter_iter = counter_iter + 1, go to Step 4.
Step 11. If S*< S**then S** S*, m** m*, p** p*, r** r*, NC =
NC + 1, go to Step 2; otherwise report the best solutions so far, and stop the algorithm.
Note that algorithm TSM consists of a TS procedure that will be repeatedly applied until a cell formation resulting in the best objective function values, e.g., minimization of the total intercellu-lar movement cost and the machine breakdown cost in this study, has been found. In Step 1, the initial number of cells is set at the nearest integer that is greater than m/Um; it gradually increases
by increments of 1 as long as solution improvement is observed in Step 11. Every time the number of cells is increased, another TS procedure will be started. For a specific cell size, the best routing selection and grouping plan for parts and machines will be calcu-lated iteratively and obtained in Steps 5–10. All algorithmic param-eters and counters are initialized in Step 3. Initial solutions of machine cells, routing selections, and assignments to machine cells are generated in Step 2. As long as the value of counter_mut is
smal-ler than mut_check, a new neighborhood solution is generated through the insertion-move in Step 6; otherwise, gene-by-gene mutation is applied to machines to generate a new solution with higher degree of diversification in Step 5. If the newly generated neighborhood solution results in a better objective function value, the incumbent solution will be updated and the counter_stag and counter_mut will be set to 0 in Step 8; otherwise, the counter_stag and counter_mut are increased by 1. The solution process repeats until any of the two stopping criteria in Step 4 is met. The incum-bent solution obtained at this point represents the best solution of the current cell size. If larger cell sizes are considered, it is possible that better solutions may be obtained. The incumbent solution of current cell size is thus compared to the best solution found so far in Step 11 to determine whether to increase the cell size by 1 and restart another TS procedure to continue the search or to re-port the best solution found and terminate TSM.
For users having specific preferences in cell size, the proposed algorithm can save considerable amounts of run time since it will skip the process of iteratively searching for the cell size that will result in the best objective function values. The savings in run time become even more significant as the cell size increases.
4. Computational results
To validate the quality of the solutions provided by the pro-posed algorithm TSM, we have to prepare suitable test instances. However, only a limited amount of research in the context of cell formation problem has dealt with machine breakdown or reliabil-ity issues, suitable test problem can very rarely be found from the literature. Eight test instances, as shown inTable 4, are solved in this research. Among them, two (#1 and #5) are drawn from the literature and have been solved optimally in previous studies (Jabal Ameli & Arkat, 2008). The remaining six problems are prepared by adding self-created data such as machine breakdown cost (BC), mean time between failure (MTBF), and production time (PT) to test instances chosen from the literature which have machine-part matrices and process routing data ready. Detailed data of each new test problems are available under request to the authors.
Table 4
Data of test problems.
No. Original source Size (m p r) Randomly generated data
1 Bhide et al. (2005) 9 8 20 –
2 Kim, Baek, and Baek (2004) 10 10 25 BC, MTBF
3 Sofianopoulou (1999) 12 20 26 BC, MTBF, PT
4 Sofianopoulou (1999) 14 20 45 BC, MTBF, PT
5 Kazerooni, Luong, and Abhary (1997) 17 30 63 –
6 Sofianopoulou (1999) 18 30 59 BC, MTBF, PT
7 Lee, Luong, and Abhary (1997) 30 40 89 BC, MTBF, PT
8 Hu and Yasuda (2006) 30 70 149 BC, MTBF, PT
BC: machine breakdown cost (10001700) (rand()%8 + 10) 100); MTBF: mean time between failure (8005000) (rand()%4201 + 800); PT: production times (26) (rand()%5 + 2).
Table 5
Parameters setting for TSM.
Parameter Value tls 7 mut_check m(NC 1)/2 c 0.8 Nmax 9000 stag_check Nmax/3
Table 4describes basic problem data and how the machine breakdown cost (BC), mean time between failures (MTBF), and pro-duction time (PT) data are created:
1. BC is set to be any number between 1000 and 1700. 2. MTBF is set to be any number between 800 and 5000. 3. PT is set to be any number between 2 and 6.
The proposed algorithm was coded in C++ using Microsoft Vi-sual Studio 6.0 and implemented on a Intel(R) 1.66 GHz PC with 1 GB RAM. Due to the proposed method might have stochastic fea-tures, five independent runs were performed for each test instance. The intercellular movement unit cost for all instances is assumed to be five. The computational results are compared with the opti-mal solutions obtained by the LINGO 8.0 software. Parameter set-tings for the proposed TSM are given inTable 5.
4.1. Results comparison
Eight test instances considering machine reliability are solved by the proposed TSM and compared with the branch and bound (B&B) algorithm with the LINGO 8.0 software. The maximum run time is set to be 25 h when running the LINGO. The computational results are summarized and compared inTable 6. The results show that the proposed TSM is able to achieve global optimum in seven out of eight test instances in less than 9 s. In addition, the standard deviation for all test instances is equal to 0, which indicates that TSM is able to consistently produce good solutions. As for test in-stance #8, due to its giant problem size, even the LINGO is not able to find the optimal solution after 25 h of running with the objective value of 2365 obtained. In contrast, the proposed TSM results in a final solution of 2099 in less than nine seconds, which is more than 10% better than the solution of LINGO in terms of solution quality. The final solutions for all eight test problems obtained by TSM are available under request to the authors.
Among the eight test instances, test instance #5 is a medium-sized example.Jabal Ameli and Arkat (2008)solved this problem
using LINGO 8.0. Based on the data of the optimal machine-part matrix that appeared inTable 6of their article, the authors have obtained 50,164 as the optimal objective function value. Jabal Ameli and Arkat (2008)reported that it took about 3 h to obtain the optimal solution. In contrast, the proposed TSM was able to find the optimal solution in 1.775 s, illustrating the superiority of TSM in solution efficiency over other approaches derived from the literature.
To the authors’ knowledge, test instance #8 generated in this study is the largest example that has ever been used in literature in analyzing the cell formation problem with alternative process routings and machine reliability considerations. Although it has big size, it still can be solved by the TSM in less than 9 s. The sur-prisingly good solution efficiency should be attributed to the syn-ergy effects through a proper collocation of both TS and GA techniques. It is also believed that several counters used in the TSM procedure, such as the counter_stag and counter_mut, play important roles in monitoring the situation of solution stagnancy and controlling the timing for activating the mutation strategy, which contribute to increase the quality and efficiency of the solutions.
4.2. Ignoring reliability vs. considering reliability
Table 7gives the computational results when the reliability is-sue is considered in the model. From this table, it can be seen that a greater number of intercellular movements have resulted in when the machine reliability is considered, as opposed to the situation where reliability concern is ignored. But as would be expected, the machine break down cost (TBC) decreased, and so did the total system cost (TC). Thus, the reliability consideration does have meaningful effects on reducing the total system cost.
5. Conclusions
Very limited amount of articles have simultaneously considered the issues of production volume, production sequence, machine
Table 6
Results comparison of TSM and optimal solutions by LINGO 8.0.
Test instances LINGO 8.0 software (B&B) Proposed method (TSM)
No. Source Size (m p r) Lm Um NC Inter_C TBC TC CPU (s) NC Inter_C TBC TC Std. CPU(s)
1 Bhide et al. (2005) 9 8 20 2 6 2 550 5146 5696* 30 2 550 5146 5696* 0 0.303 2 This study 10 10 25 2 5 2 380 1539 1919* 1 2 380 1539 1919* 0 0.366 3 This study 12 20 26 2 5 3 150 247 397* 21 3 150 247 397* 0 0.790 4 This study 14 20 45 2 5 3 125 213 338* 9226 3 125 213 338* 0 1.187 5 Kazerooni et al. (1997) 17 30 63 2 5 4 4300 45864 50164* 323 4 4300 45864 50164* 0 1.775 6 This study 18 30 59 2 7 3 165 305 470* 1053 3 165 305 470* 0 1.250 7 This study 30 40 89 2 7 5 2925 38192 41117* 7198 5 2925 38192 41117* 0 3.296 8 This study 30 70 149 2 8 4 1160 1205 2365 90000 4 895 1204 2099 0 8.467 *Global optimum. Table 7
Comparison of computational results for ignoring and considering machine reliability.
Test instances Ignoring reliability Considering reliability Cost decreased (%)
No. Source Size (m p r) Lm Um NC Inter_C TBC TC CPU(s) NC Inter_C TBC TC CPU (s)
1 Bhide et al. (2005) 9 8 20 2 6 2 525 5366 5891 0.315 2 550 5146 5696 0.303 3.31 2 This study 10 10 25 2 5 2 320 1667 1987 0.375 2 380 1539 1919 0.366 3.44 3 This study 12 20 26 2 5 3 145 265 410 0.765 3 150 247 397 0.790 3.08 4 This study 14 20 45 2 5 3 125 216 341 0.928 3 125 213 338 1.187 0.95 5 Kazerooni et al. (1997) 17 30 63 2 5 4 3800 46446 50246 1.593 4 4300 45864 50164 1.775 0.16 6 This study 18 30 59 2 7 3 160 323 483 1.203 3 165 305 470 1.250 2.72 7 This study 30 40 89 2 7 6 475 44152 44627 5.958 5 2925 38192 41117 3.296 7.87 8 This study 30 70 149 2 8 4 875 1316 2191 7.763 4 895 1204 2099 8.467 4.20
reliability and alternative process routings in cell formation prob-lem so far. Accounting for these factors and integrating them into one model makes the CF problem complex but more realistic. AlthoughJabal Ameli and Arkat (2008)have formulated a binary integer programming model for solving this complicated problem, the resulting long computer run time has motivated the authors of this study to propose an efficient meta-heuristic algorithm. Since both the tabu search and the genetic algorithm have had excellent performances in solving many combinatorial optimization prob-lems, the synergy effects of both heuristic approaches are antici-pated in a hybrid algorithm designed to increase the quality and efficiency of solutions. The proposed TS procedure consists of two stages, the initial solution construction and the improvements stage. The similarity coefficient-based method is adapted in the first stage to produce good initial solutions, while the tabu search continuously improves and generates more effective solutions through the neighborhood moves in the second stage. Computa-tional experiences of test problems from the literature as well as those newly generated by this study show that the reliability con-sideration has meaningful effects on reducing the total system cost. Additionally, for the test problem which took the mathemat-ical approach about 3 h to solve, this study is able to solve it opti-mally in less than two seconds. Even the largest test problem that has ever been used in the literature has been solved in less than 9 s. The superiority of the proposed algorithm, TSM, in both solution effectiveness and efficiency over other approach from the litera-ture can be easily observed.
For future research, several other factors may be added into the current model or even treat them as decision objectives. These fac-tors may include the cell layout, intracellular machine layout and the cell load variation.
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