A simulated annealing algorithm for manufacturing cell
formation problems
Tai-Hsi Wu
a,*, Chin-Chih Chang
b, Shu-Hsing Chung
baDepartment of Industrial Engineering and Systems Management, Feng Chia University, 100, Wenhwa Road,
Seatwen, Taichung 407, Taiwan
b
Department of Industrial Engineering and Management, National Chiao Tung University, 1001, Ta Hsueh Road, Hsinchu 300, Taiwan
Abstract
The cell formation problem determines the decomposition of the manufacturing cells of a production system in which machines are assigned to these cells to process one or more part families so that each cell is operated independently and the intercellular movements are minimized or the number of parts flow processed within cells is maximized. In this study, a simple yet effective simulated annealing-based approach, SACF, is proposed to solve the cell formation problem. Considerable efforts are devoted to the design of parts and machine assignment procedures to direct SACF to converge to solutions with good values of grouping efficacy. A set of 25 test problems with various sizes drawn from the literature is used to test the performance of the proposed heuristic algorithm. The corresponding results are compared to several well-known algorithms published. The comparative study shows that the proposed SACF algorithm improves the grouping efficacy for 72% of the test problems. The proposed algorithm should thus be useful to both practitioners and researchers. 2007 Elsevier Ltd. All rights reserved.
Keywords: Simulated annealing; Cell formation problem; Grouping efficacy
1. Introduction
To make manufacturing systems more efficient and pro-ductive, group technology (GT) has been applied within a manufacturing environment. GT groups parts with similar design characteristics or manufacturing characteristics into part families. One application of GT is cellular manufac-turing (CM). A number of benefits arise from adopting CM, such as: reduced inventory, reduced capacity, reduced labor and overtime costs, shorter manufacturing lead times, faster response to internal and external changes such as machine failures, product mix and demand changes (Wemmerlov & Hyer, 1989). Information such as parts to
be produced, process plans and machines to perform all the required operations is needed when designing CM. The entire production system is decomposed into produc-tion cells. Machines are then assigned to these cells to pro-cess one or more part families so that each cell is operated independently and so that the intercellular movements are minimized or the number of part flow processed within cells is maximized, i.e., parts do not have to move from one cell to the other for processing.
This cell formation process is one of the most important steps in CM. It becomes difficult to obtain optimal solu-tions in an acceptable amount of time, especially for prob-lems with large sizes. Extensive research has been devoted to cell formation (CF) problems, with many methods hav-ing been proposed for identifyhav-ing machine cells and part families. Many of them are developed on the basis of heu-ristic clustering techniques to obtain approximate solu-tions, but some of them may be far from optimum. The research of Moon and Kim (1999) takes into account the 0957-4174/$ - see front matter 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.eswa.2007.01.012
*
Corresponding author. Present address: Department of Business Administration, National Taipei University, 151, University Road, San Shia, Taipei 237, Taiwan. Tel.: +886 2 86746558; fax: +886 2 86715912.
E-mail addresses: [email protected], [email protected]
(T.-H. Wu).
www.elsevier.com/locate/eswa Expert Systems with Applications 34 (2008) 1609–1617
Expert Systems with Applications
process plans for parts and manufacturing factors such as production volume and cell size. Their process of forming manufacturing cells starts by collecting the above problem data and then converting it into a weighted graph represen-tation in which the nodes and arcs represent machines and their relationships defined as the value of total part flow between machines, respectively. Similar representations have been used byRajagopalan and Batra (1975), Harlala-kis, Nagi, and Proth (1990), Vohra, Chen, Chang, and Chen (1990), and Wu and Salvendy (1993) with different problem concerns. Some of them (Rajagopalan & Batra, 1975; Harlalakis et al., 1990) consider the situation where the size of each cell and the number of cells to be formed has to be restricted, while some of them (Vohra et al., 1990; Wu & Salvendy, 1993) did not.
Due to their excellent performance in solving combinato-rial optimization problems, meta-heuristic algorithms such as genetic algorithms, simulated annealing, neural networks and tabu search make up another class of search methods that has been adopted to efficiently solve the CF problem and its variants with good results obtained.Sun, Lin, and Batta (1995)presented a short-term tabu search-based algo-rithm for solving the CF problem with the objective of min-imizing the intercellular parts flows, while Wu, Low, and Wu (2004) maximizes the parts flow within cells using long-term tabu search-based algorithm. Aljaber, Baek, and Chen (1997)proposed a tabu search approach to deal with this problem by modeling it as a shortest spanning path problem with respect to both parts and machines. The resulting spanning paths for parts and machines are then decomposed into subgraphs representing machine groups and part families, respectively.Cheng, Gupta, Lee, and Wong (1998)formulated the CF problem as a traveling salesman problem (TSP) and proposed a solution method-ology based on genetic algorithm, while Dimopoulos and Mort (2001) presented a hierarchical clustering approach based on genetic programming. Onwubulo and Mutingi (2001) developed a genetic algorithm, which accounts for inter-cellular movements and the cell-load variation. Conc¸alves and Resende (2004) presented a hybrid algo-rithm combining a local search and a genetic algoalgo-rithm with very promising results reported.
The purpose of this study is to develop a procedure that is efficient and effective for obtaining machine–part groupings when the manufacturing system is represented by a 0–1 machine–part incidence matrix. Since simulated annealing (SA) has been applied to a number of combina-torial problems with fairly good results obtained, it was,
hence, selected in this research as the basis for developing search methods for the CF problem. SA can be viewed as a process which attempts to move from the current solu-tion to its neighborhood solusolu-tions resulting in better objective values. However, for solutions with worse objec-tive values, they are accepted with a specified probability mainly to escape from the local optima in its search for the global optima. A set of 25 test problems with various sizes drawn from the literature is used to test the perfor-mance of the proposed heuristic algorithm. The corre-sponding results are compared to several well-known algorithms published.
The remainder of this article is organized as follows. In Section 2, we describe the problem definition. The pro-posed SA heuristic is presented in Section 3. Section 4 shows the computational results on problems with various sizes, and Section5 concludes the paper.
2. Cell formation problem
Cell formation in a given 0–1 machine–part incidence matrix involves rearrangement of rows and columns of the matrix to create part families and machines cells. In this research, we attempt to determine a rearrangement so that the inter-cellular movement can be minimized and the uti-lization of the machines within a cell can be maximized. Two matrices shown inFig. 1are used to illustrate the con-cept. Fig. 1a is an initial matrix where no blocks can be observed directly. After rearrangement of rows and col-umns, two blocks can be obtained along the diagonal of the solution matrix inFig. 1b.
There have been several measures of goodness of machine–part groups in cellular manufacturing in the liter-ature. Two measures frequently used are the grouping effi-ciency (Chandrashekharan & Rajagopalan, 1986a) and the grouping efficacy (Kumar & Chandrasekharan, 1990) due to they are easy to implement. Grouping efficiency g is defined as follows:
g¼ qg1þ ð1 qÞg2
where g1is the ratio of the number of 1’s in the diagonal
blocks to the total number of elements in the diagonal blocks of the final matrix, g2 is number of 0’s in the
off-diagonal blocks to the total number of elements in the off-diagonal blocks of the final matrix, and q is a weight factor. For those 1’s outside the diagonal blocks, they are called ‘‘exceptional elements’’; while those 0’s inside the diagonal blocks are called ‘‘voids’’.
Parts P1 P2 P3 P4 P5 M1 1 0 0 1 0 M2 0 1 1 0 1 M3 1 0 0 1 0 M4 0 1 1 0 1 Machines M5 1 0 0 1 0 Parts P2 P3 P5 P1 P4 M2 1 1 1 0 0 M4 1 1 1 0 0 M1 0 0 0 1 1 M3 0 0 0 1 1 Machines M5 0 0 0 1 1
Although grouping efficiency has been used widely, it was argued for its low discriminating capability in some cases affected by the size of the matrix. To overcome this problem, Kumar and Chandrasekharan (1990) proposed another measure, the grouping efficacy C, and can be defined as:
C¼e e0 eþ ev
where e is the total number of 1’s in the matrix; e0is the
total number of exceptional elements; and ev is the total
number of voids. Grouping efficacy ranges from 1 to 0, with 1 being the perfect grouping. As grouping efficacy has been widely accepted in recent studies regarding CF problem, it is used as the performance measure for the pro-posed SA algorithm in this study.
3. Simulated annealing approach
Simulated annealing (SA) algorithm was originally pro-posed by Metropolis, Rosenbluth, and Teller (1953) to simulate the annealing process. SA starts with a high tem-perature. After generating an initial solution, it attempts to move from the current solution to one of its neighborhood solutions. The changes in the objective function values (DE) are computed. If the new solution results in better objective value, it is accepted. However, if the new solution yields worse value, it can still be accepted according to the probability function, P(DE) = exp(DE/kBT), where kB is
Boltzmann’s constant and T is the current temperature. This check is performed by first selecting a random num-ber from (0, 1). If the value is less than or equal to the probability value, the new configuration is accepted; other-wise, it is rejected. By accepting worse solutions, SA can avoid being trapped on local optima. SA repeats this pro-cess L times at each temperature to reach the thermal equi-librium, where L is a control parameter, usually called the Markov chain length. The parameter T is gradually decreased by a cooling function as SA proceeds until the stopping condition is met.
3.1. Initial solution
Since the CF problem considers the grouping of parts and machines, an intuitive solution approach is to decom-pose the entire problem into two subproblems dealing with parts assignment and machines assignment, respectively. When parts assignment is firstly determined, followed by a proper assignment of machines, generation of an initial solution is hence completed.
3.1.1. Parts assignment
As in McAuley’s research (McAuley, 1972), Jaccard’s similarity measure is used to evaluate similarity between parts as an important index for assigning parts to cells in this subproblem. The similarity measure, denoted Sij,
is defined as: Sij¼ aij
aijþbijþcij, where aij represents the num-ber of machines processing both parts i and j; while bij
is the number of machines processing part i but not part j, and cij is the number of machines processing part j
but not part i. After calculating the similarity matrix for each pair of parts, we are now able to generate the initial parts assignment by using the following greedy rule: the higher similarity measure a pair of parts has, with the higher priority they should be placed in the same cell. This process is repeated until all parts have been assigned to cells.
An example is used to illustrate this process. Consider a sample machine–part matrix inFig. 2a, the corresponding similarity matrix for parts is displayed inFig. 2b. Suppose that there are two cells to be formed. The largest coefficient in the matrix ofFig. 2b is 1, indicating that parts 2 and 3 must be assigned to the same cell, say cell 1. We proceed with the second largest coefficient in the matrix, 0.5, appearing in pairs (2, 5) and (3, 5). Part 5 is thus assigned to cell 1 with parts 2 and 3. The remaining coefficient in the matrix is 0.33 in pair (1, 4). Since these two parts do not have any relationship with any parts in cell 1, together they should be assigned to the next cell, cell 2, as shown in Fig. 3. In the case when three cells are to be formed, randomly select one part from the pair with the least coef-ficient in the matrix and assign it to cell 3; the rest arrange-ment remains the same.
3.1.2. Machines assignment
Since the number of voids and exceptional elements are major components comprising the formula of grouping efficacy, procedures considering these two elements should very possibly generate solutions with good values of group-ing efficacy for the CF problem. This provides the motiva-tion for our research to design the machines assignment procedure below: P1 P2 P3 P4 P5 M1 1 0 0 1 0 M2 0 1 1 0 1 M3 1 0 0 0 0 M4 0 1 1 0 0 M5 0 0 0 1 0 P1 P2 P3 P4 P5 P1 - 0 0 0.33 0 P2 - 1 0 0.5 P3 - 0 0.5 P4 - 0 P5 -
Fig. 2. Sample machine–part matrix and corresponding similarity matrix for parts: (a) machine–part matrix and (b) similarity matrix for parts.
Cell 1 Cell 2 P2 P3 P5 P1 P4 M1 0 0 0 1 1 M2 1 1 1 0 0 M3 0 0 0 1 0 M4 1 1 0 0 0 M5 0 0 0 0 1
Step 1. Read the results of parts assignment.
Step 2. For each machine, find the cell to which the machine assignment will result in the least sum of number of exceptional elements and voids. If a tie happens, assign the machine to a cell with the least number of voids.
Step 3. Repeat Step 2 until all machines have been assigned to cells.
Results of parts assignment shown in Fig. 3is used to demonstrate the machines assignment procedure. Fig. 4 gives the sum of number of voids and exceptional elements for each machine–cell combination. Machines are then assigned to cells that result in the least sum. As a result, machines 2 and 4 are assigned to cell 1, while machines 1, 3, and 5 are assigned to cell 2. The initial solution matrix for the CF problem can thus be obtained and shown in Fig. 5.
3.2. Solution improvement
As mentioned in Section3.1, when parts assignment is determined, machines assignment process follows. The solution quality of parts assignment, thus, plays a very critical factor in the success of entire solution quality. This section introduces strategies for searching better neighbor-hood solutions to improve the current solution and move toward the optimal solutions.
The neighborhood of a given solution is defined as a set of all feasible solutions that can be reached by a single move/transition. In this study, two types of moves are implemented interactively: (1) a single-move, and (2) an exchange-move. The single-move is an operation that moves a part j from its current cell i (source cell) to a
new cell i0 (destination cell). The new move made is
denoted (i0, j). For the single-move, a move that results in
the most improvement in the objective function value from the current solution is selected. That is,
M1ði10; j1Þ ¼ maxfobjði
0;jÞ
objcurrent; 8i02 I; i06¼ i; 8j 2 J g
where I and J are the sets for cells and parts, respectively. The exchange-move is an operation consists of two dependent single-moves. If a part j is moved from its source cell i to destination cell i0(first single-move), then one part
j0 ðj02 J
i0¼ fparts assigned to cell i0gÞ from the
destina-tion cell i0of the first move has to be moved to the source
cell i of the first move (second single-move) in exchange. Two moves, (i0, j) and (i, j0), are generated. For the
exchange-move, the pair of moves resulting in the most improvement in the objective function value from the cur-rent solution is selected. That is,
M2fði20; j2Þ; ði2; j20Þg ¼ maxfobjði
0;jÞ;ði;j0Þ
objcurrent; 8i02 I; i06¼ i; 8j 2 J ; 8j02 J
i0g
3.3. SA algorithm for CF problem
This section describes the proposed algorithm SACF in detail. It is evident that the number of cells to be formed will affect the grouping solutions obtained. In our algo-rithm, the number of cells resulting in the best grouping efficacy is generated automatically. However, the flexibility is preserved for users to specify the number of cells they prefer. In addition, several counters and indicators are used in the algorithm to speed up the solution searching process and/or escape from the local optima. Before we proceed to the algorithm, some notations are introduced first. S current solution
Sc neighborhood solution
S* best solution found in current number of cells
S** best solution found so far
T0 initial temperature
Tf final temperature
a cooling rate
L Markov chain length k iteration number C initial number of cells C* optimal number of cells
D length of period for evoking exchange-move The proposed algorithm SACF can be summarized as follows.
Algorithm SACF
Step 1. Generate an initial solution S by using parts assignment and machines assignment procedures in Section3.1. Set C = 2, S**= S*= S, C*= C. Cell 1 Cell 2 v+e v+e M1 3+2 0+0 M2 0+0 2+3 M3 3+1 1+0 M4 1+0 2+2 M5 3+1 2+0
Fig. 4. Sum of voids and exceptional elements for each machine–cell combination. Cell 1 Cell 2 P2 P3 P5 P1 P4 M2 1 1 1 0 0 M4 1 1 0 0 0 M1 0 0 0 1 1 M3 0 0 0 1 0 M5 0 0 0 0 1
Step 2. Initialize SA and other parameters: T0, Tf, a, L, D,
counter = 0, counter_MC = 0, counter_trapped = 0, counter_stagnant = 0.
Step 3. If counter_MC < L and counter_trapped < L/2, then repeat Steps 3.1 to 3.7:
Step 3.1. Generate a new parts assignment plan through neighborhood searching by per-forming single-move.
Step 3.2. Perform exchange-move move if counter equals to a multiple of D.
Step 3.3. Read parts assignment from above steps and generate corresponding machines assignment using procedure in Section 3.1.2and thus the neighborhood solution Sc.
Step 3.4. If f(Sc) > f(S*), then S*= Sc
, S = Sc, counter_ stagnant = 0, counter_MC = counter_MC + 1, go to Step 3.
Step 3.5. If f(Sc) = f(S*), then S = Sc,
counter_stagnant = counter_stagnant + 1, counter_MC = counter_MC + 1, go to Step 3.
Step 3.6. Compute D = f(Sc) f(S). Select a ran-dom variable X U(0,1). If eDT > X, set S = Sc, counter_trapped = 0; otherwise, counter_trapped = counter_trapped + 1.
Step 3.7. counter_MC = counter_MC + 1, go to Step 3.
Step 4. If Tk6Tfor counter_stagnant P check, go to Step
5; otherwise, Tk+1= Tk· a, counter_MC = 0,
counter = counter + 1, go to Step 3.
Step 5. If f(S*) > f(S**), then f(S**) = f(S*), S**= S*,
C*= C, C = C + 1, go to Step 2; otherwise report
the current f(S**), S**, C*, and stop the algorithm.
Note that SACF consists of an SA procedure that is repeat-edly applied until a cell size resulting in the best grouping efficacy has been found. Initial number of cells is set at 2 in Step 1, and is gradually increasing by 1 at a time as long as solution improvement is observed in Step 5. All algorith-mic parameters and counters are initialized in Step 2. In addition to the Markov chain length, normally used to assure the thermal equilibrium is reached in each tempera-ture, another counter recording the number of times a solu-Table 1
Levels for each parameter
Parameter Level 1 Level 2 Level 3
T0 10 30 50
a 0.7 0.8 0.9
L 10 30 70
D 6 12 18
Table 2
Results of experimental analysis on all parameter combinations
T0 a L D
6 12 18
Ratio CPU time (s) Ratio CPU time (s) Ratio CPU time (s)
10 0.7 10 1.064 5.077 1.065 5.206 1.061 3.945 30 1.066 6.489 1.066 5.476 1.065 5.250 70 1.065 10.106 1.064 8.253 1.066 7.894 0.8 10 1.067 6.868 1.066 4.726 1.066 4.650 30 1.066 8.421 1.063 5.748 1.066 4.767 70 1.066 17.373 1.066 8.711 1.066 8.166 0.9 10 1.067 6.691 1.065 5.136 1.063 5.386 30 1.065 9.902 1.067 6.244 1.067 6.382 70 1.064 10.012 1.065 8.635 1.065 7.728 30 0.7 10 1.066 6.471 1.066 4.523 1.065 4.528 30 1.067 7.758 1.065 5.693 1.066 5.052 70 1.068 12.410 1.066 8.000 1.065 11.124 0.8 10 1.064 5.272 1.066 4.641 1.060 4.377 30 1.066 8.237 1.067 5.478 1.062 5.438 70 1.067 13.025 1.068 9.169 1.064 10.446 0.9 10 1.066 5.834 1.065 4.873 1.066 5.384 30 1.065 7.220 1.067 7.215 1.067 6.939 70 1.065 14.258 1.067 10.446 1.064 8.817 50 0.7 10 1.065 5.557 1.068 5.442 1.059 4.118 30 1.067 7.678 1.067 6.782 1.067 6.756 70 1.063 9.672 1.068 11.191 1.065 7.553 0.8 10 1.064 6.770 1.060 4.189 1.063 4.614 30 1.066 7.389 1.066 5.974 1.068 6.182 70 1.065 10.321 1.066 9.325 1.067 10.428 0.9 10 1.065 8.607 1.065 5.038 1.061 4.830 30 1.068 8.936 1.065 6.284 1.062 5.385 70 1.067 13.849 1.065 8.651 1.063 8.933
tion fails in the Boltzmann’s probability test is used in Step 3 to avoid being trapped in local solutions and causing too much computational effort wasted in certain temperatures. Two types of moves, the single and exchange-move, namely, are utilized interactively in the proposed algorithm to guide the solution searching. Both moves play different roles in the process of solution improvement. From our experience of intensive testing, it is observed that single-move usually leads to better solutions smoothly and effi-ciently, but only up to certain point. Frequent use of exchange-move, however, brings too much disturbance to solution searching with too much computational effort spent. We hence use single-move as a primary tool for find-ing better neighborhood solution in Step 3.1, but employ exchange-move to add some disturbance to current solu-tion every certain period, D, in Step 3.2 to increase the probability of finding more ‘‘diversified’’ solutions to bring the searching process to a new and unexplored solution space. SACF also records the number of times when neigh-borhood solutions become stagnant. When this number reaches a pre-specify constant check in Step 4, a solution check is performed by comparing the grouping efficacy of current cell size to the best solution found so far in Step 5 to determine whether to increase the cell size by 1 and continue the procedure or report the best solution found and terminate SACF. After intensive testing, the value of check is set at 4 in this study.
For users having their preferences in cell size, the pro-posed algorithm can save lots of run time since it will skip the process of iteratively searching for the cell size resulting in the best grouping efficacy. The savings in run time become even more significant as the cell size increases. 4. Computational results
In this section, 25 test problems from the literature are used to evaluate the computational characteristics of the proposed heuristic SACF, and the results are compared with those of algorithms reported in the literature, i.e., the ZODIAC (Chandrasekharan & Rajagopalan, 1987),
the TSP-GA (Cheng et al., 1998), and the GA (Onwubulo & Mutingi, 2001). The matrices of the test problems range from 5· 7 to 40 · 100 and comprise well-structured and unstructured matrices. The proposed algorithm SACF was coded in C and implemented on a Pentium III 933 MHz personal computer with 256 MB RAM.
4.1. Parameter settings
As widely known, settings of SA parameters critically affect the solution efficiency and effectiveness. An experi-ment regarding the setting of five parameters appeared in SACF is firstly conducted.
The five SA parameters appeared in SACF: T0, Tf, a, L,
and D represents the starting temperature, final tempera-ture, cooling rate, Markov chain length, and length of per-iod for evoking exchange-move, respectively. Except Tfwas
set at 0.002, three levels are chosen and given inTable 1for each parameter.
Six test problems (#2, #7, #10, #12, #21, #24) repre-senting various problem sizes are selected and used in the experimental analysis of parameters. Due to the stochastic features of the proposed method might have, five indepen-dent runs were performed on each parameter combination for each test instance. The ratios of our results to the best of ZODIAC, TSP-GA, and GA, are calculated and the average ratios are given inTable 2. From Table 2, it can be observed that all parameter combinations perform quite well and do not differ much in terms of solution quality (more than 6% improvement over the best results of ZODIAC, TSP-GA, and GA). Since the parameter combi-nation (T0= 50, a = 0.7, L = 10, D = 12) produces the best
improvement in a relatively efficient manner, we decide to use it as the suggested parameter setting for use in the next section.
4.2. Results
The parameter setting obtained in Section4.1is used by SACF to run all the test instances in this section.Fig. 6and
10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 No. Efficacy (%) SACF ZODIAC TSP-GA GA
Performance of SACF compared to ZODIAC, TSP-GA, and GA
No. Test instances Size Other approaches Best of the three
approaches
Proposed approach SACF
Source Grouping efficacy (%) Grouping efficacy (%) Cell size CPU time (s)
ZODIAC TSP-GA GA Max. Avg. Std.
1 Waghodekar and Sahu (1984) 5· 7 56.52 68.00 62.50 68.00 69.57 69.57 0.00 2 0.002
2 Seifoddini (1989) 5· 18 77.36 77.36 77.36 77.36 79.59 79.59 0.00 2 0.014
3 Kusiak and Cho (1992) 6· 8 76.92 76.92 76.92 76.92 76.92 76.92 0.00 2 0.002
4 Kusiak and Chow (1987) 7· 11 39.14 46.88 50.00 50.00 60.87 58.84 1.01 5 0.018
5 Boctor (1991) 7· 11 70.37 70.37 70.37 70.37 70.83 69.13 1.39 4 0.012
6 Chandrashekharan and Rajagopalan (1986a) 8· 20 85.24 85.24 85.24 85.24 85.25 85.25 0.00 3 0.026
7 Chandrashekharan and Rajagopalan (1986b) 8· 20 58.33 58.33 55.91 58.33 58.41 58.41 0.00 2 0.024
8 Mosier and Taube (1985a) 10· 10 70.59 70.59 72.79 72.79 75.00 71.49 2.87 5 0.016
9 Chan and Milner (1982) 10· 15 92.00 92.00 92.00 92.00 92.00 92.00 0.00 3 0.018
10 Stanfel (1985) 14· 24 65.55 67.44 63.48 67.44 71.21 69.38 1.59 8 0.449
11 King (1980) 16· 43 53.76 53.89 86.25 86.25 52.44 52.44 0.00 5 1.095
12 Mosier and Taube (1985b) 20· 20 21.63 37.12 34.16 37.12 41.04 41.02 0.04 6 0.351
13 Kumar et al. (1986) 20· 23 38.66 46.62 39.02 46.62 50.81 47.05 1.99 7 0.597
14 Carrie (1973) 20· 35 75.14 75.28 66.30 75.28 78.40 77.78 1.23 5 1.214
15 Boe and Cheng (1991) 20· 35 51.13 55.14 44.44 55.14 56.04 56.04 0.00 4 0.789
16 Chandrasekharan and Rajagopalan (1989) 24· 40 100.00 100.00 100.00 100.00 100.00 100.00 0.00 7 1.568
17 Chandrasekharan and Rajagopalan (1989) 24· 40 85.11 85.11 85.11 85.11 85.11 85.11 0.00 7 1.819
18 Chandrasekharan and Rajagopalan (1989) 24· 40 73.03 73.03 73.03 73.03 73.51 73.51 0.00 7 1.512
19 Chandrasekharan and Rajagopalan (1989) 24· 40 20.42 49.37 37.62 49.37 52.44 52.44 0.00 8 3.405
20 Chandrasekharan and Rajagopalan (1989) 24· 40 18.23 44.67 34.76 44.67 47.13 45.59 1.13 9 5.828
21 Chandrasekharan and Rajagopalan (1989) 24· 40 17.61 42.50 34.06 42.50 44.64 43.81 0.60 9 5.005
22 Kumar and Vannelli (1987) 30· 41 33.46 53.80 40.96 53.80 62.42 61.08 1.41 13 9.626
23 Stanfel (1985) 30· 50 46.06 56.61 48.28 56.61 60.12 59.88 0.21 13 15.440
24 Stanfel (1985) 30· 50 21.11 45.93 37.55 45.93 50.51 49.60 0.68 11 17.591
25 Chandrasekharan and Rajagopalan (1989) 40· 100 83.92 84.03 83.90 84.03 84.03 84.03 0.00 10 106.934
T.-H. Wu et al. / Expert Systems with Application s 3 4 (2008) 1609–16 17 1615
Table 3show the computational results of SACF and pub-lished results in the literature including ZODIAC, TSP-GA, and GA for the 25 test instances. According toTable 3, results obtained by SACF are better than or equal to those reported results except in problem #11. To be more specific, SACF obtains for 6 (24%) problems values of the grouping efficacy that are equal to the best results found in ZODIAC, TSP-GA, and GA methods and improves the values of the grouping efficacy for the rest 18 (72%) problems. In 3 (12%) problems, i.e., problem #4, #12, and #22, the percentage improvement is higher than 10%, with the highest being 21.74%, appeared in prob-lem #4. Since five replicates are performed for each test instance, the best, average and standard deviation values of grouping efficacy are listed inTable 3as well. The stan-dard deviation is 0, in 13 out of 25 problems, and the larg-est value is never greater than 2.87. The very low standard deviation indicates that SACF is not only able to produce good solutions but also can be considered as a robust heu-ristic algorithm. In addition, the cell size resulting in the best grouping efficacy for each test problem is also given in Table 3. As to run time data, it ranges from 0.002 s to 106.934 s depending on different problem sizes.
5. Concluding remarks
A simple yet effective approach for the cell formation problem, SACF, has been proposed in this research. Con-siderable efforts have been devoted to the design of (1) parts assignment procedure in which part families are formed through the construction of similarity matrix of parts, and (2) machine assignment procedure, in which the number of voids and exceptional elements, major components com-prising the formula of grouping efficacy, are explicitly con-sidered. We believe this explicit consideration of number of voids and exceptional elements in the machine assignment procedure has directed SACF to converge to solutions with good values of grouping efficacy. In the solution improve-ment stage, two types of moves, the single and exchange-move, namely, have been utilized interactively and collocate properly in the proposed algorithm to guide the solution searching. In addition, several counters and indicators have been used in the algorithm to speed up the solution search-ing process and/or escape from the local optima.
Computational results obtained from running a set 25 test instances from the literature have shown that SACF improves the best values of the grouping efficacy found in ZODIAC, TSP-GA, and GA methods for 18 (72%) prob-lems, and obtains for 6 (24%) problems values of the grouping efficacy that are equal to the best results found in ZODIAC, TSP-GA, and GA. In 3 (12%) problems, the percentage improvement is higher than 10%, with the highest being 21.74%.
Although SACF is able to find the number of cells that can result in the best grouping efficacy, the process of iter-atively searching for the cell size, however, consumes too much run time. Developing more effective methods for
finding proper cell sizes may thus be regarded as a future research.
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