Cell formation in group technology: a combinatorial search approach

int. j. prod. res., 1997, vol. 35, no. 7, 2025± 2043

Cell formation in group technology: a combinatorial search approach A. J. VAKHARIA²

*

and Y.-L. CHANG³

This paper addresses the cell formation problem in group technology (GT). We develop two heuristic methods for generating solutions to the problem. These methods are based on two powerful combinatorial search methodsÐ simulated annealing and tabu search. The performance of the heuristics is examined using randomly generated, published and industry data. The results indicate that the simulated annealing based heuristic is the preferred technique in the context of the problem addressed in this paper. Further, we also demonstrate that the simulated annealing based heuristic generates near-optimal solutions to the cell formation model formulated in this paper.

1. Introduction Traditionally discrete parts manufacturers have organized their production process using the job shop (or process layout) con® guration. In recent years, however, such manufacturers are focusing on improving the e ciency and productivity of their production activities (due to sti€ er price competition in the domestic and international markets). Group Technology (GT) based manufacturing systems have been found to be particularly appropriate for discrete part manufacturing (see Hyer 1984, for a list of users of GT in the US) since they provide some of the strategic bene® ts of a job shop such as product customization and simultaneously provide some of the operational bene® ts of the line process such as reduced WIP inventories. GT is a rational method of organizational management based on the principle that similar things should be done similarly. In the context of manufacturing, these `things’ include product design, process planning, fabrication, assembly, control and even the use of a standard methodological approach for system design. One tenet of GT for manufacturing is to break up the shop facility into production cells with each cell being dedicated to the processing of a set of part families. The di€ erences between the process layout and cellular layout are as follows. In the process layout, all parts travel through the entire shop and thus scheduling and materials control are di cult. Further, job priorities are di cult to set and hence foremen traditionally maintain large inventories so as to ensure that ample work is always available. On the other hand, a change to a cellular layout ensures that most of the parts ¯ ow through a single cell. Thus, the materials ¯ ow is simpli® ed and the scheduling task is made much easier. Workers within a cell may be cross-trained on all the machines within the group and follow a job from start to ® nish. Since similar

Received July 1996. Decision and Information Sciences Department, College of Business Administration, University of Florida, Gainesville, FL 32611-7160, USA. ³ Ivan Allen College of Management, Policy and International A€ airs, Georgia Institute of Technology, Atlanta, GA 30332, USA. *To whom correspondence should be addressed. ²

0020± 7543/97 $12. 00

Ñ

1997 Taylor & Francis Ltd.

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A. J. Vakharia and Y.-L . Chang

parts are grouped, this leads to a reduction in the number of setups required and allows a quicker response to the changing conditions. The cost savings attributable to GT have been documented by several users. For example, the range of percentage savings reported by US manufacturers are (Hyer and WemmerloÈv 1989, WemmerloÈv and Hyer 1989): setup costs decreased 20± 60%; labour costs decreased 15± 25%; tooling costs decreased 20± 30%; rework/scrap decreased 15± 75%; machine tools costs decreased 15± 25%; and WIP costs decreased 20± 50%. In another company where GT was implemented, the reported bene® ts include a 32% increase in sales, a 44% decrease in overall inventory and an 83% decrease in the shipment of late orders ( Ransom 1972). This research focuses on the GT cell formation problem (i.e. the identi® cation of equipment which can be grouped to create a GT cell as well as the identi® cation of parts processed within each cell). More speci® cally, the objectives of this study are to: (1) propose cell formation heuristics based on combinatorial search methods; and (2) illustrate and compare the performance of the heuristic approaches. The remainder of this paper is organized as follows. In § 2, the cell design problem is discussed in more detail. Section 3 describes the mathematical formulation which is used as a basis for developing the proposed heuristics. In § 4, we present details of the two heuristics that have been formulated to obtain solutions to the model. Section 5 describes the experimental design used to compare the heuristics and the results of the comparison are presented in § 6. Finally, the implications and conclusions of this research are presented in § 7. 2.

Cell formation in group technology The basic steps that have been followed in implementing GT for industry applications are (Askin and Vakharia 1990): (1) development of a parts coding scheme; (2) cell formation; and (3) specifying the machine layout within cells. The major decisions made at the cell formation stage focus on the identi® cation of part families, identi® cation of production cells, and allocation of the families to cells or vice versa. These three decisions are interrelated and prior researchers have not always followed the same order in making these decisions. For example, Carrie (1973) identi® es part families ® rst and subsequently groups machines required to process these families into cells while McAuley (1972) identi® es machine groups ® rst and then allocates individual parts to these groups to create cells. On the other hand, Askin et al. (1991) have developed a procedure which simultaneously identi® es production cells and part families. The objectives/constraints under which the GT system design problem is typically tackled are as follows (Vakharia and WemmerloÈv 1990):

(1) Minimize the intercell materials handling cost (or maximize cell independence). The primary focus of GT is to identify cells where the between cell interaction is restricted. (2) Minimize investment in equipment. In reorganizing a job shop into a GT cell system, there is typically an increase in the number of machines required. Hence, this objective focuses on minimizing the additional investment in such equipment. (3) Maintain acceptable equipment utilization levels. A cell system design is feasible if the utilization of equipment in each cell is less than the maximum acceptable level.

Cell formation in GT: a combinatorial search approach

2027

(4) Identify cells of a `reasonable’ size. The size of the GT cell will impact how easily the cell can be managed and controlled. Hence, cells identi® ed should not contain more than a speci® ed number of machines. There appears to be some inherent con¯ ict between the objectives. For example, we can always create cells without intercell parts movements simply by adding machines as and when required. On the other hand, we can reduce the number of machines required by allowing parts to move freely between cells. Hence, one could view cell formation as a multiple-objective decision making process. The literature on cell formation has experienced a tremendous growth in recent years. WemmerloÈv and Hyer (1986) and Selim et al. (1995) give detailed overviews of alternative design procedures. Several researchers have proposed classi® cations of the numerous GT cell design techniques. For example, Burbidge (1979) has classi® ed design approaches into three groups depending upon whether the approach relied upon: (a) tacit judgement, rules of thumb or visual identi® cation; (b) formal classi® cation and coding; and (c) production ¯ ow analysis. On the other hand, a more comprehensive classi® cation has been developed by WemmerloÈv and Hyer (1986) based on a division of methods into those using part characteristics (e.g. classi® cation and coding systems which use part shape and design characteristics to identify part families) and those using production characteristics (e.g. those based on production processes and process plans). These classi® cations are useful in that they provide the researcher with an overall understanding of the cell formation problem. One of the criticisms that has been directed towards some of the cell formation research is that several methods simply focus on a single objective of `independence’ (e.g. the methods proposed by McAuley 1972 and Carrie 1973). Hence, cells identi® ed using such procedures cannot be implemented simply because they may not satisfy some of the other cell formation objectives. A second problem with other cell formation methods which do consider multiple objectives is that these methods are typically based on heuristic methods which generate one solution to the problem (e.g. Burbidge 1975). Thus, the user has no idea whether this solution quality can be improved by simple perturbations. For example, given a part and machine allocation to a set of manufacturing cells, it may be possible to improve the solution quality simply by processing a part in multiple cells since this could lead to a reduction in equipment investment. A ® nal criticism of the prior research on cell formation can be levelled at the studies which have developed mathematical programming models for cell formation (e.g. Askin and Subramaniam 1987, Choobineh 1988, Shafer and Rodgers 1991) since most of these models are computationally intractable for large problems. however, as Wei and Gaither (1990) point out, mathematical models for cell formation allow us to compare the solution quality of heuristic methods. Further, we are of the opinion that these models can also be used to provide insights into the development of near-optimal heuristic methods. More recently, a few studies have developed search heuristics for solving mathematical models for cell formation. Boctor (1990) developed a simulated annealing heuristic for minimizing the number of exceptional elements when rearranging the part-machine matrix to identify machine groups and part families simultaneously. However, the potential usefulness of this study is limited since it only addresses a single objective in cell formation. Venugopal and Narendran (1992) also developed and evaluated a simulated annealing based procedure to identify machine groups

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such that the workload between individual machines is balanced within cells. Their method, however, does not consider the availability of multiple machines of the same type which typically exist in most manufacturing environments. On the other hand, their results indicate that their procedure is fairly successful in minimizing the number of exceptional elements while balancing the workload. Finally, Harhalakis et al. (1990 ) use a simulated annealing based heuristic to identify cells for a speci® c industrial application. Based on this discussion, this paper proposes the use of combinatorial search methods to generate two heuristics for cell formation. The methods can be used in conjunction with other heuristic methods so as to improve solution quality or they can be used in isolation as cell formation procedures. However, in the context of this paper, both heuristics are structured around a mathematical programming model presented in the next section. 3.

Mathematical model for cell formation At this point we present a model for cell formation. As noted earlier, there are several models for cell formation which have been proposed in the literature. The model proposed below is a modi® cation of that presented by Vakharia et al. (1989) and incorporates all the objectives discussed in the prior section. Before presenting the model, the following notation is introduced. De® ne:

i j m c

= index for = index for = index for = index for

parts ( i = 1, . . . , N) operations on part i ( j = 1, . . . Ji ) machine types ( m = 1, . . . , M) cells ( c = 1, . . . , C)

Decision variables: Xmc = number of machines of type m required in cell c Y ijc = 1 if operation j on part i is performed in cell c 0 otherwise

{

Input parameters: Qm H di pijm Am Um S

= procurement cost per week of one machine of type m = cost to transport one batch of any part between cells = demand (in batches) per week for part i = processing time (set-up plus run time) required to process one batch of part i through operation j on machine type m (in hours) = available productive time for each machine of type m per week (in hours) = maximum acceptable utilization per machine of type m = maximum number of machines to be included in a cell

The mathematical formulation (Program ZLIP) is: M

Minimize Z =

å å

C

( Qm Xmc ) + H

m = 1 c= 1

N Ji - 1 C

å å å

i = 1 j = 1 c= 1

|Y ij+ 1c -

Y ijc |

( 1)

Subject to:

å

C

Y ijc = 1 c= 1

" i, j

( 2)

Cell formation in GT: a combinatorial search approach

å

N i= 1

å

Ji j= 1

Y ijc Pijm di £ Am Um

å

M

Xmc £

2029

Xmc

" M, c

( 3)

S

" c

( 4)

, j, c " m, c

( 5)

m= 1

Î [0, 1]

Y ijc Xmc ³

"

0 and integer

i

( 6)

In this model, the objective function (equation (1)) is the total cost of the machines required as well as the materials handling cost for loads transferred between cells. Constraint equation (2) ensures that each operation on a part is completely carried out in one cell (i.e. operation splitting is not permitted) while constraint (3) estimates the number of machines required in each cell based on the available time per week, maximum utilization levels and the load allocated to that machine. The cell size constraint for each cell is enforced in constraint set (4); and constraint sets (5) and (6) enforce the technological constraints on the decision variables. Finally, note that program ZLIP simply minimizes the total cost of equipment used to identify the manufacturing cells. If, however, we already had information on the number of machines of each type currently available in the shop and hence, were interested in minimizing the cost of procuring additional equipment, we could incorporate this as follows. First, de® ne: Nm as the total number of machines of type m available in the shop; X1m as the additional number of machines of type m procured; and X2m as the number of machines of type m unused in the current solution. M C Second, replace the ® rst term in the objective function ( i.e. å m= 1 å c= 1 ( Qm Xmc )) M by å m = 1 ( Qm X1m ). Finally, add the following constraints to the model: C

å-

c 1

Xmc - Nm = X1m - X2m

" m

( 7)

X1m , X2m ³

" m

( 8)

0 and integer

An optimal solution to program ZLIP is di cult to obtain since it includes a non-linear objective function as well as zero-one integer and strictly integer variables. Although the non-linearity could be eliminated by adding dummy variables (i.e. introduce a constraint set Y ij+ 1c - Y ijc = Bijc - Dijc with Bijc , Dijc ³ 0 and reformuN J C late the second part of the objective function as H å i= 1 å Ji=- 11 å c= 1 ( Bijc + D)ijc ) ), the problem still would contain zero-one and strictly integer variables. Further, the problem of minimizing intercellular part movements subject to the constraint that each machine and part is allocated to a single cell and there is an upper bound on the cell size, is NP-hard. Given that these are a subset of the objectives and constraints incorporated in program ZLIP, our formulation is also NP-hard. Note that the size of the program ZLIP tends to be fairly large (e.g. for a 40 part, 10 machine type, 3 cell case with each part requiring processing on 4 machine types, program ZLIP has 193 constraints, 480 zero one variables and 30 integer variables) and this can also create computational problems. Hence, we propose using two combinatorial search methods to obtain near-optimal solutions to problem ZLIP. Details of these methods are given in the next section.

[

]

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A. J. Vakharia and Y.-L . Chang

4.

Heuristic methods Based on the complexity of the cell formation problem, we propose heuristics based on two powerful combinatorial search methods (speci® cally, simulated annealing and tabu search) to obtain near-optimal solutions for the cell formation problem. We now proceed to discuss each method.

Simulated annealing heuristic for cell formation (SAHCF) Simulated annealing is a randomized local search method that has been used to derive near-optimal solutions for computationally complex optimization problems. It was originally developed as a simulation model for a physical annealing process (Metropolis et al. 1953) and hence it is referred to as `simulated annealing’. The reader interested in a comprehensive discussion of the theories and surveys on applications of this approach is referred to Van Laarhoven and Aarts (1987) and Johnson et al. (1989, 1991). Essentially, the method di€ ers from local search methods since it allows the acceptance of an inferior solution in the general neighbourhood of a current solution with some positive probability. Hence, it facilitates the search of a response surface that may have multiple local optima. In general, to implement a simulated annealing based procedure to any complex optimization problem, several decisions have to be made. These are: 4.1.

(1) (2) (3) (4)

What is a solution? What is the cost of a solution? How do we determine an initial solution? What is the neighbourhood of a solution?

In addition, the computation time for a simulated annealing based heuristic can be controlled by the number of iterations ( T ) and the number of local searches carried out at each iteration (St ). A ® nal parameter that needs to be speci® ed when implementing the heuristic is the probability of accepting an inferior solution at each iteration (APt ). Typically, the value of this probability should approach 0 as we reach the iteration limit. We now proceed to discuss each of these decisions in the context of the cell formation problem. First, in this paper, a solution is represented by a machine and parts assignment to cells such that a feasible solution to model ZLIP formulated in § 3 is obtained. Second, the cost of a solution is the value of the objective function in model ZLIP. Third, to determine an initial solution to the problem, any cell formation heuristic developed in prior research could be used or one could randomly generate a feasible solution to program ZLIP. In this paper, cell designs are generated as follows: heuristic procedure to minimize intercell materials ¯ ow without violating • Acapacity constraints (for hypothetical and published data); and A heuristic procedure developed by Vakhria and WemmerloÈv (1990) which • attempts to maximize the unidirectional within cell materials ¯ ow (for industry data).

Finally, the neighbourhood of a current solution is represented as follows. We randomly select an operation j on part i which is processed on machine m in cell c in the initial solution. We then de® ne the neighbourhood of this operation as:

Cell formation in GT: a combinatorial search approach

2031

(1) A randomly selected operation j1 for part i1 currently processed on machine type m1 in cell c1 also in the initial solution. A potential new solution can be identi® ed if we interchange operations j and j1. However, this interchange is considered feasible if and only if the cell size constraint in formulation ZLIP is not violated through the interchange. If, however, this interchange is infeasible in terms of this constraint, we consider 2. (2) A randomly selected cell c2. In this case, a potential new solution can be obtained by reassigning operation j to cell c2 provided this reassignment once again does not violate the cell size constraint in formulation ZLIP. In any one of these potential `neighbourhood’ solutions are found to be feasible, the value of Xmc in constraint set (3) is recomputed. Based on this, the associated cost (Z1 ) of the new solution due to the interchange or reassignment are computed as: Z1 =

{

Z + A1 + A2 if operations j and j1 are interchanged Z + A3 + A4 if operation j is reassigned

( 9)

where Z = cost of the initial solution A1 = change in materials handling costs by interchange of operations j and j1 = H ( Y i, j,c1 - Y i, j- 1,p |) + ( | Y i, j+ 1,q - Y i, j,c1 |) + ( | Y i1, j1,c - Y i1, j1- 1,p1 |) + ( | Y i1, j1+ 1,q1 - Y i1, j1,c |) - ( | Y i, j,c - Y i,j- 1,p |) - ( | Y i, j + 1,q - Y i, j,c |) - ( | Y i1, j1,c1 - Y i1, j1- 1,p1 |) - ( | Y i1, j1+ 1,q1 - Y i1, j1,c1 |) A2 = change in equipment investment costs by interchange of operations j and j1

[

]

= Qm ( Xmc - Xmc ) + Qm1 ( Xm1c1 - Xm1c1 ) 1

1

A3 = change in materials handling costs by reassignment of operation j = H ( | Y i, j,c1 - Y i,j- 1,p |) + ( | Y i, j + 1,q - Y i, j,c1 |) - ( | Y i, j,c - Y i,j- 1,p |) - ( | Y i, j + 1,q - Y i, j,c |) A4 = change in equipment investment costs by reassignment of operation j

[

]

[

p= q= p1 = q1 =

cell cell cell cell

to to to to

]

Xmc ) + ( Xmc1 - Xmc1 ) which operation j - 1 on part i is assigned in the initial solution which operation j + 1 on part i is assigned in the initial solution which operation j1 - 1 on part i1 is assigned in the initial solution which operation j1 + 1 on part i1 is assigned in the initial solution

= Qm ( Xmc

-

1

1

1

Xmc = number of type m machines in cell c after the interchange/reassignment In addition to these four decisions, we investigate the impact of alternative values of T (i.e. the number of iterations) with ® xed value of St (i.e. number of searches at each iteration) on solution quality and computation time (details of these values are discussed in § 5. In terms of the acceptance probability, we set this value to be 0. 50 at the start of the procedure (i.e. AP0 = 0. 5) and at each iteration t this value is decreased by ² (where ² = AP0 / T ). Hence, the value of APt ® 0 as t ® T . We now describe in detail the SAHCF algorithm. Let t

= index for iterations ( t = 1, . . . , T )

2032

A. J. Vakharia and Y.-L . Chang

s = index for local searches at iteration t( s = 1, . . . , St ) APt = acceptance probability at iteration t ² = AP0 / T v = a feasible move which represents an exchange of two operations in two cells or simply a move of one operation from one cell to another. The move is regarded as feasible if it does not violate constraint set (4) in program ZLIP E = a solution with feasible part operation and machine assignments (i.e a feasible solution to program ZLIP) Z = the objective function value (equation (1) in program ZLIP) associated with solution E E1 = a neighbouring solution to E determined by carrying out a feasible move v Z1 = the objective function value (equation (1) in program ZLIP) associated with solution E1 E * = the best solution to program ZLIP derived Z* = the objective function value (equation (1) in program ZLIP) associated with solution E* Zinit = the objective function value associated with the initial solution R = a random number in the interval {0,1} Based on this notation, the SAHCF is as follows: (1) Set T , St , " t, AP0 and ² = AP0 / T . (2) Obtain and initial feasible solution to program ZLIP. Let this be solution E with objective function value Z. Let E* = E , Z* = Z, Zinit = Z, t = 0. (3) Set t = t + 1. If t > T , goto 7, else set s = 0 and goto 4. (4) Set s = s + 1. If s > St , set APt+ 1 = APt - ² and goto 3, else goto 5. (5) Obtain a neighbouring solution E1 or E with a feasible move v such that Z1 < Z. If there exists no such E1 then goto 6 else: (a) If Z1 < Z* then set E * = E1 and Z* = Z1 ; and (b) Set Z = E1 , E = E1 and goto 4. (6) Generate a neighbouring solution of E1 of E with a feasible move v such that Z1 ³ Z. Generate a random number R. If R £ APt , then set E = E1 and Z = Z1 . Goto 4. (7) Stop. The best solution found is E * with associated objective function value Z*. Tabu search heuristic for cell formation ( TSHCF) Tabu search was introduced by Glover (1986) as a technique to overcome local optimality entrapment in complex combinatorial optimization problems. The underlying idea is to constrain the search process by `classifying certain of the moves as forbidden (i.e. tabu) and that of freeing the search for a short term memory function that provides for strategic forgetting’ (Glover 1989, p. 191). Hence, the term tabu search has been used to describe this technique. The process by which certain moves are classi® ed as tabu entails the maintenance of a `tabu list’ which keeps track of prior moves. Another feature that has been incorporated in the methodology is that of the `aspiration level’ . This notion has been suggested so as to allow selection of a `pro® table’ move even though it may be tabu. In addition to the four decisions that have to be made in implementing an annealing based procedure, Glover (1990) also speci® es two decisions which have to be made in implementing a tabu search based 4.2.

Cell formation in GT: a combinatorial search approach

2033

heuristic. These are: (1) What is the size of the tabu list ( L ) that must be maintained? (2) How is the aspiration level (AL ) to be determined? Further as with an annealing heuristic, the number of iterations ( T ) needs to be user speci® ed. These decisions in the context of the cell formation problem are discussed in detail below. First, the size of the tabu list ( L ) that must be maintained focuses on the number of prior feasible solutions to program ZLIP that should be recorded. In this paper, we explore the impact of di€ erent sizes on the tabu list and its impact of solution quality (speci® c values are reported in § 5). Second, what is the aspiration level (AL )? In our context, we dynamically change the value of AL and set it to be the lowest value of the objective function (i.e. equation (1) in program ZLIP) determined at an iteration. Third, when applying the tabu search based heuristic, we modify the de® ntion of the neighbourhood of the current solution as follows. Note that for the SAHCF, we simply de® ne the neighbourhood as one randomly selected feasible move which satis® es constraint set (4) (see steps 5 and 6 in the SAHCF in § 4.1). However, for the tabu search heuristic, we de® ne the neighbourhood of the current solution as the set of all feasible moves which satisfy constraint set (4) and we select the `best’ move to replace the current solution, we choose the `best’ replacement in the neighbourhood. Finally, we ® x the number of iterations (i.e. T ) at a constant value for purposes of this study (once again, the speci® c value of T is reported in § 5). We now describe in detail the TSHCF algorithm. In addition to the notation introduced earlier, let: = index for tabu list ( l = 1, . . . , L ) L = store v in L in a round-robin fashion (i.e. if L is full, replace the oldest element in L with v, else simply add v to L ) AL = aspiration level F = set of all neighbourhood solutions of E derived with a feasible move v. l v®

Based on this notation, the TSHCF is as follows: (1) Set T and let L = 0. (2) Obtain an initial feasible solution to program ZLIP. Let this be solution E with objective function value Z. Let E * = E, Z* = X, Zinit = Z, AL = Z, t = 0. (3) Set t = t + 1. If t > T , goto 6, else goto 4. (4) Obtain a neighbouring solution E1 of E with a feasible move v which satis® es any one of the following two conditions: (a) E1 is in F, Z1 is the minimum of all moves in F and Z1 < AL ; or (b) E1 is in F - L and Z1 is the minimum of all moves in FL . If E1 does not exist then goto 6, else goto 5. (5) Set E = E1 , Z = Z1 , v ® L . If Z1 < Z*, then set E * = E1 , Z* = Z1 , AL = Z1 . Goto 3. (6) Stop. The best solution found is E * with associated objective function value Z*. Both the cell formation heuristics described in this section are evaluated using hypothetical, published and industry data. In the next section, we describe the experimental design used to carry out such an evaluation.

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A. J. Vakharia and Y.-L . Chang

Data set

Type

Number of parts

Number of machine types

Number of cells

Maximum machines per cell

1 2 3 4 5 6 7 8 9 10

Random Random Random Random Random Random Random Random Published Industry

30 40 50 60 70 80 90 100 19 325

10 10 20 20 20 30 30 30 12 78

4 4 5 5 6 7 8 8 3 35

8 10 10 12 14 14 14 13 14 14

Table 1.

Details of the data sets.

5. Experimental design 5.1. Data sets Ten data sets (8 randomly generated data sets, 1 published data set and 1 industry data set) were used to evaluate the SAHCF and TSHCF heuristics (see Table 1). For the randomly generated and published data sets, we used the following method to identify cell designs. Assume that N ( i = 1, . . . , N) and M ( m = 1, . . . , M) are the total number of parts and machines for a particular data set, respectively. For each part i:

part demand (d ), in batches/unit time, is randomly generated using a • The discrete uniform distribution with parameters 1 and 10. length of the operation sequence (J ) is randomly generated using a dis• The crete uniform distribution with parameters 3 and 9. particular machine type used to carry out each operation is randomly • The generated using a discrete uniform distribution with parameters 1 and M. i

i

•

Hence, it is possible that a machine type may appear more than once in the operation sequence. If consecutive operations for a part are carried out on the same machine type, they are combined into a single operation. The processing time for one batch at each machine type (i.e., pijm ) included in the operation sequence is randomly generated using a discrete uniform distribution with parameters 1 and 10.

Parts and machine types were allocated to C (where C was prespeci® ed) cells in order to minimize intercell materials handling. Next, individual machines were allocated to each cell in order to obtain feasible loadings (in this case, the values of Am set to 40 time units and Um was set to 1; see equation (3) in § 3). As a ® nal step, the cell size restriction was enforced± equation (4) in § 3± by rerouting operations from one cell to another, if necessary. For the industry data set, we identi® ed a cell design using a heuristic procedure which attempts to maximize the unidirectional within-cell materials ¯ ow. This method uses a similarity measure and a clustering technique to identify parts families. Subsequently, individual machines are allocated to the families to create candidate cells. The reader interested in more details regarding this procedure is referred to Vakharia and WemmerloÈv (1990 ).

Cell formation in GT: a combinatorial search approach

2035

5.2.

Implementing the heuristic procedures In this study, the SAHCF and TSHCF procedures were operationalized as follows. The initial acceptance probability (i.e. AP ) and the number of ran• SAHCF: dom searches carried out at each iteration t (i.e., S ) were ® xed at 0. 50 and 100, 0

t

•

respectively. However, the total number of iterations (i.e. T ) carried out was set at 4 values: 5, 10, 15 and 20. As reported in a prior study (Vakharia and Chang 1990), this parameter has a direct impact on computation time and possibly some impact on solution quality. Hence, we felt that it was important to investigate alternative values of T. TSHCF: For this heuristic, the number of iterations ( T ) was ® xed at 200. However, the size of the tabu list ( L ) was variable. In this study, we investigated 4 values of this parameter: 5, 7, 10 and 20. The reason for investigating tabu list sizes is that this parameter has an impact on the number of comparisons to be carried out at each iteration and possibly the solution quality (Skorin-Kapov, 1989).

5.3.

Cost parameters To compute the cost of the solutions obtained (i.e. equation (1) in § 3), we need to specify the values of H and Cm " m. In this study, we investigated the following alternative values of these parameters: C for each m was randomly generated using a uniform distribution • Hwith= 0;parameters 0 and 1 (referred to as cost ratio A in the remainder of the m

• • • •

paper). H = 1; Cm = 0 " m (referred to as cost ratio B in the remainder of the paper). H = 1; Cm for each m was randomly generated using a uniform distribution with parameters 10 and 90 (referred to as cost ratio C in the remainder of the paper). H = 1; Cm for each m was randomly generated using a uniform distribution with parameters 10 and 390 (referred to as cost ratio D in the remainder of the paper). H = 1; Cm for each m was randomly generated using a uniform distribution with parameters 10 and 1990 (referred to as cost ratio E in the remainder of the paper).

Note that the cost ratio A corresponds to a situation where intercell materials handling costs are irrelevant and hence, we simply focus on minimizing investment in equipment. On the other hand, cost ratio B re¯ ects the prioritizing of materials handling costs over equipment investment costs. The last three cost ratios represent situations where the ratio of average H : Cm is 1 : 50, 1 : 200 and 1 : 1000, respectively. These three cases were included to assess if the relative performance of the heuristic methods was a€ ected by alternative cost parameters. 5.4.

Performance measures The following performance measures were used to evaluate the heuristic procedures: (1) GAP (%): The relative di€ erence between the best objective function value obtained (i.e. Z*) and a lower bound on this value computed as:

2036

A. J. Vakharia and Y.-L . Chang

[

GAP( %) = 100 * {Z* - L B}/Z*

]

( 10)

where: Z* = `best’ objective function value found at the termination of the heuristic L B= a lower bound on the optimal solution to the problem obtained by solving a LP relation of problem ZLIP Note that a lower GAP value is preferred. (2) CPUSEC: The computation time for the heuristic methods (in seconds) on a Macintosh II with a 68020 procesor and a 68882 math co-processor. GAP assesses the solution quality of the heuristics as compared to a lower bound on the solution. This lower bound was obtained by solving a LP relaxation of problem ZLIP using the ZOOM/XMP mathematical programming software (Singhal et al. 1989; Marsten 1981). The second measure, CPUSEC, is the computation time of the heuristic procedures. It could be argued that this time is not a major issue since we are dealing with a manufacturing system design problem. However, the measure can be used to judge the relative e€ ectiveness of the two heuristics in comparison to one another. Consequently, it has been included as another measure of performance. We now proceed to a discussion of the results.

6.

Results The results of applying the TSHCF and SAHCF heuristics presented in this section have been organized in the following manner. We ® rst present an overall comparison of the two heuristic procedures for the randomly generated and published data sets (i.e. aggregated across the cost ratios and the iteration limit or tabu list size limit); for each cost ratio (i.e. aggregated across data sets 1± 9 and the iteration limit or tabu list size limit); and for each value of the iteration limit or tabu list size (aggregated across data sets 1± 9 and the cost ratios). Second, we present some more detailed results. We choose to focus on investigating the interaction e€ ects of the cost ratio and the tabu list size limit or iteration limit. These seem to be relevant in the context of the determining the impact of costs on solution quality as well as on computation time. Comparisons between the cost ratio/data set and the data set/tabu list size/iteration limit were also carried out. However, for sake of brevity we choose to discuss only one aspect. Finally, we present detailed results of applying the two procedures to the industry data set. In this case, we were unable to obtain results for the TSHCF procedure (due to computational and memory limitations) and hence, our comparison is limited in this context. However, we present detailed results for the SAHCF heuristic.

6.1.

Summary results Tables 2, 3 and 4 contain the summary results for each data set, each cost ratio and each value of the tabu list size limit (for TSHCF) and iteration limit (for SAHCF). Before discussing these results, please note that Table 4 cannot be used to compare the SAHCF and TSHCF methods. On the other hand, this table is useful to evaluate the e€ ects of the number of iterations for SAHCF or the tabu list size for TSHCF. In general, the results in these three tables indicate that the SAHCF procedure provides close to optimal or optimal solutions for these test problems. For

Cell formation in GT: a combinatorial search approach Data set

Performance measure

TSHCF

SAHCF

1

GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC CAP(%) CPUSEC

12. 83 7738. 90 6. 51 10 091. 60 8. 84 23 311. 50 13. 49 32 160. 10 8. 49 42 506. 65 14. 20 66 962. 40 7. 82 69 565. 85 13. 21 30 357. 88 2. 26 36 362. 70

2. 98 6208. 55 4. 15 12 684. 60 8. 82 12 563. 15 9. 60 14 312. 15 7. 60 20 338. 70 9. 97 15 438. 55 6. 43 19 105. 05 8. 64 18 108. 70 1. 62 16 007. 34

2 3 4 5 6 7 8 9

Table 2.

Aggregate results for each data set.

Cost ratio

Performance measure

TSHCF

SAHCF

A

GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC

16. 34 29 621. 36 11. 45 29 930. 95 6. 63 27 162. 50 7. 39 29 529. 61 6. 88 31 843. 55

11. 53 30 238. 50 6. 86 10 671. 06 4. 60 8599. 03 5. 82 8684. 78 4. 41 8603. 47

B C D E Table 3.

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Aggregate results for each cost ratio.

example, the GAP values in Tables 2, 3 and 4 range from a low of 2. 98% (Table 2) to a high of 11. 53% (Table 3). Since this measure compares the best solution obtained using the heuristic method with a lower bound obtained by solving a LP relaxation of problem ZLIP, we can assume that the solutions obtained are near-optimal. In sum, these aggregate results clearly indicate that the SAHCF procedure outperforms the TSHCF method and that the former procedure also provides close to optimal solutions to the cell formation problem. 6.2.

Interaction e€ ects Tables 5 and 6 present a comparison of how the solution quality of the TSHCF and SAHCF procedures is a€ ected by the cost ratio and the tabu list size (for the former) and iteration limit (for the latter). Based on these tables, note the following:

2038

A. J. Vakharia and Y.-L . Chang Iteration value or tabu list size

Performance measure GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC

L= 5/T= 5 L= 7/T= 10 L= 10/T= 15 L= 20/T= 20 Table 4.

TSHCF

SAHCF

10. 00 26 749. 24 9. 81 28 708. 96 9. 57 30 630. 51 9. 57 29 981. 67

72. 09 6014. 89 6. 61 11 618. 42 6. 75 14 959. 62 6. 13 20 844. 53

Aggregate results for each iteration value/tabu list size. Tabu list size

Cost ratio

Performance measure

L= 5

L= 7

L= 10

L= 20

A

GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC

16. 71 30 403. 22 12. 37 26 911. 33 6. 75 22 314. 89 7. 40 22 343. 22 6. 79 31 773. 55

16. 71 30 449. 89 11. 61 26 902. 89 6. 60 22 416. 67 7. 34 31 889. 67 6. 78 31 885. 67

16. 43 30 436. 55 10. 70 26 970. 11 6. 57 31 972. 11 7. 34 31 896. 22 6. 78 31 877. 55

15. 53 27 195. 78 11. 13 26 939. 45 6. 60 31 946. 33 7. 46 31 989. 33 7. 15 31 837. 45

B C D E

Table 5.

TSHCF results aggregated across data sets.

(1) Impact of cost ratio: regardless of the tabu list size limit for TSHCF or the iteration limit for SAHCF, the GAP is minimum for cost ratio E. Although the summary ® gures in Table 3 also validate this result, Tables 5 and 6 show that it is consistent across alternative values of the tabu list sizes (for TSHCF) or iteration limit (for SAHCF). (2) Impact of tabu list size for TSHCF: There does not appear to be a signi® cant impact of the tabu list size on the relative e€ ectiveness of the TSHCF procedure. Given that higher values of this parameter re¯ ect greater memory storage requirements, one recommendation in this context is to use lower values of this parameter. (3) Iteration limit for SAHCF: As with the tabu list size for TSHCF, the iteration limit does not have a signi® cant impact on the solution quality for the SAHCF method. Given that this value determines the computation time for SAHCF, it is recommended that a smaller number of iterations be used if SAHCF is implemented. The results regarding the tabu list size (for TSHCF) and iteration limit for (SAHCF) are contrary to expectations. Hence, we examined the detailed results in order to explain why this occurred. It appears that in almost all cases, the best solution identi® ed (i.e. Z*) was computed in the ® rst few searches. Thus, more searches (using the tabu list or the iteration parameter) did not result in any improvements in solution quality.

Cell formation in GT: a combinatorial search approach

2039

Iteration limit Cost ratio Performance measure GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC GAP(%) CPUSEC

A B C D E

Table 6.

T= 5

T= 10

T= 15

T= 20

11. 65 14 885. 33 7. 62 4557. 78 5. 14 3477. 67 6. 32 3619. 00 4. 74 3534. 67

11. 55 28 395. 00 7. 30 8769. 44 4. 54 7139. 44 5. 84 6984. 11 3. 82 6804. 11

11. 81 31 201. 67 6. 56 12 450. 33 4. 56 10 289. 67 6. 07 10 434. 44 4. 72 10 422. 00

11. 09 46 472. 00 5. 98 16 906. 67 4. 17 13 489. 33 5. 06 13 701. 356 4. 36 13 653. 11

SAHCF results aggregated across data sets.

6.3.

Results for industry data set As indicated in Table 1, the industry data set analysed in this paper consisted of 325 parts and 78 machine types. Using the Vakharia and WemmerloÈv (1990) heuristic, we identi® ed 35 candidate cells using a maximum cell size limitation of 14 machines per cell. Before discussing the detailed results for this data set, three issues need to be clari® ed: are not reporting results for the TSHCF for this data set given that the • We SAHCF outperformed this method for all the prior data. LP relaxation for this problem was not solved due to problem size limita• The tions, and hence, we have not reported the values of GAP(%) in this section. cost ratio B, the SAHCF procedure did not result in any improvements in • For the objective function value. Since this cost ratio re¯ ects no penalty on

machine duplication and given that the heuristic procedure to identify the initial solution results in completely independent cells, this result is not surprising. Note that this result is in direct contrast to the result obtained for the randomly generated and published data sets where the signi® cant improvements were realized for this cost ratio. This points to the fact that there is an interaction e€ ect between the objectives incorporated in determining an initial solution procedure and those considered in the model ZLIP. Consequently, the results corresponding to this cost ratio are not discussed below.

We have chosen to report the results of the application of SAHCF to this data set using Figs 1 and 2. In Fig. 1, we provide a visual representation of the convergence of the SAHCF procedure to the best solution (i.e. Z*) when the number of iterations is ® xed at 5 for cost ratios A, C, D and E. The values in this ® gure are computed as follows: (1) If the solution obtained at an iteration (Ziter ) is `better’ than the initial solution (Zinit ) or Ziter < Zinit , then we compute PERC( %) - {Zinit - Ziter }/ Zinit * 100. (2) If the solution obtained at an iteration (Ziter ) is `worse’ then the initial solution (Zinit ) or Zinit < Ziter then we compute PERC( %) = 1 + ( {Ziter - Zinit }/ Zinit * 100.

[

] ]

[

2040

A. J. Vakharia and Y.-L . Chang

Figure 1.

Industry data setÐ PERC(%) values.

Figure 2.

Industry data setÐ IMPR( $) values.

Cell formation in GT: a combinatorial search approach

2041

As can be seen from Fig. 1, the best value of the objective function (i.e. Z*) is identi® ed early on in the search process for all the data sets. In fact for this particular data set for all cost ratios, this value is identi® ed when carrying out the local searches in iteration 1. This result points to the fact that we can improve the solution quality for cell formation problems with very little computational e€ ort using the SAHCF methodology. These results are consistent with recent ® ndings which indicate that the major improvements using simulated annealing are obtained with a small neighbourhood as compared to large neighbourhoods (Cheh et al. 1991) since the size of the neighbourhood is tied to the number of iterations in this paper. In Fig. 2, we report the absolute di€ erence between the initial solution and the ® nal solution (i.e., Zinit - Z*) for cost ratios A, C, D and E. The reason for reporting these results is as follows. Although Fig. 1 indicates that the best percentage improvements are approximately 20% regardless of the cost ratio, this underscores the fact that the total $ improvements can be fairly substantial if the equipment costs are high. Hence, as can be observed from Fig. 2, the application of SAHCF to the industry data set with cost ratios D and E results in approximate savings of $11 000 and $67 000, respectively. Further, note that the values of these $ savings rise as the cost of equipment increases. Hence, the application of SAHCF is perhaps more bene® cial in real $ terms when equipment costs are high relative to materials handling costs. Implications and conclusions In this paper, we have proposed two combinatorial search techniques for addressing the cell formation problem in group technology. The ® rst technique is based on tabu search (TSHCF) and the second technique is based on simulated annealing (SAHCF). In applying these techniques, to randomly generated, published and industry data sets, the following major conclusions emerged:

(1) The SAHCF procedure is preferred to the TSHCF procedure terms of solution quality and computation time. (2) For the randomly generated and published data, the SAHCF procedure provided near-optimal solutions to the cell formation model formulated in this paper. (3) In implementing SAHCF, the number of iterations used should be relatively small (perhaps 5) since the improvements in the objective function are found fairly quickly in the search process. (4) The method used to determine the initial solution could have some impact on the relative improvements resulting when applying the SAHCF procedure. (5) Although the percentage improvements in using SAHCF are fairly constant across all cost ratio percentages investigated in this paper, the absolute di€ erences in objective function values increase as the cost of equipment increases relative to the materials handling cost. In sum, this paper has demonstrated the viability of applying the SAHCF procedure to cell formation problems. Further, the SAHCF as proposed in this paper considers the objectives of simultaneously minimizing equipment investment and materials handling costs. We have also shown that the procedure can provide near-optimal solutions to the problem and can be used either as a stand-alone cell formation technique or as an improvement technique after another heuristic method has identi® ed an initial solution.

2042

A. J. Vakharia and Y.-L . Chang

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