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Manufacturing cell formation using an improved P-median model
Pergamon
Computers ind. Engng Vol. 34. No. 1, pp. 135-146, 1998 © 1998 Publishedby ElsevierScienceLtd. All rights reserved PII: S0360-8352(97)00156-3 Printed in Great Britain 0360-8352/98 $19.00 + 0.00
MANUFACTURING IMPROVED
CELL FORMATION P-MEDIAN
USING
AN
MODEL
STUART JAY DEUTSCH, SUSAN F. FREEMAN and MARY HELANDER Department of Mechanical, Industrial and Manufacturing Engineering, Northeastern University, Boston, MA 02 115, U.S.A. Abstract--The p-median model objective function is modified for the cell formation problem to minimize the variability between parts in a group by considering part similarity to all other parts in the group instead of similarity to an arbitrary median. The heuristic vertex substitution method for solution of the part grouping problem is adapted for this objective function and then modified to provide improved starting points. The theoretical lower bound for the heuristic is developed and shown to be valid for all solutions. Worst case run time is shown to be O(n 2) or O(n 3) for distance matrix or network inputs respectively. Tests on published problems show that the proposed p-median model method provides as good or better objective function value (OFV) than the OFV of a p-median model in which parts are grouped to an arbitrary median. Likewise the new p-median model is shown, for these published problems, to give as good or better OFV than the algorithms reported by the original authors of the problem. The test problems suggest that other measures of solution quality such as bottlenecks and duplicate machines in addition to OFV become important measures of solution quality for dense problems. @ 1998 Elsevier Science Ltd. All rights reserved.
INTRODUCTION
The manufacturing cell formation problem can be solved by forming part families that place parts in groups based on common machine usage. Families of parts can be formed with a model that contains an objective function that maximizes similarities between all parts or minimizes dissimilarities between all parts [1]. This approach effectively forms groups where the variability between parts in a group is to be minimized. The drawbacks of this quadratic model are due to the nonlinearity of the objective function, therefore the p-median model has been used to groups parts by minimizing the dissimilarity between parts in a group and their median [2]. The p-median problem on a general network of n nodes involves finding a subset of p nodes called medians so as to minimize the sum of the weighted distances between each of the nodes and the closest median. Klastorin [3] found that the p-median model worked well for clustering data when the similarity metric corresponded to a distance measure that was Euclidean. If the weights in the p-median decision model are set to 1, the results from the clustering model are equivalent to the unweighted p-median model. In this paper, the p-median model is adapted so that its objective function explicitly minimizes the dissimilarities between all parts in a group by summing all the pairwise distances between parts in a group versus the traditional p-median formulation that sums the distances to an arbitrary median part. Due to the underlying problem complexities, exact solution methods provided by Narula, Ogbu and Samuelson [4] and Jarvinen, Rajala and Sinervo [5] are restricted to small problem sizes. Other heuristic approaches by Francis, McGinnis and White [6], Maranzana [7], and Khumawala [8] are also limited to small problems. The Teitz and Bart vertex substitution heuristic [9] was chosen for implementation because it is known to be efficient for finding high quality, near optimal solutions to large scale p-median problems (see Rosing et al. [10], Goodchild and Noronha [11], and Densham and Rushton [12]). Since the part family decision model proposed is based on the p-median model, the use of a heuristic is necessary because the p-median problem is NP-complete [13, 14]. The vertex substitution heuristic successively substitutes vertices in the complement of the solution. If a cycle of substitutions finds an improved solution, the process continues until no improvement is found. The next section develops a cell formation model by adapting a p-median model objective function to group parts into cells effectively. In Section 3, the Teitz and Bart heuristic [9] for sol135
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ving the adapted p-median model is implemented with the adapted objective function and additional modifications to improve the heuristic's starting point. Next, the worst case run time for the heuristic is shown, to demonstrate the applicability of the heuristic to any problem size. In Section 5, a lower bound is developed that can be used to judge the quality of the solutions found and is shown to be valid. In Section 6, the modified p-median model and solution heuristic is applied to a number of published problems from the literature and the quality of the resulting solutions is compared to previously reported solutions. Bottleneck and machine duplications are computed along with the objective function values to compare the quality of solutions and graphical analysis is provided to demonstrate the relationship between problem characteristics and resulting solutions. The last section contains the summary and conclusions.
THE CELL F O R M A T I O N D E C I S I O N M O D E L
In a decision model for the part family formation problem, a set of 0-1 decision variables are used to assign parts to families. Due to the nonlinearity of the objective function, this form is adapted to the p-median model where the decision variables determine if parts are located in families together, given an assigned median for any given family. The p-median objective function is then altered to reflect the objective of cell formation by finding the sum of the pairwise distances between all parts in a group, as opposed to summing distances to an arbitrary median part. Let n be the total number of parts to be grouped into at most p families. Let ~i~ be the decision variable that describes whether or not part i should be assigned to the kth family identified. That is, ~ik =
1 0,
if part i is assigned to family k, otherwise
(1)
for i = 1. . . . . n and k = 1,...,p. The following two constraints ensure that ensure that each part is assigned to exactly one family: Z p p ~ i k = 1 for i = 1. . . . . n
(2)
k=l
aik ~ {0,1} for i = 1 , . . . , n and k = 1. . . . ,p
(3)
Note that, as long as the ~tk's are specified according to constraints (2) and (3), then for any two parts id' c { 1,2 . . . . . n} and family k e { 1,2 . . . . ,p}, ~ik~jk =
1 0
if parts i and j are both assigned to family k, if either part i or j is not in family k
(4)
if parts i and j are assigned to the same family, if parts i and j are not assigned to the same family
(5)
and
Z pp~ik~tjk = k=l
{~
Let wi be a weight associated with part i. These weights can be quantities reflecting part production volume, part cost or part priority. In this application the weights are set to 1. Let d~j denote a general dissimilarity measure for parts i and j. The value of d~j can be thought of as the cost incurred when parts i and j are placed in the same family, and can also be thought of as a measure of distance. A general decision model for the part family formation problem is: n-1
(P1)
Minimize
n
Y~PPZ Z k=l
i=lj=i+l
wido'O~ikO~jk
(6)
M a n u f a c t u r i n g cell f o r m a t i o n
~_,pp~xik=
subject to
1
137
for i = 1. . . . . n
(7)
k=l
~ik6{O,1}
fori=l
.... nandk=l
.... p
(8)
This decision model given by (P1) seeks to assign every part to exactly one family to minimize the total weighted sum of distances between all pairs of parts in the same family. This formulation is a 0-1 integer programming model with a nonlinear objective function. Since the nonlinearity of the objective function involves the product of decision variable pairs and the constraints are fully linear, this problem is also referred to as a quadraticmodel [15]. In the p-median model [2], p parts are first selected to serve as medians or seed parts for clustering and remaining parts are assigned to medians. The feasible range of p is from 1 to n, but the value o f p selected for any problem is determined from characteristics of the problem such as part-machine matrix density, part volume, machine capacity and problem size. To show how the general cell formation model relates to the p-median formulation, note that since each part must be assigned to exactly one family, then 1 _
x,~ --
1 0
if parts i and j are assigned to the same family, if parts i and j are not assigned to the same family
(9)
for i -- 1. . . . . n a n d j -- 1. . . . . n, thep-median based decision model is: //
(P2)
Minimize
n
~--~y~wid~yx~j
(10)
i=l j=i n
subject to
Zxi~=l
fori=l
..... n
(11)
j=l n
Zxi~ = p
(12)
j=l
x~
for i = 1. . . . . n and j = fori=l
1,...,n
..... hand j=
1. . . . . n
(13) (14)
Constraint sets (11) and (14) work together to ensure that each part is assigned to exactly one part family. Constraint set (12) ensures that at most p of the n parts will be chosen as seeds for creating the families. Constraint set (13) enforces the assignment of a seed identity to exactly one part per family. The objective function for the p-median based decision model given by expression (10) drives the selection of a feasible part family assignment by minimizing within each family the total weighted dissimilarity between the part assigned as the family seed and all other parts assigned to its group. In this way, the p-median formulation fails to consider the possible pairwise dissimilarity between non-seed parts assigned to the same family. To consider all pairwise dissimilarities within assigned groups, the objective function given by (10) must be replaced by Minimize
ZppZ h= 1
Z widij
iE Gh]~ Ghi'<3"
(15)
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where the set Gh denotes the subset of parts assigned to the hth family. These sets can be constructed directed from the decision variables as follows: Gh ~ {i: X,h -----1}
for h = 1. . . . ,p. Given any feasible solution to constraint sets (11)-(14), note that the sets G1 . . . . ,Gp will be disjoint and their union will cover the set of parts { 1,2 . . . . . n}. That is, each of the n parts will be assigned t o exactly one of the sets G~ . . . . . Gp. The replacement of the p-median objective function (10) by (15) results in a fully linearized decision model that is equivalent to the formulation given by the decision model (P0. The unweighted modified version is used in this paper to formulate part families that are feasible to constraint sets (11)-(14) while minimizing the total pairwise part dissimilarity within each group. This approach more explicitly considers the variability of parts assigned to the same group than does the direct use of the traditional p-median formulation.
THE VERTEX SUBSTITUTION HEURISTIC In this work, the Teitz and Bart vertex substitution heuristic [9] was adapted to the part family decision problem by using the objective function given by expression (15) instead of the standard p-median objective function given by (10). The modified vertex substitution heuristic was further adapted by incorporating an efficient method for finding an initial solution based on the add heuristic [6]. The add heuristic is a greedy search heuristic that first finds the 1-median solution and then searches the remaining parts to find the median that most decreases the objective function value. This procedure is repeated until p-medians have been selected. The alteration of the Teitz and Bart heuristic was motivated by the new objective function as well as the need for implementation on realistically large problems. The steps for the modified vertex substitution heuristic are: Initialization step:
Step 1: Select an initial solution of part groupings using the add heuristic. The steps for the add heuristic are: (a) Find the 1-median solution: total the columns of the distance matrix and select part j corresponding to minimum total in column j as the 1-median. (b) Add a median, from the set of parts not yet selected as medians, to the set of medians already selected that most decreases the objective function. (c) Repeat step (b) until p medians are selected. Step 2: Select a similarity coefficient, d~i, to use as a dissimilarity measure between all pairs of parts. Main steps: Step 3: Form part families by assigning each part to the median part that is "closest" with respect to the dissimilarity coefficient. Step 4: Compute the objective function for the current solution using expression (15). Step 5: Select a part that is not a cell seed in the current solution. This part should be one that has never been substituted before. If no part is found, go to step 7. Step 6: If the substitute improves the current solution, replace the current solution. Go to step 5. Step 7: Stop with a current solution that estimates the optimal part family assignment. Several methods for grouping problems using a dissimilarity measure have been proposed. The part-machine matrix is used to compute the metric where a one indicates a part uses a particular machine and zero's elsewhere. The Hamming distance metric used by Kusiak [2] in a p-median model for the cell formation problem uses the part-machine matrix by counting the number of non-matches between two parts. In Dutta et al. [16], a dissimilarity measure was proposed that totals the number of machines used by two parts and subtracts two times the number
M a n u f a c t u r i n g cell f o r m a t i o n
139
of machines they have in common, if they use the same machines, the dissimilarity is zero. Other measures can be used in the model. WORST-CASE
RUN TIME
Analysis of the order of the algorithm used to implement the adapted Teitz and Bart heuristic shows that it is polynomial with respect to worst case running time. In the initialization step, finding the 1-median is O(n) because it requires finding a minimum of n values [17]. Step l(b) requires n - c executions where c is the current number of medians and step l(b) must be repeated p times. Therefore the initialization step is O(np). For the first main steps of the heuristic, assignment of n nodes to p medians is O(np). In step 2, the objective function computes the distances between parts with all other parts in their groups. This requires (~) steps in the worst case where the paired distances for all n parts must be used. n . nk The upper bound for the running time of (k) is (k,) (see Cormen et al. [17]). In this case, k = 2, so the worst case running time for this step is i3(n2). Step 3 is O(n - p ) , because it selects parts not in the current solution, or ( n - p ) steps in the worst case. Step 4 is a constant and trivial in terms of effect on running time. The loop of steps 2 through 4 is repeated at most (n - p) times, since p parts are medians and there are at most ( n - p) parts to select from for substitution. Summarizing the worst case running time in terms of n and p: Initialization step: O(np). Step 1: O(np). Step 2: O(n2). Step 3: O ( n - p). Step 4: constant.
Steps 2-4 are repeated at most (n - p ) times, resulting in a running time of (n3-pn2). Since p < n, overall worst case running time of the heuristic is O(n2). The part-machine matrix is an input format for the specific application of the p-median model and heuristic to the cell formation problem. Input for the p-median model and heuristic can also be a network, making it applicable to a variety of problems such as location-allocation problems, transportation and service problems, and facility layout and location problems. The worst case run time changes with network input. The all points shortest path problem must be solved to generate the distance matrix. The Floyd-Warshall algorithm used to solve this problem is O(n 3) [18]. So for step 1, the worst case run time is O(np) or O(n3). Thus, with a network input, the worst case run time will remain polynomial. OBJECTIVE FUNCTION
LOWER
BOUND
Bounds for the p-median problem are generally found through linear programming relaxation's or dual Lagrangean relaxation's [6]. Realistic cell formation problems are generally of a size where bounds would be difficult to find with these methods. For example, a problem consisting of n = I000 parts and p = 100 cells would result in a model with 1 million decision variables and over 1 million constraints in the linear programming relaxation of Equations (10)(14). To judge the quality of heuristic solutions for these large-scale problems, a lower bound is developed in this section. The lower bound is computed by summing the minimum number of minimum distances from the distance matrix. The steps for finding the lower bound are given below followed by an outline of the proof that the computation is guaranteed to be less than or equal to the optimal objective function value. The steps for finding the lower bound are: Input: A [d~j[, where d U is the dissimilarity measure between parts i and j.
Stuart Jay Deutsch et al.
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Step 1: Calculate the average number of parts per group for all p groups by dividing n parts into p groups where number of parts per group is as close to n/p as possible. Define ah to be the number of parts in group h, for h = 1. . . . ,p. Step 2: Calculate the total number of pairwise dissimilarities to be included in the sum ¢/h r = E PPh=I(2 )" Step 3: The number r found in step 3 is the number of minimum terms to be selected from above the diagonal in the dissimilarity matrix Idol. Sum the selected distances to get the lower bound. While the objective function for any part grouping solution is the sum of dissimilarities between all pairs of parts within a group, summed for all p groups, the lower bound is the sum of r minimum distances between pairs of parts selected from all part distances in the distance matrix. The minimum number of distances that comprise the lower bound for the objective function occurs when the size of the groups is equal or nearly equal, that is close to n/p. This sum of the smallest number of minimum distances may not be a feasible solution because it does not represent distances resulting from an actual part grouping solution.The following sections show that the minimum number of distances for the problem is achieved when the size of the groups is equal or nearly equal. First note that the total number of dissimilarity terms summed in objective function (15)is always (2') + (~2)+ ... + ( f ) for a feasible solution to the grouping constraints. To show that the minimum number of distances occurs when the groups are of equal size, it is shown that shifting parts so that groups of equal size are now unequal size always increases r, the number of distances needed for the lower bound sum.
Case 1: nip is integer In Case 1, n/p is integer. All group sizes are exactly equal so that the sum of the number of distances for two groups is
(~") + (2P) = a2p - ap
(16)
If x is the number of parts shifted from one of the equal size groups to the other, then for two unequal sized groups the sum of the number of distances for two groups is ap + x
2 2 -- a~ -- ap ÷ x
(17)
The result of Equation (17) is greater than Equation (16) for all x > 0. In general, any shifts from equal sized groups to unequal sized groups will increase the number of distances needed in the objective function sum, therefore the number of distances used in the objective function calculation is minimum in the equal sized groups of Case 1.
Case 2: nip is not integer In Case 2, nip is not integer. Some groups are 1 part larger than others to make the groups sizes nearly equal. As before, using two groups, where one group has ap parts and the second group has ap + 1 parts, the sum of the number of distances for the two groups is
(.p+l~ 2 ,
2
(18)
Two kinds of part shifts must be considered, the first shifts x parts from the smaller group to the larger group. The sum of the number of distances for the two groups is now (a,~+,+x) + (q-x) = a2 + x 2 + x
(19)
For all x > 0 , this number is always larger than (18). In the second kind of shift, x parts are shifted from the larger group to the smaller group, so that the sum of the number of distances is
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141
Part Number 1 2 3 4 5 6 A Machine
B C D E
1
1
462 D E Machine B
1
1
1 1
1
Part Number
1 1
1
A
1
1
C
Initial Part-Machine Matrix
1 3
1
1
1
1
1
1
1 1
1 1
Rearranged Part-Machine Matrix
Fig. 1. A n example of a p a r t - m a c h i n e matrix showing the case of completely separable cells.
(ap+l-x~ 2
UP+x~= a 2 + x 2 - x
,' "1- ~ 2
!
(20)
If x = 1, then the number of distances is a 2, or equal to (18). For all values of x > 1, then, (20) is always greater than (18). The number of distances used in the objective function calculation is minimum in the nearly equal sized groups of Case 2. Therefore, Case 1 and Case 2 show that the minimum number of distances occurs when the part groups are of equal or nearly equal size. If the minimum number of minimum distances are summed from the distance matrix, this will be a lower bound for the heuristic. To show that the lower bound described is valid for all feasible solutions, consider the unweighted objective f u n c t i o n ~pph=lY~4eGhY~jeGh;i
~-'~J~
IdeA,
RESULTS ON SAMPLE PROBLEMS
In this section, the modified p-median model and vertex substitution method is used to compare the part groupings to those found by other techniques for eleven problems from the literature. While the objective function value is one measure of performance for manufacturing cell formation problems, often the minimum objective function value evaluation results in a multiplicity of solutions and other measures are needed to characterize the quality of each solution. Other measures may also be used to address the degree of separability the cells have achieved. The input for the part grouping problem is typically a part-machine matrix, consisting of O's and l's, showing the machines required to make a part. Figure 1 shows a part-machine matrix that has five unique machines and six parts that utilize these machines. The l's in the matrix indicate the use of a particular machine by a part, and a blank or a 0 indicates that machine is not used by that part. For this problem, two part families can be found, creating two machine
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al.
P~Numb~ 1 2 3 4 5 6 A
1
B Machine C D E
Part Number 24536 1
1 1 1 1 1 1
1 1
1
1 11
Initial Part-Machine Matrix
C E Machine B
1
1
1
1
1
1
1
1
1
A
1
D
11
1
Rearranged Part-Machine Matrix
Fig. 2. An exampleof a part-machine matrix showingthe case of partially separable cells, with a bottleneck part. cells. Undesirable intercellular moves occur when a part cannot be completely processed in its assigned cell. These cells are said to be interconnected by bottleneck machines or bottleneck parts. Figure 2 gives a second example of a part-machine matrix with six parts requiring five machines. In this case, rows and columns cannot be rearranged to form two completely separable cells. This problem results in a bottleneck part or machine. For example, part 3 is a bottleneck part because it requires one operation to be done on machine B which is not in the cell. Machine B is referred to as a bottleneck machine, because it is needed by parts in more than one cell. In addition to an objective functions value, the number of bottleneck machines and number of machine duplications are measures of solution quality for cell formation problems. The machine duplication number is the number of additional machines that must be added because in the solution they are required in more than one cell. The duplication allows the creation of completely separable cells. The number of bottlenecks is a count of the unique machine types required in more than one cell. Cell formation problems may have a number of alternate solutions, particularly if there are bottlenecks and consequently there may not be a single preferred solution for every problem in which the solution is based solely on the objective function value. A set of part family formation problems published in the cellular manufacturing literature are used to demonstrate solution quality for the part family formation problem using the modified p-median model with the vertex substitution heuristic. The problems were selected because they have been used previously in comparing cell formation methods, and because they have also been used when new cell formation methods were proposed. Table 1 compares the previous solutions and the solution found by the modified p-median model. Columns 2, 3 and 4 contain the source of the problem, the author's method for part grouping and the problem size respectively. Columns 5, 6 and 7 contain the OFV for the original solution, for the original p-median model [2] and for the proposed modified p-median model respectively. To enable direct comparison, the solutions reported for the original cell formation example and the original p-median model were used to compute an OFV using Equation (15). Thus, the objective function value, for all solutions, is based on the sum of the pairwise dissimilarity between all parts in a group. The lower bound for the objective function is reported in column 8. Columns 9, 10 and 11 contain the resulting number of machine duplications for the original solution, the original p-median model [2] and the proposed modified p-median model, respectively. Similarly, columns 12-14 give the corresponding results for the number of bottlenecks. For all of the published problems, the objective function valued for the solution for the modified p-median is the same or lower than the value associated with the solution from the original cell formation method or the original p-median model. Thus, the vertex substitution heuristic provides better results for these problems in terms of the objective function. Also, the computed lower bound illustrates that the vertex substitution heuristic provides reasonable solutions since the lower bound is equal to the objective function for non-bottlenecked problems, and the resulting objective function, for all other cases, is less than 15% above the lower bound for bottlenecked systems. When the problem is not bottlenecked, the choice of cell formation method is inconsequential since all solutions are the same quality. For the remaining eight bottlenecked
Manufacturing
143
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problems, the modified p-median model resulted in lower OFV's in six cases and equivalent value in problems 4 and 10. Distinguishing these problems from the other bottlenecked problems is that groups are readily defined since each part requires many machines with only an occasional machine being used by more than one part. Problem 10 has a higher density of ones in its part-machine matrix than does problem four. This results in an objective function shape associated with a non-bottlenecked problem. When a problem has bottlenecks and also a sharp, non-flat objective function, these problems are pseudo-bottlenecked. Pseudo-bottlenecked problems, like unbottlenecked problems, converge to one solution and the quality of the solution will be robust with regard to the choice of cell formation method. On the other hand, for more severely bottlenecked problems, such as problems 9 and 11, the solution for the modified p-median model results in significantly lower values of the objective function. The modified p-median method resulted in solutions with number of machine duplications as small or smaller than the other cell formation methods in nine of the eleven problems. Problems 3 and 6 were of lesser quality since they require a larger number of duplicates in the modified pmedian solution although the solution was of better quality in terms of a smaller OFV. Similarly, in four of the eleven problems, the modified p-median model resulted in a greater number of bottlenecks than the other cell formation methods (worse quality), but a better quality solution with lower OFV's in these problems. It should be noted that in problems eight and eleven since the number of duplicate machines are the same size or smaller and simultaneously, the number of bottlenecks are larger than other cell formation methods, the cost structure of type of bottlenecked machine that needs to be duplicated will determine the best solution. The modified p-median method does not contain explicit criterion for reducing the number of bottlenecks or machine duplications, but the objective function of the model groups parts that use the same sets of machines, so that reduction of bottlenecks is an implicit goal. Dual consideration in the model to explicitly and simultaneously minimize objective function value and number of machine duplications is a logical next step from this modified p-median model. SUMMARY AND CONCLUSIONS Many methods have been used to solve the cell formation problem using similarity coefficients, and mathematical programming models. Choobineh [30] formed cells by attempting to minimize the costs of producing a part in each cell and of purchasing new equipment, but similarity coefficients used are not known until the problem is solved, so the method is iterative. The modified p-median model with the vertex substitution heuristic proposed is not iterative, only requiring data given by the standard part-machine matrix. Co and Araar [31] minimized the deviation between workload assigned to each machine and the capacity of the machine to form cells but capacity limitations may result in two similar parts being assigned to different machine cells, and so solutions do not reflect complete divisibility. The solutions found with the modified p-median heuristic do reflect divisibility. Gunasingh and Lashkari [32] presented two 0-1 integer programming models which maximize the similarity between machines and parts, and minimize cost minus savings in intercell movements where tooling is used to assess similarity, with limits set on machine purchase costs. Askin and Chiu [33] use 0-1 integer programming and consider four cost categories, machine, group, tooling, and intercellular transfer costs. Goal programming approaches [34] have combined difficult problems such as the p-median and the TSP problem. These three models [32-34], are of greater resolution but require larger amounts of data to include these components and therefore can only handle small problem sizes. The modified pmedian method is shown to be amenable for large problem sizes. Wei and Gaither [35] also formulate a 0-1 integer programming problem that minimizes the cost of intercellular transfers with limits on the cell sizes, which may limit finding improved solutions and is also restricted to small problems. Cluster analysis methods [27, 28, 36, 37] rearrange large data sets, so that these are not efficient for large problems and assessing solution quality is difficult without an explicit objective function. The modified p-median method contains an explicit objective function, and several quantifiable measures of solution quality are used. By modifying the traditional p-median model to consider an objective function that minimizes the sum of pairwise dissimilarity between parts assigned to the same group, the cell formation/
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part family assignment decision can be improved. The result is a more explicit approach to minimizing the variability of parts assigned to the same group. In this paper, it is also shown that etficient heuristic methods are possible for large, realistically sized cellular manufacturing problems by applying an adapted version of the Teitz and Bart vertex substitution heuristic to find near optimal solutions for the proposed model formulation. Comparison to other techniques, using published problems, showed that the heuristic provides solutions of similar or better quality with respect to a combined dissimilarity measurement. As a heuristic for implicitly minimizing bottlenecks and machine duplication, the proposed approach also appears to work favorably in comparison with other published techniques. REFERENCES 1. Kusiak, A., Boe, J. W. and Cheng, C. H. Clustering Analysis: models and algorithms. Control and Cybernetics, 15, 1986, 139-154. 2. Kusiak, A. The part families problem in Flexible Manufacturing Systems. Annals of Operations Research, 3, 1985, 279-300. 3. Klastorin, T. D. The p-median problem for cluster analysis: a comparative test using the Mixture Model approach. Management Science, 31, 1985, 84-95. 4. Narula, S. C., Ogbu, U. 1. and Samuelsson, H. M. An algorithm for the p-median problem. Operations Research, 25, 1977, 709-712. 5. Jarvinen, P., Rajala, J. and Sinervo, H. A branch-and-bound algorithm for seeking the p-median. Operations Research, 20, 1972, 173-182. 6. Francis, R. L., McGinnis, Jr., L. F. and White, J. A. Facility Layout and Location: An Analytical Approach. Prentice-Hall, Englewood Cliffs, NJ, 1992. 7. Maranzana, F. E. On the location of supply points to minimize transport costs. Operations Research Quarterly. 15, 1964, 261-270. 8. Khumawala, B. M. An efficient algorithm for the p-median problems with maximum distance constraints. Geographical Analysis, 5, 1973, 309-321. 9. Teitz, M. B. and Bart, P. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16, 1968, 955-961. 10. Rosing, K. E., Hillsman, E. L. and Rosing-Vogelaar, H. A note comparing optimal and heuristic solutions to the pmedian problem. Geographical Analysis, 11, 1979, 86-89. II. Goodchild, M. F. and Noronha, V. T. Location-allocation for small computers. Monograph 8, Department of Geography, The University of Iowa, Iowa City, IA, 1983. 12. Densham, P. J. and Rushton, G. A more efficient heuristic for solving large p-median problems. Papers in Regional Science: the Journal of the RSAI, 71, 1992, 307-329~ 13. Garey, M. R. and Johnson, D. S. Computers and Intractability: A Guide to the Theory of NP-Completeness. Bell Telephone Laboratories, Inc., 1990. 14. Parker, R. G. and Rardin, R. L. Discrete Optimization. Academic Press, New York, 1988. 15. Hillier, F. S. and Lieberman, G. J. Introduction to Operations Research, 5th edn. McGraw-Hill, New York, 1990. 16. Dutta, S. P., Lashkari, R. S., Nadoli, G. and Ravi, T. A heuristic procedure for determining manufacturing families form design-based grouping for flexible manufacturing. Computers and Industrial Engineering, 16, 1986, 193-201. 17. Cormen, T. H., Leiserson, C. E. and Rivest, R. L. Introduction to Algorithms. MIT Press, Cambridge, MA, 1990. 18. Ahuja, R. K., Magnanti, T. L. and Orlin, J. B. Network Flows, Theory, Algorithms and Applications. Prentice-Hall, Englewood Cliffs, NJ, 1993. 19. King, J. R. and Nakornchai, V. Machine-component group formation in group technology: review and extension. International Journal of Production Research, 20, 1982, 117 133. 20. Kusiak, A. and Chow, W. S. Efficient solving of the group technology problem. Journal of Manufacturing Systems, 6, 1987, 117 124. 21. Seifoddini, H. and Wolfe, P. M. Application of the similarity coefficient method in group technology, liE Transactions, 9, 1986, 271-277. 22. Seifoddini, H. Single linkage versus average linkage clustering in machine cells formation applications. Computers in Industrial Engineering, 16, 1989, 419-426. 23. Chan, H. M. and Milner, D. A. Direct clustering algorithm for group formation cellular manufacturing. Journal of Manufacturing Systems, 1, 1981, 65-74. 24. Stanfel, L. E. Machine clustering for economic production. Engineering Costs and Production Economics, 9, 1985, 73-81. 25. Chandrasekharan, M. P. and Rajagopalan, R. GROUPABILITY: an analysis of the properties of binary data matrices for group technology. International Journal of Production Research, 27, 1989, 1035-1052. 26. Carrie, A. S. Numerical taxonomy applied to group technology and plant layout. International Journal of Production Research, 11, 1973, 399-416. 27. King, J. R. Machine-component grouping in production flow analysis: an approach using a rank order clustering algorithm. International Journal of Production Research, 18, 1980, 213-232. 28. Chandrasekharan, M. P. and Rajagopalan, R. An ideal seed non-hierarchical clustering algorithm for cellular manufacturing. International Journal of Production Research, 24, 1986, 451-464. 29. Shafer, S. M. and Rogers, D. F. Similarity and distance measures for cellular manufacturing: Part II. An extension and comparison. International Journal of Production Research, 31, 1993, 1315 1326. 30. Choobineh, F. A framework for the design of cellular manufacturing systems. International Journal o[ Production Research, 26, 1988, 1161-1172.
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31. Co, H. C. and Araar, A. Configuring cellular manufacturing systems. International Journal of Production Research, 26, 1988, 1511-1522. 32. Gunasingh, K. R. and Lashkari, R. S. Machine grouping problem in cellular manufacturing systems--an integer programming approach. International Journal of Production Research, 27, 1989, 1465-1473. 33. Askin, R. G. and Chiu, K. S. A graph partitioning procedure for machine assignment and cell formation in group technology. International Journal of Production Research, 28, 1990, 1555-1572. 34. Shafer, S. M. A goal programming approach to the cell formation problem. Journal of Operations Management, 10, 1991, 28-50. 35. Wei, J. C. and Gaither, N. An optimal model for cell formation decisions. Decision Sciences, 21, 1990, 416-432. 36. Peihua, G. Process-based machine grouping for cellular manufacturing systems. Computers in Industry, 17, 1991, 917. 37. Askin, R. G. and Subramanian, S. A cost-based heuristic for group technology configuration. International Journal of Production Research, 25, 1987, 101-113.
Computers ind. Engng Vol. 34. No. 1, pp. 135-146, 1998 © 1998 Publishedby ElsevierScienceLtd. All rights reserved PII: S0360-8352(97)00156-3 Printed in Great Britain 0360-8352/98 $19.00 + 0.00
MANUFACTURING IMPROVED
CELL FORMATION P-MEDIAN
USING
AN
MODEL
STUART JAY DEUTSCH, SUSAN F. FREEMAN and MARY HELANDER Department of Mechanical, Industrial and Manufacturing Engineering, Northeastern University, Boston, MA 02 115, U.S.A. Abstract--The p-median model objective function is modified for the cell formation problem to minimize the variability between parts in a group by considering part similarity to all other parts in the group instead of similarity to an arbitrary median. The heuristic vertex substitution method for solution of the part grouping problem is adapted for this objective function and then modified to provide improved starting points. The theoretical lower bound for the heuristic is developed and shown to be valid for all solutions. Worst case run time is shown to be O(n 2) or O(n 3) for distance matrix or network inputs respectively. Tests on published problems show that the proposed p-median model method provides as good or better objective function value (OFV) than the OFV of a p-median model in which parts are grouped to an arbitrary median. Likewise the new p-median model is shown, for these published problems, to give as good or better OFV than the algorithms reported by the original authors of the problem. The test problems suggest that other measures of solution quality such as bottlenecks and duplicate machines in addition to OFV become important measures of solution quality for dense problems. @ 1998 Elsevier Science Ltd. All rights reserved.
INTRODUCTION
The manufacturing cell formation problem can be solved by forming part families that place parts in groups based on common machine usage. Families of parts can be formed with a model that contains an objective function that maximizes similarities between all parts or minimizes dissimilarities between all parts [1]. This approach effectively forms groups where the variability between parts in a group is to be minimized. The drawbacks of this quadratic model are due to the nonlinearity of the objective function, therefore the p-median model has been used to groups parts by minimizing the dissimilarity between parts in a group and their median [2]. The p-median problem on a general network of n nodes involves finding a subset of p nodes called medians so as to minimize the sum of the weighted distances between each of the nodes and the closest median. Klastorin [3] found that the p-median model worked well for clustering data when the similarity metric corresponded to a distance measure that was Euclidean. If the weights in the p-median decision model are set to 1, the results from the clustering model are equivalent to the unweighted p-median model. In this paper, the p-median model is adapted so that its objective function explicitly minimizes the dissimilarities between all parts in a group by summing all the pairwise distances between parts in a group versus the traditional p-median formulation that sums the distances to an arbitrary median part. Due to the underlying problem complexities, exact solution methods provided by Narula, Ogbu and Samuelson [4] and Jarvinen, Rajala and Sinervo [5] are restricted to small problem sizes. Other heuristic approaches by Francis, McGinnis and White [6], Maranzana [7], and Khumawala [8] are also limited to small problems. The Teitz and Bart vertex substitution heuristic [9] was chosen for implementation because it is known to be efficient for finding high quality, near optimal solutions to large scale p-median problems (see Rosing et al. [10], Goodchild and Noronha [11], and Densham and Rushton [12]). Since the part family decision model proposed is based on the p-median model, the use of a heuristic is necessary because the p-median problem is NP-complete [13, 14]. The vertex substitution heuristic successively substitutes vertices in the complement of the solution. If a cycle of substitutions finds an improved solution, the process continues until no improvement is found. The next section develops a cell formation model by adapting a p-median model objective function to group parts into cells effectively. In Section 3, the Teitz and Bart heuristic [9] for sol135
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ving the adapted p-median model is implemented with the adapted objective function and additional modifications to improve the heuristic's starting point. Next, the worst case run time for the heuristic is shown, to demonstrate the applicability of the heuristic to any problem size. In Section 5, a lower bound is developed that can be used to judge the quality of the solutions found and is shown to be valid. In Section 6, the modified p-median model and solution heuristic is applied to a number of published problems from the literature and the quality of the resulting solutions is compared to previously reported solutions. Bottleneck and machine duplications are computed along with the objective function values to compare the quality of solutions and graphical analysis is provided to demonstrate the relationship between problem characteristics and resulting solutions. The last section contains the summary and conclusions.
THE CELL F O R M A T I O N D E C I S I O N M O D E L
In a decision model for the part family formation problem, a set of 0-1 decision variables are used to assign parts to families. Due to the nonlinearity of the objective function, this form is adapted to the p-median model where the decision variables determine if parts are located in families together, given an assigned median for any given family. The p-median objective function is then altered to reflect the objective of cell formation by finding the sum of the pairwise distances between all parts in a group, as opposed to summing distances to an arbitrary median part. Let n be the total number of parts to be grouped into at most p families. Let ~i~ be the decision variable that describes whether or not part i should be assigned to the kth family identified. That is, ~ik =
1 0,
if part i is assigned to family k, otherwise
(1)
for i = 1. . . . . n and k = 1,...,p. The following two constraints ensure that ensure that each part is assigned to exactly one family: Z p p ~ i k = 1 for i = 1. . . . . n
(2)
k=l
aik ~ {0,1} for i = 1 , . . . , n and k = 1. . . . ,p
(3)
Note that, as long as the ~tk's are specified according to constraints (2) and (3), then for any two parts id' c { 1,2 . . . . . n} and family k e { 1,2 . . . . ,p}, ~ik~jk =
1 0
if parts i and j are both assigned to family k, if either part i or j is not in family k
(4)
if parts i and j are assigned to the same family, if parts i and j are not assigned to the same family
(5)
and
Z pp~ik~tjk = k=l
{~
Let wi be a weight associated with part i. These weights can be quantities reflecting part production volume, part cost or part priority. In this application the weights are set to 1. Let d~j denote a general dissimilarity measure for parts i and j. The value of d~j can be thought of as the cost incurred when parts i and j are placed in the same family, and can also be thought of as a measure of distance. A general decision model for the part family formation problem is: n-1
(P1)
Minimize
n
Y~PPZ Z k=l
i=lj=i+l
wido'O~ikO~jk
(6)
M a n u f a c t u r i n g cell f o r m a t i o n
~_,pp~xik=
subject to
1
137
for i = 1. . . . . n
(7)
k=l
~ik6{O,1}
fori=l
.... nandk=l
.... p
(8)
This decision model given by (P1) seeks to assign every part to exactly one family to minimize the total weighted sum of distances between all pairs of parts in the same family. This formulation is a 0-1 integer programming model with a nonlinear objective function. Since the nonlinearity of the objective function involves the product of decision variable pairs and the constraints are fully linear, this problem is also referred to as a quadraticmodel [15]. In the p-median model [2], p parts are first selected to serve as medians or seed parts for clustering and remaining parts are assigned to medians. The feasible range of p is from 1 to n, but the value o f p selected for any problem is determined from characteristics of the problem such as part-machine matrix density, part volume, machine capacity and problem size. To show how the general cell formation model relates to the p-median formulation, note that since each part must be assigned to exactly one family, then 1 _
x,~ --
1 0
if parts i and j are assigned to the same family, if parts i and j are not assigned to the same family
(9)
for i -- 1. . . . . n a n d j -- 1. . . . . n, thep-median based decision model is: //
(P2)
Minimize
n
~--~y~wid~yx~j
(10)
i=l j=i n
subject to
Zxi~=l
fori=l
..... n
(11)
j=l n
Zxi~ = p
(12)
j=l
x~
for i = 1. . . . . n and j = fori=l
1,...,n
..... hand j=
1. . . . . n
(13) (14)
Constraint sets (11) and (14) work together to ensure that each part is assigned to exactly one part family. Constraint set (12) ensures that at most p of the n parts will be chosen as seeds for creating the families. Constraint set (13) enforces the assignment of a seed identity to exactly one part per family. The objective function for the p-median based decision model given by expression (10) drives the selection of a feasible part family assignment by minimizing within each family the total weighted dissimilarity between the part assigned as the family seed and all other parts assigned to its group. In this way, the p-median formulation fails to consider the possible pairwise dissimilarity between non-seed parts assigned to the same family. To consider all pairwise dissimilarities within assigned groups, the objective function given by (10) must be replaced by Minimize
ZppZ h= 1
Z widij
iE Gh]~ Ghi'<3"
(15)
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where the set Gh denotes the subset of parts assigned to the hth family. These sets can be constructed directed from the decision variables as follows: Gh ~ {i: X,h -----1}
for h = 1. . . . ,p. Given any feasible solution to constraint sets (11)-(14), note that the sets G1 . . . . ,Gp will be disjoint and their union will cover the set of parts { 1,2 . . . . . n}. That is, each of the n parts will be assigned t o exactly one of the sets G~ . . . . . Gp. The replacement of the p-median objective function (10) by (15) results in a fully linearized decision model that is equivalent to the formulation given by the decision model (P0. The unweighted modified version is used in this paper to formulate part families that are feasible to constraint sets (11)-(14) while minimizing the total pairwise part dissimilarity within each group. This approach more explicitly considers the variability of parts assigned to the same group than does the direct use of the traditional p-median formulation.
THE VERTEX SUBSTITUTION HEURISTIC In this work, the Teitz and Bart vertex substitution heuristic [9] was adapted to the part family decision problem by using the objective function given by expression (15) instead of the standard p-median objective function given by (10). The modified vertex substitution heuristic was further adapted by incorporating an efficient method for finding an initial solution based on the add heuristic [6]. The add heuristic is a greedy search heuristic that first finds the 1-median solution and then searches the remaining parts to find the median that most decreases the objective function value. This procedure is repeated until p-medians have been selected. The alteration of the Teitz and Bart heuristic was motivated by the new objective function as well as the need for implementation on realistically large problems. The steps for the modified vertex substitution heuristic are: Initialization step:
Step 1: Select an initial solution of part groupings using the add heuristic. The steps for the add heuristic are: (a) Find the 1-median solution: total the columns of the distance matrix and select part j corresponding to minimum total in column j as the 1-median. (b) Add a median, from the set of parts not yet selected as medians, to the set of medians already selected that most decreases the objective function. (c) Repeat step (b) until p medians are selected. Step 2: Select a similarity coefficient, d~i, to use as a dissimilarity measure between all pairs of parts. Main steps: Step 3: Form part families by assigning each part to the median part that is "closest" with respect to the dissimilarity coefficient. Step 4: Compute the objective function for the current solution using expression (15). Step 5: Select a part that is not a cell seed in the current solution. This part should be one that has never been substituted before. If no part is found, go to step 7. Step 6: If the substitute improves the current solution, replace the current solution. Go to step 5. Step 7: Stop with a current solution that estimates the optimal part family assignment. Several methods for grouping problems using a dissimilarity measure have been proposed. The part-machine matrix is used to compute the metric where a one indicates a part uses a particular machine and zero's elsewhere. The Hamming distance metric used by Kusiak [2] in a p-median model for the cell formation problem uses the part-machine matrix by counting the number of non-matches between two parts. In Dutta et al. [16], a dissimilarity measure was proposed that totals the number of machines used by two parts and subtracts two times the number
M a n u f a c t u r i n g cell f o r m a t i o n
139
of machines they have in common, if they use the same machines, the dissimilarity is zero. Other measures can be used in the model. WORST-CASE
RUN TIME
Analysis of the order of the algorithm used to implement the adapted Teitz and Bart heuristic shows that it is polynomial with respect to worst case running time. In the initialization step, finding the 1-median is O(n) because it requires finding a minimum of n values [17]. Step l(b) requires n - c executions where c is the current number of medians and step l(b) must be repeated p times. Therefore the initialization step is O(np). For the first main steps of the heuristic, assignment of n nodes to p medians is O(np). In step 2, the objective function computes the distances between parts with all other parts in their groups. This requires (~) steps in the worst case where the paired distances for all n parts must be used. n . nk The upper bound for the running time of (k) is (k,) (see Cormen et al. [17]). In this case, k = 2, so the worst case running time for this step is i3(n2). Step 3 is O(n - p ) , because it selects parts not in the current solution, or ( n - p ) steps in the worst case. Step 4 is a constant and trivial in terms of effect on running time. The loop of steps 2 through 4 is repeated at most (n - p) times, since p parts are medians and there are at most ( n - p) parts to select from for substitution. Summarizing the worst case running time in terms of n and p: Initialization step: O(np). Step 1: O(np). Step 2: O(n2). Step 3: O ( n - p). Step 4: constant.
Steps 2-4 are repeated at most (n - p ) times, resulting in a running time of (n3-pn2). Since p < n, overall worst case running time of the heuristic is O(n2). The part-machine matrix is an input format for the specific application of the p-median model and heuristic to the cell formation problem. Input for the p-median model and heuristic can also be a network, making it applicable to a variety of problems such as location-allocation problems, transportation and service problems, and facility layout and location problems. The worst case run time changes with network input. The all points shortest path problem must be solved to generate the distance matrix. The Floyd-Warshall algorithm used to solve this problem is O(n 3) [18]. So for step 1, the worst case run time is O(np) or O(n3). Thus, with a network input, the worst case run time will remain polynomial. OBJECTIVE FUNCTION
LOWER
BOUND
Bounds for the p-median problem are generally found through linear programming relaxation's or dual Lagrangean relaxation's [6]. Realistic cell formation problems are generally of a size where bounds would be difficult to find with these methods. For example, a problem consisting of n = I000 parts and p = 100 cells would result in a model with 1 million decision variables and over 1 million constraints in the linear programming relaxation of Equations (10)(14). To judge the quality of heuristic solutions for these large-scale problems, a lower bound is developed in this section. The lower bound is computed by summing the minimum number of minimum distances from the distance matrix. The steps for finding the lower bound are given below followed by an outline of the proof that the computation is guaranteed to be less than or equal to the optimal objective function value. The steps for finding the lower bound are: Input: A [d~j[, where d U is the dissimilarity measure between parts i and j.
Stuart Jay Deutsch et al.
140
Step 1: Calculate the average number of parts per group for all p groups by dividing n parts into p groups where number of parts per group is as close to n/p as possible. Define ah to be the number of parts in group h, for h = 1. . . . ,p. Step 2: Calculate the total number of pairwise dissimilarities to be included in the sum ¢/h r = E PPh=I(2 )" Step 3: The number r found in step 3 is the number of minimum terms to be selected from above the diagonal in the dissimilarity matrix Idol. Sum the selected distances to get the lower bound. While the objective function for any part grouping solution is the sum of dissimilarities between all pairs of parts within a group, summed for all p groups, the lower bound is the sum of r minimum distances between pairs of parts selected from all part distances in the distance matrix. The minimum number of distances that comprise the lower bound for the objective function occurs when the size of the groups is equal or nearly equal, that is close to n/p. This sum of the smallest number of minimum distances may not be a feasible solution because it does not represent distances resulting from an actual part grouping solution.The following sections show that the minimum number of distances for the problem is achieved when the size of the groups is equal or nearly equal. First note that the total number of dissimilarity terms summed in objective function (15)is always (2') + (~2)+ ... + ( f ) for a feasible solution to the grouping constraints. To show that the minimum number of distances occurs when the groups are of equal size, it is shown that shifting parts so that groups of equal size are now unequal size always increases r, the number of distances needed for the lower bound sum.
Case 1: nip is integer In Case 1, n/p is integer. All group sizes are exactly equal so that the sum of the number of distances for two groups is
(~") + (2P) = a2p - ap
(16)
If x is the number of parts shifted from one of the equal size groups to the other, then for two unequal sized groups the sum of the number of distances for two groups is ap + x
2 2 -- a~ -- ap ÷ x
(17)
The result of Equation (17) is greater than Equation (16) for all x > 0. In general, any shifts from equal sized groups to unequal sized groups will increase the number of distances needed in the objective function sum, therefore the number of distances used in the objective function calculation is minimum in the equal sized groups of Case 1.
Case 2: nip is not integer In Case 2, nip is not integer. Some groups are 1 part larger than others to make the groups sizes nearly equal. As before, using two groups, where one group has ap parts and the second group has ap + 1 parts, the sum of the number of distances for the two groups is
(.p+l~ 2 ,
2
(18)
Two kinds of part shifts must be considered, the first shifts x parts from the smaller group to the larger group. The sum of the number of distances for the two groups is now (a,~+,+x) + (q-x) = a2 + x 2 + x
(19)
For all x > 0 , this number is always larger than (18). In the second kind of shift, x parts are shifted from the larger group to the smaller group, so that the sum of the number of distances is
Manufacturing cell formation
141
Part Number 1 2 3 4 5 6 A Machine
B C D E
1
1
462 D E Machine B
1
1
1 1
1
Part Number
1 1
1
A
1
1
C
Initial Part-Machine Matrix
1 3
1
1
1
1
1
1
1 1
1 1
Rearranged Part-Machine Matrix
Fig. 1. A n example of a p a r t - m a c h i n e matrix showing the case of completely separable cells.
(ap+l-x~ 2
UP+x~= a 2 + x 2 - x
,' "1- ~ 2
!
(20)
If x = 1, then the number of distances is a 2, or equal to (18). For all values of x > 1, then, (20) is always greater than (18). The number of distances used in the objective function calculation is minimum in the nearly equal sized groups of Case 2. Therefore, Case 1 and Case 2 show that the minimum number of distances occurs when the part groups are of equal or nearly equal size. If the minimum number of minimum distances are summed from the distance matrix, this will be a lower bound for the heuristic. To show that the lower bound described is valid for all feasible solutions, consider the unweighted objective f u n c t i o n ~pph=lY~4eGhY~jeGh;i
~-'~J~
IdeA,
RESULTS ON SAMPLE PROBLEMS
In this section, the modified p-median model and vertex substitution method is used to compare the part groupings to those found by other techniques for eleven problems from the literature. While the objective function value is one measure of performance for manufacturing cell formation problems, often the minimum objective function value evaluation results in a multiplicity of solutions and other measures are needed to characterize the quality of each solution. Other measures may also be used to address the degree of separability the cells have achieved. The input for the part grouping problem is typically a part-machine matrix, consisting of O's and l's, showing the machines required to make a part. Figure 1 shows a part-machine matrix that has five unique machines and six parts that utilize these machines. The l's in the matrix indicate the use of a particular machine by a part, and a blank or a 0 indicates that machine is not used by that part. For this problem, two part families can be found, creating two machine
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al.
P~Numb~ 1 2 3 4 5 6 A
1
B Machine C D E
Part Number 24536 1
1 1 1 1 1 1
1 1
1
1 11
Initial Part-Machine Matrix
C E Machine B
1
1
1
1
1
1
1
1
1
A
1
D
11
1
Rearranged Part-Machine Matrix
Fig. 2. An exampleof a part-machine matrix showingthe case of partially separable cells, with a bottleneck part. cells. Undesirable intercellular moves occur when a part cannot be completely processed in its assigned cell. These cells are said to be interconnected by bottleneck machines or bottleneck parts. Figure 2 gives a second example of a part-machine matrix with six parts requiring five machines. In this case, rows and columns cannot be rearranged to form two completely separable cells. This problem results in a bottleneck part or machine. For example, part 3 is a bottleneck part because it requires one operation to be done on machine B which is not in the cell. Machine B is referred to as a bottleneck machine, because it is needed by parts in more than one cell. In addition to an objective functions value, the number of bottleneck machines and number of machine duplications are measures of solution quality for cell formation problems. The machine duplication number is the number of additional machines that must be added because in the solution they are required in more than one cell. The duplication allows the creation of completely separable cells. The number of bottlenecks is a count of the unique machine types required in more than one cell. Cell formation problems may have a number of alternate solutions, particularly if there are bottlenecks and consequently there may not be a single preferred solution for every problem in which the solution is based solely on the objective function value. A set of part family formation problems published in the cellular manufacturing literature are used to demonstrate solution quality for the part family formation problem using the modified p-median model with the vertex substitution heuristic. The problems were selected because they have been used previously in comparing cell formation methods, and because they have also been used when new cell formation methods were proposed. Table 1 compares the previous solutions and the solution found by the modified p-median model. Columns 2, 3 and 4 contain the source of the problem, the author's method for part grouping and the problem size respectively. Columns 5, 6 and 7 contain the OFV for the original solution, for the original p-median model [2] and for the proposed modified p-median model respectively. To enable direct comparison, the solutions reported for the original cell formation example and the original p-median model were used to compute an OFV using Equation (15). Thus, the objective function value, for all solutions, is based on the sum of the pairwise dissimilarity between all parts in a group. The lower bound for the objective function is reported in column 8. Columns 9, 10 and 11 contain the resulting number of machine duplications for the original solution, the original p-median model [2] and the proposed modified p-median model, respectively. Similarly, columns 12-14 give the corresponding results for the number of bottlenecks. For all of the published problems, the objective function valued for the solution for the modified p-median is the same or lower than the value associated with the solution from the original cell formation method or the original p-median model. Thus, the vertex substitution heuristic provides better results for these problems in terms of the objective function. Also, the computed lower bound illustrates that the vertex substitution heuristic provides reasonable solutions since the lower bound is equal to the objective function for non-bottlenecked problems, and the resulting objective function, for all other cases, is less than 15% above the lower bound for bottlenecked systems. When the problem is not bottlenecked, the choice of cell formation method is inconsequential since all solutions are the same quality. For the remaining eight bottlenecked
Manufacturing
143
cell formation
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problems, the modified p-median model resulted in lower OFV's in six cases and equivalent value in problems 4 and 10. Distinguishing these problems from the other bottlenecked problems is that groups are readily defined since each part requires many machines with only an occasional machine being used by more than one part. Problem 10 has a higher density of ones in its part-machine matrix than does problem four. This results in an objective function shape associated with a non-bottlenecked problem. When a problem has bottlenecks and also a sharp, non-flat objective function, these problems are pseudo-bottlenecked. Pseudo-bottlenecked problems, like unbottlenecked problems, converge to one solution and the quality of the solution will be robust with regard to the choice of cell formation method. On the other hand, for more severely bottlenecked problems, such as problems 9 and 11, the solution for the modified p-median model results in significantly lower values of the objective function. The modified p-median method resulted in solutions with number of machine duplications as small or smaller than the other cell formation methods in nine of the eleven problems. Problems 3 and 6 were of lesser quality since they require a larger number of duplicates in the modified pmedian solution although the solution was of better quality in terms of a smaller OFV. Similarly, in four of the eleven problems, the modified p-median model resulted in a greater number of bottlenecks than the other cell formation methods (worse quality), but a better quality solution with lower OFV's in these problems. It should be noted that in problems eight and eleven since the number of duplicate machines are the same size or smaller and simultaneously, the number of bottlenecks are larger than other cell formation methods, the cost structure of type of bottlenecked machine that needs to be duplicated will determine the best solution. The modified p-median method does not contain explicit criterion for reducing the number of bottlenecks or machine duplications, but the objective function of the model groups parts that use the same sets of machines, so that reduction of bottlenecks is an implicit goal. Dual consideration in the model to explicitly and simultaneously minimize objective function value and number of machine duplications is a logical next step from this modified p-median model. SUMMARY AND CONCLUSIONS Many methods have been used to solve the cell formation problem using similarity coefficients, and mathematical programming models. Choobineh [30] formed cells by attempting to minimize the costs of producing a part in each cell and of purchasing new equipment, but similarity coefficients used are not known until the problem is solved, so the method is iterative. The modified p-median model with the vertex substitution heuristic proposed is not iterative, only requiring data given by the standard part-machine matrix. Co and Araar [31] minimized the deviation between workload assigned to each machine and the capacity of the machine to form cells but capacity limitations may result in two similar parts being assigned to different machine cells, and so solutions do not reflect complete divisibility. The solutions found with the modified p-median heuristic do reflect divisibility. Gunasingh and Lashkari [32] presented two 0-1 integer programming models which maximize the similarity between machines and parts, and minimize cost minus savings in intercell movements where tooling is used to assess similarity, with limits set on machine purchase costs. Askin and Chiu [33] use 0-1 integer programming and consider four cost categories, machine, group, tooling, and intercellular transfer costs. Goal programming approaches [34] have combined difficult problems such as the p-median and the TSP problem. These three models [32-34], are of greater resolution but require larger amounts of data to include these components and therefore can only handle small problem sizes. The modified pmedian method is shown to be amenable for large problem sizes. Wei and Gaither [35] also formulate a 0-1 integer programming problem that minimizes the cost of intercellular transfers with limits on the cell sizes, which may limit finding improved solutions and is also restricted to small problems. Cluster analysis methods [27, 28, 36, 37] rearrange large data sets, so that these are not efficient for large problems and assessing solution quality is difficult without an explicit objective function. The modified p-median method contains an explicit objective function, and several quantifiable measures of solution quality are used. By modifying the traditional p-median model to consider an objective function that minimizes the sum of pairwise dissimilarity between parts assigned to the same group, the cell formation/
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part family assignment decision can be improved. The result is a more explicit approach to minimizing the variability of parts assigned to the same group. In this paper, it is also shown that etficient heuristic methods are possible for large, realistically sized cellular manufacturing problems by applying an adapted version of the Teitz and Bart vertex substitution heuristic to find near optimal solutions for the proposed model formulation. Comparison to other techniques, using published problems, showed that the heuristic provides solutions of similar or better quality with respect to a combined dissimilarity measurement. As a heuristic for implicitly minimizing bottlenecks and machine duplication, the proposed approach also appears to work favorably in comparison with other published techniques. REFERENCES 1. Kusiak, A., Boe, J. W. and Cheng, C. H. Clustering Analysis: models and algorithms. Control and Cybernetics, 15, 1986, 139-154. 2. Kusiak, A. 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