Part-machine grouping using weighted similarity coefficients

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Abstract

The first step in the transition to cellular manufacturing is part-machine grouping. In this paper, grouping parts into families and machines into cells is done in two phases: by first grouping machines and then assigning parts. Limits both on the number of machines per cell and on the number of parts per family are considered. The number of cells is not fixed. A weighted sum of within-cell voids and out-of-cell operations is used to evaluate the part-machine grouping obtained. In Phase One, weighted similarity coefficients are computed and machines are clustered using a Tabu search algorithm. In Phase Two, part types are assigned to the previously formed groups using a linear minimum cost network flow model. The proposed approach is compared with three heuristics, namely ZODIAC, GRAFICS and MST, on a large number of problems.

Introduction

Cellular manufacturing requires the identification of groups of similar parts (part families) and the corresponding manufacturing resources they require (machine cells) in order to partition the shop into several, as self-contained as possible, manufacturing cells. The similarity of parts facilitates set-up time reductions, which translate into smaller lot sizes, reduced in-process inventories, shorter lead times, higher throughput and improved product quality. In addition, the small number of machines in each cell simplifies and reduces material handling. Wemmerlöv and Johnson (1997) presents a survey of users' experiences.

The first step in the transition to cellular manufacturing is part-machine grouping, i.e. determining the part families and associated machine cells. A machine-part incidence matrix [air] whose components take a binary value to indicate whether or not machine i is required to process part r, may be used. More detailed data include part demands, sequence and duration of operations, alternative process plans, machine capacities, intercell transportation costs, machine acquisition costs, part subcontracting costs, etc. The simplest criterium used for cell formation is the minimization of intercellular moves. Other criteria may be maximization of a weighted grouping measure, cost minimization, within-cell load balancing, intercell load balancing, etc. As for constraints, simpler approaches consider none, while other approaches impose limits on cell sizes or on the number of cells, machine capacity constraints, budget constraints, etc. Part families and machine cells can be formed simultaneously or machine cells can be formed first and then parts assigned to them or part families formed first and then machines assigned.

As can be deducted from the above considerations, many different approaches have been proposed for part-machine grouping. There are a number of interesting review papers (Reisman et al., 1997, Selim et al., 1998, Singh, 1993, Venugopal, 1998). Many researchers have used the concept of similarity coefficient (or, equivalently, a distance/dissimilarity measure) between machines or between parts. In Table 1, those approaches have been classified according to the definition of similarity coefficient used and the solution technique proposed. The main classification of similarity coefficients is whether or not they use information on the sequence of operations. As it can be seen in the table, most of the proposed approaches are not sequence based. Although a variety of similarity measures have been proposed, the two most common are Jaccard (McAuley, 1972) and Hamming distance (Kusiak, 1987).

In this paper, after illustrating the problem with a simple example, we present a two-phase approach involving a quadratic integer program in Phase One and a linear minimum cost network flow program in Phase Two. A Tabu search (TS) algorithm is proposed for Phase One. Finally, computational experiences with the proposed approach are reported and conclusions are drawn.

Section snippets

A simple example

Consider the binary part-machine incidence matrix shown in Table 2.

For this simple problem, the well known SLCA approach (McAuley, 1972) would first compute the Jaccard similarity coefficients (SC) between any two machines i and j asSij=nijnij+ñij+ñjiwhere nij is the number of parts requiring processing on both machines i and j, while ñij is the number of parts requiring processing on machine i but not on machine j. Table 3 shows the Jaccard SC matrix for this problem.

The corresponding

A two-phase approach

The following notation will be used.

Input data

    i, j

    machine indexes

    r

    part type index

    k

    cell/family index

    M

    number of machines

    P

    number of part types

    Mmin

    minimum number of machines per cell

    Mmax

    maximum number of machines per cell

    Pmin

    minimum number of part types per family

    Pmax

    maximum number of part types per family

    A=[air]

    M×P binary part-machine incidence matrix

Decision variables for Phase Onexik={1ifmachineicellk0otherwiseyk={1ifcellkisformed0otherwise

Decision variables for Phase Twozrk={1ifpartrcellk0

Tabu search algorithm

TS is a metaheuristic procedure that has proven itself to be a useful optimization technique for solving a wide variety of combinatorial problems. For a thorough description of TS, we refer to books (Glover & Laguna, 1996) and tutorials (Glover, 1990, Glover and Laguna, 1995) that cover this technique.

In TS, the crucial implementation decisions include the definition of the neighborhood structure, and the way the current solution is modified according to the history of the search. For the

Computational experiences

To show the flexibility of the two-phase approach and to compare the performance of the TS algorithm, a maximum spanning tree (MST) heuristic (Lozano, Adenso-Díaz, Eguia, & Onieva, 1999) has been used. Table 8 shows a list of 22 problems from the literature, with their sizes and references. In a first set of experiments, the two methods were used to solve the part-machine grouping problem assuming no cell size limits exist. Initially, a value of q=0.1 was used. Such a low value means

Conclusions

This paper presents a two-phase approach for part-machine grouping with limits on both machine cells and part families sizes. A weighted sum of intracell voids and exceptional elements is used to evaluate the quality of the solution obtained. For Phase One and in order to be in line with that objective function, weighted SC are proposed. To solve the resulting quadratic integer programming a TS algorithm has been implemented and compared with three heuristics. Computational experiences show

Acknowledgement

This paper was partially funded by the Asturian government, contract number PC-04-08.

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