Cellular manufacturing problem - A graph theoretic approach

Abstract

The cell formation problem which arises in cellular manufacturing can be formulated in graph theoretic terms. The input for a cellular manufacturing problem consists of a set X of m machines and a set of Y of p parts and an m × p matrix A = (a_ij), where a_ij = 1 or 0 according as the part p_j is processed on the machine m_i. This data can be represented as a bipartite graph G with bipartition X,Y and m_i is joined to p_j if a_ij = 1. Let G₁,G₂,...,G_k be nontrivial connected subgraphs of G such that V(G₁),V(G₂),...,V(G_k) forms a partition of V(G). Then π = {G₁,G₂,...,G_k} is called a k-cell partition of G. Any edge of G with one end in G_i and the other end in G_j with i ≠ j represents an inter cellular movement of a part. One of the objectives in cellular manufacturing problem is to minimize the inter cellular movements of parts. Let β(G,π) denote the number of edges in G with one end in V(G_i) and other end in V(G_j). Let β(G,k) = min_πβ(G,π), where the minimum is taken over all k-cell partitions π of G. In this paper we propose a graph theoretic algorithm using Depth-First-Search to solve the cellular manufacturing problem for the case when k = 2. Comparison of the results that we have obtained with solutions obtained by other known algorithms shows that our algorithm gives a better solution.