A branch and bound algorithm for a fractional 0-1 programming problem

TL;DR

This paper introduces an exact branch and bound algorithm for the NP-hard Cell Formation Problem in manufacturing, optimizing clustering to maximize intra-cell processing and minimize inter-cell movement, outperforming many existing heuristics.

Contribution

It presents a novel branch and bound method that provides exact solutions for the fractional 0-1 programming formulation of CFP, achieving optimality on most benchmark instances.

Findings

Optimal solutions for 21 out of 35 benchmark instances.

Good approximate solutions for remaining instances.

Algorithm effectively handles fractional objective functions.

Abstract

We consider a fractional 0-1 programming problem arising in manufacturing. The problem consists in clustering of machines together with parts processed on these machines into manufacturing cells so that intra-cell processing of parts is maximized and inter-cell movement is minimized. This problem is called Cell Formation Problem (CFP) and it is an NP-hard optimization problem with Boolean variables and constraints and with a fractional objective function. Because of its high computational complexity there are a lot of heuristics developed for it. In this paper we suggest a branch and bound algorithm which provides exact solutions for the CFP with a variable number of cells and grouping efficacy objective function. This algorithm finds optimal solutions for 21 of the 35 popular benchmark instances from literature and for the remaining 14 instances it finds good solutions close to the best known.