Solving group technology problems via clique partitioning

Abstract

This paper presents a new clique partitioning (CP) model for the Group Technology (GT) problem. The new model, based on a novel 0/1 quadratic programming formulation, addresses multiple objectives in GT problems by drawing on production relationships to assign differing weights to machine/part pairs. The use of this model, which is readily solved by a basic tabu search heuristic, is illustrated by solving 36 standard test problems from the literature. The efficiency of our new CP model is further illustrated by solving three large scale problems whose linear programming relaxations are much too large to be solved by CPLEX. An analysis of the quality of the solutions produced along with comparisons made with other models and methods highlight both the attractiveness and robustness of the proposed method.

Appendix: Overview of tabu search method for CBQP

Our TS method for CBQP is centred around the use of strategic oscillation, which constitutes one of the primary strategies of tabu search. The variant of strategic oscillation we employ may be sketched in overview as follows.

The method alternates between constructive phases that progressively set variables to 1 (whose steps we call “add moves”) and destructive phases that progressively set variables to 0 (whose steps we call “drops moves”). To control the underlying search process, we use a memory structure that is updated at critical events, identified by conditions that generate a subclass of locally optimal solutions. Solutions corresponding to critical events are called critical solutions. For CBQP a critical event occurs during the solution process when exactly n variables are equal to 1.

A parameter span is used to indicate the amplitude of oscillation about a critical event. We begin with span equal to 1 and gradually increase it to some limiting value. For each value of span, a series of alternating constructive and destructive phases is executed before progressing to the next value. At the limiting point, span is gradually decreased, allowing again for a series of alternating constructive and destructive phases. When span reaches a value of 1, a complete span cycle has been completed and the next cycle is launched.

Information stored at critical events is used to influence the search process by penalizing potentially attractive add moves (during a constructive phase) and inducing drop moves (during a destructive phase) associated with assignments of values to variables in recent critical solutions. Cumulative critical event information is used to introduce a subtle long term bias into the search process by means of additional penalties and inducements similar to those discussed above.