A genetic approach to the quadratic assignment problem

Scope and Purpose There are a vast number of practical design and resource-allocation problems, in many different fields, where the decision to be made is a matching (or assignment) of items in one set to items in another, disjoint set. If the costs associated are simply constants for each possible pairing, this is the classical " Assignment Problem " , for which good algorithms have been known for more than a century. However, if the cost structure is more complex, so that the cost of a given assignment depends on two-way or higher-order interactions between pairings, difficult combinatorial problems result. The quadratic assignment problem (QAP) is perhaps the simplest in structure of these difficult problems. In QAP, a constant cost is associated with simultaneously making two particular assignments. Such cost structures arise, for example, in facility location problems, where the cost of locating facility i at site j and facility k at site l is a function of the distance between the two sites j and l, and the degree of interaction between the two facilities i and k. Genetic algorithms are a family of parallel, randomized-search optimization heuristics which are based on the biological process of natural selection [13]. They have proven to be most effective on non-convex optimization problems for which it is relatively easy to assess the quality of a given feasible solution, but difficult to 3 systematically improve solutions by deterministic iterative methods. Most NP-complete combinatorial problems fall into this category. Since QAP is the nonlinear assignment problem with the most " special structure " , it is more likely to yield good solutions to clever deterministic heuristics which take advantage of that structure. Conversely, if genetic algorithms can be shown to perform competitively on QAP, this gives us good reason to believe that extending genetic algorithms to the many more complex nonlinear assignment problems found in VLSI design, facility layout, and location problems may yield better results than deterministic heuristics can provide for these less-structured problems. In this paper, we present the results of an investigation of a particular genetic algorithm for QAP, and discuss the potential of genetic algorithms for more complex nonlinear assignment problems. We show that the GA performed consistently equal to or better than previously known heuristics without undue computational overhead. We conclude with some more general comments on the design and implementation of genetic algorithms, motivated by our results for …