Development of a method to linearize the quadratic assignment problem

Abstract

The paper presents a new powerful technique to linearize the quadratic assignment problem. There are so many techniques available in the literature that are used to linearize the quadratic assignment problem. In all these linear formulations, both the number of variables and the linear constraints significantly increase. The quadratic assignment problem (QAP) is a well-known problem whereby a set of facilities are allocated to a set of locations in such a way that the cost is a function of the distance and flow between the facilities. In this problem, the costs are associated with a facility being placed at a certain location. The objective is to minimize the assignment of each facility to a location. There are three main categories of methods for solving the quadratic assignment problem. These categories are heuristics, bounding techniques and exact algorithms. Heuristics quickly give near-optimal solutions to the quadratic assignment problem. The five main types of heuristics are construction methods, limited enumeration methods, improvement methods, simulated annealing techniques and genetic algorithms. For every formulated QAP, a lower bound can be calculated. We have Gilmore-Lawler bounds, eigenvalue related bounds and bounds based on reformulations as bounding techniques. There are four main classes of methods for solving the quadratic assignment problem exactly, which are dynamic programming, cutting plane techniques, branch and bound procedures and hybrids of the last two. The QAP has application in computer backboard wiring, hospital layout, dartboard design, typewriter keyboard design, production process, scheduling, etc. The technique proposed in this paper has the strength that the number of linear constraints increases by only one after the linearization process.