A O(n8) X O(n7) Linear Programming Model of the Quadratic Assignment Problem

The Quadratic Assignment Problem (QAP) is the problem of making exclusive assignments of n indivisible entities to n other indivisible entities in such a way that a total quadratic interaction cost is minimized. The problem can be interpreted from a wide variety of perspectives. The perspective adopted in this paper is that of the generic facilities location/layout context, as in the seminal work of Koopmans and Beckmann [8]. Specifically, there are n facilities (or departments) to be located at n possible sites (or locations). The volume of traffic going from facility i to facility j is denoted fij.The travel distance from site r to site s is denoted drs. A quadratic “material handling” cost of hirjs is incurred if facilities i and j are assigned to sites r and s, respectively. In addition, there is a fixed cost (an “operating cost”), oir, associated with operating facility i at site r. It is assumed (without loss of generality) that the units for “distance”, “volume of traffic”, and ”operating cost” have been chosen so that the hirjs’s and oir’s are commensurable. The problem is that of finding a perfect matching of the facilities and sites so that the total material handling plus facilities operating costs is minimized. Let F := {1, 2, ..., η} and T := {1, 2, ..., ς} be the sets of facilities and sites, respectively. Without loss of generality, assume η = ς = n. For i ∈ F and r ∈ T , let wir be a 0/1 binary variable that indicates whether facility i is assigned to (or located at) site r (wir = 1), or not (wir = 0). Then, a classical formulation of the QAP is as follows (see [10]):