ELSEVIER European Journal of Operational Research 104 (1998) 241-249 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Theory and Methodology Network-based formulations of the quadratic assignment problem Michael O. Ball, Bharat K. Kaku *, Andrew Vakhutinsky College of Business and Management, Universityof Maryland, CollegePark, MD 20742-1815, USA Received 16 January 1995; accepted 1 August 1996 Abstract We present two formulations of the Quadratic Assignment Problem (QAP) that result in network flow problems with integer variables and side constraints. A linearization of the QAP is obtained in both cases by considering each facility to consist of two parts--a source for all outgoing flows from that facility, and a sink for all incoming flows to the facility. Preliminary computational experience with both approaches is presented. (~) 1998 Elsevier Science B.V. Keywords: Design; Facilities; Network programming;Integer programming 1. Introduction and The QAP is generally used to model the facilities layout problem with equal-sized facilities, and can be stated as follows. Given n indivisible facilities to be assigned to n discrete locations: Minimize ~ cipyip + ~ ~fijdpqyip yjq, i,p i,p j,q subject to where Yip = 0 (1) Yip = 1 Vp, (2) i Yip = 1 Vi, (3) p if facility i is assigned to location p, otherwise, * Corresponding author. Cip =fixed (linear) cost of assigning facility i to location p, fij = flow from facility i to j, dpq = distance between locations p and q. Defining p(i) as the location to which facility i is assigned, the problem can be stated alternatively as one of finding a one-to-one mapping of the set of facilities into the set of locations so as to: Minimize ~ Ci,p(i) "1- ~ fijdp(i)p(j). (4) i ij The QAP is an NP-hard problem, and to date, branch-and-bound or implicit enumeration algorithms have been the most successful in obtaining exact so- lutions to the QAP. Gilmore [7] and Lawler [ 10] independently developed a lower bound that formed the basis for the most successful implicit enumera- tion algorithms published earlier, e.g., Bazaraa and 0377-2217/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PH S0377-2217(96) 00330-X