# Quadratic Assignment Problem

**Topics:**Optimization, Combinatorial optimization, Tabu search

**Pages:**11 (2678 words)

**Published:**October 11, 2010

Department of Industrial Engineering Koc University, Istanbul, Turkey gkirlik@ku.edu.tr

Gokhan Kirlik April 16, 2010

Abstract Quadratic assignment problem is one of the most known and challenging combinatorial optimization problems. In this study, a new tabu search algorithm is proposed to solve the quadratic assignment problem. Proposed algorithm is tested with diﬀerent tabu search elements such as neighborhood size, size of the tabu list, termination condition. The performance of the proposed approach is tested on with 25, 50 and 100-department instances which are taken from QAPLib.

1

Introduction

Quadratic assignment problem (QAP) is ﬁrstly introduced by Koopmans and Beckman in 1957 [5]. It can be described as follows: given n×n matrices A = (aij ) and B = (bij ) where matrices represent ﬂow and distance, respectively. Find a permutation π ∗ minimizing n n

min f (π) Q π∈ (n)

=

i=1 j=1

aij bπi πj

where (n) is the set of permutations of n elements [1]. Shani and Gonzalez have shown that QAP is NP-hard [8]. Solving this problem optimality for the large instances is computationally infeasible. Therefore, heuristic approaches have to be used for solving medium- and large-scale QAPs. In this 1

study, tabu search (TS) algorithm is used to solve QAP. Tabu search technique was developed by Glover [2, 3]. This method has become very popular and is widely used for a variety of problems [4]. Tabu search is based on the neighborhood search with local-optima avoidance but in a rather deterministic way. The key idea of tabu search is allowing climbing moves when no improving neighboring solution exists. However, some moves are to be forbidden at a present search iteration in order to avoid cycling. The proposed tabu search algorithm is tested with diﬀerent neighborhood sizes, tabu tenors and termination conditions. During the tests 25department [7] and 50, 100-department [9] instances are used which are taken from QAPLib. This paper will proceed as follows. In section 2, mathematical model of the QAP is given. In section 3, proposed SA is presented. Finally in section 4 computational results are given.

2

Problem Formulation

In this study, QAP with n departments and n locations with minimizing costs between placed departments is considered. The cost is obtained by product of ﬂows and distances between departments. Integer linear model of the problem is as follows. Set N : Deparment (or location) set where N = {1, 2, ..., n} Indices i, k : The depatment index used as unique identiﬁer for each department. j, l : The location index used as unique identiﬁer for each location. Parameters n : Number of departments(or locations) aik : Total ﬂow department i to department k. bjl : Distance location j to department l. Decision Variables 1 If department i assigned to location j xij = 0 otherwise

2

Model

n n n n

min

i=1 j=1 k=1 l=1

aik bjl xij xkl

n

(1)

s.t.

xij = 1 ∀i

j=1 n

(2)

xij = 1 ∀j

i=1

(3) (4)

xij ∈ {0, 1} ∀i, j

Equation (1) is the objective function that minimizes the costs. Equation (2) ensures that each department is assigned exactly one location. Similarly, equation (3) ensures that each location is assigned to exactly one department. Equation (4) is the binary integrality constraints.

3

Tabu Search

Tabu search starts from an initial solution s where s ⊆ S, S is the set of solution of combinatorial optimization problem. At each step of the procedure, a set N (s) of the neighboring solutions of the current solution s is considered and the move that improves most the objective function value is chosen. If there are no improving moves, tabu search chooses one that least degrades the objective function. In order to avoid the returning to the local optimal solution just visited, the reverse move must be prohibited. This is done by storing this move in a memory (or more precisely...

References: [1] L. M. Gambardella, E. D. Taillard, and M. Dorigo. Ant colonies for the quadratic assignment problem. Journal of the Operational Research Society, 50:167–176, 1999. [2] F. Glover. Tabu search: Part 1. ORSA Journal on Computing, 1:190–206, 1989. [3] F. Glover. Tabu search: Part 2. ORSA Journal on Computing, 1:4–32, 1990. [4] F. Glover and M. Laguna. Tabu Search. Kluwer, Dordrecht, 1997. [5] T. C. Koopmans and M. Beckmann. Assignment problems and the location of economics activities. Econometrica, 25:53–76, 1957. [6] A. Misevicius. A tabu search algorithm for the quadratic assignment problem. Computational Optimization andApplications, 30:95–111, 2005. [7] C. E. Nugent, T. E. Vollman, and J. Ruml. An experimental comparison of techniques for the assignment of facilities to locations. Operations Research, 16:150–173, 1968. [8] S. Shani and T. Gonzalez. P-complete approximation problems. Journal of the Association for Computing Machinery, 23:555–565, 1976. [9] M. R. Wilhelm and T. L. Ward. Solving quadratic assignment problems by simulated annealing. IIE Transactions, 19(1):107–119, 1987.

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