Quadratic Assignment Problem
Tabu Search Algorithm for the Quadratic Assignment Problem
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
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
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