Discrete OptimizationA simple tabu search for warehouse location
Introduction
Given a set of n warehouses and a set of m stores, the uncapacitated warehouse location problem (UWLP) consists of choosing a subset of warehouses that minimizes the fixed costs of the warehouses and the transportation costs from the warehouses to the stores.
The UWLP has attracted considerable attention in mathematical programming. (See [10], [13], [22] for surveys of some of these approaches.) Many specific branch and bound algorithms were developed, including dual and primal–dual approaches [11], [17]. Dual-based and primal–dual algorithms are very effective on the OR Library benchmarks for the UWLP [5]. However, they experience significant difficulties and exhibit exponential behaviour on the instances [21]. These instances model real situations, have a large number of suboptimal solutions, and exhibit a strong tension between transportation and fixed costs, which makes it difficult to eliminate many warehouses early in the search.
Genetic algorithms have been shown to be very successful on the UWLP. In a series of papers spanning over several years (e.g., [12], [19], [20], [21]), Kratica et al. have shown that genetic algorithms find optimal solutions on the OR Library and the instances (whenever the optimal solutions are known) with high frequencies and very good efficiency. Their final algorithm uses clever implementation techniques such as caching and bit vectors to avoid recomputing the objective function which is quite costly on large-scale problems. Ref. [21] also contains a detailed comparison with mathematical programming approaches and shows that the speed-up of the genetic algorithm over mathematical programming approaches increases exponentially with problem size on the instances.
Various heuristic search algorithms have also been proposed but with less success. Ref. [2] presents simulated annealing algorithms which produce high-quality solutions but are quite expensive in computation times. Ref. [1] presents the only tabu-search algorithm we are aware of. The algorithm generates 5n neighbors at each iteration and moves to the best neighbor which is not tabu and improves the current value of the objective function. Each of these iterations takes significant computing time, which limits the applicability of the algorithm.
This paper originated as an attempt to find out whether there exists a tabu-search algorithm, which would be robust, efficient, and competitive with state-of-the-art genetic algorithms. It presents a very simple tabu-search algorithm which performs amazingly well on the UWLP. The algorithm uses a linear neighborhood and essentially takes O(mlogn) time per iteration. It finds optimal solutions on the OR Library and the instances (whenever the optimal solution is known) with high frequencies. It also outperforms the state-of-the-art genetic algorithm of Kratica et al., both in efficiency and robustness.
The main contributions of the paper can be summarized as follows:
- 1.
It presents the first efficient and robust tabu-search algorithm which outperforms, or is competitive with, branch & bound and genetic algorithms in terms of solution quality, robustness, and efficiency.
- 2.
The algorithm is extremely simple to understand and to implement, which makes it an appealing approach for practitioners interested in uncapacitated warehouse location.
- 3.
The algorithm is easy to tune. It has a single parameter which controls the termination of the algorithm and is easy to tune in practice. The paper also describes in detail various tradeoffs between efficiency and solution quality obtained by varying this parameter.
As a consequence, we believe that the new tabu-search algorithm is a valuable addition to the repertoire of algorithms for the UWLPs. The rest of the paper is organized as follows. Section 2 defines the UWLP, Section 3 briefly describes prior work, and Section 4 presents the tabu-search algorithm. Section 5 reports the experimental results and Section 6 concludes the paper.
Section snippets
Uncapacitated warehouse location
We are given a set of n warehouses W and a set of m stores S. Each warehouse w has a fixed cost fw and the transportation cost from warehouse w to store s is given by cws. The problem is to find a subset of warehouses and an assignment of warehouses to the stores to minimize the fixed and the transportation costs. Observe that, once the warehouses are selected, it suffices to assign the stores to their closest warehouse. As a consequence, the problem consists of finding a subset Open of
Prior work
In mathematical programming, considerable attention has been devoted to the UWLP. It is beyond the scope of this paper to review the wealth of results in that area. See the excellent survey [8], [13], [22] for more information. One of the main results has been the development of linear-programming-based branch and bound algorithms. The standard reference in this area is the Dualoc algorithm of Erlenkotter [11], a branch and bound algorithm based on a dual descent. The algorithm performs very
The tabu-search algorithm
We now describe the tabu-search algorithm. Since the only combinatorial part is the selection of warehouses, it is natural to represent a state in the tabu search by a vector y=〈y1,…,yn〉, where yw is 1 if warehouse w is open and 0 otherwise. In the following, we use the notationto represent the warehouses that are opened in a state y.
Experimental results
This section describes the experimental results of the algorithm. Section 5.1 describes its efficiency, its solution quality, and its robustness with the value of its parameter stabilityLimit initialized to 500. As mentioned, this value was chosen to make the comparison with the genetic algorithm meaningful. Section 5.2 studies the robustness of the algorithm with respect to parameter stabilityLimit. Both the solution quality and the efficiency of the algorithm are studied. 5.3 Comparison with
Conclusion
The uncapacitated warehouse location problem (UWLP) has been studied heavily in combinatorial optimization, leading to excellent mathematical programming and genetic algorithms. This paper originated in an attempt to find out whether it was possible to design a tabu-search algorithm, which would be robust, efficient, and competitive with state-of-the-art genetic algorithms. It presented a very simple tabu-search algorithm which performs amazingly well on the UWLP. The algorithm uses a linear
Acknowledgements
Many thanks to Russell Bent for interesting discussions in this paper, to J. Kratica for kindly providing us with his hard instances, and to the reviewers for their comments which help improve the presentation of the results significantly. Pascal Van Hentenryck is partially supported by NSF ITR Award DMI-0121495.
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