A path relinking approach with ejection chains for the generalized assignment problem

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Abstract

The generalized assignment problem is a classical combinatorial optimization problem known to be NP-hard. It can model a variety of real world applications in location, allocation, machine assignment, and supply chains. The problem has been studied since the late 1960s, and computer codes for practical applications emerged in the early 1970s. We propose a new algorithm for this problem that proves to be more effective than previously existing methods. The algorithm features a path relinking approach, which is a mechanism for generating new solutions by combining two or more reference solutions. It also features an ejection chain approach, which is embedded in a neighborhood construction to create more complex and powerful moves. Computational comparisons on benchmark instances show that the method is not only effective in general, but is especially effective for types D and E instances, which are known to be very difficult.

Introduction

We introduce an effective metaheuristic algorithm for the generalized assignment problem (GAP), which is one of the representative combinatorial optimization problems known to be NP-hard (e.g., [29]). This problem has many important applications, notably including scheduling, supply chain, location and vehicle routing problems. Consequently, the challenge of designing good exact and/or heuristic algorithms for GAP has significant practical as well as theoretical value (e.g., [6], [24], [28], [33]).

Our algorithm features a path relinking approach [9], [10], [12], [19], [23] associated with adaptive memory programming (tabu search), which provides an “evolutionary” mechanism for generating new solutions by combining two or more reference solutions. The idea of path relinking was proposed by Glover [9], [10], and some of its basic aspects were also introduced in an earlier paper by Ibaraki et al. [16]. For more about the general principles of the path relinking approach, see e.g., [19], [23]. Preliminary results of our path relinking approach were reported in [31], [32]. To the best of our knowledge, our paper [32] (a preliminary extended abstract of this paper) was the first paper that reported computational results of a path relinking approach for the GAP, in which results for benchmark instances with up to 200 jobs were reported. Alfandari et al. [1] independently proposed another effective path relinking algorithm slightly earlier, although without reporting any computational results in the original version of their paper. (The full version [2] containing computational results appeared somewhat later, and we compare their method with ours in Section 4.) Our algorithm also features an ejection chain approach, likewise associated with tabu search [11], [33], where Lagrangian relaxation provides adjusted cost information to guide the neighborhood search to promising solutions. Rego and Glover suggested in Section 4.3 of [27] that combining ejection chain methods and path relinking would be fruitful. Moreover, we incorporate an automatic mechanism for adjusting search parameters, to maintain a balance between visits to feasible and infeasible regions.

Computational comparisons are conducted on benchmark GAP instances known as types B, C, D and E. These test problems are taken from the OR-Library,1 which is the primary repository for such problems, and are supplemented by additional test instances generated by ourselves. The proposed algorithm is compared with many existing heuristic algorithms including the recent path relinking approach by Alfandari et al. [2], [3], tabu search by Díaz and Fernández [6], a Lagrangian heuristic algorithm by Haddadi and Ouzia [14], tabu search by Yagiura et al. [33], variable depth search algorithms by Yagiura et al. [34], [35], another variable depth search algorithm by Racer and Amini [26], tabu search by Laguna et al. [18], MAX–MIN ant system by Lourenço and Serra [20], genetic algorithm by Chu and Beasley [4] and a mixed integer programming solver CPLEX 6.5. The results show that our GAP method is highly effective, especially for instances of types D and E, which have been established as the most difficult problem classes.

Our algorithm is confirmed by extensive computational experiment to be efficient and robust, both in relation to parameter settings and variations in problem structures. The outcomes indicate that useful benefits result by combining path relinking and ejection chain strategies associated with adaptive memory methods, and by making use of classical relaxation methodology. The resulting method yields a powerful and effective tool for practical applications.

Section snippets

Definition of the problem

Given n jobs J = {1,2, …, n} and m agents I = {1,2, …, m}, we undertake to determine a minimum cost assignment subject to assigning each job to exactly one agent and satisfying a resource constraint for each agent. Assigning job j to agent i incurs a cost of cij and consumes an amount aij of resource, whereas the total amount of the resource available at agent i is bi. An assignment is a mapping σ : J  I, where σ(j) = i means that job j is assigned to agent i. For convenience, we define a 0–1 variable xij

The path relinking algorithm

Our algorithm, called PREC (path relinking and ejection chains), is an extension of local search. Local search starts from an initial solution σ and repeatedly replaces σ with a better solution in its neighborhood N(σ) until no better solution is found in N(σ). The resulting solution σ is locally optimal in the sense that no better solution exists in its neighborhood. Shift and swap neighborhoods, denoted Nshift and Nswap respectively, are usually used in local search methods for GAP, whereN

Computational results

In this section, algorithm PREC is evaluated on benchmark instances. PREC was coded in C and run on a workstation Sun Ultra 2 Model 2300 (two UltraSPARC II 300 MHz processors with 1 GB memory), where the computation was executed on a single processor. SPECint95 of this workstation is 12.3 according to the SPEC site7 (Standard Performance Evaluation Corporation). The parameters in PREC were set to ρ = 20 and γ = 10.

Conclusion

The proposed path relinking approach (PREC) proves to be highly effective for the generalized assignment problem. Isolating the path relinking component of our algorithm and comparing it to the use of a uniform crossover component discloses that the outcomes from path relinking are significantly superior to those of uniform crossover. More extensive comparisons of the PREC algorithm, testing against other leading heuristic approaches for GAP, confirm the high performance of PREC; it found

Acknowledgments

The authors are grateful to anonymous referees for valuable comments to improve this paper.

This research was partially supported by Scientific Grant-in-Aid by the Ministry of Education, Culture, Sports, Science and Technology of Japan, and by Informatics Research Center for Development of Knowledge Society Infrastructure (COE program of the Ministry of Education, Culture, Sports, Science and Technology, Japan).

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