The Generalized Assignment Problem and Extensions

Abstract

The generalized assignment problem is a classical combinatorial optimization problem that models a variety of real world applications including flexible manufacturing systems [6], facility location [11] and vehicle routing problems [2]. Given n jobs J = {1, 2, . . . , n} and m agents I = {1, 2, . . . , m}, the goal is 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 c ij and consumes an amount a ij of resource, whereas the total amount of the resource available at agent i is b i . An assignment is a mapping σ: J → I, where σ(j) = i means that job j is assigned to agent i. Then the generalized assignment problem (GAP) is formulated as follows: minimize cost(σ) = ∑ j∈J c σ(j), j subject to ∑ j∈J σ(j)=i a ij ≤ b i , ∀i ∈ I. (1) The GAP is known to be NP-hard, since the partition problem of a given set of positive integers into two equal sized subsets can be reduced to GAP with m = 2. Researchers have studied the problem since the late 1960s, and computer codes for practical applications emerged in the early 1970s. Yagiura, Ibaraki and Glover [12] proposed a tabu-search algorithm for GAP. It features an ejection chain approach, which is embedded in a neighborhood construction to create more complex and powerful moves. It also incorporates an adaptive mechanism for adjusting search parameters, to maintain a balance between visits to feasible and infeasible regions. Computa-tional comparisons are conducted on benchmark GAP instances known as types C, D and E. These test problems are taken from the OR-Library, which is the primary repository for such problems, and are supplemented by additional test instances generated by the authors. Compu-tational results on small instances with up to 60 jobs show that the proposed algorithm obtains solutions that are optimal or that deviate by at most 0.16% from the best known solutions. Comparisons with other approaches from the literature show that, for instances of larger sizes with up to 1600 jobs, it obtains the best solutions among all heuristics tested. This algorithm was further improved by the authors using the path relinking approach [13]. The results show that the path relinking GAP method not only improves the previous best approach, but is especially effective for the types D and E instances, which are known as the most difficult ones among existing benchmarks. Motivated by practical applications, various generalizations of GAP have been proposed. The multi-resource generalized assignment problem (MRGAP), in which more than one resource constraint is considered for each agent, is a natural generalization of GAP and has many practical applications, e.g., in distributed computer systems and in the trucking industry [3, 4, 8]. For MRGAP, Rego et al. [10] applied a metaheuristic approach related to tabu search called RAMP (relaxation adaptive memory programming)[9], and Yagiura et al. [14] devised a tabu search algorithm incorporating very large-scale neighborhood search.