Lagrangian relaxation guided problem space search heuristics for generalized assignment problems

Abstract

We develop and test a heuristic based on Lagrangian relaxation and problem space search to solve the generalized assignment problem (GAP). The heuristic combines the iterative search capability of subgradient optimization used to solve the Lagrangian relaxation of the GAP formulation and the perturbation scheme of problem space search to obtain high-quality solutions to the GAP. We test the heuristic using different upper bound generation routines developed within the overall mechanism. Using the existing problem data sets of various levels of difficulty and sizes, including the challenging largest instances, we observe that the heuristic with a specific version of the upper bound routine works well on most of the benchmark instances known and provides high-quality solutions quickly. An advantage of the approach is its generic nature, simplicity, and implementation flexibility.

Introduction

The generalized assignment problem (GAP) has many applications and can be usually found as a subproblem within more complex problems. A non-exhaustive list of such problems includes computer network problems, vehicle routing and scheduling problems, and resource allocation and scheduling problems in manufacturing and service. We use machine scheduling terminology to introduce and present the development of the algorithm tested in the paper; this terminology and the associated approach are directly applicable to other domains with little change. In this machine scheduling environment, the GAP attempts to find the minimum-cost assignment of a set of jobs with certain resource requirements to a given set of capacitated machines, without violating the machines’ capacity limitations.

There are many exact algorithms and heuristics developed to solve the GAP. Exact algorithms include branch and bound-based algorithms, branch-and-cut and branch-and-price algorithms (e.g., Ross and Soland, 1975, Nauss, 2003, Savelsbergh, 1997). Some heuristics use the linear programming relaxation (Trick, 1992). Others use search techniques such as genetic algorithms (e.g., Chu and Beasley, 1997), and tabu search algorithms (e.g., Higgins, 2001, Díaz and Fernández, 2001, Yagiura et al., 2004). Lagrangian relaxation (LR) is arguably one of the heavily used methods to solve general integer programming problems (Fisher, 1981). Most papers on GAP also have applied Lagrangian relaxation to the GAP and relax one of the two constraint sets. Some of these papers have proposed combining LR with surrogate relaxation (e.g., Narciso and Lorena, 1999). The Lagrangian decomposition (LD) method has been also applied to GAP (e.g., Haddadi, 1999). More recently, LR and subgradient methods have been integrated in branch and bound schemes (e.g., Haddadi and Ouzia, 2001, Haddadi and Ouzia, 2004).

In this paper, we present another algorithm that combines LR and its associated subgradient optimization procedure with a search technique based on problem space search (PSS). PSS uses the problem input data perturbations (as opposed to traditional solution space search, where the search is directly over solutions). The idea of perturbation is not new and has been used in the literature before. Storer et al. (1992) present the first example of PSS within the context of scheduling. Charon and Hudry (1993) use a very similar noising method applied to clique partitioning problem. Their noising methods are based on elementary (or local) transformations applied to solution(s). Codenotti et al. (1996) use the perturbations applied to intermediate solutions of large-scale Euclidean TSP benchmark instances. Here the main difference is that PSS aims the “noising” or the “perturbations” applied to the problem data rather than intermediate solution(s). The main idea is to combine the strengths of LR and PSS to find improved solutions to the GAP.

Our goal is to present the general framework of the LR-PSS-based algorithm, especially in the context of the subgradient optimization used to solve the Lagrangian relaxation problem, and to show the effectiveness of the combined approach by applying it to an extensive set of challenging benchmark instances from the literature.

To give an overall idea, we first construct a Lagrangian relaxation of the GAP by relaxing the machine capacity constraints in the integer programming formulation of the GAP. The relaxed problem’s solution gives a lower bound on the optimal solution of the GAP, which can be improved using the subgradient optimization procedure. As the lower bound solutions are rarely feasible, the solution of the relaxed problem in each iteration of the procedure is also used to find heuristic solutions to the original GAP and produce upper bounds. As this “feasibility restoration” in fact does not guarantee capacity-feasible solutions due to its structure, a PSS-based enhancement is introduced: The problem data in each iteration is temporarily perturbed a given number of times to increase the chances of obtaining capacity-feasible solutions with potentially improved objective values. At algorithm termination, the best feasible solution and the best lower bound are reported. The two primary factors affecting the PSS enhancement are (1) the number of perturbations in each iteration, and (2) the amount of perturbation of the problem data. We test the effectiveness of the algorithm using the problem instances from the literature.