Exact and heuristic algorithms for the interval data robust assignment problem

doi:10.1016/j.cor.2010.11.009

Computers & Operations Research

Volume 38, Issue 8, August 2011, Pages 1153-1163

https://doi.org/10.1016/j.cor.2010.11.009 Get rights and content

Abstract

We consider the Assignment Problem with interval data, where it is assumed that only upper and lower bounds are known for each cost coefficient. It is required to find a minmax regret assignment. The problem is known to be strongly NP-hard. We present and compare computationally several exact and heuristic methods, including Benders decomposition, using CPLEX, a variable depth neighborhood local search, and two hybrid population-based heuristics. We report results of extensive computational experiments.

Introduction

The Assignment Problem is a classical combinatorial optimization problem whose study can be traced back to the 1930s and even earlier, see [28]. Its standard version deals with choosing an optimal way to assign n items to n other items, e.g., n tasks to n workers in the most well-known interpretation, given the costs for all possible task-worker pairs. The problem is polynomially solvable, and serves as a model in numerous application settings. We refer the interested reader to the new book by Burkard et al. [11] for further details on the problem, its algorithmic solution methods, and its variants.

We consider the case where the costs (objective function coefficients) may be uncertain. To handle uncertainty, in this paper we use the minmax regret (also known as robust deviation) approach, where it is required to find a feasible solution that is $ɛ -optimal$ for any possible scenario (realization of the vector of the objective function coefficients), with $ɛ$ as small as possible. Minmax regret combinatorial optimization problems have been studied extensively over the past two decades (see, e.g., [2], [4], [5], [18], [19], [21], [24], [25] and the references therein). The motivation for using the minmax regret approach as a modeling tool is thoroughly discussed in the book [21]; the book [18] and the survey [2] provide reviews of recent results.

In interval data minmax regret (IDMR) models, it is assumed that for each objective function coefficient (the cost of a task-worker pair in the context of the assignment problem), an uncertainty interval is specified, and the set of scenarios is the Cartesian product of all uncertainty intervals. In the robust optimization literature, there have been extensive studies of minmax regret versions of many classical combinatorial optimization problems (see, e.g., [18]). It has been observed that typically polynomially solvable classical problems with the minisum type of objective function become NP-hard in the IDMR version [5], [1], although there are some exceptions, see [4], [3]. The same holds for the IDMR assignment problem which was proven to be weakly NP-hard in [19] and, later, strongly NP-hard in [1]. Currently, significant research efforts are devoted to developing practically efficient computational methods for solving IDMR versions of classical combinatorial optimization problems, and there has been successful work in this direction [23], [24], [25]. However, computational work and experience for the IDMR Assignment Problem have been limited to testing a mixed-integer linear programming (MILP) formulation with the CPLEX solver, and a straightforward scenario-based heuristic, for relatively small problem sizes [19], [18].

The main purpose of this paper is to fill this gap and conduct a computational study of the IDMR Assignment Problem, with the goal of comparing several solution approaches for different problem sizes. We compare two exact solution methods: solving the MILP formulation from [19] with ILOG CPLEX MILP solver, and Benders decomposition which was used effectively for other robust combinatorial optimization problems [23], [25]. For larger problem sizes, we also test several heuristics, including a simple scenario-based heuristic, a regular local search, a variable depth neighborhood local search, and two hybrid population-based heuristics that use an optimization subproblem within the framework of a Genetic Algorithm, thus being consistent with the concept of a Memetic Algorithm. We conduct experiments with several families of problem instances generated by different methods.

We note that our hybrid population-based heuristics, to the best of our knowledge, are the first application of a population-based algorithm to a robust combinatorial optimization problem (along with [26] where a population-based approach is applied to the robust set covering problem). One of the two hybrid heuristics developed in this paper, according to our experiments, shows the best performance for large problems among all tested approaches.

The paper is organized as follows. The IDMR Assignment Problem is formally defined in Section 2, along with the main notation and known main properties. Two mathematical programming formulations, one from [19] and the other used as a basis for Benders decomposition, are stated in Section 3. In Section 4, the details of the Benders decomposition implementation are given, and in Section 5 the results of the computational experiments with the exact methods are discussed. Section 6 is devoted to the development of heuristic methods to solve the problem, and in Section 7 the results of computational experience with heuristic methods are discussed. In Section 8, conclusions are presented.

Section snippets

Notation and problem statement

We use notation and terminology developed in earlier work on minmax regret optimization [4], [5], [26]. Let $G = (U \cup V, E)$ be a bipartite network with $U \cup V$ the set of nodes, $| U | = | V | = n$ , and the set of edges $E = {(u, v) | u \in U, v \in V}$ . Suppose that for every edge $(u, v) \in E$ , two nonnegative real numbers c_uv⁻, c_uv⁺ are given, $c_{uv}^{-} \leq c_{uv}^{+}$ . The cost of edge (u,v) can take on any real value from its uncertainty interval [c_uv⁻, c_uv⁺], regardless of the values taken by the costs of other edges. The Cartesian product of

Formulation 1

In the remainder of the paper, $X \in A$ will denote not only a perfect matching in G, but also the n²-dimensional characteristic vector of this matching with components $x_{uv} = \{\begin{matrix} 1 & if (u, v) \in X; \\ 0 & otherwise . \end{matrix}$ Vectors will be denoted with upper-case letters, and their components with the same lower-case letters. Then, Problem ROB.ASSIGN can be reformulated as follows: $\min_{X \in A} (\max_{s \in S} (\sum_{(u, v) \in E} c_{uv}^{s} x_{uv} - \min_{Y \in A} (\sum_{(u, v) \in E} c_{uv}^{s} y_{uv})))$ or, using Lemma 1, $\min_{X \in A} (\sum_{(u, v) \in E} c_{uv}^{+} x_{uv} - \min_{Y \in A} (\sum_{(u, v) \in E} c_{uv}^{s (X)} y_{uv})) .$ Introducing a free variable $θ$ and

Implementation of the exact algorithms

In Section 3, two different mathematical programming formulations have been given. Current Mixed-Integer Linear Programming packages are capable of solving formulations such as (9), (10), (11), (12), (13), (14) with relative ease [10], but only up to a certain instance size. The results of applying the CPLEX solver to this formulation are discussed in Section 5.

To tackle the formulation (4), (5), (6), (7), (8) which has an exponential number of constraints (5), we use Benders decomposition [9],

Implementation details

All the algorithms in this paper have been coded in ANSI C++. Two external sources of code were used. The first one is a routine to solve any assignment problem. For this purpose, the recent C version of the code described in [12] was chosen. This code is available at http://www.assignmentproblems.com/APC_APS.htm and is an implementation of the Hungarian method for dense cost matrices, the case considered in this paper. Second, the ILOG CPLEX library, version 11.2, is used to solve linear

Heuristic procedures

The simplest heuristic for the Robust Assignment Problem is to solve Problem OPT.ASSIGN(s) for some natural scenario and present the obtained solution as an approximate solution. This takes just solving two regular Assignment Problems (one for obtaining a solution to Problem OPT.ASSIGN(s) and one for evaluating its value Z(X)). Natural choices for scenario s are s⁼ and s⁺. The scenario-based heuristic that consists in solving Problem OPT.ASSIGN(s) for s=s⁼ and s=s⁺ and choosing the best of the

Computational experience with heuristic methods

Section 6 outlined different heuristic approaches to the problem. In this section, we discuss the results of computational experiments with these heuristics, and the comparison with the solutions found by exact algorithms. As mentioned in Section 5.1, all algorithms have been programmed in C++ and run on the same computer. Again, each algorithm has been given a maximum computation time of 600 s as a time limit.

Table 3, Table 4, Table 5, Table 6 report the results for each family of instances

Conclusions, recommendations, and additional comments

For the interval data minmax regret assignment problem, we compared several exact and heuristic methods. Our main recommendations and findings are as follows.

1.
The difficulty of the problem depends not only on the instance size but also on the degree of uncertainty.
2.
The scenario-based heuristic obtains good solutions in a very short time (typically within a small fraction of a second for the tested problem sizes). It should be used when a good solution must be obtained very quickly, or to obtain

Acknowledgements

The research of the first author was partially supported by the Spanish CICYT Grant DPI2007-63026. The research of the second author was supported by a discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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View full text

Exact and heuristic algorithms for the interval data robust assignment problem

Abstract

Introduction

Section snippets

Notation and problem statement

Formulation 1

Implementation of the exact algorithms

Implementation details

Heuristic procedures

Computational experience with heuristic methods

Conclusions, recommendations, and additional comments

Acknowledgements

Complexity of the min-max and min-max regret assignment problems

Operations Research Letters

Min-max and min-max regret versions of combinatorial optimization problems: a survey

European Journal of Operational Research

Complexity of the min-max (regret) versions of min-cut problems

Discrete Optimization

On the complexity of a class of combinatorial optimization problems with uncertainty

Mathematical Programming

Interval data minmax regret network optimization problems

Discrete Applied Mathematics

Linear programming on Cray supercomputers

Journal of the Operational Research Society

OR-Library: distributing test problems by electronic mail

Journal of the Operational Research Society

A genetic algorithm for the set covering problem