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Design of a heuristic algorithm for the generalized multi-objective set covering problem

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Abstract

Set covering optimization problems (SCPs) are important and of broad interest because of their extensive applications in the real world. This study addresses the generalized multi-objective SCP (GMOSCP), which is an augmentation of the well-known multi-objective SCP problem. A mathematically driven heuristic algorithm, which uses a branching approach of the feasible region to approximate the Pareto set of the GMOSCP, is proposed. The algorithm consists of a number of components including an initial stage, a constructive stage, and an improvement stage. Each of these stages contributes significantly to the performance of the algorithm. In the initial stage, we use an achievement scalarization approach to scalarize the objective vector of the GMOSCP, which uses a reference point and a combination of weighted \(l_1\) and \(l_\infty\) norms of the objective function vector. Uniformly distributed weight vectors, defined with respect to this reference point, support the constructive stage to produce more widely and uniformly distributed Pareto set approximations. The constructive stage identifies feasible solutions to the problem based on a lexicographic set of selection rules. The improvement stage reduces the total cost of selected feasible solutions, which benefits the convergence of the approximations. We propose multiple cost-efficient rules in the constructive stage and investigate how they affect approximating the Pareto set. We used a diverse set of GMOSCP instances with different parameter settings for the computational experiments.

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Data availability

We have used two sets of data sets. The first set of data used in this study has been previously published. This set of data has also been used in our citations 13,35,37,53. The data set can be accessed using the following link https://github.com/vOptSolver/vOptLib/blob/master/SCP/readme.md. We have described the procedure for generating the second set of data in the manuscript.

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Acknowledgements

We are very grateful to all reviewers for the careful analysis of our text and valuable feedback.

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Authors and Affiliations

  1. Department of Mathematics, University of Tennessee at Chattanooga, 615 McCallie Avenue, Chattanooga, TN, 37403-2598, USA

    Lakmali Weerasena & Aniekan Ebiefung

  2. SIM Center, University of Tennessee at Chattanooga, 615 McCallie Avenue, Chattanooga, TN, 37403-2598, USA

    Anthony Skjellum

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  1. Lakmali Weerasena
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Correspondence to Lakmali Weerasena.

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Weerasena, L., Ebiefung, A. & Skjellum, A. Design of a heuristic algorithm for the generalized multi-objective set covering problem. Comput Optim Appl 82, 717–751 (2022). https://doi.org/10.1007/s10589-022-00379-7

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