Annals of Operations Research

, Volume 248, Issue 1–2, pp 493–514 | Cite as

An algorithm for approximating the Pareto set of the multiobjective set covering problem

  • Lakmali Weerasena
  • Margaret M. Wiecek
  • Banu Soylu
Original Paper
  • 216 Downloads

Abstract

The multiobjective set covering problem (MOSCP), a challenging combinatorial optimization problem, has received limited attention in the literature. This paper presents a heuristic algorithm to approximate the Pareto set of the MOSCP. The proposed algorithm applies a local branching approach on a tree structure and is enhanced with a node exploration strategy specially developed for the MOSCP. The main idea is to partition the search region into smaller subregions based on the neighbors of a reference solution and then to explore each subregion for the Pareto points of the MOSCP. Numerical experiments for instances with two, three and four objectives set covering problems are reported. Results on a performance comparison with benchmark algorithms from the literature are also included and show that the new algorithm is competitive and performs best on some instances.

Keywords

Multiobjective set covering problem Heuristics Local branching Tree-based search 

References

  1. Bosman, P. A. N., & Thierens, D. (2003). The balance between proximity and diversity in multiobjective evolutionary algorithms. IEEE Transactions on Evolutionary Computation, 7(2), 174–188.CrossRefGoogle Scholar
  2. Caprara, A., Fischetti, M., & Toth, P. (1999). A heuristic method for the set covering problem. Operations Research, 47, 730–743.CrossRefGoogle Scholar
  3. Corley, H. W. (1980). An existence result for maximizations with respect to cones. Journal of Optimization Theory and Applications, 31(2), 277–281.CrossRefGoogle Scholar
  4. Danna, E., Rothberg, E., & LePape, C. (2005). Exploring relaxation induced neighborhoods to improve MIP solutions. Mathematical Programming Series A, 102, 71–90.CrossRefGoogle Scholar
  5. Daskin, M. S., & Stern, E. H. (1981). A hierarchical objective set covering model for emergency medical service vehicle deployment. Operations Research Society of America, 15(2), 137–151.Google Scholar
  6. Ehrgott, M. (2001). Approximation algorithms for combinatorial multicriteria optimization problems. International Transactions in Operational Research, 7, 5–31.CrossRefGoogle Scholar
  7. Ehrgott, M. (2005). Multicriteria optimization (2nd ed.). Berlin: Springer.Google Scholar
  8. Ehrgott, M., & Gandibleux, X. (2000). A survey and annotated bibliography of multiobjective combinatorial optimization. OR Spektrum, 22, 425–460.CrossRefGoogle Scholar
  9. Fischetti, M., & Lodi, A. (2003). Local branching. Mathematical Programming, 98, 23–47.CrossRefGoogle Scholar
  10. Florios, K., & Mavrotas, G. (2014). Generation of the exact Pareto set in multi-objective traveling salesman and set covering problems. Applied Mathematics and Computation, 237, 1–19.CrossRefGoogle Scholar
  11. Hamming, R. W. (1950). Error detecting and error correcting codes. Bell System Technical Journal, 29(2), 147–160.CrossRefGoogle Scholar
  12. Hansen, M. P., & Jaszkiewicz, A. (1998). Evaluating the quality of approximations to the non-dominated set, Institute of Mathematical Modeling. Technical Report, vol. 7Google Scholar
  13. Hansen, P., Mladenovic̀, N., & Uros̀evic̀, D. (2006). Variable neighborhood search and local branching. Computers and Operations Research, 33, 3034–3045.CrossRefGoogle Scholar
  14. Hooker, J. (2011). Logic-based methods for optimization: Combining optimization and constraint satisfaction (Vol. 2). London: Wiley.Google Scholar
  15. IBM-ILOG CPLEX, 2014, 12.6 User’s Manuel. IBM.Google Scholar
  16. Jaszkiewicz, A. (2002). On the performance of multiple-objective genetic local search on the 0/1 knapsack problem—A comparative experiment. IEEE Transactions on Evolutionary Computation, 6(4), 402–412.CrossRefGoogle Scholar
  17. Jaszkiewicz, A. (2003). Do multiple-objective metaheuristics deliver on their promises? A computational experiment on the set covering problem. IEEE Transactions on Evolutionary Computation, 7, 133–143.CrossRefGoogle Scholar
  18. Jaszkiewicz, A. (2004). A comparative study of multiple objective metaheuristics on the biobjective set covering problem and the Pareto memetic algorithm. Annals of Operations Research, 131, 135–158.CrossRefGoogle Scholar
  19. Karp, R. M. (1972). Reducibility among combinatorial problems. In R. E. Miller & J. W. Thatcher (Eds.), Complexity of computer computations (pp. 85–103). New York: Plenum Press.CrossRefGoogle Scholar
  20. Liu, Y. H. (1993). A heuristic algorithm for the multicriteria set covering problems. Applied Mathematics Letters, 6, 21–23.CrossRefGoogle Scholar
  21. Lust, T., Teghem, J., & Tuyttens D., (2011). Very large-scale neighborhood search for solving multiobjective combinatorial optimization problems. In Evolutionary multi-criterion optimization: 6th international conference (Vol. 6576, pp. 254–268)Google Scholar
  22. Lust, T., & Tuyttens, D. (2014). Variable and large neighborhood search to solve the multiobjective set covering problem. Journal of Heuristics, 20(2), 165–188.CrossRefGoogle Scholar
  23. McDonnell, M. D., Ball, I. R., Cousins, A. E., & Possingham, H. P. (2002). Mathematical methods for spatially cohesive reserve design. Environmental Modeling and Assessment, 7, 107–114.CrossRefGoogle Scholar
  24. Musliu, N. (2006). Local search algorithm for unicost set covering problem. Advances in Applied Artificial Intelligence, 4031, 302–311.CrossRefGoogle Scholar
  25. Prins, C., Prodhon, C., & Calvo, R. W. (2006). Two-phase method and Lagrangian relaxation to solve the bi-objective set covering problem. Annals of Operations Research, 147(1), 23–41.CrossRefGoogle Scholar
  26. Saxena, R. R., & Arora, S. R. (1998). Linearization approach to multiobjective quadratic set covering problem. Optimization, 43, 145–156.CrossRefGoogle Scholar
  27. Soylu, B. (2015). Heuristic approaches for biobjective mixed 0–1 integer linear programming problems. European Journal of Operational Research, 245(3), 690–703.CrossRefGoogle Scholar
  28. Steuer, R. E. (1986). Multiple criteria optimization: Theory, computation, and application. New York: Wiley.Google Scholar
  29. Ulungu, E. L., & Teghem, J. (1994). Multiobjective combinatorial optimization problems: A survey. Journal of Multi-criteria Decision Analysis, 3, 83–104.CrossRefGoogle Scholar
  30. Weerasena, L., Shier, D., & Tonkyn, D. (2014). A hierarchical approach to designing compact ecological reserve systems. Environmental Modeling and Assessment, 19(5), 437–449.CrossRefGoogle Scholar
  31. Zitzler, E. (1999). Evolutionary algorithms for multiobjective optimization: Methods and applications. Ph.D. dissertation, Computer Engineering and Networks Laboratory (TIK), ETH, Zurich, Switzerland.Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Lakmali Weerasena
    • 1
  • Margaret M. Wiecek
    • 1
  • Banu Soylu
    • 2
  1. 1.Department of Mathematical SciencesClemson UniversityClemsonUSA
  2. 2.Department of Industrial EngineeringErciyes UniversityKayseriTurkey

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