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Mathematical Methods of Operations Research

, Volume 87, Issue 2, pp 251–283 | Cite as

Variants of the \( \varepsilon \)-constraint method for biobjective integer programming problems: application to p-median-cover problems

  • Jesús Sáez-Aguado
  • Paula Camelia Trandafir
Original Article
  • 100 Downloads

Abstract

We conduct an in-depth analysis of the \(\varepsilon \)-constraint method (ECM) for finding the exact Pareto front for biobjective integer programming problems. We have found up to six possible different variants of the ECM. We first discuss the complexity of each of these variants and their relationship with other exact methods for solving biobjective integer programming problems. By extending some results of Neumayer and Schweigert (OR Spektrum 16:267–276, 1994), we develop two variants of the ECM, both including an augmentation term and requiring \(N+1\) integer programs to be solved, where N is the number of nondominated points. In addition, we present another variant of the ECM, based on the use of elastic constraints and also including an augmentation term. This variant has the same complexity, namely \(N+1\), which is the minimum reached for any exact method. A comparison of the different variants is carried out on a set of biobjective location problems which we call p-median-cover problems; these include the objectives of the p-median and the maximal covering problems. As computational results show, for this class of problems, the augmented ECM with elastic constraint is the most effective variant for finding the Pareto front in an exact manner.

Keywords

Biobjective integer programming Epsilon-constraint method p-Median-cover problem 

Notes

Acknowledgements

The authors wish to thank the anonymous referees for their useful suggestions and comments that improved the paper, and to FICO for providing the Xpress Optimizer application, which has been used to obtain the computational results.

References

  1. Bérubé JF, Gendreau M, Potvin JY (2009) An exact \(\varepsilon \)-constraint method for bi-objective combinatorial optimization problems: application to the traveling salesman problem with profits. Eur J Oper Res 194(1):39–50MathSciNetCrossRefzbMATHGoogle Scholar
  2. Boland N, Charkhgard H, Savelsbergh MA (2015) Criterion space search algorithms for biobjective integer programming: the balanced box method. INFORMS J Comput 27:735–754MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chalmet LG, Lemonidis L, Elzinga DJ (1986) An algorithm for the bi-criterion integer programming problem. Eur J Oper Res 25:292–300MathSciNetCrossRefzbMATHGoogle Scholar
  4. Chankong V, Haimes Y (1983) Multiobjective decision making. Theory and methodology. Elsevier, AmsterdamzbMATHGoogle Scholar
  5. Cohon JL (1978) Multiobjective programming and planning. Academic Press, LondonzbMATHGoogle Scholar
  6. Dächert K, Gorski J, Klamroth K (2012) An augmented weighted Tchebycheff method with adaptively chosen parameters for discrete bicriteria optimization problems. Comput Oper Res 39:2929–2943MathSciNetCrossRefzbMATHGoogle Scholar
  7. Daskin MS (2013) Network and discrete location. Models, algorithms and applications, 2nd edn. Wiley, LondonzbMATHGoogle Scholar
  8. Ehrgott M (2005) Multicriteria optimization. Springer, BerlinzbMATHGoogle Scholar
  9. Ehrgott M (2006) A discussion of scalarization techniques for multiple objective integer programming. Ann Oper Res 147:343–360MathSciNetCrossRefzbMATHGoogle Scholar
  10. Ehrgott M, Gandibleux X (2000) A survey and annotated bibliography of multiobjective combinatorial optimization. OR Spektrum 22:425–460MathSciNetCrossRefzbMATHGoogle Scholar
  11. Ehrgott M, Gandibleux X (2003) Multiobjective combinatorial optimization. Theory, methodology and applications. In: Ehrgott M, Gandibleux X (eds) Multiple criteria optimization: state of the art annotated bibliographic surveys. Surveys international series in operations research and management science, vol 52. Kluwer Academic Publishers, Dordrecht, pp 369–444Google Scholar
  12. Ehrgott M, Ruzika S (2008) An improved epsilon constraint method for multiobjective programming. J Optim Theory Appl 138(3):375–396MathSciNetCrossRefzbMATHGoogle Scholar
  13. Ehrgott M, Ryan D (2002) Constructing robust crew schedules with bicriteria optimization. J Multicriteria Decis Anal 11:139–150CrossRefzbMATHGoogle Scholar
  14. FICO Xpress Optimization Suite (2016) XPRESS-MP optimizer reference manual. Fair Isaac Corporation, Leamington SpaGoogle Scholar
  15. Figueira J, Greco S, Ehrgott M (2005) Multiple criteria decision analysis: state of the art surveys. Springer, BerlinCrossRefzbMATHGoogle Scholar
  16. Filippi C, Stevanato E (2013) A two-phase method for bi-objective combinatorial optimization and its application to the TSP with profits. Algorithm Operations Research 7:125–139MathSciNetzbMATHGoogle Scholar
  17. Grandinetti L, Guerriero F, Lagana D, Pisacane O (2010) An approximate \(\varepsilon \)-constraint method for the multi-objective undirected capacitated arc routing problem. In: Paola F (ed) 9th International symposium on experimental algorithms SEA 2010. Springer, BerlinGoogle Scholar
  18. Haimes Y, Lasdon L, Wismer D (1971) On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans Syst Man Cybern Soc 1:296–297MathSciNetzbMATHGoogle Scholar
  19. Hamacher HW, Pedersen CR, Ruzika S (2007) Finding representative systems for discrete bicriterion optimization problems. Oper Res Lett 35(3):336–344MathSciNetCrossRefzbMATHGoogle Scholar
  20. Hugot H, Vanderpooten D, Vanpeperstraete JM (2006) A bi-criteria approach for the data association problem. Ann Oper Res 147:217–234MathSciNetCrossRefzbMATHGoogle Scholar
  21. Laumanns M, Thiele L, Zitzler E (2006) An efficient, adaptive parameter scheme for metaheuristic based on the \(\varepsilon \)-constraint method. Eur J Oper Res 169:932–942Google Scholar
  22. Mavrotas G (2009) Effective implementation of the \(\varepsilon \)-constraint method in multi-objective mathematical programming problems. Appl Math Comput 213:455–465MathSciNetzbMATHGoogle Scholar
  23. Nemhauser GL, Wolsey LA (1999) Integer and combinatorial optimization. Wiley, LondonzbMATHGoogle Scholar
  24. Neumayer P, Schweigert D (1994) Three algorithms for bicriteria integer linear programs. OR Spektrum 16:267–276MathSciNetCrossRefzbMATHGoogle Scholar
  25. Özpeynirci O, Köksalan M (2007) Pyramidal tours and multiple objectives. J Glob Optim 48:569–582MathSciNetCrossRefzbMATHGoogle Scholar
  26. Özlen M, Azizoǧlu M (2009) Multi-objective integer programming: a general approach for generating all nondominated solutions. Eur J Oper Res 199(1):25–35CrossRefzbMATHGoogle Scholar
  27. Ralphs TK, Saltzman MJ, Wiecek MM (2006) An improved algorithm for solving biobjective integer programs. Ann Oper Res 147:43–70MathSciNetCrossRefzbMATHGoogle Scholar
  28. Ramesh R, Karwan MH, Zionts S (1990) An interactive method for bicriteria integer programming. IEEE Trans Systems Man Cybern Soc 20(3):395–403MathSciNetCrossRefzbMATHGoogle Scholar
  29. Sáez-Aguado J, Trandafir PC (2012) Some heuristic methods for solving p-median problems with a coverage constraint. Eur J Oper Res 220:320–327MathSciNetCrossRefzbMATHGoogle Scholar
  30. Soland RM (1979) Multicriteria optimization: a general characterization of efficient solutions. Decis Sci 10:26–38CrossRefGoogle Scholar
  31. Steuer R (1985) Multiple criteria optimization: theory. Computation and application. Wiley, New YorkzbMATHGoogle Scholar
  32. Sylva J, Crema A (2004) A method for finding the set of nondominated vectors for multiple objective integer linear programs. Eur J Oper Res 158:46–55CrossRefzbMATHGoogle Scholar
  33. Ulungu EL, Teghem J (1994) Multi-objective combinatorial optimization problems: a survey. J Multi Criteria Decis Anal 3:83–104CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Statistics and Operational ResearchUniversity of ValladolidValladolidSpain
  2. 2.Department of Statistics and Operational ResearchPublic University of NavarreTudelaSpain

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