Advertisement

4OR

, Volume 10, Issue 2, pp 181–192 | Cite as

Min–max and min–max (relative) regret approaches to representatives selection problem

  • Alexandre DolguiEmail author
  • Sergey Kovalev
Research Paper

Abstract

The following optimization problem is studied. There are several sets of integer positive numbers whose values are uncertain. The problem is to select one representative of each set such that the sum of the selected numbers is minimum. The uncertainty is modeled by discrete and interval scenarios, and the min–max and min–max (relative) regret approaches are used for making a selection decision. The arising min–max, min–max regret and min–max relative regret optimization problems are shown to be polynomially solvable for interval scenarios. For discrete scenarios, they are proved to be NP-hard in the strong sense if the number of scenarios is part of the input. If it is part of the problem type, then they are NP-hard in the ordinary sense, pseudo-polynomially solvable by a dynamic programming algorithm and possess an FPTAS. This study is motivated by the problem of selecting tools of minimum total cost in the design of a production line.

Keywords

Uncertainty Min–max approach Min–max regret Computational complexity Dynamic programming 

Mathematics Subject Classification

03D15 62F35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aissi H (2006) Approximation and resolution of minmax and minmax regret versions of combinatorial optimization problems. 4OR Q J Oper Res 4(4): 347–350CrossRefGoogle Scholar
  2. Aissi H, Bazgan C, Vanderpooten D (2005) Complexity of the min-max and min-max regret assignment problem. Oper Res Lett 33: 634–640CrossRefGoogle Scholar
  3. Aissi H, Bazgan C, Vanderpooten D (2009) Minmax and minmax regret versions of combinatorial optimization problems: a survey. Eur J Oper Res 197: 427–438CrossRefGoogle Scholar
  4. Aissi H, Bazgan C, Vanderpooten D (2010) General approximation schemes for min-max (regret) versions of some (pseudo-)polynomial problems. Discret Optim 7: 136–148CrossRefGoogle Scholar
  5. Aissi H, Aloulou MA, Kovalyov MY (2011) Minimizing the number of late jobs on a single machine under due date uncertainty. J Sched 14(4): 351–360CrossRefGoogle Scholar
  6. Averbakh I (2001) On the complexity of a class of combinatorial optimization problems with uncertainty. Math Program A 90(2): 263–272CrossRefGoogle Scholar
  7. Averbakh I (2005) Computing and minimizing the relative regret in combinatorial optimization with interval data. Discret Optim 2: 273–287CrossRefGoogle Scholar
  8. Conde E (2004) An improved algorithm for selecting p items with uncertain returns according to the minmax regret criterion. Math Program 100: 345–353CrossRefGoogle Scholar
  9. Deineko V, Woeginger G (2006) On the robust assignment problem under a fixed number of cost scenarios. Oper Res Lett 34(2): 175–179CrossRefGoogle Scholar
  10. Dolgui A, Guschinsky N, Levin G, Proth JM (2008) Optimisation of multi-position machines and transfer lines. Eur J Oper Res 185: 1375–1389CrossRefGoogle Scholar
  11. Galli L (2011) Combinatorial and robust optimisation models and algorithms for railway applications. 4OR Q J Oper Res 9(2): 215–218CrossRefGoogle Scholar
  12. Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, San FranciscoGoogle Scholar
  13. Kasperski A, Zielinski P (2006a) An approximation algorithm for interval data minmax regret combinatorial optimization problems. Inf Process Lett 97: 177–180CrossRefGoogle Scholar
  14. Kasperski A, Zielinski P (2006b) The robust shortest path problem in series-parallel multidigraphs with interval data. Oper Res Lett 34: 69–76CrossRefGoogle Scholar
  15. Kasperski A, Zielinski P (2009a) A randomized algorithm for the min-max selecting items problem with uncertain weights. Ann Oper Res 172: 221–230CrossRefGoogle Scholar
  16. Kasperski A, Zielinski P (2009b) On the approximability of minmax (regret) network optimization problems. Inf Process Lett 109: 262–266CrossRefGoogle Scholar
  17. Kovalyov MY (1995) Improving the complexities of approximation algorithms for optimization problems. Oper Res Lett 17: 85–87CrossRefGoogle Scholar
  18. Kovalyov MY (1996) A rounding technique to construct approximation algorithms for knapsack and partition type problems. Appl Math Comput Sci 6: 101–113Google Scholar
  19. Kouvelis P, Yu G (1997) Robust discrete optimization and its applications. Kluwer, BostonGoogle Scholar
  20. Mirzapour Al-E-Hashem SMJ, Aryanezhad MB, Malekly H, Sadjadi SJ (2009) Mixed model assembly line balancing problem under uncertainty. International Conference on Computers and Industrial Engineering, CIE 2009 , art. no. 5223925, pp. 233–238Google Scholar
  21. Montemanni R, Gambardella LM (2005) The robust shortest path problem with interval data via Benders decomposition. 4OR Q J Oper Res 3(4): 315–328CrossRefGoogle Scholar
  22. Roy B (2010) Robustness in operational research and decision aiding: a multi-faceted issue. Eur J Oper Res 200: 629–638CrossRefGoogle Scholar
  23. Scutella MG, Recchia R (2010) Robust portfolio asset allocation and risk measures. 4OR Q J Oper Res 8(2): 113–139CrossRefGoogle Scholar
  24. Xu W, Xiao T (2009) Robust balancing of mixed model assembly line. COMPEL Int J Comput Math Electr Electron Eng 28(6): 1489–1502CrossRefGoogle Scholar
  25. Yu G, Yang J (1998) On the robust shortest path problem. Comput Oper Res 25(6): 457–468CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Ecole des MinesCNRS UMR 6158 LIMOSSaint-EtienneFrance

Personalised recommendations