Advertisement

A compromise solution method for the multiobjective minimum risk problem

  • Fatima BellahceneEmail author
  • Philippe Marthon
Original paper
  • 7 Downloads

Abstract

We develop an approach which enables the decision maker to search for a compromise solution to a multiobjective stochastic linear programming (MOSLP) problem where the objective functions depend on parameters which are continuous random variables with normal multivariate distributions. The minimum-risk criterion is used to transform the MOSLP problem into its corresponding deterministic equivalent which in turn is reduced to a Chebyshev problem. An algorithm based on the combined use of the bisection method and the probabilities of achieving goals is developed to obtain the optimal or epsilon optimal solution of this specific problem. An illustrated example is included in this paper to clarify the developed theory.

Keywords

Multiobjective programming Stochastic programming Nonlinear programming Minimum-risk criterion 

Notes

Acknowledgements

We thank the anonymous referees for their useful comments that improved the content and presentation of the paper.

References

  1. Abbas M, Bellahcene F (2006) Cutting plane method for multiple objective stochastic integer linear programming problem. Eur J Oper Res 168(3):967–984CrossRefGoogle Scholar
  2. Amrouche S, Moulai M (2012) Multi-objective stochastic integer linear programming with fixed recourse. Int J Multicrit Decis Mak 2(4):355–378.  https://doi.org/10.1504/ijmcdm.2012.050677 CrossRefGoogle Scholar
  3. Bazara M, Sherali H, Shetty C (1993) Theory and algorithms, 2nd edn. Wiley, New YorkGoogle Scholar
  4. Ben Abdelaziz F (2012) Solution approaches for the multiobjective stochastic programming. Eur J Oper Res 216:1–16CrossRefGoogle Scholar
  5. Ben Abdelaziz F, Mejri S (2001) Application of goal programming in a multi-objective reservoir operation model in Tunisia. Eur J Oper Res 133:352–361CrossRefGoogle Scholar
  6. Ben Abdelaziz F, Lang P, Nadeau N (1999) Dominance and efficiency in multicriteria decision under uncertainty. Theory Decis 47(3):191–211CrossRefGoogle Scholar
  7. Benayoun R, Montgolfier J, Tergny J, Laritchev O (1971) Linear programming with multiple objective functions: step method (STEM). Math Program 1:366–375CrossRefGoogle Scholar
  8. Bravo M, Gonzalez I (2009) Applying stochastic goal programming: a case study on water use planning. Eur J Oper Res 196(39):1123–1129CrossRefGoogle Scholar
  9. Caballero R, Cerda E, Munoz MM, Rey L (2000) Relations among several efficiency concepts in stochastic multiple objective programming. In: Haimes YY, Steuer R (eds) Research and practice in multiple criteria decision making, vol 487. Lecture notes in economics and mathematical systems. Springer, Cham, pp 57–58CrossRefGoogle Scholar
  10. Caballero R, Cerda E, Del Mar M, Rey L (2001) Efficient solution concepts and their relations in stochastic multiobjective programming. J Optim Theory Appl 110(1):53–74CrossRefGoogle Scholar
  11. Caballero R, Cerda E, Munoz MM, Rey L (2002) Analysis and comparisons of some solutions concepts for stochastic programming problems. Top 10:101–123CrossRefGoogle Scholar
  12. Chaabane D, Mebrek F (2014) Optimization of a linear function over the set of stochastic efficient solutions. CMS 11:157–178.  https://doi.org/10.1007/s10287-012-0155-1 CrossRefGoogle Scholar
  13. Fazlollahtabar H, Mahdavi I (2009) Applying Stochastic Programming for optimizing production time and cost in an automated manufacturing system. In: International conference on computers and industrial engineering, Troyes 6–9 July, pp 1226–1230Google Scholar
  14. Goicoechea A, Dukstein L, Bulfin RL (1976) Multiobjective stochastic programming, the PROTRADE-method. Operation Research Society of America, San FranciscoGoogle Scholar
  15. Kumral M (2003) Application of chance-constrained programming based on multiobjective simulated annealing to solve mineral blending problem. Eng Optim 35(6):661–673CrossRefGoogle Scholar
  16. Miettinen KM (1999) Nonlinear multiobjective optimization. Kluwer’s international series. Kluwer, DordrechtGoogle Scholar
  17. Minc H, Marcus M (1964) A survey of matrix theory and matrix inequalities. Allyn and Bacon Inc., BostonGoogle Scholar
  18. Munoz MM, Ruiz F (2009) Interest: an interval reference point based method for stochastic multiobjective programming problems. Eur J Oper Res 197:25–35CrossRefGoogle Scholar
  19. Slowinski R, Teghem J (1990) Stochastic versus fuzzy approaches to multiobjective, mathematical programming under uncertainty. Kluwer, DordrechtCrossRefGoogle Scholar
  20. Stancu-Minasian IM (1976) Asupra problemei de risk minim multiplu I: cazul a dou funcii obiectiv. II: cazul a r (r > 2) funciiobiectiv. Stud Cerc Mat 28(5):617–623Google Scholar
  21. Stancu-Minasian IM (1984) Stochastic programming with multiple objective functions. D. Reidel Publishing Company, DordrechtGoogle Scholar
  22. Teghem J (1990) Strange-Momix: an interactive method for mixed integer linear programming. In: Slowinski R, Teghem J (eds) Stochastic versus fuzzy approaches to multiobjective mathematical programming under uncertainty. Kluwer, Dordrecht, pp 101–115Google Scholar
  23. Teghem J, Kunsch PL (1985) Application of multiobjective stochastic linear programming to power systems planning. Eng Costs Prod Econ 9(13):83–89CrossRefGoogle Scholar
  24. Teghem J, Dufrane D, Thauvoye M, Kunsch P (1986) STRANGE: interactive method for multiobjective linear programming under uncertainty. Eur J Oper Res 26(1):65–82CrossRefGoogle Scholar
  25. Urli B, Nadeau R (1990) Stochastic MOLP with incomplete information: an interactive approach with recourse. J Oper Res Soc 41:1143–1152CrossRefGoogle Scholar
  26. Urli B, Nadeau R (2004) PROMISE/scenarios: an interactive method for multiobjective stochastic linear programming under partial uncertainty. Eur J Oper Res 155:361–372CrossRefGoogle Scholar
  27. Vahidinasab V, Jadid S (2010) Stochastic multiobjective self-scheduling of a power producer. Joint Energy Reserves Mark Electr Power Syst Res 80(7):760–769CrossRefGoogle Scholar
  28. Wang Z, Jia XP, Shi L (2009) Optimization of multi-product batch plant design under uncertainty with environmental considerations. Clean Technol Environ Policy 12(3):273–282CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.LAROMAD Laboratory, Faculty of SciencesMouloud Mammeri UniversityTizi-OuzouAlgeria
  2. 2.ENSEEIHT InformatiqueToulouse Cedex 7France

Personalised recommendations