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.
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References
Abbas M, Bellahcene F (2006) Cutting plane method for multiple objective stochastic integer linear programming problem. Eur J Oper Res 168(3):967–984
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
Bazara M, Sherali H, Shetty C (1993) Theory and algorithms, 2nd edn. Wiley, New York
Ben Abdelaziz F (2012) Solution approaches for the multiobjective stochastic programming. Eur J Oper Res 216:1–16
Ben Abdelaziz F, Mejri S (2001) Application of goal programming in a multi-objective reservoir operation model in Tunisia. Eur J Oper Res 133:352–361
Ben Abdelaziz F, Lang P, Nadeau N (1999) Dominance and efficiency in multicriteria decision under uncertainty. Theory Decis 47(3):191–211
Benayoun R, Montgolfier J, Tergny J, Laritchev O (1971) Linear programming with multiple objective functions: step method (STEM). Math Program 1:366–375
Bravo M, Gonzalez I (2009) Applying stochastic goal programming: a case study on water use planning. Eur J Oper Res 196(39):1123–1129
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 57–58
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):53–74
Caballero R, Cerda E, Munoz MM, Rey L (2002) Analysis and comparisons of some solutions concepts for stochastic programming problems. Top 10:101–123
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
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 1226–1230
Goicoechea A, Dukstein L, Bulfin RL (1976) Multiobjective stochastic programming, the PROTRADE-method. Operation Research Society of America, San Francisco
Kumral M (2003) Application of chance-constrained programming based on multiobjective simulated annealing to solve mineral blending problem. Eng Optim 35(6):661–673
Miettinen KM (1999) Nonlinear multiobjective optimization. Kluwer’s international series. Kluwer, Dordrecht
Minc H, Marcus M (1964) A survey of matrix theory and matrix inequalities. Allyn and Bacon Inc., Boston
Munoz MM, Ruiz F (2009) Interest: an interval reference point based method for stochastic multiobjective programming problems. Eur J Oper Res 197:25–35
Slowinski R, Teghem J (1990) Stochastic versus fuzzy approaches to multiobjective, mathematical programming under uncertainty. Kluwer, Dordrecht
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):617–623
Stancu-Minasian IM (1984) Stochastic programming with multiple objective functions. D. Reidel Publishing Company, Dordrecht
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 101–115
Teghem J, Kunsch PL (1985) Application of multiobjective stochastic linear programming to power systems planning. Eng Costs Prod Econ 9(13):83–89
Teghem J, Dufrane D, Thauvoye M, Kunsch P (1986) STRANGE: interactive method for multiobjective linear programming under uncertainty. Eur J Oper Res 26(1):65–82
Urli B, Nadeau R (1990) Stochastic MOLP with incomplete information: an interactive approach with recourse. J Oper Res Soc 41:1143–1152
Urli B, Nadeau R (2004) PROMISE/scenarios: an interactive method for multiobjective stochastic linear programming under partial uncertainty. Eur J Oper Res 155:361–372
Vahidinasab V, Jadid S (2010) Stochastic multiobjective self-scheduling of a power producer. Joint Energy Reserves Mark Electr Power Syst Res 80(7):760–769
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):273–282
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Bellahcene, F., Marthon, P. A compromise solution method for the multiobjective minimum risk problem. Oper Res Int J 21, 1913–1926 (2021). https://doi.org/10.1007/s12351-019-00493-1
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DOI: https://doi.org/10.1007/s12351-019-00493-1