Chance-Constrained Approaches for Multiobjective Stochastic Linear Programming Problems

Affiliation(s)

Department of Mathematics and Computer Science, University of Kinshasa, Kinshasa, D.R. Congo.

Department of Decision Sciences, University of South Africa, Pretoria, South Africa.

Department of Mathematics and Computer Science, University of Kinshasa, Kinshasa, D.R. Congo.

Department of Decision Sciences, University of South Africa, Pretoria, South Africa.

ABSTRACT

Multiple objective stochastic linear programming is a relevant topic. As a matter of fact, many practical problems ranging from portfolio selection to water resource management may be cast into this framework. Severe limitations on objectivity are encountered in this field because of the simultaneous presence of randomness and conflicting goals. In such a turbulent environment, the mainstay of rational choice cannot hold and it is virtually impossible to provide a truly scientific foundation for an optimal decision. In this paper, we resort to the bounded rationality principle to introduce satisfying solution for multiobjective stochastic linear programming problems. These solutions that are based on the chance-constrained paradigm are characterized under the assumption of normality of involved random variables. Ways for singling out such solutions are also discussed and a numerical example provided for the sake of illustration.

Multiple objective stochastic linear programming is a relevant topic. As a matter of fact, many practical problems ranging from portfolio selection to water resource management may be cast into this framework. Severe limitations on objectivity are encountered in this field because of the simultaneous presence of randomness and conflicting goals. In such a turbulent environment, the mainstay of rational choice cannot hold and it is virtually impossible to provide a truly scientific foundation for an optimal decision. In this paper, we resort to the bounded rationality principle to introduce satisfying solution for multiobjective stochastic linear programming problems. These solutions that are based on the chance-constrained paradigm are characterized under the assumption of normality of involved random variables. Ways for singling out such solutions are also discussed and a numerical example provided for the sake of illustration.

KEYWORDS

Satisfying Solution; Chance-Constrained; Multiobjective Programming; Stochastic Programming

Satisfying Solution; Chance-Constrained; Multiobjective Programming; Stochastic Programming

Cite this paper

J. Kampempe and M. Luhandjula, "Chance-Constrained Approaches for Multiobjective Stochastic Linear Programming Problems,"*American Journal of Operations Research*, Vol. 2 No. 4, 2012, pp. 519-526. doi: 10.4236/ajor.2012.24061.

J. Kampempe and M. Luhandjula, "Chance-Constrained Approaches for Multiobjective Stochastic Linear Programming Problems,"

References

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[4] P. Rietveld and H. Ouwersloot, “Ordinal Data in Multicriteria Decision Making, a Stochastic Dominance Approach to Sitting Nuclear Power Plants,” European Journal of Operational Research, Vol. 56, No. 2, 1992, pp. 249-262. doi:10.1016/0377-2217(92)90226-Y

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[6] B. Liu, “Theory and Practice of Uncertainly Programming,” Physical-Verley, Heidelberg, 2002.

[7] A. S. Adeyefa and M. K. Luhandjula, “Multiobjective Stochastic Linear Programming: An Overview,” American Journal of Operations Research, Vol. 1, No. 4, 2011, pp. 203-213. http://www.SciRP.org/journal/ajor

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[14] B. Liu, “Introduction to Uncertain Programming,” Tsinghua University, Beijing. 2005. (Unpublished).

[15] J. K. Sengupta, Stochastic Programming: Methods and Applications. North-Holland Publishing Company, Amsterdam, 1972.

[16] G. P. McCormick and K. Ritter, “Methods of Conjugate Directions versus Quasi-Newton Methods,” Mathematical Programming, Vol. 3, No. 1, 1972, pp. 101-116

[17] R. Flecher and C. M. Reeves, “Function Minimization by Conjugate Gradients,” The Computer Journal, Vol. 7, No. 2, 1964, pp. 149-154. doi:10.1093/comjnl/7.2.149

[18] J. P. Evans, F. J. Gould and J. W. Tolle, “Exact Penalty Functions in Nonlinear Programming,” Mathematical Programming, Vol. 4, No. 1, 1973, pp. 72-79. doi:10.1007/BF01584647

[19] A. K. Bhunia and J. Majumda, “Elitist Genetic Algorithm for Assignment Problem with Imprecise Goal,” European Journal of Operational Research, Vol. 177, No. 2, 2007, pp. 684-692. doi:10.1016/j.ejor.2005.11.034

[20] K. M. Miettinem, “Nonlinear Multiobjective Optimization,” Kluwer Academic Publishers, Massachusetts, 1999.

[21] B. C. Eaves and W. I. Zangwill, “Generalized Cutting Plane Algorithms,” SIAM Journal on Control, Vol. 9, No. 4, 1971, pp. 529-542. doi:10.1137/0309037

[22] M. K. Luhandjula, “On Fuzzy Random-Valued Optimization,” American Journal of Operations Research, Vol. 1, No. 4, 2011, pp. 259-267. doi:10.4236/ajor.2011.14030

[23] C. P. Olivier, “Solving Bilevel Linear Multiobjective Programming Problems,” American Journal of Operations Research, Vol. 1, No. 4, 2011, pp. 214-219.

[1] P. C. Fishburn, “Utility for Decision Making,” John Wiley, New York, 1970.

[2] R. Caballero, E. Cerdà, M. M. Munoz and L. Rey, “Relations among Several Efficiency Concepts in Stochastic Multiple Objective Programming,” In: Y. Y. Haines and R. E. Steuer, Eds., Research and Practice in Multiple Criteria Decision Making, 2000.

[3] R. Caballero, E. Cerdà, M. M. Mu?oz, L. Rey and I. M. “Stancu-Minasian, Efficient Solution Concepts and Their Relations in Stochastic Multiobjective Programming,” Journal of Optimization Theory and Applications, Vol. 110, No. 1, 2001, pp. 53-74. doi:10.1023/A:1017591412366

[4] P. Rietveld and H. Ouwersloot, “Ordinal Data in Multicriteria Decision Making, a Stochastic Dominance Approach to Sitting Nuclear Power Plants,” European Journal of Operational Research, Vol. 56, No. 2, 1992, pp. 249-262. doi:10.1016/0377-2217(92)90226-Y

[5] H. M. Markowitz, “Mean Variance Analysis in Portfolio choice and Capital Markets,” Basil Blackwell, Oxford, 1970.

[6] B. Liu, “Theory and Practice of Uncertainly Programming,” Physical-Verley, Heidelberg, 2002.

[7] A. S. Adeyefa and M. K. Luhandjula, “Multiobjective Stochastic Linear Programming: An Overview,” American Journal of Operations Research, Vol. 1, No. 4, 2011, pp. 203-213. http://www.SciRP.org/journal/ajor

[8] A. Charnes and W. W. Cooper, “Chance-Constrained Programming,” Management Science, Vol. 6, No. 1, 1959, pp. 73-79. doi:10.1287/mnsc.6.1.73

[9] H. A. Simon, “A Behavior Model of Rational Choice,” In: Models of Man: Social and Rational, Macmillan, New York, 1957.

[10] I. M. Staincu-Minasian, “Overview of Different Approaches for Solving Stochastic Programming Problems with Multiple Objective Functions,” In: S.-Y. Huang and J. Teghem, Eds., Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertaintly, Kluwer Academic Publisher, Dordrecht, 1990.

[11] I. M. Stancu-Minasian, “Stochastic Programming with Multiple Objective Functions,” D. Reidel Publishing Company, Boston, 1984.

[12] P. Kall, “Stochastic Linear Programming,” Springer-Verlag, Berlin, 1972.

[13] S. Kataoka, “A Stochastic Programming Model,” Econometrica, Vol. 31, No. 1-2, 1963, pp. 181-196.

[14] B. Liu, “Introduction to Uncertain Programming,” Tsinghua University, Beijing. 2005. (Unpublished).

[15] J. K. Sengupta, Stochastic Programming: Methods and Applications. North-Holland Publishing Company, Amsterdam, 1972.

[16] G. P. McCormick and K. Ritter, “Methods of Conjugate Directions versus Quasi-Newton Methods,” Mathematical Programming, Vol. 3, No. 1, 1972, pp. 101-116

[17] R. Flecher and C. M. Reeves, “Function Minimization by Conjugate Gradients,” The Computer Journal, Vol. 7, No. 2, 1964, pp. 149-154. doi:10.1093/comjnl/7.2.149

[18] J. P. Evans, F. J. Gould and J. W. Tolle, “Exact Penalty Functions in Nonlinear Programming,” Mathematical Programming, Vol. 4, No. 1, 1973, pp. 72-79. doi:10.1007/BF01584647

[19] A. K. Bhunia and J. Majumda, “Elitist Genetic Algorithm for Assignment Problem with Imprecise Goal,” European Journal of Operational Research, Vol. 177, No. 2, 2007, pp. 684-692. doi:10.1016/j.ejor.2005.11.034

[20] K. M. Miettinem, “Nonlinear Multiobjective Optimization,” Kluwer Academic Publishers, Massachusetts, 1999.

[21] B. C. Eaves and W. I. Zangwill, “Generalized Cutting Plane Algorithms,” SIAM Journal on Control, Vol. 9, No. 4, 1971, pp. 529-542. doi:10.1137/0309037

[22] M. K. Luhandjula, “On Fuzzy Random-Valued Optimization,” American Journal of Operations Research, Vol. 1, No. 4, 2011, pp. 259-267. doi:10.4236/ajor.2011.14030

[23] C. P. Olivier, “Solving Bilevel Linear Multiobjective Programming Problems,” American Journal of Operations Research, Vol. 1, No. 4, 2011, pp. 214-219.