Many Optimization problems in engineering and economic involve the challenging task of pondering both conflicting goals and random data. In this paper, we give an up-to-date overview of how important ideas from optimization, probability theory and multicriteria decision analysis are interwoven to address situations where the presence of several objective functions and the stochastic nature of data are under one roof in a linear optimization context. In this way users of these models are not bound to caricature their problems by arbitrarily squeezing different objective functions into one and by blindly accepting fixed values in lieu of imprecise ones.
KeywordsLinear ProgrammingMultiobjective ProgrammingStochastic ProgrammingExpected Value Optimality/EfficiencyMinimum Risk Solution/EfficiencyVariance Optimality/EfficiencyOptimality/Efficiency with Given Probabilities.
A. Andreas and L. Wu-Sheng, “Practical optimization: Algorithm and engineering applications”, Springer, New York, 2007.
D.W. Bunn, “Applied decision analysis”, New York, McGraw Hill, 1984.
A. Charnes and W.W. Cooper, “Chance-constrained programming”, Management Sciences, 6, pp. 73-79, 1959.
A. Charnes and W.W. Cooper, “Management models and industrial applications of linear programming”, volume 1 and II, John Wiley and Sons Inc., 1961.
B.S. Gottfried and J. Weisman, “Introduction to optimization theory”, Prentice-Hall, Inc., Englewood, 1973.
D.G. Luenberger, “Introduction to linear and nonlinear programming”, Addison-Wesley, Menlo Park, (CA), 1984.
S.S. Rao, “Optimization theory and applications”. John Wiley and Sons Inc., New York (NY), 2nd edition edition, 1984.
S.S. Rao, “Engineering optimization: Theory and Practice”, John Wiley & Sons Inc., New York (NY), 3rd edition edition, 1996.
M. Ehrgott, “A discussion of scalarization techniques for multiobjective integer programming”, Annals of Operations Research, 147 no 1, pp. 343--360, 2006.
H.A. Eiselt, G. Pederzoli, and C-L. Sandblom, “Continuous optimization models” Walter deGruyter and Company, Berlin, 1987.
A. Goicoechea, D.R. Hansen, and L. Duckstein, “Multiobjective decision analysis with engineering and business application”, John Wiley and Sons Inc., New York (NY), 1982.
J.P. Ignizio, “Linear programming in single- and multiple- objective systems”, Prentice-Hall Inc. Englewood Cliffs, (NJ), 1982.
M. Zeleny, “A concept of compromise solutions and the method of the displaced ideal”, Computers and Operations Research, 1, pp. 479-496, 1974.
I.M. Stancu-Minasian, “Stochastic programming with multiple objective functions”, D. Reidel Publishing Com- pany, 1984.
K.J. Arrow, “A difficulty in the concept of social welfare”, Journal of Political Economy, 58(4), pp. 328-346, 1950.
M. Ehrgott, “Multicriteria optimization”, Springer-Verlag, Berlin, 2nd edition edition, 2005.
J. Jahn, “Vector optimization: Theory, applications and extensions”. Springer-Verlag, Berlin, 2004.
B. Rustem, “Algorithms for nonlinear programming and multiple-objective decisions”, John Wiley and Sons Inc., New York (NY), 1998.
R. D. Shapiro, “Optimization models for planning & allocation: Text & cases in mathematical programming”, Wiley & Sons Inc. New York (NY), 1984.
M. Zeleny, “Multiple criteria decision making”, Springer- Verlag, Berlin, 1976.
A.S. Adeyefa and M.K. Luhandjula. “Risk management in multiobjective programming under uncertainty for managing sustainable development”, Accepted for publication, 2010.
S.B. Graves and J.L. Ringuest, “Probabilistic dominance criteria for comparing uncertain alternatives: A tutorial”, Omega, 37, pp. 346-357, 2009.
B. Liu, “Theory and Practice of Uncertainty Programming”, Physica-Verlag, Heidelberg, 2002.
M.K. Luhandjula, “Multiple objective programming pro- blems with possibilistic coefficients”, Fuzzy Sets & Systems, 2, pp. 135-145, 1987.
M.K. Luhandjula, “Fuzzy optimization: An appraisal”, Fuzzy sets & Systems, 30, pp. 257-282, 1989.
M.K. Luhandjula, S.S. Ruzibiza, and A.S. Adeyefa, “Solving multiobjective programming problems with fuzzy coefficients”, Advances in Fuzzy sets & Systems, 8, pp. 13-33, 2011.
M.K. Luhandjula and M. Sakawa, “Multiple-objective linear programming problems in the presence of fuzzy coefficients”, In R. Slowinski and J. Teghem, editors, Stochastic versus fuzzy approaches to multiobjective mathematical programming under uncertainty. Kluwer Academic Publishers, 1990.
A.D. Roy, “Safety first & the holding of assets”, Econometrica, 20, pp. 431-49, 1952.
S. Vadja, “Probabilistic programming”, Academic Press, New York (NY), 1972.
H-J. Zimmerman, “Description and optimization of fuzzy systems”, International Journal of General Systems, 2, pp. 209-215, 1976.
H. Simon, “A behavioral model of rational choice in models of man, social and rational”, Macmillan, 1957.
A.S. Adeyefa, “Methodological approaches for multiobjective stochastic linear programmimg problems”, IFORS Conference Sandton, South Africa, 2008.
Levy and J. Paroush, “Multiperiod stochastic dominance”, Management Science, 21, pp. 428-435, 1974.
M.K. Luhandjula, A.S. Adeyefa, and S. Mukeru, “Satisficing solutions for multiobjective stochastic linear programming problems”, Submitted, 2011.
D. Panda, S. Kar, K. Maity, and M. Maiti, “A single period inventory model with imperfect production and stochastic demand under chance and imprecise constraints”, European Journal of Operational Research,, 188(1), pp. 121-139, 2008.
S. Hulsurkar, M.P. Biswal, and S.B. Sinha, “Fuzzy programming approach to multi-objective stochastic linear programming problem”, Fuzzy sets and Systems, 88(2), pp. 173-181, 1997.
S.B. Sinha, S. Hulsurkar, and M.P. Biswal, “Fuzzy programming approach to multi-objective stochastic programming problems when bi’s follow joint normal distribution”, Fuzzy Sets and Systems, 109(1), pp. 91-96, 2000.
M. Abbasi and M. Houshmand, “Production planning and performance optimization of reconfigurable manufacturing systems using genetic algorithm”, International Journal of Advanced Manufacturing Technology, pages 1-20, 2010.
X.R. Gandibleux, M. Sevaux, K. S?rensen, and V. Tkindt, “Metaheuristics for multiobjective optimization”, Sprin- ger-Verlag, Berlin, 2004.
T. Ibaraki, K. Nonobe, and Yagiura. Metaheuristics: Progress as real problem solvers. Springer. New York (NY), 2005.
K.F. Doerner, W.J. Gutjahr, R.F. Hartle, and Strauss, “Nature-inspired metaheuristics for multiobjective activity crashing”, Omega, 36(6), pp. 1019--1037, 2008.
J. Dréo, A. Pétrowski, P. Siarry, and E. Taillard, “Metaheuristics for hard optimization”, Springer-Verlag, Berlin, 2006.
F. Dugardin, F. Yalaoui, and L. Amodeo, “New multi- objective method to solve re-entrant hybrid flow shop scheduling problem”, European Journal of Operational Research, 203(1), pp. 22-31, 2010.
X.R. Gandibleux and M. Ehrgott, “(1984-2004)-20 years of multiobjective metaheuristics, but what about the solution of combinatorial problems with multiple objectives?”, Lecture Notes in Computer Science 3410. Springer-Ver- lag, 2005.
H. Katagiri, M. Sakawa, K. Kato, and I. Nishizaki, “A fuzzy random multi-objective 0-1 programming based on the expectation optimization model using possibility and necessity measures”, Mathematical and Computer Modelling, 40, pp. 530-539, 2004.
J. Dupa?ová, “Applications of stochastic programming: Achievements and questions”, European Journal of Operational Research, 140(2), pp. 281-290, 2002.
J. Mayer, “Stochastic linear programming algorithms”, Overseas Publishers Association, 1984.
R. Slowinski and J. Teghem, “Stochastic versus fuzzy approaches to multiobjective mathematical programming under uncertainty”, Kluwer Academic Publishers, 1990.
I.I. Croley, E. R. Thomas, and N.R. Kuchibhotla, “Multiobjective risks in reservoir operation”, Water Resources Research, 15(4), pp. 807-814, 1979.
A. Goicoechea, L. Duckstein, and M. Fogel, “Multiple objectives under uncertainty: An illustrative application of protrude”, Water Resources Research, 15(2), pp. 203- 210, 1979.
H. Fazlollahtabar and I. Mahdavi, “Applying stochastic programming for optimizing production time & cost in an automated manufacturing system”, IEEE 978-1-4244- 4136-5/09, 2009.
A. Alarcon-Rodriguez, G. Ault, and S. Galloway, “Multi- objective planning of distributed energy resources, pp. A review of the state-of-the-art”, Renewable & Sustainable Energy Reviews, 14, pp. 1353-1366, 2010.
M. Bravo and I. Gonzalez, “Applying stochastic goal programming, pp. A case study on water use planning”, European Journal of Operational Research, 196(3), pp. 1123-1129, 2009.
T.Z. Caner and ü. A. Tamer, “Tactical level planning in float glass manufacturing with co-production, random yields & substitutable products”, European Journal of Operational Research, 199(1), pp. 252-261, 2009.
J. Teghem, “Strange an interactive method for multiobjective stochastic linear programming and strange-momix its extension to integer variables”, In R. Slowinski and J. Teghem, editors, Stochastic versus fuzzy approaches to multiobjective mathematical programming under uncertainty. Kluwer Academic, Dordrecht, 1990.
J. Teghem, D. Dufrane, M. Thauvoye, and P.L. Kunsch, “Strange: an interactive method for multi-objective linear programming under uncertainty”, European Journal of Operational Research, 26(1), pp. 65-82, 1986.
J. Teghem and P. Kunsch, “Application of multiobjective stochastic linear programming to power systems planning”, Engineering Costs and Production Economics, 9, pp. 83-89, 1985.
V. Vahidinasab and S. Jadid, “Stochastic multiobjective self-scheduling of a power producer in joint energy & reserves markets”, Electric Power Systems Research, 80(7), pp. 760-769, 2010.
W. Zheng, J. Xiao-Ping, and Sh. Lei, “Optimization of multi-product batch plant design under uncertainty with environmental considerations”, Clean Techn Environ Policy. Springer-Verlag, 2009.
J.R. Birge and F. Louveax, “Introduction to stochastic programming”, Springer. New York (NY), 1997.
P. Kall and S.W. Wallace, “Stochastic programming”, John Wiley & Sons Inc. New York (NY), 1994.
A. Prekopa, “Stochastic programming”, Kluwer Academic Publishers, 1995.
J. vonNeumann and O. Morgenstern, “Theory of games and economic behavior”, Princeton University Press, Princeton (NY), 1953.
K. Tammer, “Relations between stochastic and parametric programming for decision problems with a random objective function”, Optimization, 9(4), pp. 523-535, 1978.
B.O. Bereanu, “On stochastic linear programming i: Distribution problems, a single random variable”, Revue Roumaine de Mathe atiques Pures et Appliqueés, 8(4), pp. 683-697, 1963.
S. Kataoka, “On stochastic programming II: A preliminary study on a stochastic programming model”, Hitotsubashi Journal of Arts Sciences, 2, pp. 36-44, 1963.
I.M. Stancu-Minasian and S. Tigan, “The vectorial minimum risk problem”, Proceedings of the Colloquium on Approximation and Optimization, Cluj-Napoca, Romania, pages 321-328, 1984.
M.S. Bazaraa, J.J. Goode, and C.M. Shetty, “Constraint qualifications revisited”, Management Science, 18, pp. 567-573, 1972.
P. Kall, “Stochastic linear programming”, Springer-Ver- lag. Berlin, 1972.
R. Caballero, E. Cerdà, M.M. Mu?oz, and L. Rey, “Stochastic approach versus multiobjective approach for obtaining efficient solutions in stochastic multiobjective programming problems”, European Journal of Operational Research, 158, pp. 633-648, 2004.
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 & Applications, (110)1, pp. 53-74, 2001.
S.P. Canto, “Application of benders' decomposition to power plant preventive maintenance scheduling”, European Journal of Operational Research, 184(2), pp. 759- 777, 2011.
I. Deák, “Two stage stochastic problems with correlated normal variables: Computational experiences”, Annals of Operations Research, 142(1), pp. 79--97, 2006.
J.L. Higle and S. Sen, “Stochastic decomposition”, Kluwer Academic Publishers, 1996.
A. Charnes and W.W. Cooper, “Deterministic equivalents for optimizing and satisfying under chance constraints”, Operations Research, 11, pp. 18-39, 1963.
J. Linderoth, A. Shapiro, and S. Wright, “The empirical behaviour of sampling methods for stochastic programming”, Annals of Operations Research, 142, pp. 215-241, 2006.
G.Ch. Pflug, “Optimization of stochastic models”, Kluwer Academic Publishers., 1996.
G.Ch. Pflug, “Optimization of stochastic models, pp. The interface between simulation and optimization”, Kluwer Academic Publishers., 1996.
J. Castro, “A stochastic programming approach to cash management in banking”, European Journal of Operational Research, 192(3), pp. 963-974, 2009.
B. Aouni and D.L. Torre, “A generalized stochastic goal programming model”, Applied Mathematics & Computation, 215, pp. 4347-4357, 2010.
E. Ballestero, I. Gonzalez, A. Benito, and M. Bravo, “Stochastic goal programming approach to solve a portfolio selection with multiple uncertain scenarios”, Mopgp ‘06, 7th Conference on Multi-objective Programming and Goal Programming, Toures, France, 2006.
A. Chen, J. Kim, S Lee, and Y. Kim, “Stochastic multi- objective models for network design problem”, Expert Systems with Application, 3792, pp. 1608-1619, 2010.
R. Minciardi, M. Robba, and R. Sacile, “Decision models for sustainable groundwater planning and control”, Control Engineering Practice, 15(8), pp. 1013-1029, 2007.
M.M. Mu?oz and F. Ruiz, “ISTMO, pp. An interval reference point-based method for stochastic multiobjective programming problems”, European Journal of Operational, 197, pp. 25-35, 2009.
L. Paquete and T. Stützle, “Design and analysis of stochastic local search for the multiobjective traveling salesman proble”, Computers and Operations Research, 36(9), pp. 2619-2631, 2009.
A. Shing, B.S. Minsker, and A.J. Valochi, “An interactive multi-objective optimization framework for groundwater inverse modeling”, Advances in Water Resources, 31(10), pp. 1269-1283, 2008.
M. Mortazavi, “A goal programming model with stochastic goal constraints”, Station #18, Department of Mathematical Sciences, Eastern New Mexico University, Portales, 2003.
B. Aouni, F. Ben Abdelaziz, and J-M. Martel, “Decision- maker’s preferences modelling in the stochastic goal programming”, European Journal of Operational Research, 162(3), pp. 610-618, 2005.
E. Erdo?an and Iyengar, “On two-stage convex chance- constrained problems”, Mathematical Methods of Operations Research, 65(1), pp. 115-140, 2007.
W.K. Klein Haneveld and M.H. van der Vlerk, “Integrated chance constraints: Reduced forms and an algorithm”, Computational Management Science, 3(4), pp. 245-269, 2006.
R. Caballero, E. Cerdà, M.M. Mu?oz, and L. Rey, “Analysis and comparison of some solution concepts for stochastic multiobjective programming problems”, Sociedad de Estadistica Investigacion Operativa Top, 10(1), pp. 101-123, 2002.
J. Teghem and P.L. Kunsch, “Multiobjective decision making under uncertainty: an example for power system”, In Y.Y. Haimes and V. Chankong, editors, Decision making with multiple objectives, pages 443-456. Sprin- ger-Verlag, Berlin, 1985.
F. Ben Abdelaziz, B. Aouni, and R.E. Fayedh, “Multiobjective stochastic programming for portfolio selection”, European Journal of Operational Research, 177, pp. 1811-1823, 2007.
L.F. Escudero, P.V. Kamesam, A.J. King, and R.J-B. Wets, “Production planning via scenario modelling”, Annals of Operations Research, 43(6), pp. 309-335, 2009.
J. Aghaei, H.A. Shayanfar, and N. Amjady, “Joint market clearing in a stochastic framework considering power system security”, Applied Energy, 86(9), pp. 1675-1682, 2009.
N. Geng, Z. Jiang, and F. Chen, “Stochastic programming based capacity planning for semiconductor wafer fab with uncertain demand & capacity”, European Journal of Operational Research, 198, pp. 899-908, 2009.
C.S. Nembou and B.A. Murtagh, “A chance-constrained programming approach to modelling hydro-thermal electricity generation in Papua-New Guinea”, Asia-Pacific Journal of Operational Research, 13(2), pp. 105-114, 1996.
R. Pal and I. Bose, “An optimization based approach for deployment of roadway incident response vehicles with reliability constraints”, European Journal of Operational Research, 198(2), pp. 452-463, 2009.
A. Castillo, T.B. Joro, and Y.Y.C. Li, “Workforce scheduling with multiple objectives” European Journal of Operational Research, 196(1), pp. 162-170, 2009.
P.L. Kunsch, “Application of strange to energy studies”, In R. Slowinski and J. Teghem, editors, Stochastic versus fuzzy approaches to multiobjective mathematical programming under uncertainty. Kluwer Academic Publishers, 1990.
U.K. Bhattacharya, “A chance constraints goal programming model for the advertising planning problem”, European Journal of Operational Research, 192(2), pp. 382-395, 2009.
C. Changchit and P. Terrell, “A multiobjective reservoir operation model with stochastic inflows”, Computers and Industrial Engineering, 24(2), pp. 303-313, 1993.
X. Ji, S. Zhu, S. Wang, and S. Zhang, “A stochastic linear goal programming approach to multistage portfolio management based on scenario generation via linear programming”, IIIE Transactions, 37(10), pp. 957-969, 2005.
A.J. Keown, “A chance-constrained goal programming model for bank liquidity management”, Decision Sciences, 9(1), pp. 93-106, 1978.
A.J. Keown and J.D. Martin, “A chance-constrained goal programming model for working capital management”, The Engineering Economist, 22(3), pp. 153-174, 1977.
M. Tamiz, R. Hasham, s D.F. Jone, B. Hesni, and E.K. Fargher, “A two staged goal programming model for portfolio selection”, Lecture Notes in Economics and Mathematical Systems, 432, pp. 286-299, 1996.
R.A. Howard, “Decision analysis, pp. Practice and promis”, Management Science, 34(6), pp. 679-695, 1998.
M.W. Kirby, “Paradigm change in operations research: Thirty years of debate”, Operations Research, 55(1), pp. 1-13, 2007.
M.K. Luhandjula, “Optimization under uncertainty”, Fuz- zy sets and Systems, 146, pp. 187-203, 2004.
M.K. Luhandjula and A.S. Adeyefa, “Multiobjective problems with fuzzy random coefficients”, Advances in Fuzzy sets & systems, 7(1), pp. 1-16, 2010.