A general concept for solving linear multicriteria programming problems with crisp, fuzzy or stochastic values

Abstract

For modelling imprecise data the literature offers two different methods: either the use of probability distributions or the use of fuzzy sets. In our opinion these two concepts should be used in parallel or in combination, dependent on the real situation. Moreover, in many economic problems, the well-known probabilistic or fuzzy solution procedures are not suitable because the stochastic variables do not have a simple classical distribution and the fuzzy values are not fuzzy intervals. For example, in case of investment problems the coefficients may often be described by means of more complex distributions or more general fuzzy sets. In this case we propose to distinguish several scenarios and to describe the parameters of the different scenarios by fuzzy intervals. For solving such stochastic linear programs with fuzzy parameters we propose a new method, which retains the original objective functions dependent on the different states of nature. It is based on the integrated chance constrained program introduced by Klein Haneveld [On integrated chance constraints, in: Gargnano (Ed.), Stochastic Programming, Springer, Berlin, 1986, pp. 194–209] and the interactive solution process FULPAL, see Rommelfanger [Fuzzy Decision Support-Systeme - Entscheiden bei Unschärfe, second ed., Springer, Berlin, Heidelberg, 1994; FULPAL: an interactive method for solving multiobjective fuzzy linear programming problems, in: R. Slowinski, J. Teghem (Eds.), Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty, Reidel Publishing Company, Dordrecht, 1990, pp. 279–299; FULPAL 2.0—an interactive algorithm for solving multicriteria fuzzy linear programs controlled by aspiration levels, in: D. Scheigert (Ed.), Methods of Multicriteria Decision Theory, Pfalzakademie Lamprecht, 1995, pp. 21–34; The advantages of fuzzy optimization models in practical use, Fuzzy Optim. Decision Making 3 (2004) 295–310] and Rommelfanger and Slowinski [Fuzzy linear programming with single or multiple objective functions, in: R. Slowinski (Ed.), Fuzzy Sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic Publishers, Norwell, MA, 1998, pp. 179–213]. An extensive numerical example illustrates the efficiency and the generality of the proposed new method.