The real-world decision making situations can be often modeled in the framework of mathematical programming problems that involve multiple, noncommensurable, and conflicting objectives to be considered simultaneously (Kaiser and Messer, 2012, Miettinen, 1999, Steuer, 1985 and Tamiz, 1996). Moreover, the objective functions and/or the constraints usually involve many parameters which are often uncertain. By extending stochastic programming (Birge and Louveaux, 2011, Dantzig, 1955 and Infanger, 2011) to multiobjective programming, goal programming approaches (Contini, 1968 and Stancu-Minasian, 1984) and interactive approaches (Goicoecha et al., 1982, Teghem et al., 1986 and Urli and Nadeau, 2004) were presented. An overview of models and solution techniques for multiobjective stochastic programming problems were summarized in the context of Stancu-Minasian (1990). On the other hand, by considering the vague nature of the DM’s judgments in multiobjective linear programming, fuzzy programming approaches have been studied (Kahraman, 2008, Lai and Hwang, 1994, Li and Hu, 2007, Li and Hu, 2008, Lodwick and Kacprizyk, 2010, Rommelfanger, 1990, Tsuda and Saito, 2010, Verdegay, 2003, Werners, 1987, Zimmermann, 1978 and Zimmermann, 1985).
For the purpose of considering not only fuzziness but also randomness in decision making problems, fuzzy stochastic optimization has been studied (Hulsurkar et al., 1997, Kato et al., 2004, Liu, 2004, Luhandjula, 1996, Luhandjula, 2006, Luhandjula and Joubert, 2010, Luhandjula and Gupta, 1996 and Wang, 2011) together with the development of mathematical basis on fuzzy probability and statistics (Buckley, 2006). In order to handle the situation that the realized values of random variables are fuzzy numbers, fuzzy random programming has been developed (Katagiri and Sakawa, 2011, Katagiri et al., 2005, Katagiri et al., 2005, Katagiri et al., 2004 and Katagiri et al., 2008) by using the concept of fuzzy random variables (Kwakernaak, 1978, Puri and Ralescu, 1986 and Wang and Qiao, 1993). These models handled decision making situations where the realized values of random variables in objective functions and/or constraints become fuzzy sets or fuzzy numbers.
However, we are faced with the situation where the mean of a random variable is estimated as a fuzzy set due to a lack of information. Such a parameter can be represented with a random fuzzy variable (Liu, 2002 and Liu, 2004), instead of a fuzzy random variable.
Recently, a random fuzzy variable draws attention as a new tool for decision making problems under random fuzzy environments, such as random fuzzy programming (Katagiri et al., 2002 and Liu, 2002), portfolio selection (Hasuike, Katagiri, & Ishii, 2009), investment decision (Sakalli & Baykoc, 2010), project selection (Huang, 2007), facility location (Uno, Katagiri, & Kato, 2012) and cooperative bi-level programming (Katagiri, Niwa, Kubo, & Hasuike, 2010).
Under these circumstances, this article tackles a MOLPP with random fuzzy variables. Especially, we focus on the case where the uncertain parameters involved in problems are assumed to be estimated by Gaussian random variables with fuzzy mean, which can be represented with random fuzzy variables. In order to overcome the difficulty of obtaining an optimal solution of the problem due to the nonlinearity caused by simultaneous consideration of randomness and fuzziness, we shall propose a new decision making model in which the original random fuzzy problem can be transformed into an exactly-solvable deterministic MOLPP, using the ideas of possibility theory (Dubois and Prade, 2001 and Zadeh, 1978) and value at risk (VaR) criterion (Jorion, 1996 and Pritsker, 1997) (or fractile criterion Geoffrion, 1967 and Kataoka, 1963) together with techniques of stochastic programming and possibilistic programming.
This paper is organized as follows. Section 2 introduces a definition of random fuzzy variables and provides a fuzzy set-based simple definition. Section 3 formulates a MOLPP with random fuzzy variables and proposes a decision making model optimizing possibilistic values at risk, called pVaR. Furthermore, we show the original random fuzzy MOLPP is transformed into a deterministic nonlinear MOLPP through the proposed model. The proposed model has an advantage that each Pareto optimal solution which is a candidate for a satisficing solution can be solved by convex programming techniques under realistic assumptions. In Section 5, we provide an interactive algorithm in order to obtain a satisficing solution for a DM from among a set of Pareto optimal solutions. After providing a simple numerical example in Section 4, We conclude this paper in Section 5.
