In the real world, we often encounter situations where there are two or more decision makers in an organization with a hierarchical structure, and they make decisions in turn or at the same time so as to optimize their objective functions. Decision making problems in decentralized organizations are often modeled as Stackelberg games (Simaan and Cruz, 1973), and they are formulated as two-level mathematical programming problems (Shimizu et al., 1997 and Sakawa and Nishizaki, 2009). In the context of two-level programming, the decision maker at the upper level first specifies a strategy, and then the decision maker at the lower level specifies a strategy so as to optimize the objective with full knowledge of the action of the decision maker at the upper level. In conventional multi-level mathematical programming models employing the solution concept of Stackelberg equilibrium, it is assumed that there is no communication among decision makers, or they do not make any binding agreement even if there exists such communication.
Computational methods for obtaining Stackelberg solutions to two-level linear programming problems are classified roughly into three categories: the vertex enumeration approach (Bialas and Karwan, 1984), the Kuhn–Tucker approach (Bard and Falk, 1982, Bard and Moore, 1990, Bialas and Karwan, 1984 and Hansen et al., 1992), and the penalty function approach (White and Anandalingam, 1993). The subsequent works on two-level programming problems under noncooperative behavior of the decision makers have been appearing (Colson et al., 2005, Faisca et al., 2007, Gümüs and Floudas, 2001, Nishizaki and Sakawa, 2000 and Nishizaki et al., 2003) including some applications to aluminium production process (Nicholls, 1996), pollution control policy determination (Amouzegar and Moshirvaziri, 1999), tax credits determination for biofuel producers (Dempe and Bard, 2001), pricing in competitive electricity markets (Fampa et al., 2008), supply chain planning (Roghanian et al., 2007) and so forth.
In order to deal with multiobjective problems (Sakawa, 1993 and Sakawa, 2001) in hierarchical decision making, two-level multiobjective linear programming problems were formulated and a computational method for obtaining the corresponding Stackelberg solution was also developed (Nishizaki and Sakawa, 1999). Considering stochastic events related to hierarchical decision making situations, on the basis of stochastic programming models, two-level programming problems with random variables were formulated and algorithms for deriving the Stackelberg solutions were developed (Nishizaki et al., 2003). Furthermore, considering not only the randomness of parameters involved in objective functions and/or constraints but also the experts’ ambiguous understanding of realized values of the random parameters, fuzzy random two-level linear programming problems were formulated, and computational methods for obtaining the corresponding Stackelberg solutions were also developed (Sakawa and Katagiri, 2012, Sakawa and Kato, 2009, Sakawa et al., in press and Sakawa et al., 2011).
From a viewpoint of ambiguity and randomness different from fuzzy random variables (Kwakernaak, 1978, Puri and Ralescu, 1986 and Wang and Qiao, 1993), by considering the experts’ ambiguous understanding of means and variances of random variables, a concept of random fuzzy variables was proposed, and mathematical programming problems with random fuzzy variables were formulated together with the development of a simulation-based approximate solution method (Liu, 2002).
Under these circumstances, in this paper, assuming noncooperative behavior of the decision makers, we formulate random fuzzy two-level linear programming problems. To deal with the formulated two-level linear programming problems involving random fuzzy variables, we assume that the decision makers concerns about the probabilities that their own objective function values are smaller than or equal to certain target values. By considering the imprecise nature of the human judgments, we introduce the fuzzy goals of the decision makers for the probabilities. Then, assuming that the decision makers are willing to maximize the degrees of possibility with respect to the attained probability, we consider the possibility-based probability model for random fuzzy two-level programming problems. Extended concepts of Stackelberg solutions are introduced. Computational methods for obtaining approximate Stackelberg solutions through particle swarm optimization are also presented. An illustrative numerical example demonstrates the feasibility and efficiency of the proposed method.
