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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 & Cruz Jr., 1973), and they are formulated as two-level mathematical programming problems (Shimizu et al, 1997; Sakawa & 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 & Karwan, 1984), the Kuhn-Tucker approach (Bard & Falk, 1982; Bard & Moore, 1990; Bialas & Karwan, 1984; Hansen et al, 1992), and the penalty function approach (White & Anandalingam, 1993). The subsequent works on two-level programming problems under noncooperative behavior of the decision makers have been appearing (Nishizaki & Sakawa, 1999; Nishizaki & Sakawa, 2000; Gumus & Floudas, 2001; Nishizaki et al., 2003; Colson et al., 2005; Faisca et al., 2007) including some applications to aluminium production process (Nicholls, 1996), pollution control policy determination (Amouzegar & Moshirvaziri, 1999), tax credits determination for biofuel producers (Dempe & Bard, 2001), pricing in competitive electricity markets (Fampa et al, 2008), supply chain planning (Roghanian et al., 2007) and so forth. However, processing time of solution methods for noncooperative two-level linear programming problems, for example, Kth Best method by Bialas et al. (1982) and Branchand-Bound method by Hansen et al. (1992), may exponentially increases at worst as the size of problem increases since they are strict solution methods based on enumeration. In order to obtain the (approximate) Stackelberg solution with a practically reasonable time, approximate solution methods were presented through genetic algorithms (Niwa et al., 1999) and particle swarm optimization (PSO) (Niwa et al., 2006).

Stackelberg Solutions to Noncooperative Two-Level Nonlinear Programming Problems through Evolutionary Multi-Agent Systems