Abstract
This chapter describes a Genetic Algorithm (GA) based Fuzzy Goal Programming (FGP) model to solve a Multiobjective Bilevel Programming Problem (MOBLPP) with a set of chance constraints within a structure of decentralized decision problems. To formulate the model, the chance constraints are converted first to their crisp equivalents to employ FGP methodology. Then, the tolerance membership functions associated with fuzzily described goals of the objective functions are defined to measure the degree of satisfaction of Decision Makers (DMs) with achievement of objective function values and also to obtain the degree of optimality of vector of decision variables controlled by upper-level DM in the decision system. In decision-making process, a GA scheme is adopted to solve the problem and thereby to obtain a proper solution for balancing execution powers of DMs in uncertain environment. A numerical example is provided to illustrate the method.
TopIntroduction
In a bilevel programming problem (BLPP), two DMs (leader and follower), are involved at two hierarchical decision levels and each independently controls a vector of decision variables to optimize individual objective functions, where objective functions frequently conflict among themselves in making decision. Actually, it is a special case of multilevel programming problem (MLPP) having multiple objectives in a decentralized decision system.
The modeling aspect of a BLPP was first studied by Fortuny-Amat, & McCarl (1981) and Candler, & Townsley (1982). Then, different methods for BLPP have been presented by Bard (1981), Bialas, & Carwan (1982, 1984) and others.
However, it may be noted that leader’s decision is dominated by follower’s decision is paradoxically raised for the uses of classical approaches. To avoid such situations, fuzzy programming (FP) method (Zimmermann, 1978, 1987), based on fuzzy sets (Zadeh, 1965), was extended (Slowinski, 1986; Hulsurkar, Biswal, & Sinha,1997) to solve problems with imprecise data. The FP approaches to hierarchical optimization problems have also been studied by Tiryaki (2006). But, the difficulty with the use of FP method to BLPP is that re-evaluation of the problem with elicited membership values of objectives to be made repeatedly to reach optimal solution owing to conflict nature of objectives with regard to optimizing them. To avoid such difficulty, FGP (Pal, Moitra, & Maulik, 2003) approaches to hierarchical decentralized problems have been presented by Pal, & Biswas (2007), Pal, & Chakraborti (2013) among others in the area of study.
Now, most of the methodological developments made for BLPPs and MLPPs in the past are mainly concerned with optimization of hierarchical problems with one objective function at each level. But, it is to be observed that most of the hierarchical decision problems involve multiple objectives at each decision level. An iterative method for solving classical MOBLPP has been presented by Shi, & Xia (1997). The use of FGP to solve MOBLPPs have been proposed by Biswas,& Pal (2007), Pal, & Biswas (2007) in the past. But, in contrast to single-objective BLPP (Malhotra, & Arora, 2000), methodologies for solving MOBLPPs are yet to be studied deeply to solve practical problems.
However, in a practical decision situation, it may be mentioned that DM is frequently faced with problem of uncertainty of incorporating values of model parameters, which is inherent to resource utilization in a decision environment.
The most prominent method to solve problems with probabilistic data is stochastic programming (SP). In SP method, the uncertain model parameters are described by random variables rather than crisp description of them (Rao, 1979). Actually, SP is based on theory of probability (Charnes, & Cooper, 1959) and called chance constrained programming (CCP). The field of SP has been studied (Liu, 2000, 2002) extensively and applied to practical problem (Bravo, & Ganzalez, 2009). Again, SP methods for solving real-world optimization problems like economics, industry, military operations have become increasingly important in the current multiobjective decision making (MODM) world.
Key Terms in this Chapter
Chance Constrained Programming: In a stochastic programming approach, certain probability as a chance factor is introduced to an objective and / or a system constraint of a problem to satisfy as constraint in the process of searching solution in an uncertain environment.
Stochastic Programming: In a certain programming environment, the model parameters are found to be random in nature (i.e., not exact) and certain probability distributions of occurrence of various events are considered in modeling and solving problems in an uncertain environment.
Fuzzy Programming: Modeling aspects of optimization problems in which model parameters are defined in an imprecise way owing to inexactness in human judgments as well as inherent impressions in parameters themselves. Here, instead of optimizing the objectives directly, achievements of membership values called grades in the range [0, 1] of the objectives are measured.
Fuzzy Goal Programming: In terms of achieving the highest membership value (unity) as the aspiration levels of the defined fuzzy goals are considered here in contrast to consideration of achieving the aspired levels (called goal levels) of objectives in conventional goal programming approach.
Goal Programming: In a certain programming environment, optimization of a set of objectives is involved there in the decision situation. Here, instead of optimizing them directly, achievement of the assigned target values called aspiration levels of them is considered. In goal programming method, the unwanted deviations (under and / or over) from the aspired levels are minimized in the goal achievement function (objective function) to reach a satisfactory solution in a crisp decision environment.
Multiobjective Bilevel Programming: It is a special field study in mathematical programming for stage-by-stage decision analysis in a hierarchical decision system. Here, more than one objective function is involved with two decision levels for optimizing them by two decision makers.
Multiobjective Programming: A multiplicity of objectives is involved with an optimization problem, where the objectives are generally incommensurable and conflict each other for optimizing them in a decision environment.