This chapter provides a literature review of optimization problems in the context of grey system theory, as proposed by various authors. The chapter explains the binary interactive algorithm approach as a problem-solving method for linear programming and quadratic programming problems with uncertainty and a genetic-algorithm-based approach as a second problem-solving scheme for linear programming, quadratic programming, and general nonlinear programming problems with uncertainty. In the chapter, details on the computation procedures involved for solving the aforementioned optimization problems with uncertainty are presented and results from these two approaches are compared and contrasted. Finally, possible future work area in the subject is suggested.
TopIntroduction
In traditional deterministic optimization problems, it is assumed that all the relevant data to the problem are known. However, this assumption is not always realistic. In many practical situations, it is required to deal with optimization problems in which the relevant data to the problem are not fully available and decisions have to be made in the face of uncertainty.
Several methods for dealing with uncertainty in optimization problems have been thoroughly studied so far. These methods mainly relate to stochastic optimization and fuzzy optimization.
Stochastic optimization methods are based on probability theory; and they are useful to deal with the uncertainty in input parameters when their probability distributions are known. The major problems of these methods are the difficulty in the availability of probability distributions of model parameters and the computational difficulty in implementation, since the methods may lead to large or complicated intermediate models (Huang, 1994).
Fuzzy optimization methods, derived from fuzzy set theory, contain two major categories: fuzzy possibilistic programming (FPP) and fuzzy flexible programming (FFP) (Inuiguchi et. al. 1990 as cited in Huang, 1994). In FPP methods, fuzzy parameters are introduced into the modelling frameworks, which represent the fuzzy regions where the parameters possibly lay (Zadeh 1978 as cited in Huang, 1994). The major problems with the FPP methods are the difficulty in obtaining the possibility information and the computational difficulty when applied to practical problems. In FFP methods, the flexibilities in the constraints and fuzziness in the system objective are expressed as fuzzy sets with their membership grades corresponding to the degrees of satisfaction of the constraints/objective (Tanaka et al. 1974; Zimmermann 1985 as cited in Huang, 1994). The major problem with the FFP methods is that only the stipulation uncertainties are reflected. In addition, the methods are indirect approaches containing intermediate control parameters which may be difficult to determine by certain criteria (Inuiguchi et. al. 1990 as cited in Huang, 1994).
The concept of grey systems and grey decisions are introduced into the existing optimization approaches and grey optimization models (Huang et al. 1992) are developed in order to address the aforementioned limitations.