Algorithms of discrete optimization and their application to problems with fuzzy coefficients

An approach to solving optimization problems with fuzzy coefficients in objective functions and constraints is described. It consists in formulating and solving one and the same problem within the framework of mutually related models with constructing equivalent analogs with fuzzy coefficients in objective functions alone. It enables one to maximally cut off dominated alternatives “from below” as well as “from above”. Since the approach is applied within the context of fuzzy discrete optimization problems, several modified algorithms of discrete optimization are discussed. These algorithms are associated with the method of normalized functions, are based on a combination of formal and heuristic procedures, and allow one to obtain quasi-optimal solutions after a small number of steps, thus overcoming the computational complexity posed the NP-completeness of discrete optimization problems. The subsequent contraction of the decision uncertainty regions is associated with reduction of the problem to multiobjective decision making in a fuzzy environment with using techniques based on fuzzy preference relations. The techniques are also directly applicable to situations in which the decision maker is required to choose alternatives from a set of explicitly available alternatives. The results of the paper are of a universal character and can be applied to the design and control of systems and processes of different purposes as well as the enhancement of corresponding CAD/CAM systems and intelligent decision making systems. The results of the paper are already being used to solve problems of power engineering.