About some methods for solving fuzzy linear programming problems

Abstract

The article provides an overview of approaches to solving various types of fuzzy linear programming problems, including those with many objective functions. Classical linear programming problems belong to deterministic decision-making models. However, in conditions of uncertainty, which has the character of fuzziness (vagueness), it is advisable to use approximate models for representing information in the form of fuzzy sets. The article presents formulations of the main types of fuzzy mathematical programming problems. The main attention is paid to problems in which approximate inequalities are used in the restrictions, and the target function can be transferred to the category of restrictions. One of the most well-known methods for solving problems with fuzzy goals and restrictions is the Bellman-on-Zade approach, according to which the solution is the maximum point of the lower envelope of the intersection of fuzzy goals and fuzzy restrictions. Zimmerman’s approach consists in moving to a special lambda problem, in which the parameter determines the degree of admissibility of the found optimal solution. For a problem with a clear objective function in which the constraints are partially or completely fuzzy, two approaches are considered, one of which is to define a fuzzy set of solutions, and the second is to define a crisp set of “maximizing solutions”. The article also considers a problem with clear constraints and several fuzzy goals. There are several approaches to finding its solution, but of particular interest is the approach that takes into account the importance of goals based on weighting coefficients. The method allows for the most important purposes to obtain a greater value of estimates of the achieved level. Most of the approaches considered are accompanied by illustrative examples. The approaches to solving fuzzy linear programming problems discussed in the article can be used to find the optimal solution when solving various applied problems under conditions of fuzzy uncertainty (decision making, planning, control, etc.).