Fuzzy ideal cone: A method to obtain complete fuzzy non-dominated set of fuzzy multi-criteria optimization problems with fuzzy parameters

Abstract

This paper is the first which attempts to capture complete fuzzy non-dominated set of fuzzy multi-criteria optimization problems with fuzzy parameters. A proper mathematical formulation of fuzzy non-dominated set and its generation are primary aims of the proposed study. Present work is mainly focused on visualizing the considered problem from fuzzy geometrical viewpoint. Constraint set of a fuzzy multi-criteria optimization problem is viewed from two different spaces - decision space and criterion space. Fuzzy decision feasible region or the constraint set on decision space is formulated using the alpha-cuts of the parameters present in the problem. Under the assumption that fuzzy criteria are fuzzy number valued, it is shown that a fuzzy point will be obtained on the criterion space corresponding to each point on the decision feasible region. Union of all these fuzzy points determines fuzzy criteria feasible region or constraint set in the criterion space. To capture entire fuzzy non-dominated set of criteria feasible region, a method, hereby named fuzzy ideal cone method, has been proposed. The method essentially uses the cone of non-positive hyperoctant of the criteria space to generate complete fuzzy non-dominated set. Proposed methodology is supported by several numerical examples and pictorial illustrations.