Solving Atanassov's I-fuzzy Linear Programming Problems Using Hurwicz's Criterion

ABSTRACT Dubey et al. [40] have shown that solving an Atanassov's I-fuzzy Linear Programming Problem represented by Atanassov's I-fuzzy sets with linear membership and non-membership functions is equivalent to solving an appropriate fuzzy optimisation problem with piecewise linear S-shaped membership functions. The equivalence is established using Hurwicz optimism–pessimism criterion [38] and indeterminacy resolution in Atanassov's I-fuzzy sets. Moreover, in case of convex break points in the piecewise linear membership function, the crisp counterpart of the equivalent optimisation problem involves binary variables. Here, in this paper we first convert the resulting fuzzy optimisation problem having convex break points into an equivalent fuzzy optimisation problem having concave break points on the lines of Inuiguchi et al. [34], before formulating its crisp equivalent. The advantage of this strategy is that the resulting crisp equivalent problem has no binary variables. Further, we also make use of the indeterminacy factor resolution principle to establish a duality relation which can be interpreted as a Atanassov's I-fuzzy variant of the (crisp) weak duality theorem.