An Approximation Approach for Representing S-Shaped Membership Functions

In general, to formulate a fuzzy-linear-programming problem with
n
S-shaped utility (membership) functions, traditional methods require
n
or more extra binary variables because S-shaped curves are neither convex nor concave in all places. Adding binary variables does not improve the bound of the linear-programming relaxation. On the contrary, added binary variables increase the computational burden in the solution process if problems get large. Therefore, a formulation without binary variables should be more efficient. Accordingly, this study proposes a piecewise-linear approach to formulate an S-shaped membership function (MF) without adding any extra binary variables, which improves the efficiency of fuzzy-linear programming in solving decision/management problems with S-shaped MFs. Finally, a computational experiment is provided to demonstrate the superiority of the proposed models. An illustrative example is also provided to show the usefulness of the proposed method.