The Set of Optimal Solutions of Geometric Programming Problem with Max-Product Fuzzy Relational Equations Constraints

In this paper, we consider how to get the set of optimal solutions of geometric programming problem with single-term exponents subject to a system of fuzzy relational equations about max-product composition. The feasible domain of this problem is nonconvex. Firstly, we propose some algorithms to illustrate how to get the set of optimal solutions based on three cases. Secondly, we show that the set of all optimal solutions is fully determined by one maximum optimal solution and a finite number of optimal lower solutions. If we know the optimal value, the maximum optimal solution can be easily computed. However, solving all optimal lower solutions remains as a challenging problem, since finding all the potential minimal solutions of max-product fuzzy relational equations is an NP-hard problem. Finally, four numerical examples are provided to illustrate validity of the proposed method.