Min-product fuzzy relation inequalities with absent variables and their weighted max-min optimization in supply chain system

We mainly investigate the min-product fuzzy relation inequalities (FRIs) with arbitrary l absent variables to address the pricing issue and account for sudden factors in the supply chain system. We first define the $l+1$ dimensional path (abbreviated as $(l+1)-D$ path), and provide some properties related to the corresponding solutions. An algorithm based on the $(l+1)-D$ path is proposed to solve the FRIs system. It is verified that the complete solution set of the FRIs system is entirely determined by the unique minimum solution and the finite $(l+1)-D$ path solution. Then we also establish a weighted max-min optimization problem subject to min-product FRIs with arbitrary l absent variables. An algorithm with polynomial time complexity based on the weighted optimal $(l+1)-D$ path is developed to obtain the optimal solution and objective value. More, we further characterize the complete optimal solution set to our established weighted max-min optimization problem. Several numerical examples are presented to illustrate the feasibility and simplicity of our developed algorithm.