A polymorphic uncertain equilibrium model and its deterministic equivalent formulation for decentralized supply chain management

Supply chain management is a multidisciplinary engineering problem. In this paper, a polymorphic uncertain equilibrium model (PUEM) is constructed to capture the joint maximization of the profits for the manufacturers and retailers in a supply chain network. To ensure applicability of the model in practice, the demand of the consumers is regarded as a continuous random variable, the holding cost of the retailer and the transaction cost between the manufacturer and retailer are described by fuzzy sets. For the PUEM, a deterministic equivalent formulation (DEF) is first derived by compromise programming approach such that the existing powerful algorithms in the standard smooth optimization are employed to find an approximate equilibrium point for the uncertain problem. Actually, the DEF turns out to be a nonlinear complementarity problem (NCP), a special variational inequality. Thus, a modified partially Jacobian smoothing algorithm is developed to solve the corresponding NCP, where the gradient information of the model is used to efficiently generate search direction. Sensitivity analysis offers a number of useful managerial implications based on practical applications of the model.