A complementarity model and algorithm for multi-commodity flow supply chain network equilibrium with random demands

Abstract

This research aims to establish a nonlinear complementarity model and method for multi-commodity supply chain network equilibrium problems consisting of manufacturers and retailers in which the demands associated with the retail outlets are random. This work is different from previous research in that it focuses on a condition in which a manufacturer cannot sell all of its products. Based on this, the optimizing behaviour of the various decision-makers is modelled, the equilibrium conditions of the manufacturers and the retailers are derived, respectively, and nonlinear complementarity model of this problem is established. Qualitative properties of the equilibrium pattern in terms of existence, boundedness and global uniqueness are provided under milder conditions. Neither strict complementarity nor global Lipschitz continuity are required, and non-singular assumption is replaced by local error bound. A modified smoothing Levenberg–Marquardt algorithm to solve this model is proposed, and its global convergence and quadratic convergence rate are established, respectively. Finally, the model is illustrated through several numerical examples for which equilibrium prices and product shipments are computed under a variety of scenarios. Numerical results indicate the validity of the model, the availability of the algorithm, as well as the variation characteristics of equilibrium price and product shipment pattern.