Closed-loop supply chain network equilibrium model and its Newton method

Abstract

Purpose – The purpose of this paper is to develop a closed-loop supply chain (CLSC) network equilibrium model which consists of manufactures, retailers and consumer markets engaged in a Cournot pricing game with heterogeneous multi-product. Design/methodology/approach – The authors model the optimal behavior of the various decision makers and CLSC network equilibrium, and derive the equilibrium conditions based on variational inequality approach. The authors present a new Newton method to solve the proposed model. Findings – The authors find that the algorithm converges to the solution rapidly for most cases. Besides, the authors discuss the effect of some parameters on the equilibrium solution of the model, and give some insights for policy makers, such as improving the technology level of the manufacturer, reducing the cost of waste disposal and increase the minimum ration of used product to total quantity. Originality/value – The authors derive the network equilibrium conditions by the variational inequality formulation in order to obtain the computation of the equilibrium flows and prices. The authors present a new Newton method to solve the proposed model. The authors discuss the effect of some parameters on the equilibrium solution of the model, and give some managerial insights