Modeling and analysis of material flows in re-entrant supply chain networks using modified partial differential equations

The basic partial differential equation (pde) models for supply chain networks with re-entrant nodes and their macroscopic are proposed. however,through numerical examples,the basic continuum models do not perform well for multiple re-entrant systems. then,a new state equation considering the re-entrant degree of the products is introduced to improve the effectiveness of the basic continuum models. the applicability of the modified continuum models for re-entrant supply chains is illustrated through numerical examples. finally,based on the modified continuum model,numerical examples of different re-entrant degrees are given,meanwhile,the changes in the wip and outflux are analyzed in details for multiple re-entrant supply chain systems. copyright 2011 fenglan he et al.