A Random Variational Inequality Method For Nash Game Models Solving Supply Chain Problems

In recent years,the supply chain problem in an uncertain environment has gradually attracted the attention of the academic and business circles,especially the operation and coordination of the supply chain.The study of the supply chain structure is a complex composed of suppliers,manufacturers and demand markets.Structure,which includes how the manufacturer allocates the market share of the commodity to the supplier and the supplier obtains the raw material and converts the raw material into a specific commodity.In the process of supply chain optimization,many random factors are the biggest challenges in the system,ensuring that the interests of the participants in the supply chain are the greatest,and it is necessary to match the supply of the upper layer and the needs of the lower layers.This has always been a hot spot in the research field of supply chain problems.Based on the theory of stochastic variational inequality approach,we study the Nash equilibrium model of Manufacturer Supplier game under uncertainty.The main results may be summarized as follows:The chapter 2 establish a two-stage manufacturer supplier game model under uncertainly.This model contain N-suppliers and 1 manufacture.The objective of each suppliers is to maximize his profit,while the manufacture aims to minimize the total cost incurred.At the first stage,the suppliers compete each other on delivery frequencies in other to maximize their share of demand allocation from the manufacture,thereby achieving their maximum profits.At the second stage,the suppliers have to decide how to minimum their until cost of production and unit cost of delivery.First we formulate this problem as a bi-level non-cooperate game under uncertainty.At the upper level,the suppliers compete on delivery frequencies,while at the level,the manufacture determines the demand allocations to the suppers in response to the delivery frequencies given by the suppers.Then we transfer the bi-level non-cooperative game as a stochastic optimization problem with share constraints.In chapter 3,we formulate the stochastic optimization problem with share constraints,which obtained in chapter 2 as a stochastic variational inequality problem.Specifically,first we establish a function space and introduce respond function on it.Then we can deduce a nonanticipativity constraint in the stochastic optimization problem.If the constraint qualification is not emoty we can get the KKT system of the stochastic optimization.This KKT system is a special stochastic variational inequality-stochastic complementarity problem.Since this stochastic variational inequality has a special structure,we can use progressive hedging algorithm to find an equilibrium.In chapter 4,we design a progressive hedging algorithm to find an equilibrium.This algorithm contain two update steps.In this primal update,we makes a projection to the nonanticipativity subspace.In the dual update,we makes a projection to the subspace of nonanticipativity complementary.In chapters,we conduct some numerical experiments to test the efficiency of algorithm.