A STOCHASTIC VARIATIONAL INEQUALITY APPROACH TO THE NASH EQUILIBRIUM MODEL OF A MANUFACTURER-SUPPLIER GAME UNDER UNCERTAINTY

We study a two-stage game model of multi-manufacturers and multi-suppliers in supply-chain optimization, where N suppliers compete to provide M products for manufacturers in an environment of uncertainty. At the first stage (the contracting stage) the suppliers compete on the basis of delivery frequency to the manufacturers, while at the second stage (the production stage) the suppliers minimize their respective production costs, which could be influenced by other suppliers' decisions and the uncertainty in the market and their production processes. We formulate this problem as a two-stage quadratic stochastic Nash game. The Nash game is converted to a possibly nonmonotone stochastic variational inequality problem. A recently developed progressive hedging algorithm is proposed for finding a Nash equilibrium of the game. This algorithm is particularly suitable for stochastic variational inequality problems due to its decomposable structure in terms of the scenarios. Numerical results are presented to show the effectiveness of the proposed algorithm and to provide suggestions on the choice of the algorithmic parameters.