Optimal inventory control policies of a two-stage push–pull production inventory system with lost sales under stochastic production, transportation, and external demand

An analytical model based on Markov processes is proposed for the analysis of a linear, horizontally integrated, two stages, push–pull inventory system. Uncertainty about both supply and demand is taken into consideration. Exponentially distributed lead times, compound Poisson external demand and lost sales are assumed. An algorithm that creates the infinitesimal generator matrix of the system is developed and an exact numerical solution of the system performance measures is also provided. The proposed model can be either used to evaluate what if scenarios exploring the behavior of the system or to optimize performance measures of the considered system. As an example, the model is used to analyze and get insights of the behavior of a supply–demand balanced system.