Optimal order decisions for product components considering after-sales service

In a productive system with new generation and old generation product, one hand, producer faces the needs for new components, and the other hand, also needs consider the after-sales service needs for components of sold products. The after-sales service needs can be satisfied by ordering the old components, and also by new components. For determining the optimal order quantities for new and old components, a single period stochastic inventory model was formulated. The uniqueness and existence of optimal decisions were proved, and the conditions that should satisfy were given. And moreover, some properties for optimal decisions were obtained.