Design of a quality system for multi-products with the fixed costs for products servicing

In this paper, we design sampling inspections and service capacities simultaneously for multi-products. Particularly, we extend the work of Kim by introducing the fixed cost of providing services. We show that, due to the fixed cost considered, the cost function of a product is no longer linear or convex in terms of the level of service provision, and the total inspection is preferred to the small level of service capacity which results in high burden of the fixed cost. And we develop a simple framework to deal with this joint design problem for a product. Also we consider the problem of allocating the given number of the total service capacities among products. A dynamic programming algorithm is developed to determine the optimal allocation which minimizes the overal total cost of the system and the optimal allocation can be obtained with the considerably smaller computations than the total number of possible allocations. The results can be used to support planning decisions and to aid the joint design of inspections and service capacities for products.