Centralized effect on expected costs in a multi-location newsboy problem

This is a single-period single-product inventory model with several individual sources of demand. It is a multi-location problem with an opportunity for centralization. The holding and penalty cost functions, at each location, are assumed to be linear and identical. Two types of inventory system are considered in this paper: The decentralized system and the centralized system. The decentralized system is a system in which a separate inventory is kept to satisfy the demand at each source of demand and there is no reinforcement between locations; i.e., the surplus supplied location is not allowed to supply the deficient supplied location. Therefore, the total expected cost of the decentralized system is the sum of the expected cost of individual locations, as the holding cost or penalty cost of each location depends only on its own inventory level and is unrelated to any other location’s inventory level. The centralized system is a system in which the surplus location is allowed to supplement the deficient one by transportation. Therefore, the holding cost or penalty cost is calculated by the net surplus or net deficient after the reinforcement. The total expected cost of the centralized system should therefore include the expected holding cost, the expected penalty cost and the expected transportation cost. Let nh>0,np>0 and nt≥0 be the unit holding cost, the unit penalty cost and the unit transportation cost, respectively. This paper demonstrates that, for any probability distribution of a location’s demand, the following two properties are always true: (1)ℝ17