This work considers the distribution of goods with stochastic shortages from factories to stores. It is assumed that in the process of shipping the goods to various stores, some proportion of the goods will be damaged (which will lead to shortage of goods in transit). The cost of the damaged goods is added to the cost of the shipment. A proportion of the total expected cost of the shortage goods is assumed to be recovered and should be deducted from the total cost of the shipment. In order to determine the minimum transportation costs for the operation, we adopt dynamic optimization principles. The optimal transportation cost and optimal control policies of shipping the goods from factories to stores were obtained. We find that the optimal costs of the goods recovered could be determined. It was further found that the optimum costs of distributing the goods with minimum and maximum error bounds coincide only at infinity. 1. Introduction A distribution company plans to minimize the cost of distributing kinds of products from number of factories to number of stores. For the case of a company distributing a particular product from one factory to all the stores (a single product), that is, at factory , product is distributed to all the stores and factory produce products and are distributed to all the stores, and so on; see Nwozo and Nkeki [1]. It is also expected that the goods that leave the factories will not come back to the factories (in the case of defective, damaged, etc., items). The company considered number of control policies to determine which of them will yield the optimum control policy. They also estimated that certain percentage of the products are to reach their final destination successfully at minimum costs. It is assumed that some proportion of the shortage goods should be recovered. The recovered costs of the shortage goods should be deducted from the total costs continuously over time horizon. In the related literature, Powell [2] considered the problem of a stochastic fleet assets management problem and used the postdecision state variable implicitly to determine this problem. Closely related to this on the extensive work on stochastic fleet assets management problem is the work by Cheung and Powell [3]. Godfrey and Powell [4] further used postdecision state variable explicitly to determine similar problem on a stochastic fleet management problem. Van Roy et al. [5] proposed the idea of using a parsimonious sufficient statistics in an application of approximate dynamic programming to inventory management. Mulvey and Vladimirou [6]
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