Finding The Solution Of The Budget Constrained Multi-product Newsboy Problem With Reactive Production Via Homotopy Method

The classical newsboy problem is a single period stochastic inventory model. The original work in this area can be traced back to Harris[l] in 1913. At that time, Harris presented a formula for the optimal economic order quantity(EOQ) in simple inventory models. Afterwards, much more attention was paid to the character of the single period product, and this problem was called single-period problem(SPP) or newsboy problem(NP). In section 2, we will look back this classical model, which minimizes the expected cost function of each product. In the past several decades there are many extensions to this initial model.. Among them, Hadley and Whitin[2] developed the constrained newsboy problem in 1963. Many papers have been published in this direction, see[4,5,6]. Khouja[3] classified the extensions into 11 categories in 1999. The newsboy problem with reactive production is not in the 11 extensions. Chung and Flynn[4] first developed this extension to the newsboy problem in 2001. They divided the production into two stages, an anticipatory stage and a reactive stage. But there are two aspects of localization in their paper [4], only one product is concerned and there are no budget constraints to the production. In today's supply chain management, the most frequent instance we have to face are the newsboy problem which contains multiple products, and the budget have to be constrained. If the products are seasonal products with a long-selling season and highly volatile stochastic demand, we have to divide the production into two parts, anticipatory stage and a reactive stage. So we extend the newsboy problem to deal with this situation.In this paper we will consider the newsboy model which contains multiple products, the production occurs in two stages, an anticipatory stage and a reactive stage. The budgets in both stages have to be constrained. It can be explained as follows. There are several products which are all the seasonal ones, and each one has a highly volatile stochastic demand. The demand of each product is very high at the peak period, and is negligible at the off-peak period. Chung and Flynn said that if this situation was solved with the classical newsboy problem model, it will lead to a low service level or a highly volatile demand. Both of these two results are not what we want to see. So Chung and Flynn modified the classical model, and introduced two production periods, which are the anticipatory stage and the reactive stage. In the real world, the budgets in both of the two production periods should be constrained. Our main work is to find that whether the optimal order quantities for each product in the two stages exist or not. To solve our problem, we can follow the classical procedure which is used to solve the constrained multi-product newsboy problem. But we find it is too complex to get the exact solution of our model. So we introduce a new method, the homotopy method, to deal with our problem. We illustrate our model in both the classical procedure and the homotopy method. Comparing the two methods, we can see that the homotopy method is more exact and efficient.This paper is organized as follows.In Section 2, we give a brief review of the Newsboy Problem. We will uniform the formula into the same parameter, and give the solution procedure.In Section 3, we give the model formulation of our problem. We use the classical procedure and the combined homotopy interior point method to deal with our problem. We will compare the two procedures, and find which one is better.In Section 4, we give a numerical example to illustrate our solution procedure.In Section 5, we draw the conclusion to our problem.