可替代产品库存模型的研究

市场上，很多产品之间可相互替代，某种产品缺货时，可用另一种产品替代，也可以重新进货以满足顾客的需求。我们的目的是：从销售商的角度，讨论这两个因素对库存策略的影响。我们建立了这类问题有两个产品的单调期的利润最大化模型。证明了问题的解的存在性，给出了目标函数是凹函数和子模函数的充分条件，讨论了求解的方法和各参数对库存的影响。通过对几种特殊情况的讨论和比较，证明了替代和再订货可以提高利润并且可减少库存总量。

One-period-two-product Inventory Model with substitution and reorder

In the market,many same or products of similar nature can substitue for each other,and excess demand for products can also be met by reorder.Our purpose is to find how the substitution and reorder influence inventory from the retailer's viewpoint.For one-period-two-product inventory of this kind with stochastic demands and proportional costs and revenues,we develop the profit maximixation model.We show that the objective function is bounded,and therefore the optimal policy exists.We present the sufficient conditions for the objective function to be concave and submodular.We discuss the optimal conditions for order quantity and provide some properties of the optiml order quantities.Through the discussion of special cases,we prove that both substitution and reorder can bring profits and reduce the general stock level.