A Class of Expected Value Bilevel Programming Problems with Random Coefficients Based on Rough Approximation and Its Application to a Production-Inventory System

and Applied Analysis 3 CS(D i , x i ): total costs including shortage costs and storage costs for item i P(D i , x i ): profit for item i. For the ith item, some assumptions are introduced as follows. (i) The time horizon is infinite. (ii) The production rate is instantaneous. (iii) The lead time is zero. (iv) There is no fixed ordering cost and there is no initial inventory. (v) The demandD i of the ith item in a cycle is a normally distributed random variable with a mean μ i and a variance σ i . (vi) The shortage cost depends on shortage level, that is, for the ith item, the shortage cost isp i ⋅max{0, D i −x i }. (vii) The holding cost is dependent on the inventory, that is, for the ith item, the holding cost is h i ⋅max{0, x i − D i }. 2.2. Model Formulation. As shown in Figure 1, the whole decision process includes two sections for some marketoriented commodities. As the upper level, the retailer has to consider the optimal inventory level to satisfy customer’s demand and achieve the maximal profit. However, it is scarcely possible to accurately estimate the customer’s demand, especially in some holidays. Hence, the market information usually impacts the retailer’s decision on how much the inventory level should be and further impacts her/his order quantity. As the lower level, the producer has to make the production plan according to the retailers’ feedback due to the fierce competition. The optimal production quantity satisfying the retailer’s demand and some resource limitations should be considered. To quickly grab the market share, the producer usually considers theminimal production as the first objective. Above all, the retailer and producer alternatively interactwith each other by transfering the inventory and production information. Therefore, a bilevel model, in which the production is considered the lower level and the retailer is considered the lower level, is mathematically formulated to find the optimal production and inventory strategy. 2.2.1. Lower Level. Production should satisfy the following constraints and achieve the following objective. (i) All producers share a common resource: