Formulate and solve a discrete stochastic model for a single product with lead time zero. The storage and shortage costs are independent of time. The set-up cost is constant

In operations research, a discrete stochastic model for inventory management is commonly employed to determine optimal ordering policies under uncertainty. In this case, we focus on a single product with zero lead time, and we assume that the storage and shortage costs are independent of time, while the set-up cost is constant. The goal is to formulate this problem and solve it for optimal decision-making.

Problem Formulation

Let's consider a scenario where the inventory level fluctuates over time due to stochastic demand, and we need to decide when to place an order for replenishment and how much to order. The key components of the model are as follows:

1. Demand: The demand for the product is random and follows a discrete probability distribution. Let the demand during each period be represented by a random variable DtD_t, where tt refers to the time period.

2. Inventory Level: The inventory level at the beginning of each period is denoted by ItI_t. The inventory level evolves according to the following equation:

   It=It−1+Qt−1−DtI_t = I_{t-1} + Q_{t-1} - D_t

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