Consider an inventory system, where at the beginning of period k, the inventory level is S_k, and we can order x_k units of goods. The available units of goods are then used to serve a random demand W_k, and the amount of inventory carried over to the next period is S_k+1 = max {0, S_k + x_k W_k}. We assume that S_k, x_k, W_k are non-negative integers, and that the random demand W_k follows the probability distribution

Pr(W_k = 0) = 0.1, Pr(W_k = 1) = 0.7, Pr(W_k = 2) = 0.2 for all k = 1, . . . , N.

The cost incurred in period k is _k (S_k, x_k, W_k) = (S_k + x_k W_k)² + x_k. Furthermore, there is a storage constraint in each period k, which is given by S_k + x_k 2. The terminal cost is given by _N(S_N) = 0.

Now, consider a 2period problem, i.e., N = 3, where we assume that S₁ = 0, and our goal is to find the optimal ordering quantities x₁ and x₂ to minimize the total cost. We assume S_k, x_k and W_k are all nonnegative integers.