Problem 1. (30 points) Consider an inventory system, where at the beginning of period k, the inventory level is Sk, and we can order xk units of goods. The available units of goods are then used to serve a random demand Wk, and the amount of inventory carried over to the next period is Sk+1=max{0,Sk+xkWk}. We assume that Sk,xk,Wk are non-negative integers, and that the random demand Wk follows the probability distribution Pr(Wk=0)=0.1,Pr(Wk=1)=0.7,Pr(Wk=2)=0.2forallk=1,,N. The cost incurred in period k is k(Sk,xk,Wk)=(Sk+xkWk)2+xk. Furthermore, there is a storage constraint in each period k, which is given by Sk+xk2. The terminal cost is given by N(SN)=0. Now, consider a 2 -period 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 Sk,xk and Wk are all nonnegative integers