Problem 3 (Grain Storage in the Winter, 20 pts) An attack is expected at time t +1 on the small villages surrounding the city of Winterfell. The villagers are expected to flee and take refuge in Winterfell (they do not have the ability to bring food with them). Therefore, Sansa, the leader of Winterfell, is concerned that once this happens, Winterfell will not have enough food to support the extra population. Her plan is to preemptively ship grain from the villages before the attack happens. Grain at timet Winterfell Villages Population: Grain: k People seeking protection at time t+1 The villages have a total population of k people and k units of grain. At time t, Sansa has to decide to ship x4 amount of grain to Winterfell, leaving k x; in the villages. D++1 ~ U(k/2, k) (continuous uniform distribution) people will flee to Winterfell, leav- ing a population of k D++1 remaining in the villages. Each person needs to eat 1 unit of grain. How much grain should she ship to Winterfell to maximize the expected total number of people that are fed (i.e., including people who fled and those who did not flee)? Hint: follow the process we used to solve the newsvendor problem. First define an appropriate G(X+, D4+1) and then we want to maximize E G(X+, D4+1) Problem 3 (Grain Storage in the Winter, 20 pts) An attack is expected at time t +1 on the small villages surrounding the city of Winterfell. The villagers are expected to flee and take refuge in Winterfell (they do not have the ability to bring food with them). Therefore, Sansa, the leader of Winterfell, is concerned that once this happens, Winterfell will not have enough food to support the extra population. Her plan is to preemptively ship grain from the villages before the attack happens. Grain at timet Winterfell Villages Population: Grain: k People seeking protection at time t+1 The villages have a total population of k people and k units of grain. At time t, Sansa has to decide to ship x4 amount of grain to Winterfell, leaving k x; in the villages. D++1 ~ U(k/2, k) (continuous uniform distribution) people will flee to Winterfell, leav- ing a population of k D++1 remaining in the villages. Each person needs to eat 1 unit of grain. How much grain should she ship to Winterfell to maximize the expected total number of people that are fed (i.e., including people who fled and those who did not flee)? Hint: follow the process we used to solve the newsvendor problem. First define an appropriate G(X+, D4+1) and then we want to maximize E G(X+, D4+1)