Problem 1: An Expected Journey (30 points) Bilbo Baggins is journeying to The Lonely Mountain. During his trip, he will stop at 10 encampments and he can buy any integer number of units of provisions for his journey at each encampment. At any given time, Bilbo can hold at most 15 units of provisions and must have a non-negative quantity of provisions with him at all times, otherwise his quest fails. Table 1 summarizes the per-unit cost of buying provisions at each encampment, which he visits in the order in which they are numbered. Bilbo uses provisions while journeying from one encampment to the next. However, the quantity that he uses depends upon the load that he is carrying with himself. If Bilbo departs an encampment with 6 or fewer units of provisions, he uses 3 units on his way to the next encampment. If he departs an encampment with more than 6 but fewer than 10 units of provisions, he uses 5 units on his way to the next encampment. If he departs an encampment with 10 or more units of provisions, he uses 7 units of provisions on his way to the next encampment. Bilbo would like to minimize the cost of his journey. ISE 3210 - Autumn, 2020 - 1 Table 1: Per-Unit Cost of Provisions at each Encampment Formulate a dynamic optimization problem for Bilbo by identifying its stages, decision variables, state variables, state-transition functions, constraints, and objective-contribution functions. Problem 1: An Expected Journey (30 points) Bilbo Baggins is journeying to The Lonely Mountain. During his trip, he will stop at 10 encampments and he can buy any integer number of units of provisions for his journey at each encampment. At any given time, Bilbo can hold at most 15 units of provisions and must have a non-negative quantity of provisions with him at all times, otherwise his quest fails. Table 1 summarizes the per-unit cost of buying provisions at each encampment, which he visits in the order in which they are numbered. Bilbo uses provisions while journeying from one encampment to the next. However, the quantity that he uses depends upon the load that he is carrying with himself. If Bilbo departs an encampment with 6 or fewer units of provisions, he uses 3 units on his way to the next encampment. If he departs an encampment with more than 6 but fewer than 10 units of provisions, he uses 5 units on his way to the next encampment. If he departs an encampment with 10 or more units of provisions, he uses 7 units of provisions on his way to the next encampment. Bilbo would like to minimize the cost of his journey. ISE 3210 - Autumn, 2020 - 1 Table 1: Per-Unit Cost of Provisions at each Encampment Formulate a dynamic optimization problem for Bilbo by identifying its stages, decision variables, state variables, state-transition functions, constraints, and objective-contribution functions