Problem 2. An elementary school in Lexington has to come up with a replacement policy for its minibus over the next 6 years. A new minibus should be kept in service for at least for 2 years (unless it is bought at the beginning of the last period, in which case it can be sold at the end of that period). Moreover, the school cannot keep a minibus that is older than 5 years (if you enter a period with a 5-year-old bus, then you have to sell it at the beginning of that period). The price of a new minibus in period 1 is $50,000, and it increases at rate of 5% per year. The Salvage value of a minibus -as a function of its age- is given by the following table: Age(year) 1 2 3 4 5 6 or more Salvage value (in thousands of $) 30 25 20 18 18 0 The per year maintenance cost of a minibus -as a function of its age- is given by the following table: Age(year) 0 1 2 3 4 5 or more Maintenance in thousands of $) 0.5 1 1 2. 3 10 where age O means a new minibus. Assume that the School has to buy a new minibus at the beginning of period 1. For periods 2 through 6, the decision to keep or to replace the minibus is made at the beginning of every period. At the beginning of period 7, the school has to salvage the minibus. a) Let f(t,x) be the smallest cost of having the minibus from period t to the end of the period 6, if we start period t with a minibus of age x. Write the Bellman equation for f(t,x). Please define any variables you use. b) Find the numerical value of f(4,3). Problem 2. An elementary school in Lexington has to come up with a replacement policy for its minibus over the next 6 years. A new minibus should be kept in service for at least for 2 years (unless it is bought at the beginning of the last period, in which case it can be sold at the end of that period). Moreover, the school cannot keep a minibus that is older than 5 years (if you enter a period with a 5-year-old bus, then you have to sell it at the beginning of that period). The price of a new minibus in period 1 is $50,000, and it increases at rate of 5% per year. The Salvage value of a minibus -as a function of its age- is given by the following table: Age(year) 1 2 3 4 5 6 or more Salvage value (in thousands of $) 30 25 20 18 18 0 The per year maintenance cost of a minibus -as a function of its age- is given by the following table: Age(year) 0 1 2 3 4 5 or more Maintenance in thousands of $) 0.5 1 1 2. 3 10 where age O means a new minibus. Assume that the School has to buy a new minibus at the beginning of period 1. For periods 2 through 6, the decision to keep or to replace the minibus is made at the beginning of every period. At the beginning of period 7, the school has to salvage the minibus. a) Let f(t,x) be the smallest cost of having the minibus from period t to the end of the period 6, if we start period t with a minibus of age x. Write the Bellman equation for f(t,x). Please define any variables you use. b) Find the numerical value of f(4,3)
