A repairman moves between 4 sites, with site 1 denoting home office, and 2, 3, and...

 

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A repairman moves between 4 sites, with site 1 denoting home office, and 2, 3, and 4 denoting remote sites. Work at sites 2,3, and 4 requires the use of an equipment trailer. The cost of relocating the equipment trailer between the sites is d(j.k) = 300 for kj. The cost c(k, j) of using the trailer is 100 if the work is at site k > 1 and trailer is at site jk with j> 1; 50 if j = k and j>1, and 200 if the work is at k > 1 and the trailer is at j = 1 (home office). If the repairman is at the home office, no work is done and cost is zero. At any time, the repairman knows his present job location, observes the location of the next job (or 1 if no job is to be done) and decides whether to move the trailer. The transition matrix between job locations is [0.1 0.3 0.3 0.3] 0.0 0.5 0.5 0.0 P 0.0 0.0 0.8 0.2 0.4 0.0 0.0 0.6] For example, if the current location (job) is 2, the probability that the next job is at site 3 is 0.5. Assume the discount factor of 0.95. (Q2.1) Write the dynamic programming equations for the optimal value function. (Q2.2) Solve these equations by the value iteration method (starting from 0) and calculate the lower and upper bounds at each step. (Q2.3) Solve the problem by the policy iteration method starting from "do dot move trailer from base" at each state. (Q2.4) Solve the problem by linear programming. A repairman moves between 4 sites, with site 1 denoting home office, and 2, 3, and 4 denoting remote sites. Work at sites 2,3, and 4 requires the use of an equipment trailer. The cost of relocating the equipment trailer between the sites is d(j.k) = 300 for kj. The cost c(k, j) of using the trailer is 100 if the work is at site k > 1 and trailer is at site jk with j> 1; 50 if j = k and j>1, and 200 if the work is at k > 1 and the trailer is at j = 1 (home office). If the repairman is at the home office, no work is done and cost is zero. At any time, the repairman knows his present job location, observes the location of the next job (or 1 if no job is to be done) and decides whether to move the trailer. The transition matrix between job locations is [0.1 0.3 0.3 0.3] 0.0 0.5 0.5 0.0 P 0.0 0.0 0.8 0.2 0.4 0.0 0.0 0.6] For example, if the current location (job) is 2, the probability that the next job is at site 3 is 0.5. Assume the discount factor of 0.95. (Q2.1) Write the dynamic programming equations for the optimal value function. (Q2.2) Solve these equations by the value iteration method (starting from 0) and calculate the lower and upper bounds at each step. (Q2.3) Solve the problem by the policy iteration method starting from "do dot move trailer from base" at each state. (Q2.4) Solve the problem by linear programming.