# Dynamic Programming and Reinforcement Learning

Question 1. We have a tree farm. At any time, the size s of a tree is within six categories: 0, 1, …, 5, where 0 means that the tree has died, category 4 is about the average size of a mature tree and 5 indicated above average size of a mature tree. We need to decide when to harvest a given tree. Each year it costs about \$ 10+s to maintain a tree, and \$ 30+5s to harvest a tree. The sales price of a tree of each size is as follows:

The transition probability matrix for the size of the tree is as follows:

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 sizes 0 1 2 3 4 5 0 1 0 0 0 0 0 1 0.05 0.15 0.7 0.1 0 0 2 0.05 0 0.2 0.6 0.1 0.05 3 0.05 0 0 0.5 0.4 0.05 4 0.1 0 0 0 0.85 0.05 5 0.1 0 0 0 0 0.9

(Q1.a) Describe a dynamic programming problem to determine an optimal harvesting policy. (Q1.b) Solve the problem numerically. What numerical methods are applicable to this problem and why.

Question 2. Each quarter the marketing manager of a retail store divides the customers into two groups based on their purchase behavior in the previous quarter. The classes are denoted by L and H . The manager wishes to determine to which group of customers he should sent a catalog. The cost of sending a catalog is \$ 15 per customer. If a customer from group L receives a catalog, then the expected purchase in the current quarter is \$ 20, otherwise it is \$ 10. If a customer from group H receives a catalog, then the expected purchase in the current quarter

is \$ 50, otherwise it is \$ 25. Furthermore, if a customer from group L receives a catalog, then the probability that he will stay in group L for the next quarter is 0.3, otherwise, it is 0.5. If a customer from group H receives a catalog, then the probability that s/he will stay in group H for the next quarter is 0.8, otherwise, it is 0.4.

(Q2.a) Formulate an average reward problem to help the manager. (Q2.b) Determine an optimal policy using policy iteration method. (Q2.c) Solve the problem using linear programming.

(Q2.d) For what discount factor is the discounted infinite-horizon problem equivalent to the average reward problem in this context?

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