Each quarter the marketing manager of a retail store divides customers into two classes based on…

Each quarter the marketing manager of a retail store divides customers into two classes based on their purchase behavior in the previous quarter. Denote the classes as L for low and H for high. The manager wishes to determine to which classes of customers he should send quarterly catalogs. The cost of sending a catalog is $15 per customer and the expected purchase depends on the customer’s class and the manager’s action. If a customer is in class L and receives a catalog, then the expected purchase in the current quarter is $20, and if a class L customer does not receive a catalog his expected purchase is S 10. If a customer is in class H and receives a catalog, then his expected purchase is $50, and if a class H customer does not receive a catalog his expected purchase is $25.

The decision whether or not to send a catalog to a customer also affects the customer's classification in the subsequent quarter. If a customer is class L at the start of the present quarter, then the probability he is in class L at the subsequent quarter is 0.3 if he receives a catalog and 0.5 if he does not. If a customer is class H in the current period, then the probability that he remains in class H in the subsequent period is 0.8 if he receives a catalog and 0.4 if he does not. Assume a discount rate of 0.9 and an objective of maximizing expected total discounted reward.

a. Formulate this as an infinite-horizon discounted Markov decision problem.

b. Find an optimal policy using policy iteration starting with the stationary policy which has greatest one-step reward.

c. For E = 0.1, find an &-optimal policy using ordinary value iteration, GaussSeidel value iteration, and relative value iteration starting with 00' = (0,O). In each, compare the number of iterations required under the supremum norm (6.3.3) and span seminorm (6.6.1 1) stopping criteria.

d. For E = 0.1, find an &-optimal policy using modified policy iteration and Gauss-Seidel modified policy iteration. Be sure to choose an appropriate starting value.

e. Formulate the problem as a linear program, giving its primal and dual. Solve both and interpret the solutions.

f. Find an optimal policy under the constraint that the total expected discounted cost of sending catalogs cannot exceed 99.

g. Apply the action elimination algorithm of Sec. 6.7.3. Record the total number of actions eliminated at each iteration.

h. What is the turnpike horizon for this problem? Compare it with the bounds in (6.8.1) and (6.8.5).

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