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the following year the chain reaped a record $54 million in revenues and realized $26 million in profit Though not a record, in the depths of the Depression, any profit was noteworthy The fair s popularity defied the skeptics, generating $10 million in gate receipts alone and $26 million in gross revenues for the vendors, enough to convince the sponsors to re-up for a second equally successful summer in 1934 For Walgreens, the fair was especially fruitful The Century of Progress drew 887 vendors, yet Walgreens alone pulled in fully 4 percent of gross revenues about $1 million of the $26 million the fair raised, or 36 times more than the average vendor yielded Charles believed that the fair s real benefit to the chain, however, would be longer lasting than a few more dollars in the till While the Pepper Pod allowed that Walgreens would make a satisfactory profit from the fair, the more important statistic to Charles was the 4 million customers served, many of them for the first time, gaining the company substantial goodwill and advertising value, in Charles s words These millions of people are going to be your everyday customers from now on What they think of Walgreens in the future will depend entirely on your treatment of them 30 Walgreens knew how to hit the iron when it was hot or even just warm enough to make a good dent While Walgreens expansion during the Depression couldn t come close to the stratospheric growth it enjoyed in the Roaring Twenties from 20 to 397 stores, a leap of almost 2,000 percent Walgreens grew from 397 stores in 1929 to 493 in 1939, a rate of 24 percent, which could fairly be called robust and, given the times, even astounding Walgreens Depression era expansion was harder than those numbers suggest because it wasn t simply a matter of taking the 397 stores with which it started the decade and adding 96 In 1933, for example, Walgreens opened 24 units but closed 25; the next year the chain opened 44 but closed 31 Thus, increasing the total fleet by 96 probably required opening 200-some stores and closing 100 or so, with all the extra effort and tough decision making that goes with such a strategy.

The Markov decision model has many potential applications in inventory control, maintenance, manufacturing and telecommunication among others Perhaps this versatile model will see many more signi cant applications when it becomes more familiar to engineers, operations research analysts, computer science people and others To that end, s 6 and 7 focus on the algorithmic aspects of Markov decision theory and illustrate the wide applicability of the Markov decision model to a variety of realistic problems The presentation is con ned to the optimality criterion of the long-run average cost (reward) per time unit For many applications of Markov decision theory this criterion is the most appropriate optimality criterion The average cost criterion is particularly appropriate when many state transitions occur in a relatively short time, as is typically the case for stochastic control problems in computer systems and telecommunication networks Other criteria are the expected total cost and the expected total discounted cost These criteria are discussed in length in Puterman (1994) and will not be addressed in this book This chapter deals with the discrete-time Markov decision model in which decisions can be made only at xed equidistant points in time The semi-Markov decision model in which the times between the decision epochs are random will be the subject of the next chapter In Section 61 we present the basic elements of the discrete-time Markov decision model A policy-improvement procedure is discussed in Section 62 This procedure is the key to various algorithms for computing an average cost optimal polity The so-called relative values of a given policy play an important role in the improvement procedure The relative values and their interpretation are the subject of Section 63 In Section 64 we present the policy-iteration algorithm which generates a sequence of improved policies Section 65 discusses the linear programming formulation for the Markov decision model, including a formulation to handle probabilistic constraints on the state-action frequencies The policy-iteration algorithm and the linear programming formulation both require the solving of a system of linear equations in each iteration step In Section 66 we discuss the alternative method of value iteration which avoids the computationally burdensome solving of systems of linear equations but involves only recursive computations The value-iteration algorithm endowed with quickly converging lower and upper bounds on the minimal average cost is usually the most effective method for solving Markov decision problems with a large number of states Section 67 gives convergence proofs for the policy-iteration algorithm and the value-iteration algorithm.

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In 3 we have considered a dynamic system that evolves over time according to a xed probabilistic law of motion satisfying the Markovian assumption This assumption states that the next state to be visited depends only on the present state of the system In this chapter we deal with a dynamic system evolving over time where the probabilistic law of motion can be controlled by taking decisions Also, costs are incurred (or rewards are earned) as a consequence of the decisions

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