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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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that are sequentially made when the system evolves over time An in nite planning horizon is assumed and the goal is to nd a control rule which minimizes the long-run average cost per time unit A typical example of a controlled dynamic system is an inventory system with stochastic demands where the inventory position is periodically reviewed The decisions taken at the review times consist of ordering a certain amount of the product depending on the inventory position The economic consequences of the decisions are re ected in ordering, inventory and shortage costs We now introduce the Markov decision model Consider a dynamic system which is reviewed at equidistant points of time t = 0, 1, At each review the system is classi ed into one of a possible number of states and subsequently a decision has to be made The set of possible states is denoted by I For each state i I , a set A(i) of decisions or actions is given The state space I and the action sets A(i) are assumed to be nite The economic consequences of the decisions taken at the review times (decision epochs) are re ected in costs This controlled dynamic system is called a discrete-time Markov model when the following Markovian property is satis ed If at a decision epoch the action a is chosen in state i, then regardless of the past history of the system, the following happens: (a) an immediate cost ci (a) is incurred, (b) at the next decision epoch the system will be in state j with probability pij (a), where pij (a) = 1,
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Note that the one-step costs ci (a) and the one-step transition probabilities pij (a) are assumed to be time homogeneous In speci c problems the immediate costs ci (a) will often represent the expected cost incurred until the next decision epoch when action a is chosen in state i Also, it should be emphasized that the choice of the state space and of the action sets often depends on the cost structure of the speci c problem considered For example, in a production/inventory problem involving a xed set-up cost for restarting production after an idle period, the state description should include a state variable indicating whether the production facility is on or off Many practical control problems can be modelled as a Markov decision process by an appropriate choice of the state space and action sets Before we develop the required theory for the average cost criterion, we give a typical example of a Markov decision problem Example 611 A maintenance problem At the beginning of each day a piece of equipment is inspected to reveal its actual working condition The equipment will be found in one of the working conditions i = 1, , N , where the working condition i is better than the working condition
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i + 1 The equipment deteriorates in time If the present working condition is i and no repair is done, then at the beginning of the next day the equipment has working condition j with probability qij It is assumed that qij = 0 for j < i and j i qij = 1 The working condition i = N represents a malfunction that requires an enforced repair taking two days For the intermediate states i with 1 < i < N there is a choice between preventively repairing the equipment and letting the equipment operate for the present day A preventive repair takes only one day A repaired system has the working condition i = 1 The cost of an enforced repair upon failure is Cf and the cost of a pre-emptive repair in working condition i is Cpi We wish to determine a maintenance rule which minimizes the long-run average repair cost per day This problem can be put in the framework of a discrete-time Markov decision model Also, since an enforced repair takes two days and the state of the system has to be de ned at the beginning of each day, we need an auxiliary state for the situation in which an enforced repair is in progress already for one day Thus the set of possible states of the system is chosen as I = {1, 2, , N, N + 1} State i with 1 i N corresponds to the situation in which an inspection reveals working condition i, while state N + 1 corresponds to the situation in which an enforced repair is in progress already for one day De ne the actions 0 if no repair is done, a = 1 if a preventive repair is done, 2 if an enforced repair is done The set of possible actions in state i is chosen as A(1) = {0}, A(i) = {0, 1} for 1 < i < N, A(N ) = A(N + 1) = {2}
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The one-step transition probabilities pij (a) are given by pij (0) = qij for 1 i < N,
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pi1 (1) = 1 for 1 < i < N, pN,N +1 (2) = pN +1,1 (2) = 1, and the other pij (a) = 0 The one-step costs ci (a) are given by ci (0) = 0, Stationary policies We now introduce some concepts that will be needed in the algorithms to be described in the next sections A rule or policy for controlling the system is a prescription for taking actions at each decision epoch In principle a control rule ci (1) = Cpi , cN (2) = Cf and cN +1 (2) = 0
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