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problem involving a randomly changing state, a constraint may be placed on the frequency at which a certain inoperative state occurs The following illustrative example taken from Wagner (1975) shows that for control problems with probabilistic constraints it may be optimal to choose the decisions in a random way rather than in a deterministic way Suppose the daily demand D for some product is described by the probability distribution P {D = 0} = P {D = 1} = 1 , 6 P {D = 2} = 2 3

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The demands on the successive days are independent of each other At the beginning of each day it has to be decided how much to order of the product The delivery of any order is instantaneous The variable ordering cost of each unit is c > 0 Any unit that is not sold at the end of the day becomes obsolete and must be discarded The decision problem is to minimize the average ordering cost per day, subject to the constraint that the fraction of the demand to be met is at least 1 This 3 probabilistic constraint is satis ed when using the policy of ordering one unit every day, a policy which has an average cost of c per day However, this deterministic control rule is not optimal, as can be seen by considering the randomized control rule under which at any given day no unit is ordered with probability 4 and two 5 units are ordered with probability 1 Under this randomized rule the probability 5 that the daily demand is met equals ( 4 )( 1 ) + ( 1 )(1) = 1 and the average ordering 5 6 5 3 cost per day equals ( 4 )(0) + ( 1 )(2c) = 2 c It is readily seen that the randomized 5 5 5 rule is optimal So far we have considered only stationary policies under which the actions are chosen deterministically A policy is called a stationary randomized policy when it is described by a probability distribution { a (i), a A(i)} for each state i I Under policy action a A(i) is chosen with probability a (i) whenever the process is in state i If a (i) is 0 or 1 for every i and a, the stationary randomized policy reduces to the familiar stationary policy choosing the actions in a deterministic way For any policy , let the state-action frequencies fi,a ( ) be de ned by fia ( ) = the long-run fraction of decision epochs at which the process is in state i and action a is chosen when policy is used Consider now a Markovian decision problem in which the goal is to minimize the long-run average cost per time unit subject to the following linear constraints on the state-action frequencies:.

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