LINEAR PROGRAMMING APPROACH

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program; see Derman (1970) and Hordijk and Kallenberg (1984): Minimize

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i I a A(i)

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ci (a)xia

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subject to xja

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a A(j ) i I a A(i)

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pij (a)xia = 0, xia = 1,

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i I a A(i) (s) ia xia (s) , i I a A(i)

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s = 1, , L, a A(i) and i I

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xia 0,

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Denoting by {xia } an optimal basic solution to this linear program and letting the set S0 = {i | a xia > 0}, an optimal stationary randomized policy is given by a (i) = xia / d xid , arbitrary,

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a A(i) and i S0 , otherwise

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Here the unichain assumption is essential for guaranteeing the existence of an optimal stationary randomized policy Example 611 (continued) A maintenance problem Suppose that in the maintenance problem a probabilistic constraint is imposed on the fraction of time the system is in repair It is required that this fraction is no more than 008 To handle this constraint, we add to the previous linear program for the maintenance problem the constraint

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N 1

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xi1 + xN 2 + xN +1,2 008

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The new linear program has the optimal solution

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x10 = 05943, x20 = 02971, x30 = 00286, x31 = 00211, x41 = 00177, x52 = x62 = 00206 and the other xia = 0

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The minimal cost is 04423 and the fraction of time the system is in repair is exactly 008 The LP solution corresponds to a randomized policy The actions 0, 0, 1, 2 and 2 are prescribed in the states 1, 2, 4, 5 and 6 In state 3 a biased coin is tossed The coin shows up heads with probability 00286/(00286 + 00211) = 0575 No preventive repair is done if heads comes up, otherwise a preventive repair is done

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DISCRETE-TIME MARKOV DECISION PROCESSES

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A Lagrange-multiplier approach for probabilistic constraints A heuristic approach for handling probabilistic constraints is the Lagrange-multiplier method This method produces only stationary non-randomized policies To describe the method, assume a single probabilistic constraint ia fia ( )

i I a A(i)

on the state-action frequencies In the Lagrange-multiplier method, the constraint is eliminated by putting it into the criterion function by means of a Lagrange multiplier 0 That is, the goal function is changed from i,a ci (a)xia to ci (a)xia + ( i,a ia xia ) The Lagrange multiplier may be interpreted as i,a the cost to each unit that is used from some resource The original Markov decision problem without probabilistic constraint is obtained by taking = 0 It is assumed that the probabilistic constraint is not satis ed for the optimal stationary policy in the unconstrained problem; otherwise, this policy is optimal for the constrained problem as well Thus, for a given value of the Lagrange multiplier > 0, we consider the unconstrained Markov decision problem with one-step costs

ci (a) = ci (a) + ia

and one-step transition probabilities pij (a) as before Solving this unconstrained Markov decision problem yields an optimal deterministic policy R( ) that prescribes always a xed action Ri ( ) whenever the system is in state i Let ( ) be the constraint level associated with policy R( ), that is, ( ) =

i,Ri ( ) fi,Ri ( ) (R( ))

If ( ) > one should increase , otherwise one should decrease Why The Lagrange multiplier should be adjusted until the smallest value of is found for which ( ) Bisection is a convenient method to adjust How do we calculate ( ) for a given value of To do so, observe that ( ) can be interpreted as the average cost in a single Markov chain with an appropriate cost structure Consider the Markov chain describing the state of the system under policy R( ) In this Markov process, the long-run average cost per time unit equals ( ) when it is assumed that a direct cost of i,Ri ( ) is incurred each time the process visits state i An effective method to compute the average cost ( ) is to apply value iteration to a single Markov chain; see Example 661 in the next section The average cost of the stationary policy obtained by the Lagrangian approach will in general be larger than the average cost of the stationary randomized policy resulting from the linear programming formulation Also, it should be pointed out that there is no guarantee that the policy obtained by the Lagrangian approach is the best policy among all stationary policies satisfying the probabilistic constraint, although in most practical situations this may be expected to be the case In spite