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s = 0, where s is an arbitrarily chosen state
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ALGORITHMS FOR AN OPTIMAL POLICY
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Step 2 (policy-improvement step) For each state i I , determine an action ai yielding the minimum in min ci (a) g(R) i (a) + a A(i)
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The new stationary policy R is obtained by choosing R i = ai for all i I with the convention that R i is chosen equal to the old action Ri when this action minimizes the policy-improvement quantity Step 3 (convergence test) If the new policy R = R, then the algorithm is stopped with policy R Otherwise, go to step 1 with R replaced by R In the same way as for the discrete-time Markov decision model, it can be shown that the algorithm converges in a nite number of iterations to an average cost optimal policy Also, as a consequence of the convergence of the algorithm, there exist numbers g and i satisfying the average cost optimality equation i = min ci (a) g i (a) + a A(i) pij (a) j , i I (722)
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The constant g is uniquely determined as the minimal average cost per time unit Moreover, each stationary policy whose actions minimize the right-hand side of (722) for all i I is average cost optimal The proof of these statements is left as an exercise for the reader Value-iteration algorithm For the semi-Markov decision model the formulation of a value-iteration algorithm is not straightforward A recursion relation for the minimal expected costs over the rst n decision epochs does not take into account the non-identical transition times and thus these costs cannot be related to the minimal average cost per time unit However, by the data transformation method from Section 71, we can convert the semi-Markov decision model into a discrete-time Markov decision model such that both models have the same average cost for each stationary policy A value-iteration algorithm for the original semi-Markov decision model is then implied by the valueiteration algorithm for the transformed discrete-time Markov decision model In the discrete-time model it is no restriction to assume that all ci (a) = ci (a)/ i (a) are positive; otherwise, add a suf ciently large positive constant to each ci (a) The following recursion method results for the semi-Markov decision model: Step 0 Choose V0 (i) such that 0 V0 (i) mina {ci (a)/ i (a)} for all i Choose a number with 0 < mini,a i (a) Let n := 1
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Step 1 Compute the function Vn (i), i I , from ci (a) Vn 1 (i) + Vn (i) = min pij (a)Vn 1 (j ) + 1 a A(i) i (a) i (a) i (a)
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(723) Let R(n) be a stationary policy whose actions minimize the right-hand side of (723) Step 2 Compute the bounds mn = min{Vn (j ) Vn 1 (j )],
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Mn = max{Vn (j ) Vn 1 (j )}
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The algorithm is stopped with policy R(n) when 0 (Mn mn ) mn , where is a prespeci ed accuracy number Otherwise, go to step 3 Step 3 n := n + 1 and go to step 1 Let us assume that the weak unichain assumption from Section 65 is satis ed for the embedded Markov chains {Xn } associated with the stationary policies It is no restriction to assume that the Markov chains {X n } in the transformed model are aperiodic Then the algorithm stops after nitely many iterations with a policy R(n) whose average cost function gi (R(n)) satis es 0 gi (R(n)) g , g i I,
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where g denotes the minimal average cost per time unit Regarding the choice of in the algorithm, it is recommended to take = mini,a i (a) when the embedded Markov chains {Xn } in the semi-Markov model are aperiodic; otherwise, = 1 2 mini,a i (a) is a reasonable choice Linear programming formulation The linear program for the semi-Markov decision model is given under the weak unichain assumption for the embedded Markov chains {Xn } By the data transformation and the change of variable uia = xia / i (a), the linear program (631) in Section 65 becomes: Minimize
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