POLICY-ITERATION ALGORITHM in Visual Studio .NET

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v1 = 0 g + 09v1 + 01v2 v2 = 0 g + 08v2 + 01v3 + 005v4 + 005v5 v3 = 0 g + 07v3 + 01v4 + 02v5 v4 = 5 g + v1 v5 = 10 g + v6 v6 = 0 g + v1 v6 = 0 The solution of these linear equations is given by g(R (3) ) = 04338, v1 (R (3) ) = 04338, v2 (R (3) ) = 47717, v3 (R (3) ) = 65982, v4 (R (3) ) = 50000, v5 (R (3) ) = 95662, v6 (R (3) ) = 0 Step 2 (policy improvement) The test quantity Ti (a, R (3) ) has the values T2 (0, R (3) ) = 47717, T2 (1, R (3) ) = 7, T3 (0, R (3) ) = 65987, T3 (1, R (3) ) = 70000, T4 (0, R (3) ) = 68493, T4(1) (1, R (3) ) = 50000 This yields the new policy R (4) = (0, 0, 0, 1, 2, 2) Step 3 (convergence test) The new policy R (4) is identical to the previous policy R (3) and is thus average cost optimal The minimal average cost is 04338 per day Remark 642 Deterministic state transitions For the case of deterministic state transitions the computational burden of policy iteration can be reduced considerably Instead of solving a system of linear equations at each step, the average cost and relative values can be obtained from recursive calculations The reason for this is that under each stationary policy the process moves cyclically among the recurrent states The simpli ed policy-iteration calculations for deterministic state transitions are as follows: (a) Determine for the current policy R the cycle of recurrent states among which the process cyclically moves (b) The cost rate g(R) equals the sum of one-step costs in the cycle divided by the number of states in the cycle (c) The relative values for the recurrent states are calculated recursively, in reverse direction to the natural ow around the cycle, after assigning a value 0 to one recurrent state
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(d) The relative values for transient states are computed rst for states which reach the cycle in one step, then for states which reach the cycle in two steps, and so forth It is worthwhile pointing out that the simpli ed policy-iteration algorithm may be an ef cient technique to compute a minimum cost-to-time circuit in a deterministic network
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The policy-iteration algorithm solves the average cost optimality equation (643) in a nite number of steps by generating a sequence of improved policies Another way of solving the optimality equation is the use of a linear program for the average cost case The linear programming formulation to be given below allows the unichain assumption in Section 64 to be weakened as follows Weak unichain assumption For each average cost optimal stationary policy the associated Markov chain {Xn } has no two disjoint closed sets This assumption allows non-optimal policies to have multiple disjoint closed sets The unichain assumption in Section 64 may be too strong for some applications; for example, in inventory problems with strictly bounded demands it may be possible to construct stationary policies with disjoint ordering regions such that the levels between which the stock uctuates remain dependent on the initial level However, the weak unichain assumption will practically always be satis ed in realworld applications For the weak unichain case, the minimal average cost per time unit is independent of the initial state and, moreover, the average cost optimality equation (643) applies and uniquely determines g as the minimal average cost per time unit; see Denardo and Fox (1968) for a proof This reference also gives the following linear programming algorithm for the computation of an average cost optimal policy Linear programming algorithm
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Step 1 Apply the simplex method to compute an optimal basic solution (xia ) to the following linear program:
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