## Optimal Policies

The interval costs and the expected length of an interval can be used to construct an optimal average cost solution for a fixed maintenance interval with policy n = \kr\k = 1, 2n}.

### 5.1 Average Cost Criterion

If we take a fixed policy with n = \kr\k = 1, 2,...,n} the sequence of entries into the critical set defines an embedded renewal process and the average cost per cycle can be obtained using the renewal-reward theorem. For this we need the expected length of an interval. For perfect restoration the expected interval length satisfies iT (x,y) = J [1 - Gx'y (s)] ds + J J iT (u,w)fx'y (u,w)dudw

0 Ao Bo and for partial restoration iT(x, y) = J [1 - GD(x,y) (s)J ds + J J iT(u,w)fD(x'y)(u,w)dudw

0 Ao Bo

Ao Bo where 1 — GD(x'y)(s) is the interval survival function.

On applying the renewal-reward theorem [Bat00], the average cost per unit time is r< S vT (x,y)

The optimum policy for a system starting in state Xo = x can then be determined as t* = argmin {C(x, t)} .

### 5.2 Total Cost Criterion

If the inspection intervals are allowed to change with the evolution of the system state, a non-periodic policy n = {ti, T2,...,Tn} can be defined. The construction of the interval cost functions with the inspection and action at the beginning, makes the step from one interval to the next straightforward. With this construction the function vT (x, y) is the value function for a dynamic programming problem [Bat00]. The optimality equation for the perfect repair version is

A0 B0

For partial restoration the programming problem becomes

Ao Bo

If costs are discounted with rate r the value function is modified and the dynamic programming problem becomes e-rTc(D(x, y)) + {vt(0, 0) + CF}pDpy) ..

Ao Bo where u,w I —rs uw i \ i pr'F = e rsgu,w(s)ds .

### 5.3 Obtaining Solutions

The optimization problems above contain integral equations of the Volterra type so that discretization of the state space and application of quadrature rules produce equivalent matrix equations with the general form vecv = vecc + vecMv which are readily solved numerically as long as care is taken in dealing with singularities [PTVF92]. The dynamic programming problems translate in the same way and allow a policy improvement algorithm [Bat00] to be applied to develop the optimal policy. Convergence proofs for the algorithms are given by Dagg in his thesis [Dag00].

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