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Appendix: Opening the Black Box

 

 

probability distributions on the asset condition. We write the likelihood that the test concludes that the asset is in condition clwhen the actual condition of the assets is cjas the probability {cicj}. (The quotations around the condition clsuggest that the test is making a claim that need not be true.)

As a result of the test outcome, the probability distribution on condition is revised. This revision is accomplished by application of Bayes’ Theorem, which determines the (posterior) probability that the assets is in state cjgiven that the test says that it is in any state cl:

.  

This equation, valid for all states { cj} and all ages t and all times k, permits the hazard rate to be updated as a result of the test. Therefore, the test-outcome-based hazard rate can be found:

where h(t|cj) is the conditional hazard rate for an asset of age t in condition cj.

Minimization of Expected Costs

The costs associated with the asset inventory are the replacement cost, the failure cost, the cost of testing, the maintenance (or overhaul) cost, and operating costs that are associated with the unobservable states. In general, such a cost is based on the probability of the asset occupying any of the unobservable conditions. Let R = the cost of replacing an asset. Then the cost of replacements in year k is

Let F = the cost of an asset failure. Let (t, f) be the age and condition of any asset. Then the total cost of failures in year k is

.

Let τ = the cost of testing an asset. Then the cost of testing in year k is

where the third argument in the decision function , d(t, f,”c”), which is the test outcome, is null and the summation is over all states (t,f) such that the value of the decision function .

Let V = the cost of maintenance or repair. Maintenance or repair can be chosen before or after a test is done. There are two terms in the cost of maintenance or repair. Then the cost of maintenance or repair in year k is

where the summations are over all pairs (t,f) such that

Let  = the operating cost of a asset if it is in (unobservable) condition cj. Then the operating cost in year k is

where the probability distribution on the unobservable condition is modified by the test outcome if testing were chosen. Then, the present worth of the policy that makes decisions  d(t, f,”c”)  is, for annual interest rate r: 

.  

The decision problem is to choose the decision rule  d(t, f,”c”)  that minimizes the present value of the policy. The solution to this optimal control problem is the optimal policy for managing aging assets.

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