Given a system that evolves in time, a mission time,TM, which is the time until when the
system is required to function as specified, and a number of inspections,N, performed at times, tinsp PRN, the maintenance problem is formulated as an optimisation task,
where both objective and constraints require the evaluation of the reliability,rptq.
Three main different costs can be identified: – manufacturing (or initial) costs, C0,
– costs of inspection,CI,
– costs of repairCR
– costs of failure,CF.
It is assumed that manufacturing costs are deterministic, as they are linked to construc- tion and usage of materials. Note that, as pointed out in [71], the costs of repair and failure will be obtained as expected values, Er¨s, as they are linked to the repair and
failure probability, respectively.
6.2.1 Costs due to inspections and repair
The cost due to inspections depends on inspection quality, q, and on the inspection times,tinsp, and can be expressed as
wherecI is a fixed unit cost, and q, see Eq. (6.3), quantifies the quality of inspections.
In the Eq. (6.1) the function
ηptq “ 1
p1`sqt; (6.2)
actualises the costs to the time of the analysis. As inspection activities do not reveal damage with certainty, the probability of repair, pRpq, tq, is linked to the probability
of detecting the damage within an inspection, P OD, which in turns depends on the inspection quality, q, the level of damage, Dptq, and the technique used to spot the
flaw. For example, as a means of controlling damage associated with crack propagation, in fatigue-prone metallic components, non-destructive inspection (NDI) techniques can be used. NDI techniques have an associated probability of detection [81], which can be modeled as
P ODptq “ p1´p0q p1´e qpf1´f2 Dptqqq; (6.3)
where, p0, is the probability of not detecting a large crack, while f1 and f2 are param-
eters that depend on the specific NDI technique. Note the probability of detection is calculated based on two factors: the first one, 1´p0, measures the probability of de-
tecting a large crack; while the second factor,1´eq pf1´f2 Dptqq, can be interpreted as a coefficient between 0 and 1 that is a function of the state of damage (or crack length for the specific case), Dptq. The expected cost of repair can be expressed as
ErCRpq,tinspqs “cR pRpq,tinspq ηptinspq; (6.4)
where, cR is a fixed unit cost, and pR is, clearly, a function of the inspection times.
Note that the unit cost of repair can be very small or sometimes negligible compared to the cost of inspection.
6.2.2 Costs of failure
The cost of failure depends on the state of damage,Dptq, as it is assumed proportional
to the failure probability of the system, as well as on the inspection quality,q, . Here, failure cost is expressed as
ErCFpq,tinsp, tqs “cF pFpq,tinsp, tq; (6.5)
where,cF is a fixed unit cost associated with failure, partial collapse, or unavailability,
and pFpq,tinsp, tq is the failure probability, calculated as in Section 6.3. Note that the
failure probability depends on both the inspection times, tinsp, and on the time when the reliability is assessed, t, as will be explained in the next section.
6.2.3 Total costs
The expected total cost is the sum of all expected costs
ErCTs “C0`CIpq,tinspq `ErCRpq,tinspqs `ErCFpq,tinsp, tqs; (6.6)
including the initial manufacturing costC0, for simplicity assumed to be deterministic.
The remaining combination of costs, including repair, inspection and failure equalises the total cost of maintenance
ErCMs “ErCIpq,tinspqs `ErCRpq,tinspqs `ErCFpq,tinsp, tqs. (6.7)
6.2.4 Formulation of the optimisation problem
The maintenance problem is formulated as a constrained optimisation problem, where the constraint is the limit state safety level that the system has to comply with. Here, the following formulation of the optimisation problem is considered
minimize
qPR`,tinspPr0,TMsN
ErCMpq,tinsp, tqs
subject to pFpq,tinsp, tq ďpcriticF ; (6.8)
where, pcriticF is determined by a prescribed limit state safety level. Eq. (6.8) is solved using the penalty function
ψpcq “1´eα|minp0,cq|; (6.9)
which is a function of the constraint
c“ ´log10`pFpq,tinsp, tq
˘
`log10`pcriticF ˘; (6.10)
where, the constraint is satisfied if c ą 0. The problem of Eq. (6.8) can, thus, be
reformulated into an approximate unconstrained problem, as minimize
qPR`,tinspPr0,TMsN
ErCMpq,tinsp, tqs `g ψpcq; (6.11)
where,gis a penalty factor, whose value can be chosen knowing the order of magnitude of the minimum value of the objective function.