# Numerical Examples

In the optimization of inspection interval with respect to the cost rate, the following cost data are used: Ci = 1, CP = 10, CP = 50 and the PM ratio c = 0.8. The parameters of the RV and GP models are the same as calibrated in Section 3.1.

Figure 9: Results for equivalent RV and GP models (vT = 0.3): Inspection interval versus (a) Mean cycle cost, and

(b) mean cycle length

Figure 10: Mean cost rate as a function of inspection interval in equivalent RV and GP models (vT = 0.3)

Figure 9 compares the mean cycle cost and mean cycle length obtained from the RV and GP models (for vT = 0.3), whereas the mean cost rates, and its different component, are plotted in Figure 10. These results show remarkable differences between the two equivalent models of deterioration.

The cost curve for RV model has two plateaus, the lower one extends from t, =0 – 30 and the

corresponding cost is Cj + CP. The upper limit of the cost is CF. Because of the lack of temporal uncertainty, the RV model favors a short inspection interval. The GP model shows a distinct optimum of inspection interval (topt = 7 units) in Figure 10(b). For a small inspection interval, the cycle cost is fairly high due to increase in cost associated with frequent inspections. This aspect is qualitatively different from the RV model, which associates a smaller cost with a short inspection interval. For a large interval, the GP cycle cost approaches to the failure cost as expected. In general, the GP cycle cost is higher than that for RV model due to the effect of temporal uncertainty.

(a) inspection interval age, and (b) mean cost rate

The optimization results for RV and GP models for other values of the lifetime COV are compared in Figure 11. It is remarkable in the RV model that both the optimum cost rate and inspection interval are insensitive to the variance of lifetime distribution, which is due to the lack of consideration of temporal uncertainty. In contrast, the minimum cost rate in the GP model increases with lifetime uncertainty. The cost rate increases from 0.38 to 0.97 as the COV of lifetime (vT) is increased from 0.1 to 0.8. The optimal inspection interval increases from 7 to 30 units, as vT is increased from 0.1 to 0.8.

3 Conclusion

The purpose of infrastructure life cycle management is to mitigate the risk of structural failures caused by unchecked deterioration. The uncertainty associated with deterioration can be incorporated through probabilistic models that can be classified in two broad categories, namely the random variable (RV) model and stochastic process model. The paper introduces a versatile stochastic gamma process (GP) model and compares it with an equivalent random variable (RV) model with respect to distributions of lifetime, deterioration magnitude and the life cycle cost.

The paper underscores the points that a RV model cannot capture temporal variability associated with evolution of degradation. As a consequence, the deterioration along a specific sample path is deterministic in the RV model, whereas it varies probabilistically in the GP model. This distinction has profound implications to the optimization of age-based and condition-based maintenance strategies. The results presented in the paper show that the optimum cost and inspection interval obtained from stochastic process model are qualitatively different than that obtained from the RV model. The RV model appears to underestimate the life cycle cost due to lack of consideration of temporal uncertainty.

In summary, a careful consideration of the nature of uncertainties associated with deterioration is important for a meaningful time-dependent reliability analysis and life-cycle management of structures. If the deterioration process is affected by temporal uncertainty, it is mandatory to model it as a stochastic process.

Engineering Mechanics, 19(4): 345-359.