Comparison of Lifetime Distributions
The model calibration scheme is based on the method of moments that yields the parameters of RV and GP models such that the mean and COV of the lifetime distribution are identical in both models. In spite of the identical first two moments, the tails of lifetime distribution in RV and GP models can be remarkably different, as shown in Figure 4. Figure 4(a) shows that the lifetime distribution of the RV model is more skewed that that of the GP model. The difference between distribution tails can be seen more clearly through a comparison of survival functions in Figure 4(b). The GP model is more pessimistic than RV model about the prospect of survival due to uncertainty associated with evolution of deterioration that takes place in form of independent gamma distributed increments. In contrast, the deterioration in the RV model is fully correlated over the lifetime, which results in an overestimation of the survival probability.
Figure 4: Comparison of lifetime distributions in equivalent RV and GP models for vT = 0.3: (a) probability density,
and (b) survival function
The age-based replacement is the simplest policy for the renewal of aged fleet of structures and components. In this policy, a component is replaced when it reaches to a specific age (t0) regardless of its condition. The component is of course replaced, if failure occurs before the replacement time, t0.
Denote the total cost associated with all the consequences of a structural failure as CF, and the cost of a preventive replacement as CP. According to the renewal theory, the average cost per unit time in long term, also known as the mean cost rate K, can be computed as a function of the replacement age (Barlow and Proschan 1965):
K 4 Ft (toC + [1 – Ft (to)]CP….
K (to) = (14)
Jo [1 – Ft (t)]dt
It is easy to check that K(to) ^ CF / f! T as t0 ^ ^. Using Eq.(14), an optimal age of preventive replacement (t0) can be found that would minimize the mean cost rate.
Since the calculation of the cost rate is sensitive to the lifetime distribution Fft), it would be of interest to examine the impact of RV and GP model on the replacement policy. For an illustration, the cost data are assumed as CP = 10 and CF = 50. The basic data given in Tables 1 and 2 are used in the calculation of lifetime distribution. Figure 5 shows the variation of mean cost rate with the replacement age when the lifetime COV is vT = 0.3. The optimal replacement age for both models is about the same, 29 units, but corresponding mean cost rate for the GP model (K(29) = 0.45 units) is greater than that of the RV model (0.38). The reason is that the GP model involves higher uncertainty than the RV model, as discussed in Section 3.
Figure 5: Mean cost rate
A comprehensive comparison of optimal cost rate and replacement age results for other values of vT in the RV and GP models are shown in Figure 6. The results of optimal replacement age obtained from the two models are qualitatively different. In the RV model, the optimal replacement age decreases continuously as the lifetime COV vT increases. This observation is somewhat intuitive in the sense that when faced with increased uncertainty it is prudent to reduce the replacement age. The results of GP model exhibit two distinct trends. Initially, with increase in lifetime COV the replacement age decreases. However, as vT increases beyond 0.5, the trend reverses and the replacement age begins to increase. It means that in case of large uncertainty associated with component lifetime, the life-cycle cost can be optimized by extending the replacement age.
Figure 6: Comparison of age replacement policy in equivalent RV and GP models: (a) replacement age, and (b)
mean cost rate
The comparison of the cost rate in Figure 6(b) shows that mean cost rate obtained from GP model is always higher than that of the RV model as a result of additional temporal uncertainty associated with GP model. The difference between the optimum cost rates increases with increase in lifetime uncertainty.