General information
Description
The approach is based on Kijima model 1 [1] of imperfect repair (renewal) process. It allows, for example, modeling both perfect and imperfect repairs through the use of the so–called restoration factor q, which defines virtual age. Brief description of the model can be found here. We consider the Weibull lifetime distribution function with shape parameter β and scale parameter η:

To find optimal maintenance, the Cumulative Intensity Function W(t) versus time is needed. It is done using the approximate formulae [2] and the advanced Monte Carlo method [3]. The Monte Carlo method is described briefly here. The algorithm of calculating optimal solution with examples and 3 maintenance policies are described in [4]. Here we just explain 3 policies briefly.

We consider a system deteriorating with age in the infinite time horizon with underlying lifetime Weibull distribution function which shape parameter is greater than 1. It can be repaired according to Kijima model 1 with cost Cq or replaced by a new one with cost C0. To prevent the entire degradation of the item, we have to introduce periodic replacement in the maintenance process to set the age of the item to 0 at the end of each cycle. The optimization criterion is the expected cost per unit time

The cost CT can be minimized by scheduling preventive maintenance and selecting different maintenance policies.

Policy 1: Perform repairs up to age T and replace at age T. In this case, the length of a cycle is constant, and the expected total cost per unit time is

According to this policy, replacements are performed periodically. The objective is to find an optimal value T=T∗, which minimizes (2).

Policy 2: Perform repairs up to the first n-1 failures and replace at the nth failure

where Tn=E(length of cycle). The total coast is minimized with respect to n in this model.

Policy 3: Perform repairs up to age T3∗ and replace at the first failure after T3∗. Cost (1) is minimized with respect to T3∗.

Policies 2 and 3 are more efficient compared to policy 1 under condition that costs are the same in all these policies. However in policy 1 the replacement is scheduled and can be less expansive [4].

1. M. Kijima and N. Sumita, "A useful generalization of renewal theory: counting process governed by nonnegative Markovian increments," Journal of Applied Probability, No. 23, pp. 71-88, 1986.M.P.
2. O. Yevkin and V. Krivtsov “An approximate solution to the g-renewal equation with an underlying Weibull distribution”, IEEE Transactions on Reliability, Vol. 61, No. 1, 2012, pp. 68-73.
3. O. Yevkin. “A Monte Carlo approach for evaluation of availability and failure intensity under g-renewal process model”. In "Advances in Safety, Reliability and Risk Management". Proceedings of ESREL conference, France, Troyes, 2011, pp. 1015-1021.
4. O. Yevkin and V. Krivtsov “Comparative analysis of optimal maintenance policies under general repair with underlying Weibull distribution”, IEEE Transactions on Reliability, Vol. 62, No. 1, 2013, pp. 82-91.