Bayesian Nonparametric Analysis of Single Item Preventive Maintenance Strategies

This work addresses the problem of finding the minimal-cost preventive maintenance schedule for a single item. We develop an optimization algorithm that reduces the computational effort to find the optimal schedule. This approach relies on the item having an increasing failure rate, which is typical, and employs a Gibbs sampling algorithm to simulate from the failure rate distribution using real data. We also analyze the case when the item has a “bathtub” failure rate; we develop techniques that lead to an algorithm that finds an optimal schedule for this case as well. We then analyze the effectiveness of our approach on both artificial and real data sets from the South Texas Project nuclear power plant. NOMENCLATURE PM Preventive maintenance (restore functioning item to “new” condition) CM Corrective maintenance (restore failed item to its condition just before failure) Ptrip Probability of item failure causing a loss of production Cpm Cost of preventive maintenance Ccm Cost of corrective maintenance Ctrip Downtime cost INTRODUCTION Proper and safe operation is a major concern for any nuclear power plant. An inefficiently functioning plant costs more to operate and produces less energy than it potentially could. One of the main factors that greatly influences a plant’s efficiency is the maintenance of its subsystems and components. Should some components fail too frequently, the plant will incur losses resulting from a drop in production. On the other hand, replacing or repairing components too often may result in a substantial increase to the cost of operating the plant. A maintenance policy 1 Copyright c © 2009 by ASME is a decision-making rule that defines when and how, an item, or system, will be maintained. We seek an optimal maintenance policy from a class of time-based maintenance policies. In other words, an optimal maintenance policy for a plant balances safe and consistent operations with cost-effective upkeep by using a pre-specified utility function that may consider operating cost, production levels, safety standards, as well as many other factors. Finding an optimal maintenance policy may be difficult. Decision makers have to deal with the fact that the behavior of many components may be stochastic. Also, the system requiring maintenance may be very complex and involve many interdependencies, which may be hard to model and even harder to optimize its maintenance. This paper addresses modeling the stochasticity of a single component without delving too much into modeling component interdependencies. Specifically, we will discuss modeling a component’s failure distribution and making optimal maintenance decisions. BACKGROUND Consider the problem of developing the optimal maintenance policy for a single item. We say that the item has failure distribution f if the time to failure from when the item is “new” has probability density f , i.e. the probability that the item fails before time t is Prob(Failure Time≤ t) = ∫ t 0 f (u)du = F(t). (1) In this paper we assume that the item’s failure distribution is time-stationary, meaning the failure distribution does not change with time. Then, since PM restores the item to a “new” condition and CM restores the item to a “good-as-old” condition, the item’s failure distribution f depends only on the last time the item was “new”. The assumption of a stationary failure rate implies that the failure distribution f does not change throughout the time horizon L, which may not hold true if L is very long. The focus of this study, however, is to provide the optimal maintenance interval for a number of months in advance. As the plant ages and more failures are recorded, the maintenance intervals will be re-estimated using newly available data to accommodate the changing failure distribution. Our plans for future research include the possibility that this failure time is not stationary so that we can incorporate plant aging in our model. The failure rate function, z(t), is defined as z(t) = f (t) ∫ ∞ t f (u)du . (2) A more intuitive definition of the failure rate is a conditional probability of an item’s failure in a time interval. Say that an item has survived until some time t; the probability that the item fails before t +∆t is approximately z(t)∆t. In other words, z(t)∆t ≈Prob(t < Failure Time < t +∆t|Failure Time > t), (3) with a more precise approximation as ∆t decreases. Now, let N(t1, t2) represent the number of item failures between times t1 and t2, and let E[N(t1, t2)] be the expected number of such failures. An interesting property of the failure rate function (see [1]) is that ∫ t2 t1 z(u)du = E[N(t1, t2)]. (4) For more in-depth background information, see a textbook on reliability theory, e.g. [2]. 1 INCREASING FAILURE RATE Consider the problem of scheduling exactly one PM for an item with a strictly increasing, stationary, and continuous failure rate between times 0 and L, where L is the time horizon (i.e., we don’t care what happens to the item after time L). We assume that the item is in a “new” state at time 0, and is again restored to a “new” state by PM. Should an item fail, it is restored to its condition just before the failure (i.e., “good-as-old” state) by corrective maintenance (CM). During a failure, an item may trigger a larger system failure or “trip” with probability Ptrip. Should this happen, instead of a simple CM we must address the entire system “trip” and incur the cost of downtime, which is more expensive than CM. We also assume that PM, CM, and restoration of a downed system are performed instantaneously. Suppose that we wish to optimize our maintenance schedule by minimizing the expected cost, and we can only perform a single maintenance. Letting C = [PtripCtrip +(1−Ptrip)Cm], we can express this problem as: min T∈A [ X(T ) = Cpm +C (E[N(0,T )]+E[N(0,L−T )]) ] , (5) where, nominally, A = [0,L]. However, the set A can also impose constraints on the optimal maintenance time, for instance, it can contain only time periods corresponding to planned outages. In our current presentation we do not assume any restrictions. Model (5) requires that we perform a single maintenance. If we find the optimal T ∗, which solves model (5) then we can compare X(T ∗) with C̄E[N(0,L)], the expected cost of not performing a PM and choose the alternative with smaller expected cost. The objective function in model (5) is convex in T. We can 2 Copyright c © 2009 by ASME Figure 1. PM OVER INTERVAL [0,L]. show this by rewriting it in terms of the failure rate, z(t). The objective function in (5) becomes: