Optimal Preventive Maintenance under Decision Dependent Uncertainty

We analyze a system of N components with dependent failure times. The goal is to obtain the optimal block replacement interval (different for each component) over a finite horizon that minimizes the expected total maintenance cost. In addition, we allow each preventive maintenance action to change the future joint failure time distribution. We illustrate our methodology with an example from South Texas Project Nuclear Operating Company. INTRODUCTION In this paper we address two important problems for the maintenance and operation of nuclear power plants: proper modeling and analysis of dependent failures and the impact of maintenance interventions on the future failure behavior of the system being analyzed. Event/fault trees is the common methodology used in practice to model internal dependencies and estimate the probabilities of the ”top” events. One disadvantage of this approach is that it is a static model of the system and time-dynamics (i.e. how the failure behavior of the system evolves over time) cannot be incorporated. The use of Markov processes to describe the time evolution of the system, makes the assumption that the history of the failure process is not important, and only the current state of the system is sufficient to forecast the future. There is an enormous literature on optimal maintenance policies for a single item that dates back to the early 1950s. The majority of the work covers maintenance optimization over an infinite horizon, see Valdez-Flores and Feldman [1] for an extensive review. The problem that we address in this report is over a finite planning horizon, which comes from the fact that every nuclear power plant has a license to operate that expires in a finite predefined time. In addition the form of the policy is effectively predefined by the industry as a combination of preventive and corrective maintenance, as we describe below. Marquez and Heguedas [2] present an excellent review of the more recent research on maintenance policies and solve the problem of periodic replacement in the context of a semiMarkov decision processes methodology. Su and Chang [3] find the periodic maintenance policies that minimize the life cycle cost over a predefined finite horizon. A review of the Bayesian approaches to maintenance intervention is presented in Wilson and Popova [4]. Chen 1 Copyright c © 2006 by ASME and Popova [5] propose two types of Bayesian policies that learn from the failure history and adapt the next maintenance point accordingly. They find that the optimal time to observe the system depends on the underlying failure distribution. A combination of Monte Carlo simulation and optimization methodologies is used to obtain the problem’s solution. In Popova [6], the optimal structure of Bayesian group-replacement policies for a parallel system of n items with exponential failure times and random failure parameter is presented. The paper shows that it is optimal to observe the system only at failure times. For the case of two items operating in parallel the exact form of the optimal policy is derived. The reliability literature on models and policies that allow for a change of the future failure behavior of the system is limited. There are several papers that could be classified as either models where repair actions reduce the rate of failures, or models where the repair action reduce the (virtual) age of the system, see Rausand and Høyland, page 287, [7], for details. Such problems where the decisions made influence the future stochastic nature of the system are referred as decision-dependent-randomness, hence the title of the paper. For a general overview of the existing literature that relates to this class of problems, see Morton and Popova, [8]. Models with decision dependent uncertainty are discussed by Jonsbråten [9], and Jonsbråten et. al. [10]. We analyze a system of N components that could be in any dependent structure. The time horizon T is finite. At the beginning of each time period, t ∈ /1,2, . . . ,T/ we observe the state of the system and must decide whether or not to perform preventive maintenance (PM) to each of the N items. Each PM restores the state of the item to ”as good as new”. At the end of each time period we investigate system and if any of the items have failed during the past time interval, we perform corrective maintenance (CM) that keeps the age of the repaired item the same (i.e. it is repaired to “as good as old” state). We will assume that the time increases in increment of 1 unit. Each item i, i = 1, . . . ,N can fail independently and in addition can trigger the failure of any of the other N − 1 items. The collection of all N items constitute the system and it’s failure will be a function of the items’ failures. We introduce the notion of failure pattern below and show that the system failure is a subset of the set of all failure patterns. NOMENCLATURE Sets: t ∈ {1,2, . . . ,T} optimization horizon i ∈ {1,2, . . . ,N} number of items in the system p ∈ {1,2, . . . ,P} enumerated failure patterns a ∈ {1,2, . . . ,A} enumerated ages of the system Constants: Pmi t preventive maintenance cost of item i in the time period t Cmi t corrective maintenance cost of item i in the time period t Up failure pattern, it shows which items have failed and which are still working Bp additional aftermath costs associated with failure pattern Up fa,p the probability of getting failure pattern Up with age structure a D large constant used in the optimization model Resourcest the available resources to perform preventive maintenance in time period t MaxAgei an upper bound on the age of item i