Optimal Preventive Maintenance Policy under a Budget Constraint

We design an optimal preventive maintenance policy for a system of N items that minimizes the total expected maintenance cost. We assume that the budget for preventive maintenance is limited and constrained. The problem has a finite time horizon and we consider constant inter-preventive maintenance times for every item. The resulting nonlinear optimization problem is reformulated as a binary integer program and computation results are presented on a real data set from South Texas Project Nuclear Operating Company in Bay City, Texas, USA. INTRODUCTION In this paper we model and solve the problem of designing an optimal maintenance policy for a system of N items, from a particular class of maintenance policies that minimizes total expected maintenance cost over a finite planning horizon under a budget constraint. We present computation results for a sample problem from Feed Water (FW) system at the South Texas Project (STP) in Bay City, Texas, USA. STP is one of the newest and the largest nuclear power plant in the USA and an industry leader in safety, reliability and efficiency. It has two nuclear reactors that together can produce 2500 megawatts of electric power. The first reactor became operational in August 1988 and the second in June 1989; the former is the sixth and latter is the fourth youngest nuclear reactor of the 103 licensed nuclear power plants in the USA. The two reactors (first and second) have the license to operate for another 23 and 25 years respectively, therefore all * Address all the communication to this author. the decisions taken by the board of directors have finite time horizons. The literature on replacement policies is enormous. The earlier work on the reliability was done in the early 1960s by Barlow and Proschan [1]. An excellent survey of the literature is presented by Valdez-Florez and Feldman [2] and Dekker [3]. Rausand and Høyland [4] is one of the contemporary textbooks on reliability. Marquez and Heguedas [5] present a review of the recent research on maintenance policies and solve the problem of periodic replacement in the context of a semiMarkov decision processes methodology. We will review some of the most recent work on replacement policies and refer the readers to the above references for earlier research. The reliability literature on replacement models and policies that allow for a change of the future failure behavior of the system is limited. There are several papers that can be classified as either models where replacement actions reduce the rate of failures, or models where the replacement actions reduce the (virtual) age of the system (see Rausand and Høyland [4] for details). Such problems where the replacement decisions may influence the future stochastic nature of the system are referred to as decision-dependent-randomness problems. For a general overview of the existing literature that relates to this class of problems see Morton and Popova [6]. Models with decision dependent uncertainty are discussed by Jonsbråten [7] and Jonsbråten and Woodruff [8]. Lai et al. [9] analyze a single-unit system subjected to external shocks. The unit can fail due to ageing or shock. This is an example in which the failure rate of the system increases after a lethal shock or with ageing. The policy is to replace the system after the n shock or failure, whichever occurs first. They minimize 1 Copyright © 2008 by ASME the long run expected cost per unit time to obtain the optimal value of n. Lai and Tang [10] consider a two-unit system where the failure of each unit either increases the failure rate of the other or brings it to an instantaneous failure. The system is replaced at age T or at failure whichever occurs first. The value of T is obtained by minimizing the long run expected cost per unit time. Galenko et al. [11] present optimal maintenance policy for multiple items with decision dependent uncertainty over a finite time horizon (divided into a discrete set) and in the absence of a budget constraint. We address the problem of designing an optimal preventive maintenance policy for a portfolio of items over a finite and continuous time horizon under a constrained total preventive maintenance budget. Galenko et al. [12] forms the basis of our work. They present optimal maintenance policy for a single item case and do not consider a budget constraint. The items considered in this work can fail in different failure modes and there is a failure cost associated with every item-failure mode combination. The time to failure of each item is governed by a known probability distribution with increasing failure rate. An item once failed can induce failure in other items and it is assumed that the item failing first in one of its failure modes (natural failure) cannot result in multiple types of failures (induced failures) in other items. Induced failures of items are governed by a known failure model. When an item fails unexpectedly in service, it is repaired to ‘as good as old’ condition by performing corrective maintenance (CM). Each item can undergo preventive maintenance (PM) that brings the item to ‘as good as new’ condition. We assume that the time to perform CM and PM is small, and it can be ignored. It is also assumed that every item undergoes preventive maintenance at the end of the planning period. There are three types of positive maintenance costs: (i) preventive maintenance cost (ii) corrective maintenance cost and (iii) downtime cost. Failure of an item may lead to production loss due to plant trip, which is estimated by the downtime cost. The corrective maintenance and downtime costs are associated with item failures and therefore these two can be combined together. The total system cost is the sum of total expected cost associated with item failures and total preventive maintenance cost. We assume constant inter-preventive maintenance time for an item. Inter-preventive maintenance times of two different items may be different. The objective of this problem is to utilize the limited preventive maintenance budget over a finite time horizon so as to minimize the total expected maintenance cost. NOMENCLATURE Indices: N i∈ : Index of items. i F m∈ : Index of distinct failure modes of item N i∈ . Data: i pm C : Cost of performing preventive maintenance on item i . m i cm C , : Cost of performing corrective maintenance on item i given that it failed in failure mode i F m∈ . m i trip C , : Cost of downtime given that item i failed in failure mode i F m∈ and resulted in a plant trip. i m p : Probability of item i failing in failure mode i F m∈ given that it has failed. m i m j p , , ′ : Probability that item i having failed naturally in failure mode i F m∈ causes item j to fail in failure mode j F m ∈ ′ . m i trip p , : Probability that item i having failed in failure mode i F m∈ results in plant trip. (.) i q : Failure rate function of item i . ) (t Z i : Counting process for the number of natural failures of item i in time interval ) , 0 ( t . )] ( [ t Z E i : Expected number of natural failures of item i in time interval ) , 0 ( t . L : Length of preventive maintenance planning period. B : Total available preventive maintenance budget over the planning period L . Decision Variable: i T : Inter preventive maintenance time for item i . MAINTENANCE MODELS WITH A BUDGET CONSTRAINT In this section we present two preventive maintenance optimization models for a system of N items under a budget constraint based on the following policy, denoted by (P): (P): Bring every item N i∈ to ‘as good as new’ state (preventive maintenance) after every i T units of time by incurring a cost pm C . If an item N i∈ has a natural failure meanwhile in mode i F m∈ then repair it to ‘as good as old’ state (corrective maintenance) for a cost m i cm C , or m i trip C , depending upon whether its failure resulted in a plant trip. In addition repair every item j that has failed in mode j F m ∈ ′ as a result of failure of item i , by incurring a cost of m j cm C ′ , or m j trip C ′ , , depending upon whether j ’s failure caused a plant trip. 2 Copyright © 2008 by ASME The objective is to find values of N i T i ∈ ∀ that minimize the total expected maintenance cost ) , , , ( ) ( | | 2 1 N T T T C L L = T C subject to a limited preventive maintenance budget B . Barlow and Hunter [13] show that if an item is repeatedly repaired to ‘as good as old’ state then the failure-event process ) (t Z i is a non-homogenous Poisson process with expected number of item failures in interval ) , 0 ( t given by the relation, ∫ = t i i du u q t Z E 0 ) ( )] ( [ , where (.) i q is the associated failure rate function of item i . NONLINEAR MAINTENANCE OPTIMIZATION MODEL The maintenance policy (P) naturally leads to the following nonlinear maintenance optimization model (Model 1). ) ( min T C T Subject to B T L C i N i i pm ≤ ⎥⎥ ⎤ ⎢⎢ ⎡ ∑ ∈ (1) N i L T i ∈ ∀ ≤ ≤ 0 (2) Where