A maintenance optimization model for mission-oriented systems based on Wiener degradation

Abstract

Over the past few decades, condition-based maintenance (CBM) has attracted many researchers because of its effectiveness and practical significance. This paper deals with mission-oriented systems subject to gradual degradation modeled by a Wiener stochastic process within the context of CBM. For a mission-oriented system, the mission usually has constraints on availability/reliability, the opportunity for maintenance actions, and the monitoring type (continuous or discrete). Furthermore, in practice, a mission-oriented system may undertake some preventive maintenance (PM) and after such PM, the system may return to an intermediate state between an as-good-as new state and an as-bad-as old state, i.e., the PM is not perfect and only partially restores the system. However, very few CBM models integrated these mission constraints together with an imperfect nature of the PM into the course of optimizing the PM policy. This paper develops a model to optimize the PM policy in terms of the maintenance related cost jointly considering the mission constraints and the imperfect PM nature. A numerical example is presented to demonstrate the proposed model. The comparison with the simulated results and the sensitivity analysis show the usefulness of the optimization model for mission-oriented system maintenance presented in this paper.

Introduction

A mission-oriented system is a system that is required to perform certain missions with fixed durations at various times in its lifetime [1]. Examples of such systems can be found in the ground systems of the aerospace launch center such as the fuel feed system; and in the military systems such as the avionics portion of an airborne weapon system. If the system fails during the mission phase, the mission must be aborted. A mission-oriented system is usually monitored by condition monitoring after each mission, and if the system is found to be in an unacceptable state, some PM will be carried out, which may return the system to an intermediate state between an as-good-as new state and an as-bad-as old state, i.e., the PM is not perfect and only partially restores the system. In addition, a mission can have constraints on availability/reliability, the opportunity for PM actions, and the monitoring type (continuous or discrete). The system will eventually be replaced by a new one when the following events occur: the system fails; the system availability/reliability does not satisfy the mission requirement or the system service time reaches a pre-specified number of years. This pre-specified number of years of service is usually determined by the design of the system or other factors such as safety, obsolescence etc.

Like other systems, many mission-oriented systems suffer deterioration in operation. When the deterioration of the system reaches a failure threshold, the system should be repaired or replaced. If the degradation of a mission-oriented system can be measured and modeled appropriately, we can take maintenance actions on the basis of the observed degradation data before the system has a failure. This is what the condition-based maintenance (CBM) aimed for, which is to achieve a cost-effective maintenance based on the system condition information. Over the past few decades, CBM has attracted many researchers’ interest because of its effectiveness and practical significance. For a comprehensive literature review in CBM, see Scarf [2], Wang [3], Jardine et al. [4], van Noortwijk [5]. The decisions made from CBM depend on the system condition data obtained by monitoring techniques among others. The framework of CBM mainly includes three parts, i.e., condition data acquisition, reliability and/or lifetime estimation from condition monitoring data, and CBM decision-making [4]. Degradation modeling is a way to estimate the remaining life time based on monitored data and it is commonly used when degradation measure is available.

Degradation modeling provides an important means for maintenance decision-making [6]. Many types of degradation models have been developed in the literature. General path models and stochastic process models are two kinds of frequently used models to describe the degradation processes [7]. A general path model focuses on the inter-item variability and thus can be used to estimate the lifetime distribution for the population, see Lu and Meeker [8], Park and Bae [9]. However, the degradation uncertainty for an individual is not taken into account in this kind of models. A stochastic process model focuses on the individual item behavior and can remedy the shortage of the general path model [10].

Stochastic-process-based models such as Markov chains [11], Gamma processes [12], [13] and Wiener processes [14], [15] have been widely used to model degradation processes, and have shown advantages over the general path model in CBM modeling. For a comprehensive review, see Si et al. [6]. Among these stochastic models, a Wiener process model can describe a non-monotonic degradation process with Gaussian noise, and provide a good description for some system behavior [16]. Crowder and Lawless [17] incorporated a random effect into the Wiener process to specify the random variation among the individuals and proved that CBM can offer much cost saving over a failure replacement policy. Whitmore and Schenkelberg [18] used a Wiener process with a time scale transformation to model the performance degradation of the self-regulating heating cable. The degradation data collected from sensors usually do not show a monotonic trend due to the noise influence. By regarding every degradation increment as an additive superposition of a large number of small effects, the degradation process can be assumed to follow a normal distribution. Therefore, in this paper, a Wiener process is chosen to describe mission-oriented system degradation.

Since the deterioration for a mission-oriented systems is described by a degradation model such as a Wiener process, the degradation threshold to initiate the PM (called the PM threshold) is the decision variable constituting the PM policy. In order to optimize the PM policy, some criteria, such as cost, availability and risk, are usually chosen as the optimization objectives in practice. Marseguerra et al. [19] used Genetic Algorithm and Monte Carlo simulation to optimize CBM with two objectives including profit and availability. Grall et al. [20] assumed a replacement policy with a multi-level control-limit rule to optimize maintenance policy based on a cost criterion. Further, Castanier et al. [21] assumed a repair/replacement policy based on a multi-level control limit rule and obtained the optimal thresholds based on both the cost and availability criteria. Amari and McLaughlin [22] utilized a Markov chain to describe a deterioration system subject to periodic inspection for the CBM model. The optimal inspection frequency and the maintenance threshold were found to maximize the system availability. Liao et al. [23] investigated a realistic maintenance policy which achieved the maximum availability level for the continuously degrading system. Li et al. [24] developed a maintenance model with reliability constraints for the mission-oriented avionic system based on the lifetime distribution. However, very few CBM models integrated the mission constraints together with an imperfect nature of the PM into the course of optimizing the PM policy with an objective function of the related maintenance cost.

The main objective of this paper is to develop a model to optimize the PM policy for a mission-oriented system based on a Wiener degradation process so that the mission requirements can be achieved with a minimum cost. Compared with the similar studies by Liao et al. [23] and Li et al. [24], the proposed model in this paper has the following innovations: (a) considering the mission constraints to PM occasions; (b) involving the imperfect PM effects and (c) taking the system availability requirement imposed by the mission into account.

Specifically, we consider a system with a mission availability constraint subject to periodic PM that is imperfect. For modeling the PM process, we use a geometric process for the residual damage and an exponential function for the mean PM duration, respectively. The residual damage is the left-over damage uncovered by the imperfect PM. Through classifying the system lifecycle, we present a probabilistic model for the PM process. An enumeration algorithm is used to find the optimal PM threshold. By analyzing the results, it is found that ignoring the mission constraints can lead to unsatisfactory results for a mission-oriented system.

The remaining parts are organized as follows. In Section 2, the system degradation is described by a Wiener stochastic process model, and the remaining useful life (RUL) distribution is obtained. Section 3 is devoted to describe the PM policy of the mission-oriented system. In Section 4, the influences of imperfect PM are modeled. In Section 5, the system renewal process is classified and a probabilistic analysis is presented in order to establish the optimization model. Section 6 formulates the optimization model and provides the optimization procedure. Section 7 provides a numerical example to illustrate the proposed model and validates usefulness of the proposed model. Conclusions are drawn in Section 8.