A Bayesian method on adaptive preventive maintenance problem

Abstract

In this paper we consider a Bayesian theoretic approach to determine an optimal adaptive preventive maintenance policy with minimal repair. By incorporating minimal repair, major repair, planned replacement, unplanned replacement and periodic scheduled maintenance in the model, the mathematical formulas of the expected cost per unit time are obtained. When the failure density is Weibull with uncertain parameters, a Bayesian approach is established to formally express and update the uncertain parameters for determining an optimal adaptive preventive maintenance policy. Furthermore, various special cases of our model are discussed in detail.

Introduction

It is important to preventively maintain a system to avoid operational failures, especially when such failures can be costly and/or dangerous. The problem that arises in reliability studies is the question of when to maintain the system. The number of failures during actual operation should be reduced to as few as possible by means of an effective maintenance policy. In most maintenance models, it is commonly assumed that a maintenance action regenerates the system. For a complex system, the maintenance action is not necessarily the replacement of the whole system, but often the repair or replacement of a part of the system. Hence, the maintenance action may not renew the system completely. In this case, Barlow and Hunter [2] considered two types of preventive maintenance policies––one type for single-item systems and another for multi-item systems. These two types of policies have been studied extensively in the literature [3], [13], [14]. In particular, Beichelt [5] considered a general failure model and maintenance policies according to the failure type. Type 1 (minor) failures are removed by minimal repairs whereas Type 2 (catastrophic) failures are removed by replacement. After n−1 successive minimal repairs a preventive replacement is carried out at the nth failure. Furthermore, Nguyen and Murthy [17], Nakagawa [15] and Sheu and Liou [25] investigate a different set of policies for repairable systems, called sequential preventive maintenance policies, which considered the case where Type 2 failures are removed by preventive maintenance of the failure system. With these policies, there is a basic assumption that the life distribution of the system changes after each maintenance in such a way that its failure rate function increases with the number of maintenance actions.

In practice, replacement policies such as the age replacement policy or the block (periodic) replacement policy are usually adopted to preventively maintain a system. In addition to replacement actions, Barlow and Hunter [2] generalize the replacement policy by incorporating minimal repairs at failures. This model has been intensively investigated for various cases when the failure distribution of the system is known with certainty [6], [8], [12], [14], [15], [16], [24], [31]. However, in practical applications the failure distribution of a system is usually either unknown or contains uncertain parameters. In this case, it is necessary to select an appropriate estimation method to accurately calculate the parameter(s) of a given distribution and the expected mean life of the system. Researchers in this field include Gibbons and Vance [9], Lawless [10], Mann [11], Pan and Chen [18], Sinha and Sloan [27], Soland [28], Thoman et al. [29], and Varde [30]. In particular, Sathe and Hancock [21] adopt a Bayesian approach by considering prior distributions on the shape and scale parameters of a Weibull failure distribution. They use these to derive the optimal replacement policy such that the expected long-run average cost is minimized.

Taking a further step, Wilson and Benmerzouga [31] investigate Bayesian group replacement policies for the case where the failure times are exponentially distributed. Bassin [4] introduced a Bayesian block replacement policy for a Weibull restoration process and derived the optimal overhaul interval when the expected repair cost is known. Moreover, when the repair cost is constant, Mazzuchi and Soyer [12] employ the Bayesian decision theoretic approach and develop a Weibull model for both the block replacement protocol with minimal repair and the traditional age replacement protocol. However, the repair cost for system failures may be random and unknown in practice. In this paper, we extend Mazzuchi and Soyer’s model by allowing the minimal repair cost to be random. Furthermore, we propose an adaptive preventive maintenance model for a repairable system and develop a Bayesian technique to derive the optimal maintenance policy.

Our model incorporates five possible maintenance actions: minimal repair, major repair, planned replacement, unplanned replacement and periodic scheduled maintenance. A scheduled maintenance is carried out as soon as T time units have elapsed since the last major maintenance action, which includes a system replacement, major repair or previous scheduled maintenance. At the Nth scheduled maintenance, the system is replaced rather than maintained. Furthermore, when the system fails before age T, it either receives a major repair (or replaced after (N−1) maintenance) or minimally repaired depending on the random repair cost at failure. The objective is to determine the optimal plan (in terms of N and T) which minimizes expected cost per unit of time. When the failure density is Weibull, a Bayesian approach is proposed to derive the optimal adaptive preventive maintenance policy with minimal repair such that the expected cost per unit time is minimized.

The remainder of this paper is organized as follows. In the second section, the extended adaptive preventive maintenance policy is described in detail and the expected cost per unit time is formulated. In the third section, a Bayesian decision theoretic approach is established for situations where the failure density is Weibull. Finally, some special cases of our model are discussed in detail.