Linking component importance to optimisation of preventive maintenance policy, Reliability Engineering and System Safety, http://dx.Abstract. In reliability engineering, time on performing preventive maintenance (PM) on a component in 14 a system may affect system availability if system operation needs stopping for PM. To avoid such an 15 availability reduction, one may adopt the following method: if a component fails, PM is carried out on a 16 number of the other components while the failed component is being repaired. This ensures PM does not 17 take system's operating time. However, this raises a question: Which components should be selected for 18 PM? This paper introduces an importance measure, called Component Maintenance Priority (CMP), that is 19 used to select components for PM. The paper then compares the CMP with other importance measures 20 and studies the properties of the CMP. Numerical examples are given to show the validity of the CMP. 21 importance 23 1 Introduction 24 1.1 Motivation 25 To improve the availability of engineered systems such as production lines and electricity 26 transmission networks is the common pursuit of many firms. To achieve a high availability level, 27 preventive maintenance may be used. However, performing preventive maintenance (PM) on a 28 component in a system takes time and can therefore reduce the availability of the system if system 29 operation needs stopping for the PM. To avoid such a dilemma, one may adopt the following method: if a 30 component in the system fails, PM is carried out on a number of the other components while the failed 31 * Corresponding author. Email address: s.m.wu@kent.ac.uk; telephone: 0044 1227 827 940 2 component is being repaired. However, this raises another question: Which components should be 32 selected for PM? 33 Reliability importance measures are developed to prioritise the components of a system in the light 34 of a given criterion and can offer guidance to improve system reliability/availability, reduce maintenance 35 cost, and improve system safety. For example, the Birnbaum importance is the partial derivative of the 36 system reliability with respect to the reliability of an individual component and measures the effect of the 37 reliability improvement of individual components on the improvement of the system reliability [1]. In the 38 reliability literature, many importance measures have been developed for various purposes, see [2] for an 39 excellent paper that reviews recent advances on importance measures. However, few importance 40 measures can be used to select such components for the above-mentioned purpose. 41 This paper extends the Birnbaum importance measure to an importance measure, called CMP 42 (Component Maintenance Priority), with which components can be selected for PM and the number of 43 components for PM may further be optimised. 44 1.2 Related work 45 Various component importance measures for binary coherent systems and state importance 46 measures for multi-state systems have been introduced in the literature. For example, the Fussell and 47 Vesely importance of a component is the probability that at least one minimal cut set containing the 48 components has failed, given that the system has failed; the criticality importance of a component is the 49 probability that the component has caused system failure, when the system is failed. The reader is 50 referred to the monograph by Kuo and Zhu [3] for detailed accounts of the theory and surveys of 51 commonly used importance measures. 52 The Birnbaum importance is probably the first importance measure introduced in the literature, for 53 the purpose of reliability improvement [1]. The Birnbaum importance measures the extent of the change 54 in the reliability of the system resulted from a change in the reliability of a component. It has been 55 extended to many variants considering different scenarios and applications, for example, cost-based 56 importance measures [4] that considers the lifecycle cost of maintaining each component in a system. 57 Other variants include performance based importance measures [5], joint component importance [6-8], 58 and joint component importance for multistate systems [9]. 59 Recent development in importance measures also include: importance measure for systems with 60 degrading components [10], importance measure that estimates the effect of a component residing at 61 certain states on the performance of the entire multi-state systems [11], importance measure for 62 components when the system may be reconfigured [12], among many others ([13-15], for example). 63 Importance measures in the literature may be inter-related. Borgonovo shows that the Fussell-64 Vesely importance, the criticality importance, the Birnbaum importance, the risk achievement worth and 65 the differential importance measure (DIM) are linked by simple relations [16]. Vaurio shows that DIM 66 and the criticality-importance yield the same ranking in realistic examples [17,18]. Furthermore, 67 3 according to Birnbaum [1], importance measures can be categorised into three classes according to the 68 knowledge needed for determining them: structural importance measures, reliability importance 69 measures, and lifetime importance measures. For example, Fig. 1 shows the process of available knowledge 70 and the corresponding importance measures that can be used. 71 72 Here goes Fig. 1 73 74 A common drawback of the Birnbaum importance measure and its variants is that they rank only 75 individual components and are not directly applicable to groups of components. The differential 76 importance measure (DIM), introduced by Borgonovo and Apostolakis in [19], overcomes this drawback 77 by defining the importance of a group of components using a first-order Taylor expansion, but it does not 78 account for the effects of interactions among components. Zio and Podofillini then extended the DIM 79 including both the first order and the second order Taylor expansion, which has a merit that is account of 80 the interactions of pairs of components [20]. 81 The existing reliability literature, however, lacks an importance measure for solving the following 82 question: suppose that a component is failed and during the time when the component is being repaired, 83 other components in the system can be selected for PM. This raises an interesting question: which 84 components should be selected? This paper develops a new importance measure, the CMP, for answering 85 this question. The CMP can also be applied to schedule PM policy. Conventionally, optimisation of PM has 86 been centred on seeking the optimal intervals between consecutive PM activities. This paper, however, 87 optimises the number of maintenance personnel needed to minimise the expected cost in a given time 88 horizon. As such, in addition to its novelty of introducing a new importance measure, the paper also 89 creates the novelty of proposing a new method of optimising PM. 90 1.3 Summary 91 This rest of this paper is structured as following. Section 2 lists the notation and assumptions, and 92 discusses the justification of the assumptions. Section 3 introduces the CMP and discusses its relationship 93 with some existing importance measures. Section 4 gives upper and lower bounds of the expected 94 number of PM under two maintenance policies. Section 5 discusses issues relating to the CMP. Section 6 95 gives numerical examples. Section 7 concludes the findings of this paper. 96 2 Notation and Assumptions 97 The following notation and assumptions are used. 98 99 This paper uses the notation shown in Table 1 . 100 101 4 Here goes Table 1 102 2.2 Assumptions 103 A1. The system considered in this paper is a coherent system, which implies: each component is relevant, 104 and the structure function is increasing (non-decreasing) if the number of components increases. A2. Once a component is failed, a certain symptom immediately appears and can be noticed. The failed 106 component can then be located. A3. There are two types of PM: PM during system's downtime and PM during system's uptime, which are 108 denoted by PMD and PMU, respectively. 109 A4. Performing a PM (either PMD or PMU) on a component requires that the component stops working. 110 A5. A PM (either PMD or PMU) on a component can only be triggered when another component has failed. 111 A6. Component failures can have two different situations. 112 (A) When a critical component has failed, the system fails. Then the failed component is repaired. In 113 the meantime, PMD is performed on other selected components. 114 (B) When a non-critical component has failed, the system does not fail. Then the failed component is 115 repaired. In the meantime, PMU is performed on other selected components. The repair and the 116 PMU will not affect the system operating. 117 A7. All of the components in the system are statistically independent. 118 Remark 1. From Assumption A3, the PMU policy is: PM is performed when the system is still 119 working. This implies that a binary system (for example, a parallel system) under the PMU policy may 120 never fail. Take the system in Fig. 2 as an example, under Assumptions A3, A4, and A5, the subsystem 121 constituted by components 5, 6, and 7 may never fail. This is because: once one of the three components 122 fails, the failed component will be immediately repaired while the subsystem is working. This will ensure 123 that the subsystem will never fail. However, if we use the PMU policy on multistate systems, performing PMU can improve the 125 performance of the system. For example, for a water pumping station that is composed of three pumps, if 126 pump 1 degrades from a higher state to a lower one (or is failed as termed in this paper), its performance, 127 which is the amount of water a pump can pump, degrades and the pump may need repairing. The PMU 128 that is performed on pump 2 can improve the performance of the pump, for example. In essence, a jump from a higher state to a lower one in a multistate system (component) is the same 130 as the failure of a binary system (component). As such, in what follows, we simply focus our discussion on 131 binary system (component) cases. 132 3 A new importance measure: component maintenance priority 133 The main effort of this paper focuses on the development and analysis of a new importance measure 134 for binary systems. General cases are also discussed in this section. 135 157 performed on unfailed components only when the number of the components is larger than 2. 158 This is because PMU is only performed on a component when the component stops working 159 and the system is working. When a 2-component parallel system includes one failed 160 component, the unfailed component must be working and cannot be stopped for PMU. 161 From the above analysis, one can see that the measure ( ) can only be used for the scenarios when 162
Also, Wu et al. REF introduced an importance measure, called CMP (component maintenance priority), which was extended from Birnbaum importance measure and used to select components for preventive maintenance.