On the development and application of maintenance

IMA Journal of Mathematics Applied in Business & Industry (1998) 9, 91-107

On the development and application of maintenance policies for a two-component system with failure dependence P. A. SCARF AND M. DEARA

Centre for Operational Research and Applied Statistics, University ofSalford, Salford M5 4WT, UK

1. Introduction This paper considers a number of different maintenance policies for a two-component system. These policies are evaluated and compared on the basis of cost. The two-component system (hereafter referred to as the system) described in the paper exhibits failure dependence in that the probability of failure of a component depends on the current state of the other component. The system also exhibits so-called economic dependence, in that replacement (and failure) costs may be shared when both components are replaced simultaneously. Most papers to date on maintenance of multi-component systems only consider economic dependence; see Dekker et al. (1997) for a recent review. Published applications are very few (Dekker 1996). A number of models of failure dependence have been proposed. For a discussion see Murthy & Wilson (1994). In this paper we consider failure dependence in that, whenever one component fails, it can induce the failure of the other component (or one or more of the other components in the case of multi-component systems). This is called type I failure interaction by Murthy & Nguyen (1985). In particular, if we label the two components 1 and 2, then failure of component 1 induces a failure of component 2 with probability p, and has no effect on component 2 with probability 1 — p. Failure of component 2 induces a failure of component 1 with probability q, and has no effect on component 1 with probability 1 — q, where 0 ^ p, q ^ 1. If there is no failure interaction between the components, then p and q are zero and the components fail independently of one another. Other dependent-failure models include: the induced-shock model in which failure of a component acts as a shock to the other component(s) (Nakagawa & Murthy, 1993); the combination of the type I failure-interaction model and the induced-shock model; the use of a multivariate distribution to characterize joint failure times. For example, in the latter, a simple bivariate exponential distribution might be used to model joint failure-time data. Economic dependence is modelled on the basis that the cost of replacement of a component or components includes a one-off set-up cost whose magnitude does not depend on the number of components replaced. © Oxford University Press 1998

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Maintenance policies for two-component systems with failure dependence and economic dependence are considered in this paper. Policies considered are of the age-based replacement and opportunistic age-based replacement type. Where tractable, long-run costs per unit time are calculated using renewal-theory-based arguments; otherwise simulation studies are carried out. The management implications for the adoption of the various policies are discussed. The usefulness of the results in the paper is illustrated through application to a particular two-component system.

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P. A. SCARF AND M. DEARA

2. Maintenance policies for a two-component system 2.1

Notation

First we define events relating to failure and replacement •^I. -^2. fn'- failure of component(s) 1, 2, both. KF, TZF, TZF2F- replacement on failure of component(s) 1, 2, both. ~R.P, TZ2, ~R-P2P: preventive replacement of component(s) 1, 2, both. HF2 ,li.p2 : failure and preventive replacement of components 1 and 2, and vice versa. Now we define the costs: CF, CF: cost of failure replacement of component 1, 2. Cp, Cp: cost of preventive replacement of component 1, 2. CFF, Cpp: cost of failure, preventive replacement, of both components. CFP: cost of failure and preventive replacements of components 1 and 2 respectively. CP2F: cost of preventive and failure replacements of components 1 and 2 respectively. X\,T2: age of component 1, 2, since last renewal. 2.2

The policies

Consider a two-component system with type I failure interaction. On failure of the system either (i) both components have failed or (ii) just one of the components has failed. All maintenance policies we consider replace failed components immediately. Furthermore,


The maintenance policies that we consider in the paper are of the age-based replacement type: replace a component on failure or at age T, whichever is sooner (Barlow & Proschan 1965). Failure-based maintenance is viewed as the limiting case (T -*• oo) of age-based replacement. As there is also economic dependence between components, it is natural to consider opportunistic age-based replacement policies: replace a component on failure or at age T or at age T (7"' < 7") if an opportunity exists. These simple concepts are extended in the paper to the case of two components. There is scope also to consider policies of the block replacement and modified block replacement type (Berg & Epstein, 1976) for twocomponent systems but these will be considered elsewhere. The application described in the paper considers the clutch system used in a bus fleet operated by Ekspres Nasional Berhad in Malaysia; a simple engineering model considers this system comprising the clutch assembly (component 2) and the clutch controller (component 1). It was of interest to the company to investigate various maintenance policies for this system; it would be necessary and important to compare the likely returns from adopting some new near-optimal maintenance policy with the effort required (in cost terms) to implement such a policy. Maintenance practitioners are well aware that adopting a cost-optimal policy will have far reaching implications for maintenance management if the implementation is to be successful. The application itself is considered in some detail in Sections 3 and 4. However we start by describing some maintenance policies of the agebased replacement type for a two-component system in the next section. Section 3 deals with the determination of long-run cost per unit time for these policies. It should be noted that, although we imply that time measures calendar time throughout, there is no reason why it should not measure use.

TWO-COMPONENT SYSTEM WITH FAILURE DEPENDENCE

93

all but the first of the policies described in this paper replace unfailed components preventively; simple policies only consider preventive replacement at opportunities arising as a result of failure of the system; more complex policies also consider preventive replacement and opportunistic preventive replacement. As mentioned earlier, we consider, for the most part, preventive replacement in the sense of age-based replacement (Barlow & Proschan 1965); that is, replace on failure or preventively at some so-called critical age, T. We assume that preventive replacement or failure replacement restores the respective component to as new condition. The system itself is only renewed when both components are replaced simultaneously. Policy la: Independent failure-based maintenance. On failure of the system, replace the failed component(s). Using the notation of subsection 2.1, we have: Policy Ib: Combined failure-based maintenance. On failure of the system either (i) replace both failed components or (ii) replace the failed component and preventively replace the unfailed component. That is, if Tx then TZ({; if T2 then Up{; if Tn then ftf/. Policy Ic: Opportunistic failure-based maintenance. On failure of the system either (i) replace both failed components or (ii) replace the failed component and preventively replace the unfailed component if the unfailed component's age is greater than its opportunistic replacement age. That is, if T\ then: if r2 > T2' then 1lF2p; otherwise TlF; if T2 then: if n > T[ then HPX2F; otherwise 7£f; Policy Ha: Independent age-based replacement. As policy la plus: preventively replace the component if its age is greater than its critical replacement age. That is, if ri > T\ thenftf, if T2 > r 2 then7ef; if T\ then TZF; if T2 then UF; if JF,2 then UF2F. Policy lib: Combined age-based replacement. As policy Ib plus: replace both components (system) if the system's age is greater than its critical replacement age. Note here that, under this policy, components will always have same age, so we can talk about system age. Thus if Ti = r2 > T thenfcf/; if.Fi then UFP; if F2 then TlPF; \iTn then UFF. Policy IIcl: Opportunistic age-based replacement. On failure of the system either (i) replace both failed components or (ii) replace the failed component and preventively replace the unfailed component if the age of the unfailed component is greater than its opportunistic replacement age. Also, preventively replace component 1 (2) if its age is greater than its critical replacement age, and simultaneously replace component 2(1) if its age is greater than its opportunistic replacement age. Thus,

if Tn then TlF{;

if T\ then: if T2 > T2' then nF{; otherwise 1lF;


if T\ then 1ZF; if T2 then UF; \iT\i then HF{.

94

P. A. SCARF AND M. DEARA Comp. 2 2

1,2

0 1 Comp. 1

FIG. l. Policy IIcl.

1,2

Comp. 1

FIG. 2. Policy Hc2.

if T2 then: if ri > T[ then 7lf/; otherwise 7£f; if T, > 7, then: if T2 > T{ then ftf/; otherwise ftf; if r2 > ^2 then: if n > T[ then TZ(2P; otherwise ftf. This policy can be illustrated graphically as in Fig. 1. Policy IIc2: Opportunistic age-based replacement (version 2). Policy IIcl may be adapted slightly so that both components are replaced preventively if the component ages satisfy the inequality Ti>T2-

where ri and r 2 are the ages of components 1 and 2 at time T respectively. This is illustrated in Fig. 2. Policy IIc3: Opportunistic age-based replacement (version 3). As policy IIcl except that both components are preventively replaced together if either component reaches its opportunistic age of replacement. That is, if both ri > T[ and x2 > 7*2', then ftf/. This is illustrated in Fig. 3.


Comp. 2

TWO-COMPONENT SYSTEM WITH FAILURE DEPENDENCE

95

Comp. 2 2 1,2 T

2

Comp. 1

FIG. 3. Policy IIc2.

Other policies

In addition to the policies above, other policies may be considered. The age replacement policy may be modified in that the preventive replacement is performed not when the component reaches the critical age, but at the first subsequent maintenance 'window'; this window occurs when there is no demand of the system. This policy is called the modified age replacement (Barlow & Proschan 1965). The block replacement policy (BRP) performs preventive replacement of a component at fixed time epochs kT (k = 1,2, 3,...) irrespective of its failure history in the preceding interval. Under this policy there is no need to keep detailed records about the ages of the components, and this simplicity is the main advantage of the use of the BRP; it has the drawback that nearly-new components may be replaced at the planned replacements. Berg & Epstein (1976) proposed the modified block replacement policy to avoid this drawback; this policy replaces components on failure and at the fixed time epochs kT (k = 1,2,3,...) only if the component age is greater than some critical age b (0 < b < T). These policies may be extended to two-component systems in the manner of the policies in subsection 2.2.

3. Determining optimal policies Here we discuss how the optimal policy among each class of policies may be determined. The long-run cost per unit time is used as our optimality criterion. For some classes of policies we can determine the optimal policy analytically; for the other policies we use simulation. In all calculations the type I failure-interaction model is assumed—recall from section 1 that a failure of component 1(2) induces a failure of component 2(1) with probability p(q) (0 < p, q ^ 1). Also throughout we take Fi(/) and F2(t) to be the distribution functions of the lifetimes, X\ and X2, of components 1 and 2 respectively. Policy la: Independent failure-based maintenance. Under this policy, the system is renewed when both components fail. Murthy & Nguyen (1985) derive the long-run expected cost per unit time as

f h(x)dx]/ f

Jo

Jo

(3.1)


2.3

96

P. A. SCARF AND M. DEARA

where G(x) = 1 - [1 - G,(x)][l - G J O O L

(3-2)

h(x) = C,f(i—^)(1 -G2{x))gl{x) + C${]—3-)(\ - G,0O)ft0O. P 1 with

1 - G,0O = n=O

and

1 - G2(x) = £ (

f

2

Here F,(0)(x) = 1 and F{n\x) is the n-fold Stieltjes convolution of F\ (x), and likewise for Fi(x). The first term in the numerator of equation 3.1 is the cost of system renewal, and the second term is the total cost of natural failures (where a component fails and does not induce a failure in the other component) over the system renewal cycle. The denominator is just the expected length of the system renewal cycle. G(x) in equation 3.2 is the distribution function of time to system failure induced by a failure of component i (1 = 1, 2). Policy Ib: Combined failure-based maintenance. For this policy, whenever the system fails, both components are replaced regardless of whether both have failed, and the system is renewed. Thus the long-run cost per unit time is, from the renewal theorem (Cox 1970), f/ + (1 - p)CF2p] /0°° F,OQdFaOO + [qCtf + (1 -