DU Jingyu , LI Jianping , HU Yimfun,GUAN Xin , SI Min, LIU Bin

(1. Faculty of Engineering and Informatics,University of Bradford, Bradford , UK;2. School of Electrical Engineering, Shenyang University of Technology, Shenyang 110870, China)

Abstract:Effective maintenance is a key for infrastructures′ high operational reliability. The integration of corrective repairs and schedule-based failure preventions has been a mainstream of modern maintenance, and an associated policy-making technique, delay-time modelling, is overviewedin this paper for optimising the maintenance cost-efficiency in different practical scenarios, including imperfect, opportunistic and nested maintenance. A few typical examples of its applications in minimising maintenance operating expenses are discussed in this paper and their results are explained to better demonstrate the benefits of the technique. This work aims to prepare for the future applications of the delay-time modelling in railway maintenance policy making.

Keywords: maintenance; failure prevention; delay-time modelling; cost minimisation

1 Introduction

Maintenance includes a series of activities to keep equipment functioning during its lifecycle and is usually designed according to the equipment′s failure rate, which is usually in the shape of a ‘bathtub’ curve including ‘infant mortality’, ‘useful life’ and ‘wearout’ periodsduring the equipment′s lifecycle[1]. Many work, e.g. maintaining electronic equipment[2], predicting machines′ availability[3], and analysing the reliability of executing software[4], consider failure rate as a constant in equipment′s ‘useful life’ period. A constant failure rate can be characterised with exponential distributions[5-7], and failures are assumed to be independent of each other. The failures in the ‘wearout’ period may also follow an exponential distribution, but it is usually modelled by either a normal distribution or a Weibull distribution[8].

Minimising equipment′s operating expenses (OPEX) has been one of the most important objectives for many maintainers. Common maintenance policies are dominated by two strategies-corrective repairs and schedule-based failure preventions, with a tendency towards more condition-based. An overview of the maintenance strategies and focuses on the delay-time modelling to optimise the cost-efficiency of the two dominating strategies′ integration is presented. It aims to study the delay-time modelling and its methodology as the preparation for the future research into railway maintenance optimisation.

The Section 2 present the maintenance strategies. The delay-time concept is presented in Section 3 and three key delay-time models and their applications are discussed in Section 4. Finally the conclusions are given in Section 5.

2 Maintenance strategies

Generally, maintenance is either corrective or preventive.

2.1 Corrective maintenance

Corrective maintenance requiresactions to be carried out when equipment′s performance is worse than its users′ expectations or the equipment has broken[9]. The effectiveness of this strategy is measured generally by how fast the system′s performance is restored after an underperformance or a failure is identified. Without remotely condition monitoring (RCM), failures are hardly known a priori and corrections are therefore performed unpredictably[10]. The nearly stochastic occurrence of failures usually disrupts productions and operations and results in penalties and reputational damages. Therefore, the focus of many maintenance policies has been changed from corrective to preventive.

Corrective maintenance activities are classified into three types based on the outcome of the maintenance activates, namely perfect repair, imperfect repair and minimal repair[11]. Perfect repair restores equipment′s condition to its initial ‘entry-into-service’ state (‘as good as new’). Minimal repair restores equipment′s condition to the state nearly immediately prior to the failure and has little improvement on equipment′s failure rate (‘as bad as old’). Imperfect repair is between the above two. These three levels of maintenance are also similarly applicable to other maintenance strategies.

2.2 Preventive maintenance

Preventive maintenance is usually either schedule-based or condition-based. The former has been a traditional mainstream in many industries to control failure rate at a low level (e.g. the ‘sawtooth bathtub’ curve is shown in Figure 1). The schedule is usually based on the manufacture′s recommendations[12], subjective data e.g. engineering opinions obtained by questionnaires[13-14], or objective data e.g. maintenance and performance records[15].

Figure 1 A ‘sawtooth bathtub’ curve showing failure rate being managed by scheduled maintenance[16]

The OPEX associated to schedule-based preventive maintenance was on average three times less than it of corrective maintenance in engineering[17], and the total downtime was reduced by more than 20% for a factory process of copper and copper alloys extrusion press[15]. Schedule-based preventive maintenance requires a schedule and carefully-designed maintenance activities for every piece of equipment with the considerations of four typical types of dependencies namely economic dependence[18], structural dependence that affects performance or increases the complication of repairs, stochastic dependence of failures, and, occasionally but more and more importantly, resource dependence as a part of strategic spares management[19].

Many researchers have improved the maintenance scheduling by using the advanced mathematical approaches, e.g. developing an aggregate model to integrate maintenance scheduling and production planning[20], using a heuristic method based on the nested general variable neighbourhood search[21], and designing flexible warranty plans for future services[22-23]. In particular, many schedule-based maintenance policies are based on the concept of delay-time[24], i.e. the duration from a fault′s occurrence leading to an operational failure.

As illustrated in Figure 2, inspecting an equipment everyt, maintainers can prevent 3 failures, whereas prevent 4 failures if inspect at every Δ(t/2). In addition, maintainers can manage (extend) defect′s delay time to avoid expensive emergency repairs; the extension enables multiple maintenance activities to be carried out in one access[25].

Figure 2 Delay time and inspection period for defect prevention[26]

RCM is often aimed at maximising the effectiveness of preventive maintenance decision making[12], relying on emerging technologies of sensors and Internet of Things for data acquisition[27-28]. Sensors measuring quantities that indicate the health and actual state of equipment[29]. The indications are generally better than time of operation or running hours can offer for prognoses. Based on the historical records, experiments and maintenance knowledge, asset conditions can be classified into different levels of severity according to prognostic indicators such as rusty surface, abnormal noise and vibration, pressure difference, smoke emission, slow response to human instructions, etc[30-32]. Real-time monitoring or theoretical study is usually carried out to understand the asset′s deterioration and remaining lifespan, and the cost of whole-life maintenance, unexpected failures and end-of-life renewal can be estimated[28]. Markov chain analysis has been widely used for theoretical study, e.g. in the asset management for railway overhead line stanchions in Japan[33]. Long-term condition records are very useful to predict equipment′s failure occurrence, e.g. an deteriorating trend (Figure 3) indicates emerging problems[29]. However, ‘maintenance-free’ equipment rarely exists, especially the sensors monitoring the condition of national infrastructures. Delicate sensors′ exposure to all weathers renders traditional corrective and preventive maintenance important for RCM.

Figure 3 Trending equipment′s condition assists in failure prevention[29]

3 Delay-time concept

The delay-time concept is very helpful for scheduling the routine activities and prioritising non-instantaneous failures. It is initiated by Christer[34]and pioneeringly trialled for maintaining the machinery of a production line at a pet food factor, its high-speed product canning process[13], and then the success was replicated in another factory for scheduling its routine inspections for an extrusion press machine of copper and copper alloys production process[15].

For repairable equipment with defect occurring at timeubut are not instantaneously disruptive, the equipment′s condition deteriorates over a periodhand finally fails at timeu+h. If an examination had been carried out in the interval (u,u+h) and the defect had been identified by the examination and rectified before a timeu+h, then the unexpected failure would have been prevented[31].his therefore defined as the delay time of defect. The models and studies being based on the delay-time concept require two important pieces of knowledge, the probability density function (PDF) of each defect′s arisingλi(u) and defects likelihood PDFfi(h) of failing the equipment at delay timeh. As illustrated in Figure 4, with scheduled perfect maintenance interval beingT, a defect with the delay timehwill lead to a failure if its arising timeu∈(t,t+T-h).

Figure 4 A defect leads to failure if it arises between t and (t+T-h), illustrated by the delay time concept

The above is a typical delay-time example with the three stages of healthy, defective and breakdown. More stages can be added to the consideration, if a research team′s computational capability allows, to better replicate the equipment′s deterioration and to enable the analysis and the preventive maintenance scheduling being more cost-efficient. An example is splitting the defective stage into original defect and serious defect[20].

The expected number of defects that will lead to a failure in a scenario of perfect maintenance is calculated as

(1)

The assumptions are that a defect arises in the interval of (t,t+T) has a constant rate of occurrencekand the defect′s occurrence is independent to the defect′s delay timeh, the probability of which isf(h)dh. The defect will lead to a failure repair in the period of (t+T-h,t+T). With the constant failure rate and a further assumption that a defect will arise, the probability of it arises before (t+T-h) is (T-h)/T, and the defect arising in the period Δtis approximated askΔt. The expected equipment downtime in the periodTis

D(t,T)=A(t,T)db

(2)

Wheredbis the failure′s mean down time (MDT).

Then, the maintenance cost during the periodTis

C(t,T)=cbA(t,T)+cr[kT-A(t,T)]+ce

(3)

Where:cbis the failure cost;cris the cost of defect rectification;ceis the routine maintenance cost.

For the general cases, when the defect arising (failure) rateλ(u) is not a constantk, equation (1) is updated as

(4)

The above three formulae ofA(T),C(T) andD(T) are the foundation of delay time modelling[35]. Many of the early delay-time models are based on the perfect maintenance i.e. any inspections and repairs are 100% effective and faultless[13,36]. The modelling results in Ref.[13] are plotted in Figure 5a) which indicates the optimal inspection period is 30 hours. The study also considers the benefits from vulnerable components being re-designed and becoming more reliable, Figure 5b). The results are in line with the engineering expectation that with reliability being higher, the optimal inspection period becomes longer.

Figure 5 The total downtime at different inspection periods[13]

4 Delay-time models

The applicational research on delay-time modelling in the past decades has developed it more mature and popular, and recent studies include the maintenance for traction locomotive[37], airport runway[38]and second-hand components in manufacturing machinery[39]. This chapter presents a few carefully selected examples of using delay-time modelling in the scenarios of imperfect, nested and opportunistic maintenance to optimise the maintenance policies that have the integration of corrective repairs and schedule-based failure preventions. Most of the examples are motivated by cost saving, minimising the per unit time OPEX.

4.1 Imperfect maintenance

Many maintenance activities are imperfect, and this leads to additional parameters being introduced to delay-time models to incorporate the effect of imperfection, mostly in terms of the probabilities[7,14].

(5)

Andwheni=n,wecanget

(6)

Withn→∞andeachmaintenanceintervalbeingfixedasdefectsariseinanhomogeneousPoissonprocess(HPP),andQrepresentstheproportionoffailureswithnodelaytime,themodifiedA(t,T)forimperfectmaintenanceinthefollowing[15].

(7)

ThemodellingresultsareplottedinFigure6,whichshowsthatiftheexpecteddowntimeduetomaintenanceis120min,thentheroutinemaintenanceshallbescheduledevery2-3weeks[14].

Figure 6 Expected downtime per press hour for different maintenance intervals[14]

Regardinginspection-inducedfailures,Ref.[7]incorporatesfailuresinmodellingbyusingthetwoparametersr1andr2torepresentthelikelihoodofinspection-inducedfailures,whicharerepairedwithnodelay,occurringduringaroutineinspectionbeforeadefectarisesandduringadefect′sdelaytimeh,respectively.Thestudyalsousesmaintenanceeffectivenesslikelihoodpandq,whereprepresentsthelikelihoodof‘falsepositive’andqrepresentsthelikelihoodof‘falsenegative’.Aroutineinspectionis(1-p-r1)likelytobeeffectiveforhealthyequipment,whereas(1-q-r2)effectivefordefectiveequipmentduringdelaytimeh.Thestudymodelstwoscenariosofdefectarisingthatareintheformatofexponentialdistributionfornon-agingequipment(Scenario1)andofWeibulldistributionforagingequipment(Scenario2).Thedelaytimehdistributesexponentially.Theparametersarevaluedasp=0.1, q=0.7, r1=0.1andr2=0.2.Thestudyistofindouttheoptimalcostperunittime.

Scenario1includesthreecases:(1)noplannedreplacementonlyroutineinspectionsastheequipmentisnotaging, (2)planningreplacementateverttimeTwithoutinspection(1, T)fortheconsiderationthattheprobabilityofinspectioninducingfailureishigh, (3)runningtofailure(1,∞),aspecialcaseof(1, T).

Figure7showsthemodellingresultsofanumericalexample.Theresultssuggesttheoptimaldecisioncomingfromcase,notplanningreplacementbutinspectingatevery3.6timeunits.Scenario2solvestheoptimisationwithtwoparameters(N, T),planningreplacementatNthinspectionandplanninginspectionateverytimeT.ThemodellingresultsareplottedinFigure8.Theresultssuggesttheoptimalplanbeinginspectionatevery3.8timeunitsandreplacementatthe3rdinspection.

Figure 7 Cost per unit time for the three cases of the Scenario 1[7]

Figure 8 Cost per unit time for the Scenario 2[7]

4.2 Nested maintenance

Figure 9 Nested inspections of a schedule-based maintenance policy

The expected costEC(N) is the summation of three constituents, the cost of inspection, failures, and defect rectification without a failure. WithCI,i,CF,iandCR,ibeing the cost of anith inspection, a failure due to an undetectedith or (N+1)thdefect, and defect rectification for a detected ith defect,λiandλN+1being the constant rate of the ith or the (N+1) th defect′s occurrence, andFibeing the cumulative distribution function (CDF) of the ith defect′s delay time, the expected per unit time cost over a periodTNis calculated as

(8)

Another example of nested maintenance is presented in Ref.[41]. It models another maintenance optimisation problem having two failure modes (a minor and major defect) and two maintenance actions (a scheduled minor and major inspection) of a factory′s production line. The products′ quality is immediately affected by the existence of the minor defect. The impact on product quality is reflected as the factory′s per unit time profit loss fromP1toP2. The minor defect is more likely to occur with the existence of the major defect, which has the delay-time nature, with initiation timeX1having a PDFf1(x1) and a CDFF1(x)=Pr(X1

The minor inspection is scheduled every periodtand is nested in the major inspection which is scheduled every periodTthatT=kt,kbeing an integral parameter. Bothtandkare parameters to be optimised. The minor defect is identified by the minor inspection which is perfect but cannot reveal the existence of the major defect. The major defect can be identified by the production failure or by the major inspection, which is imperfect with the probability (1-r). Both the minor and major defect are rectified immediately after being identified. The minor defect′s rectification time and both inspections′ duration are considered negligible. The maintenance problem has been pictured with only the major defect in Figure 10.

Figure 10 Delay-time of the major defect with imperfect inspections[41]

The optimisation is to search for the optimal (k*,t*) so that the expected per unit OPEX will be the minimal for the period between two major defect rectifications. The modelling mathematics are complicated, and to simplify the calculation, based on the practical experience thatt≪T, it is approximated that the major defect initiates and fails the production line only immediately after a minor inspection. The study provides a numerical example that demonstrates the effectiveness of the model. The optimisation results are plotted in Figure 11 and the optimal results is at (k*,t*)=(4,9).

Figure 11 Revenue calculation results for different t and k[42]

4.3 Opportunistic maintenance

Unexpected equipment failures bring the opportunities for checking other ‘healthy’ components of the equipment during the failure repair. As the equipment is usually dismantled, the inspect and repair for other components at the time can be cheaper and timesaving. Opportunistic maintenance[43]and the potential improvement in reliability is proved theoretically.

5 Conclusion and future work

This review starts from introducing common maintenance strategies for controlling the equipment′s whole lifecycle failure rate, and then focuses on the applications of delay-time modelling for scheduling the preventive maintenance activities cost-efficiently. Many successful applications being published so far are in factories, workshops, and other places where accesses are direct and generous.

The future research of delay-time modelling can extend the success to national infrastructures e.g. railway that the access is generally remote and more limited, and explore the areas of scheduled maintenance activities being delayed, delay-time models′ parameter estimation, calibration, and computational feasibility and/or approximation, varying maintenance intervals, and connecting to spare parts management.

Another possible research will be the combination of the delay-time models with artificial intelligence techniques to improve the efficiency of maintaining railway equipment and to providea higher reliability of train operations.