Reliability analysis of a maintenance network with repair and preventive maintenance.pdf

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International Journal of Quality & Reliability ManagementReliability analysis of a maintenance network with repair and preventive maintenance

Gauri Shankar Vandana SahaniArticle in format ion:

To cite this document:Gauri Shankar Vandana Sahani, (2003),"Reliability analysis of a maintenance network with repair andpreventive maintenance", International Journal of Quality & Reliability Management, Vol. 20 Iss 2 pp. 268 -280Permanent link to this document:http://dx.doi.org/10.1108/02656710210429627

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A. Crespo Mrquez, P. Moreu de Len, J.F. Gmez Fernndez, C. Parra Mrquez, M. LpezCampos, (2009),"The maintenance management framework: A practical view to maintenancemanagement", Journal of Quality in Maintenance Engineering, Vol. 15 Iss 2 pp. 167-178 http://

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R el ia bi li ty a na ly si s o f a

m a in t en a nc e n e tw o rk w i thr ep ai r a nd p re ve nt iv emaintenance

Gauri Shankar and Vandana SahaniPt. Ravishankar Shukla University, Raipur, India

KeywordsWear, Preventive maintenance, Failure, Weibull analysis, Reliability

AbstractThis paper considers the maintenance of an operational system consisting of a numberof independent and identical units. These units are required to be in service at all times and are

supported by a maintenance float. Reliability analysis is used to study the effect of preventivemaintenance and repair on the float factor determination. Numerical examples have also beenincluded to illustrate the mathematical findings.

NomenclatureN The number of units initially in operationn The number of free floating units (those on reserve plus those undergoing and/or

waiting for repair/maintenance)MTBFM Mean time between failure/preventive maintenanceWM Weighted mean of times to repair and/or maintenance

Nt Number of units functioning at time ttn The time the nth failure/preventive maintenance occurs

R(t) The system reliability is defined as the proportion of units functioning at time t

f The maintenance float factor. It is defined as the proportion of units that have failedup to time tnF The maintenance total float required to support the number of units in operation:

F Nfp Proportion of units sent for repairq Proportion of units sent for preventive maintenance: q 1 p

Greek charactersm1(m2) Repair (maintenance) rates for exponential distributionh1h2 Scale parameters of mixed Weibull distributionb1b2 Shape parameters of mixed Weibull distributiong Location parameter of mixed Weibull distribution

IntroductionDuring the last few decades, the study of maintenance float models of an

operating system have attracted the attention of several research workers. This

class of problem was originally described as a queuing model by Barlow et al.

The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at

http://w w w .emeraldinsight .com/researchregister http://w w w .emeraldinsight.com/0 2 6 5 -6 7 1 X.htm

Financial assistance from MAPCOST, Bhopal (Grant No. Maths 13/90) to Dr Gauri Shankarand SRF from CSIR, New Delhi to Ms. Vandana Sahani is gratefully acknowledged. The authorsare also thankful to the referee for a number of valuable suggestions.

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Reliability Management

Vol. 20 No. 2, 2003

pp. 268-280

q MCB UP Limited

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DOI 10.1108/02656710210429627
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(1965) and is referred to as repairmans problem. One of a few analyticalsolutions to the total float determination problem was given by Levine (1965).They introduced a reliability factor that is based on the ratio (MTTR/MTBF) asa variable and derived the maintenance float policy for an operational systemassuming exponential failure and repair distributions. Later on, themaintenance float problem was investigated respectively for Weibull (Loweand Lewis, 1983), Gamma (Madu, 1987; Shankar and Sahani, 1994a),normal/lognormal (Madu and Chanin, 1989), Rayleigh (Shankar and Sahani,1993), Burr (Shankar and Sahani, 1994b), and inverse Gaussian (Shankar andSahani, 1998) failure distributions. The objective of the maintenance floatproblem is to determine the minimum maintenance float to minimize theaverage equipment downtime or to maximize the machine availability. Themachine availability is defined as the ratio of uptime to the total operating time:

AvailabilityUptime=UptimeDowntime:

Most of the models in the maintenance float literature have, however, focusedonly upon the machine failure and its repair. The failure models discussed sofar comprised a single distribution characterising either a random (time-independent) or a wear-out (time-dependent) failure. In practice, an unit cansuffer either one of these failures, and a statistical model characterising thefailure process should provide for both the eventualities.

Mann et al. (1974) remarked that mixed distribution models can beconsidered as a special case of a general time to failure distribution. Kao (1959)considered mixed Weibull distributions in the reliability study of electrontubes. Mixed models have also been used by several workers (Cohen, n.d.;

Harris and Singpurwalla, 1968) though in different context. Mann et al.(1974)further motivated the use of mixed Weibull distributions to describe the failurepattern (of the devices) classified as types:

. catastrophic or sudden; or

. wear-out or delayed.

The purpose of this paper is to study the maintenance float problem as a modelto describe both the catastrophic and/or wear-out failures. Here, an attempt hasbeen made to study, through some numerical calculations, the effect of wear-out preventive maintenance and repair on the float factor evaluation. Themaintenance and repair times are assumed to follow exponential distribution.

Numerical examples have also been given to illustrate the potential applicationof the model. The procedure discussed here provides a decision tool for systemdesigners and maintenance managers. Finally, curves have also been drawn toexplore the implications of the model.

D e sc r ip t io n o f t h e s y st e mThe operating system described consists of a number, N, of independent andidentical units. TheseNunits are required to be in service and are supported by

Reliability of amaintenance

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a maintenance float. Whenever, any of the initial units in simultaneousoperation fails it is released for repair. The failed unit goes from operationcentre (node A) to repair centre (node B) while a unit from maintenance reserve(node D) is sent in to replenish the operating system. Similarly, the units whichare exposed to wear-out failures are sent to node C for preventive maintenancebecause wear-out commences after a specific time period has elapsed and a unitfrom reserve again restores the number of operating units to N. At thecompletion of repair/maintenance, the unit goes to reserve and awaits the callfor further service. It is further assumed that the probability of any unit beingexposed to sudden failure is constant (p) and the probability of any unit underpreventive maintenance isq(= 1p). The maintenance float network is shownin Figure 1).

M o d el f o r m ul a t i on

Consider devices whose failures can be classified as types:. catastrophic or sudden; or. wear-out or delayed.

Catastrophic failures occur as soon as the device is exposed to risk and suchunits are sent to repair centre for repair. On the other hand, the units subject towear-out or delayed failures are sent for preventive maintenance after a specifictime period has elapsed. It is generally believed that the wear-out failures aredue to an aging or a decaying of the material or the device.

F i g u re 1 .Maintenance floatnetwork

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Here, the maintenance float factor (MFF) is derived for mixed Weibulldistribution subject to repair/preventive maintenance. The followingassumptions are made:

.

the units are completely rejuvenated after repair/maintenance;. the waiting time for repair/maintenance is negligible; and. preventive maintenance of an operating units is made only when there is

at least one unit in the reserve.

The distribution functionF(t) for mixed Weibull distribution (Mann et al., 1974)is given by:

Ft p12 exp{2 t=h1b1 } 12p12 exp{2 t2 g=h2

b2 }:

Mann et al. (1974) pointed out that a Weibull distribution with a locationparameteroand a shape parameter b1(, 1) would be an appropriate model fordescribing catastrophic failures. Since wear-out commences after a particulartime, the Weibull distribution with a non-zero location parametergand shapeparameter b2(. 1) would be an appropriate model for preventive maintenance.

The reliability function is, thus, obtained as:

Rt 1 2Ft;

or:

Rt p exp{2 t=h1b1 } 12pexp{2 t2 g=h2}

b2 }: 1

We know that, for an exponential repair distribution with parameter m1,MTTR 1/m1. Similarly, for an exponential maintenance distribution withparameterm2, the mean time to preventive maintenance, MTPM 1/m2. Now,following Shankar and Sahani (1994a), the weighted mean (WM) of time torepair/maintenance is defined as follows:

WM p MTTR 12PMTPM:

The failure and repair/preventive maintenance patterns of individual unitswithin an equipment configuration can be analysed in terms of renewal theory.This presumes perfect renewal i.e. there is no distinction between theperformance of repaired/maintained and new equipment. Also, there is no

queue for repair/preventive maintenance and the repaired/maintained unitsreturns immediately into service. Given an extended operating period, expectedtime for the nth renewal of an unit can be expressed as T(n), where:

Tn n MTBFWM: 2

Now, withNunits initially to operation and nunits in the float, the criterion forsizing the float (Levine, 1965) is given as:


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tn $ t1 WM; 3

that is, the point where the time for the nth failure and/or maintenance equalsor exceeds the first renewal time on average.

The reliability at timet, is the fraction of functioning units at timet. If thereare Nunits initially and at time t, only Ntare functioning, then:

Rt Nt

N p exp{2 t=h1

b1 } 12pexp{2 t2 g=h2b2 :

Thus, the time to first failure/maintenance, t1, can be obtained as:

Rt N2 1

N p exp{2 t1=h1

b1 } 12pexp{2 t1 2 g=h2b2 }: 4

This complex hyperbolic function cannot be solved directly for t1. Therefore,

numerical method is utilised to find the solution.Now, on solving equation (4) for t1, we have:

t1 fN;b1;h1;h2;g;p say 5

wheref1(N,b1,b2,h1,h2,g,p) is some function ofb1,b2,h1,h2,g,p and N,andt1can be obtained uniquely for given values of parameters. Consequently,on substituting t1and WM in equation (3), one gets:

tn fN;b1;h1;h2;g;p WM: 6

For given values of parameters and MTTR and MTPM, equation (5) can be

solved for tnby numerical methods.Therefore:

Rtn p exp{2 tn=h1b1 } 12pexp{2 tn 2 g=h2

b2 }:

Finally, MFF can be determined as:

f 1 2Rtn:

The total maintenance float, F, required to support an operational group N, isfound out as:

F f:N:

Here, it may be noted that Fprovides an estimate ofn.

RemarkIt may be noted thatfis characterised by six parameter namely, b1, b2, h1, h2, gandp. In order to simplify the derivation, some relation between the parametershave been assumed as follows.

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Here, let us assume that the parameters of the distribution are such that:

e2y1 k e2y2

wherey1 t=h1b1

and t2 g=h2b2

.Thus, we have:

Rt Nt

N p k e2y2 12pe2y;

or:

Nt

N p:k12pe2y2 ;

or:

exp2y2 Nt=Np:k12p;

2y2 lnNt={Np:k12p};i.e.:

t2 g=h2b2 ln{Np:k12p}=Nt;

th2ln{Np:k12p=Nt}1=b2 g:

The time to first failure/maintenance, thus comes out to be:

t1

h1

ln{Np:k12p

=N2 1}1=b2 g

:Therefore:

tn h2ln{Np:k12p=N2 1}1=b2 WM:

Hence,R(tn) andfcan be expressed as:

Rtn p kexp{2 tn 2 g=h2b2 } 12pexp{2 tn 2 g=h2

b2 };

and:

f 1 2Rtn:

For some selected values of parameters,k and fare given in Table I.

k 1 3 5 7 9 11 13 15

f 0.322 0.602 0.688 0.703 0.725 0.742 0.754 0.764

Not es:N 30; h2 200; b2= 2; MTTR 100 hours; MTPM 80 hours; p 0.4

T a bl e I .kand ffor some

selected values ofparameters


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M o d el i l l u st r a ti o nThe mean of mixed Weibull distribution is expressed in terms of the gammafunction as follows:

MTBFM ph1 G11=b1 12pg h2G11=b2:

The float factorfis calculated for various values of MTBFM, WM andpkeepingother factors constant. As a number of factors affect the system, It is convenientto vary each in turn to establish its effect on the maintenance float factor.

A computer program in BASIC is developed to find t1and hencefby usingNewton-Raphson modified method. In order to explore the implications of themodel, some of the results are shown in Figures 2-6.

Figure 2 shows the relationship between f and mean time to preventivemaintenance (MTPM) for differentN. It is seen from the graph that fincreasesfor increasing values of MTPM. Furthermore, the float requirement is higherfor operating systems with smaller number of machines. Figure 3 studies theeffect ofp on MFF for different weighted means (WM). Here, as the proportionpfor failure increases, that is, the proportion of units under preventive

F i g u re 2 .

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F i g u re 3 .

F i g u re 4 .


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F i g u re 5 .

F i g u re 6 .

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maintenance decreases, the number of float required increases. Figure 4represents the relation offand MTBFM for some values of weighted means.This points to the analysis of repair task so as to reduce float levels. Figure 5plots the value offand WM for varying MTBFM. Thus, slower the service, themore the number of units that will either be receiving repair/maintenance orwaiting in line for repair/maintenance to commence. The joint effect of MTTRand MTPM on float factor is shown graphically in Figure 6.

Managerial applicationsExample 1A service company requires 25 machines to be in operational status to satisfyits heavy demand load. Based on historical data, it has been determined thatthese machines fail according to the Weibull distribution. Now, in order to havemaximum availability of the system the concept of preventive maintenance is

introduced. Here, the proportion 1p of the machines are taken for preventivemaintenance and the time at which a machine is taken for maintenance isdistributed according to the three parameter Weibull distribution. TheMTBFM for the machines is 350 hours and the average time required for repairand maintenance is 250 and 200 hours respectively. The maintenanceexpenditure is Rs10,000 per day. The company is considering an investmentalternative that will reduce MTTR to 167 hours and MTPM to 125 hours. Thisplan will require the firm to increase its maintenance expenditure to Rs14,000per day. The cost for each unit in float is Rs300 per unit per hour. Whichalternative is preferable? Assume an eight-hour work day.

Solution.For numerical illustration the parameters for the mixed Weibull

distribution are taken as follows:

h1 100;h2 200;b1 0:5;b2 2;p0:2 andg33:49:

Alternative 1. Here, N 25, MTBFM 350 hours, MTTR 250 hours andMTPM 200 hours:

f 1 2 pexp{2 tn=h1}b1 12pexp{2 tn 2 g=h2}

b2 0:5933:

Therefore, total floatF 25 * 0.593 14.8 15 units; the total float cost per

hoursRs4,500; maintenance expenditure per hourRs10,000/8hour Rs1250; thus, total cost Rs5,750/hourAlternative 2.Here, alsoN 25, MTBFM 350 hours, MTTR 167 hours

and MTPM 125 hours. Maintenance float factor f 0.3209. Hence, totalfloat 25 * 0.320 8 units; float cost per hour Rs2,400; maintenance costper hourRs1,750; the total cost for the alternative Rs4,150/ hour.

The decision is, therefore, to select the second alternative since by increasingthe maintenance expenditure by Rs550/hour, the company will reduce its float


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by 7 units. This will lead to a cost saving of Rs1,600/hour. The model thusoffers a decision tool to maintenance managers.

Example 2Suppose that there are 20 units and we have the cost schedule related to MTPMand MTTR in Table II.

Let float cost per unit per hour be Rs300 and MTBFM 1,100 hours. Whichcombination of repair and maintenance time will give the minimum cost?

Solution.The objective here, is to minimise the total cost which is sum ofrepair and maintenance costs of float. In Table III, the decision is to selectMTPM 300 hours and MTTR 500 hours since this gives the minimumcost. In Table IV, the combination of MTPM 300 hours and MTTR 700hours gives the minimum cost. In Table V, the decision is to selectMTPM 100 hours and MTTR 900 hours.

MTPMMaintenance

cost/hour (Rs) f F Float cost/hour (Rs) Total cost

100 5,000 0.148 3 900 13,900300 4,000 0.311 6 1,800 13,800500 3,000 0.501 10 3,000 14,000700 2,000 0.688 14 4,200 14,200

Not es:N 20; MTTR 500 hoursT a b l e I I I .

MTPMMaintenance


100 5,000 0.184 4 1,200 12,200300 4,000 0.362 7 2,100 12,100500 3,000 0.551 11 3,300 12,300700 2,000 0.727 15 4,500 12,500

Not e:MTTR 700 hoursT a b l e I V .

MTPM Maintenance cost/hour (Rs) MTTR Repair cost/hour (Rs)

100 5,000 500 8,000300 4,000 700 6,000500 3,000 900 4,000700 2,000

T a b l e I I .Cost schedulerelated to MTPMand MTTR

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From Tables III-V, we conclude that the best decision will be to select MTPM of100 hours and MTTR of 900 hours since this combination gives the minimumtotal cost. If this decision is implemented, a float size of four units only isrequired.

ConclusionThis paper introduces the maintenance float problem as a model to describecatastrophic and/or wear-out failures. Subsequently, it can be applied forcomputing the size of float (F) required for any givenN. This model provideson easy approach to maintenance management decision making. Furthermore,the approach provided here would be of assistance to reliability engineers andsystem designers to increase the reliability of an operating system.

R eferences

Barlow, R.E., Proschan, F. and Hunter, L.C. (1965),Mathematical Theory of Reliability, John Wiley& Sons, New York, NY.

Cohen, A.C. (n.d.), Estimation of mixture of poisson and mixtures of exponential distributions,NASA Technical Memorandum TMX 53245, NASA.

Harris, C.M. and Singpurwalla, N.D. (1968), Life distributions derived from stochastic hazardrates, IEEE Trans. on Reliability, Vol. R-17 No. 2, pp. 70-9.

Kao, J.H.K. (1959), A graphical estimation of mixed Weibull parameters in life testing of electrontubes, Technometrics,Vol. 1 No. 4, pp. 389-407.

Levine, B. (1965), Estimating maintenance float factors on the basis of reliability theory, Ind.Qual. Control, Vol. 21 No. 2, pp. 401-5.

Lowe, P.H. and Lewis, W. (1983), Reliability analysis based on the Weibull distribution: anapplication to maintenance float factors, Int. Jour. of Prod. Res., Vol. 21 No. 4, pp. 461-70.

Madu, C.N. (1987), The study of a maintenance float model with gamma failure distribution,Int. Jour. of Prod. Res., Vol. 25 No. 9, pp. 1305-23.

Madu, C.N. and Chanin, M.N. (1989), Reliability ratio analysis of a maintenance float policy,SCIMA, Vol. 18 No. 3, pp. 95-111.

Mann, N.R., Schaffer, R.E. and Singpurwalla, N.D. (1974), Methods for Statistical Analysis ofReliability and Life Data, John Wiley & Sons, New York, NY.

MTPMMaintenance


100 5,000 0.222 4 1,200 10,200300 4,000 0.409 8 2,400 10,400

500 3,000 0.595 12 3,600 10,600

700 2,000 0.759 15 4,000 10,500

Not es:MTTR 900 hours T a bl e V .


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Shankar, G. and Sahani, V. (1993), The study of a maintenance float model with Rayleigh failuredistribution, IJOMAS, Vol. 9 No. 2, pp. 207-14.

Shankar, G. and Sahani, V. (1994a), The study of a maintenance float model with gamma failureand two repair centres, Microelectronics and Reliability, Vol. 34 No. 6, pp. 1127-32.

Shankar, G. and Sahani, V. (1994b), The study of a maintenance float model with Burr failuredistribution, Microelectronics and Reliability, Vol. 34 No. 9, pp. 1513-7.

Shankar, G. and Sahani, V. (1998), The study of a maintenance float model with inverseGaussian failure distribution, IAPQR Trans., Vol. 23 No. 1, pp. 73-9.

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