Int. J. Production Economics 81–82 (2003) 589–595

Prediction of the demand of the railway sleepers:A simulation model for replacement strategies

Won Young Yuna,*, Luis Ferreirab

aDepartment of Industrial Engineering, Pusan National University, San 30 Changjeon-Dong Kumjeong-Ku, Pusan 609-735, South KoreabSchool of Civil Engineering, Queensland of University of Technology, Brisbane, Australia

Abstract

This paper describes the development of a simulation model to assess the inventory requirements of alternative rail

sleeper replacement strategies. The main aim of the model is to determine the optimal replacement strategy, given

replacement costs and resultant train operating cost benefits. We consider the replacement problem under the following

assumptions: The time to failure under constant stress follows a Weibull distribution and the scale parameter is a

function of stress level and the three stress levels under normal (all adjacent units are good), medium-stress (one

adjacent unit has failed) and high-stress conditions (two adjacent units are failed) are considered. The cumulative

exposure model is used to model the failure distributions.

The operational cost per unit time depends on the maximum number of consecutive failed units. The replacement

cost consists of the fixed cost and variable cost proportional to the number of units replaced. A finite horizon is

considered and total expected cost is a criterion for comparing the proposed policies. The model has been tested using

rail system data and the results are presented in this paper.

r 2002 Elsevier Science B.V. All rights reserved.

Keywords: Railway sleepers; Replacement; Simulation; Weibull distribution; Cumulative exposure

1. Introduction

In Australian freight operations, 25–35% oftotal train operating expenses is track maintenancerelated (Ferreira and Higgins, 1998). Exclusive ofrail costs, sleeper replacement represents the mostsignificant maintenance cost for the railways(Hagman and McAlpine, 1991). For efficientmaintenance of sleepers, a railway authority orcompany needs to plan for a specific amount, andthen store the sleepers at regional sites in advance.

For efficient inventory management, firstly it isnecessary to predict the demand for sleepers atmain location. To forecast future sleeper demand,we need to analyze the trends in demand. Ingeneral, the demand for sleepers consists ofamounts used with two different activities inrailway maintenance, namely: unplanned main-tenance (unscheduled replacement of sleepers) andplanned maintenance (scheduled replacement ofsleepers) (refer Carter, 1986).

Unscheduled replacement is done when thefailed sleepers are replaced after accidents or atirregular detection. Scheduled replacement isundertaken on regular schedule times (periodicreplacement) or when the railway is in imminent

*Corresponding author. Tel.: +82-51-510-2421; fax: +82-

51-512-7603.

E-mail address: wonyun@hyowon.pusan.ac.kr (W.Y. Yun).

0925-5273/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 5 - 5 2 7 3 ( 0 2 ) 0 0 2 9 9 - 2

danger of train accident (condition-based replace-ment). Thus, scheduled replacement interval orstrategy affects the demand process of sleepers andis an important factor for the prediction of thedemand of the sleepers. Therefore, inventorymanagement and replacement strategy of sleepersshould be optimized simultaneously.

In this paper, we consider unscheduled andscheduled replacement strategies for sleepers anddevelop a simulation procedure to test theefficiency of proposed replacement policies.

Thus, our replacement problem for sleepersbelongs to that for replacement policies forsystem with multi-units or complex systems. Forthe replacement problem for a complex system,there are some existing studies (Ozekici, 1996;Wildeman and Dekker, 1997). Most of pastresearch assumes that failures of units are inde-pendent and only economic dependency is con-sidered. In our case, dependency between sleeperfailures is an important characteristic peculiar torailway track maintenance. Finally, we study somefield data and compare several replacementpolicies.

Assumptions:

(1) The system consists of n sleepers.(2) The time to failure under constant stress

follows a Weibull distribution with distribu-tion function

F ðtÞ ¼ 1� exp �t

b

� �a� �; ð1Þ

where a is a shape parameter and b is a scaleparameter.

The Weibull distribution is known toproduce good results when representing slee-per failure (Tucker, 1985).

(3) The scale parameter for sleeper failure dis-tribution is a function of the stress level loadedonto sleepers. The scale parameters of undernormal (all adjacent units are good), medium-stress (one adjacent unit is failed) and high-stress conditions (two adjacent units are failed)are b0;b1; and b2; respectively.

(4) After a sleeper is used under normal stress tilltime t, a sleeper adjacent to it is failed at t andthe actual age of the sleeper s is obtained as

follows:

F1ðsÞ ¼ F0ðtÞ; s ¼ tb1b0: ð2Þ

The cumulative exposure model is usedto model the failure distributions (Nelson,1990).

(5) Replacement time is negligible.(6) The operational cost per unit time depends on

the maximum number of consecutive failedunits.

(7) The replacement cost consists of the fixed costcf and variable cost cv which is proportionateto the number of units replaced.

(8) The finite horizon, T is considered and thesalvage cost of sleeper is negligible.

(9) The status of sleepers (good or failed) isknown at no cost.

2. Model

We consider a track with n sleepers of which thefailure times follow Weibull distributions. Thefailure of each sleeper is assumed to be dependenton condition (two adjacent sleepers fail or areworking). This dependency is very important andpeculiar characteristic in sleeper’s failure ordeterioration. What makes failure modeling ofsleepers more complex is how we deal with failurephenomena after the replacement of an adjacentfailed sleeper. Thus, a relation model in accelera-tion life testing is adopted to represent thedependency (Nelson, 1990).

In this paper, we consider replacement cost ofsleepers and operation cost. The replacement costis a cost to replace some failed sleepers. The latterrepresent the train operation cost which isdependent on track condition. Total cost forsleepers during time horizon consists of the totalreplacement cost and the total operation costduring the time horizon. Total replacement costduring the time horizon increases as we replacefrequently but the total operation cost during thetime horizon decreases when failed sleepers arefew. One critical aspect in determining the condi-tion of track with respect to sleepers is dispersionof failed sleepers in the track. A section of railway

W.Y. Yun, L. Ferreira / Int. J. Production Economics 81–82 (2003) 589–595590

track with about 20% failed sleepers may still besafe to operate if each failed sleeper lies betweentwo sound ones, yet the same section of track withonly 1% failed sleepers all adjacent to one anotherwould be unusable. Thus, we assume that theoperation cost depends on the maximum numberof consecutive failed sleepers.

Under previous assumptions about failure pro-cesses and cost models, we consider several replace-ment strategies for comparison by simulation.

2.1. Replacement policies

We can consider four types of replacementstrategies for sleepers: Policy A is a kind ofunscheduled replacement and policies B–D arekinds of scheduled replacement. However, theseare only a subset of all sleeper replacementoptions.

Replacement policy A: We replace the failedsleeper just at failure.

Replacement policy B: We replace all the failedsleepers periodically. In this case, replacementinterval is a decision variable.

Replacement policy C: We replace all the failedsleepers when the k consecutive sleepers are failedfirst. In this case, k is a decision variable.

Replacement policy D (combination of policies B

and C): We replace all the failed sleepers when thek consecutive sleepers are failed or when time t iselapsed from the last replacement, whicheveroccurs first. In this case, k and t are decisionvariables.

In the replacement policies B–D, we replace allthe failed sleepers at scheduled replacement pointbut we can extend this restriction to more generalpolicy. For example, we can delay the replacementpoint to the first failure time of three consecutivesleepers and then replace all consecutive failedsleepers except isolated ones. Until specified time,there are no three consecutive failed sleepers; wereplace all consecutive failed sleepers.

To obtain the optimal replacement policyminimizing the total expected cost during the finitetime horizon, we must derive the total expectedcost for a given replacement policy. It is verydifficult to derive this analytically because of thedependency between failures of sleepers and finite

time horizon (stochastic dynamic programmingproblem). We formulate the optimization pro-blems by simulation under the given set ofassumptions. The total expected cost during timehorizon is a criterion for optimization. For givenreplacement strategies, the total expected costsduring time horizon are estimated by simulation.Therefore, we summarize the simulation procedureto find total expected cost and compare somereplacement policies with field data by simulation.

2.2. The procedure of simulation

In this section, we explain the simulationprocedure by pseudo-code. As a typical example,we consider replacement policy B with replace-ment interval Tm (Fig. 1). It is possible to addother policies by changing some steps. Before weexplain the psuedo-code, we summarize twomethods to obtain random variates related toWeibull distribution.

2.2.1. Method for generating Weibull failure times

A sleeper failure time follows Weibull distribu-tion with scale parameter b; and shape parametera:

1. If we want to obtain failure times, thefollowing function is used:

NTF ¼ b½�ln U �1=a; ð3Þ

where U is uniform variate [0,1].2. If we know the age of sleeper, x and we want

to obtain failure time t then the failure time has thefollowing distribution:

PrfTpt=T > xg ¼PrfxoTptg

PrfTXxg¼

F ðtÞ � F ðxÞ1� F ðxÞ

;

where F ðtÞ is the distribution function undercurrent stress level. To obtain the next failure timefor the used sleepers, we use an inverse transfor-mation method. Let U be the random variate ofuniform [0,1], then

T ¼ b1 or 2½ðx=b1 or 2Þa � ln U �1=a; ð4Þ

where the conditions of two adjacent sleepersdetermine the scale parameter.

Input variables: Number of sleepers n; shapeparameter a (>1); scale parameters under normal,

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Step 0 (Initialize) : NT=0;TC=0;MCF=0;NF=0;NTF=0; Age(j)=0;NAF(j)=0

Step 2 (determine next time to which simulation time is moved): Find the

minimum value among NFT(j) for all j ; next replacement time, NRT= Min {NFT,

NRT};

Step 3 (Change Parameters): INT=Min{NFT,NRP}-NT; Age(j) =Age(j) +INT, for all

j; Change actual ages of the two sleepers adjacent to the failed sleeper using

Equation 2, Age(j)=Age(j);

For the sleepers of which stresses should be changed, obtain next failure times as

follows: When the actual age, Age (j) = x, the distribution function for next failure

is obtained using equation (4). Finally, the next failure times for the two sleepers of

which stress levels are changed are

NFT (j) = Min {NFT, NRP} + { T − Age (j)}

Update total cost and total number of failed sleepers: NT = Min { NFT, NRP};

NT=T

Step 4 (replace all the failed sleepers): INT = NRT − NT; NT = NRT

For failed sleepers, Age (j) = 0 and generate new ones using Equation 3 and

NFT (j) = T + NT

For working sleepers adjacent to replaced sleepers, change actual ages using

Equation 3 and generateααββ /1

00 ]ln)/)([()( UjAgejT −=

Finally, the next failure times are NFT (j) = NT + { T(j) − Age (j)};Update total cost

and total number of failed sleepers; Next replacement time, NRT, is changed

Min{NFT,NRP}= nextreplacement time

Step1 (Generate Next Failure Times): Generate n random numbers ofWeibull(α,β0) under normal stress level using Equation 3; let them be NFT(j)

Step 5 (Terminate simulation): stop simulation and give total expected cost

yesno

no

yes

Fig. 1. Simulation step.

W.Y. Yun, L. Ferreira / Int. J. Production Economics 81–82 (2003) 589–595592

medium and high stresses b0; b1; and b2; horizonlength T ; simulation replication number M; costparameters for operation cost function and repla-cement cost; parameters for replacement policy,for this policy B, replacement interval Tm:

Definition of general variables:

NT current timeTC total cost to current timeMCF maximum number of consecutive failed

sleepers at current timeNF number of the failed sleepers at current

timeNTF total number of the failed sleepers to

current timeAge(j) age of sleeper, i at current timeNAF(j) number of the failed sleepers adjacent to

sleeper, j at current timeNRT next replacement time after current time

For each replacement type, we can construct thesimulation procedures and run the simulation withseveral values of decision variable. For example,we can consider 1–5 years for replacement policyB. Then we compare the total costs and decide ona good scheduled replacement interval.

3. Field study

In this section, we consider a replacementproblem with typical field data related to timbersleepers. It was estimated in 1991 that 75% of theworld’s railway consists of timber sleepers(Adams, 1991). Despite the increasing reliabilityand effectiveness of alternatives such as steel andconcrete, Sonti et al. (1995) state that timber hasbeen and will continue to be the most popularmaterial for railway sleepers in the United States.In Gruber (1998), it is stated that well over 90% ofmaintenance and construction of railway tracksutilize timber sleepers based on cost and benefit.

We consider a railway track with 1000 timbersleepers and the failure distribution is given bya Weibull distribution with shape parameter 3 andscale parameter 20 because the average life isabout 20 year in Australia. If an adjacent sleeper is

failed, the decay rate for a sleeper increases andusually reduces the remaining life to a half of theoriginal remaining life. So we assume that theshape parameter is 10 and 5 after failure ofadjacent sleeper. The planning period is assumedto be 20 years.

Replacement cost is estimated as follows:

Cost per sleeper replaced:

replacing o5% : installation cost $38+cost of sleeper $22

5–10% : installation cost $33+cost of sleeper $22

10–50% : installation cost $28+cost of sleeper $22

15–20% : installation cost $23+cost of sleeper $ 22

>20% : installation cost $ 18+cost of sleeper $22

Operation cost per time unit is very difficult toestimate exactly and we assume approximatelythat it is a function

cðmÞ ¼ fmd ;

where, d ¼ 2 and f ¼ 400; then if a sleeperfails, then operation cost is $400 per sleeper ayear.

We consider four replacement policies. Replace-ment every 0.5, 1, 2, 3 years are simulated, thelevels of failed sleepers before intervention of>5%, 10%, 15%, 20% are used, and twoconsecutive failed sleepers are considered as areplacement criterion.

Table 1 shows the simulation results. In thistable, the best policy is 10% policy where wedelay replacement until 10% sleepers fail andthen replace all the failed ones. And one-yearpolicy is also an alternative. We consideredthe additional policies which consider two-con-secutive failures as replacement, that is, whenwe replace all the failed sleepers, we do not replacethe isolated and failed ones. 10% and two-consecutive and 0.5-year and two-consecutivepolicies have large values for total cost, 54 697and 27 467.

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4. Conclusion and discussion

This paper considers replacement policies forprediction of demand for inventory managementof sleepers in railway: ‘‘What is the optimalreplacement strategy?’’ We define the problem bythe previous assumption and total expected cost toT ; time horizon is the optimization criterion. Weconsider various replacement types in this case. Topropose replacement policies, information forreplacement should be checked. We are able toknow which sleepers fail at any time, the conditionof rail system and remaining time to T ; life cycle(Which unit is good or failed? And failuredistributions of good units), Therefore, we pro-posed four kinds of replacement policies. Simula-tion procedure is proposed and a case study hasbeen shown using field data.

In this paper, we proposed four policies but forfurther study, we can consider other types ofreplacement: We can consider emergency replace-ment from the safety prospective. For example, ifthree adjacent sleepers are failed, then it is unsafe

to operate a train. This case belongs to replace-ment policy C.

We can also add replacement condition basedon the number of total failed sleepers and for moregeneral cases, we can also consider preventivereplacements. In general, when replacement con-dition based on the proposed replacement policiesis satisfied and we replace a failed sleeper, we canconsider replacement of working but old sleepersnear the failed sleeper (For opportunistic replace-ments, refer Wildeman and Dekker (1997)). Asanother case, when the replacement condition isnot satisfied, then we can also consider preventivereplacement for old one and some consecutivefailed one.

References

Adams, J.C.B., 1991. Cost effective strategy for track

stability and extended asset life through planned sleeper

retention. Demand Management of Assets National Con-

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pp. 145–152.

Table 1

Total costs for replacement policies

0

10000

20000

30000

40000

50000

60000

70000

80000

5% 10%

15%

20%

0.5y

ear

1yea

r

2yea

rs

3yea

rs

Policies

Total cost

Replacement cost

Operation cost

Cos

t

W.Y. Yun, L. Ferreira / Int. J. Production Economics 81–82 (2003) 589–595594

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