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Track geometry defect rectification based ontrack deterioration modelling and derailment riskassessment

Qing He, Hongfei Li, Debarun Bhattacharjya, Dhaivat P Parikh & ArunHampapur

To cite this article: Qing He, Hongfei Li, Debarun Bhattacharjya, Dhaivat P Parikh & ArunHampapur (2015) Track geometry defect rectification based on track deterioration modelling andderailment risk assessment, Journal of the Operational Research Society, 66:3, 392-404, DOI:10.1057/jors.2014.7

To link to this article: https://doi.org/10.1057/jors.2014.7

Published online: 21 Dec 2017.

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Track geometry defect rectification basedon track deterioration modelling and derailmentrisk assessmentQing He1*, Hongfei Li2, Debarun Bhattacharjya2, Dhaivat P Parikh3 and Arun Hampapur21University at Buffalo, The State University of New York, Buffalo, NY, USA; 2IBM T J Watson Research Center,Yorktown Heights, NY, USA; and 3IBM Global Business Service, Coppell, TX, USA

Analysing track geometry defects is critical for safe and effective railway transportation. Rectifying the appro-priate number, types and combinations of geo-defects can effectively reduce the probability of derailments. In thispaper, we propose an analytical framework to assist geo-defect rectification decision making. Our major con-tributions lie in formulating and integrating the following three data-driven models: (1) A track deteriorationmodel to capture the degradation process of different types of geo-defects; (2) A survival model to assess thedynamic derailment risk as a function of track defect and traffic conditions; (3) An optimization model to plantrack rectification activities with two different objectives: a cost-based formulation (CF) and a risk-based for-mulation (RF). We apply these approaches to solve the optimal rectification planning problem for a real-worldrailway application. We show that the proposed formulations are efficient as well as effective, as compared withexisting strategies currently in practice.Journal of the Operational Research Society (2015) 66(3), 392–404. doi:10.1057/jors.2014.7Published online 19 February 2014

Keywords: rail transportation; track geometry defects; track deterioration model; track derailment risk; track defectrectification

1. Introduction

Rail is a crucial mode of transportation in the United States.According to the National Transportation Statistics report fromthe Bureau of Transportation Statistics (Association of AmericanRailroads, 2013), 42.7% of the US freight revenue ton-mileswere carried by railroad; this represents the largest portion of theinter-city freight market.Detection and rectification of track defects are major issues in

the railway industry. The existing literature categorizes thesedefects into one of two groups: track structural defects andtrack geometry defects (Sadeghi and Askarinejad, 2010). Trackstructural defects are generated from the structural conditions ofthe track, which include the condition of the rail, sleeper,fastening systems, subgrade and drainage systems. On the otherhand, track geometry defects (referred to as geo-defects in theremainder of this paper) indicate severe ill-conditioned geo-metry parameters such as profile, alignment, gage, etc, as shownin Figure 1 (ARC-TECH.NET, 2012).Track defects have become the leading cause of train accidents

in the United States since 2009. In all, 658 of 1890 (34.8%) trainaccidents were caused by track defects in 2009, incurring a

$108.7 million loss (Peng, 2011). Therefore, it is imperative tounderstand the deterioration process of railway track systems andfacilitate track maintenance planning. Previous studies on trackdeterioration divide track segments into several shorter sectionsfor analysing summary statistics of raw geometry measurements(El-Sibaie and Zhang, 2004; Federal Railroad Administration,2005; Berawi et al, 2010; Sadeghi and Askarinejad, 2010). Theoverall statistics provide a measure of segment quality, calledTrack Quality Indices (TQIs). TQIs have been widely used forrailway maintenance scheduling, since they provide a high-levelassessment of railway track performance (Tolliver and Benson,2010). However, TQIs only provide an aggregate level pictureand they cannot identify individual severe geo-defects for trackrectification.Maintaining the existing tracks through rectification and

renewal is critical to railroad operation and safety. In 2011,Class I railroads, defined as line haul freight railroads withoperating revenues of $433.2 million or more (Association ofAmerican Railroads, 2013), spent $7.52 billion on track main-tenance (Tolliver and Benson, 2010). Track maintenanceactivities can be categorized into two main groups: preventivemaintenance and corrective maintenance (Andersson, 2002).Preventive maintenance is pre-planned and carried out to avoidfuture defects, whereas corrective maintenance rectifies existingdefects in the infrastructure. Most of the literature in this areadescribes preventive or planned maintenance (Beichelt and

*Correspondence: Qing He, Department of Civil, Structural and Environ-mental Engineering and Department of Industrial and Systems Engineering,University at Buffalo, 313 Bell Hall, Buffalo, NY 14260, USA.E-mail: qinghe@buffalo.edu

Journal of the Operational Research Society (2015) 66, 392–404 © 2015 Operational Research Society Ltd. All rights reserved. 0160-5682/15

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Fischer, 1980; Higgins, 1998; Cheung et al, 1999; Higginset al, 1999; Simson et al, 2000; Budai et al, 2006; Oyama andMiwa, 2006; Van Zante—de Fokkert et al, 2007; Peng et al,2011) due to its large scale of operation and high complexity,whereas very few studies have addressed the problems incorrective or unplanned maintenance (Zhao et al, 2007),referred to as track rectification (or track repair) in this paper.Track rectification is subsumed by the broader area of railway

project selection and planning, which uses prioritization andoptimization techniques. Prioritization is essentially performedin a sequential manner by first enlisting the maintenance projectsrequired to be executed. Once projects are identified, the nextstep is to prioritize the projects based on their relative perceivedurgency. Projects with the highest priority are executed until allfinances are expended. Remaining projects are re-prioritizedtogether with the new projects upon availability of funds.A popular multi-criteria decision-making methodology for

prioritization is the Analytic Hierarchy Process (AHP), firstpresented by Saaty (1980). After interviewing six trackmanagers, Nyström and Söderholm (2010) applied AHP tech-niques to understand different criteria from decision-makersand then rank maintenance actions. Cheng et al (2012) com-bined the AHP with artificial neural network methodologiesto formulate a ranking predication model by connectingobjective measures with subjective judgments. However, theAHP suffers from a number of drawbacks and is often critici-zed for the way the criteria weights are elicited, for rankreversal problems, and for fundamental issues with the theory(Boucher et al, 1997). Typically, prioritization involves rank-ing rectification activities based on the ‘worst first’ principle,and therefore fails to account for the change in benefit for thefunds expended; it may produce a maintenance strategy whichcould be far from optimal.Unlike prioritization approaches, optimization-based methods

have the ability to develop a maintenance plan while capturingthe effect of deferring a maintenance activity on the trackcondition. An ideal optimization approach is one that evaluatesall possible repair strategies at the network level without

imposing unnecessary constraints or subjective judgments.Niemeier et al (1995) constructed five optimization models,each building on a basic linear programming formulation, forselecting an optimal subset of projects submitted for a statewideprogramming process. Ahern and Anandarajah (2007) deve-loped a weighted integer goal-programming model for selectingrailway projects for investment while maximizing the objectivesand meeting the budget limit for capital investment. In thispaper, we extend the literature on optimization-based methodsby proposing a data-driven formulation that incorporates trackdeterioration and derailment risk models.

2. Background and model preliminaries

2.1. Background

Track rectification decisions are typically made by the localtrack master in the network. A local track master is responsiblefor the railway infrastructure in a certain area, containing oneor several track sections, typically from dozens of miles toa couple of hundred miles (Nyström and Söderholm, 2010).Among other responsibilities, she/he is responsible for rectify-ing track geometry defects, thereby ensuring high standards ofsafety and cost effectiveness.According to the US Federal Railroad Administration (FRA)

track safety standards, individual defects whose amplitudesexceed a certain tolerance level must be treated properly. Tradi-tionally, geometry cars classify each defect into two severitylevels, denoted in this paper as either ‘Red tags’ or ‘Yellow tags’.Red tags are defects that are in violation of FRA track safetystandards, and railroads must fix these defects as soon as possibleafter their discovery or else they risk being fined. Yellow tags aredefects whose amplitudes are currently below FRA limits, andthey may or may not meet the particular railroad’s own standardsfor rectification. According to current practice, railroads fix Redtags within a due date after inspection and they examine theYellow tags, repairing very few of them based on their severityas estimated from field experience. Whether to fix a Yellow tagor not may depend on several factors, such as the state of thetrack geometry, defect history, curvature of the track, MGT(Million Gross Tons), consequential derailment cost, etc. It maybe particularly prudential to rectify extreme severe Yellow taggeo-defects that are likely to soon become Red tag defects.To our best knowledge, there is not much prior literature on theindividual geo-defect rectification process.

2.2. Data summary

Our models are based on field data sets from a Class I railroad,including 3-year traffic data, derailment data and geo-defectdata from January 2009 to December 2011. Since main linetracks carry most of the traffic, and derailments associated withthese tracks usually cost much more than other track types, wefocus our analysis on about 2000 miles of main line tracks inthis study. In total, there are approximately 4000 Red tagdefects and 27 000 Yellow tag defects. The data set contains

Figure 1 Some track geometry parameters (ARC-TECH.NET,2012).

Qing He et al—Track geometry defect rectification 393

around 40 different types of geo-defects. The top 12 major geo-defect types pertaining to our analysis are described in Table 1.For modelling purposes, the data sets are processed along

both spatial and temporal dimensions:

● Spatially, the rail network is defined by line segments (theyusually connect two cities), track numbers (0–8 for main linetracks) and mile post locations. Constructed in such afashion, the rail lines range from a few miles to hundreds ofmiles. To generate consistent spatial units and accommodatedifferent modelling purposes, we divide the main line net-work further into two different levels of smaller segments,called lots and sections. Each lot is 0.02 mile (about 100 ft) inlength, used for track deterioration analysis. At a higher level,a continuous track segment is divided into 2 mile longsections, used for track derailment risk and geo-defectrectification modelling.

● Temporally, regular track geometry inspection is performed1–4 times per year according to the characteristics of eachtrack segment. Geo-defects are reported and updated after eachinspection run. When they occur in the same inspection runwindow, different types of geo-defects are aggregated to thelevel of an inspection run, mainly for derailment risk analysis.

2.3. Model summary

In order to improve current track rectification decisions, thisstudy aims to help existing railroads address the following three

questions: (1) HowYellow tags of each type deteriorate into Redtags; (2) How unrectified Yellow tags affect derailment risk; (3)How to prioritize and rectify Yellow tags within a limited repairhorizon. The main objective of this study is to propose aframework for making optimal track geo-defect rectificationdecisions, in order to appropriately reduce the probability of aderailment as well as its associated costs. In addition, effectivetrack geometry maintenance reduces dynamic vehicle and trackinteraction, thus reducing the stress state of the railroad.Compared with the existing state-of-the-practice decision modeldepicted in Figure 2, our major contributions lie in formulatingand integrating the following three models:

● A track deterioration model to capture the degradationprocess of different types of geo-defects.

● A survival model to assess the dynamic derailment risk as afunction of the current track and traffic condition.

● An optimization model to plan track rectification activitiesthat either minimizes the highest derailment risk or the totalexpected costs, including potential derailment costs andrectification costs.

To make the optimization model tractable, we make thefollowing assumptions:

● Assumption 1: Red tags may be rectified individually, butYellow tags are rectified in bulk, at the section and defecttype level. This means that the decision of rectifying Yellowtags are made for each defect type in each section level.

Table 1 Geo-defect summary (in alphabetical order)

Defect type Description

ALIGN ALIGN is the projection of the track geometry of each rail or the track centre line onto the horizontal plane, alsoknown as ‘straightness’ of the tracks.

CANT Rail cant (angle) measures the amount of vertical deviation between two flat rails from their designed value.(1 degree= 1/8″ for all rail weights, approximately)

DIP DIP is the largest change in elevation of the centreline of the track within a certain moving window distance. Dip mayrepresent either a depression or a hump in the track and approximates the profile of the centreline of the track.

GAGE_C Gage Change is the difference in two gage readings in a certain distance measurement interval.GAGE_TGHT GAGE_TGHT measures how much tighter from standard gage (56-1/2″).GAGE_W1 Gage is the distance between right and left rail measured 5/8″ below the railhead. GAGE_W1 for wood ties (sleepers)

measures how much wider from standard gage (56-1/2″). The amplitude of GAGE_W1 plus 56-1/2″ is equal to theactual track gage reading.

GAGE_W2 Same as GAGE_W1, but for concrete ties.HARM_X Harmonic cross-level defect is two cross-level deviations a certain distance apart in a curve.OVERELEV Over-elevation occurs when there is an excessive amount of elevation in a curve (overbalance) based on the degree of

curvature and the board track speed.REV_X Reverse cross-level occurs when the right rail is low in a left-hand curve or the left rail is low in a right-hand curve.SUPER_X Super cross-level is cross-level, elevation or super-elevation measured at a single point in a curve.SURF Uniformity of rail surface measured in short distances along the tread of the rails. Rail surface is measured over

a 62-foot chord, the same chord length as the FRA specification.TWIST Twist is the difference between two cross-level measurements a certain distance apart.WARP Warp is the difference between two cross-level or elevation measurements up to a certain distance apart.WEAR The Automated Rail Weight Identification System (ARWIS) identifies the rail weight while the car is testing and

measures the amount of head loss. The system measures for vertical head wear (VHW) and gage face wear (GFW)per rail.

XLEVEL Cross-level is the difference in elevation between the top surfaces of the rails at a single point in a tangent tracksegment.

394 Journal of the Operational Research Society Vol. 66, No. 3

● Assumption 2: The repair horizon, from a few days up to afew weeks, is much smaller than the track inspection interval,typically ranging from 3 months to 1 year. Therefore, oncethe repair activity is determined, we assume that derailmentrisk will take effect immediately according to the repairactivities scheduled.

● Assumption 3: The travel time between two defects isnegligible as compared with the time spent at the defectrectification location.

● Assumption 4: Defects to be repaired at the same time stepconstitute one cluster. The centre of each cluster is defined asthe defect with median post mile. The total distance travelledis approximated by the total distance from the defects to theircluster centres.

3. The track deterioration model

We develop a statistical track deterioration model for Yellowtag defects, representing the causes and consequences of trackdeterioration. The model takes various factors into account,including the current track conditions and traffic information,and has the capability to predict future track conditions. Thetrack deterioration process is captured by studying geo-defectamplitude changes, measured during each geometry inspectionrun. The statistical model constructs the relationships betweenthe effective parameters and the track deterioration rate, whileincorporating uncertainty arising from environmental factorsand measurement noise. The statistical model is able to predictthe deterioration of each geo-defect and the probability ofwhether a Yellow tag geo-defect will become a Red tag withina given duration.To model track deterioration, we track the evolution of track

defects. However, due to the lack of geo-defect indices, it is notpossible to track any particular geo-defect over time. We handle

this situation by tracking the condition of small track segments,where each segment contains very few geo-defects for eachinspection run. First, we divide the tracks into non-overlappinglots of equal length 0.02 miles (105.6 ft). Then we aggregate thedefects within each lot by inspection run for each defect type.We take the 90th percentile of the amplitudes to representthe track segment condition for the inspection run underconsideration.Exploratory analysis suggests fitting different models for

different defect types, since the model parameters have varyingeffects on deterioration rate for each defect type. For example,GAGE_W1 and GAGE_W2 (see Table 1) exhibit differentdeterioration rates with respect to traffic in MGT. We assumethat geo-defects get worse over time, that is, defect amplitudesincrease when there is no maintenance work. For each defecttype, let yk(t) denote the aggregated geo-defect amplitude(the 90th percentile of the defect amplitudes) of the track lotk at inspection time t. The deterioration rate or the amplitudechange rate over time Δt can be represented by (yk(t+Δt)−yk(t))/Δt. We model the deterioration rate (for any given defecttype) as follows:

logykðt +ΔtÞ - ykðtÞ

ΔtykðtÞ� �

¼ α0 + α1X1kðtÞ + � � �

+ αpXpkðtÞ + εkðtÞ8k ¼ 1 ¼ N ð1Þ

where N is the total number of track lots. Xpk(t) is thepth external factor or predictor for kth track lot at inspec-tion time t. Based on our exploratory data analysis, we findthat the distribution of (yk(t+Δt)− yk(t))/Δtyk(t) is highlyskewed and becomes close to normal after making a logtransformation. Therefore, we choose to use an exponential

Track geometry car inspection

Red tag defects Yellow tag defects

Fix immediately Inspect within a month, fixif required

Existingdecisionmodel

Fix within due datesOptimization model to prioritizeand schedule Yellow tag repair

Track deterioration model Derailment risk model

Proposeddecisionmodel

Figure 2 Flow chart comparing the existing and proposed decision models for geo-defect rectification.

Qing He et al—Track geometry defect rectification 395

relationship between the external factors or predictorsX1k(t),… , Xpk(t) and the deterioration rate in our model, similarto Sadeghi and Askarinejad (2010). The random error εk(t)is assumed normally distributed with mean 0 and standarddeviation σ2.The factors included in our model are: monthly traffic MGT

travelling through track lot k (X1k(t)), monthly total numberof cars (X2k(t)), monthly total number of trains (X3k(t)) andnumber of inspection runs in sequence since the last observedRed tag geo-defect (X4k(t)). Model fitting shows that the factorshave different impacts on deterioration rates for each defecttype. The estimated coefficients, α0, α1,… , αp, are presented inTable 2, where α0 depicts the intercept of the model, and αirepresents the coefficient for ith X factor. Mean squared errors(MSE), as shown in Table 2, are used for model selection:A model is chosen based on minimizing MSE. We test differentmodels with different nonlinear functions of the factors, Xpk(t).The model in (1) has the smallest MSE.Table 2 confirms that most defects deteriorate faster when the

traffic load (either MGT, number of cars or number of trains)increases (Sadeghi and Askarinejad, 2010). Contrary to tradi-tional deterioration models for aggregated TQI, our proposedmodel aims to predict the deterioration rate for each individualYellow tag defect. It is therefore important to leverage theYellow tag existing duration, indicated by the sequence numberof inspections since the last Red tag (α4). The estimated set ofcoefficients for α4 also demonstrates that Yellow tags deterio-rate at an increasing rate. Owing to limited data, some defecttypes, such as ALIGN, HARM_X, REV_X and SUPER_X, donot have any significant predictive factors in the model. Thismeans that the deterioration rates for these defect types onlydepend on the current amplitude and not on traffic and otherfactors.To compute the probability of a Yellow tag defect becoming

a Red tag in the future, we predict the defect amplitude forthe next inspection run according to Equation (1). Based oncurrent practice in geometry inspection, we choose Δt as90 days. Let the threshold for a Yellow tag defect becoming aRed tag for a certain defect type be r. Assuming that both r and

current amplitude yk(t) are positive, and that yk(t) is less than r,we define

hkðtÞ ¼ logr - ykðtÞΔtykðtÞ

� �

as the log-transformation of the deterioration rate thresholdat current amplitude yk(t). According to the proposed modelin (1), the log-transformation of deterioration rate is assumed tobe normally distributed. We can calculate the probability ofa Yellow tag at time t on track k becoming a Red tag in Δt, bycomputing the upper quantile of the transformed quantity,z= log((yk(t+Δt)− yk(t))/(Δtyk(t))), which follows a normaldistribution. Then the probability Pk

R(t) of a Yellow tag geo-defect at time t on track lot k becoming a Red tag in Δt is

PRk ðtÞ ¼

Z1hkðtÞ

zdz (2)

4. The track derailment risk model

State-of-practice track geometry analysis and risk estimationsystems mainly focus on current static track derailment riskutilizing mechanic models, such as ZETA-TECH’s TrackSafemodel (Bonaventura et al, 2005) and TTCI’s PerformanceBased Track Geometry model (Li et al, 2004). Instead, thisstudy analyses large amounts of historical data and predicts thefuture derailment risk given both traffic and defect conditions.Survival analysis is the field of study dealing with the

analysis of data regarding the occurrence of a particular event,within a time period after a well-defined time origin (Collett,2003). Analysing survival times is common in many areas, forinstance, in biomedical computation, engineering and the socialsciences. In our railway application, each inspection run will‘refresh’ the track segment since all Red tag geo-defects will berepaired. If there is no derailment between two scheduledinspection runs on a track segment, the track can be consideredto have ‘survived’ from one inspection to the next. If any

Table 2 Parameters for the track deterioration model (for selected defect types)

Defect type α0-intercept α1-Traffic(MGT)

α2-Traffic(number of cars)

α3-Traffic(number of trains)

α4-Sequencenumber

Meansquared

error (MSE)

CANT − 7.66 − 7.01E-02 6.05E-06 6.52E-04 0.067 0.242DIP − 7.58 7.21E-02 — — — 0.099GAGE_C − 8.53 8.62E-02 — — 0.113 0.046GAGE_W1 − 7.19 3.55E-02 4.59E-06 — 0.068 0.021GAGE_W2 − 8.08 1.90E-02 — 2.05E-04 0.080 0.061OVERELEV − 7.58 2.45E-01 — — 0.699 0.102SURF − 6.99 2.00E-01 — − 1.33E-03 0.044 0.007WEAR − 8.22 2.95E-02 — 4.73E-04 0.075 0.080XLEVEL − 7.66 — 2.64E-06 3.23E-04 0.092 0.019

‘—’ means that the coefficient is not significant.

396 Journal of the Operational Research Society Vol. 66, No. 3

derailment occurs, the track segment is said to have failed in thetime period since the last inspection run.We refer to derailment on a particular track section as a

hazard. In survival theory, there are three basic functions: thedensity function f (t), survival function S(t) and hazard functionλ(t). For a derailment, density function f(t) expresses thelikelihood that the derailment will occur at time t. The survivalfunction represents the probability that the track section willsurvive until time t:

SðtÞ ¼ProbðT⩾tÞ ¼Z1t

f ðxÞdx ¼ 1 -Zt0

f ðxÞdx

¼1 -FðtÞ ð3Þ

where T denotes the survival time of a track segment after inspec-tion and F(t) denotes the cumulative function of variable T.The hazard function represents the instantaneous rate of failureprobability at time t, given the condition that the event hassurvived to time t. By definition, the relationships between thesethree functions are:

λðtÞ ¼ f ðtÞSðtÞ ¼ -

d ln SðtÞdt

SðtÞ ¼ exp

"-Zt0

λðxÞdx#

f ðtÞ ¼ λðtÞSðtÞ ¼ λðtÞ exp"-Zt0

λðxÞdx#

Parametric models such as the exponential, Weibull, log-logistic or log-normal distributions may be used to specify thedensity distribution f(t), but such pre-defined distributions maybe inappropriate for real-world data. Without having to specifyany assumptions about the shape of the baseline function, Cox(1972) proposed a semi-parametric method for estimating thecoefficients of covariates in the model using the method ofpartial likelihood (PL) rather than maximum likelihood. Thismodel assumes that the covariates multiplicatively shift thebaseline hazard function and is by far the most popular choicein practice due to its elegance and computational feasibility(Cleves et al, 2004). Furthermore, unlike non-parametricanalysis such as the Kaplan-Meier method and the rank test, itallows both nominal and continuous variables. The hazardfunction form of the Cox model is:

λðt; β;XÞ ¼ λ0ðtÞeβ0X (4)

where λ0(t) is an unspecified nonnegative function of time calledthe baseline hazard, β is a column vector of coefficients to beestimated, and β′X=β0 +β1x1 +β2x2 +…+βkxk. Since the hazardratio for two subjects with fixed covariate vectors Xi and Xj,

λiðtÞλjðtÞ ¼

λ0ðtÞeβ0Xi

λ0ðtÞeβ0Xj;

is constant over time, the model is also known as theproportional hazards (PH) model. In order to estimate β, Cox(1972) proposed a conditional (or partial) likelihood func-tion which depends only on the parameter of interest, provingthat the resulting parameter estimators from the PL functionwould have the same distributional properties as full maxi-mum likelihood estimators (Cox, 1975). The PL function isdescribed as

LpðβÞ ¼Yni¼1

eβ0XiP

j2RðtðiÞÞeβ

0Xj

2664

3775δi

;

and the maximum PL estimator is found by solving the equation,

∂ lnðLpðβÞÞ∂β

¼ 0:

One may refer to Therneau and Grambsch (2000) for detailsof implementing a Cox model using existing statistical soft-ware. As described in Section 2, all the raw geo-defects arespatially aggregated to the section level (2 mile), and temporallyinto each inspection level. In any particular aggregated record,the dependent variable is either the time duration between twoinspection runs (the censored survival time), or time durationbetween the derailment and the last inspection run beforederailment. Selected candidate predictors are listed as follows:

● Monthly traffic in MGT.● Number of Yellow tag geo-defects (starting with ‘numYEL’

in Table 3) in each defect category.● 90th percentile amplitude (starting with ‘amp90’ in Table 3)

of Yellow tag geo-defects in each defect category.

The final Cox model fit to the censoring derailment data isillustrated in Table 3. An efficient way to evaluate the fittedmodel is to use Cox-Snell residuals (Cox and Snell, 1968),

rCSi ¼ eβ0Xi H0ðtiÞ;

where H0ðtiÞ is the estimated integrated baseline hazard (orcumulative hazard). These residuals are based on the obser-vation that, for a random time to event T with survivor func-tion S(T), the random variable −logS(T) is distributed expo-nential with mean one. If the model is calibrated correctly, theCox-Snell residuals should show a standard exponential distri-bution with hazard function equal to one, and thus the cumu-lative hazard of the Cox-Snell residuals should follow a straight45 degree line. The plot in Figure 3 confirms that most of thestep lines are close to the dashed straight line, except for a fewtail large ones. As a result, we feel there is no evidence to rejectthe model.The model shown in Table 3 includes 10 simple covariates,

where each significant covariate represents a particular geo-defect type. A positive coefficient implies that the hazard ishigher (hazard ratio is greater than 1.0), whereas a negative one

Qing He et al—Track geometry defect rectification 397

indicates a lower hazard (hazard ratio is less than 1.0). In thefitted model, all covariates have positive coefficients, indica-ting that all geo-defect types listed in Table 3 have a strongpositive impact on derailment risk. The higher the values of thecovariates are, the higher is the derailment risk. Initially, geo-defect data of each type is aggregated into two types ofcovariates: the number-based and amplitude-based covariates.The number-based covariates count the number of defectsin each section for each type, whereas the amplitude-basedcovariates calculate 90th percentiles of geo-defect amplitudes.After careful model selection, we restrict the number-basedgroup to GAGE_W1, REV_X and GAGE_W2; on the otherhand, the amplitude-based group includes SUPER_X, GAGE_C,DIP, WARP, HARM_X, WEAR and ALIGN.Recall from Equation (3) that we can derive the derailment

probability PiD(t) on section i prior to time t as:

PDi ðtÞ ¼ ProbiðT⩽tÞ ¼ 1 - SiðtÞ; (5)

where Si(t) indicates the survival probability on section iprior to time t. After some defects are rectified, the derailment

probability will decrease, based on modifications to the hazardfunction λi(t) on section i at time t, as determined byEquation (4). (The covariate vector X is specified in Table 3.)To accommodate different rectification alternatives (or acti-vities) a, we denote λia(t) and Sia(t) as the hazard function andsurvival probability with respect to time t, respectively, afterrectification activity a is performed on section i. Each recti-fication activity corresponds to fixing a single type or combina-tion types of geo-defects, equivalent to setting zeros for therelevant covariates. The derailment probability after rectifica-tion activity a is taken, Pia

D(t), is therefore:

PDiaðtÞ ¼ 1 - SiaðtÞ (6)

5. Optimal track rectification

Given both the Yellow tag deterioration probability PkR(t) and

the track derailment probability PiaD(t), we are ready to formulate

the track rectification optimization model. When the rectifica-tion problem involves repairing a large amount of Yellow tags,it makes the problem difficult to solve. To reduce the computa-tional complexity, we assume that although rectification deci-sions are made for individual defects with Red tags, multipleYellow tags are rectified together at the section and defect typelevel. In summary, the proposed optimization model aims tooptimize the timing to rectify Red tags within their repair duedates, whereas optimize both selection and rectification timingfor Yellow tags before they deteriorate to Red. Defects that areto be repaired in the same time step constitute a cluster. Thecentre of each cluster is defined as the defect with medianpost mile.Let Δt be the time interval between two inspection runs,

and let TR be the total number of discrete rectification periods(eg days, weeks). The model parameters are as follows:

i∈ I Index for track sectionsa∈A Index for rectification activitiesk, l∈Ka Indices for all defects, including both

Yellow and Red tagsk'∈Ky Index for Yellow tags, Ky⊆Ka

Table 3 Estimation results of Cox PH model

covariates coef exp(coef) (hazard ratio) se(coef) z Pr(> |z|)

numYEL_GAGE_W1 1.01E-01 1.11E+00 1.96E-02 5.165 2.40E-07numYEL_GAGE_W2 2.28E-01 1.26E+ 00 1.17E-01 1.948 0.05139numYEL_REV_X 6.66E-01 1.95E+ 00 3.21E-01 2.077 0.03782amp90_ALIGN 4.17E+00 6.45E+ 01 2.40E+ 00 1.736 0.08261amp90_DIP 9.58E-01 2.61E+ 00 4.67E-01 2.052 0.04016amp90_GAGE_C 6.33E-01 1.88E+ 00 2.10E-01 3.013 0.00259amp90_HARM_X 5.43E+00 2.29E+ 02 2.78E+ 00 1.958 0.05028amp90_SUPER_X 5.10E-01 1.67E+ 00 1.68E-01 3.028 0.00246amp90_WARP 7.44E-01 2.10E+ 00 3.69E-01 2.014 0.04404amp90_WEAR 1.54E+00 4.65E+ 00 8.34E-01 1.843 0.06533

Figure 3 Cumulative hazard of Cox-Snell residuals.

398 Journal of the Operational Research Society Vol. 66, No. 3

t∈ {1,… , TR} Index for defect rectification periodsLk Due period to repair defect kτk Repair time for defect kCk Repair cost for defect kC'kl Travel cost from defect k to lCD Derailment costPiaD(Δt) Probability of a derailment in the inspection

time interval Δt, if alternative a is chosenfor section i

Pk′R(Δt) Probability that Yellow tag defect k′ will

progress to a Red tag defect in the inspec-tion time intervalΔt, if it is not chosen to berectified

λk If λk= 1, defect k has to be rectified inhorizon TR, including all the Red tags andYellow tags with Pk′

R(Δt)⩾ δ, where δrepresents predefined probability; If λk= 0,we have the flexibility to decide whether torectify defect k or not

μiak' If μiak′= 1, Yellow tag defect k′ can berectified by choosing repair activity a forsection i

Wtmin Minimal allowed work time at time period t

Wtmax Maximal allowed work time at time period t

The following model decision variables are defined:

xiat∀ i, a, t binary variables that denote whether repairactivity a is chosen for section i at time stept (xiat= 1), or not (xiat= 0)

ylt∀ l, t binary variables that denote whether defectl is the centre of a cluster to be rectified attime step t (ylt= 1), or not (ylt= 0)

Zkt∀ k, t binary variables that denote whether defectk is rectified at time step t (zkt= 1), or not(zkt= 0)

wkl∀ k, l binary variables that denote whether defectk is rectified within a cluster centred at l attime step t (wkl= 1), or not (wkl= 0)

B Budget including both rectification costsand travel costs

(1) ObjectiveThe optimization model denoted as the Cost-based Formula-

tion (CF) can now be formulated as follows:

CFð ÞMinB +Xi

Xa

Xt

xiatCDPDiaðΔtÞ (7)

Objective (7 and 8) aims to minimize the total expectedcost, which is the sum of budget and derailment costs in theinspection interval. Budget is further defined in Constraint (9),composed of rectification costs and travel costs. The merit oftreating budget as a decision variable is that the optimization

model can produce the best needed budget to achieve systemoptimality. If budget is given and limited, the objective can bemodified as below,

MinXk

Xl

wklC0kl +

Xk

Xt

zktCk

+Xi

Xa

Xt

xiatCDPDiaðΔtÞ ð8Þ

Note that travel costs and rectification costs are for bothRed and Yellow tag defects, while derailment costs are onlycaused by Yellow tags, since Red tags are always repairedsoon after they are discovered. The rectification of Red tagsis considered in the formulation since this affects the totaltravel costs.

(2) Constraints(i) Budget constraints:

Xk

Xl

wklC0kl +

Xk

Xt

zktCk⩽B (9)

If budget is given, constraints (9) bound the allowable totalrepair costs. Otherwise, the constraints calculate needed budgetwhen the inequality will be tight.(ii) General rectification constraints:

λk⩽Xt

zkt⩽1 8k 2 Ka (10)

Xt

tzkt⩽Lk 8k 2 Ka (11)

Constraints (10) ensure that each defect can only be rectifiedonce, and all the Red tags and a portion of severe Yellow tags(Pk′

R(Δt)⩾ δ) should be chosen in the repair horizon. Constraints(11) guarantee that the defect will be rectified before its duedate. After a regular inspection run, each Red tag will beassigned a repair due date, from a couple of days up to a fewweeks, according to its severity. In this paper, the due periods Lkfor Yellow tags are obtained as the minimal values to satisfyinequality Pk′

R(Lk)⩾ δ.(iii) Relation constraints among decision variables:

zkt⩽Xl

ylt 8k 2 Ka; t 2 f1; ¼ ; TRg (12)

ylt⩽zlt 8l 2 Ka; t 2 f1; ¼ ; TRg (13)

wkl⩽Xt

ylt 8k; l 2 Ka (14)

ylt⩽Xk

wkl 8l 2 Ka; t 2 f1; ¼ ; TRg (15)

zkt⩽Xl

wkl 8k 2 Ka; t 2 f1; ¼ ; TRg (16)

Qing He et al—Track geometry defect rectification 399

Constraints (12) and (13) describe the relationship betweenzkt and ylt. zkt can be 1 only if any defect is rectified at time t, thatis: ∑l ylt= 1. And ylt= 1 will lead to zlt= 1. Similarly, Con-straints (14) and (15) specify the relationship between ylt andwkl. Constraints (16) ensure that zkt can be 1 only if defect k isrectified within any cluster, that is:∑l wkl= 1.(iv) Cluster repair constraints:

Xt

ylt⩽1 8l 2 Ka (17)

Xl

ylt⩽1 8t 2 f1; ¼ ; TRg (18)

Constraints (17) and (18) ensure that a defect can only be thecentre of at most one cluster and each time step has at most onecluster, respectively.(v) Yellow tag rectification constraints:

Xa

Xt

xiat ¼ 1 8i 2 I (19)

μiak0xiat⩽zk0t 8k0 2 Ky; i 2 I; a 2 A; t 2 f1; ¼ ; TRg (20)

zk0t⩽Xi

Xa

μiak0xiat 8k0 2 Ky; t 2 f1; ¼ ; TRg (21)

According to constraints (19), each track section containingYellow tags should be assigned to an activity at each time step.Activities are defined as a combination of defect types to berepaired at a track section. Grouping the Yellow tags togetherfor rectification seems reasonable for the following two prac-tical reasons: (1) the work team doesn’t need to switch repairequipment and materials while handling multiple defects of thesame type, thereby reducing the potential total costs, and (2) it iscompatible with the proposed derailment risk model, which isestablished based on aggregated value for certain defect typesrather than individual defects. Suppose that a section isobserved to contain Yellow tags of three defect categories.In this case, there are 23= 8 activities available to the decisionmaker, because the local track master can choose to repair none,single type, mixed types, or all of the defects of each type in anypossible combination. Constraints (20) and (21) build therelationship between individual defect repair zk't and sectionrepair xiat. In (20), applying a repair activity on a section willrectify all the associated defects. Constraints (21) guarantee thateach chosen Yellow tag corresponds to a repair activity.(vi) Work time constraints:

Wmint

Xl

ylt⩽Xk

τkzklt⩽Wmaxt

8l 2 Ka; t 2 f1; ¼ ; TRg(22)

Constraints (22) implement work time limitations for eachrectification period.

The optimization model is a mixed integer program (MIP)which is linear in the objective function as well as constraints,and it can be solved using standard commercial solvers.Note that the cost of derailment CD is a random variable.

Figure 4 presents the histogram of the derailment cost dataand a fitted probability density function. The derailment costdistribution follows a strongly heavy-tailed distribution, rang-ing from hundreds of US dollars to millions of US dollars.The expected derailment cost based on our data set is around$510K.Owing to the structure of our proposed formulation, CD is

only present in the objective function. When trying to minimizethe total expected costs, we can simply replace derailment costCD with the expected derailment cost CD in the model. This isbecause the total expected cost is given as:

E B +Xi

Xa

Xt

xiatCDPDiaðΔtÞ

" #

¼ B +Xi

Xa

Xt

xiatE½CD�PDiaðΔtÞ

¼ B +Xi

Xa

Xt

xiatCDPDiaðΔtÞ ð23Þ

since the expectation of a sum is the sum of expectations.According to expression (23), the stochastic objective willeventually be derived in the same structure as the deterministicone (7). Therefore, the proposed MIP handles uncertainderailment costs.Instead of minimizing total expected costs, an alternate

objective is to minimize a function of the maximal derailmentprobability across all track sections.

MinfβB + Maxi

Xa

Xt

xiatPDiaðΔtÞg

where β denotes a weighting factor, which should be sufficien-tly small so as to avoid overwhelming the derailment risk. Suchmin-max objectives try to hedge the worst-case track conditionin terms of safety and increase the robustness of track infra-structure from a systems point of view. We denote this model as

Figure 4 Probability density of derailment cost.

400 Journal of the Operational Research Society Vol. 66, No. 3

the Risk-based Formulation (RF), described as follows:

RFð ÞMin βB +F (24)

subject to

Xa

Xt

xiatPDiaðΔtÞ⩽F 8i 2 I (25)

And Constraints (9)-(22) where F represents the maximalderailment probability after rectification.

6. Numerical examples

In this study, the proposed rectification model is applied to geo-defects detected from inspection runs occurring in December2011. Among US nation-wide line segments, five main linefreight tracks denoted A, B, C, D and E, ranging from 50 milesto 500 miles, are selected as case studies of track rectificationplanning (see Table 4). In the preprocessing stage, each tracksegment is further divided into no more than 2 mile sectionsaccording to network structure and traffic data. Red tags are rareevents, while the occurrences of Yellow tags are much larger.On average, we can detect Yellow tags within every 1∼2 miles.However, the number of Yellow tags is not evenly distributedthrough the track. After a complete geometry inspection run, thetrack sections with defects only account for 15∼20% of totalsections, and among these track sections, each contains around3∼5 defects. This means that the derailment risk is notuniformly distributed in the entire track. Therefore, for purposesof efficient Yellow tag repair, it seems reasonable to considerrectifying sections rather than individual tags. Note that thenumber of binary variables and number of constraints increaseexponentially when the size of the track increases from around50 miles to 500 miles. Track segments longer than 500 milesare therefore not considered in this paper.The number of Yellow tags are aggregated at the section

level in order to build the set of repair activities as well asparameters μiak′. Considering a 90-day inspection interval Δt,we obtain the derailment probability Pia

D(Δt) before the nextinspection run for each section and each repair activity based onthe proposed Cox PH model. By applying the defect deteriora-tion model, we calculate the probability Pk′

R(Δt) of Yellow tagsconverting to Red. If Pk′

R(Δt) is greater than a predefined

threshold δ, the corresponding Yellow defect k' will be labelledas ‘must repair’. The rectification horizon TR is set to be lessthan 22, representing typical repair due in work days of amonth. Owing to lack of data, the Red tag repair due periodsLkk∈Ka\KY are randomly sampled from a uniform distributionin the interval [1, 22], and the Yellow repair due periodsLkk∈KY are obtained as the minimal values that satisfy inequal-ity Pk′

R(Lk)⩾ δ.Accurately estimating the exact repair costs is very difficult,

due to variations in equipment, fleet and personnel at eachlocal maintenance centre. For simplicity, we only take intoaccount three different defect groups and assume that costs forboth Yellow and Red tags stay constant across differentmaintenance scenarios, as shown in Table 5. In this paper,we first take the average of rectification costs for Red tags, andthen scale by 0.9 to obtain the corresponding Yellow tagrectification costs.All five rectification optimization instances were solved by

CPLEX 12.3 on a personal computer with a Quad-Core 3.5GHzCPU and 16 GB RAM. Table 6 summarizes the computationalresults of rectification planning across five tracks, with boththe CF and RF optimization models. As shown in Table 6, wecould obtain exact optimal integer solutions for A, B, C, Dwithin a few minutes. CF model for Track E took about 900 sto be solved and no exact optimal integer solutions could befound for RF model of track E within 1 h. However, comparingthe best feasible solutions (obtained within 1 h) with a contin-uous linear programming objective function value that providesan upper bound for the original discrete optimization problem,the gap is found to be less than 0.01%. This implies that ourcurrent best solution may be close enough to the exact optimalinteger solution.

Table 4 Summary of test track segments

Trackname

Length(mile)

No. of sections withdefects

No. of defecttypes

No. of Yellowtags

No. of Redtags

Rectificationhorizon TR

No. of binaryvariables

No. ofconstraints

A 50 13 3 60 4 5 4886 14 974B 100 21 6 77 2 10 8441 58 300C 200 34 7 100 18 15 18 760 149 090D 300 49 8 122 37 20 34 801 426 754E 500 94 8 343 38 22 157 185 1 157 160

Table 5 Average rectification cost for individual geo-defects

Geo-defect types Averagerectification costfor a Yellow tagdefect (USD)

Averagerectification costfor a Red tagdefect (USD)

Gage related: GAGE_W1,GAGE_W2 and GAGE_C

1539 1710

Surface related: DIP and SURF 1125 1250Other defect types 1534.5 1705

Qing He et al—Track geometry defect rectification 401

Note that the two models share most of the constraints incommon, but have different objectives: CF reduces totalexpected costs, whereas RF minimizes the maximal possiblederailment risk. Compared with the CF solutions, RF requiresmore budget for short and medium tracks (A, B and C), aimingto repair track sections with high derailment risk. For longertracks (D and E), both CF and RF provide similar values for theoptimal budget. One reason could be that the probability ofobserving track sections with ‘high base risk’, representing therisk after rectify all the defects, is much higher for long tracks.However, RF minimizes the worse situation for track sectionswith high derailment risks. In that sense, RF is more conserva-tive than CF, but it may be more suitable for users requiring amore risk-averse maintenance strategy.To further examine the differences between CF and RF

solutions, we present the details of track repair activity solutionsfor track A, shown in Table 7. Each track has 2n repair acti-vities to be conducted, where n is the number of defect types.For each repair activity a at section i, derailment probabilityPiaD(Δt) is obtained using Equation (6). Rectification costs

of a repair activity at each section are also provided in Table 7.CF aims to balance the costs from derailment and rectification,while RF considers minimizing the maximal derailment pro-bability as the top priority. For example, section 1235 (see

Table 6 Summary of computational results

Trackname

CF RF

Optimal budget (USD) A 19 136.29 37 612.71B 23 475.49 60 373.72C 94 947.81 112 029.82D 117 148.79 111 465.19E 212 250.15 208 995.63

Optimal objective A 64 141.26 0.0125B 127 086.22 0.01976C 186 058.50 0.0121D 291 994.38 0.0279E 493 349.31 0.0268

Maximal derailmentprobability

A 0.0145 0.00877B 0.0343 0.0137C 0.0124 0.0110D 0.0320 0.0270E 0.0273 0.0247

Solution time (s) A 2.15 3.03B 4.46 20.70C 14.087 34.77D 54.27 128.22E 895.25 3600*

*CPLEX didn’t give optimal solutions in 3600 s.

Table 7 Solutions to rectification optimization for Track A

Section id (i) Repair activity (a) PiaD(Δt) Rectification costs

of a activityMaximal probability

of Yellow tagsdeteriorating to Red

∑txiat (CF) ∑txiat (RF)

1216 NONE 0.0099 0 0.457 1 01216 GAGE_W1 0.0058 4617 0.457 0 11217 NONE 0.0050 0 0.254 1 11217 GAGE_W1 0.0035 10 773 0.254 0 01218 NONE 0.0047 0 0.076 1 11218 GAGE_W1 0.0045 1539 0.076 0 01225 NONE 0.0057 0 0.277 1 11225 XLEVEL 0.0057 3069 0.277 0 01227 NONE 0.0072 0 0.974 0 01227 XLEVEL 0.0072 3069 0.974 1 11232 NONE 0.0054 0 0.991 0 01232 GAGE_W1 0.0044 6156 0.991 1 11234 NONE 0.0063 0 0.648 1 11234 GAGE_W1 0.0045 10 773 0.648 0 01235 NONE 0.0145 0 0.268 1 01235 GAGE_W1 0.0088 15 390 0.268 0 11235 XLEVEL 0.0145 4603.5 0.268 0 01235 GAGE_W1+XLEVEL 0.0088 19 993.5 0.268 0 01237 NONE 0.0064 0 0.378 1 11237 GAGE_W1 0.0043 12 312 0.378 0 01238 NONE 0.0076 0 0.446 1 11238 GAGE_W1 0.0045 13 851 0.446 0 01240 NONE 0.0109 0 0.382 0 01240 GAGE_C 0.0061 1539 0.382 1 11240 GAGE_W1 0.0091 1539 0.382 0 01240 GAGE_C+GAGE_W1 0.0051 3078 0.382 0 01241 NONE 0.0084 0 0 0 11241 GAGE_C 0.0048 1539 0 1 01242 NONE 0.0046 0 0.049 1 11242 GAGE_W1 0.0044 1539 0.049 0 0

402 Journal of the Operational Research Society Vol. 66, No. 3

emboldened figures in Table 7) has the highest derailmentprobability 0.0145. The repair activity ‘GAGE_W1’ is chosenin RF model since it reduces the probability from 0.0145to 0.0088. However, no repair activity is applied for section1235 in the CF model, due to its high rectification cost (over$15 000). To illustrate the deterioration probability fromYellow to Red, we also present the maximal deteriorationprobability across each track i. We set the threshold δ as 0.85.If the deterioration probability for any Yellow tag in a section ishigher than 0.85, tags of the same type are labelled as ‘mustrepair’ tags, for example, XLEVEL in section 1227.For the sake of comparison, we generate severity-based

manual plans to mimic the rectification strategy in practice.We define the empirical severity index as:

Sij ¼ exp

maxðπkijÞπ95thj

!ln ðnijÞ 8i; j

where πkij is the amplitude of Yellow tag defect k in section iand type j, πj

95th denotes the 95th percentile of amplitude of geo-defect type j, and nij represents the number of Yellow tagdefects in section i and type j. Given the same amount of budgetspecified by the CF model, severity measures Sij are ranked andselected in descending order, until the budget is consumed.Then the selected sections are simply assigned to their repairdue periods, and further adjusted to satisfy the work time limitat each period. Finally, considering additional travel costs,sections with lowest Sij are excluded until the budget constraintis satisfied again.Figure 5 compares the total costs of CF models to the manual

plan for the five track segments. The percentage changes of totalcosts from the manual plan as compared with the optimal CFmodel are labelled at the top of the bars in the bar chart. As therectification problem gets larger in scale from track A to E, thepercentage of reduced total costs gained by applying CF againstthe manual plan increases dramatically, from 12.2 to 35.4%.On average, the CF model reduces total costs by around 20%.

7. Concluding discussion

This paper presents an analytical framework to address the trackgeo-defect rectification problem by integrating three models:(i) a statistical deterioration model to predict the amplitude ofYellow tag defects in the future; (ii) a track derailment riskmodel to dynamically predict the derailment risk as a functionof the different types of geo-defects; and (iii) an optimizationmodel to make geo-defect rectification decisions with twodifferent objectives: a cost-based formulation (CF) and a risk-based formulation (RF).According to our data-driven study, different geo-defect

types deteriorate at different speeds. Most of them exhibit largerdeterioration rates with higher traffic, but some of them arefound not to be very sensitive to traffic data. Track derailmentrisk is associated with each section of track in 2 miles and eachtype of geo-defects. We observe that the number of defects ineach section is very significant for some types of geo-defectsand that the 90th percentiles of geo-defect amplitudes play animportant role in predicting derailment risk.Comparing the CF and RF optimization models, CF aims to

reduce total expected costs, while RF minimizes the maximalderailment risk. Risk-averse track local masters may wish toadopt RF model since it hedges against the worst-case situationof derailment risk at each section. The CF model may be moresuitable if the railroad company would prefer to maintaina certain level of total costs. Real-world case studies on anexisting US railway network show that the proposed metho-dology yields reliable and economical solutions, as comparedwith current rail industry practices.The rectification problem involves optimizing a large number

of integer decision variables and constraints, which is difficultto solve in general. We tackle this issue by assuming thatYellow tags are rectified in bulk, at the section and defect typelevel. Based on the computational results from our numericalcase studies, we learn that we can obtain an exact optimalsolution within a short span of time, such as a few minutes, incases where the railway track is up to a couple of hundred mileslong. This is therefore able to handle most tracks operated by alocal track master. In cases where we cannot find an exactoptimal solution within, for example, 1 h, the best feasiblesolution obtained is acceptable since its corresponding objectivefunction value lies close to the upper bound, as computed by therelaxed continuous linear programming problem.Our proposed model assumes that the budget is a decision

variable. Instead, we could also consider the case where thebudget is fixed and provided to the user. Track possession costsare usually one of the biggest concerns while conducting railwaypreventive maintenance activities. However, in corrective main-tenance activities, safety is always the top priority. In this paper,we consider track possession costs indirectly by minimizing thetravel distance between defects in each repair period. In futurework, one may also wish to consider repair job scheduling andcrew scheduling based on existing train timetables to minimizetravel, as well as handling and possession costs. As a final

Figure 5 Comparisons of the total cost of CF and manualsolutions.

Qing He et al—Track geometry defect rectification 403

remark, we would like to note that our track rectification modelcould be easily extended to solve other practical problems,particularly those involving further constraints that a local trackmaster may need to take into account.

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Received 11 February 2013;accepted 13 January 2014 after one revision

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