Squat maintenance at the Dutch railways

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Squat maintenance at the Dutch railways

P. Peereboom

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Master’s Thesis Econometrics, Operations Research and Actuarial Studies Specialization: Operations Research

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An optimal maintenance schedule to remove squats

Peter Peereboom

August 26, 2013

Abstract

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3.2 Data processing . . . 18 4 Model 21 4.1 Distributions . . . 21 4.2 Dynamic programming . . . 22 4.2.1 State space . . . 23 4.2.2 Transition Probabilities . . . 24 4.2.3 Maintenance decisions . . . 27 4.3 Objective function . . . 28 4.4 Constraints . . . 29 4.5 Value function . . . 31 4.6 Final model . . . 32 5 Truncation values 33 5.1 Moderate origination rate . . . 33

5.2 High origination rate . . . 37

5.3 Low origination rate . . . 37

5.4 Poisson truncation . . . 38

5.5 Comparison of different truncations . . . 39

6 Results 41 6.1 Optimal decisions . . . 41

6.2 Comparison with other maintenance strategies . . . 43

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7 Summary and conclusions 49

References 51

A Figures 53

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Nomenclature

λ µA µB st st0 nA t nB_t nC_t p q f (i) P ois(·) B(·) Cm Cg Ccr Csr xg_l,t xcr_l,t xsr_l,t V (st) xl,t St+1(st, xt+1) X Y Z w Yi(w)

Expected number of squats originating per month Expected time a squat is small

Expected time a squat is moderate State of the system at time t

State after maintenance action of time t Number of small squats in stage t Number of moderate squats in stage t Number of severe squats in stage t

Probability that a moderate squat grows into a severe squat Probability that a small squat grows into a moderate squat Function denoting the probability that i squats originate Probability mass function of the Poisson distribution Probability mass function of the Binomial distribution Total maintenance costs

Grinding costs per meter Renewal costs per meter

Replacement costs per replacement

Binary variable denoting whether track segment l has been ground at time t or not

Binary variable denoting whether track segment l has been replaced completely at time t or not

Number of squats removed by sporadic replacements on track segment l at time t

Value function of state st (xg_l,t, xsr_l,t, xcr_l,t)

Random function denoting the state of the system at time t + 1 given xt and st

Random variable denoting the number of originating squats Random variable denoting the number of small squats that grows into moderate squats

Random variable denoting the number of moderate squats that grows into severe squats

Window of Welch’s method

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1 Introduction

The Dutch railway network is the busiest network of the European Union (Ramaekers, de Wit, and Pouwels, 2009), and one of the busiest networks in the world (d’Epinay and Meyer, 2007). The intense use of the Dutch railway network causes substantial wear and tear of the tracks. To limit the costs and prevent high safety risks caused by track defects, a good maintenance policy is essential. A classification of maintenance strategies is described in Kothamasu, Huang, and Verduin (2006). A schematic overview of a selection of these maintenance strategies is given in Figure 1.

Figure 1: Schematic overview of maintenance strategies based on Olde Keizer (2012) and Kothamasu et al. (2006)

1.1 Maintenance strategies

Maintenance strategies can be classified in proactive maintenance strategies and reactive maintenance strategies. As the name already suggests, reactive maintenance strategies react on failures of the system. If a maintenance strategy of this type is applied, maintenance is only performed when system defects occur. On the other hand, proactive maintenance strategies try to prevent system failures. If proactive maintenance is used, maintenance is aimed to prevent defects from occurring and therefore one does not wait for defects to occur. A more detailed description of the different maintenance strategies, based on Kothamasu et al. (2006) is given below.

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preven-1 INTRODUCTION 1.2 Track defects

tive maintenance for all systems, functioning or failed takes place at fixed time intervals.

Age-based maintenance is a maintenance strategy in which the system is re-paired when it reaches a specific age, which we will call a. If the system fails before it has reached this age, the system will be repaired and its age will be set to 0. A new preventive maintenance action will be planned when this repaired system has reached age a. The difference between age-based main-tenance and constant interval mainmain-tenance is that age-based mainmain-tenance does reschedule its maintenance actions in case of a failure, while constant interval maintenance does not.

Condition-based maintenance is a maintenance strategy in which the deci-sion to perform maintenance or not depends on the condition of the system. To determine the condition of the system, some system specific parame-ters are monitored. Statistical analysis is used to determine the condition of the system from those parameters. When the system reaches a prede-termined critical condition, maintenance will be initiated. By monitoring system parameters and determining the condition of the system, one can give a prediction of the moment of failure of the system. As this mainte-nance strategy uses data to predict the moment of failure, we call it a pre-dictive maintenance strategy. Together with constant interval maintenance and age-based maintenance, condition-based maintenance forms a group of proactive maintenance strategies. However, since machines sometimes break down unexpectedly, before proactive maintenance has taken place, there is still need for reactive failure-based maintenance actions.

This classification of maintenance strategies is far from complete, as many more maintenance strategies are possible. Nevertheless, the four mainte-nance strategies that have been described above are among the most used strategies and they are the most relevant strategies for the remainder of this thesis. In fact, in our study a hybrid strategy is called for. As will become clear later on, the condition of the rails does play a role in the maintenance decisions. But properties of failure-based maintenance will also be used in this thesis. The maintenance strategy that will be developed in this thesis will be compared to some of the previously described maintenance strategies.

1.2 Track defects

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1 INTRODUCTION 1.3 Problem description

of the track. Different types of track defects require different maintenance policies, therefore we will focus on only type of track defects in this thesis. The track defects that will be treated in this thesis are rail top defects, more specifically squats. Squats are small rail top defects that are found on all types of track (Li, 2011). An example of a squat is shown in Figure 2. Squats gradually grow over time and can cause safety risks if they become too large. Timely maintenance should be performed to prevent squats from reaching a dangerously large size. Different maintenance methods are available to remove squats from tracks. However, the size of a squat determines whether certain methods can be used. In this Master’s thesis it will be determined when maintenance actions should be performed and what methods should be used for these maintenance actions. A more detailed description of squats and the different maintenance strategies that can be used to remove them, will be given in a later section.

Figure 2: Example of a squat (Li, 2011)

1.3 Problem description

In this Master’s thesis an optimal maintenance strategy for the removal of squats will be developed. The research question that should be answered in order to find this optimal strategy is:

When should squats be removed and which methods should be used to minimize costs?

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1 INTRODUCTION 1.4 Thesis outline

The method that will be used to solve this problem is dynamic program-ming. Dynamic programming can be used in mathematical problems where a number of decisions have to be made sequentially (Bellman, 1952). This corresponds to the problem of this thesis in which after each inspection it has to be decided whether or not to perform maintenance actions and if a maintenance actions has to be performed it has to be decided which main-tenance action is optimal. Solving a dynamic programming problem can be very time consuming, therefore we need good computer software to ensure fast calculations. The software package that will be used in this thesis is R (R Core Team, 2013).

1.4 Thesis outline

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2 Squats

Squats are a relatively new phenomenon in railway maintenance. In the Netherlands, investigation in squats started only in the mid 1990s (Smulders, 2003). However, decades before squats were investigated in the Netherlands for the first time, they had already been reported elsewhere. In Europe, they have been reported first in the UK in the 1970s (Li, Zhao, Molodova, and Dollevoet, 2010). But the country where squats have been reported for the very first time is Japan. In Japan, squats were already reported in the 1950s, when they were described as the “black spot” (Clayton, 1996).

The International Union of Railways has defined a squat as a visible widening and localized depression of the contact surface of the rail head, accompanied by a dark spot containing cracks with circular arc or V shape (UIC, 2002).

In the remainder of this section an extensive description of the life cycle of squats will be given. In the first part of this section it will be explained how squats can originate. This will be followed by a description of their growth process and the different stages of their development. Finally the different methods that can be used to remove squats will be discussed.

2.1 Origination of squats

2.1.1 Short pitch corrugation

The origination of squats can have many different causes. One of the most frequently encountered causes of squats is short pitch corrugation. A field survey conducted by Li (2011) showed that 33% of the squats detected in this survey were caused by short pitch corrugation. Short pitch corrugation is a wave pattern on the rail top with a wavelength in the range of 20-80mm (Afferrante and Ciavarella, 2009). An example of a squat caused due to corrugation can be found in Figure 3a.

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2 SQUATS 2.1 Origination of squats

(a) Squat caused by corrugation

(b) Corrugation caused by squat

Figure 3: Comparison of different types of corrugation (Li, 2011)

2.1.2 Welds

Different field surveys have shown that in the Netherlands 10 – 15% of the squats occur at welds. When rails are welded, the material surrounding the weld is affected by the heat of the welding process. The region where the material has been affected is called the heat affected zone. In the heat affected zone, the microstructure and properties of the material change, which can result in a lower hardness of the material. The weld itself may also have a lower hardness than the rail material. The lower hardness of the material can lead to the initiation of squats (Li, 2011). An example of a squat at a weld is given in Figure 4.

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2 SQUATS 2.2 Growth process of squats

2.1.3 External factors

The two causes of squats that have been described so far are internal causes. Squats were caused by material imperfections. Squats can also be caused by external factors. We will give some examples of external factors below.

If a small hard object ends up between rails and a wheel of a train, a shallow indentation may originate. The contact forces at such an indenta-tion can be higher, which results in increasing wear. If the wear at a shallow indentation is much higher than the wear at the rest of the rails, the inden-tation can grow into a squat. However not all indeninden-tations will eventually become squats. Only indentations larger than a critical size will grow into squats, while smaller indentations will disappear due to surface wear.

Larger and harder objects on the tracks can indent in wheels of trains and cause periodic deep indentations. At deep indentations squats can grow easier and faster than small indentations and follow a different growth mech-anism (Li, 2011).

The final cause of squats that will be described in this thesis is a wheel burn. Sometimes the wheels of trains may skid or slide, for example when trains break. Wheel burns are damages caused by skidding and sliding of wheels. Some authors consider wheel burns as a distinct damage, but since they bear all characteristics of squats (Li, 2011) we will consider them as a cause of squats. In Figure 5 examples of a shallow indentation, a deep indentation and a wheel burn are shown.

(a) Shallow indenta-tion

(b) Deep indentation (zoomed) (c) Wheel burn

Figure 5: Examples of external causes for squats (Li, 2011; Li et al., 2010, 2011)

2.2 Growth process of squats

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2 SQUATS 2.2 Growth process of squats

(a) Class A (b) Class B (c) Class C

Figure 6: Examples of squats from different classes (Li et al., 2010)

According to the definition of UIC (2002) only class B squats and class C squats can be defined as squats. Both class B squats and class C squats are also called mature squats. These squats have all characteristics of squats. When squats are in class B cracks start to initiate and grow slowly. When squats are in class C the growth of these cracks accelerates and they can eventually lead to catastrophic rail break.

Class A squats are also named initiating squats, since they do not have all the characteristics of squats yet. According to the definition of UIC (2002), these defects are not defined as squats, although they are referred to as called early squats, young squats or squat seeds. In this thesis we will extend the definition of squats to include small squats.

All squats start as class A squats. These light squats are hard to identify. Track indentations larger than a certain size are called class A squats, but only squats that grow into class B squats can be called squats. Many small rail defects disappear due to surface wear and these defects should not be called squats. Correctly identified class A squats gradually grow following a growth process that consists of three phases (Li et al., 2010). The growth process that holds for a small squat that is currently 10mm long will be explained below.

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2 SQUATS 2.3 Removal of squats 0 10 20 30 40 0 10 20 30 40 50 Growth of squats Number of months Length of squat in mm

Figure 7: Growth curve of a squat (Li et al., 2010)

2.3 Removal of squats

Depending on their size, squats can be removed in different ways. According to Li et al. (2010), three different methods can be used to remove squats. These methods are complete rail renewal, detection based grinding and spo-radic rail replacement. We will discuss these three methods in more detail below.

Complete rail renewal means, like the name of the method suggests, that the rails of a track segment, which has a length of 200 meters, are completely replaced. This method might be a good option if there are many squats close to each other. An advantage of this method is that since the rail is completely replaced, all squats are removed with certainty from the replaced track segment.

A second method is detection based grinding. This method is cheaper than complete rail renewal, but it cannot always be used. Only small squats can be removed by grinding. Larger squats have cracks that cannot be completely removed by grinding. Small cracks remain and will cause new squats. Another limitation of grinding is that, eventually, the rail track always needs to be replaced. Due to grinding and surface wear, the railhead will become thinner and thinner and after a number of years, the rail has to be replaced, because the railhead becomes too thin. If grinding does not take place too often, the service life of rails can be quite high.

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2 SQUATS 2.3 Removal of squats

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3 Data

This section contains a description of the data that can be used in the future to determine origination rates and growth processes of squats. The Delft University of Technology is working on methods to determine the locations of squats. This method is currently used, but unfortunately data is not available yet. Axle box acceleration data and visual inspections are used to determine locations and classes of squats. This section consists of two parts. It will start with an explanation of axle box acceleration data. Furthermore it will contain an explanation of methods that are used to process these data.

3.1 Axle box acceleration data

Currently, ultrasonic and eddy current measurements are used to find defects on rail. These methods have an important disadvantage, which is that they cannot detect defects without cracks (Molodova, Li, and Dollevoet, 2011). Since small squats do not always have cracks yet, these methods are not suitable for finding these defects. Therefore, in this thesis axle box acceler-ation (ABA) data, processed by the university of Delft, will be discussed. This method has substantial advantages compared to the currently used de-tection methods. In contrast with the currently used methods, ABA can detect rail defects without cracks, such as small squats. Another advantage of ABA data is that it can be easily collected. No complicated instrumenta-tion is needed. To collect ABA data, only accelerometers are needed. These accelerometers can be installed on standard operating vehicles (Molodova et al., 2011). In Figure 8 the position of the accelerometer on the train is shown. In the next section it will be explained how the data that has been measured by the accelerometer is related to the presence of squats.

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3 DATA 3.2 Data processing

3.2 Data processing

The ABA data that can be collected consists of four vertical ABA signals, GPS coordinates and the train speed. The ABA signals are used to find the squats. An acceleration peak in the ABA measurement data indicates the presence of a squat. This data is then combined with the train speed data. The magnitude of the acceleration levels depends heavily on the train speed. In Molodova et al. (2011) an example is given of a dataset that has been deleted, because the slow train speed resulted in unusable data. Finally, the GPS coordinates are used to find the locations of the squats. In Figure 9 an example is given of raw data that has been collected by an accelerometer. In this graph, there are clearly some peaks visible. The peaks that are numbered as peak 1, peak 2 and peak 3 are locations of three small initiating squats. These squats have been verified by visual inspections. There is also a clear peak between the numbers 2 and 3, but inspection showed that this peak did not correspond to a squat defect (Li et al., 2010).

Figure 9: Measured ABA data (Li et al., 2010)

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3 DATA 3.2 Data processing

(a) Impact index by existing ABA technology

(b) Impact index by new ABA technology

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4 Model

In this section the model that will be used to determine the optimal main-tenance schedule will be derived and explained. This section starts with an explanation of the distributions that have been used. The next part of this section consists of an explanation of the dynamic programming concept. In this part the state space and transition probabilities will also be determined. Additionally, the effects of the maintenance actions will be described. This will be followed by parts where the objective function and the constraints of the model will be derived. Finally the complete model will be presented.

4.1 Distributions

Unfortunately, a lack of data about origination and growth rates of squats, forces us to make assumptions about the relevant distributions. The orig-ination rate is the number of squats that originate between two successive track inspections. Based on information given by the University of Delft, in the remainder of this thesis it will be assumed that inspections take place every three months. The distribution of the origination rate should meet two important requirements. The distribution should be discrete and only nonnegative values should have probability mass. These requirements are important, since they prevent that the model includes half squats or nega-tive squats, which would not make any sense. One of the most well known probability distributions, that fits both requirements is the Poisson distri-bution. According to Law (2007) the Poisson distribution can be used to determine “the number of events that occur in an interval of time when the events are occurring at a constant rate”. This description matches the appli-cation of the distribution, so it is decided to use a Poisson distribution with rate λ for the origination rate. Here λ is positive parameter that is equal to the expected number of originating squats every three months. Since the value of λ is unknown, the optimal maintenance schedule and corresponding optimal costs will be determined for a wide range of possible values.

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4 MODEL 4.2 Dynamic programming

close to the mean. However, there is no evidence that other distributions give a better description of the growing behavior of squats. Exponential distributions with parameters µA and µB will be used to model the time squats spend in class A and B respectively. In Section 2.2 it has already been stated that squats spend on average 15 months in both class A and class B. Since the parameter of the exponential distribution is equal to its mean distributions with parameters µA= µB= 15 will be used.

4.2 Dynamic programming

The maintenance problem of this thesis will be solved using dynamic pro-gramming. Dynamic programming is an optimization method, where a large problem is splitted into several smaller subproblems. Solving these subprob-lems sequentially yields a solution for the original problem. In the mainte-nance problem of this thesis, we have the problem of performing the right maintenance methods at the right moments. It is assumed that inspection of the track takes place only at predetermined moments with fixed intervals. Maintenance decisions take only place after inspection, which means that they also take place at fixed intervals. Therefore this problem can also be seen as a set of sequential subproblems. In these subproblems it should after inspection be decided if maintenance is necessary or not and if maintenance is needed, it should be determined which type of maintenance should be performed.

After each inspection, the current situation of the track will be called the state of the system. The collection of all possible states will be called the state space. The state space of this problem will be described in Section 4.2.1. The state of the system at the inspection of time t will be denoted by st. In the computations in this thesis, we also need to know the state after the maintenance actions of time t. This is an unobserved state that is used to simplify calculations. The state after a maintenance action will be denoted by st0.

In order to determine the probability of being in a specific state at time t, the probabilities of going from state s(t−1)0 to state s_t should be known. These probabilities are called transition probabilities. They can be calcu-lated for each state transition. The calculations of these transition proba-bilities will be described in Section 4.2.2.

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4.2.1 State space

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4.2.2 Transition Probabilities

The probability of a transition from a state to another state is a transition probability. These probabilities can be different for all transitions and de-pend on three random processes. These processes are the number of squats that originate between two inspections, the time a squat stays in class A and the time a squat stays in class B. An assumption that will be made is that a squat cannot “skip” a class. This is a realistic assumption, since the probability that two transitions take place between two consecutive inspec-tions is negligible. Squats that are classified as small squats at an inspection cannot grow into severe squats before the next inspection. Squats also can-not originate as moderate or severe squats, they always originate as small squats. Because of these assumptions, many transitions are impossible, so without calculating a transition probability of 0 can be assigned to them. A transition from stage s(t−1)0 to stage s_t is not possible if one or more of the following conditions are satisfied.

• The total number of squats in stage s_tis smaller than the total number of squats in stage s_(t−1)0.

• The number of severe squats in stage st is smaller than the number of severe squats in stage s_(t−1)0.

• The number of severe squats in stage st is larger than the combined number of moderate squats and severe squats in stage s_(t−1)0.

• The number of moderate and severe squats in stage stis smaller than the combined number of moderate squats and severe squats in stage s_(t−1)0.

• The number of moderate squats in stage s_tis larger than the combined number of small squats and moderate squats in stage s_(t−1)0.

• The number of small squats in stage stis smaller than the total number of squats in stage s_(t−1)0 minus the combined number of moderate squats and severe squats in stage st.

If none of these conditions is satisfied, the transition probability is pos-itive and should be calculated. The probability of a transition from state (nA_(t−1)0, nB_(t−1)0, nC_(t−1)0) to state (nA_t, nB_t , nC_t ) is a product of the following three probabilities.

• The probability that nA

t + nBt + nCt − nA(t−1)0− nB_(t−1)0− nC_(t−1)0 squats originate

• The probability that nB

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• The probability that nC

t − nC(t−1)0 moderate squats grow into severe squats.

Below an example is given where transition probabilities for all states that can be reached from state (1, 1, 0) are shown.

Example 4.1. Consider the state (1, 1, 0) at an inspection at time t − 1 and assume that no maintenance actions take place. The next inspection will take place at time t. At the next inspection the moderate squat can either grow into a severe squat with probability p or remain of moderate size with probability 1 − p. Likewise, the small squat can either grow into a moderate class with probability q or remain a small squat with probability 1 − q. Finally squats can originate. Theoretically an infinite number of squats can originate, but in this example we will limit the maximum number of originating squats to 5, where f (i) is the probability that i squats originate (i = 0, ..., 5). The following states can now be reached from state (1, 1, 0).

• (1, 1, 0) • (2, 1, 0) • (3, 1, 0) • (4, 1, 0) • (5, 1, 0) • (6, 1, 0) • (0, 2, 0) • (1, 2, 0) • (2, 2, 0) • (3, 2, 0) • (4, 2, 0) • (5, 2, 0) • (1, 0, 1) • (2, 0, 1) • (3, 0, 1) • (4, 0, 1) • (5, 0, 1) • (6, 0, 1) • (0, 1, 1) • (1, 1, 1) • (2, 1, 1) • (3, 1, 1) • (4, 1, 1) • (5, 1, 1)

The probability of going to state (nA_t, nB_t, nC_t) is given in Equation 1.

P((1, 1, 0) → (nA_t, nB_t , nC_t ))

= f (nA_t + nB_t + n_tC− 2) · pnCt · (1 − p)1−nCt

· qnBt +nCt−1· (1 − q)2−nBt−nCt ₍₁₎

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states that can be reached is even higher. This shows that some limits are necessary to keep the problem manageable.

In Example 4.1 the probability of going to state (nA_t, nB_t , nC_t ) if the cur-rent state is (1, 1, 0) is given. This function can be generalized in a probabil-ity function of going from state (nA_(t−1)0, nB_(t−1)0, nC_(t−1)0) to state (nA_t, nB_t , nC_t ). Here the assumption that at most 5 squats can originate per period is re-laxed and the assumption that in the beginning of this section has been made, namely that the origination process follows a Poisson distribution with parameter λ, is used. When the Poisson distribution is used, there is no limit on the number of squats originating per period. The Poisson dis-tribution is given in (2). Furthermore the binomial disdis-tribution is needed, which is given in (3). P ois(k; λ) = λ k_e−λ k! (2) B(k; n, p) =n k pk(1 − p)n−k (3)

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4 MODEL 4.2 Dynamic programming P((nA_(t−1)0, nB_(t−1)0, nC_(t−1)0) → (nA_t, nB_t , nC_t )) = P ois(nA_t + nB_t + nC_t − (n_(t−1)A 0+ nB_(t−1)0+ nC_(t−1)0); λ) · B(nC_t − nC_(t−1)0; nB_(t−1)0, p) · B(nB_t + nC_t − (nB_(t−1)0+ nC_(t−1)0); nA_(t−1)0, q) (4) 4.2.3 Maintenance decisions

When maintenance is performed, the state of the rails also changes. The effects of the different maintenance actions will now be described.

The first maintenance action that will be discussed is detection based grinding. This type of maintenance can only remove small squats, since the larger squats are already too deep to be ground away. When condition based grinding is used, all small squats are removed with certainty. For the state of the rails this means that all squats of class A will be removed, so if rails are ground at time t, we have nA_t0 = 0. The number of moderate and severe squats remains the same.

Next we have sporadic rail replacement. It is assumed that at most one squat can be removed when sporadic rail replacement is performed. It is also assumed that the most severe squats are removed when this maintenance action is performed. For the state of the rails this means that if there are squats of class C, the number of class C squats will decrease by one. If there are no squats of class C, but there are squats of class B, the number of class B squats will decrease by one. If there are no squats of class C and no squats of class B, but there are squats of class A, the number of class A squats will decrease by one. The last option is not very realistic, since grinding a complete track segment is cheaper than one rail replacement. It is possible to do multiple rail replacement at the same time, which results in the removal of multiple squats.

Finally a complete rail renewal can be performed. It is assumed that the renewed rail initially does not contain any squats, which means that after a rail renewal, all squats have been removed. This means that if complete rail renewal is performed at time t, we have nA_t0 = nB_t0 = nC_t0 = 0.

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4 MODEL 4.3 Objective function

initial thickness of 10 millimeter, which is reduced by 0.2 millimeter each year due to surface wear. The initial lifespan of a track segment is 50 years. Grinding reduces the lifespan of a grinded track segment with 2 years, which is 4% of the lifespan. Therefore 4% of the costs of a track renewal are added to the grinding costs, which results in total grinding costs ofe8 per meter. The total cost of grinding a track segment is thereforee3200.

Clearly, eight or more sporadic replacements are at least as expensive as complete rail renewal. Since rail renewal removes the squats from the com-plete track segment and restores the rail top, it is assumed that the number of sporadic rail replacements does not exceed seven. It is also assumed that small squats are never removed by sporadic rail replacements, since grind-ing is less expensive and removes all small squats at the same time. Takgrind-ing these assumptions into account, there are 17 different maintenance plans possible as we will explain in what follows. The possible maintenance plans are numbered from 1 to 17. The first one is a complete rail renewal. In this case grinding and sporadic replacements are not necessary. Furthermore it can be decided to grind and do in addition up to seven rail replacements. These plans are numbered from 2 to 9. Finally it can be decided to do only up to seven rail replacements. These plans are numbered from 10 to 17. Not all 17 maintenance plans are always relevant. If there are no small squats for example, all maintenance plans that decide to grind are irrelevant. It is also possible to determine the state of the system directly after the maintenance action, if the current state is known. Below, for each maintenance plan, the state of the system directly after maintenance, given that the state be-fore maintenance is (nA_t , nB_t , nC_t), is given. For the maintenance plans, the numbers between brackets denote the number of grinding operations, the number of sporadic rail replacements and the number of complete rail re-newals respectively. 1. 2.-9. 10.-17. (0, 0, 1) → st0 = (0, 0, 0) (1, i, 0) → st0 = (0, (nB_t − (nC_t − i)−)+, (nC_t − i)+)) (0, i, 0) → st0 = (nA_t, (nB_t − (n_tC− i)−)+, (nC_t − i)+)

Here (x)+ _{= max(x, 0) is called the positive part of x, and (x)}− ₌ − min(x, 0) is called the negative part of x.

4.3 Objective function

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4 MODEL 4.4 Constraints

defects or minimize the costs. In this thesis, we will do the latter, but safety implications of recommended strategies will also be discussed. We will only consider the costs of the maintenance operations (Cm), which means that we have three cost variables. We have costs of grinding the tracks, which are Cg per track segment of 200 meters. Next, we have the costs of the renewal of the rail, which are Ccr per track segment of 200 meters. Finally, we have the costs of the replacements, which are Csr per replacement of 6 meters. Together these costs result in the following objective function.

min Cm= min ( _L X l=1 T X t=1 xg_l,t· Cg+ L X l=1 T X t=1 xcr_l,t· Ccr+ L X l=1 T X t=1 xsr_l,t· Csr ) (5) Here, xg_l.t, xcr_l,t and xsr_l,t denote respectively whether track segment l has been ground at time t, whether track segment l has completely been renewed at time t and the number of squats that have been removed by sporadic replacements on track segment l at time t. The variables xg_l.t and xcr_l,t are binary variables, and the variable xsr_l,t is an integer variable.

The optimization process can be simplified by separately solving this op-timization function for each segment. Since we assume the costs, constraints and random processes are independent for different track segments, this is allowed. The objective function of (6) can then be optimized to determine the optimal maintenance schedule for an arbitrary track segment. In the remainder of this thesis the location index will therefore be dropped.

min Cm = min ( _T X t=1 xg_t · Cg+ T X t=1 xcr_t · Ccr+ T X t=1 xsr_t · Csr ) (6) 4.4 Constraints

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4 MODEL 4.4 Constraints

(9) are redundant and may be omitted, since (8) and (10) already imply nonnegativity. xg_t ≥ 0 (7) xg_t ∈ (0, 1) (8) xcr_t ≥ 0 (9) xcr_t ∈ (0, 1) (10) xsr_t ≥ 0 (11) xsr_t integer (12) There are also constraints that limit the values of xsr. As explained before, the maximum number of sporadic rail replacements is 7. However sporadic rail replacements are never used to remove small squats, so the maximum number of sporadic rail replacements can also never be larger than the sum of the number of moderate and severe squats. Those two rules can be merged into one constraint which is given in (13). There is also a rule that all severe squats should be immediately removed. Therefore the number of sporadic rail replacements should be as least as large as the number of severe squats. This constraint is given in (14). Since this constraint also limits xsr from below, the (11) has also become redundant and can be omitted.

xsr_t ≤ max(nB_t + nC_t , 7) (13)

xsr_t ≥ nC_t (14)

To decrease the number of possible decisions, some additional logical constraints can be implemented. If a complete rail renewal is performed, it is not necessary to grind or perform sporadic rail replacements, since all squats have already been removed. This condition leads to the following two constraints.

xcr_t · xg_t = 0 (15) xcr_t · xsr

t = 0 (16)

The effects of the maintenance actions can also be written as constraints. The number of severe squats after maintenance is always 0. Constraint (17) ensures that this is the case. The number of moderate squats after mainte-nance depends on the maintemainte-nance action. It is equal to zero after complete rail renewal and it decreases by xsr_t − nC

t after sporadic rail replacements, in (18) this is written in constraint form. The number of light squats is equal to zero after either grinding or complete rail renewal and does not change after sporadic rail replacements. this constraint is showed in (19).

nC_t0 = 0 (17)

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4 MODEL 4.5 Value function

nA_t0 = nA_t − xcr_t · nA_t − xg_t · nA_t (19) Finally, the state of the system before maintenance can be written in terms of the state after the maintenance action of the previous period and the random effects. The number of severe squats before maintenance is equal to the number of squats that became severe during the previous period, since all severe squats are removed after maintenance. The number of squats that have grown into severe squats follows a binomial distribution. The number of moderate squats before maintenance equals the number of moderate squats after the maintenance action of the previous period, where the number of moderate squats that have grown into severe squats is subtracted and the number of small squats that have grown into moderate squats is added. These numbers are both determined by a binomial distribution. The number of small squats is equal to the number of small squats after the maintenance action of the previous period, plus the number of squats that have originated during the previous period, minus the number of small squats that have grown into moderate squats during the previous period. (20), (22) and (24) denote the number of small, moderate and severe squats before maintenance. (21), (23) and (25) denote the corresponding distributions.

nA_t+1= nA_t0 − Y + X (20) X ∼ Pois(λ) (21) nB_t+1= nB_t0 − Z + Y (22) Y ∼ B(nA_t0, µ_A) (23) nC_t+1= Z (24) Z ∼ B(nB_t0, µ_B) (25)

4.5 Value function

In order to fit this problem into a dynamic programming framework, the objective function that has been derived in Section 4.3 should be translated into a value function. A value function Vt(st) is a function that gives the optimal value of the current and subsequent stages, given that the current state is st and there are T − t stages to go (Bradley, Hax, and Magnanti, 1973). Here T is the end of the planning horizon. Since maintenance is done on a track segment of 200 meters, the cost parameters have the following values.

• Cg_{= 3200} • Ccr _{= 40000} • Csr_{= 5000}

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4 MODEL 4.6 Final model

Since the state of the system after maintenance is known, the distribution of the state in the next period can also be computed using the formulas discussed in a previous section. If the current state of the system is st, the value function is given in (26). Here n is the number of stages to go.

V (st) = min

xt {3200 · x g

t + 5000 · xsrt + 40000 · xcrt + E[V (St+1(st, xt)]} (26) Here S is a stochastic function denoting the current state. This depends on the previous state, the maintenance actions of the previous period and a random element. The value function can be iteratively solved starting at time T , which is the end of the planning horizon when there are 0 stages to go.

Now the value function and the relevant constraints can be combined to form a final model. This model is given in section 4.6.

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5 Truncation values

In this section track segments with different origination rates will be ana-lyzed to find effective ways to reduce the running time of the model. The goal of this analysis is to formulate rules to reduce the number of states in the state space. This will be done by limiting the number of squats in each class that is considered. The upper limits that will be derived are called truncation values. Models with three different origination rates will be an-alyzed. First a model where squats originate at a moderate frequency, then a model with a very low origination rate and finally a model with a high origination rate. These rates are chosen to represent a broad range of orig-ination rates covering realistic values (Molodova, 2013). In the subsequent part of this section, another method to reduce the size of the model will be analyzed, namely Poisson truncation. Here a truncated Poisson distribution will be used instead of the original Poisson distribution. Finally the results of the different models will be used to formulate general truncation rules.

5.1 Moderate origination rate

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5 TRUNCATION VALUES 5.1 Moderate origination rate Yi(w) =              w P s=−w Yi+s 2w+1 if i = w + 1, . . . , m − w i−1 P s=−(i−1) Yi+s 2i−1 if i = 1, . . . , w (27)

Here w is the window, which is a positive integer such that w ≤ bm/4c. Furthermore n is the number of replications, which is equal to 10 in this case, and m is the length of a replication, or the number of periods, which is equal to 400 here. This moving average is computed and plotted for increasing values of w, until a smooth plot can be constructed. It turns out that a relatively smooth plot can be created if the window is chosen to have a value of 50. Plots for lower values of w, which did not give smooth plots, can be found in the appendix in Figure 19. Based on Figure 11, which is the plot of the moving average for w = 50, a warm-up period of 75 periods, which is equal to 225 months is chosen. In this graph a vertical line is drawn at period 75, where the line has flattened and the warm-up period ends.

0 50 100 150 200 250 300 350

0

500

1000

1500

Warm−up period determined by Welch's method

Period

A

v

er

age cost per per

iod

Figure 11: Graph used to determine the warm-up period

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5 TRUNCATION VALUES 5.1 Moderate origination rate

0 1 2 3 4 5 6 7 8 9 11 13 15

Number of small squats

Number of small squats

Propor tion 0.0 0.1 0.2 0.3 0.4 0.5

Figure 12: Bar chart of the number of small squats

From this bar chart it turns out that the number of small squats is rarely larger than four. Based on the results of the simulation states with five or six small squats are possible, but they are very uncommon. Among the 325,000 observations of the simulation, the number of small squats never exceeds 6, so it can be safely assumed that states with more than six severe squats have negligible effects on the results of the analysis and can therefore be neglected.

0 1 2 3 4 5 6 7 8 9 10

Number of moderate squats

Number of moderate squats

Propor tion 0.0 0.2 0.4 0.6 0.8

Figure 13: Bar chart of the number of moderate squats

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5 TRUNCATION VALUES 5.1 Moderate origination rate

with more than five moderate squats can certainly be removed from the model and all states with five moderate squats are likely to be removed from the model after further analysis.

0 1 2 3 4

Number of severe squats

Number of severe squats

Propor tion 0.0 0.2 0.4 0.6 0.8 1.0

Figure 14: Bar chart of the number of Severe squats

Situations where the number of severe squats is larger than one, don’t occur regularly. Once in a while tracks with two severe squats may be found, and very rarely it may occur that a track has three severe squats. However, our simulation indicates that tracks with more than four severe squats are not likely to occur. Therefore the model will be restricted to states with at most three severe squats. After some further analysis this upper bound on the number of severe squats may be lowered to two, since track segments with three severe squats are only found once every 6000 years.

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5 TRUNCATION VALUES 5.2 High origination rate

If a model is considered with a different origination rate, the truncation values might differ as well. Therefore track segments where squats originate frequently (Section 5.2) or rarely (Section 5.3) are also considered.

5.2 High origination rate

We will first consider the case where the origination rate is high, namely five squats per period. The methods used for the moderate origination rate can also be used in this case, so we start with the determination of the warm-up period. It turns out that again a window of 50 yields a smooth graph that has also flattened after 75 periods. So the warm-up period that will be used is again 75 periods. A graph of Welch’s method with w = 50 can be found in the Appendix in Figure 21. The same simulation procedure is applied to create bar charts of the number of small, moderate and severe squats. These bar charts can be found in Figures 22, 23 and 24 respectively in the Appendix. It turns out that the number of small squats takes on higher values than when the origination rate was one squat per period. This is intuitive, since the number of originating squats can take on higher values. Even track segments with 15 small squats can occur. However this event is extremely rare, since it happens less than once every 1000 years. The truncation value that is used for an origination rate of five squats per period is 15, although 13 of 14 might also be suitable, since tracks with more than 13 squats are highly unusual. The number of moderate squats also takes slightly higher values, but still does not exceed seven, so the number of moderate squat can also in case of a high origination rate be truncated at this value. The distribution of the number of severe squats when the origination rate is five squats per period is pretty similar to the distribution of the number of severe squats when the origination rate is one squat per period. Severe squats occur more often, but still three severe squats is the maximum number that has been found in the analysis. More squats originate per period, so maintenance actions are also needed more regularly. This increase in maintenance actions almost doubles the costs. Also the optimal decisions change. If squats originate at a rate of one squat per period, tracks are usually grinded when there are at least two small squats. If squats originate at a rate of five squats per period, tracks are usually grinded when there are at least four small squats. Waiting one more period and reducing the number of grinding operations is worth the risk of letting small squats grow into moderate squats. The average costs per period are e2968.

5.3 Low origination rate

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5 TRUNCATION VALUES 5.4 Poisson truncation

but the window is increased to 100, which is the maximal value it can take. Due to the low origination rate, the number of squats on the track is very low and therefore there is also a low number of maintenance actions. This causes much fluctuation in the costs and the necessity of a larger window. A graph of Welch’s method can be found in the Appendix in Figure 20. Since there are less maintenance actions, the average costs also decreases. The costs when the origination rate is low are just e613 per period. This is less than half of the costs when the origination rate is one squat per period and almost five times lower than the costs when the origination rate is five squats per period. Nevertheless, the optimal decisions do not differ much from the decisions when the origination rate is moderate. Truncation values are again found by simulating a track segment during 100 years 1000 times. The resulting bar charts of the number of small, moderate and severe squats can be found in Figures 25, 26 and 27 in the appendix. It turns out that the maximum number of small squats is somewhat lower than when the origination rate is one squat per period. The maximum number of moderate squats is also slightly lower than when the origination rate is moderate. The distribution of the number of severe squats is almost similar to the distribution when the origination rate is moderate.

5.4 Poisson truncation

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5 TRUNCATION VALUES 5.5 Comparison of different truncations λ Truncation value 0.25 3 0.5 4 0.75 4 1 5 1.5 6 2 7 3 9 4 11 5 13 6 14 7 16 8 17 9 19 10 20

Table 1: Truncation values of the Poisson distribution

λ Expected costs Expected costs Difference original Poisson truncated Poisson

distribution distribution

0.25 613.34 613.09 0.04%

1 1533.49 1532.79 0.05%

5 2967.66 2967.66 0.00%

Table 2: Difference in expected costs between ordinal and truncated Poisson distributions

5.5 Comparison of different truncations

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5 TRUNCATION VALUES 5.5 Comparison of different truncations

Therefore it can be concluded that a truncation value of 7 is a good choice for moderate squats. For severe squats, the truncation value should be 3. For the number of small squats it is not possible to determine a uniform truncation value that can be used for all origination rates. Since the num-ber of small squats depends heavily on the origination rate, the truncation value should depend on the origination rate as well. However, there is a clear relationship between the truncation value of the Poisson distribution and the truncation value of the number of small squats. When the num-ber of small squats exceeds three, tracks are always grinded, irrespective of the origination rate. Therefore the truncation value of the number of small squats can be set to the truncation value of the Poisson distribution plus 3. In Tables 3, 4 and 5 the costs of all models are compared.

Model Costs

Untruncated model, Untruncated distribution 613.34 Untruncated model, Truncated distribution 613.09 Truncated model, Untruncated distribution 613.34 Truncated model, Truncated distribution 613.09

Table 3: Costs and calculation times for different models with a low origi-nation rate

Model Costs

Table 4: Costs and calculation times for different models with a moderate origination rate

Model Costs

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6 Results

In this section the optimal decisions will be discussed. The cost of a strategy where the optimal decisions are compared with strategies that are actually employed by ProRail and some of the maintenance strategies that have been discussed in the introduction.

6.1 Optimal decisions

The optimal decisions have already been discussed shortly in the previous section. In this section, the optimal decisions will be given in more detail and they will be given for more different values of λ. In the previous section it has already been stated that the effects of the origination rate on the optimal decisions are not very large. For all origination rates, the number of sporadic replacements is usually equal to the number of severe squats. Also, only when there are both severe squats and many small and moderate squats, it is sometimes decided to perform a complete rail renewal. In Figures 15, 16 and 17 flowcharts that can be used to determine the optimal maintenance actions are presented.

λ =0.25 0  C t n 7  B t n  C8 t B t n n  C8 t B t n n 1  A t n A2 t n A 1 t n A0 t n

Grind Do nothing Grind Grind and

replacements C t n replacements C t n _Renewal Yes No Yes Yes Yes Yes Yes Yes No No _No No Yes No No No

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6 RESULTS 6.1 Optimal decisions λ =1 0  C t n 6  B t n  C8 t B t n n  C8 t B t n n 1  A t n A 7 t n A1 t n A0 t n

replacements C t n replacements C t n _Renewal Yes No Yes Yes Yes Yes Yes Yes No No No No Yes No No No 6  B t n No Yes 2  A t n No Yes

Figure 16: Flowchart to determine the optimal maintenance action if λ = 1

λ =1 0  C t n 6  B t n  C8 t B t n n  C8 t B t n n 3  A t n A 3 t n A 0 t n

replacements C t n replacements C t n _Renewal Yes No Yes Yes Yes Yes Yes No No No Yes No No No 6  B t n Yes 4  A t n No Yes No

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6 RESULTS 6.2 Comparison with other maintenance strategies

differences if nC_t = 0, but this concerns cases with many moderate squats. As these cases rarely occur, these differences are not very relevant. The values in the boxes of the flow charts do differ among the different origination rates. If the origination rate is higher, the number of small squats required for a maintenance action is also higher.

If we want to determine the maintenance action that should be taken we should first know the origination rate. We will now discuss the flow chart where λ = 0.25. First we look at the number of severe squats. If there are no severe squats, we already know that we only have to decide whether we want to grind or not. This depends on the number of small and moderate squats. If there are seven moderate squats and more than two small squats, it is optimal to grind the track. If there are seven moderate squats and up to two small squats, it is better to do nothing. If we answer the question whether there are seven squats in the negative, we know that there are less than seven squats, since states with more than seven moderate squats do not exist in the truncated model. If there are less than seven squats and at least two small squats, the optimal decision is to grind the track. If there are less than seven moderate squats and at most one small squat, it is better to do nothing. If there are severe squats, these squats should be replaced either by sporadic replacements or by a rail renewal. If they are replaced, additional grinding can be performed to remove the small squats as well. The decision process works in a similar way, except from the fact that we now look at the total number of mature squats.

6.2 Comparison with other maintenance strategies

The optimal maintenance strategy that has been determined will be com-pared with some other maintenance strategies. The maintenance strategies that will be compared are the following:

1. The optimal maintenance strategy based on the dynamic programming model that has been described in this thesis.

2. A maintenance strategy where the track is grinded every year. In addition severe squats will be removed by sporadic rail replacements when they occur, and complete rail renewal will be performed if the number of mature squats is at least 8. This is the maintenance strategy that is currently used by ProRail.

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6 RESULTS 6.2 Comparison with other maintenance strategies

squats occur. When this strategy is used, small squats can never grow into mature squats.

The truncation values that have been determined for the optimal main-tenance strategy, do not hold for strategies 2 and 3. The number of small squats can be much larger when these strategies are used, and therefore us-ing the same truncation rates would lead to an underestimation of the costs. Therefore, new truncation values have to be determined for these strategies. The new truncation values have been selected such that the fraction of time that the number of squats in a certain class is equal to the truncation value of that class is at most 0.1%. Similar strategies have been used to determine truncation values in the previous sections, and it has been shown that these strategies work well. In Table 6 the costs of the four strategies are given for different origination rates.

Strategy λ 1 2 3 4 0.25 613 1109 (67) 1232 (133) 701 (74) 0.5 975 1416 (85) 2378 (170) 1253 (82) 0.75 1274 1700 (105) 3278 (198) 1689 (80) 1 1533 2001 (137) 4115 (188) 2019 (85) 1.5 1883 2630 (154) 5358 (241) 2482 (73) 2 2150 3184 (184) 6200 (247) 2758 (64) 3 2556 4215 (201) 7289 (258) 3035 (46) 4 2810 5056 (192) 7940 (264) 3141 (24) 5 2968 5731 (205) 8305 (252) 3178 (15) 6 3070 6308 (196) 8729 (261) 3192 (9)

Table 6: Costs of different maintenance strategies for different origination rates

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6 RESULTS 6.2 Comparison with other maintenance strategies 0 1 2 3 4 5 6 0 2000 4000 6000 8000 lambda

Expected costs per per

iod

Strategy 1 Strategy 2 Strategy 3 Strategy 4

Figure 18: Expected maintenance costs per period for different maintenance policies

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6 RESULTS 6.3 Safety

in many unnecessary grinding actions and larger cost differences. The dif-ference between the different strategies might be smaller if it turns out that the time squats remain in a class is more constant. In this case the strategy that is currently used by ProRail will perform much better. Squats need more time to grow, and therefore the number of moderate and severe squats will be lower.

6.3 Safety

Another important factor that should be considered when taking mainte-nance decisions is safety. In this section, the safety of the different main-tenance strategies will be compared. The safety of the track is determined by the number of squats. In Tables 7, 8 and 9 the average number of small, moderate and severe squats at an inspection and before possible maintenance actions are given for the four previously discussed maintenance strategies.

Strategy λ 1 2 3 4 0.25 0.55 0.51 1.28 0.25 0.5 0.82 1.02 2.13 0.50 0.75 1.07 1.49 2.71 0.75 1 1.29 2.05 3.32 0.99 1.5 2.13 3.09 4.39 1.45 2 2.55 4.11 5.44 1.99 3 3.34 6.07 7.54 2.99 4 4.61 7.92 9.64 3.99 5 5.41 9.67 11.68 4.99 6 6.07 11.37 13.70 5.99

Table 7: Average number of small squats for maintenance strategies for different origination rates

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influ-6 RESULTS 6.3 Safety

ence the safety level of the track, since they are too small to affect the trains. Squats start to influence the safety level when they become mature squats.

Strategy λ 1 2 3 4 0.25 0.36 0.33 1.23 0.00 0.5 0.41 0.66 1.59 0.00 0.75 0.41 0.96 1.61 0.00 1 0.37 1.30 1.56 0.00 1.5 0.81 1.95 1.62 0.00 2 0.68 2.47 1.78 0.00 3 0.52 3.19 2.13 0.00 4 0.76 3.53 2.48 0.00 5 0.53 3.71 2.80 0.00 6 0.10 3.87 3.16 0.00

Table 8: Average number of moderate squats for maintenance strategies for different origination rates

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6 RESULTS 6.3 Safety Strategy λ 1 2 3 4 0.25 0.07 0.06 0.22 0.00 0.5 0.07 0.12 0.27 0.00 0.75 0.07 0.17 0.26 0.00 1 0.07 0.23 0.23 0.00 1.5 0.15 0.35 0.22 0.00 2 0.12 0.45 0.22 0.00 3 0.09 0.56 0.25 0.00 4 0.14 0.58 0.27 0.00 5 0.10 0.59 0.29 0.00 6 0.02 0.59 0.31 0.00

Table 9: Average number of severe squats for maintenance strategies for different origination rates

Similar to the average number of moderate squats for the dynamic pro-gramming policy, the average number of severe squats also is fairly constant for different origination rates, and sharply decreases when the origination rate is high. For the other policies we can also see similar patterns. However, for policy 3 we see a different pattern. The number of severe squats is quite stable for this policy, which was not the case for the number of moderate squats. Since complete rail renewals are not uncommon for high origination rates, when this policy is used, the number of severe squats remains limited. Again, policy 4 is the safest one, severe squats cannot exist when this policy is used, due to its nature. Policy 2, the strategy that is currently used by ProRail turns out to be the least safe strategy again.

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7 Summary and conclusions

In this thesis an optimal maintenance schedule for the removal of squats from the Dutch railway tracks is constructed using dynamic programming. This optimal maintenance schedule consists of two types of decisions. It finds the optimal moment of maintenance and it decides which of three possible maintenance methods is optimal. The optimal maintenance decisions are not very sensitive to changes in the origination rate. Rail renewal is barely used, only in very extreme cases where the number of moderate and large squats is extremely large. Sporadic rail replacement is always used when squats grow into severe squats. Grinding is, depending on de origination rate and the number of moderate and severe squats, mostly used when there are between two and four squats.

The optimal moment to remove squats and the corresponding optimal maintenance method depend on the number of squats and the class they are in. The optimal maintenance decision can be taken using a flowchart, which is presented in Section 6.1. From the previous section it can be concluded that based on costs, the maintenance policy that uses these flowcharts is su-perior to the policy that is currently used by ProRail. Cost reductions of up to 45% can be established. It also turned out that ProRail has already made a big step in reducing maintenance costs. The old maintenance policy that had been used before the grinding technology was invented, was for some origination rates more than twice as expensive as the currently used pol-icy. The policies have also been compared to a pure preventive maintenance strategy, which might be interesting considering safety issues. This policy is also cheaper than the currently used strategy for most origination rates. The dynamic programming policy is not the safest policy. A preventive policy is safer, but also more expensive. However, the dynamic programming policy is considerably safer than the policy that is currently used by ProRail.

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7 SUMMARY AND CONCLUSIONS

once for multiple reparations, which reduces the total costs. Since there was no data available for the set-up costs and the model and the computation times would have grown even faster, especially since a multi-item approach is then required, these costs have not been considered in this thesis. Fi-nally penalty costs for the presence of moderate and severe squats can be added to the objective function. Unsafe policies would become suboptimal and the dynamic programming model would result in a model that is both inexpensive and safe.

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REFERENCES REFERENCES

References

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Bradley, S.P., A.C. Hax, and T.L. Magnanti (1973). Applied mathematical programming. Addison-Wesley.

Clayton, P. (1996, December). Tribological aspects of wheel–rail contact: a review of recent experimental research. Wear 191, 170–183.

d’Epinay, T.L. and A. Meyer (2007). SBB 2007 mit guter leistung g¨uterverkehr belastet gesamtergebnis. Technical report, SBB.

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Law, A.M. (2007). Simulation, modeling analysis (4 ed.). McGraw-Hill. Li, Z. (2011, October). A guideline to best practice of squat treatment.

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Li, Z., R. Dollevoet, M. Moldova, and X. Zhao (2011, May). Squat growth— Some observations and the validation of numerical predictions. Wear 271, 148–157.

Li, Z., X. Zhao, M. Molodova, and R. Dollevoet (2010, April). Squat treat-ment by way of minimum action based on early detection to reduce life cycle costs. In Proceedings of the 2010 Joint Rail Conference, Volume 1, pp. 305–311.

Molodova, M. (2013, January). Detection of early squats by axle box accel-eration. Ph. D. thesis, Delft University of Technology.

Molodova, M., Z. Li, and R. Dollevoet (2011, May). Axle box acceleration: Measurement and simulation for detection of short track defects. Wear 271, 349–356.

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R Core Team (2013). R: A Language and Environment for Statistical Com-puting. Vienna, Austria: R Foundation for Statistical ComCom-puting. Ramaekers, P., T. de Wit, and M. Pouwels (2009, February). Hoe druk is

het nu werkelijk op het Nederlandse spoor? Technical report, CBS. Smulders, J. (2003, July). Management and research tackle rolling contact

fatigue. Railway gazette international 158 (7), 433–436.

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A FIGURES

A

Figures

A.1 Warm-up period

0 100 200 300 400 0 2000 4000 w=1 A v er

age cost per per

iod 0 100 200 300 400 0 1000 w=5 A v er

age cost per per

iod 0 100 200 300 400 0 1000 w=10 A v er

age cost per per

iod 0 100 200 300 0 500 1500 w=20 A v er

age cost per per

iod

Figure 19: Welch’s plot for moderate origination rate and different windows

0 50 100 150 200 250 300

0

200

400

600

Period

A

v

er

age costs per per

iod

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A FIGURES A.2 Squat distribution 0 50 100 150 200 250 300 350 0 1000 2000 3000

Period

A

v

er

age cost per per

iod

Figure 21: Graph used to determine the warm-up period for a high origina-tion rate

A.2 Squat distribution

0 2 4 6 8 10 12 14 16 18 20

Propor tion 0.0 0.1 0.2 0.3 0.4 0.5

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A FIGURES A.2 Squat distribution

0 1 2 3 4 5 6 7 8 9 10

Propor tion 0.0 0.2 0.4 0.6 0.8

Figure 23: Number of moderate squats when the origination rate is 5 squats per period

0 1 2 3 4

Propor tion 0.0 0.2 0.4 0.6 0.8 1.0

(58)

0 1 2 3 4 5 6 7 8 9 11 13 15

Propor tion 0.0 0.2 0.4 0.6

Figure 25: Number of small squats when the origination rate is 0.25 squats per period

0 1 2 3 4 5 6 7 8 9 10

Propor tion 0.0 0.2 0.4 0.6 0.8

(59)

0 1 2 3 4

Propor tion 0.0 0.2 0.4 0.6 0.8 1.0

Squat maintenance at the Dutch railways