Railway traffic control and train scheduling based oninter-train conflict management

Transportation Research Part B 33 (1999) 511±534

Railway trac control and train scheduling based on inter-train con¯ict management _ Ismail SËahin* Yõldõz Technical University, Department of Civil Engineering, Transportation Research Group, Besiktas, Istanbul 80750, Turkey Received 2 May 1997; received in revised form 19 November 1998; accepted 13 December 1998

Abstract This research deals with analyzing dispatchers' decision process in inter-train con¯ict resolutions and developing a heuristic algorithm for rescheduling trains by modifying existing meet/pass plans in con¯icting situations in a single-track railway. We described the railway trac management brie¯y to establish a sucient ground for the problem de®nition. Train dispatchers currently carry out the rescheduling process. In order to model decision behaviour of train dispatchers, we assumed that they use a utility function of some weighted attributes of each con¯icting train to determine (dynamic) priorities pair wise, and that he/ she resolves con¯icts according to the calculated values of dynamic priorities of trains. We determined the weights by analyzing the previous decisions of train dispatchers. This analysis is important to determine the e€ectiveness of decisions of train dispatchers respecting other solution techniques and is usually omitted in studies of railway trac control. We used a systems approach in construction of the heuristic algorithm, which is based on inter-train con¯ict management. The kernel of this algorithm is the immediate con¯ict and its two alternative resolutions. The algorithm chooses the best alternative resolution, which causes less total consequential delay in the system due to the con¯icting train being stopped. One of the most important features of the algorithm is to consider the e€ects of potential con¯icts by using a look-ahead method. In the end we tested the methods for hypothetical problem instances and evaluated the results. These tests showed that the algorithm produced ``good enough'' schedules eciently and e€ectively in con¯icting situations. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Railway trac control; Train scheduling; Con¯ict management; Multi-attribute decision making; Heuristic algorithm; Dynamic priority; Look-ahead method

* Corresponding author. Tel: +90-212-259-7070 ext. 2732; fax: +90-212-259-6762; e-mail: [email protected] 0191-2615/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.

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1. Introduction: goal and content 1.1. Goal The railway industry plays a vital role in many countries' transportation. The competitive edge of the railways over other available modes depends on the quality of service it provides. Amongst the most important parameters that a€ect this service quality are the eciency and the e€ectiveness with which railway operations are conducted. This provides the motivation for an in-depth study of trac control process, an integral part of the railway's day-to-day operations. One of the most challenging and dicult problems in railway trac control is to resolve intertrain con¯icts. In a schedule-responsive railway, train movements are regulated with respect to a pre-established and con¯ict-free schedule. Any disruption of railway trac in such an environment may cause the system to be unstable with a high operating cost due to increasing interference delays. Therefore, any con¯ict arising due to a disruption has to be resolved timely and e€ectively. Trac control in a single-track railway is a combinatorial problem, which is NPcomplete with time complexity of O(2n), where n is the number of con¯icts. This means that the computation time and memory requirement to solve such a problem increases exponentially with the problem size (Garey and Johnson, 1979). The size of the problem depends on the number of meet/passes (i.e. number of inter-train con¯icts), which in turn depends on the length of line, trac intensities, train speeds and their diversity and the length of time horizon considered. Because of such diculties, only small size problem instances can be optimally solved. It is not feasible (even possible) to obtain exact solutions for even medium size instances due to enormous computation time and computer memory requirement. The con¯ict resolution process is still dominantly in the realm of human expertise relying on train dispatcher's intuitive knowledge, experience and judgment. However, the solutions found by dispatchers may not necessarily be optimal from the standpoint of ecient operations. These decisions are ad hoc and do not usually consider alternative competing solutions. The optimum control problem determines, by solving these con¯icts, where meets and passes should occur and on which track each train should be at any given time to achieve certain predetermined objectives such as minimum overall train delay, and safety of train movement. Given the above mentioned diculties, we develop a heuristic algorithm, which can solve railway trac control problem by producing ``good enough'' solutions (modi®ed schedules) in polynomial times. 1.2. Content In this study, we deal with trac control problem for a schedule-responsive single-track railway in which operations are carried out based on a pre-established schedule. Railway trac control is a large-scale, complex and dynamic problem. Therefore, it is crucial to de®ne the problem in detail. In the earlier sections of this study, a literature search is presented and the railway trac management is outlined. We then de®ne the railway system considered and the notation used for the mathematical model of the trac control problem. Presently, train dispatchers carry out railway trac control. They use their knowledge and skill obtained over many years and their reasoning ability to perform the duty. Therefore, we studied this problem in a particular section

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of TCDD (Turkish State Railways) and developed a choice (decision) model of train dispatchers in con¯ict resolutions. A heuristic algorithm for railway trac control based on inter-train con¯ict management is presented in the subsequent section. We have used systems approach to establish the algorithm. In the last two sections of this study, we present some applications to test the algorithm for di€erent problem instances, and the conclusions and future prospects, respectively. 2. Literature search The railway trac control problem has been drawing the attention of researchers for decades. The e€ort has mainly been spent to establish the mathematical models and to develop solution procedures. The techniques used to realize these e€orts can be categorized in three groups: Simulation, Mathematical Programming and Expert Systems. Assad (1980) provides an excellent review of analytical models developed for rail transportation. Amongst the most important studies using Simulation are Frank (1965), Peat, Marwick, Mitchell (1975) and Petersen and Taylor (1982). The Mathematical Programming technique was ®rst applied to this problem by Amit and Goldfarb (1971); later, Szpigel (1973), Sauder and Westerman (1983), Petersen et al. (1986), Jovanovic (1989), and Kraay et al. (1991) used the Mathematical Programming technique to establish the mathematical model and to develop solution procedures of the problem. A relatively new technique, the Expert Systems has recently been applied to the railway trac control problem in Japan: Iida (1988), Komaya and Fukuda (1989) and Komaya (1992) are amongst the studies using the Expert Systems. Additionally, we have found valuable information contributing to the solution of the problem in the following literature: Cherniavsky (1972), Rudd and Story (1976) and Wong and Rosser (1978), Eisele (1985), Harker (1990), Jovanovic and Harker (1990). To the best of our knowledge, some of these studies have been used in practice. However, providing satisfactory solutions for such a problem is still drawing the attention of researchers. 3. Railway trac management In order to solve the railway trac control and the associated rescheduling problem, it is necessary to review the problem from a broader perspective. Therefore, we will outline the railway trac management, which covers the logical and hierarchical interrelationship amongst its elements. Two fundamental elements of railway trac management are the trac management center and trac management territory as shown in Fig. 1 (SËahin, 1996). Territories are established by dividing the railway network into subsections. Trac moving within a territory is regulated by the associated trac management center. Although these centers are independent of one another, there exists a certain coordination, especially between the centers of adjacent territories. The train dispatcher of a center is responsible for the regulation of railway trac within his/her territory. The dispatcher carries out his/her duty through trac control system, observes the status of the territory (i.e. occupation of line sections, location of trains, position of switches and aspects of signals) in a continuous manner and collects information; meanwhile, he/she communicates with the upper level decision-makers and the personnel in the territory in order to exchange decisive information. In case of an unplanned event and emergency he/she makes a

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Fig. 1. Interrelationship amongst elements of railway trac management in the centralized trac control.

decision and takes necessary actions in accordance with the rules and regulations pre-de®ned by the railway authority. According to Steiner (1978), a train dispatcher using a conventional CTC (Centralized Trac Control) system spends only 4% of his/her shift to make con¯ict resolution plans (rescheduling process), and 16% to implement these plans; the rest of time is devoted to gather information and to keep records. In other words, the train dispatcher has to deal with ad hoc decisions and tries to manage the con¯icts by reducing the diculty, e.g. using pre-assigned basic priority of each train. As mentioned above, the schedule-responsive operation is carried out according to a preestablished schedule. This con¯ict-free schedule is applicable under presumed conditions of operation; however, unplanned events (e.g. an unscheduled stop, a slow order and introducing a new train in service) always occur and cause trains to deviate from their schedule, consequently. Such deviations, which occur either at the beginning of the territory (e.g. due to late arrivals from

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Fig. 2. An abnormal situation and inter-train con¯icts developed.

adjacent territory) or within the territory (e.g. due to unplanned movements), may cause the railway system to be unstable resulting in propagating and increasing delay amongst trains. It is crucial to detect and de®ne such abnormal situations precisely in order to limit the damage likely to be given to the trac moving in the system. As soon as the dispatcher detects an abnormality and/or is informed about such an event, he/she tries immediately to reach the information such as the exact location and type of the abnormality, and the exact or expected time when the abnormality is going to end. Fig. 2 shows an abnormal situation in which train 1 makes an unscheduled stop between stations/sidings B and C, deviating from the pre-established schedule. As seen in the ®gure, previously con¯ict-free schedule has now several potential con¯icts developed by train 1. Let us assume that the unplanned stop of train 1 is detected and de®ned at time `present' (02:40). The next information to be gathered is the time when it may be removed, that is, when the train will start resuming its journey remaining (02:50 in this case). As we can see in the ®gure, train 1 would not be in any con¯ict if it were not disrupted. However, after the abnormality is removed at 02:50, the projected trace of train 1 interferes with those of trains 2, 3 and 4. Now, previously con¯ict-free schedule becomes an unfeasible schedule, which has to be modi®ed to be applicable. A rescheduling process is applied to get rid of these con¯icts and to make a feasible plan. It should be noted that the future con¯ict resolutions of train 1 in the case of scheduled operation (before the abnormality) are not valid anymore. This contributes an additional diculty to the solution of the problem. 4. Notation used for and mathematical model of railway trac control problem A railway system consists of three fundamental elements: rail line, vehicles (e.g. trains, single engines and work trains) operating over this line, and trac control system authorizing movement

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of these vehicles. Trains (hereafter, the word `train' will be used to represent a vehicle) moving over the lines altogether constitute the railway trac. The process of trac control carried out with the help of carefully prepared rules that integrate these elements to provide safe and ecient movement of trains. In this study we will be using the notation as in Jovanovic (1989). A single-track railway line is a guided way over which the movement of trains is restricted in one-dimension. The end and intermediate stations and sidings along the line are called meetpoints. Meetpoints are numbered consecutively in the outbound direction, with the index value, m, of the ®rst station is equal to 1 and the index value of the last station is equal to N (where, N is also the number of meetpoints). Each meetpoint m has two or more tracks (main track+side tracks) whose lengths are denoted by m . Let I be the set of outbound trains and J the set of inbound trains. Some of the attributes of outbound train i are as follows: The index i of the departure (origin) meetpoint and the index "i of the arrival (destination) meetpoint on train's itinerary; the set Si that contains indices of all points where train i has a scheduled arrival and/or dwell for a speci®ed time, ordered in the direction of travel (i , "i 2 Si ); the scheduled arrival time, m i , and the scheduled departure time, m , (the minimum dwell time at a scheduled point is given by m m i i ÿ i ), for all m 2 Si ; the actual m m arrival time, ai , and the actual departure time, di , (the actual stop time at meetpoint m is i m m dm i ÿ ai ); the length of train li ; and the minimum free running times i ˆ …i ; . . . ; i ; "i ÿ1 m‡1 i ; . . . ; i †, where im represents the minimum running time from meetpoint m to m ‡ 1 (or at m  m m meetsection m). The respective attributes of inbound train j are denoted by j , "j , Sj , m j , j , dj , lj , "j Jÿ1 m‡1 m m and j ˆ …j ; j ; j ; . . . ; j †, where j is the free running time from meetpoint m ‡ 1 to m. The indices associated with trac control system are as follows: Let m ik be the minimum following headway that allows outbound train k to follow outbound train i between meetpoints m and m ‡ 1, without speed restriction; and let  m js be the minimum following headway between be the minimum meeting (safety) headway at meetpoint m inbound trains j and s. Also, let m ij between the arrival of outbound train i and the departure of inbound train j, and let  m ji be the minimum meeting headway between the arrival of inbound train j and the departure of outbound train i. Time necessary to clear a signal and/or to align a route for a train depends on the features of the trac control system and included in the headway. A deterministic method to calculate headway values can be found in detail in SËahin (1987). The mathematical model of rescheduling process in railway trac control problem is similar to that of job-shop scheduling problem. In the formulation, line sections between adjacent stations/ sidings (or block segments between signals) and trains running in two directions correspond to machines and jobs in the job-shop scheduling, respectively. While the objective is to minimize the sum of job completion times in the job-shop scheduling, the sum of running times (or deviation from scheduled arrival times) of trains are minimized in the rescheduling process. The mathematical model of such a problem includes the following sets of constraints: departure and arrival time constraints, running time constraints, minimum dwell time constraints, following and overtaking constraints. meet constraints, station/siding capacity constraints and line blockage constraints. This is a zero-one mixed-integer-programming model. As mentioned earlier, this problem falls in the NP-complete class and even medium size of which cannot be solved in polynomial time. We provide the results of the optimal solutions of some small size problem instances in Section 7 for comparison with the other two solution methods given in the following sections.

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In order to reach these results, we prepared the mathematical programs of the instances, and obtained the solutions by using a commercial software package (SËahin, 1996). 5. How does a train dispatcher solve the problem?: Multi-attribute decision making in railway trac control In this section our goal is to model choice (decision) behaviour of train dispatchers at intertrain con¯ict resolution during their shifts. We de®ne each train pair i and j con¯icting between m‡1 m‡1 m‡1 m m meetpoints m and m ‡ 1 as vectors of n attributes, …Vm i1 ; Vi2 ; . . . ; Vin † and …Vj1 ; Vj2 ; . . . ; Vjn †, m‡1 m respectively, where Vin and Vjn are the values of attribute n; i 2 I and j 2 J; meetpoints, m and m ‡ 1 are in the itinerary of both trains; and the set of attributes is C ˆ f1; 2; . . . ; c; . . . ; ng. The vector of a con¯icting outbound train is de®ned with respect to meetpoint m and that of an inbound train to m ‡ 1. The attributes considered in the model should be selected amongst the ones that represent the state of each train properly. We considered the following four attributes this study. Accordingly, the attributes of a con¯icting train i with respect to meetpoint m are as follows: 1. 2. 3. 4.

basic priority, Vm i1 , critical ratio, Vm i2 , delay at myopic resolution, Vm i3 , number of potential con¯ictsÐafter current con¯ict resolutionÐ,Vm i4 .

Note that except basic priority, the other attributes are evolutionary, taking di€erent values along the journey of trains. The basic priority of a train is a number assigned according to its speed class by railway authorities; and the smaller the basic priority number, the more favorable it is to the train. The critical ratio is usually used in production planning as one of the common priority rules (Adam and Ebert, 1986). The critical ratio for train i at meetpoint Ai is that VAi i2

…"i i ÿ dAi i † ˆ P min im m2fAi ;...;"i ÿ1g

The critical ratio is a dynamic rule that updates the train's status at the meetpoint Ai with respect to time remaining to its scheduled time at destination and minimum free running time remaining, where Ai is current meetpoint or next meetpoint if the train is between meetpoints, or origin meetpoint. A low critical ratio indicates a high urgency for running the train to avoid getting further behind the schedule. Delay at myopic resolution is the stop time of the con¯icting train due to con¯ict resolution. The number of potential con¯icts can be extracted from the existing schedule with respect to the train's current location and projected trace up to its destination. In model building we assume that the train dispatcher uses a utility function of weighted attributes, in order to model his/her choice behavior. Following is three stages of multi-attribute choice process given in Oral and Kettani (1989) and Potvin et al. (1992):

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m m m . Valuation stage: The vector of attributes, …Vm i1 ; Vi2 ; . . . ; Vic ; . . . ; Vin †, represents the state of a con¯icting train i at meetpoint m. The dispatcher uses these attributes to resolve con¯icts; however, he/she is assumed to evaluate attribute score, Vm ic , and then assigns a non-negative to it. These subjectively assigned values are called partial utilities and subjective number, dm ic are not known to the analyst modeling the choice behaviour. This transformation may be m m shown as dm ic ˆ fc …Vic † ˆ wc Vic , where wc , is the weight of attribute c. . Integration stage: The dispatcher is assumed to combine partial utilities to produce an intefor train i. The integration process is that gratedP or overall utility, pm i , P m m m pi ˆ c dic ˆ c wc Vic . We will call this overall utility function, pm i , as dynamic priority of train i with respect to meetpoint m. . Conclusion stage: We assume that an inter-train con¯ict occurs between two trains. Therefore, the train with higher utility value (or dynamic priority) is determined by comparing the utility values of the con¯icting train pair, which were calculated in the previous step. For m‡1 , it is assumed that train i has higher priority over train j. In the next instance, if pm i > pj section we present a linear programming model to estimate weights of the attributes.

5.1. Linear programming model Let ˆ ‰…i; j†; mŠ be the set of dispatcher's preferences for the con¯icting trains and the location of con¯icts, which is established based on the previous decisions of the train dispatcher. This representation indicates that the dispatcher chooses train i to pass and train j to stop for con¯ict resolution. We will use a linear programming model proposed by Srinivasan and Shocker (1973a, b) utilizing the vector of attributes of con¯icting train pairs, in order to estimate weight of each attribute. The model is that X Zij …1† Min …i;j†2

subject to m‡1 c wc …Vm ic ÿ Vjc † ‡ Zij 50; 8‰…i; j†; mŠ 2

…2†

m‡1 ‰…i;j†;mŠ2 wc …Vm ic ÿ Vjc † ˆ 1

…3†

Zij 50; 8…i; j† 2

…4†

wc 50; 8c 2 C where the variables, Zij show the deviation of the model from the choice made by the dispatcher. In case the objective value is equal to 0 (i.e. all Zij s are 0), the set of estimated weights, wc is said to exactly reproduce the preferences of the train dispatcher. If the value of the objective function is greater than zero, then, we can conclude that there are one or more positive Zij s, and the corresponding preferences are not satis®ed in constraint (2). In this case, the train, which has been preferred by the dispatcher, has a smaller dynamic priority than that of the other; hence, the model does not reproduce the decision problem. Constraint (3) is used to prevent trivial solutions (wc ˆ 0, 8c).

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In order to test the validity of the model, we obtained some real-life data from TCDD (Turkish State Railways). We have chosen a single-track dispatch territory between Eskisehir and Ari®ye, an intermediate section of Istanbul±Ankara railway connection, which is about 163 km with 17 intermediate stations/sidings. The time interval chosen for the test is between 00:00 and 06:00. The reason why we considered this particular line section and time interval is that a relatively high intensity of train trac (six trains in each direction) running in two directions meets in this territory and time interval. In order to establish the set of con¯icts, we have extracted the con¯icting train pairs, location of each con¯ict, dispatcher's preferences at resolutions and attribute values of each con¯icting train from train-graphs, which are drawn automatically by an equipment of centralized trac control system. Since a train-graph shows actual movement of trains, we can easily de®ne the state of each train operating in the system at any time. We chose traingraphs of four di€erent operating days, and there were 90 actual inter-train con¯icts in total in the interval selected (SËahin et al., 1995). Since we have attribute values of each con¯icting train and decision for each con¯ict resolution, we need, at this point, to perform the model given above in order to reproduce the choice (or decision) behavior of the train dispatchers. The output of the linear programming model is the objective function (1), which indicates the total magnitude of preference errors (violations) of the pair wise preference constraints (2), which have been formulated based on observed con¯ict resolutions, and the weights w1 , w2 , w3 and w4 . The output of the model for the input data mentioned above is as follows: The value of the objective function : 0:1226831 and w1 ˆ 0:1547; w2 ˆ 0:6200; w3 ˆ 0:2253 and w4 ˆ 0:0000: Based on the input data, the critical ratio occupies the greatest part of the mental model with 62.00% of weight. This result seems quite reasonable for a schedule-responsive railway such as TCDD. The basic priority has considerable weight of 15.47% and shows an impact of operational decisions of railway authority on train dispatchers. The output of this analysis shows also that remaining two attributes, delay in myopic resolution and the number of potential con¯icts, are being important indicators of the mental model of train dispatchers. It is non-trivial that the myopic resolution occupies a considerable part with 22.53% in the mental model of train dispatchers. While this helps dispatchers arrive at a feasible resolution quickly; it also indicates a desire for a practical outcome. As we explored from the result, train dispatchers do not take into account the potential con¯icts at their resolutions (w4 ˆ 0:00). We may accept this result due to contradiction between the dynamic and complex nature of trac control problem and the limitation of mental capability of train dispatchers as human decision makers; however, we have to stress that this attribute is one of the most important elements of railway trac control problem for its optimal (or near optimal) solution, and interactions amongst trains operating in the system and consequences of a con¯ict resolution ought to be considered. Using the set of weights, it is possible to calculate the overall utility values (or dynamic priorities) of con¯icting train i and train j with respect to meetpoint m using the P outbound P inbound m‡1 m‡1 m m  ˆ c wc Vjc . The choice behaviour was modeled such that equations: pi ˆ c wc Vic and pj the train, which is allowed to pass, has higher dynamic priority. Therefore, the attribute, which is desired to have smaller value, is multiplied by `negative' weight. A sample of inter-train con¯icts

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and respective dispatcher's resolutions are shown in Table 1. In the table, train pairs in the ®rst column indicate the con¯icting train numbers. This column also consists of dispatcher's resolution: con¯icting train, which is allowed to proceed, is indicated above and, which is stopped, is below. The sign of the weights of basic priority and critical ratio is negative because the smaller the original value of these attributes, the greater their contribution to the dynamic priority. The software package used to solve the linear programming model does not allow using extremely small or extremely large numbers in a formulation. In order to comply with this scaling rule, we used scaling coecients to reduce/increase the magnitude of attribute values in the model and in computation of dynamic priorities. The overall success of the model is quite encouraging with 82% harmonious decisions; that is, the model could reproduce 74 over 90 preferences made by the dispatcher. There could be some special conditions at the time of con¯ict, which may force the dispatcher to make ¯uctuating decisions. However, much has to be done to improve the model such as introduction of better identifying attributes and/or their combination, incremental weight scheme dependent on attribute values, etc. 6. An algorithm for railway trac control based on inter-train con¯ict management A heuristic algorithm has been developed in order to obtain better con¯ict solutions than train dispatchers and optimal or near optimal solutions in reasonable length of time. Due to largescale, complex and dynamic nature of the problem, we have chosen a systems approach to de®ne Table 1 Sample of inter-train con¯icts and their resolutions Con¯icting train no.

Basic priority

Critical ratio

Delay at myopic resolution

Number of potential con¯icts

Dynamic priority

Explanation

0.0769

4.5390

0.2000

1.0000

125 170

20 70

1.075 1.028

9 12

0 0

ÿ2.86 ÿ3.19

Importance of basic priority

161 124

60 20

1.037 1.600

8 14

1 5

ÿ3.27 ÿ4.11

Importance of critical ratio

151 126

50 20

0.963 1.109

6 12

2 2

ÿ3.03 ÿ2.82

Resolution that does not match with the model

125 170

20 70

1.143 1.000

12 3

0 0

ÿ2.91 ÿ3.51

Relation between basic priority and myopic resolution

161 156

60 50

1.134 1.132

16 3

3 3

ÿ3.18 ÿ3.65

E€ect of myopic resolution

161 170

60 70

1.167 1.211

8 6

0 5

ÿ3.64 ÿ3.97

Potential con¯icts should have been e€ective

126 151

20 50

1.000 1.092

12 6

3 2

ÿ2.51 ÿ3.40

Easy resolution

151 124

50 20

0.854 0.682

11 11

1 3

ÿ2.50 ÿ1.66

Adverse resolution concerning model assumption

Scaling coecients

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the complex interrelationship and dynamics of the railway system. Additionally, using systems approach helps to ®nd the consequences of a con¯ict resolution in a section of the railway line on other sections (in turn on trains operating on those sections). The algorithm is constructed in a similar manner as in VoÈlckers (1986). We assume in this study that a con¯ict occurs between two trains. The algorithm detects the trains involved in the immediate con¯ict. Then it stops each train sequentially for the resolution and searches for the consequences. For this, it calculates the expected time of arrival at scheduled points of every train in the problem domain by considering their potential con¯icts and associated average delays. The sum of deviation (tardiness or delay) of the expected arrival times of trains from their scheduled times within a pre-speci®ed time horizon has been chosen as the ``measure of e€ectiveness''. The objective is to choose the train to be the stopped one, which minimizes the measure of e€ectiveness. The algorithm proceeds in this manner until all of the con¯icts are resolved. Fig. 3 shows the process of trac control based on inter-train con¯ict management in a schedule-responsive railway (SËahin, 1995). The process is explained in detail in the following sections. 6.1. De®nition of state of the railway system Some data of trains operating in the system constitutes the necessary state information of the system. The following data of outbound train i…i 2 I† at any point of time is used to de®ne the state of the railway system (the associated data of inbound train j…j 2 J† is also gathered for completion of state de®nition): Ai "i i xA i Ai pi

respective meetpoint indicating current location; Ai 2 fi ; . . . ; "i g destination meetpoint (normally ®rst or last meetpoint in the system) expected time of arrival at next meetpoint (or actual arrival time if train is at Ai ) dynamic priority at meetpoint Ai , which established by using some evolutionary attribute(s) of the train.

6.2. Detection of disturbed trains We assume that there exists a pre-established schedule in hand. Having de®ned state of the i railway system, the state parameter, xA i , of outbound train i is compared to its scheduled arrival Ai time, i at meetpoint Ai so as to detect any existing deviation from the schedule. Let Id be the set of disturbed outbound trains …Id 2 I†. Since the same process is applied to inbound trains as well, let Jd be the set of disturbed inbound trains …Jd 2 J†; where  Ai i …5† Id ˆ i : xA i 6ˆ i ; i 2 I; Ai 2 fi ‡ 1; . . . ; "i g n o  i i j ÿ 1; . . . ; "j A Jd ˆ j : x A …6† j 6ˆ  j ; j 2 J; Aj 2  6.3. Detection of immediate con¯ict and its parameters An inter-train con¯ict simply occurs when two opposing trains move on a single-track section between neighboring meetpoints, or if a faster train catches a slower one moving in the same

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Fig. 3. Process of railway trac control based on inter-train con¯ict management in a schedule-responsive railway.

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direction. The proposed heuristic algorithm detects and resolves one con¯ict at a time: It is the immediate con¯ict. Although railway trac evolves with time, we can analytically anticipate the trains, which are going to be involved in the immediate con¯ict, and the location and occurrence time of this con¯ict with a high con®dence. In order to explain why the immediate con¯ict has been selected as a starting point of resolution, we need to look at the potential con¯icts of a train in the immediate con¯ict (note that the immediate con¯ict itself is also a potential con¯ict). It is clear that even if we resolve the potential con¯icts primarily before the immediate con¯ict, the resolution of the immediate con¯ict will most likely cancel the other potential con¯ict resolutions out, resulting in wasting a lot of computational e€ort. Hence the immediate con¯ict of any train is independent from the potential con¯icts of the same train while they are dependent contrarily on the immediate con¯ict. There could be more than one immediate con¯ict of di€erent disturbed trains in the system. In that case, we select the earliest one amongst them. The algorithm proceeds by detecting and resolving the immediate con¯icts, consecutively, one at a time. We will use discrete-event and continuous simulation techniques (Banks and Carson, 1984), sequentially, in order to determine the parameters of the immediate con¯ict: trains being in con¯ict and location of the con¯ict. We determine the con¯icting trains and the meetsection, where the immediate con¯ict is going to occur by using discrete-event simulation; and the exact location and time of occurrence by continuous simulation. 6.3.1. Discrete-event simulation We need to make some de®nitions before explaining the simulation procedure. These are the entity, event, input variable and the future event list (FEL). The entities in our model are trains, meetpoints and line sections between meetpoints. An arrival and a departure of a train are called events. An input variable maintains a relationship between two consecutive events, i.e. arrival±departure (dwelling) or departure±arrival (running). For instance, the events of train i are as follows: m m m m m ˆ dm am‡1 i i ‡ i and di ˆ ai ‡ …i ÿ i †

…7†

m where, the arrival event is represented by am‡1 and the departure event by dm i i ; running time, i , m m and dwelling time, …i ÿ i †, represent the input variables. The future event list is prepared for each disturbed train in its direction of movement. Events for other trains considered in the system are also prepared for comparison. Table 2 shows the FEL of disturbed train i and those of others, m‡1 . Note that the FEL is and the existence information of a con¯ict between the events dm i and ai completed when the ®rst con¯ict (if any) is detected of each disturbed train. Therefore, we need to

Table 2 Future event list of trains Meetpoint/meetsection

Disturbed train

Future events list of disturbed train

FEL of other trains in the system

Con¯ict exist?

m

i…i 2 Id †

m‡1 dm i ; ai

m‡1 dm k 2 I; i 6ˆ k k ; ak m‡1 m d ; a j 2 J

Yes/no

j

j

Yes/no

524

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evaluate the future events of trains, in order to detect the existence of any con¯ict. This process is carried out by checking the feasibility of movements of trains considered. For example, the feasibility check for disturbed train i…i 2 Id † and other trains k and j…k 2 I, and j 2 J† is shown below. A con¯ict occurs between outbound trains i and k if both of the following inequalities are violated simultaneously. m m m m m dm i ÿ dk 5ki and dk ÿ di 5ik ;

…8†

Let us use a special indicator for the existence of such a con¯ict as follows: i==k ˆ 1, m: there exists a con¯ict between outbound train i and outbound train k at meetsection m, i==k ˆ 0; m: no con¯ict, otherwise. Similarly, a con¯ict occurs between opposing trains i and j if both of the following inequalities are violated simultaneously. m m‡1 ÿ am‡1 5m‡1 ; i:e: m dm i ÿa j 5ji and dj i ij

…9†

i  j ˆ 1; m: there exists a con¯ict between outbound train i and inbound train j at meetsection m, i  j ˆ 0; m: no con¯ict, otherwise. Let Si …k; j; m† and Sj …i; s; m† be the set of immediate con¯ict parameters for each disturbed train i …i 2 Id † and j…j 2 Jd †, respectively:  …10† Si ˆ …k; j; m† ˆ k; j : i==k ˆ 1; m; ixj ˆ 1; m; i 2 Id ; k 2 I; i 6ˆ k; j 2 J  Sj ˆ …i; s; m† ˆ i; s : jxi ˆ 1; m; jnns ˆ 1; m; i 2 I; j 2 Jd ; s 2 J; j 6ˆ s : …11† 6.3.2. Continuous simulation In the previous section we determined the set of immediate con¯ict parameters for each disturbed train as Si …k; j; m† and Sj …i; s; m†. Now, we need to select the earliest con¯ict and its parameters amongst these sets. Simulation starts at the departure time of one of the con¯icting trains and continuous with a pre-speci®ed time increment,
t. At each simulation time, the location of each train pair is calculated and compared. When both trains coincidentally interfere at the same location, time of the simulation is recorded and simulation is completed for this pair of trains. Finally, the immediate con¯ict with the earliest occurrence time amongst the candidate con¯icts is selected as the one, which should be resolved. 6.4. Establishing alternative resolutions of immediate con¯ict The inter-train con¯icts occur between trains moving either in the same or opposite direction. There could be two resolutions for a con¯ict between opposing trains i and j, between meetpoints m and m ‡ 1: either train i or train j is stopped at the nearest meetpoint before the con¯ict until the passing train clears the line. Similarly, there could be two resolutions for a con¯ict between outbound trains i and k (or inbound trains j and s) between meetpoints m and m ‡ 1: either train i…j† or train k…s† is stopped at the meetpoint, that they have just left until the sucient following headway is maintained.

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Here, we assume that there is at least one unoccupied track with sucient length at the meetpoint, where the train is going to be stopped. 6.5. Evaluation of alternative resolutions: a look-ahead method The most important part of the heuristic algorithm is the evaluation component for the alternative resolutions. The success of the algorithm is directly related with the selection of the right alternative, which respects to the objective, which is the minimization of the overall tardiness of trains in the system. The superiority of the algorithm is to employ a look-ahead feature for the selection of best alternative. 6.5.1. Length of time horizon It is clear that a too narrow time horizon has an important weakness, which could lead the algorithm to make a myopic decision in which the consequences of the selected alternative over other trains may not be considered. Too wide a time horizon, on the other hand, could be costly in terms of computation load. Therefore, we need to ®nd a reasonable length of time horizon. We propose two rules: 1. time horizon with the length that should cover at least a few scheduled times of trains after current time or after immediate con¯ict, or 2. time horizon with an end time that coincides with the latest arrival time of trains at their destination. We utilized this rule in the algorithm. 6.5.2. Expected time of arrivals We can calculate the expected time of arrival, x"i i of outbound train i currently at meetpoint Ai , at a scheduled point or destination, "i as follows: X X X X Ai Ai m i im ‡ …m Dij ‡ Dik : …12† x"i i ˆ aA i ‡ …di ÿ ai † ‡ i ÿ i † ‡ m2fAi ;...;"i ÿ1g

m2fAi ‡1;...;"i ÿ1g

j2Si …j†

k2Si …k†

Ai i In the above equation, given the actual arrival, aA i , and departure, di , times of train i at the current meetpoint Ai ; the second term represents the immediate (actual) delay at meetpoint Ai Ai i (where, aA i ˆ i if Ai ˆ 1); the third term is the sum of free running time between Ai and "i ; the fourth term represents sum of the minimum scheduled dwell times from meetpoint Ai to "i ; the ®fth term is the sum of average meeting delays due to interference with inbound trains j; and the last term is the sum of average overtaking delays due to interference with outbound trains k, where Si …j† and Si …k† represent the sets of inbound and outbound trains, respectively, which are expected to be in potential con¯ict with train i within time horizon. The expected time of arrival of train j is established in the same manner.

6.5.2.1. Set of potential conflicts. We will create the set of potential con¯icts at macro level, which only takes into account whether a con¯ict exists between train pairs. Trains' actual departure times and expected time of arrivals at respective meetpoints are considered for comparison to detect the existence of any potential con¯icts.

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Hence, the set of potential con¯icts of outbound train i is that ( ) A ÿ1 A ‡1 j : diAi ÿ1 < x jAi ÿ1 and dj j < xi j ;  ; Si …j† ˆ i 2 I; j 2 J; m 2 ft ‡ 1; . . . ; "i ÿ 1g \ j ÿ 1; . . . ; "j ‡ 1  Si …k† ˆ

 < dmÿ1 and x"i i < x"kk or dmÿ1 < dmÿ1 and x"kk < x"i i ; k : dmÿ1 i i k k : m ˆ min…Ai ; Ak †; i 2 I; k 2 I; i 6ˆ k; m 2 fi ‡ 1; . . . ; "i g \ fk ‡ 1; . . . ; "k g

…13†

…14†

The set of potential con¯icts of inbound train j can be obtained in the same manner. It is important to note that detection of the potential con¯icts is carried out in an iterative manner in conjunction with computation of the expected time of arrivals as shown in Fig. 4.

Fig. 4. Iterative computation of the expected time of arrival of each train in the system.

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6.5.2.2. Average delay times. At this point, we need to determine average meeting and overtaking delay times due to potential con¯icts. We may use analytical models provided in Petersen (1975) or in English and Schwier (1977) for more detailed formulation. In this study we have used the delay models given in the former and the derivation of these models can be found in the Appendix. In the average interference delay formulae, Pij represents waiting probability (priority) for train pairs; i.e. Pij is the probability that train i is chosen to be delayed (by being stopped) by the dispatcher due to interference with train j or the fraction of time that train i is delayed by train j. Similarly, the probability that train j is delayed is that Pji ˆ 1 ÿ Pij . We have established two models for discrete choice to determine the probabilities for train pairs: binary logit model and linear model (Ben-Akiva and Lerman, 1985). The models of the waiting probabilities were established in a dynamic manner that they include the actual and scheduled timing information of con¯icting train pairs through dynamic priorities of trains. We used the critical ratio, the evolui tionary attribute, in order to establish dynamic priorities of trains (e.g. pA i for train i at meetpoint Ai Ai ). The values of critical ratio attribute (e.g. Vi for train i) were extracted from train-graphs used in Section 5. Having applied maximum-likelihood method for binary logit model (Kanafani, 1983) and least square method for linear model, we determined the weights (coecients). The following are the models established for con¯icting train i and train j (moving in the same or opposite direction) (SËahin, 1996): Ai

Binary logit model : pij ˆ

eÿpi Ai

ÿpi

e

Aj

ÿpj

‡e

A

;

j i Linear model : Pij ˆ 0:735…VA i ÿ Vj † ‡ 0:5

A

A

Aj

j j i pA i ˆ ÿ4:832Vi ; Pj ˆ ÿ4:832Vj

…15†

…16†

6.5.3. Selection of the better alternative At this last stage of the algorithm, one of the alternatives of the immediate con¯ict resolution is selected based on the measure of e€ectiveness. Up to now, we have calculated the expected time of arrival of each train operating in the system within a speci®ed time horizon. The measure of e€ectiveness becomes the sum of deviation of expected time of arrival of each train from its scheduled arrival time. The formal expression of the measure of e€ectiveness, MoE, with respect to trains' ®nal destination, is given below: i X   Xh " " max i …x"i i ÿ "i i †; 0 ‡ j …x j j ÿ  j j †; 0 …17† MoE ˆ i2I

j2J

Where i and j are the relative importance of train i and train j, respectively. We have set s to one in the numerical test because we introduced the relative importance with schedule slack, the di€erence between the scheduled arrival time and the minimum arrival time (which is the sum of scheduled departure time at origin, scheduled dwell times and minimum free running times between origin and destination meetpoints) at destination. The smaller the schedule slack is, the greater the relative importance becomes. Since we have produced two distinct alternative resolutions for the immediate con¯ict, there should also be two MoEs; e.g. MoE of con¯icting trains i and j are shown as (MoE)i and (MoE)j ,

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respectively. Now, we have sucient information to make a choice between the alternatives: train, which is stopped for the immediate con¯ict resolution, and which causes less consequential overall delay in the system, is selected: n o …18† train stopped : i or j : min‰…MoE†i ; …MoE†j Š : After providing and applying this proposal, the algorithm returns to the beginning and rede®nes the state of the system to detect the existence of any other con¯ict; and the process continues in a similar manner. 7. Numerical tests and results We tested three di€erent solution methods provided in the previous sections: optimal (exact) solution, dispatcher's solution and heuristic's solution. We have implemented the software for dispatcher's and heuristic's solution in BASIC programming language with a prototype user interface; and optimal solutions were found by the software called LINDO (Schrage, 1991). The tests were carried out on a PC with Pentium 90 MHz microprocessor. Problem instances with di€erent size and input data were established and used in the tests, in which we assumed that the number of tracks and length of meetpoints are adequate for trains to be stopped and that there is no intermediate scheduled stop. These assumptions were made basically to reduce the size of the mathematical programs and to reach the optimal solutions in a reasonable time. Minimum following and safety headway were set to 10 and 2 min, respectively, and switching times to 2 min. We tested the solution methods for 35 di€erent problem instances. We found optimal (exact) solution for 26 of these instances. The size of the problem instances ranges between six trains (three trains per direction in 3-h domain), ®ve meetpoints and 20 trains (10 trains per direction in 8-h domain), 19 meetpoints. The number of trains involved in 35 problem instances is 305. The running times of trains in the instances range between 36 and 162 min, and 74 min on average. The comparison of the solution methods is carried out with the following three criteria: . Measure of e€ectiveness: Sum of the deviation of the expected time of arrival of each train from its scheduled arrival time at destination. The values of the relative importance are set to one. h i X   X " " max i …x"i i ÿ "i i †; 0 ‡ max j …x j j ÿ  j j †; 0 …19† MoE ˆ i2I

j2J

. Total waiting times: Total unscheduled waiting times of trains in the system. X

X

m m m ‰…dm i ÿ i † ÿ …i ÿ i †Š ‡

i2I m2fi ;...;"i ÿ1g

X

X

m  m †Š m ‰…dm j ÿ j † ÿ …j ÿ  j j2J m2f j ;...;"j ‡1g

…20†

. Computation time: The di€erence between the end and the start time of a method to ®nd a solution for a particular problem instance.

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7.1. Some results of the numerical tests 1. The heuristic algorithm has found the optimal solution for 19 and 13 times over 26 problem instances by employing linear model and binary logit model, respectively. And the linear model has performed better than the binary logit model in terms of three criteria: measure of e€ectiveness, total waiting times due to interference and computation time. The following results have been obtained by using the linear model in the algorithm. 2. Maximum deviation of the heuristic algorithm from the respective optimal solution is 13 min. On average, the heuristic deviated only 3.24% (with 96.76% success) from the exact solution with respect to measure of e€ectiveness, and it performed 13.54% better than the exact solution method with respect to total waiting times. This result implies that the algorithm performs almost as well as the exact solution method in selecting the better con¯icting train to stop. The heuristic has reached the solutions (on average) in 0.72% of time of the exact solution method. 3. Dispatcher's solution method has found the optimal solution for 13 times over 26 problem instances. 4. Maximum deviation of the dispatcher's solution method from respective optimal solution is 30 min. On average, the dispatcher's solution method deviated 70.65% from the exact solution with respect to measure of e€ectiveness, and it performed 3.96% worse than the exact solution method with respect to total waiting times. In terms of total waiting times of trains, the result of both solution methods is close, but measure of e€ectiveness criterion unfavorably di€ers 70.65% for the dispatcher's solution. This means that the dispatcher has a weakness in selecting the better con¯icting train to stop. The dispatcher's solution method reached solutions (on average) in 0.40% of time of the exact solution method. 5. In comparison the heuristic algorithm with the dispatcher's solution method, we found out that the heuristic has performed better than the dispatcher. The dispatcher's solution method deviated 67.12% from the heuristic's solution in terms of measure of e€ectiveness, and 21.81% in terms of total waiting times. The algorithm has over performed for two criteria. It causes trains to experience not only less amount of wait but also less deviation from scheduled times. This implies that the algorithm chooses the better con¯icting train to stop. The dispatcher's solution method has been slightly faster than the heuristic due to heavy search and computation process for potential con¯icts of the heuristic. 6. The overall deviation from schedule (MoE) is 1409 min and total waiting times of trains is 4928 min in the heuristic's solutions. For the same problem instances the same measures become 1765 and 5699 min, respectively in the dispatcher's solution method. This result shows that the heuristic saves 356 min in terms of MoE and 771 min in terms of total waiting times. 7. Comparing the heuristic algorithm to the dispatcher's solution method, the savings in the MoE is 1.2 min per train and in the total waiting times is 2.5 min per train. 8. Conclusions and future prospects We have studied the railway trac control and train scheduling problem in this research. The algorithm proposed has been constructed in a manner that is di€erent from those of the existing

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ones. We de®ned the problem by using systems approach. The algorithm is basically constructed on immediate inter-train con¯ict resolution; and it reaches a decision by employing a look-ahead method to ®nd out the consequences of each alternative solution. In the look-ahead method, we used analytical models for the computation of average interference delays. The better the delay models used are, the better the solution reached is. We introduced another new approach applied for the railway trac control problem: A multiattribute choice for dynamic priority determination. The results seem promising. As we know, train dispatchers in many railways still mainly perform the process of trac control. Therefore, it is crucial to identify their decision behavior, in order to develop advanced control models. The application of the multi-attribute choice model presented in this study seems to be a powerful tool for the learning process to make this identi®cation. The heuristic algorithm may be improved to solve control problems for double and multiple track railways with bi-directional signaling system. The ability of the algorithm should be extended from line to network level. The algorithm developed could be a good start toward developing a Decision-Support System for train dispatchers. We are currently working on implementing such a system. And it may also be used as a con¯ict resolution procedure in a real-time trac control system such as ATCS (Advanced Train Control Systems), which is being developed in North America. The early results obtained from the algorithm for hypothetical instances are encouraging. We are preparing to apply our model to real-life problem instances in the Turkish State Railways (TCDD), in order to observe its performance and to make necessary adjustments in the model. Acknowledgements The author wishes to express his gratitude to the sta€ members of the Canadian Institute of Guided Ground Transport (which does not exist any more) for their hospitality during my stay at the Institute; to W. D. (Bill) Burgell, former Chief Dispatcher in the Union Paci®c Railroad, USA, for his informative suggestions about train dispatching rules; and ®nally to the sta€ members of the Department of Operations of TCDD for their useful suggestions and for providing the necessary data for the application carried out in this research. Appendix The following average meeting and average overtaking delay models (Petersen, 1975) have been used in the average delay module of the heuristic algorithm. A.1. Average meeting delays for meetsections Fig. 5 shows the timing convention utilized where outbound train i meets inbound train j between meetpoints m and m ‡ 1 (i.e. meetsection m). In the ®gure  ˆ 0 indicates the time that train i arrives at meetpoint m and im and jm represent the free running time of train i and train j in this section of track, respectively. A meet con¯ict will occur in this section if train j arrives at

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531

Fig. 5. Timing convention.

meetpoint m in the interval of time  ˆ 0 and  ˆ
, where
ˆ im ‡ jm . The arrival time of train j is the time  in the case of free running (i.e. if no interference occurs). Thus, 044
, and  could equally likely occur at any point in this interval. It is assumed that average transit time for the section in a single-track line is at least one half the average switching time: im ‡ jm 5…SWi ‡ SWj †=2, where, SWi and SWj stand for switching times of train i and train j, respectively. The switching time is to switch into and out of a siding (meetpoint) at the rated speed, and accelerate back up to the speed limit. Four di€erent meet con®gurations are possible, and each of which is examined in order of increasing  as shown in the reference. The following model (A1) is the combination of each con®guration. In the model, priority Pij (waiting probability for train pairs) is the fraction of time that train i is delayed by train j. Dm ij

! P2ij
SWi 1 SW2i SW2j ‡ Pij ˆ ‡ ‡ : 2
2 8 8

…A1†

In order to obtain delay to train j due to train i…Dji †, Pij is replaced with Pji ˆ 1 ÿ Pij in (A1). A.2. Average overtaking delays for meetsections We will provide the results of the expected average overtaking delays for train i and train k running in the outbound direction. The delays for inbound trains are similarly obtained. The delay to the overtaking train i between meetpoints m and m ‡ 1 is that Dm ik ˆ

P2ik m … ÿ im † 2 k

…A2†

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and delay to the overtaken train k is that Dmki ˆ

P2ki m … ÿ im † ‡ 2m ik ‡ SWk : 2 k

…A3†

The above given expressions are restricted to the single-track sections with equal length along a railway line; however, assuming the siding spacing to be equal is not realistic. These expressions should be modi®ed for unequal siding spacing because it is more likely that interference will occur on the longer sections. These longer sections also have the greatest interference delays. In order to do such a modi®cation, depending on the probability that a meet or an overtake occurs on a particular section within the time horizon proposed in Section 6.5.1 point 2, the interference delays should be weighted as follows: Weight for average meeting delays for meetsection m : im ‡ jm ˆ wm P P m jm im ‡ m2fi ;...;"i ÿ1g   m2fjÿ1 ;...;"j g

…A4†

Weight for average overtaking delays for meetsection m : km ÿ im wm ˆ P P o km ‡ im

…A5†

m2fk ;...;"k ÿ1g

m2fi ;...;"i ÿ1g

Following the average meeting and overtaking delay computation for each section, the expected values of the interference delays for the line are shown below. Average meeting delays for the line Dij ˆ

N ÿ1 X

m wm m Dij and

…A6†

mˆ1

Dji ˆ

Nÿ1 X

m wm m Dji :

…A7†

mˆ1

Average overtaking delays for the line (for outbound trains) Dik ˆ

Nÿ1 X

m wm o Dik

and

…A8†

mˆ1

Dki ˆ

Nÿ1 X

m wm o Dki :

mˆ1

Average overtaking delays for inbound trains can similarly be obtained.

…A9†

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References Adam Jr, E.E., Ebert, R.J., 1986. Production and Operations Management Concepts, Models and Behavior. Prentice Hall, Englewood, Cli€s, NJ. Amit, I., Goldfarb, D., 1971. The timetable problem for railways. In: Developments in Operations Research, vol. 2, Gordon and Breach, New York, pp. 379±387. Assad, A.A., 1980. Models for rail transportation. Transportation Research 14A, 205±220. Banks, J., Carson II, J.S., 1984. Discrete-Event System Simulation. In: Fabrycky W.J., Mize J.H. (Eds.), Prentice Hall International Series in Industrial and Systems Engineering, Prentice Hall, Englewood Cli€s, NJ. Ben-Akiva, M., Lerman, S.T., 1985. Discrete Choice Analysis. The MIT Press, Cambridge, MA. Cherniavsky, A.L., 1972. A program for timetable compilation by a look-ahead method. Arti®cial Intelligence 3, 61± 76. Eisele, D.O., 1985. Interface between passenger and freight operations. Transportation Research Record 1029, 17±22. English G.W., Schwier C., 1977. Eastern Canada Railway Line Capacity Study. CIGGT report no. 77±16. Canadian Institute of Guided Ground Transport, Queens University at Kingston, Ontario, Canada. Frank, O., 1965. Two-way trac on a single line of railway. Operations Research 14, 801±811. Garey, M.R., Jonhson, D.S., 1979. Computers and Intractability: A Guide to the Theory of NP-completeness. W.H. Freeman, San Francisco, CA. Harker, P.T., 1990. Use of advanced train control systems in scheduling and operating railroads: models, algorithms, and applications. Transportation Research Record 1263, 101±110. Iida, Y., 1988. Timetable preparation by A.I. approach. Proceeding of European Simulation Multiconference, Nice, France, pp. 163±168. Jovanovic, D., 1989. Improving railroad on-time performance: models, algorithms and applications. Ph.D. dissertation in systems, The University of Pennsylvania, Philadelphia, PA. Jovanovic, D., Harker, P.T., 1990. A decision support system for train dispatching: an optimization-based methodology. Journal of the Transportation Research Forum XXXI, 25±37. Kanafani, A.K., 1983. Transportation Demand Analysis. McGraw Hill, New York. Komaya, K., 1992. An integrated framework of simulation and scheduling in railway systems. In: Murthy, T.K.S., Allan, J., Hill, R.J., Sciutto, G., Sone, S. (Eds.), Computers in Railways III, Vol. 1: Management. Computational Mechanics Publications, Southampton, Boston, pp. 611±622. Komaya, K, Fukuda T., 1989. ESTRAC-III: an expert system for train trac control in disturbed situations. In: Perrin, J.P. (Ed.) IFAC Control, Computers, Communications in Transportation, IFAC/IFIP/IFORS Symposium, Paris, France, 19±21 September 1989. Published for the International Federation of Automatic Control by Pergamon Press, Paris, France, pp. 147±153. Kraay, D., Harker, P.T., Chen, B., 1991. Optimal pacing of trains in freight railroads: model formulation and solution. Operations Research 39, 82±99. Oral, M., Kettani, O., 1989. Modeling the process of multiattribute choice. Journal of Operational Research Society 40, 281±291. Peat, Marwick, Mitchell & Co. Train Dispatching Simulation Model: Capabilities and Description. USA. Report No. DOT-FR-4-5014-1, March 1975, Prepared for Federal Railroad Administration, Department of Transportation, Washington DC. Petersen, E.R., 1975. Interference delays on a partially double-tracked railway with intermediate signalling. In: Petersen, E.R., Fullerton, H.V. (Eds), The Railcar Network Model. CIGGT Report No. 75±11, Canadian Institute of Guided Ground Transport, Queens University at Kingston, Ontario, Canada, pp. 33±59. Petersen, E.R., Taylor, A.J., 1982. A structured model for rail line simulation and optimization. Transportation Science 16, 192±206. Petersen, E.R., Taylor, A.J., Martland, C.D., 1986. An introduction to computer-assisted train dispatch. Journal of Advanced Transportation 20, 63±72. Potvin, J-Y., Dufour, G., Rousseau, J-M., 1992. Simulating Vehicle Dispatching with Linear Programming Models. Publication no. 816, Centre for Research on Transportation, University of Montreal, Montreal, Quebec, Canada. Rudd, D.A., Story, A.J., 1976. Single track railway simulation new models and old. Rail International, June, 1976 335± 342.

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_ S I. CËahin / Transportation Research Part B 33 (1999) 511±534

_ 1987. A deterministic model for railway line capacity analysis. M.Sc. thesis, Yõldõz Technical University, SËahin, I., Istanbul, Turkey (in Turkish). _ 1995. Railway trac control based on inter-train con¯ict management. Graduate Third Prize Winner, SËahin, I., Transportation Research Forum 37th Annual Meeting, 19±21 October, Chicago, IL, pp. 608±626. _ 1996. A Decision Support System for Trac Control Based On Inter-Train Con¯ict Management. Ph.D. SËahin, I., Thesis, Yõldõz Technical University, Turkey, ISBN 975-461-024-X (Published in Turkish). _ SoÈgÆuÈtluÈ F.S., Erel A., 1995. Multi-attribute decision making in railway trac control. XVIIth National SËahin, I, Conference on Operations Research, Middle East Technical University, 10±11 July, Ankara, Turkey. Sauder, R.L., Westerman, W.W., 1983. Computer aided train dispatching: decision support through optimization. Interfaces 13, 24±37. Schrage, L., 1991. LINDO, An Optimization Modelling System, 4th ed. The Scienti®c Press, South San Francisco, CA. Srinivasan, V., Shocker, A.D., 1973a. Linear programming techniques for multidimensional analysis of preferences. Psychometrica 38, 337±369. Srinivasan, V., Shocker, A.D., 1973b. Estimating the weights of multiple attributes in a composite criterion using pairwise judgements. Psychometrica 38, 473±493. Steiner, F.T., 1978. Computer Aided Dispatching, Bulletin 334. Union Switch & Signal Division, Westinghouse Air Brake Company, Swissrale, PA. Szpigel B., 1973. Optimal train scheduling on a single track railway. In: Ross, M. (Ed.). OR'72, North Holland Publishing Co. pp. 343±352. VoÈlckers U., 1986. Dynamic planning and time-con¯ict resolution in air trac control. Arti®cial Intelligence and ManMachine Systems. Proceedings of an International Seminar Organized by Deutsche Forschungs-und Versuchsanstalt fuÈr Luft-und Raumfahrt (DFVLR), Bonn, Germany, pp. 175±197. Wong, O., Rosser, M., 1978. Improving System Performance for a Single Line Railway with Passing Loops. New Zealand Operational Research 6, 137±155.