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The improvement of an existing fuzzy logic rule base for the treatment and simulation of conflicts in

the dispositional tasks of a railway traffic control center

Mark Farkas

Department of Telecommunications and Media Informatics

Budapest University of Technology and Economics Budapest, Hungary

Tibor Héray Department of Automation, Faculty of Engineering

Sciences Széchenyi István University

Győr, Hungary

László T. Kóczy Institute of Informatics, Electrical and Mechanical

Engineering, Faculty of Engineering Sciences Széchenyi István University

Győr, Hungary

Abstract— The paper considers a railway timetable related problem in a simplified form, generated by the delay of one or several incoming trains at a given station. These incoming train delays are either automatically generated or manually entered. Usually there are connecting trains in the timetable, especially in up to date periodic timetables and thus incoming delays might indicate the necessity of introducing a delay with connecting outgoing trains. A hierarchical fuzzy rule base is applied in order to determine the optimal outgoing delay, taking also usual restrictions into consideration. The delays of the incoming trains are modeled by independent exponential distributions and a software simulation is built around the existing rule base. The behavior of the hierarchical fuzzy rule base and the pertinence of the outgoing delay is a subject of further investigation. The delays are also updated periodically that makes the recalculation of outgoing delays necessary from time to time.1

Keywords-periodic railway timetables, connection conflicts, fuzzy hierarchical rule base, automated simulation

I. INTRODUCTION A simplified model of a railway traffic control situation

and conflict was introduced in our previous article [1]. A railway junction point with eight track stations was examined, which is to be found in the connection point of a central-structured railway network (Fig. 1).

This paper was supported by a Széchenyi University Main Research Direction Grant 2009 and National Scientific Research Fund Grants OTKA T048832 and K75711.

Fig. 1. A typical railway junction point In this model, each incoming line at the inbound side of the station is either a double track line A1, A2 or a single track line B. At the outbound side of the station C1 and C2 are double tracked and D, E lines are single tracked.

Fast and slow trains are operated on these tracks as well but further refinement of the railway model may introduce new types of trains. These trains are denoted with F and S in their unique identifier.

An incoming and an outgoing train is in a connected relation if the respective time of arrival and departure lie within one hour. In every other case trains are considered to be unrelated.

II. THE STATIC AND DYNAMIC MODEL OF A RAILWAY JUNCTION POINT

A railway junction point is a vertex v in a directed graph G, where each vertex has an associated capacity c. The capacity represents the number of trains that can stay at the railway station simultaneously. The edges in this directed graph represents lines and tracks.

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1

, ; :

: , ,

: , ,n

G V E c V

TR train train

RW G c TR

(1)

This model only describes the static behavior of the

junction point. In order to model connection conflicts a timetable time data is required that contains the planned time of arrival and departure of each train. In our examples, only a partial timetable is given, which only contains relevant trains for our investigations.

A unique identifier is associated to every train in TR and line, and each train has a corresponding line and platform associated to only this train at a given time.

For each vertex v, let dep denote the time of departure of a train and arr denote the time of arrival respectively. These can be regarded as a periodical function with a certain period T (hence the timetable is regarded as a periodical timetable).

: ,

: ,

( , ) :

i

j

i j

i d

j a

i j d a

dep k T kT k

arr k T kT k

CT i j dep k arr k T T

(2)

To describe a possible connection conflict situation

incoming delays (ID) must be available for each train at a given moment. The incoming delay of a train may change at any time before the arrival of the corresponding train and it can only be estimated with a given probability and uncertainty. Therefore it is essential to re-evaluate incoming delays on a regular basis. Re-evaluation plays a crucial role in the accuracy of the decision support system and the effect of the re-evaluation may be a subject of further examination.

III. A HIERARCHICAL FUZZY MODEL FOR CONFLICT RESOLUTION In order to determine the aggregated outgoing delay (ODi)

of a train i, incoming delay IDj is evaluated for each connected train j. From CT(i,j) and IDj, ODi,j (outgoing delay of the train i based on only one incoming train j with an incoming delay IDj) can be obtained using a fuzzy rule base system (S1) described in [1]. The fuzzy system is hierarchical, thus ODi,j fuzzy values are used to evaluate ODi by the S2 system (Fig. 2 and Tab. II.).

Fig. 2. The hierarchical fuzzy rule base (S1 and S2)

The crisp connection times, incoming delays and outgoing delays are fuzzified according to CT, ID and OD fuzzy membership functions that contain trapezoidal membership functions with their associated linguistic labels. [1]

Figure 2 shows the way of resolving the conflict of connecting trains from the aspect of the outgoing train i, where each input symbol is a crisp value (a time value to be precise) and the only output symbol ODi is a defuzzified time value that determines the amount of delay in the time of departure of the train i.

A. S1 rule base [1]

TABLE I.

ID [min] – incoming delay

CT [min] – connection time VS 1…2

S 4…8

M 10…15

L 18…25

VL > 30

AZ < 1.5 0 0 0 0 0 VS 3…5 M S 0 0 0 S 8…11 VL L M 0 0 M 14…19 VVL VVL VL L 0 L 22…27 0 0 VVL VL 0 VL > 30 0 0 0 VVL RE

B. S2 rule base [1]

TABLE II.

Rule base S2 of delays for departing trains No. of occur.

Biggest ODi,j on the grounds of S1 AZ S M L VL VVL

1 AZ S M L VL VVL 2 AZ M L VL VVL VVL 3 S M VL VL VVL VVL ≥4 S L VL VVL VVL VVL

IV. MEASURING THE PERFORMANCE OF THE SYSTEM The hierarchical fuzzy system S introduced in [1] is

capable to resolve possible timetable conflicts using a partial timetable data extracted from real-life timetables and the estimated incoming delays. In the followings a possible

S1CT(i,j0)

IDj0

OD(i,j0)

S1CT(i,j1)

IDj1

OD(i,j1)

S1CT(i,jn)

IDjn

OD(i,jn)

S2ODi

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Mark Farkas, Tibor Héray, László T. Kóczy • The Improvement of an Existing Fuzzy Logic Rule Base for the Treatment and Simulation...

goodness measure and simulation technique is shown that can be used as an objective function in order to evaluate the effectiveness of the decision support system.

A. Total aggregate waiting cost It is always crucial to measure the efficiency of a system,

but ‘efficiency’ is defined in accordance to the model itself. One idea could be to measure the effectiveness of our system using the total aggregate waiting cost ( Tw) [3].

Before going into further details about this concept one should define the value of time (VOT):

The value of time is always expressed relatively to the driving time, because of psychological reasons waiting is always more embarrassing than driving for instance therefore certain commonly used factors (f) are used to calculate the value of time: eg. 1 minute of waiting is equivalent to 2 minutes of driving. [3]

The total aggregate waiting cost in a railway conflict is the aggregate time spent with waiting for each passenger involved in that particular conflict situation.

B. Train priorities and passenger preferences Understanding passenger preferences is a cornerstone in

the development of a successful performance measure as passengers are the real users of the railway network. There are three kinds of passengers and each category has different needs: [4]

1) Arriving passengers: Arriving passengers are passengers, who have reached

their destination when the train arrived to the railway station. This subclass is not important in the examination of our model as in this case the cost of waiting comes from the incoming delay of the train, which is an external variable and it is not affected by the system.

2) Remaining passengers: Remaining passengers are passengers, who remain on the

train after the arrival and continue their journey with the same train. This train may have to wait for connecting trains. This is the only time interval, which is affected by the decision support system, thus this waiting must be included in the total aggregate cost of waiting.

3) Transfer passengers: Transfer passengers also have to wait for the departure of

the connection train but they are not necessarily change to the same departing train. Transfer passengers has a minimal waiting time which is equivalent to CT, this value cannot be affected by the system, thus only the outgoing delay OD counts as an effective waiting time in that case.

This aggregate waiting cost can be used to measure the effectiveness of the discussed hierarchical fuzzy system. One has to note however that this is not the only possible measure.

The incoming delays of the connecting trains do not affect the total aggregate waiting cost directly but through the outgoing delay, which is calculated by the hierarchical fuzzy rule base system.

C. simulating incoming delays A simplified yet usable approach is to model every

incoming delay IDi as a random variable with an exponential distribution; these variables are considered to be independent in this simplified model.

1 , 0

1( ) ( ) 1 , 0

d

d D D

f d e d

F D P D d e dD e d

The possibility of missing a transfer can be expressed as

the following in the case of transfer passengers:

1iOD

i iP d OD P d OD e If T is the period time in the timetable then the passengers

have to wait T - (d - ODi) minutes for the next connection. (d-ODi) is approximately 0 therefore the cost of missing a connection –viewed from the aspect of the passenger- can be estimated using the formula below:

iOD

mc mcC VOT T e When transfer passengers arrive earlier than the departure

of the transfer train the cost of waiting for the connection is approximately:

1iOD

wc i wc iC OD VOT OD e

Similarly the waiting time of the remaining passengers is

estimated by:

1iOD

st i st iC OD VOT OD e

Where VOT values are value of time multipliers in different

scenarios (mc stands for missed connection, wc means waiting for connection and st is staying on the train and waiting for the arrival of connecting trains). The total aggregate cost of waiting can be expressed as the following:

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1

i

i

OD

i mc t

OD

wc t st r i

C OD VOT N T e

VOT N VOT N OD e

The objective is to minimize the total aggregate cost of

waiting by chosing the ODi properly. This can be achieved by deriving the C function and finding a zero place in the derived function:

arg min

ii

OD iOD

opt C OD

The optimum output delay of train i (optODi) depends on

the number of transfer (Nt) and remaining (Nr) passengers therefore two different trains may have different optimum output delay. In order to calculate this optimum point statistical datas are required about the passenger traffic.

It is obvious at a first glance that it is unfeasible to design a separate fuzzy rule base and fuzzy membership functions for each train so the effectiveness of the hierarchical fuzzy system has to be measured on the whole railway system. From the aspect of a railway junction the mean squared error (MSE) seems to be a good measure for this task.

2

1i

n

i ODi

C OD C optMSE

n

A lower mean squared error means a lower average total

aggregate cost of waiting. Therefore the above definition of the MSE serves as a good measure of performance and efficiency of the hierarchical fuzzy system used in the disposition tasks of the railway junctions.

V. IMPROVING THE FUZZY SYSTEM The trapezoidal membership functions of the hierarchical

fuzzy decision support system can be improved using the MSE definition as an objective function.

There are numerous heuristics that can be used to achieve this goal however in our later studies the bacterial memetic algorithms will be utilized.

This method is described in [5] and it is assumed that only trapezoidal membership functions are utilized and the model constists of a rule base

R = {Ri}

where the ith rule (Ri) is

Ri: IF (x1 is Ai1) AND … AND (xn is Ain) THEN (y is Bi)

where xi is the ith dimension of the input variable and Aij is the fuzzy sets of the jth antecedent in the ith rule. Bi are the fuzzy sets of the ith consequence and y is the output. [5]

Trapezoidal membership functions are described by their four respective breakpoints and this class of membership functions are considered to be sufficiently general and widely used. The COG method is used for defuzzification.

Fuzzy rules are encoded in the following way:

Fig. 3. Encoding of the fuzzy rules

Levenberg-Marquardt method is used as a local searcher

step in the bacterial memetic algorithm. More details about the creation of the initial population, the bacterial mutation and gene transfer operators are found in the referenced article [5].

With the help of this heuristics one need not to predefine trapezoidal membership functions for the linguistic variables in the hierarchical fuzzy system and almost optimal results can be achieved.

This step will improve our existing hierarchical fuzzy

system and leads to a more efficient decision support in the treatment of conflicts in the dispositional tasks of railway traffic control centers.

REFERENCES [1] T. Héray - G. Rózsa - L. T. Kóczy: The application of fuzzy logic for the

treatment of conflicts in the dispositional tasks of a railway traffic control center, 9th IEEE AFRICON Conference 2009, Nairobi, Kenya

[2] L. T. Kóczy, K. Hirota and L. Muresan: Interpolation in hierarchical fuzzy rule bases, J. Intell. Fuzzy Syst 1 (1999) pp. 77-84.

[3] P. Vansteenwegen, D. Van Oudheusden: Developing railway timetables which guarantee a better service, European Journal of Operational Research 173 (2006) pp. 453-463.

[4] Karl Nachtigall, Stefan Voget: A Genetic Algorithm Approach to Periodic Railway Synchronization, ELSEVIER Computers Ops Res. Vol. 23, No. 5 (1996) pp. 337-350.

[5] J. Botzheim, C. Cabrita, L. T. Kóczy, A. E. Ruano: Fuzzy Rule Extraction by Bacterial Memetic Algorithms, 11th World Congress of International Fuzzy Systems Association (2005), Beijing, China

[6] T. Héray - G. Rózsa - L. T. Kóczy: The possible applications of fuzzy logic for the treatment of conflicts in the dispositional tasks of railway traffic control center, 8th IEEE AFRICON Conference 2007, Windhoek, Namibia, 6p.

[7] L. A. Zadeh: Fuzzy Sets, Inf. Control 8 (1965) pp. 338-353.

Ri Ri+1Ri-1… …

Ai,1 Ai,2 Bi

ai,1bi,1ci,1 di,1 ai,2bi,2ci,2 di,2 ai bi ci di

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[8] L. A. Zadeh: Outline of a New Approach to the Analysis of Complex Systems and Decision Processes, IEEE Trans. Syst. Man and Cybernetics SMC-1 (1973), pp. 28-44.

[9] K. Nachtigall - J. Opitz: Taktfahrlagen auf komplexen Infrastrukturen mittels iterativer lokaler Konfliktauflösung, Eisenbahntechnische Rundschau (2008) pp. 369-373.

[10] K. Nachtigall – S. Voget: Minimizing waiting times in integrated fixed interval timetables by upgrading railway tracks, European journal of operational research 103 (1997) 3 pp 610-627.

[11] J. Opitz - K. Nachtigall: Automatische Erzeugung von konfliktfreien Taktfahrlagen, Eisenbahningenieur 58 (2007) 7 pp. 50-55,

[12] U. Grabs: Konflikterkennung und -lösung für dispositive Aufgaben in Betriebszentralen, Signal und Draht, 87(1995) pp. 255-256.

[13] R. Hundt: Automatische Konflikterkennung und -lösung dispositiver Aufgaben in der Betriebsleittechnik, Signal und Draht, 87(1995) pp. 437-438.

[14] P. Hille: Konfliktlösungsmodelle, Signal und Draht, 91(1999) pp. 15-18. [15] L. T. Kóczy and A. Zorat: Fuzzy systems and approximation, Fuzzy Sets

and Systems 85 (1995) pp. 203-222.