MODELLING OF ELECTRICAL TRAIN (ET) NETWORK SYSTEM USING MAX-PLUS ALGEBRA

Transcription

1 Proceedings of The International Conference on Numerical nalysis and Otimization ICeMTH 0. Modelling of Electrical Train ET) Networ System Using Ma-Plus lgebra MODELLING OF ELECTRICL TRIN ET) NETWORK SYSTEM USING MX-PLUS LGEBR Siti lfiah Subiono bstract. Transortation is imortant thing for society to do their daily mobilities. In the city society which has high mobility characteristics lie Jaarta there are many comle roblems in transortation. One of the roblem is traffic jam. In order to reduce the traffic jam in Jaarta the electrical train ET) K Commuter Jabodetabe is built. s a chea and a fast transortation facility ET is widely used by society. One of imortant issues related to ET networ system is often occurrence of inaccuracies arriving and dearture times of trains at each station so it is comlained by society. This research is urosed to modelling ET networ system using ma-lus algebra which the duration of driving time between station is given as an interval of times with elements in ma-lus algebra. This ET networ system can be formed into adjacency matri with element in the form of interval in ma-lus algebra. Keywords and Phrases : Ma-Plus lgebra Model of ET Networ System.. INTRODUCTION Discussion about how reairing the ublic transortation system must be imroved considerably. Related to this roblem the theory of ma-lus algebra is one of theory that can used to modelling analysis and control of transortation networ system. In had used ma-lus algebra to scheduling bus line networ in the city with case-study of TransJaarta bus networ. The model is constructed based on the number of lines that have been active oerated the maimum numbers of busses that allocated and the rule of synchronization for each line. From this research the design of eriodic schedule of dearture time is obtained. In had discussed about ma-lus algebra aroach to transortation system esecially railway system. scheduled railway system can efficiently be modelled as a discrete event dynamic system DEDS) using ma-lus algebra. There are lines of intercity Dutch railway system is modelled here. In had discussed about modelling whole connection of trains in Dutch railway system. The constructed model more comle than. In had discussed about modelling scheduled railway system and analysis stability and realizability timetable using ma-lus algebra. In this aer ma-lus algebra is used to modelling KRL networ system with timetable. The driving times and the timetable are given by interval form in R ma.. MX-PLUS LGEBR The ma-lus algebra is defined by the oerations addition and maimization alied to the real numbers etended with minus infinity. M -

2 Siti lfiah Subiono Definition The ma-lus algebra R ma ) is defined as follows : def a) R ma R {- } where R is the set of real numbers; b) is maimization in the usual ordering of R ma ; c) is the usual addition where a - def - a def - for all a R ma. Ma-lus algebra having 0 as neutral element with resect to will be denoted by e and - as neutral element with resect to and absorbing element for will be denoted by.. INTERVL MX-PLUS LGEBR Interval ma-lus algebra is etention of ma-lus algebra. Definition Interval ma-lus algebra is defined as IR) ma { R m m } { }. In IR) ma oerations and are defined as: y y y and y y y y IR) ma Eamle - and - 0. Definition Let IR) n m ma be the set of n m matrices in interval ma-lus algebra. The interval where matri is set of matrices that have interval value and be written as ) R nm ma and. Eamle Let 0 so 0 and Oeration and in IR) ma ma-lus algebra. can be etended for oeration-oeration matri in interval. MODELLING OF RILWY SYSTEM. Desired Model train networ can be modelled as a system of the form : ) y ) C ) 0) 0 The vector ) contains the th dearture times of all trains including auiliary ones. Vector M -

3 Siti lfiah Subiono ) ' )... n ) n )... n r ) with referring to the real trains and related to auiliary variables to be interreted as dummy trains. The vector is the timetable for the st deartures. The initial state is 0 although the 0 th dearture of a train does not have a clear interretation. Since we only can loo at what haens with the outut matri is. So the outut is the observation of the dearture times of the actual trains.. Timetable The vector contains the scheduled th dearture times for all trains. Because trains are scheduled modulo we obtain 0 holds for all. This can also be written as d ) d0). T) η where η ee e the column vector that all the element are e in ma-lus algebra.. MODELLING OF ELECTRICL TRIN ET) NETWORK SYSTEM. K Commuter Jabodetabe Networ System The railroad K Commuter Jabodetabe networ system is taen from. In this aer will be modelled the railroad from Jaarta Kota to Bogor Jaarta Kota to Beasi and Jaarta Kota to Tangerang with latform are Jaarta Kota Gondangdia Bogor Beasi and Tangerang. The data about a fied number of trains on each line and the driving times of the trains from one station to the net are taen from with resect to a normal woring day from 0:00 m until 0:00 m. These data are used to derive the driving times the difference between the arriving and dearture time of the trains from one station to the net). Here assumed that the cyclicity of the timetable is 0 minutes. Distribution number of trains that oerating on each line is determined using reference time 0: m. The directed grah of K Commuter Jabodetabe as follow: JK 9 GD T BE BO Grah K Commuter Jabodetabe Networ M -

4 Siti lfiah Subiono JK BO : Line JK : Jaarta Kota BO : Bogor JK BE : Line GD : Gondangdia BE : Beasi JK T : Line T : Tangerang. Physical Secification The hysical secification is defined as follow: i) The lines in K Commuter Jabodetabe networ ii) The number and distribution of trains on each line iii) The synchronization rules between the trains There are three lines that modelled here. The driving times of trains and the timetable of dearture times are given below: Line Dearture Station Sto Station Timetable Driving time minutes) The number of trains Jaarta Kota Gondangdia 9 Gondangdia Bogor Bogor Gondangdia 0 0 Gondangdia Jaarta Kota 9 Jaarta Kota Beasi Beasi Jaarta Kota 0 9 Jaarta Kota Tangerang 0 Tangerang Jaarta Kota The synchronization rules are given as follows: On line the trains that deart for the -st time from Jaarta Kota to Gondangdia should wait for the -th arrival of trains which dearted from Beasi to Jaarta Kota and wait for the -th arrival of trains which dearted from Tangerang to Jaarta Kota. On line the trains that deart for the -st time from Jaarta Kota to Beasi should wait for the --th arrival of trains which dearted from Gondangdia to Jaarta Kota and wait for the -th arrival of trains which dearted from Tangerang to Jaarta Kota. On line the trains that deart for the -st time from Jaarta Kota to Tangerang should wait for the --th arrival of trains which dearted from Gondangdia to Jaarta Kota and wait for the -th arrival of trains which dearted from Beasi to Jaarta Kota.. Model of K Commuter Jabodetabe Networ System Before desire model needed definition variable that will be used in system. Vector ) contains dearture times for the -th of trains in each station as follows: Line Variable Dearture Station Sto Station Jaarta Kota Gondangdia Gondangdia Bogor Bogor Gondangdia Gondangdia Jaarta Kota M -

5 Siti lfiah Subiono Jaarta Kota Beasi Beasi Jaarta Kota Jaarta Kota Tangerang Tangerang Jaarta Kota The model of networ system before synchronization as follows: i) Line 9 ) ) ) ii) Line iii) Line ) 9 ) 0 99 ) ) Based on synchronization rules in. can be constructed model of whole railway system as follows: i) Line 9 99 )) ) 9 ) ) ) )) ) ) ) ii) Line iii) Line 9 99 )) 9 )) )) )) 9 ) ) ) ) ) Then model ) ) and ) above can be stated in standard form ma-lus algebra model as follows: M ) ) M -

6 Siti lfiah Subiono M - There are matrices with { } and size of each matri. Matrices and as follows:

7 Siti lfiah Subiono M - Therefore model ) ) and ) can be written as: ) ) ) ) d. CONCLUSIONS Ma-lus algebra can be used to constructed model of electrical train ET) networ system. The model can be written as ) ) ) d M REFERENCES BRKER J.G lgorithms and lications Timed Discrete Even Systems Ph.D Thesis Deartment of Technical Mathematics an Informatics Delft University of Technology 99. GOVERDE R.M.P Railway Timetable Stability nalysis Using Ma-Plus System Theory Transortation Research Part Bvol. hal RUDHITO. WHYUNI S. SUPRWNTO. and SUSILO F ljabar Ma-Plus Interval Prosiding Seminar Nasional Mahasiswa S Matematia - UGM 00. SUBIONO On Classes of Min-Ma-Plus System and Their lications Thesis Ph.D. Technische Universiteit Delft Delft000. SUBIONO Ma-lus lgebra Toolbo ver..0 htt:// 00. WINRNI Penjadwalan Jalur Bus dalam Kota dengan ljabar Ma-Plus Tesis Magister ITS Surabaya 009. Jadwal K Commuter tanggal ases: Januari 0 Info-Jadwal-K-Commuter htt:// Rute Perjalanan K Commuter tanggal ases: Januari 0 htt:// SITI LFIH: ITS Surabaya. s: alfiah09@mhs.matematia.its.ac.id sitialfiah@yahoo.com SUBIONO: ITS Surabaya. s: Subiono00@matematia.its.ac.id