Optimal Design of Timetables to maximize schedule reliability

and minimize energy consumption,

rolling stock and crew deployment

DDDr B.Nag,Director,RDSO

Timetable Design & its impact

Present Method of Timetable Design

done manually by feasible path construction and mapping train services

No method available for construction of optimal timetables with respect to energy consumption, schedule reliability or crew & rolling stock deployment

TIME

STATIONS

Problems of Timetable Design

Large number of trains Large number of sections Many constraints No metrics developed for schedule

reliability, energy consumption, rolling stock & crew requirements

Mathematical model not solvable because of size

Resources for Train Services Energy Sectional capacity Rolling Stock Crew

Minimum Headway Time

1k

1a1g,kg,a1g,a

mk11g pppMax

1 m

gt 1gt

kg,kp

Slack Time

1g1gg1g dd

route

1 2 xj

gt

1gt delayed on route 1

by

1gt

gt will be delayed on route 1

by

0,Max 1,1g

gt will be delayed on route 2

by

0,Max 2,1g1,1g

n

1g

x

1j j,g

x

jii,g

Index(TSI)Slack Total

Resources for Train Services Energy Sectional capacity Rolling Stock Crew

Power Requirement

0

2000

4000

6000

8000

10000

12000

0 50 100 150 200

Speed kmph

Po

we

r k

W

Level 1:500 Grade

1:300 Grade 1:100 Grade

1 m

gtg,kp

k route

Energy Consumption depends on train speed, section terrain (ruling gradient & curvature), rolling stock resistance & locomotive efficiency, and regeneration.

(TCE)Energy ofn Consumptio Totalx

1k

n

1gk,gdk,gu

Resources for Train Services Energy Sectional capacity Rolling Stock Crew

Rolling Stock Maintenance

Primary Maintenance Station A

Secondary Maintenance

Station B

Rolling Stock DeploymentThe number of rakes required for a train service depends on

the following factors: the cycle time maintenance requirements, mp hours for primary

maintenance and ms hours for secondary maintenance

running time of a trip, say ug hours each way

scheduled lie-over of say lp,g hours at the primary maintenance station and ls,g hours at the secondary maintenance station;

n

1g g

sg,spg,p

u

mlml

(RSDI)Index DeploymentStock Rolling

Resources for Train Services Energy Sectional capacity Rolling Stock Crew

Labour Regulations Maximum duty of 8 –10 hours Out-station Rest for minimum of 8

hours Base-station Rest for minimum of

16 hours

Crew Requirements0900 Hrs

1630 Hrs

0100 Hrs

0830 Hrs

Train 1

Train 11

Crew Deployment Index (CDI)

x

1j

n

1g j,g

j,gj,gj,gj,g l4201020l

nx

1

j routeon train tof timerunning is

otherwise 0 & 420l if 1

otherwise 0 & 1020l if 1

j routeat train tofover -lie theis l

g jg,

jg,jg,

jg,jg,

gj,g

Objective FunctionsMinimize Total Consumption of Energy (TCE)

Maximize Total Slack Index (TSI)

Minimize Rolling Stock Deployment Index (RSDI)

Minimize Crew Deployment Index (CDI)

Mathematical Model- Constraints

Minimum section traversal time

Cycle TimeBounds on departure times, slack times, total running time

Solution Methodology Analytic Hierarchy Process (AHP)

to determine relative weights of each objective

Compute distance functions for distance from PIS, NIS & Closeness Rating(CR)

Global Criteria Approach (GCA) to Minimize dPIS or

Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to Maximize CR

Relative Importance of Objective Functions for the Decision

Maker Schedule Reliability

Customer Perception Line Capacity creation

Energy Consumption Rolling Stock

cost of rake lead time for new rake procurement Creation of facilities for maintenance

Crew Cost of crew recruitment, training and

operation lead time for recruitment & training

Analytic Hierarchy Process(AHP))

Energy Consumed

Schedule Reliability

Rolling Stock

Deployment

Crew Deployment

Raw weight

RW

Normalized weight

NW

Energy Consumed 1 3 5 6 3.08 0.57

Schedule Reliability 1/3 1 4 2 1.28 0.23

Rolling Stock

Deployment1/5 1/4 1 1/4 0.33 0.06

Crew Deployment 1/6 1/2 4 1 0.76 0.14Column Total 1.7 4.75 14 9.25 5.45 1.00

082.09.0

3422.4

RCN

1q

qNWC

RCN

1q

q

RCN

CI

(COR) Ratioy Consistencq

1i iimax

Solution Methodology Analytic Hierarchy Process (AHP)

to determine relative weights of each objective

Compute distance functions for distance from PIS, NIS & Closeness Rating(CR)

Global Criteria Approach (GCA) to Minimize dPIS or

Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to Maximize CR

Distance Functions

minmax

min24

minmax

min23

minmax

max22

minmax

min21

PIS

CDICDI

CDICDIw

RSDIRSDI

RSDIRSDIw

TSITSI

TSITSIw

TCETCE

TCETCEwd

minmax

max24

minmax

max23

minmax

min22

minmax

max21

NIS

CDICDI

CDICDIw

RSDIRSDI

RSDIRSDIw

TSITSI

TSITSIw

TCETCE

TCETCEwd

Global Criteria Approach (GCA)

PISd MinimizeTechnique for Order Preference by Similarity to Ideal Solution(TOPSIS)

NISPIS

NIS

dd

dCR

(CR) Rating Closeness Maximize

Test Problem

GAMS Program of 9083 equations & 9123 variables required 6 sec compilation and execution time on a 600 Mhz computer

Test Case Results

Cost Benefits Will vary on the section, train density,

rolling stock characteristics, operating & maintenance practices, crew management

Results for the test case indicate 10% savings in energy, 2% less requirement of crew and 12% less requirement of rolling stock when comparing the best & worst scenarios

Summary Identification of decision variables for

time table design- slack times and minimum headways

Developing the concept of route to reduce problem size

Developing metrics for robustness & rolling stock deployment

Exploring the aspects of speed differentials and throughput

Formulation and Demonstration of mathematical model to optimally design timetables

Applications Formulation of timetables which are robust to

disturbances Formulation of timetables which maximize

resource utilization and minimize energy consumption

Identification of bottleneck sections Compare timetables for robustness Analysis of implications of speed changes: CBA Analysis of resource augmentations : CBA Identify optimal timetables for maintenance

blocks Can be integrated with Driving Advice System for

pacing of trains

Strategic opportunities offered by

optimal timetable design Take advantage of new technologies such

as regenerative braking Use optimal pacing of trains Reduce capital investment on new lines

and rolling stock Methodologies can be applied for on-line

scheduling and integration with on-board driving advice systems

Thank You