Lingyun MengBeijing Jiaotong University

Beijing, ChinaEmail: lymeng@bjtu.edu.cn

Train dispatching model with stochastic capacity breakdowns on an N-tracked railroad network

October 15th 2012 INFORMS, Phoenix, U.S.A.

Xuesong ZhouUniversity of Utah

Utah, U.S.A.Email: zhou@eng.utah.edu

Outline

IntroductionMathematical formulationsSolution algorithmsExperimental results

Task of Train Dispatching

Goal: Recover impacted train schedules from . Measures: Re-timing Re-ordering Re-routing Re-servicing

Dispatching in a Dynamic & Stochastic Environment

Dispatching schedules are updated when new information are available.

Uncertain disturbance information: e.g. stochasticincident duration.

Stochasticity

Rolling horizon

Dynamicity

Problem Description

1.It refers to a blockage of one track. It’s a strong perturbation.

2. It has a relatively longer duration compared to minor disturbances.

Characteristics of disruptions in this study

Re-routing and Re-Servicing become strongly necessary, because Re-timing and Re-ordering are too week to deal with disruptions.

Block the track

Capacity loss

1. When can the capacity be fully restored ?2. How to reschedule trains so that the system-wide

performance can be optimized ?

In a word, the key question is how to generate a train dispatching plan?

State of the art A wide range of studies are devoted to optimization model

formulation and algorithm development, e.g. Kraft (1983) Jovanovic (1989), Carey (1994) and D’Ariano (2008) .

The majority of previous optimization models for train dispatching primarily assume certain and perfect information of disruptions, e.g. Adenso-Diaz et al. (1999) and Chikara et al. (2009).

Meng and Zhou (2011) has proposed an approach for robust train dispatching on a SINGLE-TRACK line.

This study tries to extend the model to the N-TRACKED network context.

Lingyun Meng, Xuesong Zhou, 2011. Robust single-track train dispatching model under a dynamic and stochastic environment: a scenario-based rolling horizon solution approach. Transportation Research Part B, 45(7): 1080-1102.

Solution approach

General ideas and contributions to literature

1. Use cumulative flow count-based variables to represent train arrival/departure times at stations/blocks

3. Use capacity aggregation mechanism to capture the stochasticity of capacity

2. Use lagrangian relaxation method to simultaneously re-route and re-schedule trains

Mathematical model

1. Objective function

Minimize the expected train exit(completion) time for all trains

2. Constraints

Capacity (breakdown)/headway times constraintsDeparture time constraintsSegment running time constraintsDwell time constraints

Notation

1 General subscripts

Flexible path-based

2 Input variables

Cell capacity

3 Decision variables

By and At

Cell e2

Station A

Station B

train a

t = 0 5 10 15 20 25

1

000000 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10

000000 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Out

boun

d

Time

Space

Cumulative flow count-based decision variables (CFCD)

( , , )pfa i t k ( , , )p

fd i t k

( , , )pfa i t k ( , , )p

fd i t k- = 0 0 1 1 1 0 0 0 0 0 0t: 5 6 7 8 9 10 11 12 13 14 15

Cell occupancy time:

3 time units are used

Cell e4

Cell e3

Cell e2

Cell e1Station A

Station B

train a train b

Out

boun

d

Time

Space

Link l

t = 0 5 10 15 20 25

1 (1)ag

1 (4)ah

Headway time constraints represented by CFCD

=( , , ) ( , ( ), ), , , , ,p p pf f fa i t k a i t g i k f p i t k

=( , , ) ( , ( ), ), , , , ,p p pf f fd i t k d i t h i k f p i t k

Occupancy time shift constraints:

arrival timeOccupancy starting time

Capacity issue for multiple trains by CFCD

, : ( ) ,

[ ( , , ) ( , , )] ( , , ), , ,p pf f

p pf f

f p i e i N

a i t k d i t k cap e t k e t k

Whether cell e is

occupied by train f along path p at time t under scenario k

Avoid if-then / big “M” constraints

Capacity(resource)-oriented train scheduling model, compared to conflict-oriented model

Cell e4

Cell e3

Cell e2

Cell e1Station A

Station B

train a train b

Outb

ound

Time

Space

Link l

t = 0 5 10 15 20 25

Cell decomposition of one directional link l corresponding to a double track

Station A

Station B

Out

boun

d

Time

Space

train a

train b

train c

Cell e

t = 0 5 10 15 20 25

Link l

Cell decomposition of bi-directional link l corresponding to a single track

N-track issue

Please also see Steven Harrod (2010) on block-based scheduling by Hypergraph

Network issue

(1, , ) 1, , ( ),f

pf

p P

a t k f t est f k

Ensure that one train only selects one path from the corresponding set of possible paths at its starting cell

1

3

2

4

1

2

3

4

(1)

(3)

(2)(3)

(1)

0 1 2 3 4 5 6 Time Axis

Spa

ce

Segment traveling arc Ground holding arcat origin station

Physical railroad network Space-time extended network for train rerouting and scheduling

railroad linkWaypoint station Dummy arc at destination station

earliest starting time

Capacity aggregation technique to deal with Uncertainty of Disruptions. See Luh (1999) for Job shop scheduling under uncertainty.

, : ( ) ,

[ ( , , ) ( , , )] [ ( , , )], ,p pf f

p pf f k

k f p i e i N

a i t k d i t k E cap e t k e t

Cell capacity constraints is satisfied in an expected manner, rather than for each scenario.

Note that we will further deduce the solutions to feasible solutions under each scenario

Stochastic capacity issue

, : ( ) ,

[ ( , , ) ( , , )] ( , , ), , ,p pf f

p pf f

f p i e i N

a i t k d i t k cap e t k e t k

Lagrangian relaxation based solution algorithm

, : ( ) ,

[ ( , , ) ( , , )] [ ( , , )], ,p pf f

p pf f k

k f p i e i N

a i t k d i t k E cap e t k e t

Cell capacity constraints (side constraints) are relaxed.

Scenario case

Aggregated case

Lagrangian relaxation based solution algorithm

Algorithm 1 Subgradient algorithm to update lagrangian multipliers

Algorithm 2 Time-dependent shortest path algorithm to find optimal solution (Lower Bound) for the relaxed problem

Algorithm 3 Priority rule-based algorithm to deduce solutions into problem feasible solutions under given scenario

Simultaneously and flexibly rerouting and rescheduling trains on an N-tracked network

Numerical experiments results

D

CH

I

M

L

N

O

G

F

J

K

A

B

Origin station

Intermediatestation

E

MOW

7 major stations144 trains belonging to 4 railway companiesown about 40%, 30%, 20% and 10% of total trains

# of variables 165,601# of equations 85,971# of non-zero elements

859,393

Solution time (seconds)

122.507 by Integer programming 2.48 by LP relaxation

Prelimiary results by LR algorithm

For RAS data set 1 within a network of 13 nodes ,14 links and 3 trains.

Optimality: 83.7%Within less than 1 second

Recall the modification of objective function to total completion time

Performance of the lagrangian relaxation algorithm

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 960

50

100

150

200

250

300

350

400

Lower boundUpper bound

Number of lagrangian iteration

System cost

Ongoing work

(3) More experiments with larger number of trains, larger network size

(2) Lagrangian relaxation decomposition technique

(1) Algorithm fine tuning

Thanks for your attention!Any questions?