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Train dispatching model with stochastic capacity breakdowns on an N- tracked railroad network. Xuesong Zhou University of Utah Utah, U.S.A. Email: [email protected]. Lingyun Meng Beijing Jiaotong University Beijing, China Email: [email protected]. October 15th 2012 - PowerPoint PPT Presentation
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Lingyun MengBeijing Jiaotong University
Beijing, ChinaEmail: [email protected]
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: [email protected]
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?