2019-2020 research findings in Public Transit from the Centre for Transport Studies, University of TWENTE. The presented findings at the Transportation Research board include overcrowding, operational control, electric buses, and train assignment.

1. Bus Operations Optimization: A Literature Review on Bus
Holding, Rescheduling and Stop-skipping
99th TRB Annual Meeting — Paper No: 20-00414
Konstantinos Gkiotsalitis, Assistant Professor, University of Twente
Centre for Transport Studies, Faculty of Engineering Technology
Research Questions of the Literature Review
• which are the side-effects of common real-time control methods on the ac-
tors of the system?
• is there potential to combine different real-time control methods?
• which are the most common models of the different real-time control meth-
ods? Do they offer an accurate representation of reality and can they be
efficiency solved (i.e., to global optimality) in real-time?
• which are the most common solution methods?
• what should be the direction of future research?
Bus Holding
Time
Control stop, s
𝐻𝑠
𝑑 𝑛−1,𝑠 𝑡
𝑛 − 1
𝑛
𝑑 𝑛,𝑠
𝑛 + 1
Bus trajectory
realized trajectory
expected trajectory
𝑛 − 1
𝑛
𝑛 + 1
Space
ሚ𝑑 𝑛+1,𝑠
Figure 1: Time-space diagram of the realized and expected trajectories of three
successive bus trips
Bus Holding Methods
Multivariable
Mathematical Programs
Single-variable
Mathematical Programs
Rule-based control logics
• No globally optimal
solutions
• Many of them can be
solved in quasi-real-
time
• Focus on the holding
time of one bus at a
time (myopic)
• Easily solved to
global optimality
• Analytic Equations
• Provide holding
advice in real-time
• Optimality is not
strictly defined
Figure 2: Bus holding approaches
Table 1: Summary of selected bus holding methods
Study Decision Variables Mathematical Program Capacity
Sanchez-Martinez et al. (2016) multivariable non-convex considered
Delgado et al. (2012) multivariable non-convex considered
Hickman (2001) single variable convex ignored
Eberlein et al. (2001) multivariable non-convex ignored
Sun et al. (2008) multivariable convex ignored
Zolfaghari et al. (2004) multivariable convex considered
Puong et al (2008) multivariable non-convex considered
Zhao et al. (2003) single variable non-convex ignored
Wu et al. (2017) single variable non-convex considered
Bartholdi et al. (2012) single variable no program ignored
Gkiotsalitis et al. (2019) multivariable non-convex ignored
Saez et al. (2012) multivariable non-convex considered
Xuan et al. (2011) single variable non-convex ignored
Hernandez et al. (2015) multivariable non-convex considered
Rescheduling Methods
Table 2: Summary of selected rescheduling works
Study Problem Mathematical Program Solution Method
Gkiotsalitis et al. (2016) rescheduling integer nonlinear genetic algorithm
Li et al. (2008) rescheduling integer linear parallel action
Li et al. (2009) rescheduling integer linear Lagrangian
relaxation
Mirchandani et al. (2010) rescheduling macroscopic model heuristic
and signal priority
Gkiotsalitis et al. (2016) rescheduling integer nonlinear evolutionary
and bus holding optimization
Cevallos et al. (2006) rescheduling with integer program genetic algorithm
passenger transfers
Coffey et al. (2012) rescheduling with linear program CPLEX
passenger transfers
Stop-skipping
Bus stop 1
Bus stop 2
serve skip
skip
Bus stop 3
serve
Bus stop |S|
Possible
Combinations:
2|S|
1
|S|
|S|-1 |S|-2
s’+1
s’
2 3
direction 1
direction 2
Table 3: Summary of selected stop-skipping works
Study Problem Considered trips Solution Method
Fu et al. (2003) stop-skipping one brute-force
Cortes et al. (2010) stop-skipping & bus holding multiple genetic algorithm
Gkiotsalitis (2019) stop-skipping multiple genetic algorithm
Liu et al. (2013) stop-skipping one genetic algorithm
Li et al. (1991) stop-skipping & short-turning one heuristic
Lin et al. (1995) stop-skipping & bus holding one −
Eberlein et al. (1998) stop-skipping multiple analytic solution
Saez et al. (2012) stop-skipping & bus holding multiple genetic algorithm
Sun et al. (2005) stop-skipping one brute-force
Jamili et al. (2015) stop-skipping multiple simulated annealing
Key Findings & Future Research directions
• there is a lack of mathematical models for stop-skipping control in rolling
horizons since most works determine the skipped stops of each bus when
it is about to be dispatched;
• integration of control methods, such as rescheduling and bus holding, has
shown promising results; however, only a few works have considered such
integrations;
• most works derive dynamic control measures with the use of single-
objective functions that combine multiple objectives with the use of weight
factors; however, there is little attention on multi-objective approaches that
derive Pareto frontiers;
• the capacity limits of buses are typically not considered in dynamic control
methods for simplifying the solution of the problem; thus, there is a lack of
models that offer a realistic representation of the system dynamics;
• there is a lack of bus holding, rescheduling and stop-skipping works that
consider stochastic travel times and passenger demand in real-time control.

2. A Model for Real-Time Bus Holding subject to Vehicle Ca-
pacity Limits
99th TRB Annual Meeting — Paper No: 20-00102
Konstantinos Gkiotsalitis, Assistant Professor, University of Twente
Eric C. van Berkum, Professor, University of Twente
Research Contribution
• An easy-to-solve mathematical program for the bus holding problem under
capacity limitations that can determine (immediately) the holding time of a
bus trip upon its arrival at a bus stop.
• The proposed model is the first of its kind and is based on the modeling of
the real-time bus holding problem as a regularity-based optimization prob-
lem under bus load variations and capacity limitations.
Assumptions
• In high-frequency services, passengers who cannot board a bus will wait for
the next trip of the same bus line because their waiting times are relatively
small.
• Passengers cannot coordinate their arrivals at stops to the arrival times
of buses at high-frequency services. Thus, we assume a demand-based
passenger arrival rate, λs, at any stop s.
• The allowed holding time of buses at stops has an upper (maximum) limit,
ζ, due to the inconvenience caused to on-board passengers.
Problem Formulation
Time
Control stop, s
𝐻𝑠
𝑑 𝑛−1,𝑠 𝑡
𝑛 − 1
𝑛
𝑑 𝑛,𝑠
𝑛 + 1
Bus trajectory
realized trajectory
expected trajectory
𝑛 − 1
𝑛
𝑛 + 1
Space
ሚ𝑑 𝑛+1,𝑠
Figure 1: Realized and expected trajectories of the preceding, n−1, and follow-
ing, n + 1, bus trip of n for which we need to decide its holding time at stop
s.
( ˜Q) min
x,ν1,ν2
f(x, ν1, ν2)
s.t. (f) (f) satisfies Eq.(2)
ν1 ≥ 0
ν1 ≥ φn + xλs − cn
ν2 ≥ 0
ν2 ≥ ˜ln+1 − ˜βn+1 − cn+1 + ˜βn+1taλs + ν1 + (˜an+1,s − t − x)λs k
0 ≤ x ≤ ζ
(1)
where
f(x, ν1, ν2) := t + x − dn−1,s − Hs
2
+ ˜an+1,s + ˜βn+1ta + ˜βn+1taλs + ν1 + (˜an+1,s − t − x)λs ktb
− ν2tb − t − x − Hs
2
+ M1ν1 + M2ν2
(2)
• our reformulated mathematical program strives to minimize the difference
between the actual and the target headways while reducing the number of
refused passenger boardings due to capacity limitations.
• Program ( ˜Q) is proven to be convex and a locally optimal solution is also a
globally optimal one.
• this formulation is the first that produces a globally optimal solution when
considering the vehicle capacity limits.
Numerical Experiments
Choa Chu Kang
Loop - Choa Chu
Kang Int (44009)
Yew Tee
Stn (45321)
Opp Blk
666 (45421)
Bus Line 302
Bus stop
Control point stop
Terminal
bus line
direction
Figure 2: Topology and bus stops of bus line 302 in Singapore
dn−1,s −
(t + x)
(t + x) −
˜dn+1,s
80
120
160
200
240
280
120
292.74
headway(s)
Headways at do-nothing
dn−1,s −
(t + x)
(t + x) −
˜dn+1,s
80
120
160
200
240
280
target headway, Hs
198.86 203.6
headway(s)
Headways after holding
trip n trip n+1
10
20
30
40
50
60
70
80
47 50
do-nothing
passengers
Bus load when departing stop s
trip n trip n+1
10
20
30
40
50
60
70
80 vehicle capacity
52
45
holding
passengers
Bus load when departing stop s
Figure 3: Headways and bus loads at stop s in the do-nothing case (left sub-
figures) and in the case where we apply the holding suggested by our model
(right sub-figures)
I II III IV V VI VII VIII
40
50
60
70
80
90
100
scenario
Busload(passengers)
trip n
our holding logic
classical holding logic of Fu and Yang (2002)
vehicle capacity
I II III IV V VI VII VIII
40
50
60
70
80
90
100
scenario
Busload(passengers)
trip n + 1
our holding logic
alg.1
vehicle capacity
Figure 4: Bus load of trip n when it departs from stop s with the implementation
of our holding logic and the classical holding logic of Fu and Yang 2002.
Key Findings
• our proposed bus holding approach can improve the squared headway de-
viation by up to 82% compared to the case of no holding.
• we reduce the number of refused boardings due to vehicle capacity limita-
tions compared to classical bus holding logics, such as Fu and Yang (2002).

3. Robust Bus Scheduling considering Transfer Synchroniza-
tions
99th TRB Annual Meeting — Paper No: 20-00085
Konstantinos Gkiotsalitis, Assistant Professor, University of Twente
Oskar A.L. Eikenbroek, PhD Candidate, University of Twente
Oded Cats, Associate Professor, Delft University of Technology
Research Contribution
Our work considers the potential variability in the travel times and dwell times
of daily trips and has the following additional features:
• is concerned with tactical planning, in particular bus timetabling (i.e., offline
optimization of the dispatching times of the daily trips);
• it has a dual objective and minimizes the regularity of individual bus lines
while ensuring the synchronization of trips at the transfer stops;
• considers operational regulatory constraints such as schedule sliding pre-
vention and layover time limits.
Assumptions
• the number of bus trips per line is decided during the frequency settings
phase;
• bus trips from the same line are not expected to overtake one another;
• the actual travel times and dwell times of bus trips can deviate from their
expected values.
Problem Formulation
min
x
max
ξ,ζ
f(x, ξ, ζ) (1)
s.t.: x ∈ F(ξ, ζ) = {satisfy layover and schedule sliding constraints}
(2)
xl,1 = δmin
l , ∀l ∈ L (3)
ξmin
l,s ≤ ξl,n,s ≤ ξmax
l,s , ∀l ∈ L, ∀n ∈ N(l), ∀s ∈ S(l) {1} (4)
ζmin
l,s ≤ ζl,n,s ≤ ζmax
l,s , ∀l ∈ L, ∀n ∈ N(l), ∀s ∈ S(l) (5)
where
• The minimax problem states that we want to find the scheduling option with
the best performance at the worst-case scenario.
• x are the bus scheduling changes at different bus lines (decision variables).
• ξ, ζ are the travel time and dwell time noises which are the adversary of our
system and appear as variables with bounded values.
• f(x, ξ, ζ) is our objective function which is convex and indicates (i) the
network-wide excessive waiting times of passengers, and (ii) the waiting
times at transfer stops.
Problem Solution with Alternating Optimization
For a given noise instance (ξk
, ζk
), the corresponding optimization problem
( ˜P(ξk
, ζk
)) min
x
f(x, ξk
, ζk
) s.t. {x | x satisfies all related constraints}
is an easy to solve convex program and has always a feasible solution.
The worst-performing scenario for such solution can be determined from the
maximization problem with parameter xk
:
(P(xk
)) : max
ξ,ζ
˜f(xk
, ξ, ζ)
s.t.: ξmin
l,s ≤ ξl,n,s ≤ ξmax
l,s , ∀l ∈ L, ∀n ∈ N(l), ∀s ∈ S(l) {1}
ζmin
l,s ≤ ζl,n,s ≤ ζmax
l,s , ∀l ∈ L, ∀n ∈ N(l), ∀s ∈ S(l)
(6)
This can be summarized in the following algorithm:
Step 0: Choose initial solution guess x1
that satisfies all constraints and set
k := 1;
Step 1: Solve P(xk
) and obtain (ξk
, ζk
);
Step 2: Solve ˜P(ξk
, ζk
) for (ξk
, ζk
) and obtain xk+1
, set k := k + 1;
Step 3: If the performance of the most recent solutions is stabilized, STOP.
Else, go to Step 1.
,WHUDWLRQV
9DOXHRIWKHSHQDOL]HGREMHFWLYHIXQFWLRQVHF2

4. H
%RXQGHGZRUVWFDVHSHUIRUPDQFHRIPRVWUREXVWVROXWLRQ
6ROXWLRQSHUIRUPDQFHRQZRUVWFDVHQRLVH
3HUIRUPDQFHRIQHZVROXWLRQVXEMHFWWRWKHSUHYLRXVZRUVWFDVHQRLVH
Figure 1: Convergence of the alternating optimization. The robust solution re-
duces the worst-case objective function value to 0.701E+10
Case Study
Gullmarsplan
Essingetorget
Frihamnen
Radiohuset
common
bus stops
--- bus line 1
--- bus line 4
Figure 2: Bus lines 1 and 4 in Stockholm
median upper whisker
−5
0
5
10
15
5.17 5.52
−1.72
7.13
Improvement(%)
(a) Average Excessive
Waiting Time per passenger
Design (ii) Design (iii)
median upper whisker
0
20
40
28.18
3.4
−0.92
11.31
Improvement(%)
(b) Average Waiting Time
for Transferring
Design (ii) Design (iii)
Figure 3: Validation results: investigating the potential improvement of robust
designs to mild (ii) and extreme (iii) disruptions compared to the designed
schedule for the average case.
Key Findings
• it is clear that there is a trade-off between: (a) robust designs that impose
stricter limits to the adversary and result in solutions that perform better at
common-case scenarios, and (b) robust designs that prepare for a wide-range
of values of the adversary and overperform at extreme-case scenarios.
• designed schedules for the average case can be improved by 5.17% in terms
of regularity and 28.2% in terms of transfer times.

5. Periodic Stop-skipping: NP-Hardness and Computational
Limitations
99th TRB Annual Meeting — Paper No: 20-00069
Konstantinos Gkiotsalitis, Assistant Professor, University of Twente
Centre for Transport Studies, Faculty of Engineering Technology
Research Contribution
• the modeling, for the first time, of the dynamic stop-skipping problem as a
rolling horizon optimization problem by expanding the classical formulation
of Fu et al. (2003)1
;
• the mathematical analysis of the resulting integer nonlinear program and
the proof of its NP-hardness;
• the in-depth investigation of the scalability of the periodic stop-skipping
problem with respect to the number of trips in the rolling horizon and the
stop-skipping candidate stops.
Assumptions
• Buses that serve the same line do not overtake each other. This is a com-
mon assumption in bus operations;
• The passenger arrivals at stops are random because passengers cannot
coordinate their arrivals with the arrival times of buses at regularity-based
services;
• The passenger demand at skipped stops is accommodated by the next bus
trip of the same line;
• Passengers use different door channels for boardings and alightings.
Problem Objective
f(x) := c1
|N|
n=2
|S|
s=1
(un,s − mn−1,s)
hn,s
2
+ mn−1,s
hn−1,s
2
+ hn,s
+ c2
|N|
n=2
|S|−1
s=1
|S|
y=s+1
bn,sy
y
z=s+1
(tn,z + (kn,z + δ)xn,z)
+ c3
|N|
n=2
|S|
s=2
(tn,s + (kn,s + δ)xn,s)
(1)
where the generalized cost of the objective function includes three terms.
• The first term includes two components: (un,s − mn−1,s)
hn,s
2 computes
the total waiting time of the passengers who arrive after the departure (or
passing) of bus n − 1 at stop s, assuming random arrivals with an average
passenger waiting time equal to half the headway. The second compo-
nent represents the total waiting time of those passengers who have been
stranded by bus n − 1 (mn−1,s) and have to wait for an average amount of
time equal to mn−1,s
hn−1,s
2 + hn,s .
• The second term of the objective function calculates the total in-vehicle time
of passengers summed over all O-D pairs
• and the final term computes the total bus trip time.
Complexity and Solution Method
• The rolling horizon stop-skipping problem is an NP-Hard decision problem.
• Solution method: brute-force with an exponential computational complex-
ity that requires to explore 2|N|×|S|
potential solutions to obtain a globally
optimal one, where |N| is the number of bus line trips and |S| the number
of bus line stops.
• Note: Other exact optimization approaches for combinatorial optimization
include branch and bound (BB); however, in our case, BB is reduced to
an exhaustive search because our 0-1 problem does not have a continuous
relaxation.
1
Fu, L., Q. Liu, and P. Calamai, Real-time optimization model for dynamic scheduling of transit
operations. Transportation Research Record, No. 1857, 2003, pp. 4855.
Numerical Experiments
Choa Chu Kang
Loop - Choa Chu
Kang Int (44009)
Yew Tee
Stn (45321)
Opp Blk
666 (45421)
Bus Line 302
Bus stops
Terminal
bus line
direction
Figure 1: Topology and bus stops of bus line 302 in Singapore
Table 1: Computational costs in CPU minutes for different numbers of trips in
the rolling horizon subject to the number of stop-skipping candidate stops.
Stops Trips in the rolling horizon
1 2 3 4 5 6 7 8
3 0.00 0.00 2E-04 3E-03 0.02 0.26 2.05 25.61
4 0.00 0.00 0.01 0.37 18.23 121.71 10.00 10.00
5 0.00 0.01 0.42 82.36 10.00 10.00 10.00 10.00
6 0.00 0.13 315.66 10.00 10.00 10.00 10.00 10.00
7 0.00 1.74 10.00 10.00 10.00 10.00 10.00 10.00
8 0.00 23.72 10.00 10.00 10.00 10.00 10.00 10.00
9 0.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00
10 0.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00
11 2E-04 10.00 10.00 10.00 10.00 10.00 10.00 10.00
12 7E-04 10.00 10.00 10.00 10.00 10.00 10.00 10.00
13 2E-03 10.00 10.00 10.00 10.00 10.00 10.00 10.00
14 3E-03 10.00 10.00 10.00 10.00 10.00 10.00 10.00
15 7E-03 10.00 10.00 10.00 10.00 10.00 10.00 10.00
16 0.01 10.00 10.00 10.00 10.00 10.00 10.00 10.00
17 0.03 10.00 10.00 10.00 10.00 10.00 10.00 10.00
18 0.05 10.00 10.00 10.00 10.00 10.00 10.00 10.00
19 0.11 10.00 10.00 10.00 10.00 10.00 10.00 10.00
20 0.24 10.00 10.00 10.00 10.00 10.00 10.00 10.00
21 0.51 10.00 10.00 10.00 10.00 10.00 10.00 10.00
22 1.06 10.00 10.00 10.00 10.00 10.00 10.00 10.00
4 6 8 10 12 14 16 18 20 22
1
2
3
4
5
6
7
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
bus stops
allowedtripsinthe
rollinghorizon
Figure 2: Maximum number of trips that can be in a rolling horizon subject to
the number of stop-skipping candidates.
Key Findings
• To obtain a globally optimal solution within a reasonable time (i.e., less than
2 minutes) we need to consider only one trip in the rolling horizon.
• For two trips or more, there should be a compromise with respect to the
number of stop-skipping candidates.
• Even if we use heuristics, the scalability issues persist (i.e., if we have 2 trips
in the rolling horizon we can consider up to 12 stop-skipping candidates,
instead of up to 7).

6. A Dynamic Model for Real-Time Track Assignment at Railway
Yards
99th TRB Annual Meeting — Paper No: 20-00661
Bram B.W. Schasfoort, MSc, University of Twente
Konstantinos Gkiotsalitis, Assistant Professor, University of Twente
Oskar A.L. Eikenbroek, PhD Candidate, University of Twente
Eric C. van Berkum, Professor, University of Twente
Research Contribution
• model the real-time track assignment problem (RT-TAP)
• investigate real-time mathematical optimization techniques that aim to min-
imize the total delays of outbound trains at a railway yard
Assumptions
• there is unlimited storage at a temporary yard that can temporarily accom-
modate trains which cannot be assigned at the main yard,
• trains do not need to be shunted,
• there exists a feasible path to and from the yard,
• each train is assigned to one and only one track and cannot be pre-empted,
• each track is assigned at most one train, which does not exceed the track
length.
Problem Formulation
RT-TAP is formulated as follows (the nomenclature can be found in our
manuscript):
min
i∈N
wi( ¯di − di) (1)
subject to:
k∈K j∈V
xk
ij = 1, i ∈ N (2)
j∈V
xk
0j = 1, k ∈ K (3)
i∈V
xk
i,n+1 = 1, k ∈ K (4)
i∈V
xk
ij −
i∈V
xk
ji = 0, k ∈ K, j ∈ N (5)
yk
i ≥ ai, k ∈ K, i ∈ V (6)
zk
i = yk
i + pi, k ∈ K, i ∈ N (7)
zk
j ≥ zk
i + H − M(1 − xk
ij), k ∈ K, (i, j) ∈ E (8)
¯di ≥ zk
i , k ∈ K, i ∈ N (9)
¯di ≥ di, i ∈ N (10)
li
j∈V
xk
ij ≤ Lk, k ∈ K, i ∈ N (11)
RT-TAP is an integer program (IP) with the objective to minimize the sum of the
total weighted delay wi( ¯di − di) (non-negative by (10) so that - implicitly - ¯di is
minimized, i.e., di = maxk∈K zk
i ).
Our formulation is a multi-commodity flow problem. Constraint (2) assures
that each train i is only assigned once to a track. Constraints (3) and (4) assure
that a schedule starts at (dummy) node 0 and ends at dummy train n + 1. (5)
is so that for every train a (dummy) train is scheduled before and after it (on
the same track). (6) ensures that we can only schedule trains after arrival. (7)
is the latent departure time of train i on track k. (8) says that trains can only
be scheduled after the previous train has departed, and requires a minimum
headway H ≥ 0.
Complexity and Solution Method
We prove that (the static) TAP is NP-Hard by a polynomial-time reduction from
the (1|ri| i Ci)-scheduling problem, which, on its turn, has a polynomial-time
reduction from 3-PARTITION. Hence, we investigate different solution methods:
• solving RT-TAP with CPLEX;
• solving RT-TAP with a problem-specific Genetic Algorithm;
• solving RT-TAP with a First-scheduled, First-served (FSFS) heuristic.
Numerical Experiments
Master Thesis – Bram Schasfoort – Concept 22.07.2019 31
Numerical Experiments.
The following chapter aims to describe the application of the model on a numerical case study. The
proposed solution method for efficient allocation of trains to tracks while reducing the delay is
applied on the railway yard Waalhaven-zuid (Whz) which is located in the western part of the
Netherlands. The method presented in previous chapter is written in the programming language
C++. Reason for choosing C++ is the computational speed it has. Computational speed is important
as the model needs to be run at real-time level.
This chapter is structured as followed. In the first section a general case study description is given.
This section is followed up by a discussion of which data was received, used, and which
assumptions are made in the model input. In section 3, the established input is shown. In the 4th
section, the model performance is measured at real-time level followed by a stochastic optimization
of the scheduling problem. The final section describes the results measured in section 5 and 6.
Case study description.
The railway yard consists of about 100 tracks with each a function. The function for the tracks
varies between processing, renting, passing through, shunting, repair and/or parking. At Whz, only
the process tracks consist of electrical wiring and are Central Operated (in Dutch; Centraal Bediend
Gebied (CBG)) and therefore can be controlled by the TD. Because main capacity problems related
to the RT-TAP only consisted at the process tracks, only these tracks are considered for the case
study. A schematic overview of the process tracks can be found in figure 8.1 (a more extensive
overview can be found in appendix 5, 6 and 7).
The characteristics of the process tracks at Whz can be divided in two different categories. (1) the
drive through tracks, and (2) process tracks. The drive though tracks need to remain clear all time
as they are purposed for entering and exiting the process tracks, and therefore the yard. The main
purpose of trains arriving at Whz has to do with the container terminal (Rail Service Centre; RSC).
In this terminal trains are getting unloaded from their cargo and reloaded again. In the sequence,
trains first arrive at Whz from either one of the two sides (dependent on the train’s origin) and is
then assigned to one of the 12 process tracks. After the necessary operations, the train gets assigned
to another track at RSC and the unloading and loading operations start there. When the necessary
operations at RSC is done, the train continues again to one of the 12 tracks at Whz. After again a
sequence of operations, the train exits the railway yard from one of the two exit tracks (dependent
L1 = 610 m
 connection to main track
(Two directional)
connection to RSC 
(Two directional)Process tracks
Drive through
L2 = 630 m
L3 = 585 m
L4 = 710 m
L5 = 800 m
L6 = 760 m
L7 = 675 m
L8 = 678 m
L9 = 722 m
L10 = 516 m
L12 = 455 m
L11 = 410 m
Figure 1: Schematic overview of Waalhaven Zuid process tracks
The tracks at Waalhaven Zuid are divided into two different categories:
• the drive-through (solid lines);
• the process tracks (dashed lines).
Table 1: Performance of CPLEX, Genetic Algorithm (GA), and First-scheduled
First-served (FSFS) in different scenarios
4 tracks Total delay of Running time 3 tracks Total delay of Running time
trains (h:mm:ss) trains (h:mm:ss)
CPLEX 1:48:36 121 sec CPLEX 4:41:48 255 sec
FSFS 3:29:10 1 sec FSFS 6:24:45 1 sec
GA 1:48:36 1 sec GA 4:41:48 1 sec
0 2 4 6 8 10 12 14 16 18 20 22 24
Time of the day (in hours)
0
30
60
90
120
150
180
210
240
270
Totaldelay(min)
FSFS
GA
Figure 2: Daily performance
Key Findings
• GA is capable of converging to the global optimum, whereas the FSFS
exhibits a substantial optimality gap.
• the reassignments produced by the GA algorithm result in reduced delays
by 4 minutes and 42 seconds (on average) compared to the assignments
of the FSFS method.
• the employment of heuristics can reduce significantly the computational
costs and enable the reassignment of trains to tracks in near real-time