Dugald C. Jackson ProfessorDepartment of Electrical Engineering and Computer Science
Co-Director, Operation Research Center
2
Probabilistic Models and Optimization Algorithms for Large-scale
Transportation Problems
by
Jing Lu
Submitted to the Sloan School of Managementon January 17, 2020, in partial fulfillment of the
requirements for the degree ofDoctor of Philosophy in Operation Research
AbstractThis thesis tackles two major challenges of urban transportation optimization problems: (i)high-dimensionality and (ii) uncertainty in both demand and supply. These challenges areaddressed from both modeling and algorithm design perspectives.
The first part of this thesis focuses on the formulation of analytical transient stochasticlink transmission models (LTM) that are computationally tractable and suitable for large-scale network analysis and optimization. We first formulate a stochastic LTM based on themodel of Osorio and Flötteröd (2015). We propose a formulation with enhanced scalabil-ity. In particular, the dimension of the state space is linear, rather than cubic, in the link’sspace capacity. We then propose a second formulation that has a state space of dimensiontwo; it scales independently of the link’s space capacity. Both link models are validatedversus benchmark models, both analytical and simulation-based. The proposed models areused to address a probabilistic formulation of a city-wide signal control problem and arebenchmarked versus other existing network models. Compared to the benchmarks, bothmodels derive signal plans that perform systematically better considering various perfor-mance metrics. The second model, compared to the first model, reduces the computationalruntime by at least two orders of magnitude.
The second part of this thesis proposes a technique to enhance the computational ef-ficiency of simulation-based optimization (SO) algorithms for high-dimensional discreteSO problems. The technique is based on an adaptive partitioning strategy. It is embeddedwithin the Empirical Stochastic Branch-and-Bound (ESB&B) algorithm of Xu and Nelson(2013). This combination leads to a discrete SO algorithm that is both globally convergentand has good small sample performance. The proposed algorithm is validated and used toaddress a high-dimensional car-sharing optimization problem.
Thesis Supervisor: Carolina OsorioTitle: Associate Professor of Civil and Environmental Engineering
3
4
Acknowledgments
Firstly, I would like to thank Carolina Osorio for not only being an advisor, but also a
mentor and role model to me during my time at MIT. It is my pleasure to work with her
for last 5 years. I am constantly inspired by her vision, ambitions, and positivity. Her
expertise, her advice, her critical thinking, and her positive attitude towards everything
have an undeniable influence on me. Carolina is always thoughtful and encouraging, which
makes our meetings not only fruitful but also enjoyable. She helped to keep me motivated
and gave me courage to keep trying after failures. Without her help, I could not have been
overcome all the difficulties during this journey.
I would like to thank Professor Patrick Jaillet and Professor Saurabh Amin for serving
as the other two members of my thesis committee, and for their useful feedback and insights
during my committee meetings and multiple 1-1 meetings. In addition, I would like to
thank Professor Saurabh Amin for his help and advice in choosing career path. I would
also like to thank Professor Richard Larson to serve on my general exam committee and to
share his vision, life experiences and amazing stories with me during meeting in his fancy
office.
Additionally, many thanks to Roberta Pizzinato for being so helpful for scheduling
meetings and travel arrangements. I would also like to thank ORC stuff Laura Rose and
Andrew Carvalho for their administrative assistance.
I am extremely grateful to my colleagues at MIT. I would especially thank the fellow
labmates in the Osorio lab: Linsen Chong, Tianli Zhou, Chao Zhang, Kevin Zhang, Timo-
thy Tay, and Evan Fields, without whom I would not have gone so far. Special thanks also
go to my friends at MIT: Manxi Wu, Haizheng Zhang, Daisy Zhou, Fangzhou Lu, Haihao
Lu, Yuchen Wang, Shujing Wang, Yiqun Hu, Shu Ma, Junbin Huang, Li Wang, Rong Yuan,
and many others for accompanying me during this long journey. Special thanks also go to
my friends outside MIT: Ning Hua, Ju Wang, Xiaoli Xu, Yun Zhang, and many others for
supporting me through ups and downs. All of you are really like my families. Our explo-
ration of fun places and restaurants around the great Boston area has become an integral
part of my life. My thanks also go to colleagues I have worked with during my summer
5
internship at Cruise Automation, John Khawam, Sean Skwerer, Chuoran Wang, Michael
McCoy, and many others for the insightful discussions in the future of autonomous vehi-
cles. I learned a lot during this precious opportunity.
I am much obliged to Professor Joel Spencer, my mentor during the undergraduate years
at New York University, who brought me into the world of mathematics and continuously
supports me until today. Without him, I would not have had my achievements today.
Last, but certainly not least, I would like to acknowledge and thank my family for their
unconditional trust, support, and encouragement. A special thank must be given to my
mother Jue Sun. This thesis is dedicated to you.
This work is partially supported by the U.S. National Science Foundation under Grant
No. 1562912. Any opinions, findings, and conclusions or recommendations expressed in
this material are those of the authors and do not necessarily reflect the views of the National
µ(k) 1440 veh/h = 0.4 veh/secλ(k) varies by experiment
ℓ, L, kfwd, kbwd varies by experiment
Table 3.1: Link parameters
0 50 100 150 200 250 300time T
0
0.2
0.4
0.6
0.8
1
1.2
P(D
Q(T
)=0)
SimProposedMultivariateMixture
0 50 100 150 200 250 300time T
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
P(U
Q(T
)=L)
SimProposedMultivariateMixture
Figure 3-1: Experiment 1: impact of the temporal variation of demand on the link’s up-stream and downstream boundary conditions
the y-axis displays the corresponding probability. The simulated estimates are displayed as
a red line with asterisks, those of the proposed model are the black solid line, those of the
the multivariate model are the black dot-dashed line, and those of the mixture model are the
black dashed line. The simulated estimates are displayed with 95% confidence intervals,
which are barely visible.
Recall that in experiment 1, there is a sharp increase in demand at time T = 125 seconds
and a sharp decrease at time T = 175 seconds. All analytical models yield similar temporal
trends for both P(DQ(T) = 0) and P(UQ(T) = ℓ). More specifically, as congestion
increases, we expect P(DQ(T) = 0) to decrease and P(UQ(T) = ℓ) to increase. Similarly,
as congestion decreases, we expect P(DQ(T) = 0) to increase and P(UQ(T) = ℓ) to
decrease. All models exhibit these trends. They all capture the sharp decrease and increase
94
0 50 100 150 200 250 300time T
0
0.2
0.4
0.6
0.8
1P
(DQ
(T)=
0)
SimProposedMultivariateMixture
0 50 100 150 200 250 300time T
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
P(U
Q(T
)=L)
SimProposedMultivariateMixture
Figure 3-2: Experiment 2: impact of the temporal variation of demand on the link’s up-stream and downstream conditions
trends of the simulator. The multivariate model yields the most accurate approximation.
The proposed model tends to overestimate the stationary spillback probability during the
highly congested regime, whereas the mixture model tends to underestimate the stationary
spillback probability during both the congested and the highly congested regimes. Overall,
the proposed model yields a good approximation of both probabilities P(DQ(T) = 0) and
P(UQ(T) = ℓ).
The results of experiment 2 are displayed in Figure 3-2. These plots have the same
layout as those of Figure 3-1. The simulated estimates are displayed with 95% confidence
intervals, which are barely visible. Recall that experiment 2 considers a sharp decrease in
demand at T = 100 seconds and a sharp increase at T = 200 seconds. The left plot shows
an increase in P(DQ(T) = 0) after T = 100 seconds and a decrease after T = 200 seconds.
The right plot shows that the spillback probability (P(UQ(T) = ℓ)) decreases after T = 100
seconds and increases after T = 200 seconds. All analytical models capture these sharp
changes in probability mass. The proposed model yields a less accurate approximation of
the stationary spillback probability during the highly congested regime, while the mixture
model yields a less accurate approximation for both the congested and the highly congested
regimes. Overall, all three models approximate well the dynamics of the link’s boundary
conditions for sudden and significant changes in congestion levels.
95
Next, we benchmark the accuracy of the proposed model over a set of 21 experiments.
All experiments start with an empty link and have a duration of TF = 300 seconds. For each
experiment, the arrival rate changes at 150 seconds. The experiments consider all combi-
nations of the following time-varying arrival rates (λ ∈ {0.2 → 0.1, 0.3 → 0.1, 0.3 →0.2} veh/sec) and space capacities (ℓ ∈ {10, 20, 30, 40, 60, 80, 100}). The space capac-
ity values considered correspond to link lengths L ∈ {50, 100, 150, 200, 300, 400, 500}
(in meters), forward lags kfwd ∈ {5, 10, 15, 20, 30, 40, 50} and backward lags kbwd ∈
{10, 20, 30, 40, 60, 80, 100}. For all experiments the arrival rates are such that congestion
builds up during the first 150 seconds and the link reaches a stationary regime, then the
arrival rate decreases, congestion dissipates and the link reaches a less congested stationary
regime. For each experiment and each model, we set a maximum computation runtime of
40 hours. Model evaluations that have not concluded within the 40 hours are terminated.
The error metric used to evaluate the accuracy of a given analytical model is the average
absolute difference between the simulated estimate and the analytical approximation:
eDQ =1
TF
TF∑T=1
|PA(DQ(T) = 0) − PS(DQ(T) = 0)| (3.33)
eUQ =1
TF
TF∑T=1
|PA(UQ(T) = ℓ) − PS(UQ(T) = ℓ)|, (3.34)
where PA denotes the probability approximated by an analytical model (proposed, mixture
or multivariate) and PS denotes the simulated estimate.
Figure 3-3 displays the average absolute difference for the 21 experiments. The top
(resp. bottom) three plots consider the spillback probability P(UQ(T) = ℓ) (resp. P(DQ(T) =
0)). The first, second and third column of plots consider the experiments with arrival rate
0.2 → 0.1 veh/sec, 0.3 → 0.1 veh/sec and 0.3 → 0.2 veh/sec, respectively. Each plot
compares the three models: the proposed model (circles), the mixture model (asterisks)
and the multivariate model (triangles). The x-axis displays the space capacity (i.e., ℓ) and
the y-axis displays the average absolute difference (i.e., eUQ or eDQ). The top three plots,
which consider the spillback probability, have a logarithmic-scaled y-axis. For the exper-
iments with space capacity greater than 30 (i.e., ℓ > 30), the multivariate model does not
96
1020
3040
5060
7080
9010
010
-30
10-2
5
10-2
0
10-1
5
10-1
0
10-5
100
Pro
pose
dM
ixtu
reM
ultiv
aria
te
(a)A
rriv
alra
teλ(k)=
0.2
→0.1
veh/
sec
1020
3040
5060
7080
9010
010
-20
10-1
5
10-1
0
10-5
100
Pro
pose
dM
ixtu
reM
ultiv
aria
te
(b)A
rriv
alra
teλ(k)=
0.3
→0.1
veh/
sec
1020
3040
5060
7080
9010
010
-20
10-1
5
10-1
0
10-5
100
Pro
pose
dM
ixtu
reM
ultiv
aria
te
(c)A
rriv
alra
teλ(k)=
0.3
→0.2
veh/
sec
1020
3040
5060
7080
9010
010
-3
10-2
10-1
Pro
pose
dM
ixtu
reM
ultiv
aria
te
(d)A
rriv
alra
teλ(k)=
0.2
→0.1
veh/
sec
1020
3040
5060
7080
9010
010
-3
10-2
10-1
Pro
pose
dM
ixtu
reM
ultiv
aria
te
(e)A
rriv
alra
teλ(k)=
0.3
→0.1
veh/
sec
1020
3040
5060
7080
9010
010
-3
10-2
10-1
Pro
pose
dM
ixtu
reM
ultiv
aria
te
(f)A
rriv
alra
teλ(k)=
0.3
→0.2
veh/
sec
Figu
re3-
3:C
ompa
riso
nof
the
aver
age
abso
lute
erro
rsfo
rthe
21ex
peri
men
tsw
ithtim
e-va
ryin
gde
man
d
97
conclude within 40 hours, thus these runs are terminated and are not displayed in the plots.
The main insights from Figure 3-3 are as follows. For most experiments, the multivari-
ate model gives the lowest errors for both P(UQ(T) = ℓ) and P(DQ(T) = 0), followed
by the mixture model. For the spillback probabilities (i.e., top three plots), both the pro-
posed model and the mixture model have errors that decrease exponentially as the space
capacity ℓ increases. As the space capacity increases, the error in the approximation of
P(DQ(T) = 0) (bottom three plots) for the proposed model preserves the same order of
magnitude. The numerical values of the errors displayed in Figure 3-3 are provided in Ap-
pendix B.6 (Tables B.1 and B.2). The average (over the 21 experiments) eUQ and eDQ of the
proposed model are 0.0006 and 0.0180, respectively, whereas those of the mixture model
are 0.0023 and 0.0073. Compared to the mixture model, the proposed model, on average,
gains accuracy in approximating the upstream boundary conditions and loses accuracy in
approximating the downstream boundary conditions.
Overall, the multivariate model has the highest accuracy, yet is computationally inef-
ficient for large space capacity values. The proposed model and the mixture model have
comparable accuracy. They both perform well for all experiments.
We now compare the multivariate model, the mixture model and the proposed model in
terms of computational runtime. Figure 3-4 compares the runtimes for the 21 experiments.
Figure 3-4(a), 3-4(b) and 3-4(c) consider the experiments with time-varying arrival rate
λ(k) estimates from detector 1µ(k) varies within signal cycle given by Eq. (3.36)
Table 3.4: Link parameters used in the proposed model
0 50 100 150 200 250
Time T
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Exp
ecte
d in
flow
[veh
/sec
]
Microscopic simulator 95% CIProposed
0 50 100 150 200 250
Time T
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7E
xpec
ted
outfl
ow [v
eh/s
ec]
Microscopic simulator 95% CIProposed
Figure 3-11: Comparison of the expected inflow and outflow for the tandem link experiment
and the performance metrics are estimated from 200 simulation replications.
The left (resp. right) plot of Figure 3-11 shows the expected inflow (resp. outflow). For
both plots, the analytical approximations of both the expected inflow and of the expected
outflow fall within the 95% confidence intervals of the simulated estimates. This shows
the ability of the proposed model to capture the impact of platoon arrivals on the links’
boundary conditions.
In summary, the comparisons to a microscopic traffic simulator indicate that the pro-
posed model can accurately approximate the links’ boundary conditions for realistic traffic
107
situations such as signalized links and platoon arrival patterns.
3.5 Optimization case study
In this section, we evaluate and benchmark the computational efficiency of the proposed
model. We use it to tackle a signal control optimization problem for the Swiss city of Lau-
sanne. The signal control problem considered is the same as that studied in Lu and Osorio
(2018) (cf. Chapter 2). Section 3.5.1 formulates the problem. Section 3.5.2 compares the
performance of the proposed model with that of three other approaches: (i) the mixture
model Lu and Osorio (2018) (cf. Chapter 2), (ii) the deterministic intelligent link transmis-
sion model (ILTM) (Himpe et al.; 2016), and (iii) a widely used commercial signal control
software. The ILTM model is a network loading model that combines an efficient iterative
link transmission model with the node model of Tampère et al. (2011).
3.5.1 City-scale signal control
The Lausanne network consists of 603 links, 902 lanes and 231 intersections. The network
model of the stochastic microscopic simulator is shown in Figure 3-12. We consider a
fixed-time signal control problem in which we determine the signal plans of 17 intersections
distributed throughout the network (displayed as squares in Figure 3-12). The signal plans
of the 17 intersections are determined jointly. The problem is a fixed-time signal control
problem for the evening peak period 5:00-5:30pm. The decision variables are the green
splits of the signal phases of the 17 intersections. All other control variables (e.g., cycle
times and offsets) are fixed. This leads to a total of 99 endogenous signal phase variables
108
Figure 3-12: Lausanne network model
(i.e., the decision vector is of dimension of 99). We use the following notation.
bd ratio of available cycle time to total cycle time for intersection d;
x vector of green splits;
x(j) green split of signal phase j;
xLB vector of lower bounds for green splits;
D set of intersection indices;
PD(d) set of endogenous signal phase indices of intersection d;
L set of all lanes;
T total number of one-minute time intervals;
N1 number of lanes, i.e., cardinality of L.
The problem is formulated as follows:
minx
f(x) =1
TN1
∑i∈L
T∑t=1
P(UQi(t; x) = ℓi) (3.37)
109
subject to
∑j∈PD(d)
x(j) = bd, ∀d ∈ D (3.38)
x ≥ xLB. (3.39)
The decision vector, x, is the green splits of the signal controlled lanes. Constraint (3.38)
ensures that, for every intersection, the sum of the green times equals the available cycle
time. Constraint (3.39) sets lower bounds, which are set to 4 seconds in this case study.
P(UQi(t; x) = ℓi) denotes the spillback probability of lane i at integer time t under signal
plan x. Therefore, the objective function is the average (over space and over time) spill-
back probability. The goal is to find a signal plan that minimizes the spatial and temporal
occurrence of spillbacks.
To evaluate the performance of the signal plans proposed by the various methods, we
use a calibrated microscopic model of the Lausanne network, which embeds realistic link
and node models. Thus, when evaluating the performance of a signal plan, the simula-
tor accounts for how route choices and link demand can vary with signal plan changes.
The deterministic ILTM model is also a complete network loading model that embeds the
well-established node model of Tampère et al. (2011). However, the probabilistic analyti-
cal models (i.e., the proposed model and the mixture model) are link models. In order to
use them for network optimization, they assume demand for each link is exogenous and
describe the within-link dynamics, they do not describe the across-link (i.e., node) dynam-
ics. Implementation details, including the computation of the exogenous link demand, are
given in Section 4.1 of Lu and Osorio (2018) (cf. Chapter 2). The above problem is solved
with the proposed model and with the mixture model using the interior-point algorithm of
the fmincon routine of Matlab (MATLAB; 2016).
Since the ILTM is a deterministic model and it does not approximate the spillback
probabilities that define the objective function (3.37). Thus, we use 2 alternate objective
functions for ILTM:
1. The average, over time and over the network lanes, proportion of lane that is occupied
110
by vehicles:
f(x) =1
TN1
∑i∈L
T∑t=1
(cvn
upi (t) − cvndown
i (t))/ℓi. (3.40)
where cvnupi (t) (resp. cvndown
i (t)) is the cumulative number of vehicles that passes
the upstream (resp. downstream) end of link i up until time t, which is the output
metric produced by the deterministic ILTM model. This objective function reflects
the average saturation degree of each link in the network over time, which is between
0 and 1. A larger objective function value reflects the network is more likely to be
congested and thus more likely for spillback to happen within the network.
2. The average, over time and over the network lanes, link travel time (given by Eq. (3.41)).
f(x) =1
TN1
∑i∈L
T∑t=1
TTi(t). (3.41)
where TTi(t) is the travel time of link i at time t, which is an output metric produced
by the deterministic ILTM model. This objective function reflects average trip travel
time through the network. This is a common choice of objective function for deter-
ministic models. A larger objective function value reflects that a longer travel time
through the network which suggests a higher chance of spillback.
A natural deterministic approximation of (3.37) would be the average, over all lanes in the
network, proportion of time the lane is full. However, this function is not continuously
differentiable. Thus, we could not use it with the derivative-based interior-point algorithm
used with the other models.
For all methods, the maximum runtime is set to 24 hours. If the algorithm does not
converge to a local optimal solution within the time limit, the algorithm is terminated and
the current iterate is used as the final solution.
111
Initial point 1 2 3 4Mixture 145.0 146.1 144.4 149.4
ILTM 10.6 10.7 10.7 10.8Proposed 1.3 1.3 1.3 1.3
Table 3.5: Average runtime (in min) per iteration of the signal control optimization algo-rithm
3.5.2 Numerical results
Problem (3.37)-(3.39) for the proposed and mixture model, and Problems (3.40) and (3.41)
with constraints (3.38) and (3.39) for the ILTM model are solved considering four different
initial points. The initial points are drawn uniformly randomly from the feasible region
(Equations (3.38)-(3.39)) using the sampling code of Stafford (2006).
For the proposed model and the ILTM model, all four optimization runs (i.e., one for
each initial point) conclude within the time limit. Actually, they all finish within 2.5 hours
for the proposed model and within 5.5 hours for the ILTM model. For the mixture model,
the algorithms do not converge within the time limit. Table 3.5 compares the average com-
putation time (in minutes) per algorithmic iteration. Each column of Table 3.5 corresponds
to a different initial point. Since we consider 2 different objective functions for ILTM, Table
3.5 displays the average computation time, averaged over the two optimization problems.
Table 3.5 indicates the computation time of all models does not vary significantly across
initial points. For the proposed model, the average runtime per iteration is in the order of 1
minute. For ILTM model, it is in the order of 10 minutes, and for the mixture model it is in
the order of 2.4 hours (i.e., 146 minutes). Thus, the proposed model improves the runtime
by 1 order of magnitude compared to ILTM and by 2 orders of magnitude compared to the
mixture model. Note however, that, unlike the probabilistic models (proposed or mixture)
the ILTM model is a full network model that embeds an endogenous node model. This
added realism comes, of course, with an increase in the computational runtime.
We now compare the performance of the derived signal plans. To evaluate the perfor-
mance of a given signal plan, we use a microscopic traffic simulation model of Lausanne
(Dumont and Bert; 2006), which is calibrated for the evening peak period demand. It is im-
plemented in the Aimsun software (TSS; 2014). For a given signal plan, we embed it within
112
the microscopic simulation software, and evaluate 50 simulation replications. Each repli-
cation consists of a warm-up period of 15 minutes followed by a simulation period of 30
minutes. For each simulation replication, we estimate the objective function (Eq. (3.37)),
which is the average (over lanes) proportion of time (over 30 minutes) a lane is full. For
each signal plan, we construct a cumulative distribution function (cdf) of these 50 objective
function observations.
Each plot of Figure 3-13 considers a different initial signal plan and plots five cdf
curves: one for the initial signal plan (dashed line), one for the solution derived by the
proposed model (solid line), one for the solution derived by the mixture model (dot-dashed
line) and two for the solutions derived by the deterministic ILTM model with different
objective functions (circle-dotted lines). The x-axis displays the objective function realiza-
tions (i.e., average (over all the lanes in the network) proportion of time a lane is full). The
y-axis displays the proportion of the 50 simulation replications that have objective function
realizations smaller than x. Thus, the more a cdf curve is shifted to the left, the better the
performance of the corresponding signal plan.
For all 4 plots (Fig. 3-13(a)-3-13(d)), the cdf curves of the derived signal plans, from
the mixture model, from the proposed model and from the ILTM model, are to the left of
the initial signal plan. Thus, all models identify signal plans that outperform the initial
signal plans. For all 4 plots, the cdf curves of the derived signal plans from the stochastic
models (i.e, the proposed and mixture model) are to the left of the derived signal plans
from ILTM model. Thus, the stochastic models can identify signal plans that outperform
the signal plans derived by ILTM model (with both objective functions).
Figure 3-14 compares the average proportion of time a lane is full for the ILTM signal
plans (with objective (3.40), i.e., ILTM-1) and the proposed plan. Each plot of Figure 3-14
considers a different initial point and each dot in the plot represents a lane in the network.
The x-axis (resp. y-axis) displays the proportion of time the lane is full averaged over all
50 simulation replications with the ILTM (resp. proposed) signal plan. A reference line of
y = x is displayed in each plot. Note that for both ILTM and the proposed model, even
after optimization, there remain lanes with an average that is greater than 0.4, which means
that there remain highly congested lanes. Approximately 20% of the lanes have improved
113
0.010.02
0.030.04
0.050.06
x: average proportion of time a lane is full
0
0.2
0.4
0.6
0.8 1
Cumulative distribution function F(x)
InitialP
roposedM
ixtureILT
M-1
ILTM
-2
(a)Initialpoint1
0.010.015
0.020.025
0.030.035
0.040.045
x: average proportion of time a lane is full
0
0.2
0.4
0.6
0.8 1
Cumulative distribution function F(x)
InitialP
roposedM
ixtureILT
M-1
ILTM
-2
(b)Initialpoint2
0.010.02
0.030.04
0.050.06
0.070.08
x: average proportion of time a lane is full
0
0.2
0.4
0.6
0.8 1
Cumulative distribution function F(x)
InitialP
roposedM
ixtureILT
M-1
ILTM
-2
(c)Initialpoint3
0.010.02
0.030.04
0.050.06
0.07x: average proportion of tim
e a lane is full
0
0.2
0.4
0.6
0.8 1
Cumulative distribution function F(x)InitialP
roposedM
ixtureILT
M-1
ILTM
-2
(d)Initialpoint4
Figure3-13:C
umulative
distributionfunctions
oftheaverage,overalllanes
inthe
network,proportion
oftime
alane
isfullconsidering
differentinitialsignalplans
114
performance under the proposed signal plan, while 10% have worse performance. The
mean difference in performance per lane between the ILTM signal plans and the proposed
plan for each plot is 0.006, 0.007, 0.004 and 0.009, respectively. Thus, on average, the
proposed method yields signal plans that mitigate the occurrence of spillbacks. Note that
this occurs, even if the proposed model, unlike ILTM, is merely a link model, it lacks a
node model.
Figure 3-15 compares the performance of the signal plans in terms of the average (over
all lanes in the network) proportion of the lane that is occupied by vehicles, which is equiv-
alent to the first objective function used by the ILTM optimization. This figure has a similar
layout as Figure 3-13. The figure compares the cdf curves of the different signal plans. As
before, the more a cdf curve is shifted to the left, the better its performance (i.e., the higher
the proportion of simulation replications, out of the 50, that have low average proportion of
the lane that is occupied by vehicles). All four plots in Figure 3-15 indicate that all derived
signal plans outperform their corresponding initial signal plans. The derived signal plans
from both probabilistic models (mixture and proposed) have similar performance and they
outperform the signal plans derived by the deterministic ILTM model with both objective
functions.
Figure 3-16 compares the performance of the signal plans in terms of the average trip
travel time, which is equivalent to the second objective function used by the ILTM opti-
mization. For all initial points, all models yield signal plans that outperform the initial
points. The signal plans derived by the proposed model and by the mixture model have sim-
ilar performance and they outperform the signal plans derived by the deterministic ILTM
model with both objective functions.
We compare the performance of the proposed signal plans with that of a signal plan de-
rived by the widely used commercial signal control software Synchro (Trafficware; 2011).
The Synchro software is based on a deterministic macroscopic traffic model, it does not
solve the same optimization problem (3.37)-(3.39). For details on how the Synchro signal
plan is derived, we refer the reader to Section 5.3 of Osorio and Chong (2015). Figures 3-
17, 3-18 and 3-19 display, respectively, the three performance metrics of interest: average
proportion of time a lane is full, average proportion of the lane that is occupied by vehicles
115
00.1
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e lane is full for ILTM
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e lane is full for ILTM
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(d)Initialpoint4
Figure3-14:
Average
proportionoftim
elane
isfullforILT
Msignalplans
andproposed
signalplans
116
0.04
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vera
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ccup
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ehic
les
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Cumulative distribution function F(x)
Initi
alP
ropo
sed
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ture
ILT
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(a)I
nitia
lpoi
nt1
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ccup
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Cumulative distribution function F(x)
Initi
alP
ropo
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ture
ILT
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ILT
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nitia
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nt2
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Cumulative distribution function F(x)
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ropo
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ture
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nitia
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nt3
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ccup
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Cumulative distribution function F(x)In
itial
Pro
pose
dM
ixtu
reIL
TM
-1IL
TM
-2
(d)I
nitia
lpoi
nt4
Figu
re3-
15:
Cum
ulat
ive
dist
ribu
tion
func
tions
ofth
eav
erag
epr
opor
tion
ofth
ela
neth
atis
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byve
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esco
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s
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45
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10x: average trip travel tim
e [min]
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InitialP
roposedM
ixtureILT
M-1
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(a)Initialpoint1
45
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e [min]
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Cumulative distribution function F(x)
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roposedM
ixtureILT
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1011
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in]
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Cumulative distribution function F(x)
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roposedM
ixtureILT
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45
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1011
x: average trip travel time [m
in]
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Cumulative distribution function F(x)InitialP
roposedM
ixtureILT
M-1
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-2
(d)Initialpoint4
Figure3-16:C
umulative
distributionfunctions
oftheaverage
triptraveltim
econsidering
differentinitialsignalplans
118
and average trip travel time normalized by free-flow travel time. Each figure contains 9
cdf curves: four black dashed lines for the four initial points, four solid thin lines for the
four solutions derived by the proposed model and one solid thick line for the signal plan
proposed by Synchro. For all figures, the left-most curves are the ones corresponding to
the signal plans derived by the proposed model. In other words, for all three performance
metrics, the signal plans derived by the proposed model outperform both the initial signal
plans and the signal plan derived by Synchro. For each figure, the performance of the four
initial points varies significantly, while that of the proposed signal plans is similar. In other
words, the proposed method is robust to the quality of the initial points.
In summary, compared to the mixture model, the proposed model improves the average
runtime by 2 orders of magnitude, and yields signal plans with either improved or simi-
lar performance. Compared to the deterministic ILTM model, both the proposed and the
mixture model are able to find signal plans with improved performances. This case study il-
lustrates the scalability and computational efficiency of the proposed model. The proposed
model is suitable for large-scale network analysis and optimization.
3.6 Conclusions
This chapter formulates an analytical probabilistic stochastic model that is scalable and
suitable for large-scale network optimization. The main idea of the proposed model is to
describe the link’s boundary conditions with only two key probabilities instead of tracking
the full marginal, or full joint, distributions. More specifically, while the dimension of
the state space of the models of Osorio and Flötteröd (2015) and of Lu and Osorio (2018)
(cf. Chapter 2) scales cubically and linearly, respectively, with the link’s space capacity,
the proposed model has a constant dimension of 2. Thus, it scales independently of the
link attributes such as the link’s space capacity. This makes it suitable for large-scale
network analysis and optimization. The model is validated versus stochastic simulation
results from a simulation-based implementation of a stochastic link transmission model.
The model’s accuracy is comparable to that of Osorio and Flötteröd (2015) and of Lu and
Osorio (2018) (cf. Chapter 2), while being more computationally efficient. The model is
119
0.010.02
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0
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Cumulative distribution function F(x)
Synchro signal plan
Initial signal planP
roposed signal plan
Figure3-17:
Cum
ulativedistribution
functionsof
theaverage
pro-portion
oftime
alane
isfull
0.040.06
0.080.1
0.120.14
x: average proportion of the lane that is occupied by vehicles
0
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Cumulative distribution function F(x)
Synchro signal plan
Initial signal planP
roposed signal plan
Figure3-18:
Cum
ulativedistribution
functionsof
theaverage
pro-portion
ofthelane
thatisoccupied
byvehicles
45
67
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in]
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Cumulative distribution function F(x)
Synchro signal plan
Initial signal planP
roposed signal plan
Figure3-19:
Cum
ulativedistribution
functionsof
theaverage
triptraveltim
e
120
also validated versus a microscopic traffic simulator, and it can accurately approximate the
link’s boundary conditions for realistic traffic situations such as platoon-like arrival. The
proposed model is then used to address a signal control problem for the city of Lausanne
(Switzerland). The derived solutions are benchmarked with those derived by the mixture
model of Lu and Osorio (2018) (cf. Chapter 2) and ILTM model of Himpe et al. (2016).
The derived signal plans from both the proposed model and the mixture model have similar
performance, considering various performance metrics. They both outperform the initial
plans, signal plans derived by ILTM model and a signal plan proposed by a widely used
commercial software. Compared to the model of Lu and Osorio (2018) (cf. Chapter 2), the
proposed model reduces computational runtime by 2 orders of magnitude.
Future work focuses on the formulation of scalable stochastic network models, the goal
is to be able to recover the joint distribution of a path or a network. First, there is a need
to formulate scalable probabilistic node models that are consistent with their deterministic
counterparts. Osorio et al. (2011) includes a two-link probabilistic node model that pro-
vides the dependencies of the links’ boundary conditions across a node. It yields the joint
distribution of the downstream boundary conditions of the upstream link and the upstream
boundary conditions of the downstream link. The extension of this formulation to nodes
with multiple incoming and outgoing links is part of ongoing work. Second, scalable and
computationally efficient network model formulations are required. Consider a network of
n links, directly coupling the proposed link model with the node model of Osorio et al.
(2011) would yield a complexity of O(2n), which is not scalable. Possible techniques
to achieve scalability include network decomposition (Flötteröd and Osorio; 2017) and
aggregation-disaggregation (Osorio and Yamani; 2017; Osorio and Wang; 2017).
121
122
Chapter 4
Adaptive Partitioning Strategy for
High-Dimensional Discrete
Simulation-based Optimization
Problems
In this chapter, we introduce a technique to enhance the computational efficiency of SO
algorithms for high-dimensional discrete SO problems. The technique is based on an inno-
vative adaptive partitioning strategy. It is integrated with the Empirical Stochastic Branch-
and-Bound (ESB&B) framework proposed by Xu and Nelson (2013). This combination
leads to a general-purpose discrete SO algorithm that is both globally convergent and has
good small sample (finite-time) performance.
4.1 Introduction
Simulation-based optimization (SO), also referred as optimization via simulation (OvS),
is the optimization of the performance of a stochastic system, where the objective func-
tion and/or constraints can only be estimated through stochastic simulations. Detailed
overviews of SO literatures are provided by Fu (2002); Amaran et al. (2016); Bhosekar
and Ierapetritou (2018). Based on the feasible region structures, Hong and Nelson (2009)
123
divide SO problems into three categories: continuous SO, ranking and selection, and dis-
crete SO. For a continuous SO problem, the decision variables are continuous, and methods
developed to tackle such problems include stochastic approximation methods (e.g., Rob-
bins and Monro (1951); Bhatnagar et al. (2011)), direct search methods (e.g., Andradóttir
(2006)), surrogate-based methods (e.g., Angün et al. (2009); Regis and Shoemaker (2013);
Wang and Ierapetritou (2018)). For ranking and selection problems, all feasible solutions
can be simulated at least once and the (near) best solution is chosen with a given confi-
dence interval (e.g., Kim and Nelson (2006)). A discrete SO problem is one with discrete
decision variables and the number of feasible solutions is usually too large for each one
to be simulated. The existing algorithms mainly focus on global convergence to the opti-
mal solution(s) asymptotically (e.g.,Xu and Nelson (2013); Tsai and Fu (2014)). Methods
that aim at identifying solutions with good performances within small sampling budgets
include Heuristic Constrained Genetic Algorithm (HCGA) (Tsai and Fu; 2014), and Indus-
trial Strength COMPASS (ISC) (Xu et al.; 2010). Nevertheless, developing discrete SO
algorithms that can efficiently tackle high-dimensional problems remains a challenge.
Xu and Nelson (2013) propose an Empirical Stochastic Branch-and-Bound (ESB&B)
framework for discrete SO problems based on the nested partitions method of Shi and Ólaf-
sson (2000) and stochastic branch-and-bound (Norkin et al.; 1998). It takes advantage of
the partitioning structure of stochastic branch-and-bound method and empirically estimates
the bounds based on sampled solutions. The ESB&B algorithm also uses improvement
bounds to represent the potential of each subregion to guide the sampling strategy in the
next iteration. The ESB&B algorithm is globally convergent, i.e., given infinite simulation
budget, it converges asymptotically to the global optimum. As mentioned in Xu and Nelson
(2013), there are many valid partitioning strategies; however, a good partitioning strategy
can usually improve the algorithm efficiency. Most partitioning strategies developed in the
literature are generic and heuristic, for example, dividing the feasible region equally into k
subregions along a randomly chosen dimension (Xu and Nelson; 2013).
In this chapter, we propose an innovative adaptive partitioning strategy. It is embedded
within the ESB&B framework (Xu and Nelson; 2013) and forms a globally convergent al-
gorithm for discrete SO problems. The proposed partitioning strategy iteratively divides
124
the feasible region in a fashion such that previously sampled solutions with similar per-
formances are located in the same subregion. It is an adaptive sample-based partitioning
strategy that enhances small sample (finite-time) performances of the ESB&B framework
of Xu and Nelson (2013). This proposed partitioning strategy can take on problem-specific
structures known a priori, such as clustering effect in car-sharing fleet assignment prob-
lems, to further enhance the algorithm efficiencies without significant modifications.
This chapter is organized as follows. Section 4.2 reviews the ESB&B algorithm by Xu
and Nelson (2013). Section 4.3 presents the proposed adaptive partitioning strategy and its
solving method developed by Dunn (2018). Section 4.4 validates the proposed partitioning
strategy on one synthetic and one real-world discrete SO problems. Section 4.5 concludes
this chapter.
4.2 ESB&B framework
Let us first define the discrete SO problem. The goal is to find x that solves
maxx∈X
E[Y(x)] (4.1)
where x = [x1, ..., xp] are the discrete decision variables, and X is a convex feasible region
that contains finite but a large number of feasible solutions, which can be represented by
constraints of the form:
li ≤ xi ≤ ui, i = 1, ..., p (4.2)
gj(x) ≤ 0, j = 1, ..., q (4.3)
li, xi, ui ∈ Z, i = 1, ..., p. (4.4)
The objective function E[Y(x)] is the expected performance at point x, which can only be
estimated by generating observations of Y(x) via simulation.
To solve the discrete SO problems of the above format, Xu and Nelson (2013) propose
the Empirical Stochastic Branch-and-Bound (ESB&B) framework, which converges to the
125
globally optimal solution(s) asymptotically (i.e., under unlimited sampling budgets and
simulation efforts). The ESB&B algorithm is detailed in Algorithm 5. The algorithm ter-
minates when the total sampling budget is used up. Whenever the algorithm is terminated,
the final solution is the one with the maximum cumulative sample average.
There are two important steps within each iteration: i) sampling and bounding (Step
2), ii) and partitioning (Step 3). For details regarding the sampling and bounding step, we
refer to Section 3 of Xu and Nelson (2013). We discuss in details the partitioning step.
Partitioning divides the estimated best subregion into a set of smaller subregions that are
disjoint and nonempty. In the ESB&B implementation of Xu and Nelson (2013),
the embedded generic partitioning strategy chooses (either deterministically or ran-
domly) a variable xi and divide along this dimension the best subregion into ω dis-
joint subregions. A detailed description of this partitioning strategy is given by the
online Appendix A of Xu and Nelson (2013).
However, there exists many valid partitions for a given subregion. A good partition-
ing strategy can help locate the most promising subregion more efficiently, and hence
allocate the sampling budget more efficiently. Let us take the following illustrative
example to discuss the pros and cons of the current generic partitioning strategy used
in ESB&B and potential directions of improvement.
Illustrative example:
Consider the maximization of a 2-dimensional deterministic function f(x1, x2), for
which neither the closed-form nor any structural information about the function is
known. The ground truth of f(x1, x2) in the current best subregion is shown in Fig-
ure 4-1. However, the values of the function can only be known through sampling.
Figure 4-2 shows samples that have already been evaluated in the current best subre-
gion.
We need to further partition this region and sample from each new subregion. Figure
4-3(a) shows the generic partitioning strategy used in the ESB&B framework, which
divides this region into equal parts along the chosen x1 dimension. This partitioning
strategy requires the users to predefine the number of subregions to be divided into
126
Algorithm 5 ESB&B framework
Step 0. Initialization: set iteration counter k = 0, initial partition P0 = {X }, best subregion R0 = X
Step 1. Partitioning:If the best subregion Rk is singleton:
(a) set P ′k = Pk
else:
(a) construct a partition of the best subregion P(Rk)
(b) define the new full partition by P ′k = (Pk\{Rk}) ∪ P(Rk)
(c) denote X P the elements of P ′k
Step 2. Sampling and bounding:
2.1 Solution sampling:
i. for each subregion X P ∈ P(Rk), randomly sample νR solutionsii. if k > 0, for each subregion X P ∈ Pk\{Rk}, sample θ(X P) solutions, where
θ(X P) is computed in Step 2.3 of iteration k− 1 based on information in Φk−1
iii. aggregate all of the sampled solutions into set Sk, and set Φk = Φk−1 ∪ Sk
2.2 Bound estimation:
i. for each x ∈ Sk, simulate ∆nF replications if x /∈ Φk−1, simulate ∆nA additionalreplications if x ∈ Φk−1
ii. for X P ∈ P ′k, calculate estimated upper bound ηk+1(X P)
2.3 Sample allocation: compute the number of solutions to be sampled, θ(X P), for allX P ∈ P ′
k for iteration k+ 1 based on information in Φk
Step 3. Updating partition and best subregion:
(a) update the best subregion Rk+1 = arg maxXP∈P ′k{ηk+1(X P)}
(b) partition Pk+1 = P ′k
(c) set k = k+ 1, go to Step 1.
127
(denoted ω), and it does not utilize any information from the solutions that have
already been sampled. After partitioning, each new subregion gets an equal amount
of sampling budget. Based on the performances of the sampled solutions, a new best
subregion will be selected and further explored. In this case, the middle and right
subregions both have the chance to be selected as the next best subregion as they
both contains peaks of the underlying function. If the right subregion is chosen as
the next best subregion, it can take a while for the algorithm to return to explore the
middle subregion where the true global maximum solution locates.
On the other hand, there is another potential partition of the current best subregion
given in Figure 4-3(b). Note that this partition divides the subregion into three parts:
two that contains the underlying function’s basins and one contains all the peaks.
This partition is better in the sense that it successfully identifies the patterns of the
underlying function. It is obtained by grouping sampled solutions with similar per-
formances in the same subregion.
In Section 4.3, we propose an adaptive partitioning strategy which is defined by the
previously sampled solutions in the subregion.
4.3 Adaptive partitioning strategy
The main idea underlying the proposed partitioning strategy is to find a partition of the
current best subregion Rk in which the sampled solutions with similar performances are
divided in the same subregion. In this section, we introduce two sets of adaptive partitioning
strategies: (i) parallel partition in Section 4.3.1, which applies to any problems, and (ii)
hyperplane partition in Section 4.3.2, which applies when splitting features in the form
of linear combinations of decision variables can be obtained from prior knowledge (e.g.,
clustering effect in car-sharing fleet assignment problem).
128
-3 -2 -1 0 1 2 3
x1
-3
-2
-1
0
1
2
3x 2
-6
-4
-2
0
2
4
6
8
Figure 4-1: The ground truth values of f(x1, x2) in the current best subregion.
-3 -2 -1 0 1 2 3
x1
-3
-2
-1
0
1
2
3
x 2
-6
-4
-2
0
2
4
6
8
Figure 4-2: The sampled solutions of f(x1, x2) in the current best subregion.
129
-3 -2 -1 0 1 2 3
x1
-3
-2
-1
0
1
2
3x 2
-6
-4
-2
0
2
4
6
(a) A naive partition of the current best subregion.
-3 -2 -1 0 1 2 3
x1
-3
-2
-1
0
1
2
3
x 2
-6
-4
-2
0
2
4
6
(b) A better partition of the current best subregion.
Figure 4-3: Different partitions of the current best subregion.
130
4.3.1 Parallel partition
A parallel partition is one that only contains cuts of the form aTx < b or aTx ≥ b where
ai ∈ {0, 1} and∑p
i=1 ai = 1. The subregions resulting from a parallel partition are in the
form of the intersection of a hyperbox with the best subregion. This family of partitions are
favorable in the sense that it directly maps to the bounds of each decision variable and hence
is easy to interpret and draw feasible uniform random samples from. A parallel partitioning
strategy can be applied to any discrete SO problem, thus it is a generic partitioning strategy.
We formulate the search for such a parallel partition as a minimization problem:
minP(Rk)
∑XP∈P(Rk)
∑x∈XP∩Φk
(Y(x) − Y)2 (4.5)
s.t. |X P ∩Φk| ≥ Nmin, ∀ X P ∈ P(Rk), (4.6)
|P(Rk)| ≤ d, (4.7)
where P(Rk) is a valid parallel partition of the current best subregion Rk, Y(x) is the
cumulative sample mean of all n(x) observations at solution x:
Y(x) =1
n(x)
n(x)∑s=1
Ys(x). (4.8)
Y is average estimated mean performances of all sampled points in subregion X P:
Y =1
|X P ∩Φk|
∑x∈XP∩Φk
Y(x) (4.9)
Therefore, the objective function is the sum over all subregions the squared deviation from
the mean. Constraint (4.6) states that at least Nmin already sampled solutions must be
clustered in one subregion of the partition, otherwise the optimization problem (4.5) is
solved to minimum by a partition in which each subregion contains exactly one sample
and the objective function value will be zero. An alternative is to limit the number of
subregions to be divided into, denoted d (Constraint (4.7)). The partition derived by solving
problem (4.5) with constraints on the partition structure (e.g., Nmin or/and d), denoted
131
P∗(Rk), is one that optimally groups sampled points with similar performances. Next, we
discuss the solution algorithm for the proposed minimization problem (4.5)-(4.7).
The proposed partition problem (4.5)-(4.7) is similar to the underlying optimization
problem of the decision tree model (e.g., Breiman et al. (1984); Dunn (2018)) with deci-
sion variables as input variables (features) and performances as dependent variables. The
decision tree model is mainly used for prediction purposes. The underlying optimization
problem of training a decision tree is to find both a tree-structured split of the training
samples based on input variables and labels for classification (or constant values for re-
gression) of leaf nodes so that the prediction error is minimized, given some constraints on
the tree structures (e.g., minimum leaf size and/or maximum tree depth) and restrictions
against overfitting. The decision tree method CART by Breiman et al. (1984) is a top-down
greedy algorithm that does not guarantee a globally optimal decision tree. A recent work
of Dunn (2018) proposes an algorithm that solves the decision tree problem at least locally
optimally, and by initiating with multiple starting points, it attempts for global optimality.
They formulate the decision tree problem as an mixed integer programming problem and
the objective function can be written as follows:
minT
R(T ) + α|T | (4.10)
s.t. N(ℓ) ≥ Nmin, ∀ ℓ ∈ leaves(T ), (4.11)
depth(T ) ≤ D, (4.12)
where R(T ) represents the prediction error of the tree T made on the training data. |T |
denotes the number of branch node in T . It is a representation of the tree complexity.
N(ℓ) represents the number of samples in each leave node ℓ of the tree T , and hence the
constraint (4.11) restricts each leaf node to contain at least Nmin samples. One can also
restricts the maximum depth of the tree (Constraint (4.11)). It sets an upper bound on the
number of leaf nodes (i.e., 2D). In other words, the goal is to find a decision tree model
that balances the prediction accuracy and the model complexity. The optimal decision tree
method is shown to perform robustly to noisy training data (both feature and label noises).
Note that a given tree T creates a unique partition on the feature space and hence a
132
unique division of the samples. For regression tasks, the constant prediction value of a leaf
node that minimizes the mean squared error is the mean performance of the samples in that
leaf node. Thus, by setting α = 0 and choosing the mean squared error as the error metric
(i.e., R(·)), we retrieve an optimization problem that is equivalent to (4.5). In other words,
the proposed partitioning problem (4.5) is equivalent to the optimal parallel regression
tree optimization (Dunn; 2018, Chapter 4) with complexity penalty coefficient α = 0.
The optimization problem (4.10)-(4.12) can be solved efficiently with their local search
algorithm coupled with multiple start points for sample sizes up to hundreds of thousands.
This is more than enough for discrete SO problems, in which samples are usually more
time-consuming to simulate and hence limited in size.
4.3.2 Hyperplane partition
A hyperplane partition is one that contains cuts of the form aTx < b or aTx ≥ b where
ai ∈ R. The subregions resulting from a hyperplane partition are polyhedrons. The current
sampling strategy MIX-D algorithm can efficiently sample uniformly from such subregion.
Hence, although more complicated than parallel partition, this family of partitions are also
practical, especially when problem-specific structures of such form is known a priori.
Assume that variables yj = aTj x for j = 1, ..., q are known in advance to be potentially
important splitting factors other than the decision variables xi for i = 1, ..., p. Let us take
the car-sharing fleet assignment problem as an example. The car-sharing service provider
wants to find a fleet assignment to stations across the network that maximizes their profit.
Geographically nearby stations often share demands among one another, since customers
are often willing to walk short distances for available vehicles if their target station runs out
of vehicle. Thus, the total number of vehicles in a cluster of nearby stations is potentially
a more important factor that influences the profit generated than the number of vehicles
in each individual station. Therefore, a partition based on the total number of vehicles in
clusters of nearby stations can more effectively divide the feasible region and lead to a more
effective search for subregions with higher profits. This can result in a further improved
finite-time performances and algorithm efficiency. In this example, yj is the total number
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of vehicles assigned in the cluster of nearby stations Cj as yj =∑
i∈Cjxi where xi are the
number of vehicle assigned to station i.
The modification in order to incorporate such a hyperplane cut is minor. We simply treat
yj as candidates that the partitioning algorithm can split on together with all the decision
variables xi. The proposed partitioning strategy by solving the optimization (4.5)-(4.7)
is robust to correlated splitting features such as yj and xi, since it naturally includes a
splitting feature selection process. One other benefit the proposed adaptive partitioning
strategy brings is that each subregion constructed contains at least one previously sampled
solution, which can be used directly to initiate the MIX-D sampling scheme.
4.3.3 Adaptive partitioning ESB&B algorithm
The proposed adaptive partitioning ESB&B algorithm is given by Algorithm 6. The pa-
rameters related to partitioning strategy that need to be specified are the maximum tree
depth D and/or minimum number of points each subregion contains Nmin, both are related
to the number of subregions the current best subregion can be divided into. The algorithm
chooses the number of subregions to divide the region into optimally, given the predeter-
mined parameters D and/or Nmin. Different from other generic partitioning strategies, the
derived partition from the proposed adaptive partitioning strategy ensures that each new
subregions contains at least one sampled points, or at least Nmin if this parameter is stated.
This benefits the MIX-D sampling scheme in the sense that these interior sampled points
can be used directly to initiate random walk. The termination of the algorithm is usually
when the simulation budget is exhausted. We select the final solution x∗ as the one with the
maximum cumulative sample average.
4.4 Numerical examples
In this section, we consider two sets of numerical examples to illustrate the proposed adap-
tive partitioning ESB&B algorithm. The first example, Griewank function, has many local
minima which makes it a challenging test function. This low-dimensional example illus-
trates how the proposed method can escape tricky local minima by adaptively partitioning
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Algorithm 6 Adaptive partitioning ESB&B algorithm
Step 0. Initialization:
(a) set iteration counter k = 0, initial partition P0 = {X }, best subregion R0 = X(b) sample uniformly at random the initial n0 solutions in the feasible region X , simulate
∆nF replications of each sample, and record them in set Φ0.
(c) set training set Ψ0 = Φ0
Step 1. Partitioning:If the best subregion Rk is singleton:
(a) set P ′k = Pk
else:
(a) construct a partition of the best subregion P(Rk) using the proposed adaptive parti-tioning strategy with training set Ψk, predetermined parameters D and/or Nmin and/orother splitting features yj than decision variables xi
(b) define the new full partition by P ′k = (Pk \ {Rk}) ∪ P(Rk)
(c) denote X P the elements of P ′k
Step 2. Sampling and bounding:
2.1 Solution sampling:
i. for each subregion X P ∈ P(Rk), randomly sample νR solutionsii. if k > 0 ,for each subregion X P ∈ Pk \ {Rk}, sample θ(X P) solutions, where
θ(X P) is computed in Step 2.3 of iteration k− 1 based on information in Φk−1
iii. aggregate all of the sampled solutions into set Sk, and set Φk = Φk−1 ∪ Sk
2.2 Bound estimation:
i. for each x ∈ Sk, simulate ∆nF replications if x /∈ Φk−1, simulate ∆nA additionalreplications if x ∈ Φk−1
ii. for all X P ∈ P ′k, calculate estimates ηk+1(X P)
2.3 Sample allocation: compute the number of solutions to be sampled, θ(X P), for allX P ∈ P ′
k for iteration k+ 1 based on information in Φk
Step 3. Updating partition and best subregion:
(a) update the best subregion Rk+1 = arg maxXP∈P ′k{ηk+1(X P)}
(b) partition Pk+1 = P ′k
(c) training set Ψk+1 = {x : x ∈ Rk+1 ∩Φk}
(d) set k = k+ 1, go to Step 1.
135
the feasible region with problem structure inferred from previously sampled points. The
second example is a real-world car-sharing case study in Zhou et al. (2019) for which the
optimal solutions are not known, it is used to explore the performance of the proposed al-
gorithm on high-dimensional discrete SO problems. In this example, we also demonstrate
the use of hyperplane partition resulting from additional splitting variables yj, which are
derived from prior knowledge on the geographical locations of the stations. For both ex-
amples, we benchmark the proposed algorithm against the original ESB&B algorithm with
the generic partitioning strategy.
4.4.1 The Griewank function
We consider the Griewank function, which is commonly used as a test case for optimization
algorithms (L. Salemi et al.; 2019). This function has a single global minimum and many
local minima, which makes it a challenging test for optimization algorithms. Figure 4-4
displays the contour plot of the two-dimensional Griewank function on domain [−5, 5] ×
[−5, 5]. The global minimum of the function is at the origin (0, 0) with response value 0
and there are four local minima near the four corners of the domain with response values
0.0086.
We first consider a minimization of the Griewank function with feasible region [−5, 5]×
[−5, 5], where the globally optimal solution (0, 0) is at the center of the feasible region. To
make it a discrete SO problem, the feasible region is divided into a 101 × 101 lattice,
and the function value at each solution is given by the Griewank function plus a normally
distributed noise with mean zero and variance σ2. In this numerical example, we take σ =
0.01, which is relative to the difference between local minima 0.0086 and global minimum
0. The number of replications for each non-encountered sample is set at ∆nF = 10, and
encountered sample ∆nA = 2. For both algorithms, we initiate with the same uniformly
randomly sampled pool of size 10 (asterisks in Figure 4-4). At each iteration, the total
sampling budget for subregions other than the current best (denoted νO) is 5, and that
for the current best subregion (denoted |P(Rk)|νR) is 10; the budget limit is 40 iterations.
Thus, at the end of each run, roughly 5% of all feasible solutions are sampled and evaluated.
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-5 -4 -3 -2 -1 0 1 2 3 4 5
x1
-5
-4
-3
-2
-1
0
1
2
3
4
5
x 2
2D Griewank Function
global minimum
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Figure 4-4: The contour plot of two-dimensional Griewank function on [−5, 5]× [−5, 5].
For the generic partitioning scheme used in the original ESB&B algorithm, the current best
subregion Rk is divided into ω = 2 subregions equally along the longest dimension of Rk.
For the proposed adaptive partitioning scheme, the maximum tree depth D is set at 2, i.e.,
Rk can be at most divided into 4 subregions, and the minimum number of sampled points
grouped in one subregion is set at 2, i.e., Nmin = 2. We run each algorithm 50 times.
Figure 4-5 shows the current best estimate of the objective value across iterations of
five randomly selected runs of each algorithm (i.e., five sample paths of each algorithm).
The solid black lines display the results for the proposed algorithm, and the dashed red
lines for the original ESB&B algorithm. As the iteration advances, the current best esti-
mate of the objective value has a general decreasing trend for both algorithms, although
there are temporary increases due to stochasticity. Figure 4-6 (resp. Figure 4-7) plots a
zoomed version of the current best estimate of the objective value with 95% confidence
intervals, so as to clearly show the performance of the ESB&B algorithm (resp. proposed
algorithm) close to the global minimum function value at zero (solid blue line). Note that
at each iteration, the current best solution is simulated at least 10 replications, since each
non-encountered sample will be simulated ∆nF = 10 replications, and if it is sampled
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again, an additional ∆nA = 2 replications will be simulated. There is only one run of the
ESB&B algorithm that ends up with an estimated objective value that cannot be rejected at
confidence level 95% to be different from the true global minimum value 0. On the other
hand, four runs of the proposed algorithm end up with estimated objective values that are
statistically indifferent from the true global minimum after iteration 15 at confidence level
95%.
Figure 4-8 shows the distance between current best solution and the global minimum
solution across iterations. The solid black lines display the results for the proposed algo-
rithm and dashed red lines for the ESB&B algorithm. Four runs of the proposed algorithm
end up with solutions close to the true global minimum at (0, 0), whereas only one run of
the ESB&B algorithm ends up close to (0, 0). Based on these five experiment runs, the
proposed method tends to find solutions that are closer to the global minimum.
5 10 15 20 25 30 35 40
Iteration
-0.05
0
0.05
0.1
0.15
0.2
Cur
rent
est
imat
e of
obj
ectiv
e va
lue
2D Griewank Function
ESBBProposed
Global minimum objective = 0
Figure 4-5: Objective function estimate of the current iterate across iterations.
Figures 4-9 and 4-11 show the experiment results for the first run of the original ESB&B
algorithm, which is a typical run of ESB&B getting trapped at a local minimum solution. In
both figures, the final partition of the feasible region and the contour plot of the Griewank
138
5 10 15 20 25 30 35 40
Iteration
-0.01
0
0.01
0.02
0.03
0.04C
urre
nt e
stim
ate
of o
bjec
tive
valu
e
2D Griewank Function
ESBB algorithm95% confidence intervalTrue global objective value
Figure 4-6: Objective function estimate of the current iterate with 95% confidence intervalacross iterations of ESBB algorithm (zoomed-in results).
function are displayed. Figure 4-9 plots the path of the best solution at current iterate
across iterations in the feasible domain, the path is plotted with blue arrows, and Figure
4-10 displays the zoomed-in results. The asterisk points plotted are the initial sampled set
Φ0. The generic partitioning strategy used in the ESB&B algorithm missed the subregion
that contains the global minimum in the first iteration and left it near the boundary of a
subregion. The algorithm gets trapped to the lower right local minimum. Figure 4-11
displays the sampling budget allocation in the feasible domain at the end of iteration 40,
in which each black dot represents a sampled solution. As discussed, much less sampling
budget is allocated to subregions that have been discarded earlier.
Figures 4-12 and 4-14 show the experiment results for first run of the proposed al-
gorithm. As before, the final partition of the feasible region and the contour plot of the
Griewank function are displayed in both figures. Figure 4-12 plots the path of the current
best solution across iterations in the feasible domain of the proposed algorithm, and Figure
4-13 displays the zoomed-in results. Figure 4-14 displays the sampling budget allocation in
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5 10 15 20 25 30 35 40
Iteration
-0.01
0
0.01
0.02
0.03
0.04C
urre
nt e
stim
ate
of o
bjec
tive
valu
e
2D Griewank Function
Proposed algorithm95% confidence intervalTrue global objective value
Figure 4-7: Objective function estimate of the current iterate with 95% confidence intervalacross iterations of the proposed algorithm (zoomed-in results).
5 10 15 20 25 30 35 40
Iteration
0
1
2
3
4
5
6
7
Dis
tanc
e to
glo
bal o
ptim
al s
olut
ion
2D Griewank Function
ESBBProposed
Figure 4-8: Distance between current best solution and the global minimum solution acrossiterations.
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-5 -4 -3 -2 -1 0 1 2 3 4 5
x1
-5
-4
-3
-2
-1
0
1
2
3
4
5x 2
ESBB algorithm
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Figure 4-9: The path of best solution at current iterate across iterations in the feasibledomain of the original ESB&B algorithm.
2.5 3 3.5 4
x1
-5
-4.5
-4
-3.5
x 2
ESBB algorithm (zoomed in)
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Figure 4-10: The path of best solution at current iterate across iterations in the feasibledomain of the original ESB&B algorithm (zoomed-in results).
141
-5 -4 -3 -2 -1 0 1 2 3 4 5
x1
-5
-4
-3
-2
-1
0
1
2
3
4
5
x 2
ESBB algorithm
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Figure 4-11: Allocation of sampling budget in the feasible domain of the original ESB&Balgorithm.
the feasible domain of the proposed algorithm. These figures have similar layouts as Figure
4-9 and 4-11, respectively. Different from the generic partitioning strategy, the proposed
adaptive partitioning strategy divides the feasible region along x2 initially and note that the
globally optimal solution is placed in the interior of one of the subregion. At iteration 24,
the algorithm escapes the lower right local minimum and starts to explore the middle sub-
region. The final partition of the feasible region identifies the underlying function’s peaks
and basins successfully. The sampling budgets are allocated more to the subregions that
contain the basins than those containing the peaks as expected.
An average sample-path performance (over the 50 runs) for each algorithm is con-
structed; this performance metric is used to compare against other algorithms, e.g., Xu et al.
(2010, 2013); Xu and Nelson (2013). Figure 4-15 plots the objective value of the estimated
optimal solution across iterations (averaged over 50 algorithm runs) for the ESB&B algo-
rithm (red line with shaded 95% confidence boundary) and the proposed algorithm (black
line with shaded 95% confidence boundary). Figure 4-16 zooms in on the average perfor-
142
-5 -4 -3 -2 -1 0 1 2 3 4 5
x1
-5
-4
-3
-2
-1
0
1
2
3
4
5x 2
Adaptive partitioning ESBB algorithm
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Figure 4-12: The path of best solution at current iterate across iterations in the feasibledomain of the proposed algorithm.
0 0.5 1 1.5 2 2.5 3 3.5 4
x1
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
x 2
Adaptive partitioning ESBB algorithm
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Figure 4-13: The path of best solution at current iterate across iterations in the feasibledomain of the proposed algorithm (zoomed-in results).
143
-5 -4 -3 -2 -1 0 1 2 3 4 5
x1
-5
-4
-3
-2
-1
0
1
2
3
4
5
x 2
Adaptive partitioning ESBB algorithm
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Figure 4-14: Allocation of sampling budget in the feasible domain of the proposed algo-rithm.
mance close to the global minimum function value at zero. Since this is a minimization
problem, the lower the curve, the better the algorithm performance in terms of estimated
objective value. Note that the curve for the proposed algorithm is lower than that for the
ESB&B algorithm except for iterations 10 to 25 where they overlap with each other. This
indicates: (i) at early stage, the proposed algorithm find solutions with improved perfor-
mances faster; (ii) during iterations 10 to 25, both algorithm reaches solutions with similar
performances around local optimal objective values; (iii) as the iteration number increases,
the proposed algorithm continues to find solutions with improved performances whereas
the ESB&B algorithm mostly gets stuck around the locally optimal solutions. At the termi-
nation of the algorithms (i.e., iteration 40), the proposed algorithm ends up with an average
estimated objective value that is statistically lower than that of the ESB&B algorithm. Ad-
ditionally, the proposed algorithm finds the true globally optimal solution (i.e., (0, 0)) for
27 out of 50 runs, whereas the ESB&B algorithm finds it for 3 out of 50 runs. The propose
algorithm finds a final solution with mean performances statistically indifferent from the
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true optimal value (i.e., 0) for 41 out of 50 runs at significance level 0.05, whereas the
ESB&B algorithm finds it for 4 out of 50 runs.
Next, we consider a minimization of the Griewank function with feasible region [−1, 9]×
[−1, 9], where the globally optimal solution (0, 0) is no longer at the center of the feasible
region. In this example, the generic partitioning strategy of ESB&B algorithm does not end
up with an initial cut close to globally optimal solution. Figure 4-17 displays the contour
plot of the two-dimensional Griewank function on domain [−1, 9]× [−1, 9] together with a
uniformly random generated initial sample set (displayed in asterisks) for both algorithms.
The algorithms parameters are set the same as in the previous experiment. We run each
algorithm 50 times.
Figure 4-18 plots the current best estimate of objective value across iterations of five
randomly selected runs of each algorithm (i.e., five sample paths of each algorithm). As
before, we observe that the current best estimate of the objective value inherits a decreasing
trend for both algorithms. Figure 4-19 (resp. Figure 4-20) plots the current best estimate
of objective value with 95% confidence interval of ESB&B algorithm (resp. proposed
algorithm) and zooms in on the performance close to the global minimum function value
at zero (solid blue line). There is only one run of the ESB&B algorithm that ends up with
estimated objective values that cannot be rejected at confidence level 95% to be different
from the true global minimum value 0. Meanwhile, all five runs of the proposed algorithm
end up with estimated objective values that are statistically indifferent from the true global
minimum after iteration 25 at confidence level 95%.
Figure 4-21 shows the distance between current best solution and the global minimum
solution across iterations. This figure has a similar layout as Figure 4-8. All runs of the pro-
posed algorithm end up with solutions close to the true global minimum at (0, 0), whereas
only one run of the ESB&B algorithm ends up close to (0, 0). Based on these five ex-
periment runs, the proposed methods tends to find solutions that are closer to the global
minimum.
Figure 4-22 plots the average estimate of the objective value across iterations for the
ESB&B algorithm (red line with shaded 95% confidence boundary) and the proposed al-
gorithm (black line with shaded 95% confidence boundary). Figure 4-23 zooms in on the
145
Figure 4-15: Objective function estimate of the current iterate across iterations averagedover 50 algorithm runs.
Figure 4-16: Objective function estimate of the current iterate across iterations averagedover 50 algorithm runs (zoomed-in results).
146
-1 0 1 2 3 4 5 6 7 8 9
x1
-1
0
1
2
3
4
5
6
7
8
9
x 2
2D Griewank Function
global minimum
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Figure 4-17: The contour plot of two-dimensional Griewank function on [−1, 9]× [−1, 9].
5 10 15 20 25 30 35 40
Iteration
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Cur
rent
est
imat
e of
obj
ectiv
e va
lue
2D Griewank Function
ESBBProposed
Global minimum objective = 0
Figure 4-18: Objective function estimate of the current iterate across iterations.
147
5 10 15 20 25 30 35 40
Iteration
-0.01
0
0.01
0.02
0.03
0.04C
urre
nt e
stim
ate
of o
bjec
tive
valu
e
2D Griewank Function
ESBB algorithm95% confidence intervalTrue global objective value
Figure 4-19: Objective function estimate of the current iterate with 95% confidence intervalacross iterations of ESBB algorithm (zoomed-in results).
average performance close to the global minimum function value at zero. As before, the
curve for the proposed algorithm is lower than that of the ESB&B algorithm. In other
words, the proposed algorithm has a better performance than the ESB&B algorithm on av-
erage. The initial decrease in estimated objective value of the proposed algorithm is faster
than that of the ESB&B algorithm, this indicates a faster exploring speed of solutions with
improved performances. At termination (i.e., iteration 40), the proposed algorithm ends up
with an average estimated objective value that is statistically lower than that of the ESB&B
algorithm. Additionally, 35 runs of the proposed algorithm end up with true globally op-
timal solution at (0, 0), whereas only 14 runs for the ESB&B algorithm obtain the true
globally optimal solution. 46 runs of the proposed algorithm end with a solution with mean
performance statistically indifference from the true optimal value 0 at significance level
0.05, whereas only 14 runs for the ESB&B algorithm achieve this.
In summary, the proposed adaptive partitioning ESB&B algorithm is a globally conver-
gent algorithm, meanwhile it has an improved finite-time (limited sample budget) perfor-
148
5 10 15 20 25 30 35 40
Iteration
-0.02
-0.01
0
0.01
0.02
0.03
0.04C
urre
nt e
stim
ate
of o
bjec
tive
valu
e
2D Griewank Function
Proposed algorithm95% confidence intervalTrue global objective value
Figure 4-20: Objective function estimate of the current iterate with 95% confidence intervalacross iterations of the proposed algorithm (zoomed-in results).
5 10 15 20 25 30 35 40
Iteration
0
2
4
6
8
10
Dis
tanc
e to
glo
bal o
ptim
al s
olut
ion
2D Griewank Function
ESBBProposed
Figure 4-21: Distance between current best solution and the global minimum solutionacross iterations.
149
Figure 4-22: Objective function estimate of the current iterate across iterations averagedover 50 algorithm runs.
Figure 4-23: Objective function estimate of the current iterate across iterations averagedover 50 algorithm runs (zoomed-in results).
150
mance in terms of finding improved solutions faster, better estimated final objective values
and higher likelihood of finding the true global optimal solutions.
4.4.2 The car-sharing fleet assignment problem
In this section, we apply the proposed algorithm to a real-world car-sharing fleet assignment
problem. This case study problem is adopted from Zhou et al. (2019). It considers a two-
way car-sharing system from the perspective of the service operator. Essentially, it consists
of finding an assignment of a fleet of vehicles across the network of stations that maximizes
the expected profit over a given finite time horizon, denoted as the planning period. Instead
of using a simplified description of demand and of demand-supply interactions, this case
study relies on a demand simulator of Fields et al. (2017) developed based on the rich
high-resolution reservation data from Zipcar’s Boston market. It is formulated as a discrete
SO problem in Zhou et al. (2019), in which the objective function value (i.e., the expected
profit of a given fleet assignment) can only be obtained via simulation.
The discrete SO problem is formulated as follows:
maxx
E[R(x;q1)] −∑i∈I
cixi (4.13)
s.t.∑i∈I
xi ≤ N (4.14)
xi ≤ Ni, ∀ i ∈ I (4.15)
xi ∈ Z≥0, ∀ i ∈ I (4.16)
where x = [xi] are the decision variables and xi is the number of vehicles assigned to station
i. R(x;q1) is the random variable representing the revenue with fleet assignment x and
exogenous simulation parameter vector (e.g., reservation pricing) q1, ci is the exogenous
cost, over the planning period, of a parking space at station i, N is the total number of
vehicles to assign, Ni is the capacity of station i, and I is the set of all stations. The
objective function (4.13) represents the expected profit for a given fleet assignment x as the
difference between the expected revenue E[R(x;q1)] and costs∑
i∈I cixi. The estimates of
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the expected revenue E[R(x;q1)] can only be obtained via simulations. Constraint (4.14)
bounds the total number of vehicles assigned across all the stations. Constraint (4.15)
bounds the number of vehicles assigned to each individual station i by the capacity of the
station. Finally, the number of vehicles assigned to each station must be an nonnegative
integer (Constraint (4.16)).
As discussed in Section 4.3.2, in the car-sharing fleet assignment problem, the total
number of vehicles assigned to a cluster of nearby stations (i.e., yj =∑
i∈Cjxi) may form
a more efficient cut of the feasible region, since stations nearby usually share demands
as customers will search nearby stations for substitutions if the target station runs out of
vehicles. Therefore, to address the formulated discrete SO problem (4.13), we apply the
following three algorithms: the original ESB&B algorithm, the proposed algorithm with
parallel partition, and the proposed algorithm with hyperplane partition. The potential
splitting factors yj for hyperplane partition are generated as follows: for each stations i,
form a cluster of stations center at station i with a given radius, and eliminate replications.
In this section, we choose the radius of 1 in the distance unit used in the simulator when
calculating the spillover effect from one station to another.
We consider the fleet assignment problem of Boston south end area which contains 23
stations, i.e., the decision vector is of dimension 23 (shown in Figure 4-24). Each station
i has a space capacity Ni = 16 and the total number of cars to assign is N = 211. All
other exogenous variables (e.g., ci) other than demand level of the simulator are set the
same as in Zhou et al. (2019). We tested the algorithms on one low-demand level and one
high-demand level case. The maximum number of algorithm iterations is set to 40. At every
iteration, the number of solutions to be sampled from subregions other than the current best
subregion is set to 10 (i.e., νO = 10), and the total number of solutions to be sampled from
the current best subregion is set to 20 (i.e., |P(Rk)|νR = 20). The number of replications
for each non-encountered sample is set at ∆nF = 5, encountered sample ∆nA = 2. For all
algorithms, we initialize with 20 randomly uniformly sampled solutions plus one solution
with relatively good performance, which can be considered as a warm start. This warm-start
solution is obtained by solving the analytical car-sharing fleet assignment model formulated
as an Mixed Integer Programming (MIP) in Zhou et al. (2019, Eq.(8)-(13)). For the original
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ESB&B algorithm, the best subregion is divided into 3 new subregions at each iteration
(i.e., ω = 3). For the proposed algorithm with both parallel partition and hyperplane
partition, the maximum tree depth is set at 2 (i.e., D = 2), and hence the current best
subregion can be divided into at most 4 subregions. The minimum number of sampled
points grouped in one subregion is set at 2, i.e., Nmin = 2. Note that the computational
time spend on simulation per iteration in this experiment setting is roughly 500 seconds, the
proposed partitioning strategy with both parallel and hyperplane partitions finishes within
2 seconds, which is comparably trivial. We run each algorithm five times.
Figure 4-24: Zipcar stations in Boston South End neighborhood (Google Maps; 2017)
Figure 4-25 compares the objective estimate of the current iterate across iterations for
each algorithm run of the low-demand experiment. The x-axis displays the iteration index
and the y-axis displays the performance estimate of the current iterate (i.e., simulation-
based estimate of the objective function of the best point). The red dashed lines displays
the results for the ESB&B algorithm, the black solid lines for the proposed algorithm with
parallel partition, and the blue asterisk lines for the proposed algorithm with hyperplane
partition. Figure 4-26 displays the aggregated average performance for each algorithm. It
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plots the objective value of estimated optimal solution at each iteration (averaged over 5
algorithm runs) for the ESB&B algorithm (red line with shaded 95% confidence bound-
ary), the proposed algorithm with parallel partition (black line with shaded 95% confi-
dence boundary), and the proposed algorithm with parallel partition (blue asterisk line with
shaded 95% confidence boundary). In both figures, we observe the following trends. Ini-
tially, all algorithms start at a position similar to each other, which is the warm-start so-
lution. As iteration advances, both the proposed algorithms with parallel and hyperplane
partition identify solutions with improved performance faster than the ESB&B algorithm.
To further evaluate the performances of the solutions derived, we simulate each derived
solution for 50 replications. Table 4.1 displays the performance statistics of the derived
solutions from different algorithm runs. The first and second columns show the algorithm
name and run id. The third and fourth columns show the mean and standard deviation
of profits generated by each derived solution. We conduct a two-sample t-test between
each derived solution from the proposed algorithm (with parallel and hyperplane partitions)
and each derived solution from the ESB&B algorithm. The null hypothesis is that the
average profit generated by the derived solutions from both algorithms are the same. The
corresponding alternative hypothesis is that the average profit generated by the solution
derived from the proposed algorithm is higher. Table 4.2 and 4.3 display the p-values
for the 50 tests. The null hypothesis cannot be rejected if the p-value is greater than the
significant level 0.05 (colored red in the tables). The null hypothesis is rejected for all tests,
except for the one which compares the derived solution from run 4 of the ESB&B algorithm
and that from run 3 of the proposed algorithm with parallel partition. In other words,
at the end of last iteration, the performances of the solutions derived from the proposed
algorithm are mostly better than those derived from the ESB&B algorithm. To investigate
the added value of hyperplane cuts in the proposed algorithm, we conduct a two-sample
t-test for each solution derived from the proposed algorithm with parallel partition and
each solution derived from the proposed algorithm with hyperplane partition. As before,
our null hypothesis is that the average profits generated by both solutions are the same,
and the alternative hypothesis is that the average profit generated by the solution derived
from the proposed algorithm with hyperplane partition is higher. Table 4.4 displays the
154
5 10 15 20 25 30 35 40Iteration
6.28
6.29
6.3
6.31
6.32
6.33
6.34
6.35E
stim
ated
pro
fit o
f cur
rent
iter
ate
($)
104
ESBBProposed-ParallelProposed-Hyperplane
Figure 4-25: The objective function estimate of the current iterate across iterations forlow-demand experiment.
Figure 4-26: Objective function estimate of the current iterate across iterations(averagedover the 5 algorithm runs) for the low-demand experiment.
155
p-values for all 25 two-sample t-tests. The red cell in the table represents the test in which
the null hypothesis cannot be rejected at significance level 0.05. There are 15 tests that
reject the null hypothesis and suggest that solution derived by the proposed algorithm with
Table 4.2: P-values of the two-sample t-test comparing the solutions derived by ESB&Balgorithm and the proposed algorithm with parallel partition in the low-demand experiment.
Figure 4-27 compares the objective estimate of the current iterate across iterations for
each algorithm run of the high-demand experiment. This figure has a similar layout as Fig-
ure 4-25. Figure 4-28 displays the aggregated average performance for each algorithm. It
has a similar layout as Figure 4-26. From the beginning to the end, the ESB&B algorithm
does not find any solution with improved performance for four algorithm runs. On the other
156
Proposed algorithm with hyperplane partitionRun 1 2 3 4 5
Table 4.3: P-values of the two-sample t-test comparing the solutions derived by ESB&Balgorithm and the proposed algorithm with hyperplane partition in the low-demand exper-iment.
Proposed algorithm with hyperplane partitionRun 1 2 3 4 5
Table 4.4: P-values of the two-sample t-test comparing the solutions derived by the pro-posed algorithm with parallel and hyperplane partition in the low-demand experiment.
hand, both the proposed algorithm with parallel and hyperplane partition do make improve-
ment as iteration advances. The proposed algorithm with hyperplane partition starts to find
better solutions earlier than the one with parallel partition.
To further evaluate the performances of the solutions derived, we simulate each derived
solution for 50 replications. Table 4.5 displays the performance statistics of the derived
solutions from different algorithm runs. This table has a similar layout as Table 4.1. We first
conduct a two-sample t-tests between each solution derived from the ESB&B algorithm and
each solution derived from the proposed algorithm (with parallel and hyperplane partition).
As before, the null hypothesis is that average profits generated by the solutions derived from
both algorithms are the same, and the alternative hypothesis is that the solution derived
from the proposed algorithm generates a higher profit. Table 4.6 and 4.7 display the p-
values for the 50 tests. For all tests, the null hypothesis is rejected at significance level
0.05. In other words, the proposed algorithm ends up with solutions with higher profits.
To validate the added value of hyperplane cuts in the proposed algorithm, we conduct a
157
5 10 15 20 25 30 35 40Iteration
1.38
1.385
1.39
1.395
1.4E
stim
ated
pro
fit o
f cur
rent
iter
ate
($)
105
ESBBProposed-ParallelProposed-Hyperplane
Figure 4-27: The objective function estimate of the current iterate across iterations forhigh-demand experiment.
Figure 4-28: Objective function estimate of the current iterate across iterations (averagedover the 5 algorithm runs) for the high-demand experiment.
158
two-sample t-test between each solution derived with parallel partition and each solution
derived with hyperplane partition. The null hypothesis is that the average profits generated
by both solutions are the same. The alternative hypothesis is that the solution derived by the
proposed algorithm with hyperplane partition generates a higher profit. Table 4.8 displays
the p-values for the 25 two-sample t-tests. For all the 25 t-tests, we reject the null hypothesis
at significance level 0.05. Hence, the proposed algorithm with hyperplane partition ends
up with derived solutions with better performances than those derived by the proposed
algorithm with parallel partition. This obvious improvement in the proposed algorithm’s
finite-time performance may be because under high demand condition, customer spillover
from one station to another happens frequently and hence a hyperplane cut based on the
total number of vehicles in a cluster of nearby stations may form a more efficient way of
Table 4.6: P-values of the two-sample t-test comparing the solutions derived by ESB&Balgorithm and the proposed algorithm with parallel partition in the high-demand experi-ment.
Proposed algorithm with hyperplane partitionRun 1 2 3 4 5
Table 4.7: P-values of the two-sample t-test comparing the solutions derived by ESB&Balgorithm and the proposed algorithm with hyperplane partition in the high-demand exper-iment.
Proposed algorithm with hyperplane partitionRun 1 2 3 4 5
Table 4.8: P-values of the two-sample t-test comparing the solutions derived by the pro-posed algorithm with parallel and hyperplane partition in the high-demand experiment.
160
4.5 Conclusion
In this chapter, we propose an adaptive partitioning strategy and combine it with the ESB&B
framework developed by Xu and Nelson (2013). This proposed partitioning strategy is a
sample-driven approach that explores the structure of the underlying objective function
by iteratively dividing the feasible region into subregions such that sampled points within
the same subregion have similar performances. The proposed partitioning strategy can be
integrated with prior knowledge of specific problem structures to form more efficient hy-
perplane partitions. Solving the proposed partitioning problem is fast and efficient through
solving the formulated MIP via the local search algorithm developed by Dunn (2018).
The proposed algorithm combines the proposed adaptive partitioning strategy within the
ESB&B framework. It is a general-purpose algorithm that converges globally for discrete
SO problems with finite and convex feasible region, and it can be applied to general SO
problems without significant amount of modification. The proposed algorithm improves
the finite-time performances of the original ESB&B algorithm with generic partitioning
strategy.
For SO problems with tight sampling budget at each iteration, a smart sampling strat-
egy other than uniform sampling is important. This can be a potential research direction
to explore. Currently, the adaptive partitioning strategy supports parallel partition and only
hyperplane partitions with potential hyperplane cuts known in advance, it can be an in-
teresting research direction to automatically generate hyperplane cuts for any problems
without such prior knowledge.
161
162
Chapter 5
Conclusions
This chapter concludes the thesis by reviewing its main contents and contributions. It also
provides potential directions for future researches.
Chapter 2 formulates an analytical stochastic link transmission model that is both com-
putationally efficient and consistent with Newell’s simplified kinetic theory of traffic
flow. The proposed model builds upon the multivariate model of Osorio and Flöt-
teröd (2015). The model has a complexity that is linear, rather than cubic, in the
link’s space capacity. This makes the model suitable for large-scale network analy-
sis.
The model is validated versus a simulation-based implementation of the stochastic
link transmission model. The proposed model yields significant gains in compu-
tational efficiency while preserving accuracy. The proposed model is then used to
address a signal control problem for the city of Lausanne. It yields signal plans that
systematically outperform initial random plans for various performance metrics. The
experiments illustrate the robustness of the model to the quality of the initial points.
The proposed plans also outperform a signal plan derived from a widely used com-
mercial signal control software.
Chapter 3 proposes a relaxation approximation of the stochastic link transmission model
formulated in Chapter 2. It proposes a formulation with a constant model complexity,
whereas the past formulations have a complexity that scales linearly or cubically with
163
link length. This makes it suitable for large-scale network optimization with time
budgets or real-time optimization problems.
The model is validated versus a simulation-based implementation of the stochastic
link transmission model. Its performance is also benchmarked with other past analyt-
ical formulations. The proposed model yields estimates with comparable accuracy,
while the computational efficiency is enhanced by at least one order of magnitude.
The proposed model is also validated versus a microscopic traffic simulator and it
can accurately approximate the link’s boundary conditions for realistic traffic situ-
ations. The model is then used to address the same city-wide traffic signal control
problem with time budget. Compared to a benchmark probabilistic analytical model,
the proposed model enhances computational efficiency by two orders of magnitude,
while deriving signal plans with similar performance. Compared to a benchmark
deterministic network loading model, the proposed model derives signal plans with
better performance. It also yields signal plans that outperform those obtained from a
widely used commercial signal control software.
Chapter 4 proposes a technique to enhance the computational efficiency of SO algorithms
for high-dimensional discrete SO problems. The technique is based on an adaptive
partitioning strategy, which divides iteratively the feasible region along the decision
variables into subregions in a fashion such that previously sampled solutions with
similar performance are located in the same subregion. The proposed partitioning
strategy can take on problem-specific structures known a priori to form more ef-
ficient hyperplane cuts. The proposed adaptive partitioning strategy is integrated
in the ESB&B framework by Xu and Nelson (2013). The resulting algorithm is a
general-purpose discrete SO algorithm that converges globally for problems with a
finite and convex feasible region, and it can be applied to deterministic problems
without a significant amount of modification. Two numerical studies show that the
proposed algorithm outperforms the original ESB&B algorithm in small sampling
budget (finite-time) performance. The advantage is greater when prior knowledge of
the problem-specific structures are available.
164
Future research directions The proposed stochastic link transmission model (Chapter 2)
and its relaxation approximation (Chapter 3) are link models that describe the within-
link dynamic and produce the changes in link states (links boundary conditions),
given arrival and departure profiles. Another key component of a complete network
loading model is the node model, which describes the between-link dynamics and
produces arrival and departure profiles, given the states of its connected links.
In order to formulate a complete probabilistic network model, there is a need to for-
mulate probabilistic and scalable node models. The probabilistic model of Osorio
et al. (2011) includes a tandem-link node model that provides a higher-order descrip-
tion of the across-node dependencies. The extension of this formulation to nodes
with multiple upstream and downstream links remains for future work.
Second, there is a need to formulate scalable network models. For a network with n
links, each with space capacity ℓ, directly coupling the proposed link model of Chap-
ter 3 with the node model of Osorio et al. (2011) would yield a model complexity in
the order of O(ℓn). Such a model is inappropriate for large-scale network analysis.
Potential research directions include but are not limited to network decomposition
technique (e.g., Flötteröd and Osorio (2014)) and aggregate-disaggregate techniques
(e.g., Osorio and Yamani (2017); Osorio and Wang (2017)).
The proposed algorithm in Chapter 4 for discrete SO problems combines an adap-
tive partitioning strategy within the ESB&B framework of Xu and Nelson (2013).
The proposed adaptive partitioning strategy uncovers the structure of the underlying
function by iteratively dividing the feasible region into subregions such that sam-
pled points within the same subregion have similar performances. The resulting
algorithm concentrates the limited computational effort (sampling budgets) on sub-
regions where good solutions appear to be. The proposed partitioning strategy can
also make more efficient hyperplane cuts by using prior knowledge of the problem-
specific structures. An extension of this approach consists of automatic detection of
such hyperplane cuts.
165
166
Appendix A
Appendices of Chapter 2
A.1 Estimation of the weight parameter w
This section describes the procedure followed to formulate, and to fit the coefficients of,
the weight parameter of Equation (40). Recall that the goal of the mixture model is to
accurately approximate the upstream and the downstream boundary conditions of the link.
In other words, it should yield an accurate approximation of the distribution of UQ and
of DQ. We consider a single isolated link and conduct a total of 180 experiments with
varied combinations of the space capacity (ℓ ∈ {5, 10, 15, . . . , 100}), the traffic inten-
sity (ρ = λ/µ ∈ {0.25, 0.5, 0.75}) and the service rate (or downstream flow capacity)
(µ ∈ {0.2, 0.4, 0.6}). Each experiment considers a time period of duration 250 seconds.
For each experiment we compare the approximation of the UQ and of the DQ distribu-
tions, over time T , to the distributions estimated via stochastic simulation with a discrete-
event simulator of the stochastic link transmissions model. Based on the results of these
experiments, we first observed that the parameters that most impact the quality of the ap-
proximation are ℓ, µ and kfwdδ. This lead us to formulate the following expression for the
weight parameter:
w(ℓ, µ, kfwdδ;β) = e− ℓ2
βµkfwdδ , (A.1)
167
where β is a scalar coefficient. The coefficient is fit such as to minimize, over all 180
experiments, the following error function:
1
2
[1
250
250∑T=1
JSD(PUQ1 (T) ∥ P
UQ2 (T))
]+1
2
[1
250
250∑T=1
JSD(PDQ1 (T) ∥ P
DQ2 (T))
], (A.2)
where PUQ1 (T) (resp. PDQ
1 (T)) is the UQ (resp. DQ) distribution obtained from the mixture
model at time T and PUQ2 (T) (resp. PDQ
2 (T)) is the UQ (resp. DQ) distribution estimated
via stochastic simulation at time T . The two summations of (A.2) consider the error in the
UQ distributions and in the DQ distributions, respectively. This leads to β = 70. This
results in the final weight parameter expression defined in Equation (40).
A.2 Tables of time-average JSD metric
Tables A.1 and A.2 display, respectively, the time-average JSD metric of the UQ and DQ
distributions.
168
Experiment Time-average JSD of the UQ distribution
λ(k) ℓ Mixture Multivariate DetDet DetExp ExpDet
0.1
10 0.0010 0.0000 0.3539 0.2600 0.0003
20 0.0012 0.0000 0.3988 0.3177 0.0001
30 0.0013 0.0000 0.4248 0.3543 0.0001
40 0.0014 NaN 0.4402 0.3779 0.0000
60 0.0013 NaN 0.4590 0.4111 0.0000
80 0.0011 NaN 0.4692 0.4320 0.0000
100 0.0010 NaN 0.4753 0.4489 0.0000
0.2
10 0.0054 0.0000 0.4261 0.2434 0.0020
20 0.0068 0.0000 0.4644 0.3080 0.0013
30 0.0070 0.0000 0.4839 0.3476 0.0008
40 0.0070 NaN 0.4961 0.3767 0.0005
60 0.0062 NaN 0.5105 0.4189 0.0003
80 0.0054 NaN 0.5181 0.4476 0.0001
100 0.0045 NaN 0.5223 0.4713 0.0001
0.3
10 0.0081 0.0000 0.4654 0.1615 0.0036
20 0.0206 0.0000 0.5071 0.2387 0.0045
30 0.0237 0.0000 0.5214 0.2863 0.0041
40 0.0223 NaN 0.5294 0.3215 0.0032
60 0.0182 NaN 0.5384 0.3772 0.0016
80 0.0145 NaN 0.5434 0.4215 0.0008
100 0.0115 NaN 0.5458 0.4602 0.0004
Table A.1: Time-average JSD metric of the UQ distribution. The value NaN denotes caseswhere the evaluation of the multivariate model exceeded the limit of 40 hours.
169
Experiment Time-average JSD of the DQ distribution
λ(k) ℓ Mixture Multivariate DetDet DetExp ExpDet
0.1
10 0.0007 0.0000 0.1550 0.0486 0.0028
20 0.0002 0.0000 0.1530 0.0476 0.0028
30 0.0000 0.0000 0.1486 0.0468 0.0027
40 0.0000 NaN 0.1466 0.0460 0.0026
60 0.0000 NaN 0.1401 0.0441 0.0025
80 0.0000 NaN 0.1335 0.0422 0.0024
100 0.0000 NaN 0.1270 0.0402 0.0022
0.2
10 0.0030 0.0000 0.2679 0.0527 0.0115
20 0.0008 0.0000 0.2640 0.0538 0.0112
30 0.0002 0.0000 0.2584 0.0538 0.0107
40 0.0000 NaN 0.2528 0.0518 0.0105
60 0.0000 NaN 0.2414 0.0497 0.0099
80 0.0000 NaN 0.2301 0.0476 0.0095
100 0.0000 NaN 0.2188 0.0456 0.0089
0.3
10 0.0077 0.0000 0.3677 0.0347 0.0262
20 0.0033 0.0000 0.3811 0.0507 0.0217
30 0.0007 0.0000 0.3760 0.0544 0.0202
40 0.0002 NaN 0.3680 0.0539 0.0196
60 0.0000 NaN 0.3512 0.0516 0.0184
80 0.0000 NaN 0.3343 0.0492 0.0174
100 0.0000 NaN 0.3173 0.0467 0.0164
Table A.2: Time-average JSD metric of the DQ distribution. The value NaN denotes caseswhere the evaluation of the multivariate model exceeded the limit of 40 hours.
170
Appendix B
Appendices of Chapter 3
B.1 Property: τDQ(k) = λDQ(k) when µ(k) = 0
In this section, we derive the property of τDQ(k) when the service rate of DQ(k) becomes
zero. When µ(k) = 0, DQ(k) is a pure (Poisson) arrival process. The only possibility for
DQ being empty at the end of the time interval k (i.e., DQ(k) = 0) is that DQ is empty at
the beginning of the time interval k, which happens with probability P(DQ(k − 1) = 0),
and there is no arrival to DQ during time interval k of length δ. Since the arrival to DQ
during time interval k is Poisson process with rate λDQ(k), the number of arrivals in a
time interval length of δ follows a Poisson distribution with parameter λDQ(k)δ and thus
no arrival to DQ during time interval k of length δ happens with probability e−λDQ(k)δ.
B.5 Estimation of the scalar coefficients in τUQ(k)
This section describes the procedure to fit the exogenous coefficients α3,1 and α3,2 of Equa-
tion (3.30). The same set of 126 simulation experiments as described in Appendix A.2 are
used. The coefficients α3,1 and α3,2 are fit by minimizing, over all 126 experiments, the
following error function given by Equation (3.34) and rewritten here:
eUQ =1
TF
TF∑T=1
|PA(UQ(T) = ℓ) − PS(UQ(T) = ℓ)|, (B.30)
where PS(UQ(T) = ℓ) is the estimate from the simulator and PA(UQ(T) = ℓ) is the ana-
lytical approximation, which is obtained from Algorithm 4 with the following adjustments.
At every time step k,
• P(DQ(k) = 0) is obtained from the simulator;
• qUQ(k) and qLLO(k) are obtained from the simulator;
• τUQ(k) is obtained from Equation (3.30), which depends on scalar parameters α3,1
and α3,2.
In other words, perfect information about the link’s downstream boundary conditions is
assumed in the calculation of PA(UQ(T) = ℓ). Thus, PA(UQ(T) = ℓ) only depends on
the choice of α3,1 and α3,2 and thus the error function eUQ depends only on α3,1 and α3,2.
The scalars are estimated jointly and the numerical values obtained are α3,1 = 25 and
α3,2 = 7.5.
176
B.6 Tables of mean absolute differences
Experiment eUQ
λ(k) ℓ Mixture Multivariate Proposed
0.2 → 0.1
10 3.11e− 3 7.80e− 5 1.80e− 3
20 5.75e− 5 4.02e− 6 4.74e− 5
30 7.57e− 7 3.92e− 7 7.35e− 7
40 9.25e− 12 NaN 3.25e− 10
60 5.20e− 17 NaN 8.45e− 15
80 2.41e− 22 NaN 1.92e− 19
100 2.49e− 28 NaN 4.72e− 25
0.3 → 0.1
10 1.48e− 2 4.50e− 4 2.10e− 3
20 4.67e− 3 1.23e− 4 1.80e− 3
30 7.03e− 4 3.90e− 5 4.38e− 4
40 8.02e− 5 NaN 6.07e− 5
60 3.11e− 7 NaN 2.91e− 7
80 1.82e− 13 NaN 3.59e− 10
100 3.08e− 17 NaN 8.60e− 14
0.3 → 0.2
10 1.87e− 2 4.44e− 4 4.30e− 3
20 5.19e− 3 1.29e− 4 2.22e− 3
30 8.36e− 4 3.79e− 5 5.49e− 4
40 1.00e− 4 NaN 7.81e− 5
60 6.59e− 7 NaN 2.91e− 7
80 2.04e− 13 NaN 4.59e− 10
100 6.07e− 17 NaN 1.23e− 13
Table B.1: Mean absolute difference eUQ of P(UQ(k) = ℓ). The value NaN denotes caseswhere the evaluation of the multivariate model exceeded the limit of 40 hours. (Startingempty with time-varying demand over time)
177
Experiment eDQ
λ(k) ℓ Mixture Multivariate Proposed
0.2 → 0.1
10 5.00e− 3 2.93e− 3 4.60e− 3
20 2.76e− 3 3.00e− 3 4.35e− 3
30 2.25e− 3 3.02e− 3 4.67e− 3
40 2.51e− 3 NaN 5.12e− 3
60 2.63e− 3 NaN 5.51e− 3
80 2.68e− 3 NaN 5.82e− 3
100 2.74e− 3 NaN 6.16e− 3
0.3 → 0.1
10 1.52e− 2 0.44e− 2 0.69e− 2
20 0.73e− 2 0.46e− 2 1.54e− 2
30 0.40e− 2 0.47e− 2 1.82e− 2
40 0.33e− 2 NaN 1.88e− 2
60 0.40e− 2 NaN 1.93e− 2
80 0.42e− 2 NaN 1.97e− 2
100 0.42e− 2 NaN 2.00e− 2
0.3 → 0.2
10 1.62e− 2 0.35e− 2 1.27e− 2
20 0.69e− 2 0.35e− 2 1.36e− 2
30 0.36e− 2 0.36e− 2 1.33e− 2
40 0.27e− 2 NaN 1.38e− 2
60 0.32e− 2 NaN 1.43e− 2
80 0.34e− 2 NaN 1.46e− 2
100 0.34e− 2 NaN 1.49e− 2
Table B.2: Mean absolute difference eDQ of P(DQ(k) = 0). The value NaN denotes caseswhere the evaluation of the multivariate model exceeded the limit of 40 hours. (Startingempty with time-varying demand over time)
178
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