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Robust Optimization Model for a Dynamic Network
Design Problem Under Demand Uncertainty
Byung Do Chung &Tao Ya o &Chi Xie &
Andreas Thorsen
Published online: 4 September 2010
#Springer Science+Business Media, LLC 2010
Abstract This paper describes a robust optimization approach for a network design
problem explicitly incorporating traffic dynamics and demand uncertainty. In
particular, we consider a cell transmission model based network design problem of
the linear programming type and use box uncertainty sets to characterize the demand
uncertainty. The major contribution of this paper is to formulate such a robust
network design problem as a tractable linear programming model and demonstrate
the model robustness by comparing its solution performance with the nominal
solution from the corresponding deterministic model. The results of the numerical
experiments justify the modeling advantage of the robust optimization approach and
provide useful managerial insights for enacting capacity expansion policies under
demand uncertainty.
Keywords Network design .Dynamic traffic assignment .Robust optimization .
Data uncertainty
Netw Spat Econ (2011) 11:371–389
DOI 10.1007/s11067-010-9147-2
B. D. Chung :T. Yao ( *):A. Thorsen
Department of Industrial and Manufacturing Engineering,
The Pennsylvania State University,
University Park, PA, USA
e-mail: taoyao@psu.edu
B. D. Chung
e-mail: buc139@psu.edu
A. Thorsen
e-mail: aht105@psu.edu
C. Xie
Center for Transportation Research,
Department of Civil,
Architectural and Environmental Engineering,
The University of Texas at Austin, Austin, TX, USA
e-mail: chi.xie@mail.utexas.edu
1 Introduction
Network design consists of a broad spectrum of problems, each corresponding to
different sets of objectives, decision variables and resource constraints, implying
different behavioral and system assumptions, and possessing varying data require-
ments and capabilities in terms of representing network supplies and demands.
Network design models have been extensively used as various types of strategic,
tactical and operational decision-making tools and spanned over a variety of
applications in, for example, transportation, production, distribution, and communi-
cation fields. In a transportation network, traffic congestion has long been a major
concern of the network operator, which occurs when traffic volumes exceed the road
capacity. Network design problems (NDP) for transportation networks in general aim
at minimizing network traffic congestions (or minimizing some general network-wide
traveler costs) through implementing an optimal capacity expansion policy in the
network.
An optimal capacity expansion policy, however, may not be reached without
properly considering the behavioral nature of travel demands, which are inherently
time-variant and uncertain. Travel demands are an aggregate result of individual
travel activities, which are determined by various observed and unobserved
socioeconomic factors and subject to geographical, technological and temporal
constraints. The vast body of the literature has focused on static deterministic NDPs
(see, for example, Magnanti and Wong 1984; Minoux 1989; Yang and Bell 1998). A
major limitation of static network design models is the inability to capture traffic
dynamics, such as traffic shockwave propagation and the build-up and avoidance of
queues. Dynamic models, on the other hand, allow us to model the time-dependent
variation of traffic flows and travel behaviors and hence better describe traffic
evolution and interaction phenomena over the network (Peeta and Ziliaskopoulos
2001). Travel demand uncertainty is not only the underlying characteristic of travel
activities but also a likely result of our inaccurate or inconsistent travel demand
estimation procedures. Without explicit and rigorous recognition of uncertainty in
travel demands, any transportation network development plans and policies may take on
unnecessary risk and even result in misleading outcomes (Zhao and Kockelman 2002).
In terms of their mathematical functional forms, dynamic traffic assignment
(DTA) based NDPs can be classified into two major groups: single-level models and
bi-level models (see the discussion in Lin et al. 2008). The focus of this paper is on
an application of robust optimization (RO) for dynamic NDPs under demand
uncertainty, or more succinctly, a robust dynamic NDP (RDNDP), which has a
single-level strcture. The single-level structure provides an easier way to manipulate
robust counterpart and make RDNDP to be computationally tractable.
The research community has observed a number of recent network design studies
that explicitly incorporate demand uncertainty into NDPs with time-varying flows
(see Waller and Ziliaskopoulos 2001; Karoonsoontawong and Waller 2007;
Ukkusuri and Waller 2008; Karoonsoontawong and Waller 2008). The common
feature of these problems is that time-varying flows are described by the cell
transmission model (CTM) (Daganzo 1994,1995) and the network flow pattern is
then characterized by CTM-based DTA methods, under either the system-optimal
(Ziliaskopoulos 2000) or user-optimal assignment mechanism (Ukkusuri and Waller
372 B.D. Chung et al.
2008). The demand uncertainty of these problems is accommodated by a chance
constraint setting, a two-stage recourse model, or a scenario-based simulation
method. These techniques, however, suffer from deficiencies related to lack of data
availability and problem tractability, which limit their applicability to a broad range
of applications. Resulting models from these stochastic modeling methods are often
computationally intractable and require known probability distributions.
We follow a similar fashion to form our RDNDP using the CTM-based system-
optimal DTA model, but employ the RO approach to account for demand
uncertainty. Given the fact that the CTM-based DTA model has a linear
programming (LP) formulation, we use the set-based RO method (Ben-Tal and
Nemirovski 1998,1999,2000,2002) to form a tractable LP model for the RDNDP,
which overcomes the limitations of previous stochastic optimization methods.
Specifically, in our RDNDP, no probability distribution is presumed; instead, we
only need to simply specify an uncertain set, which is readily available in most
applications. The solution feasibility is guaranteed by the RO method through the
use of the prescribed uncertainty set and can be readily made computationally
tractable through an appropriate reformulation.
We highlight the main contributions of our work at a glance below:
&We develop an RO framework for the SO-DTA based RDNDP. For simplicity,
we present our RO model only for single-destination, system-optimal networks.
However, the basic RO counterpart formulation method can be readily
transferred to the multi-destination problem case. This work adds to the body
of knowledge in the dynamic network design by presenting an emerging method
related to the solution robustness.
&An appealing feature of our robust counterpart problem is that it still has an LP
formulation, so it is in general computationally tractable and can be solved in
polynomial time by a few well-known solution algorithms.
&Our numerical experiments demonstrate the value of RO in the context of
dynamic traffic assignment and network design problems. The computational
viability is illustrated for the proposed modeling framework. The numerical
analysis for the impact of the investment budget bound and the demand
uncertainty level on network design solutions justifies the solution robustness.
The remaining part of this paper is structured as follows. Section 2provides a
discussion of the relevant literature. In Section 3, we generalize the formulation
given by Ukkusuri and Waller (2008) as a CTM-based deterministic dynamic NDP
(DDNDP). We then in Section 4propose a robust counterpart formulation of the
DDNDP to account for demand uncertainty, which we name the RDNDP.
Computational experiments and result analyses from applying the RDNDP model
to a few numerical examples are elaborated in Section 5. Finally, Section 6
concludes the paper and indicates some future research directions.
2 Literature review
Numerous NDPs for transportation applications have been presented in the past three
decades (see Magnanti and Wong 1984; Minoux 1989; Yang and Bell 1998). These
Robust Optimization Model for a Dynamic Network Design Problem 373
NDPs are distinguished by a variety of problem settings and supply and demand
assumptions. The literature review presented below by no means provides a
comprehensive survey to general network design problems or to network design
applications in the transportation field; instead, our discussion is focused on those
network design models and solution methods with data uncertainty, particularly
network design problems with time-varying flows.
A great amount of attention has been paid to NDPs with data uncertainty in past
years and various modeling techniques are used for dealing with uncertain input data
and parameters. The main approaches can be classified into two groups: stochastic
programming (SP) and robust optimization (RO). The SP approach requires known
probability distributions of the uncertain data and includes techniques such as the
Monte Carlo sampling approach and chance-constrained programming. For example,
Waller and Ziliaskopoulos (2001) solved a NDP under uncertain demands where the
probability distributions of demand rates are known a priori. They used a CTM-
based system-optimal NDP formulation with chance constraints. Ukkusuri and
Waller (2008) extended the CTM to model both the system-optimal and user-optimal
NDPs and presented the formulations of a chance-constrained NDP model and a
two-stage resource NDP model to account for demand uncertainty.
Mulvey et al. (1995) proposed a scenario-based RO approach for general LP
problems. Karoonsoontawong and Waller (2007) applied this approach to a CTM-
based dynamic NDP with stochastic demands under both the system-optimal and
user-optimal conditions. A similar RO model formulation approach was employed
by Ukkusuri et al. (2007), in which a scenario-based robust NDP with discrete
decision variables was tackled by a genetic algorithm. The limitations of scenario-
based RO approach are similar to stochastic programming in that we must know the
probability of each scenario in advance and it is computationally expensive when
there are a large number of scenarios.
Recently, a variety of papers have used the set-based RO technique to characterize
optimization models with data uncertainty. Interested readers are referred to Ben-Tal
and Nemirovski (2002) and Bertsimas et al. (2007) for reviews of the set-based RO
methods. For NDPs with uncertain demands, Yin and Lawpongpanich (2007)
considered a static continuous equilibrium NDP under demand uncertainty. Ordonez
and Zhao (2007) formulated and solved a static multicommodity NDP with demand
and travel time uncertainties bounded by polyhedral sets. Mudchanatongsuk et al.
(2008) extended the work by Ordonez and Zhao by considering some generalized
assumptions on demand uncertainty, in which they discussed a path-constrained
NDP and introduced a column generation method to solve the robust NDP with
polyhedral uncertainty sets. Ban et al. (2009) considered a robust road pricing
problem (which is an NDP in the broader definition) that contains multiple traffic
assignment solutions. Atamturk and Zhang (2007) formulated and solved a NDP by
using the two-stage RO method and taking advantage of the network structure for its
solutions. To characterize their uncertainty sets, they used a budget of uncertainty
which limits the number of observed demand values that can differ from nominal
values. They also discussed the numerical results for a simple location-transportation
problem and compared the two-stage robust approach with the single-stage robust
approach as well as two-stage scenario-based stochastic programming.
374 B.D. Chung et al.
There have also been approaches where the set-based RO approach is used to
construct discrete network design models. For example, Lou et al. (2009) described a
discrete NDP with user-equilibrium flows based on the concept of uncertainty
budget and proposed a cutting-plane method for problem solutions; Lu (2007)
addressed a discrete user-equilibrium NDP with polyhedral uncertainty sets using the
RO approach and used an iterative solution algorithm to solve the problem.
To the best of our knowledge, no work has been done in applying the set-based RO
technique to investigate a NDP with dynamic flows and uncertain demands. In this
paper, our effort is given to analytically developing and numerically analyzing the robust
counterpart model of such an NDP in the context of transportation network design.
3 Deterministic model
This section presents the deterministic version of the dynamic NDP model we have
discussed, or the DDNDP model in abbreviation, which provides the basic modeling
platform and functional form for the RDNDP model we will introduce in the next
section. For discussion convenience, let us first present the notation used throughout
these models (see Table 1).
Table 1 The notation
Sets Description
τSet of discrete time intervals, {1,...,T}
CSet of cells, {1,...,I}, including the set of sink cells
(C
S
) and the set of source cells (C
R
)
C
E
Set of cells that can be expanded, C
E
⊂C\C
S
AAdjacency matrix, A={a
ij
}, where each (i,j) component,
a
ij
, equals 1 if cell iis connected to cell j, and equals 0 otherwise
Parameters Description
dt
iDemand generated in cell iat time t
ct
iTravel cost in cell iat time t
Nt
iCapacity in cell iat time t
Qt
iInflow/outflow capacity of cell iat time t
dt
iRatio of the free-flow speed over the backward propagation
speed of cell iat time t
b
xiInitial number of vehicles of cell i
BTotal investment budget available for capacity expansion
f
i
Conversion coefficient of investment cost of cell ifor a unit increase of b
i
χ
i
Increase in capacity of cell ifor a unit increase of b
i
8
i
Increase in inflow/outflow capacity of cell ifor a unit increase of b
i
Variables Description
b
i
Investment cost spent on cell i
xt
iNumber of vehicles staying in cell iat time t
yt
iNumber of vehicles moving from cell ito cell jat time t
Robust Optimization Model for a Dynamic Network Design Problem 375
The network design problem aims at minimizing the sum of the total system
travel cost and the capacity expansion cost. To penalize the unmet demand by the
end of planning horizon, the travel cost in cell iat time t
,
ct
i, is set as follows:
ct
i¼1i2CnCS;t6¼ T
Mi2CnCS;t¼T
(
where Mis a sufficiently large, positive number. The big-Mvalue can also simply
serve as a penalty cost, for example, in emergency evacuation networks, representing
the potential loss of life and property caused by vehicles that do not arrive at the
destination by the end of the time horizon. Use of the penalty cost has the effect of
minimizing the number of vehicles staying in an evacuation network.
By assuming the system-optimal principle and the linear relationship between
investment and capacity increase, the DDNDP model can be written as a LP program
with the notation listed in Table 1:
min
x;y;bX
t2tX
i2CnCS
ct
ixt
iþX
i2CE
fibi
subject to
xt
ixt1
iX
k2C
akiyt1
ki þX
j2C
aijyt1
ij ¼dt1
i8i2C;8t2tð1Þ
X
k2C
akiyt
ki Qt
i8i2CnCE;8t2tð21Þ
X
k2C
akiyt
ki Qt
iþbiϕi8i2CE;8t2tð22Þ
X
k2C
akiyt
ki þdt
ixt
idt
iNt
i8i2CnCE;8t2tð31Þ
X
k2C
akiyt
ki þdt
ixt
idt
iðNt
iþbi#iÞ8i2CE;8t2tð32Þ
X
j2C
aijyt
ij Qt
i8i2CnCE;8t2tð41Þ
X
j2C
aijyt
ij Qt
iþbiϕi8i2CE;8t2tð42Þ
X
j2C
aijyt
ij xt
i08i2C;8t2tð5Þ
X
i2CE
biBð6Þ
376 B.D. Chung et al.
x0
i¼b
xi8i2Cð7Þ
y0
ij ¼08ði;jÞ2CCð8Þ
xt
i08i2C;8t2tð9Þ
yt
ij 08ði;jÞ2CC;8t2tð10Þ
bi08i2CEð11Þ
The objective function includes both the travel cost and expansion cost.
1
The
coefficient f
i
converts the investment cost (money measure) to the travel cost (time
measure). In fact, such coefficient is the reciprocal of value of time, which can be
measured by empirical methods (Wardman (1998)). Note here that the expansion
cost appears in the objective function and is subject to the investment budget
constraint, which makes this formulation different from the traditional charge design
problem (where the expansion cost term is only included in the objective function)
and budget design problem (where the expansion cost terms only appears in the
investment budget constraint).
The constraint set of the DDNDP model specifies the capacity expansion limit,
flow conservation and propagation relationships, initial network conditions and flow
non-negativity conditions. The flow conservation constraint (i.e., Eq. (1)) for cell iat
time tcan be generalized by setting dt
ito be zero in ordinary and sink cells.
Constraints (2-1) and (2-2) are the bounds for the total inflow rate of non-expandable
and expandable cell iat time t, respectively. Similarly, the total outflow rate of cell i
at time tis restricted by constraints (4-1) and (4-2). Constraints (3-1) and (3-2)
bounds the total inflow rate into a cell by its remaining space. Constraint (5) bounds
the total outflow rate of a cell by its current occupancy, and constraint (6) sets the
upper bound on the sum of capacity investments over all cells. The remaining
constraints from Eq. (7) to Eq. (11) set initial network conditions and flow non-
negativity conditions.
4 Robust formulation
Now we develop the robust counterpart of the DDNDP model, which incorporates
the demand uncertainty into a LP program via the RO approach. In the deterministic
1
Costs in general do not vary linearly with respect to the transportation facility capacity or size. Typically,
scale economies or diseconomies exist. Abdulaal and LeBlanc (1979) discussed the cases of linear
relationship, scale economies and scale diseconomies in the context of transportation network design
problems. If the average investment cost per unit of capacity is declining, then scale economies exist.
Empirical data are needed to establish the economies of scale for road construction. This paper assumes a
linear relationship between the investment cost and the capacity, for the reasons of simplicity and the
requirement of the linear model. The linear case can be regarded as an approximation to the case of scale
economies in an expected capacity-increasing range.
Robust Optimization Model for a Dynamic Network Design Problem 377
version, Eq. (1) is the only set of constraints related to the demand generation. This
equality constraint can be rewritten as an inequality constraint (Waller and
Ziliaskopoulos 2006),
xt
ixt1
iX
k2C
akiyt1
ki þX
j2C
aijyt1
ij dt1
i:ð12Þ
It is assumed that all possible demand instances of dt
ibelong to a box uncertainty
set,
Udt
i¼dt
i1qt
i
;dt
i1þqt
i
hið13Þ
where dt
iis the nominal demand level and qt
iis the demand uncertainty level. Then,
the robust counterpart of the Eq. (12) with demand uncertainty becomes
xt
ixt1
iX
k2C
akiyt1
ki þX
j2C
aijyt1
ij dt1
i;dt1
i2Udt1
i:ð14Þ
This is equivalent to the following inequality,
xt
ixt1
iX
k2C
akiyt1
ki þX
j2C
aijyt1
ij max
dt1
i2Udt1
i
dt1
i;ð15Þ
which becomes the flow conservation constraint for the RDNDP model. The above
conversion of the flow conservation constraint leads the RDNDP to be in a
deterministic functional form with the maximum possible demand in the box
uncertainty set. Given that other constraints can be directly transferred from the
DDNDP model to the RDNDP model, the RDNDP formulation can be written into
the following LP form:
min
x;y;bX
t2tX
i2CnCS
ct
ixt
iþX
i2CnCS
fibi
subject to
xt
ixt1
iX
k2C
akiyt1
ki þX
j2C
aijyt1
ij ¼dt1
ið1þqt1
iÞ8i2C;8t2tð16Þ
X
k2C
akiyt
ki Qt
i8i2CnCE;8t2t
X
k2C
akiyt
ki Qt
iþbi
8
i8i2CE;8t2t
X
k2C
akiyt
ki þdt
ixt
idt
iNt
i8i2CnCE;8t2t
378 B.D. Chung et al.
X
k2C
akiyt
ki þdt
ixt
idt
iðNt
iþbi#iÞ8i2CE;8t2t
X
j2C
aijyt
ij Qt
i8i2CnCE;8t2t
X
j2C
aijyt
ij Qt
iþbiϕi8i2CE;8t2t
X
j2C
aijyt
ij xt
i08i2C;8t2t
X
i2CE
biB
y0
ij ¼08ði;jÞ2CC
xt
i08i2C;8t2t
yt
ij 08ði;jÞ2CC;8t2t
bi08i2CE
In Eq. (16), the value of dt1
i1þqt1
i
is the maximum possible demand in cell
iat time t–1
,
accordingtotheuncertaintysetUdt1
i, which represents the worst-
case scenario. Therefore, the optimal solution will remain feasible for all instances
of demand. In other words, we will obtain an optimal solution with the cell capacity
values that are adequate for any realized demand scenarios within the uncertainty set
Udt1
i.
We make the following observation between the optimal objective value and the total
budget level Bfrom the RDNDP model. The implication of this property is that the
network designers should consider the budget level as large as possible even if the
objective function minimize the money used for network expansion together with
the travel cost.
Property 1 The optimal objective function value of the RDNDP monotonically
decreases with respect to the investment budget level.
Proof Let the objective function of RDNDP be z»
rðBÞ, given the total budget level B.
Without loss of generality, we assume that two budget levels B
1
and B
2
are given as
B
1
<B
2
. Since the RDNDP with B
2
has a larger feasible region than the RDNDP with
B
1
,z»
rB1
ðÞis smaller than or equal to z»
rB2
ðÞ, i.e. z»
rB1
ðÞz»
rB2
ðÞ.■
When other types of uncertainty sets such as an ellipsoidal uncertainty set or a
polyhedral uncertainty set are assumed, different deterministic formulations are
derived. For example, the equivalent tractable robust counterpart with an ellipsoidal
uncertainty set is a conic quadratic problem; if a polyhedral uncertainty set is
assumed, it becomes a linear problem (see Yao et al. 2009 for details).
Robust Optimization Model for a Dynamic Network Design Problem 379
5 Numerical analysis
The purpose of presenting computational experiments in this section is twofold: 1) to
demonstrate the difference between robust network design solutions and corresponding
nominal solutions from DDNPP; and 2) to illustrate the advantage of the RO approach
for network design under demand uncertainty. Two numerical examples are selected
from the literature for the experiments: 1) a smaller network with 16 cells and 15 time
intervals; and 2) a larger network with 167 cells and 300 time intervals. For each
example, under the assumption that all cells except destination cells can be invested, we
derived the optimal capacity investment solutions and the objective function values
from the DDNPP and RDNDP models with various demand uncertainty levels. To
evaluate the solution robustness, we also conducted a parallel simulation experiment to
randomly generate 100 demand instances within the given box uncertainty set. The
objective function values from the simulation experiment are also evaluated by solving
the embedded SO DTA problem based on the same capacity expansion scheme as the
one derived by the RDNDP model.
5.1 A toy network
The first experiment uses the test network shown in Fig. 1and the data set in
Table 2, which are from Ukkusuri and Waller (2008). Since they considered a set of
deterministic demands, it is assumed that the demand data in their paper are nominal
values of the network design problem under uncertainty. Let us assume that
uncertain demands from source cell 1 and 14 are [2(1−θ), 2(1+ θ)] at time 0 and 1,
and [1(1−θ), 1(1+ θ)] at time 3. Note that when θis equal to 0, the uncertainty sets
become the nominal values. The investment cost coefficient (f
i
) and penalty cost (M)
for this example are set to 0.1 and 10, respectively. The resulting RDNDP model has
748 constraints and 344 variables, which has been solved within 3 seconds on a PC
with an Intel 1.87 GHz CPU and 2 GB RAM using GAMS/CPLEX.
5.1.1 Optimal solutions under different uncertain levels
The objective function value is calculated and plotted as the total budget level is
varied from 0 to 80 in the interval of 1 unit. Figure 2shows the change of the
objective function values of the DDNDP and RDNDP models with three different
uncertainty levels (including, θ=0.1, 0.2 and 0.3). As the budget level increases, the
objective function value of the RDNDP model decreases and it converges to a
certain value. (see Property 1) Robust solutions are the best worst-case solutions and
Fig. 1 Cell representation of the Toy network (Ukkusuri and Waller 2008)
380 B.D. Chung et al.
thus their objective function values are greater than those of the corresponding
deterministic cases. Note that any nominal solution is equivalent to its robust
solution with the zero uncertainty level (θ= 0).
In all the above cases, the same cells (including cells 7, 9, 11, 15 and 16) are
chosen for capacity expansion, which indicates that they are bottleneck cells in the
network. However, the proportions of the investment on the cells are dependent on
the investment budget level and the demand uncertainty level. Figure 3shows the
investment distribution over the cells. The implication behind these distribution
curves is that the investment strategy should be changed depending on the budget
bound we set and the demand uncertainty degree we expect to face.
It is readily observed that there is a critical/maximum investment point associated
with the investment budget level, beyond which a higher investment does not reduce
the travel cost, or a higher investment even increases the objective function value if
it is used for capacity expansion in the network. For example, this maximum
investment point is between 30 and 40 monetary units in the DDNDP case, and the
point is about 70 monetary units in the RDNDP case with θ= 0.3. The critical
investment point can be interpreted as the threshold for investment: when the budget
is less than this threshold, the marginal travel cost (reduction) is greater than the
marginal construction cost (increase); when the budget is greater than the threshold, the
marginal travel cost (reduction) is less than the marginal construction cost (increase).
5.1.2 Worst-case analysis
After obtaining the investment solutions from the DDNDP and RDNDP models, we
then evaluated the relative improvement of robust solutions from their corresponding
Table 2 Cell characteristics of the Toy network (Ukkusuri and Waller 2008)
Cell 2 3 4 5 6 7 8 9 10 11 12 15 16
Nt
i44444244 4 4 4 4 4
Qt
i12221212 1 2 1 1 1
^
xi00000000 0 0 0 0 0
Fig. 2 The objective-budget
relationship under different
demand uncertainty levels
Robust Optimization Model for a Dynamic Network Design Problem 381
nominal solutions under the worst-case scenario. The relative improvement (RI)in
this study is defined as:
RI ¼TCdTCr
TCr
where TC
d
is the total travel cost from the nominal solution and TC
r
is the total
travel cost from the robust solution.
(a) The DDNDP Model (b) The RDNDP Model (
θ
= 0.1)
(c) The DDNDP Model (
θ
= 0.2) (d) The DDNDP Model (
θ
= 0.3)
Fig. 3 Optimal investment distributions over the network
Table 3 Total travel cost of robust and nominal solutions in worst-case scenarios
Budget θ=0.1 θ= 0.2 θ= 0.3
DDNDP RDNDP DDNDP RDNDP DDNDP RDNDP
0 86.7 86.7 97.4 97.4 109.1 109.1
10 79.7 79.7 89.4 89.4 99.6 99.6
20 77.2 76.7 86.4 85.77 95.7 95.35
30 75.03 74.87 83.73 83.65 92.85 92.8
40 74.7 73.53 83.4 82.15 92.35 90.98
50 74.7 72.9 83.4 80.73 92.35 89.48
60 74.7 72.9 83.4 79.8 92.35 87.98
70 74.7 72.9 83.4 79.8 92.35 86.7
80 74.7 72.9 83.4 79.8 92.35 86.7
382 B.D. Chung et al.
The following worst-case analysis consists of two parts. First, we fixed the
demand uncertain level θand increased the investment budget level B. The
computation results are shown in Table 3and Fig. 4. When the budget level is low, it
is natural that there is little difference between the nominal and robust solutions.
Moreover, when the investment budget is less than 10 monetary units, the model
always selects cell 7 as the site for capacity expansion, in that it is a merging cell and
the bottleneck of the network. The total travel cost associated from the robust design
solutions is slightly lower than that of the corresponding nominal solutions when the
total budget is between 10 and 35 units. We can also see that the robust solutions
significantly outperform the nominal solutions when the budget is large enough and
the demand uncertainty is on a sufficiently high level. However, the relative
improvement of the robust solution against the nominal solution shown in Fig. 4is
not necessarily a monotonically increasing function with respect to the investment
budget level. Though TC
r
and TC
d
both decrease as the investment budget level
increases, the travel cost reduction rates of the two terms change over the budget
level, which is a result of the tradeoff between marginal investment costs and
marginal travel costs in the two different problem cases.
Next, we fixed the total budget level Bat four different levels (including 30, 40,
50 or 60 monetary units) with the demand uncertainty level θranging from 0 to 0.5.
The computation result is depicted in Fig. 5. We can see that with a lower budget
level, the demand uncertainty has a weaker affect on the performance of the RDNDP
model. However, the solution of the RDNDP model may be largely different from
Fig. 4 Relative improvement of
travel cost in worst-case scenar-
ios under different
demand uncertainty levels
Fig. 5 Relative improvement
of travel cost in worst-case
scenarios under different invest-
ment budget levels
Robust Optimization Model for a Dynamic Network Design Problem 383
the solution of the corresponding DDNDP model when the budget level is relatively
high. Similar to Fig. 4, we can also observe that the relative improvement of the total
travel cost of the robust solution against the nominal solution is not always a
monotonically increasing function with respect to the demand uncertainty level.
5.1.3 Simulation results
Finally, we evaluated the objective function by implementing the robust network
design solutions and nominal solution with random demands generated by the given
Table 4 Comparison of simulation results
Budget Mean Standard deviation Maximum
DDNDP RDNDP DDNDP RDNDP DDNDP RDNDP
(a) θ=0.1
0 80.66 80.66 1.52 1.52 83.55 83.55
10 74.36 74.36 1.27 1.27 77.10 77.10
20 72.01 71.78 1.35 1.29 75.16 74.81
30 70.06 70.18 1.38 1.30 72.78 72.72
40 69.82 68.95 1.07 1.01 72.55 71.55
50 69.62 68.34 1.20 1.07 72.14 70.56
60 69.64 68.35 1.16 1.04 71.88 70.39
70 69.62 68.34 1.09 0.98 71.61 70.02
80 69.64 68.36 1.32 1.18 72.79 71.18
(b) θ=0.2
0 85.27 85.27 2.96 2.96 93.02 93.02
10 78.22 78.22 2.51 2.51 83.35 83.35
20 76.16 75.69 2.50 2.41 82.14 81.51
30 73.90 73.83 2.47 2.44 78.84 78.76
40 73.56 72.62 2.49 2.34 78.58 77.33
50 73.51 71.49 2.35 2.18 78.82 76.48
60 72.92 70.39 2.68 2.44 77.92 74.89
70 73.41 70.83 2.20 1.96 78.75 75.57
80 73.65 71.05 2.63 2.35 80.03 76.78
(c) θ=0.3
0 90.21 90.21 4.24 4.24 99.54 99.54
10 82.88 82.88 4.20 4.20 90.37 90.37
20 79.86 79.44 3.64 3.61 87.50 86.97
30 77.30 77.24 3.99 3.98 86.49 86.58
40 77.42 76.34 3.57 3.52 84.98 83.85
50 76.83 74.57 3.45 3.23 84.96 82.33
60 76.88 73.72 3.92 3.57 85.75 81.93
70 76.88 73.72 3.92 3.57 85.75 81.93
80 77.57 73.64 3.52 3.19 84.35 79.85
384 B.D. Chung et al.
box uncertainty sets. Specifically, 100 sets of random data generated from a beta
distribution (i.e., beta(5, 2)) are used for this evaluation. Note that we only know the
support of the primitive uncertain data and accordingly use box uncertainty sets to
characterize the bounded uncertain demand. The beta distribution is a reasonable
choice for simulating bounded uncertain data. The mean, standard deviation, and
maximum values of the objective function values generated from the simulation
experiment are shown and compared in Table 4. It can be seen that, in almost every
case, the mean objective function value of the robust solutions is better than that of
the nominal solutions; in all cases, the standard deviation and maximum values of
the robust solutions are less than or equal to those of the nominal solutions.
5.2 The nguyen-dupis network
Now we present a second numerical example to show the computational tractability
and the performance consistency of with the RDNDP model in larger networks. The
Nguyen-Dupis network with 13 nodes in total (including 2 source nodes and 1 super
sink node) is considered here (see Fig. 6). An equivalent cell network with 167 cells
is created from the original node-link version. The resulting RDNDP model from the
5 6 7
4 8
9 10 11 2
13 3
1 12 r
r
s
s
14
Fig. 6 The node-link topology
of the Nguyen-Dupis network
Fig. 7 The objective-budget
relationship under different
demand uncertainty levels
Robust Optimization Model for a Dynamic Network Design Problem 385
Nguyen-Dupis network has 96,794 constraints and 190,694 variables, which has
been solved in about 60 s on a PC with an Intel 1.87 GHz CPU and 2 GB RAM
using GAMS/CPLEX.
Figure 7shows the optimal objective function values of the DDNDP and RDNDP
models with three different demand uncertainty levels. As similar to the previous
example, there is a set of cells that are chosen for capacity expansion (where, in this
case, there are 36 cells in total) in the Nguyen-Dupis network, which delivers a
similar objective-budget relationship to the previous toy example. Investment
decisions vary with different demand uncertainty levels.
The relative improvement of the robust solutions from the corresponding nominal
solutions in the worst-case scenarios is aggregated in Figs. 8and 9. It is shown from
Fig. 8that the robust solution significantly improves the nominal solution when the
investment budget level is greater than 2,200 monetary units, in particular when the
demand uncertainty level is high. A similar phenomenon can be observed from Fig. 9.
Finally, the simulation results are compared in Table 5. It is found that the
simulated objective function values from DDNDP and RDNPD are comparable
when the investment budget level is less than 1,500 monetary units. However, the
robust solutions provide a lower travel cost when the investment budget goes higher.
Our computational results show that the robust solution is more attractive than the
Fig. 8 Relative improvement
of travel cost in worst-case
scenarios under different
demand uncertainty levels
Fig. 9 Relative improvement of
travel cost in worst-case scenar-
ios under different investment
budget levels
386 B.D. Chung et al.
nominal solution from the simulation experiment. We note that the total travel cost is
affected by 1) the network capacity expansion policy and 2) the underlying traffic
flow pattern. Since our focus is the network design problem, we only tested the
impact of robust network capacity expansion solution in simulation with the
deterministic traffic assignment solutions. We expect the improvement be more
significant when both a robust capacity expansion policy and a robust traffic
assignment procedure are used.
6 Conclusion and further research
In this paper, we formulated and solved the RDNDP, a robust network design
problem for dynamic and uncertain demands, and numerically evaluated its solution
performance. The appealing LP formulation of the RDNDP model is rooted from the
underlying LP-based DTA model―CTM. A box uncertainty set is assumed for
modeling uncertain demands. Through this NDP example, we demonstrate how the
constraints affected by uncertain parameters can be manipulated to derive a tractable
mathematical program.
Since it becomes particularly important to provide a solution which is robust to
extreme events and reduce the variance of cost after the realization of uncertain
parameters (Waller and Ziliaskopoulos 2006), we choose a beta distribution (which
is an asymmetric distribution) to model random demands and conduct a worst-case
analysis. The RO approach can provide better network design solutions that produce
Table 5 Comparison of the robust optimization results and simulation results
Budget Mean Standard deviation Maximum
DDNDP RDNDP DDNDP RDNDP DDNDP RDNDP
(a) θ=0.1
0 7,631.63 7,631.63 111.42 111.42 7,846.08 7,846.08
1,500 6,488.78 6,488.61 91.84 91.42 6,685.74 6,684.25
2,500 6,391.82 6,366.79 78.54 75.08 6,549.64 6,517.81
3,500 6,380.82 6,353.72 78.88 75.25 6,530.00 6,497.45
(b) θ=0.2
0 7,976.02 7,976.02 204.57 204.57 8,395.07 8,395.07
1,500 6,788.14 6,784.49 182.44 179.68 7,192.98 7,185.69
2,500 6,671.00 6,645.71 149.01 143.96 7,030.38 6,990.74
3,500 6,666.09 6,614.52 154.87 146.25 7,006.21 6,937.27
(c) θ=0.3
0 8,389.44 8,389.44 369.82 369.82 9,046.43 9,046.43
1,500 7,096.40 7,089.95 272.83 269.47 7,847.98 7,838.24
2,500 6,963.85 6,931.39 251.54 241.73 7,494.99 7,441.00
3,500 6,957.23 6,878.45 297.76 278.49 7,520.06 7,414.78
Robust Optimization Model for a Dynamic Network Design Problem 387
lower objective function values than the corresponding deterministic approach,
especially at a high demand uncertainty level and a high investment budget level.
Numerous future research directions remain. First, the RDNDP model with
various types of uncertainty set including a polyhedral uncertainty set or an
ellipsoidal uncertainty set should be investigated to find a less conservative solution.
Second, the ambiguous chance-constrained programming can be applied to the
model when we have more information about the uncertain data. For example, this
approach may be particularly interesting when we only know the support and mean
of uncertain parameters or when we know that demand can arise from a set of
distributions. Third, while we dealt with the RDNDP with a single-destination,
system-optimum network setting, which has potential applications in emergency
evacuation planning, optimal traffic detouring, lower-bound evaluation of traffic
systems, etc., the user-optimal and multi-destination versions of the same problem
are worth further investigation and evaluation along the track of RO.
Acknowledgment This work was partially supported by the grant awards CMMI-0824640 and CMMI-
0900040 from the National Science Foundation.
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