Sustainable Transportation Network Design Incorporating Environment Disruption under Strategic User Equilibrium

Transcription

1 Sustainable Transportation Network Design Incorporating Environment Disruption under Strategic User Equilibrium Xiang Zhang (Corresponding Author) School of Civil and Environmental Engineering, University of New South Wales Sydney, NSW,, Australia, S. Travis Waller School of Civil and Environmental Engineering, University of New South Wales, and National ICT Australia (NICTA) Sydney, NSW,, Australia, David Rey School of Civil and Environmental Engineering, University of New South Wales Sydney, NSW,, Australia, Melissa Duell School of Civil and Environmental Engineering, University of New South Wales Sydney, NSW,, Australia, Submitted to be considered for Presentation at the th Annual Meeting of the Transportation Research Board and Publication in the Journal of the Transportation Research Board: Transportation Research Record Word count:, words text + tables x words + figure x words =, words Submission Date: July,

2 Zhang, Waller, Rey, Duell ABSTRACT This study addresses a sustainable network design problem (NDP), where sustainability is considered from the aspects of uncertainty and environment disruption. Few previous works integrated both factors into the NDP. In this study, the mathematical framework of the strategic user equilibrium (StrUE) traffic assignment under volatility of both total travel demand and link capacity is analysed, which reflects disequilibrium observed in traffic networks. We incorporate the StrUE traffic assignment model into a network design project. We propose a multiobjective bi-level program for the NDP and formulate two objective functions of minimizing the expected total system travel time and minimizing the expected total system off-gas emissions in the uncertainty-based NDP. To solve the sustainable NDP, a tailored exact solution approach is developed based on the ε-constraint method to find global and exact Pareto optimal solutions. Finally, a series of computational experiments are conducted to test the efficiency of the proposed approach. The results show that ignoring StrUE with uncertainty considerations can result in sub-optimal design solutions in terms of expected network performance in long-term transportation planning decisions. The results also report the trade-offs between travel time and off-gas emissions in the uncertainty-based NDP. Keywords: network design problem, strategic user equilibrium, sustainability, environment disruption, uncertainty

3 Zhang, Waller, Rey, Duell. INTRODUCTION The network design problem (NDP) identifies the optimal set of arcs in a network for addition or improvement. Most NDPs aim to maximize social benefits subject to resource constraints (e.g. budget), while maintaining the requirement that flow patterns satisfy the user equilibrium conditions (). One of the challenges in NDPs arises from its bi-level property that captures the interactions between network planners and users in the decision process. In the upper level, the planners seek the optimal capacity allocation policies. In the lower level, users choose their routes to minimize their own travel costs and this traffic assignment process can be represented by network equilibrium conditions. This work seeks to address two research gaps identified in the field of NDPs. First, most previous studies assume fixed travel demands and link capacities, interpreted as expectations, based on which traffic assignment is conducted and design options are evaluated. The traffic assignment models employed in previous works primarily adhere to the Wardropian user-equilibrium (UE) principles, but the results may not be consistent with the reality that the equilibrium conditions are not always observed on a day-to-day basis. Further, the design solutions derived from a deterministic demand are not resilient to scenario volatility as compared to those based on demand uncertainty (). Even so, the traffic assignment models under deterministic user equilibrium (DUE) conditions have been adopted frequently in NDP formulations because of computational efficiency and solution consistency. Unlike DUE, the strategic user equilibrium (StrUE) traffic assignment was proposed accounting for the uncertainty of traffic properties in the process of route choice behaviour (). One of the strengths of the StrUE assignment is that it is consistent with the lack of network equilibrium observed in reality, and due to core assumptions, it can be applied on practically sized networks and applications. However, it has not been extensively used in NDPs. Second, previous studies have not paid enough attention to sustainability concerns in the NDPs, which should include uncertainty consideration and environment conservation (). Specifically, ignoring sustainability considerations might underrate network-wide impacts and potentially, undermine the performance of the transportation system. For example, it has been reported by Ozan et al. () that the exclusion of environmental degradation in the NDP might lead to higher network-wide emissions and lower acceptability of system-wide policies. However, the integration of the environment concern with the uncertainty-based NDP has rarely been explored. Therefore, this study is proposed to investigate the behaviour of sustainable NDPs incorporating both uncertainty and environment disruption, where the StrUE traffic assignment and multiobjective optimization are employed. This is the primary contribution of this study. The remainder of this paper is organized as follows. Section summarizes some of relevant literature. Section proposes the model formulation of multiobjective sustainable NDP under uncertainty. Section develops a tailored solution methodology. In Section, a series of computational experiments are conducted over the test network. Conclusions and future work are discussed in Section.. LITERATURE REVIEW The network design problem is primarily formulated as a bi-level program because of its ability to describe the interactions between network planners and users. The upper level represents a network-wide design resource allocation problem and the lower level represents travellers route choice behaviour. Various advanced bi-level programming formulations for the NDP have been proposed. Yang et al. () and Farahani et al. () presented comprehensive reviews of research outcomes in NDPs. Chen et al. () reviewed literature about uncertainty considerations in NDPs.

4 Zhang, Waller, Rey, Duell In terms of network design objectives, most models emphasized the minimization of an indicator related to travel time (-). Previous studies accounted for multiple objectives in NDPs from different aspects including environmental justice (), value of time (), traffic emissions (), and degree of robustness (). In terms of solution methodologies, due to the intrinsic complexity of the general nonconvex bi-level formulation of the NDP, heuristic algorithms are primarily employed, including genetic algorithms (GA) (), ant colony optimization (), simulated annealing algorithms (), and particle swarm optimization (). As an exception, Fontaine et al proposed an exact approach for an approximated NDP using Benders Decomposition (). Despite the considerable time consumption of exact solution algorithms, these methods are of vital importance to investigate model behaviour, since they can generate global and exact optimal solutions for the proposed NDP models. With the global concern in stable development and environmental conservation, scholars have paid more attention to sustainability considerations in the decision-making process (). Sustainability is a comprehensive concept and associated to considerations regarding uncertainty and environment (-). To address a NDP under demand uncertainty, Ukkusuri et al. () took trip matrices as random variables, proposed a robust transportation network design. Yin et al. () applied different versions of the mean variance model to the capacity enhancement problem under demand uncertainty. Both studies drew the conclusion that exclusion of uncertainty could lead to sub-optimal investments. Additionally, in uncertainty-based NDPs, reliability analysis is frequently involved. Chen et al. () proposed an alternative reserve capacity model, which aims at minimizing the probabilities of link failures. Additionally, measuring the impact of network design policies on environmental disruption has emerged as a rich topic (). Sushant Sharma et al. () explored multiobjective NDPs considering emission and travel time, and infer that design results show respectable improvements by applying multiple objectives compared with minimizing travel time alone. Ramani et al. () presented a framework for sustainability assessment of transportation planning by taking environment as a major concern. However, in the context of sustainable NDPs, few previous studies considered demand uncertainty and capacity variability simultaneously within a NDP. Duell et al. () integrated both the aforementioned uncertainties into the evaluation of network design projects. In this study, the strategic user equilibrium (StrUE) traffic assignment model under uncertainty was adopted as the lower-level program of the NDP formulation, based on which the total system travel time under uncertainty was evaluated at the upper level. StrUE assignment was originally proposed by Dixit et al. () and then extended by Wen et al. () to include the link capacity uncertainty to StrUE assignment. The model will be further employed in this study. This study extends the StrUE traffic assignment with both demand and capacity uncertainty to the multiobjective sustainable NDP incorporating environment disruption. Some previous work has focused on uncertainty and environment considerations in the NDP, but rarely in combination or aggregation. This study addresses this unresolved issue.. MODEL FORMULATION In this section, we analyse the StrUE traffic assignment model considering both demand and capacity uncertainties. We then propose a multiobjective sustainable NDP incorporating uncertainty and environment disruption under StrUE conditions.. Notation Notations used throughout this paper are listed in Table unless otherwise specified.

5 Zhang, Waller, Rey, Duell N A A O D Z Π Π rs (i, j) c ij c ij c ij x ij TABLE Mathematical Notations Node set Link set Set of potentially expanded or added links Origin node set Destination node set OD pair set: Z N N Path set Path set for OD pair (r, s) Link with upstream node i N and downstream node j N Capacity of link (i, j) Initial capacity of link (i, j) A unit capacity expansion for link (i, j) Flow on link (i, j) x Link flow vector [x ij ] t ij Travel time on link (i, j) t ij Free flow travel time on link (i, j) S ij Travel speed on link (i, j) FFS ij Free flow speed on link (i, j) T Total travel demand q rs Proportion of total travel demand between OD pair (r, s), where (r,s) q rs = p π,rs Proportion of total travel demand on path π connecting OD pair (r, s) f ij Proportion of total travel demand on link (i, j), where (i,j) A f ij = f Link proportion vector, f = [f ij ] h π Flow on path π Π rs d rs Travel demand for OD pair (r, s) π δ ij Link-path incidence coefficient, which equals one if link (i, j) is on path π, and zero otherwise g Probability distribution for total travel demand φ ij Probability distribution for capacity of link (i, j) y ij The degree to which link (i, j) is expanded y Link capacity expansion vector, y = [y ij ] L ij Length of link (i, j) γ ij The unit expenditure of addition or improvement for link (i, j) B Total budget available TSTT Total system travel time TSOE Total system off-gas emission E() Expected value of a variable Var() Variance of a variable. StrUE Traffic Assignment under Uncertainty The primary assumption of StrUE traffic assignment is that every network user chooses their own route in order to minimize their expected travel time between an origin and destination, where their travel time varies based on the conditions they encounter during travel. Furthermore, every user makes their route choice decision by considering the uncertainties of traffic properties, i.e. travel demand uncertainty and link capacity uncertainty, but will adhere to the predetermined strategy in their daily trip. The StrUE model assumes that users know the distribution of traffic properties, but do not know with perfect knowledge the conditions they will experience during daily travel. Thus,

6 Zhang, Waller, Rey, Duell users need a strategy to facilitate their travel behaviour and here, the strategy from users is to choose the paths with least expected costs. The result is a network equilibrium based on link proportions (not link flows), which represents the proportions of total demand travelling on links in the network. Under this circumstance, link proportions remain the same when the total demand and link capacities change day to day. In this context, link flows and congestion conditions (indexed by flow-capacity ratios) fluctuate as the result of the random variables for total travel demand and link capacity. In contrast to the deterministic user equilibrium (DUE), the total travel demand and link capacities are not deterministic in the context of StrUE. The StrUE traffic assignment can be formulated into the optimization problem as follows: P: Subject to f ij min z(f) = t ij (f T) g(t) φ ij (c)dt dc df f (i,j) A q rs = p π,rs () (r, s) Z () rs p π,rs π Π rs () f ij = δ π ij p π,rs (i, j) A () (r,s) Z π Π rs where g(t) is the probability density function (PDF) of the total travel demand; φ ij (c) is the PDF of capacity of each link (i, j). In order to prove that the above optimization problem is equivalent to StrUE conditions, which state that users always choose their expected shortest paths under flow conservations, the Lagrangian Function of the above problem is explored. It can be obtained by incorporating constraint () into the objective function with a set of Lagrangian multipliers τ = {τ rs } as follows: L(p, τ) = z[f(p)] + τ rs (q rs p π,rs ) () π Π rs π Π rs The general optimality conditions of the optimization problem P are: L(p, τ) p π,rs π Π rs () p π,rs L(p, τ) p π,rs = π Π rs () L(p, τ) = τ τ rs τ () rs p π,rs π Π rs () By simplifying Formulas () through () using Constraints (), the optimality conditions can be transformed as follows: E(t π,rs ) τ rs π Π rs () p π,rs (E(t π,rs ) τ rs ) = π Π rs () q rs = p π,rs (r, s) Z () π Π rs p π,rs, π Π rs () In this context, τ rs serves as the least expected path travel time with regards to the uncertainties of

7 Zhang, Waller, Rey, Duell the travel demand and link capacity. Formulas () and () indicate that any used path must have the minimum expected travel time. The proportion of any path where expected travel time is greater than the minimum one will not be used. Formula () is link-path proportion conservation relationship. Formula () indicates non-negative path proportions. Formulas () through () derived from the optimization problem P demonstrate the equivalence between P and the StrUE conditions as stated earlier. In these conditions, t π,rs means the expected travel time of path π connecting OD pair (r, s), considering volatilities of both total travel demand and link capacity, which is formulated as follows: E(t π,rs ) = E ( δ π ij t ij ) = δ π ij E(t ij ) E(t ij ) = (i,j) A (i,j) A t ij (f ij T) g(t) φ ij (c)dt dc π Π rs () (i, j) A () For each scenario with certain T and c ij, (i, j) A, the widely used Bureau of Public Roads (BPR) function for link travel time is employed in StrUE (): t ij = t ij ( + α ( f β ij T ) ) (i, j) A () c ij The total demand T and link capacity c ij, (i, j) A are random variables. It can be assumed that the total demand is distributed lognormally (). The link capacity can be assumed as a uniformly distributed random variable, and the distribution for each link is independent. One of the strengths of uniform distribution is that we can define the nonzero probability density interval of uniform distribution so that the possible values of the link capacity can be limited to a certain domain and cannot be too large. It coheres with the reality that the link capacity is impossible to become infinite. Another strength is that the inverse of the capacity, denoted as c inv adheres to the inverse uniform distribution, whose analytical expression is obtainable and tractable along with the capacity expansion in NDPs. Previous works applied Gamma distribution to link capacity, which has a drawback in mathematical restrictions where the coefficient of variance of the capacity must be lower than % so as to get feasible solutions (). Instead, the uniform distribution can relax this hypothesis, which will be shown later in computational experiments. Thus, the expected link travel time E(t ij ) in equation () can be re-written as follows: E(t ij ) = t ij (f ij T) g(t) ω ij (c inv )dt dc inv (i, j) A () g(t) follows a lognormal distribution with parameters μ and σ, which are respectively the mean and standard deviation of the variable ln(t) normally distributed. ω ij (c inv ) is the PDF of inverse of the capacity of each link (i, j). It follows an inverse uniform distribution on the interval [b ij, a ij ], with respect to uniform distribution φ ij (c) on the interval [a ij, b ij ] showing the upper and lower bounds of capacity on link (i, j). With these predetermined parameters, the analytical expression of link travel time t ij in Equation () can be solved: E(t ij ) = t ij + α (f ij ) β G β Ω ij,β () where, G β is the βth moment of the lognormal distribution for the total travel demand T: G β = exp (βμ + β σ ) () Ω ij,β is the βth moment of the inverse uniform distribution for inverse of link capacity c inv,ij,

8 Zhang, Waller, Rey, Duell (i, j) A: (b ij a ij )(β ) (a β ij b β ij ) β >, β, β Ω ij,β = ln ( b ij ) b ij a ij a ij β = { a ij b ij β =. Sustainable NDP Formulation The network design problem is, in most cases, formulated as a bi-level nonlinear programming model. The objective of the NDP is to optimize planning objectives with the requirement that flow patterns satisfy the equilibrium conditions. This work extends the sustainability considerations into the framework of network design problem. A sustainable network should have following primary characteristics. First, the network should have sufficient capacity to satisfy traffic demands and cope with substantial volatility in terms of both demand and capacity. Second, traffic off-gas emissions should be controlled or lowered in order to maintain appropriate standards regarding air quality. Thus, network sustainability can be summarized by the following two major factors: i) uncertainty; ii) environment disruption. In our study, we use the total system off-gas emissions, denoted as TSOE, to evaluate the environment disruption of transport system. Various sorts of off-gas are emitted by motor vehicles, including hydrocarbons (HC), nitrous oxides (NO x ), carbon monoxide (CO), carbon dioxide (CO ), etc. Among them, CO emission accounts for the vast majority of all emission volumes and incurs harmful global warming (), which this study cares about. Based on the above analysis, our sustainable NDP is formulated as follows: P: Upper level program: subject to () min z TT = E(TSTT(f)) y () min z OE = E(TSOE(f)) y () y ij =,,, y ij,max (i, j) A () γ ij y ij B () (i,j) A var(tstt) R TT () var(tsoe) R OE () where the link proportion pattern f = [f ij ] can be obtained by solving the StrUE traffic assignment problem below: Lower level program (StrUE assignment by specifying the inverse of link capacity): f ij min z(f) = t ij (f T) g(t) ω ij (c inv )dt dc inv df f (i,j) A subject to Constraints () through () In Formulation P, the objective is to minimize the expected TSTT and TSOE with respect to all the scenarios with different total travel demands and link capacities under StrUE conditions. y is the discrete decision vector of the optimization problem, which indicates the extent to which each ()

9 Zhang, Waller, Rey, Duell link is expanded or added. Constraint () limits the extent to which a link can be expanded. Constraint () limits budget allocations. Constraints () and () are concerned with reliability, where the variance can be interpreted as the measure of reliability in a network design setting. R TT and R OE are the upper thresholds of var(tstt) and var(tsoe) respectively, which transport planners can consider as part of the decision-making process. Therefore, Formulation P integrates sustainability into the NDP from the aforementioned perspectives, where StrUE under uncertainty is applied, and the expected TSTT and TSOE are incorporated. Remark Besides consideration of environment conservation, one of the reasons why it is significant to consider both travel time and off-gas emissions in design objectives is that an essential difference exists between these two factors: travel time is cared about by both travellers and network planners, while travellers determine their route choice behaviour primarily based on travel time and ignore other factors including off-gas emissions. TSTT and TSOE for each scenario with deterministic travel demand and link capacities can be calculated as follows: TSTT = f ij T t ij = f ij T t ij ( + α ( f β ij T ) ) () c ij TSOE = (i,j)εa f ij T LTOE ij (i,j)εa (i,j)εa where LTOE ij is the link transport off-gas (CO ) emissions for link (i, j), which can be calculated based on the CO Emission Functions from Motor Vehicle Emission Simulator (MOVES) (): LTOE ij = TOE ij L ij = S. ij L ij = [t ij ( + α ( f β ij T ) )] c ij. L ij. where TOE ij is the transport off-gas emission rate on link (i, j). TOE ij is measured in gram per kilometre if t ij is measured in hours and L ij in kilometres. The expectations of TSTT and TSOE with respect to the uncertainties of travel demand and link capacities are as follows: E(TSTT) = f ij T t ij ( + α(f ij T) β c β inv ) g(t) ω ij (c inv )dt dc inv (i,j)εa = (f ij t ij G + α t ij (f ij ) β+ G β+ Ω ij,β ) (i,j)εa E(TSOE) = f ij T [t ij ( + α(f ij T) β c inv β )]. L ij. (i,j)εa g(t) ω ij (c inv )dt dc inv = f ij (L ij ). (t ij ). T ( + α(f ij T) β c inv β ). (i,j)εa g(t) ω ij (c inv )dt dc inv () Equation () provides the analytical expression of the expected total system travel time. By contrast, the expectation of total system off-gas emissions is not analytically calculated due to its nonlinear relationship with the link travel time, but it can be evaluated by using a simulation-based method, e.g. numerical integration. () () ()

10 Zhang, Waller, Rey, Duell For the reliability constraints, the variances of TSTT and TSOE are determined as follows: var(tstt) = E(TSTT ) (E(TSTT)) () var(tsoe) = E(TSOE ) (E(TSOE)) () Based on the accessibility of analytical expressions of the expectations, var(tstt) can be solved by analytical equations, while var(tstt) can be measured by simulation-based procedure. Additionally, in the context of NDP under StrUE conditions, c ij represents the modified capacity on link (i, j), which has the following relationship with the initial capacity c ij : c ij = c ij + y ij c ij () c ij and c ij (i, j) A are independent random variables with the uniform distribution in the StrUE assumptions. Additionally, it is assumed that c ij is a constant, which indicates that after capacity expansion, the mean value of capacity is increased by y ij c ij while the variance remains the same. Mathematically, if the domain [a, b ] of the PDF φ ij (c ij ) is non-empty, c ij and c inv,ij respectively adhere to uniform distribution on [a + y ij c ij, b + y ij c ij ] and inverse uniform distribution on [(b + y ij c ij ), (a + y ij c ij ) ].. SOLUTION METHODOLOGY In this section, we propose a tailored exact solution methodology to our proposed sustainable NDP problem. For the multiobjective Formulation P, we seek to estimate the Pareto front composed of all the nondominated solutions of the form (z TT, z OE ). In this work, a solution s = (z TT, z OE ) is said to be nondominated if no solution s with cost (z TT, z OE ) is found such that: z TT > z TT and z OE z OE, or z TT z TT and z OE > z OE ; otherwise s is said to be dominated. In order to obtain various Pareto optimal solutions, we employ the ε-constraint method (). Using this approach, one of the objective functions is passed as a constraint to form a single objective program and a cost parameter ε is used to limit the amount of resources used for the constrained objective function. In this study, one of the objective functions of the expected TSTT and TSOE can be converted to the additional ε-constraint in the upper-level program, and we take the former (i.e. TSTT) in the rest of this section. The resulting single-objective bi-level program can be solved by an enumeration method. The major steps of the solution procedure are summarized as follows: Step Transformation to a single-objective problem Convert the upper-level multiobjective optimization to a single-objective program using the ε-constraint method: P: min z OE = E(TSOE(f)) () y subject to: E(TSTT(f)) ε () and Constraints () through () Step Determine reliability constraints Solve the StrUE traffic assignment problem on the initial unmodified network to obtain the link proportion f, and evaluate the expectations and variances of TSTT(f ) and TSOE(f ). The thresholds R TT and R OE can be determined as follows:

11 Zhang, Waller, Rey, Duell R TT = ρ var(tstt(f )) () R OE = ρ var(tsoe(f )) () where ρ and ρ are nonnegative parameters. Step Determine ε values Set the lower and the upper bounds on parameter ε: ) The upper bound ε u is set to be E(TSTT(f )) of the initial network; ) Solve the NDP with single-objective E(TSTT(f)) under StrUE, which is the problem by removing the objective function of E(TSOE(f)) from P, and obtain link proportion f. The lower bound ε l is set to be E(TSTT(f )). The solution method can be found in Step. Substitute any selected ε j within the domain [ε l, ε u ] into Formulation P. Values of ε j can be determined in many ways, for example: Divide [ε l, ε u ] into k equidistant intervals and set each ε j as a dividing point: { ε j = ε u j step j =,,,, k () step = (ε u ε l )/k Step Solve problems P and obtaining Pareto optimal solutions Determine the feasible region D with regards to Constraint () with regards to link expansion. Narrow domain D to D based on Constraint () with regards to budget limitation. Implement the StrUE assignment with each link capacity expansion vector y within D. Evaluate the expectations of TSTT(f) for each design option y within domain D and narrow D to D based on the ε-constraint (). Evaluate the variances of TSTT(f) and TSOE(f) for each y within D and narrow D to D based on Constraints () and () with regards to reliability. Compute the objective functions of TSOE(f) for each design option y within D, select the one with the lowest TSOE(f) and take it as a Pareto optimal solution y j along with (z TT,j, z OE,j ). Find out the Pareto optimal solutions under different ε j using the same method. Step Extract final results Output the NDP decision vectors, flow proportions, and values of objective functions. Terminate. Remark The StrUE traffic assignment problem can be solved by MSA method or Frank-Wolfe algorithm. Unlike the DUE assignment problem, the link travel time function considering uncertainties should be used, i.e. Equation (). Another point is that the link proportion vector f, rather than the link flow pattern x, is updated and identified by the algorithm. Remark In Step, the values of parameters ρ and ρ depends on the planners preference. When they are equal to one, it means that the reliability of the modified network should not be worse than that of the initial network. If the reliabilities are of high importance in the decision process, the values should fluctuate between zero and one. Remark In Step, two alternative intelligent methods can be used to determine the domain D, provided that the equal maximum capacity expansion for each potentially expanded or added link is used, i.e. y ij,max = y max (i, j) A : ) Base-number-conversion method: Convert each number within domain [, (y max + ) A ] in the decimal system to the respective number in the base (y max + ) system,

12 Zhang, Waller, Rey, Duell where A denotes the number of potentially expanded or added links. If the digit of the derived number is less than A, then substitute zeros into all the digits before the first non-zero digit. As a result, the number of each digit represents the value y ij for each link. ) Matrix-based algorithm. The majors steps are summarized in Figure : () Input: A = A ; y = y max ; () Initialize: Matrix F = zeros(y^a, A); () for every column k (from to A) in Matrix F () r = A k; () Update of column vector v: () v = [ zeros(y^r, ) ; ones(y^r, ); ; (y ) ones(y^r, ) ]; () Assignment of column k of F: F(:, k) = repmat(v, y^(k ), ); () Extract: Each row vector of Matrix F represents a feasible solution under Constraint () () End Note: notations are in line with MATLAB FIGURE Matrix-based algorithm Unlike some other exact methods, e.g. Benders Decomposition, the proposed approach does not have to transform the bi-level program to a single-level program or make a linear approximation before solving it. In addition, the proposed approach sorts out the feasible solutions before invoking the traffic assignment or evaluating objective functions, so that no penalty term is required. Further, the proposed approach improves the computational efficiency by evaluating each design option only once. By contrast, some heuristic algorithms inevitably make repeated calculations on some solutions at different iterations, such as genetic algorithms.. COMPUTATIONAL RESULTS AND DISCUSSION In this section, the above model and solution approach are tested. We conduct a series of computational experiments on a well-known synthesized medium network Nguyen-Dupuis Network, which has been used for extensive studies on NDPs before ()()(). The topology of the network can be found in Reference (). The parameters used are summarized in Table. In this case, a binary decision vector y is adopted, wherein y ij equals if link (i, j) is expanded, or otherwise. The model and algorithm are coded and implemented in the programming software Python.. TABLE Nguyen-Dupuis Network Parameters I. Link Information Link ID Free Flow Free Flow Initial Parameters in BPR Length Speed Travel time Capacity a function (mi) (mi/h) (min) (veh/h) α β

13 Zhang, Waller, Rey, Duell II. Origin OD Demands (veh/h) b Destination III. NDP Parameters Unit capacity expansion c ij =, (i, j) A Cost of link capacity expansion γ ij = $ million, (i, j) A Budget B = $ million a. Expectations of initial capacity b. Expectations of demands Throughout the experiments, we adjust the uncertainty degrees of both the total travel demand and link capacity, which are respectively indexed by the standard deviation of total travel demand (Sd t ) and the size of capacity interval [a ij, b ij ] (I ij = b ij a ij ). Here, in order to facilitate the implementation, we assume the harmonious I ij = I for all the links of the network, which means that each link has the same standard deviation of capacity as per the assumed uniform distribution. Additionally, for reliability concerns, the thresholds R TT and R OE in Constraints ()() respectively equal var(tstt) and var(tsoe) of the initial unmodified network, which means that the modified network cannot deteriorate either one as compared to the base case. We use the proposed exact solution methodology to solve the problem. In the first experiment, we convert the objective E(TSOE) to the additional ε-constraint in the upper-level program, and set the ε as the E(TSOE) of the initial network, resulting in a StrUE-NDP with single objective E(TSTT). In the second experiment, we do the same work but switch the roles of E(TSTT) and E(TSOE). Some of the results that cover varying combinations (Sd t, I) regarding uncertainty levels are summarized in Table. TABLE Evaluation of StrUE-NDP Design Solutions as compared to the Base Case a I. The st Experiment Reductions Reductions Problems E(TSTT) (min) E(TSOE) (g) Problems E(TSTT) (min) E(TSOE) (g) No. Sd t I Abs % Abs % Abs % Abs % Dev b Dev c No. Sd Dev Dev t I Dev Dev Dev Dev

14 Zhang, Waller, Rey, Duell II. The nd Experiment Reductions Reductions Problems E(TSTT) (min) E(TSOE) (g) Problems E(TSTT) (min) E(TSOE) (g) No. Sd t I Abs % Abs % Abs % Abs % No. Sd Dev Dev Dev Dev t I Dev Dev Dev Dev a. Design solutions of StrUE-NDP in the st and nd experiments are respectively the expanded link sets {,,,,,} and {,,,,,} b. Abs Dev: Absolute deviation c. % Dev: Percentage deviation From Table, it can be observed that there is a significant improvement in the expected network performance in terms of both travel time and off-gas emissions for the StrUE-NDP solutions, as compared to the base case. Specifically, the travel time reduction and the CO off-gas emission reduction with regards to the base case are respectively, minutes per hour and,, grams per hour on average. The percentage values of above reductions are respectively around % and % on average. Actually, these magnitudes highly depend on the network topology and construction budget. Further, in each experiment the project rankings of the StrUE-NDP, represented by the expanded link sets, remain the same under various uncertainty parameters, which means that in this example the design solutions derived from the proposed StrUE-NDP model are compatible with different uncertainty levels. A potential reason is that we assume an equal standard deviation for all the links in this case study and because the network is not large enough to display significant route switching behaviour. Another insight from Table is that the reductions in expected TSTT and TSOE tend to increase with the increase in the uncertainty degrees. This suggests that the variability of both travel demand and link capacity have impacts on the evaluation of network design projects. In addition, we solve the StrUE-NDP model and the DUE-NDP model separately in both experiments. Then we evaluate the design solutions of the StrUE-NDP and the DUE-NDP with regards to E(TSTT) and E(TSOE). This is because in the real world, traffic conditions among a

15 Zhang, Waller, Rey, Duell network vary with volatility in day-to-day travel demands and road capacities and the expected network performance makes sense in the evaluation of design projects. In order to quantify the percentage difference between the performance of StrUE-NDP solutions and DUE-NDP solutions, we introduce the indicator R as follows: R = E Str E D () where E Str represents reduction in expected TSTT or TSOE for StrUE-NDP solutions, with regards to the base case; E D represents the aforementioned reduction for DUE-NDP solutions. The greater the value of R is, the larger extent to which the StrUE-NDP model achieves more reduction in E(TSTT) or E(TSOE) than the DUE-NDP model is implied. The results are summarized in Table. TABLE Comparison of StrUE-NDP Solutions and DUE-NDP Solutions a I. The st Experiment Evaluation of design solutions Problem StrUE-NDP DUE-NDP b R (%) No. E(TSTT) E(TSOE) E(TSTT) E(TSOE) Sd t I (min) (g) (min) (g) E(TSTT) E(TSOE) II. The nd Experiment Evaluation of design solutions Problem StrUE-NDP DUE-NDP R (%) No. E(TSTT) E(TSOE) E(TSTT) E(TSOE) Sd t I (min) (g) (min) (g) E(TSTT) E(TSOE)....

16 Zhang, Waller, Rey, Duell a. Design solutions of DUE-NDP in the st and nd experiments are both expanded link set {,,,,,} b. For evaluation of DUE-NDP design solutions, we solve the StrUE assignment with the above expanded link set, obtain the link proportion vector f and calculate E(TSTT) and E(TSOE) under uncertainty. Table presents the solutions of the DUE-NDP with the expected values of travel demand and link capacities. Although the design solutions under DUE conditions improve the expected network performance as compared to the base case, there exist significant divergencies between solutions of the DUE-NDP and the StrUE-NDP. First, different network design project rankings are derived from these two sorts of problems. Specifically, for the StrUE-NDP, the dertermined expanded link sets are {,,,,,} and {,,,,,} in the st and nd experiments respectively; while for the DUE-NDP, the expanded link set is {,,,,,} in both experiments. Second, the design solutions of the StrUE-NDP achieve more reductions in both expected TSTT and expected TSOE as compared to those of the DUE-NDP. Especially, as per R values, the StrUE-NDP solutions achieve around % percent more reduction in the expected TSTT than the DUE-NDP solutions. For the expected TSOE, this percentage is between % and %. The absolute values for above differences are approximate, minutes for E(TSTT) and, grams off-gas emissions for E(TSOE) during a specific period of time ( hour) on average. It means that the StrUE-NDP is more applicable to the long-term transportation planning under uncertainty, despite that the DUE-NDP can generate the best solution for the scenario where both total travel demand and link capacity equal their respective expectations. The results report the impacts of uncertainty considerations on the design solutions and the strengths of the StrUE-NDP over the DUE-NDP. Further, both Table and Table reveal the trade-offs between travel time and off-gas emissions in the StrUE-NDP, through each pair of the same numbered problems in two experiments. Each pair of problems provides two different decision vectors and demonstrates that a benefit of reducing the expected total system travel time can lead to a sacrifice in terms of the increase in the expected total system off-gas emissions, and vice versa. The expected TSTT and TSOE do not necessarily reach the optimal conditions simultaneously. In order to more closely examine the relationship between total travel time and off-gas

17 Zhang, Waller, Rey, Duell emissions, we conduct a third experiment. In the third experiment, the values of ε in the employed ε-constraint method are modified, and we solve the proposed problem. It turns out that for each of the above problems, only two Pareto optimal solutions are found, which are listed in Table, i.e. (E(TSTT), E(TSOE)) of two same numbered problems. This behaviour is primarily because both travel time and off-gas emissions are assumed to increase with traffic flow, which means that improving either of them will probably result in the improvement in the other. However, the travel time and off-gas emissions do not increase at the same rate. Hence a trade-off may exist between these two metrics, as demonstrated in the experiments. Future work will examine this relationship for other harmful emissions, which may not exhibit the same behaviour.. CONCLUSIONS AND FUTURE DIRECTIONS This study proposes a bi-level programming formulation for the multiobjective sustainable network design problem (NDP), incorporating uncertainty and environment disruption. The strategic user equilibrium (StrUE) traffic assignment model considering variability of both travel demand and link capacity is analysed and employed in the lower-level program. Two objective functions of expected total system travel time and expected total system off-gas emissions under uncertain realizations are formulated in the upper-level program, together with the reliability constraints. In order to solve the proposed model to a guaranteed global optimal value, a tailored exact solution approach is developed, based on an enumeration method and the ε-constraint method. Finally, the efficacy of the proposed approach is tested via computational experiments using the Nguyen-Dupuis Network. We report that the proposed approach is capable of generating high-quality design solutions in terms of expected network performance under uncertainty. Further, the computational results demonstrate that accounting for the StrUE under uncertainty in NDPs has significant impacts on the network expansion policies. It can help yield design solutions which achieve more reductions in expected TSTT and TSOE with regards to the volatility of scenarios, as compared to the NDP under deterministic user equilibrium (DUE). Moreover, the trade-off between travel time and off-gas emissions is found and analysed in uncertainty-based NDPs. This study can be extended in multiple directions. First, relationships between expected network performance and reliability metrics under StrUE conditions will be further explored. Second, more comprehensive evaluation of environment disruption of the stochastic transport system will be researched in sustainable NDPs. Third, assumptions regarding fixed proportions and independent link capacity distributions within the StrUE traffic assignment should be relaxed. Finally, more efficient exact solution methodologies will be developed to address large-scale networks. REFERENCES. Chiou, S.W. Bilevel Programming for the Continuous Transport Network Design Problem. Transportation Research Part B: Methodological, Vol., No.,, pp.-.. Sharma, S., S. Ukkusuri, and T. Mathew. Pareto Optimal Multi-objective Optimization for Robust Transportation Network Design Problem. In Transportation Research Record: Journal of the Transportation Research Board, No., Transportation Research Board of the National Academies, Washington, D.C.,, pp.-.. Dixit, V., L. Gardner, and S.T. Waller. Strategic User Equilibrium Assignment under Trip Variability. Presented at nd Annual Meeting of the Transportation Research Board, Washington, D.C.,.. Cantarella, G. E., and A. Vitetta. The Multi-criteria Road Network Design Problem in an Urban Area. Transportation, Vol., No.,, pp.-.

18 Zhang, Waller, Rey, Duell. Ozan, C., S. Haldenbilen, and H. Ceylan. Estimating Emissions on Vehicular Traffic Based on Projected Energy and Transport Demand on Rural Roads: Policies for Reducing Air Pollutant Emissions and Energy Consumption. Energy Policy, Vol., No.,, pp.-.. Yang, H., and M.G.H. Bell. Models and Algorithms for Road Network Design: a Review and Some New Developments. Transport Reviews, Vol., No.,, pp.-.. Farahani, R.Z., E. Miandoabchi, W.Y. Szeto, and H. Rashidi. A Review of Urban Transportation Network Design Problems. European Journal of Operational Research, Vol., No.,, pp... Chen, A., Z. Zhou, P. Chootinan, S. Ryu, C. Yang, and S.C. Wong. Transport Network Design Problem under Uncertainty: a Review and New Developments. Transport Reviews, Vol., No.,, pp.-.. Luathep, P., A. Sumalee, W.H.K. Lam, Z.C. Li, and H.K. Lo. Global Optimization Method for Mixed Transportation Network Design Problem: A Mixed-integer Linear Programming Approach. Transportation Research Part B: Methodological, Vol., No.,, pp.-.. Wang, D.Z., and H.K. Lo. Global Optimum of the Linearized Network Design Problem with Equilibrium Flows. Transportation Research Part B: Methodological, Vol., No.,, pp.-.. Duthie, J., and S.T. Waller. Incorporating Environmental Justice Measures into Equilibrium-based Network Design. In Transportation Research Record: Journal of the Transportation Research Board, No., Transportation Research Board of the National Academies, Washington, D.C.,, pp.-.. Szeto, W.Y., and H.K. Lo. Transportation Network Improvement and Tolling Strategies: The Issue of Intergeneration Equity. Transportation Research A: Policy and Practice, Vol., No.,, pp.-.. Ferguson, E.M., J. Duthie, and S.T. Waller. Comparing Delay Minimization and Emissions Minimization in the Network Design Problem. Computer-Aided Civil and Infrastructure Engineering, Vol., No.,, pp.-.. Karoonsoontawong, A., and S.T. Waller. Robust Dynamic Continuous Network Design Problem. In Transportation Research Record: Journal of the Transportation Research Board, No., Transportation Research Board of the National Academies, Washington, D.C.,, pp.-.. Zhang, X., H. Wang, and W. Wang. Bi-level Programming Model and Algorithms for Stochastic Network with Elastic Demand. Transport, Vol., No.,, pp.-.. Poorzahedy, H., and F. Abulghasemi. Application of Ant System to Network Design Problem. Transportation, Vol., No.,, pp... Friesz, T.L., H. Cho, N.J. Mehta, R.L. Tobin, and G. Anandalingam, A Simulated Annealing Approach to the Network Design Problem with Variational Inequality Constraints. Transportation Science, Vol., No.,, pp.-.. He, Y., S. Yang, and Q. Xu. Short-term Cascaded Hydroelectric System Scheduling Based on Chaotic Particle Swarm Optimization using Improved Logistic Map. Communications in Nonlinear Science and Numerical Simulation, Vol., No.,, pp.-.. Fontaine, P., and S. Minner. Benders Decomposition for Discrete Continuous Linear Bilevel Problems with application to traffic network design. Transportation Research Part B: Methodological, Vol.,, pp.-.. Barrella, E., and A. Amekudzi. Backcasting for Sustainable Transportation Planning. In Transportation Research Record: Journal of the Transportation Research Board, No., Transportation Research Board of the National Academies, Washington, D.C.,, pp.-.

19 Zhang, Waller, Rey, Duell. Ukkusuri, S.V., and G. Patil. Multi-period Transportation Network Design under Demand Uncertainty. Transportation Research Part B: Methodological, Vol., No.,, pp.-.. Zietsman, J., L.R. Rilett, and S.J. Kim. Sustainable Transportation Performance Measures for Developing Communities. No. SWUTC//-,.. Ukkusuri, S.V., T.V. Mathew, and S.T. Waller. Robust Transportation Network Design under Demand Uncertainty. Computer-Aided Civil and Infrastructure Engineering, Vol., No.,, pp.-.. Yin, Y., S.M. Madanat, and X. Lu. Robust Improvement Schemes for Road Networks under Demand Uncertainty. European Journal of Operational Research, Vol., No.,, pp.-.. Chen, A., P. Chootinan, and S.C. Wong. New Reserve Capacity Model of a Signal-controlled Road Network. In Transportation Research Record: Journal of the Transportation Research Board, No., Transportation Research Board of the National Academies, Washington, D.C.,, pp.-.. Cantarella, G.E., and A. Vitetta. The Multi-criteria Road Network Design Problem in an Urban Area. Transportation, Vol., No.,, pp.-.. Sharma, S., and T.V. Mathew. Multiobjective Network Design for Emission and Travel-time Trade-off for a Sustainable Large Urban Transportation Network. Environment and Planning B: Planning and Design, Vol., No.,, pp.-.. Ramani, T., J. Zietsman, H. Gudmundsson, R. Hall, and G. Marsden. Framework for Sustainability Assessment by Transportation Agencies. In Transportation Research Record: Journal of the Transportation Research Board, No., Transportation Research Board of the National Academies, Washington, D.C.,, pp.-.. Duell, M., and S.T. Waller. The Implications of Volatility in Day-to-day Travel Flow and Road Capacity on Traffic Network Design Projects. In Transportation Research Record: Journal of the Transportation Research Board, No., Transportation Research Board of the National Academies, Washington, D.C.,, pp.-.. Wen, T., L. Gardner, V. Dixit, M. Duell, and S.T. Waller. A Strategic User Equilibrium Model Incorporating Both Demand and Capacity Uncertainty. Presented at rd Annual Meeting of Transportation Research Board, Washington, D.C.,.. National Research Council. Highway capacity manual, th Edition, Washington D.C.: Transportation Research Board,, pp Brilon, W., J. Geistefeldt, and M. Regler. Reliability of Freeway Traffic Flow: a Stochastic Concept of Capacity. In Proceedings of the th International Symposium on Transportation and Traffic Theory. Vol.,.. Carey, J. Global Warming: Faster Than Expected?. Scientific American, Vol., No.,, pp.-.. U.S. Environmental Protection Agency (U.S. EPA). Motor Vehicle Emission Simulator (MOVES), User Guide for MOVES, EPA--B--, July, pp Ehrgott, M. Multicriteria Optimization, nd Edition. Springer Science & Business Media, Berlin,, pp.-.