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Robust Dynamic Continuous Network Design Problem Ampol Karoonsoontawong and S. Travis Waller A robust optimization model is presented for the dynamic traffic assignment–based continuous network design problem, which accounts for a bilevel objective and long-term origin–destination demand uncertainty. The model also embeds Daganzo’s cell transmission model. The objective minimizes the trade-off between expected total system travel time (TSTT) and expected risk. As such, the robust model provides the optimal solution that is least sensitive to the variation of travel demand, given the degree of robustness by transportation planners. The new robust model is compared with the existing network design models on a simple cell transmission test network. The robust model with greater degree of robustness yields less expected risk with the sacrifice of higher expected TSTT. The robust model yields the most robust solution, and no other model provides a satisfactory solution across the budget range. In addition, how a visualized graph may be used to elicit the preference information from transportation planners on the desired degree of robustness is illustrated.

LITERATURE REVIEW Typically, the NDP objective function minimizes the total system travel time (TSTT) subject to the conservation of flow conditions and budget constraint or minimizes the TSTT plus the cost converted to equivalent time unit subject to the conservation of flow conditions. The NDP formulations can be classified according to the following four criteria: (a) system-optimal (SO) or user-optimal (UO) behavior, (b) static or dynamic traffic assignment, (c) discrete or continuous investment variable, and (d) deterministic or stochastic parameters. SO behavior is mathematically tractable but unrealistic, whereas UO behavior typically complicates the problem but is more realistic. The static traffic assignment assumes the steady-state condition, whereas the dynamic traffic assignment (DTA) accounts for time dynamics. The discrete investment variable allows entire-lane or new link addition, whereas the continuous investment permits a fraction of lane addition. The continuous investment variable has been used extensively in the literature, with the justification that because most roads in the urban area are already constructed, discrete variables are not practical. The continuous link expansion can be implemented by altering lane width, median, and shoulder area. Also, the continuous NDP can be considered as a possible heuristic for the discrete NDP. Last, the problem parameters typically have been considered deterministic as opposed to stochastic. There is little literature on the DTA-based NDP. Janson showed that the DTA-based NDP model is more desirable than the static model (2). Waller et al. proposed a continuous NDP formulated as a linear programming model (3), in which the users’ route choices are based on the single-destination SO DTA linear programming model with fixed-departure-time O-D demands introduced by Ziliaskopoulos (4). The SO DTA linear programming model propagates traffic according to the cell transmission model (CTM), a traffic flow theoretical model by Daganzo (5). Ukkusuri and Waller proposed a continuous NDP formulated as a linear programming model (6), in which the users’ route choices are based on the UO DTA linear programming model by Ukkusuri (7), also employing Daganzo’s CTM. Jeon et al. employed a genetic algorithm to solve the discrete NDP by allowing either one lane addition or none (8). Waller and Ziliaskopoulos introduced the stochastic SO DTA-based NDP with long-term O-D demand uncertainty (9), formulated as a two-stage stochastic linear program with recourse (SLP2) and a chance-constrained program (CCP). Ukkusuri et al. introduced the UO versions of SLP2 and CCP models (10). Karoonsoontawong and Waller conducted a comprehensive comparison of the SO and UO versions of SLP2 models (11). Karoonsoontawong and Waller also proposed the formulation and exact solution methods for the linear bilevel program of continuous NDP based on the multiorigin single-destination UO DTA, and they developed three metaheuristics for the multiorigin, multidestination, larger-size problem: simulated annealing, genetic algorithm, and

The network design problem (NDP) is bilevel by nature and can be seen as a static version of the noncooperative, two-person game introduced by Von Stackelberg in the context of unbalanced economic markets (1). With the assumption of perfect information, the static game means that each player has only one move. The leader (transport planners) goes first, attempts to minimize the total costs, and anticipates all possible responses of his opponent, the follower (road users). The follower observes the leader’s decision and reacts such that the follower achieves an optimal benefit (individual’s least travel time) regardless of external effects (systemwide costs). Furthermore, the NDP relies on several input parameters, especially a long-term origin– destination (O-D) demand matrix, which is traditionally treated as deterministic. As estimated future population, demographics, and land use patterns are used to estimate the long-term O-D demand matrix, the O-D demands are indeed random variables. This paper proposes a new robust optimization model based on dynamic traffic assignment linear programming models when accounting for bilevel nature and long-term O-D demand uncertainty. The proposed model yields a solution that is the least sensitive to the variation of O-D demands, given the planner’s preference on the degree of robustness. The major contributions of this paper are two new formulations of dynamic continuous NDPs: a two-stage stochastic linear bilevel program and a robust optimization model.

A. Karoonsoontawong, 6.512 ECJ Hall, and S. T. Waller, 6.204 ECJ Hall, Department of Civil Engineering, University of Texas, Austin, TX 78712. Corresponding author: A. Karoonsoontawong, [email protected]. Transportation Research Record: Journal of the Transportation Research Board, No. 2029, Transportation Research Board of the National Academies, Washington, D.C., 2007, pp. 58–71. DOI: 10.3141/2029-07

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Karoonsoontawong and Waller

59

random search (12). Ukkusuri addressed the robust NDP based on static user equilibrium under O-D demand uncertainty by using a genetic algorithm (13).

CONCEPT OF ROBUST OPTIMIZATION Robust optimization integrates the methods of multiobjective programming into stochastic optimization (14). It can be seen as an extension of stochastic optimization by introducing the higher moments of the distribution of objective value and explicitly handling the violation of constraints. The importance of controlling variability of the solution as opposed to just optimizing its first moment is well recognized in other applications, such as portfolio management (15). Robust optimization models have two distinct components: a structural component and a control component (14 ). The structural component (or the first-stage component) is fixed and free of any noise in input data, and the control component (or the second-stage component) is subjected to noisy input data. First consider a standard form of linear program: min ξ = cT x + d T y n n

(1)

redundancies built in (14). Mulvey et al. proposed the concepts of solution robust and model robust. An optimal solution is said to be solution robust if it remains close to optimal for any realization s ∈ Ω; that is, the solution is robust with respect to optimality. An optimal solution is said to be model robust if it remains almost feasible for any realization s ∈ Ω; that is, the solution is robust with respect to feasibility. Thus, they proposed a robust optimization model that can measure the trade-off between solution and model robustness and take a multicriteria objective form: min σ ( x , y1 , . . . , ys ) + ωρ ( z1 , . . . , z s )

(5)

subject to Ax = b

(6)

Bs x + Cs ys + z s = es x ≥ 0, ys ≥ 0

∀s ∈Ω

∀s ∈ Ω

(7) (8)

where

Ax = b

(2)

Bx + Cy = e

(3)

x, y ≥ 0

(4)

ys = vector of control variable associated with the realization s ∈ Ω; zs = vector of errors that measures the infeasibility allowed in the control constraints for the realization s ∈ Ω; σ(.) = objective that is a measure of optimality robustness; ρ(.) = feasibility penalty function that is a measure of model robustness; and ω = goal programming weight used to find a range of answers that trade off solution robustness for model robustness.

There are two sets of decision variables. The first, x ∈ R n1, is the vector of design variables. Its optimal value does not depend on the realization of the uncertain parameters, and it cannot be adjusted when the uncertain data are realized. The second, y ∈ R n2, is the vector of control decision variables. Its optimal value is conditioned on both the realization of uncertain parameters and the optimal value of the design variables, and it is subjected to adjustment when the uncertain parameters are observed. Constraint 2 is the set of structural constraints whose coefficients are fixed and free of noise. Constraint 3 is the set of control constraints whose coefficients are subject to uncertainty. Define a set of scenarios Ω = {1, 2, 3, . . . , S}. Each scenario, s ∈ Ω, is associated with the set {ds, Bs, Cs, es} of realizations of uncertain parameters with the probability of the scenario ps and Σs∈Ω ps = 1. Mulvey et al. pointed out that it is unlikely that any solution to the linear program of 1 through 4 will remain both feasible and optimal for all scenario s ∈ Ω unless the modeled system has substantial

In stochastic optimization, the recourse variables or second-stage decision variables, which can be adjusted to account for the data realizations, are the same as the control variables in robust optimization. Also, the first-stage decision variables in stochastic optimization are the same as the design variables in robust optimization. In a stochastic environment, the objective function ξ becomes a random variable. For scenario s, the associated ξ is ξs = cT x + d Ts ys with probability ps. The measure of optimality robustness can set differently depending on applications. One possible functional form of σ(.) is the mean value as used in stochastic linear programming [i.e., σ(.) = Σs∈Ω ps ξs]. Another is the maximum value as used in the worst-case analysis [i.e., σ(.) = maxs∈Ω ξs]. Mulvey et al. suggested that σ(.) can include the higher moments of the distribution of ξs that is a novelty of robust optimization model (14 ), for example, a utility function that represents a trade-off between mean value and variability. Since it may not always be possible to obtain a feasible solution to a problem under all scenarios, this is normally handled outside the optimization model. As another novelty, the robust optimization model generates solutions with the least amount of infeasibilities to be dealt with outside the model. The second term in the objective function is used to penalize violations of the control constraints under some scenarios. The appropriate penalty functional form is problem dependent, and it also implies the proper solution algorithm. Mulvey et al. considered a quadratic penalty function (ρ(z1, . . . , zs) = Σs∈Ω pszTszs), which is applicable to equality constrained problems where both positive and negative violations of the control constraints are equally unfavorable (14 ). The resulting model is a quadratic program.

x ∈R 1 , y ∈R

2

subject to

where x ∈ R n1, y ∈ R n2, c ∈ R n1, d ∈ R n 2, b ∈ R m1, e ∈ R m2, A ∈ R m1 × n1, B ∈ R m2 × n1, and C ∈ R m2 × n2 .

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Although robust optimization has some advantages over stochastic optimization, it does have limitations. First, a robust optimization model is a parametric program, and there is no a priori procedure for specifying a correct choice of the goal programming weight ω. This problem is prevalent in multicriteria programming (16). Second, robust optimization as well as stochastic optimization do not provide a mechanism for specifying the best set of realizations of the uncertain parameters. Mulvey et al. (14 ) suggested that the methods for integrating the variance reduction techniques into stochastic optimization (17) also apply to robust optimization. The next section reviews the relevant existing dynamic continuous network design models.

linear programming formulation developed by Ziliaskopoulos (4). Then, Ukkusuri (7) formulated the deterministic user-optimal network design model (UONDP) and the two-stage stochastic UONDP model (SLP2-UONDP) based on the UO DTA linear programming model by Ukkusuri (7). Karoonsoontawong and Waller proposed the deterministic bilevel programming NDP model (BLPNDP) (12). This paper formulates its two-stage stochastic linear bilevel version (SLP2-BLPNDP) and the robust bilevel programming NDP model (RO-BLPNDP). For the stochastic and robust models, only the O-D demands are considered uncertain with known discrete probability distribution.

REVIEW OF EXISTING FORMULATIONS OF DTA-BASED CONTINUOUS NDP

Deterministic Network Design Models

There are five relevant, existing DTA-based NDP models. These models, together with the two new models, are limited to fixeddeparture-time, single-destination O-D demands. Continuous investment variables are assumed to change practical capacity but not free-flow travel time, and the improvement cost is a linear function of the improvement levels. It is noted that the convex piecewise linear improvement cost functions can easily be incorporated. Notation for sets is as follows: C CS CR T E ES FS(i) RS(i)

= = = = = = = =

set of cells, set of sink cells, set of source cells, set of discrete time intervals, set of cell connectors, set of sink cell connectors, set of cell connectors emanating from cell i, and set of cell connectors emanating to cell i.

Notation for deterministic parameters is as follows: TAB = total available budget, δit = ratio of link free-flow speed and backward propagation speed for each cell and time interval (=1 in this paper), Mt = cost per time interval that yields UO flows, ξ i = initial number of vehicles in cell i (= 0 in this paper), dit = demand at cell i in time interval t, Nit = maximum number of vehicles in cell i at time interval t, Q it = maximum number of vehicles that can flow into or out of cell i during time interval t, χi = increase in N ti per one unit of bi (= χ in this paper), φi = increase in Q ti per one unit of bi (= φ in this paper), dit = demand at cell i in time interval t, and ~ ξ = vec( dit ∀i ∈ C, t ∈ T) = vector of stochastic parameters. Any parameter with a tilde is a stochastic parameter. In this paper, ~ only one uncertain parameter is considered, namely, dit (stochastic time-dependent demand), which is assumed independent across time intervals and cells. Notation for variables is as follows: bi = amount of budget allocated to cell i, xit = number of vehicles in cell i at time interval t, and y ijt = number of vehicles moving from cell i to cell j at time interval t. Waller (18) and Waller et al. (3) formulated the deterministic system-optimal network design model (SONDP) and the two-stage stochastic SONDP model (SLP2-SONDP) based on the SO DTA

The three deterministic NDP models (SONDP, UONDP, and BLPNDP) share the same set of constraints. The first set is Budget Constraints 9 and 10, and the second is the CTM Constraints 11 through 20. Let budget_constraint(b) denote the following constraints:

∑b

i

≤ TAB

(9)

i ∈C \ CS

bi ≥ 0

∀i ∈ C \ C S

(10)

The network improvements are constrained by a user-specified TAB. Variable bi is defined as the amount of the budget spent on cell i. This allows the model to compute improvements for street segments smaller than a link. However, this degree of flexibility is not considered in this study. It will be seen in the computational results that the budgets spent throughout a particular link are always identical because of the underlying traffic flow theoretical CTM model; otherwise, a bottleneck that causes traffic congestion can take place on the link. Next, let CTM(x, y, b) represent the following constraints: xit − xit −1 +

∑

(i , j )∈FS (i )

yijt−1 −

∑

y tji−1 = dit

∀i ∈ C \ CS , t ∈ T (11)

∀i ∈ C \ C S , t ∈ T

(12)

( j ,i )∈RS (i )

∑

yijt − xit ≤ 0

∑

y tji ≤ δ it ( N it + χ i i bi − xit )

∑

y tji ≤ Qit + φi i bi

∀i ∈ C \ ( C R ∪ C S ) , t ∈ T

(114)

∑

yijt ≤ Qit + φi i bi

∀i ∈ C \ C S , t ∈ T

(15)

(i , j )∈FS (i )

( j ,i )∈RS (i )

( j ,i )∈RS (i )

(i , j )∈FS (i )

∀i ∈ C \ (C R ∪ CS ) , t ∈ T (13)

xi0 = ζi

∀i ∈ C \ C S

(16)

yij0 = 0

∀ (i, j ) ∈ E

(17)

xi|T | = 0

∀i ∈ C \ C S

(18)

xit ≥ 0

∀i ∈ C \ C S , t ∈ T

(19)


∀ (i, j ) ∈ E , t ∈ T

yijt ≥ 0

61

(20)

Parameter χi specifies the increase in the jam density of cell i for one unit of TAB, and φi denotes the increase in the saturation flow of cell i for one unit of TAB. Unlike the traditional static NDP models (nonconvex nonlinear programs), the DTA-based NDP models have only linear terms, that is, Nit and Qit in the DTA models are replaced with Nit + χi bi and Qit + φi bi. Two basic traffic flow relationships are embedded in the CTM constraints. The first is the cell mass conservation (Constraint 11). The second states that the traffic flow between two cells is constrained by the number of vehicles occupying the upstream cell (Constraint 12), the remaining capacity of the downstream cell (Constraint 13), and the maximum flow that can get out of the upstream cell and into the downstream cell (Constraints 14 and 15). One important constraint (Constraint 18) enforces that all vehicles must reach the destination by the last time interval. Without this constraint, the optimal solution is erroneous; that is, all vehicles do not leave the origin. For each cell, the constants Nit, Qit, and δit can vary with time to capture time-dependent capacity and flow variations. Daganzo showed that these constants together with the length of the cell can reasonably capture the road characteristics (5). Typically, these constants should be obtained from the calibration by using loop detector data (19). However, for simplicity, arbitrarily assume the values of Nit and Qit for the test problems, and set all δ ti values equal to 1.

System Optimal Network Design Model

subject to budget_constraint(b)

(25)

CTM ( x , y, b )

(26)

The same drawback of SONDP and UONDP models is that they cannot represent the problem correctly. As discussed previously, the nature of the NDP is bilevel. Thus, the better deterministic model is the BLPNDP described next. It is noted that the derivation of Mt is referred to by Ukkusuri (7 ).

Bilevel Network Design Model The objective for the BLPNDP (12) is to determine the optimal network improvements such that TSTT is minimized under UO DTA behavior. The linear continuous bilevel model is min b

∑ ∑(t i y ) ( )

b ,x , y i , j

t ij

(21)

∈ES t ∈T


(22)

CTM ( x , y, b )

(23)

(27)

budget_constraint(b)

(28)

∑ ∑( M

(29)

x ,y

min

t

ij

t ∈T

subject to

min

The objective for SONDP is to determine the optimal network improvements so as to minimize TSTT under SO DTA behavior. The linear continuous SONDP model is

∑ ∑(t i y )

(i , j )∈ES

(i , j )∈ES

t ∈T

t

i

yij ) t

subject to CTM ( x , y, b )

(30)

The three deterministic NDP models assume all parameters are known and deterministic. These are obviously unrealistic as the parameters, especially O-D demands, are uncertain. The improved approach to model the NDP is the two-stage stochastic linear programming (SLP2) described in the next section.

Two-Stage Stochastic Network Design Models TSTT basically equals the difference between the summation of arrival times (arrival) at the sink cell and the summation of departure times (departure) from the source cells. Arrival is Σ(i,j)∈Es Σt∈T (t yijt ). Since the underlying DTA linear programming model assumes fixed departure-time O-D demands, departure is constant. Thus, departure can be dropped from the optimization model without an effect on the optimal solution. In this paper, TSTT is referred to as Σ (i,j)∈Es Σt∈T (t yijt ).

The three two-stage stochastic programs (SLP2-SONDP, SLP2UONDP, and SLP2-BLPNDP) and the robust model (RO-BLPNDP) use Budget Constraints 9 and 10 and the following CTM constraints corresponding to each realization ω of uncertain parameters (i.e., O-D demands). Let CTMω(x, y, b) ∀ ω ∈ Ω denote Constraints 31 through 40: xit ,ω − xit −1,ω +

∑

(i , j )∈FS (i )

yijt−1,ω −

UO Network Design Model

b ,x , y

∑ ∑( M

(i , j )∈ES

t ∈T

t

i

yijt )

y tji−1,ω = dit ,ω

∀i ∈ C \ C S , t ∈ T

The objective for UONDP is the same as that for the UO DTA linear programming model. Thus, it is to determine the optimal network improvements such that the individual users minimize their own travel time. It is noted that the TSTT is not optimized in this model. The linear continuous UONDP model is min

∑

( j ,i )∈RS (i )

(24)

∑

yijt,ω − xit ,ω ≤ 0

∑

y tji,ω ≤ δ it ( N it + χ i i bi − xit ,ω )

(i , j )∈FS (i )

( j ,i )∈RS (i )

: π i0 ,t ,ω

∀i ∈ C \ CS , t ∈ T : π1i ,t ,ω

∀i ∈ C \ ( C R ∪ C S ) , t ∈ T

(31) (32)

: π 2 ,t ,ω

(33)

62


∑

y tji,ω ≤ Qit + φi i b

∑

yijt,ω ≤ Qit + φi i b

( j ,i )∈RS (i )

(i , j )∈FS (i )

∀i ∈ C \ ( C R ∪ C S ) , t ∈ T

∀i ∈ C \ C S , t ∈ T

: π i4 ,t ,ω

: π 3i ,t ,ω (34)

(35)

x 0,ω = ζi i

∀i ∈ C \ C S

(36)

yij0 ,ω = 0

∀ (i, j ) ∈ E

(37)

∀i ∈ C \ C S

(38)

x

|T |,ω ij

=0

min ∑ b ,x , y

ω ∈Ω

∑ ∑( p

(i , j )∈ES

ω

i

t ,ω

t i yij

t ∈T

)

(43)


(44)

CTMω ( x , y, b ) ∀ω ∈Ω

(45)

Two-Stage Stochastic UO Network Design Model

xit ,ω ≥ 0

∀i ∈ C \ C S , t ∈ T

(39)

The recourse function of SLP2-UONDP (10) is~the UO DTA corresponding to the realized uncertain parameters ( ξω ) and the assigned budgets (b):

yijt,ω ≥ 0

∀ (i, j ) ∈ E , t ∈ T

(40)

h b, ξ ω = min

Next, the standard two-stage linear stochastic network design model can be written as

( )

(

)

ω ω min E ⎡⎣ h b, ξ ⎤⎦ = ∑ ⎡⎣ p i h b, ξ ⎤⎦ b ω ∈Ω

(41)

(42)

The timing of what one knows and when one knows it is of fundamental importance for two-stage stochastic programming and robust optimization. Here, the order of the events is articulated for clarity. The first-stage decisions are the amount of budget allocated to each cell i (bi) and are made with the knowledge of the distribution of stochastic parameters. Thereafter, a realization of stochastic ~ ~ parameters (d it,ω of dit ) unfolds. After that, the scenario-dependent, second-stage decisions are determined from corresponding DTA; that is, the number of vehicles assigned to cell i at time t for scenario ω (xit,ω) and the number of vehicles traversing on cell connectors between cell i and cell j at time t for scenario ω (y ijt,ω ). The first-stage decisions cannot depend on the scenario ω because this would correspond to knowing the future. ~ The recourse function h(b, ξω) varies depending on the problems as described in the next subsections.

Two-Stage Stochastic System Optimal Network Design Model The recourse function of SLP2-SONDP (18) is~the SO DTA corresponding to the realized uncertain parameters ( ξω) and the assigned budgets (b):

(

)

h b, ξ ω = min x ,y

∑ ∑t i y

( i , j )∈ES

)

x ,y

∑ ∑M y

( i , j )∈ES

t

t ,ω

ij

t ∈T

subject to CTMω ( x , y, b ) After substituting this recourse function into the standard SLP2 (Constraints 41 and 42), the resulting linear program of SLP2-UONDP is


(

t ,ω

ij

t ∈T

subject to CTMω ( x , y, b ) After substitution of this recourse function into the standard SLP2 formulation (Constraints 41 and 42), the resulting linear program of SLP2-SONDP for a single destination problem is

min ∑ b ,x , y

∑ ∑( p

ω ∈Ω ( i , j )∈ES t ∈T

ω

i

t ,ω

M t i yij

)

(46)


(47)

CTMω ( x , y, b ) ∀ω ∈Ω

(48)

Like the deterministic case, the drawback of the SLP2-SONDP and SLP2-UONDP models is that these single-level models cannot correctly characterize the stochastic NDP as the nature of the problem requires the bilevel structure. The next section proposes the two new dynamic continuous network design models.

TWO NEW FORMULATIONS OF DTA-BASED CONTINUOUS NDP A two-stage stochastic bilevel NDP program (SLP2-BLPNDP) and a robust optimization NDP model (RO-BLPNDP) are formulated on the basis of BLPNDP (12) when accounting for the long-term O-D demand uncertainty. SLP2-BLPNDP minimizes the expected TSTT, whereas RO-BLPNDP minimizes the weighted summation of the expected TSTT and the expected risk. The basic idea behind ROBLPNDP is as follows. It is possible that the solution of the SLP2BLPNDP yields the least-expected TSTT, but with wildly different TSTT for various O-D demand realizations. This can be handled by incorporating the higher moments of the distribution of objective value in RO-BLPNDP. However, according to the nature of the problem, the higher moments are not considered appropriate, and a risk term is adopted, which is a squared up-side deviation of TSTT from a fixed target TSTT. The rationale will be described later. These two new models share the same standard two-stage stochastic model (Constraints 41 and 42) with their recourse functions described in the next subsections.


63

Two-Stage Stochastic Bilevel Network Design Model The recourse function of SLP2-BLPNDP is TSTT calculated from the ~ UO DTA corresponding to the realized uncertain parameters ( ξω) and the assigned budgets (b):

(

h b, ξ

ω

) = ( ∑) ∑ t

i

BLPNDP), since only the first-stage decision variables are used to optimize the objective. Next, the objective function of the robust optimization model is the trade-off between the expected cost and the expected risk term (14), and there are certain possible objective functions: min z + λ i ∑ pω ( z ω − z ) b

t ,ω ij

y

ω ω min z + λ i ∑ p z − z

i , j ∈ES t ∈T

b

ω ∈Ω

subject to x,y

(i , j )∈ES

(

min z + ∑ pω λ u ( z ω − z ) + λ d ( z ω − z )

∑ ∑M y

min

2

ω ∈Ω

t ,ω t ij

b

t ∈T

ω ∈Ω

+

−

)

min z + λ i max ω∈Ω

b

subject to where

CTMω ( x , y, b )

zω =

After substitution of this recourse function into Constraints 41 and 42, the linear continuous bilevel SLP2-BLPNDP model is min ∑ b

∑ ∑( p

ω

i

ω ∈Ω ( i , j )∈ES t ∈T

t i yijt,ω )

z=

(z (z

(49)

subject to budget _ constraint ( b ) min x ,y

∑ ∑M y

(i , j )∈ES

t

t ,ω ij

∀ω ∈Ω

(51)

subject to CTMω ( x , y, b )

∀ω ∈Ω

(52)

Robust Bilevel Network Design Model First it is shown that the robust NDP has to be bilevel. Consider SLP2-SONDP and SLP2-UONDP. The objective functions of these two stochastic programs are minimizing their expected costs. A naive application of robust optimization could be simply the addition of the risk term. For example, the risk term of SLP2-SONDP could be min ∑

∑ ∑( p

ω

i

b , x , y ω ∈Ω ( i , j )∈E t ∈T S

t i yijt,ω )

⎛ ⎞ + λ i ∑ p ⎜ ∑ ∑ ( t i yijt,ω ) − ∑ ∑ ∑ ( pω i t i yijt,ω )⎟ ⎝ (i , j )∈ES t ∈T ⎠ ω ∈Ω ω ∈Ω ( i , j )∈ES t ∈T

2

where λ is a nonnegative goal programming weight (or a degree of robustness). Clearly, this quadratic program does not have a meaningful implication, since the second-stage decision variables ( yijt ) can play a role to reduce the risk. In other words, transportation users make route choice decisions such that the weighted summation of the expected cost and risk is minimized; this destroys the assumption of SO DTA condition. Therefore, the robust optimization cannot be applied to the single-level models such as SLP2-SONDP and SLP2-UONDP. It can be applied only to the bilevel model (SLP2-

t ∈T

∑ ∑ ∑( p

ω

i

ω ∈Ω ( i , j )∈ES t ∈T

t i yijt,ω ),

− z ) = max ( 0, z ω − z ) ,

ω

− z ) = max ( 0, − ( z ω − z )) , and

+

−

λ u , λ d = differential weights on upside and downside risk. The last objective function includes the worst-case of risk term, and it is considered too conservative. For the first three objective functions involving a moving target (i.e., the target TSTT is the expected TSTT, –z ), it is possible that the optimal solution yields very low risk but very high expected cost. This is indeed not desirable. Thus, the moving target in the risk term should not be used, and the fixed target (i.e., the target TSTT is a constant) is used in this study. If transportation planners have the information on the ideal expected TSTT, then it can be set to the target TSTT. Otherwise, in this study, the optimal objective function of the SLP2-BLPNDP is used as the target TSTT (TSTTtarget). Further, because of the nature of the problem, the upside and downside risk should not be considered symmetric. That is, it is preferred that the realized TSTT of a scenario is less than the target TSTT. Thus, the downside risk is ignored, and only the upside risk term is taken into account. Specifically, the adopted objective function of the robust model is

(

ω ω target min z + λ i ∑ p ( z − TSTT ) b

ω

(i , j )∈ES

t ,ω ij

ω

(50)

t ∈T

∑ ∑ ( t i y ),

ω ∈Ω

)

+ 2

Formally, the robust optimization model can be viewed as a twostage stochastic program, Constraints 41 and 42. The recourse function of RO-BLPNDP is the trade-off between TSTT and risk term, which are calculated from the UO DTA corresponding to the realized uncertain parameters (ξω) and the assigned budgets (b):

(

)

h b, ξ ω = min ω D

∑ ∑(t i y ) + λ i ( D )

(i , j )∈ES

ω 2

t ,ω ij

t ∈T

subject to Dω ≥

∑ ∑ ( t i y ) − TSTT

(i , j )∈ES

Dω ≥ 0

t ,ω ij

t ∈T

target

64

min x ,y


∑ ∑M y

(i , j )∈ES

t

grams in SLP2-BLPNDP (Constraints 51 and 52) can be represented as a single nested program:

tω ij

t ∈T

min ∑

subject to

∑ ∑M y

x , y ω ∈Ω ( i , j ) ∈E t ∈T S

CTMω ( x , y, b )

t ,ω ij

t

subject to It can be noticed that the objective function of the recourse function together with the first two constraints implies that only the upside deviation from TSTTtarget is included in the risk term. After substituting this recourse function into Constraints 41 and 42, the resulting continuous quadratic-linear bilevel program of RO-BLPNDP is

∑ ∑( p

minω ∑

ω

i

b , D ω ∈Ω ( i , j )∈ES t ∈T

(

t i yijt,ω ) + λ ∑ pω ω ∈Ω

i

(D ) ) ω 2

(53)

subject to budget_constraint ( b ) ω

D ≥

∑

(i , j )∈ES

∑(t i y

t ,ω ij

t ∈T

(54)

) − TSTT

target

∀ω ∈ Ω

CTM ( x , y, b )

However, the modified Kth best algorithm is a heuristic and cannot guarantee an exact optimal solution as discussed by Karoonsoontawong and Waller (12). The mixed integer programming (MIP) reformulation based on Karush–Kuhn–Tucker (KKT) (12) is still applicable to SLP2BLPNDP by replacing the follower’s programs (51 and 52) with their KKT conditions. This yields a single-level nonlinear program, which can be further linearized. The final program is an MIP: min ∑ b

(555)

∀ω ∈ Ω

∑ ∑( p

ω ∈Ω ( i , j )∈ES t ∈T

ω

i

t i yijt,ω )

(59)

subject to ω

D ≥0 min x ,y

∀ω ∈ Ω

∑ ∑M y

(i , j )∈ES

t

t ,ω ij

∀ω ∈Ω

(56)

budget_constraint ( b )

(57)

KKT_UODTA ω ( x , y, π , z , b )

t ∈T

subject to CTMω ( x , y, b )

∀ω ∈Ω

(58)

Only the solution robustness in RO-BLPNDP is considered, and the model robustness is not, since the model is feasible for every scenario ω. In the next section, the solution methods for the two new models are described. It can be seen that RO-BLPNDP with λ = 0 is SLP2-BLPNDP.

(60) ∀ω ∈Ω

(61)

KKT_UODTAω(x, y, π, z, b) represents three sets of linear constraints: primal feasibility (Constraints 31 through 40), dual feasibility (Constraints 62 through 68), and linearized complementary slackness (Constraints 69 through 74).

Dual Feasibility ∀i ∈ C R , t ∈ T \ { T } : xit ,ω

π i0 ,t ,ω − π i0 ,t +1,ω + π1i ,t ,ω ≤ 0

(62)

π i0 ,t ,ω − π i0 ,t +1,ω + π1i ,t ,ω − δ it π i2 ,t ,ω ≤ 0 SOLUTION METHODS The single-level models (SONDP, UONDP, SLP2-SONDP, and SLP2-UONDP) are linear programs and can be directly solved by using any linear programming solver (CPLEX in this study). The solution methods for BLPNDP are discussed by Karoonsoontawong and Waller (12), where BLPNDP is an instance of a linear bilevel programming problem, which is strongly NP-hard (20). There is not an efficient algorithm to find an exact solution. To illustrate the usefulness of the proposed models when compared with the existing NDP models, a reformulation technique is used, similar to that used by Karoonsoontawong and Waller (12), to find exact solutions. The solution methods for SLP2-BLPNDP and RO-BLPNDP are discussed next.

Solution Method for SLP2-BLPNDP The modified Kth best algorithm of Karoonsoontawong and Waller (12) can be applied to SLP2-BLPNDP, since the multiple nested pro-

∀i ∈ C \ (C R ∪ CS ) , t ∈ T \ { T } : xit ,ω

(63)

π i0 ,t +1,ω − π 0j ,t +1,ω − π1i ,t ,ω − δ it π 2j ,t ,ω − π 3j ,t ,ω − π i4 ,t ,ω ≤ 0 ∀ ( i , j ) ∈ E \ ES , t ∈ T \ { T } : yijt,ω

(64))

π i0 ,t +1,ω − π1i ,t ,ω − π i4 ,t ,ω ≤ M t

∀ ( i , j ) ∈ Es , t ∈ T \ { T } : yijt,ω

(65)

π i0 ,t ,ω unrestricted in sign

∀i ∈ C \ C S , t ∈ T

(66)

π1i ,t ,ω , π i4 ,t ,ω ≥ 0

∀i ∈ C \ C S , t ∈ T

(67)

π i2 ,t ,ω , π 3i ,t ,ω ≥ 0

∀i ∈ C \ ( C R ∪ C S ) , t ∈ T

(68)

Note that π i0,t,ω, π 1,t,ω , π 2,t,ω, π 3,t,ω , and π 4,t,ω are dual variables i i i associated with Constraints 31, 32, 33, 34, and 35.


65

Complementary Slackness

Solution Method for RO-BLPNDP

Let gi1,t,ω ≥ 0 ∀i ∈ C \ CS, t ∈ T, denote the inequality 32; let gi2,t,ω ≥ 0 ∀i ∈ C \ (CR ∪ CS), t ∈ T, denote 33; let gi3,t,ω ≥ 0 ∀i ∈ C \ (CR ∪ CS), t ∈ T, denote 34; and let gi4,t,ω ≥ 0 ∀i ∈ C \ CS, t ∈ T, denote 35. Let git,ω ≥ 0 ∀i ∈ C \ CS, t ∈ T \ {⎟ T⎟}, denote the inequality of 62 and 63 and gijt,ω ≥ 0 ∀(i, j) ∈ E, t ∈ T \ {⎟ T⎟}, denote 64 and 65. The complementary slackness conditions are

The continuous quadratic-linear bilevel program (53 through 58) contains all linear terms except the quadratic term in the upper-level objective function. It can be shown that the upper-level objective function is convex quadratic as follows. Since the weight λ, the probability pω ∀ ω ∈ Ω, and the squared terms (Dω)2 ∀ ω ∈ Ω are nonnegative, these can be rearranged as yTQy, where y is the vector of variables and Q is the vector of coefficients. Clearly, the quantity y TQy is nonnegative (yTQy ≥ 0) for all nonzero real vector y; thus Q is positive semidefinite and the quadratic upper-level objective function is convex. The nested linear programs of 57 and 58 are replaced with their linearized KKT conditions, yielding the convex mixed-integer nonlinear program (MINLP):

gi1,t ,ω i π1i ,t ,ω = 0

∀i ∈ C \ C S , t ∈ T

(69)

gi2 ,t ,ω i π i2 ,t ,ω = 0

∀i ∈ C \ ( C R ∪ C S ) , t ∈ T

(70)

gi3,t ,ω i π i3,t ,ω = 0

∀i ∈ C \ ( C R ∪ C S ) , t ∈ T

(71)

4 ,t ,ω i

g

i

π

4 , t ,ω i

=0

∀i ∈ C \ C S , t ∈ T

(72)

git ,ω i xit ,ω = 0

∀i ∈ C \ C S , t ∈ T \ { T }

(73)

gijt,ω i yijt,ω = 0

∀ (i, j ) ∈ E , t ∈ T \ { T }

(74)

The nonlinear equations 69 through 74 can be linearized by first defining the following binary variables: z1,t,ω ∀i ∈ C \ CS, t ∈ T; z2,t,ω i i ∀i ∈ C \ (CR ∪ CS), t ∈ T; zi3,t,ω ∀i ∈ C \ (CR ∪ CS), t ∈ T; z4,t,ω ∀i ∈ C \ i CS, t ∈ T; z it,ω ∀i ∈ C \ CS, t ∈ T \ {⎟ T⎟}; and z ijt,ω ∀(i, j) ∈ E, t ∈ T \ {⎟ T⎟}. With sufficiently large constant MM, the linearized complementary slackness conditions are π1i ,t ,ω ≤ MM i zi1,t ,ω

∀i ∈ C \ C S , t ∈ T

gi1,t ,ω ≤ MM i (1 − zi1,t ,ω ) π i2 ,t ,ω ≤ MM i zi2 ,t ,ω 2 ,t ,ω i

g

≤ MM i (1 − z

2 ,t ,ω i

∀i ∈ C \ C S , t ∈ T

(76)

∀i ∈ C \ ( C R ∪ C S ) , t ∈ T

(77)

)

gi3,t ,ω ≤ MM i (1 − zi3,t ,ω ) π i4 ,t ,ω ≤ MM i zi4 ,t ,ω

∀i ∈ C \ ( C R ∪ C S ) , t ∈ T ∀i ∈ C \ C S , t ∈ T

gi4 ,t ,ω ≤ MM i (1 − zi4 ,t ,ω ) xit ,ω ≤ MM i zit ,ω

y

≤ MM z i

t ,ω ij

(78) (79) (80) (81)

∀i ∈ C \ C S , t ∈ T

(82)

∀i ∈ C \ C S , t ∈ T \ { T }

(83)

git ,ω ≤ MM i (1 − zit ,ω ) t ,ω ij

∀i ∈ C \ ( C R ∪ C S ) , t ∈ T

∀i ∈ C \ ( C R ∪ C S ) , t ∈ T

π i3,t ,ω ≤ MM i zi3,t ,ω

(75)

∀i ∈ C \ C S , t ∈ T \ { T }

∀ (i, j ) ∈ E , t ∈ T \ { T }

gijt,ω ≤ MM i (1 − zijt,ω )

∀ (i, j ) ∈ E , t ∈ T \ { T }

(84) (85) (86)

minω ∑

∑ ∑( p

ω

i

b , D ω ∈Ω ( i , j )∈ES t ∈T

(

t i yijt,ω ) + λ ∑ pω ω ∈Ω

i

(D ) ) ω 2

(87)

subject to budget_constraint ( b ) Dω ≥

∑ ∑ ( t i y ) − TSTT t ,ω ij

(88) ∀ω ∈ Ω

TARGET

(899)

( i , j )∈ES t ∈T

Dω ≥ 0

∀ω ∈ Ω

KKT_UODTA ω ( x , y, π , z , b )

(90) ∀ω ∈Ω

(91)

The only nonlinearity in 87 through 91 is the upper-level objective function, which is convex quadratic. Thus, the global optimum can be found by any MINLP solver (GAMS/DICOPT in this study) (21).

COMPUTATIONAL EXPERIENCE Consider a test problem that is a multiorigin single-destination cell transmission network. This simple network helps one intuitively understand how the network performance will vary when designing the network improvement based on different optimization models. The MIP models are implemented in GAMS/CPLEX and MINLP in GAMS/DICOPT on a machine with Windows OS, a Pentium 4-M CPU, 1.80 GHz, and 640 MB RAM.

Cell Transmission Test Network Figures 1a through 1e show the test network, composed of six cells and six cell connectors and associated data (jam density, saturation flow, and network-improvement unit costs). All cells except the sink cell are considered for link expansion. The simulation period is eight time intervals. The vector Mt in Figure 1c satisfies the following inequality to enforce the UO behavior (7): Mt − Mt −1 > ( M|T | − Mt ) TD where TD is the total demands. The Mt values for time intervals t1 and t2 can be set at any value because it is impossible for a vehicle to arrive earlier than time interval t3.

66


c3

c4

c1 c6 c2

c5 (a) Nit +inf +inf 2 2 1 +inf

Cell c1 c2 c3 c4 c5 c6

Qit 0.5 0.5 1 1 1 +inf

Tim e Interval t1 t2 t3 t4 t5 t6 t7 t8

(b)

Mt Value 1 1 86 98 99.8 99.96 99.995 100 (c)

φi = 0.05; ∀i ∈C \CS χi = 0.03; ∀i ∈C \CS (d) Cell c1 c2

t1 2, 0.5 0.5, 2

t2 1, 0.5 0.5, 1

Note: 1. All other cells and time intervals have zero demand. ω1 ω2 2. The two scenarios are equally likely (p = p = 0.5).

(e)

TAB

37 38 39 40 41 42 43 44 45 46

TSTT of SONDP* 12.362 12.325 12.287 12.25 12.212 12.175 12.137 12.1 12.062 12.025

E[TSTT] of SLP2SONDP** 13.717 13.617 13.517 13.417 13.317 13.217 13.117 13.017 12.917 12.817

TSTT of UONDP* 12.612 12.575 12.537 12.5 12.462 12.425 12.387 12.35 12.312 12.275

E[TSTT] of SLP2UONDP** 13.732 13.677 13.622 13.568 13.514 13.459 13.405 13.35 13.296 13.241

TSTT of BLPNDP* 12.3625 12.325 12.2875 12.25 12.2125 12.175 12.1375 12.1 12.0625 12.025

E[TSTT] of SLP2BLPNDP** 13.475 13.4 13.325 13.25 13.175 13.1 13.025 12.95 12.875 12.8

Note: * using mean values of demands ** using stochastic demands

(f) FIGURE 1 CTM test network and TSTT and E[TSTT] of deterministic and stochastic NDP models (* using mean values of demands; ** using stochastic demands).

Comparison of Seven Network Design Models The experiment is conducted to compare all seven NDP models by using the CTM test network in Figure 1. For the deterministic NDP models, the mean values of time-dependent demands in Figure 1e are used. The seven NDP models are solved to obtain exact solutions by using solution methods discussed previously. The single-level NDP linear programming models (SONDP, UONDP, SLP2-SONDP, and SLP2-UONDP) are coded in GAMS and solved by CPLEX. The linear bilevel NDP models (BLPNDP and SLP2-BLPNDP)

are transformed to equivalent single-level MIPs that are then solved by GAMS/CPLEX. The RO-BLPNDP model is transformed to an equivalent single-level MINLP, which contains all linear terms except the convex quadratic objective function. This is solved by GAMS/DICOPT to obtain a global optimum. For RO-BLPNDP, since information on preferred expected TSTT of the system is not available, TSTTTARGET is set equal to the optimal expected TSTT of SLP2-BLPNDP solution. Furthermore, two values of the goal programming weight (i.e., λ = 10 and 100) are used in the robust NDP model.


67

In the SLP2-SONDP and SLP2-UONDP models, the time-varying demands are stochastic; these correspond to the parameters in the right-hand side of the constraints are random. As discussed by ~ Karoonsoontawong and Waller (11), h(x, ξ) is convex ~ ~ on the sup~ port of ξ (Ξ∼ξ), and Jensen’s inequality results in Eh(x, ξ) ≥ h(x, E ξ) (22, Proposition 3.1). Consequently, the SONDP and UONDP models using the mean values of demands yield underestimated solutions (in terms of their objective functions) to SLP2-SONDP and SLP2-UONDP models. That is, the SONDP solution yields TSTT less than expected TSTT from SLP2-SONDP as shown in Figure 1f. The UONDP model yields its objective value (Σ(i,j)∈EsΣt∈T (Mt yijt )) less than Σω∈ΩΣ(i,j)∈EsΣt∈T pω Mt y ijt,ω) of SLP2-UONDP, but the UONDP solution may not necessarily yield less TSTT than expected TSTT of SLP2-UONDP solutions. From this experiment, Figure 1f shows that the UONDP does underestimate TSTT when compared with E[TSTT] of SLP2-UONDP. For the bilevel problem, Jensen’s inequality cannot be guaranteed. Consider the recourse function of SLP2-BLPNDP and denote ~ by hp(b, ξ ω) the nested program, which is the same as the recourse function of SLP2-UONDP. By applying Jensen’s inequality to the ~ ~ ~ nested program, hp(b, E ξ) ≤ E[hp(b, ξ)]. Note that hp(b, E ξ) = Σ(i,j)∈EsΣt∈T Mt yijt*, where xit*, yijt* are the optimal solution ~ to the nested program for the mean value of demands, and E[hp(b, ξ)] = Σω∈ΩΣ(i,j)∈Es Σt∈T pω Mt y ijt,ω*, where xit,ω*, yijt,ω* are the optimal solution to the nested program for demand scenario ω. As such, Σ(i,j)∈EsΣt∈T Mt yijt* ≤ Σω∈Ω Σ(i,j )∈EsΣt∈T pω Mt y ijt,ω*; this does not necessarily imply that Σ(i,j)∈Es Σt∈T yijt* ≤ Σω∈Ω Σ(i,j)∈EsΣt∈T pω yijt,ω*. Therefore, the recourse ~ function of SLP2-BLPNDP may not be convex on the support of ξ and Jensen’s inequality may not be guaranteed. The BLPNDP model using the mean values of demands does not necessarily yield underestimated solutions to the SLP2-BLPNDP. However, the results of the experiment as shown in Figure 1f show that the BLPNDP model with the mean values of demands underestimates the TSTT when compared with the expected TSTT from SLP2-BLPNDP over the budget range between 37 and 46. For a fair comparison, the solutions of the seven models at 10 TAB levels are evaluated under the stochastic condition to obtain TSTT for the two demand scenarios (TSTTω1 and TSTTω2). Then, the expected TSTT (E[UO_TSTT]), the expected risk (E[Risk]), and the robustness measure are calculated from Equations 92 through 94: E [ UO_TSTT ] = E [ Risk ] =

∑( p

ω ∈Ω

ω

i

TSTT ω )

∑ p i ( max ( 0, TSTT ω

ω ∈Ω

ω

(92) − TSTT TARGET ))

Robustness ( λ ) = E [ UO_TSTT ] + λ × E [ Risk ]

2

(93)

(94)

For the test network, Ω = {ω1,ω2} with pω1 = p ω2 = 0.5. Figure 2 shows the expected TSTT and the expected risks of all NDP solutions. As can be seen from the graphs in Figures 2a and 2c, and the numerical solutions in Figures 2e and 2f, SONDP, UONDP, BLPNDP, SLP2-SONDP, and SLP2-UONDP yield worse expected TSTT and worse expected risks than SLP2-BLPNDP and RO-BLPNDP. From Figures 2b and 2d, SLP2-BLPNDP yields the lowest expected TSTT over the budget range, but it gives higher expected risks when compared with RO-BLPNDP solutions. The lower degree of robustness in RO-BLPNDP implies that the model views the risk term (E[Risk]) as less important; that is, the expected TSTT is more favored.

RO-BLPNDP (λ = 100) yields the least expected risks across the budget range, as shown in Figure 2d, and RO-BLPNDP (λ = 10) and SLP2-BLPNDP the second and third least expected risks. Recall that SLP2-BLPNDP is simply RO-BLPNDP with λ = 0. When evaluating the robustness of all solutions with λ = 10 and λ = 100 over the budget range [37, 46], Figures 3a through 3d portray the results, and Figure 3e provides the rankings of all NDP solutions. The SLP2-UONDP and UONDP solutions perform the worst, the SONDP and BLPNDP solutions the second worst, and the SLP2SONDP solutions the third worst. Figures 3a through 3b show that RO-BLPNDP yields the most robust solution, and no other model provides a satisfactory solution across the budget range. This shows the worthiness of RO-BLPNDP. Moreover, to illustrate the solution robustness and the worthiness of RO-BLPNDP, the risks of different NDP solutions are depicted for the two demand scenarios at the total available budget of 43 in Figure 4a. When the slope of the line in Figure 4a is steeper, the TSTT of the two scenarios are more wildly different. The SLP2-UONDP yields the steepest line in Figure 4a, SONDP and BLPNDP the second, and UONDP the third. RO-BLPNDP (λ = 100) yields the flattest line, RO-BLPNDP (λ = 10) the second, and SLP2-BLPNDP the third. This reiterates the virtue of RO-BLPNDP. Next, how the change of weight (or degree of robustness), which represents the trade-off between the expected TSTT and the expected risk, plays a role in robust network design is illustrated. Figures 4c through 4d show the effects of different weight values on the expected TSTT and the expected risk at the total budget of 43. When the weight is increased from zero to a critical weight value, the expected TSTT rapidly increases and the expected risk drastically decreases. After this critical weight value, the expected TSTT and the expected risk gradually change with the increase of the weight. This kind of visualized graph can be used by transportation planners to learn their preferences on the value of the goal programming weight by comparing the expected TSTT and the expected risk in the system at different weight values. Further, Figure 4b shows the risks of RO-BLPNDP solutions with different weight values for all demand scenarios with the budget of 43. It shows that the higher weight values are associated with the flatter line; that is, the more robust solutions are less sensitive to the variation of O-D demand. Finally, the budget allocation policies based on different NDP models at the total budget of 43 are shown in Figure 4e. Apparently, different NDP models recommend different policies, except the SONDP and BLPNDP models, which provide the same solution for this test network.

CONCLUSIONS This paper proposed two new optimization models based on dynamic traffic assignment linear programming models when accounting for the inherent bilevel nature of the problem and long-term O-D demand uncertainty. The first model is a two-stage stochastic programming model, SLP2-BLPNDP, which is extended from the linear bilevel programming DTA-based network design model, BLPNDP, by Karoonsoontawong and Waller (12). This model minimizes the expected TSTT subject to a budget constraint and DTA conditions. The second model is a robust optimization model, RO-BLPNDP, that is built on the SLP2-BLPNDP by incorporating the risk term into the objective function. That is, RO-BLPNDP minimizes a weighted summation of the expected TSTT and the expected risk. Given a degree of robustness (weight) from transportation planners, RO-BLPNDP provides the most robust network design solution that is the least

68


13.5

SLP2-BLPNDP

UONDP

14

13.4

13.8

BLPNDP

E[UO_TSTT]

E[UO_TSTT]

13.6

SONDP

14.2

13.6 SLP2-SONDP

13.4 SLP2-UONDP

13.2 13

13.3 13.2

RO-BLPNDP (lambda=10)

13.1 13

SLP2-BLPNDP

12.9

12.8 RO-BLPNDP (lambda=10)

Budget


12.8 37 38 39 40 41 42 43 44 45 46

37 38 39 40 41 42 43 44 45 46

12.6


Budget

(a)

(b)

0.7

SONDP

0.6

UONDP

SLP2-BLPNDP

0.02

BLPNDP

0.4

E[Risk]

E[Risk]

0.5

0.025

SLP2-SONDP

0.3

SLP2-UONDP

0.2

SLP2-BLPNDP



0


37 38 39 40 41 42 43 44 45 46

37 38 39 40 41 42 43 44 45 46


0.01 0.005

0.1 0

0.015

Budget

Budget (d)

(c) TAB

SONDP

37 38 39 40 41 42 43 44 45 46

13.5438 13.4875 13.4313 13.375 13.3375 13.3 13.2625 13.225 13.1875 13.15

UONDP 14.0719 14.0063 13.9407 13.625 13.5594 13.4938 13.4281 13.3625 13.2969 13.2313

BLPNDP 13.5438 13.4875 13.4313 13.375 13.3375 13.3 13.2625 13.225 13.1875 13.15

SLP2-SONDP

SLP2-UONDP

SLP2-BLPNDP

RO-BLPNDP (λ =10)

13.7167 13.6167 13.5167 13.4167 13.3167 13.2167 13.1167 13.0167 12.9167 12.8167

13.7318 13.6773 13.6227 13.5682 13.5136 13.4591 13.4045 13.35 13.2955 13.2409

13.475 13.4 13.325 13.25 13.175 13.1 13.025 12.95 12.875 12.8

13.5056 13.4 13.325 13.25 13.175 13.1 13.0509 12.9858 12.9 12.8769

RO-BLPNDP (λ =100)

13.5131 13.4252 13.3372 13.25 13.1845 13.1194 13.0543 12.9892 12.9356 12.8878

(e) TAB; TARGET TSTT 37; 38; 39; 40; 41; 42; 43; 44; 45; 46;

13.475 13.400 13.325 13.250 13.175 13.100 13.025 12.950 12.875 12.800

SONDP 0.075 0.09 0.107 0.125 0.144 0.165 0.188 0.211 0.236 0.263

UONDP 0.517 0.568 0.624 0.156 0.157 0.16 0.164 0.17 0.178 0.188

BLPNDP 0.075 0 .09 0.107 0.125 0.144 0.165 0.188 0.211 0.236 0.263

SLP2-SONDP

0.459 0.347 0.251 0.17 0.105 0.056 0.022 0.005 0.008 0.02

SLP2-UONDP

0.498 0.491 0.484 0.478 0.471 0.464 0.458 0.451 0.445 0.438

SLP2-BLPNDP

0.006 0.003 0.0007031 0 0.000703 0.002112 0.006 0.011 0.018 0.025

RO-BLPNDP (λ =10)

0.00224 0.003 0.0007031 0 0.000703 0.002112 0.001 0.00192 0.008 0.011

(f) FIGURE 2

Expected total system travel times and expected risks of different NDP solutions at 10 budget levels.

RO-BLPNDP (λ =100)

0.00175 0.000779 0.000197 0 0.000108 0.000426 0.000957 0.001698 0.005 0.01

69

21 20 19 18 17 16 15 14 13 12

Robustness (lambda = 100)

SONDP UONDP BLPNDP SLP2-SONDP SLP2-UONDP SLP2-BLPNDP

Budget

90

SONDP

80 UONDP

70 60

BLPNDP

50 SLP2-SONDP

40 30

SLP2-UONDP

20

SLP2-BLPNDP

10 0 37 38 39 40 41 42 43 44 45 46

RO-BLPNDP (lambda=10) RO-BLPNDP (lambda=100)

37 38 39 40 41 42 43 44 45 46



Budget

(a)

(b) 15.5

13.5


13.6 Robustness (lambda = 10)

RO-BLPNDP (lambda=10) RO-BLPNDP (lambda=100)

SLP2-BLPNDP

13.4 13.3 13.2


13.1 13 RO-BLPNDP (lambda=100)

14.5


14 RO-BLPNDP (lambda=100)

13.5 13 37 38 39 40 41 42 43 44 45 46

37 38 39 40 41 42 43 44 45 46

12.9

SLP2-BLPNDP

15

Budget

Budget

(d)

(c) TAB

37 38 39 40 41 42 43 44 45 46

SONDP

4, 4 3, 3 3, 3 2, 2 4, 4 4, 5 6, 6 6, 6 6, 6 6, 6

UONDP

7, 7 6, 6 6, 6 4, 3 5, 5 5, 4 5, 5 5, 5 5, 5 5, 5

BLPNDP

SLP2SONDP

4, 4 3, 3 3, 3 2, 2 4, 4 4, 5 6, 6 6, 6 6, 6 6, 6

SLP2UONDP

5, 5 4, 4 4, 4 3, 4 3, 3 3, 3 4, 4 4, 3 3, 3 3, 3

6, 6 5, 5 5, 5 5, 5 6, 6 6, 6 7, 7 7, 7 7, 7 7, 7

SLP2BLPNDP 3, 3 1, 2 1, 2 1, 1 1, 2 1, 2 3, 3 3, 4 4, 4 4, 4

ROBLPNDP (λ =10) 1, 2 1, 2 1, 2 1, 1 1, 2 1, 2 1, 2 1, 2 1, 2 1, 2

ROBLPNDP (λ =100) 2, 1 2, 1 2, 1 1, 1 2, 1 2, 1 2, 1 2, 1 2, 1 2, 1

(e) FIGURE 3

Robustness evaluation with 10 and 100 of all NDP solutions at 10 total available budget levels.

sensitive to the variation of O-D demand uncertainty. In fact, SLP2BLPNDP is RO-BLPNDP with no degree of robustness. Further, the two new models were compared to the existing five optimization models by using a simple cell transmission test network. With this complexity, one is limited to solving a small network, allowing insight into the problem and intuitive understanding of the effects of alternative improvement policies. It is expected that this work will facilitate the progressive development toward useful practical methodologies. Although the findings from the computational experience cannot necessarily be generalized, they provide insightful and interesting information. It was found that among the five existing optimization models, when evaluated under stochastic condition,

SLP2-UONDP and UONDP perform the worst, SONDP and BLPNDP the second worst, and SLP2-SONDP the third worst. As expected, SLP2-BLPNDP always yields the least expected TSTT. In a comparison of the three models (SLP2-BLPNDP, RO-BLPNDP (λ = 10), and RO-BLPNDP (λ = 100)), RO-BLPNDP (λ = 100) yields the least expected risk as it places higher weight on the risk term in the objective function, and the SLP2-BLPNDP yields the highest expected risk. It was shown that RO-BLPNDP yields the most robust solution, and no other model provides a satisfactory solution across the budget range. In addition, the paper illustrated how a visualized graph may be used to elicit the preference of transportation planners on the desired degree of robustness.

70


1

SONDP

1400

0.9

Risk

0.7

1200

BLPNDP

1000

0.6 0.5

SLP2-SONDP

0.4 SLP2-UONDP

0.3 0.2

Risk (x10-5)

0.8

UONDP

SLP2-BLPNDP

800 RO-BLPNDP (lambda=10)

600 400

SLP2-BLPNDP

200

0.1 RO-BLPNDP (lambda=10)

0 1

2

1 2 Demand Scenario


Demand Scenario (a)

(b)

13.06

0.007

13.055 13.05

0.006 0.005

13.045

E[Risk]

E[UO_TSTT]


0

13.04 13.035

0.004 0.003

13.03

0.002

13.025

0.001

13.02

0 0

20

40 60 Lambda (c)

80

100

0

20

40 60 Lambda (d)

30

Assigned Budget

25 20 15 10 5

DP

N SO

FIGURE 4

P

ND

P

ND

P P ND ) ND DP O 10 SO 0) PN U 2 (l= L 10 2 P B = P l P L ( D S P2 SL P PN SL ND BL P L -B RO Model RO (e)

UO

c1

Ce

c3

ll

c5 0

P BL

Exploration of different NDP solutions at total available budget of 43 (TSTT target 13.025).

80

100


This analytical work is expected to attract more interest to the robust dynamic NDP. Indeed, the robust optimization model cannot handle the real-size network and is limited to a single destination problem. However, it will facilitate the development of a heuristic based on math programming and metaheuristics. Also, it can provide a benchmark for a comparison of future practical heuristics.

71

10.

11.

ACKNOWLEDGMENTS The authors acknowledge the discussions with David P. Morton, Department of Mechanical Engineering, University of Texas at Austin. This research is based on work supported by the National Science Foundation (NSF) and the NSF Mid-America Earthquake (MAE) Center.

12.

13. 14.

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All the opinions, findings, and conclusions expressed in this research are those of the authors and do not necessarily reflect the views of either the NSF or the MAE Center. The Transportation Network Modeling Committee sponsored publication of this paper.