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Linear Programming Models for the User and System Optimal Dynamic Network Design Problem: Formulations, Comparisons and Extensions Satish V. Ukkusuri1 and S. Travis Waller2
Abstract: In this paper we formulate a network design model in which the traffic flows satisfy dynamic user equilibrium conditions for a single destination.
The model presented here
incorporates the Cell Transmission Model (CTM); a traffic flow model capable of capturing shockwaves and link spillovers. Comparisons are made between the properties of the Dynamic User equilibrium Network Design Problem (DUE NDP) and an existing Dynamic System Optimal (DSO) NDP formulation. Both network design models have different objective functions with similar constraint sets which are linear and convex. Numerical demonstrations are made on multiple networks to demonstrate the efficacy of the model and demonstrate important differences between the DUE and DSO NDP approaches. In addition, the flexibility of the approach is demonstrated by extending the formulation to account for demand uncertainty. This is formulated as a stochastic programming problem and initial test results are demonstrated on test networks. It is observed that not accounting for demand uncertainty explicitly, provides suboptimal solution to the DUE NDP problem.
1
Assistant Professor, Transportation Systems Engineering, Department of Civil & Environmental Engineering, 4032 JEC, Rensselaer Polytechnic Institute, Troy, NY, 12180. Email:
[email protected] 2 Assistant Professor, 6.204 ECJ Hall, Department of Civil Engineering, University of Texas at Austin, Austin, TX, 78705. Email:
[email protected]
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2 Introduction The opportunities of new technologies and improved computational resources have encouraged considerable research in the development of new methodologies for dynamic transportation networks. Transportation investments form the core of decision making in current and future projects. Decisions made in the current stage of capacity improvements must take into account the dynamic nature of demand.
Optimal use of existing transportation systems and efficient
capacity improvements is an effective way to improve the efficiency of the transportation network. However, for optimal transportation network investments, transportation planners need easy to use analytical tools that will help them in this decision making process. Only then will there be a significant improvement in transportation throughput in terms of travel times and user’s route choice. This research develops a model to help network planners to make optimal capacity improvements given a resource constraint. Instead of developing a non-tractable complex bilevel formulation of the problem (Jeon et al., 2005), this research develops a linear model which accounts for time dependent variation of the traffic demand. Specifically, this paper formulates the Dynamic User equilibrium Network Design Problem (DUE NDP) for a single destination using a Linear Programming (LP) approach. Results are compared with the known Dynamic System Optimal Network Design Problem (DSO NDP) formulation (Waller and Ziliaskopoulos, 2001).
Initial tests are made on a small test network.
The differences between capacity
improvements under DUE and DSO behavior are analyzed using tests on small network. An approximation technique is described to solve the model for larger networks. An experimental test is performed on a network similar to the Nguyen and Dupis (1984) test network for various levels of traffic demand, pattern and budget. The results for the two models are compared. Finally, the proposed DUE NDP formulation is extended to account for long term demand uncertainty to demonstrate flexibility of the proposed linear programming formulation. Further,
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3 this extension demonstrates the importance of incorporating uncertainty in network design problems. While single-destination DTA problems are infrequently found in actual applications, the approach can be used to evaluate and calibrate other models and may potentially be used as a subroutine in large-scale multi-destination solution approaches (as discussed in Golani and Waller, 2004). Furthermore, analytical approaches, despite their limitations, contribute to the better understanding of the problem and the development of operational analytical approaches. Moreover, since this model can propagate traffic according to the Cell Transmission Model (CTM) (Daganzo, 1994, 1995), it can capture the queue evolution along an arc, which is not generally possible with models relying on link exit functions. Although the proposed analytical model is limited to a single destination, it can capture realities of actual networks better than many other analytical formulations, because its underlying structure is based on traffic flow theoretical models. Note that with this formulation, the single level dynamic user equilibrium network design problem is a linear program, when even the simplest static traffic network design problem is an extremely difficult nonlinear (commonly non-convex) mathematical program. Numerous discrete and continuous formulations have been proposed in the literature over the last thirty years, mostly based on non-linear mathematical programming formulations. Virtually all existing NDP approaches assume that static traffic flow patterns on the improved network prevail.
This section is limited to only a few models, representative of the
methodologies used to approach the problem, and is by no means comprehensive. An extensive review of the NDP literature can be found in Magnanti and Wong (1984) and more recently in Yang and Bell (1998). UE based formulations are discussed in LeBlanc and Abdulaal (1976), Abdulaal and LeBlanc (1979), Marcotte (1983), LeBlanc and Boyce (1986), Suwansirikul et al. (1987), and Friesz et al. (1992).
The solution methodologies are based on non-linear
optimization approaches, which can handle small sized problems. Abdulaal and LeBlanc (1979)
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4 used Hooke-Jeeves solution approach, while Suwansirikul et al. (1987) proposed a heuristic solution approach based on the decomposition of the original problem to a set of sub-problems and demonstrated that this performs better than Abdulaal and LeBlanc’s (1979) approach. Friesz et al. (1992) presented a promising heuristic approach based on simulated annealing and a tabusearch based heuristic by Mouskos (1992). More recently, Davis (1994) presented a formulation and a heuristic algorithm based on a stochastic UE model, which adds more to the realism of drivers’ behavior. Friesz et al. (1992) formulated a multi-objective design which was further extended by Tzeng and Tsaur (1997). Yang and Bell (1998) presented a formulation that accounts for the elasticity of the demand, for the mixed continuous and discrete NDP. The only approaches known to the authors that attempt to capture traffic dynamics in network design are by Janson (1995), Waller and Ziliaskopoulos (2001), and Heydecker (2002). While the first approach is limited to using DTA to only evaluate alternatives rather than compute them, the paper provides various insights on the effect of traffic dynamics on network design decisions. In addition, the approach clearly demonstrates that accounting for the traffic dynamics provides more flow consistent solutions than static approaches. The second approach uses a similar CTM LP as in this paper, but accounts for only dynamic SO network flows. Heydecker (2002) formulated the continuous equilibrium network design problem to incorporate elastic dynamic assignment with departure time choice. Rather than giving final implementations the paper focuses on deriving optimal conditions for the NDP and the sensitivity analysis of the NDP process. Uncertainty in transportation has been long identified as an important issue in transportation planning (Mahmassani, 1984). However, past research indicates relatively little work regarding methodological approaches to deal with long term demand uncertainty in transportation network design problems. Mahmassani et al. (1993) developed a real-time DTA framework that accounts for origin-destination demand uncertainties. This is based on the rolling
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5 horizon approach, a look ahead policy to capture uncertainty into the future. It is well recognized that this method predicts well in the short term while there is significant uncertainty into the long term. The stochastic extension of the DUE NDP accounts for the long term demand explicitly in the formulation. Birge and Ho (1993) presented a multi-stage optimal control model and solution algorithm for a single destination real-time system optimal routing problem that accounted for the stochasticity of demand using a model based on link exit functions. Most of the approaches described here account for uncertainty in transportation networks in real-time (short horizons), rather than long term demand uncertainty which is the focus here. Waller and Ziliaskopoulos (2001) proposed a stochastic network design model accounting for long term demand uncertainty, but their model is limited to system optimal conditions. A major limitation of the static models is that it is difficult to capture the traffic interaction among adjacent links, which could lead to erroneous solutions.
For example, static NDP
algorithms may suggest expansion of two non-consecutive links lying on the same freeway, because the optimal solution is based on the addition of the travel times on all links as determined by the link performance functions, resulting in a “bottleneck”. The introduced DTA based model captures traffic propagation among adjacent links, thereby avoiding potential “bottlenecks”. Another drawback of the static traffic assignment based NDPs is that they assume steady-state time-invariant origin-destination demand; which is obviously unrealistic during the peak period and could lead to sub-optimal solutions. A DTA based NDP overcomes this deficiency by capturing traffic dynamics and by providing recommendations that optimize the network for the duration of the assignment period. Another advantage of the introduced model is that the capacity expansions for each road segment can be captured by a single parameter on the right hand side of a linear constraint. In contrast, in static NDP models the capacity expansions appear in the denominator of the link performance function, which creates non-linearities that are difficult to handle in a mathematical program. The next section introduces the notation used
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6 throughout the paper. Further, formulation of the DUE NDP is presented in detail. The detailed computational implementation of the NDPs on multiple networks is discussed in the section following that. In the section after that the comparison and discussion of the results on a network similar to the Nguyen and Dupuis network are presented. The proposed DUE NDP is extended to incorporate demand uncertainty. This is formulated as a stochastic programming problem and two solution methods are discussed in the section following the DUE NDP model implementation.
The final section concludes the paper with possible directions for future
research.
The Model: Notation and Problem Formulation This section describes the notation used for the DUE NDP, and briefly defines the DUE Network Design Problems (NDP) and SUO NDP models.
It then discusses issues relevant to the
formulation of these problems. The description below summarizes the notation used in the network design formulations.
UO Dynamic Network Design Notation __________________________________________________________________________ q : qmax k : kj : D: Mt: C :
Link flow : Maximum flow Density Jam density Sum of the total demand within the network during the duration of analysis Cost per time interval that will yield user equilibrium flows The set of cells--ordinary cells (CO), diverging cells (CD), merging cells (CM), source cells (CR) and sink cells (CS). T : The set of discrete time intervals S : The maximum time interval in the set T
xit : Number of vehicles in cell i at time interval t The maximum number of vehicles in cell i at time interval t, N it : t yij :The number of vehicles moving from cell i to cell j at time interval t E : The set of cell connectors—ordinary cell connectors (EO), merging cell connectors (EM), diverging cell connectors (ED), source cell connectors (ER), and sink cell connectors (ES)
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7
Qit :
The maximum number of vehicles that can flow into or out of cell i during time interval t
it : The ratio of forward to backward propagation speed for each cell and time interval ( i ) : The set of successor cells to i 1(i) The set of predecessor cells to cell i : Discretization time interval
dit The demand (inflow) at cell i in time interval t bi : Budget assigned to cell i B : Total Budget
Dynamic System Optimal Network Design Problem (DSO NDP) Formulation The objective function of the DSO NDP as formulated by Waller and Ziliaskopoulos (2001) is mentioned here for the sake of completeness. The formulation is based on the SO DTA LP of Ziliaskopoulos (2000) which uses a relaxation of the CTM (Daganzo 1994, 1995). The objective of the DSO NDP is to minimize the total system travel time and therefore produce DSO cell densities and flows, as well as an optimal set of cell capacity improvements. The objective function can be stated mathematically as Minimize
tT ( i , j )ES
previously employed Minimize
tT iC / C s
t yijt which is equivalent to the
xit (Ziliaskopoulos, 2000) as proven in Ukkusuri
(2002). The constraint set for the DSO NDP is the same as for DUE NDP and is shown in Equations 2-12. This DSO NDP formulation will be used for comparison with the DUE NDP in the following two sections.
Derivation and Interpretation of the Cost Vector Mt The formulation of the DUE NDP is based on the User equilibrium Dynamic Traffic Assignment Linear Program (UODTA LP) formulation, given in Waller and Ukkusuri (2003), Karoonsoontawong and Waller (2006) and Ukkusuri (2002). A brief description of the User equilibrium DTA objective is provided before developing the DUE NDP formulation. For DUE conditions, costs must be assigned so that the objective will value an individual’s earlier arrivals
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8 to such an extent that it will sacrifice system-wide costs. Alternatively, if a vehicle has the opportunity to arrive at some minimum time t, it will do so even at the cost of all flow which arrives after time t. Under the stated strategy, the benefit for an arrival flow at time 1 versus time 2 should be equal to the maximum possible value in the formulation (i.e., if any flow can leave the network at time 1 it will do so regardless of any other conditions). After the relative cost difference between times 1 and 2 have taken a value, the same argument then applies to time intervals 3 and 4 and so on. The vector of costs which yield this behavior is denoted by Mt , t T . The vector Mt must reward vehicles for taking shortest time paths by overcoming the natural system costs which exist due to the single objective required for an LP. For example, take a system optimal solution which has a vehicle arriving at time 6. Assume the vehicle could arrive at time 5 in a DUE solution but would cause a delay for many vehicles arriving after it. To capture DUE conditions the LP objective should view a transfer from arriving at time 6 to time 5 as being so great that it outweighs all else within the network occurring at a later time. Definition 1 t A vector of link flows y ij is at user equilibrium, if the objective function (Equation 1) Mt vector
satisfies:
Mt – Mt-1 > (MS – Mt) D
(1)
where, D is the total demand within the network and S is the maximum time interval in the modeling period. From complementary slackness conditions, this bound was proven to satisfy the dynamic user equilibrium flows, i.e., for a given flow if there exists a time dependent shortest path, the user has no incentive to deviate from this minimum travel time path and all the unused time dependent shortest paths have zero flow. This definition is consistent with earlier work on DUE (Mahmassani et al, 1993; Peeta and Ziliaskopoulos, 2001) as discussed in Waller and Ukkusuri (2003) and Ukkusuri (2002). This will be used in the first example demonstrated in this
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9 paper. An approximation scheme for this Mt vector will be used later to solve a more reasonable sized test network.
Dynamic User equilibrium Network Design Problem (DUE NDP) Formulation The objective of the DUE NDP is to determine the optimal decision for capacity improvements of links (street segments) continuously, so as to minimize the total travel cost of each vehicle (the UO objective), while accounting for the User Optimum (UO) DTA behavior. The total amount spent on a network is constrained by B, which is a user-specified budget. The variable b j is defined as the amount of budget spent on each cell j and this influences both the physical capacity available on the link (captured by the density) and the throughput on the link (captured by the saturation flow). Parameter j specifies the increase in the jam density of cell j for one unit of increase of budget b j , while j represents the cost parameter specifying the increase in saturation flow of cell j for one unit in increase of b j Constraint (3) governs the cell mass conservation relationship for all cells excluding the source and sink cells. Constraint (4) is the relaxation that the flow between two cells is constrained by the number of vehicles occupying the beginning cell, the remaining capacity at the ending cell, and the minimum of the maximum flow that can get out of the beginning cell and into the ending cell. Constraint (5) is the equivalent of constraint (4) for sink cells. Constraints (6) and (7) regulate the amount of flow transmitted out of diverging cells i for time t. Similarly, constraints (8) and (9) regulate the flow transmitted into a merging cell j; Constraint (10) expresses the cell mass balance for the source cells and the initial cell volumes; and constraint (11) specifies the initial flow conditions; demands dit and initial occupancies
i are given and constraint (12) is a special case of the mass balance constraint specifying a required final state where all traffic has left the network. A new single constraint (13) is added requiring that the sum of b j must be less than the total budget B. In the NDP model, tj N tj which
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10 appears in constraints 4, 6 and 9 is replaced with tj ( N tj b j j ) , Q tj which appears in constraints 4 to 9 is replaced with Qtj b j j . single destination is shown in 2 to 13
The complete formulation of the DUE NDP for a
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11
(2)
Objective Function : Minimize
tT ( i , j )Es
M t yijt
subject to:
xit xit 1
1
ykit 1
k ( i )
j ( i )
yijt 1 0, i C \ {CR , CS }, t T
yijt xit 0, yijt Qtj b j j , yijt Qit bii , yijt tj x tj tj ( N tj b j j ),
(3)
(4)
(i, j ) Eo ER , t T
yijt xit 0, yijt Qit bii , (i, j ) Es , t T
(5)
yijt Qtj b j j , yijt tj x tj tj ( N tj b j j ), (i, j ) ED , t T
(6)
j ( i )
yijt xit 0,
j ( i )
yijt Qit bii , i CD , t T
yijt xit 0, yijt Qit bii , (i, j ) EM , t T
1
yijt Qtj b j j ,
i ( j )
1
yijt tj xtj tj ( N tj b j j ), j CM , t T
(7)
(8)
(9)
( j )
xit xit 1 yijt 1 dit 1 , j (i ), i CR , t T , xi0 i
(10)
yij0 0, (i, j ) E ,
(11)
xiT 0, i C / CS ,
xit 0, i C , t T , yijt 0, (i, j ) E , t T
(12)
b B;
(13)
i
i
bi 0
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12 Computational Analysis of the Dynamic Network Design Problem This section discusses the computational experience to solve the NDP’s mentioned in the previous sections. As a starting point, the fundamental difference between the DUE NDP and DSO NDP is shown by computing the optimal link investments for different levels of budget. The spending differences under DUE and DSO conditions are also demonstrated. Results from the DUE NDP The DUE NDP model is tested for different levels of budget. The cell version of the network is shown in Figure 1 and the time invariant cell saturation flows and jam densities are shown in Table 1. The value of the model saturation flow rate Qi depends on the link saturation flow rate and the size of the time interval. The time-dependent demand from source cells 1 and 14 is (2, 2, 1) vehicles when t = 1, 2 and 3 respectively. The optimal Mt vector used for this network is from the optimality conditions derived in Ukkusuri (2002) and is shown in Table 2. In the latter subsection, an approximation Mt vector will be presented to solve a more reasonable network. The proposed cell expenditures translate into increases in jam density and saturation flow rates for the specified cells. This could be interpreted as a lane addition, if the expenditure warrants such an increase or other smaller capacity increasing measures such as lane clearance changes, addition of median etc.
The model is implemented for different budget levels,
increasing at the level of 0.1 units. The budget is increased from 0 to 200 units, thus giving 20,000 data points for analysis. This solves 20,000 linear programs with budget increment of 0.1 each time. The total system travel time (TSTT) is calculated for each budget, and this is plotted with the budget increase. A sample calculation for a budget value of 120 units is shown in Table 3, where all the other cells not shown have no improvements. The results show that at a total budget value of 120 units, it should be spent completely on the merge ramp, comprising of cells 14-15-16-7-9-11-13 and the optimal values of bi is as shown in Figure 2.
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13 Further, Figure 2 and Figure 3 show the change in total system travel time as a function of the total budget B. The free-flow conditions are achieved when B is approximately 134 units. The 20,000 budget points give a plot of the TSTT vs B. The marginal benefit of adding additional budget is greater after 45 units, until about 120 units. But adding an extra unit after 120 units has a lesser marginal benefit than before. This benefit is in fact less than that in the 0-45 units range. Further, it is important to note that at a budget of around 134 units the network experiences freeflow conditions and any further addition of budget will have a diminishing benefit. This can be observed in the dual variable of the formulation as well (Ukkusuri, 2002). Comparison of the DUE and DSO NDP A sample result of the DSO NDP is given in Table 3, where the total budget B is the same as before (120 units) for comparison purposes. This table shows the budget is spent entirely on the merging ramp, but it is important to note that the values of capacity improvements in each cell are significantly different. It is also interesting to note that the capacity improvement on the bottleneck cell 7 is higher in the DSO case as compared to the UO case and vice versa on the freeway cells 9 and 11. The plot from the 20,000 data points for the DSO NDP is shown in Figure 2. It can be observed that the improvement in TSTT in the initial region is greater for the same increase in budget under DSO conditions as compared to under DUE conditions Further, as one would expect it is observed that at large budgets both the DUE and DSO solution solutions converge to the same solution since they reach the free flow conditions.
Implementing the DUE NDP on Nguyen Dupis Test Network As discussed in Waller and Ukkusuri (2003), one of the limitations of the DUE NDP formulation is the nature of Mt cost vector to solve large-scale networks. The rate of growth of the values of the Mt vector decreases exponentially and it becomes difficult to capture this for larger networks.
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14 In this section, an approximation procedure for calculating Mt’s is presented and computational experience on a reasonable size test network is presented. By the definition of the Mt vector, all the flows arriving at a later time interval are delayed and priority of route choice is given to the vehicles in the time interval in consideration. However, in real traffic networks, not all the flows may be delayed. For example, a vehicle while choosing its route in the central business district (CBD) area may have little/no effect on the travel time of a vehicle/s in the later time intervals far away from the CBD. This allows the relaxation of the definition of Mt and allows the development of better approximation techniques, which can still capture DUE conditions defined before.
The method employed here uses
approximation techniques to get a closed form expression for a tractable Mt vector.
This
functional form of Mt is used on larger networks and equilibration of flows is analyzed.
The
behavior of the Mt is a strictly monotonic and nearly concave in nature. Different functional forms to capture Mt could potentially exist depending on the assumptions. Note that Mt is independent of the capacity improvements. In other words, the network design decisions are demand inelastic. Initial values of Mt are derived from the original definition of Mt (Waller and Ukkusuri, 2003), but an approximation (in terms of the Mt vector, the DUE flows are found to be optimal for this example, not approximate) is used to capture the values of Mt vector in the higher time intervals.
The higher values of Mt are chosen such that the difference between the
successive Mt values follow an arithmetic progression and the difference is such that it is decreasing with increasing Mt values. For instance, for an assignment time of 70 time intervals in the following example, the value of M1 to M8 is 1 because no vehicle reaches the destination cell until t = 13.
M9 to M12 are derived from the optimal Mt vector whereas M13 to M70 are
approximated so that they capture UO conditions. The magnitude of M13 to M70 is increasing monotonically such that the difference between them is decreasing rapidly. For instance, M21 is
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15 1.91 1023 and M22 is 1.92 1023 , the difference is 1021 whereas the difference between M70 and
M69 tapers off to 1012 . The plot of the Mt is shown in Figure 4. Given this tractable vector, the DUE NDP is solved. While the employed Mt vector does not follow the bound proved in Waller and Ukkusuri (2003), it follows a similar pattern. The basic process adopted here for the larger network is that the NDP LP is solved with the tractable vector, then the flows are verified to be user equilibrium by solving the combinatorial approach for dynamic UE DTA presented in Waller and Ziliaskopoulos (2006) and Golani and Waller (2004). It should be noted that the combinatorial approach restricts flows to take integral values, though, so the behavior was only approximately verified as user equilibrium (as the flows generated from the LP take continuous values).
In general, for any given example, an
approximate Mt vector is derived, the NDP model solved, then the flows verified with the combinatorial algorithm. If the flows are not found to be DUE in nature, a new approximate Mt vector could be calculated. Finally, the DUE and DSO NDP models are tested on network similar to the Nguyen and Dupis (1984) network (as shown in Figures 5 and 6). The network has 13 nodes, 20 links and 2 O-D pairs. The cell representation of this network consisting of 63 cells is shown in Figure 6. The length of each cell in the network is 440 ft. Each cell has two lanes in the network with a speed limit of 30 mi/hr, a saturation flow of 1800 vphpl, and jam density of 200 veh/mi. The network has two O-D pairs and the demand is 8.5 and 12.3 respectively during the peak congestion for each time interval for ODs 1-2 and 4-2.
Comparison and Discussion of the Results The models are tested with three demand scenarios; a light congestion scenario, where a total of 116 vehicle-trips are generated over all the origins, a moderate congestion case with 186 vehicle trips, and a heavy congestion case with 308 vehicle-trips.
For each congestion level, two
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16 demand-loading patterns are tested: a uniform and a peak pattern. In the uniform pattern, the demand is uniformly distributed over the assignment period, while in the peak pattern 50% of the demand is generated in the middle assignment period and the rest is uniformly distributed within the remaining assignment period. The TSTT’s of the various scenarios are presented in Tables 5, 6, and 7 with the amount of budget used. The analysis shows that the budget level, demand-loading pattern, and congestion level significantly influence the DUE and DSO NDP recommended spending policies. The differences and effects of each of these factors on the spending policy are discussed next. Budget Level The network is tested at different levels of budget values. Figure 7 shows the change in total system travel time as a function of the budget spent. The analysis tested thirteen different budget sizes for the same demand level and pattern (308 vehicles, peak hour distribution) for the DUE and DSO NDP. The maximum achievable improvement for the DUE NDP is 65.6% and for the DSO NDP is 64.8 %. It is worth noting, however, that the marginal benefit of an additional thousand dollars diminishes rapidly after the first twelve hundred is spent. It was observed that after a sufficiently high budget both the DUE and DSO TSTT’s converge to a TSTT of 4130.4 seconds because the network experiences free flow conditions. Congestion Level Table 6 shows that the heavier the congestion, the higher the benefits are in the total travel time savings for the same amount spent. At a very low congestion level (116 vehicles), there is only a 14.8% improvement in the TSTT observed, compared to 36.4% for 308 vehicles at the same budget level. These results are intuitive, but still useful in performing a “benefit-cost” analysis for various scenarios of demand realization.
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17 Demand Pattern It was observed that the uniform demand shows a higher TSTT percentage improvement (41.6%) at significantly lower budget than the peak demand for the DUE NDP. The difference between the uniform and peak demand decreases at higher budgets. A similar trend is observed for the DSO NDP. This is intuitive since congestion is lower under uniformly generated demand patterns as against the peak demand scenario. Note further that the spending policies under the two scenarios are substantially different as shown in Table 6. Spending Policy The spending policies of the DUE NDP and DSO NDP for the uniform and peak scenarios at a budget of 2 million dollars and a demand level of 308 vehicles are shown in Figure 8. The insights obtained from this are discussed in this section. Firstly, we note that all the freeway cells are improved in all the scenarios. Secondly, in the peak demand case both the DUE and DSO cases improve capacities on different routes. It is observed that the DUE NDP improves along the least travel time paths which are the freeway segments (cells 1 through 26) and the path 4-910-11-2. Note that cell 18 is the bottleneck cell and DUE spends a small portion on path 4-9-1011-2 so that it can equilibrate the flow onto the freeway and this path thereby minimizing the travel time of each vehicle. On the other hand, the DSO NDP spends the entire budget on the freeway links as it benefits the overall system by improving the marginal least costs. Thirdly, it can be observed that in the uniform case both the DUE and DSO NDP spend the budget on the same routes but at different levels. This is because of the difference in flows in both the DUE and DSO assignments. Further, it can be observed that the improvement applied on all the cells is consistent for a particular link. For example, cells 26, 61, 62, 63 that correspond to link 21 (Figures 5 and 6) are expanded equally in all scenarios. Cells 20, 21, 22, 23, 24 corresponding to links 19 and 9 receive a smaller amount, most likely because they carry less traffic under all the scenarios. Furthermore, all the cells in both the DUE and DSO cases that
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18 receive resources for a given budget level, receive at least the same amount of resources at higher budget levels.
Evaluation of DSO capacity improvements with DUE flows Both the proposed models (DUE and DSO NDP) make simplifying assumptions for modeling tractability. It would be beneficial to assess which would perform better for proposing capacity improvements. Ideally, a bi-level approach would be desirable where the planner takes the system’s perspective and follows a DSO approach while the flows in the network follow DUE behavior. However, this is beyond the scope of this paper. An alternate approach to study an approximate benefit of the NDP models would be to observe how the UO DTA behaves under DUE and DSO NDP capacity improvements. The former is simply the DUE NDP and the latter is the evaluation of the UO DTA under DSO NDP capacity improvements. This analysis shows that the evaluation of the UO DTA with DSO capacity improvements yields better total system travel time as compared to the DUE NDP. For example at a budget, level of 800,000 Dollars, the DUE NDP and DSO NDP gives TSTTs of 5568.80 and 5113.58 seconds respectively. However, the evaluation of the UO DTA with DSO NDP capacity improvements gives a TSTT of 5456.55 seconds. Similar results were observed at different budget levels. It was observed that the TSTT under UO DTA evaluation moves farther away from the DUE NDP TSTT to a value in between the DUE and DSO NDP TSTTs. This result shows that in planning applications, if no bi-level model were available, it is better to use DSO NDP rather than DUE NDP because this would give a better TSTT even under the DUE flows. But it is important to note that this result was observed for the Nguyen Dupis network studied here and the results may not be generalized for all traffic networks.
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19 Extension of the DUE NDP Model One of the advantages of the linear formulation of the DUE NDP problem is the potential extension of the model to incorporate stochastities at different levels of decision making. Many models in transportation over the past several decades have been solved under deterministic conditions. Notwithstanding the success of both the static and dynamic models, however, the assumption that all model parameters are known with certainty limits its usefulness in planning under uncertainty. In this section, we demonstrate the impact of this uncertainty in a dynamic traffic network environment. We propose models that include the long term demand as a random variable within the DUE NDP optimization problem discussed in the previous sections. The emphasis here is on formulation and potential solution methodologies rather than in depth analysis of the value of capturing demand uncertainty. This will remain an active area of future research. The Stochastic DUE NDP Models We propose two methods to handle demand uncertainty in dynamic networks: Chance constrained model and the two stage recourse model. Each of them is discussed briefly in the following section. Chance Constraint DUE NDP Model Most of the Stochastic Linear Programming (SLP) methods eliminate the possibility of secondstage infeasibility completely. However, in some situations it may be more appropriate to accept the possibility of infeasibility under some circumstances, provided the probability of this event is restricted below a given threshold. These lead to mathematical programs with probabilistic constraints or Chance constraints where the infeasibility is accepted but only with a finite probability. The two major advantages of Chance constraint formulations are (1) the ability to introduce reliability constraints explicitly and (2) to derive robust solutions with exogenously specified functional forms. A good review of the SLP methods can be found in Higle and Sen
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20 (1999).
The general method of solving Chance constrained formulations is to formulate the
deterministic equivalents to the chance constraints and the objective function, and to solve the resulting mathematical program with appropriate solution techniques. This is demonstrated for the DUE NDP model. In the deterministic DUE NDP, there is a single set of constraints that generate the demand at each origin for each time period. The demand constraint can be written as an inequality without the loss of generality as:
xit xit 1 yijt 1 dit 1 ,
j (i ), i CR , t T
(14)
If d it 1 is a random variable with the probability distribution F
dit 1
(which can be discrete or
continuous), the equivalent chance constraint of (7) can be written as:
Pr[ xit xit 1 yijt 1 dit 1 ] ,
j (i ), i CR , t T
(15)
Further, it can be seen that by the definition of the distribution function: t 1
Pr[ xit xit 1 yijt 1 d it 1 ] F di ( xit xit 1 yijt 1 ),
(16)
which in combination with (8) results in: t 1
F di ( xit xit 1 yijt 1 )
(17)
or t 1
( xit xit 1 yijt 1 ) ( F di )1 ( )
(18)
Relation (18) is the deterministic equivalent of the chance constraint (15). Including it in the DUE NDP instead of the constraint (14) produces the equivalent deterministic LP for the Chance constrained UO DTA. Constraint (18) contains parameter , which corresponds to the network reliability factor. t 1
The higher the value of
chosen, the greater the demand value
( F di ) 1 ( ) becomes and the more reliable the solution becomes, perhaps, at the expense of optimality.
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21
By using the chance constraint and increasing the confidence level , the linear program solves the NDP with demands greater than the expected value. The design strategies inferred from the solution should then be acceptable for any demand up to the value used and some confidence level exists for these control strategies. Further, by solving for higher demand levels, potential “bottlenecks” can be found. These bottlenecks might manifest themselves with any variation in demand either spatially or temporally, but are not present at the expected value demand case. While a single program execution may be sufficient for getting a solution when confidence level is exogenously provided, additional computations are necessary if probabilistic system performance data is required. This is demonstrated on the network in Figure 1 to highlight the importance of capturing demand uncertainty. Two Stage DUE NDP Recourse Model Recourse models are dynamic models that capture the decision maker’s ability to take corrective action into the future. The infeasibility of the solution is corrected in the second stage at a certain cost once the uncertainty is realized. A review of the recourse models and the associated solution methodologies can be found in Birge and Louveaux (1997). The general multi-stage SLP with recourse can be formulated as shown below.
Let SP i denote the problem solved at the i th stage (i=1, 2, 3…).
Then the mathematical
statement of the two-stage stochastic LP (SLP) problem with recourse is:
T min x c x E [Q ( x , )] s.t : Ax b ( SP 1 ) x0 where T Q ( x , ) min y q ( ) y SP 2 s.t : T ( ) x W ( ) y h ( )
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22 where represents the number of scenarios (system realization), x and y are variables, T ( ) represents the technology matrix and W represents the recourse matrix. Further, W, A and b are parameters; T and h are realization dependent parameters. For the DUE NDP, the second-stage classical T parameter is not realization dependent and corresponds to the NDP parameters Q and N. The classical H parameter corresponds to the realized NDP demand d, the classical W parameter corresponds to both φ and χ, and b corresponds to the NDP parameter B. The first-stage classical variables x, correspond to the design budgets b, and the second-stage classical variables y correspond to the basic DUE NDP density and flow variables. Finally, there are no first-stage costs in the objective function and the second-stage costs correspond to the expected value of the DUE NDP objective over all realizations.
If discrete values are taken for potential demand realizations, the two-stage
stochastic formulation for the dynamic DUE NDP is as shown:
Minimize
E M t y ( )tij tT iC
(19)
subject to:
x( )ti x( )ti 1
k 1 ( i )
y ( )tki1
j ( i )
y ( )tij1 0,
i C \ {CR , CS }, t T ,
(20)
y()tij x()ti 0, y()tij Qtj bj j , y()tij Qit bii , y()tij tj x()tj tj ( N tj bj j ), (i, j ) Eo ER , t T ,
(21)
y ( )ijt x( )it 0, y ( )ijt Qit bii , (i, j ) Es , t T ,
(22)
y ( )ijt Qtj b j j , y ( )tij tj x( )tj tj ( N tj b j j ), (i, j ) ED , t T ,
(23)
j ( i )
y ( )ijt x( )it 0,
j ( i )
y ( )ijt Qit bii , i CD , t T ,
y ( )ijt x( )it 0, y ( )ijt Qit bii , (i, j ) EM , t T ,
(24)
(25)
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23
y()tij Qtj bjj ,
1
( j) i
1
y()tij tj x()tj tj (Ntj bj j ), j CM , t T,
(26)
( j )
x()it x()it 1 y()ijt 1 d ()it 1 , j (i), i CR , t T , xi0 i , i C,
(27)
y ( )0ij 0, (i, j ) E ,
(28)
x( )Ti 0, i C / CS ,
(29)
x( )ti 0, i C , t T ,
(30)
y ( )tij 0, (i, j ) E , t T ,
(31)
b B
(32)
iC
i
Based on the formulation (19-32), general SLP solution methods can be applied (see Birge and Louveaux, 1997) such as L-shaped Methods, Nested decomposition, Monte Carlo sampling or stochastic decomposition. The effect of online information can be formulated as two stage recourse problem using the formulation presented before. The intuition for the formulation is based on realization that the information is used as a recourse variable, i.e., information used in the first stage is used in the decision making (or route choice by UO DTA pattern) in the second stage.
Example Demonstration In this subsection, the chance constrained model is implemented on the small four node- four link network in Figure 1. The value of the Mt vector used for the test network is the same as in Table 1. All the other parameters are the same as used in the demonstration in dynamic case in the previous section. The model is run at different values of alpha. Table 7 gives the expansion policies for the improved cells. There are only a few cells which consistently have improvements for all demand scenarios. It is important to note that the capacity improvements presented here are for the base case without sampling for different values of alpha. An interesting trend
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24 observed in the capacity improvements is that the freeway merging ramp is improved in almost all the scenarios. All the cells on the merging ramp have identical expansion policies for different levels of reliability considered here. This trend indicates that these cells would be the candidates for expansion under all scenarios even under the realization of higher demands and different values of alpha. Figure 9 represents the approximate expected system performance as a function of the αlevel used to generate the respective solutions. From the graph we can observe that the use of the expected value of the demand does not perform optimally under all scenarios. By using an αvalue of 0.75, a gain of 12.4% is realized. However, it is important to realize that the results presented here do not sample the entire sample space and only a few alpha values are considered. Hence, there might be some other value of alpha which could perform better under all demand scenarios, but we can conclusively say that it is not the expected value of demand. The key result from the Chance constraint analysis is primarily in showing that under demand uncertainty planning for the expected value of demand will not yield optimal solutions. The results are similar to the System Optimal case as demonstrated by Waller and Ziliaskopoulos (2001) but gaining additional insight into the DUE variation is critical.
Conclusions With the maturity of new modeling techniques, Dynamic and Stochastic Network Design Models deserve greater attention. So far, most of the modeling approaches pertain to static network design models. There has been very little effort in accounting for time dependent demand and stochastic demand. The following contributions can be drawn from the analysis, formulation and model development in this paper: (1) The models presented in this research are a first step towards building NDP models that adequately account for dynamic traffic propagation. Some benefits of the models include the incorporation of the Cell Transmission Model as the traffic flow model rather than using link performance functions, capturing the time dependent traffic characteristics and
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25 in presenting a model for dynamic planning applications. (2) For practitioners interested in network investment decisions considering time dependent traffic, the developed models can be used as an initial tool in identifying bottlenecks and in the optimal investment of limited resources (3) For the DNDP models presented it was observed that instead of developing models which account for user equilibrium conditions it can be better to use models which satisfy DSO conditions as these capacity improvements were found to give better network wide benefits. (4) Accounting for uncertainty was demonstrated to have significant impact in terms of the transportation planning benefits. Rather than planning for a single point estimate into the future, minimizing the expected costs of all the different realizations provide more resilient network investments. This insight can help transportation planners in accounting for uncertainty in their decision making better. Other recent work which has extended the problem formulation and results for the stochastic network design is by Karoonsoontawong and Waller (2006) Additionally, we extend the dynamic network design model to account for demand uncertainty. The major difference between the DUE and DSO NDP model is that; in the former it is observed that the capacity improvements are made so that vehicles take the least cost path where as in the latter it was observed vehicles prefer the least marginal cost path. Further, for the examples considered in this research, it was observed that the marginal benefit from capacity improvements is greater during the initial spending policies and this benefit decreases with increase in budget. It is also observed that for the networks examined under a non bi-level model, it would be better to make planning decisions based on DSO NDP since this gives a better overall system improvement than the DUE NDP. As mentioned earlier the formulations presented in this paper are limited to a single destination. Further, due to the relaxation of the flow conservation relationship, it could result in the ‘holding back of traffic’. However, it is important to note that
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26 in the single-destination DUE version of the model no unit of flow will be held if it causes any delay in that unit’s arrival time. The holding, therefore, only occurs in the case of a non-unique solution and correct flows can be found in post-processing. Although, these are clear limitations of this formulation, the models presented here are an initial step in modeling NDPs that account for dynamic and stochastic traffic conditions. Numerous future research efforts remain. Because the proposed formulation is linear, extensions to the model can be developed to study fixed-arrival time demand and numerous other realities made possible with the beneficial mathematical structure.
Furthermore, specialized
decomposition approaches (Li et al., 2001) may be developed to exploit the special structure of the problem to solve the dynamic network design problem more efficiently. The proposed DUE NDP models and the insights obtained from them can be used in developing approaches that are more efficient by exploiting related optimization techniques. Acknowledgements This research is based on work supported by the National Science Foundation (NSF) project CMS-0349846 “Multiple Stage Optimization of Stochastic Dynamic Transportation Networks” and the NSF Mid-America Earthquake (MAE) Center project “Dynamic Traffic Networks Models”. All the opinions, findings and conclusions expressed in this research are those of the authors and do not necessarily reflect the views of the NSF or the MAE Center.
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27 References Abdulaal, M., and LeBlanc, L.J. (1979). “Continuous Equilibrium Network Design Models.” Transportation Research, 13B, 19-32. Birge, J.R., and Ho, J.K. (1993). “Optimal Flows in Stochastic dynamic networks with congestion.” Operations Research, 41, 203-216. Birge, J.R., and Louveaux, K. (1997). Introduction to Stochastic Programming, Springer-Verlag, New York, NY. Daganzo, C.F. (1994). “The Cell Transmission Model: A simple dynamic representation of highway traffic consistent with the hydrodynamic theory.” Transportation Research, 28B(4), 269-287. Daganzo, C.F. (1995). “The Cell Transmission Model, Part II: Network Traffic.” Transportation Research, 29B(2), 79-93. Davis, G.A. (1994). “Exact Local Solution of the Continuous Network Design Problem via Stochastic User Equilibrium Assignment.” Transportation Research, 28B, 61-75. Friesz, T., H. Cho, N. Mehta, R. Tobin, and G. Anandalingam (1992). “A Simulated Annealing Approach to the Network Design Problem With Variational Inequality Constraints.” Transportation Science, 26, 18-25. Golani, H. and Waller, S.T. (2004). “A combinatorial approach for multi-destination Dynamic Traffic Assignment.” Transportation Research Record, in press. Heydecker, B. “Dynamic Equilibrium Network Design.” Proc., 15th ISTTT Conference (ed Michael P. Taylor), Adelaide: Permagon, 349-369, 2002. Higle, J.L. and Sen, S. (1999). “An introductory tutorial on Stochastic Linear Programming Models.” Interfaces, 29, 33-61.
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28 Hoang, H.H. (1982). “Topological Optimization of Networks: A Non-linear Mixed Integer Model Employing Generalized Benders Decomposition.” IEEE Trans. Automat. Control, 27, 164-169. Janson, B.N. (1995). "Network Design Effects of Dynamic Traffic Assignment." Journal of Transportation Engineering, 121 Issue 1, 1-13. Jeon, K., Ukkusuri, S.V. and Waller, S.T. Selectorecombinative Genetic Algorithm to Relax computational Complexity of Discrete Network Design Problem. Journal of the Transportation Research Board, No. 1964, Transportation Research Board of the National Academies, Washington, D.C., pp. 91–103, 2006. Karoonsoontawong, A. and S. T. Waller (2005). “A Comparison of System- and User-Optimal Stochastic Dynamic Network Design Models Using Monte Carlo Bounding Techniques." Journal of Transportation Research Board, pp. 91-102, 2006. LeBlanc, L.J., and Abdulaal, M. (1979). “An Efficient Dual Approach to the Urban Road Network Design Problem.” Comp. Math. Appl. 5. , 11-19. LeBlanc, L.J., and Boyce, D. (1986).“A Bi-level Programming Algorithm for the Exact Solution of the Network Design Problem with User-Optimal Traffic Flows.” Transportation Research, 20B, 259-265. LeBlanc, L.J. and Abdulaal, M. (1984). “A Comparison of the User-Optimum versus SystemOptimum Traffic Assignment in Transportation Network Design.” Transportation Research, 18B, 115-121. Li, Y., Waller, S.T. and A.K. Ziliaskopoulos (2001). "A Decomposition Method for SystemOptimal Dynamic Traffic Assignment." Networks and Spatial Economics, in press. Magnanti, T.L., and R. T. Wong (1984). “Network Design and Transportation Planning: Models and Algorithms.” Transportation Science, 18, 1-55.
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29 Mahmassani, H.S. (1984). “Uncertainty in Transportation System Evaluation: Issues and Approaches.” Transportation Planning and Technology, Vol. 9, No. 1, 1-12. Mahmassani, H.S., Peeta, S., Hu, T.Y., and Ziliaskopoulos, A.K. (1993). “Dynamic Traffic Assignment with multiple user classes for real-time ATIS/ATMS applications.” Proc., Advanced Traffic Management Conference, FHWA, US DOT, Washington DC, 91-114. Marcotte, P. (1983). “Network Optimization with Continuous Control Parameters.” Transportation Science 17, 181-197. Mouskos, K. (1992). “A Taboo-Search Based Approach for Network Design.” Ph.D. Dissertation, The University of Texas at Austin, Austin, TX. Nguyen, S. and Dupis, C. (1984). “An efficient method for computing traffic equilibria in networks with asymmetric transportation costs.” Transportation Science, 18, 185-202. Peeta, S. and Ziliaskopoulos, A. “Foundations of Dynamic Traffic Assignment: The Past, the Present and the Future” Networks and Spatial Economics, Vol. 1, No. 3/4, pp. 233-266. Sheffi, Y. (1985).Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods, Prentice-Hall, Englewood Cliffs, New Jersey. Suwansirikul,C., Friesz, T.L., and Tobin,R.L. (1987). “Equilibrium Decomposed Optimization: A Heuristic for the Continuous Equilibrium Network Design Problem.” Transportation Science 21, 254-263. Tzeng, G.H. and Tsaur (1997). “Application of multiple criteria decision making for network improvement plan model” Journal of Advanced Transportation, 31, 48-74. Ukkusuri, S. (2002). “Linear Programs for User Optimal Dynamic Traffic Assignment Problem.” Master’s Thesis, University of Illinois at Urbana Champaign. Waller, S.T. and A.K. Ziliaskopoulos (2001). "A Dynamic and Stochastic Approach to Network Design." Journal of the Transportation Research Board, No. 1771, pp. 106-114.
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30 Waller, S.T. and Ukkusuri, S. (2003). “A linear programming formulation for the user optimal dynamic traffic assignment problem”. Submitted to Transportation Science. Waller, S.T. and A. Ziliaskopoulos , “A Combinatorial User Optimal Dynamic Traffic Assignment Algorithm,” Annals of Operations Research, 1572-1581, 2006. Yang, H. and Bell, M.G.H. (1998). “Models and Algorithms for road network design: a review and some new developments.” Transport Reviews, 18, 257-278. Ziliaskopoulos, A.K. (2000). “A Linear programming model for the single destination system optimum dynamic traffic assignment problem.” Transportation Science, 34(1), 37-49.
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31 LIST OF TABLES Table 1 Time Invariant parameters of the test network Table 2 Value of Mt with time Table 3 Optimal values of bi for B = 120 in DUE NDP Table 4 Baseline Case with no improvements and Table 5 Effect of Budget Levels for Peak distribution, Demand = 308 vehicle trips Table 6 Effect of Distribution Types and Congestion Levels Table 7 Cell Expansion policies for chance constraint Model in $ Millions for different values of alpha
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32
Table 1 Time Invariant parameters of the test network Cell
2
3
4
5
6
7
8
9
10
11
12
15
16
N tj (Jam Density)
4
4
4
4
4
2
4
4
4
4
4
4
4
Qit (Capacity)
1
2
2
2
1
2
1
2
1
2
1
1
1
xi0 (Number of vehicles in
0
0
0
0
0
0
0
0
0
0
0
0
0
cell i at time 0)
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33 Table 2 Value of Mt with time Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14
M value 1 1 1 1 1 9375001.063 9960938.505 9997559.595 9999848.413 9999991.464 9999999.405 9999999.964 9999999.999 10000000.000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
34 Table 3 Optimal values of bi for B = 120 in DUE NDP Cell Number
7 9 11 13 14 15 16
Optimal Budget spent (bi) under DUE conditions 40 20 20 10 10 10 10
Optimal Budget spent (bi) under DSO conditions 48.8889 15.5556 15.5556 10 10 10 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
35 Table 4 Baseline Case with no improvements and Demand Level (Vehicle Trips)
Distribution Type
Budget in $ Thousand (B)
308 186 116 308 186 116
Uniform Uniform Uniform Peak Peak Peak
0 0 0 0 0 0
Total System Travel Time UO(TSTT in vehicle-hrs) 3.31 1.07 0.52 3.33 1.09 0.54
DSO Total System Travel Time 3.31 1.07 0.52 3.27 1.07 0.53
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
36 Table 5 Effect of Budget Levels for Peak distribution, Demand = 308 vehicle trips Max Budget in $ Thousand (B) 0 100 200 300 500 800 1200 1600 2000 2500 3000 4000 4500 5500 5700 5800 5900 6000
DUE Total System Travel Time (vehicle-hrs) 3.333 2.121 2 1.763 1.604 1.547 1.399 1.341 1.306 1.247 1.222 1.195 1.165 1.15 1.149 1.148 1.147 1.147
Percentage of Improvement 0 36.4 40.0 47.1 51.9 53.6 58.0 59.8 60.8 62.6 63.3 64.1 65.0 65.5 65.5 65.6 65.6 65.6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
37
Table 6 Effect of Distribution Types and Congestion Levels
Demand Type 308 308 308 308 308 186 116 308 186 116
Distribution Type Peak Uniform Peak Uniform Peak Peak Peak Peak Peak Peak
Maximum Budget in $ Thousand (B) 5800 5800 200 200 100 100 100 200 200 200
Budget Spent in$ Thousand (
bi ) i
5800 5523 200 200 100 100 100 200 200 200
DUE Total System Travel Time 1.148 1.133 2 1.951 2.121 0.903 0.461 2 0.802 0.442
Percentage of Improvement 65.6 66.0 40.0 41.6 36.4 17.5 14.8 40.0 26.7 18.3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
38 Table 7 Cell Expansion policies for chance constraint Model in $ Millions for different values of alpha Cell No 7 9 11 13 14 15 16
α = 0.5 40.52 20 20 10 10 10 10
α = 0.75 53.23 19.89 19.89 5 7.5 7.5 7.5
α=1 36.52 21.91 21.91 11.91 9.41 9.41 9.41
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39 LIST OF FIGURES Figure 1 Cell Representation and Time Invariant parameters of the test network Figure 2 DSO TSTT with the increase in total budget B Figure 3 DUE TSTT with the increase in total budget B Figure 4 Tractable Mt vector for the DUE NDP (a) Figure 5 Link-node version of Nguyen and Dupis’s Test Network
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Figure 6 Cell Version of Nguyen and Dupis’s Test Network Figure 7 TSTT as a function of budget for DUE and DSO NDP Figure 8 Spending policies for the uniform and peak loading demand patterns for UO and DSO NDP, 308 trips and 2000 Thousand Dollars Figure 9 Expected System Performance for the DUENDP
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Figure 2 DSO TSTT with the increase in total budget B
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Figure 3 DUE TSTT with the increase in total budget B
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Figure 4 Tractable Mt vector for the DUE NDP
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Figure 5 Link-node version of Nguyen and Dupis’s Test Network
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* Response to Reviewer Comments
Comment by EIC: The reviews are not as specific or as helpful as I would like to see. However, I invite you to prepare a revised paper that address all the points raised by the referees. Please give a foundation/arguments for any differences of opinion that you may hav with the reviewers. Regards, TLF Reviewer #2: Since the dynamic system optimum has been modeled by an LP and cells, the attempt by the authors to model the dynamic user equilibrium by such cells and an LP should be a very important issue in DTA research. As many parts of the formulation share the same components with the system optimum formulation, the most important discussion should be the proposed definition of the user equilibrium, the costs vector M_t, and the objective function in the formulation. Although most ideas in the discussion are well presented, there are some issues to be pointed out. First of all, the definition of the vector of costs, M¬_t, is mathematically vague. As I see the time-profiled values of M_t provided in the appendix, M_t seems not have physical meaning such as total travel time or arc/cell delay. It will help readers if more careful explanation such as a definition in a function form is provided. In addition it is noted that M_t is computed from the optimality condition, while the authors missed whose optimality it is. >> A more detailed discussion of the Mt vector is provided in the revised draft. While it is true that Mt, does not have an intuitive meaning it is proved that this provides an optimal DUE solution. The optimality relates to the dynamic user equilibrium problem. In order not to compromise the material in another publication in second round of review at the Transportation Science, the authors provide the references at http://www.rpi.edu/~ukkuss/pubs/index.html. The main details of the Mt vector and the optimality conditions are provided in the M.S. Thesis and the transportation science paper in review. Secondly, the cost in the dynamic traffic network usually changes as the traffic patterns and the network capacity change, while the authors seem use the constant values of M_t. This inconsistency would be removed when the definitions become clearer. >> That is a great observation. However, a potential misunderstanding should be clarified. It is assumed that the network improvements do not cause induced demand (elastic demand) which will change the dynamic equilibrium flows in the transportation network. Instead it is assumed that the capacity improvements and hence the network flows are demand inelastic. In this case the Mt are not varying with the network capacity. This is a reasonable assumption similar to the many papers considered in literature. Finally, it would be good, if we could see the resulting traffic patterns in a graphical form at a given budget and we could check whether the patterns indeed satisfy the user equilibrium conditions or not. For example, the authors may want to see the travel cost and traffic flow plots on the same time scale to see if the minimum of the travel cost and the maximum of the traffic occur at the same time interval.
>> Yes, indeed the authors did verify that the flows satisfy user equilibrium conditions. For instance in the small network (figure 1), the possible travel times that could be obtained by taking other paths were enumerated and found that the paths taken by all vehicles is user equilibrium. The detailed time expanded network is presented in Appendix A of the M.S. Thesis of S. Ukkusuri which can be obtained at http://www.rpi.edu/~ukkuss/pubs/index.html. For the Nguyen Dupius network, such an enumeration is cumbersome; hence the authors verify the flows obtained by comparing it with a combinatorial algorithm developed by Waller and Ziliaskopoulos (2006) for the dynamic user equilibrium (please see reference below). I understand many of discussion and results are presented in the previous papers by the authors. However most of them are in review and in sequence are not available to read. I believe it will be very helpful for reviewers if the authors could complement the paper with the related material in appendix. >> Instead of adding an appendix to an already lengthy paper the authors have provided web sources where the material is available for download. For the editorial review, the authors seem missed many of important references in the bibliography, especially literatures about the dynamic user optimum as an LP including their own preprints. >> The references and paper organization have been revised in this version.
Reviewer #3: General Comments This paper claimed to propose a linear programming formulation for network design problem with dynamic traffic assignment using cell-transmission-model representation (CTM). The paper then extended the formulation to the case with uncertain demand .The topic is obviously very interesting and challenging. In general, the paper is not very well written. The paper is not very well organised in which it contains several long-winded paragraphs (the authors may consider editing these long paragraphs to help improving the structure of the paper). >> The paper has been revised to make the sentences shorter and the paragraphs more crisp. In addition, from the contents of the paper the referee thinks the paper is more like a summary report of the authors' various previous technical works without giving sufficient background materials and explanations. For instance, the concept of the reformulation of the DUO as a linear program through the cost-vector matrix (Mt) is the key to the rest of the development in this paper. However, the referee, who is not very familiar with the authors' previous works, cannot find any clear explanation of this reformulation. The referee attempted to find the "key" references cited for the derivation of this formulation
(Ukkusuri, 2002), but found it very difficult to track this down as it is a Master thesis work. Several key references are also missing from the list (e.g. Waller and Ukkusuri, 2003). >> The present work is an independent body of work dealing only with a linear network design model. This work was not published by the authors elsewhere; however a dynamic user equilibrium formulation based on a LP approach was developed earlier which is currently in review. To provide more detail, a discussion of the Mt vector is provided in the revised draft. While it is true that Mt, does not have an intuitive meaning it is proved that this provides an optimal DUE solution. The optimality relates to the dynamic user equilibrium problem. In order not to compromise the material in another publication in second round of review at the Transportation Science, the authors provide the references at http://www.rpi.edu/~ukkuss/pubs/index.html. The main details of the Mt vector and the optimality conditions are provided in the M.S. Thesis and the transportation science paper in review. These references are available for download. On the technical side, the paper does not provide a clear definition of the model considered. The term DUO or DUE has been regularly used for different definitions in literature. Do the authors refer to DUO as in the concept of user optimised route considering "instantaneous travel time"? This is quiet different from the DUE principle in which the driver chooses his/her route so as to minimise the "actually experienced" travel time. In addition, does the model include both route and departure time choice. >> The problem considered in the paper is dynamic user equilibrium (DUE) where the entire experienced travel time is minimized for each driver. This is clarified in the revised draft on page 3 of the paper. In addition, the model only accounts for route choice. There is no choice dimension along the departure time. In other words, the departure times for each demand slice are already provided. For the main part of the paper which is about the DUO-NDP, it is not very clear from the beginning that the authors in fact do not solve the "standard" NDP but actually adopted a relaxed formulation using the DUO objective function as the objective of the NDP itself. This actually may mislead the readers and invalidate the claims made early on the paper. Actually, the paper does not really deal with the bilevel-network design problem which was made clear later on in the last section of the paper. >> The clarification that the NDP problem solved is not the bi-level problem is clarified early on in the problem definition in this revised draft. The model considered is a linear network design model considering user optimal behavior without any system wide impacts implicit in the problem (thereby not truly accounting for the bi-level nature of the problem). In relation to the fact that DUO's objective is adopted for the NDP, I am wondering if the Mt is a function of the design parameters itself. For instance, if the network improvement is made somewhere in the network, the shortest-path and associated travel time will
obviously changed accordingly. Does the Mt matrix is a fixed matrix derived from the optimality condition of the DUO as priori? If so, this sounds like a very strong assumption in which the users will not actually change the route even if the network is modified as Mt is not a function of the design parameter itself in the DUO-NDP. Again, the referee may not understand this completely but the derivation or calculation of Mt should be explained carefully in the paper. >> That is a great observation. However, a potential misunderstanding should be clarified. It is assumed that the network improvements do not cause induced demand (elastic demand) which will change the dynamic equilibrium flows in the transportation network. Instead it is assumed that the capacity improvements and hence the network flows are demand inelastic. In this case the Mt are not varying with the network capacity. This is a reasonable assumption similar to the many papers considered in literature. More details of the Mt vector are provided at http://www.rpi.edu/~ukkuss/pubs/index.html The extension of this model to the case with uncertain demand is also somewhat unclear. Do the authors actually solve this problem by using some sort of Monte-Carlo simulation to generate different scenarios of demand level and then solve each scenario? >> Yes, the authors solve the stochastic LP formulation using a Monte-Carlo approach for the generated demand scenarios and a single NDP problem accounting for the expected value demand. This difference shows the value of stochastic solution. I think the authors may wish to revise this paper significantly before the re-submission. Apart from the general comments above, I also list some minor comments which may help the authors revising their paper. >> The revised version has been significantly revised in the following respects: reorganization of the introduction section, long sentences/paragraphs are revised, clear problem definition of the NDP problem, reduce the material related to the dual formulation, additional information on Mt vector and adding the references which were cited. Specific Comments Background: I think the authors spent too much time describing the work on NDP with static network whereas the main focus in this paper is the case with dynamic traffic assignment. Several key references in relation to a specific characteristic of the one-tomany or many-to-one OD network were left out (see the works by Akamatsu and Kuwahara). I am also not sure why the authors split the background and introduction sections making them too long-winded. >> the background on static NDP is reduced and included with the introduction for better flow of the manuscript. Page 7: What is SUO NPD model?
>> This is an error and is revised. This should be DSO NDP (dynamic system optimal network design problem) instead. Thank you. Section 3 (The Model): The authors may consider including specific definition of DUO in this section with some brief explanation of the assumption and operation of CTM (particularly for those who are not familiar with these works). >> Done. Page 8: The reference of Waller and Ukkusuri (2003) is missing. >> This reference is included. Pages 8-9: I found the example and explanation of the DUO formulation here very difficult to follow and not very helpful. The authors may consider a simpler way to clarify this concept. >> Done. Page 9: At the bottom of this page, the references of Mahmassani and Peeta (1995) and Peeta and Ziliaskopoulos (2001) are missing. >> These references are included. Page 10: Delete section number (3.3) from the section title. >> Done. Page 10 (Section 3.3): This paragraph is too long and very difficult to follow. >> This section is modified for clarity. Page 13: Delete section number (3.4) from the section title. >> Done. Page 13: I found Section 3.4 a bit distracting and does not add much to the paper. >> This is revised for clarity. Page 17, results from the DUO NDP: Is TSTT the objective function of the DUO-NDP or the calculation of the actual total travel cost in the network? >> TSTT is the actual total network travel time and not the DUO NDP objective. Page 18: If the model proposed is actually LP as claimed by the author, the solution
should be one of the extreme points of the feasible solution. Is this somehow shown in Table 3? >> This is not directly relevant to the problem at hand. However, as we use CPLEX which is a standard LP solver and as the formulation is an LP it is clear that the solution would be one of the extreme points of the polyhedron. Page 19: The explanation of the approximation method for the Mt is very vague. I found it very hard to follow and understand what is actually done here. >> This is revised for clarity in the revised version. References: Please check the reference list carefully. Some of the cited papers are missing from the reference. >> The missing references are added in the revised version. References
Waller, S.T. and A. Ziliaskopoulos , “A Combinatorial User Optimal Dynamic Traffic Assignment Algorithm,” Annals of Operations Research ISSN: 02545330 (paper), 1572-9338 (online), 2006.
