# Analysis of a joint entry- and distance-based cordon pricing scheme: a dynamic modeling approach

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## Abstract

Transportation demand management (TDM) covers strategies for reducing traffic congestion within the affected urban areas. Congestion pricing includes a branch of TDM strategies; among them, the entry-based cordon pricing, i.e., applying charge on entry, is the most popular because of practicality and social acceptance. Many researchers have investigated different second-best approaches for evaluations of cordon pricing plans, mostly by applying static traffic assignment methods. In this paper, a joint entry- and distance-based scheme is proposed to circumvent the deficiencies intrinsic to each. The optimal joint design is considered as the solution to an optimization problem, in which an equilibrium dynamic traffic assignment model is used to take account of flow variations and represent congestion effects more realistically. The problem is solved for the network of Sioux Falls by using an enumeration algorithm, and the solution is compared with those obtained for distinct entry- and distance-based schemes. Based on the results, the joint tolling has the best performance in reducing the total travel time of the travelers and in alleviating the congestion level inside the cordoned area, while generating a higher level of revenue from tolls. Furthermore, the numerical experiments show the unreliability of the results by static against dynamic modeling approach.

## Keywords

Transportation network Traffic demand management Cordon-based congestion pricing Dynamic user equilibrium## 1 Introduction

High levels of traffic in metropolis areas have become a critical social problem of the twenty-first century. Congestion pricing, as part of traffic demand management (TDM) strategies, is a significant economic tool which has been implemented in several cities around the world [1]. It mitigates the congestion level by encouraging the commuters enough to alter their behavior, such as their departure time, route or mode of transport [2, 3]. Congestion pricing schemes can be categorized into two groups: the first-best and the second-best strategies. In the first-best strategy, the well-known Pigouvian formula [4] is used to calculate the negative externalities which should be paid by the drivers. It is applied by charging tolls over all links of the network, so that the user equilibrium flow pattern shifts toward a socially preferred pattern. However, this is not a practical strategy, because of the political and social resistance against all roads tolled. On the other hand, the second-best policy is more acceptable to the public since it settles charges on a limited subset of the links. First applied in Singapore in 1975, cordon pricing is the most appropriate second-best policy for network authorities due to its practicality, effectiveness and social acceptance. Cordon pricing follows three main objectives in general: alleviating congestion, enhancing the environmental indices and generating revenue [5]. In the conventional form of cordon pricing, the commuters are charged on entry into a specific zone, enclosed by a hypothetical cordon line. Mostly implemented on the central business districts, this second-best strategy can reduce the intensity of congestion within the cordoned area by deflecting the routes from entry or changing the mode of transport. May et al. [6], Shepherd et al. [7, 8], Santos et al. [9], May et al. [10], Zhang and Yang [11] and Gholami et al. [12] are among those who investigated the impact of the entry-based cordon pricing on the performance of transportation systems. Although a single layer cordon has a positive effect on reducing the congestion inside, it may cause unwilling congestion outside the cordoned area. In order to tackle this deficiency, some researchers suggested that using multilayer cordons can improve the geographical coverage of cordon pricing [10, 12, 13]. May et al. [10], Zou et al. [14] and Gholami et al. [12] argued against cordon pricing by showing that there may exist more effective toll points in the whole network than on the cordon line, providing that the total number of tolled links is not changed. Based on their experiments, relaxing the constraint of keeping the cordon line closed can increase the social benefits of community. However, implementation of such a strategy encounters practical issues since it does not form a coherent geographical district and is mentally hard to be perceived and accepted by the users. May et al. [6] pointed out that cordon pricing can affect only the trips with the origins outside and the destinations inside the cordoned area. They implemented a bidirectional charging plan which can enhance the overall performance of the pricing.

Area pricing is known as an appropriate surrogate for cordon pricing, as it can influence the incoming, outgoing and inner traffic flows within a specific area. Maruyama and Sumalee [15] compared cordon and area pricing strategies in terms of social welfare, demonstrating that the latter is superior to the former. Fujishima [16] investigated these two toll designs and concluded that the cordon pricing is better if long-distance commuting is prevalent, while area pricing is dominant where central urban area is relatively large. Zhang et al. [17] compared them and found out that area pricing is superior to cordon pricing. They showed numerically that the larger area or the higher tolls do not necessarily result that either of them performs better.

- (1)
It encourages the drivers to reduce their toll by using shorter routes, even if they are highly congested. This logical response is in contrary to the objective of congestion pricing. To cope with the problem, Liu et al. [27] proposed a model for a joint nonlinear distance and time-based toll design, in which the latter compensate for the deficiency of the former. Despite the novelty, the time-based tolling evokes safety issues in the real world instances.

- (2)
It may destroy the desired function of cordoned area by creating congestion inside the cordon around its boundary. This phenomenon is logically expected because of the low charges assigned to low mileage travels within the cordon.

Subsequently, various researchers carried out studies on the congestion pricing in dynamic paradigms with taking account of time-varying traffic flows. Bearing in mind the complexity and computational difficulties of using a dynamic model, the literature is mainly focused on the first-best rather than the second-best strategies [31, 32, 33]. Interested readers are referred to Cheng et al. [34] for further information on the subject. There are a few researches dedicated to the second-best tolling design based on a dynamic traffic assignment model. De Palma et al. [35] analyzed road pricing schemes using the dynamic network equilibrium simulator METROPOLIS. They presented an iterative algorithm to solve the problem for the area and cordon pricing plans. Lin et al. [36] proposed a bi-level programming model which applies the cellular transmission model (CTM) for the dynamic tolled user equilibrium problem. They used a dual variable approximation method to solve the model for second-best scenarios. Chung et al. [37] developed a robust bi-level cellular particle swarm optimization model, where the lower-level problem is a dynamic loading model, and the upper level one is a dynamic system optimum problem which aims at adjusting tolls on predefined locations. Furthermore, there are some studies about dynamic congestion pricing where the traffic flows are based on the macroscopic fundamental diagrams instead of traffic equilibrium laws [38, 39, 40, 41].

The main contributions of this paper include: (1) presenting a hybrid entry- and distance-based cordon charging plan which can enhance the performance of the system compared with the distinct entry-based and distance-based policies; (2) developing a dynamic tolled traffic assignment algorithm capable of dealing with our congestion pricing designs; (3) showing by example that the dynamic approach is more realistic for modeling the cordon pricing problem than the static approach. The paper is organized as follows. In the next section, the hybrid pricing scheme is introduced and its main properties are noted. Section 3 describes the dynamic tolled network loading model and algorithm. The proposed method for determining the optimum toll levels is presented in Sect. 4. Section 5 provides the numerical results for the test network, and the final section gives the concluding remarks.

## 2 Hybrid pricing scheme

As mentioned above, entry-based cordon pricing is not completely equitable in a sense that it does not consider the real externalities imposed by vehicles to the whole network. In this strategy, every vehicle is charged with the same toll while passing certain points at the cordon boundary, regardless of what the route of the travel is. On the other side, applying the distance-based pricing inside a cordoned area may cause unwanted congestion within the cordon around the boundary. These drawbacks limit the applicability and performance of this policy. In this paper, a hybrid entry- and distance-based strategy is proposed to circumvent these issues. In the hybrid layout, the toll consists of two parts: (1) a certain amount of charge for entering the cordon, and (2) a fee proportional to the distance travelled within the cordoned area. The first part prevents unnecessary trips into the cordoned area, while the second part provides more equity to the users traveling inside. This is further to note that the vehicle traveling outside the cordon line is free of charge.

*y*-intercept showing the entry toll and a positive slop representing the toll per unit distance travelled. In other words, the line of the hybrid toll (dashed line in Fig. 1) can be considered as a generalized toll design, as it can be converted to the entry-based and distance-based designs with setting to zero the line's slope and intercept, respectively.

In a mathematical perspective, the objective of congestion pricing is to minimize the total travel time of travelers in an optimization model, subjected to the constraint that the route choice behavior of drivers is affected by tolling. According to the following theorem, we can easily see that the hybrid regime dominates the others.

### **Theorem 1**

*The objective function value of the optimum solution to the hybrid pricing model (*\(Z_{\text{h}}\)*) is a lower bound of those to the corresponding entry*-*based (*\(Z_{\text{e}}\)*) and distance*-*based (*\(Z_{\text{d}}\)*) ones.*

### *Proof*

It is obvious that the hybrid pricing model is a relaxation of each entry-based and distance-based pricing models. So, the solution spaces of the latter two problems are subsets of that of the hybrid model. Considering each space contains a finite number of solutions, we should have \(Z_{\text{h}} \le Z_{\text{e}}\) and \(Z_{\text{h}} \le Z_{\text{d}}\).□

Theorem 1 shows that using the proposed joint design may improve the system performance compared with the distinct designs of entry- and distance-based strategies. This result is further explained below using a simple example.

Entry-based tolling: vehicles are charged on entry into the cordoned area. The entry links (1, 2) and (5, 3) activate this toll.

Distance-based tolling: vehicles driving within the cordoned area are charged with tolls proportional to their distance travelled. The cordoned links (2, 4), (3, 2) and (3, 4) count these tolls.

Joint tolling: vehicles entering the cordon area are charged with both the entry- and distance-based tolls.

As can be noticed, the entry-based layout can affect OD pair B by deflecting flows from path 1–5–3–6 crossing the cordon line to path 1–5–6 which is external to the cordon. However, it cannot directly alter flow distribution of OD pair A, since all the paths between the OD pair are charged equally at the cordon stations. As a result, this plan is not fully efficient in diverting the traffic flow from the cordoned area. The distance-based scheme can obviously change the flow pattern among the routes of OD pair A, following their different traveling lengths inside the cordoned zone. On the other hand, it is unable to affect flow distribution between the paths of OD pair B, because none of them traverse the tolled links inside the zone. Hence, this toll design cannot optimally prohibit vehicles from entering the cordoned area. On the contrary, the hybrid regime yields the optimal state by diverting the flows of both OD pairs from passing over the cordon line.

## 3 Dynamic tolled network loading

Javani et al. [42] proposed a path-based capacity-restrained dynamic traffic assignment model, whose specific feature is allowing for an evaluation of many TDM strategies within the strategic transportation planning framework. An important extension of this model would be the inclusion of the tolls applied to the cordoned area, so that the resulting model suits for analyzing the cordon pricing strategies included in the paper. Such an extended model is discussed below.

### 3.1 Mathematical model

*A*. Each link \(a \in A\) corresponds to a pair \((n,\,m)\) of the nodes in

*N*, where

*n*is the tail and

*m*is the head node of the link. Assume

*G*is strongly connected, and let \(A_{p} \subseteq A\) be the set of links and \(N_{p} \subseteq N\) be the set of nodes on each path

*p*in

*G*. Assume \(A^{\prime} \subseteq A\) is the set of entry links to and \(A^{\prime\prime} \subseteq A\) is the set of tolled links within the cordoned area. Also, denote by \(r(p)\) the origin node of path \(p\), and let \(A_{pn} \subseteq A_{p}\) be the set of links which connect \(r(p)\) to each node \(n \in N_{p}\). Considering that the demand rate is known for each OD pair \(i\) and departure interval \(d\), the equilibrium dynamic tolled traffic assignment problem for the given parameters \(\delta\) and \(\gamma\) can be stated as the following model. (The definitions of other needed variables and the parameters are given in Table 1.)

Variables and parameters used in problem *P*

Symbol | Definition |
---|---|

\(x_{a}^{t}\) | Flow of link |

| Vector of link flows \(x_{a}^{t}\) |

\(f_{a}^{t} (x_{a}^{t} )\) | Travel time function of link |

\(D\) | Set of time intervals in the full analysis period |

\(I\) | Set of OD pairs |

\(P_{i}\) | Set of paths from the origin to the destination of OD pair |

\(q_{i}^{d}\) | Travel demand (rate) between OD pair |

\(h_{p}^{d}\) | Flow of path |

| Vector of path flows \(h_{p}^{d}\) |

\(\alpha_{pa}^{dt}\) | Path-link incidence variable, taking 1 if the flow of path |

\(\Delta t\) | Length of time intervals |

\(T_{pn}^{d}\) | Travel time on path |

\(\phi\) | Value of time (VOT) applied to convert monetary value to travel time |

\(l_{a}\) | Length of link |

\(\delta\) | Toll charged on each entry link to the cordoned area |

\(\gamma\) | Toll charged per unit distance of travel on the links inside the cordoned area |

\(D^{\prime}\) | Set of time intervals in which the tolls are charged, \(D^{\prime} \subseteq D\) |

Equations (1–9) represent a dynamic tolled user equilibrium traffic assignment model which is appropriate for analyses of the cordon pricing strategies mentioned above. Based on the work of Bliemer et al. [44], the proposed model is categorized as the class of the capacity-restrained dynamic traffic assignment models in which link physical capacities may be exceeded, and queues and spillbacks are not considered explicitly.

### 3.2 Solution method

The path-based dynamic traffic assignment algorithm proposed by Javani et al. [42] is used to solve the optimization problem *P* given the parameters \(\delta\) and \(\gamma\). The idea is to divide the model into subproblems SP1 and SP2 corresponding to Eqs. (1–5) and (6–10), respectively. It is to note that by setting the path-link variable \(\alpha_{pa}^{dt}\) to 0 or 1, Eqs. (1–5) become similar to the conventional static traffic assignment problem, so that they can be solved with ordinary traffic assignment algorithms. Besides, Eqs. (6–10) provide the temporally continuous path flows with considering \(\alpha_{pa}^{dt}\) as variable.

The solution algorithm initiates with an arbitrary set of variables \(\alpha_{pa}^{dt}\), in such a way that Eqs. (6–10) are satisfied. It follows an iterative procedure: solving problem SP1 with the current values of \(\alpha_{pa}^{dt}\) to obtain the equilibrium path flows and solving the SP2 with the last calculated path and link flows to adjust the incidence variables. This loop is repeatedly executed until a convergence criterion is met. Brief descriptions of how the subproblems are solved are given in the following subsections.

#### 3.2.1 Solving subproblem SP1

#### 3.2.2 Solving subproblem SP2

The subproblem SP2 is solved for the path-link variables \(\alpha_{pa}^{dt}\), assuming that the link flows \(x_{a}^{t}\) are fixed at their current values. Because the information of active paths is stored in the random access memory (RAM), a simple path tracing scheme can be used to update the variables \(\alpha_{pa}^{dt}\) based on the current values of path travel times \(T_{pn}^{d}\). The following explains this in further detail.

Consider \(P_{i}^{d + } \subseteq P_{i}\) as the set of active paths pertaining to OD pair \(i\) and departure interval \(d\) in the current iteration, and we are tracing path \(p \in P_{i}^{d + }\) and have now reached node \(n \in N_{p}\). At this point, the travel time \(T_{pn}^{d}\) has been computed by the process using Eq. (5), and we are moving forward on link \(k = (n,m) \in A_{p}\). To update \(\alpha_{pk}^{dt}\) for each time interval \(t \in D\), the time interval \(\hat{t} \in D\) that the flow on path \(p\) reaches node \(n\) is first computed as below:

Javani et al. [42] suggested applying an averaging technique along with the tracing scheme to speed up the convergence of the variables \(\alpha_{ps}^{dt}\) to their optimal values. This is performed by substituting \(T_{pn}^{d}\) in Eq. (5) with a weighted average of its current value and the value it had at the previous iteration. More details are available in [42].

## 4 Optimal cordon tolls

*S*is the total travel time experienced by travelers and \(\delta\) and \(\gamma\) are the entry toll and the rate of distance toll (i.e., toll per unit distance), respectively, which are bounded from above by \(\delta_{\hbox{max} }\) and \(\gamma_{\hbox{max} }\) in that order.

## 5 Numerical results

^{1}[46]. Besides, to fulfill the purpose of the paper, a cordon line is defined around the central part of the network. Figure 4 illustrates this network along with the determined cordoned area. As can be seen, there are 10 exit or entry stations on the cordon line, surrounding 6 links within the cordoned area.

Parameters of cordon pricing problem for Sioux Falls

Parameter | Value | Unit |
---|---|---|

\(\gamma_{\hbox{max} }\) | 1 | $ |

\(\delta_{\hbox{max} }\) | 3 | $ |

\(\phi\) | 10 | $/h |

\(\left| D \right|\) | 6 | – |

\(\left| {D^{\prime}} \right|\) | 4 | – |

\(\Delta t\) | 0.25 | h |

Links parameters for Sioux Falls (\(A_{a}\) in hours; \(B_{a}\) in h/[100 vehicles/h]^{4})

Link | Parameters | Link | Parameters | ||||
---|---|---|---|---|---|---|---|

\(A_{a}\) (× 10 | \(B_{a}\) (× 10 | \(l_{a}\) (km) | \(A_{a}\) (× 10 | \(B_{a}\) (× 10 | \(l_{a}\) (km) | ||

(1, 2), (2, 1) | 6 | 0.02 | 3 | (11, 12), (12, 11) | 6 | 15.5 | 3 |

(1, 3), (3, 1) | 4 | 0.02 | 2 | (11, 14), (14, 11) | 4 | 10.61 | 2 |

(2, 6), (6, 2) | 5 | 12.41 | 2.5 | (12, 13), (13, 12) | 3 | 0.01 | 1.5 |

(3, 4), (4, 3) | 4 | 0.07 | 2 | (13, 24), (24, 13) | 4 | 8.93 | 2 |

(3, 12), (12, 3) | 4 | 0.02 | 2 | (14, 15), (15, 14) | 5 | 10.85 | 2.5 |

(4, 5), (5, 4) | 2 | 0.03 | 1 | (14, 23), (23, 14) | 4 | 10.2 | 2 |

(4, 11), (11,4) | 6 | 15.5 | 3 | (15, 19), (19, 15) | 3 | 0.1 | 1.5 |

(5, 6), (6, 5) | 4 | 10.01 | 2 | (15, 22), (22, 15) | 3 | 0.53 | 1.5 |

(5, 9), (9, 5) | 5 | 0.75 | 2.5 | (16, 17), (17, 16) | 2 | 4.01 | 1 |

(6, 8), (8, 6) | 2 | 5.21 | 1 | (16, 18), (18, 16) | 3 | 0.03 | 1.5 |

(7, 8), (8, 7) | 3 | 1.19 | 1.5 | (17, 19), (19, 17) | 2 | 5.54 | 1 |

(7, 18), (18, 7) | 2 | 0.01 | 1 | (18, 20), (20, 18) | 4 | 0.02 | 2 |

(8, 9), (9, 8) | 10 | 23.06 | 5 | (19, 20), (20, 19) | 4 | 9.58 | 2 |

(8, 16), (16, 8) | 5 | 11.57 | 2.5 | (20, 21), (21, 20) | 6 | 13.73 | 3 |

(9, 10), (10, 9) | 3 | 0.12 | 1.5 | (20, 22), (22, 20) | 5 | 11.3 | 2.5 |

(10, 11), (11, 10) | 5 | 0.75 | 2.5 | (21, 22), (22, 21) | 2 | 4.01 | 1 |

(10, 15), (15, 10) | 6 | 0.27 | 3 | (21, 24), (24, 21) | 3 | 7.9 | 1.5 |

(10, 16), (16, 10) | 4 | 10.8 | 2 | (22, 23), (23, 22) | 4 | 9.6 | 2 |

(10, 17), (17, 10) | 8 | 19.3 | 4 | (23, 24), (24, 23) | 2 | 4.51 | 1 |

*i*and departure time

*d*and \(\varGamma_{p}^{d}\) is the generalized travel time on path \(p\) as given in Eq. (11).

In the following subsections, first, the performance of the proposed joint cordon pricing scheme is analyzed and compared to its entry-based and distance-based counterparts. Second, the efficiency of the three toll designs in reducing the level of congestion in cordoned area is investigated. Finally, it is demonstrated that using the proposed dynamic tolled traffic assignment model is more suited and reliable for assessing the pricing plans than its static counterpart. In all experiments, the dynamic assignment algorithm is terminated after an RG of 10^{−6} or less is achieved. The total enumeration algorithm was coded in MATLAB 7.1 [48] and linked to the available C++ code of the assignment algorithm. All results reported below were produced on a laptop with an Intel 2.50 GHz CPU and 8 GB RAM.

### 5.1 Entry-based, distance-based or hybrid regime?

Apart from the validity of the congestion pricing, there is a serious debate about which of the cordon and distance-based pricing schemes or combinations of them would be the best treatment for the congestion problems within the urban areas. Having their own efficiencies and deficiencies, the distance-based pricing has won more favor in the previous researches mostly under the static loading conditions (see, e.g., [5, 21, 22]). To compare these policies in the more realistic case of dynamic loading, the proposed model is solved considering wide ranges of entry- and distance-based tolls for the cordoned test network.

The enumeration algorithm was performed by varying \(\delta\) from 0 to 3 and \(\gamma\) from 0 to 1 with increments of 0.01, resulting in 30,401 regularly distributed grid points within the problem space. This is a grid search on the feasible hybrid toll point, covering also the points of entry-based tolls (\(\gamma = 0\)) and distance-based tolls (\(\delta = 0\)) as bounded solutions.

Optimal designs of cordon pricing schemes for Sioux Falls

Pricing scheme | Optimal toll design ($) | Total travel time (h) | Reduction of total travel time (%) | Total revenue ($) |
---|---|---|---|---|

No pricing | \(\delta = 0.00,\,\gamma = 0.00\) | 8281 | – | – |

Entry-based | \(\delta = 0.23,\,\gamma = 0.00\) | 8249 | 0.39 | 2554 |

Distance-based | \(\delta = 0.00,\,\gamma = 0.15\) | 8257 | 0.29 | 1804 |

Hybrid | \(\delta = 0.17,\,\gamma = 0.08\) | 8223 | 0.70 | 2865 |

Optimal designs of cordon pricing schemes with revenue restriction of 2 × 10^{3} $ for Sioux Falls

Pricing scheme | Optimal toll design ($) | Total travel time (h) | Reduction of total travel time (%) | Total revenue ($) |
---|---|---|---|---|

Entry-based | \(\delta = 0.06,\,\gamma = 0.00\) | 8249 | 0.39 | 675 |

Distance-based | \(\delta = 0.00,\,\gamma = 0.03\) | 8264 | 0.21 | 374 |

Hybrid | \(\delta = 0.08,\,\gamma = 0.03\) | 8238 | 0.52 | 1267 |

### 5.2 Congestion reduction in cordoned area

Volume-to-capacity ratios within cordon area for different toll designs for Sioux Falls

Link | Volume-to-capacity ratio | |||
---|---|---|---|---|

No pricing | Entry-based pricing | Distance-based pricing | Hybrid pricing | |

(9, 10) | 1.87 | 1.88 | 1.76 | 1.78 |

(10, 9) | 1.88 | 1.83 | 1.79 | 1.80 |

(10, 15) | 1.97 | 1.99 | 1.89 | 1.90 |

(15, 10) | 1.94 | 1.99 | 1.88 | 1.90 |

(15, 22) | 2.16 | 2.13 | 2.04 | 2.05 |

(22, 15) | 2.18 | 2.16 | 2.07 | 2.05 |

Average | 1.99 | 1.98 | 1.89 | 1.88 |

Percent reduction (%) | 0 | 0.50 | 5.03 | 5.53 |

### 5.3 Change in input flow to cordoned area

Input flow to cordoned area for different toll designs for Sioux Falls

Pricing scheme | Entering volume (vehicles) | Reduction (%) |
---|---|---|

No pricing | 11,322 | – |

Entry-based | 11,106 | 1.91 |

Distance-based | 11,262 | 0.53 |

Hybrid | 11,052 | 2.38 |

Diverted flow from cordoned area for different toll designs for Sioux Falls

Pricing scheme | Diverted volume (vehicles) | Reduction (%) |
---|---|---|

No pricing | 0 | – |

Entry-based | 216 | 6.46 |

Distance-based | 60 | 1.79 |

Hybrid | 270 | 8.08 |

### 5.4 Static versus dynamic approach

As explained above, dynamic traffic assignment models are generally more appropriate than static models to assessing the TDM plans, considering their ability in reproducing traffic flow more realistically. Aside from the computational efficiency of static models, the important question arises is how reliable these models are, and if they may cause detrimental results.

*D*includes only a single time interval of 60-min duration. In this special case, \({P} (\delta ,\gamma )\) becomes a static tolled traffic loading model (notice that the variable \(\alpha_{pa}^{dt}\) and the corresponding constraints (1–5) are removed from the problem in this case). Applying the total enumeration technique, the optimum solution to the static model of the pricing problem is given by \(\delta = 0.15\) and \(\gamma = 0.03\), which are obviously lower than those established by the dynamic formulation. This toll design can be evaluated by the dynamic tolled assignment model to distinguish whether or not it would be really effective. Table 9 shows the comparison of the performance of the optimal hybrid tolls from the static and dynamic approaches evaluated under both the static and dynamic loading conditions. According to Table 9, the following conclusions can be remarked:

Optimal hybrid toll designs from static and dynamic modeling approaches for Sioux Falls

Pricing scheme | Modeling approach | Optimal toll design ($) | Total travel time (h) | Total revenue ($) | ||
---|---|---|---|---|---|---|

Static loading | Dynamic loading | Static loading | Dynamic loading | |||

No pricing | – | \(\delta = 0.00,\,\gamma = 0.00\) | 7477 | 8281 | – | – |

Hybrid | Static | \(\delta = 0.15,\,\gamma = 0.03\) | 7436 | 8247 | 2054 | 2045 |

Dynamic | \(\delta = 0.17,\,\gamma = 0.08\) | 7484 | 8223 | 2883 | 2865 |

- (1)
For all different toll levels, including the no-toll state, the static traffic assignment model underestimates the total travel time of the travelers by around 9%–10% compared with its dynamic counterpart. In other words, the static model cannot reflect the network congestion precisely. This is in accordance with the results provided in [42].

- (2)
The optimum toll design obtained by the static approach cannot perform adequately well under dynamic loading conditions. As can be seen in the table, it attains a realistic total travel time of 8247 h, which is well above the minimum of 8283 h provided by the dynamic approach. Therefore, the static formulation of the problem may not be reliable as a standard modeling approach.

- (3)
Evaluating a toll design with the static loading model may yield detrimental results. It is demonstrated that the total travel time of the optimum toll design (i.e., \(\delta = 0.17\) and \(\gamma = 0.08\)) under static loading conditions equals 7484, which is ridiculously more than the same value in the no-toll state (i.e., 7477 h). In other words, using the static traffic assignment model may cause some really beneficial pricing alternatives to be rejected.

- (4)
Static modeling slightly overestimated the total amount of collected tolls when comparing to its dynamic counterpart.

## 6 Conclusions

- (1)
The hybrid regime is superior in reducing the total travel time of the travelers in the whole analysis period and generates the highest amount of collected tolls within the peak hour as well. Also, it performs the best in terms of mitigating the congestion within the cordoned area of the network during the peak hour.

- (2)
Comparing the distinct entry-based and distance-based tolling systems showed that the former outperforms the latter in total travel time saving and revenue collection, but fails to perform better when focusing on the congestion level within the cordoned area.

- (3)
The distance-based tolling system is strongly outperformed by the entry-based and hybrid schemes in diverting drivers from entry into the cordoned area.

- (4)
A comparison between the results from static and dynamic modeling approaches revealed some surprising points. First, the static network loading model significantly underestimates the total travel time of the travelers and slightly overestimates the total collected tolls, compared with its dynamic counterpart. Second, the static approach provides a solution that is away from the optimality under dynamic loading conditions and therefore may not be considered as a reliable approach. Third, the static assignment cannot produce a realistic evaluation of the pricing policies, so that some beneficial alternatives may be rejected.

The above results vouch for the applicability and advantages of the proposed method for the cordon tolling problem at planning levels. Although the results are limited to the scale of our test network, they strongly encourage further investigations on the effects of such policies on real life networks. However, designing a heuristic or metaheuristic approach capable of dealing with large-scale problems may be required. In addition, using more sophisticated dynamic traffic assignment model which can consider the queue formation phenomena may be helpful in future researches.

## Footnotes

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