*Reliability-Based Marginal Cost Pricing Problem DOI: http://dx.doi.org/10.5772/intechopen.92844*

network (see **Figure 7**), which consists of 24 nodes and 76 links. The link design capacity and link free-flow travel time can be found in [25]. Degradation parameter *θ<sup>a</sup>* for each link is 0.95. In this example, we also assume *VMRw* ¼ 1*:*5, and the perception error distribution of a unit travel time follows *N*ð Þ 0*:*1, 0*:*2 . Forty-two OD pairs are considered in the Sioux Falls network and the mean of the lognormal demand for all OD pairs is given in **Table 4**. The stopping tolerance criterion is set at 0.001. Convergence is achieved in 48 iterations as depicted in **Figure 8**.

In this example, we compare two scenarios. One is the toll free case, and the other one is the PRSN-MCP toll scheme. **Table 5** presents the link toll under the PRSN-MCP scenario. By levying these tolls on each link, the network becomes smooth and efficient. At the equilibrium state, the expected total perceived travel time achieved by the toll free case and PRSN-MCP toll scheme is 345,749 and 324,636, respectively. Therefore, the proposed PRSN-MCP model is an efficient method in reducing traffic congestion.


#### **Table 4.**

under heavier congestion levels. On the other hand, we can see that *U TT* ½ �� SS�SD

*Difference of the expected total perceived travel time between PRSN-MCP and RSN-MCP under different OD*

*Linear and Non-Linear Financial Econometrics - Theory and Practice*

**6.4 Application to the Sioux Falls network in the PRSN-MCP (SS-SD) case**

The final example illustrates the calculation of the PRSN-MCP (SS-SD) toll in a larger network. This example network is the well-known medium-scale Sioux Falls

 is increasing while the *VoR* level increases. This is to be expected, since a higher travel time reliability requires a larger time buffer. Therefore, ignoring the travelers' perception error may significantly reduce the performance of the RSN-MCP tolls, especially when the network congestion level is heavy and travelers

*<sup>U</sup> <sup>T</sup>*~*T*<sup>~</sup> SS�SD

**Figure 7.** *Sioux Falls network.*

**70**

*demand multiplier* z *and* VoR *levels.*

**Figure 6.**

require a higher travel time reliability level.

*Means of the stochastic demand for all OD pairs in the Sioux Falls network.*

**Figure 8.** *Convergence of the MSA for the Sioux Falls network.*


*MT*<sup>~</sup>*T*<sup>~</sup> ðÞ¼ *<sup>s</sup>* <sup>X</sup>

<sup>¼</sup> <sup>X</sup>

*Reliability-Based Marginal Cost Pricing Problem DOI: http://dx.doi.org/10.5772/intechopen.92844*

<sup>¼</sup> <sup>X</sup>

<sup>¼</sup> <sup>X</sup>

<sup>¼</sup> <sup>X</sup>

<sup>¼</sup> <sup>X</sup>

derivative evaluated at*s* ¼ 0,

*<sup>E</sup> <sup>T</sup>*~*T*<sup>~</sup> � �<sup>2</sup> h i <sup>¼</sup> <sup>X</sup>

*Var <sup>T</sup>*~*T*<sup>~</sup> � � <sup>¼</sup> *<sup>E</sup> <sup>T</sup>*~*T*<sup>~</sup> � �<sup>2</sup> h i � *<sup>E</sup> <sup>T</sup>*~*T*<sup>~</sup> � �<sup>2</sup>

<sup>¼</sup> <sup>X</sup>

<sup>¼</sup> <sup>X</sup>

**Acknowledgements**

No. DUT20GJ210).

**Author details**

Shaopeng Zhong

**73**

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup><sup>E</sup>* exp *sVaT*<sup>~</sup> *<sup>a</sup>* � � � �

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>E*f g exp ½ � *sVa*ð Þ *Ta* <sup>þ</sup> *<sup>ε</sup><sup>a</sup>*

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>ETa* exp ð Þ *sVaTa E<sup>ε</sup>a*<sup>j</sup>

*<sup>E</sup> <sup>T</sup>*~*T*<sup>~</sup> � � <sup>¼</sup> <sup>X</sup>

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>χ</sup>*

2

Then we can obtain the variance of *T*~*T*~ as follows:

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>χ</sup>*

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>χ</sup>*

Dalian University of Technology, Dalian, China

provided the original work is properly cited.

\*Address all correspondence to: szhong@dlut.edu.cn

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>ETa* exp ð Þ *sVaTa M<sup>ε</sup>a*j*Ta*

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>ETa* f g exp ð Þ *sVaTa* exp ð Þ *sVaε<sup>a</sup>*

n o

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>ETa* exp ð Þ *sVaTa* exp *sVaTa <sup>χ</sup>* <sup>þ</sup> *<sup>ϖ</sup>*<sup>2</sup>

Similarly, the second-order moment of *T*~*T*~ can be derived from the second

2

*Var TT* ½ �þ *<sup>ϖ</sup>*<sup>2</sup>

n o � �

This research has been supported by the National Natural Science Foundation of China (Project No. 71701030 and 71971038), the Humanities and Social Sciences Youth Foundation of the Ministry of Education of China (Project No. 17YJCZH265), and the Fundamental Research Funds for the Central Universities of China (Project

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

The first-order moment is, from the first derivative evaluated at *s* ¼ 0,

*Ta* exp *sVaεa Ta* ½ � ð Þ j

*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>χ</sup> E V*½ � *aTa* (69)

*E V*<sup>2</sup> *aTa*

> *E V*<sup>2</sup> *aTa*

n o � � (70)

(68)

(71)

ð Þ *sVa*

*sVa=*2 � � � � � �

*E V*ð Þ *aTa* <sup>2</sup> h i <sup>þ</sup> *<sup>ϖ</sup>*<sup>2</sup>

<sup>2</sup> *E V*ð Þ *aTa* <sup>2</sup> h i � *E V*½ � *aTa* <sup>2</sup> n o <sup>þ</sup> *<sup>ϖ</sup>*<sup>2</sup>

n o � �

*E V*<sup>2</sup> *aTa*

n o

**Table 5.** *PRSN-MCP tolls for each link at equilibrium state.*
