**4. Risk-based MCP (RSN-MCP) in a stochastic network**

## **4.1 Analysis of risk-based SN-MCP**

In the previous section, we know that the Stochastic Network-User Equilibrium (SN-UE) flow pattern can be driven toward a SN-SO flow pattern by charging a toll equal to the SN-MCP. Meanwhile, the expected total travel time can be minimized. In this section, we consider the risk-based (averse or prone) case. The objective function of the RSN-MCP model is to minimize the weighted sum of the mean and the variance of the total travel time, not simply to minimize the expected total travel time. Therefore, the RSN-MCP toll can be determined as

$$\text{RSN} - \text{MCP} = \left\{ \partial E[TT]/\partial v\_a - E[T\_a] \right\} + \text{VaR} \cdot \left\{ \partial V a r [TT]/\partial v\_a - V a r [T\_a] \right\} \tag{36}$$

### **4.2 Calculation of RSN-MCP**

In this section, we discuss the most complete and realistic situation in which travelers consider both stochastic fluctuations in supply (or link capacity) and demand in their route choice decision-making process. From Eqs. (32) and (36), we can see that the difference between SN-MCP and RSN-MCP is the term in the second parentheses of Eq. (36). This second term reflects the congestion toll on travel time reliability due to travelers' risk-based behavior. Let us now turn our attention to *<sup>∂</sup>Var TT* ½ �*=∂va*. The variance of the total travel time is described by

$$\begin{split} Var[TT] &= E\left[TT^{2}\right] - \left(E[TT]\right)^{2} \\ &= \sum\_{a \in A} \left\{ \left(t\_{a}^{0}\right)^{2} \cdot Var[V\_{a}] + \left(\beta t\_{a}^{0}\right)^{2} \frac{Var\left[V\_{a}^{n+1}\right]}{Var\left[C\_{a}\right]} + 2\beta \left(t\_{a}^{0}\right)^{2} \frac{\left(E\left[V\_{a}^{n+2}\right] - E\left[V\_{a}^{n+1}\right]E\left[V\_{a}\right]\right)}{E\left[C\_{a}\right]} \right\} \\ &= \sum\_{a \in A} \left\{ \left(t\_{a}^{0}\right)^{2} \cdot VarR \cdot \nu\_{a} + \left(\beta t\_{a}^{0}\right)^{2} \left\{ \frac{\left(1-\theta\_{a}^{0-2a}\right)}{t\_{a}^{0}\left(1-\theta\_{a}\right)\left(1-2n\right)} \nu\_{a}^{2a} \mathcal{Y}\_{a}^{4a+6n+2} - \left[\frac{\left(1-\theta\_{a}^{0-n}\right)}{t\_{a}^{n}\left(1-\theta\_{a}\right)\left(1-n\right)} \nu\_{a}^{n+1} \mathcal{Y}\_{a}^{1+n}\right]^{2} \right\} \right\} \\ &= \sum\_{a \in A} \left\{ \nu\_{a}^{0}\right\}^{2} \frac{\left(1-\theta\_{a}^{0-n}\right)}{\mathcal{T}\_{a}^{n}(1-\theta\_{a})(1-n)} \nu\_{a}^{n+2} \mathcal{Y}\_{a}^{n+2} \left(\nu\_{a}^{2a+2} - 1\right) \end{split} \tag{37}$$

Differentiating Eq. (37) with respect to the mean link flow yields

$$\begin{aligned} \frac{\partial Var[TT]}{\partial v\_{a}} &= \left(t\_{a}^{0}\right)^{2} \cdot \text{VMR} + \left(\theta\_{a}^{0}\right)^{2} \left\{ \begin{aligned} &\frac{\left(1-\theta\_{a}^{1-2\mathfrak{u}}\right)}{\Xi\_{a}^{2}\left(1-\theta\_{a}\right)\left(1-2n\right)} \left\{v\_{a}^{2n}\rho\_{a}^{4n^{2}+4n}\left[(2n+2)v\_{a} - \left(2n^{2}+n-1\right)\cdot \text{VMR}\right]\right\} \\ &- \left(\frac{\left(1-\theta\_{a}^{1-2}\right)}{\Xi\_{a}^{2}\left(1-\theta\_{a}\right)\left(1-n\right)}\right)^{2} \left\{v\_{a}^{2n}\rho\_{a}^{2n^{2}+2n-2}\left[(2n+2)v\_{a} - \left(n^{2}-n-2\right)\cdot \text{VMR}\right]\right\} \\ &+ 2\theta\left(t\_{a}^{0}\right)^{2} \frac{\left(1-\theta\_{a}^{1-n}\right)}{\Xi\_{a}^{2}\left(1-\theta\_{a}\right)\left(1-n\right)} \left\{\begin{aligned} &\left\{v\_{a}^{n}\rho\_{a}^{n^{1}+3n}\left[(n+2)v\_{a} - \frac{\left(n^{2}+n-2\right)}{2}\cdot \text{VMR}\right]\right\} \\ &- \left\{v\_{a}^{n}\rho\_{a}^{n^{1}+n-2}\left[(n+2)v\_{a} - \frac{\left(n^{2}-n-4\right)}{2}\cdot \text{VMR}\right]\right\} \end{aligned} \end{aligned} \tag{38}$$

*MT*<sup>~</sup> *<sup>a</sup>*

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

*MT*<sup>~</sup> *<sup>a</sup>*

Eq. (40) in Eq. (41), we can get

*Var T*~ *<sup>a</sup>*

can be given by

**61**

where *<sup>T</sup>*~*T*<sup>~</sup> <sup>¼</sup> <sup>P</sup>

as long as *∂E T*~*T*~ � �*=∂va*, *E T*~ *<sup>a</sup>*

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

PRSN � MCP <sup>¼</sup> *<sup>∂</sup><sup>E</sup> <sup>T</sup>*~*T*<sup>~</sup> � �*=∂va* � *<sup>E</sup> <sup>T</sup>*<sup>~</sup> *<sup>a</sup>*

*<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>VaT*<sup>~</sup> *<sup>a</sup>*.

PRSN � MCP <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>χ</sup>* f g *<sup>∂</sup>E TT* ½ �*=∂va* � *E T*½ � *<sup>a</sup>* <sup>þ</sup> *VoR* � ð Þ <sup>1</sup> <sup>þ</sup> *<sup>χ</sup>* <sup>2</sup>

ðÞ¼ *<sup>s</sup> <sup>E</sup>* exp *sT*<sup>~</sup> *<sup>a</sup>* � � � �

¼ *E*½ � exp *s T*ð Þ *<sup>a</sup>* þ *ε<sup>a</sup>*

¼ *E* exp ð Þ *sTa E<sup>ε</sup>a*<sup>j</sup>

¼ *ETa* exp ð Þ *sTa M<sup>ε</sup>a*<sup>j</sup>

*Ta* exp *sεa Ta* ½ � ð Þ j

2

� � � � (42)

� � <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>χ</sup> E T*½ � *<sup>a</sup>* (43)

*Var T*½ �þ*<sup>a</sup> <sup>ϖ</sup>*<sup>2</sup>

*E T*½ � *<sup>a</sup>* (44)

� � � �

� � are known. From the

*aTa*

� � and *Var T*~ *<sup>a</sup>*

� �*=∂va* � *E T*½ � *<sup>a</sup>*

*E T*½ � *<sup>a</sup>* (45)

(46)

� �.

(47)

(41)

n o

n o

where *Ex*½� denotes the expectation with respect to random variable *x*. Substituting

ðÞ¼ *<sup>s</sup> ETa* exp *sTa* <sup>1</sup> <sup>þ</sup> *<sup>χ</sup>* <sup>þ</sup> *<sup>ϖ</sup>*2*<sup>s</sup>*

From the first derivative of the equation above and evaluating at *s* ¼ 0, we can

where *E T*½ � *<sup>a</sup>* denotes the mean of the random travel time. Likewise, the second-

2

� �<sup>2</sup> <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>χ</sup>*

Using these equations, we can analyze the RSN-MCP model with travelers' perception errors. When taking travelers' perception error into consideration, the objective function of the PRSN-MCP model is to minimize the weighted sum of the mean and the variance of the total perceived travel time. Thus, the PRSN-MCP toll

The variance of the perceived travel time can be expressed as follows:

<sup>¼</sup> *MTa <sup>s</sup>* <sup>1</sup> <sup>þ</sup> *<sup>χ</sup>* <sup>þ</sup> *<sup>ϖ</sup>*2*<sup>s</sup>*

obtain the first moment of the perceived travel time distribution

*E T*~ *<sup>a</sup>*

order moment is derived from the second derivative evaluated at

� �<sup>2</sup> h i <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>χ</sup>*

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

*E T*~ *<sup>a</sup>*

*Ta* ð Þ*s*

� � � � � �

2

*E T*ð Þ*<sup>a</sup>* <sup>2</sup> h i <sup>þ</sup> *<sup>ϖ</sup>*<sup>2</sup>

2

� � � � <sup>þ</sup> *VoR* � *<sup>∂</sup>Var <sup>T</sup>*~*T*<sup>~</sup> � �*=∂va* � *Var <sup>T</sup>*<sup>~</sup> *<sup>a</sup>*

f g *<sup>∂</sup>Var TT* ½ �*=∂va* � *Var T*½ � *<sup>a</sup>* <sup>þ</sup> *<sup>ϖ</sup>*<sup>2</sup> *<sup>∂</sup>E V*<sup>2</sup>

n o � �

According to Eq. (46), it is clear that the value of PRSN-MCP can be determined

� �, *∂Var T*~*T*~ � �*=∂va*, and *Var T*~ *<sup>a</sup>*

Moreover, based on the moment analysis, we can derive the mean and variance of *T*~*T*~ (see Appendix for the derivations). Substituting Eqs. (43), (45), (A2), and (A4)

conditional moment analysis above, we have already obtained *E T*~ *<sup>a</sup>*

into Eq. (46) and performing some derivation, we have

By substituting Eqs. (31), (35), and (38) into Eq. (36), the value of RSN-MCP in case of SS-SD can be determined. In the same way, by neglecting the degradation of link capacity, the RSN-MCP in case of SS-SD degenerates into the classical RSN-MCP model proposed by [18], which considers only the stochastic travel demand.
