**5. Formulation of perceived RSN-MCP (PRSN-MCP)**

#### **5.1 Model incorporating the travelers' perception error**

Up to this point, we have studied the SN-MCP model and RSN-MCP model based on the assumption that all the travelers have perfect knowledge about the network condition. However, in real life, due to the limitations of their own condition, travelers' perception errors have to be incorporated into their route choice decision process. In view of this, it is necessary to investigate the RSN-MCP model with travelers' perception errors. In order to develop such a model, we need to make some additional assumptions on the perception error as follows:

A5. The perception error distribution of an individual traveler for a segment of road with unit travel time equals *<sup>N</sup> <sup>χ</sup>*, *<sup>ϖ</sup>*<sup>2</sup> ð Þ, where *<sup>N</sup> <sup>χ</sup>*, *<sup>ϖ</sup>*<sup>2</sup> ð Þ represents a normal distribution with predefined and deterministic mean *χ* and variance *ϖ*2.

A6. Traveler's perception errors are independent for nonoverlapping route segments.

A7. Traveler's perception errors are mutually independent over the population of travelers.

In order to compute the value of PSN-MCP of each link in the stochastic network, we need to derive the perceived link travel time, based on moment analysis. According to Assumption A5, the perception error for unit travel time, denoted by *ε*j *<sup>t</sup>*¼1, is a sample from. Besides, travel time on link *a* is the sum of independent unit travel times (see Assumption A6). Therefore, the conditional perception error for link with deterministic travel time *t* 0 *<sup>a</sup>* is normally distributed as

$$\left. \varepsilon\_{a} \right|\_{T\_{a} = t\_{a}^{0}} \sim N\left( \chi t\_{a}^{0}, \varpi^{2} t\_{a}^{0} \right) \tag{39}$$

with conditional moment generating function (MGF)

$$\left.M\_{\varepsilon\_d}\right|\_{T\_a = t\_a^0}(\varsigma) = \exp\left(\chi t\_a^0 \varsigma + \frac{\varpi^2 t\_a^0 \varsigma^2}{2}\right) = \exp\left[st\_a^0 \left(\chi + \frac{\varpi^2 \varsigma}{2}\right)\right] \tag{40}$$

where *s* is a real number. Following [22], the MGF of the perceived travel time *T*~ *<sup>a</sup>* of link for an individual traveler can be derived as follows:

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

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

<sup>1</sup> � *<sup>θ</sup>*<sup>1</sup>�2*<sup>n</sup> <sup>a</sup>* � �

� <sup>1</sup>�*θ*1�*<sup>n</sup>* ð Þ *<sup>a</sup> c n <sup>a</sup>*ð Þ 1�*θ<sup>a</sup>* ð Þ 1�*n* � �<sup>2</sup>

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**5. Formulation of perceived RSN-MCP (PRSN-MCP)**

**5.1 Model incorporating the travelers' perception error**

*vn*

� *<sup>v</sup><sup>n</sup>*

*<sup>a</sup>* ð Þ <sup>1</sup> � *<sup>θ</sup><sup>a</sup>* ð Þ <sup>1</sup> � <sup>2</sup>*<sup>n</sup> <sup>v</sup>*<sup>2</sup>*<sup>n</sup>*

*<sup>a</sup> <sup>y</sup>*<sup>4</sup>*n*2þ6*<sup>n</sup>*

*ayn*<sup>2</sup>þ3*<sup>n</sup> <sup>a</sup>* ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *va* � *<sup>n</sup>*ð Þ <sup>2</sup> <sup>þ</sup> *<sup>n</sup>* � <sup>2</sup>

� � � �

*ay<sup>n</sup>*2þ*n*�<sup>2</sup> *<sup>a</sup>* ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *va* � *<sup>n</sup>*ð Þ <sup>2</sup> � *<sup>n</sup>* � <sup>4</sup>

� � � �

*v*<sup>2</sup>*<sup>n</sup>*

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.

Up to this point, we have studied the SN-MCP model and RSN-MCP model based on the assumption that all the travelers have perfect knowledge about the network condition. However, in real life, due to the limitations of their own condition, travelers' perception errors have to be incorporated into their route choice decision process. In view of this, it is necessary to investigate the RSN-MCP model with travelers' perception errors. In order to develop such a model, we need to

A5. The perception error distribution of an individual traveler for a segment of road with unit travel time equals *<sup>N</sup> <sup>χ</sup>*, *<sup>ϖ</sup>*<sup>2</sup> ð Þ, where *<sup>N</sup> <sup>χ</sup>*, *<sup>ϖ</sup>*<sup>2</sup> ð Þ represents a normal

A6. Traveler's perception errors are independent for nonoverlapping route seg-

A7. Traveler's perception errors are mutually independent over the population of

In order to compute the value of PSN-MCP of each link in the stochastic network, we need to derive the perceived link travel time, based on moment analysis. According to Assumption A5, the perception error for unit travel time, denoted by

*<sup>t</sup>*¼1, is a sample from. Besides, travel time on link *a* is the sum of independent unit travel times (see Assumption A6). Therefore, the conditional perception error for

> 0 *a s* 2 2

where *s* is a real number. Following [22], the MGF of the perceived travel time

*<sup>a</sup>* is normally distributed as

<sup>¼</sup> exp *st*<sup>0</sup>

� � (39)

*<sup>a</sup> <sup>χ</sup>* <sup>þ</sup> *<sup>ϖ</sup>*<sup>2</sup>*<sup>s</sup>* 2

(40)

� � � �

0 *<sup>a</sup>* , *ϖ*<sup>2</sup> *t* 0 *a*

make some additional assumptions on the perception error as follows:

distribution with predefined and deterministic mean *χ* and variance *ϖ*2.

0

� �

*εa*j *Ta*¼*t* 0 *<sup>a</sup>* � *N χt*

0 *<sup>a</sup> <sup>s</sup>* <sup>þ</sup> *<sup>ϖ</sup>*<sup>2</sup>*<sup>t</sup>*

*T*~ *<sup>a</sup>* of link for an individual traveler can be derived as follows:

with conditional moment generating function (MGF)

ðÞ¼ *s* exp *χt*

*<sup>a</sup>* ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *va* � <sup>2</sup>*n*<sup>2</sup> <sup>þ</sup> *<sup>n</sup>* � <sup>1</sup> � � � *VMR* n o � �

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(38)

*<sup>a</sup> <sup>y</sup>*<sup>2</sup>*n*2þ2*n*�<sup>2</sup> *<sup>a</sup>* ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup> *va* � *<sup>n</sup>* ½ � ð Þ� <sup>2</sup> � *<sup>n</sup>* � <sup>2</sup> *VMR* n o

<sup>2</sup> � *VMR*

<sup>2</sup> � *VMR*

*<sup>∂</sup>Var TT* ½ � *∂va*

ments.

*ε*j

**60**

travelers.

link with deterministic travel time *t*

*M<sup>ε</sup><sup>a</sup>* j *Ta*¼*t* 0 *a*

¼ *t* 0 *a*

> þ2*β t* 0 *a*

� �<sup>2</sup> � *VMR* <sup>þ</sup> *<sup>β</sup><sup>t</sup>*

� �<sup>2</sup> <sup>1</sup> � *<sup>θ</sup>*<sup>1</sup>�*<sup>n</sup> <sup>a</sup>* � �

*cn*

0 *a* � �<sup>2</sup>

*<sup>a</sup>*ð Þ 1 � *θ<sup>a</sup>* ð Þ 1 � *n*

*c*2*n*

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

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$$\begin{split} \mathcal{M}\_{\tilde{T}\_{s}}(s) &= E\left[\exp\left(s\tilde{T}\_{a}\right)\right] \\ &= E\left[\exp s(T\_{a} + e\_{a})\right] \\ &= E\left\{\exp\left(sT\_{a}\right)E\_{e\_{a}\mid\_{T\_{a}}}\left[\exp\left(s e\_{a}\mid\_{T\_{a}}\right)\right]\right\} \\ &= E\_{T\_{a}}\left\{\exp\left(sT\_{a}\right)\mathcal{M}\_{e\_{a}\mid\_{T\_{a}}}(s)\right\} \end{split} \tag{41}$$

where *Ex*½� denotes the expectation with respect to random variable *x*. Substituting Eq. (40) in Eq. (41), we can get

$$\begin{split} M\_{\hat{T}\_a}(\varsigma) &= E\_{T\_a} \left\{ \exp \left[ sT\_a \left( 1 + \chi + \frac{\varpi^2 \varsigma}{2} \right) \right] \right\} \\ &= M\_{T\_a} \left[ s \left( 1 + \chi + \frac{\varpi^2 \varsigma}{2} \right) \right] \end{split} \tag{42}$$

From the first derivative of the equation above and evaluating at *s* ¼ 0, we can obtain the first moment of the perceived travel time distribution

$$E\left[\tilde{T}\_a\right] = (\mathbf{1} + \chi)E[T\_a] \tag{43}$$

where *E T*½ � *<sup>a</sup>* denotes the mean of the random travel time. Likewise, the secondorder moment is derived from the second derivative evaluated at

$$E\left[\left(\tilde{T}\_a\right)^2\right] = \left(\mathbf{1} + \boldsymbol{\chi}\right)^2 \mathbf{E}\left[\left(T\_a\right)^2\right] + \sigma^2 \mathbf{E}[T\_a] \tag{44}$$

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

$$\operatorname{Var}\left[\tilde{T}\_a\right] = \operatorname{E}\left[\left(\tilde{T}\_a\right)^2\right] - \operatorname{E}\left[\tilde{T}\_a\right]^2 = \left(\mathbf{1} + \boldsymbol{\chi}\right)^2 \operatorname{Var}[T\_a] + \sigma^2 \operatorname{E}[T\_a] \tag{45}$$

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 can be given by

$$\text{PRSN} - \text{MCP} = \left\{ \partial E\left[ \tilde{T}\tilde{T} \right] / \partial v\_d - E\left[ \tilde{T}\_d \right] \right\} + \text{VaR} \cdot \left\{ \partial \text{Var}\left[ \tilde{T}\tilde{T} \right] / \partial v\_d - \text{Var}\left[ \tilde{T}\_d \right] \right\} \tag{46}$$

where *<sup>T</sup>*~*T*<sup>~</sup> <sup>¼</sup> <sup>P</sup> *<sup>a</sup>*<sup>∈</sup> *<sup>A</sup>VaT*<sup>~</sup> *<sup>a</sup>*.

According to Eq. (46), it is clear that the value of PRSN-MCP can be determined as long as *∂E T*~*T*~ � �*=∂va*, *E T*~ *<sup>a</sup>* � �, *∂Var T*~*T*~ � �*=∂va*, and *Var T*~ *<sup>a</sup>* � � are known. From the conditional moment analysis above, we have already obtained *E T*~ *<sup>a</sup>* � � 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) into Eq. (46) and performing some derivation, we have

$$\begin{aligned} \text{PRSN}-\text{MCP} &= \left(\mathbf{1} + \boldsymbol{\chi}\right) \{ \partial \mathbb{E}[\text{TT}]/\partial \boldsymbol{v}\_{a} - \mathbb{E}[\boldsymbol{T}\_{a}] \} \\ &+ \text{VoR} \cdot \left\{ \left(\mathbf{1} + \boldsymbol{\chi}\right)^{2} \{ \partial \text{Var}[\text{TT}]/\partial \boldsymbol{v}\_{a} - \text{Var}[\text{T}\_{a}] \} + \sigma^{2} \left\{ \partial \mathbb{E}\left[\mathbf{V}\_{a}^{2} \boldsymbol{T}\_{a}\right]/\partial \boldsymbol{v}\_{a} - \text{E}[\text{T}\_{a}] \right\} \right\} \end{aligned} \tag{47}$$
