## 4 Mathematical Formulation
### Objective
#### 4.2.2 Total Passenger Travel Time Perception  
In the space-time network corresponding to the integrated variable \(\left(\mathbf{\Psi}^{T},\mathcal{J}_{1}\left|\mathbf{\Psi}\right.\right)^{T}\), the TTTP is equal to the inner product of the path equilibrium flow \(\left.\mathbf{F}^{*T}\right|_{\mathcal{J}_{1}\left|\mathbf{\Psi}\right. }\) and path time perception \(\mathbf{C}(\mathbf{F}^{*}\Big{|}_{\mathcal{J}_{1}\left|\mathbf{\Psi}\right. })\):  
\[TTTP=\mathbf{F}^{*T}\Big{|}_{\mathcal{J}_{1}\left|\mathbf{\Psi}\right.} \cdot\mathbf{C}(\mathbf{F}^{*}\Big{|}_{\mathcal{J}_{1}\left|\mathbf{\Psi}\right.}) \tag{30}\]  
One form of path time perception \(\mathbf{C}(\mathbf{F}):R_{+}^{\left|\mathcal{P}\right|}\to R_{+}^{\left| \mathcal{P}\right|}\) is given by (31), which is composed of travel time \(\Delta^{T}\mathbf{t}\), in-vehicle congestion penalty time \(\mathbf{g}(\mathbf{\widetilde{t}},\mathbf{V}):R_{+}^{\left|\mathcal{B}\right| }\to R_{+}^{\left|\mathcal{P}\right|}\), and in-vehicle fatigue penalty time \(\mathbf{h}(\Delta^{T}\mathbf{\Gamma}^{T}\mathbf{\widetilde{t}}):R_{+}^{\left| \mathcal{B}\right|}\to R_{+}^{\left|\mathcal{P}\right|}\), where \(\mathbf{\widetilde{t}}=\mathbf{\Gamma}\) is the in-vehicle arc time vector. \[\mathbf{C}(\mathbf{F})=\Delta^{T}\mathbf{t}+\mathbf{g}(\mathbf{\widetilde{t }},\mathbf{V})+\mathbf{h}(\Delta^{T}\mathbf{\Gamma}^{T}\mathbf{\widetilde{t}}) \tag{31}\]  
The path \(p\in\mathcal{P}\) component of \(\mathbf{g}(\mathbf{\widetilde{t}},\mathbf{V})\) is given by (32), where the in-vehicle arc \(\mathbf{\widetilde{\alpha}}\in\mathcal{B}\) equilibrium flow and capacity are \(\mathbf{V}^{*}=\mathbf{\Gamma}\Delta\mathbf{F}^{*}=(\dots,f_{\mathbf{\widetilde {a}}}^{*},\dots)^{T}\) and \(cap_{\mathbf{\widetilde{a}}}\), respectively. Passengers will feel crowded if the ratio of \(f_{\mathbf{\widetilde{a}}}^{*}\) to \(cap_{\mathbf{\widetilde{a}}}\) is greater than the congestion penalty threshold \(\theta\in[0,1]\). We construct a vector \(\mathbf{\theta}:=(\dots,\theta_{\widetilde{a}},\dots)^{T}\in R_{+}^{\left| \mathcal{B}\right|},\theta_{\widetilde{a}}=\theta,\forall\widetilde{a}\in \mathcal{B}\), and define the congestion coefficient as \(\mathbf{L}:=\mathbf{V}^{*}\oslash\mathbf{cap}-\theta\); then, \(\mathbf{g}(\mathbf{\widetilde{t}},\mathbf{V})\) is expressed by the vector of (33). It is worth noting that the symbols \(\circ\) and \(\oslash\) represent the Hadamard product and quotient, respectively, and that the elements of the new vector after the operation are composed of the product and quotient of the corresponding elements of the two vectors. The symbol \(P_{R_{+}^{\alpha}}(\bullet)\) denotes the projection operator of Euclidean \(n\)-dimensional space on the nonnegative orthant, that is, the elements of the new vector after the operation are the maximum values of 0 and the corresponding elements of the original vector. \[g_{p}=\mu^{(c)}\sum_{a\in\mathcal{E}}\sum_{\widetilde{a}\in\mathcal{B}}\delta_{ap }\gamma_{\widetilde{a}a}t_{\widetilde{a}}\cdot\max(0,f_{\widetilde{a}}^{*}/cap_ {\widetilde{a}}-\theta),\forall p\in\mathcal{P} \tag{32}\]  
\[\mathbf{g}(\widetilde{\mathbf{t}},\mathbf{V})=\mu^{(c)}\Delta^{T}\mathbf{ \Gamma}^{T}P_{R_{+}^{[\widetilde{\mathbf{t}}]}}(\widetilde{\mathbf{t}}\circ \mathbf{L}) \tag{33}\]  
The path \(p\in\mathcal{P}\) component of \(\mathbf{h}(\Delta^{T}\mathbf{\Gamma}^{T}\widetilde{\mathbf{t}})\) is given by (34), where the in-vehicle time is \(\widetilde{t}_{p}=\sum_{a\in\mathcal{E}}\sum_{\widetilde{a}\in\mathcal{B}} \delta_{ap}\gamma_{\widetilde{a}a}t_{\widetilde{a}}\). Passengers will feel fatigue if \(\widetilde{t}_{p}>t_{0}\). We construct a vector \(\mathbf{t}_{0}:=(\ldots,t_{p},\ldots)^{T}\in R_{+}^{[\widetilde{\mathbf{t}}] },t_{p}=t_{0},\forall p\in\mathcal{P}\), and \(\mathbf{h}(\Delta^{T}\mathbf{\Gamma}^{T}\widetilde{\mathbf{t}})\) is then expressed by the vector of (35).