## 4 Mathematical Formulation
### Equilibrium Space-Time Network Design Model  
Assuming that the passenger path choice behavior in the space-time network conforms to [PERSON]'s first principle, the integrated optimization of the line-planning and timetabling problem can be described as the urban rail transit ESTNDP. Then, it can be expressed as a generalized bi-level optimization problem named an equilibrium space-time network problem design model (ESTNDM) as follows:  
\[\min_{\boldsymbol{\Psi},\mathcal{J}_{1}|\boldsymbol{\Psi}}LCC|_{T^{(b)}}^{T^{ (c)}} \tag{36}\]  
\[\min_{\mathcal{J}_{1}|\boldsymbol{\Psi}}TTTP \tag{37}\]  
\[\mathbf{C}(\mathbf{F}^{*})^{T}(\mathbf{F}-\mathbf{F}^{*})\geq 0 \tag{38}\]  
\[\Phi^{\top}\mathbf{F}=\mathbf{Q} \tag{39}\]  
\[F_{p}^{\omega}\geq 0,\forall w\in\mathcal{W},p\in\mathcal{P}_{w} \tag{40}\]  
The equilibrium constraints (38)-(40) of the ESTNDM are used to describe the path choice behavior of passengers. We assume that the time when passengers arrive at the platform is the same as when they swipe their card, i.e., passenger walking time at the station is not considered. Passenger \(q_{w},\forall w\in\mathcal{W}\) faces the same path selection problem with the inherent attribute representing their total waiting time before train \(i\) arrives. Thus, the passenger routing choice problem under \((\boldsymbol{\Psi}^{T},\mathcal{J}_{1}|\boldsymbol{\Psi})^{T}\) can be considered as equivalent to a static traffic assignment problem with a fixed demand. Passengers \(q_{w},\forall w=(r,s,i)\in\mathcal{W}\), have a fixed origin point \(v_{ri}\) in the STN shown in Figure 6 and can freely choose their destination point \(v_{sj},\forall j\in\mathcal{I}\). With the virtual network, by adding a virtual destination point \(D_{i}^{rs}\) and virtual links \((v_{sj},D_{i}^{rs})\), the problem can be transformed into a static assignment problem with a fixed OD flow. Assuming that the cost of the virtual links is 0, the virtual network is equivalent to the original problem [54]. Because the in-vehicle fatigue penalty time \(\mathbf{h}_{\mu}(\mathbf{\Delta}^{T}\mathbf{\Gamma}^{T}\mathbf{t})\) is a non-additive function, the path time perception is not equal to the sum of the link time perception, i.e., \(h_{pu}=\)\(h_{pu}(t_{p}-t_{0})\
eq\sum_{\alpha\in\mathcal{E}}\delta_{\mu p}h_{pu}(t_{a}-t_{0}), \forall p\in\mathcal{P}\). Therefore, we must build a variational inequality model with path flow as a variable [55], as in (38)-(40). Next, we prove the existence and uniqueness of the equilibrium flow. First, we introduce the following theorem. Suppose that \(\mathbf{t}\) is differentiable and (strictly) monotone, \(g^{\prime}_{\mu i}(v_{i})\geq 0\) and \(h^{\prime}_{\mu i}(v_{i})\geq 0\) for all \(i=1,\dots,n_{\mathcal{P}}\). Then, the path cost function \(\mathbf{C}\) is both differentiable and (strictly) monotone. Suppose that \(\mathbf{C}(\mathbf{F})\) is continuous on the compact convex set \(\Omega\). Then the variational inequality problem \(VI(\mathbf{C},\Omega)\) admits at least one solution \(\mathbf{F}^{*}\)  
Suppose that \(\mathbf{C}(\mathbf{F})\) is strictly monotone on the compact convex set \(\Omega\). Then, the solution is unique if one exists. Because the link time \(\mathbf{t}\) is differentiable and strictly monotone, functions \(\mathbf{g}_{\mu}\) and \(\mathbf{h}_{\mu}\) are differentiable and monotone, and \(\mathbf{C}=\mathbf{\Delta}^{T}\mathbf{t}+\mathbf{g}_{\mu}(\mathbf{t},\mathbf{v}) +\mathbf{h}_{\mu}(\mathbf{\Delta}^{T}\mathbf{\Gamma}^{T}\mathbf{t})\) is both differentiable and strictly monotone based on Lemma 1. Moreover, because the OD flow is fixed, which means that the feasible set \(\Omega=\{f|(\ref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: equsing (2). For instance, decoding chromosome \(\widetilde{Q}_{2}(Q_{m})=(2,1,2)\) results in timetable \(\mathcal{J}_{1}=<3,1,4,5,2,6>\).