## 3 Model Formulation
### Objective Function and Constraints of the Model  
The total system cost of the intermodal transit model consists of user parameters (_i.e._, the value of the time for user's access, wait and in-vehicle cost, _etc._), operation parameters, social parameters and the number of decision variables (_i.e._, number of routes).  
This transit network model must satisfy users, operators and social terms. Thus, the objective function is defined as the sum of the user, operator and social costs, which is given in the following equation:  
\[\begin{array}{cc}\textit{Minimize}&C_{T}=\sum\limits_{k=1}^{K}\left[ \frac{user}{\left(C_{a}+C_{w}+C_{ui}\right)}+\overbrace{\left(C_{f}+C_{oi}+C_{ m}+C_{p}\right)}^{Operating}+\overbrace{C  
subject to:  
\[\sum_{k=1}^{K}\sum_{h=1}^{H}X_{ihk}=1\hskip 28.452756 pti=1,\ldots,I \tag{21}\] \[\sum_{i=1}^{I}\sum_{j=l+1}^{H}X_{ijk}\leq 1\hskip 28.452756 ptk=1,\ldots,K\] (22) \[\sum_{h=1}^{H}X_{ihk}-\sum_{m=1}^{I}X_{mik}\geq 0\hskip 28.452756 pti=1, \ldots,I\hskip 28.452756 ptk=1,\ldots,K\] (23) \[\sum_{i\
otin H}\sum_{h\in H}\sum_{k=1}^{K}X_{ihk}\geq 1\hskip 28.452756 pt \forall H\] (24) \[\sum_{h=1}^{H}X_{ihk}+\sum_{m=1}^{I}X_{mik}-Y_{ij}\leq 1\hskip 28.452756 pti= 1,\ldots,I\hskip 28.452756 ptj=I+1,\ldots,I+J\hskip 28.452756 ptk=1,\ldots,K\] (25) \[l_{\min}\leq L_{K}\leq l_{\max}\hskip 28.452756 ptk=1,\ldots,K\] (26) \[f_{\min}\leq F_{k}\leq f_{\max}\hskip 28.452756 ptk=1,\ldots,K\] (27) \[\sum_{k=1}^{K}\left[\left(\frac{2F_{k}}{V_{k}}\times L_{K} \right)+\left(Q_{k}\times t_{dF}\right)+\left(F_{k}\times S_{kj}\right) \right]\leq N\] (28) \[\frac{Q_{K}}{LFC}\leq F_{k}\hskip 28.452756 ptk=1,\ldots,K \tag{29}\]  
where decision variables contain two binary variables, called \(Y_{ij}\) and \(X_{ihk}\), which stand for the transit network, and a continuous variable for the feeder bus frequency (\(F_{k}\)). The determination of \(F_{k}\), as one of the decision variables, depends on the transit network configuration. Thus, the optimal feeder bus frequency using the analytical solution can be determined by setting the first derivative of the total cost function (\(C_{TK}\)) with respect to the feeder bus frequency, equating it to zero and solving it. Therefore, the optimal bus frequency can be taken as:  
\[F_{opt,K}=\sqrt{\frac{\mu_{w}Q_{k}}{4I_{k}\left[\left(\lambda_{l}+\lambda_{m} +\lambda_{s}\right)+\frac{1}{V_{k}}\left(\lambda_{f}+\lambda_{p}\right) \right]+\left(2S_{kj}\times\lambda_{p}\right)}} \tag{30}\]  
Furthermore, the minimum required frequency for route \(k\) is given as follows:  
\[F_{req,K}=\frac{Q_{k}}{LF\times C} \tag{31}\]  
Thus, the given frequency for route \(k\) is obtained by selecting the maximum value for the optimum frequency (\(F_{opt,K}\)) and required frequency (\(F_{req,K}\)). Some limitations are considered for the proposed improved model to represent an effective transit network model satisfying route feasibility, frequency, and so forth. Equations (21)-(25) correspond to the route feasibility in the network design. Several researchers used these constraints in their studies [2; 11; 13]. Equation (21) explains that each bus stop should be placed in a single route (many-to-one pattern). Furthermore, Equation (22) ensures that each generated route must be connected to only one railway station. Accordingly, in Equation (23), each bus is assumed to pass all of the stops in its route node. Equation (24) describes how each feeder bus route should be linked to only one railway station. The constraint given in Equation (25) specifies that a bus stop can be assigned to a station in which the corresponding route terminates at one of the rail stations. Constraints on the minimum and maximum length of feeder routes are given in Equation (26). Similarly, limitations for the minimum and maximum frequencies are indicated in Equation (27). Equation (28) shows the allowable maximum number of vehicles in the fleet, and Equation (29) represents the restriction for the minimum frequency in order to satisfy the demand.