## 2 Model Description
### Criteria of Lines Layout  
A metro line is a series of stations. There are no preliminary assumptions determining the network topology, except for one requirement: the network must be coherent. In other words, it must be possible to travel from any origin station to any destination station. The aim of the optimization is to find such a layout of metro lines that the total cost of the network is minimized and the total travel time of all the passengers is minimized too. The model does not take into account the time needed for transfers; what is more, the travel time is assumed to be dependent only on journey length, neglecting the contribution of transfers entirely. Such an assumption could be partially justified when the frequency of trains is relatively high. The above criteria are contradictory: if the travel time is expected to be minimized, the topology of the network has to be somewhat like a grid. This, in turn, means that some connections are duplicated, which causes the increase of the costs. Generally, some basic patterns ofmetro network topology can be listed: a triangle, a star, and a cartwheel [28], but having in mind the irregular shape of the populated areas and the irregular distribution of the population density in the considered case, no such assumption was made.  
The construction costs of a line segment depend on its length; additionally, it is possible to take into account the cost dependency on some local conditions like rivers, lakes, built up areas, or industry zones. For this purpose, a map describing the cost distribution is introduced into the model, and the building cost of \(i\)-th segment of \(j\)-th line between two stations can be expressed as  
\[c_{ij}=\int\limits_{L}kdL \tag{5}\]  
where \(k\)--local unit building cost and \(L\)--segment length.  
The second criterion taken into account is the overall network efficiency. There are many different measures developed to asses it [29; 30; 31], but from the passengers' point of view the expected travel time is the most important. The total travel time of all potential passengers calculated below is only an estimation, but it reflects the passengers' expectations. In the optimization process, the shortest distance between each pair of stations is calculated (\(d_{ij}\)); it could include transfers, but the transfer time is not taken into account. Then, the total travel time between a pair of origin and destination stations for all their potential passengers can be estimated in the following way:  
\[t_{ij}\sim d_{ij}\big{(}s_{i}+s_{j}\big{)}. \tag{6}\]  
If the total travel time is taken into account, the constructed objective should result in a highly interconnected network topology. It will lead to the construction's cost increase, but the passengers' demands will be satisfied in a higher degree [32]. This criterion will remain valid if a lesser number of passengers is taken into account, as long as the population distribution is the same. Just the proportional factor will change. Of course, considering the explicit costs, the investment will appear less beneficial because the costs per a single passenger will increase.  
The applied optimization method requires an increasing objective function; thus, both components of the objective function are expressed as inverts  
\[f^{2}_{cost}=\frac{1}{\sum_{j}\sum_{i}c_{ij}} \tag{7}\]  
\[f^{2}_{time}=\frac{1}{\sum_{j}\sum_{i}t_{ij}}. \tag{8}\]  
Finally, the objective function is a product of both factors  
\[f^{2}_{obj}=f^{2}_{cost}f^{2}_{time}\overset{opt}{\rightarrow}\text{max}. \tag{9}\]  
Such formulation of the objective function allows for simultaneous optimization in relation to both criteria without the use of arbitrarily adopted weights.