## 3 Methodology
### Bi-Level Formalization in Traffic Control Problems  
The bi-level formalization is a comparatively new way of extension and definition of optimization problems. The classical optimization problem contains an analytically defined goal function. The target of the optimization is to find an extreme of the goal function towards a set of variables. However, the latter has to belong to a feasible region, defined by a set of analytically defined constraints. The bi-level formalization makes an extension to the definition of the feasible area of the initial optimization problem. Itincludes an additional optimization problem, which is a part to the feasible domain of the initial problem. Thus, two optimization problems are interconnected in hierarchical order: upper- and lower-level optimization problems. The solutions of the upper-level problem change parameters in the goal function and/or constraints of the lower-level problem and vice-versa. The benefit of this bi-level problem definition comes from the extended set of optimal variables, which are given from the upper- and lower-level problems. For technical reasons, the extended set of controls allows achieving more goals in the control process and respectively satisfying an extended set of constraints. This is beneficial for the behavior of the control object. Particularly, for the domain of transportation, this can improve the traffic dynamics by increasing the intensity of the traffic flows, decreasing the waiting time of vehicles, to influence driver behavior. The bi-level formalization is currently intensively applied for the solution of different tasks, related to the effective management of transportation resources and for the identification of transport parameters. In [40], the bi-level optimization framework is applied for the estimation of the origin-destination matrix, which is needed for the definition of the inflows in urban networks. In [41], bi-level formalization was used for coordinating the vehicle motion at roundabouts. In [42], the signals for traffic control were evaluated in a bi-level problem, according to the requests of crossing pedestrians. The bi-level optimization program has been applied to cope with the dynamically changes in time of the urban traffic flows [43]. An attempt for traffic signal modeling and optimization using bi-level programming was made in [44]. Because the defined problem is quite complex, heuristics were developed for finding a suboptimal solution. Bi-level optimization was used in minimizing the pricing and carbon emissions in green transportation [45]. The policy for transport exploitation has been evaluated under multi-objective bi-level optimization [46]. The traffic signal optimization with bi-level formalization was used in [47]. The signal control for the case of transport of hazardous goods is also formalized by definition in a bi-level framework [48]. The bi-level optimization graphically can be presented as a two-level hierarchical system, of which the levels solve appropriate optimization problems. Both hierarchical levels influence the parameters of the other problem; see Figure 2  
Using the notations for the upper-level problem for the arguments \(\mathbf{y}\), the goal function \(f_{y}(\mathbf{x},\mathbf{y})\), and the domain of the constraints \(\mathbf{S}_{y}(\mathbf{x},\mathbf{y})\), both sets of relations \(f_{y}(\mathbf{x},\mathbf{y})\) and \(\mathbf{S}_{y}(\mathbf{x},\mathbf{y})\) depend on a set of parameters \(\mathbf{x}\). The lasts are given as problem solutions of the lower-level optimization problem:  
\[\min_{\mathbf{x}}f_{x}(\mathbf{x},\mathbf{y})\]  
\[\mathbf{x}\in\mathbf{S}_{\mathbf{x}}(\mathbf{x},\mathbf{y}),\]  
which in its turn is also parameterized by the set of solutions \(\mathbf{y}\) of the upper-level problem. Hence, both upper- and lower-level problems are interconnected by their solutions, and  
Figure 2: A bi-level hierarchical optimization problem. they change each other's parameters in the goal functions and constraints.