## 1 Introduction  
Transport is a complex process that aims to move people and goods from the point of origin to the destination, within the existing infrastructure. The efficiency of transport processes can be determined by the value of the goal function of the optimization problem [1; 2]. In practice, the fundamental indices considered when optimizing a transport process are its costs [3; 4] and travel time [5; 6; 7]. Road congestion is a widespread global phenomenon resulting from high population density and increased number of motor vehicles [8]. The effects of congested road networks are a significant burden in terms of time loss, pollution, increased industrial costs, and road maintenance costs. The management of the causes and effects of congestion will become more urgent as the population grows, as congestion increases travel times and causes significant delays. From an industrial perspective, delays reduce productivity and consequently increase operating costs [9; 10]. Many modern vehicles are equipped with navigation devices that can determine the shortest route from origin to destination taking into account current road conditions. To reduce congestion or even avoid congestion, it is necessary for all users of the system to coordinate the route and for users to follow suggested routes. In the future, with the proliferation of autonomous vehicles, this could become a realistic solution to road congestion problems [11]. The scientific approach to traffic problems focuses on developing complex models that determine the conditions and constraints of processes in transportation systems. A model is a qualitative and quantitative representation of a certain reality fragment (studied system) presented using a different material basis. It should reflect selected features and system properties, which are important both in terms of the studies conducted and, above all, consistent with the modelling objective(s) [12]. Creating a model reflecting selected functional aspects of a system makes studying the system become easier and more beneficial from the economic perspective. Moreover, in the case of designed systems, this enables its detailed analysis [13]. A mathematical model is a finite set of mathematical symbols and relations and strict rules for their operation, in which the symbols and relations contained in the model have an interpretation related to specific elements of the modelled part of reality [14; 15]. It describes a phenomenon studied using variables that can take numerical values. Studying such a model is possible due to the use of a mathematical apparatus, which allows accurate estimation of system behaviour in assumed and allowed scenarios [16; 17; 18]. Mathematical models can be classified according to the several different criteria. Taking into consideration time, models are divided into: static--reflecting the system and its surroundings in a given state (not taking time into account) and dynamic (so-called process models)--reflecting system operation as a function of time. Another criterion is the cause-and-effect relationship, which classifies models as deterministic and stochastic. The deterministic models reflect a system in the event of complete certainty of information regarding the analyzed system. Stochastic (probabilistic) models take into account the risk and lack of information on the structure of the characteristics of the system studied, where a mathematical description of the process is reflected by random variables. The form of relationships and relations classifies models as linear or nonlinear. The linear models describe evaluation criteria, conditions, and restrictions through linear equations (inequalities) with respect to decision variables, while the nonlinear models contain decision variable functions in the form of curves of at least the second degree. Mathematical modelling involves describing phenomena using the language of mathematics and formal logic. It is widely applied in exact sciences (including mathematics and natural sciences) and economics.