## 1 Introduction
Mathematical modelling is also used in the field of transport systems when designing and optimizing transport processes and systems [19; 20; 21]. The subject matter and the objective of the research are the most important factors in selecting an appropriate tool to create a model. A transport system model should reflect the modelling objective(s) and an appropriate accuracy level adapted to it. Simple models are easy to create and analyse but may provide incomplete information about the studied system and consequently lead to incorrect conclusions. On the other hand, too complex models are very time-consuming (also cost-consuming), and excessive volumes of information can alter the reliable analysis of processes and phenomena [22; 23]. The increase in traffic volume in the Warsaw transport network is permanently monitored in the traffic measurements carried out by the General Directorate of National Roads and Motorways. The results of the conducted research have led to the authors' interest in transport optimization. [PERSON] and [PERSON] [24] conducted an analysis of traffic flows in six largest cities in Poland and showed that during the peak of the morning the highest congestion occurs in Warsaw, with an average of approximately 65%. Improving road network and reducing travel time are important components of the sustainable development of large urban agglomerations. The research carried out indicates the complex nature of the problem in both scientific and practical terms. The aim of this article is to determine the impact of the changing traffic flow on the optimal solutions, taking into consideration the travel time. The Nash equilibrium and the Stackelberg approach have been compared in a real case study. A new algorithm was proposed, on the basis of which a model of the transport system was developed. The model includes a graph of the transport network with link characteristics. Based on traffic measurements, the volume of traffic flows was determined for the three O-D pairs. The study was carried out for three scenarios: a baseline defined on the basis of traffic measurements and alternative scenarios assuming a 10% deviation. Optimal traffic flow distributions according to Nash equilibrium and the Stackelberg approach were calculated using nonlinear programming and advanced IT software. Additionally, the proposed approaches and algorithms were used to optimize the classic Nguyen-Dupuis network, and the results obtained were compared with other approaches. The novelty of this paper is the development of the new algorithm and its implementation. The real case was conducted for the optimization of the existent transport system. Process optimization using two methods (tools) allows estimating the improvement of the efficiency of the transport system operating under current and future technical conditions. Based on the results of the real traffic case study, the Price of Anarchy (PoA) was determined. Furthermore, it has been observed that an increase in the volume of traffic flows in all O-D pairs can cause a decrease in traffic volumes on certain links in the road network. The published papers did not refer to this phenomenon, which may occur during the optimization of real traffic flows. The article is divided into seven sections. In the next section, a literature review has been presented that takes into account the most relevant studies from recent years. Section 3 presents the mathematical model of the transport system and a description of the Nash equilibrium and the Stackelberg approach. Section 4 shows the results of the case study research in the three assumed scenarios. Section 5 contains a discussion of the results obtained, while Section 6 describes the limitations in applying the proposed method. Finally, the conclusions of the study were presented and directions for further research have been indicated. All notations, abbreviations and acronyms used in the text are described in Tables A1 and A2 in Appendix A.