On the optimization of the bus network design: An analytical approach based on the three-dimensional macroscopic fundamental diagram

Igor Dakic, Ludovic Leclercq, Monica Menendez

Research output: Contribution to journalArticlepeer-review


Multiple factors can influence the public transport level of service. All take root in the network structure and the operating regime, i.e. how the bus lines are arranged atop the street network and how the service frequency is adjusted to meet urban mobility patterns. This is known as the bus network design problem and has been the subject of several studies. The problem is so challenging that most studies until now resort to strong assumptions such as a static description of the peak hour demand, homogeneous user behavior, and equal trip lengths. Potential effects of different types of user behavior and trip lengths patterns on the user and/or operator cost have not been investigated whatsoever. Moreover, none of the existing studies have considered the effects of bus network structure on private car users, the level of interactions between the modes, and the passenger mode choice that depends on traffic conditions. This paper aims to close this gap and provide a general framework considering multiple trip length patterns, two types of user behavior, and the effects that the bus network structure might have on the traffic performance and passenger mode choice. For modeling different trip length patterns, we use the trip length distribution as an intermediate level of abstraction. To capture complex modal interactions and quantify the operating speeds, we apply the recently proposed three-dimensional macroscopic fundamental diagram. We use the operating speed for each mode to determine the mode choice at the trip length level. This way, we are able to solve the optimal bus network design problem under the free-flow/saturated traffic conditions in an analytical way, while considering more realistic settings including a dynamic description of the peak hour demand, mixed traffic, and different mode choice decisions depending on trip lengths and walking preferences. Numerical analysis reveals that all the tested factors, including demand intensity, user behavior, and trip length patterns, have significant effects on the operator and user cost function. Results show that the probability of choosing any given mode follows certain distribution that varies across the trip length patterns, indicating the importance of modeling the mode choice at the trip length level. Furthermore, the results indicate that users can benefit if they are willing to adjust the number of transfers to minimize the walking distance at the origin and the destination. Moreover, we show that the optimal bus network design determined for the uniform trip pattern underestimates the number of required buses, which leads to passenger congestion at stops during the peak period. This, however, does not happen when we take into account the actual trip length distribution for the bus network design. A comparison with a simplified approach that considers the bus system only, reveals the value of accounting for the complex bi-modal interactions, especially for higher demand levels. Finally, we show that by allowing the design parameters to vary across cardinal directions we provide more flexibility for the bus system to serve the passenger demand while reducing the operator cost compared to the existing approaches.

Original languageEnglish (US)
Pages (from-to)393-417
Number of pages25
JournalTransportation Research Part B: Methodological
StatePublished - Jul 2021


  • Bi-modal MFD
  • Bus network design
  • Bus operations
  • Space allocation
  • Trip length distribution

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Transportation


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