Capturing network properties with a functional form for the multi-modal macroscopic fundamental diagram

Allister Loder, Igor Dakic, Lea Bressan, Lukas Ambühl, Michiel C.J. Bliemer, Monica Menendez, Kay W. Axhausen

Research output: Contribution to journalArticlepeer-review


In urban road networks, the interactions between different modes can clearly impact the overall travel production. Although those interactions can be quantified with the multi-modal macroscopic fundamental diagram; so far, no functional form exists for this diagram to explicitly capture operational and network properties. In this paper, we propose a methodology to generate such functional form, and we show its applicability to the specific case of a bi-modal network with buses and cars. The proposed functional form has two components. First, a three dimensional lower envelope limits travel production to the theoretical best-case situation for any given number of vehicles for the different modes. The lower envelope's parameters are derived from topology and operational features of the road network. Second, a smoothing parameter quantifies how interactions between all vehicle types reduce travel production from the theoretical best-case. The smoothing parameter is estimated with network topology and traffic data. In the case no traffic data is available, our functional form is still applicable. The lower envelope can be approximated assuming fundamental parameters of traffic operations. For the smoothing parameter, we show that it always hold similar values even for different networks, making its approximation also possible. This feature of the proposed functional form is an advantage compared to curve fitting, as it provides a reasonable shape for the multi-modal macroscopic fundamental diagram irrespective of traffic data availability. The methodology is illustrated and validated using simulation and empirical data sets from London and Zurich.

Original languageEnglish (US)
Pages (from-to)1-19
Number of pages19
JournalTransportation Research Part B: Methodological
StatePublished - Nov 2019


  • Bus
  • Congestion
  • MFD
  • Multi-modal

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Transportation


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