Modeling the α-max capacity of transportation networks: a single-level mathematical programming formulation

Zhaoqi Zang, Xiangdong Xu, Anthony Chen, Chao Yang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

4 Citations (Scopus)


Network capacity, defined as the largest sum of origin–destination (O–D) flows that can be accommodated by the network based on link performance function and traffic equilibrium assignment, is a critical indicator of network-wide performance assessment in transportation planning and management. The typical modeling rationale of estimating network capacity is to formulate it as a mathematical programming (MP), and there are two main approaches: single-level MP formulation and bi-level programming (BLP) formulation. Although single-level MP is readily solvable, it treats the transportation network as a physical network without considering level of service (LOS). Albeit BLP explicitly models the capacity and link LOS, solving BLP in large-scale networks is challenging due to its non-convexity. Moreover, the inconsideration of trip LOS makes the existing models difficult to differentiate network capacity under various traffic states and to capture the impact of emerging trip-oriented technologies. Therefore, this paper proposes the α-max capacity model to estimate the maximum network capacity under trip or O–D LOS requirement α. The proposed model improves the existing models on three aspects: (a) it considers trip LOS, which can flexibly estimate the network capacity ranging from zero to the physical capacity including reserve, practical and ultimate capacities; (b) trip LOS can intuitively reflect users’ maximum acceptable O–D travel time or planners’ requirement of O–D travel time; and (c) it is a convex and tractable single-level MP. For practical use, we develop a modified gradient projection solution algorithm with soft constraint technique, and provide methods to obtain discrete trip LOS and network capacity under representative traffic states. Numerical examples are presented to demonstrate the features of the proposed model as well as the solution algorithm.

Original languageEnglish
Pages (from-to)1211-1243
Number of pages33
Issue number4
Publication statusPublished - Aug 2022


  • Convex programming
  • Flexibility
  • Gradient projection
  • Network capacity
  • Trip level of service

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Development
  • Transportation


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