Solving school bus routing problem with mixed-load allowance for multiple schools
Journal article, 2021

The school bus routing problem (SBRP) is a challenging real-world problem that affects many citizens on a daily basis. This study considers several important classes of practical requirements for SBRP that include multiple schools, mixed-loads for students boarding the same bus but from different stops and commuting to different schools, heterogeneous fleets, various student-pickup time windows, and school-bell-ring constraints. Accordingly, a time-discretized multi-commodity network flow model is proposed based on a student-loading state-oriented space-time network. To enable optimization of both the student-bus assignment and bus routing, we introduce an extended-state dimension to represent the number of students commuting to different schools by buses. By implementing an augmented Lagrangian relaxation approach, the primal SBRP is reformulated as a quadratic 0–1 programming model with linear flow balance constraints. Furthermore, the augmented Lagrangian model can be decomposed and linearized as a series of linear multi-commodity network flow sub-problems that can be successively solved using dynamic programming algorithms in a cyclic block coordinate descent framework. The proposed model and approach to the solution are implemented on a nine-node test network, a Sioux Falls network under different scenarios, and a large-scale Chicago sketch network.

Augmented Lagrangian relaxation

Multi-commodity network flow

School bus routing

Cyclic block coordinate descent method


Pan Shang

Beijing Jiaotong University

Liya Yang

Renmin University of China

Ziling Zeng

Chalmers, Architecture and Civil Engineering, GeoEngineering

Lu (Carol) Tong

Beihang University

Computers and Industrial Engineering

0360-8352 (ISSN)

Vol. 151 106916

Subject Categories

Computational Mathematics

Other Mathematics

Computer Systems



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