Solving school bus routing problem with mixed-load allowance for multiple schools

Abstract 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.