  In this article, we propose algorithms to address two critical transportation
system problems: the Generalized Real-Time Line Planning Problem (GRLPP) and
the Generalized Budgeted Multi-Visit Team Orienteering Problem (GBMTOP). The
GRLPP aims to optimize high-capacity line plans for multimodal transportation
networks to enhance connectivity between passengers and lines. The GBMTOP
focuses on finding optimal routes for a team of heterogeneous vehicles within
budget constraints to maximize the reward collected. We present two randomized
approximation algorithms for the generalized budgeted multi-assignment problem
(GBMAP), which arises when items need to be assigned to bins subject to
capacity constraints, budget constraints, and other feasibility constraints.
Each item can be assigned to at most a specified number of bins, and the goal
is to maximize the total reward. GBMAP serves as the foundation for solving
GRLPP and GBMTOP. In addition to these two algorithms, our contributions
include the application of our framework to GRLPP and GBMTOP, along with
corresponding models, numerical experiments, and improvements on prior work.
