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Recently, automation, shared use, and electrification are viewed as the ''three revolutions'' in the future transportation sector, and the traditional scheduled public transit system will be greatly enhanced with flexible services and autonomous vehicle scheduling capabilities. Many emerging scheduled transportation applications include the fully automatic operation system in urban rail transit, joint line planning, and timetabling for high-speed rail as well as emerging self-driving vehicle dispatching. The vehicle routing problem (VRP) holds promise for seeking an optimal set of vehicle routes and schedules to meet customers' requirements and plays a vital role in optimizing services for feature scheduled transportation systems. Due to the difficulty of finding optimal solutions for large-scale instances, enormous research efforts have been dedicated to developing efficient algorithms, while our paper presents a unique perspective based on a timedependent and state-dependent path searching framework. An open-source and light-weight VRP with pickup and delivery with time windows (VRPPDTW) modeling package, namely VRPLite, has been developed in this research to provide a high-quality and computationally efficient solution engine for transportation on demand applications. This paper describes the space-time-state

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The rapid development in the catering services and urban logistics industries has significantly promoted the prosperity of business-to-consumer (B2C) e-commerce urban logistics distribution, which is gaining increasing interest from food producers, distribution platforms, and consumers. Unlike traditional logistics supply chains, catering distribution services have strong timeliness and consideration of delivery delay. Traditional distribution platforms usually group commodities by their origins or destinations and then transport each group with one logistics vehicle. However, for a logistics distribution area with limited commodities, the vehicle capacity cannot be fully utilized if one vehicle can only transport commodities with the same origin or destination. Therefore, a mixed-load strategy is proposed in which commodities with different origins or destinations in a distribution area could be transported by the same vehicle to improve vehicle capacity utilization. A mixed-load strategy would further cause delivery sequencing problems, leading to different delivery delays for customers. This study proposed an equity-oriented vehicle routing problem for food distribution services with timeliness requirements considering a mixed-loading strategy and vehicle capacity constraints. For the above problem, a multi-commodity flow optimization model was constructed for the equity-oriented vehicle routing problem and a mixed-load strategy based on a time-discretized space-time-state network representation. An augmented Lagrangian relaxation approach was utilized to reformulate the original model and thus effectively solve the proposed model. Furthermore, the augmented Lagrangian model was decomposed and linearized into a series of shortest path searching subproblems and iteratively solved by a dynamic programming algorithm using an alternating direction method of multipliers (ADMM)-based solution framework. Finally, the proposed model and solution approaches were tested on numerous networks.