Multi-trip time-dependent vehicle routing problem with time windows

In this study, we investigate a routing problem in urban transportation which considers time-dependent travel time, multiple trips per vehicle, and loading time at the depot simultaneously. Its objective is to minimize the total travel distance while satisfying the time windows, vehicle capacity, and maximum trip duration constraints. We model the problem as a multi-trip time-dependent vehicle routing problem with time windows (MT-TDVRPTW). We formulate the time-dependent ready time function and duration function for any segment of consecutive nodes as piecewise linear functions and develop an iterative algorithm to derive them efficiently. Then, these two functions are embedded in the segment-based evaluation scheme to accelerate the local search operators. Based on them, we design a hybrid meta-heuristic algorithm to solve the problem, leveraging the adaptive large neighborhood search (ALNS) for guided exploration and the variable neighborhood descend (VND) for intensive exploitation. Moreover, we propose problem-specific local search operators and removal operators to enhance the effectiveness of the algorithm. Extensive experiments are conducted to assess the performance of the algorithm on instances of varied sizes. The algorithm is shown to be robust and efficient under different speed profiles and maximum trip duration limits. Finally, we evaluate the performance of the algorithm on a special case: the multi-trip vehicle routing problem with time windows.