A branch-and-cut algorithm for the time-dependent vehicle routing problem with time windows and combinatorial auctions

The urban logistics industry typically faces traffic congestion problems that result in variation of travel times across the day and impose a great challenge in routing design. While companies may either use a private fleet of delivery vehicles or outsource the tasks to the third-party logistics (3PL) providers to fulfill their logistics demand, some companies might employ a combination of both when their resource is unable to cope with demand or if demand fluctuates significantly over time. To tackle both challenges, we focus on a new variant of the vehicle routing problem known as the time-dependent vehicle routing problem with time windows and combinatorial auctions (TD-VRPTWCA), which considers time-dependent travel times in the routing design of the private fleet while selecting competitive bids from the 3PLs to serve a subset of the customers economically. The goal is to minimize the sum of the travel cost incurred by the private fleet and the outsourcing cost charged by the 3PLs for the chosen bids. To solve this problem, we present an arc-flow model with nine families of valid inequalities to strengthen the linear relaxation of the model. Based on this, a branch-and-cut approach is developed and evaluated on instances adapted from the well-known Solomon's benchmark data. Extensive computational results demonstrate the effectiveness of the proposed method.