A Benders Decomposition Algorithm for a Bid Generation Problem in the Procurement of Three-echelon Transportation Services

In this paper, we address a bid generation problem in a combinatorial auction for the procurement of transportation services with time windows in a three-echelon transportation network. By considering the interconnectivity of transportation operations in different echelons, a path-based mathematical programming model is proposed for this problem. To solve the model with complicated linking constraints, a Benders decomposition algorithm in a heuristic manner is designed. The algorithm iteratively and alternatively solves a master problem that is a selective routing problem with time windows for the second echelon and two subproblems that are routing problems with time windows for the first and third echelon respectively. The basic Benders decomposition algorithm is enhanced by applying the acceleration techniques of multiple cuts, Pareto-optimal cuts and multiple cuts combined with Pareto-optimal cuts. Extensive computational experiments were performed to evaluate the effectiveness of the acceleration techniques and the efficiency of the algorithm. The computational results indicate that the enhanced algorithm significantly outperforms CPLEX in terms of solution quality and CPU time. Moreover, our sensitivity analysis on some key parameters of the model show that the lower the proportion of reserved requests, the larger of the capacity of each vehicle, and the higher the percentage of profit increase of a carrier after bidding.