Abstract
We propose a novel optimization problem for a vehicle delivery planning and its effective heuristics based on dynamic programming (DP). Novelty of the problem is that each delivery item has a designated delivery date with a certain relaxation window. Then, the minimum delivery cost can be decreased compared with the rigorous delivery date problem. The optimization of the actual delivery dates is formalized as 0-1 integer programming, which can be reduced to a resource allocation problem. DP is then applicable for the rigorous optimization, though the computational cost is large. Here we proposed a heuristic approach based on DP. First, the value function, which is obtained through recursive steps in DP, is approximated by a linear programming relaxation based on the method proposed by Bertsimas et al. Next, a search of the optimal solution at each recursive step is bounded by setting a threshold. Furthermore, the number of recursive steps can be shortened according to a characteristic time scale of a target problem. The proposed heuristics are applied to the target delivery problem, and detailed numerical calculations on various size problems with various parameter settings show the effectiveness with a largely reduced computational time.
