MIC’2001 - 4th Metaheuristics International Conference 293 Effective Local Search Algorithms for the Vehicle Routing Problem with General Time Window Constraints Toshihide Ibaraki ∗ Mikio Kubo † Tomoyasu Masuda ∗ Takeaki Uno ‡ Mutsunori Yagiura ∗ ∗ Department of Applied Mathematics and Physics Graduate School of Informatics Kyoto University, Kyoto 606-8501, Japan Email: {ibaraki, masuda, yagiura}@amp.i.kyoto-u.ac.jp † Logistics and Information Engineering Tokyo University of Mercantile Marine Tokyo 135-8533, Japan Email: kubo@ipc.tosho-u.ac.jp ‡ Algorithm Foundation Research National Institute of Informatics National Center of Sciences, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan Email: uno@me.titech.ac.jp 1 Introduction The vehicle routing problem with time windows (VRPTW) is the problem to minimize the sum of the distances traveled by a fixed number of vehicles, which visit every customer exactly once under capacity and time window constraints. This problem is a well-known combinatorial optimization problem and has a wide range of applications such as bank deliveries, postal deliveries, school bus routing and so on. Since it is known to be NP-hard, no exact algorithm for VRPTW exists unless P = NP, and a number of heuristics have been proposed in the literature. The problem VRPTW usually allows only one time window for each customer [3, 2, 5]. In this paper, we allow a general time window constraint (VRPGTW), in the sense that one or more time windows are allowed for each customer. In our formulation, the time window constraint for each customer is treated as a penalty function, which can be non-convex and discontinuous as long as it a piecewise linear function. Hence, after fixing the order of customers for a vehicle to visit, we must determine the optimal start times to serve the customers so that the total time penalty of the vehicle is minimized. We show that this problem can be efficiently solved by using dynamic programming. This dynamic programming is incorporated in the local search algorithms. In our local search, in addition to standard neighborhoods, we use a new type of neighborhood called the cyclic exchange neighborhood, whose size generally grows exponentially with the input size. To overcome this difficulty, we propose an efficient heuristic to find an improved solution in the cyclic exchange neighborhood via the improvement graph. The computational results for various benchmark instances exhibit good prospects of the proposed algorithms. Porto, Portugal, July 16-20, 2001