1. Toshihide IBARAKI Mikio KUBO Tomoyasu MASUDA Takeaki UNO Mutsunori YAGIURA Effective Local Search Algorithms for the Vehicle Routing Problem with General Time Window Constraints

2. Problem Input: Output: minimum cost vehicle routes Constraints: capacity and time window constraints

3. General Time Windows Each customer indicates the time to be serviced time windows of a customer can be non-convex and discontinuous as long as it is a piecewise linear function penalty penalty function

4. Objective function the total distance the total time penalty the total capacity excess a vehicle schedule time penalty and capacity constraints soft constraints

5. Problem structure We have to determine: ・（ a ） and （ b ） simultaneously done by the local search procedure （ a ） the assignment of customers to the vehicles （ b ） the visiting order of customers for each vehicle （ c ） the optimal start times of services of each vehicle ・ (c) determined by using dynamic programming

6. Local search （ LS ） LS repeats replacing with a better solution In its neighborhood a locally optimal an initial solution

7. Neighborhoods the CROSS exchange neighborhood the ２ -opt* exchange neighborhood the Intra-Route exchange neighborhood the cyclic exchange neighborhood

8. The cross exchange neighborhood

9. The ２ -opt* exchange neighborhood

10. The intra-route neighborhood

11. The cyclic exchange neighborhood a set of solutions obtained by exchanging paths of length at most among several routes of at most The neighborhood size grows exponentially with and 以下 Effective search via an improvement graph

12. an arc belong to different routes The improvement graph An improvement graph is defined with respect to the current solution corresponds to a path customer a node exists if paths and customer

13. The improvement graph ・ a cycle C is subset-disjoint ： all paths corresponding to nodes in C belongs to different routes ・ valid cycle ： subset-disjoint cycle with a negative cost Identifying a valid cycle is NP-hard Effective heuristic is proposed a valid cycle a corresponding operation is cost-decreasing

14. Find the optimal start times Dynamic Programming Approach Problem Input: the customer order of the vehicle (which is denoted by ) Output: the start times of services that minimize the total time penalty of the vehicle objective function

15. DP Algorithm : the minimum penalty value if customers of the vehicle are serviced before time : a time penalty function for customer : traveling time from the (h-1) st to the h th customer : the departure time of the vehicles from the depot

16. penalty

17. Time complexity of DP time : the total pieces of piecewise linear functions for customers in route : the number of customers in route … time Optimal penalty obtained

18. Iterated Local Search （ ILS ） The operation that repeats LS more than once. Initial solutions are generated using the information of the previous search. Final output is the best solution of the entire search.

19. Adaptive Multi-start Local Search （ AMLS ） LS is repeatedly applied. a set of locally optimal solutions obtained in the previous search is maintained. an initial solution for LS is generated by combining Final output is the best solution of the entire search. … P

20. Computational experiments Solomon’s benchmark instances Instance only one time window is given. both capacity and time window constraints are treated as hard constraints. Experiment’s method ILS and AMLS are run for 15000 seconds. Compare the costs of the best solutions output by ILS and AMLS with those of the best known solutions.

21. Computational experiments （ type1 ） improved tie infeasible 3 improved 11 tie

22. Computational experiments （ type2 ） improved tie infeasible 6 improved 8 tie

23. Product and Inventory Scheduling Application of VRPGTW Collaboration Research with KOKUYO Co.,Ltd.

24. Problem Input : the number of machines, product demands, setup costs, inventory costs, Output : minimum cost schedule

25. An example of a schedule Machine 1 Machine 2 Machine time

26. Inventory (total inventory)(inventory) accumulated consumption line of product inventory

27. Formulation to VRPGTW (1) is divided by the parameter customers represent the product and each customer corresponds to producing the amount.

28. Formulation to VRPGTW (2) ｛ ｝ setup time setup cost

29. Formulation to VRPGTW (3) desired produce start time yen customer

30. Computational experiments Compare the costs of the best solutions output by the ILS with those of the current real schedule. Compare also the costs with different values of.

31. Computational experiments

32. Conclusion We proposed the local search heuristic for the Vehicle Routing Problem with General Time Windows Constraints. Our general algorithm produced 9 improved solutions and19 tie solutions out of 48 instances. The effectiveness of our algorithm was confirmed through the application to the KOKUYO problem. DP algorithm is incorporated.