1. A tabu search heuristic to solve the split delivery Vehicle Routing Problem with Production and Demand Calendars (VRPPDC) Marie-Claude Bolduc Gilbert Laporte, Jacques Renaud, Fayez Boctor CORS/Optimization Days – May 12, 2008 mc@mcbolduc.com

2. Presentation outline Problem definition (VRPPDC); Literature; Solutions; – Mathematical formulation; – Tabu search heuristic. Computational results; Conclusion.

3. Split delivery Vehicle Routing Problem with Production and Demand Calendars (VRPPDC) Network: – P products, T periods – 1 distribution center (DC) – n customers with demand calendars (expressing due dates) – Private limited fleet of m homogeneous vehicles – Common carrier – Production calendar (a priori set by the factory) Objective: Served all customers – Private vehicle Start and end at the DC; Capacity Q, length L, 1 route. – Availability of the products (according to the production calendar) – Respect of the due date (according to the demand calendar) Question: How to determine the distribution calendar (the routes of the private fleet, as well as the customers served by a common carrier) in order to minimize the overall transportation and inventory costs over a given time horizon? Problem definition

4. Literature Most case: production calendars + 3 rd -party logistic for transportation Objective: determine both distribution and production calendars (adapted to feed the former) – Fumero & Vercellis (1999) Mathematical VRPPDC model with private fleet only (no common carrier) “On time" deliveries only – Boudia, Louly & Prins (2006) Single product problem (no common carrier) GRASP with a reactive mechanism

5. Mathematical formulation

6. Tabu search heuristic: Step-by-step 1.Initial solution phase (two-phase method) 2.Tabu phase (with tabu iterations) – Adjust the aspiration criterions  ikt – Evaluate all the neighbors (i,k’,t’) with respect of the due dates – Adjust the diversitification criterion  i – If the solution of the best neighbor is feasible and improve s*, save it and update the aspiration criterions  ikt. – Update  (penalty quantity factor),  penalty distance factor) and the tabu list 3.Improvement phase

7. Tabu search heuristic: Initial feasible solution Period decomposition method based on VRPPC (Vehicle Routing Problem with Private fleet and Common carrier) Two-phase method using SRI (Bolduc, Renaud & Boctor, 2006) and RIP (Bolduc et al., 2006) For each period… Phase 1: – List the customer-product (= a unique customer) with a demand due in-time (current period, demand calendars). (  split) – Common carrier cost. – Solve using SRI or RIP heuristic. Phase 2: – List the customers (single unified demand) with a demand due in current period. – Common carrier cost. – Solve using SRI or RIP heuristic. Solution: Best of phase 1 and 2.

8. Tabu search heuristic: Tabu search phase Penalized cost function: Attribute set: (customer-truck-period) Neighbor: Movable quantity: – Complete switch (moved everything in respect to inventory + due dates) – Partial switch (complete switch | space used in truck) (  split)

9. Tabu search heuristic: Neighbor reduction strategy Random strategy – Setting a priori proportion. – Randomly selected some neighbors for evaluation. Distance strategy – Setting a priori the longest distance between customer and route. – Customer is evaluated only if he is within the fixed distance.

10. Tabu search heuristic: Improvement phase Advance the delivery of – complete blocks of product demand; – partial blocks of product demand; Switch an internal customer (i.e., one served by a private vehicle) with an external customer (i.e., one served by a common carrier). – Osman (1993) 1-1-exchange only between private fleet and common carrier

11. Results: SRITabu Solution procedure A Solution procedure B   and  updated considering iterations with infeasible or without improvement

12. Conclusion Our tabu search (procedure A, random 50% and 100 000 iterations) : deviation of 1.12% from the best-known solution in about 48 min. Good results, efficient neighbor reduction Solution to a rich variant of VRP Thanks! Questions?

13. References Bolduc M.-C., Renaud J. & Boctor F.F. "A heuristic for the routing and carrier selection problem." European Journal of Operational Research, 183, 2007, 926–932. Bolduc M.-C., Renaud J., Boctor F.F. & Laporte G. "A perturbation metaheuristic for the vehicle routing problem with private fleet and common carriers." Journal of the Operational Research Society, 2006, to appear. Boudia M., Louly M.A.O. & Prins C. "A reactive GRASP and path relinking for a combined production–distribution problem." Computers & Operations Research, 34, 2007, 3402–3419. Cordeau J.-F., Gendreau M. & Laporte G. "A tabu search heuristic for periodic and multi-depot vehicle routing problem." Networks, 30, 1997, 105–119. Fumero F. & Vercellis C. "Synchronized development of production inventory, and distribution schedules." Transportation Science, 33, 1999, 330–340. Osman I.H. "Metastrategy simulated annealing and tabu search algorithms for the vehicle routing problem." Annals of Operations Research, 41, 1993, 421–451.