The Dynamic Vehicle Routing Problem with A-priori Information

1. The Dynamic Vehicle Routing Problem with A-priori Information ROUTE2000 Thursday August 17th 2000 Allan Larsen The Department of Mathematical Modelling, The Technical University of Denmark.

2. Outline • The Dynamic Traveling Repairman Problem (DTRP). • Simulation of the Partially DTRP (PDTRP). • Using a-priori information in a Dynamic Traveling Salesman Problem with Time Windows (ADTSPTW). • Closing comments.

3. Real-life routing issues • The traditional VRP does not consider: • Customers calling in requesting service during the day of operation. • Time-dependent travel times. • Varying customer demands and on-site service times.

4. Static Contra Dynamic Vehicle Routing • Static Vehicle Routing: • All informations relevant to the planning of the routes are known to the planner before the routing process begins. • Informations relevant to the routing do not change after the routes have been constructed. • Dynamic Vehicle Routing: • Not all informations relevant to the planning of the routes are known by the planner when the routing process begins. • Informations can change after the initial routes have been constructed.

5. 9:34 A simple example • A single vehicle serves 5 advance request customers and…? Depot

6. Dynamic vehicle dispatching problems • Serves one customer at the time. • Examples: • Emergency services (police, fire and ambulance services). • Taxi cab services. • Low response time are important - i.e. minimize the waiting time.

7. Dynamic Vehicle Routing Problems • Services a pool of customers. • Queueing often occurs. • Examples: • Pick-up and delivery of long-distance courier mail (UPS, FedEx, DHL etc.) • Distribution of heating oil to private households. • Transportation of elderly and handicapped people. • Keeping routing costs low is important - i.e. minimize the route length.

8. Traditional solution approaches • Re-optimization - i.e. solve a static VRP each time new information is received. • Try to find a feasible spot in the routes to insert the new request. • Defer the inclusion of the new request until the latest possible moment in time.

9. Research issues • How does the level of dynamism influence the performance of the solution methods? • Is it possible to increase the performance of the solution methods if we obtain a-priori information on future requests?

10. The Dynamic Traveling Repairman Problem • Introduced by Bertsimas & Van Ryzin (1989). • All requests are dynamic and generated according to a Poisson process. • The requests are independently and uniformly distributed over a quadratic service region. • The repairman travels at constant speed. • Find routing policies so that the expected system time (waiting time + service time) is minimized.

11. The Partially Dynamic Traveling Repairman Problem • Modification of the DTRP • We assume a subset of the requests are known in advance. • The travel costs are minimized. • Issues addressed: • What is the relations between system performance and the level of dynamism?

12. Measuring the dynamism • The degree of dynamism (dod) measure (Lund et. al 1996): • I.e. in the example from before - dod = 1/(5+1) = 16 %

13. Simulation of the PDTRP • The requests are dispersed over a 10 x 10 km service region. • The vehicle travels at 40 km/h. • Generated 100 instances of problems with {0, 5, 10, …, 100} % dynamism each with an average of 40 customers. • On-site service times were generated using a log-normal distribution (average of 3 min. & variance of 5 min.) • All results are average values of these 100 instances.

14. Simulation results - PDTRP FCFS -SQM FCFS NN-FCFS-SQM PART Nearest Neighbor

15. Motivated by the pick-up and delivery of long-distance courier mail. Modelled as a Dynamic Traveling Salesman Problem with Time Windows. The service area is divided into a number subregions. We assume that the arrival intensity (λ) of each sub-region is known in advance. A-priori DTSPTW (1) • =0.1 • =0.2 • =0.25 • =0.5

16. We assume that a set of idle points, IP, is given. Each idle point serves as a “resting location” for the vehicle to go to when it is idle. A-priori DTSPTW (2) The objective is to minimize a weighted sum of the distance and the lateness.

17. A-priori DTSPTW (3) • Propose three simple repositioning policies: • NEAREST-IP: Go to the closest IP. • BUSIEST-IP: Go to the IP with the highest λ-value. • HI-REQ: Go to the IP with the highest expected number of requests. • A threshold parameter is chosen in order to avoid unnecessary traveling. • Performed extensive simulation with various levels of dynamism and temporal characteristics.

18. A-priori DTSPTW (4)

19. A-priori DTSPTW (5)

20. Closing comments • Findings: • Linear relationship between the degree of dynamism and the route costs for PDTRP. • Modest performance improvements were achieved for a-priori information based repositioning policies. • Further research: • Multiple vehicles. • Diversion.