1. Bekkjarvik, 04.12.20031 A Heuristic Solution Method for a Stochastic Vehicle Routing Problem Lars Magnus Hvattum

2. Bekkjarvik, 04.12.2003 2 Outline of the Presentation The Real-World Problem Vehicle Routing Concepts  VRP  Dynamic VRP  Stochastic VRP Our Problem Solution Approach Behaviour Results Conclusions

3. Bekkjarvik, 04.12.2003 3 The Real-World Problem Linjegods A/S, Heimdal Pick-up customers 133 customers per day (on average) About half of the customers are known in advance The rest is revealed during execution  Unknown location, demand, time-window

4. Bekkjarvik, 04.12.2003 4 Vehicle Routing Concepts (1) The Vehicle Routing Problem (VRP) A set of customers and a central depot A set of vehicles, located at the depot Design minimum cost routes visiting all customers Additional constraints  Capacity  Time windows

5. Bekkjarvik, 04.12.2003 5 Vehicle Routing Concepts (2) Dynamic VRPs  A problem is dynamic when inputs to the problem are made known or updated to the decision maker concurrently with the determination of the solution (Psaraftis, 1995) No plan is generated a priori Events are handled as they are revealed over time Need a policy for how the routes should evolve in time as a function of the inputs

6. Bekkjarvik, 04.12.2003 6 ? ? ? ? ? ? Vehicle Routing Concepts (3) Stochastic VRPs Some elements of the problem are stochastic  Travel times, demands, customers,... Typically formulated as a two-stage stochastic programming problem The solution is an a priori plan  Recourse actions SVRPs are usually static problems!

7. Bekkjarvik, 04.12.2003 7 Our Problem A mix of stochastic and dynamic VRPs Divides the time horizon (e.g. 08:00 to 16:00) into m time slots (stages) Make a plan at the beginning of each time slot for how to service the currently known customers May change the plan at the beginning of the next time slot (recourse) Information is received dynamically Modelled as a multi-stage stochastic programming problem with recourse

8. Bekkjarvik, 04.12.2003 8 Solution Approach (1) Must create a plan for the rest of the day at the start of each stage (time slot) Attempt 1 (pure dynamic)  Ignore stochastic information  Solve a static VRP based on the currently known information using a heuristic local search  Would produce good solutions if new customers do not appear

9. Bekkjarvik, 04.12.2003 9 Solution Approach (2) Attempt 2 (sampling based)  Exploit stochastic information by use of sampling  Create possible future scenarios based on the distribution of the random variables (customer locations, demands, time windows, call-in time...)  Solve the set of sample scenarios (static VRPs) by using quick heuristics  Search for common features among the sample scenario solutions, and implement these in the final plan (iteratively, based on ideas from progressive hedging)

10. Bekkjarvik, 04.12.2003 10 The main loop is: Solution Approach (3)

11. Bekkjarvik, 04.12.2003 11 Solution Approach (4) The SSBHH sub-procedure is:

12. Bekkjarvik, 04.12.2003 12 Behaviour – an example Pure dynamic Sample based08:00

13. Bekkjarvik, 04.12.2003 13 Behaviour – an example Pure dynamic Sample based09:00

14. Bekkjarvik, 04.12.2003 14 Behaviour – an example Pure dynamic Sample based10:00

15. Bekkjarvik, 04.12.2003 15 Behaviour – an example Pure dynamic Sample based11:00

16. Bekkjarvik, 04.12.2003 16 Behaviour – an example Pure dynamic Sample based12:00

17. Bekkjarvik, 04.12.2003 17 Behaviour – an example Pure dynamic Sample based13:00

18. Bekkjarvik, 04.12.2003 18 Behaviour – an example Pure dynamic Sample based14:00

19. Bekkjarvik, 04.12.2003 19 Behaviour – an example Pure dynamic Sample based15:00

20. Bekkjarvik, 04.12.2003 20 Behaviour – an example Pure dynamic Sample basedEnd of day

21. Bekkjarvik, 04.12.2003 21 Results

22. Bekkjarvik, 04.12.2003 22 Conclusions Have formulated a problem as a mix between a stochastic and a dynamic VRP Presented a Sample Scenario Based Hedging Heuristic (SSBHH) Have shown that taking stocastic information into account can improve solution quality