The Dynamic Multi-Period Vehicle Routing Problem. Jean-François Cordeau, Gilbert Laporte, Jesper Larsen, Min Wen

Transcription

1 The Dynamic Multi-Period Vehicle Routing Problem Jean-François Cordeau, Gilbert Laporte, Jesper Larsen, Min Wen

2 Outline Introduction to the Lantmännen case study Problem description Mathematical formulation Solution method Computational results Conclusion 2

3 Outline Introduction to the Lantmännen case study Problem description Mathematical formulation Solution method Computational results Conclusion 3

4 Lantmännen case study -1 Lantmännen one of the largest groups within the food, energy and agricultural in the Nordic region. owned by 44,000 Swedish farmers has 13,000 employees has a total revenue of 3.2 billion Euro Distribution task in agricultural business area Deliver fodders from the distribution terminal to farmers all over Sweden Farmers call for service at any time during the day ( demand, preferable service days) Route plans are made at the beginning of each day and are excuted during the day Case study - terminal Västerås 4

5 Lantmännen case study -2 Data Planning horizon: 10/15 days Cusomer: approx. 800/1200 in total Vehicles: 23 5

6 Outline Introduction to the Lantmännen case study Problem description Mathematical formulation Solution method Computational results Conclusion 6

7 Problem Description -1 Problem definition Given: Task: A planning horizon (days) A set of customers with demands and specified feasible visiting days (revealed gradually) A set of homogeneous vehicles with common capacity decide which customers to be visited on the day determine routes Objective: a) minimize the total distance traveled over the planning horizon b) minimize customer waiting over the planning horizon c) minimize the deviation of daily workload over the planning horizon Constraints: a) each customer is visited once within its preferable visiting days b) truck capacity cannot be exceeded c) route duration cannot be exceeded (24 hours) 7

8 Problem Description -2 Example a small planning problem with 2-day horizon At beginning of day1: 6 customers can be visited on day 1 or day2 must be visited on day 1 Plan the routes for day 1 and execute ( ) New customer revealed during day 1: 2 customers must be visited on day2 Plan the routes for day 2 and execute ( ) Challenge- dynamic How to select the best customers for the planning day without knowing future customers? 8

9 Outline Introduction to the Lantmännen case study Problem description Mathematical formulation Solution method Computational results Conclusion 9

10 Mathematical Formulation -1 Static case is a Periodic Vehicle Routing Problem with visiting frequency 1 Sets: N L K 0 [ a b ] : the set of customers and depot : the plannning horizon : the set of vehicles Parameters: d q i i i, : the feasible visiting days of customer i i : the duration of service at customer i : the demand of customer i c : the distance travelling from i to j ij D : the limit on route duration Q : the vehicle capacity capacity Variables: x :1 if vehicle k travel from i to j on day l ijkl 10

11 Mathematical Formulation -2 Multiple objectives for day t (t = 1,,T) in the dynamic case Minimize total customer waiting time Waiting of customer i: y i a i where a i =max(a i,t) is the adjusted earliest day of visiting i, y i is the day customer is visited. yi ai' 2 Objective function: min α ( ) i N' b a ' i i Minimize average deviation of workload Workload deviation of day t: w t W where W is the average total daily duration from the historical data, w t is the total duration on day t Objective function: min β w W t 11

12 Outline Introduction to the Lantmännen case study Problem description Mathematical formulation Solution method Computational results Conclusion 12

13 Solution Method -1 Three phase heuristic Purpose of the planning on each day: Make a route plan for the future t days; execute the routes on the current day. Preprocessing: remove the customers already visited; adjust the time windows for the customers not visited; include the new customers revealed on previous day. Phase 1: select customers to be included in the planning (over a t day period) (location and time windows correlation analysis) Phase 2: route the customers as a PVRP with frequency 1 (Variable Neighborhood Search) Phase 3: improve the routes to be executed on the current day (Unified Tabu Search) 13

14 Solution Method -2 Correlation analysis select customer Variable Neighborhood Search solve PVRP with frequency 1 14

15 Outline Introduction to the Lantmännen case study Problem description Mathematical formulation Solution method Computational results Conclusion 15

16 Computational Result-1 Effectiveness of the correlation analysis Number of days planned: t =1,2 or infinite Select customers using correlation analysis 16

17 Computational Result-2 Multiple objectives Minimizing total customer waiting i N yi ai' α ( ) ' b a ' i i 2 Minimizing deviation of daily workload β w W t Why Flat? Dynamic problem W is an estimation of average daily workload based on historical data 17

18 Computational Result-3 Our results vs Company s results Total distance: slightly better Customer waiting is improved by 24.2% Deviation of daily workload is improved by 33.4% 18

19 Outline Introduction to the Lantmännen case study Problem description Mathematical formulation Solution method Computational results Conclusion 19

20 Conclusion Dynamic multi period multi objective routing problem Three phase heuristic method Correlation analysis Converges fast Handle multiple objectives 20

21 Discussion Thank you for your attention Questions? 21