Branch-and-Price algorithm for Vehicle Routing Problem: tutorial

Transcription

1 Branch-and-Price algorithm for Vehicle Routing Problem: tutorial Kyuree AHN Department of Industrial and Systems Engineering KAIST, Republic of Korea Friday, May 17, 2017

2 Presentation Overview Problem description Vehicle Routing Problem with Time Window (VRPTW) Column generation Set partitioning problem Master Problem Generating Columns Subproblem Overall Algorithm Branch-and-Bound Why we need branch-and-bound? Branching Decisions Branch-and-price General problem Implement on split job Results Concluding Remark 2017 winter Xs3d lab seminar - Branch-and-Price 2

3 Presentation Overview Problem description Vehicle Routing Problem with Time Window (VRPTW) Column generation Set partitioning problem Master Problem Generating Columns Subproblem Overall Algorithm Branch-and-Bound Why we need branch-and-bound? Branching Decisions Branch-and-price General problem Implement on split job Results Concluding Remark 2017 winter Xs3d lab seminar - Branch-and-Price 3

4 Vehicle Routing Problem with Time Window (VRPTW) VRPTW 2 [2,4] 3 [3,5] 2 [2,4] 3 [3,5] 1 [1,3] Depot 1 [1,3] Depot 4 [4,6] 4 [4,6] 2017 winter Xs3d lab seminar - Branch-and-Price 4

5 Vehicle Routing Problem with Time Window (VRPTW) VRPTW 2017 winter Xs3d lab seminar - Branch-and-Price 5

6 Vehicle Routing Problem with Time Window (VRPTW) VRPTW minimize total distance Each customer served once Capacity constraint Flow constraints Time constraints 2017 winter Xs3d lab seminar - Branch-and-Price 6

7 Presentation Overview Problem description Vehicle Routing Problem with Time Window (VRPTW) Column generation Set partitioning problem Master Problem Generating Columns Subproblem Overall Algorithm Branch-and-Bound Why we need branch-and-bound? Branching Decisions Branch-and-price General problem Implement on split job Results Concluding Remark 2017 winter Xs3d lab seminar - Branch-and-Price 7

8 Set partitioning Problem Set Partitioning problem Master Problem 2 [2,4] 3 [3,5] 4 [4,6] 9 feasible routes (d-1-d), (d-2-d), (d-3-d), (d-4-d), (d-1-2-d), (d-2-3-d), (d-3-4-d), Generating Columns Subproblem Overall Algorithm 1 [1,3] Depot C = 1, 2, 3, 4 SP (d d), (d d) 2017 winter Xs3d lab seminar - Branch-and-Price 8

9 Set partitioning Problem Set Partitioning problem Master Problem 2 [2,4] 3 [3,5] 4 [4,6] 9 feasible routes (d-1-d), (d-2-d), (d-3-d), (d-4-d), (d-1-2-d), (d-2-3-d), (d-3-4-d), Generating Columns Subproblem Overall Algorithm 1 [1,3] Depot C = 1, 2, 3, 4 SP (d d), (d d) 2017 winter Xs3d lab seminar - Branch-and-Price 9

10 Set partitioning Problem Set Partitioning problem Master Problem Generating Columns Subproblem Overall Algorithm One column = One route = One subset y r = 1, if route r is used 0, otherwise c r = cost(distance) of route r R = set of all feasible routes δ ir = 1 if customer i is serviced by route r SP 2017 winter Xs3d lab seminar - Branch-and-Price 10

11 Set partitioning Problem Set Partitioning problem Master Problem Generating Columns SP Subproblem Overall Algorithm Relax Integer Condition MP 2017 winter Xs3d lab seminar - Branch-and-Price 11

12 Master Problem(MP) Set Partitioning problem Master Problem Generating Columns Linear relaxation of SP: y r no longer binary MP Subproblem Overall Algorithm 0 y i 1 Part Ⅰ : Initial columns Part Ⅱ : Generated by subproblem 2017 winter Xs3d lab seminar - Branch-and-Price 12

13 Master Problem(MP) Set Partitioning problem Master Problem Generating Columns Linear relaxation of SP: y r no longer binary MP Subproblem Overall Algorithm 0 y i 1 Part Ⅰ : Initial columns Part Ⅱ : Generated by subproblem 2017 winter Xs3d lab seminar - Branch-and-Price 13

14 Master Problem(MP) Set Partitioning problem Master Problem Generating Columns Linear relaxation of SP: y r no longer binary MP Subproblem Overall Algorithm 0 y i 1 Part Ⅰ : Initial columns Part Ⅱ : Generated by subproblem How do we generate new column? 2017 winter Xs3d lab seminar - Branch-and-Price 14

15 Set Partitioning problem Master Problem Generating Columns Subproblem Overall Algorithm Generating Columns Find new column with maximum reduced cost MP Reduced cost Dual variable π i, i C If reduced cost negative, solution can be improved 2017 winter Xs3d lab seminar - Branch-and-Price 15

16 Generating Columns Set Partitioning problem Master Problem Generating Columns Find new column with maximum reduced cost MP Dual variable π i, i C Subproblem Overall Algorithm Reduced cost Minimize! 2017 winter Xs3d lab seminar - Branch-and-Price 16

17 Subproblem Set Partitioning problem Master Problem Generating Columns c ij = c ij π i : how much arc(i,j) can improve current MP Subproblem Overall Algorithm Find path with negative reduced cost When minimum reduced cost > 0, the algorithm ends 2017 winter Xs3d lab seminar - Branch-and-Price 17

18 Overall Algorithm Overview Master Problem Generating Columns START Generate initial routes Subproblem Overall Algorithm Solve Master Problem and get dual Run Subpbroblem Add routes to the problem Reduced cost negative? YES NO End 2017 winter Xs3d lab seminar - Branch-and-Price 18

19 Presentation Overview Problem description Vehicle Routing Problem with Time Window (VRPTW) Column generation Set partitioning problem Master Problem Generating Columns Subproblem Overall Algorithm Branch-and-Bound Why we need branch-and-bound? Branching Decisions Branch-and-price General problem Implement on split job Results Concluding Remark 2017 winter Xs3d lab seminar - Branch-and-Price 19

20 Why we need branch-and-bound? Why B&B? B&B MP Branch-and- Price Solution of relaxed master problem may fractional(non-integer) If column generation end but solution is fractional Branch-and-bound occurs 2017 winter Xs3d lab seminar - Branch-and-Price 20

21 Branching Decisions Why B&B? B&B Branch-and- Price Bounding Global upper bound(gub, current best solution) GUB = initially Local lower bound(llb, optimal solution of current node) If LLB < GUB, branch and update GUB If LLB > GUB, discard Branching on arc (x ij = k K x ijk ) When x ij fractional, branching occurs. x ij = 0 : arc i, j is removed from the subproblem x ij = 1 : arcs (i, j) are removed from the current master problem x ij = 0 x ij = winter Xs3d lab seminar - Branch-and-Price 21

22 Branch-and-Price Why B&B? B&B START Generate initial routes Branch-and- Price Solve Master Problem Run Subpbroblem Add routes to the problem Reduced cost negative? N Solution Integral? N Branch Y Y Done 2017 winter Xs3d lab seminar - Branch-and-Price 22

23 Presentation Overview Problem description Vehicle Routing Problem with Time Window (VRPTW) Column generation Set partitioning problem Master Problem Generating Columns Subproblem Overall Algorithm Branch-and-Bound Why we need branch-and-bound? Branching Decisions Branch-and-price General problem Implement on split job Results Concluding Remark 2017 winter Xs3d lab seminar - Branch-and-Price 23

24 Implement on split job x ijk = 1, if vehicle k serve split job j after split job i, 0, otherwise s ik starting time of vehicle k service customer i. c ij distance between end location if ith job and start location of jth job s ik c ij 2017 winter Xs3d lab seminar - Branch-and-Price 24

25 Results 10 Tasks 2 Depots : (195, 47) and (587, 145) Speed : 150m/min Maximum flight time = 10min Start End Task a Location Location i b i q i winter Xs3d lab seminar - Branch-and-Price 25

26 Results 2017 winter Xs3d lab seminar - Branch-and-Price 26

27 Results 2017 winter Xs3d lab seminar - Branch-and-Price 27

28 Concluding Remarks Initial location of vehicle Reuse of vehicle Another SP Subproblem is often NP-hard Dynamic programming 2017 winter Xs3d lab seminar - Branch-and-Price 28

29 References Gamrath, Gerald. "Generic branch-cut-and-price." (2010). Larsen, Jesper. Parallelization of the vehicle routing problem with time windows. Diss. Technical University of DenmarkDanmarks Tekniske Universitet, Department of Informatics and Mathematical ModelingInstitut for Informatik og Matematisk Modellering, Kallehauge, Brian, et al. "Vehicle routing problem with time windows." Column generation. Springer US, Azi, Nabila, Michel Gendreau, and Jean-Yves Potvin. "An exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles." European Journal of Operational Research (2010): winter Xs3d lab seminar - Branch-and-Price 29

30 Thank you

31 References Gamrath, Gerald. "Generic branch-cut-and-price." (2010). Larsen, Jesper. Parallelization of the vehicle routing problem with time windows. Diss. Technical University of DenmarkDanmarks Tekniske Universitet, Department of Informatics and Mathematical ModelingInstitut for Informatik og Matematisk Modellering, Kallehauge, Brian, et al. "Vehicle routing problem with time windows." Column generation. Springer US, Azi, Nabila, Michel Gendreau, and Jean-Yves Potvin. "An exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles." European Journal of Operational Research (2010): winter Xs3d lab seminar - Branch-and-Price 31