An Exact Algorithm for the Vehicle Routing Problem with Backhauls

1. An Exact Algorithm for the Vehicle Routing Problem with Backhauls A Thesis Submitted to the Department of Industrial Engineering and the Institute of Engineering and Science of Bilkent University in Partial Fulfillment of the Requirements For the Degree of Master of Science by Cumhur Alper GELOĞULLARI Supervisor Assoc. Prof. Osman OĞUZ 28.08.2001

2. Outline • Importance of Routing Problems • Problem Statement • Literature Review • The Algorithm • Computational Experiments • Conclusion

3. Motivation • Logistics: “That part of the supply chain process that plans, implements and controls the efficient, effective flow and storage of goods, services, and related information from the point of origin to the point of consumption in order to meet customers’ requirements” • Logistics: a means of cost saving • Distribution costs constituted 21% of the US GNP in 1983. • VRPs play a central role in logistics.

4. Problem Statement The basic Vehicle Routing Problem (VRP): Customers D

5. Problem Statement The basic Vehicle Routing Problem (VRP): Minimizetotal distance traveled subject to each customer is serviced each route starts and ends at the depot capacity restrictions on the vehicles

6. Problem Statement The VRPs exhibit a wide range of real world applications. • Dial-a-ride problem • House call tours by a doctor • Preventive maintenance inspection tours • Collection of coins from mail boxes • Waste Collection • School Bus Routing

7. Problem Statement

8. D Linehaul customer Backhaul customer Problem Statement The Vehicle Routing Problem with Backhauls (VRPB): • linehaul (delivery) customers • backhaul (pick up) customers

9. Problem Statement • The VRP replaces deadhead trip back to the depot with a profitable activity. • Yearly savings of $160 millions in USA grocery industry.

10. Literature Review Related Problems: The TSP and m-TSP • Traveling Salesman Problem (TSP) • Multiple Traveling Salesman Problem (m-TSP) • m-TSP is a special case of the VRP.

11. Literature Review Exact Algorithms for the VRPB • Vehicles are assumed to be rear-loaded. • Two exact algorithms for the VRPB: • Toth & Vigo (1997) • Mingozi & Giorgi (1999)

12. The Algorithm The VRPB under consideration is • Asymmetric • Linehaul and Backhaul customers can be in any sequence in a vehicle route • Both homogenous and heterogenous fleet

13. The Algorithm PRELIMINARIES: • L : # of linehaul customers • B : # of backhaul customers • di : demand of (or amount supplied by) customer i • m : # of vehicles • Qk : capacity of vehicle k • cij : distance from customer i to customer j • a route is denoted by Rk = {i1=0, i2, i3......., ir=0} • q(Rk) = capacity required by route Rk

14. The Algorithm • VRPB = m-TSP subject to capacity constraints • m-TSP is a relaxation of the VRPB. • A feasible solution to the m-TSP is not necessarily a feasible solution for the VRPB.

15. The Algorithm The Default Algorithm • Step 1: Solve the corresponding m-TSP. Let be its solution. • Step 2: Check whether is feasible for the VRPB. • Step 3: If feasible, stop optimal solution for the VRPB is obtained. else add inequalities valid for the VRPB but violated by goto step 1.

16. The Algorithm Solution of the m-TSP • Solve m-TSP with branch & bound • Bektaş’ s Formulation • decision variable xij

17. The Algorithm Feasibility Check Computation of q(Rk): Consider the route: {0,4,1,2,3,5,0} where

18. The Algorithm Feasibility Check & Cuts 1) Route Elimination Constraints: Qmax : maximum vehicle capacity : # of edges in Rk If for a route, Rk , q(Rk) >Qmax then Rk is infeasible for the VRPB. is valid for the VRPB but violates Rk .

19. The Algorithm Feasibility Check & Cuts For the previous example: Let Qmax=30 The route {0,4,1,2,3,5,0} is infeasible for the VRPB, then add to the m-TSP formulation. Addition of this constraint prohibits the formation of this infeasible route ONLY . 1 2 3 4 5 D

20. We add The Algorithm Feasibility Check & Cuts 2) Multiple Routes Elimination Constraints: Consider the example: Route Route #q(Rk)Qk Vehicle # {0,1,2,3,4,0} 1 25  30 1 {0,5,6,0} 2 22  20 2 {0,7,0} 3 12  15 3

21. The Algorithm Acceleration Procedures Local search: • Begin with an initial solution and improve it • For the TSP: a 2-exchange

22. The Algorithm Acceleration Procedures iteration 0: cost=200 iteration 5: cost=207 iteration 1: cost=202 iteration 6: cost=207 iteration 2: cost=202 iteration 7: cost=208 iteration 3: cost=205 iteration 8: cost=209 iteration 4: cost=206 iteration 9: cost=210

23. The Algorithm Acceleration Procedures Representation of the set of routes: D D D D

24. The Algorithm Acceleration Procedures Local Search Operators: Swap Operator: i i j j

25. The Algorithm Acceleration Procedures Local Search Operators: Relocate Operator: j j i j j j

26. The Algorithm Acceleration Procedures Local Search Operators: Crossover Operator: i i D D D D j j

27. Computational Experiments • C code using CPLEX Callable Library Routines • A total of 720 instances are tested. • Two sets of AVRPB instances

28. Computational Experiments • Homogenous Fleet (identical vehicles) (540 instances) • Problem size: 10 - 90 with increments of 10 • For a given problem size, 3 instances for %B=0, %B=20 and %B=50 • cij~U[0,100] di~U[0,100] • Common vehicle capacity: • Number of vehicles: where   [0,1].  = 0.25,  = 0.50,  = 0.75 and  = 1.00

32. Computational Experiments • Observations • As  , the problem gets harder to solve • For a given value of  , the problem gets easier as %B • Acceleration Procedures work well

33. Computational Experiments • Acceleration Procedures work well

34. Computational Experiments • Heterogenous Fleet (different vehicles) (180 instances) • Q=100 m=4 Q1=125 Q2=113 Q3=87 Q4=75 •  = 0.25,  = 0.50 %B=0, %B=50

35. Computational Experiments • For Homogenous Fleet: • Time to solve the hardest problem took 42 min. • Acceleration procedures provide • max improvement of 66% in time • min improvement of -4.95% in time • For Heterogenous Fleet: • Time to solve the hardest problem took 33 min. • Acceleration procedures provide • max improvement of 28% in time • min improvement of -10.48% in time

36. Conclusion • First Exact Algorithm for the VRPB such that • Asymmetric • Linehaul and Backhaul customers can be in any sequence in a vehicle route • Both homogenous and heterogenous fleet • The algorithm can be used for both AVRP and AVRPB

37. Further Research • VRPB with time and distance restrictions • VRPB with time windows • Other local search procedures