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

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 each route starts and ends at the depot
The basic Vehicle Routing Problem (VRP):
Minimize total 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. Problem Statement The Vehicle Routing Problem with Backhauls (VRPB):
linehaul (delivery) customers
backhaul (pick up) customers D
Linehaul customer
Backhaul customer

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, i , 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 For the previous example: Let Qmax=30 1 2 3 4 5 D
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. The Algorithm Consider the example: Feasibility Check & Cuts
2) Multiple Routes Elimination Constraints:
Consider the example:
Route Route # q(Rk) Qk Vehicle #
{0,1,2,3,4,0} 
{0,5,6,0} 
{0,7,0} 
We add

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 i i j j Local Search Operators:
Swap Operator: i i j j

25. The Algorithm Acceleration Procedures j j i j j j
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: 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= Q1= Q2= Q3=87 Q4=75
 = 0.25,  = %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.
max improvement of 28% in time
min improvement of % 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