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An Exact Algorithm for the Vehicle Routing Problem with Backhauls

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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

- Importance of Routing Problems
- Problem Statement
- Literature Review
- The Algorithm
- Computational Experiments
- Conclusion

- 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.

The basic Vehicle Routing Problem (VRP):

Customers

D

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

The VRPs exhibit a wide range of real world applications.

- Dial-a-ride problem
- House call tours by a doctor

D

Linehaul customer

Backhaul customer

The Vehicle Routing Problem with Backhauls (VRPB):

- linehaul (delivery) customers
- backhaul (pick up) customers

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

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.

Exact Algorithms for the VRPB

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

The VRPB under consideration is

- Asymmetric
- Linehaul and Backhaul customers can be in any sequence
in a vehicle route

- Both homogenous and heterogenous fleet

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

- 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.

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.

Solution of the m-TSP

- Solve m-TSP with branch & bound
- Bektaş’ s Formulation
- decision variable xij

Feasibility Check

Computation of q(Rk):

Consider the route: {0,4,1,2,3,5,0} where

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 .

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

We add

Feasibility Check & Cuts

2) Multiple Routes Elimination Constraints:

Consider the example:

Route Route #q(Rk)QkVehicle #

{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

Acceleration Procedures

Local search:

- Begin with an initial solution and improve it
- For the TSP:
a 2-exchange

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

Acceleration Procedures

Representation of the set of routes:

D

D

D

D

Acceleration Procedures

Local Search Operators:

Swap Operator:

i

i

j

j

Acceleration Procedures

Local Search Operators:

Relocate Operator:

j

j

i

j

j

j

Acceleration Procedures

Local Search Operators:

Crossover Operator:

i

i

D

D

D

D

j

j

- C code using CPLEX Callable Library Routines
- A total of 720 instances are tested.
- Two sets of AVRPB instances

- 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

- Observations
- As , the problem gets harder to solve
- For a given value of , the problem gets easier as %B
- Acceleration Procedures work well

- Acceleration Procedures work well

- 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

- 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

- 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

- VRPB with time and distance restrictions
- VRPB with time windows
- Other local search procedures