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
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An Exact Algorithm for the Vehicle Routing Problem with Backhauls
An Exact Algorithm for the Vehicle Routing Problem with Backhauls. A Thesis Submitted to the Department of I ndustrial 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 - PowerPoint PPT Presentation
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An Exact Algorithm for the Vehicle Routing Problem with Backhauls
A ThesisSubmitted to the Department of Industrial Engineering
and the Institute of Engineering and Scienceof Bilkent University
in Partial Fulfillment of the RequirementsFor the Degree ofMaster of Science
byCumhur Alper GELOĞULLARI
SupervisorAssoc. Prof. Osman OĞUZ
28.08.2001
Outline
• Importance of Routing Problems
• Problem Statement
• Literature Review
• The Algorithm
• Computational Experiments
• Conclusion
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.
Problem StatementThe basic Vehicle Routing Problem (VRP):
D
Customers
Problem StatementThe basic Vehicle Routing Problem (VRP):
Minimize total distance traveledsubject to
each customer is servicedeach route starts and ends at the depotcapacity restrictions on the vehicles
Problem StatementThe 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
Problem Statement
PARAMETER DOMAINObjective Minimize distance/time, # of vehiclesFleet size one vehicle / multiple vehiclesFleet type homogenous / heterogenous# of depots single depot / multiple depots
Type of demand deterministic / stochastic
Total time / distance constraints imposed / not imposed
• The VRP replaces deadhead trip back to the depot with a
profitable activity.
• Yearly savings of $160 millions in USA grocery industry.
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.
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)
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
The AlgorithmPRELIMINARIES:• 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
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.
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, stopoptimal solution for the VRPB is obtained.
else add inequalities valid for the VRPB but violated by
goto step 1.
*TSP mx
*TSP mx
*TSP mx
The Algorithm
Solution of the m-TSP• Solve m-TSP with branch & bound
• Bektaş’ s Formulation– decision variable xij
The AlgorithmFeasibility Check
Computation of q(Rk):
Consider the route: {0,4,1,2,3,5,0} where
Route: 0 4 1 2 3 5 0
Type of customer: - B L L L B -
Demand / Supply: 0 15 10 5 5 10 0
Total Load on the vecicle: 20 35 25 20 15 25 0
The AlgorithmFeasibility 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 .
)( kRl
1)( ),(
kRji
ij Rlxk
The AlgorithmFeasibility Check & Cuts
For the previous example: Let Qmax=30The 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 .