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

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

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

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

Problem Statement

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

Problem Statement 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 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

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, stop optimal solution for the VRPB is obtained. else add inequalities valid for the VRPB but violated by goto step 1.

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

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

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 .

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

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} 1 25  30 1 {0,5,6,0} 2 22  20 2 {0,7,0} 3 12  15 3 We add

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

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

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

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

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

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

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

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

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

Computational Experiments Acceleration Procedures work well

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

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 -10.48% in time

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

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