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The Dynamic Vehicle Routing Problem

with A-priori Information ROUTE2000

Thursday August 17th 2000

Allan Larsen

The Department of Mathematical Modelling,

The Technical University of Denmark.

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Outline

• The Dynamic Traveling Repairman Problem (DTRP).

• Simulation of the Partially DTRP (PDTRP).

• Using a-priori information in a Dynamic Traveling Salesman Problem with Time Windows (ADTSPTW).

• Closing comments.

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Real-life routing issues

• The traditional VRP does not consider:

– Customers calling in requesting service during the day of operation.

– Time-dependent travel times.

– Varying customer demands and on-site service times.

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Static Contra Dynamic Vehicle Routing

• Static Vehicle Routing:

– All informations relevant to the planning of the routes are known to the planner before the routing process begins.

– Informations relevant to the routing do not change after the routes have been constructed.

• Dynamic Vehicle Routing:– Not all informations relevant to the planning of the routes are known by the planner when the routing process begins.– Informations can change after the initial routes have been

constructed.

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A simple example

• A single vehicle serves 5 advance request customers and…?

9:34

Depot

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Dynamic vehicle dispatching problems

• Serves one customer at the time.

• Examples:

– Emergency services (police, fire and ambulance services).

– Taxi cab services.

• Low response time are important - i.e. minimize the waiting time.

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Dynamic Vehicle Routing Problems

• Services a pool of customers.

• Queueing often occurs.

• Examples:

– Pick-up and delivery of long-distance courier mail (UPS, FedEx, DHL etc.)

– Distribution of heating oil to private households.

– Transportation of elderly and handicapped people.

• Keeping routing costs low is important - i.e. minimize the route length.

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Traditional solution approaches

• Re-optimization - i.e. solve a static VRP each time new information is received.

• Try to find a feasible spot in the routes to insert the new request.

– Defer the inclusion of the new request until the latest possible moment in time.

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Research issues

• How does the level of dynamism influence the performance of the solution methods?

• Is it possible to increase the performance of the solution methods if we obtain a-priori information on future requests?

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The Dynamic Traveling Repairman Problem

• Introduced by Bertsimas & Van Ryzin (1989).

– All requests are dynamic and generated according to a Poisson process.

– The requests are independently and uniformly distributed over a quadratic service region.

– The repairman travels at constant speed.

– Find routing policies so that the expected system time (waiting time + service time) is minimized.

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The Partially Dynamic Traveling Repairman Problem

• Modification of the DTRP

– We assume a subset of the requests are known in advance.

– The travel costs are minimized.

• Issues addressed:

– What is the relations between system performance and the level of dynamism?

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Measuring the dynamism

• The degree of dynamism (dod) measure (Lund et. al 1996):

requestsofnumbertotalrequestsimmediateofnumber

dod • I.e. in the example from before - dod = 1/(5+1) = 16 %

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Simulation of the PDTRP

• The requests are dispersed over a 10 x 10 km service region.

• The vehicle travels at 40 km/h.

• Generated 100 instances of problems with {0, 5, 10, …, 100} % dynamism each with an average of 40 customers.

• On-site service times were generated using a log-normal distribution (average of 3 min. & variance of 5 min.)

• All results are average values of these 100 instances.

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Simulation results - PDTRP

FCFS -SQM

FCFS

PARTNN-FCFS-SQM

Nearest Neighbor

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A-priori DTSPTW (1)

• Motivated by the pick-up and delivery of long-distance courier mail.

– Modelled as a Dynamic Traveling Salesman Problem with Time Windows.

• The service area is divided into a number subregions.

• We assume that the arrival intensity (λ) of each sub-region is known in advance.

λ=0.1 λ=0.2

λ=0.5 λ=0.25

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A-priori DTSPTW (2)

• We assume that a set of idle points, IP, is given.

– Each idle point serves as a “resting location” for the vehicle to go to when it is idle.

The objective is to minimize a weighted sum of the distance and the lateness.

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A-priori DTSPTW (3)

• Propose three simple repositioning policies:

– NEAREST-IP: Go to the closest IP.

– BUSIEST-IP: Go to the IP with the highest λ-value.

– HI-REQ: Go to the IP with the highest expected number of requests.

• A threshold parameter is chosen in order to avoid unnecessary traveling.

• Performed extensive simulation with various levels of dynamism and temporal characteristics.

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A-priori DTSPTW (4)

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A-priori DTSPTW (5)

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Closing comments

• Findings:

– Linear relationship between the degree of dynamism and the route costs for PDTRP.

– Modest performance improvements were achieved for a-priori information based repositioning policies.

• Further research:

– Multiple vehicles.

– Diversion.