Snow Removal Algorithms for the city of Regina.

• Norberto FloresCIMAT, Mexico

• Nikolas KaralisNational Technical University of Athens, Greece

• Notice RingaUniversity of Guelph

• Ortho FlintUniversity of Western Ontario

Under the supervision of :

Dr. Edward Doolittle

Historical PerspectiveThe Konigsberg Problem Given the city of Konigsberg with its seven bridges, is it possible to go for a walk, starting and ending the same place and passing each of the bridges exactly once?

or equivalently :The Euler Tour ProblemGiven a connected graph G = (N,E) find a tour that visits every edge in E exactly once, or determine that no such tour exists.

The Chinese Postman Problem (CPP). Given a connected graph G = (N,E,C) with distances on the edges, find a tour, which passes through every edge at least once and does this in the shortest possible way.

The Capacitated Arc Routing Problem (CARP). Given a connected undirected weighted graph G = (N,E,Q), where Q is a demand matrix, and given a number of identical vehicles each with capacity W (where W max qij), find a number of tours such that 1) Each arc with positive demand is serviced by exactly one vehicle, 2) The sum of demand of those arcs serviced by each vehicle does not exceed W, and3) The total cost of the tours is minimized.

Solution Attempts

Heuristics• Construct-Strike Algorithm,

Christofides 1973

• Augment-Merge Algorithm, Golden and Wong 1981

• Path-Scanning Algorithm, Baker et al. 1983

• Parallel-Insert Algorithm, Chapleau et al. 1984

• Augment-Insert Algorithm, Pearn 1991

Meta Heuristics•Simulated Annealing, Eglese 1994

•Tabu Search, Hertz et al. 2000

•Memetic Algorithm, Lacomme et al. 2001

•Ant Colony System, Doerner et al. 2003

•Guided Local Search, Buellens et al. 2003

Optimal Solutions

• Branch and Bound, Hirabayashi et al. 1992• Cutting Plane (LP Relaxation), Belenguer and

Benavent 2003• Branch, Cut and Price algorithm (not applied to the CARP, but useful for

combinatorial optimization problems such as Vehicle Routing Problems (VRP) ).

Calculations•5 Plow Machines •50.8 km. in total

•106.4 km. will be traversed.

Average Speed (worst case estimation) when plowing : 1km/h

Average Speed (worst case estimation) when traversing a clean road : 20km/h

Generalization for the whole city of ReginaRoad Categories Colour Code Time Frame Length in km. % of Total

Category 1 Red 24 h 121.9 13.5

Category 2 Blue 36 h 84.2 9.3

Category 3 Black 48 h 162.6 18.0

Category 4 Green 60 h 45.9 5.1

Category 5 Grey 486.6 54.0

Total 901.1 100

REALLY ROUGH estimations Worst Case Senario based on the simplest algorithm

We take into account the first 4 categories.415 km to be cleaned.They will traverse about 415 km.

20 km to be serviced by each plow machine.20 km to be traversed.

20 hours of servicing + 1 hour of traversing per plow.

Cost Estimations

• 20 plow machines X 1 hour of traversing X 10 times per year = 200 hours of traversing per year

• 40 people X 200 hours per year = 8,000 hour per year.

• 50-100 $/hour X 8000 = 40,000 – 80,000 $ per year.

• In a more realistic concept, the actual cost is about 10,000 $ per year.

Another approach… Softcomputing

Set of computational techniques of computer science, artificial intelligence, machine learning and some engineering disciplines.

Study, model, and analyze very complex phenomena: those for which more conventional methods have not yielded low cost, analytic, and complete solutions.

More complex systems from biology, medicine, the humanities, management, etc, often remained intractable to conventional mathematical and analytical methods.

Areas of softcomputing include:

Neural networks (NN)Fuzzy systems (FS)Evolutionary computation (EC)

Evolutionary algorithms (Genetic A.)Harmony searchMemetic algorithms

Agents theory (Ant colony)Simulated annealing

• Soft computing techniques resemble biological processes more closely than traditional techniques, which are largely based on formal logical systems, such as sentential logic and predicate logic, or rely heavily on computer-aided numerical analysis (as in finite element analysis).

• Soft computing techniques often complement each other.

Main idea:Softcomputing techniques exploit the tolerance of imprecision, partial truth, and uncertainty for a particular problem.

Agents

Environment Individuals

rising order of complexity →

Observable Partially observableDeterministic StochasticEpisodic SequentialStatic DynamicDiscrete ContinuousSingle-agent Multiple agent

Rules of behaviorCommunication protocolGOALS

Plow problem…

Environment Individuals

Avoid visit streets already plowed

Be aware of othersInclude all streets with hierarchy

Communicate status to others

Communication protocol

Decision making module

Sensing module

Knowledge

etc………

etc………

LanesRoad rules

priorityExperience data

Map(graph)

EnvironmentOther agents

Human issues

Thank you