VEHICLE ROUTING PROBLEM

דרור חבליןאליאור זיברט

Classical Vehicle RoutingClassical Vehicle Routing

n customers must be served from a single depot utilizing vehicle with capacity Q for delivering goods

Each customer requires a quantity qi ≤ Q of goods

Customer orders cannot be split

Additional FeaturesAdditional FeaturesDepotsDepots– Multiple locationsMultiple locations

VehiclesVehicles– Multiple vehicle types Multiple vehicle types

and capacitiesand capacities– Release, maximum and Release, maximum and

down timesdown timesCustomersCustomers– Time windows (soft or Time windows (soft or

hard)hard)– Accessibility restrictionsAccessibility restrictions– PriorityPriority– Pickup and deliveryPickup and delivery

RoutesRoutes– Maximum timeMaximum time– Link costsLink costs

Objective Functions– Minimize total traveled

distance– Minimize total traveled

time– Minimize number of

vehicles– Maximize quality of

service – Multiple objective

functions

How Can It Be SolvedHow Can It Be Solved??? ??? Heuristics that Grow Fragments– Nearest neighbor– Double-ended nearest

neighbor– Multiple fragment

heuristicHeuristics that Grow Tours– Nearest addition– Farthest addition– Random addition

Heuristics Based on Trees– Minimum spanning tree– Christofides heuristic– Fast recursive

partitioning

AND MANY MORE

Ant Colony Optimization

(ACO)

OUR CHOICE OF ALGORITHEM

Ants (blind) navigate from nest to food sourceShortest path is discovered via pheromone trails– each ant moves at random– pheromone is deposited on path– ants detect lead ant’s path, inclined to follow– more pheromone on path increases probability

of path being followed

ACO Concepts

ACO SystemACO SystemVirtual “trail” accumulated on path segmentsStarting node selected at randomPath selected at random– based on amount of “trail” present on possible

paths from starting node– higher probability for paths with more “trail”

Ant reaches next node, selects next pathContinues until reaches starting nodeFinished “tour” is a solution

ACO System, cont.ACO System, cont.A completed tour is analyzed for optimality“Trail” amount adjusted to favor better solutions– better solutions receive more trail– worse solutions receive less trail– higher probability of ant selecting path that is part

of a better-performing tourNew cycle is performedRepeated until most ants select the same tour on every cycle (convergence to solution)

ANT ALGORITHEM

The AlgorithmAt the beginning of the search process, a constant amount of pheromone is assigned to all arcs. When located at a node i an ant k uses the pheromone trail to compute the probability of choosing j as the next node:

α - is a weight function based on arc cost etc..β – is a weight function base on arc lengthi When all ants have comleted a tour each ant compute the quantity per unit of length , the pheromone value changes as follows:

By using this rule, the probability increases that forthcoming ants will use this arc.

kiN

ijijij tnt )()(

m

k

kijij

1

Our Code Design :

resultsresults

1, 5

850

900

950

1000

1050

1100

1150

137474711201493186622392612298533583731

1, 5

1, 5

800

850

900

950

1000

1050

1100

1413825123716492061247328853297370941214533

1, 5

More resultMore result

0

20

40

60

80

100

120

140

11032053074095116137158179191021

1, 5

5, 0