03 Operations Research

8/6/2019 03 Operations Research

1/66

The role of Operations

ResearchSolving Decisional Models for EnvironmentalManagement


2/66

The search process

x(t+1)=f(x(t),u(t))

Control Variables

yes

Modification of

control variable

u(t)

Time series

Maps

Simulation Model

Simulation Results

J=J(x,u)

Performance Criterion

J

max?

Search algorithm

StopAdapted from Seppelt, 2003


3/66

What is search

To search means to explore possible valuesfor the control variable u, to obtain thevalue of the state x caused by u, and toevaluate the performance J


4/66

The policy

The policy defines the value of u to bechosen in front of the value of x at eachtime step t and in each spatial location z

P {Mt() ; t= 0,1, . . .


5/66

Finding the policy

The problem is hard effects are uncertain

decisions are dynamic and recursive


6/66


7/66

Scenario analysis

Scenario definition requires knowledge onthe modelled system, which might not be

available, due to complexity (Jakeman andLetcher, 2003)

In some (most) cases Scenarios are toolimited

Too much weight on past experience


8/66

Optimisation

Given the goal, the computer automaticallygenerates scenarios, by combination ofparameters, controls, exogenous inputs

A numerical optimisation algorithm is incharge of evaluating the performance and

directing exploration in the most promisingdirection


9/66

Simulation explained


10/66

The simulation model

It is a constraint for the optimisationproblem all model variables may vary in space

boundary and initial conditions are given

xt+!t(z) = M!t(xt(z),ut(z),"(z),z)


11/66

The performance

criterion

state x and control u depend on time andlocation

the performance criterion maps a trajectoryfrom state and policy space to a real

number

J(xt(z),ut(z))


12/66

The problem

xt+!t(z) = M!t(xt(z),ut(z),"(z),z)

J

(xt

(z

),ut

(z

))Performance criterion:

Model:

Problem: find u so that

x(t) = M!t(x,u,")

J(x,u) J(x,u)

for all t in [0, T] and u in U


13/66

Solving the problem

Exact methods Approximation algorithms

Heuristic and metaheuristic algorithms


14/66


15/66

Exact methods

The algorithm depends on the problemformulation

Linear programming, simplex Integer programming, branch & bound Dynamic programming


16/66

Approximation

algorithms Find a sub-optimal solution

It is possible to say how far from the

optimum (e.g. 95%)

Formally proven

Often is very specific to the problem athand

E.g: neuro-dynamic programming


17/66

Heuristics

Apply rules-of-thumb to the solution of theproblem

No guarantee to find an optimum No guarantee on the quality of the solution Yet, they are fast and generic


18/66

Metaheuristics

set of concepts that can be used to defineheuristic methods that can be applied to awide set of different problems.

general algorithmic frameworks which canbe applied to different optimization

problems with relatively few modificationsto make them adapted to a specific problem


19/66

Metaheuristics are...

Simulated annealing

Evolutionary computation Tabu search

Ant Colony Systems


20/66

Ant Colony Systems


21/66

Ants

Leaf cutters,fungi growers

Breeding otherInsects

Weaver ants

I i l


22/66

Insects, socialinsects, and ants

1018 living insects (estimate)

~2% of all insects are social

Examples of social insects:

All ants

All termites

Some bees

Some wasps

50% of all social insects are ants

Average body weight of an ant: 1 - 5 mg

Total mass of ants ~ total mass of humans

Ants have colonized earth for 100 million yrs; Homosapiens sapiens for 50000 years


23/66

Ant colony society

Size of a colony:from a few (30) to millions of worker ants

Labour division:

Queen: reproduction

Soldiers: defense

Specialised workers: food gathering, offspring tending,nest grooming, nest building and maintenance


24/66

How do ants

coordinate theiractivities?

Basic principle: stigmergy stigmergy is a kind of indirect

communication, used by social insects to

coordinate their activities


25/66

StigmergyStimulation of workingants due to the results

they have reachedGrass P. P., 1959


26/66


27/66

Artificial Stigmergy

Indirect communication by changes on theenvironment accessible only locally tocommunicating agents

Features of artificial stigmergy : Indirect communication Local accessibility

(Dorigo and Di Caro, 1999)


28/66

The bridge experiment

Goss et al., 1989, Deneubourg et al., 1990


29/66

Pheromone trail

Ants and termites follow pheromone trails


30/66

Asymmetric bridge

experiment

Goss et al., 1989


31/66

Ant System for TSP

Pheromonetrail deposition

Probabilistic rule tochoose the path

Travelling Salesman Problem

memory

Dorigo, Maniezzo, Colorni, 1991Gambardella & Dorigo, 1995

Ph i ti


32/66

Pheromone, euristicsand memory to visit the

next citMemory of

visited cities

tij ;hiji

k

j

tik ;hik

tir;hir

r

pijk

t( )= f ij t( ),ij t( )( )

r


33/66

Pheromone trail

depositionij

kt+1( ) 1 ( ) ij

kt( ) + ij

kt( )

ijk

t( ) = quality kwhere (i,j) are the links visited by ant k, and

where qualityk

is inversely proportional to the lengthof the solution found by ant k


34/66


35/66

3 main components: TIME: a short route

gets pheromone more quickly

QUALITY: a short routegets more pheromone

COMBINATORICS: a short pathgest pheromone more frequentlysince it (usually) has a smaller numberof decision points

Why does it works?

i

j

tijd

origin

destination

d

Some results on the


36/66

Some results on theTravelling Salesman

Problem

A i h i i


37/66

A constructive heuristics

with local search The best strategy to approximate thesolution of a combinatorial optimisation

problem is to couple a constructive heuristics and local search

It is hard to find the best coupling: Ant System (and similar algorithms)

appear to have found it


38/66

The ACO

Metaheuristics Ant System has been extended, to be appliedto any problem formulated as a shortest pathproblem

The extension has been named Optimisationmetaheuristics based on Ant Colonies (ACO)

The main application:

NP complex combinatorial optimisationproblems


39/66

The ACO-metaheuristic

procedureprocedure ACO-metaheuristic()

while (not-termination-criterion)

planning subroutines

generate-and-manage-ants()

evaporate-pheromone()

execute-daemon-actions() {opzionale}

end planning subroutinesend while

end procedure

E.g. local search


40/66

ACO: Applications

Sequential ordering in a production line

Vehicle Routing of trucks goodsdistributions Job-shop scheduling

Project scheduling Water distribution problems


41/66

Genetic Algorithms


42/66

Evolutionary computing

EC (Evolutionary computing) = GA (Genetic Algorithms - Holland, 1975) ES (Evolution Strategies - Rechenberg

1973)

EP (Evolutionary Programming - Fogel,

Owens, Walsh, 1966)

GP (Genetic Programming - Koza, 1992)


43/66

Biological basis

Evolution operates on chromosomes, whichencode the structure of a living being

Natural selection favours reproduction ofthe most efficient chromosomes

Mutations (new chromosomes) and

recombination (mixing chromosomes) occurduring reproduction


44/66

Use

GA are very well suited when the structureof the search space is not well known

In other words, if the model of our systemis rough, we can use GA to find the best

policy


45/66

How it works

A genetic algorithm maintains a populationof candidate solutions for the problem athand, and makes it evolve byiteratively applying a set of stochastic

operators

Th k l f h


46/66

The skeleton of the

algorithm Generate initial population P(0) t:=0

while not converging do evaluate P(t)

P(t) select best individuals from P(t)

P(t) apply reproduction on P(t)

P(t+1)replace individuals (P(t),P(t)

end while


47/66

Components of a GA

Encoding principles (gene, chromosome) Initialization procedure (creation)

Selection of parents (reproduction) Genetic operators (mutation,

recombination)

Evaluation function (environment) Termination condition

R


48/66

Representation

(encoding) Possible individuals encoding

Bit strings (0101 ... 1100)

Real numbers (43.2 -33.1 ... 0.0 89.2)

Permutations of element (E11 E3 E7 ... E1 E15)

Lists of rules (R1 R2 R3 ... R22 R23)

Program elements (genetic programming)

... any data structure ...


49/66

Choosing the encoding

Use a data structure as close as possible tothe natural representation

Write appropriate genetic operators asneeded If possible, ensure that all genotypes

correspond to feasible solutions

If possible, ensure that genetic operatorspreserve feasibility


50/66

Initialization

Start with a random population a previously saved population a set of solutions provided by a human

expert

a set of solutions provided by anotherheuristic


51/66

Selection

Purpose: to focus the search in promisingregions of the space

Inspiration: Darwins survival of the fittest

Trade-off between exploration and

exploitation of the search space

Li R ki


52/66

Linear Ranking

SelectionBased on sorting of individuals by decreasing

fitness. The probability to be extracted for the ith

individual in the ranking is defined as

where b can be interpreted as the expectedsampling rate of the best individual

L l T


53/66

Extracts k individuals from the populationwith uniform probability (without re-insertion) and makes them play atournament,

where the probability for an individual towin is generally proportional to its fitness

Selection pressure is directly proportional tothe number k of participants

Local Tournament

Selection

R bi i


54/66

Recombination

(Crossover)

Enables the evolutionary process to movetoward promising regions of the searchspace

Matches good parents sub-solutions toconstruct better offspring


55/66

Mutation

Purpose: to simulate the effect of errors thathappen with low probability during

duplication

Results: Movement in the search space

Restoration of lost information to thepopulation

E l i (fi


56/66

Evaluation (fitness

function)

Solution is only as good as the evaluation

function; choosing a good one is often thehardest part

Similar-encoded solutions should have asimilar fitness


57/66

Termination condition

Examples:

A pre-determined number of generations or

time has elapsed

A satisfactory solution has been achieved

No improvement in solution quality hastaken place for a pre-determined number ofgenerations


58/66

Acknowledgements

Part of the material extracted fromIntroduction to Genetic Algorithms byAssaf Zaritsky, Ben Gurion University,


59/66

Simulated Annealing


60/66

The origin

It is the oldest metaheuristic

Originated in statistical mechanics(Metropolis Monte Carlo algorithm)

First presented to solve combinatorial

problems by Kirkpatrick et al. 1983


61/66

The idea

It searches in directions which result insolutions that are of worse quality than the

current solution

It allows to escape local minima

The probability of the exploration of these

unpromising directions is decrementedduring the search


62/66

The algorithm s

GenerateInitialSolution()

Temp:=InitialTemp

while not converging loop

sPickAtRandomFromNeighbor(s)

if J(s) < J(s)

ss else

Accept s as new solution with prob p(T,s,s)

Update(T)

end while


63/66

Boltzmann probability

if s is worse than s, then s might still bechosen as the new solution

the probability depends on d=f(s)-f(s) andon temperature T

the higher d the lower the probability

the higher T the higher the probability


64/66


65/66

Pros and Cons

Pros

proven to converge to the optimum

easy to implement Cons

converges in infinite time the cooling process must be slow


66/66

End of Part II

Documents

03 Operations Research