A decision aiding system based on a decision rule approach and fuzzy logic

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Decision Rule Apporach and Fuzzy Logic

Jarle Skjørestad Heldstab

June 15th, 2010

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1 Introduction 8

1.1 The issue . . . 8

1.2 Problemstatement . . . 8

1.3 Thesis overview . . . 9

2 Background 11 2.1 Multiple criteriadecision analysis . . . 11

2.1.1 Classication problems . . . 11

2.1.2 Sorting problems . . . 12

2.2 Decision aiding system . . . 12

2.3 Real-world decision aiding inaccordance tohuman preferences 13 2.4 Information matrix . . . 14

2.5 Decision rule approach . . . 15

2.6 Related systems . . . 15

3 Theory 17 3.1 Classicalrough set theory . . . 17

3.1.1 Indiscernibility relation . . . 17

3.1.2 Lower and upper approximation ofdecision classes . . 17

3.1.3 Decision rules . . . 18

3.2 Dominancerough set approach . . . 18

3.2.1 Decision class unions . . . 19

3.2.2 Dominance principle . . . 19

3.2.3 Rough Approximations . . . 20

3.2.4 Decision rules . . . 21

3.3 Fuzzy Logic . . . 21

3.3.1 Fuzzy membershipfunctions . . . 21

3.3.2 Fuzzy rules . . . 22

4 System setup and how it works 23 4.1 Technologies and Logicaldesign . . . 23

4.2 Training process. . . 23

4.2.1 Algorithmfor induction of decision rules . . . 24

4.3 Decision making process . . . 27

4.4 Decision making process in real-world decision problems in accordance tohuman preferences . . . 30

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4.4.2 Fuzzy membershipfunctions . . . 31

4.4.3 Fuzzy inferenceprocess . . . 33

5 Results 37 5.1 Example of decisionmaking based onhuman preferences . . . 37

5.2 Exampleofreal-worlddecisionaidinginaccordancetohuman preferences . . . 40

5.3 Example of decisionmaking in the oiland gas industry . . . . 46

5.4 Articial neuralnetworks . . . 47

5.4.1 Expectations . . . 48

5.4.2 Case Study . . . 48

5.4.3 Results . . . 50

5.4.4 Comments onexpectations . . . 51

5.4.5 Number of neurons in the hidden layer . . . 52

6 Conclusions 53 6.1 Summary . . . 53

6.2 Future work . . . 54

References 56

Appendix A - Presentation given at IRIS 58

Appendix A - Decision rules from section 5.3 64

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1 Logical designof decision aiding system . . . 24

2 Training process. . . 24

3 Decision making process . . . 31

4 Presenting the fuzzy membershipfunctions tothe system . . . 32

5 Membership functions for A . . . 32

6 Membership functions for B . . . 33

7 Fuzzication . . . 34

8 Implication . . . 35

9 Aggregation and defuzzication . . . 36

10 Membership functions for Location . . . 42

11 Membership functions for Size . . . 42

12 Membership functions for Standard . . . 43

13 Membership function for Buildyear . . . 43

14 Membership function for Value . . . 44

15 Graphical representation of the participation of each input reected on the output . . . 45

16 Defuzzication applied onthe resulting output . . . 45

17 Articial neuralnetwork trainingperformance . . . 49

18 IRIS presentation slide 1 . . . 58

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1 Electricalcars described by a human preference model . . . . 13

2 Electricalcars described by a real-world model . . . 14

3 Human preference model describing residentialproperties . . . 37

4 Description of aresidential property without value. . . 39

5 Description of aresidential property with a value . . . 40

6 Numericaldescription of a residentialproperty withouta value 41 7 Samplesadapted fromactual drillingoperations . . . 47

8 Crab data example . . . 50

9 Comparison marginof error case 1 . . . 50

10 Comparison marginof error case 2 . . . 51

11 Hidden layerneurons . . . 52

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The rst problem discussed in this thesis studied the theory of

using a softwaresystem for decision aiding indecision problems with

multiple criteria. Two such problem types were considered, namely

classication and sorting problems. A second problem in this thesis

studied a method to allowa humanpreference model asa basisfor a

real-world decision aiding problem. A problem number three looked

at the possibilites of improving decision making in the oil and gas

industry.

Threeexperimentswasperformedtotestthethreeproblemsinthis

thesis. For the rst problem, an implementation of a decision aiding

system utilizing a decision rule approach, namely classical rough set

theoryanddominanceroughsetapproach,wasdone. Thesystemwas

able to make decisions inclassication and sorting problems. For the

second problem, fuzzy logic was combined into the implementation.

Theresultsshowedthatcombiningadecisionruleapproachwithfuzzy

logic made itpossible to usea human preferencemodel asa basisfor

real-world decision aiding problems. For the third problems, it was

found that in a real-world decision making problem from the oil and

gasindustry,utilization ofan autonomousdecision aiding systemcan

improve the qualityoftheresults.

Lastly, the implemented decision aiding system was compared to

an articial neural network, and the results showed that the system

had some advantages over the neuralnetwork.

Keywords: Decisionaiding,classicalroughset theory,dominanceroughset

approach, fuzzylogic, decision rules, decisionmaker.

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First of all, I would like tothank my supervisors during this process, Asso-

ciate ProfessorHeinMeling atthe Universityof Stavanger,and NejmSadal-

lahat the International Research Instituteof Stavanger.

I would also like tothank Ingrid Tjøstheim Eek, my dear girlfriend, and

our one year old son, Sebastian. Thanks for putting up with me in this

period. Withoutyou, this thesis would not have been the same.

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In this study, two decision rule approaches, namely the classical rough set

theory and dominance rough set approach, are utilized to build a decision

aiding system. Thesystem has the abilityto makedecisionsinclassication

and sortingproblems. The study alsoproposes tocombine the decision rule

approachtechniqueswithfuzzylogic,andthususethedecisionruleapproach

intheprocessofconstructingafuzzylogiccontroller,whiletakingadvantage

of the decision making capabilities of the fuzzy controller to further extend

the system's area of application.

The rest of this chapter gives abrief overview of the work carriedout in

this thesis. Firstly,ashortstatementofthe issueofthe workisgiven. Then,

the problemstatement is pinpointed. Lastly, ashort outlineof the contents

of the thesisis given.

1.1 The issue

The natureof the decisionproblems that humansingeneral faceare ofmul-

tiple criteria [1]. According tothe autors of [2], when people make decisions

they search for rules which provide good justication of their choices. The

process of making such decisions can vary greatly in accordance to several

factors. One such factor includes human preferences, or for example in the

oilandgasindustry,thequalityofdecisionmakingmayvarypersuanttothe

experience of the drillingsta. The idea behind a decisionaiding system is

toautomate the process ofdecisionmaking, and thusimprovethe quality of

the decisionmaking results.

1.2 Problem statement

Three main problems are discussed inthis thesis:

1. The rst stepwith this workis topropose the designand implementa-

tion ofadecisionaidingsystem thathas the abilitytoassista decision

maker by recommending decisions in classication and sorting prob-

lems. The basis of the recommendations that the system gives, stem

from example decisions originating from historical data or from the

preferences of the decisionmaker.

2. For the second step with this work, two assumptions are made:

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•

^Firstly, ît îs âssumed ^that ^people ^prefer ^to ^make qualitative examples of how they make their decisions, thus example decisions

thart stem frompeople have aqualitative form

•

^Second, ît îs âssumed ^that ^the ^nature ôf ^most ôf ^the ^multiple

criteriadecision problems inthe real world is quantitative.

In the lightof these assumptions,already existing decisionaiding sys-

tems [ref]assumes thatqualitative exampledecisionscan onlybe used

as a basis in decision problems with a qualitativecharacteristic, mak-

ing them not suitable for using a qualitative human preference model

as a basis for recommending decisions in quantitative real-world deci-

sion problems. The second step with this work is thereforeto propose

a solution to the problem of decision making in real-world quantita-

tiveproblemsonthe basisofqualitativedecisionexamplesprovided by

people.

3. Step three of this work is to nd out study if such a decision aiding

system could be used to improve decision making problems in the oil

and gas industry by employing the it in a real-world decision making

problem.

These problems will be examined by using example case studies for each

problem. In addition, for problems 1 and 2, the proposal of the design of a

decisionaidingsystemwillbepresented,and thushowthesystem cantackle

the problems mentioned inthis section.

1.3 Thesis overview

Chapter 2, Background willgive anintroduction tomultiple criteria de-

cision analysis,and describe conceptswithin the domainofthis thesis.

Chapter 3, Theory willintroduce the relevant theory used inthe thesis.

Chapter 4, System setup will describe the high-level design and imple-

mentationof the decisionaiding system. The chapteralsoincudes one

sectionthatdescribesthetrainingprocessofthesystem,anthetwolast

sections are dedicatedto two decisionmaking processes inthe system.

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solves the problems discussed in this thesis. The chapter also reviews

a case study comparison of articial neural networks and the decision

aiding system.

Chapter 6, Conclusions summarizes the thesis and points topossible fu-

ture work

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Thischapterwilldescribesomewellknownconceptsusedinthisthesis. . The

rst section briey introduces multiple criteria decision analysis. Then the

second section describes the purpose of a decision aiding system. The third

section introduces information matrices. The explanation of decision rule

approach is introduced fth, before the related systems section is discussed

last.

2.1 Multiple criteria decision analysis

Manycomplexreal-worldproblemsarecharacterizedasdecisionmakingwith

multiple,conicting and non commensurate objectives. The nature of deci-

sionproblemsthatadecisionmaker,andhumansingeneral,usuallyfacesare

based on multiple attributes [2]. When making decisions in such problems,

it is necessary totakeintoconsideration several pointsof view, for example

human preferences. Thepointsofviewcan berepresented inaninformation

matrix that can be used asabasis for making adecision. The characteristic

ofthe pointsofviewinsuchinformationmatricesmay beeitherquantitative

or qualitative, and the corresponding value set describing a point of view

mighthave anominaloran ordinalscale. Multiplecriteria decisionanalysis

[1] is a research eld that provides logical and well structured theories and

techniques for dealing with such complex decision problems. The basis of

the problems are simple however. They consist of one nite or innite set

of alternatives, and at least two criteria, and at least one decision maker is

involved.

Twosuchcomplexproblemsare lookedatinthisthesis,namelyclassica-

tionand sortingproblems[3]. Thoseproblemsdealwithmakingdecisionsby

the assignment of objects toone class of a set of predened decisionclasses,

on the basis of points of viewregarding the decision.

2.1.1 Classication problems

Classicationproblems[3]aremultipleattributedecisionproblems, meaning

that objects in classication problems are described by a set of regular at-

tributes, and the valueset of the regularattributes constitutes adescription

ofthe object. Thenalaimof classicationproblems istoassignthe objects

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Classication problems are alsoknown asnominal classicationproblems.

2.1.2 Sorting problems

Sorting problems [3] are closely related to classication problems, however,

indistinction,the nalaimofsortingproblemsistosortobjectsfrombestto

worse, or vice versa. This implies that a preference relationship among the

decisionclasses must beconsidered. Forthisreason,objectsinsortingprob-

lems are described by a set of criteria,not regularattributes, and the value

onthe object's criteriaconstitutes thedescription ofthe object. Criteriaare

special attributes where preference relationship is taken into consideration.

Due to the preference ordering on the criteria, improvement on the values

that describe the object should not worsen the sorting rank of the object.

Sorting problems are alsoknown as ordinalclassication problems.

2.2 Decision aiding system

Decision aiding can be dened as being the activity of the person who,

through theuse ofexplicitbutnot necessarilycompletelyformalizedmodels,

helps obtain elements of responses to the questions posed by a stakeholder

in a decision process. These elements work towards clarifying a decision

and usually towards recommending, orsimplyfavoring,a behavior that will

increase the consistency between the evolution ofthe process andthis stake-

holder's objectives and value system [2].

Based on the description of an object, decision aiding systems thereby

can be used in classication problems to recommend the assignment of an

objectto exactlyone decisionclass from aset of predened decisionclasses,

wherethebasisoftherecommendationisbasedonalreadyexistingexamples.

In a sorting problem, the system has the ability to recommend the sorting

of an object to exactly one preference ordered decision class from a set of

predened preference ordered decisionclasses, wherethe basis ofthe sorting

is based onalready existing examples.

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preferences

Real-worlddecisionaidinginaccordancetoahumanpreferencemodelmeans

that the basis of the assignment of anobject toadecisionclass, made infor

example a real-world decision making problem, stem from a human prefer-

encemodel. Thisindicates thatahumanpreference modelmust beprovided

beforehand of the decision aiding process. One way of understanding the

human preferencemodelistorequest aset ofexamplesofhowthey preferto

maketheirdecisions. The examplesprovided canbeanalyzed,thusresulting

in aknowledge base that can beused as the basis ofthe recommendation of

decisionsin real world decisionmaking problems. Further,an assumptionis

made regarding the characteristic of the examples provided: It is assumed

thatpeopleprefertoprovidequalitativeexamplesofhowtheymaketheirde-

cisions. Thisassumptioncorresponds welltothe factthatpreference models

are formalrepresentations of comparison of objects established through the

use of aformaland abstract language[4]. Anotherassumptionis madewith

respecttoreal-worlddecisionmakingproblems: Itisassumedthatreal-world

classication problems usually have anumericalcharacter. This assumption

makesreal-worlddecisionaidinginaccordancetoahumanpreference model

a challenge in a decisionaiding system, because linguistic examples have to

betransformed intonumericalnumbers.

Example This exampleshows inTable 1 ahumanpreference modelof two

electrical cars. The cars are described on two criteria, namely Range

and Top speed, andthusbased onthevalueonthe criteria,assignedto

a decisionclass stating the Price of the two cars.

Car Range Top speed Price

1 good high high

2 low low low

Table 1: Electricalcars described by ahuman preference model

Table 2 presents the areal-worldversion of the the two electricalcars.

They aredescribedbynumericalvaluesetsontheconditionattributes,

and the decisionattribute Price is alsonumerical.

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1 160 km 120 km/h 300.000

2 40 km 80 km/h 120.000

Table 2: Electricalcars described by a real-world model

Real-world decision aiding in accordance to human preferences means

takingareal-worldobjectdescribedbynumericalattributesasinTable

2asinput,andthenassigntheobjecttoapredenednumericaldecision

class,basedontheinformationinahumanpreferencemodelasinTable

1.

2.4 Information matrix

Problems within the multiple criteria decision analysis domain are usually

structured within information matrices. The separate rows of the informa-

tion matrix refer to distinct objects, where every object store some associ-

ated information. Simply stated, the information matrix is an

i × j

^ma-

trix, where the rows corresponds to objects, and columns corresponds to

attributes. More formallyas presented in [5], an informationmatrix can be

dened asa 4-tuple

S =< U, Q, V, f >

^, ^where ^each ^tuple ^has ^the ^meaning:

• U

îs^the^nite^setôfôbjects,alternativesoractions,alsocalledUniverse, of interest.

• Q = {q1, q2, ..., q i }

îs â ^nite ^set ôf

i

attributes. The set

Q

^is ^further

dividedintotwodisjointclasses,

C

^and

D

^,^called^condition^and^decision

attributes.. Bot sets

C

^and

D

^are ^not ^empty^,

C 6=

^Ø,

D 6=

^Ø, ^and

both sets are unique,

C ∩ D

^=Ø, ^hence

C ∪ D = Q

^. ^Furthermore, condition attributes are those used to describe the characteristics of

the objects. The decision attributes dene a partition of the objects

into groups according to the condition attributes. The distinction of

the sets ismade withthe aimof explainingthe evaluationson

D

^using

the evaluationson

C

^.

• V q

îs ^the ^domainôf ^the âttribute

q ∈ Q

^and

V = ∪ q∈Q V q

^.

•

^The ^function

f : U × Q → V

^is ^such ^that

f(x, q) ∈ V q

^, ^where

q ∈ Q

and

x ∈ U

^. ^The ^function ^f ^is^called ^informal ^function.

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A technique within multiple criteria decision analysis for dealing with clas-

sicationand sortingproblems, is the decision rule approach[5]. Adecision

rule approach analyzes existing exemplary decisions and computes a set of

logicaldecisionrulesontheformif-then. Thelefthandsideofadecisionrule

is called the condition part, and the right hand side of the rule is called the

decisionpart. Thelefthandsideoftherulemayhaveseveralconditions. The

conditions of a decision rule are dened as

f(x) relation to constant

^, ^where

relation

^to îs â ^relationalôperator ^from ^the ^set ^{

=, ≤, ≥

^}, ^and ^the ^constant

beingavalueofattribute

f(x)

^. ^An^example^decision^rule^with^two^conditions

can beas follows:

If A = 2 and B is ≥ 2 then P roduct ≥ 4.

An induction of decision rules from a universe of decision examples can

be compared to articial intelligence, dened in [6] as being the study of

intelligent behavior. Thereby, the resulting set of decision rules constructs

a knowledge base that can be utilized by a decision aiding system to make

intelligent decisions. The decision rules induced from examples covers the

whole set of objects in the example set, and are able to assign all of the

example objects, and neverbeforeseen objects, totheir decisionclass based

ononlythedescriptionoftheobject. Thisisdonebymatchingthecondition

of a rule to the description of an object, thus if it is a match, the decision

partoftheruleholdsfortheobject. Theprocessofmatchingrulestoobjects

is described morethorough inchapter 4.

2.6 Related systems

Two related decision aiding systems implementing a decision rule approach

to classication and sorting problems is ROSE [7, 8] and jMAF [9]. Both

systems takes information matrices as input and use it as a basis for deci-

sion making problems. Firstly, ROSE is a software written in C++ imple-

menting basic elements of the classical rough set theory and rule discovery

techniques. Thesystem containsseveral toolsfor roughset basedknowledge

discovery. Among these are the ability to induce sets of decision rules from

rough approximations of decision classes, and use the sets of decision rules

as classiers.

jMAF is a multiple-criteria and attribute analysis framework written in

the Javalanguage. The system implementsmethodsof analysis provided by

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multiplecriteria sortingproblems.

None of these software systems are open source material, hence it was

necessary tomakeownimplementationsofthe techniquesthatthesesystems

oer tobe able to perform the necessary experiments.

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This chapter describes the fundamental theories used in this thesis, namely

classical rough set theory,dominance rough set approach, and fuzzy logic.

3.1 Classical rough set theory

Theclassicalroughsettheory(CRST)[10,11,12]wasdevelopedbyZdzislaw

Pawlakin1982. The theory dealswith describing the dependencies between

attributes, the signicance of attributes, as well as inconsistent data. The

theory waschosen inthisthesisbecauseithasthe abilitytosupportnominal

classication problems.

3.1.1 Indiscernibilityrelation

The indiscernibilityrelationisamathematicalbasisconcept ofthe roughset

theory. Givenaninformationmatrix

S =< U, Q, V, f >

^,^two^objects

x, y ∈ U

are said tobe indiscernible(similar) if and onlyif they are described by the

same information,hence they represent redundant data. More formally, the

function

f (x, q) = f (y, q)

^for^every

q ∈ P ⊆ Q

^. ^Any^subset

P

^of

Q

^determines

a binary relation

I p

^on

U

^. ^This ^relation ^is ^called ^anindiscernibility relation and is dened as

(x, y) ∈ I p

^.

I p

^is ^anequivalence relationfor any P.

Any set of all indiscernible objects is called an elementary set, and it

constitutes a basic granule of knowledge about the data in the universe.

Equivalence classes of the relation

I p

^are ^referred ^to ^asP-elementary sets in

S

^, ^and

Ip(x)

^denotes ^the P-elementary set containingobject

x ∈ U

^.

3.1.2 Lower and upper approximation of decision classes

The principle of rough approximation of decision classes in classical rough

set theory is allowingtotake inconsistencyintothedata analysis process by

using the introduced indiscernibility relation. For each decision class, two

rough approximations, namely the lower approximation and upper approx-

imation, are calculated. The aim is to include in the lower approximation

onlythose objectswhichare consistent, meaningthat they certainlybelongs

to the decisionclass, and inthe upperapproximationsobjectsthat possibly

belong to the decision class. The dierence between the lower and upper

approximationsof decisionclasses denes a regionof objects that cannotbe

certainly classiedintoone decisionclass.

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More formally, if

P ⊆ Q

^and

Y ⊆ U

^, ^then ^the ^P-lower approximation and P-upperapproximation of Ycan be dened as:

• P Y = {x ∈ Y : I p (x) ∈ Y }

• P Y = ^S _x∈Y I p (x)

The P-boundary, whichmeansthe doubtfulregion ofY,isdened asfollows

• Bn p (Y ) = P Y − P Y

Theaccuracyofaroughset

Y

^,^denoted

α y (Y )

^,^can^be^estimated^by^calculat-

ingthe ratioof the numberof objectsbelonging tothe lowerapproximation

to the number of objects belongingto the upperapproximation:

• α y (Y ) = ^{|P Y} _{|P Y} ^| _|

The subsequent steps of the analysis of the approximation of rough sets in-

volvethedevelopmentofasetofrulesfortheclassicationofthealternatives

intothe groups that they actually belong.

3.1.3 Decision rules

The lower and upper approximation of decision classes are sets that can be

used in decision rule algorithms. Certain decision rules are induced from

the lower approximations, and possible rules are induced from the upper

approximations. Onestrategytogeneratingthedecisionrules,istogenerate

the minimal set of decision rules that satisfy the correct classication of

example objects from an information matrix. Minimal means that no rule

covers asubset of objects of anotherrule using weaker orthe same strength

on conditions, given that they both cover the same approximation. For

extractingthedecisionrulesfromtheinformationmatrix,analgorithmcalled

Modlem [13]is used. The procedureof the algorithmis shown in Algorithm

3.

3.2 Dominance rough set approach

Dominanceroughsetapproach(DRSA)wasproposedbytheauthorsof[5]as

anextension ofclassicalrough set theory. The theory has the abilitytodeal

with preference order in the value sets that describe objects, in comparison

to classical rough set theory that cannot. From this, dominance rough set

theory was chosen in this thesis to support ordinal classicationproblems.

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Givenaninformationmatrix

S =< U, Q, V, f >

^,^decision^attributes

D

^makes

a partition of

U

înto â ^nite ^number ôf ^classes

Cl = {Cl t , t ∈ T }

^, ^with

T = {1, ..., n}

^and

Cl t = {x ∈ U : f (x, d) = t}

^, ^with

x ∈ U

^belonging ^to

one and onlyone class

Cl t ∈ Cl

^. ^The ^classes^from

Cl

^are ^preference ^ordered

according to increasing order of class indices, i.e. for all

r, s ∈ T

^, ^such ^that

r > s

^,^each^object^from

Cl r

^are^preferred^to^the^objects^from

Cl s

^. ^Given ^this

denition, twosets usedindominanceroughset approachforapproximation

of the unions

Cl _t ^≥

^and

Cl _t ^≤

^can ^be^dened:

•

Ûpward ûnions ôf ^classes,^dened

Cl ^≥ t = ∪ s≥t Cl s

^.

An object

x ∈ Cl ^≥ _t

^means ^that ^x ^belongs ^to^class

Cl t

^or^better.

•

^Downward ^unions ^of ^classes, ^dened

Cl t ^≤ = ∪ s≤t Cl s

^.

An object

x ∈ Cl ^≤ t

^means ^that ^x ^belongs ^to^class

Cl t

^or^worse.

3.2.2 Dominance principle

Using the denitions fromthe previous sectionwith respect to criteria from

set

C

^, ^sets ôf ôbjects ^dominating ôr ^dominated ^by â ^particular ôbject ^can

bedened. This denition isthe dominance principle. It issaid that object

x

P-dominates object

y

^, îf ând ônly îf

x q y

^for ^all

q ∈ P

(denotation

xD p y ⇔ x q y, ∀q ∈ P

^), ^where

P ⊆ C

^, ^then ^object

x

^should ^have ^a

comprehensive descriptionat least asgoodas object

y

^.

•

P-dominating set:

D ⁺ _p (x) = {y ∈ U : yD p x}

Representing the set of objects that outrank x.

•

P-dominated set:

D ⁻ _p (x) = {y ∈ U : xD p y}

Representing the set of objects that x outranks.

The dominance principle hencerequires that anobject

x

^dominating ^object

y on all attributes (

x q y

^), âlso ^dominate ^the ^decision ôf ôbject ^y. ^These

objectsare calledconsistent,and thoseobjectsnot satisfyingthe dominance

principleare calledinconsistent. Becausetheremightbeinconsistentobjects

in aninformationmatrix, the concept of rough approximations isdened.

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The concept of rough approximations in DRSA deals with inconsistencies

with respect to the dominance principle. The formal expression of rough

approximationsis stated:

•

^P-lower approximation of Upward union:

P (Cl ^≥ _t ) = {x ∈ U : D _p ⁺ (x) ⊆ Cl _t ^≥ }

^.

This denition meansthat anobject

x

^certainly ^belongs^to

Cl t

^or^bet-

ter, if there is noobject belongingto

Cl _t−1

^that ^P-dominate ^x.

•

^P-upperapproximationof Upward union:

P (Cl ^≥ t ) = {x ∈ U : D _p ⁻ (x) ∩ Cl t ^≥ 6=

^Ø

}

^.

Thisdenitionmeansthatanobject

x

^possibly^belongs^to

Cl t

^or^better,

if there exist an object that belongs to

Cl t−1

^or ^better, ^and ^that

x

^P-

dominates.

•

^P-lower approximation of Downward union:

P (Cl ^≥ _t ) = {x ∈ U : D _p ⁻ (x) ⊆ Cl _t ^≤ }

^.

x

^certainly^belongs^to

Cl t

^or^worse,

if all the objects that

x

P-dominates alsobelong to

Cl t

^or^worse.

•

^P-upperapproximationof Downward union:

P (Cl ^≥ t ) = {x ∈ U : D _p ⁺ (x) ∩ Cl t ^≤ 6=

^Ø

}

^.

x

^possibly^belongs^to

Cl t

^or^worse,

ifthereexistanobjectthatbelongsto

Cl t−1

^or^worse,^thatP-dominates

x

^.

In the case of inconsistencies, the boundaries between the upper and lower

approximations

Bn p (Cl ^≥ t )

^and

Bn p (Cl ^≤ t )

^are ^dened. Inconsistency means thattheexamplescannotbecertainlyclassied(alsocalleddoubtfulregions).

The denotion of the boundaries are:

•

^P-boundary ôf Ûpward ûnion:

Bn p (Cl ^≥ t ) = P (Cl ^≥ t ) − P (Cl ^≥ t )

^.

•

^P-boundary ^of ^Downward ^union:

Bn p (Cl ^≤ t ) = P (Cl ^≤ t ) − P (Cl ^≤ t )

^.

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The lowerand upperapproximations of decisionclasses are sets that can be

used to extract knowledge in terms of decision rules. Certain decision rules

are induced from the lower approximations, and possible rules are induced

from the upper approximations. In this thesis, the minimal set of decision

rules that satisfy the correctclassication of example objects froman infor-

mation matrix are extracted using apopular algorithmcalled Domlem [14].

The generalscheme ofthe algorithmispresented in Algorithm1. Generally,

the main procedure of the algorithmis repeated for a rough approximation

set, generatinga minimalset of decisionrules.

3.3 Fuzzy Logic

Reasoning in fuzzy logic [15, 16? ] is a a matter of generalizing the fa-

miliar two-valued logic statement that is either true or false, but not both.

However, in fuzzy logic, a proposition may be either true or false, or have

an intermediate truth-value, such as maybe true. Consider the question: Is

Friday aweekend day? Ifthe number1is anumericalvalue foryes, and 0is

forno, usingfuzzylogicitispossibletoanswerthe questionby avalueoffor

example0.8,meaningthatFridayisaweekenddayforthemostpart,butnot

completely. Fromthis,fuzzylogic isamethodappropriatetomakedecisions

where the boundaries of the basis of the decisions are not clearly identied.

These properties of fuzzy logic makes it possible to use linguistic terms as

the basis of a numerical decision, and thus the main reason why the fuzzy

logic theory was chosen in this thesis. Also, since the fuzzy logic controller

processes user-dened decision rulesfor makingdecisions, combining itwith

a decision rule approaches such as classical rough set theory or dominance

rough set theory seems natural and straight forward.

3.3.1 Fuzzy membership functions

Fuzzy membership functions [16] are used togeneralize the value of the de-

gree of truth in fuzzy logic. The function itself can be an arbitrary curve

whose shape can bedene asa functionthat suits usfromthe point ofview

of simplicity, convenience, speed, and eciency. The simplest membership

functions are formed using straight lines, and the only condition a member-

ship function must really satisfy is that it must vary between 0 and 1. The

(22)

set is an extension of a classical set. If

X

îs ^the ûniverse ôf ^discourse ând

its elements are denoted by

x

^, ^then ^a^fuzzy ^set

A

ⁱⁿ

X

îs ^dened âs â ^set ôf

ordered pairs. More formally:

A = {x, µ A (x) | x X}

^,

where

µ A (x)

^is ^called ^the ^membership ^function ^of

x

ⁱⁿ

A

^. ^The ^membership

function maps each element of

X

^to ^a ^membership ^value ^between ⁰ ^and ^1.

Consider again the question fromthe previouschapter;Is Fridaya weekend

day? Fuzzy membershipfunctionsare used tomodeltowhichdegreeFriday

is a weekend day.

There are many ways toassign membership functions to fuzzy variables

[13]. This thesis relies on human intuition, which is simply derived from

the capacity of humans todevelop membershipfunctions through their own

innate intelligence and understanding. Intuition involves contextual and se-

manticknowledgeaboutanissue,thesecurvesarethen afunctionofcontext

and the analyst developing them. For example, considering a temperature

scale, if the temperatures are referred to the range of human comfort, one

set ofcurvesispresent,andifthey arereferredtothe rangeof safeoperating

temperatures for a steam turbine, anotherset willbe present. However, the

important character of these curves for purposes of use in fuzzy operations

is the fact that they overlap.

3.3.2 Fuzzy rules

Decisions in fuzzy logic are based on matching the decription of objects to

every rule in the fuzzy knowledge base. The knowledge base in fuzzy logic

is a set of fuzzy rules [16] that assumes the form If x is A then y is B, or

possibly with multiple inputs as follows: If x is A and z is C then y is B.

On the left hand side of the rule,

A

^and

B

^are ^fuzzy ^sets ^included ⁱⁿ ^the

condition part of the rule, while

x

^and

y

^are ^both ^numerical ^inputs ^to ^the

rule. Ontherighthandsideoftherule,Cisthefuzzysetofthedecisionpart

of the rule, while z is the overall conclusion of the rule. Fuzzy membership

functionsare used todetermine ifaninputx belongstothe fuzzyset A,and

thusthe conclusion part z ofbelongs tothe fuzzyset C by the same degree.

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Therst stepwiththiswork istobuildadecisionaidingsystemthathas the

ability to make decisions inclassication and sorting problems. The second

stepwith thisworkistobuildthedecisionaidingsystemsuchthat ishas the

ability to approach the problem of decision making in real-world problems

on the basis of human preferences.

Thisrstpartofthis chapterpresentsthetechnologiesthatisusedinthe

system. The second part describes the logical design and implementation

details, and then the third part of the chapter focus on how the system

is trained, and the forth on how is it used. The last part of the chapter

presentshowthesystemapproachestheproblemofreal-worlddecisionaiding

in accordanceto a human preference model.

4.1 Technologies and Logical design

Thedecisionaidingsystemusestwodecision-ruleapproachtechniques,namely

the classical rough set theory and dominance rough set approach for creat-

ing aknowledge base. Then, to approachthe problemintroduced insection

2.3, the thesis proposes that the system takes advantage of combining the

decision rule approachwith fuzzy logic.

Theimplementationof the decisionaidingsystem ismeanttobe usedas

a proof of concept, thus, the emphasize onthe graphicaluser interface is on

thefunctionalandinformativeside,ratherthanbeingawellthought-through

human machine interface. Dominancerough set approach functionality, the

fuzzy logic controller,and the decision rule matcher have been implemented

in Microsoft language C#. The fuzzy logic implementationused is from an

open sourcelibrary calleds[17]. The classicalrough set theoryfunctionality

used in the system stem from a software called Rose2 [7, 8]. The logical

design of the system can beseen inFigure1.

4.2 Training process

First of all, the decision aiding system needs training in order to learn how

to make decisions. During the training process, the system rst takes an

informationmatrixasinputandperformsadecisionruleanalysisthatresults

in the knowledge base of the system in terms of decision rules on the form

if-then. Figure2 presents agraphicalpresentation of the process.

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Figure 2: Training process

Theknowledgebaseisthe basisofthe decisionmaking capabilitiesof the

system, and itrepresents the minimalset of rulespossible tocoverthe set of

exemplaryobjectsintheinformationmatrix. Accordingto[17], minimalsets

of decision rules represent the most concise and non-redundant knowledge

representations.

Fromtheroleoftheuserofthesystem,minimaleortisrequiredandthus

simplied to present to the system the information matrix with exemplary

decisions. Byminimal eort, it is meant that the user doesnot have to get

familiarwithwiththeory basisofusedanalysismodelinordertopresentthe

informationmatrix.

4.2.1 Algorithm for induction of decision rules

Twoalgorithmsforinductionof decisionrules areused. Themain procedure

of the two algorithmsis iteratively repeated over sets of lower or upper ap-

proximation of decision classes. Foreach loop, a best condition, orpossibly

(25)

becomesthe condition part of a decisionrule.

Algorithm1[14]demonstratestheprocedureusedtoinducedecisionrules

in sorting problems. In the algorithm,

P ⊆ C

^and

E

^denotes ^a ^complex

(conjunction of elementary conditions

e

⁾ ^being ^a ^candidate ^for ^a ^condition

partoftherule. Moreover,

[E]

^denotesâ^setôfôbjects^matching^the^complex

E

^. ^Complex

E

îs âccepted âs â ^condition ^part ôf ^the ^rule îf ând ônly îf

Ø

6= [E] = ∩ e∈E [e] ⊆ B

^, ^where

B

^is ^the ^considered approximation.

Algorithm 1 Rule induction proceduresotring problems

1: Procedure DOMLEM

2: (input :

L upp

^- ^a ^family ^of ^lower approximations of upward unions of decision classes : {

P (Cl ^≥ _t , P (Cl ^≥ _t−1 , ..., P (Cl ₂ ^≥ )

^}^; ^output ^:

R ≥

^set ^of ^- ^a

set of

D ≥

^-decision ^rules)^;

3: begin

4:

R ≥ :=

^Ø^;

5: for each decisionrule

B ∈ L upp

^do

6: begin

7: E :=nd_rules(B);

8: for each rule

E ∈

^E ^do

9: if

E

^is^a ^minimal^rule ^then

R ≥ := R ≥ ∪ E

^;

10: end

11: end.

(26)

1: Function nd_rules

2: (input : a set

B

^; ôutput ^: â^set ôf ^rules Ê ^covering ^set

B

3: begin

4:

G := B

^; ^{a^set ^of ^objects ^from^the ^given approximation}

5: E

:=

^Ø^;

6: while

G 6=

^Ø ^do

7: begin

8:

E :=

^Ø^; ^{starting^complex}

9:

S := G

^; ^{set ^of ^objects^currently ^covered ^by

E

^}

10: while

E 6=

^Ø^or ^not ⁽

E ⊆ B

⁾ ^do

11: begin

12:

best :=

^Ø^; ^{best ^candidate ^for ^elementary ^condition}

13: for each criterion

q i ∈ P

^do ^begin

14:

Cond := {(f(x, q i ) ≥ r q i ) : ∃x ∈ S (f(x, q i ) = r q i )}

^;

15: {foreachpositiveobjectfrom

S

^create^an^elementary^condition}

16: for each

elem ∈ Cond

^do

17: if

evaluate({elem} ∪ E)

is_better_than

evaluate({best} ∪ E)

18: then

best := elem

^;

19: end; {for}

20:

E := E ∪ {best}

^; ^{add ^the ^best ^condition ^to^the ^complex}

21:

S := S ∩ [best]

^;

22: end; {while not (

E ⊆ B

^)}

23: for each elementary condition

e ∈ E

^do

24: if

[E − {e}] ⊆ B

^then

E := E − {e}

^;

25: create a rule onthe basis of E;

26: E

:=

^E

∪{E}

^; ^{add ^the ^induced ^rule}

27:

G := B − ∪ E∈E [E];

^{remove ^examples ^covered ^by ^the ^rule}

28: end; {while

G 6=

^Ø}

29: end; {function}

Infunction

evaluate(E)

^,^the ^complex

E

^with ^the^highest^ratio

| [E] ∩ G | / | [E] |

^is ^chosen. ^The ^complex

E

^with ^the ^highest ^value ^of

| [E] ∩ G |

^is

chosenin case of a tie.

Algorithm 3 [13] demonstrates the procedure for induction of rules in

classication problems.

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1: Procedure MODLEM

2: (input :

B

^- â ^family ôf ^lower ôr ûpper approximations; output : P - single localcovering of

B

⁾

3: begin

4:

G := B

^; ^{examples^not ^covered ^by conjunction fromP}

5: P

:=

^Ø^;

6: while

G 6=

^Ø ^do

7: begin

8:

P :=

^Ø^; ^{starting ^complex} ^{candidate ^for ^condition ^part ^of ^the

rule}

9:

S := U

^; ^{set ^of ^objects^currently ^covered ^by

P

^}

10: while

P =

^Ø ^or ^not ⁽

[P ] ⊆ B

⁾ ^do

11: begin

12:

best :=

^Ø^; ^{candidate ^for^elementary^condition}

13: for each attribute

a ∈ C

^do ^begin

14: new_p := Find_best_condition(a, S);

15: if Better(new_p, best, criterion) then best :=new_p;

16: {evaluate ifnew condition new_p isbetter than previousbest}

17: end;

18:

P := P ∪ {best}

^;^{add ^the ^best ^condition ^to ^the ^condition ^part}

19:

S := S ∩ [best]

^;

20: end; {while not (

P ⊆ B

^)}

21: for each elementary condition

best ∈ P

^do

22: if

[E − {best}] ⊆ B

^then

P := P − {best}

^; ^{test ^minimality ^of

the rule}

23: P

:=

^P

∪{P }

^; ^{add ^P ^to^the ^local^covering}

24:

G := B − ∪ P ∈P [P ];

^{remove ^examples ^covered ^by ^the ^rule}

25: end; {while

G 6=

^Ø}

26: for each

P ∈

^P ^do

27: if

∪ _P ⁰ _∈P _−P [P ⁰ ] = B;

^then ^P^:= ^P ^- ^P

28: end; {procedure}

4.3 Decision making process

Thedecisionmakingprocessisthesubsequentstepafterthetrainingprocess.

From the user of the system's perspective, an object is sent as input to

(28)

one predened decision class. From the internal workings of the system's

perspective, an input object is matched to each of the decision rules in the

knowledgebaseinordertoassignittoexactlyonedecisionclass. Algorithm4

isusedtomatchtheinputobjecttothedecisionruleset. Thealgorithmloops

through each decisionrule, and compares the description of the inobject to

the condition part of the rules. If there is a match, the rule supports the

object, and the decision part of the rule states the recommending of the

decision class of the object.

Algorithm 4 Matching decisionrules to object description

1: Function match_object_to_rules

2: (input :

z

^- ^an ^object,

D

^- ^nite ^set ^of ^decision ^rules^; ^output ^:

R

^- ^a

set of decisionrules that matches the description of object

z

⁾

3: begin

4:

R :=

^Ø^;

5: for each decisionrule

Rule ∈ D

^do

6: for each condition of

Rule

^do

7: if (allconditions matchdescription of object

z

⁾^;

8: then

R := R ∪ Rule

^;

9: end; {for}

10: end; {for}

11: end; {function}

Fromtheprocedure ofAlgorithm1and 2,Let

Cov z

^be^the ^set^of ^decision

rules covering a given object z. Three situations can occur when matching

the descriptionof objectz to the set of decision rules:

1. no rule covers object z,

Cov z =

^Ø

2. one rule covers object z,

Cov z =

¹

3. several rules cover objectz,

Cov z >

¹

Therstsituationmeansthatthedecisionaidingsystemisnotabletoassign

the object to a predened decisionclass, because there are no decision rule

tojustify the decision. Theobjectthereforemay beassignedtoanydecision

class. Forthe secondand thirdsituation,classicationand sortingproblems

tackle them dierently:

(29)

(a) The second situation for classication problems is straight for-

ward. The object is assigned to the decision class that is recom-

mended by the one rule that covers the object.

(b) The third situation where several rules, indicating dierent clas-

sications, matches an object, the rule with the highest strength

is considered as the conclusive decision. The strength of the rule

isa quotientof the support to allobjects inthe training set.

2. For sortingproblems, situation2 and 3 are tackledby usinga method

proposed in [18] that takes into account the strength of the rules sug-

gestingan assignment toaclass

Cl t

âs ârguments ⁱⁿ^favor ôf

Cl t

^, ^and

all othercovering rules as argumentsagainst

Cl t

^as ^following:

(a) For the second situation, a score value,

Score r (Cl t , z)

^, ^is ^calcu-

latedforeachdecisionclasstodeterminethemostcertaindecision

class

Cl t

^from

Cl

^according ^to^the ^followinig ^formula:

Score p (Cl t , z) = ^|Cond _|Cond ^p ^∩Cl ^t ^| ²

p ||Cl t |

^.

Fromtheformula,

Cond r

^denotes^the^set ^of^objects^that^the^rule

r

supports, and

|Cond r |

^,

|Cl t |

^and

|Cond r ∩ Cl t |

^denotecardinalities of the correspondingsets: the set of objects verifying

Cond r

^, ^the

set of objectsbelonging toclass

Cl t

ând ^the ^set ôf ôbjects^verify-

ing

Cond r

^and ^belonging^to ^class

Cl t

^. ^F^rom ^this ^the ^denition ^of

Score r (Cl t , z)

^can ^be interpreted asa product of credibility

CR r

and relativestrength

RS r

^of ^rule

r

^:

CR r = ^|Cond _|Cond ^r ^∩Cl _p _| ^t ^| ²

^,

RS r = ^|Cond _|Cl ^r ^∩Cl ^t ^|

t | .

Anew objectwillbeassigned tothe class

Cl t

^for ^which ^the ^value

of

Score r (Cl t , z)

^is^the ^greatest.

(b) The third situationis tackled similarly to situationtwo. A score

value is calculated for each decision class to determine the most

certain

Cl t

^from

Cl

^.

Score R (Cl t , z) = Score ⁺ _R (Cl t , z) − Score ⁻ _R (Cl t , z)

^.

Score ⁺ _R (Cl t , z)

încludes^the ^decision^rules^that âgrees^with ^the âs-

signmentof the new object toclass

Cl t

^. ^The ^following ^formula^is

dened:

(30)

Score ⁺ _R (Cl t , z) = ^|(Cond ^pl ^∩Cl ^t )∪...∪(Cond pk ∩Cl t )| ²

|Cond pl ∪...∪Cond pk ||Cl t | ,

where

Cond pl , ..., Cond pk

^are ^the ^objects^that ^the ^given ^rules ^sup-

port, and is analogus to

Score r (Cl t , z)

^for ^situation ^2. ^F^orther-

more, the following formulais dened:

Score ⁻ _R (Cl t , z) =

|(Cond pk+1 ∩Cl ^≥ _pk+1 )∪...∪(Cond pl ∩Cl _pl ^≥ )∪(Cond pl+1 ∩Cl ^≤ _pl+1 )∪...∪(Cond pk ∩Cl ^≤ _pk )| ²

|Cond pk+1 ∪...∪Cond pl ∪Cond pl+1 ∪...∪Cond pk ||Cl ^≥ _pk+1 ∪...∪Cl ^≥ _pl ∪Cl ^≤ _pl+1 ∪...∪Cl _pk ^≤ |

,

where

Cl ^≥ _pk+1 , ..., Cl _pl ^≥

^and

Cl _pl+1 ^≤ ∪ ... ∪ Cl ^≤ _pk

âre âll ^the ûpward

and downward unions of decision classes that the rules that do

not support the new object has suggested for assignment. Sim-

ply stated,

Score ⁻ _R (Cl t , z)

îs â ^product ôf credibility and relative strength of the rules that suggest that the decision class should

not be

Cl t

^.

Similarly to situation 2, the greatest score value will determine

the nal decisionclass of the new object.

The decision aiding process can be used to measure how well the decision

aiding system can perform. By presenting it to all the objects from the

trainingset,and thencomparethe originaldecisionclassofthe objecttothe

decision class of the object recommended by the system.

4.4 Decisionmakingprocess inreal-worlddecisionprob-

lems in accordance to human preferences

The decision aiding system approaches the type of problems explained in

section2.3bytakingadvantageofthe decisionmakingcapabilitiesofafuzzy

logic controller. The fuzzy logic controller has the ability to make numerial

decisions froma knowledge base with alinguistic character.

Figure3showsagraphicalpresentationoftheowoftwodecisionmaking

processes. From top tobottom, the rst process isdiscussed inthis section,

while the other process was discussed in the previous section.

(31)

4.4.1 Fuzzy knowledge base

Therststep inthesetup processofthefuzzylogiccontrollerforthesystem

istodealwithinductionofaknowledgebaserequiredbythefuzzycontroller

[15]. The main disadvantage of fuzzy logic systems is the possible diculty

in preparing the knowledge base forthe system [16]. The knowledgebase in

a fuzzy logic controller usually consists of a set of human determined fuzzy

rules, which can be complex to determine in case of many rules. However,

for the fuzzy controller used inthe decisionaiding system in this thesis, the

knowledgebase is formed by taking advantage of the decisionrule approach

used during training of the system. That means using the set of linguistic

decision rules resulting from the training process as the knowledge base in

the fuzzycontroller.

4.4.2 Fuzzy membership functions

Thenextstepinthesetupofthefuzzycontrollerafterdeningtheknowledge

base is to include fuzzy membership functions. This is a role that the user

of the decision aiding system must act upon. In addition to providing a

humanpreference modelasabasis ofthe decisionmaking process,the fuzzy

membershipfunctionsrequiredbythefuzzylogiccontrollermustbeprovided.

Figure4shows howthefuzzymembershipfunctionspresented tothe system

during the trainingprocess.

(32)

This means that the user constructs the membership functions for the

linguistictermsusedinthepreferencemodelaccoringtohishumanintuition.

The membership functions are then used in the fuzzylogic controller.

Consider two linguisticterms, A and B, used inthe example fuzzy rule:

If A

is

^High ^then ^B

∈

^Medium.

A typicalform offor the membership functionfor the proposition A is High

ispresented inFigure5. The samegurealsoshowsthe formsforthe propo-

sitions A is Medium and A is Bad.

Figure 5: Membershipfunctions for A

From the membershipfuntions of A inFigure5,the x-axis shows that A

is high to some degree if the numericalvalue of A corresponds to a value in

the range of approximately

[6, 10]

^. ^The ^membership ^function ^of ^the ^term ^B

(33)

6.

Figure6: Membership functionsfor B

4.4.3 Fuzzy inference process

The decisionmaking process inafuzzy logiccontrollerrequires foursteps of

the fuzzy controller, calledthe fuzzy inference process [16]. The input tothe

fuzzy inferenceprocess isanobjectthat has been given anumericaldescrip-

tion. The result of the process is a crisp output number that corresponds

to the decision class of the object. The basis of the knowledge used in the

inference process is alinguistic human preference model.

Step 1: Fuzzication

Firstly, the numerical values that describe the input object are fuzzi-

ed. The fuzzifying process transforms the values intoa number that

represents to which degree they belong to a corresponding linguistic

set, resulting ina number between 0and 1.

Consider againthe fuzzyrule If A

is

^High ^then ^B îs ^High. Îfânôbject

was described by the linguistic term A with a value corresponding to

the numerical value of 8, then the membership function in Figure 7

indicates thatthe propositiononthe lefthandsideof therule ispartly

true in accrodance to the description of the object. This can be seen

in Figure 5,which presents the result of the fuzzication process, cor-

responding to a value of 0.7. The value indicates that proposition of

the rule supports the object toa degreeof 0.7.

(34)

Ifarule has several conditions,suchasIf A

is

^High ^and ^B ^is^Bad ^then

C is Medium, the fuzzication process applies a logical operator cor-

responding tothe logical AND-operand, and asa result, the condition

that has the lowest degreeof support is used.

Step 2: Implication method

The implication method is performed for each rule in the knowledge

base. This means reecting the results from the fuzzication step on

the output for each rule. The implicationmethod used inthe decision

aiding system is amethodthat truncates the outputfuzzy set.

Consider again the fuzzy rule used in the fuzzication step, in which

supported an object to a degree of 0.7. The shaded area of Figure 8

demonstrates how the implicationmethod truncates the output fuzzy

set B inaccordance to the fuzzication result.

(35)

Step 3: Aggregation

The aggrecation step is used to derive an overall conclusion regarding

themembershipofanobjectintothefuzzysetbasedonthedescription

of the object. According to the description of an object, several rules

may support the object, resulting in several fuzzy sets as a result of

the implicationmethod. Simply stated,the aggregationstep combines

allthese sets into one singlefuzzy set by joiningthe maximum of each

set.

Step 4: Defuzzication

Lastly, thedefuzzicationsteptransformsthe singlefuzzyset fromthe

aggregation step into a single crisp value. This is done by applying a

methodcalledthe center of gravitymethod. Thereare several defuzzi-

cation methods, however the center of gravity method is the most

prevalent defuzzication method [16]. The center of gravity can be

found using the following formula:

Center of gravity = ^´ ^´ ^µ _µ ^B ^(z)×zdz

B (z)×dz

^,

where

´ µ B (z)

^denotes ^the întegral ôf êvery ^resulting ^fuzzy ^set, ând

z

is the corrersponding x-axis value. Simply stated, if anarea of a plate

is considered as equal density, then the centre of gravity is the point

alongthe xaxisaboutwhichthisshapewouldbalance. Figure9shows

(36)

shaded area is the result of the aggregationstep inFigure6.

Figure9: Aggregation and defuzzication

FromFigure9,thecenterofgravityindicatesthattheresultofthefuzzy

inference process is a numericalvalue of 9. Thereby, using a linguistic

preference model in terms of the decision rules as a knowledge base

in a fuzzy logic controller, makes the system able to make numerical

decisions.

(37)

This chapterpresent the resultsof the case studiesperformedin this thesis.

The rst part of the chapter presents a case study about making decisions

in accordance to human preferences. The next part presents a case study

where a human preference modelis used as a basis of a real-world problem.

Then, a case study fromthe oiland gas industry is presented. Lastly, the a

case study onrelated systems are described.

5.1 Example of decision making based on human pref-

erences

This example introduces acase study wherethe value of a residentialprop-

erty is to be estimated without knowing the exact numerical details of the

property. The role of the user in this case study is therefore to provide a

preference modeldescriblingexamplevalues thatconstitutesthe basisof the

problem. It is assumed that a specialist in the domain of the real estate

market has provided his preference modelof the value of 10exemplary res-

identialproperties described by multiple criteria. The preference modelcan

beseen in Table 3.

Location Size Standard Buildyear Value

1 urban big good recently high

2 urban small good old medium

3 urban medium bad old medium

4 suburban medium excellent recently high

5 suburban big bad recently medium

6 urban big bad new high

7 urban big excellent new high

8 suburban small good old low

9 countryside small good old low

10 suburban medium Bad old medium

Table 3: Human preference modeldescribing residential properties

More formally, the preference model in represented in an information

matrix inculding anite set of 10objectsU, described by the set of criteria

Q:

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• Q = {Location, Size, Standard, V iew, V alue}

The set Q is further divided into a set

C

ôf ^condition ^criteria, ând â ^set

D

of decision criteria:

• C = {Location, Size, Standard, V iew}

• D = {V alue}

^.

EveryobjectinUisassignedtoapreference orderedandpredeneddecision

class belonging to the domain of set D, according to the evaluation on the

condition classes inset C. The domainof the classes incondition set C and

the decisionclass in set D is asfollows:

• V Location = {countryside, suburban, urban}

• V Size = {small, medium, big}

• V Standard = {bad, good, excellent}

• V Age = {old, average, new}

• V V alue = {low, medium, high}

There are monotonicrelationships between the criteriameaningthatfor ex-

ample a residentialproperty inanurban area should have at least the same

orahighervaluethanthatofaresidentialpropertyinasubrubanarea,hence

it follows that preference order isfrom left toright.

Training process The rst step that is performed during training of the

system is empolying a decision rule approach to analyse the information

matrix, ormore precisely dominance roughset approach. The analysis pro-

cess denotes knowledge discovery by using the decision attribute

V alue = {Low, Medium, High}

^. ^From^this ^set, ^three^classes ^can ^be^identied:

Cl 1 = {Low}

^,

Cl 2 = {Medium}

^and

Cl 3 = {High}

^. ^Furthermore, four unions of classes denoted

Cl ^≤ ₁

^,

Cl ^≤ ₂

^,

Cl ^≥ ₂

^and

Cl ₃ ^≥

^are introduced. From the basis of the four unions of classes, the lower approximations of every union can be

found:

• P (Cl ^≤ ₁ ) = {2, 8, 9}

• P (Cl ^≤ ₂ ) = {2, 3, 5, 8, 9, 10}

(39)

• P (Cl ^≥ ₂ ) = {1, 3, 4, 5, 6, 7, 10}

• P (Cl ^≥ ₃ ) = {1, 4, 6, 7}

The lower approxiamtions of classes are input to the algorithm from (x),

resulting in a minimal set of decision rules. The decision rules can be seen

in the following list, with the objects number that the corresponding rule

supports stated inparenthesis:

IfBuild year

≤

^Old ^then ^V^alue

≤

^Medium

IfLocation

≤

^Suburban ^and ^Standard

≤

^Bad ^then ^V^alue

≤

^Medium

IfSize

≥

^Medium ^then ^V^alue

≥

^Medium

IfLocation

≥

^Urban ^then ^Value

≥

^Medium

IfSize

≤

^Small ^and ^Location

≤

^Suburban ^then ^V^alue

≤

^Low

IfStandard

≥

^Excellent^then ^Value

≥

^High

IfBuilt

≥

^New ^then ^Value

≥

^High

IfLocation

≥

^Urban ^and ^Size

≥

^Big ^then ^Value

≥

^High

As already stated, such rules corresponds to how people try to justify

theirdecisions,hencetheycanbeusedtomakedecisionsaccordingtohuman

preferences.

Using the system Aftertrainingofthedecisionsupportsystem, resulting

in the set of induced decision rules, it is possible toinfer the value of a new

residential properties based on the linguistic description of the property by

usingthe decisionrules. Following,anewresidentialproperty ascan beseen

in Table 4 is presented to the decision aiding system. It is of interest to

know the value of the property according to the specialistin the real estate

marked's preferences.

Location Size Standard Build year Price

1 suburban medium good recently

Table 4: Description of a residential property without value

Theobjectis inputtothe decisionaiding system. Thedescription of the

objectis thenmatched tothe set of decisionrules, resultinginone rule that

matches the object's description:

1. If Size

≥

^medium^then ^Value

≥

^medium

A decision aiding system based on a decision rule approach and fuzzy logic

•

•

i × j

S =< U, Q, V, f >

• U

• Q = {q1, q2, ..., q i }

i

Q

C

D

C

D

C 6=

D 6=

C ∩ D

C ∪ D = Q

D

C

• V q

q ∈ Q

V = ∪ q∈Q V q

•

f : U × Q → V

f(x, q) ∈ V q

q ∈ Q

x ∈ U

f(x) relation to constant

relation

=, ≤, ≥

f(x)

If A = 2 and B is ≥ 2 then P roduct ≥ 4.

S =< U, Q, V, f >

x, y ∈ U

f (x, q) = f (y, q)

q ∈ P ⊆ Q

P

Q

I p

U

(x, y) ∈ I p

I p

I p

S

Ip(x)

x ∈ U

P ⊆ Q

Y ⊆ U

• P Y = {x ∈ Y : I p (x) ∈ Y }

• P Y = S x∈Y I p (x)

• Bn p (Y ) = P Y − P Y

Y

α y (Y )

• α y (Y ) = |P Y |P Y | |

S =< U, Q, V, f >

D

U

Cl = {Cl t , t ∈ T }

T = {1, ..., n}

Cl t = {x ∈ U : f (x, d) = t}

x ∈ U

Cl t ∈ Cl

Cl

r, s ∈ T

r > s

Cl r

Cl s

Cl t ≥

Cl t ≤

•

Cl ≥ t = ∪ s≥t Cl s

x ∈ Cl ≥ t

Cl t

•

Cl t ≤ = ∪ s≤t Cl s

x ∈ Cl ≤ t

Cl t

C

x

y

• P Y = ^S _x∈Y I p (x)

• α y (Y ) = ^{|P Y} _{|P Y} ^| _|

Cl _t ^≥

Cl _t ^≤

Cl ^≥ t = ∪ s≥t Cl s

x ∈ Cl ^≥ _t

Cl t ^≤ = ∪ s≤t Cl s

x ∈ Cl ^≤ t

D ⁺ _p (x) = {y ∈ U : yD p x}

D ⁻ _p (x) = {y ∈ U : xD p y}

P (Cl ^≥ _t ) = {x ∈ U : D _p ⁺ (x) ⊆ Cl _t ^≥ }

Cl _t−1

P (Cl ^≥ t ) = {x ∈ U : D _p ⁻ (x) ∩ Cl t ^≥ 6=

P (Cl ^≥ _t ) = {x ∈ U : D _p ⁻ (x) ⊆ Cl _t ^≤ }

P (Cl ^≥ t ) = {x ∈ U : D _p ⁺ (x) ∩ Cl t ^≤ 6=

Bn p (Cl ^≥ t )

Bn p (Cl ^≤ t )

Bn p (Cl ^≥ t ) = P (Cl ^≥ t ) − P (Cl ^≥ t )

Bn p (Cl ^≤ t ) = P (Cl ^≤ t ) − P (Cl ^≤ t )

P (Cl ^≥ _t , P (Cl ^≥ _t−1 , ..., P (Cl ₂ ^≥ )