MOLNAR M ad alina – Florina [620801]

UNIVERSITATEA ,,AUREL VLAICU" DIN ARAD
FACULTATEA DE S TIINT  E EXACTE
SPECIALIZAREA MODELARE MATEMATIC A^IN CERCETARE S I
DIDACTIC A
LUCRARE DE DISERTAT  IE
Fuzzy Analytic Hierarchy Process
^INDRUM ATOR S TIINT  IFIC:
Lector dr. N ADABAN SorinABSOLVENT: [anonimizat] ad alina – Florina
ARAD
Iulie 2017
1

Contents
Introduction 2
1 Fuzzy set theory 4
2 Fuzzy AHP technique 6
2.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 AHP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Conclusions 11
2

Preface
3

Chapter 1
Fuzzy set theory
Fuzzy set theory (-zadeh,1965-) has been used to model decision making
processes (junior 2014{). In (zimmerman{) it's trying to accomodate fuzzi-
ness in meaning it is contained in human language, evaluation, judgment,
and decision. Linguistic terms lead to lionguistics variables that is expressed
qualitatively and quantitatively by a fuzzy set and respective membership
function (junior 2014{), In fuzzy set theory the set of elements belongs to a
space with a boundaries that is not exactly dened. In crisp set case, the
object may belong or not to a set of elements, whereas in fuzzy set theory,
the objects can take on membership values in an interval [0 ;1] that deliberate
a degree of membership (-zadeh,1965-). To model the qualitatives terms in
form of fuzzy numbers (TFN's), which is widely used to show the linguistic
evaluations.(khazaeni,2012{)
The membership function for a triunghiular fuzzy number ~Ais presented
in Fig 1 on space R![0;1] with bases noted as ( l;m;u ) which is dened
as in Eq 1.0.1:
(1.0.1) ~A(x) =8
>><
>>:0; x1
x1
ml; lxm
xu
mu; mxu
0; otherwise:
The lowest possible value, the most favorable value, are the parameters
(l;m;u ) which they are real numbers and the highest posible value ( l <
m < u ), which describe a fuzzy case. For two TFN's ~A= (a1;a2;a3) and
~B= (b1;b2;b3) the operational rules (algebraic operations) are:
~AM~B= (a1;a2;a3)M
(b1;b2;b3) = (a1+b1;a2+b2;a3+b3)
4

Figure 1.1: Illustration of fuzzy triangular number (TFN).
~AO~B= (a1;a2;a3)O
(b1;b2;b3) = (a1b1;a2b2;a3b3)
~A1= (1
a3;1
a2;1
a3)
where withLis noted the extended summation of two TFN's and withN
is noted the extended multiplication.
5

Chapter 2
Fuzzy AHP technique
AHP is widely used across industries for dealing with multiple criteria decision-
making problems involving subjective judgment (huang 2008).Therefore, fuzzy
AHP method, which combines traditional AHP with fuzzy set theory, was
developed for coping with uncertain judgments and to express preferences as
fuzzy sets or fuzzy numbers which re
ect the vagueness of human thinking.
The basic idea of fuzzy set theory is that an element has a degree of mem-
bership in a fuzzy set .The traditional AHP is very used in MCDA, that is
an eective tool for dealing with complex decision making, and could help
the decision maker to set priorities and make the best decision. By reducing
complex decisions to a series of pairwise compatisons, and then synthesizing
the results, the AHP help to capture both subjective and objective aspects
of a decision.({saaty 1980-)
It is more easily understood in context of mathematical calculations, in
comparison with other methods. In spite of its large and successful aplica-
tions, it has been criticized for its inability in managing uncertainty result-
ing from relating whole numbers to DMs(decisions makers) understanding.
Therefor is ineective in aplications to ambiguous decision problems which
are wide-spread in real word(-javanbarg{2002-). The integration of fuzzy
set theory with traditional AHP, which uses the principles of decomposition,
pairwise comparisons, priority vector aggregation and synthesis({thaylan-) is
vital in dealing with sources of uncertaintly. It has a potential to map the
perceptions, incomplte information and approximations to produce decisions
by employing the membership functions.
The use of fuzzy AHP, fuzzy extension of AHP, is suitable in solving
the hierarchical rating of fuzzy decision problemes. It aords the decision
makers to concentrate on specic sub-criteria to make pairwise comparison
among the criteria that having the same root depending on its position on
the hierarchy structure which produces the relative trade-o in the form of
6

comparison matrices. The measure analysis Fuzzy AHP (|-Chang1996) can
be used to drift the priority weights from fuzzy comparison. The following
steps explain the procedure: Step 1: The linguistic terms used by DMs are
translated to express the comparative judgments among the main criteria
with respect to the overall goal, and evaluation criteria with respect to their
main criterion ioto TFNs. The structure of the comparison matrices will be
as below:
(2.0.1) ~A(~aij)nn0
BBB@(1;1;1) (l12;m 12;u12)


(l1n;m 1n;u1n)
(l21;m 21;u21) (1;1;1)


(l2n;m 2n;u2n)
…


……
(ln1;mn1;un1) (ln2;mn2;un2)


(1;1;1)1
CCCA
Step 2 : In order to build nal pairwise comparison matrix for the aggrega-
tion of preferences of tdecision makers, the following methods are:
1. Weight aggregation 1 (WA1) (Yazdani-chamzini…)
~wij= (Lwij;Mw ij;Uw ij)
Lwij= min
t
Lwij;Mw ij=1
TTP
t=1Mw ijt;Uw ij= max tfUw ijtg
~wij=the triunghiular fuzzy weight of the ith criterion in comparison
with the jth criterion
2. Weight aggregation 2 (WA2) (khazaeni 2012) based on arithmetic op-
erations
Lwij=1
TTX
t=1Lwijt;Mw ij=1
TTX
t=1Mw ijt;Uw ij=1
TTX
t=1Uw ijt
3. Weight aggregation 3 (WA3) based on geometric mean of preferences
(jaiswal..)
Lwij=TQ
t=1Lwijt1
t
;Mw ij=TQ
t=1Mw ijt1
t
;Uw ij=TQ
t=1Uw ijt1
t
7

Figure 2.1: Illustration of intersection between two fuzzy triangular numbers.
Step 3 : SupposeX=fx1;x2;:::;x ngas an obiect set, and U=fu1;u2;:::;u ng
as a goal set. Each object is considered, by this methodology, and later the ex-
tent analysis for each object (gi) is performed.Further, mextent analysis val-
ues for each object can be secured with the following sings: M1
gi;M2
gi;:::;Mm
gi,
all values of the extent analysis are fuzzy triangular number, ( i= 1;2;:::;n ),
and (j= 1;2;:::;m ). In the following equations are illustrated the value of
fuzzy synthetic extent with respect to ith object:
Si=mX
j=1Mj
giOnP
i=1mP
j=1Mj
gi1
mX
j=1Mj
gi=mP
j=1li;mP
j=1mi;mP
j=1ui
nX
i=1mX
j=1Mj
gi=nP
i=1li;nP
i=1mi;nP
i=1ui
nP
i=1mP
j=1Mj
gi1
=1
(nP
i=1ui);1
nP
i=1mi;1
nP
i=1li
Between two fuzzy synthetic, the degree of possibility extent values e.g.,
S2= (l2;m 2;u2)S1= (l1;m 1;u1) as illustrated in Fig 2 is dened as:
V(S2S1) =hgt(S2\S1) =(d) =8
<
:1; ifm 2m1
0; ifl 2u2
l1u2
(m2u2)(m1l1); otherwise:
where dis the ordinate of highest intersection point between two fuzzy num-
bers. The two values of V(S2S1) andV(S1S2) are computed by the
comparison between S1andS2. The degree of possibility for a convex fuzzy
8

number to be greater than kconvex fuzzy numbers Si(i= 1;2;:::;k ) is
explainded as:
V(SS1;S2;S3;:::;S k) =V[(SS1)and(SS2)and:::and (SSk)] =minV (SSi)
By assuming d0(Ci) = minV(SiSkfork= 1;2;:::;n (i6=k), the weight
vector can be given by:
W0= [d0(C1);d0(C2);d0(C3);:::;d0(Cn)]T
whereCi(i= 1;2;:::;n ) are n elements. In the end a procedure of normaliz-
ing the weight vectors should be achieved, and the result will be a non-fuzzy
number.
W= [d(C1);d(C2);d(C3);:::;d (Cn)]T
The extent analysis Fuzzy AHP method, in comparison to the most widely
applied extended MCDA methods, is currently one of the most popular meth-
ods despite its deciency of producing zero values in case of presence of ex-
treme assessments in the pair-wise comparison proposed a modied metodol-
ogy based on modied normalization formula that has been suggested by
(Wang and elhag 2006).
9

2.1 Literature review
Fuzzy set theory was developed by Zadeh in 1960 and thus fact has opened
doing various studies based on fuzzy, on situations that cannot be established
quantitatively. In the following it will be described the studies that have been
done recently and used the fuzzy AHP: In (Somsuk 2014) fuzzy AHP method
is used in order to identify the factors aecting the success of the university
business incubators in Thailand. The most eective factor among 14 factors,
that are detained from questionnaires, is administrative and political sug-
gestions. Jung has used fuzzy AHP method in the problem of an integrated
manufacture planning. (jung 2011). In (CAkir and Cabolat) is proposed a
system with the inventory classication, that is based on fuzzy AHP, in a
company producing electical small household applices. Chanetal et al. have
pointed out that the environment business in the context of increasing the
competitiveness of the environment and they have proposed an integrated
fuzzy AHP approach which is to decide on the choice of alternative green
design (Chan 2014). To evaluate the job security in hot and humid places
Zheng et al. have used the fuzzy AHP method.
2.2 AHP
2.3 Fuzzy TOPSIS
2.4 Hybrid methods
2.4.1 ANN and Fuzzy AHP
2.4.2 Fuzzy AHP and SWOT
2.5 Fuzzy AHP and Fuzzy Moora
2.6 Comparison between Fuzzy AHP and AHP
10

Chapter 3
Applications
3.1 Water management
11

Chapter 4
Conclusions
12