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Thesis number:
Eigen Fuzzy Sets of Fuzzy Relation withApplications
Saleem Muhammad Naman
This thesis is presented as part of Degree of Master of Sciences inMathematical Modeling and Simulation.
Blekinge Institute of Technology2010
School of EngineeringDepartment of Mathematics and SciencesBlekinge Institute of Technology, SwedenSupervisors: Elisabeth Rakus-Andersson Examiner: Elisabeth Rakus-Andersson
Contact Information:
Author:
Saleem Muhammad Namanemail: [email protected]
Supervisors:Elisabeth Rakus-Andersson
Professor in Applied MathematicsBlekinge Institute of TechnologySchool of EngineeringDepartment of Mathematics and ScienceS-37179 Karlskrona, SwedenTel. +46455385408Fax +46455385460
Examiner:
Elisabeth Rakus-AnderssonProfessor in Applied MathematicsBlekinge Institute of TechnologySchool of EngineeringDepartment of Mathematics and ScienceS-37179 Karlskrona, SwedenTel. +46455385408Fax +46455385460
Abstract
Eigen fuzzy sets of fuzzy relation can be used for the estimation of highest andlowest levels of involved variables when applying max-min composition on fuzzyrelations. By the greatest eigen fuzzy sets (set which can be greater anymore)maximum membership degrees of any fuzzy set can be found, with the help ofleast eigen fuzzy set (set which can be less anymore) minimum membership de-grees of any fuzzy sets can be found as well.The lowest and highest level, impactor effect of anything can be found by applying eigen fuzzy set theory. The impli-cational aspect of this research study is medical and customer satisfaction levelmeasurement. By applying methods of eigen fuzzy set theory the effectiveness ofmedical cure and customer satisfaction can be found with high precision.
iv
Acknowledgements
All thanks to God, Who gave me strength to do this degree successfully. I wouldlike to express my deep and sincere gratitude to Elisabeth for her guidance,support and inspiration under her supervision of this thesis. Her wide knowledgeand supportive comments have been great values for me. I would like to sayspecial thanks to Raisa Khamitova for her valuable time, suggestions, and effortsto support in my studies throughout the degree. This thesis would not have beensuccessful unless my friends and family who helped and encouraged me to do thisthesis. They supported me in whole thesis work.
Muhammad Naman Saleem2010, Sweden
Contents
Abstract iii
Acknowledgements v
1 Basic Definitions 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Fuzzy Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.3 Max-Min composition of relations [1, 2] . . . . . . . . . . . 3
1.4 Fuzzy Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Fuzzy set in L-R form . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Contents of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Eigen Fuzzy Sets 92.1 Eigen Fuzzy Sets [2–4] . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Greatest Eigen fuzzy set(GEFS) . . . . . . . . . . . . . . . . . . . 9
2.2.1 First Method of finding the GEFS . . . . . . . . . . . . . . 92.2.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Second method of the GEFS determination . . . . . . . . . 112.2.4 Third Method of the GEFS creation . . . . . . . . . . . . 12
2.3 Least Eigen Fuzzy set [2] . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Eigen Fuzzy Sets with Fuzzy Numbers 173.1 Minimum for two fuzzy Number . . . . . . . . . . . . . . . . . . . 17
3.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Maximum for Two fuzzy numbers . . . . . . . . . . . . . . . . . . 19
3.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Eigen Fuzzy set with Fuzzy Numbers [2] . . . . . . . . . . . . . . 20
3.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
vii
viii
4 Applications of Eigen Fuzzy sets 234.1 Example: Mobile users satisfaction . . . . . . . . . . . . . . . . . 234.2 A Genetic Algorithm Based on Eigen Fuzzy Sets for Image Recon-
struction [7–9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Bibliography 29
Chapter 1
Basic Definitions
1.1 Introduction
In everyday life most of the problems involved imprecise concept. In order tohandle such concept the conventional methods of set theory and numbers areinsufficient and need to be extended to some other concepts. Fuzzy set theoryand fuzzy logic focuses on such concept.
According to Professor. Lotfi A.Zadeh (the founder of fuzzy logic) ” Fuzzylogic is precise logic of imprecision”.
The set A defined on the basis of a generalized characteristic function, is calleda Fuzzy set. In Fuzzy set theory classical sets are called crisp set.
1.2 Fuzzy set
The membership function µA of fuzzy set A is a function defined as µA : X →[0, 1]. Every element x ∈ X has a membership degreey = µA(x) ∈ [0, 1]. The fuzzy set A is finally completely determined by set ofpairs
A = {(x, y) = (x, µA(x))} , x ∈ X (1.1)
where X is a crisp universe set.
The important part of fuzzy set A is its support denoted by Supp(A) whichis non-fuzzy set.
Supp(A) = {x ∈ X : µA(x) > 0} (1.2)
1.2.1 Example
Let us consider a non-Fuzzy finite set ”Male−Heightin−cm” = {153, 165, 173, 178, 183, 188, 198, 213}and let us intuitively decide strength of the relationship between the set and each
1
2 Chapter 1. Basic Definitions
value belonging to its support.
′′V ery−tall−man“ = {(153, 0)(165, 0.3)(173, 0.5)(178, 0.7)(183, 8)(188, 0.9)(198, 1)(213, 1)}
Now it becomes the Fuzzy set and can be written in another form also like
′′V ery−tall−man“ = 0/153+0.3/165+0.5/173+0.7/178+0.8/183+0.9/188+1/198+1/213
Figure 1.1: Fuzzy set “very tall man”
1.3 Fuzzy Relation
Fuzzy relations join two non-fuzzy sets in the common set of pairs on conditionthat each pair has the membership degree assigned.
1.3.1 Definition
X and Y are two non-fuzzy Universes. A Fuzzy relation R ⊂ X × Y is a set ofpairs (x, y), where each pair has now a membership degree µR(x, y) assigned
µR(x, y) ∈ [0, 1] (1.3)
µR : X × Y → [0, 1] (1.4)
1.3.2 Example
x = {3, 5, 8, 10}
Chapter 1. Basic Definitions 3
XRY= “ x is considerably less than y”
X ×X = {(3, 3), (3, 5), (3, 8), (3, 10), . . . (10, 10)}
XRY= “ x is considerably less than y”
x1 x2 x3 x4
R =
x1x2x3x4
0 0.5 0.8 10 0 0.5 0.80 0 0 0.30 0 0 0
1.3.3 Max-Min composition of relations [1, 2]
Let us consider three finite universes
X = {xi} , i = 1, 2, . . . , n (1.5)
Y = {yi} , j = 1, 2, . . . ,m (1.6)
Z = {zi} , k = 1, 2, . . . , p (1.7)
We introduce relations
R ∈ X × Y,R = {(xi, yi), µR(xi, yi)} ;xi ⊂ X, yi ⊂ Y, µR(xi, yi) ∈ [0, 1]
Q ∈ Y × Z,Q = {(yi, zk), µQ(yi, zk)} ; yi ⊂ Y, zk ⊂ Z, µQ(yi, zk) ∈ [0, 1]
We compose the relation R and Q by using a relation
S = RoQ
where the sign of ′o′ denotes the max-min composition of R with Q. Relation Swill be a Fuzzy relation.
S = RoQ ={[
(xi, zk), µs(xi, zk) = maxyj∈Y {min {µR(xi, yj), µQ(yj, zk)}}]}
For better understanding of max-min composition, let us consider an example.
Example
X= The power of Air Conditioner(AC)={x1=Low power AC, x2=High powerAC}Y= The cooling capacity={y1=poor cooling, y2= medium cooling, y3=good cool-ing}Z= The consumption of electricity= {z1=low, z2=high } Now inserting the mem-bership degree to each pair of R
R ∈ X × Y
4 Chapter 1. Basic Definitions
y1 y2 y3
R =x1x2
[0.8 0.4 0.10.2 0.6 0.7
]The next relation Q settles the relationship between Y and Z
Q ∈ Y × Z
z1 z2
Q =y1y2y3
0.9 0.20.5 0.50.1 0.8
By composing the relation R and Q we obtain the result that reveals the associ-ation between X and Z
S = RoQ
S = RoQ
[0.8 0.4 0.10.2 0.6 0.7
]o
0.9 0.20.5 0.50.1 0.8
As
µs(x1, z1) = max(min(0.8, 0.9),min(0.4, 0.5),min(0.1, 0.1))
= max(0.8, 0.4, 0.1) = 0.8
z1 z2
S =x1x2
[0.8 0.40.5 0.7
]By comparing any assigned membership degree in S, we can see a truthful as-sociation between the examined parameter “power of AC” and “consumption ofelectricity”. For example the value“0.8” assigned to pair(low power AC and lowconsumption of electricity), which is practically and intuitively true.
1.4 Fuzzy Number
A fuzzy number ‘N ′ is a fuzzy set which satisfies the following conditions,
• At least one element x ∈ N must have µA(x) = 1
• The left membership function define for x which goes to the peak withmembership degree 1, must be increasing.
Chapter 1. Basic Definitions 5
Figure 1.2: A fuzzy number Trapezoidal
• The right membership function defined for x which goes from the peak withthe membership degree 1, must be decreasing.
1.5 Fuzzy set in L-R form
A Fuzzy number ‘N ′ with a continuous support being a subset in Z = R is calledL-R if there are two reference function L(z) and R(z) and there two scalars whichcreate the following membership function,
µN(z) =
{L(m−z
α) m− α ≤ z ≤ m
R( z−mβ
) m ≤ z ≤ m+ β(1.8)
where m is called mean value of number N , α is called left spread and β is calledits right spread. The figure 1.4 shows the definition.
µN(mN) = 1
µN(mN − αN) = 0
µN(mN + βN) = 0
1.6 Contents of Thesis
The first chapter of the thesis includes some basic concepts and definitions aboutfuzzy set theory and operations on fuzzy sets. In Chapter 2 and 3, discussions
6 Chapter 1. Basic Definitions
Figure 1.3: A fuzzy number Triangle
Figure 1.4
Chapter 1. Basic Definitions 7
about main topics of eigen fuzzy sets and fuzzy relations are added. To findthe customer’s maximum satisfaction level about any product I applied thesemethods which are discussed in chapter 4.
Chapter 2
Eigen Fuzzy Sets
By perusing the contents of chapter 1 we have approached the conception ofcomposition of two relations, now on the base of these concepts we can understandthe definition of eigen fuzzy sets.
2.1 Eigen Fuzzy Sets [2–4]
Let R be a fuzzy relation between the elements of a finite set X and let A bea fuzzy subset of X. The Max-Min composition of R and A gives B, (AoR =B,B ⊆ X). When B becomes equal to A, we say that A is an eigen fuzzy setassociated with given relation R.
R is a fuzzy relation determine R ⊆ X × X with membership function µR :X ×X → [0, 1], µR(x, x′) ∈ [0, 1], x, x′ ∈ X and the eigen fuzzy set A ⊆ X,µA :X → [0, 1], µA(x) ∈ [0, 1], x ∈ X,satisfying AoR = A should exist.
2.2 Greatest Eigen fuzzy set(GEFS)
By using Max-Min composition we can find the greatest eigen fuzzy set(GEFS)associated with R, as we are dealing with finite case, some interpretation can betaken in matrix form.
There exist three fundamental algorithms of determining GEFS, we will dis-cuss all three methods one by one.
2.2.1 First Method of finding the GEFS
Let A1, be the fuzzy subset of X in which the grades of membership are equal tothe greatest element for each column of R
µA1(x′) = maxx∈XµR(x, x′),∀x′ ∈ X
9
10 Chapter 2. Eigen Fuzzy Sets
If we define A0, a constant fuzzy set with minimum of these values, it is easy toverify that A0 is an eigen fuzzy set, but it is not always the GEFS.
Let us define a sequence of fuzzy sets (An)n
A1oR = A2
A2oR = A1oR2 = A3
... . . . . . ....
AnoR = A1oRn = An+1
The sequence (An)n is decreasing and bounded by A0 and A1
A0 ⊆ . . . ⊆ An+1 ⊆ An ⊆ . . . ⊆ A3 ⊆ A2 ⊆ A1
2.2.2 Algorithm
A relation R ⊆ X × Y with membership function µR(x, x′) is given
1. Find the set A1 identified by
µA(x′) = maxx∈XµR(x, x′),∀x′ ∈ X
2. set the index n=1
3. calculate An+1 = AnoR
4. An+1= An→No→n=n+1→go−to−step−3→Y es→An=An+1
Example 2.1
Let X={x1, x2, x3, x4} and R be given by following representation
x1 x2 x3 x4
R =
x1x2x3x4
0.1 0.5 0.9 0.60.3 0.6 0.7 0.40.3 0.8 0.4 0.10.2 0.5 0.3 0.2
The set A1 has the membership degree of xj found as the largest value in columnj, j=1,2,3,4 and if thus determine as
A1 =[
0.3 0.8 0.9 0.6]
A0 =[
0.3 0.3 0.3 0.3]
A0oR = A0(Trivial − solution)
Chapter 2. Eigen Fuzzy Sets 11
For n = 1 we obtained
A2 = A1oR =[
0.3 0.8 0.9 0.6]o
0.1 0.5 0.9 0.60.3 0.6 0.7 0.40.3 0.8 0.4 0.10.2 0.5 0.3 0.2
A2 =
[0.3 0.8 0.7 0.4
]Since A2 6= A1, we set n=2 in step 4, compose A2 with R to get A3
A3 = A2oR =[
0.3 0.8 0.7 0.4]o
0.1 0.5 0.9 0.60.3 0.6 0.7 0.40.3 0.8 0.4 0.10.2 0.5 0.3 0.2
A3 =
[0.3 0.7 0.7 0.4
]For n=3 we get
A4 = A3oR =[
0.3 0.7 0.7 0.4]
A4 is accepted as greatest eigen fuzzy set (GEFS) of relation R as equality A3=A4
holds. We can observe that
A4 ⊆ A3 ⊆ A2 ⊆ A1
2.2.3 Second method of the GEFS determination
In this method we do not need to evaluate any composition. We only have tofind the invariant element from successive reduction of R.
At each step of this method, the invariant elements are exactly the ones ofprevious method
Example 2.2
R =
0.1 0.5 0.9 0.60.3 0.6 0.7 0.40.3 0.8 0.4 0.10.2 0.5 0.3 0.2
Steps:
1. Find the greatest element in each column of R, [0.3, 0.8, 0.9, 0.6]
2. The smallest of these element is denoted by r, in this case r=0.3 and thecolumn containing is x1th column.
12 Chapter 2. Eigen Fuzzy Sets
3. Delete this 1st column and the same number row (1st) from R to get R′
R′ =
0.6 0.7 0.40.8 0.4 0.10.5 0.3 0.2
4. Set the value of r=0.3 in An( n is not known yet) at position of the deleted
column
An =[
0.3 − − −]
5. Repeat all previous steps for R′
R′ =
0.6 0.7 0.40.8 0.4 0.10.5 0.3 0.2
where r′ =0.4 > 0.3 and from R′ if we get r′ < 0.3 then instead of value ofr′ we will take value of r at r′. Set 0.4 at x4th position
An =[
0.3 − − 0.4]
R′′ =
[0.6 0.70.8 0.4
]Where r′′ =0.7 which is greater than r′( r′′ > r′) so at x3rd position we have0.7
An =[
0.3 − 0.7 0.4]
At the end we left with
R′′′ =[
0.6]
and r′′′=0.6 which is less than r′′ so we will take 0.7 instead of 0.6 at x2thposition
A4 =[
0.3 0.7 0.7 0.4]
2.2.4 Third Method of the GEFS creation
We can also obtain GEFS by evaluating at most successive power of R in Max-Min composition sense. The following method is not simple but it is straightforward method to find GEFS.
Chapter 2. Eigen Fuzzy Sets 13
Steps
1. Produce A1 from R
2. Compose R with itself to get R2
R2 = RoR
and produced A2 from R2.
3. Compose R2 with R to get R3
R3 = R2oR
and produced A3 from R3.When one gets such that An = An+1 then An is accepted GEFS associatedwith R.
Example 2.3
R =
0.1 0.5 0.9 0.60.3 0.6 0.7 0.40.3 0.8 0.4 0.10.2 0.5 0.3 0.2
From R we get A1 = [0.3, 0.8, 0.9, 0.6]
composing R with R
R2 = RoR =
0.1 0.5 0.9 0.60.3 0.6 0.7 0.40.3 0.8 0.4 0.10.2 0.5 0.3 0.2
o
0.1 0.5 0.9 0.60.3 0.6 0.7 0.40.3 0.8 0.4 0.10.2 0.5 0.3 0.2
R2 =
0.3 0.8 0.5 0.40.3 0.8 0.6 0.40.3 0.6 0.7 0.40.3 0.5 0.5 0.4
From R2 we get A2 = [0.3, 0.8, 0.7, 0.4]
R3 = R2oR =
0.3 0.6 0.5 0.40.3 0.6 0.7 0.40.3 0.7 0.7 0.40.3 0.5 0.5 0.4
14 Chapter 2. Eigen Fuzzy Sets
where A3=[0.3, 0.7, 0.7 , 0.4]
R4 = R3oR =
0.3 0.7 0.6 0.40.3 0.7 0.6 0.40.3 0.7 0.7 0.40.3 0.5 0.5 0.4
A4 = A3 =
[0.3 0.7 0.7 0.4
]2.3 Least Eigen Fuzzy set [2]
To evaluate the least eigen fuzzy set of a relation R. There is a little change inthe Algorithm of GEFS
2.3.1 Algorithm
Let A relation R ⊆ X ×X with membership function µR(x, x′) is given
1. Find the set A1 defined by
µA1(x′) = minx∈X(x, x′), ∀x′ ∈ X
2. Set the index n=1
3. calculate An+1 = AnoR
4. An+1 = An→No→n=n+1→go−to−step3→yes→An=An+1
Example 2.4
Let continue with same relation R we used in previous examples
R =
0.1 0.5 0.9 0.60.3 0.6 0.7 0.40.3 0.8 0.4 0.10.2 0.5 0.3 0.2
We want to calculate LEFS, so take the smallest membership degree from eachcolumn of R to get
A1 =[
0.1 0.5 0.3 0.1]
For n=1 we create A2 by composing A1 with R
A2 = A1oR =[
0.1 0.5 0.3 0.1]o
0.1 0.5 0.9 0.60.3 0.6 0.7 0.40.3 0.8 0.4 0.10.2 0.5 0.3 0.2
=[
0.3 0.5 0.5 0.4]
Chapter 2. Eigen Fuzzy Sets 15
We find µA2(x1) by Max-Mix composition, as the quantity
µA2(x1) = max(min((0.1, 0.1),min(0.5, 0.3),min(0.3, 0.3),min(0.4, 0.2)))
= max(0.1, 0.3, 0.3, 0.2)
= 0.3
Since A2 6= A1, we set n=2 and compose R with A2 to get A3
A3 = A2oR =[
0.3 0.5 0.5 0.4]o
0.1 0.5 0.9 0.60.3 0.6 0.7 0.40.3 0.8 0.4 0.10.2 0.5 0.3 0.2
=[
0.3 0.5 0.5 0.4]
That satisfies the equality A3=A2. The set A3 accepted as the least eigen fuzzyset(LEFS) of relation R. We can see that
A1 ⊆ A2 ⊆ A3
which confirm the proper choice of least set.
Chapter 3
Eigen Fuzzy Sets with FuzzyNumbers
We have discussed about fuzzy number in chapter 1(1.4) and before discussingeigen fuzzy sets with fuzzy number we need to explain order operations on fuzzynumbers or comparison of two fuzzy numbers.
3.1 Minimum for two fuzzy Number
3.1.1 Definition
Let N1 = (mN1 , αN1 , βN1) and N2 = (mN2 , αN2 , βN2) are two fuzzy numbers, then
min(N1, N2) = (mN1 , αN1 , βN1)
If mN1 < mN2 and
supp(N1) ∩ supp(N2) = 0 (3.1)
or
min(N1, N2) = (min(mN1 ,mN2),max(αN1 , αN2),min(βN1 , βN2))
if mN1 6= mN2 or mN1 = mN2 and
supp(N1) ∩ supp(N2) 6= 0 (3.2)
Example 3.1
Let us consider N1 = (30, 2, 3) and N2 = (50, 1, 5). The supp(N1) = [28, 33] andsupp(N2) = [49, 55] have no common elements, we easily can compare and decidethat min(N1, N2) = N1 as shown in figure 3.1.
17
18 Chapter 3. Eigen Fuzzy Sets with Fuzzy Numbers
Figure 3.1: Minimum for N1 = (30, 2, 3) and N2 = (50, 1, 5) according to 3.1
Example 3.2
Again let us consider N1 = (30, 2, 3) and N2 = (32, 1, 4) so we have supp(N1) =[28, 33] and supp(N2) = [31, 36]. Here we can see from the formula (3.2)
supp(N1) ∩ supp(N2 6= 0 (3.3)
so we decide
min(N1, N2) = (min(30, 32),max(2, 1),min(33, 36)) = (30, 2, 33)
Figure 3.2 shows the result of the last operation. In the same way we can deter-
Figure 3.2: Minimum for N1 = (30, 2, 3) and N2 = (32, 1, 4)
mine the largest fuzzy number chosen for two number from the pair (N1, N2).
Chapter 3. Eigen Fuzzy Sets with Fuzzy Numbers 19
3.2 Maximum for Two fuzzy numbers
3.2.1 Definition
Let N1 = (mN1 , αN1 , βN1) and N2 = ((mN2 , αN2 , βN2) are two fuzzy numbers, then
max(N1, N2) = (mN1 , αN1 , βN1)
if mN1 > mN2 and
supp(N1) ∩ supp(N2) = 0 (3.4)
or
max(N1, N2) = (max(mN1 ,mN2),min(αN1 , αN2),max(βN1 , βN2))
if mN1 6= mN2 or mN1 = mN2 and
supp(N1) ∩ supp(N2) 6= 0 (3.5)
Example 3.3
Consider two fuzzy numbers as we did in example 3.1,
Figure 3.3: Maximum for N1 = (30, 2, 3) and N2 = (32, 1, 4)
Example 3.4
Again we set N1 = (30, 2, 3) and N2 = (32, 1, 4) and applying formula 3.4 we get
max(N1, N2) = (32, 1, 4) (3.6)
20 Chapter 3. Eigen Fuzzy Sets with Fuzzy Numbers
Figure 3.4: Maximum for N1 = (30, 2, 3) and N2 = (32, 1, 4)
3.3 Eigen Fuzzy set with Fuzzy Numbers [2]
By using the operations on fuzzy numbers that we discussed in (3.1), (3.2) and(3.3)-(3.4) we can propose a new concept of composition of relation with a fuzzyset. But in this case the relation and the set both will have membership degreesformed as fuzzy numbers in (L-R) form.
3.3.1 Definition
As we know the eigen fuzzy set of fuzzy relation R ⊆ X × X is a set A ⊆ X,X = {x} that satisfies the condition AoR = A.R is the fuzzy relation with membership function µR : X × X → FN(LR),µR(x, x′) ∈ FN(LR), x, x′ ∈ X. Now we also want to prove that the greatesteigen fuzzy set (with fuzzy Numbers) A ⊆ X of relation R exists, where µA :X → FN(LR), µA(x) ∈ FN(LR), x ∈ X. Theoretically we need to follow allthe step that we described in 2.2.2, so first we will verify that set A0 with itsmembership function defined by
µA0(x) = (mµA0(x), αµA0
(x), βµA0(x)) = (mN0 , αN0 , βN0) : ∀x ∈ X
where
(mN0 , αN0 , βN0) = minx′∈X(max(mµR(x,x′), αµR(x,x′), βµR(x,x′)))
is an eigen set of R. When we compose A0 with R we notice that
µAoR(x′) = µA0(x′)
Chapter 3. Eigen Fuzzy Sets with Fuzzy Numbers 21
so(mN0 , αN0 , βN0)
is a constant fuzzy number.Further we define A1 by its membership function
µA1(x′) = (mµA1
(x′), αµA1(x′), βµA1
(x′)) = maxx∈XµR(x, x′)
= maxx∈X(mµR(x,x′), αµR(x,x′), βµR(x,x′))
For all x′ ∈ X, and we introduce the sequence (An)n of fuzzy numbers
A2 = A1oR = A1oR1
A3 = A2oR = A1oR2
A4 = A3oR = A1oR3
...
An+1 = AnoR = A1oRn
for all integers n > 1, and we conclude
A0 ⊆ . . . ⊆ An+1 ⊆ An . . . ⊆ A2 ⊆ A1
and it can be compared with (2.2.2).
Example 3.5
Let us consider a fuzzy relation R that have membership degrees formed as fuzzynumber in L-R form
R =
(5, 2, 6) (40, 1, 3) (50, 1, 5)(27, 1, 7) (7, 4, 5) (15, 2, 3)(10, 4, 5) (20, 2, 5) (35, 1, 3)
Set A1 will consist of greatest fuzzy number from each column
A1 =[
(27, 1, 7) (40, 1, 3) (50, 1, 5)]
A2 = A1oR =[
(27, 1, 7) (27, 1, 7) (35, 1, 3)]
A3 = A2oR =[
(27, 1, 7) (27, 1, 7) (35, 1, 3)]
A2 = A3 so A3 is accepted as greatest eigen fuzzy set with fuzzy numbers.
Chapter 4
Applications of Eigen Fuzzy sets
Eigen fuzzy set theory is applicable in many areas of sciences and engineering,for example we can find the effectiveness level of a medicine by using eigen fuzzysets [2] and can find customers satisfaction level about a particular product.
4.1 Example: Mobile users satisfaction
By applying eigen fuzzy set theory we can find maximum and minimum satis-faction level of (particular model) mobile customers or users. In this example Ichoose Nokia Express Music 5800 and asked from 10 users about different featuresof this model These features are listed as:
• F1 = User friendly or usability
• F2 = Touch screen response
• F3 = Battery timing
let we denote the set of features by F as:
F = {F1, F2, F3}
The estimation of the maximum and minimum level is possible by employinga fuzzy relation, created due course to the definition formulated by a sentence: “ The satisfaction of the jth feature is equal or stronger than kth feature ina customer j, k = 1, . . . n. Each pair of relation Rmax(Fj ,FK) has a membershipdegree from range [0, 1] , indicated the grade to which the statement definingRmax is true for jth and kth feature [?].
we use formula
µRmax(Fj, Fk) =b
m(4.1)
where j, k = 1, . . . , n
23
24 Chapter 4. Applications of Eigen Fuzzy sets
Table 4.1: Sign configurations for features F1, F2 in U1 to U10
Users F1 F2
U1 + -U2 + -U3 + +U4 + -U5 + -U6 - -U7 - +U8 + +U9 + -U10 - +
Table 4.2: Sign configurations for features F1, F3 in U1 to U10
Users F1 F3
U1 + +U2 + +U3 + -U4 - +U5 - +U6 + -U7 + -U8 - +U9 - -U10 - -
where m is total number of users in this case m=10 and b stands for a numberof users for whom for our example 1 asked from 10 users about 3 different featuresof Nokia express music 5800. suppose that ′′−′′ is assigned for low satisfactionand ′′+′′ is for good or
greater satisfaction is to count b, we should consider two configuration pat-terns of these signs. ” + ”, ” + ” that is interpreted that customer is fully satisfiedwith both the features. ”+”, ”−”that means the customer is more satisfied withfeature 1(F1) thanF2. So the membership degree of the pair (F1, F2) = 7
10= 0.70
and membership degree of pair (F2, F1) = 410
= 0.40From the table 4.2 membership degree of pair (F1, F3) = 5
10= 0.5 and member-
ship degree of pair (F3, F1) = 510
= 0.5From the table 4.3 membership degree of pair (F2, F3) = 3
10= 0.3 and member-
ship degree of pair (F3, F2) = 810
= 0.8. The fuzzy relation Rmax can be written
Chapter 4. Applications of Eigen Fuzzy sets 25
Table 4.3: Sign configurations for features F2, F3 in U1 to U10
Users F2 F3
U1 - +U2 - +U3 - +U4 + +U5 - -U6 - +U7 - -U8 + +U9 - +U10 + +
asF1 F2 F3
Rmax =F1
F2
F3
µRmax(F1, F1) µRmax(F1, F2) µRmax(F1, F3)µRmax(F2, F1) µRmax(F2, F2) µRmax(F2, F3)µRmax(F3, F1) µRmax(F3, F2) µRmax(F3, F3)
Now putting all values of membership degree in above matrix and for the
diagonal entry µRmax(F1, F1),µRmax(F2, F2) and µRmax(F3, F3) we will estimate bas a total number of ”+” signs and taking its average,so we have
µRmax(F1, F1) =1
2[
7
10+
5
10] = 0.6 (4.2)
µRmax(F2, F2) =1
2[
4
10+
3
10] = 0.35 (4.3)
µRmax(F3, F3) =1
2[
5
10+
8
10] = 0.65 (4.4)
Now setting all these values in Matrix we get:
Rmax =
0.6 0.7 0.50.4 0.35 0.30.5 0.8 0.65
For GEFS we produce
A1 =[
0.6 0.8 0.65]
using algorithm 2.2.2 :
A2 = A1oR =[
0.6 0.65 0.65]
26 Chapter 4. Applications of Eigen Fuzzy sets
A3 = A2oR =[
0.6 0.65 0.65]
A2 = A3 so A3 is accepted as the greatest eigen fuzzy set. In the same way wecan find Rmin by calculating membership degree from (4.1) with the association′′+′′ ′′+′′ and ′′−′′ ′′+′′ by the value of m and b. so
Rmin =
0.6 0.4 0.50.7 0.35 0.80.5 0.3 0.65
For least eigen fuzzy set we produce
A1 =[
0.5 0.3 0.5]
A2 = A1oRmin =[
0.5 0.4 0.5]
A3 = A2oRmin =[
0.5 0.4 0.5]
A3 is least eigen fuzzy set.By interpreting the membership degrees of Amin and Amax in percentage scale,we can say that the customer’s satisfaction level of Nokia Express music 5800 isabout F1 50% - 60% and F2 40 % - 65 % and in F3 min satisfaction 50 % andmax is 65 %. We can verify that RT
max = Rmin
4.2 A Genetic Algorithm Based on Eigen Fuzzy
Sets for Image Reconstruction [7–9]
Fuzzy relation calculus for image compression is a natural tool for a geneticalgorithm based on eigen fuzzy sets for image reconstruction. If we normalize thevalues of the pixels of any image (of size m×m ) with respect to the length of thegray scale used, it can be interpreted as a square fuzzy relation R. For instance,the usage of the greatest eigen fuzzy set of R w.r.t. the maxmin composition(GEFS) and the smallest eigen fuzzy set of R w.r.t. the minmax composition(LEFS) are generally applied to problems of image information retrieval, imageanalysis and image reconstruction. Genetic algorithms (GA) can be used forcoding/decoding images.
The GA method encodes a potential solution to a specific problem on a simplechromosome like data structures and applies recombination operators to thesestructures. Genetic algorithms are often viewed as optimization functions al-though the range of problems to which they have been applied is quite broad.The GA approach is used for the reconstruction of an image by using its GEFS
Chapter 4. Applications of Eigen Fuzzy sets 27
and SEFS in the fitness function of a chromosome. It is well known that theprincipal step in developing a GA is the definition of a solution coding: in otherwords, a chromosome must contain information about the solution that it repre-sents. In the case of random images; the gene of a chromosome is a pixel. Theallele value of a gene is an integer value in the set X = {0, 1, . . . , 255} since weare giving the example of gray images of sizes 256×256.
The two types of eigen fuzzy set, the max-min and min-max compositionsare used to calculate the fitness value in a GA used for image reconstructionscopes. The GA is mostly applied over many random gray images by using severalpopulation dimensions and generation numbers and assumed as reconstructedimage that one with the greatest value of fitness that is having GEFS and SEFSvery close to those ones of the original image.
4.3 Conclusion
The results of this thesis study and survey showed that how the highest customersatisfaction level can be measured by applying eigen fuzzy set theory. By the useof these concepts of eigen fuzzy sets of fuzzy relation, we reached the results thathow customer satisfaction can be increased for any product, and which attributesof any product quality needs to improve.
In future the eigen fuzzy sets theory can be used for image enhancementtechniques to improve perceptual image quality as well as it could be used forspeech enhancement and quality. Also these methods of eigen fuzzy sets canbe used for improving the quality of any product based on the statistical dataachieved by any mean rather than approaching through scientific research.
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[3] E. Sanchez, “Resolution of eigen fuzzy set equations.” Fuzzy Sets and Sys-tems 1, 1978, pp. 69–74.
[4] G. R and V. W, “Eigen fuzzy number sets,” in Fuzzy Sets and Systems, vol. 16,1985, pp. 75–85.
[5] E. Sanchez, “Eigen fuzzy sets and fuzzy relations,” Journal of MathematicalAnalysis and Applications, vol. 81, pp. 399–421, 1981.
[6] R. E. Gerstenkorna, T., “An application of fuzzy set theory to differentiatingthe effectiveness of drugs in treatment of inflammation of genital organs.”Fuzzy Sets and Systems, vol. 68, pp. 327–333, 1994.
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[8] S. S. F. Di. Martino and H. Nobuhara, “Eigen fuzzy sets and image informa-tion retrieval,” IEEE, pp. 1385–1390, 2004.
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