we study to find out the major causes that are hindering/affecting the happiness in family using Combined Disjoint Block Fuzzy Cognitive Maps (CDBFCM). We find that happiness is strongly correlated with perceived good health. This method is introduced by W.B. Vasantha Kandasamy, is analyzed in this paper. The Combined Disjoint Block FCM is defined in this method becomes effective when the number of concepts can be grouped and are in large numbers. This paper has five sections. First section gives the information about development of Fuzzy Cognitive Maps and happiness in the family life. Second Section gives basic notations and definitions of Fuzzy Cognitive maps and Combined Disjoint Block Fuzzy Cognitive Maps. In Section three, we explain method of determining the hidden pattern. In the fourth section, we give the concepts of problem. Final section gives the conclusion based on our study and a brief discussion of implications for further researches close the paper.

1. Integrated Intelligent Research(IIR) International Journal of Business Intelligent
Volume: 03 Issue: 02 December 2014,Pages No.23- 27
ISSN: 2278-2400
23
The Role of Happiness in the Family Using
Combined Disjoint Block Fuzzy Cognitive Maps
(CDBFCMS)
A.Victor Devadoss1
, M. Clement Joe Anand2
, A. Anthuvan Rozario3
, M. Sagaya Bavia4
1
Head & Associate Professor, Department of Mathematics, Loyola College, Chennai-34, India
2
Ph.D Research Scholar, Department of Mathematics, Loyola College, Chennai-34, India
3,4
M.Sc Mathematics, Department of Mathematics, Loyola College, Chennai-34, India
Email:hanivictor@ymail.com, arjoemi@gmail.com, rozariosvd@gmail.com, sagayam24@ymail.com
Abstract-In this paper, we study to find out the major causes
that are hindering/affecting the happiness in family using
Combined Disjoint Block Fuzzy Cognitive Maps (CDBFCM).
We find that happiness is strongly correlated with perceived
good health. This method is introduced by W.B. Vasantha
Kandasamy, is analyzed in this paper. The Combined Disjoint
Block FCM is defined in this method becomes effective when
the number of concepts can be grouped and are in large
numbers. This paper has five sections. First section gives the
information about development of Fuzzy Cognitive Maps and
happiness in the family life. Second Section gives basic
notations and definitions of Fuzzy Cognitive maps and
Combined Disjoint Block Fuzzy Cognitive Maps. In Section
three, we explain method of determining the hidden pattern. In
the fourth section, we give the concepts of problem. Final
section gives the conclusion based on our study and a brief
discussion of implications for further researches close the
paper.
Key Words: Combined Disjoint Fuzzy Cognitive Maps,
Family, Fuzzy Cognitive Maps, Happiness, Health.
I. INTRODUCTION
A mathematical model called Fuzzy Cognitive Maps,
introduced by L.A. Zadeh in the year 1965 and Political
scientist R. Axelord in the year 1976, is used to study decision
making in social and political systems.FCMs can successfully
represent knowledge and human experience, introduced
concepts to represent the essential elements and the cause and
effect relationships among the concepts to model the behavior
of any system. It is a very convenient, simple and powerful
tool, which is used in numerous fields such as social,
economical, medical and so on.A family (from Latin: familia),
in human context, is a group of people affiliated by
consanguinity, affinity, or co-residence. In most societies it is
the principal institution for the socialization of children.
Anthropologists most generally classify family organization as
matrilocal (a mother and her children), conjugal (a wife,
husband, and children, also called nuclear family), and
consanguinal (also called an extended family) in which parents
and children co-reside with other members of one parent's
family. Happiness is a fuzzy concept and can mean many
things to many people. Part of the challenge of a science of
happiness is to identify different concepts of happiness, and
where applicable, split them into their components.Emotional
states such as happiness and attitudes towards life are seen as a
key determinant of feelings of stress and anxiety related to life
events of family. Findings from medicine and psychology have
shown that emotional reactions to life events can affect
physiology in ways that are potentially damaging or beneficial
for health. Happiness is a mental or emotional state of well-
being characterized by positive or pleasant emotions ranging
from contentment to intense joy. In recent years, a number of
studies have advanced the claim that happiness, more
generally, positive attitudes towards life can predict longevity
and other indicators of physical well-being among healthy
populations. Happiness forms a central theme of Buddhist
teachings. In Christianity, the ultimate end of human existence
consists in felicity. Human complexities, like reason and
cognition, can produce well-being or happiness, but such form
is limited and transitory. In temporal life, the contemplation of
God, the infinitely Beautiful, is the supreme delight of the will.
Perfect happiness, as complete well-being, is to be attained not
in this life, but the next.
II. BASIC NOTATION AND DEFINITIONS
Fuzzy Cognitive Maps (FCMs) are more applicable when the
data in the first place is an unsupervised one. The FCMs work
on the opinion of experts. FCMs model the world as a
collection of classes and causal relations between classes.
2.1 Definition
When the nodes of the FCM are fuzzy sets then they are called
as fuzzy nodes.
2.2 Definition
FCMs with edge weights or causalities from the set {-1, 0, 1}
are called simple FCMs.

2. Integrated Intelligent Research(IIR) International Journal of Business Intelligent
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2.3 Definition
An FCMs is a directed graph with concepts like policies,
events etc, as nodes and causalities as edges, It represents
causal relationships between concepts.
2.4 Definition
Consider the nodes/concepts C1, C2,…, Cn of the FCM.
Suppose the directed graph is drawn using edge weight eij ∈ {-
1, 0, 1}. The matrix E be defined by E = (eij) where eij is the
weight of the directed edge CiCj. E is called the adjacency
matrix of FCM, also known as the connection matrix of the
FCM.
It is important to note that all matrices associated with an FCM
are always square matrices with diagonal entries as zero.
2.5 Definition
Let C1, C2,…, Cn be the nodes of an FCM.
A=(a1, a2,…,an) where eij ∈ {-1, 0, 1}. A is called the
instantaneous state vector and it denotes the on-off position of
the node at an instant. ai = 0 if ai is off and ai = 1 if ai is on for i
= 1, 2,…, n.
2.6 Definition
Let C1, C2,…, Cn be the nodes of and FCM. Let
1 2 2 3 3 4
, , ,..., i j
C C C C C C C C be the edges of the FCM (i≠j).
Then the edges form a directed cycle. An FCM is said to be
cyclic if it possesses a directed cycle. An FCM is said to be
acyclic if it does not possess any directed cycle.
2.7 Definition
An FCM is said to be cyclic is said to have a feedback.
2.8 Definition
When there is a feedback in an FCM, i.e, when the causal
relations flow through a cycle in a revolutionary way, the FCM
is called a dynamical system.
2.9 Definition
Let 1 2 2 3 3 4 1
, , ,..., n n
C C C C C C C C
 be a cycle. When Ci is
switched on and if the causality flows through the edges of a
cycle and if it again causes Ci, we say that the dynamical
system goes round and round. This is true for any node Ci for i
=1,2,…,n. The equilibrium state for this dynamical system is
called the hidden pattern.
2.10 Definition
If the equilibrium state of a dynamical system is a unique state
vector, then it is called a fixed point. Consider a FCM with C1,
C2,…, Cn as nodes. For example let us start the dynamical
system by switching on C1. Let us assume that the FCM settles
down with C1 and Cn on i.e., in the state vector remains as (1,
0, 0,…, 0) is called fixed point.
2.11 Definition
If the FCM settles down with a state vector repeating in the
form A1→A2→…→Ai→A1 then this equilibrium is called a
limit cycle.
2.12 Definition
Finite number of FCMs can be combined together to produce
the point effect of all the FCMs. Let E1, E2,…,Ep be the
adjacency matrices of the FCMs with nodes C1, C2,…, Cn then
the combined FCM is got by adding all the adjacency matrices
E1, E2,…, Ep. We denote the combined FCM adjacency matrix
by E = E1+E2+…+Ep.
2.13 Definition
Let C1, C2,…, Cn be n distinct attributes of a problem n very
large and a non prime. If we divide n into k equal classes i.e.,
k/n=t which are disjoint and if we find the directed graph of
each of these k classes of attributes with t attributes each, then
their corresponding connection matrices are formed and these
connection matrices are joined as blocks to form a n x n
matrix. This n x n connection matrix forms the combined
disjoint block FCM of equal classes. If the classes are not
divided to have equal attributes but if they are disjoint classes
we get a n x n connection matrix called the combined disjoint
block FCM of unequal classes/size.
2.14 Definition
Suppose A = (a1, a2,…,an) is a vector which is passed into a
dynamical system E. Then AE = (a1’, a2’,…, an’) after
thresholding and updating the vector suppose we get (b1,
b2,…,bn) we denote that by (a1’, a2’,…, an’)  (b1, b2,…,bn).
Thus the symbol ‘ ’ means the resultant vector has been
threshold and updated.
FCMs have several advantages as well as some disadvantages.
The main advantages of this method it is simple. It functions
on expert’s opinion. When the data happens to be an
unsupervised one the FCM comes handy. This is the only
known fuzzy technique that gives the hidden pattern of the
situation. As we have a very well known theory, which states
that the strength of the data depends on, the number of
experts’s opinions. At the same time the disadvantages of the
combined FCM is when the weightages are 1 and -1 for the
same CiCj, we have the sum adding to zero thus at all times the
connection matrices E1,E2,…,Ek may not be conformable for
addition.
Combined conflicting opinions tend to cancel out and assisted
by the strong law of large numbers, a consensus emerges as the
sample opinion approximates the underlying population
opinion. This problem will be easily overcome if the FCM
entries are only 0 and 1.
III. METHOD OF DETERMINING THE HIDDEN
PATTERN
Let C1, C2,…, Cn be the nodes of an FCM, with feedback, Let

3. Integrated Intelligent Research(IIR) International Journal of Business Intelligent
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E be the associated adjacency matrix. Let us find the hidden
pattern when C1 is switched on. When an input is given as the
vector A1 = (1, 0,…, 0), the data should pass through the
relation matrix E. This is done by multiplying Ai by the matrix
E. Let AiE = (a1, a2,…, an) with the threshold operation that is
by replacing ai by 1 if ai > k and ai by 0 if ai < k ( k is a suitable
positive integer). We update the resulting concept; the concept
C1 is included in the updated vector by making the first
coordinate as 1 in the resulting vector. Suppose AiE→ A2 then
consider A2E and repeat the same procedure. This procedure is
repeated till we get a limit cycle or a fixed point.
4.Concepts of the Problem
Using the linguistic questionnaire and the expert‘s opinion we
have taken the following sixteen attributes {A1, A2,…, A16}.
A1 - Loneliness
A2 - Frustration
A3 - Economic conditions
A4 - Conflicting thoughts / misunderstanding
A5 - Social and religious values
A6 - Faithfulness /fidelity
A7 - Physical Illness (heart attack, diabetics, ulcer, etc)
A8 - Loss of dear ones /significant other
A9 - Faultlessness
A10 - Negligence
A11 - High expectations
A12 - Family pressure
A13 - Family Status /background
A14 - Child rearing issues
A15 - Lack of tolerance
A16 - Adamant character
These 16 attributes are divided into 4 classes C1, C2, C3 and C4
with 4 in each class. Let C1 = {A1, A7, A12, A13}, C2 = {A3, A5,
A11, A16}, C3 = {A2, A6, A8, A14}, and C4 = {A4, A9, A10, A15}.
Now we take the expert opinion for each of these classes and
take the matrix associated with the combined disjoint block
FCMs. The experts opinion for the class C1 = {A1, A7, A12,
A13} in the form of the directed graph.
A1 A7
A12
A13
According to this expert, the attributes are Loneliness, Physical
Illness (heart attack, diabetics, ulcer, etc), Family pressure, and
Family Status /background. The related connection matrix M1
is given by
A1 A7 A17 A13
1
7
17
13
0 0 0 1
1 0 1 1
1 1 0 1
1 0 1 0
A
A
A
A
é ù
ê ú
ê ú
ê ú
ê ú
ê ú
ê ú
ë û
The directed graph is given by the expert on {A3, A5, A11, A16}
which forms the class C2.
A3 A5
A16
A11
According to this expert, the attributes are Economic
conditions, Social and religious values, High expectations and
Adamant character. The related connection matrix M2 is given
below:
A3 A5 A11 A16
3
5
11
16
0 1 1 1
0 0 1 0
1 0 0 0
1 0 0 0
A
A
A
A
é ù
ê ú
ê ú
ê ú
ê ú
ê ú
ê ú
ë û
Now we give the directed graph for the class C3 as given by the
expert C3 = {A2, A6, A8, A14}
A2 A6
A14
A8
According to this expert, the attributes are Frustration,
Faithfulness /fidelity, Loss of dear ones /significant other and
Child rearing issues. The related connection matrix M3 is given
below:
A2 A6 A8 A14
2
6
8
14
0 0 0 0
1 0 0 0
1 0 0 0
1 0 0 0
A
A
A
A
é ù
ê ú
ê ú
ê ú
ê ú
ê ú
ê ú
ë û

4. Integrated Intelligent Research(IIR) International Journal of Business Intelligent
Volume: 03 Issue: 02 December 2014,Pages No.23- 27
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The directed graph is given by the expert on {A4, A9, A10, A15}
which forms the class C4
A4 A9
A15
A10
According to this expert, the attributes are the expert
Conflicting thoughts misunderstanding, Faultlessness,
Negligence, and Lack of tolerance.
A4 A9 A10 A15
4
9
10
15
0 0 1 1
1 0 0 1
1 1 0 1
0 0 1 0
A
A
A
A
é ù
ê ú
ê ú
ê ú
ê ú
ê ú
ê ú
ë û
Now Combined Disjoint Block connection matrix of the fuzzy
cognitive maps B is given by
A1 A7 A12 A13 A3 A5 A11 A16 A2 A6 A8 A14
A4 A9 A10 A15
1
7
12
13
3
5
11
16
2
6
8
14
4
9
10
15
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Suppose we consider the ON state of the attribute loneliness
and all other states are OFF the effect of X =
(1000000000000000) on the CDBFCM is given by
XB = (1001000000000000) = X1 (Say)
X1B = (1011000000000000) = X2 (Say)
X2B = (1111000000000000) = X3 (Say)
X3B = (1111000000000000) = X4
X3 is a fixed point of the dynamical system. Thus when one
experiences loneliness in the family, he/she gets frustrated.
Misunderstanding and conflicting thoughts are also very much
present in the person. Economic conditions of the person or
family leads the person to feel loneliness in the human
society.Suppose we consider the on state of the attributes social
and religious values and the loss of dear ones or significant
other and all other nodes are in the off state. Now we study the
effect on the dynamical system B.Let T =
(0000100100000000) state vector depicting social and
religious values and the loss of dear ones or significant other,
passing the state vector T into the dynamical system B.
TB = (0000111100000000) = T1 (Say)
T1B = (0000111100000000) = T2
Here T1 is a fixed point of the dynamical system. Thus social
and religious values lead the person to be faithful in the family
and the experience of the loss of dear ones or significant other
leads to physical illnesses. When the social and religious
values are not followed, fidelity to the life partner and children
is lost.
Suppose we consider the ON state of the attributes conflicting
thoughts / misunderstanding, physical illness, family pressure
and family background and all other nodes are in the OFF
state. Now we study the effect on the dynamical system B. Let
G = (0001001000011000) state vector depicting conflicting
thoughts / misunderstanding, physical illness, family pressure
and family background, passing the state vector G into the
dynamical system B.
GB = (1011101010011011) = G1 (Say)
G1B = (1111111110011111) = G2 (Say)
G2B = (1111111110011111) = G3
Here G2 is a fixed point of the dynamical system. Thus
conflicting thoughts / misunderstanding leads the person to
experience loneliness and frustration though all are present in
the family. This shows us the lack of tolerance among the
family people. Faithfulness and fidelity is lost once the
misunderstanding takes the upper hand in the family. The
physical illnesses scar the economic conditions of the family.
Child rearing issues give pressure to the family.
IV. CONCLUSION
We analyzed what are the causes that are hindering/affecting
the happiness of the family using CDBFCM model. The limit
point of the dynamical system reveals that the attributes A1, A2,
A3, A4, A5, A6, A7, A8, A11, A12, A13, A14, A15, and A16 are the
major causes that are hindering/affecting the happiness of the
family. This means, loneliness, frustration, economic
conditions, conflicting thoughts / misunderstanding, social and
religious values, faithfulness /fidelity, physical Illness (heart
attack, diabetics, ulcer, etc), loss of dear ones /significant other,
high expectations, family pressure, family Status /background,
child rearing issues, lack of tolerance and adamant character.
From the above observation and study, we suggest that in
happy families, husbands and wives do not stop being a couple
once they become a mother and a father. Successful families
recognize and accept that getting angry with each other is
normal. They know that a bunch of people of different ages
living under one roof are bound to get on each other's nerves
now and then, so they are quick to forgive and forget and to

5. Integrated Intelligent Research(IIR) International Journal of Business Intelligent
Volume: 03 Issue: 02 December 2014,Pages No.23- 27
ISSN: 2278-2400
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make up and apologize. Happy families talk to each other as
they just do. They find time to have discussions and time to
have family meetings. They listen to each other and they
express their feelings to each other. Families that eat together,
stay together. It's that simple. Family dinners are essential and
it's a time to connect. Have a minimum of four family dinners
per week. Have one or two unifying activities that the family
does together on a nightly basis. We suggest bedtime stories
for young children or reading a chapter from a novel to an
older child. Pacing and timing of events can make a world of
difference for older relatives. Let us not forget that a small
modifications can make a big difference.
ACKNOWLEDGEMENTS
We are indebted to Dr. T. Bharathi, Associate Professor,
Department of Mathematics, Loyola College for her timely
guidance and encouragement in presenting this paper.
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