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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 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 hanivictor@ymail.com, arjoemi@gmail.com, rozariosvd@gmail.com, sagayam24@ymail.com 1

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. 1. 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 International Journal of Business Intelligents (IJBI) www.ijbui.com

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 229

Indo-Bhutan International Conference On Gross National Happiness Pages: 229-235 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. 2.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. 2.3 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 International Journal of Business Intelligents (IJBI) www.ijbui.com

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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 C1C2 , C2 C3 , C3C4 ,..., Ci C j 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 C1C2 , C 2C3 , C3C 4 ,..., Cn 1Cn 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.

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Indo-Bhutan International Conference On Gross National Happiness Pages: 229-235 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 International Journal of Business Intelligents (IJBI) www.ijbui.com

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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. 3.Method of determining the hidden pattern Let C1, C2,…, Cn be the nodes of an FCM, with feedback, Let 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 231

Indo-Bhutan International Conference On Gross National Happiness Pages: 229-235 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.

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

A1 0 A7 1 A17 1 A13 1

0 0 1 0 1 1 1 0 1 0 1 0

Vol 02, October 2013 A3 A5 A11 A16

A3 0 A5 0 A11 1 A16 1

1 0 0 0

1 1 0 0

1 0 0 0

Now we give the directed graph for the class C3 as given by the expert C3 = {A2, A6, A8, A14}

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

0 1 A8 1 A14 1

A2 A6

0 0

0 0

0 0

0 0

0 0 0 0

The directed graph is given by the expert on {A4, A9, A10, A15} which forms the class C4

The directed graph is given by the expert on {A3, A5, A11, A16} which forms the class C2.

According to this expert, the attributes are the expert Conflicting thoughts misunderstanding, Faultlessness, Negligence, and Lack of tolerance. A4 A9 A10 A15 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: International Journal of Business Intelligents (IJBI) www.ijbui.com

A4 0 A9 1 A10 1 A15 0

0 0 1 0

1 0 0 1

1 1 1 0

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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

A1 0 A7 1 A12 1 A13 1 A3 0 A5 0 A11 0 A16 0 A2 0 A6 0 A8 0 A14 0 A4 0 A9 0 A10 0 A15 0

0 0 1 0 0 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

1 1 1 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 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 0 0 1 0 0 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 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 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 0 0 0 1 0

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 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. International Journal of Business Intelligents (IJBI) www.ijbui.com

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 233

Indo-Bhutan International Conference On Gross National Happiness Pages: 229-235 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. 5 Conclusion and Suggestions 5.1 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. 5.2 Suggestions 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 International Journal of Business Intelligents (IJBI) www.ijbui.com

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forgive and forget and to 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. 6. 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. References [1] Fabio Sabatini, “The relationship between happiness and health: evidence from Italy”, Health, Econoomic and Data Group (HEDC), May 2011. [2] W. B. Vasantha Kandasamy and Smarandache Florentin, “Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps”, Indian Institute of Technology, Madras, 2003. [3] W. B. Vasantha Kandasamy, Florentin Smarandache, and K. Ilanthenral, “Applications of Bimatrices to Some Fuzzy and Neutrosophic Models”, HEXIS Phoenix, Arizona, 2005. [4] Klir George J. and Floger Tina A., “Fuzzy Sets, Uncertainty, and Information”, PHI Learning Private Limited, New Delhi, 2010. [5] Kosko, “Fuzzy Cognitive Maps‖, International Journal of man-machine studies”, January, 1988, 62-75. [6] A.Victor Devadoss, M.Clement Joe Anand, “A Solution to Control Suicide in the Domestic Violence using Combined Disjoint Block Fuzzy 234

Indo-Bhutan International Conference On Gross National Happiness Pages: 229-235 Cognitive Maps (CDBFCM)”, International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June- 2012. [7] W. B. Vasantha Kandasamy and Smarandache Florentin, “Analysis of social aspects of migrant labours living with HIV/AIDS using Fuzzy Theory and Neutrosophic Cognitive Maps”, Xi-quan, Phoenix, 2004. [8]A. Victor Devadoss, M. Clement Joe Anand, A. Felix, “A Study on the Impact of

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Violent Video-Games playing among Children in Chennai using Neutrosophic Cognitive Maps (NCMs)”, International Journal of Scientific & Engineering Research, Volume 3, Issue 8, August-2012. [9]A. Victor Devadoss, M. Clement Joe Anand, A. Felix, “A Study of Divorce using Combined Overlap Block Neutrosophic Cognitive Maps (COBNCMs)”, International Journal of Computer Information Systems, Vol. 5, No.1, 2012.

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