Diese Präsentation wurde erfolgreich gemeldet.

# Using New Triangular Fuzzy Cognitive Maps (TRFCM) to Analyze Causes of Divorce in Family

Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige   Anzeige
×

1 von 6 Anzeige

# Using New Triangular Fuzzy Cognitive Maps (TRFCM) to Analyze Causes of Divorce in Family

Our main objective of this paper is to find out the new fuzzy tool called Triangular Fuzzy Cognitive Maps to analyze the social problem. Usually in FCM we analyze the causes and effects of the relationships among the concepts to model the behavior of any system. But this new model gives the causes and effect of the relationships among the concepts to model behavior with ranking of any system. In this paper, we analyze the divorce problem using Triangular Fuzzy Cognitive Maps, it has five sections. In first section, we give the brief introduction to Fuzzy Cognitive Maps (FCM), Section two gives the basic definitions of FCM. In section three we derive the definitions for Triangular Fuzzy Cognitive Maps (TrFCM) and Hidden pattern of the dynamical system. In Fourth section we analyzed the concept of the problem using Triangular Fuzzy Cognitive Maps (TrFCM).In Final section we give the conclusion based on our study.

Our main objective of this paper is to find out the new fuzzy tool called Triangular Fuzzy Cognitive Maps to analyze the social problem. Usually in FCM we analyze the causes and effects of the relationships among the concepts to model the behavior of any system. But this new model gives the causes and effect of the relationships among the concepts to model behavior with ranking of any system. In this paper, we analyze the divorce problem using Triangular Fuzzy Cognitive Maps, it has five sections. In first section, we give the brief introduction to Fuzzy Cognitive Maps (FCM), Section two gives the basic definitions of FCM. In section three we derive the definitions for Triangular Fuzzy Cognitive Maps (TrFCM) and Hidden pattern of the dynamical system. In Fourth section we analyzed the concept of the problem using Triangular Fuzzy Cognitive Maps (TrFCM).In Final section we give the conclusion based on our study.

Anzeige
Anzeige

### Using New Triangular Fuzzy Cognitive Maps (TRFCM) to Analyze Causes of Divorce in Family

1. 1. Integrated Intelligent Research(IIR) International Journal of Business Intelligent Volume: 03 Issue: 01 June 2014,Pages No.4-9 ISSN: 2278-2400 4 Using New Triangular Fuzzy Cognitive Maps (TRFCM) to Analyze Causes of Divorce in Family M.Clement Joe Anand1 , A.Victor Devadoss2 1 Ph.D Research Scholar, Department of mathematics, Loyola College, Chennai- 34, India. 2 Head & Associate Professor, Department of mathematics, Loyola College, Chennai- 34, India. E-mail: arjoemi@gmail.com , hanivictor@ymail.com Abstract-Our main objective of this paper is to find out the new fuzzy tool called Triangular Fuzzy Cognitive Maps to analyze the social problem. Usually in FCM we analyze the causes and effects of the relationships among the concepts to model the behavior of any system. But this new model gives the causes and effect of the relationships among the concepts to model behavior with ranking of any system. In this paper, we analyze the divorce problem using Triangular Fuzzy Cognitive Maps, it has five sections. In first section, we give the brief introduction to Fuzzy Cognitive Maps (FCM), Section two gives the basic definitions of FCM. In section three we derive the definitions for Triangular Fuzzy Cognitive Maps (TrFCM) and Hidden pattern of the dynamical system. In Fourth section we analyzed the concept of the problem using Triangular Fuzzy Cognitive Maps (TrFCM).In Final section we give the conclusion based on our study. Keywords: Fuzzy Cognitive Maps (FCMs), Triangular Fuzzy Numbers, Divorce, Family. I. INTRODUCTION Lotfi. A. Zadeh (1965) has introduced a mathematical model called Fuzzy Cognitive Maps. After a decade, Political scientist Axelord (1976) used this fuzzy model to study decision making in social and political systems. Then Kosko (1986, 1988 and 1997) enhanced the power of cognitive maps considering fuzzy values for the concepts of the cognitive map and fuzzy degrees of interrelationships between concepts. 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 and medical etc. Usually we analyze the number of attributes ON-OFF position. But the thing is here, this gives the weightage of the attributes we call ranking of the attributes. Now we see the basic definitions for FCMs to develop the Triangular Fuzzy Cognitive Maps (TrFCM). II. PRELIMINARIES In this section, some concepts and methods used in this paper are briefly introduced. 2.1. Fuzzy Set theory The fuzzy set theory is to deal with the extraction of the primary possible outcome from a multiplicity of information that is expressed in vague and imprecise terms. Fuzzy set theory treats vague data as probability distributions in terms of set memberships. Once determined and defined, sets of memberships in probability distributions can be effectively used in logical reasoning. 2.2. Triangular Fuzzy number and the Algebraic Operations 2.2.1 Triangular Fuzzy number It is a fuzzy number represented with three points as follows:   1 2 3 , , A a a a  a1 a2 a3 1  The membership function defined as 1 1 1 2 2 1 3 2 3 3 2 3 0 ( ) 0 A for x a x a for a x a a a x a x for a x a a a for x a                        2.2.2. Operation of Triangular Fuzzy Number The following are the four operations that can be performed on triangular fuzzy numbers: Let   1 2 3 , , A a a a  and   1 2 3 , , B b b b  then, (i) Addition (+): 1 1 2 2 3 3 ( , , ) A B a b a b a b      (ii) Subtraction (-): 1 3 2 2 3 1 ( , , ) A B a b a b a b      (iii) Multiplication (  ) : (a) 1 2 3 ( , , ), , 0, k A ka ka ka k R k     (b) 1 2 1 2 1 2 1 2 ( , , ), 0, 0 A B a a bb c c a a     . (iv) Division (  ): 1 1 1 1 1 1 1 1 1 1 1 1 ( ) ( , , ) , , , 0 A a b c a c b a            , 1 1 1 1 2 2 2 2 , , , 0, 0. a b c A B a a c b a           2.2.3 Degrees of the Triangular Fuzzy Number The linguistic values of the triangular fuzzy numbers are Very Low (0, 0, 0.25) Low (0, 0.25, 0.50) Medium (0.25, 0.50, 0.75) High (0.50, 0.75, 1)
2. 2. Integrated Intelligent Research(IIR) International Journal of Business Intelligent Volume: 03 Issue: 01 June 2014,Pages No.4-9 ISSN: 2278-2400 5 Very High (0.75, 1, 1) 2.3. Fuzzy Cognitive Maps (FCMs) 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. 2.3.1 Definition When the nodes of the FCM are fuzzy sets then they are called as fuzzy nodes. 2.3.2. Definition FCM with edge weights or causalities from the set {-1, 0, 1} are called simple FCM. 2.3.3. Definition An FCM is a directed graph with concepts like policies, events etc, as nodes and causalities as edges, It represents causal relationships between concepts. 2.3.4. 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. 2.3.5. 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. 2.3.6. 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. 2.3.7. Definition An FCM is said to be cyclic is said to have a feedback. 2.3.8. 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. 2.3.9Definition 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 ges 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. 2.3.10. 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. 2.3.11. 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. 2.4. METHOD OF DETERMINING THE HIDDEN PATTERN OF FUZZY COGNITIVE MAPS (FCMs) 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. III. PROPOSED TRIANGULAR FUZZY COGNITIVE MAPS (TrFCMs) Triangular Fuzzy Cognitive Maps (TrFCM) are more applicable when the data in the first place is an unsupervised one. The TrFCM works on the opinion of three experts. TrFCM models the world as a collection of classes and causal relations between classes. It is a different process when we compare to FCM. Usually the FCM gives only the ON-OFF position. But this Triangular Fuzzy Cognitive Maps is more precise and it gives the ranking for the causes of the problem by using the weightage of the attribute it is main advantage of the new Triangular Fuzzy Cognitive Maps. 3.1. BASIC DEFINITIONS OF TRIANGULAR FUZZY COGNITIVE MAPS 3.1.1. Definition When the nodes of the TrFCM are fuzzy sets then they are called as fuzzy triangular nodes. 3.1.2. Definition Triangular FCMs with edge weights or causalities from the set {-1, 0, 1} are called simple Triangular FCMs. 3.1.3. Definition An TrFCM is a directed graph with concepts like policies, events etc, as nodes and causalities as edges, It represents causal relationships between concepts. 3.1.4. Definition Consider the nodes/concepts TrC1, TrC2,…,TrCn of the Triangular FCM. Suppose the directed graph is drawn using edge weight Treij∈ {-1, 0, 1}. The triangular matrix M be defined by Tr(M) = (Treij) where Treij is the triangular weight of the directed edge
3. 3. Integrated Intelligent Research(IIR) International Journal of Business Intelligent Volume: 03 Issue: 01 June 2014,Pages No.4-9 ISSN: 2278-2400 6 TrCiTrCj. Tr(M) is called the adjacency matrix of Triangular Fuzzy Cognitive Maps, also known as the connection matrix of the TrFCM. It is important to note that all matrices associated with an TrFCM are always square matrices with diagonal entries as zero. 3.1.5. Definition Let TrC1, TrC2,…,TrCn be the nodes of an TrFCM. A=(a1, a2,…,an) where Treij∈ {-1, 0, 1}. A is called the instantaneous state vector and it denotes the on-off position of the node at an instant. Instantaneous vector 1 ( ) 0 Tr i Tr i a Maximum weight a Otherwise       3.1.6. Definition Let TrC1, TrC2,…,TrCn be the triangular nodes of and TrFCM. Let 1 2 2 3 3 4 , , ,..., Tr Tr Tr Tr Tr Tr Tr i Tr j C C C C C C C C be the edges of the TrFCM (i≠j). Then the edges form a directed cycle. An TrFCM is said to be cyclic if it possesses a directed cycle. An TrFCM is said to be acyclic if it does not possess any directed cycle. 3.1.7. Definition An TrFCM is said to be cyclic is said to have a feedback. 3.1.8. Definition When there is a feedback in an TrFCM, i.e, when the causal relations flow through a cycle in a revolutionary way, the TrFCM is called a dynamical system. 3.1.9. Definition Let 1 2 2 3 3 4 1 , , ,..., Tr Tr Tr Tr Tr Tr Tr n Tr n C C C C C C C C  be a cycle. When TrCi is switched ON and if the causality flows through the triangular 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 triangular node TrCi fori =1,2,…,n. The equilibrium state for this dynamical system is called the hidden pattern. 3.1.10. Definition If the equilibrium state of a dynamical system is a unique state vector, then it is called a fixed point. Consider a TrFCM with TrC1, TrC2,…,TrCn as nodes. For example let us start the dynamical system by switching on TrC1. Let us assume that the TrFCM settles down with TrC1 and TrCn ON i.e., in the state vector remains as (1, 0, 0,…, 0) is called fixed point. 3.1.11. Definition If the TrFCM settles down with a state vector repeating in the form A1→A2→…→Ai→A1 then this equilibrium is called a limit cycle. 3.2. METHOD OF DETERMINING THE HIDDEN PATTERN OF TRIANGULAR FUZZY COGNITIVE MAPS (TrFCMs) Step 1: Let TrC1, TrC2,…,TrCn be the nodes of an TrFCM, with feedback, Let Tr(M) be the associated adjacency matrix. Step 2: Let us find the hidden pattern when TrC1 is switched ON. When an input is given as the vectorA1 = (1, 0,…, 0), the data should pass through the relation matrix M. This is done by multiplying Ai by the triangular matrix M. Step 3: Let AiTr(M) = (a1, a2,…, an) will get a triangular vector. Suppose A1Tr(M) = (1, 0,…, 0)it gives a triangular weight of the attributes, we call it as Ai Tr(M)weight. Step 4: Adding the corresponding node of the three experts opinion, we call it as Ai Tr(M)sum. Step 5: The threshold operation is denoted by( ) ie., A1Tr(M)Max(weight). That is by replacing ai by 1 if aiis the maximum weight of the triangular node (ie.,ai=1), otherwise ai by 0(ie., ai=0). Step 6: Suppose A1Tr(M)→ A2 then consider A2Tr(M)weight is nothing but addition of weightage of the ON attribute and A1 Tr(M)weight . Step 7: Find A2 Tr(M)sum(ie., summing of the three experts opinion of each attributes). Step 8: The threshold operation is denoted by( ) ie., A2Tr(M)Max(weight). That is by replacing ai by 1 if ai is the maximum weight of the triangular node (ie.,ai=1), otherwise ai by 0 (ie., ai=0). Step 9: If the A1Tr(M)Max(weight). =A2Tr(M)Max(weight). Then dynamical system end otherwise repeat the same procedure. Step 10: This procedure is repeated till we get a limit cycle or a fixed point. IV. CONCEPT OF THE PROBLEM we have taken the following Twelve concepts {TrC1, TrC2,…,TrC12} To analyze of the major reasons for divorce happening in the society using linguistic questionnaire and the expert’s opinion The following concepts are taken as the main nodes of our problem. TrC1-High Expectations TrC2- Adultery (Illegal relationship) TrC3- Incompatibility (Sexual dissatisfaction) TrC4- Low Tolerance and Rigidity (Rough and Stiff Manner) TrC5- Dowry and Harassment TrC6- Lack of Commitment TrC7- Lack of Physical Attraction/Boredom in Marriage TrC8- Family Pressure TrC9- Child rearing issues TrC10- Lack of Communication TrC11- Family Background TrC12- Religious and Cultural Strains Now we give the connection matrix related with the FCM. TrC1 TrC2 TrC3 TrC4 TrC5 TrC6 TrC7 TrC8 TrC9 TrC10 TrC11 TrC12
4. 4. Integrated Intelligent Research(IIR) International Journal of Business Intelligent Volume: 03 Issue: 01 June 2014,Pages No.4-9 ISSN: 2278-2400 7 ( ) 1 2 3 4 5 6 7 8 9 10 11 12 0 0 0 0 0 0 0 0 Tr Tr Tr Tr Tr Tr Tr Tr Tr Tr Tr Tr C VL M VL L VL L L H VL VH VL C L VH H L VL M VL L VL L VL C VL L VH H M L VL M VL L L C L VL VH M VL L L L VL VL L C L L VL VH M VL VH VL L L VL C VL L VL VL L VH VH VL VL L VL Tr M C M VL L VL VL VL L VL L L VL C VL L VL VL VH VL VH L VL VL L C L C C C = 0 0 0 0 VL L VL L L M VH VH H VL M L VL L VL VL VH L VL L L VL H L VL L L VL VL L M VH L VL VL L VL L L VL L H VH é ù ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ë û TrC1 TrC2 TrC3 TrC4 TrC5 TrC6 TrC7 TrC8 TrC9 TrC10 TrC11 TrC12 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5 6 7 8 9 10 11 12 0 0,0,0.25 0.25,0.50,0.75 0,0,0.25 0,0.25,0.50 0,0,0.25 0,0.25,0.50 0,0.25,0.50 0.50,0.75,1 0,0,0.25 0.75,1,1 0,0,0.25 0,0.25,0.50 0 0.75,1,1 0.50,0.75,1 0,0.25, Tr Tr Tr Tr Tr Tr Tr Tr Tr Tr Tr Tr C C C C C C Tr M C C C C C C = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0.50 0,0,0.25 0.25,0.50,0.75 0,0,0.25 0,0.25,0.50 0,0,0.25 0,0.25,0.50 0,0,0.25 0,0,0.25 0,0.25,0.50 0 0.75,1,1 0.50,0.75,1 0.25,0.50,0.75 0,0.25,0.50 0,0,0.25 0.25,0.50,0.75 0,0,0.25 0,0.25,0.50 0,0.25,0.50 0,0.25,0.50 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,0,0.25 0.75,1,1 0 0.25,0.50,0.75 0,0,0.25 0,0.25,0.50 0,0.25,0.50 0,0.25,0.50 0,0,0.25 0,0,0.25 0,0.25,0.50 0,0.25,0.50 0,0.25,0.50 0,0,0.25 0.75,1,1 0 0.25,0.50,0.75 0,0,0.25 0.75,1,1 0,0,0.25 0,0.25,0.50 0,0.25,0.50 0,0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,0.25 0,0,0.25 0,0.25,0.50 0,0,0.25 0,0,0.25 0,0.25,0.50 0 0.75,1,1 0.75,1,1 0,0,0.25 0,0,0.25 0,0.25,0.50 0,0,0.25 0.25,0.50,0.75 0,0,0.25 0,0.25,0.50 0,0,0.25 0,0,0.25 0,0,0.25 0 0,0.25,0.50 0,0,0.25 0,0.25,0.50 0,0.25,0. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 50 0,0,0.25 0,0,0.25 0,0.25,0.50 0,0,0.25 0,0,0.25 0.75,1,1 0,0,0.25 0.75,1,1 0 0,0.25,0.50 0,0,0.25 0,0,0.25 0,0.25,0.50 0,0.25,0.50 0,0,0.25 0,0.25,0.50 0,0,0.25 0,0.25,0.50 0,0.25,0.50 0.25,0.50,0.75 0.75,1,1 0 0.75,1,1 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .50,0.75,1 0,0,0.25 0.25,0.50,0.75 0,0.25,0.50 0,0,0.25 0,0.25,0.50 0,0,0.25 0,0,0.25 0.75,1,1 0,0.25,0.50 0,0,0.25 0 0,0.25,0.50 0,0.25,0.50 0,0,0.25 0.50,0.75,1 0,0.25,0.50 0,0,0.25 0,0.25,0.50 0,0.25,0.50 0,0,0.25 0,0, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0.25 0,0.25,0.50 0.25,0.50,0.75 0 0.75,1,1 0,0.25,0.50 0,0,0.25 0,0,0.25 0,0.25,0.50 0,0,0.25 0,0.25,0.50 0,0.25,0.50 0,0,0.25 0,0.25,0.50 0.50,0.75,1 0.75,1,1 0 é ù ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ë û Attribute TrC1 is ON: A(1) = (1 0 0 0 0 0 0 0 0 0 0 0) A(1) Tr(M)Weight = (0, (0,0,0.25), (0.25,0.50,0.75), (0,0,0.25), (0,0.25,0.50), (0,0,0.25), (0,0.25,0.50), (0,0.25,0.50), (0.50,0.75,1), (0,0,0.25), (0.75,1,1), (0,0,0.25)) A(1) Tr(M)Average = (0, 0.0833, 0.5, 0.0833, 0.25, 0.0833, 0.25, 0.25, 0.75, 0.0833, 0.9167, 0.0833) A(1) Tr(M)Max(Weight)  (0 0 0 0 0 0 0 0 0 0 1 0) =A1 (1) A1 (1) Tr(M)Average = (0.2292, 2.0626, 0.6875, 0.2292, 0.6875, 0.6875, 0.2292, 0.2292, 0.6875, 1.3751, 0, 2.5209) A1 (1) Tr(M)Max(Weight)  (0 0 0 0 0 0 0 0 0 0 0 1) =A2 (1) A2 (1) Tr(M)Average = (1.8907, 0.6302, 0.6302, 1.8907, 0.6302, 1.8907, 1.8907, 0.6302, 1.8907, 5.6720, 6.9325, 0) A2 (1) Tr(M)Max(Weight)  (0 0 0 0 0 0 0 0 0 0 1 0) =A2 (1) =A1 (1) Attribute TrC2 is ON: A(2) = (0 1 0 0 0 0 0 0 0 0 0 0) A(2) Tr(M)Weight = ((0,0.25,0.50), 0, (0.75,1,1), (0.50,0.75,1), (0,0.25,0.50), (0,0,0.25), (0.25,0.50,0.75), (0,0,0.25), (0,0.25,0.50), (0,0,0.25), (0,0.25,0.50), (0,0,0.25)) A(2) Tr(M)Average = (0.25, 0, 0.9167, 0.75, 0.25, 0.0833, 0.5, 0.0833, 0.25, 0.0833, 0.25, 0.0833) A(2) Tr(M) Max(Weight)  (0 0 1 0 0 0 0 0 0 0 0 0)= A1 (2) A1 (2) Tr(M)Average =(0.2292, 0.6875, 0, 2.5209, 2.0626, 1.3751, 0.6875, 0.2292, 1.3751, 0.2292, 0.6875, 0.6875) A1 (2) Tr(M) Max(Weight)  (0 0 0 1 0 0 0 0 0 0 0 0)= A2 (2) A2 (2) Tr(M) Average =(1.8907, 0.6302, 6.9325, 0, 3.7814, 0.6302, 1.8907, 1.8907, 1.8907, 0.6302, 0.6302, 1.8907) A2 (2) Tr(M) Max(Weight)  (0 0 1 0 0 0 0 0 0 0 0 0)= A3 (2) = A1 (2) Attribute P3 is ON: A(3) = (0 0 1 0 0 0 0 0 0 0 0 0 ) A(3) Tr(M)weight = ((0,0,0.25), (0,0.25,0.50), 0, (0.75,1,1), (0.50,0.75,1), (0.25,0.50,0.75), (0,0.25,0.50), (0,0,0.25), (0.25,0.50,0.75), (0,0,0.25), (0,0.25,0.50), (0,0.25,0.50)) A(3) Tr(M) Average = (0.0833, 0.25, 0, 0.9167, 0.75, 0.5, 0.25, 0.0833, 0.5, 0.0833, 0.25, 0.25) A(3) Tr(M) Max(Weight)  (0 0 0 1 0 0 0 0 0 0 0 0)= A1 (3) A1 (3) Tr(M) Average = (0.6875, 0.2292, 2.5209, 0, 1.3751, 0.2292, 0.6875, 0.6875, 0.6875, 0.2292, 0.2292, 0.6875) A1 (3) Tr(M) Max(Weight)  (0 0 1 0 0 0 0 0 0 0 0 0)= A2 (3) A2 (3) Tr(M)Average =(0.6302, 1.8907, 0, 6.9325, 5.6720, 3.7814, 1.8907, 0.6302, 3.7814, 0.6302, 1.8907, 1.8907) A2 (3) Tr(M) Max(Weight)  (0 0 0 1 0 0 0 0 0 0 0 0 )= A2 (3) = A1 (3) Attribute P4 is ON: A(4) = (0 0 0 1 0 0 0 0 0 0 0 0) A(4) Tr(M)weight = ((0,0.25,0.50), (0,0,0.25), (0.75,1,1), 0, (0.25,0.50,0.75), (0,0,0.25), (0,0.25,0.50), (0,0.25,0.50), (0,0.25,0.50), (0,0,0.25), (0,0,0.25), (0,0.25,0.50)) A(4) Tr(M)Average = (0.25, 0.0833, 0.9167, 0, 0.5, 0.0833, 0.25, 0.25, 0.25, 0.0833, 0.0833, 0.25) A(4) Tr(M Max(Weight)  (0 0 1 0 0 0 0 0 0 0 0 0) = A1 (4) A1 (4) Tr(M)Average = (0.2292, 0.6875, 0, 2.5209, 2.0626, 1.3751, 0.6875, 0.2292, 1.3751, 0.2292, 0.6875, 0.6875) A1 (4) Tr(M) Max(Weight)  (0 0 0 1 0 0 0 0 0 0 0 0) = A2 (4) A2 (4) Tr(M)Average = (01.8907, 0.6302, 6.9325, 0, 3.7814, 0.6302, 1.8907, 1.8907, 1.8907, 0.6302, 0.6302, 1.8907) A2 (4) Tr(M) Max(Weight)  (0 0 0 1 0 0 0 0 0 0 0 0) = A3 (4) = A2 (4) Attribute P5 is ON: A(5) = (0 0 0 0 1 0 0 0 0 0 0 0) A(5) Tr(M)Weight = ((0,0.25,0.50), (0,0.25,0.50), (0,0,0.25), (0.75,1,1), 0, (0.25,0.50,0.75), (0,0,0.25), (0.75,1,1), (0,0,0.25), (0,0.25,0.50), (0,0.25,0.50), (0,0,0.25))