163 HAPTER 8 TRIANGULAR NEUTROSOPHI OGNITIVE MAPS
164 8.1 INTRODUTION The main objective in this chapter is the study at length of the various reasons for Miracles with incidents from the Holy Bible adopting the new fuzzy tool called iangular Neutrosophic ognitive Maps to analyze the social problem. Usually in NM we analyze the measure not only the existence of causal relation between concepts or the absence of causal relations between two concepts but also gives representation to the indeterminacy of relations between any two concepts.. But this new model gives ranking for the causes and indeterminacy of relations between any two concepts of the problem. The present study deals with studying at length the various reasons for miracles with incidents from the Holy Bible using iangular Neutrosophic ognitive Maps (NMs). 8.2 Faith used in four ways in Bible. 1. Mental acceptance, personal belief and acknowledgement of God s existence 2. A conduct or work inspired by the complete surrender of total acceptance 3. A trusting or trustworthiness. 4. uth and morality based on hristian faith 29. The Holy Bible clearly outlines how an individual s faith index can be enhanced. Faith comes from hearing and hearing the word of hrist ( Romans 1:17). As an individual continues to hear God s word, Faith sets in ( Romans 1:14). This leads to the edifying of good conscience which holds this faith ( 1Timothy 1:19).
165 This planted faith starts growing and flourishing within as one continuously seeks the Maker wholeheartedly (Jeremiah 29:13). A diligent search will lead to the Maker ( Proverbs 8:19). This search enriches knowledge of God (Proverbs 2:5). Growing in His knowledge enables an individual to produce the fruit of good deeds ( olossians 1:1). This results in the perfecting of Faith ( James 2:22). The key is in never wavering, never having misgivings but to be continually deeply rooted and grounded in our faith ( olossians 1:23). Faith in the Almighty is essential in the growing wake of personal struggles, conflicts, differences within a household and trying and impossible situations. The seed of Faith sown in the hearts of individuals will help them stand the test of times to come. It is therefore necessary that Faith be nurtured which in turn, builds in us a strong relationship with the Master establishing a strong foundation for the dispensation of miracles from humanly difficult and impossible situations. It is absolutely essential to keep hearing testimonies of church goers, believers and people from good hristian homes. It is well known that hearing the good news will enrich the soul and stands us up strong in dire situations. Total surrender to the maker and listening for His Voice will help us charter the route that He has planned for us. Remembering our difficult times and how we surmounted them will enrich the quality of faith in us. Meditation on the truth and reading encouraging real life encounters would further serve to enhance our Faith quotient. Self-enrichment is the ladder left for us to climb at a pace that is entirely in the hands of the individual and aided by the heavenly power, will result in signs and wonders and Miracles.
166 8.3 Degrees of the iangular Fuzzy Number The linguistic values of the triangular fuzzy numbers are Very (,,.1) Low(VL) Low(L) (,.1,.3) Medium (.1,.3,.5) Low(ML) Medium (M) (.3,.5,.7) Medium (.5,.7,.9) High(MH) High(H) (.7,.9, 1) Table 8.1 Intermediate fuzzy linguistic values Intermediate medium (.3,.5,.7) Lower intermediate (.1,.3,.5) Higher intermediate (.5,.7,.9) Table 8.2 8.4 Proposed iangular Neutrosophic ognitive Maps (NM) iangular Neutrosophic ognitive Maps (NM) are more applicable when the data in the first place is an unsupervised one. The NM works on the opinion of three experts. NM models the world as a collection of classes and
167 causal relations between classes. It is a different process when we compare to NM. Usually the NM gives only the ON-OFF position and intermediate conditions. But this iangular Neutrosophic ognitive Maps is more precise and it gives the ranking for the causes and indeterminacy of relations between any two concepts of the problem by using the weightage of the attribute it is main advantage of the New iangular Neutrosophic ognitive Maps. 8.5 BASI DEFINITIONS OF TRIANGULAR NEUTROSOPHI OGNITIVE MAPS Definition 8.5.1 A NM is a directed graph with concepts like policies, events, etc, as nodes and causalities as edges. It represents casual relationship between concepts. Definition 8.5.2 When the nodes of the NM are fuzzy sets then they are called as fuzzy nodes. Definition 8.5.3 A NM with edge weights or causalities from the set{-1,, 1, I} are called simple NM. Definition 8.5.4 Let i and j denote the two nodes of the NM. The directed edge from i to j denotes the casuality i on j called connections. Every edge in the NM is weighted with a number in the set {-1,, 1, I}. Let e ij be the weight of the directed edge i j, e ij Є {-1,, 1, I}. e ij = if i does not have any effect on j, e ij = 1 if increase (or decreases) in i causes decrease(or increase) in j, e ij = I if the relation or effect of i on j is an indeterminate.
168 Definition 8.5.5 Let 1, 2,., n be nodes of a NM. Let the neutrosofic matrix N (E) be defined as N (E) = (e ij ) where e ij is the weight of the directed edge i j, where e ij Є {-1,, 1, I}. N (E) is called the netrosofic adjacency matrix of the NM. Definition 8.5.6 Let 1, 2,., n be nodes of an NM. Let A= {a 1, a 2,., a n } where a i Є {, 1, I}. A is called the instantaneous state neutrosophic vector and it denotes the on-off indeterminate state position of the node at an instant a i = if a i is OFF(no effect) a i = 1 if a i is ON(has effect) a i = I if a i is indeterminate(effect cannot be determined) for i=1,2,.,n. Definition 8.5.7 Let 1, 2,., n be nodes of a NM. Let 1, 2, 2 3, i j be the edges of NM. Then the edges form a directed cycle. An NM is said to be cyclic if it possesses a directed cycle. An NM is said to be acyclic if it does not possess any directed cycle. Definition 8.5.8 An NM with cycles is said to have a feedback. When there is a feedback in the NM.i.e when the casual relations flow through a cycle in a revolutionary manner the NM is called a dynamical system. Definition 8.5.9 Let 1 2, 3, n-1, n be cycle, when i is switched on and if the casuality flow through the edges of a cycle and if it again causes i, say that the
169 dynamical system goes round and round. This is true for any node i, for i= 1,2,..n the equilibrium state for this dynamical system is called the hidden pattern. Definition 8.5.1 If the equilibrium state of a dynamical system is a unique state vector, then it is called a fixed point. onsider the NM with 1, 2,., 11 as nodes. For example let us start the dynamical system by switching ON 1. Let us assume that the NM settles down with 1 and n ON i.e. the state vector remain as (1,,,,1) this neutrosophic state vector (1,,,.,1) is called the fixed point. Definition 8.5.11 If the NM settles with a neutrosophic state vector repeating in the form A1 A2. A i A 1 then this equilibrium is called a limit cycle of the NM. 8.6 Method of Determining the Hidden Pattern of iangular Neutrosophic ognitive Maps (NM) Step 1: Let 1, 2,, n be the nodes of an NM, with feedback, Let (M) be the associated adjacency matrix. Step 2: Let us find the hidden pattern when 1 is switched ON. When an input is given as the vectora 1 = (1,,, ), the data should pass through the relation matrix M. This is done by multiplying A i by the triangular matrix M. Step 3: Let A i (M) = (a 1, a 2,, a n ) will get a triangular vector. Suppose A 1 (M) = (1,,, )it gives a triangular weight of the attributes, we call it as A i
17 (M) weight. Step 4: Adding the corresponding node of the three experts opinion, we call it as A i (M) sum. Step 5: The threshold operation is denoted by( ) ie., A 1 (M) Max(weight). That is by replacing a i by 1 if a i is the maximum weight of the triangular node (ie.,a i =1), otherwise a i by (ie., a i =). Step 6: Suppose A 1 (M) A 2 then consider A 2 (M) weight is nothing but addition of weightage of the ON attribute and A 1 (M) weight. Step 7: Find A 2 (M) sum (ie., summing of the three experts opinion of each attributes). Step 8: The threshold operation is denoted by( ) ie., A 2 (M) Max(weight). That is by replacing a i by 1 if a i is the maximum weight of the triangular node (ie.,a i =1), otherwise a i by (ie., a i =). Step 9: If the A 1 (M) Max(weight). =A 2 (M) Max(weight). Then dynamical system end otherwise repeat the same procedure. Step 1: This procedure is repeated till we get a limit cycle or a fixed point. Using the linguistic questionnaire and the expert s opinion we have taken the following nine concepts { 1, 2,..., 9 } 8.7 oncept of the Problem The following eleven concepts { 1, 2,, 11 } were taken to analyze and find the miracles through Holy Bible using linguistic questionnaire and the expert s opinion The following concepts are taken as the main nodes of our problem. 1 - Perseverance through prayer 2 - Patience
171 3 - Faith 4 - Authority in the Spiritual Realm 5 - Gods ompassion 6 - Forgiveness 7 - Hearing the Word of God 8 - Humility 9 - Love 1 - Obedience 11 - Repentance Now we give the connection matrix related with the NM.Given by the expert(pasteur)
168 LINGUISTI VARIABLES FOR THE TRIANGULAR FUZZY NODES 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 1 11 H H MH MH MH MH H H VH H H M M VH MH H H H ML M H M M VH H H H M H H MH M M M LI M M IM VH H MH VH H M H MH MH H MH H MH MH H LI H VH H IM H HI MH H H M MH VH H M M M MH H H MH MH H H VH M M 9 1 11 M H H M IM H IM VH M M VH ML H VH MH H M M M M H M H H VH HI M M M M
169 Linguistic values of the triangular fuzzy nodes 1 2 3 4 5 6 7 8 9 1 11 1 2 3 4 5 6 7 8 9 1 11.7,.9,1.7,.9,1.5,.7,.9.5,.7,.9.5,.7,.9.5,.7,.9.7,.9,1.7,.9,1.9,1,1.7,.9,1.7,.9,1.3,.5,.7.3,.5,.7.9,1,1.5,.7,. 9.7,.9,1.7,.9,1.7,.9,1.1,.3,.5.3,.5,.7.7,.9,1.3,.5,.7.3,.5,.7.9,1,1.7,.9,1.7,.9,1.7,.9,1.3,.5,.7.7,.9,1.7,.9,1.5,.7,.9.3,.5,.7.3,.5,.7.3,.5,.7. 1,.3,.5.3,.5,.7.3,.5,.7.3,.5,.7.9,1,1.7,.9,1.5,.7,.9.9,1,1.7,.9,1.3,.5,.7.7,.9,1.5,.7,.9.5,.7,.9.7,.9,1.5,.7,.9.7,.9,1.5,.7,.9.5,.7,.9.7,.9,1.1,.3,.5.7,.9,1.9,1,1.7,.9,1.3,.5,.7.7,.9,1.7,.9,1.5,.7,.9.7,.9,1.7,.9,1.3,.5,.7.5,.7,.9.9,1,1.7,.9,1.3,.5,.7.3,.5,.7.5,.7,.9.5,.7,.9.7,.9,1.7,.9,1.5,.7,.9.5,.7,.9.7,.9,1.7,.9,1.9,1,1.3,.5,.7.3,.5,.7.7,.9,1.7,.9,1.3,.5,.7.3,.5,.7.7,.9,1.3,.5,.7.3,.5,.7.9,1,1.3,.5,.7.3,.5,.7.9,1,1.1,.3,.5. 7,.9,1.7,.9,1.7,.9,1.7,.9,1.3,.5,.7.3,.5,.7.3,.5,.7.3,.5,.7.7,.9,1.3,.5,.7.7,.9,1.7,.9,1.9,1,1.5,.7,.9.3,.5,.7.3,.5,.7.3,.5,.7.3,.5,.7
17 Attribute 1 is ON: A (1) = (1 ) A (1) (M) Weight = (, (.7,.9,1), (.7,.9,1), (.5,.7,.9), (.5,.7,.9), (.5,.7,.9), (.5,.7,.9), (.7,.9,1), (.7,.9,1), (.9,1,1),(.7,.9,1) A (1) (M) Average = (,.8666,.8666,.7,.7,.7,.7,.8666,.8666,.9666,.8666) A (1) (1) (M) Max(Weight) ( 1 ) =A 1 A (1) 1 (M) Average = (.9343,.2899,.8376,.8376,.6766,.8376,.4833,.4833,.4833,,.4833) A (1) (1) 1 (M) Max(Weight) (1 ) =A 2 A (1) 2 = A (1) Step:2 Attribute 2 is ON: A (2) = { 1 } A (2) (M) Weight = (.7,.9,1), (), (.3,.5,.7), (.3,.5,.7), (.9,1,1), (.5,.7,.9), (.7,.9,1), (.7,.9,1), (.7,.9,1), (.1,.3,.5),(.3,.5,.7) A (2) (M) Average = {.8666.5.5.9666.7.8666.8666.8666.3.5} A (2) (M) Max(Weight) = { 1 (2) }= A 1 A (2) 1 (M) Average ={.6766,.9343,.8376,.4833,,.8376,.6766,.6766.6766,.6766,.83765} A (2) 1 (M) Max(Weight) {, 1, }
171 = A 2 (2) A 1 (2) =A 2 (2) STEP:3 Attribute 3 is ON A (3) = { 1 } A (3) (M) Weight = (.7,.9,1), (.3,.5,.7), (), (.3,.5,.7), (.9,1,1), (.7,.9,1), (.7,.9,1), (.7,.9,1), (.3,.5,.7), (.7,.9,1),(.7,.9,1) A (3) (M) Average ={.8666,.5,,.5,.9666,.8666,.8666,.8666,.5,.8666,.8666 } A (3) (M) Max(Weight) { 1 }= A 1 (3) A (3) 1 (M) Average ={.6766,.9343,.8376,.4833,,.8376,.6766,.6766,.8376,.6766,.8376} A (3) 1 (M) Max(Weight) { 1 (3) }= A 2 A (3) 2 (M) Average ={.896,.9343,.8376, 4833,,.8376,.6766,.6766,.8376, 6766,.83765} A (3) 2 (M) Max(Weight) ={ 1 (3) }= A 3 A (3) 3 (M) Average ={.896,,.46715,.46715,.939,.654,.896,.896,.896,.282,.4671} A (3) 3 (M) Max(Weight) = { 1 (3) } = A 3 (3) (3) A 1 = A 3 Step:4 Attribute 4 is ON:
172 A (4) ={ 1 } A (4) (M) Weight = (.5,.7,.9), (.3,.5,.7), (.3,.5,.7), (), (.3,.5,.7), (.1,.3,.5), (.3,.5,.7), (.3,.5,.7), (.3,.5,.7),(.7,.9,1)(.7,.9,1). A (4) (M) Average ={.7,.5,.5,,.5,.3,.5,.5,.5,.9666,.8666} A (4) (M) Max(Weight) ={ 1 }=A 4 (1) A (4) 1 (M) average ={.9343,.2899,.8376,.8376,.6766,.8376,.4833,.4833,.4 833,,.4833} A (4) 1 (M) Max(Weight) { 1 }=A 4 (2) A (4) 2 (M) average ={,.896,.896,.654,.654,.654,.654,.896,.896,.93,.896} A (4) 2 (M) Max(Weight) { 1 4 }= A (3) A (4) 3 (M) average = {.8728,.279,.7825,.7825,.6321,.7825,.4515,.4515,.4515,,.4515} A (4) 3 (M) Max(Weight) ={ 1 (4) }= A (4) (4) A (1) 4 = A (4) Step:5 Attribute 5 is ON: A (5) = { 1 }
173 A (5) (M) Weight = (.5,.7,.9), (.9,1,1), (.7,.9,1), (.3,.5,.7), (), (.7,.9,1), (.5,.7,.9), (.5,.7,.9), (.7,.9,1), (.5,.7,.9),(.7,.9,1) A (5) (M) average ={.7,.9666,.8666,.5,,.8666,.7,.7,.8666,.7,.8666} A (5) (M) Max(Weight) { 1 (5) }= A 1 A (5) 1 (M) average ={.8376,,.4833,.4833,.9343,.6766,.8376,.8376,.8376,.2899,.4833} A (5) 1 (M) Max(Weight) { 1 (5) }= A 2 A (5) 2 (M) average ={.654,.9343,.896,.46715,,.896,.654,.654,.896,.654,.896} A (5) 2 (M) Max(Weight) { 1 (5) }= A 3 (5) A 1 = A (5) 3 (fixed point) Step:6 Attribute 6 is ON: A (6) ={ 1 } A (6) (M) Weight = (.5,.7,.9), (.5,.7,.9), (.7,.9,1), (.1,.3,.5), (.7,.9,1), (), (.9,1,1), (.7,.9,1), (.3,.5,.7), (.7,.9,1),(.7,.9,1) A (6) (M) average ={.7.7.8666,.3,.8666,,.9666,.8666,.5,.8666,.7} A (6) (M) Max(Weight) { 1 (6) }= A 1
174 A (6) 1 (M) average ={.6766,.8376,.8376,.4833,.6766,.9343,,.8376,.4833,.4833,.4833} A (6) 1 (M) Max(Weight) { 1 (6) }= A 2 A (6) 2 (M) average ={.654,.654,.886,.282,.896,,.9343,.896,.4671,.896,.654} A (6) 2 (M) Max(Weight) { 1 (7) } = A 3 (6) (7) A 3 = A 3 (Fixed Point) Step:7 Attribute 7 is ON: A= { 1 } A (7) (M) Weight = {(.5,.7,.9), (.7,.9,1), (.7,.9,1), (.3,.5,.7), (.5,.7,.9), (.9,1,1), (), (.7,.9,1), (.3,.5,.7), (.3,.5,.7),(.5,.7,.9)} A (7) (M) average ={.7,.8666,.866,.5,.7,.9666,,.8666,.5,.5,.5} A (7) (M) Max(Weight) { 1 (7) }=A 1 A (7) 1 (M) average ={.6766,.6766,.8376,.2899,.8376,,.9343,.8376,.4833,.8376,.6766} A (7) 1 (M) Max(Weight) { 1 (7) }= A 2 A (7) {7) = A 2
175 Step:8 Attribute 8 is ON: A (8) = { 1 } A (8) (M) Weight = (.5,.7,.9), (.7,.9,1), (.7,.9,1), (.5,.7,.9), (.5,.7,.9), (.5,.7,.9), (.7,.9,1), (.7,.9,1), (), (.3,.5,.7),(.3,.5,.7) A (8) (M) average ={.7,.8666,.8666,.7,.7,.8666,.8666,,.9666,.5,.5 } A (8) (M) Max(Weight) { 1 (8) } = A 1 A (8) 1 (M) average ={.8376,.8376,.4833,.4833,.8376,.4833,.4833,.9343,,.4833,.4833} A (8) 1 (M) Max(Weight) { 1 (8) }= A 2 A (8) =A 2 (8) (fixed point) Step:9 Attribute 9 is ON: A (9) = { 1 } A (9) (M) Weight = (.7,.9,1), (.7,.9,1), (.3,.5,.7), (.3,.5,.7), (.7,.9,1), (.3,.5,.7), (.3,.5,.7), (.7,1,1), () (.3,.5,.7), (.3,.5,.7) } A (9) (M) average ={.8666,.8666,.5,.5,.8666,.5,.5,.9666,,.5,.5} A (9) (M) Max(Weight) ={ 1 (9) } = A 1
176 A (9) 1 (M) average ={.6766,.8376,.8376,.6766,.6766,.8376,.8376,,.9343,.4833,.4833} A (9) 1 (M) Max(Weight) { 1 (9) } = A 2 A (9) (9) =A 2 Step:1 Attribute 1 is ON: A (1) ={ 1 } A (1) (M) Weight = (.9,1,1), (.1,.3,.5), (.7,.9,1), (.7,.9,1), (.7,.9,1), (.3,.5,.7), (.3,.5,.7), (.3,.5,.7), (), (.3,.5,.7) } A (1) (M) average ={.9666,.3,.8666,.8666,.7,.8666,.5,.5,.5,,.5}. A (1) (M) Max(Weight) {1 (1) } = A 1 A (1) 1 (M) average ={,.8376,.8376,.6766,.6766,.6766,.6766,.8376,.8376,.9343,.8376} A (1) 1 (M) Max(Weight) = { 1 (1) } = A 2 A (1) (!) =A 2 Step:11 Attribute 11 is ON: A (11) ={ 1}. A (11) (M) Weight = {(.7,.9,1), (.3,.5,.7), (.7,.9,1), (.7,.9,1), (.9,1,1), (.5,.7,.9), (.3,.5,.7), (.3,.5,.7), (.3,.5,.7), (.3,.5,.7),() }
177 A (11) (M) average ={.866,.5,.8666,.8666,.966,.7,.5,.5,.5,.5, } A (11) (M) Max(Weight) { 1 (11) } = A 1 A (11) 1 (M) average ={.6766,.9343,.8376,.4833,,.8376,.6766,.6766,.8376,.6766,.8376} A (11) 1 (M) Max(Weight) {1 (11) } = A 2 A (11) 2 (M) average ={.866,,.4671,.4671,.4671,.93,.654,.896,.896,.282,.4671} A (11) 2 (M) Max(Weight) { 1 (11) } = A 3 A (11) (11) 1 = A 3 (fixed point).
178 Weightage of the attributes.4833.2899.8376.8376.8376.6766.9343.4833.4833.8376 1.8376.9343.8376.8376.6766.6766.6766.6766.8376.8376 1.6766.6766.9343.8376.8376.6766.6766.8376.8376.6766 1.4833.4833.9343.4833.4833.8376.4833.4833.8376.8376 1.6766.8376.4833.8376.9343.8376.2899.8376.6766.6766 1.654.896.4671.896.9343.896.282.896.654.654 1.896.654.896.654.654.896.4671.896.9343.654 1.4515.4515.4515.4515.7825.6321.7825.7825.279.8728 1.4671.282.896.896.896.654.93.4671.4671.896 1.6766.6766.8376.6766.6766.8376.4833.8376.9343.6766 1.4833.4833.4833.4833.8376.6766.8376.8376.2899.9343 1 11 1 9 8 7 6 5 4 3 2 1 c c c c c c c c c c c Attributes.69.5129.6319.593.771.5999.6349.5388.7294.57.6936 6.6995 5.6421 6.9515 6.4941 7.7787 6.5954 6.984 5.9275 8.234 6.2728 7.6297 Totalaverageweight Totalweight
179 8.9 ONLUSION Using A new fuzzy model iangular Fuzzy ognitive Maps (NMs) enabled the ranking of the Miracles through Holy Bible as Faith(.7294), Hearing the word of God(.771), Persistence/ Perseverance in prayer(.6936), God,s ompassion(.6349), Love (.6319),Repentance(.69), Forgiveness(.5999), Humility(.593), Patience(.57) Authority in the spiritual realm (.5388).Obedience(.5129).When Neutrosophic ognitive Maps (NM) was used the above causes are ON stage and indeterminacy of relations between any two concepts.. But this new model gave the ranking of the causes of the problem. This is the beauty of this iangular Neutrosophic ognitive Maps (NM).