OPSEARCHDOI 10.1007/s12597-014-0194-1

APPLICATION ARTICLE

Arithmetic operations on generalizedintuitionistic fuzzy number and its applicationsto transportation problem

Dipankar Chakraborty ·Dipak Kumar Jana ·Tapan Kumar Roy

Accepted: 18 October 2014© Operational Research Society of India 2014

Abstract Intuitionistic fuzzy has always been a subject of keen interest, and a rig-orous research has also been done on it. However, those research works were mainlybased on normal intuitionistic fuzzy- a generalized approach to it could hardly beseen. So in this paper, we have developed a generalized intuitionistic fuzzy numberand its arithmetic operations. It is a unique attempt made by us in which for the firsttime two basic generalized intuitionistic fuzzy numbers namely generalized trape-zoidal and generalized triangular intuitionistic fuzzy numbers have been consideredto serve the purpose. All arithmetic operations have been formulated on the basisof (α, β)-cut method, vertex method and extension principle method. Comparisonamong those three methods using an example is given and numerical results havebeen presented graphically. A new method is proposed to solve generalized intu-itionistic fuzzy transportation problem (GIFTP) using ranking function. To validatethe proposed method we have solved a GIFTP by assuming transportation cost, sup-ply and demand of the product in generalized intuitionistic fuzzy numbers and the

D. ChakrabortyDepartment of Mathematics, Heritage Institute of Technology,Anandapur, Kolkata 700107, West Bengal, Indiae-mail: dipankar1920@gmail.com

D. K. Jana (�)Department of Applied Science, Haldia Institute of Technology,Haldia Purba Midnapur 721657, West Bengal, Indiae-mail: dipakjana@gmail.com

T. K. RoyDepartment of Mathematics, Indian Institute of Engineering Science and Technology,Shibpur, Howrah 711103, West Bengal, Indiae-mail: roy t k@yahoo.co.in

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optimum results have been compared with the results of normal intuitionistic fuzzytransportation problem.

Keywords Generalized intuitionistic fuzzy number · (α · β)-cut · Vertex method ·Extension principle · Ranking function · Transportation problem

1 Introduction

In reality decision-making problems display some level of imprecision and vague-ness in estimation of parameters. However in modelling such problems fuzzy setshave proved to be very helpful. Applications of fuzzy set theory in decision mak-ing (c.f. Jana et al. [17–19]) and in particular to optimization problems have beenwidely studied ever since the introduction of fuzzy sets by Zadeh [24]. Recent yearshave witnessed a growing interest in the study of decision making problems withintuitionistic fuzzy sets/numbers. The intuitionistic fuzzy set (IFS) is an extensionof fuzzy set. IFS was first introduced by Atanassov [11]. Fuzzy sets is characterizedby the membership function only but IFS is characterized by a membership functionand a non-membership function so that the sum of both values is less than one [12].Presently IFSs are being studied and used in different fields of sciences and tech-nologies. Several researchers have shown their interest in the field of IFS (c.f. [1–3,5, 6, 13, 15, 29]).

The ranking of fuzzy numbers is an important factor in the study of fuzzy set the-ory. In order to rank fuzzy numbers, one fuzzy number needs to be compared withthe others by using ranking function. Rezvani [16], Kaur and Kumar [14] have showntheir interest in the applications of ranking function. Recently different definitionsof intuitionistic fuzzy number (IFN) have been proposed along with the correspond-ing ranking function of IFNs. The arithmetic operations and the ranking function ofIFNs have been implemented by many researchers ( c.f. [8, 26, 27]). Li [25] pro-posed extension principles for interval-valued intuitionistic fuzzy sets and algebraicoperations. Wang and Zhang [7] presented aggregation operators on intuitionistictrapezoidal fuzzy number and its application to multi-criteria decision making prob-lems. Farhadinia and Ban [30] developed new similarity measures of generalizedintuitionistic fuzzy numbers and generalized interval-valued fuzzy numbers fromsimilarity measures of generalized fuzzy numbers. Seikh et al. [31] discussed gen-eralized triangular fuzzy numbers in intuitionistic fuzzy environment. Zhang andLiu [32] presented the method for aggregating triangular fuzzy intuitionistic fuzzyinformation and its application to decision making. Wang et al. [4] proposed newoperators on triangular intuitionistic fuzzy numbers and their applications in systemfault analysis.

Recently, the IFN has also found its significance in linear programming, trans-portation problem etc. Parvathi and Malathi [20] have proposed intuitionistic fuzzysimplex method. Several algorithms ([21, 23]) are proposed for solving the trans-portation problems in intuitionistic fuzzy environment but in all the algorithms theparameters are represented by normal intuitionistic fuzzy numbers. Mahapatra and

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Roy [9] proposed normal intuitionistic fuzzy number and its arithmetic operationsby extension principle and its application to system failure. So our endeavour in thispaper has been to make the following improvements on intuitionistic fuzzy and itsapplication.

• First time we have defined generalized triangular intuitionistic fuzzy number andgeneralized trapezoidal intuitionistic fuzzy number i.e. the present paper dealswith generalized IFNs.

• The arithmetic operations on generalized intuitionistic fuzzy number using threeprinciple methods like (α, β)-cut method, vertex method and extension principlemethod have been discussed extensively.

• To validate the accuracy of those three methods a comparative study has beenmade and the desired results have been deduced numerically and graphically.

• To show the importance of generalized intuitionistic fuzzy in real life, a trans-portation problem has been solved and optimum results have been compared withthe normal intuitionistic fuzzy transportation problem.

The rest of this paper is organized as follows. In Section 2, we proposed general-ized intuitionistic fuzzy number and recall some preliminary knowledge. Section 3provided arithmetic operation based on (α, β)-cut method, vertex method and exten-sion principle method. In Section 4, we have given a comparative study among thosethree methods discussed in Section 3. The ranking function of generalized IFNsusing mean of (α, β)-cut has been provided in Section 5. In Section 6, a methodis proposed to solve generalized intuitionistic fuzzy transportation problem and anumerical example is provided in Section 7 to validate the proposed method. Section8 summarizes the paper and also discusses about the scope of future work.

2 Preliminaries

Definition 2.1 IFS([11, 12]): Let E be a given set and let A ⊂ E be a set. An IFS A∗in E is given by A∗ = {< x, μA(x), νA(x) >; x ∈ E} where μA : E → [0, 1] andνA : E → [0, 1] define the degree of membership and the degree of non-membershipof the element x ∈ E to A ⊂ E satisfy the condition 0 ≤ μA(x) + νA(x) ≤ 1.

Definition 2.2 IFN[9]: An IFN ˜AI is

(i) intuitionistic fuzzy subset on real line,(ii) there exist m ∈ � , μ

˜AI (m) = 1 , ν˜AI (m) = 0.

(iii) convex for the membership function μ˜AI i.e. μ

˜AI (λx1 + (1 − λ)x2) ≥min(μ

˜AI (x1), μ˜AI (x2)), x1, x2 ∈ R, λ ∈ [0, 1].

(iv) concave for the non-membership function ν˜AI i.e. ν

˜AI (λx1 + (1 − λ)x2) ≤max(ν

˜AI (x1), ν˜AI (x2)), x1, x2 ∈ R, λ ∈ [0, 1].

Definition 2.3 [10] An interval number is a closed and bounded set of real numbers[a, b] = {x : a ≤ x ≤ b∀x, b, x ∈ �}.

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(i) The addition of two interval numbers A = [a1, a2] and B = [b1, b2] denotedby A(+)B and is defined by A(+)B = [a1 + b1, a2 + b2].

(ii) The subtraction of two interval numbers A = [a1, a2] and B = [b1, b2]denoted by A(−)B and defined by A(−)B = [a1 − b2, a2 − b1].

(iii) The scalar multiplication of interval number A = [a1, a2] is denoted by kA

where k is scalar and defined by

kA ={

[ka1, ka2] ; if k ≥ 0,

[ka2, ka1] ; if k < 0

(iv) The product of two interval numbers A = [a1, a2] and B = [b1, b2]denoted by A(.)B and is defined by A(.)B = [p, q], where p =min(a1b1, a2b1, a1b2, a2b2) and q = max(a1b1, a2b1, a1b2, a2b2).

(v) The division of two interval numbers A = [a1, a2] and B = [b1, b2] denotedby A(÷)B and is defined by

A(÷)B =

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

[a1, a2] (.)[

1b1

, 1b2

]

, if 0 /∈ [b1, b2] ;emptyinterval, if b1 = b2 = 0;[a1, a2] (.)

[

1b2

, ∞]

, if b1 = 0, b2 �= 0;[a1, a2] (.)(−∞, 1

b1], if b1 �= 0, b2 = 0;

�a1,a2 [b1,0)∪ �a1,a2

(0,b2] , otherwise.

Definition 2.4 Vertex Method [22]: When y = f (x1, x2, · · · , xn) is continuous inthe n−dimensional rectangular region, and also no extreme point exists in this region(including the boundaries), then the value of interval function can be obtained by

Y = f (X1, X2, · · · , Xn) =[

minj

(

f (cj ))

, maxj

(

f (cj ))

]

, j = 1, 2, · · · , N

where cj is the ordinate of the j − th vertex and X1, X2, · · · , Xn are interval of realnumbers.

Definition 2.5 Extension Principle for Intuitionistic Fuzzy Sets [28]: Let f :X → Y be a mapping from a set X to a set Y ., then the extension principle allows usto define the IFS ˜BI in Y induced by IFS ˜AI in X through f as follows

˜BI = {

< y, μ˜BI (y), ν

˜BI (y) >: y = f (x), x ∈ X}

with μ˜BI (y) =

{

supy=f (x) μ˜AI (x), y ∈f (X);

0, y /∈f (X).

and ν˜BI (y) =

{

infy=f (x) ν˜AI (x), y ∈ f (X);

1, y /∈ f (X).where f −1(y) �= φ is the inverse

image of y.

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3 Generalized intuitionistic fuzzy number

In the previous section we have already discussed the definition of the IFN. Now inthis section we are going to introduced the definition of the generalized intuitionisticfuzzy number (GIFN).

Definition 3.1 GIFN: An intuitionistic fuzzy number ˜AI = {< x, μ˜AI , ν

˜AI >} ofthe real line � is called GIFN, if the followings holds

(i) there exist m ∈ � , μ˜AI (m) = w , ν

˜AI (m) = 0, 0 < w ≤ 1.(ii) μ

˜AI is continuous mapping from � to the interval (0, w] and x ∈ �, the relation0 ≤ μ

˜AI (x) + ν˜AI (x) ≤ w holds.

The membership function and non-membership function of ˜AI is of the followingform

μ˜AI (x) =

⎧

⎪

⎪

⎨

⎪

⎪

⎩

wf1(x), m − α ≤ x ≤ m;w, x = m;wh1(x), m ≤ x ≤ m + β;0, otherwise.

The non-membership function is of the following form

ν˜AI (x) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

wf2(x), m − α′ ≤ x ≤ m, 0 < w(f1(x) + f2(x)) ≤ w;0, x = m;wh2(x), m ≤ x ≤ m + β ′, 0 < w(h1(x) + h2(x)) ≤ w;w, otherwise.

Here f1(x) and h1(x) are strictly increasing and decreasing function in [m − α, m]and [m, m + β] respectively and f2(x) and h2(x) are strictly decreasing and increas-ing function in [m − α′, m] and [m, m + β ′] respectively, where m is the meanvalue of ˜AI . α and β are called the left and right spreads of membership functionμI˜A(x) respectively. α′ and β ′ are called the left and right spreads of non-membership

function νI˜A(x) respectively.

Now we will consider the two particular cases of the above definition as a gen-eralized trapezoidal intuitionistic fuzzy number(GTIFN) and generalized triangularintuitionistic fuzzy number(GTrIFN).

Definition 3.2 GTIFN: Let a′1 ≤ a1 ≤ a2 ≤ a3 ≤ a4 ≤ a′

4. A GTIFN˜AI in � written as (a1, a2, a3, a4; w)(a′

1, a2, a3, a′4; w) has membership function

(in Fig. 1)

μ˜AI (x) =

⎧

⎪

⎪

⎨

⎪

⎪

⎩

w x−a1a2−a1

, a1 ≤ x ≤ a2;w, a2 ≤ x ≤ a3;w

a4−xa4−a3

, a3 ≤ x ≤ a4;0, otherwise.

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

a’1

a1

a2

a3

a4

a’4

Fig. 1 Membership and non-membership function of GTIFN

and non-membership function

ν˜AI (x) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

w a2−x

a2−a′1, a′

1 ≤ x ≤ a2;0, a2 ≤ x ≤ a3;w

x−a3a′

4−a3, a3 ≤ x ≤ a′

4;w, otherwise.

Definition 3.3 GTrIFN: Let a′1 ≤ a1 ≤ a2 ≤ a3 ≤ a′

3. A GTrIFN ˜AI in � writtenas (a1, a2, a3; w)(a′

1, a2, a′3; w) has membership function (in Fig. 2)

μ˜AI (x) =

⎧

⎪

⎪

⎨

⎪

⎪

⎩

w x−a1a2−a1

, a1 ≤ x ≤ a2;w, x = a2;w

a3−xa3−a2

, a2 ≤ x ≤ a3;0, otherwise.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a’1

a1

a2

a3

a’3

Fig. 2 Membership and non-membership function of GTrIFN

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and non-membership function

ν˜AI (x) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

w a2−x

a2−a′1, a′

1 ≤ x ≤ a2;0, x = a2;w

x−a2a′

3−a2, a2 ≤ x ≤ a′

3;w, otherwise.

Some basic properties of GTIFN are given below which can directly follow fromthe IFN.

Definition 3.4 A GTIFN ˜AI = (a1, a2, a3, a4; w)(a′1, a2, a3, a′

4; w) is said to bepositive iff a′

1 ≥ 0.

Definition 3.5 Two GTIFN ˜AI = (a1, a2, a3, a4; w1)(a′1, a2, a3, a′

4; w1) and˜BI = (b1, b2, b3, b4; w2)(b

′1, b2, b3, b′

4; w2) are said to be equal iff a1 = b1, a2 =b2, a3 = b3, a4 = b4, a′

1 = b′1, a′

4 = b′4 and w1 = w2.

Definition 3.6 α-cut set: A α-cut set of ˜AI = (a1, a2, a3, a4; w)(a′1, a2, a3, a′

4; w)

is a crisp subset of � which is defined as follows

Aα = {

x : μ˜AI (x) ≥ α

} = [A1(α), A2(α)]=[

a1 + α

w(a2 − a1), a4 − α

w(a4 − a3)

]

Definition 3.7 β-cut set: A β-cut set of ˜AI = (a1, a2, a3, a4; w)(a′1, a2, a3, a′

4; w)

is a crisp subset of � which is defined as follows

Aβ = {

x : ν˜AI (x) ≤ β

}= [

A′1(β), A′

2(β)]=

[

a2 − β

w(a2 − a′

1), a3 + β

w(a′

4 − a3)

]

Definition 3.8 (α, β)-cut set: A (α, β)-cut set of ˜AI =(a1, a2, a3, a4; w)(a′

1, a2, a3, a′4; w) is given by

Aα,β = {[A1(α), A2(α)]; [A′1(β), A′

2(β)], 0 < α + β ≤ w, α, β ∈ (0, w]}

4 Arithmetic operations of GIFN

In this section we are going to discussed the arithmetic operations of GTIFN on thebasis of (α, β)-cut method, vertex method and extension principle method.

Property 4.1 Let ˜AI = (a1, a2, a3, a4; w1)(a′1, a2, a3, a′

4; w1) and ˜BI =(b1, b2, b3, b4; w2)(b

′1, b2, b3, b′

4; w2) be two positive GTIFN, then the addition oftwo GTIFN is given by (in Fig. 3)

˜AI⊕˜BI =(a1+b1, a2+b2, a3+ b3, a4+ b4; w)(

a′1+b′

1, a2+b2, a3+b3, a′4+b′

4; w)

where 0 < w ≤ 1, w = min(w1, w2).

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

a’1+b’

1a1+b

1a2+b

2a3+b

3a4+b

4a’4+b’

4

Fig. 3 Addition of two GTIFN(˜AI = −−, ˜BI = .. and ˜AI ⊕ ˜BI = −))

Proof Addition of two GTIFN based on (α, β)-cut method: Let ˜AI ⊕ ˜BI = ˜CI ,where Cα = [C1(α), C2(α)] and Cβ = [C′

1(β), C′2(β)], α, β ∈ (0, w], 0 < w ≤ 1

and w = min(w1, w2).

Now Cα = [C1(α), C2(α)]= [A1(α), A2(α)] + [B1(α), B2(α)]= [A1(α) + B1(α), A2(α) + B2(α)]=

[

a1+b1+ α

w{(a2−a1)+(b2−b1)}, a4+b4− α

w{(a4−a3)+(b4 − b3)}

]

Let a1 +b1 + αw

{(a2 −a1)+ (b2 −b1)} ≤ z ≤ a4 +b4 − αw

{(a4 −a3)+ (b4 −b3)}.Now a1 + b1 + α

w{(a2 − a1) + (b2 − b1)} ≤ z ⇒ w z−(a1+b1)

(a2+b2)−(a1+b1)≥ α

Let μLc (z) = w

z−(a1+b1)(a2+b2)−(a1+b1)

. Now dμLc (z)

dz= w

(a2+b2)−(a1+b1)> 0, if (a2 + b2) >

(a1 +b1). Therefore μLc (z) is an increasing function. Also μL

c (a1 +b1) = 0, μLc (a2 +

b2) = w and μLc (a1+b1+a2+b2

2 ) > w2 . Again a4 + b4 − α

w{(a4 − a3) + (b4 − b3) ≥

z ⇒ w(a4+b4)−z

(a4+b4)−(a3+b3)≥ α Let μR

c (z) = w(a4+b4)−z

(a4+b4)−(a3+b3). Therefore dμR

c (z)

dz=

− w(a4+b4)−(a3+b3)

< 0, if (a4 + b4) > (a3 + b3). Therefore μRc (z) is a decreasing

function. Also μRc (a4 + b4) = 0, μR

c (a3 + b3) = w and μRc

(

a3+b3+a4+b42

)

< w2 . So

the membership function of ˜C = ˜A ⊕ ˜B is

μ˜CI (z) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

wz−(a1+b1)

(a2+b2)−(a1+b1), a1 + b1 ≤ z ≤ a2 + b2;

w, a2 + b2 ≤ z ≤ a3 + b3;w

(a4+b4)−z(a4+b4)−(a3+b3)

, a3 + b3 ≤ z ≤ a4 + b4;0, otherwise.

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Hence addition rule is proved for membership function. Now for non-membershipfunction

Cβ = [C′1(β), C′

2(β)]= [A′

1(β), A′2(β)] + [B ′

1(β), B ′2(β)]

= [A′1(β) + B ′

1(β), A′2(β) + B ′

2(β)]=

[

a2+b2− β

w

{

(a2−a′1)+(b2 −b′

1)}

, a3 + b3 + β

w

{

(a′4 − a3) + (b′

4 − b3)}

]

Let a2 + b2 − βw

{(a2 − a′1) + (b2 − b′

1)} ≤ z ≤ a3 + b3 + βw

{(a′4 − a3) + (b′

4 − b3)}.Now a2 + b2 − β

w{(a2 − a′

1) + (b2 − b′1)} ≤ z ⇒ w

(a2+b2)−z

(a2+b2)−(a′1+b′

1)≤ β

Let νLc (z) = w (a2+b2)−z

(a2+b2)−(a′1+b′

1). Now dνL

c (z)

dz= − w

(a2+b2)−(a′1+b′

1)< 0, if (a2 + b2) >

(a′1 + b′

1). Therefore νLc (z) is a decreasing function. Also νL

c (a2 + b2) = 0, νLc (a′

1 +b′

1) = w and μLc (

a2+b2+a′1+b′

12 ) < w

2 . Again a3 + b3 + βw

{(a′4 + b′

4) + (a3 + b3) ≥z ⇒ w

z−(a3+b3)

(a′4+b′

4)−(a3+b3)≤ β Let νR

c (z) = wz−(a3+b3)

(a′4+b′

4)−(a3+b3). Therefore dνR

c (z)

dz=

w(a′

4+b′4)−(a3+b3)

> 0, if (a′4 + b′

4) > (a3 + b3). Therefore νRc (z) is an increasing

function. Also νRc (a3 + b3) = 0, νR

c (a′4 + b′

4) = w and νRc

(

a3+b3+a′4+b′

42

)

> w2 . So

the non-membership function of ˜C = ˜A ⊕ ˜B is

ν˜CI (z) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

w(a2+b2)−z

(a2+b2)−(a′1+b′

1), a′

1 + b′1 ≤ z ≤ a2 + b2;

0, a2 + b2 ≤ z ≤ a3 + b3;w

z−(a3+b3)

(a′4+b′

4)−(a3+b3), a3 + b3 ≤ z ≤ a′

4 + b′4;

w, otherwise.

Hence addition rule is proved for non-membership function. Thus we have

˜AI ⊕˜BI = (a1+b1, a2+b2, a3+b3, a4+b4; w)(a′1+b′

1, a2+b2, a3+b3, a′4+b′

4; w)

where 0 < w ≤ 1, w = min(w1, w2).

Addition of two GTIFN based on vertex method: Let ˜CI = ˜AI + ˜BI =f (˜AI , ˜BI). Now ordinate of the vertices for membership function are

c1 =(

a1 + α

w(a2 − a1), b1 + α

w(b2 − b1)

)

c2 =(

a1 + α

w(a2 − a1), b4 − α

w(b4 − b3)

)

c3 =(

a4 − α

w(a4 − a3), b1 + α

w(b2 − b1)

)

c4 =(

a4 − α

w(a4 − a3), b4 − α

w(b4 − b3)

)

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Therefore

f (c1) = a1 + b1 + α

w(b2 + a2 − a1 − b1)

f (c2) = a1 + b4 + α

w(a2 − b4 − a1 + b3)

f (c3) = a4 + b1 + α

w(b2 − a4 + a3 − b1)

f (c4) = a4 + b4 + α

w(a3 + b3 − a4 − b4)

Now since a1 ≤ a2 ≤ a3 ≤ a4 and b1 ≤ b2 ≤ b3 ≤ b4 so f (c1) < f (c2) < f (c3) <

f (c4). Hence

Y = [min (f (c1), f (c2), f (c3), f (c4)) , max (f (c1), f (c2), f (c3), f (c4))]

= [f (c1), f (c4)]

=[

a1 + b1 + α

w(b2 + a2 − a1 − b1) , a4 + b4 + α

w(a3 + b3 − a4 − b4)

]

As explained in (α, β)-cut method, the addition rule is obviously proved for mem-bership function. Now the ordinate of the vertices for nonmembership functionare

c′1 =

(

a2 − β

w(a2 − a′

1), b2 − β

w(b2 − b′

1)

)

c′2 =

(

a2 − β

w(a2 − a′

1), b3 + β

w(b′

4 − b3)

)

c′3 =

(

a3 + β

w(a′

4 − a3), b2 − β

w(b2 − b′

1)

)

c′4 =

(

a3 + β

w(a′

4 − a3), b3 + β

w(b′

4 − b3)

)

Therefore

f (c′1) = a2 + b2 − β

w

(

b2 + a2 − a′1 − b′

1

)

f (c′2) = a2 + b3 + β

w

(

a′1 − a2 + b′

4 − b3)

f (c′3) = a3 + b2 + β

w

(

a′4 − a3 − b2 + b′

1

)

f (c′4) = a3 + b3 + β

w

(

a′4 + b′

4 − a3 − b3)

Now since a′1 ≤ a2 ≤ a3 ≤ a′

4 and b′1 ≤ b2 ≤ b3 ≤ b′

4 so f (c′1) < f (c′

2) < f (c′3) <

f (c′4). Hence

Y ′ = [

min(

f (c′1), f (c′

2), f (c′3), f (c′

4))

, max(

f (c′1), f (c′

2), f (c′3), f (c′

4))]

= [

f (c′1), f (c′

4)]

=[

a2 + b2 − β

w

(

b2 + a2 − (a′1 + b′

1))

, a3 + b3 + β

w

(

a′4 + b′

4 − (a3 + b3))

]

OPSEARCH

As explained in (α, β)-cut method, the addition rule is proved for nonmembershipfunction. Thus we have

˜AI ⊕˜BI = (a1+b1, a2+b2, a3+b3, a4+b4; w)(a′1+b′

1, a2+b2, a3+b3, a′4+b′

4; w)

Addition of two GTIFN based on extension principle: Let ˜AI ⊕ ˜BI =˜CI where μCI (z) = sup(min(μAI (x), μBI (y)); x + y = z) and ν

˜CI (z) =inf(max(νAI (x), νBI (y)); x + y = z). Let w = min(w1, w2), then

μ˜CI (z)=

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

μL˜CI

(z)=sup(

min(w1x−a1a2−a1

, w2y−b1b2−b1

) : x+y =z)

, a1 ≤ x ≤ a2, b1 ≤ y ≤ b2;sup(min(w1, w2) : x + y =z), a2 ≤ x ≤ a3, b2 ≤ y ≤ b3;μR˜CI (z)=sup

(

min(w1a4−xa4−a3

, w2b4−yb4−b3

) : x+y =z)

, a3 ≤ x ≤ a4, b3 ≤ y ≤ b4;0, otherwise.

Hence

μ˜CI (z)=

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

μL˜CI

(z) = sup(

min(w x−a1a2−a1

, w z−x−b1b2−b1

))

, a1 ≤ x ≤ a2, a1 + b1 ≤ z ≤ a2 + b2;w, a2 ≤ x ≤ a3, a2 + b2 ≤ z ≤ a3 + b3;μR˜CI

(z) = sup(

min(wa4−xa4−a3

, wb4−z+xb4−b3

))

, a3 ≤ x ≤ a4, a3 + b3 ≤ z ≤ a4 + b4;0, otherwise.

Now let min(

w x−a1a2−a1

, w z−x−b1b2−b1

)

= α s.t.

wx − a1

a2 − a1≥ α, w

z − x − b1

b2 − b1≥ α

⇒ x ≥ α

w(a2 − a1) + a1, z ≥ α

w(b2 − b1) + x + b1

⇒ z ≥ α

w(b2 − b1) + α

w(a2 − a1) + a1 + b1

⇒ wz − a1 − b1

(a2 + b2) − (a1 + b1)≥ α

⇒ μL˜CI (z) = w

z − a1 − b1

(a2 + b2) − (a1 + b1)= sup α.

Similarly we can prove that μR˜CI

(z) = w a4+b4−z(a4+b4)−(a3+b3)

= sup α. So the membership

function of ˜CI = ˜AI ⊕ ˜BI is

μ˜CI (z) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

wz−(a1+b1)

(a2+b2)−(a1+b1), a1 + b1 ≤ z ≤ a2 + b2;

w, a2 + b2 ≤ z ≤ a3 + b3;w

(a4+b4)−z(a4+b4)−(a3+b3)

, a3 + b3 ≤ z ≤ a4 + b4;0, otherwise.

OPSEARCH

Hence addition rule is proved for membership function. Now from definition [3.1]and [2.5](for non-membership function), we get

ν˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

νL˜CI (z) = inf

(

max(wa2−x

a2−a′1, w

b2−z+x

b2−b′1

))

, a′1 ≤ x ≤ a2, a

′1 + b′

1 ≤ z ≤ a2 + b2;0, a2 ≤ x ≤ a3, a2 + b2 ≤ z ≤ a3 + b3;νR˜CI

(z) = inf(

max(wx−a3a′

4−a3, w

z−x−b3b′

4−b3))

, a3 ≤ x ≤ a′4, a3 + b3 ≤ z ≤ a′

4 + b′4;

w, otherwise.

Now let max(

wa2−x

a2−a′1, w

b2−z+x

b2−b′1

)

= β s.t.

wa2 − x

a2 − a′1

≤ β, wb2 − z + x

b2 − b′1

≤ β

⇒ −x ≤ β

w(a2 − a′

1) − a2, −z ≤ β

w(b2 − b′

1) − x − b2

⇒ −z ≤ β

w(b2 − b′

1) + β

w(a2 − a′

1) − a2 − b2

⇒ w(a2 + b2) − z

(a2 + b2) − (a′1 + b′

1)≤ β

⇒ μL˜CI (z) = w

(a2 + b2) − z

(a2 + b2) − (a′1 + b′

1)= inf β.

Similarly we can prove that μR˜CI

(z) = wz−(a3+b3)

(a′4+b′

4)−(a3+b3)= inf β. So the non-

membership function of ˜CI = ˜AI ⊕ ˜BI is

ν˜CI (z) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

w(a2+b2)−z

(a2+b2)−(a′1+b′

1), a′

1 + b′1 ≤ z ≤ a2 + b2;

0, a2 + b2 ≤ z ≤ a3 + b3;w

z−(a3+b3)

(a′4+b′

4)−(a3+b3), a3 + b3 ≤ z ≤ a′

4 + b′4;

w, otherwise.

Hence addition rule is proved for non-membership function. Thus we have

˜AI ⊕˜BI = (a1+b1, a2+b2, a3+b3, a4+b4; w)(a′1+b′

1, a2+b2, a3+b3, a′4+b′

4; w)

Property 4.2 Let ˜AI = (a1, a2, a3, a4; w1)(a′1, a2, a3, a′

4; w1) and ˜BI =(b1, b2, b3, b4; w2)(b

′1, b2, b3, b′

4; w2) be two positive GTIFN,then the subtraction oftwo GTIFN is given by (in Fig. 4)

˜AI �˜BI = (a1−b4, a2−b3, a3−b2, a4−b1; w)(a′1−b′

4, a2−b3, a3−b2, a′4−b′

1; w)

where 0 < w ≤ 1, w = min(w1, w2).

Proof Subtraction of two GTIFN based on (α, β)-cut method: Let ˜AI �˜BI = ˜C,where Cα = [C1(α), C2(α)] and Cβ = [C′

1(β), C′2(β)], α, β ∈ (0, w], 0 < w ≤ 1

OPSEARCH

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

a’1−b’

4

a1−b

4a

2−b

3a

3−b

2a

4−b

1a’

4−b’

1

Fig. 4 Subtraction of two GTIFN (˜AI = − − −, ˜BI = · · · and ˜AI � ˜BI = −)

and w = min(w1, w2).

Now Cα = [C1(α), C2(α)]= [A1(α), A2(α)] − [B1(α), B2(α)]= [A1(α) − B2(α), A2(α) − B1(α)]=

[

a1−b4+ α

w{(a2−a1)−(b4−b3)}, a4−b1− α

w{(a4−b1)−(a3 − b2)}

]

Let a1 − b4 + αw

{(a2 − a1) − (b4 − b3)} ≤ z ≤ a4 − b1 − αw

{(a4 − b1) − (a3 − b2)}.Now a1 − b4 + α

w{(a2 − a1) − (b4 − b3)} ≤ z ⇒ w

z−(a1−b4)(a2−b3)−(a1−b4)

≥ α

Let μLc (z) = ww

z−(a1−b4)(a2−b3)−(a1−b4)

. Now dμLc (z)

dz= w

(a2−b3)−(a1−b4)> 0, if (a2 − b3) >

(a1 −b4). Therefore μLc (z) is an increasing function. Also μL

c (a1 −b4) = 0, μLc (a2 −

b3) = w and μLc

(

a1−b4+a2−b32

)

> w2 . Again a4 − b1 − α

w{(a4 − b1) − (a3 − b2)} ≥

z ⇒ w(a4−b1)−z

(a4−b1)−(a3−b2)≥ α Let μR

c (z) = w(a4−b1)−z

(a4−b1)−(a3−b2). Therefore dμR

c (z)

dz=

− w(a4−b1)−(a3−b2)

< 0, if (a4 − b1) > (a3 − b2). Therefore μRc (z) is a decreasing

function. Also μRc (a4 − b1) = 0, μR

c (a3 − b2) = w and μRc

(

a4−b1+a3−b22

)

< w2 . So

the membership function of ˜C = ˜A � ˜B is

μ˜CI (z) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

wz−(a1−b4)

(a2−b3)−(a1−b4), a1 − b4 ≤ z ≤ a2 − b3;

w, a2 − b3 ≤ z ≤ a3 − b2;w

(a4−b1)−z(a4−b1)−(a3−b2)

, a3 − b2 ≤ z ≤ a4 − b1;0, otherwise.

OPSEARCH

Hence subtraction rule is proved for membership function. Now for non-membership function

Cβ = [C′1(β), C′

2(β)]= [A′

1(β), A′2(β)] − [B ′

1(β), B ′2(β)]

= [A′1(β) − B ′

2(β), A′2(β) − B ′

1(β)]=

[

a2−b3− β

w{(a2−b3)−(a′

1 − b′4)}, a3 − b2 + β

w{(a′

4 − b′1) − (a3 − b2)}

]

Let a2 − b3 − βw

{(a2 − b3) − (a′1 − b′

4)} ≤ z ≤ a3 − b2 + βw

{(a′4 − b′

1) − (a3 − b2)}.Now a2 − b3 − β

w{(a2 − b3) − (a′

1 − b′4)} ≤ z ⇒ w

(a2−b3)−z

(a2−b3)−(a′1−b′

4)≤ β

Let νLc (z) = w

(a2−b3)−z

(a2−b3)−(a′1−b′

4). Now dνL

c (z)

dz= − w

(a2−b3)−(a′1−b′

4)< 0, if (a2 − b3) >

(a′1 − b′

4). Therefore νLc (z) is a decreasing function. Also νL

c (a2 − b3) = 0,

νLc

(

a′1 − b′

4

) = w and μLc (

a2−b3+a′1−b′

42 ) < w

2 . Again a3 − b2 + βw

{(a′4 − b′

1)− (a3 −b2)} ≥ z ⇒ w

z−(a3−b2)

(a′4−b′

1)−(a3−b2)≤ β

Let νRc (z) = w

z−(a3−b2)

(a′4−b′

1)−(a3−b2). Therefore dνR

c (z)

dz= w

(a′4−b′

1)−(a3−b2)> 0, if (a′

4 −b′

1) > (a3 − b2). Therefore νRc (z) is an increasing function. Also νR

c (a3 − b2) = 0,

νRc (a′

4 − b′1) = w and νR

c

(

a3−b2+a′4−b′

12

)

> w2 . So the non-membership function of

˜C = ˜A � ˜B is

ν˜CI (z) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

w(a2−b3)−z

(a2−b3)−(a′1−b′

4), a′

1 − b′4 ≤ z ≤ a2 − b3;

0, a2 − b3 ≤ z ≤ a3 − b2;w

z−(a3−b2)

(a′4−b′

1)−(a3−b2), a3 − b2 ≤ z ≤ a′

4 − b′1;

w, otherwise.

Hence subtraction rule is proved for non-membership function. Thus we have

˜AI �˜BI = (a1−b4, a2−b3, a3−b2, a4−b1; w)(a′1−b′

4, a2−b3, a3−b2, a′4−b′

1; w)

where 0 < w ≤ 1, w = min(w1, w2).Subtraction of two GTIFN based on vertex method: Let ˜CI = ˜AI � ˜BI =f (˜AI , ˜BI). Now ordinate of the vertices for membership function are

c1 =(

a1 + α

w(a2 − a1), b1 + α

w(b2 − b1)

)

c2 =(

a1 + α

w(a2 − a1), b4 − α

w(b4 − b3)

)

c3 =(

a4 − α

w(a4 − a3), b1 + α

w(b2 − b1)

)

c4 =(

a4 − α

w(a4 − a3), b4 − α

w(b4 − b3)

)

OPSEARCH

Therefore

f (c1) = a1 − b1 + α

w(a2 − b2 − a1 + b1)

f (c2) = a1 − b4 + α

w(a2 + b4 − a1 − b3)

f (c3) = a4 − b1 + α

w(b1 − a4 + a3 − b2)

f (c4) = a4 − b4 + α

w(a3 + b4 − a4 − b3)

Now since a1 ≤ a2 ≤ a3 ≤ a4 and b1 ≤ b2 ≤ b3 ≤ b4 so f (c2) < f (c1) < f (c4) <

f (c3). Hence

Y = [min (f (c1), f (c2), f (c3), f (c4)) , max (f (c1), f (c2), f (c3), f (c4))]

= [f (c2), f (c3)]

=[

a1 − b4 + α

w(a2 + b4 − a1 − b3) , a4 − b1 + α

w(b1 − a4 + a3 − b2)

]

Now as explained above ((α, β)-cut method) subtraction rule is proved for mem-bership function. Now the ordinate of the vertices for nonmembership functionare

c′1 =

(

a2 − β

w(a2 − a′

1), b2 − β

w(b2 − b′

1)

)

c′2 =

(

a2 − β

w(a2 − a′

1), b3 + β

w(b′

4 − b3)

)

c′3 =

(

a3 + β

w(a′

4 − a3), b2 − β

w(b2 − b′

1)

)

c′4 =

(

a3 + β

w(a′

4 − a3), b3 + β

w(b′

4 − b3)

)

Therefore

f (c′1) = a2 − b2 − β

w

(

a2 − b2 − a′1 + b′

1

)

f (c′2) = a2 − b3 − β

w

(

a2 − a′1 + b′

4 − b3)

f (c′3) = a3 − b2 + β

w

(

a′4 − a3 + b2 − b′

1

)

f (c′4) = a3 − b3 + β

w

(

a′4 − b′

4 − a3 + b3)

Now since a′1 ≤ a2 ≤ a3 ≤ a′

4 and b′1 ≤ b2 ≤ b3 ≤ b′

4 so f (c′2) < f (c′

1) < f (c′4) <

f (c′3). Hence

Y ′ = [

min(

f (c′1), f (c′

2), f (c′3), f (c′

4))

, max(

f (c′1), f (c′

2), f (c′3), f (c′

4))]

= [

f (c′2), f (c′

3)]

=[

a2 − b3 − β

w

(

a2 − a′1 + b′

4 − b3)

, a3 − b2 + β

w

(

a′4 − a3 + b2 − b′

1

)

]

OPSEARCH

As explained in (α, β)-cut method, the subtraction rule is obviously proved fornonmembership function. Thus we have

˜AI �˜BI = (a1−b4, a2−b3, a3−b2, a4−b1; w)(a′1−b′

4, a2−b3, a3−b2, a′4−b′

1; w)

Subtraction of two GTIFN based on extension principle: Let ˜AI � ˜BI =˜CI where μ

˜CI (z) = sup(min(μ˜AI (x), μ

˜BI (y)); x − y = z) and ν˜CI (z) =

inf(max(ν˜AI (x), ν

˜BI (y)); x − y = z). Let w = min(w1, w2), then

μ˜CI (z)=

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

μL˜CI (z)=sup

(

min(w1x−a1a2−a1

, w2b4−yb4−b3

) : x−y =z)

, a1 ≤ x ≤ a2, b3 ≤ y ≤ b4;sup(min(w1, w2) : x − y =z), a2 ≤ x ≤ a3, b2 ≤ y ≤ b3;μR˜CI (z)=sup

(

min(w1a4−xa4−a3

, w2y−b1b2−b1

) : x−y =z)

, a3 ≤ x ≤ a4, b1 ≤ y ≤ b2;0, otherwise.

Hence

μ˜CI (z)=

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

μL˜CI (z) = sup

(

min(wx−a1a2−a1

, wb4+z−xb4−b3

))

, a1 ≤ x ≤ a2, a1 − b4 ≤ z ≤ a2 − b3;w, a2 ≤ x ≤ a3, a2 − b3 ≤ z ≤ a3 − b2;μR˜CI

(z) = sup(

min(w a4−xa4−a3

, w x−z−b1b2−b1

))

, a3 ≤ x ≤ a4, a3 − b2 ≤ z ≤ a4 − b1;0, otherwise.

Now let min(

wx−a1a2−a1

, wb4−z−xb4−b3

)

= α s.t.

wx − a1

a2 − a1≥ α,w

b4 + z − x

b4 − b3≥ α

⇒ x ≥ α

w(a2 − a1) + a1, z ≥ α

w(b4 − b3) + x − b4

⇒ z ≥ α

w(b4 − b3) + α

w(a2 − a1) + a1 − b4

⇒ wz − (a1 − b4)

(a2 − b3) − (a1 − b4)≥ α

⇒ μL˜CI (z) = w

z − (a1 − b4)

(a2 − b3) − (a1 − b4)= sup α.

Similarly we can prove that μR˜CI

(z) = w(a4−b1)−z

(a4−b1)−(a3−b2)= sup α. So the membership

function of ˜CI = ˜AI � ˜BI is

μ˜CI (z) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

wz−(a1−b4)

(a2−b3)−(a1−b4), a1 − b4 ≤ z ≤ a2 − b3;

w, a2 − b3 ≤ z ≤ a3 − b2;w

(a4−b1)−z(a4−b1)−(a3−b2)

, a3 − b2 ≤ z ≤ a4 − b1;0, otherwise.

Hence subtraction rule is proved for membership function. Now from definition [3.1]and [2.5] (for non-membership function), we get

ν˜CI (z)=

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

νL˜CI

(z)= inf(

max(w a2−xa2−a1

, wx−z−b3b′

4−b3))

, a′1 ≤ x ≤ a2, a

′1− b′

4 ≤ z≤ a2− b3;0, a2 ≤ x ≤ a3, a2− b3 ≤ z≤ a3− b2;μR˜CI (z)= inf

(

max(

wx−a3a′

4−a3, w

b2+z−x

b2−b′1

))

, a3 ≤ x ≤ a′4, a3− b2 ≤ z≤ a′

4− b′1;

w, otherwise.

OPSEARCH

Now let max(

wa2−xa2−a1

, wx−z−b3b′

4−b3

)

= β s.t.

wa2 − x

a2 − a1≤ β,w

x − z − b3

b′4 − b3

≤ β

⇒ −x ≤ β

w(a2 − a′

1) − a2,−z ≤ β

w(b′

4 − b3) − x + b3

⇒ −z ≤ β

w(b′

4 − b3) + β

w(a2 − a′

1) − a2 + b3

⇒ w(a2 − b3) − z

(a2 − b3) − (a′1 − b′

4)≤ β

⇒ μL˜CI (z) = w

(a2 − b3) − z

(a2 − b3) − (a′1 − b′

4)= inf β.

Similarly we can prove that μR˜CI

(z) = wz−(a3−b2)

(a′4−b′

1)−(a3−b2)= inf β. So the non-

membership function of ˜CI = ˜AI � ˜BI is

ν˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

w(a2−b3)−z

(a2−b3)−(a′1−b′

4), a′

1 − b′4 ≤ z ≤ a2 − b3;

0, a2 − b3 ≤ z ≤ a3 − b2;w

z−(a3−b2)

(a′4−b′

1)−(a3−b2), a3 − b2 ≤ z ≤ a′

4 − b′1;

w, otherwise.

Hence subtraction rule is proved for non-membership function.

Property 4.3 Let ˜AI = (a1, a2, a3, a4;w)(a′1, a2, a3, a

′4;w) be a positive GTIFN, then ˜CI =

k˜AI is GTIFN and

k˜AI ={

(ka1, ka2, ka3, ka4;w)(ka′1, ka2, ka3, ka′

4;w), if k > 0;(ka4, ka3, ka2, ka1;w)(ka′

4, ka3, ka2, ka′1;w), if k < 0.

where 0 < w ≤ 1.

Proof Scalar multiplication of a GTIFN based on (α, β)-cut method:

Case-I: k > 0 Let k˜AI = ˜CI , where Cα = [C1(α), C2(α)] and Cβ = [C′1(β), C′

2(β)],α ∈ (0, w], 0 < w ≤ 1.

Now Cα = [C1(α), C2(α)]= k[A1(α),A2(α)]= [kA1(α), kA2(α]= [ka1 + k

α

w(a2 − a1), ka4 − k

α

w(a4 − a3)]

Let ka1 + k αw

(a2 − a1) ≤ z ≤ ka4 − k αw

(a4 − a3).Now ka1 + k α

w(a2 − a1) ≤ z ⇒ w z−ka1

ka2−ka1≥ α

Let μLc (z) = w

z−ka1ka2−ka1

. Now dμLc (z)

dz= w

(ka2−ka1> 0, if ka2 > ka1.

Therefore μLc (z) is an increasing function. Also μL

c (ka1) = 0, μLc (ka2) = w and

μLc

(

ka1+ka22

)

> w2 .

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Again ka4 − k αw

(a4 − a3) ≥ z ⇒ wka4−z

(ka4−ka3)≥ α Let μR

c (z) = wka4−z

(ka4−ka3). Therefore

dμRc (z)

dz= − w

ka4−ka3< 0, if ka4 > ka3. Therefore μR

c (z) is a decreasing function. Also

μRc (ka4) = 0, μR

c (ka3) = w and μRc

(

ka3+ka42

)

< w2 . So the membership function of

˜C = ˜A ⊕ ˜B is

μ˜CI (z) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

w z−ka1ka2−ka1

, ka1 ≤ z ≤ ka2;w, ka2 ≤ z ≤ ka3;w ka4−z

ka4−ka3, ka3 ≤ z ≤ ka4;

0, otherwise.

Hence addition rule is proved for membership function. Now for non-membershipfunction

Cβ = [C′1(β), C′

2(β)]= k[A′

1(β),A′2(β)]

= [kA′1(β), kA′

2(β)]=

[

ka2 − kβ

w(a2 − a′

1), ka3 + kβ

w(a′

4 − a3)

]

Let ka2 − kβw

(a2 − a′1) ≤ z ≤ ka3 + k

βw

(a′4 − a3).

Now ka2 − kβw

{(a2 − a′1) ≤ z ⇒ w

ka2−z

ka2−ka′1β

Let νLc (z) = w ka2−z

ka2−ka′1. Now dνL

c (z)

dz= − w

ka2−ka′1

< 0, if ka2 > ka′1. Therefore νL

c (z)

is a decreasing function. Also νLc (ka2) = 0, νL

c (ka′1) = w and μL

c

(

ka2+ka′1

2

)

< w2 .

Again ka3 + kβw

(a′4 − a3) ≥ z ⇒ w

z−ka3ka′

4−ka3≤ β Let νR

c (z) = wz−ka3

ka′4−ka3

. ThereforedνR

c (z)

dz= w

ka′4−ka3

> 0, if ka′4 > ka3. Therefore νR

c (z) is an increasing function. Also

νRc (ka3) = 0, νR

c (ka′4) = w and νR

c

(

ka3+ka′4

2

)

> w2 . So the non-membership function of

˜C = ˜A ⊕ ˜B is

ν˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

wka2−z

ka2−ka′1, ka′

1 ≤ z ≤ ka2;0, ka2 ≤ z ≤ ka3;w

z−ka3ka′

4−ka3, ka3 ≤ z ≤ ka′

4;w, otherwise.

Hence scalar multiplication rule is proved for non-membership function. Thus wehave

k˜A = (ka1, ka2, ka3, ka4;w)(ka′1, ka2, ka3, ka′

4;w)if if k > 0

where 0 < w ≤ 1.

Case-II: k < 0 Let k˜AI = ˜CI , where Cα = [C1(α), C2(α)] and Cβ = [C′1(β), C′

2(β)],α ∈ (0, w], 0 < w ≤ 1.

Now Cα = [C1(α), C2(α)]= k[A1(α),A2(α)]= [kA2(α), kA1(α]=

[

ka4 − kα

w(a4 − a3), ka1 + k

α

w(a2 − a1)

]

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Let ka4 − k αw

(a4 − a3) ≤ z ≤ ka1 + k αw

(a2 − a1).Now ka4 − k α

w(a4 − a3) ≤ z ⇒ w z−ka4

ka3−ka4≥ α

Let μLc (z) = w z−ka4

ka3−ka4. Now dμL

c (z)

dz= w

(ka3−ka4> 0, if ka3 > ka4.

Therefore μLc (z) is an increasing function. Also μL

c (ka4) = 0, μLc (ka3) = w and

μLc

(

ka3+ka42

)

> w2 .

Again ka1 + k αw

(a2 − a1) ≥ z ⇒ w ka1−z(ka1−ka2)

≥ α Let μRc (z) = ww ka1−z

(ka1−ka2). Therefore

dμRc (z)

dz= − w

ka1−ka2< 0, if ka1 > ka2. Therefore μR

c (z) is a decreasing function. Also

μRc (ka1) = 0, μR

c (ka3) = w and μRc

(

ka1+ka32

)

< w2 . So the membership function of

˜C = k˜A is

μ˜CI (z) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

wz−ka4

ka3−ka4, ka4 ≤ z ≤ ka3;

w, ka3 ≤ z ≤ ka2;w ka1−z

ka1−ka2, ka2 ≤ z ≤ ka1;

0, otherwise.

Hence scalar multiplication rule is proved for membership function. Now for non-membership function

Cβ = [C′1(β), C′

2(β)]= k[A′

1(β),A′2(β)]

= [kA′2(β), kA′

1(β)]=

[

ka3 + kβ

w(a′

4 − a3), ka2 − kβ

w(a2 − a′

1)

]

Let ka3 + kβw

(a′4 − a3) ≤ z ≤ ka2 − k

βw

(a2 − a′1).

Now ka3 + kβw

(a′4 − a3) ≤ z ⇒ w

ka3−z

ka3−ka′4β

Let νLc (z) = w

ka3−z

ka3−ka′4. Now dνL

c (z)

dz= − w

ka3−ka′4

< 0, if ka3 > ka′4. Therefore νL

c (z)

is a decreasing function. Also νLc (ka3) = 0, νL

c (ka′4) = w and μL

c

(

ka3+ka′4

2

)

< w2 .

Again ka2 − kβw

(a2 − a′1) ≥ z ⇒ w z−ka2

ka′1−ka2

≤ β Let νRc (z) = w z−ka2

ka′1−ka2

. ThereforedνR

c (z)

dz= w

ka′1−ka2

> 0, if ka′1 > ka2. Therefore νR

c (z) is an increasing function. Also

νRc (ka2) = 0, νR

c (ka′1) = w and νR

c

(

ka2+ka′1

2

)

> w2 . So the non-membership function of

˜C = k˜A is

ν˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

wka3−z

ka3−ka′4, ka′

4 ≤ z ≤ ka3;0, ka3 ≤ z ≤ ka2;w

z−ka2ka′

1−ka2, ka2 ≤ z ≤ ka′

1;w, otherwise.

Hence scalar multiplication rule is proved for non-membership function. Thus wehave

k˜AI = (ka4, ka3, ka2, ka1;w)(ka′4, ka3, ka2, ka′

1;w) if k < 0where 0 < w ≤ 1.Scalar multiplication of a GTIFN based on vertex method: Let ˜CI = k˜AI = f (˜AI ). Nowordinate of the vertices for membership function are

c1 =(

a1 + α

w(a2 − a1)

)

, c2 =(

a4 − α

w(a4 − a3)

)

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Therefore

f (c1) = k(

a1 + α

w(a2 − a1))

)

f (c2) = k(

a4 − α

w(a4 − a3)

)

Case-I: When k > 0f (c1) < f (c2). Hence

Y = [min (f (c1), f (c2)) , max (f (c1), f (c2))]

= [f (c1), f (c2)]

=[

k(

a1 + α

w(a2 − a1)

)

, k(

a4 − α

w(a4 − a3)

)]

Now as explained above ((α, β)-cut method) we get

k˜AI = (ka1, ka2, ka3, ka4;w)(ka′1, ka2, ka3, ka′

4;w)

Case-II When k < 0f (c1) > f (c2). Hence

Y = [min (f (c1), f (c2)) , max (f (c1), f (c2))]

= [f (c2), f (c1)]

=[

k(

a4 − α

w(a4 − a3)

)

, k(

a1 + α

w(a2 − a1)

)]

Now as explained above ((α, β)-cut method) we get

k˜AI = (ka4, ka3, ka2, ka1;w)(ka′4, ka3, ka2, ka′

1;w)

Scalar multiplication of two GTIFN based on extension principle: Let k.˜AI = ˜CI whereμCI (z) = sup(min(μAI (x)); kx = z) and ν

˜CI (z) = inf(max(νAI (x)) : kx = z).

Case-I: when k ≥ 0

μ˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

μL˜CI (z) = sup

(

min(w1x−a1a2−a1

: kx = z)

, a1 ≤ x ≤ a2;w : kx = z), a2 ≤ x ≤ a3;μR˜CI

(z) = sup(

min(w1a4−xa4−a3

: kx = z)

, a3 ≤ x ≤ a4;0, otherwise.

Let w1 = w hence

μ˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

μL˜CI (z) = sup

(

min(wzk −a1a2−a1

))

, a1 ≤ zk

≤ a2;w, a2 ≤ z

k≤ a3;

μR˜CI (z) = sup

(

min(wa4− z

k

a4−a3))

, a3 ≤ zk

≤ a4;0, otherwise.

Now let min(

wzk −a1a2−a1

)

= α s.t.

w

zk

− a1

a2 − a1≥ α ⇒ w

z − ka1

ka2 − ka1≥ α

⇒ μL˜CI (z) = w

z − ka1

ka2 − ka1= sup α.

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Similarly we can prove that μR˜CI (z) = w

ka4−zka4−ka3

= sup α.

Case-II: when k < 0

μ˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

μL˜CI (z) = sup

(

min(wa4− z

k

a4−a3))

, a3 ≤ zk

≤ a4;w, a2 ≤ z

k≤ a3;

μR˜CI (z) = sup

(

min(wzk −a1a2−a1

))

, a1 ≤ zk

≤ a2;0, otherwise.

Now let min(

wa4− z

k

a4−a3

)

= α s.t.

wa4 − z

k

a4 − a3≥ α ⇒ w

ka4 − z

ka4 − ka3≥ α

⇒ μL˜CI (z) = w

ka4 − z

ka4 − ka3= sup α.

Similarly we can prove that μR˜CI (z) = w

ka1−zka2−ka1

= sup α.

Hence scalar multiplication rule is proved for membership function.

Now from definition [3.1] and [2.5](for non-membership function), we getCase-I: when k ≥ 0

ν˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

νL˜CI

(z) = inf(

max(wa2− z

k

a2−a′1))

, a′1 ≤ z

k≤ a2;

w, a2 ≤ zk

≤ a3;νR˜CI (z) = inf

(

max(wzk −a3

a′4−a3

))

, a3 ≤ zk

≤ a′4;

0, otherwise.

Now let max(wa2− z

k

a2−a′1) = β s.t.

wa2 − z

k

a2 − a′1

≤ β ⇒ wka2 − z

ka2 − ka′1

≤ β

⇒ νL˜CI (z) = w

a2 − zk

a2 − a′1

= inf β.

Similarly we can prove that νR˜CI

(z) = wz−ka3

ka′4−ka3

= inf β.

Case-II: when k < 0

ν˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

νL˜CI (z) = inf

(

max(wa3− z

k

a3−a′4))

, a3 ≤ zk

≤ a4;w, a2 ≤ z

k≤ a3;

μR˜CI

(z) = inf(

max(wzk−a2

a′1−a2

))

, a1 ≤ zk

≤ a2;0, otherwise.

Now let max(

wa3− z

k

a3−a′4

)

= β s.t.

wa3 − z

k

a3 − a′4

≥ β ⇒ wka3 − z

ka3 − ka′4

≥ β

⇒ νL˜CI (z) = w

ka3 − z

ka3 − ka′4

= inf β.

Similarly we can prove that νR˜CI

(z) = w z−ka2ka′

1−ka2= inf β.

Hence scalar multiplication rule is proved for non-membership function also.

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Property 4.4 Let ˜AI = (a1, a2, a3, a4;w1)(a′1, a2, a3, a

′4;w1) and ˜BI =

(b1, b2, b3, b4;w2)(b′1, b2, b3, b

′4;w2) be two positive GTIFN, then the division of two GTIFN

is given by (in Fig. 5)

˜AI ÷ ˜BI =(

a1

b4,a2

b3,a3

b2,a4

b1;w

)(

a′1

b′4,a2

b3,a3

b2,a′

4

b′1;w

)

where 0 < w ≤ 1, w = min(w1, w2).

Proof Division of two GTIFN based on (α, β)-cut method: Let ˜AI ÷˜BI = ˜CI , where Cα =[C1(α), C2(α)] and Cβ = [C′

1(β), C′2(β)], α, β ∈ (0, w], 0 < w ≤ 1 and w = min(w1, w2).

Now Cα = [C1(α), C2(α)]= [A1(α),A2(α)] ÷ [B1(α), B2(α)]=

[

A1(α)

B2(α),A2(α)

B1(α)

]

=[

a1 + αw

(a2 − a1)

b4 − αw

(b4 − b3),a4 − α

w(a4 − a3)

b1 + αw

(b2 − b1)

]

Leta1+ α

w (a2−a1)

b4− αw (b4−b3)

≤ z ≤ a4− αw (a4−a3)

b1+ αw (b2−b1)

.

Nowa1+ α

w(a2−a1)

b4− αw (b4−b3)

≤ z ⇒ w zb4−a1(a2−a1)+z(b4−b3)

≥ α

Let μLc (z) = w

zb4−a1(a2−a1)+z(b4−b3)

. Now dμLc (z)

dz= w

a2b4−a1b3{(a2−a1)+z(b4−b3)}2 > 0, for (a2b4) >

a1b3i.e.a2b3

> a1b4

. Therefore μLc (z) is an increasing function. Also μL

c ( a2b3

) = w,

μLc ( a1

b4) = 0 and μL

c (

a2b3

+ a1b4

2 ) = wb4b4+b3

> w2 [since b3 < b4]. Again

a4− αw (a4−a3)

b1+ αw

(b2−b1)≥

z ⇒ wa4−zb1

(a4−a3)+z(b2−b1)≥ α Let μR

c (z) = wa4−zb1

(a4−a3)+z(b2−b1). Therefore dμR

c (z)

dz=

wa3b2−a4b2

{(a4+b4)−(a3+b3)}2 < 0, for a3b1 < a4b2i.e.a3b2

< a4b1

. Therefore μRc (z) is a decreasing func-

tion. Also μRc (

a3b2

) = w, μRc (

a4b1

) = 0 and μRc (

a3b2

+ a4b1

2 ) < w2 . So the membership function of

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

a’1/b’

4

a1/b

4

a2/b

3

a3/b

2

a4/b

1 a’4/b’

1

Fig. 5 Division of two GTIFN (˜AI = −−, ˜BI = .. and ˜AI ÷ ˜BI = −)

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˜C = ˜A ÷ ˜B is

μ˜CI (z) =

⎧

⎪

⎪

⎨

⎪

⎪

⎩

w zb4−a1(a2−a1)+z(b4−b3)

, a1b4

≤ z ≤ a2b3

;w,

a2b3

≤ z ≤ a3b2

;w a4−zb1

(a4−a3)+z(b2−b1),

a3b2

≤ z ≤ a4b1

;0, otherwise.

Hence division rule is proved for membership function.Now for non-membership function

Cβ = [C′1(β), C′

2(β)]= [A′

1(β),A′2(β)] ÷ [B ′

1(β), B ′2(β)]

=[

A′1(β)

B ′2(β)

,A′

2(β)

B ′1(β)

]

=[

a2 − βw

(a2 − a′1)

b3 + βw

(b′4 − b3)

,a3 + β

w(a′

4 − a3)

b2 − βw

(b2 − b′1)

]

Let a2− βw (a2−a′

1)

b3+ βw (b′

4−b3)≤ z ≤ a3+ β

w (a′4−a3)

b2− βw (b2−b′

1).

Now a2− βw (a2−a′

1)

b3+ βw (b′

4−b3)≤ z ⇒ w

a2−zb3(a2−a′

1)+z(b′4−b3)

≤ β

Let νLc (z) = w

a2−zb3(a2−a′

1)−(b′4−b3)

. Now dνLc (z)

dz= w

a′1b3−a2b′

4{(a2−a′

1)−(b′4−b3)}2 < 0, if a′

1b3 <

a′2b

′4i.e.

a′1

b′4

<a2b3

. Therefore νLc (z) is a decreasing function. Also νL

c

(

a′1

b′4

)

= w,

νLc

(

a2b3

)

= 0 and μLc

⎛

⎝

a′1

b′4+ a2

b3

2

⎞

⎠ < w2 .

Again a3+ βw (a′

4−a3)

b2− βw (b2−b′

1)≥ z ⇒ w

zb2−a3(a′

4−a3)+z(b2−b′1)

≤ β Let νRc (z) = w

zb2−a3(a′

4−a3)+z(b2−b′1)

.

Therefore dνRc (z)

dz= w

a′4b2−a3b′

1{(a′

4−a3)+z(b2−b′1)}2 > 0, for a′

4b′

1) >

a3b2

. Therefore νRc (z) is an

increasing function. Also νRc (

a3b2

) = 0, νRc (

a′4

b′1) = w and νR

c (

a3b2

+ a′4

b′1

2 ) > w2 . So the

non-membership function of ˜C = ˜A ÷ ˜B is

ν˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

wa2−zb3

z(b′4−b3)+(a2−a′

1),

a′1

b′4

≤ z ≤ a2b3

;0,

a2b3

≤ z ≤ a3b2

;w

zb2−a3(a′

4−a3)+z(b2−b′1)

,a3b2

≤ z ≤ a′4

b′1;

w, otherwise.

Hence division rule is proved for non-membership function. Thus we have

˜AI ÷ ˜BI =(

a1

b4,a2

b3,a3

b2,a4

b1;w

)(

a′1

b′4,a2

b3,a3

b2,a′

4

b′1;w

)

where 0 < w ≤ 1, w = min(w1, w2).Division of two GTIFN based on vertex method: Let ˜CI = ˜AI

˜BI= f (˜AI , ˜BI ). Now

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ordinate of the vertices for membership function are

c1 =(

a1 + α

w(a2 − a1), b1 + α

w(b2 − b1)

)

c2 =(

a1 + α

w(a2 − a1), b4 − α

w(b4 − b3)

)

c3 =(

a4 − α

w(a4 − a3), b1 + α

w(b2 − b1)

)

c4 =(

a4 − α

w(a4 − a3), b4 − α

w(b4 − b3)

)

Therefore

f (c1) = a1 + αw

(a2 − a1)

b1 + αw

(b2 − b1)

f (c2) = a1 + αw

(a2 − a1)

b4 − αw

(b4 − b3)

f (c3) = a4 − αw

(a4 − a3)

b1 + αw

(b2 − b1)

f (c4) = b1 + αw

(b2 − b1)

b4 − αw

(b4 − b3)

Now since a1 ≤ a2 ≤ a3 ≤ a4 and b1 ≤ b2 ≤ b3 ≤ b4 so f (c2) < f (c1) < f (c4) < f (c3).Hence

Y = [min (f (c1), f (c2), f (c3), f (c4)) , max (f (c1), f (c2), f (c3), f (c4))]

= [f (c2), f (c3)]

=[

a1 + αw

(a2 − a1)

b4 − αw

(b4 − b3),a4 − α

w(a4 − a3)

b1 + αw

(b2 − b1)

]

As explained in (α, β)-cut method, the division rule is proved for membershipfunction. Now the ordinate of the vertices for nonmembership function are

c′1 =

(

a2 − β

w(a2 − a′

1), b2 − β

w(b2 − b′

1)

)

c′2 =

(

a2 − β

w(a2 − a′

1), b3 + β

w(b′

4 − b3)

)

c′3 =

(

a3 + β

w(a′

4 − a3), b2 − β

w(b2 − b′

1)

)

c′4 =

(

a3 + β

w(a′

4 − a3), b3 + β

w(b′

4 − b3)

)

Therefore

f (c′1) = a2 − β

w(a2 − a′

1)

b2 − βw

(b2 − b′1)

f (c′2) = a2 − β

w(a2 − a′

1)

b3 + βw

(b′4 − b3)

f (c′3) = a3 + β

w(a′

4 − a3)

b2 − βw

(b2 − b′1)

f (c′4) = a3 + β

w(a′

4 − a3)

b3 + βw

(b′4 − b3)

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Now since a′1 ≤ a2 ≤ a3 ≤ a′

4 and b′1 ≤ b2 ≤ b3 ≤ b′

4 so f (c′2) < f (c′

1) < f (c′4) < f (c′

3).Hence

Y ′ = [

min(

f (c′1), f (c′

2), f (c′3), f (c′

4))

, max(

f (c′1), f (c′

2), f (c′3), f (c′

4))]

= [

f (c′2), f (c′

3)]

=[

a2 − βw

(a2 − a′1)

b3 + βw

(b′4 − b3)

,a3 + β

w(a′

4 − a3)

b2 − βw

(b2 − b′1)

]

Now as explained above ((α, β)-cut method) division rule is proved for nonmember-ship function. Thus we have

˜AI ÷ ˜BI = (a1

b4,a2

b3,a3

b2,a4

b1;w)(

a′1

b′4,a2

b3,a3

b2,a′

4

b′1;w)

where 0 < w ≤ 1, w = min(w1, w2).Division of two GTIFN based on extension principle method Let ˜AI

˜BI= ˜CI where

μCI (z) = sup(min(μAI (x), μBI (y)); xy

= z) and ν˜CI (z) = inf(max(νAI (x), νBI (y)); x

y=

z). Let w = min(w1, w2), then

μ˜CI (z)=

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

μL˜CI (z)= sup

(

min(

w1x−a1a2−a1

, w2b4−yb4−b3

)

: xy

=z)

, a1 ≤ x ≤ a2, b3 ≤ y ≤ b4;sup(min(w1, w2) : x

y=z), a2 ≤ x ≤ a3, b2 ≤ y ≤ b3;

μR˜CI (z)= sup

((

w1a4−xa4−a3

, w2y−b1b2−b1

)

: xy

=z)

, a3 ≤ x ≤ a4, b1 ≤ y ≤ b2;0, otherwise.

Hence

μ˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

μL˜CI (z) = sup

(

min(

wx−a1a2−a1

, wb4− x

z

b4−b3

))

, a1 ≤ x ≤ a2,a1b4

≤ z ≤ a2b3

;w, a2 ≤ x ≤ a3,

a2b3

≤ z ≤ a3b2

;μR˜CI (z) = sup

(

min(

wa4−xa4−a3

, wxz −b1

b2−b1

))

, a3 ≤ x ≤ a4,a3b4

≤ z ≤ a4b3

;0, otherwise.

Now let min(

wx−a1a2−a1

, wb4− x

z

b4−b3

)

= α s.t.

wx − a1

a2 − a1≥ α,w

b4 − xz

b4 − b3≥ α ⇒ x ≥ α

w(a2 − a1) + a1, zb4 ≥ α

w(zb4 − zb3) + x

⇒ zb4 ≥ α

w(zb4 − zb3) + α

w(a2 − a1) + a1

⇒ wzb4 − a1

(a2 − a1) + z(b4 − b3)≥ α

⇒ μL˜CI (z) = w

zb4 − a1

(a2 − a1) + z(b4 − b3)= sup α.

Similarly we can prove that μR˜CI

(z) = wa4−zb1

(a4−a3)+z(b2−b1)= sup α. So the membership

function of ˜C = ˜A ÷ ˜B is

μ˜CI (z) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

w zb4−a1(a2−(a1)+z(b4−b3)

, a1b4

≤ z ≤ a2b3

;w,

a2b3

≤ z ≤ a3b2

;w

a4−zb1(a4−a3)+z(b2−b1)

,a3b2

≤ z ≤ a4b1

;0, otherwise.

Hence division rule is proved for membership function. Now from definition [3.1]and [2.5](for non-membership function), we get

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ν˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

νL˜CI

(z) = inf(

max(

w a2−x

a2−a′1, w

xz −b3

b′4−b3

))

, a′1 ≤ x ≤ a2,

a′1

b′4

≤ z ≤ a2b3

;0, a2 ≤ x ≤ a3,

a2b3

≤ z ≤ a3b2

;νR˜CI (z) = inf

(

max(

wx−a3a′

4−a3, w

b2− xz

b2−b′1

))

, a3 ≤ x ≤ a′4,

a3b2

≤ z ≤ a′4

b′1;

w, otherwise.

Now let max(w a2−x

a2−a′1, w

xz−b3

b′4−b3

) = β s.t.

wa2 − x

a2 − a′1

≤ β,w

xz

− b3

b′4 − b3

) ≤ β ⇒ −x ≤ β

w(a2 − a′

1) − a2,−zb3 ≤ β

w(zb′

4 − zb3) − x

⇒ −zb3 ≤ β

w(zb′

4 − zb3) + β

w(a2 − a′

1) − a2

⇒ wa2 − zb3

(a2 − a′1) + z(b′

4 − b3)≤ β

⇒ νL˜CI (z) = w

a2 − zb3

(a2 − a′1) + z(b′

4 − b3)= inf β.

Similarly we can prove that νR˜CI (z) = w

zb2−a3(a′

4−a3)+z(b2−b′1)

= inf β. Hence division rule isproved for non-membership function.

Property 4.5 Let ˜AI = (a1, a2, a3, a4;w1)(a′1, a2, a3, a

′4;w1) and ˜BI =

(b1, b2, b3, b4;w2)(b′1, b2, b3, b

′4;w2) be two positive GTIFN, then the multiplication of two

GTIFN is given by (in Fig. 6)

˜AI ⊗ ˜BI = (a1b1, a2b2, a3b3, a4b4;w)(a′1b

′1, a2b2, a3b3, a

′4b

′4;w)

where 0 < w ≤ 1, w = min(w1, w2).

Proof Multiplication of two GTIFN based on (α, β)-cut method: Let ˜AI ⊗ ˜BI = ˜CI ,where Cα = [C1(α), C2(α)] and Cβ = [C′

1(β), C′2(β)], α, β ∈ (0, w], 0 < w ≤ 1 and

w = min(w1, w2).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

a’1b’

1

a1b1

a2b2

a3b3

a4b4

a’4b’

4

Fig. 6 Multiplication of two GTIFN˜AI = −−, ˜BI = .. and ˜AI ⊗ ˜BI = −)

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Now Cα = [C1(α), C2(α)]= [A1(α),A2(α)].[B1(α), B2(α)]= [A1(α)B1(α),A2(α)B2(α)]=

[

α2

w2(a2 − a1)(b2 − b1) + α

w{a1(b2 − b1) + b1(a2 − a1)} + a1b1,

× α2

w2(a4 − a3)(b4 − b3) − α

w{a4(b4 − b3) + b4(a4 − a3)} + a4b4

]

Let α2

w2 (a2 −a1)(b2 −b1)+ αw

{a1(b2 −b1)+b1(a2 −a1)}+a1b1 ≤ z ≤ α2

w2 (a4 −a3)(b4 −b3) − α

w{a4(b4 − b3) + b4(a4 − a3)} + a4b4.

Let P1 = (a2 − a1)(b2 − b1), Q1 = {a1(b2 − b1) + b1(a2 − a1)} Now

α2

w2P1 + α

wQ1 + a1b1 ≤ z ⇒ α2

w2P1 + α

wQ1 + a1b1 − z ≤ 0

⇒−Q1 −

√

Q21 − 4P1(a1b1 − z)

2P1≤ α

w

≤−Q1 +

√

Q21 − 4P1(a1b1 − z)

2P1

Let μLc (z) = w

−Q1+√

Q21−4P1(a1b1−z)

2P1. Now dμL

c (z)

dz=

w√{a1(b2−b1)+b1(a2−a1)}2−4(a2−a1)(b2−b1)(a1b1−z)

> 0. Therefore μLc (z) is an increasing

function. Also μLc (a1b1) = 0, μL

c (a2b2) = w and μLc ( a1b1+a2b2

2 ) > w2 . Again let

P2 = (a4 − a3)(b4 − b3), Q2 = {a4(b4 − b3) + b4(a4 − a3)} then

α2

w2P2 + α

wQ2 + a4b4 ≥ z ⇒ α2

w2P2 + α

wQ2 + a4b4 − z ≥ 0

⇒Q2 −

√

Q22 − 4P2(a4b4 − z)

2P2≥ α

wor

≤Q2 +

√

Q22 − 4P2(a4b4 − z)

2P2≤ α

w

Let μRc (z) = w

Q2−√

Q22−4P2(a4b4−z)

2P2. Therefore dμR

c (z)

dz=

− w√{a4(b4−b3)+b4(a4−a3)}2−4(a4−a3)(b4−b3)(a4b4−z)

< 0, if (a4 + b4) > (a3 + b3). There-

fore μRc (z) is a decreasing function. Also μR

c (a4b4) = 0, μRc (a3b3) = w and

μRc (

a3b3+a4b42 ) < w

2 . So the membership function of ˜C = ˜A ⊗ ˜B is

μ˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

w−Q1+

√

Q21−4P1(a1b1−z)

2P1, a1b1 ≤ z ≤ a2b2;

w, a2b2 ≤ z ≤ a3b3;w

Q2−√

Q22−4P2(a4b4−z)

2P2, a3b3 ≤ z ≤ a4b4;

0, otherwise.

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Hence multiplication rule is proved for membership function. Now for non-membership function

Cβ = [C′1(β), C′

2(β)]= [A′

1(β),A′2(β)].[B ′

1(β), B ′2(β)]

= [A′1(β)B ′

1(β),A′2(β)B ′

2(β)]=

[

β2

w2(a2 − a′

1)(b2 − b′1) − β

w{a2(b2 − b′

1) + b2(a2 − a′1)} + a2b2,

× β2

w2(a′

4 − a3)(b′4 − b3) + β

w{a3(b

′4 − b3) + b3(a

′4 − a3)} + a3b3

]

Let β2

w2 (a2 − a′1)(b2 − b′

1) − βw

{a2(b2 − b′1) + b2(a2 − a′

1)} + a2b2 ≤ z ≤ β2

w2 (a′4 − a3)(b

′4 −

b3) + βw

{a3(b′4 − b3) + b3(a

′4 − a3)} + a3b3.

Let P ′1 = (a2 − a′

1)(b2 − b′1), Q′

1 = {a2(b2 − b′1) + b2(a2 − a′

1)} Now

β2

w2P ′

1 − β

wQ′

1 + a2b2 ≤ z

⇒ β2

w2P ′

1 + β

wQ′

1 + a2b2 − z ≤ 0

⇒Q′

1 −√

Q′21 − 4P ′

1(a2b2 − z)

2P ′1

≤ β

w≤

Q′1 +

√

Q′21 − 4P ′

1(a2b2 − z)

2P ′1

Let νLc (z) = w

Q′1−

√

Q′21 −4P ′

1(a2b2−z)

2P ′1

. Now dνLc (z)

dz=

− w√{a2(b2−b′1)+b2(a2−a′

1)}2−4(a2−a′1)(b2−b′

1)(a2b2−z)< 0. Therefore νL

c (z) is a decreasing

function. Also νLc (a′

1b′1) = w, νL

c (a2b2) = 0 and νLc (

a′1b′

1+a2b22 ) < w

2 . Again letP ′

2 = (a′4 − a3)(b

′4 − b3), Q′

2 = {a3(b′4 − b3) + b3(a

′4 − a3)} then

β2

w2P ′

2 + β

wQ′

2 + a3b3 ≥ z ⇒ α2

w2P ′

2 + α

wQ′

2 + a3b3 − z ≥ 0

⇒−Q2 −

√

Q′22 − 4P ′

2(a3b3 − z)

2P ′2

≥ β

wor

−Q′2 +

√

Q′22 − 4P ′

2(a3b3 − z)

2P ′2

≤ β

w

Let νRc (z) = w

−Q′2+

√

Q′22 −4P ′

2(a3b3−z)

2P ′2

.

Therefore dνRc (z)

dz= w√{a3(b′

4−b3)+b3(a′4−a3)}2−4(a′

4−a3)(b′4−b3)(a3b3−z)

> 0. Therefore νRc (z) is

an increasing function.

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Also νRc (a3b3) = 0, νR

c (a′4b

′4) = w and νR

c (a3b3+a′

4b′4

2 ) > w2 . So the non-membership

function of ˜C = ˜A ⊗ ˜B is

ν˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

wQ′

1−√

Q′21 −4P ′

1(a2b2−z)

2P′1

, a′1b

′1 ≤ z ≤ a2b2;

0, a2b2 ≤ z ≤ a3b3;w

−Q′2+

√

Q′22 −4P ′

2(a3b3−z)

2P ′2

, a3b3 ≤ z ≤ a′4b

′4;

w, otherwise.

Hence multiplication rule is proved for non-membership function. Thus we have

˜AI ⊗ ˜BI = (a1b1, a2b2, a3b3, a4b4;w)(a′1b

′1, a2b2, a3b3, a

′4b

′4;w)

where 0 < w ≤ 1, w = min(w1, w2).Multiplication of two GTIFN based on vertex method: Let ˜CI = ˜AI · ˜BI = f (˜AI , ˜BI ).Now ordinate of the vertices for membership function are

c1 =(

a1 + α

w(a2 − a1), b1 + α

w(b2 − b1)

)

c2 =(

a1 + α

w(a2 − a1), b4 − α

w(b4 − b3)

)

c3 =(

a4 − α

w(a4 − a3), b1 + α

w(b2 − b1)

)

c4 =(

a4 − α

w(a4 − a3), b4 − α

w(b4 − b3)

)

Therefore

f (c1) ={

a1 + α

w(a2 − a1)

}{

b1 + α

w(b2 − b1)

}

f (c2) ={

a1 + α

w(a2 − a1)

}{

b4 − α

w(b4 − b3)

}

f (c3) ={

a4 − α

w(a4 − a3)

}{

b1 + α

w(b2 − b1)

}

f (c4) ={

b1 + α

w(b2 − b1)

}{

b4 − α

w(b4 − b3)

}

Now since a1 ≤ a2 ≤ a3 ≤ a4 and b1 ≤ b2 ≤ b3 ≤ b4 so f (c1) < f (c2) < f (c3) < f (c4).Hence

Y = [min (f (c1), f (c2), f (c3), f (c4)) , max (f (c1), f (c2), f (c3), f (c4))]

= [f (c1), f (c4)]

=[{

a1 + α

w(a2 − a1)

}{

b1 + α

w(b2 − b1)

}

,{

b1 + α

w(b2 − b1)

}

×{

b4 − α

w(b4 − b3)

}]

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As explained in (α, β)-cut method multiplication rule is proved for membershipfunction. Now the ordinate of the vertices for nonmembership function are

c′1 =

(

a2 − β

w(a2 − a′

1), b2 − β

w(b2 − b′

1)

)

c′2 =

(

a2 − β

w(a2 − a′

1), b3 + β

w(b′

4 − b3)

)

c′3 =

(

a3 + β

w(a′

4 − a3), b2 − β

w(b2 − b′

1)

)

c′4 =

(

a3 + β

w(a′

4 − a3), b3 + β

w(b′

4 − b3)

)

Therefore

f (c′1) =

{

a2 − β

w(a2 − a′

1)

}{

b2 − β

w(b2 − b′

1)

}

f (c′2) =

{

a2 − β

w(a2 − a′

1)

}{

b3 + β

w(b′

4 − b3)

}

f (c′3) =

{

a3 + β

w(a′

4 − a3)

}{

b2 − β

w(b2 − b′

1)

}

f (c′4) =

{

a3 + β

w(a′

4 − a3)

}{

b3 + β

w(b′

4 − b3)

}

Now since a′1 ≤ a2 ≤ a3 ≤ a′

4 and b′1 ≤ b2 ≤ b3 ≤ b′

4 so f (c′1) < f (c′

2) < f (c′3) < f (c′

4).Hence

Y ′ = [

min(

f (c′1), f (c′

2), f (c′3), f (c′

4))

, max(

f (c′1), f (c′

2), f (c′3), f (c′

4))]

= [

f (c′1), f (c′

4)]

=[{

a2 − β

w(a2 − a′

1)

}{

b2 − β

w(b2 − b′

1)

}

,

{

a3 + β

w(a′

4 − a3)

}

×{

b3 + β

w(b′

4 − b3)

}]

Now as explained above (in interval method) multiplication rule is proved fornonmembership function. Thus we have

˜AI ⊗ ˜BI = (a1b1, a2b2, a3b3, a4b4;w)(a′1b

′1, a2b2, a3b3, a

′4b

′4;w)

where 0 < w ≤ 1, w = min(w1, w2).Multiplication of two GTIFN based on extension principle: Let ˜AI ⊗ ˜BI = ˜CI whereμCI (z) = sup(min(μAI (x), μBI (y)); xy = z) and ν

˜CI (z) = inf(max(νAI (x), νBI (y)); xy =z). Let w = min(w1, w2), then

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μ˜CI (z)=

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

μL˜CI (z)= sup

(

min(w1x−a1a2−a1

, w2y−b1b2−b1

) : xy = z)

, a1 ≤ x ≤ a2, b1 ≤ y ≤ b2;sup(min(w1, w2) : xy = z), a2 ≤ x ≤ a3, b2 ≤ y ≤ b3;μR˜CI (z)= sup

(

min(w1a4−xa4−a3

, w2b4−yb4−b3

) : xy = z)

, a3 ≤ x ≤ a4, b3 ≤ y ≤ b4;0, otherwise.

Hence

μ˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

μL˜CI

(z) = sup(

min(

w x−a1a2−a1

, wzx −b1b2−b1

))

, a1 ≤ x ≤ a2, a1b1 ≤ z ≤ a2b2;w, a2 ≤ x ≤ a3, a2b2 ≤ z ≤ a3b3;μR˜CI

(z) = sup(

min(

w a4−xa4−a3

, wb4− z

x

b4−b3

))

, a3 ≤ x ≤ a4, a3b3 ≤ z ≤ a4b4;0, otherwise.

Now let min(

min(w x−a1a2−a1

, wzx−b1

b2−b1

)

= α s.t.

wx − a1

a2 − a1≥ α,w

zx

− b1

b2 − b1≥ α

⇒ x ≥ α

w(a2 − a1) + a1, z ≥ x

( α

w(b2 − b1) + b1

)

⇒ z ≥( α

w(a2 − a1) + a1

) ( α

w(b2 − b1) + b1

)

⇒ z ≥( α

w

)2(a2 − a1)(b2 − b1) + α

w((a2 − a1)b1 + a1(b2 − b1)) + a1b1

As explained earlier in (α, β)-cut method, the we can say multiplication rule is provedfor membership function. Now from definition [3.1] and [2.5](for non-membershipfunction), we get

ν˜CI (z) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

νL˜CI

(z) = inf(

max(

w a2−x

a2−a′1, w

b2− zx

b2−b′1

))

, a′1 ≤ x ≤ a2, a

′1b

′1 ≤ z ≤ a2b2;

0, a2 ≤ x ≤ a3, a2b2 ≤ z ≤ a3b3;νR˜CI (z) = inf

(

max(

wx−a3a′

4−a3, w

zx −b3

b′4−b3

))

, a3 ≤ x ≤ a′4, a3b3 ≤ z ≤ a′

4b′4;

w, otherwise.

Now let max(

wa2−x

a2−a′1, w

b2− zx

b2−b′1

)

= β s.t.

wa2 − x

a2 − a′1

≤ β,wb2 − z

x

b2 − b′1

≤ β ⇒ −x ≤ β

w(a2 − a′

1) − a2,− z

x≤ β

w(b2 − b′

1) − b2

⇒ β2

w2(a2−a′

1)(b2−b′1)−

β

w{a2(b2 − b′

1) + b2(a2 − a′1)}

+ a2b2 ≤ z

As explained earlier in (α, β)-cut method, the multiplication rule is proved for non-membership function.

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5 Comparative study among (α, β)−cut method, vertex method and extensionprinciple method

Let us consider the expression mI = f (xI , yI , zI ) = xI ⊗ (yI � zI ), wherexI = (1, 2, 3, 4; 0.7)(0, 2, 3, 5; 0.7), yI = (10, 12, 13, 14; 0.6)(8, 12, 13, 15; 0.6) and zI =(5, 6, 7, 8; 0.8)(4, 6, 7, 8; 0.8).The α−cut and β−cut of xI , yI and zI are given by

Xα =[

1 + α

0.7, 4 − α

0.7

]

Xβ =[

2 − 2β

0.7, 3 + 2β

0.7

]

Yα =[

10 + 2α

0.6, 14 − α

0.6

]

Yβ =[

12 − 4β

0.6, 13 + 2β

0.6

]

Zα =[

5 + α

0.8, 8 − α

0.8

]

Zβ =[

6 − 2β

0.8, 7 + β

0.8

]

Now we use (α, β)-cut method, vertex method and extension principle methodon the above expression to compare among those three methods. Here w =min(0.7, 0.6, 0.8) = 0.6.

5.1 Result by (α, β)−cut method

For the expression mI = xI ⊗ (yI � zI ), (α, β)−cut areMα =

[

(1 + α0.6 )(2 + 3α

0.6 ), (4 − α0.6 )(9 − 2α

0.6 )]

and Mβ =[

(2 − 2β0.6 )(5 − 5β

0.6 ), (3 + 2β0.6 )(7 + 4β

0.6 )]

,

where α, β ∈ (0, w] and 0 < α + β ≤ w.

5.2 Result by vertex method

For the expression mI = f (xI , yI , zI ) = xI ⊗ (yI � zI ), the ordinates of the verticesfor membership function are

m1 =(

1 + α

0.6, 10 + 2α

0.6, 5 + α

0.6

)

m2 =(

1 + α

0.6, 10 + 2α

0.6, 8 − α

0.6

)

m3 =(

1 + α

0.6, 14 − α

0.6, 8 − α

0.6

)

m4 =(

1 + α

0.6, 14 − α

0.6, 5 + α

0.6

)

m5 =(

4 − α

0.6, 10 + 2α

0.6, 5 + α

0.6

)

m6 =(

4 − α

0.6, 10 + 2α

0.6, 8 − α

0.6

)

m7 =(

4 − α

0.6, 14 − α

0.6, 8 − α

0.6

)

m8 =(

4 − α

0.6, 14 − α

0.6, 5 + α

0.6

)

So the α−cut of mI is

Mα = [min (f (m1), f (m2), · · · , f (m8)) , max (f (m1), f (m2), · · · , f (m8))]

= [f (m2), f (m8)]=

[

(

1 + α

0.6

)

(

2 + 3α

0.6

)

,(

4 − α

0.6

)

(

9 − 2α

0.6

)]

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Now for the expression mI = f (xI , yI , zI ) = xI ⊗ (yI � zI ), the ordinates of thevertices for non-membership function are

m′1 = (2 − 2β

0.6, 12 − 4β

0.6, 6 − 2β

0.6) m′

2 = (2 − 2β

0.6, 12 − 4α

0.6, 7 + β

0.6)

m′3 = (2 − 2β

0.6, 13 + 2β

0.6, 6 − 2β

0.6) m′

4 = (2 − 2β

0.6, 13 + 2β

0.6, 7 + β

0.6)

m′5 = (3 + 2β

0.6, 12 − 4β

0.6, 6 − 2β

0.6) m′

6 = (3 + 2β

0.6, 12 − 4β

0.6, 7 + β

0.6)

m′7 = (3 + 2β

0.6, 13 + 2β

0.6, 6 − 2β

0.6) m′

8 = (3 + 2β

0.6, 13 + 2β

0.6, 7 + β

0.6)

So the β−cut of mI is

Mβ = [

min(

f (m′1), f (m′

2), · · · , f (m′8))

, max(

f (m′1), f (m′

2), · · · , f (m′8))]

= [f (m′2), f (m′

7)]=

[(

2 − 2β

0.6

)(

5 − 5β

0.6

)

,

(

3 + 2β

0.6

)(

7 + 4β

0.6

)]

5.3 Result by extension principle method

Let Mα = [MLα ,MR

α ], where

MLα = min

{

x(y − z) : x ∈[

1 + α

0.6, 4 − α

0.6

]

, y ∈[

10 + 2α

0.6, 14 − α

0.6

]

, z ∈

×[

5 + α

0.6, 8 − α

0.6

]}

=(

1 + α

0.6

)

(

2 + 3α

0.6

)[

since∂f

∂x> 0,

∂f

∂y> 0and

∂f

∂z< 0

]

and

MRα = max

{

x(y − z) : x ∈[

1 + α

0.6, 4 − α

0.6

]

, y ∈[

10 + 2α

0.6, 14 − α

0.6

]

, z ∈

×[

5 + α

0.6, 8 − α

0.6

]}

=(

4 − α

0.6

)

(

9 − 2α

0.6

)

Now let Mβ = [MLβ ,MR

β ], where

MLβ = min

{

x(y − z) : x ∈[

2 − 2β

0.6, 3 + 2β

0.6

]

, y ∈[

12 − 4β

0.6, 13 + 2β

0.6

]

, z ∈

×[

6 − 2β

0.6, 7 + β

0.6

]}

=(

2 − 2β

0.6

)(

5 − 5β

0.6

)

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and

MRβ = max

{

x(y − z) : x ∈[

2 − 2β

0.6, 3 + 2β

0.6

]

, y ∈[

12 − 4β

0.6, 13 + 2β

0.6

]

, z ∈

×[

6 − 2β

0.6, 7 + β

0.6

]}

=(

3 + 2β

0.6

)(

7 + 4β

0.6

)

The membership and non-membership function of expression mI = xI ⊗ (yI � zI ) isgiven in Fig. 7. But if we consider the expression f (xI , yI , zI ) = xI ⊗ yI � xI ⊗ zI ,then its (α, β)− cut using (α, β)− cut method areMα =

[

(1 + α0.6 )(10 + 2α

0.6 ) − (4 − α0.6 )(8 − α

0.6 )(4 − α0.6 )(14 − α

0.6 )

−(1 + α0.6 )(5 + α

0.6 )]

and Mβ =[

(2 − 2β0.6 )(12 − 4β

0.6 ) − (3 + 2β0.6 )(7 + 2β

0.6 ),

(3 + 2β0.6 )(13 + 2β

0.6 ) − (2 − 2β0.6 )(6 − 2β

0.6 )]

But using vertex method and extension principle method, the (α, β)-cut areMα=

[

(1+ α0.6 )(2+ 3α

0.6 ), (4 − α0.6 )(9 − 2α

0.6 )]

and Mβ=[

(2 − 2β0.6 )(5− 6β

0.6 ), (3 + 2β0.6 )(7 + 4β

0.6 )]

So we get a unique value of the expression using vertex method and extension princi-ple method but (α, β)-cut method gives two possible values for the same expression.So in view of this example we can conclude that vertex method and extension princi-ple method gives best result than (α, β)-cut method in some cases. The membershipand non-membership function of the expression mI = xI ⊗ yI � xI ⊗ zI using(α, β) − cut method is given in Fig. 8.

0 10 20 30 40 50 60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 7 Membership and non-membership function of xI ⊗ (yI � zI )

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−40 −22 3 27 51 750

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 8 Membership and non-membership function of xI ⊗ yI � xI ⊗ zI

6 Ranking function of GTIFN

Let ˜AI = (a1, a2, a3, a4;w)(a′1, a2, a3, a

′4;w) be a GTIFN. There are many methods for

defuzzification such as Centroid Method, Mean of Interval Method, Removal AreaMethod etc. In this paper we have used Mean of Interval method to find the value ofthe membership and non-membership function of GTIFN.

6.1 Mean of (α, β)-cut method

The (α, β)-cut of the GTIFN is given by

Aα,β = {[A1(α),A2(α)]; [A′1(β),A′

2(β)], α + β ≤ w,α, β ∈ (0, w]}

where A1(α) = a1 + αw

(a2 − a1), A2(α) = a4 − αw

(a4 − a3), A′1(β) = a2 − β

w(a2 − a′

1),A′

2(β) = a3 + βw

(a′4 − a3). Now by Mean of (α, β)-cut method the representation of

membership function is

Rμ(˜AI ) = 1

2

w∫

0

(A1(α) + A2(α))dα

= 1

2

w∫

0

[a1 + a4 + α

w{(a2 − a1) − (a4 − a3)}]dα

= 1

2(a1w + a4w + w

2{(a2 − a1) − (a4 − a3)})

= w(a1 + a2 + a3 + a4)

4.

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Now by Mean of (α, β)-cut method the representation of non-membership function is

Rν(˜AI ) = 1

2

w∫

0

(A1(β) + A2(β))dβ

= 1

2

w∫

0

[

a2 + a3 − β

w{(a2 − a′

1) − (a′4 − a3)}

]

dβ

= 1

2

(

a2w + a3w − w

2{(a2 − a′

1) − (a′4 − a3)}

)

= w(a′1 + a2 + a3 + a′

4)

4.

Let ˜AI = (a1, a2, a3, a4;w1)(a′1, a2, a3, a

′4;w1) and ˜BI =

(b1, b2, b3, b4;w2)(b′1, b2, b3, b

′4;w2) be two GTIFN then [8] proposed that

(i) ˜AI ≺ ˜BI iff H(˜AI ) < H(˜BI )

(ii) ˜AI � ˜BI iff H(˜AI ) > H(˜BI )

(iii) ˜AI = ˜BI iff H(˜AI ) = H(˜BI )

where

H(˜AI ) = Rμ(˜AI ) + Rν(˜AI )

2= w(a1 + 2a2 + 2a3 + a4 + a′

1 + a′4)

8,

H(˜BI ) = Rμ(˜BI ) + Rν(˜BI )

2= w(b1 + 2b2 + 2b3 + b4 + b′

1 + b′4)

8

where w = min(w1, w2).

7 Generalized intuitionistic fuzzy transportation problem

Consider the a transportation problem with two sources and two destination as:

Minimizem⊕

i=1

n⊕

j=1

˜cIij xij

subject ton∑

j=1

xij ≈ ˜aIi for i = 1, 2, · · · m (1)

m∑

i=1

xij ≈ ˜bIj for j = 1, 2, · · · n

xij ≥ 0, ∀ i, j

where aiI is the approximate availability of the product at the ith source, ˜bj

I is theapproximate demand of the product at the j th destination, cij

I is the approximate

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cost for transporting one unit of the product from the ith source to the j th des-tination and xij is the number of units of the product that should be transportedfrom the ith source to j th destination taken as a fuzzy decision variables. Ifm∑

i=1

˜aIi =

n∑

j=1

˜bIj then GIFTP is said to balanced transportation problem otherwise it is

called an unbalanced GIFTP. Let cIij = (cij1, cij2, cij3, cij4;w)(c′

ij1, cij2, cij3, c′ij4;w),

aIi = (ai1, ai2, ai3, ai4;w)(a′

i1, ai2, ai3, a′i4;w) and bI

j =(bj1, bj2, bj3, bj4;w)(b′

j1, bj2, bj3, b′j4;w). The steps to solve the above GIFTP are as

follows

Step 1: Substituting the value of cIij , aI

i and bIj in Eq. 3, we get

Minimizem⊕

i=1

n⊕

j=1

(cij1, cij2, cij3, cij4;w)(c′ij1, cij2, cij3, c

′ij4;w)xij

subject ton∑

j=1

xij ≈ (ai1, ai2, ai3, ai4;w)(a′i1, ai2, ai3, a

′i4;w) for i = 1, 2, · · · m (2)

m∑

i=1

xij ≈ (bj1, bj2, bj3, bj4;w)(b′j1, bj2, bj3, b

′j4;w) for j = 1, 2, · · · n

xij ≥ 0, ∀i, j

Step-2: Now by the arithmetic operation and definition presented in Sections 4 and2 (3) converted to crisp linear programming(CLP)

MinimizeH

⎛

⎝

m⊕

i=1

n⊕

j=1

(xij cij1, xij cij2, xij cij3, xij cij4;w)

× (xij c′ij1, xij cij2, xij cij3, xij c

′ij4;w)

)

subject to

H

⎛

⎝

n∑

j=1

xij

⎞

⎠ = H(

(ai1, ai2, ai3, ai4;w)(a′i1, ai2, ai3, a

′i4;w)

)

for i = 1, 2, · · · m

H

(

m∑

i=1

xij

)

= H(

(bj1, bj2, bj3, bj4;w)(b′j1, bj2, bj3, b

′j4;w)

)

for j = 1, 2, · · · n

xij ≥ 0, ∀i, j (3)

Step-3: Find the optimal solution xij by solving the linear programming problem.

Step-4: Find the fuzzy optimal value by putting xij inm⊕

i=1

n⊕

j=1

˜cIij xij .

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8 Numerical example

Let us considered the following GIFTP (in Table-1)Using Step-1 of the method explain in Section 7 the above GIFTP can be written

as

Minimize (2, 4, 5, 6; 0.5)(1, 4, 5, 6; 0.5)x11 ⊕ (4, 6, 7, 8; 0.2)(3, 6, 7, 9; 0.2)x12

⊕(3, 7, 8, 12; 0.3)(2, 7, 8, 13; 0.3)x13 ⊕ (1, 3, 4, 5; 0.6)(0.5, 3, 4, 5; 0.6)x21

⊕(3, 5, 6, 7; 0.6)(2, 5, 6, 8; 0.6)x22 ⊕ (2, 6, 7, 11; 0.4)(1, 6, 7, 12; 0.4)x23

⊕(3, 4, 5, 8; 0.7)(2, 4, 5, 9; 0.7)x31 ⊕ (1, 2, 3, 4; 0.8)(0.5, 2, 3, 5; 0.8)x32 (4)

⊕(2, 4, 5, 10; 0.2)(1, 4, 5, 11; 0.2)x33

subject to x11 + x12 + x13 ≈ (4, 6, 8, 9; 0.6)(2, 6, 8, 10; 0.6)

x21 + x22 + x23 ≈ (0, 0.5, 1, 2; 0.5)(0, 0.5, 1, 5; 0.7)

x31 + x32 + x33 ≈ (8, 9.5, 10, 11; 0.8)(6.5, 9.5, 10, 11; 0.8)

x11 + x21 + x31 ≈ (6, 7, 8, 9; 1)(5, 7, 8, 11; 1)

x12 + x22 + x32 ≈ (4, 5, 6, 7; 0.8)(3, 5, 6, 8; 0.8)

x13 + x23 + x33 ≈ (2, 4, 5, 6; 0.6)(0.5, 4, 5, 7; 0.6)

xij ≥ 0, ∀i, j

Table 1 Input data for GIFTP

˜D1I

˜D2I

˜D3I

Availability(aiI )

˜S1I

(2,4,5,6;0.5) (4,6,7,8;0.2) (3,7,8,12;0.3) (4,6,8,9; 0.6)

(1,4,5,6;0.5) (3,6,7,9;0.2) (2,7,8,13;0.3) (2,6,8,10;0.6)˜S2

I(1,3,4,5;0.6) (3,5,6,7;0.6) (2,6,7,11;0.4) (0,0.5,1,2;0.5)

(0.5,3,4,5;0.6) (2,5,6,8;0.6) (1,6,7,12;0.4) (0,0.5,1,5;0.7)˜S3

I(3,4,5,8;0.7) (1,2,3,4;0.8) (2,4,5,10;0.2) (8,9.5,10,11;0.8)

(2,4,5,9;0.7) (0.5,2,3,5;0.8) (1,4,5,11;0.2) (6.5,9.5,10,11;0.8)

Demand(˜bjI) (6,7,8,9;1) (4,5,6,7;0.8) (2,4,5,6;0.6)

(5,7,8,11;1) (3,5,6,8;0.8) (0.5,4,5,7;0.6)

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Now using Step-2 of the method explain in Section 7 the above GIFTP convertedinto crisp linear programming

Minimize1

40(33x11 +50x12 +60x13 +25.5x21 +42x22 +52x23 + 50x31 + 20.5x32 + 42x33)

subject to x11 + x12 + x13 = 3.975

x21 + x22 + x23 = 0.875

x31 + x32 + x33 = 7.55

x11 + x21 + x31 = 7.625

x12 + x22 + x32 = 4.4

x13 + x23 + x33 = 2.5125

xij ≥ 0, ∀ i, j

Solving the above crisp linear programming using Lingo-11.0, we get x11 = 3.97,x12 = 0, x13 = 0, x21 = 0.875, x22 = 0, x23 = 0, x31 = 2.775, x32 = 4.4 and x33 = 2.512.Now the minimum intuitionstic fuzzy optimal cost is

cIw = (26.56, 48.45, 62.98, 93.11; 0.2)(14.66, 48.45, 62.98, 102.80; 0.2).

If we consider the above transportation problem in normal intuitionistic fuzzy envi-ronment (i.e. taking w = 1). Then the optimal solution we get x11 = 6.62, x12 = 0,x13 = 0, x21 = 1, x22 = 0, x23 = 0.25, x31 = 0, x32 = 5.5 and x33 = 3.93. Now theminimum intuitionstic fuzzy optimal cost is

cI1 = (28.1, 57.7, 75, 108.7)(14.05, 57.7, 75, 118.45).

Using ranking function discussed in Section 6 we conclude that H(c1I ) > H(cw

I ).Since in transportation problem our goal is to minimize the transportation cost, weget minimum optimum result by using generalized intuitionistic fuzzy number.

9 Conclusion

In this paper, thus we have worked on generalized IFNs. Different arithmetic oper-ations like addition, subtraction, multiplication etc. of generalized IFNs based on(α, β)-cut method, vertex method and extension principle method have been pre-sented elaborately. To show the exactness between this three methods an exampleis presented and the results have been discussed numerically and graphically. TheRanking function of the generalized intuitionistic fuzzy numbers has been developedhere. A method using ranking function is also proposed to deal the transportationproblem in generalized intuitionistic fuzzy environments. To validate the proposedmethod a numerical example is presented and solved using Lingo-11.0. This idea canbe extended as intuitionistic fuzzy fault tree analysis for starting failure of automobilesystem.

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