Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—12

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Pacific Science Review A: Natural Science and Engineering

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Novel arithmetic operations on type-2 intuitionistic fuzzy and its applications to transportation problem

Dipak Kumar Jana*

Department of Applied Science, Haldia Institute of Technology, Haldia, Purba Midnapur, 721657, West Bengal, India

ARTICLE INFO ABSTRACT

Type-2 intuitionistic fuzzy sets possess many advantages over type-1 fuzzy sets because their membership functions are themselves fuzzy, making it possible to model and minimize the effects of uncertainty in type-1 intuitionistic fuzzy logic systems. This paper presents generalized type-2 intuitionistic fuzzy numbers and its different arithmetic operations with several graphical representations. Basic generalized trapezoidal intuitionistic fuzzy numbers considered for these arithmetic operations are formulated on the basis of (a, b)-cut methods. The ranking function of the generalized trapezoidal intuitionistic fuzzy number has been successively calculated. To validate the proposed arithmetic operations, we solved a type-2 intuitionistic fuzzy transportation problem by the ranking function for mean interval method. Transportation costs, supplies and demands of the homogeneous product are type-2 intuitionistic fuzzy in nature. A numerical example is presented to illustrate the proposed model. Copyright © 2016, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University. Production and hosting by Elsevier B.V. This is an open access article under the CC

BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article history: Received 18 July 2016 Received in revised form 9 September 2016 Accepted 12 September 2016 Available online xxx

Keywords:

Type-2 intuitionistic fuzzy Generalized intuitionistic fuzzy number Ranking function a, b-cut

Transportation problem

1. Introduction

To make a transportation plan for the next day, the supply capacity at each origin, the demand at each destination and the conveyance capacity often need to be estimated by the professional judgement of experts or probability statistics because no precise a priori information exists. Certain hidden costs, such as toll tax and service tax, must be considered during transport. It is appropriate to investigate this problem by using fuzzy or stochastic optimization methodologies. The applicable theoretical methods can be referred to as fuzzy set theory and type-2 intuitionistic fuzzy sets. Real-life decision making problems display some level of imprecision and vagueness in estimation of parameters. Results have been captured by fuzzy sets modelling the problems. Applications of fuzzy set theory in decision making and in particular optimization problems have been widely studied since the introduction of fuzzy sets cf. Zadeh [13]. Recently, many papers have shown growing interest in the study of decision making problems using intuition-istic fuzzy sets/numbers [1,2]. The intuitionistic fuzzy set (IFS) is an extension of fuzzy set. IFS was first introduced by Atanassov [4]. The

* Fax: +91 3224 252800. E-mail address: dipakjana@gmail.com.

Peer review under responsibility of Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University.

conception of IFS can be viewed as an approach where given data are not sufficient to define the fuzzy set. Fuzzy sets are characterized by the membership function only, but IFS is characterized by a membership function and a non-membership function so that the sum of both values is less than one [5]. Presently IFSs are being studied and used in different fields of science and technology for decision making problems. Several researchers have formulated and solved optimization problems in the field of intuitionistic fuzzy [6,8,9,14,20].

In the study of fuzzy set theory for optimization, the ranking of fuzzy numbers is a significant factor. To rank fuzzy numbers, one fuzzy number needs to be compared with the others by using a ranking function. Some researchers have formulated and solved optimization problems in the application of a ranking function [3,7]. Recently, the IFN received wide attention. Different definitions of IFNs have been proposed with corresponding ranking functions. Some research has also shown interest in the arithmetic operations and the ranking functions of IFNs [1,2].

Recently, IFNs have been used in fields, such as fuzzy linear programming and transportation problems. Parvathi et al. [10] have proposed an intuitionistic fuzzy simplex method. Pramanik et al. [16], Chakraborty et al. [17], Jana et al. [18], Jana et al. [19], Hussain et al. [11] and Nagoor Gani [12] proposed a method for solving intuitionistic fuzzy transportation problems. None of them introduced the generalized intuitionistic fuzzy number and its application to transportation problems.

http://dx.doi.org/10.1016/j.psra.2016.09.008

2405-8823/Copyright © 2016, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

D.K. Jana / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—12

The membership functions of type-2 fuzzy sets are themselves fuzzy. Type-2 fuzzy sets are nowhere near as widely used as type-1 fuzzy sets. In 1975, the basic concept of type-2 fuzzy sets (T2FSs) was proposed by professor Zadeh [24] that is an extension of ordinary fuzzy sets i.e., type-1 fuzzy sets, whose truth values are ordinary fuzzy sets, i.e., fuzzy truth values. The overviews of type-2 fuzzy sets were given in Mendel et al. [21]. Since ordinary fuzzy sets and interval-valued fuzzy sets are special cases of type-2 fuzzy sets, Takac [22] proposed that type-2 fuzzy sets are very practical in circumstances where there are more uncertainties. Pramanik et al. [16] proposed type-2 fuzzy Gaussian fuzzy sets from the view of type reduction and centroid [23].

This paper presents generalized intuitionistic fuzzy arithmetic operations and their application to a transportation problem. Trapezoidal type-2 intuitionistic fuzzy numbers (TrT2IFN) are defined, and their arithmetic operations based on the type-2 intuitionistic fuzzy extension principle and (a, b) - cut method are presented. To illustrate the proposed method, a numerical example is presented and solved as a type-2 intuitionistic transportation problem.

2. Preliminaries

In this section, we first discuss the arithmetic operations on intuitionistic type-2 fuzzy sets with graphical representation. Next, we provide a number of definitions and notations for convenience of explaining general concepts concerned with intuitionistic type-2 fuzzy sets.

Definition 2.1. Generalized Intuitionistic Fuzzy Number (GIFN):

An Intuitionistic fuzzy number A' = { < x, ma , va > } of the real line < is called GIFN, if the following hold

4 (x) =

x- al al < x < a2

a2 - al'

wz, a2 < x < a3

x a3 < x < a^

a4 - Z ' a3

0; otherwise.

and non-membership function

nA (x) = <

a2 - a2 - x a'f a 1z < x < a2

0; a2 < x < a3

x - aZ3; a3 < x < a'z

a '4 - a3

wZ; otherwise.

Definition 2.3. Generalized Triangular Intuitionistic Type-2 Fuzzy Number (GTrIT2FN): Let {L, U} and af < a\ < a\ < a'. A

GTIT2FN A = [m\(x), mU(x), vA (x), vU(x)]

in < written as

(a^, a|, a|; wz)(a'1z, a|, a'3; w) has membership function:

(i) there exists xeK, m~(m) = w, v~~(m) =0,0 <w < 1.

(ii) is continuous mapping from < to the closed interval [0, w] and xeS, the relation 0 < mA(x) + va(x) < w holds.

The membership function and non-membership function of A' is of the following form

m~(x) =

wfi (x), w;

whi(x), 0,

m - a < x < m;

x = m; m < x < m + b; otherwise.

The non-membership function is of the following form

vA(x) =

wf2(x), m - a < x < m, 0 < w(fi(x) +f2(x)) < w;

0; x = m;

wh2(x), m < x < m + b , 0 < w(h^ (x) + h2(x)) < w; w, otherwise.

In this equation, f1(x) and h1(x) are strictly increasing and decreasing functions in [m - a, m] and [m, m + b], respectively, and f2(x) and h2(x) are strictly decreasing and increasing functions in [m -

a , m] and [m, m + b' ] respectively, where m is the mean value of A1. The left and right spreads of membership function ^a(x) are called a and b. The left and right spreads of non-membership function n(x) are called a and b .

Definition 2.2. Generalized Trapezoidal Intuitionistic Type-2 Fuzzy Number (GTIT2FN): Let Z^fL, U} and

X < al < a2 < a3 < a4 < a¡. A GTIT2FN A1 = [m\(x), mUA (x),

VA(x)' (x)] in < written as (al, a2, a|, a^; wZ)(a1Z, a2, a|, a4Z; wz) has membership function (in Fig. 1).

mA (x) = <

zx - ai a2- a1

al < x < a2;

Z a3 - x

a3 - a2

a2 < x < a3;

otherwise.

and non-membership function

-6 -4 -2 0 2 4

Fig. 1. Membership and non-membership function of GTIT2FN.

D.K. Jana / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—12

nA (x) = '

a2 - ai 0,

zx - a2

a£ - a2

ai < x < a2; x = a2;

a2 < x < a3z ;

otherwise.

Definition 2.4. A GTIFN A' = (a^ a|, a3, a4; wz)(a'1, a|, a3, a4z; w?) is said to be non-negative iff a^ > 0.

Definition 2.5. Two GTIFN A~ = (a^, a|, a3, a4; w1) (a'.?, a|, a3, a4z; w^) and ~ = (b\, b|, b3, b4; w2 Xbf, b|, bK3, b4z; w|) are said to be equal iff a\ = b1, a^ = b|, a3 = b3, a4 = b4, a^ = b?, a4z = b4 and w1 = w2.

Definition 2.6. An interval number is a closed and bounded set of real numbers [az, bz] = fx : az < x < bzVx, bz, xes}. (i) The addition of two interval numbers A = [a^, a2] and B = [b1, b2] denoted by A(+)B and is defined byA(+)B = [a^ + b1, a2 + b|].

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

(iii) The scalar multiplication of interval number A = a, aj] is denoted by l<A where k is scalar and defined by

[feai, ka2] ; if k > 0, [ka2, kai] ; if k < 0

(iv) The product of two interval numbers A = [a1, a2] and B = [b-j, b2] is denoted by A( • )B and is defined byA(• )B =[p, q], where p = min(a1b1, a|b1, a\b2, a|b|) and q = max

(a1b1;a2b1;a1b2;a2b2).

(v) The division of two interval numbers A = [a^, a2] and B =[b1, b2] is denoted by A(+)B and is defined by

A( + )B :

empty interval,

[ai,a^ (■)

[ai,a^ (■)

az ; a2

ai ' a2 ua i ' a2

[bi ,0) (0 ,62]'

if 0;[b 1, b2]; if b 1 = b2 = 0; if bi = 0 , b2S0;

if bi S0 , b2 = 0; otherwise.

Definition 2.7. a-cut set: A a-cut set of A = (a^, a2, a3, a4; w) (a\, a2 , a3 , a j; w) is a crisp subset or < that is defined as follows

Aa = {x : mAi (x) > a} = [A1 (a), A2 (a)]

= [a1 + ww (a2 - aD' a4 - £ (a4 - a3)]

Definition 2.8. b-cut set. A b-cut set of A' = (a^, a2, a3, a4; w) (a\, a2, a3, ; w) is a crisp subset or < that is defined as follows

Aa = {x : (x)<b} = [Ai (b) , A2

b I - \ b I %

a2 - W a2 - ai , a3 + W a4Z - a3

Definition 2.9. (a ,b)-cut set: A (a ,ß)-cut

A' = (ai, a2,a3,a4; w)(ai,a2,a3,a^; w) is given by

Aa ,b = {lAi (a) , A2 (a)] ; [A (b) , A2 (b)] , a + b < w, a , b 2 [0, w]}

3. Arithmetic operations

Property 3.1. Let A' = (a1, a2, a3, a4; w1)(a1, a2 , a3, ; w^) and

b' = (b1, b2 , b3 , b4; w2)(b1z , b2 , b3 ,b4z;w2), where ?2(L ,U) be two

GTIFN, then the addition of two GTIFN is given by

~ ®B' = (ai + bi, a2 + b2 , a3 + b3 , a4 + b4; wa + b2 , a3 + b3 , a4z + ; w)

where 0<wa < i, w? = min(wi, w2).

Proof: Let A

cj = [c<(b), C2z(b)],

wz = min(w1, w2).

i, "2)

= C1, where Ca = [Cf (a), C?(a)] and az , ba e [0 , w?], 0 < wÇ < i and

Now Ca = [CZ (a) , C2 (a^ = [A|(a) , A2(a)] + [b^ (a), B2(a)] = [Ai(a)+Bi(a) , A2(a)+B2(a)]

i + bi + wz{(a2 - ai) + (b2 - bi)},a4 + b4

- S (a4 - a3) + (b4 - b3)}

ai + bi + wZ f(a2 - ai) + (b2 - bi)} < z < a4 + b4 - wZ

f(a4 - a3) + (b4 - b3)}.

Now ai + bi + wZ f(a2 - ai) + (b2 - bi)} <

z-(ai +bi) (a2+b2)-(ai +bi) - L w-> 0, if

Let uL(z)=w . z-z(a1 +bJ) z . Now ^iLiZl^^-^^

(a2+b2)-(a1 +b1) dz (a2 +b2)-(a1 +b\)

(a2 + b2) > (ai + bi). Therefore, ^¿(z) is an increasing function. Additionally, mL(a1 + b1) =0, m^(a2 + b2)= w and

mL^^ > w. Again a4 + b4 - wz{(a4 - a3) + (b4 - b3)>z

(a4+b4)-z 0w, a (,4a+ 4/c > a.

Let mR(z) = w

(aj+bj )-z (a4+b4)-(a|+b|)-

Therefore, ^

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

is a decreasing function. Additionally, and

Therefore, mR(z)

mR(a4 + b4) = 0, mR(a3 + b3)=w?

Therefore, the membership function of C = ,A©B is

D.K. Jana / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—12

a\ + b\)

(4 + b\) - (a\ + b\y

mC (z) = -

(a4 + b4) - (a| + b|)' 0;

■1 + bl <. r< aI + bI;

'I+bI r < a? + b?;

>?+b? <z : < a' + b';

otherwise.

Hence, the addition rule is proven for the membership function. For the non-membership function

Cb = [Ci(b), C2(b)] = [A1(b), A2(b)] + [b'i (b), B2(b)] = [Ai(b) + B 1(b), A2(b) + B2(b)] b

a? + b2 - W{(a2 - a'D + (b| - bf)},a? + b? + ^(a

(b4z - b| )}

Let a? + b? - W f(a| - afz) + (b? - bfz)g < z < a? + b? + W

{(a' - a?) + (b'4 - b?)}.

(a?+b?)-z (a?+b? )-(a f+b'-Z )

> 0, if

Let a = ((-5, -2,0,3,0.7)(-6, -2,0, 5,0.7), (-5, -2,0,3,0.6) (-6, -2,0,5,0.6)), and b = (( 1, 2,3,4,0.7)( 1, 2,3,5,0.7), ( 1, 2,3, 4,0.6)( 1, 2,3,5,0.6)) be T2IFNs, the addition of these two numbers is shown in Fig. 2.

Property 3.2. Let A~ = (a1, a|, a3, a4; w1)(a'1z, a|, a3, a4Z;w1) and

~ = (b1, b2, b3, b4;w|)(b!z, b2, b3, b4Z;w|) be two GTIFN, then the subtraction of two GTIFN is given by

A'qB' = (ai - b4, a2 - b|,a3 - b2,a4 - b\; w) (a* - b'4, a\ - b\, a3 - b2, a'Z - bf; w)

where 0<wz < 1, w? = min(wf,w|).

Proof: Let A qB =a, where Ca = [C1(a), C2 (a)] and Cb = [C1 (b), C2(b)], a, be[0, w], 0 <w < 1 and w = min(w1, w|).

Now Ca = [C1(a), C2(a)] = [A^a), A2(a)] - [B1(a), B2(a)] = [A1(a)-B2(a), A2(a)-B1(a)]

a1 - b4 + wZ{(a2 - a1) - (b4 - b3)} a4 - b1

Now a? + b? - WW {(a? - af) + (b? - b'f )} < z^w

{(a' - b\) - (a? - b|)}

Let nC(^.aJ^^. Now ^=

' (aE+bE)-(a'E+b'E) dz (a2+b|)-(a1Z+b1Z)

(a| + b|) > (a'1 + b'1). Therefore, nL(z) is a decreasing function.

Additionally,

vLc(a\ + b?)=0, nL(af + bf)=w

; ^a1^ <w. Again a? + b? + W{ (a'E + b') + (a? + b?) > z

(a4E+b4E)-(a|+b|) dnR(z) __w

< b Let nR(z) = w

z-(a?+b?) (a4E+b4E)-(a? +bE)'

Therefore,

Let a1 - b4 + wZ{(a2 - a1)-(b4 - b3)g < z < a4 - b\ - w

f(a4 - b1)-(a3 - b|)>. Now a1 - b4 + wZ {(a2 - a1) - (b4 - b3)} < z^w ^-j^^)

Let (z)=ww,!z-'°1;b4' ,t^. Now ^dr1 = , c bw b, >0, if

(aE -bE)-(a1-bE) dz (aE-bE)-(a1-bE)

(a| - b3) > (a1 - b4). Therefore, m£(z) is an increasing function.

(a'' +b'4)-(aE+bE) '

> 0, if (a4 + b4) > (a? + b?). Therefore, nR(z) is Additionally, ^(ai - b') = 0,

-b?)=i

an increasing function. Additionally, nR(a3 + b3) = 0, nR(a4Z + ) = w and nR^+^+.a+i^j >The non-membership function of

C = A®B is

mC (VM+I-!j > w. Again a' - bl - wZ{(a' - bf) - (a? - b?)} > z

(a'-bf)-z , ^^ (a'-b')-z c

0w (a4-b'1)-(aE-bI) > a Let mR(z) = w (a4-b'1)-(aE-bI). Therefore,

dmdzzl =4 -(aE-bE)w(aE-bZ) <0, if (a' - b1) > (a? - bi). Therefore, mR(z)

(aI + bi) - (a1

(z) = <

■b!)'

a'1 + b'1 < z < aI + bI;

a? + b? < z < a? + b?; a? + b? < z < a' + b4Z ; otherwise.

a decreasing function. Additionally, mR(a' - b')= 0,

mR(a! - b?) = w and MR ^-M+i-1 j <w. The

membership func-

tion of C = AOB is

Hence, the addition rule is proven for the non-membership function. Thus, we have

mC' (z) =

z - )a 1 - b4)

A~ ®B' = (a ' + b', a? + bI, a? + b?, a' + b';i + bI, a? + bi, a'E + b' ; w)

where 0 < w? < 1 , w = min(w', w?).

- (a i - b4)'

)a4 - bl ) - z

(a' - bl ) - (a? - bl)' 0;

ai - b' < z < a\ - b?;

aI - bE < z < aE - bI;

a? - bI < z < a' - bl ; otherwise.

Hence, the subtraction rule is proven for the membership function. For the non-membership function

Cb = [C1(b),C2(b)] = [A1(b) ,A2(b)] - [B1 (b),B2(b)] = [A1(b)-B2(b), A2(b) -B't(b)]

«2 - b3 - W{(«2 - b3) - «- )},«3 - b2 + W{(«4

bf)- («3 - ^2)}

Let a2 - b3 - W{(a2 - b3) - (< - b^)} < z < a3 - b\ + W

{(«4 - b'1)-(a3 - b2)}-

Now «2 - b3 - W{(a2 - b3) - («1 - b4 )} < Z0W(az) < b. 2 3 1 4 Let vL(z)-w_(«2-b3)-z_ Now ^nM-__w_< 0 if

Let Vc(Z)_W (a2-b3)-«-bi)• Now -3T- — (a2-bi)-«-b4z) < 0 11

(«2 - b3) > («/Z - b4zTherefore, nC(z) is a teCTeasmg fonrtfon.

Additionally,

nL(«2 - b3) - 0,

nL(a<-

-b4z)-!

^«dS+of-^ <w. Agam «3 - b2 + W{(«4z - b'1z)-(«3 - b2)}

> Z0W—f—M—z2 z

- (a4Z-b'1Z)-(û3-b2)

Let nR(z) = w ■ z-,(°3-b2) . Therefore, ^ =

c (a4z-b1z)-(a3-b2) dz

■ («4z-b1Z)-(û3-b2)

if (a4 - bf) > (a3 - b2). Therefore, nR(z) is an increasing function.

Additionally,

nR(«3 - b2) - 0, nR (o4Z - bz) — wz

; -b2+q4z - b1z ^ .

r i az- bz + a'z—b'z \ z

nR | _2——l | > w. The non-membership function of C = A OB is

(z) —

(«2 - b3) - («f - b4z)' 0,

Z - [Q

- fa2)

«4z - tfl - («3 - b2)

«1 - b4z < z < «2 - b3;

«2- b3 <z < «3- b2; «3 - b2 < z < «4z -

otherwise.

Hence, the subtraction rule is proven for the non-membership function. Thus, we have

A ob1 — («! - b4 , «2 - b3 , «3 - b2 , «4 - f ; W («1 - b4z, «2 - b3 , «3 - b2 , o4Z - f ; w)

where 0 < wz < 1, w? = min(w1 w2).

Property 3.3. Let A' = (a1, a2 , a3 , a4; wz)(a1z , a2 , a3 , a^"; wz) be a GTIFN, then C' = kA' is GTIFN and

~i | ika! ka2,ka3,ka4;w )(kaf , ka2 , ka3 , ka'4z ; w), if k > 0; kaz4 kaz3 kaz2 ka1; w ka4z kaz3 kaz2 ka 1z ; w), if k < 0.

where 0 < wz < 1.

Proof: Case-I: k> 0 Let kA = C, where Ca = [C1(a) , C2(a)] and Cb = [C1(b), C2(b)], ae[0, w], 0 < w < 1.

Now Ca = [Q (a), C2(a)] = k^a),A2(a)] = [kA^a) , kA2(a]

kot + k —z (Ox - a1), ko4 - k —z (Q - a'

Let k«! + kWZ(«2 - «1) < z < k«4 - kWZ(«4 - «3).

lWZ (" 2

• —L/«z

Now ka1 + k —Z («Z - ûi) < z0W,z- k«1 -

wZ ^ 2 ' — kai-ka, ~

Let mC(z) — w

-w z-k«1

ka2-kai'

dmL(z) _ "dC^ — 1

> 0, if k«2 > kû!.

Therefore, m^(z) is an increasing function. Additionally, mL(ka1) = 0, mL(ka2)=w and mL > w. Again ka« - kwZ(a4 - a3)

> z^^-^O4-^ > a Let mR(z)=w-k0^. Therefore, dm|(Zl =

_ (ka'-ka^) _ c w (ka'-kaj) dz

-k zwk z <0, if ka4 > ka3. Therefore, mR(z) is a decreasing function.

Also mR(ka4) = 0, mR(ka3) = w and mR ( <if. Therefore, the

membership function of C = A®C is

mC (z) =

z- kai

ka? - kai

ka? -z

ka? - -ka?

kai < z < ka? ; ka? < z < ka?; ka? < z < ka?; otherwise.

Hence, the addition rule is proven for the membership function. For the non-membership function

Cb = [Ci(b), C?(b)] = k[A'i(b), A?(b)] = [kA'i(b) , kA?(b)]

D.K. Jana / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—12

Let mLc(z)=w-^. Now 7-?w-r > 0, if ka? > ka[.

ka\ - kW (a| - af),ka| + kW (a?? - a? Let ka? - kW(a? - a* ) < z < ka? + fcjW(a?? - a?) Now ka? - kW| (a| - af ) < z^w^b

Let nL^^w^. Now < 0, if 'a? > /<.

Therefore, ^(z) is a decreasing function. Additionally, (fca|) = 0, nL(ka 1 )=w and m^15^) < W. Again ka? + kwW (a* - a?)

> z^wkz-^ < b Let vR(z)=W^°i

rTTT. Therefore, ^ =

kdz-kai ' dz

ka'Wkal > 0, if ka4 > ka?. Therefore, nR(z) is an increasing function.

Additionally, nR(ka?)=0, nR(ka4? )=w and nRi ^ non-membership function of C = A4~ is

( kal+ka-z\

> f The

v? (z)

r ka? - z i ?_2_

ka? - kaf '

r z - ka?

vr-3—

ka? - ka?'

kaf < z < ka?;

ka? < z < ka?; ka? < z < ka?r ;

otherwise.

Hence, the scalar multiplication rule is proven for the non-membership function. Thus, we have

kA = (ka\, ka?, ka? , kajj; w) (kaf ,ka? , ka? , ka4?; w?)if if k>0

,_4 dz -(kai-kal^ " 4

Therefore, mL(z) is an increasing function. Also mL(kal) = 0, mL(ka?)=w and >f

Again kaf + k^ (a? - af) > z^w^^ii-tT > a Let mR(z)

(ka,-ka2)

< 0, if kai > ka?. There-

= ww ,ka'-zc . Therefore, ^ = ---

(kai -ka?) dz kai -ka?

fore, mR(z) is a decreasing function. Additionally, mR(kaf)=0, mR(ka?) = w and mR ka^ <w. The membership function of C = kA~ is

m? (z) =

, z - ka4 ;

ka? - ka?'

ka 1 - z kaf - ka? ' 0;

ka? < z < ka?;

ka? < z < ka?; ka? < z < kaf ; otherwise.

Hence, the scalar multiplication rule is proven for the membership function. For the non-membership function

Cb = [Ci (b), C?(b)] = k[A'i (b),A?(b)] = [kA2(b) , kA'i(b)]

ka? + kW (a?? - a?), ka? - kW(a? - a'i)

Let ka? + kW(a?r - a?) < z < ka? - kW(a? - ai?). Now ka?+

kW (a4 - a?)< z0Wkfeb

Let *e(?)=wg£. Now dncízl:

-kçWaj<0, if ka?>ka4.

Therefore, nc(z) is a decreasing function. Additionally, nc (ka?) = 0, nLc(kal) = w and mR <f Again ka? - kw(a? - af) > z

.WkO-lr < b Let vR(z) = W^kO-lr. Therefore, ^^ =

ka'Z-kaZ'

' ka'.r-kaZ

if kaf > ka?. Therefore, nR(z) is an increasing function. Additionally,

nR(ka?) = 0, nR(kaf) = w and membership function of C~ = kA is

> ^W. Therefore, the non-

where 0 < w? < 1.

Case-II: (k < 0) Let kA1 = C1, where Ca = [Cf(a), C?(a)] and Cb = [C1(b); C?(b)], ae[0, w], 0 < w < 1.

Now Ca = [Ci(a), C?(a)] = k[Ai(a), A?(a)] = [kA?(a), kA!(a]

ka4- kwwz (a4- a9; ka1+kwwz (a?- a1)

Let ka4 - kw?(a4 - ai)<z < kai + k^(a? - ai). Now ka4

kwz (a4 - a?)< Z0w1O-a4 > a

r ka? - z

WZ -^-7 ;

ka? - ka'Z

r z - ka?

vr-2—

ka7? - ka?'

ka'4 < z < ka?;

ka? < z < ka?; ka2 < z < ka 1 ;

otherwise.

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

D.K. Jana / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—12

•' •1 ■ ' / * / / .'.' i * t / .' • \ / / '// /-H-V—\\ //

■ A:-, \ / /' \\ // \ A //'■'■ •'••' \\ //

V •."• \ \//! ' ' \\ / /

V-r-' Vf.-. j ■:■ \\ A// '.• ■.• \\ // /M'

■• " ^ " /7 \

: i i / i 'X / \ /......... • \ 1 / i N

Fig. 3. Membership and non-membership function of subtraction of two T2IFNs.

kA = (ka4 , ka|, ka|, kai; w (ka'%, ka|, ka2, kaf; wz) if k<0

where 0 <wz < 1. Let us consider a = ((1,5 ,8,10,0.7) (4,6,7 ,11,0.7), (1,5 ,8 , 10 ,0.6)(4, 6, 7 , 11,0.6)), and ~ = ((1,2 ,3 , 4 , 0.7)(0,2 ,3 ,4 ,0.7) ,(1,2 ,3 ,4 ,0.6)(0 ,2 ,3 ,4 ,0.6)) be T2IFNs, the subtraction of these two numbers is shown in Fig. 3.

Property 3.4. Let A' = (a1,a2,a3,a4;w1)(a'1z ,a2 ,a3,a4z;w1) and b' = (b\ ,b\,b3 ,b\; w|)(bf , b\,b3 ,b'Z; w2) be two GT1FN, then the division of two GT'FN is given by

si ai ai. ^ faî. ai ai aï. w)

b4 bi bi bt J [bf bi bi bf J

where 0< w? < 1, w? = min(w1, w2). Proof: Let A' +B' = C

Cb = [CKb) , Ci ( Now

where Ca = [Q(a), C2 (a)] and a,be[0,w], 0<w < 1 and w = min(wt,w2).

\ al- zb mR(z) = w-4 ■

_ Therefore diRM- w_a3b2-aib2_< 0

(a4-a3)-z(b2-b1). ST = w {(ai+bi)-(ai+bi)}2 < 0,

for a3b1) < a4b2i.e.Of <% Therefore, mR(z) is a decreasing function.

Also mR^JJ = w, = 0 and mR

membership function of C = A+B is zb\ - a1

i i i bpbl

w_zbi ~ai_ ai < z < ai.

(ai -(ai)+z(bi - bij bi bi'

< w. Therefore, the

ai - z(

ai - ai) + ^bi 0,

(bi - bi)'

zi. bi'

ai. bi'

otherwise.

Hence, the division rule is proven for the membership function. For the non-membership function

C„ = [Ci (a) , Ci (a)] = [Ai(a) , Ai(a)]-[Bi(a), Bi(a)] _ Ai (a) Ai(a)

Bi(a) Bi(a)J

ai + w (ai - ai) a\ - £ (ai - ai

bi - w (bi - bi)' bi + w (bi - bi).

ai +wZ (ai -ai)

■ ai-wz (ai-ai) : bi+4(bi-bi)'

ai+wZ (ai-ai)

Cb = [Ci(b), Ci(b) = [Ai (b) , Ai(b) * [B'i(b) , Bi(b)

["Ai (b) Ai(b)l ai -1 (ai - ai ) ai + w (ai - ai)!

[Bi(b) Bi (b)J bi +w (bi - bi) bi - w (bi - bi)J

Let mL(z)=w-^,

(ai-ai )+z(bi-i)

Now ^mP = w

ajbl-aibj {(ai-ai )+z(bi-bi)}i

for (a2b4) > a1b3i.e.a2> % Therefore, mL(z) is an increasing function.

Additionally, mL ( a2 ) = w, iL ( br ) = 0 and mL

\b3 / \b4 J

.<«, - a4-^(a4-a3)

■ bi+bi >

iffsince bi <bi]. Again

£ (bi-bi)

> Z0~w-T-pi-T-7~

> (ai-ai)-z(bi-bi) -

> a Let

0w-i—i

bi +w (biz -bi) -< b

(ai-af)-(biE-bi)

Let nC(z) = w

: bi-w (bi-b'i)•

ai -zbi

(ai -ai )-(biZ-bi) '

(ai-ai)+z(bi-b'i)

< b Let nR(z) = w

bi+w (b'T-bi)

ai+w (ai-ai) (bi-b'i)

Therefore,

(a<-ai)+z(bi-bii)-

bi-l(bi-bf) >'

dnR(z)

{(ai-ai)+z(bi-bii)}i

ai > ai bT b

Therefore, nR(z) is an

zbi-ai

;ai-ai)+z(bi-bi )

ai -w (ai-ai)

ai+w (ai1 -ai)

ai-! (ai-ai)

t \ i ai >

zbi-ai

ai bi -aibi

increasing function. Additionally, nR= 0, nR^tj

> 3j. Therefore, the non-membership function of C = A^B

D.K. Jana / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—12

-4 =w and

/JL 2!

a? - zb?

b?r - b?)

+ I a? - ai?

zb? - a?

ai? a?

-f < z < -?; b??- -b?'

-2 < z < -?-;

< z < -4 ;

(a? - a?) + z(b? - bi?)' b\~ " b'f W otherwise.

Hence, the division rule is proven for the non-membership function. Thus, we have

A -b'-Î0' a2 a? a4; wVai a2 a? a4 ; W

A = b? b? b?; w){y?; b? b? b?; W

where 0 < w? < i, w? = min(wi, w?).

Property 3.5. Let A' = (ai,a?,a?,a4;wi)(a'i?,a?,a?,a4?;wi) and

B1 = (b?,b?,b?b4;w?)(bi,b?,b?b4?;w?) be two GTIFN, then the multiplication of two GTIFN is given by

A ®B' = Ubi; a?b? ; a?b?' a?b? ; w) (af b'f ; a?b2' a?^; a?? b?? ; w

where 0 < w; < i, w; = min(wi, w?).

Proof: Let A'eB' = C1, where Ca = [Ci(a), C?(a)] and Cb = [Ci(b),C?(b)], a,be[0,w?], 0<w; < i and w? = min(w?,w?). Now

Ca = [Ci (a), C?(a)] = [Ai (a),A?(a)]-[Bi(a),B?(a)] = [Ai(a)Bi(a), A2(a)B2(a)]

a? / ?

(a? - ai) (b? - b') + £{ai(b? - b') 2

+ bi(a? - ai)} + aibi; W (a? - a?) (b? - b?)

{a? (b? - b?) + b?(a? - a?)} + a?b?

Let (a? - ai)(b? - &')+W? fai (b? - b ') + b'(a? - ai)} + Jib'

< z < W? (a? - a?)(b? - b?) - w?{a4(b4 - b?) + b?^? - a?)} + a4b4.

Let Pi = (a? - ai)(b? - b'), Q = {Ji(b? - b?)+b?(a? - J')}. Now

a ™ a _ , ? W2Pi ^Qi + Jib' <

a2 a? ?

0-TPi + —?Qi + aib' - z < 0 "'? w? '

-Qi Q2 - ?pi( Jib? - z) a?

-Qi + ^Q? - Jib' - z)

mL(z)=w V Q M-'

, r r r ■ , ■ r r r > 0. Therefore, mL(z) is an

\J {ai (b?-b i )+bi (a? -a,)}? -4(a? -a, )(b?-b i )(a , b\ -z) c'

increasing function. Additionally, mL(a-i b?) =0, mL(a2b2) =w and

-2 =(J? - J?)(b? - b?),

Qi = {а4(b4 - b?) + b?(a? - a?)} then

W?-? + W? + a?b? >

^2-? + ^Q? + a?b? - z > 0

Q? -J Q 2 - ?p2( a?b? - z) a?

Q? + ^Q2 - ^a?b? - z) a?

dmR(z) _

2-2 - w?

mR(z)=wQ?-p(ffi ffibizz).

Therefore,

__w_< 0 if

dz 7{al(bl-bi)+bl(al-ai)}2-4(al-ai)(bl-bi)(albl-z)

(a4 + b4) > (a? + b?). Therefore, mR(z) is a decreasing function.

Additionally, mR(a4b4) = 0, mR^?) = w and mR <w. The

membership function of C = A&B is

-Qi + JQ? - 4Pl{aib' - z)

W -^-^-;

mC' (z)

2-i w;

Q? Q? - ?-?(а4b4 - z)

2-? 0;

a 'b' < z < a?b?; a?b? < z < a?b|;

a?b| < z < a?b?; otherwise.

Let us consider C = (( i, 4,0.7)(0, 5,0.7), ((( i, ?, 4,0.6)(0, ?, 5,0.6)), and ~ = ((0, ?, 60.8)(0, ?, 60.8), (0, ?, 60.6)(0, ?, 60.6)) be T?IFNs, the multiplication of these two numbers is shown in Fig. 4.

Hence, the multiplication rule is proven for the membership function. For the non-membership function

D.K. Jana / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—12

Fig. 4. Membership and non-membership function of multiplication of two T2IFNs.

Cb = [Ci(b), Ci(b) = [Ai(b),Ai(b) ■ [B'i(b),Bi(b) = [Ai(b)Bi(b) , Ai(b)Bi(b)]

(ai - ai)(bi - bi) - w {ai(bi - bf) + bi (ai

- a!)} + aibi , £ (ai - 4) (bi - bi) + ^ {a\ (bi

- bi +bi ai - ai +aibi

Let w (ai - ai)(bi - b') -1 {ai(bi - b') + bi(ai - ai)} +aibi < z < &(ai - ai)(bi - bi) +|{ai(b4Z - bi) + b^ - ai)} +aibi. Let Pi = (ai - ai)(bi - bi), Q' = {ai(bi - bi) + b\(ai - ai)}.

liPi - bQi + aibi < z

âî-Pi + Qi + aibi - z < 0

Qi Q'i2 - 4P'i( aibi - z) b

Qi + ^Q'I2 - 4Pi (aibi -z

vLc (z) = w

Q'-^ Q'i2 -4P1 (aibi -z) 2p; .

dvLc(z)

< 0. Therefore,

Vfa| (b2 -bf)+b2 (a2 -af)}2 -4(a|-af)(b|-b1i)(a2b| -z) v£(z) is a decreasing function. Additionally, v^aibi) = w,

nc(a|b| ) = 0 and vR <w. Again let P2 = (a4 - a3)

(b4z - b3), Q2 = fa3(bi - b3) + b3(a< - a3)} then ■Cp + -Q2 + a3b3 > z

w2 2 w 2 3 3

0-4p2 + 4 Q2 + a3b3 - z > 0 w2 2 w 2 3 3

-Qi Qi2 - 4Pi(aibi - z)

b -Qi + JQi2 - 4pi(aibi - z) b

- — or-Ï-—;-< —

w iP w

Let vR(z) = w-(WQ2;-p4P2(aibi-z)

Therefore,

dvR(z) _

^{ai(b'i-bi)+b|(a4 -ai)}2-4(aiE -s")^-bi)(a!;b!;-z)

> 0. Therefore,

vR(z) is an increasing function.

Additionally, vR(a3b3) = 0, vR(a4zb4z) = w and vR ^M+^M.j : Therefore, the non-membership function of C = A&B is

Q1 -J Q12 - 4P1 (a3&3 - z)

1/1/_1_

v-d (z) =

-Qi + JQi2 - 4Pi(ai bi - z)

a'b'i < z < aibi.

aibi < z < aibi.

aibi < z < ai bi. otherwise.

Hence, the division rule is proven for the non-membership function. Thus, we have

Ai'5B1 = (a1fa1, a2b2, a^ , a^b"; w) ,a2fa2, a3b3 , a^; w'

where 0< w < 1, w = min(w1 w2).Let us consider a = ((2 , 4 , 6 , 8 , 0.8)(1, 4 , 6 , 12 , 0.8) , ((2 , 4,6 , 8 , 0.6)(1, 4 , 6 , 12 , 0.6)), and b=((4,6 ,8 ,10 , 0.8)(3, 6 ,8 ,12 , 0.8) ,(4 ,6 ,8 , 10 , 0.6) (3 ,6 ,8 ,12 , 0.6)) be T2IFNs, the division of these two numbers is shown in Fig. 5.

4. Ranking function of GTIFN

Let A' = (a1, a| , a3 , a[; w)(a1z, a| , a3 , d\; w) be a GTIFN. There are many methods for defuzzication, such as centroid method, mean of interval method, and removal area metho. In this paper, we have used mean of interval method to find the value of the membership and non-membership function of GTIFN.

i0 D.K. Jana / -acific Science Review A: Natural Science and Engineering xxx (2016) 1—12

Fig. 5. Membership and non-membership function of division of two T?IFNs.

4.1. Mean of interval method

The (a, b)-cut of the GTIFN is given by Aa,b = {[Ai (a),A?(a)]; [A'i(b),A?(b)], a + b < w, a, be[0, w]}

where Ai(a)=ai + £(a? - ai), A?(a)= a4 - w?(a4 - a?), a? (b) = a? - w(a? - a'i), A?^)^ + w(0? - a?). Now by the mean of interval method the representation of the membership function is

(iii) Aa = b' iff h(A') = h(b'

', Rm(A') + Rj B1) w(ji + 2a? + 2a? + a? + a'? + J

' ) Rv{A') + R„ ( B'J w (b' + 2b? + 2b? + b? + b'f + b??) 2 = 8

RmfA') = ' /(Ai(a)+A?(a))da

=U ai+a4+w?{(a?-a0-(a4-a?)} 0 L

=? (aiw+a4w+w{(a?- ai)- (a4- ai)})

w(at + a? + a? + a4)

Now, by the mean of interval method, the representation of the non-membership function is

R„(A') = ' f (Ai(b)+A?(b))db

4/ [a? + a? - W{ (a? - J?) - (a?? - a?)}

? (a?w + a?w - W {(a? - J? ) - (a?? - a?)})

w( a'? + a? + a? + a??

Let A' = ((ai; a? ; a?; a?; Wi)(j'? ; a?; a?; a? ; w?)) and B = ((bf b? ; b?;b?;w2)(b1r;b?;b?;b?;w?)) be two GTIFNs, then [i] proposed

(i) A 3 B1 iff H (A1^ < h(b'

(ii) _ B iff h (A1) > H (B'

where w? = min(wi, w?). With the help of the above formulae, we have optimized the following transportation problem in the T?IF environment.

5. Generalized type-2 intuitionistic fuzzy transportation problem

Consider a type-2 intuitionistic fuzzy transportation problem (T2IFTP) with m sources and n destinations as

Minimize 4 4 c„x.

i=ij=i

subject to zj' for i = i; 2; ■ ■ m

iZxijzb'j for j = i, ?, -n i=i

xij > 0 C i, j

where aeiI is the approximate availability of the product at the ith

source, bejI is the approximate demand of the product at the jth

destination, ceijI is the approximate cost for transporting one unit of the product from the ith source to the jth destination and xij is the number of units of the product that should be transported from the ith source to jth destination taken as fuzzy decision variables.

If S= Yljl=1bj then the intuitionistic fuzzy transportation problem (IFTP) is said to be a balanced transportation problem, otherwise it is called an unbalanced IFTP.

Let 4 = (cji, cij?, j, cj4; w)(cji, cij?, j, cj4;w),

a' = (an, ai?, aa, aM; w)(aii, a,-?, aa, ai4; w) and bj = (bji, bj?, bj?,

bj4; w)(bji, bj2, bj?, bj4;w).

The steps to solve the above IFTP are as follows: Step 1. Substituting the value of cj, a' and bj in (?), we get

D.K. Jana / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—12

Minimize 44cin, j, cy3, cij4; w) (dm, j, cp, j w) x j subject to

£xyz(an, ai2, ai3, ai4; w) (ail5 ai2, ai3, ai4; w) for i = 1,2 , ■ ■ • m (2) j=i

Exy z (bji, bj2, bj3, bj4; w) (bji, bj2, bj3,6-4; w forj = 1,2 , ■ ■ •n xij > 0 Ci j

Step 2. Now by the arithmetic operations and definitions presented in Section 3 and 2, (3) converted to crisp linear programming (CLP)

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

Step 4. Find the fuzzy optimal value by putting xij in 4 4 cjxj. 6. Numerical example

Consider a transportation problem with three origins and three destinations. The related costs are given in the following Table 1.

Minimize H Ql 4 (xycyi, xycp , xjj , xj j; w)(xyciyi, xycy2, xycy3, xydyA; w ) subject to H ^=>ij j = H ((an, aa, ai3 , ai4; w) (aii, ai2 , ^3, ai4; w)) for i = i, 2 , ■ ■ • m

^P^xij^ = h( (bji, bj2 , bj3, bj4; w) j bj2, bj3 , b/4; w)) for j = i, 2 , -n

Xij > 0 Ci ; j

Using Step-i of the method explained in Section 5 the above IFTP can be written as

Minimize (2 ,4 , 5 , 6; 0.5)(i, 4 , 5 , 6; 0.5)xii 4 (4 , 6 , 7 , 8; 0.2)(3 , 6 , 7 ,9; 0.2)xi2 4 (3 , 7 , 8 , i2; 0.3)(2, 7 , 8 , i3; 0.3)x13® (i, 3, 4 , 5; 0.6)(0.5 , 3 , 4 , 5; 0.6)x2i 4 (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 4 (3 ,4 , 5 , 8; 0.7)(2,4, 5 ,9; 0.7)x31 ® (i, 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 z (4,6 ,8 ,9; 0.6)(2 ,6 ,8 ,10; 0.6)

x21 + x22 + x23 z (0,0.5 ,1,2; 0.5)(0, 0.5,1,5; 0.7) (4)

x31 + x32 + x33z (8 , 9.5 , 10 ,11; 0.8)(6.5, 9.5 , 10 ,11; 0.8)

xii + x2i + x3i z (6 , 7 ,8 ,9; 1)(5 ,7 ,8 ,11; 1)

xi2 + x22 + x32 z (4 , 5 , 6 , 7; 0.8)(3 , 5 , 6 , 8; 0.8)

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

xij > 0Ci,j

Using Step-2 of the method explained in Section 5 the above IFTP converted into crisp linear programming

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

subject to x11 + x12 + x13 = 3.975

x21 + x22 + x23 = 6.78

x31 + x32 + x33 = 7.55

xii + x2i + x3i = 7.625

x12 + x22 + x32 = 4.465

xi3 + x23 + x33 = 2.514

xij > 0 , C i, j

D.K. Jana / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—12

Table 1

Input data for IFTP.

Availability(aj)

51 (2,4,5,6; 0.5) (1,4,5,6; 0.5)

52 (1,3,4,5; 0.6) (0.5,3,4,5; 0.6)

53 (3,4,5,8; 0.7) (2,4,5,9; 0.7) Demand (j (6,7,8,9 1) (5,7,8,i1; 1)

(4,6,7,8; 0.2) (3,6,7,9; 0.2) (3,5,6,7; 0.6) (2,5,6,8; 0.6) (1,3,3,4; 0.8) (0.5,3,3,5; 0.8; (4,5,6,7; 0.8) (3,5,6,8; 0.8)

(3,7,8,13; 0.3) (3,7,8,13; 0.3) (3,6,7,11; 0.4) (1,6,7,13; 0.4) (3,4,5,10; 0.3) (1,4,5,11; 0.3) (3,4,5,6; 0.6) (0.5,4,5,7; 0.6)

(4,6,8,9; 0.6) (3,6,8,10; 0.6) (0,0.5,1,3; 0.5) (0,0.5,1,5; 0.7) (8,9.5,10,11; 0.8) (6.5,9.5,10,11; 0.8)

Solving the above crisp linear programming using Lingo-14.0, we get x11 = 0.75, x13 = 0, x13 = 0, x31 = 6.87, x33 = 0, x33 = 0, x31 = 0, x33 = 4.4 and x33 = 3.5. The minimum cost of transportation is 395.50.

7. Conclusions

This paper proposes and investigates type-3 intuitionistic fuzzy sets on two finite universes of discourse from both constructive and axiomatic approaches. An approximate algorithm is given with a suitable numerical example. A generalized intuitionistic fuzzy number and its arithmetic operation based on (a, b)-cut method is developed. We considered basic generalized type-3 intuitionistic fuzzy numbers, namely, generalized trapezoidal type-3 intuition-istic fuzzy numbers. We discussed the ranking function of the generalized trapezoidal type-3 intuitionistic fuzzy numbers. We solved an intuitionistic fuzzy transportation problem where transportation cost, source, and demand were generalized type-3 intuitionistic fuzzy numbers. A numerical example is presented to solve the type-3 intuitionistic transportation problem. This idea can be extended as a type-3 intuitionistic fuzzy fault tree analysis for the failure of automobile system to start. The definition of type-3 intuitionistic fuzzy sets presented here is useful only for deriving theoretical results, the examples presented in this paper are simple, and the stated conclusions are all based on discrete intuitionistic type-3 fuzzy sets. Corresponding results for general intuitionistic type-3 fuzzy sets will be discussed in future work by the authors. Future work by the author will consider different reduction methods in terms of intuitionistic type-3 fuzzy sets and various approaches to knowledge discovery in complex type-3 intuition-istic fuzzy information systems.

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