Novel arithmetic operations on type-2 intuitionistic

Pacific Science Review A: Natural Science and Engineering xxx (2016) 1e12

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Pacific Science Review A: Natural Science and Engineering j o u r n a l h o m e p a g e : w w w . j o u r n a l s . e l s e v i e r . c o m / p a c i fi c - s c i e n c e review-a-natural-science-and-engineering/

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

a r t i c l e i n f o

a b s t r a c t

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

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 ða; bÞ-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/).

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 intuitionistic 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: þ91 3224 252800. E-mail address: [email protected]. 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/).

Please cite this article in press as: D.K. Jana, Novel arithmetic operations on type-2 intuitionistic fuzzy and its applications to transportation problem, Pacific Science Review A: Natural Science and Engineering (2016), http://dx.doi.org/10.1016/j.psra.2016.09.008

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D.K. Jana / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1e12

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 ða; bÞ cut method are presented. To illustrate the proposed method, a numerical example is presented and solved as a type-2 intuitionistic transportation problem.

mzA~ ðxÞ ¼

Definition 2.1. Generalized Intuitionistic Fuzzy Number (GIFN): ~ I ¼ f < x; m ~ ; n ~ > g of the real line < is An Intuitionistic fuzzy number A A

A

called GIFN, if the following hold

z

nA~ ðxÞ ¼

m a x m; x ¼ m; m x m þ b; otherwise:

az3 x az4 ; otherwise:

8 a x > > wz z2 0 ; > > > a2 a1z > > > > > > > 0; < > > x az3 > > wz 0 z ; > > > > a4 az3 > > > > : wz ;

a1z x az2 ; 0

az2 x az3 ; az3 x a4z ; 0

otherwise:

Definition 2.3. Generalized Triangular Intuitionistic Type-2 ~ I ¼ ½mL ðxÞ; mU ðxÞ; nL ðxÞ; nU ðxÞ A ~ ~ ~ ~

GTIT2FN z

z

mzA~ ðxÞ ¼

z

A

A

A

A

in

> < 1 w; mA~ ðxÞ ¼ > wh1 ðxÞ; > : 0;

az2 x az3 ;

Fuzzy Number (GTrIT2FN): Let z2fL; Ug and a01z az1 az2 a03z . A

(i) there exists x2 > > z a4 x ; > > w > > > az4 az3 > > > : 0;

az1 x az2 ;

and non-membership function

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.

8 > x az1 > > ; wz z > > > a2 az1 > > > > > > < wz ;

8 > z x a1 ; > >w z > > a2 az1 > > > > > > > < wz ; z > > > z a3 x ; > > w > > > az3 az2 > > > > : 0;

az1 x az2 ; x ¼ az2 ; az2 x az3 ; otherwise:

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

8 wf ðxÞ; > > < 2 0; nA~ ðxÞ ¼ > wh > 2 ðxÞ; : w;

0

m a x m; 0 wðf1 ðxÞ þ f2 ðxÞÞ w; x ¼ m; 0 m x m þ b ; 0 wðh1 ðxÞ þ h2 ðxÞÞ w; otherwise:

In this equation, f1 ðxÞ and h1 ðxÞ are strictly increasing and decreasing functions in ½m a; m and ½m; m þ b, respectively, and f2 ðxÞ and h2 ðxÞ are strictly decreasing and increasing functions in ½m 0 I 0 a ; m and ½m; m þ b respectively, where m is the mean value of A~ .

The left and right spreads of membership function mA~ ðxÞ are called a and b. The left and right spreads of non-membership function nA~ ðxÞ are 0

0

called a and b . Definition 2.2. Generalized Trapezoidal Intuitionistic Type-2 Fuzzy Number (GTIT2FN): Let z2fL; Ug and 0 0 I z z z z z z L U ~ a a a a a a . A GTIT2FN A ¼ ½m ðxÞ; m ðxÞ; 1

1

2

3

4

4

~ A

~ A

nLA~ ðxÞ; nUA~ ðxÞ in < written as ðaz1 ; az2 ; az3 ; az4 ; wz Þða1z ; az2 ; az3 ; a4z ; wz Þ has 0

membership function (in Fig. 1).

0

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



nA~ ðxÞ ¼

8 a x > > wz 2 0 ; > > > a > 2 a1 > > > > > 0; < z

x a2 > > > ; wz 0 z > > > a3 a2 > > > > > : wz ;

3

o h 0 n i 0 Aa ¼ x : n ~ I ðxÞ b ¼ A1 ðbÞ; A2 ðbÞ A b 0 b 0z a2 a1 ; a3 þ a4 a3 ¼ a2 w w

0

a1 x a2 ; x ¼ a2 ; az2 x a3z ; 0

Definition 2.9. ða; bÞ-cut set: A ða; bÞ-cut ~ I ¼ ðaz ; a ; a ; a ; wÞða0 ; a ; a ; a0 z ; wÞ is given by A 1 2 3 4 1 2 3 4

otherwise:

~ ¼ ðaz ; az ; az ; az ; wz Þða ; az ; az ; a z ; wz Þ is Definition 2.4. A GTIFN A 1 2 3 4 1 2 3 4 I

0

0

said to be non-negative iff a1z 0. 0

~ I ¼ ðaz ; az ; az ; a ; w Þ 2.5. Two GTIFN A 1 1 2 3 4 0 0 ~ I ¼ ðbz ; bz ; bz ; bz ; wz Þðb0 z ; bz ; bz ; b0 z ; wz Þ are ða z ; az ; az ; a z ; wz Þ and B

Aa;b ¼

n

set

of

i h 0 i o 0 ; A1 ðbÞ; A2 ðbÞ ; a þ b w; a; b2½0; w

½A1 ðaÞ; A2 ðaÞ

Definition 1

2

3

4

1

z

1

2

z

3

4

z

2

1

z

z

2

z

3

z

4

0

z

2

0

3. Arithmetic operations

z

said to be equal iff a1 ¼ b1 ; a2 ¼ b2 ; a3 ¼ b3 ; a4 ¼ b4 ; a1 ¼ b1 ; a4z ¼ b4 and w1 ¼ w2 . 0

0

Definition 2.6. An interval number is a closed and bounded set of real numbers ½az ; bz ¼ fx : az x bz cx; bz ; x2 < kaz ; kaz 1 2 i kA ¼ h > : ka2 ; kaz1

; if k 0;

p ¼ minðaz1 b1 ; az2 bz1 ; az1 b2 ; az2 bz2 Þ

and

q ¼ max

ða1 b1 ; a2 b1 ; az1 b2 ; a2 b2 Þ. (v) The division of two interval numbers A ¼ ½az1 ; a2 and B ¼ ½b1 ; b2 is denoted by Að÷ÞB and is defined by

8 h i 1 1 z > > ð$Þ ; ; a ; a > 2 1 > b1 b2 > > > > > > empty interval; > > > > > h i > > < az ; a2 ð$Þ 1 ; ∞ ; 1 b2 Að÷ÞB ¼ > > > z 1 > > > a ð$Þð ∞; ; ; a > 2 1 > b1 > > > > > > > az1 ; a2 az1 ; a2 > > : ∪ ; ½b1 ; 0Þ ð0; b2

where 0 < wz 1, wz ¼ minðwz1 ; wz2 Þ. Proof:

~ I 4B ~I , ~I ¼ C A

Let

1

; if k < 0

B ¼ ½bz1 ; bz2 is denoted by Að$ÞB and is defined byAð$ÞB ¼ ½p; q, z

~ I 4B ~ I ¼ az þ bz ; az þ bz ; az þ bz ; az þ bz ; wz a0z þ b0z ; a A 2 1 1 2 2 3 3 4 4 1 1 z z z 0z 0z þ b2 ; a3 þ b3 ; a4 þ b4 ; w

2

z

z

a fðaz az Þ þ ðbz bz Þg z 0w Now az1 þ bz1 þ w z 2 1 2 1

zðaz1 þbz1 Þ

z

Let mLc ðzÞ ¼ w z

z

z

z

z

zða1 þb1 Þ

. Now

ðaz2 þbz2 Þðaz1 þbz1 Þ

dm dz

L c ðzÞ

z

¼


a.

w > 0, ðaz2 þbz2 Þðaz1 þbz1 Þ

if

ða2 þ b2 Þ > ða1 þ b1 Þ. Therefore, mLc ðzÞ is an increasing function. Additionally,

mLc

~ ¼ ðaz ; a ; a ; a ; wÞ Definition 2.7. a-cut set: A a-cut set of A 1 2 3 4

and

fðaz4 az3 Þ þ ðbz4 bz3 Þg.

if b1 ¼ 0; b2 s0;

otherwise:

and

0 < wz 1

a fðaz az Þ þ ðbz bz Þg z az þ bz a az1 þ bz1 þ w z 2 1 2 1 4 4 wz

Let

if b1 s0; b2 ¼ 0;

Caz ¼ ½C1z ðaÞ; C2z ðaÞ

h i h i h i Now Caz ¼ C1z ðaÞ; C2z ðaÞ ¼ Az1 ðaÞ; Az2 ðaÞ þ Bz1 ðaÞ; Bz2 ðaÞ i h ¼ Az1 ðaÞ þ Bz1 ðaÞ; Az2 ðaÞ þ Bz2 ðaÞ " o az n z a2 az1 þ bz2 bz1 ; az4 þ b4 ¼ az1 þ bz1 þ z w # n o z a z z z z a4 a3 þ b4 b3 z w

if 0;½b1 ; b2 ; if b1 ¼ b2 ¼ 0;

where

az ; bz 2½0; wz ,

Cbz ¼ ½C10z ðbÞ; C20z ðbÞ, wz ¼ minðwz ; wz Þ.

(iv) The product of two interval numbers A ¼ ½az1 ; a2 and where

~ I ¼ ðaz ; az ; az ; az ; wz Þða0z ; az ; az ; a0z ; wz Þ and Property 3.1. Let A 1 2 3 4 1 1 2 3 4 1 ~ I ¼ ðbz ; bz ; bz ; bz ; wz Þðb0 z ; bz ; bz ; b0 z ; wz Þ, where z2ðL; UÞ be two B 1 2 3 4 2 1 2 3 4 2 GTIFN, then the addition of two GTIFN is given by

z

z

z

z

!

a1 þb1 þa2 þb2 2

mLc ðaz1 þ bz1 Þ ¼ 0,

mLc ðaz2 þ bz2 Þ ¼ w

and

z z z z z z az >w 2 . Again a4 þ b4 wz a4 a3 Þ þ ðb4 b3 Þ z

I

ða1 ; a2 ; a3 ; a4z ; wÞ is a crisp subset or < that is defined as follows 0

0

o

n

Aa ¼ x : m ~ I ðxÞ a ¼ ½A1 ðaÞ; A2 ðaÞ A i h a a z a2 az1 ; a4 ða4 a3 Þ ¼ a1 þ w w ~ I ¼ ðaz ; a ; a ; a ; wÞ Definition 2.8. b-cut set: A b-cut set of A 1 2 3 4

ða1 ; a2 ; a3 ; a4z ; wÞ is a crisp subset or < that is defined as follows 0

0

0w

ðaz4 þbz4 Þz


Let mRc ðzÞ ¼ w Therefore, Therefore,

a. ðaz4 þbz4 Þz


dmRc ðzÞ dz

mRc ðzÞ

mRc ðaz4 þ bz4 Þ ¼ 0,

¼ is

. if ðaz4 þ bz4 Þ > ðaz3 þ bz3 Þ.

w < 0, ðaz4 þbz4 Þðaz3 þbz3 Þ

a

decreasing

mRc ðaz3 þ bz3 Þ ¼ wz

function.

and

mRc

z

Additionally, !

z

a3 þb3 þaz4 þbz4 2

z

< w2 .

~ B ~ ¼ A4 ~ is Therefore, the membership function of C


4


8 > z az1 þ bz1 > > > z > ; w > > > > az2 þ bz2 az1 þ bz1 > > > > > > < wz ; z m ~ I ðzÞ ¼ > C > > az4 þ bz4 z > > > wz > ; > > > az4 þ bz4 az3 þ bz3 > > > > > : 0;

~ ¼ ðð5; 2; 0; 3; 0:7Þð6; 2; 0; 5; 0:7Þ; ð5; 2; 0; 3; 0:6Þ a ~ ¼ ðð1; 2; 3; 4; 0:7Þð1; 2; 3; 5; 0:7Þ; ð1; 2; 3; ð6; 2; 0; 5; 0:6ÞÞ, and b Let

az1 þ bz1 z az2 þ bz2 ;

4; 0:6Þð1; 2; 3; 5; 0:6ÞÞ be T2IFNs, the addition of these two numbers is shown in Fig. 2.

az2 þ bz2 z az3 þ bz3 ;

~ I ¼ ða ; az ; az ; az ; w Þða0z ; az ; az ; a0z ; w Þ and Property 3.2. Let A 1 2 3 4 1 1 1 2 3 4 I z z z z z 0z z z 0z z ~ B ¼ ðb1 ; b2 ; b3 ; b4 ; w2 Þðb1 ; b2 ; b3 ; b4 ; w2 Þ be two GTIFN, then the subtraction of two GTIFN is given by

az3 þ bz3 z az4 þ bz4 ; otherwise:

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

Cb ¼ C10 ðbÞ; C20 ðbÞ ¼ A01 ðbÞ; A02 ðbÞ þ B01 ðbÞ; B02 ðbÞ ¼ A01 ðbÞ þ B01 ðbÞ; A02 ðbÞ þ B02 ðbÞ o b n z b n 0z a2 a01z þ bz2 b01z ; az3 þ bz3 þ a4 ¼ az2 þ bz2 w w o az3 þ b04z bz3 z

z

b

0

z

z

0

z

z

z

fða4 a3 Þ þ ðb4z bz3 Þg.

z

b. Let nLc ðzÞ ¼ w

ðaz2 þbz2 Þz

. Now

ðaz2 þbz2 Þða01z þb01z Þ

dnLc ðzÞ dz

Additionally, az2 þbz2 þa01z þb01z 2

dnRc ðzÞ dz

z

nLc ðaz2

!

nLc ða01z

þ b2 Þ ¼ 0,

z z b 0, ða04z þb04z Þðaz3 þbz3 Þ

nRc ðzÞ

Let

wz and nRc

z

a3 þb3 þa04z þb04z 2

¼w

z

z

zða3 þb3 Þ

.

ða04z þb04z Þðaz3 þbz3 Þ

¼ ½A1 ðaÞ B2 ðaÞ; A2 ðaÞ B1 ðaÞ " o az n z a2 a1 bz4 bz3 ; az4 bz1 ¼ a1 bz4 þ z w # o az n z a4 bz1 az3 bz2 z w

if ða04 þ b04 Þ > ðaz3 þ bz3 Þ. Therefore, nRc ðzÞ is

>

wz 2

nRc ðaz3

z

þ b3 Þ ¼ 0,

nRc ða04z

þ

b04z Þ

¼

. The non-membership function of

~ B ~ ¼ A4 ~ is C

8 z z > a z þ b > 2 2 > > 0 ; > wz > 0 z z > > a2 þ b2 a1z þ b1z > > > > > > > < 0; z n ~ I ðzÞ ¼ > C z z > > z a þ b > 3 3 > z > ; > 0 0 >w z z z z > > a a þ b > 4 4 3 þ b3 > > > > : wz ;

z

z

fðaz4 bz1 Þ ðaz3 bz2 Þg.

a fðaz a Þ ðbz bz Þg z0w a1 bz4 þ w z 1 2 4 3 z

Now a.

Let mLc ðzÞ ¼ ww

zða1 bz4 Þ

Additionally,

mLc

z

z

z

a1 b4 þa2 b3 2

0w

mLc ða1 bz4 Þ ¼ 0,

!

ðaz4 bz1 Þz

ðaz4 bz1 Þðaz3 bz2 Þ

¼

Let

4

w < 0, ðaz4 bz1 Þðaz3 bz2 Þ

z

z

z

and

function.

mRc ðaz3 bz2 Þ ¼ w and mRc

z

1

3

Therefore,

2

z

if ða4 b1 Þ > ða3 b2 Þ. Therefore, mRc ðzÞ

8 > z a1 bz4 > > > > ; wz > > > > az2 bz3 a1 bz4 > > > > > > < wz ; mC~ I ðzÞ ¼ > > > az4 bz1 z > > > wz ; > > > > az4 bz1 az3 bz2 > > > > > : 0;

~ I 4B ~ I ¼ a þ bz ; az þ bz ; az þ bz ; az þ bz ; w a0z þ b0z ; az A 1 1 2 2 3 3 4 4 1 1 2 z z z 0z 0z þ b2 ; a3 þ b3 ; a4 þ b4 ; w

decreasing

z

ða4 b1 Þz mRc ðzÞ ¼ w ðaz b . z Þðaz bz Þ

az3 þ bz3 z a4z þ b4z ; otherwise:

a

a

~ B ~ ¼ A. ~ is tion of C

0

if

z

az2 þ bz2 z az3 þ bz3 ; 0

w > 0, ðaz2 bz3 Þða1 bz4 Þ

z z z z z z a >w 2 . Again a4 b1 wz fða4 b1 Þ ða3 b2 Þg z

is

0

¼

mLc ðaz2 bz3 Þ ¼ w

a1z þ b1z z az2 þ bz2 ; 0

dmLc ðzÞ dz

. Now

ðaz2 bz3 Þða1 bz4 Þ

zða1 bz4 Þ

ðaz2 bz3 Þða1 bz4 Þ

ðaz2 bz3 Þ > ða1 bz4 Þ. Therefore, mLc ðzÞ is an increasing function.

dmRc ðzÞ dz

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

where 0 < wz 1, wz ¼ minðwz1 ; wz2 Þ.

a fðaz a Þ ðbz bz Þg z az bz a a1 bz4 þ w z 1 2 4 3 4 1 wz

Let

and

Therefore,

and z

Now Ca ¼ ½C1 ðaÞ; C2 ðaÞ ¼ ½A1 ðaÞ; A2 ðaÞ ½B1 ðaÞ; B2 ðaÞ

if

n a04z þ b04z þ ðaz3 þ bz3 Þ z

¼w

an increasing function. Additionally, ! z

þ

b01z Þ

Ca ¼ ½C1 ðaÞ; C2 ðaÞ

Cb ¼ ½C1 ðbÞ; C2 ðbÞ, a; b2½0; w, 0 < w 1 and w ¼ minðw1 ; w2 Þ.

ðaz2 þbz2 Þz

w > 0, ðaz2 þbz2 Þða01z þb01z Þ

where

0

ðaz2 þbz2 Þða01z þb01z Þ

¼

~ I .B ~ ~ I ¼ C, A

Let

0

b

ðaz2 þ bz2 Þ > ða01z þ b01z Þ. Therefore, nLc ðzÞ is a decreasing function.

0w

Proof:

0

b fðaz2 a01z Þ þ ðbz2 b01z Þg z0w Now az2 þ bz2 w

mLc


a2 þ b2 w fða2 a1 Þ þ ðb2 b1 Þg z a3 þ b3 þ w

Let 0

z

z

~ I .B ~ I ¼ a bz ; az bz ; az bz ; az bz ; w a0z b0z ; az A 1 4 2 3 3 2 4 1 1 4 2 z z z 0z 0z b3 ; a3 b2 ; a4 b1 ; w

Additionally, !

az4 bz1 þaz3 bz2 2

mRc ðaz4 bz1 Þ ¼ 0,

ðaz3 bz2 Þ. Therefore, nRc ðzÞ is an increasing function. 0

8 < ka1 ; kaz ; kaz ; kaz ; w ka0z ; kaz ; kaz ; ka0z ; w ; 2 3 4 2 3 4 1 : kaz ; kaz ; kaz ; ka ; w ka0z ; kaz ; kaz ; ka0z ; w ; 1 4 3 2 4 3 2 1

if k > 0; if k < 0:

where 0 < wz 1. ~I ¼ C ~ I , where Ca ¼ ½C ðaÞ; C ðaÞ and Proof: Case-I: k > 0 Let kA 1 2 Cb ¼ ½C10 ðbÞ; C20 ðbÞ, a2½0; w, 0 < w 1.

Now Ca ¼ ½C1 ðaÞ; C2 ðaÞ ¼ k½A1 ðaÞ; A2 ðaÞ ¼ ½kA1 ðaÞ; kA2 ða " # az az ¼ ka1 þ k z az2 a1 ; kaz4 k z az4 az3 w w

0

Additionally,

nRc

~I ¼ kA

if

b az3 bz2 þ w fða04z b01z Þ ðaz3 bz2 Þg

Again


ðaz2 bz3 Þz

ðaz2 bz3 Þða01z b04z Þ

ðaz2 bz3 Þ > ða01z b04z Þ. Therefore, nLc ðzÞ is a decreasing function. Additionally,

~ I .B ~ I ¼ a bz ; az bz ; az bz ; az bz ; w a0z b0z ; az A 1 4 2 3 3 2 4 1 1 4 2 z z z 0z 0z b3 ; a3 b2 ; a4 b1 ; w

0

0

az3 bz2 þa4z b1z 2

!

nRc ðaz3 bz2 Þ ¼ 0,

nRc ða4z b1z Þ ¼ wz 0

0

and

~ B ~ ¼ A. ~ is > w2 . The non-membership function of C z

8 > az2 bz3 z > > > > ; w > > > > az2 bz3 a01z b04z > > > > > > < 0; nC~ I ðzÞ ¼ > > > z az3 bz2 > > > > ; w > > > a04z b01z az3 bz2 > > > > > : w;

a01z b04z z az2 bz3 ; az2 bz3 z az3 bz2 ; z

z

a 3 b2 z

a04z

otherwise:

b01z ;

a ðaz a Þ z kaz k a ðaz az Þ. Let ka1 þ k w z 1 2 4 4 3 wz z

Now ka1 þ k Let

mLc ðzÞ

¼

z

az wz

ðaz2 a1 Þ z0w

1 w zka . kaz2 ka1

Now

zka1 kaz2 ka1

dmLc ðzÞ dz

¼

a.

w ðkaz2 ka1

> 0, if kaz2 > ka1 .

Therefore, mLc ðzÞ is an increasing function. Additionally, mLc ðka1 Þ ¼ 0, !

mLc ðkaz2 Þ ¼ w z0w zw z ka4 ka3

kaz4 z

and

ðkaz4 kaz3 Þ

a z

mLc

ka1 þkaz2 2

Let z

>w 2.

Again z

a ðaz az Þ kaz4 k w z 4 3 z

z 4 mRc ðzÞ ¼ w ðkakaz ka Therefore, z . Þ

< 0, if ka4 > ka3 . Therefore,

4

mRc ðzÞ

Also mRc ðkaz4 Þ ¼ 0, mRc ðkaz3 Þ ¼ w and mRc

3

dmRc ðzÞ dz

¼

is a decreasing function. !

kaz3 þkaz4 2

> > ; w z > > > ka > 2 ka1 > > > > > < w;

Let mLc ðzÞ ¼ w

z

ka1 z ka2 ;

mLc ðkaz3 Þ ¼ w and mLc

kaz3 z kaz4 ;

z

Let ka2 k w ða2 ( b Now kaz2 k w

Let nLc ðzÞ ¼ w

a01z Þ

z

z ka3 þ

az2 a01z Þ z0w z

ka2 z

0z

z

ka2 ka1

dn dz

L c ðzÞ

. Now

k w ða04z b

z

ka2 z z

ka2 ka01z

¼

c

nLc ðka01z Þ ¼ w z0w w ka04z kaz3

and z

zka3

ka04z kaz3

> 0, if

b

ka04z

mLc

ka2 þka01z 2

ka01z .

z

zka

nRc ðzÞ ¼ w ka0z ka3 z .

Let

4

z

b 0z kaz3 þ k w ða4 az3 Þ

Again

Therefore,

3

dn dz

R c ðzÞ

¼

> ka3 . Therefore, nRc ðzÞ is an increasing function. !

Additionally, nRc ðkaz3 Þ ¼ 0, nRc ðka04z Þ ¼ w and nRc

kaz3 þka04z 2

>w 2.

dm dz

R c ðzÞ

¼

ka1 z ðka1 kaz2 Þ

w ka1 kaz2

a

Let

mRc ðzÞ

< 0, if ka1 > kaz2 . There-

2

2

~ is kA

a3 Þ.

w ka2 ka01z

Therefore,

c

3

Therefore, nLc ðzÞ is a decreasing function. Additionally, nLc ðkaz2 Þ ¼ 0, ! z

> 0, if kaz3 > kaz4 .

w ðkaz3 kaz4

fore, mRc ðzÞ is a decreasing function. Additionally, mRc ðka1 Þ ¼ 0, ! ka1 þkaz3 z R R ~¼ m ðka Þ ¼ w and m < w. The membership function of C

z

z

kaz3 þkaz4 2 z

ka1 z . ðka1 kaz2 Þ

¼ ww

otherwise:

Cb ¼ C10 ðbÞ; C20 ðbÞ ¼ k A01 ðbÞ; A02 ðbÞ ¼ kA01 ðbÞ; kA02 ðbÞ b z b 0z a2 a01z ; kaz3 þ k a4 az3 ¼ kaz2 k w w b

¼

a ðaz az Þ z0w kaz1 þ k w z 2 1

Again

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

z

dmLc ðzÞ dz

. Now

Therefore, mLc ðzÞ is an increasing function. Also mLc ðkaz4 Þ ¼ 0, !

kaz2 z kaz3 ;

> > kaz4 z > > > w ; > > > kaz4 kaz3 > > > > : 0;

zkaz4

kaz3 kaz4

>w 2 . The

8 > z kaz4 > > ; >w z > > ka3 kaz4 > > > > > > < w; > > > kaz z > > > w z1 ; > > > ka1 kaz2 > > > : 0;

kaz4 z kaz3 ; kaz3 z kaz2 ; kaz2 z kaz1 ; otherwise:

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

Cb ¼ C10 ðbÞ; C20 ðbÞ ¼ k A01 ðbÞ; A02 ðbÞ ¼ kA02 ðbÞ; kA01 ðbÞ b 0z b z a4 az3 ; kaz2 k a2 a01z ¼ kaz3 þ k w w

~ B ~ ¼ A4 ~ is non-membership function of C

nC~ I ðzÞ ¼

8 > kaz z > > > ; wz z 2 > > > ka2 ka01z > > > > > > < 0; > > > z kaz3 > > wz 0z ; > > > > ka4 kaz3 > > > > : wz ;

Let

b 0z ða4 az3 Þ z0w kw

ka01z z kaz2 ; z

b 0z b z kaz3 þ k w ða4 az3 Þ z kaz2 k w ða2 a01z Þ.

Let nLc ðzÞ ¼

z

kaz3 z

kaz3 ka04z

kaz z wz z 3 0z . ka3 ka4

dnLc ðzÞ dz

Now

¼

w kaz3 ka04z

< 0, if kaz3 > ka04z .

ka2 z ka3 ;

Therefore, nLc ðzÞ is a decreasing function. Additionally, nLc ðkaz3 Þ ¼ 0, !

kaz3 z ka04z ;

nLc ðka04z Þ ¼ w and mLc 0w

otherwise:

zkaz2

ka01z kaz2

ka01z

z

kaz3 þka04z 2

b Let nRc ðzÞ ¼ wz

> ka2 . Therefore,

nRc ðzÞ

z 0z b z 0 kA 1 2 3 4 1 2 3 4

~ is ~ ¼ kA membership function of C

nRc ðkaz2 Þ ¼ 0, nRc ðka01z Þ ¼ w and nRc

where 0 < wz 1. I

~I

~ ¼ C , where Ca ¼ ½C ðaÞ; C ðaÞ and Case-II: ðk < 0Þ Let kA 1 2 Cb ¼ ½C10 ðbÞ; C20 ðbÞ, a2½0; w, 0 < w 1.

nC~ I ðzÞ ¼

Now Ca ¼ ½C1 ðaÞ; C2 ðaÞ ¼ k½A1 ðaÞ; A2 ðaÞ ¼ ½kA2 ðaÞ; kA1 ða " # az az ¼ kaz4 k z az4 az3 ; kaz1 þ k z az2 az1 w w z

az ðaz az Þ z ka þ k az ðaz a Þ. Let ka4 k w z 1 1 4 3 2 wz zkaz4 z az z z

ka3 ka4

a

8 > kaz3 z > > > wz ; > z 0z > > > > ka3 ka4 > > > > < 0; > > > z kaz2 > > ; > wz 0z > > > ka1 kaz2 > > > > : wz ;

. Therefore,

dnRc ðzÞ dz

¼

wz ka01z kaz2

> 0,

is an increasing function. Additionally, !

if

z

kaz3 þ

b

Hence, the scalar multiplication rule is proven for the nonmembership function. Thus, we have

k wz ða4 a3 Þ z0w

Now

kaz2 þka01z 2

>w 2 . Therefore, the non-

ka04z z kaz3 ; kaz3 z kaz2 ; kaz2 z ka01z ; otherwise:

z

Now ka4

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



7

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

~ I ¼ kaz ; kaz ; kaz ; ka ; wz kA 1 4 3 2

ka04z ; kaz3 ; kaz2 ; ka01z ; wz

4

if k < 0

~ ¼ ðð1; 5; 8; 10; 0:7Þ Let us consider a ~ ¼ ðð1; 2; 3; ð4; 6; 7; 11; 0:7Þ; ð1; 5; 8; 10; 0:6Þð4; 6; 7; 11; 0:6ÞÞ, and b 4; 0:7Þð0; 2; 3; 4; 0:7Þ; ð1; 2; 3; 4; 0:6Þð0; 2; 3; 4; 0:6ÞÞ be T2IFNs, the subtraction of these two numbers is shown in Fig. 3.

2

3

4

2

1

2

3

!

z z z a1 a2 a3 a4 ; ; ; ;w z z z z b4 b3 b2 b1

~ I ÷B ~I ¼ A

a01z az2 az3 a04z ; ; ; ;w b04z bz3 bz2 b01z

mC~ I ðzÞ ¼

where 0 < wz 1, wz ¼ minðwz1 ; wz2 Þ. Proof:

~ I ÷B ~I , ~I ¼ C A

Let


where

and

Cb ¼ ½C10 ðbÞ; C20 ðbÞ, a; b2½0; w, 0 < w 1 and w ¼ minðw1 ; wz2 Þ. Now

Ca ¼ ½C1 ðaÞ; C2 ðaÞ ¼ ½A1 ðaÞ; A2 ðaÞ÷½B1 ðaÞ; B2 ðaÞ 3 2 az az a az az az a1 þ w az4 w z z 1 A1 ðaÞ A2 ðaÞ 2 4 3 ; 5 ¼ ; ¼4 z z B2 ðaÞ B1 ðaÞ bz a bz bz bz þ a bz bz wz

4

z

Let

a1 þ a z ðaz2 a1 Þ w

z

bz4 a z ðbz4 bz3 Þ zbz4 a1

ðaz2 a1 Þþzðbz4 bz3 Þ

Let mLc ðzÞ ¼ w

3

wz

1

z

z

w

0w

4

az4 a z ðaz4 az3 Þ w

z

2

1

z

.

bz1 þ a z ðbz2 bz1 Þ

Now

w

a1 þ a z ðaz2 a1 Þ w

z

bz4 a z ðbz4 bz3 Þ

z

az3 bz2 az4 bz2

fðaz4 þbz4 Þðaz3 þbz3 Þg2

< 0,

z

1

8 zbz4 a1 > > ; w > > > > az2 ða1 Þ þ z bz4 bz3 > > > > > > > > > < w;

az2 z ; bz4 bz3 a1 az2

az3 z ; bz3 bz2

> > > > > > az4 zbz1 > > ; > w > > > az4 az3 þ z bz2 bz1 > > > : 0;

az3

az4 z ; bz2 bz1 otherwise:

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

Cb ¼ C10 ðbÞ; C20 ðbÞ ¼ A01 ðbÞ; A02 ðbÞ ÷ B01 ðbÞ; B02 ðbÞ 3 " # 2 z b z b a04z az3 a2 w a2 a01z az3 þ w A01 ðbÞ A02 ðbÞ ; 5 ¼ 0 ; 0 ¼4 B2 ðbÞ B1 ðbÞ bz þ b b0z bz bz b bz b0z 3

w

4

3

2

w

2

1

w

a zbz4 a1

. Now

ðaz2 a1 Þþzðbz4 bz3 Þ

dmLc ðzÞ dz

¼w

az2 bz4 a1 bz3

2 > 0,

fðaz2 a1 Þþzðbz4 bz3 Þg

az

for ðaz2 bz4 Þ > a1 bz3 i:e: 2z > a1z . Therefore, mLc ðzÞ is an increasing function. b3 b4 0a z 1 ! ! 2 þ a1 z a wbz Bbz bz C Additionally, mLc 2z ¼ w, mLc a1z ¼ 0 and mLc @ 3 2 4 A ¼ z 4 z > b b b þb 3

w[since 2

¼w

~ B ~ ¼ A÷ ~ is membership function of C

vision of two GTIFN is given by

!

1

2

2

4

2

dmRc ðzÞ dz

a a for az3 bz1 Þ < az4 bz2 i:e: 3z < 4z . Therefore, mRc ðzÞ is a decreasing function. b2 b1 0a z a z 1 ! ! 3þ 4 z z a a Bbz bz C Also mRc 3z ¼ w, mRc 4z ¼ 0 and mRc @ 2 2 1 A < w 2 . Therefore, the b b

~ I ¼ ða ; az ; az ; az ; w Þða0z ; az ; az ; a0z ; w Þ and Property 3.4. Let A 1 2 3 4 1 1 1 2 3 4 I z z z z z 0z z z 0z ~ B ¼ ðb ; b ; b ; b ; w Þðb ; b ; b ; b ; w Þ be two GTIFN, then the di1

3

z

0 < wz 1.

where

z

zb1 mRc ðzÞ ¼ w ðaz aaz 4Þzðb . Therefore, z bz Þ

bz3 < bz4 . Again

4

z az4 a z w z az b1 þ z w

ðaz4 az3 Þ ðbz2 bz1 Þ

4

z0w

az4 zbz1

ðaz4 az3 Þzðbz2 bz1 Þ

3

a Let

az2 wb ðaz2 a01z Þ

Let 0w

bz3 þwb ðb04z bz3 Þ z

z

a2 zb3

ðaz2 a01z Þðb04z bz3 Þ

0w dnRc ðzÞ dz

z

z

zb2 a3

ða04z az3 Þþzðbz2 b01z Þ

¼w

az3 þwb ða04z az3 Þ

.

az2 wb ðaz2 a01z Þ

Now

bz2 wb ðbz2 b01z Þ

bz3 þwb ðb04z bz3 Þ

z

b

nLc ðzÞ ¼ w

Let

z

az2 zbz3 0

0

.

b Let nRc ðzÞ ¼ w

a04z bz2 az3 b01z

fða04z az3 Þþzðbz2 b01z Þg2

> 0, for

a04 b01z

az3 þwb ða04z az3 Þ

Again

ðaz2 a1z Þðb4z bz3 Þ

>

z

bz2 wb ðbz2 b01z Þ z

zb2 a3

z

. Therefore,

ða04z az3 Þþzðbz2 b01z Þ

az3 bz2

!

. Therefore, nRc ðzÞ is an


8


increasing function. Additionally,

nRc

0az

a

0z

nC~ I ðzÞ ¼

! ¼ 0,

bz2

nRc

a04z

!

b01z

¼ w and

1

3þ 4 z b0z b C nRc B @ 22 1A>w 2.

is

az3

~ B ~ ¼ A÷ ~ Therefore, the non-membership function of C

8 > az2 zbz3 > > > ; w > > > > z b04z bz3 þ az2 a01z > > > > > > > > > < 0; > > > > > > zbz2 az3 > > ; > w > > > a04z az3 þ z bz2 b01z > > > > > : w;

a01z

az2

z z; b04z b3 az3 z ; bz3 bz2 a04z

a1 bz1 þaz2 bz2 2

mLc

z 0z ; bz2 b1

a2

þ

P2 w2

0


0

¼ ½A1 ðaÞB1 ðaÞ; A2 ðaÞB2 ðaÞ " az n a2 z a2 a1 bz2 bz1 þ z a1 bz2 bz1 ¼ 2 w w o 2 a þ a1 bz1 ; 2 az4 az3 bz4 bz3 þ bz1 az2 a1 w # o az n z z z z z z z z þ a 4 b4 z a4 b4 b3 þ b4 a4 a3 w a2 ðaz a Þðbz bz Þ þ az fa ðbz bz Þ þ bz ðaz a Þg þ a bz Let w 2 1 1 2 1 1 1 2 2 1 1 1 2 wz

a ðaz az Þðbz bz Þ a faz ðbz bz Þ þ bz ðaz az Þg þ az bz . zw 2 4 3 4 3 4 4 3 4 4 3 4 4 wz z

w2

P1 þ

a2

wz

Q1 þ a1 bz1 z

az

0 2 P1 þ z Q1 þ a1 bz1 z 0 w w

Again

P2 ¼ ðaz4 az3 Þðbz4 bz3 Þ,

let

Q2 þ az4 bz4 z

az

Q2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Q 22 4P2 az4 bz4 z

2P2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Q2 þ Q 22 4P2 az4 bz4 z 2P2

mRc ðzÞ ¼ w

Let dmRc ðzÞ dz

¼

az

wz

or

az

wz

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi z 2

Q2

Q 2 4P2 ðaz4 b4 zÞ . 2P2

Therefore,

w ffi < 0, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z z z z z z faz4 ðb4 b3 Þþb4 ðaz4 az3 Þg2 4ðaz4 az3 Þðb4 b3 Þðaz4 b4 zÞ

if

ðaz4 þ bz4 Þ > ðaz3 þ bz3 Þ. Therefore, mRc ðzÞ is a decreasing function. ! Additionally, mRc ðaz4 bz4 Þ ¼ 0, mRc ðaz3 bz3 Þ ¼ w and mRc

az3 bz3 þaz4 bz4 2

w 2.

and

Ca ¼ ½C1 ðaÞ; C2 ðaÞ ¼ ½A1 ðaÞ; A2 ðaÞ$½B1 ðaÞ; B2 ðaÞ

a2

az

wz

a2

Cb ¼ ½C1 ðbÞ; C2 ðbÞ, a; b2½0; wz , 0 < wz 1 and wz ¼ minðwz1 ; wz2 Þ. Now

2

¼

0 2 P2 þ z Q2 þ az4 bz4 z 0 w w

where 0 < wz 1, wz ¼ minðwz1 ; wz2 Þ. 0

dmLc ðzÞ dz

Now

Q1 ¼ faz4 ðbz4 bz3 Þ þ bz4 ðaz4 az3 Þg then

~ I 5B ~ I ¼ a bz ; az bz ; az bz ; az bz ; w a0z b0z ; az bz ; az bz ; a0z b0z ; w A 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4

where

Q 21 4P1 ða1 b1 zÞ . 2P1

otherwise:

~ I ¼ ða ; az ; az ; az ; w Þða0z ; az ; az ; a0z ; w Þ and Property 3.5. Let A 1 2 3 4 1 1 1 2 3 4 ~ I ¼ ðbz ; bz ; bz ; bz ; wz Þðb0z ; bz ; bz ; b0z ; wz Þ be two GTIFN, then the B 1 2 3 4 2 1 2 3 4 2 multiplication of two GTIFN is given by

~ I 4B ~I , ~I ¼ C A

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi z

mLc ðzÞ ¼ w Q1 þ

increasing function. Additionally, mLc ða1 bz1 Þ ¼ 0, mLc ðaz2 bz2 Þ ¼ w and !


Let

2P1

fa1 ðb2 b1 Þþb1 ðaz2 a1 Þg 4ðaz2 a1 Þðb2 b1 Þða1 b1 zÞ

! 0z z z 0z ! z z z a1 a2 a3 a4 a1 a2 a3 a4 ; ; ; ;w ; ; ; ;w z z z z b4 b3 b2 b1 b04z bz3 bz2 b01z

Proof:

az

z 2P1 w rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi z 2 Q1 þ Q 1 4P1 a1 b1 z

w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > 0. Therefore, mLc ðzÞ is an z z z z z z 2

az2

az3

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Q 21 4P1 a1 bz1 z

Let

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

~ I ÷B ~I ¼ A

Q1 0

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ffi > > Q1 þ Q 21 4P1 a1 bz1 z > > > > > ; w > > 2P1 > > > > > < w; > > > > > Q2 > > > > w > > > > > :

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Q 22 4P2 az4 bz4 z 2P2 0;

;

a1 bz1 z az2 bz2 ; az2 bz2 z az3 bz3 ; az3 bz3 z az4 bz4 ; otherwise:

~ ¼ ðð1; 2; 3; 4; 0:7Þð0; 2; 3; 5; 0:7Þ; a ~ ¼ ðð0; 2; 3; 60:8Þð0; 2; 3; 60:8Þ; ððð1; 2; 3; 4; 0:6Þð0; 2; 3; 5; 0:6ÞÞ, and b ð0; 2; 3; 60:6Þð0; 2; 3; 60:6ÞÞ be T2IFNs, 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 Let

us

consider



9

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

Cb ¼ C10 ðbÞ; C20 ðbÞ ¼ A01 ðbÞ; A02 ðbÞ $ B01 ðbÞ; B02 ðbÞ ¼ A01 ðbÞB02 ðbÞ; A02 ðbÞB01 ðbÞ " bn b2 z ¼ az2 bz2 b01z þ bz2 az2 a2 a01z bz2 b01z 2 w w a01z

o

þ az2 bz2 ;

b2 w2

bz3 þ bz3 a04z az3

o

b n z

b04z bz3 þ a b0z w 3 4 #

þ az3 bz3

Now

b

w2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 0 z z Q 02 2 4P 2 ða3 b3 zÞ . 2P 0

Q20 þ

2

Therefore,

nRc ðzÞ is an increasing function. Additionally,

nRc ðaz3 bz3 Þ

¼ 0,

nRc ða04z b04z Þ

¼ w and

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi z z 0 a Q 02 4P b z 1 1 2 2

2P10 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi z z 0 a Q10 þ Q 02 4P b z 1 1 2 2 2P10

nLc ðzÞ ¼ w

Q10

w

nRc

az3 bz3 þa04z b04z 2

! >w 2.

a01z b01z þaz2 bz2 2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi z 02

Q 1 4P 01 ðaz2 b2 zÞ . 2P10

Now

!

0z z 0 ffi > > z z 02 0 0 > > a Q þ Q 4P b z > 2 2 2 3 3 > > > ; > 0 >w 2P > > 2 > > : w;

b

w ffi < 0. Therefore, ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0z z z z 0z 2 z z 0z z 0z z z

P20 þ

8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > z z 02 0 > 0 > a Q Q 4P b z > 1 1 1 3 3 > > > ; w > 0 > > 2P > 1 > > > > < 0;

nC~ I ðzÞ ¼

nLc ðzÞ is a decreasing function. Additionally, nLc ða01z b01z Þ ¼ w,

w2

b w

fa3 ðb4 b3 Þþb3 ða4 a3 Þg 4ða4 a3 Þðb4 b3 Þða3 b3 zÞ

b

Let

b2

w ffi > 0. ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z z 0z 0z z z 0z z 2 0z z z z

b

2

Q10 0

dnLc ðzÞ dz

dnRc ðzÞ dz

Q 0 þ az2 bz2 z w 1

0 2 P10 þ Q10 þ az2 bz2 z 0 w w

0

2P2

~ B ~ ¼ A5 ~ is Therefore, the non-membership function of C

P10

b

or

2P20 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi z z 0 a Q20 þ Q 02 4P b z 2 2 3 3

Therefore,

b Let w2 ða2 a01z Þðbz2 b01z Þ w faz2 ðbz2 b01z Þ þ bz2 ðaz2 a01z Þg þaz2 bz2 2 0z z 0z z z 0z z z 0z z z z b b zw 2 ða4 a3 Þðb4 b3 Þ þ w fa3 ðb4 b3 Þ þ b3 ða4 a3 Þg þa3 b3 . 0 z 0z z 0z z z 0 z z z Let P1 ¼ ða2 a1 Þðb2 b1 Þ, Q10 ¼ fa2 ðb2 b1 Þ þ b2 ða2 a01z Þg.

2

b w

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi z z 0 a Q 02 4P b z 2 2 3 3

Let nRc ðzÞ ¼ w

z

b2

a04z az3

Q2 0

~ I 5B ~ I ¼ a bz ; az bz ; az bz ; az bz ; w a0z b0z ; az bz ; az bz ; a0z b0z ; w A 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 where 0 < w 1, w ¼ minðw1 ; wz2 Þ.Let us consider ~ ¼ ðð2; 4; 6; 8; 0:8Þð1; 4; 6; 12; 0:8Þ; ðð2; 4; 6; 8; 0:6Þð1; 4; 6; 12; 0:6ÞÞ, a ~ and b¼ðð4;6;8;10;0:8Þð3;6;8;12;0:8Þ;ð4;6;8;10;0:6Þ ð3;6;8;12;0:6ÞÞ be T2IFNs, the division of these two numbers is shown in Fig. 5. 4. Ranking function of GTIFN ~ ¼ ða ; az ; az ; az ; wÞða0z ; az ; az ; a0z ; wÞ be a GTIFN. There are Let A 1 2 3 4 1 2 3 4 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. I


10


Fig. 5. Membership and non-membership function of division of two T2IFNs.

4.1. Mean of interval method

I I ~I ¼ B ~ ¼H B ~ I iff H A ~ ðiiiÞ A

The ða; bÞ-cut of the GTIFN is given by

Aa;b ¼ ½A1 ðaÞ; A2 ðaÞ; A01 ðbÞ; A02 ðbÞ ; a þ b w; a; b2½0; w a ðaz a Þ, A1 ðaÞ ¼ a1 þ w z 1 2 z

where

a ðaz az Þ, A2 ðaÞ ¼ az4 w z 4 3 z

b z b 0z A01 ðbÞ ¼ az2 w ða2 a01z Þ, A02 ðbÞ ¼ az3 þ w ða4 az3 Þ. Now by the mean of interval method the representation of the membership function is

I 1 Zw ~ ¼ Rm A ðA1 ðaÞ þ A2 ðaÞÞda 2 0

¼

1 2

Zw " 0

1

a1 þ az4 þ

az n

wz

# o az2 a1 az4 az3 da

w n

a1 w þ az4 w þ az2 a1 az4 az3 2 w a1 þ az2 þ az3 þ az4 : ¼ 4 ¼

o

2

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

I 1 Zw ~ ¼ Rn A ðA1 ðbÞ þ A2 ðbÞÞdb 2 1 ¼ 2

I I ~ þ Rm B ~ I Rm A w a1 þ 2az2 þ 2az3 þ az4 þ a01z þ a04z ~ ¼ ¼ ; H A 2 8 I I ~ þ Rn B ~ I Rn A w bz1 þ 2bz2 þ 2bz3 þ bz4 þ b01z þ b04z ~ ¼ ¼ H B 2 8 where wz ¼ minðwz1 ; wz2 Þ. With the help of the above formulae, we have optimized the following transportation problem in the T2IF 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 subject to

o b n z db a2 a01z a04z az3 az2 þ az3 w

0

o 1 z w n z a2 w þ az3 w a2 a01z a04z az3 2 2 w a01z þ az2 þ az3 þ a04z : ¼ 4 ¼

~ I ¼ ðða ; az ; az ; a ; w Þða0z ; az ; az ; a0z ; w ÞÞ and B ~ I ¼ ððbz ; bz ; Let A 1 2 3 4 1 1 1 2 3 4 1 2

m n e 4 4 cIij xij

i¼1 j¼1 n P j¼1

m P i¼1

0

Zw

where

e xij zaIi for i ¼ 1; 2; /m

(1)

e xij zbIj for j ¼ 1; 2; /n

xij 0 c i; j

I where aei is the approximate availability of the product at the ith I source, bej is the approximate demand of the product at the jth I

destination, ceij 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. P P If m aeI ¼ n beI then the intuitionistic fuzzy transportation i¼1 i

j¼1 j

bz3 ; bz4 ; wz2 Þðb01z ; bz2 ; bz3 ; b04z ; wz2 ÞÞ be two GTIFNs, then [1] proposed that

problem (IFTP) is said to be a balanced transportation problem, otherwise it is called an unbalanced IFTP. 0 0 Let cIij ¼ ðcij1 ; cij2 ; cij3 ; cij4 ; wÞðcij1 ; cij2 ; cij3 ; cij4 ; wÞ,

I I ~ I 3B ~ H B ~ _B ~ iff H A ~ ðiiÞ A I

I

0

0

0

and

bIj ¼ ðbj1 ; bj2 ; bj3 ;

0

bj4 ; wÞðbj1 ; bj2 ; bj3 ; bj4 ; wÞ. The steps to solve the above IFTP are as follows: Step 1. Substituting the value of cIij , aIi and bIj in (3), we get



m n Minimize 4 4 cij1 ; cij2 ; cij3 ;cij4 ;w c0ij1 ;cij2 ;cij3 ; c0ij4 ; w xij

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

i¼1 j¼1

subject to n P xij zðai1 ;ai2 ;ai3 ; ai4 ;wÞ a0i1 ;ai2 ;ai3 ; a0i4 ;w for i ¼ 1; 2;/m j¼1 m P i¼1

11

(2)

xij z bj1 ;bj2 ;bj3 ;bj4 ;w b0j1 ;bj2 ;bj3 ;b0j4 ; w for j ¼ 1; 2;/n

xij 0 ci;j

m n Step 4. Find the fuzzy optimal value by putting xij in 4 4 ceIij xij . i¼1 j¼1

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

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

m n H 4 4 xij cij1 ; xij cij2 ; xij cij3 ; xij cij4 ; w xij c0ij1 ; xij cij2 ; xij cij3 ; xij c0ij4 ; w i¼1 j¼1 1 n P xij A ¼ H ðai1 ; ai2 ; ai3 ; ai4 ; wÞ a0i1 ; ai2 ; ai3 ; a0i4 ; w for i ¼ 1; 2; /m subject to H Minimize

H

m P

! xij

i¼1

j¼1

(3)

¼ H bj1 ; bj2 ; bj3 ; bj4 ; w b0j1 ; bj2 ; bj3 ; b0j4 ; w for j ¼ 1; 2; /n

xij 0 ci; j

Using Step-1 of the method explained in Section 5 the above IFTP can be written as

Minimize ð2; 4; 5; 6; 0:5Þð1; 4; 5; 6; 0:5Þx11 4ð4; 6; 7; 8; 0:2Þð3; 6; 7; 9; 0:2Þx12 4ð3; 7; 8; 12; 0:3Þð2; 7; 8; 13; 0:3Þx13 4ð1; 3; 4; 5; 0:6Þð0:5; 3; 4; 5; 0:6Þx21 4ð3; 5; 6; 7; 0:6Þð2; 5; 6; 8; 0:6Þx22 4ð2; 6; 7; 11; 0:4Þð1; 6; 7; 12; 0:4Þx23 4ð3; 4; 5; 8; 0:7Þð2; 4; 5; 9; 0:7Þx31 4ð1; 2; 3; 4; 0:8Þð0:5; 2; 3; 5; 0:8Þx32 4ð2; 4; 5; 10; 0:2Þð1; 4; 5; 11; 0:2Þx33 subject to x11 þ x12 þ x13 zð4; 6; 8; 9; 0:6Þð2; 6; 8; 10; 0:6Þ x21 þ x22 þ x23 zð0; 0:5; 1; 2; 0:5Þð0; 0:5; 1; 5; 0:7Þ x31 þ x32 þ x33 zð8; 9:5; 10; 11; 0:8Þð6:5; 9:5; 10; 11; 0:8Þ x11 þ x21 þ x31 zð6; 7; 8; 9; 1Þð5; 7; 8; 11; 1Þ x12 þ x22 þ x32 zð4; 5; 6; 7; 0:8Þð3; 5; 6; 8; 0:8Þ x13 þ x23 þ x33 zð2; 4; 5; 6; 0:6Þð0:5; 4; 5; 7; 0:6Þ xij 0ci; j

(4)

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

1 ð33x11 þ 50x12 þ 60x13 þ 25:5x21 þ 42x22 þ 52x23 þ 50x31 þ 20:5x32 þ 42x33 Þ 40 subject to x11 þ x12 þ x13 ¼ 3:975 Minimize

x21 þ x22 þ x23 ¼ 6:78 x31 þ x32 þ x33 ¼ 7:55 x11 þ x21 þ x31 ¼ 7:625 x12 þ x22 þ x32 ¼ 4:465 x13 þ x23 þ x33 ¼ 2:514 xij 0; c i; j


12


Table 1 Input data for IFTP. f D 1

f D 2

f D 3

Availabilityðaei Þ

Se1 Se2 Se

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

(4,6,7,8; 0.2) (3,6,7,9; 0.2)

(3,7,8,12; 0.3) (2,7,8,13; 0.3)

(4,6,8,9; 0.6) (2,6,8,10; 0.6)

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

(3,5,6,7; 0.6) (2,5,6,8; 0.6)

(2,6,7,11; 0.4) (1,6,7,12; 0.4)

(0,0.5,1,2; 0.5) (0,0.5,1,5; 0.7)

(3,4,5,8; 0.7) (2,4,5,9; 0.7)

(1,2,3,4; 0.8) (0.5,2,3,5; 0.8)

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

(8,9.5,10,11; 0.8) (6.5,9.5,10,11; 0.8)

Demand (bej )

(6,7,8,9; 1) (5,7,8,11; 1)

(4,5,6,7; 0.8) (3,5,6,8; 0.8)

(2,4,5,6; 0.6) (0.5,4,5,7; 0.6)

3

Solving the above crisp linear programming using Lingo-14.0, we get x11 ¼ 0:75, x12 ¼ 0, x13 ¼ 0, x21 ¼ 6:87, x22 ¼ 0, x23 ¼ 0, x31 ¼ 0, x32 ¼ 4:4 and x33 ¼ 2:5. The minimum cost of transportation is 395.50. 7. Conclusions This paper proposes and investigates type-2 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 ða; bÞ-cut method is developed. We considered basic generalized type-2 intuitionistic fuzzy numbers, namely, generalized trapezoidal type-2 intuitionistic fuzzy numbers. We discussed the ranking function of the generalized trapezoidal type-2 intuitionistic fuzzy numbers. We solved an intuitionistic fuzzy transportation problem where transportation cost, source, and demand were generalized type-2 intuitionistic fuzzy numbers. A numerical example is presented to solve the type-2 intuitionistic transportation problem. This idea can be extended as a type-2 intuitionistic fuzzy fault tree analysis for the failure of automobile system to start. The definition of type-2 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-2 fuzzy sets. Corresponding results for general intuitionistic type-2 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-2 fuzzy sets and various approaches to knowledge discovery in complex type-2 intuitionistic fuzzy information systems. References [1] A. Nagoorgani, R. Ponnalagu, A new approach on solving intuitionistic fuzzy linear programming problem, Appl. Math. Sci. 6 (70) (2012) 3467e3474. [2] G.S. Mahapatra, T.K. Roy, Intuitionistic fuzzy number and its arithmetic operation with application on system failure, J. Uncertain Syst. 7 (2) (2013) 92e107.

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