Novel arithmetic operations on type-2 intuitionistic

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Pacific Science Review A: Natural Science and Engineering 18 (2016) 178e189

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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 23 December 2016

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/).

D.K. Jana / Pacific Science Review A: Natural Science and Engineering 18 (2016) 178e189

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:

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

Definition 2.3. Generalized Triangular Intuitionistic Type-2

(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 ;

179

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.

as

180

D.K. Jana / Pacific Science Review A: Natural Science and Engineering 18 (2016) 178e189

nA~ ðxÞ ¼

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

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

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

½A1 ðaÞ; A2 ðaÞ

set

of

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

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

ðaz2 þbz2 Þðaz1 þbz1 Þ

 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

ðaz4 þbz4 Þðaz3 þbz3 Þ

Let mRc ðzÞ ¼ w Therefore, Therefore,

 a. ðaz4 þbz4 Þz

dmRc ðzÞ dz

mRc ðzÞ

mRc ðaz4 þ bz4 Þ ¼ 0,

.

ðaz4 þbz4 Þðaz3 þbz3 Þ

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, !

a3 þb3 þaz4 þbz4 2

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

z

z

< w2 .

D.K. Jana / Pacific Science Review A: Natural Science and Engineering 18 (2016) 178e189

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

Additionally, az2 þbz2 þa01z þb01z 2

dnRc ðzÞ dz

z

ðaz2 þbz2 Þz

ðaz2 þbz2 Þða01z þb01z Þ

nLc ðaz2

!

. Now

dnLc ðzÞ dz

z

þ b2 Þ ¼ 0,

z

ða04z þb04z Þðaz3 þbz3 Þ

b

w > 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

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 ;

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 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

a

Let

w < 0, ðaz4 bz1 Þðaz3 bz2 Þ

z

z

z

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 4

¼ a

and

z

az3 þ bz3  z  a4z þ b4z ; otherwise:

if

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

~ B ~ ¼ A. ~ is tion of C

0

w > 0, ðaz2 bz3 Þða1 bz4 Þ

mLc ðaz2  bz3 Þ ¼ w

az2 þ bz2  z  az3 þ bz3 ; 0

¼

zða1 bz4 Þ

ðaz2 bz3 Þða1 bz4 Þ

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

is

0

dmLc ðzÞ dz

. Now

ðaz2 bz3 Þða1 bz4 Þ

a1z þ b1z  z  az2 þ bz2 ;

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

dmRc ðzÞ dz 0

and z

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

and

Therefore,

Ca ¼ ½C1 ðaÞ; C2 ðaÞ

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

if

n  a04z þ b04z þ ðaz3 þ bz3 Þ  z

¼w

an increasing function. Additionally, ! z

þ

b01z Þ

where

0

ðaz2 þbz2 Þz

w > 0, ðaz2 þbz2 Þða01z þb01z Þ

~ I .B ~ ~ I ¼ C, A

Let

0

ðaz2 þbz2 Þða01z þb01z Þ

¼

nLc ða01z

z z b ða01z þ b01z Þ. Therefore, nLc ðzÞ is a decreasing function.

0w

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

0

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

mLc

  ~ 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

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

Let 0

z

z

181

Additionally, !

az4 bz1 þaz3 bz2 2

mRc ðaz4  bz1 Þ ¼ 0,

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

~ I ¼ ða ; az ; az ; az ; wz Þða0 z ; az ; az ; a0 z ; wz Þ be a Property 3.3. Let A 1 2 3 4 1 2 3 4 ~ I is GTIFN and ~ I ¼ kA GTIFN, then C

~I ¼ kA

  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

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

if

b az3  bz2 þ w fða04z  b01z Þ  ðaz3  bz2 Þg

Again

  ~ 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

ðaz2 bz3 Þz

ðaz2 bz3 Þða01z b04z Þ

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

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

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

a3  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 ~ B ~ ¼ A4 ~ is membership function of C

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 ;

¼ ww

otherwise:

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

c

nLc ðka01z Þ ¼ w  z0w w ka04z kaz3

and z

zka3

ka04z kaz3

> 0, if

b

ka04z

mLc Let

ka2 þka01z 2

ka01z .

z

zka

nRc ðzÞ ¼ w ka0z ka3 z . 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

Therefore,

c

3

 a3 Þ.

z

>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-

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

w ka2 ka01z

¼

kaz3 þkaz4 2 z

ka1 z . ðka1 kaz2 Þ

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

> 0, if kaz3 > kaz4 .

w ðkaz3 kaz4

2

2

~ is kA

   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

183

>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

184

D.K. Jana / Pacific Science Review A: Natural Science and Engineering 18 (2016) 178e189

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

Ca ¼ ½C1 ðaÞ; C2 ðaÞ

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 a z  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 0az 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 0az az 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

ða04z az3 Þþzðbz2 b01z Þ

az3 bz2

z

. Therefore,

!

. Therefore, nRc ðzÞ is an

D.K. Jana / Pacific Science Review A: Natural Science and Engineering 18 (2016) 178e189

increasing function. Additionally, nRc 0az

a

0z

nC~ I ðzÞ ¼

¼ 0, nRc

bz2

a04z

b01z

¼ w and

~ 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

az2

az3  z  ; bz3 bz2 a04z

mLc

 z  0z ; bz2 b1

a1 bz1 þaz2 bz2 2

a2

þ

P2 w2

Ca ¼ ½C1 ðaÞ; C2 ðaÞ

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

a2

wz

Q1 þ a1 bz1  z

az

0 2 P1 þ z Q1 þ a1 bz1  z  0 w w

P2 ¼ ðaz4  az3 Þðbz4  bz3 Þ,

let

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.

a01z b01z þaz2 bz2 2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi z 02

Q 1 4P 01 ðaz2 b2 zÞ . 2P10

Now

!

0z z 0 > > > > Q20 þ > > > > > >w > > > > :

w

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

a2

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

b

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

w2

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

Q20 þ

nRc ðzÞ is an increasing function.

nC~ I ðzÞ ¼

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

Let

b2

w

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

b

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

dnLc ðzÞ dz

b



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

b

2

Q10



dnRc ðzÞ dz

Q 0 þ az2 bz2  z w 1

0 2 P10 þ Q10 þ az2 bz2  z  0 w w

0

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,

Now 2

b w

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

Let nRc ðzÞ ¼ w

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. 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

D.K. Jana / Pacific Science Review A: Natural Science and Engineering 18 (2016) 178e189

187

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 m P

m n e 4 4 cIij xij

i¼1 j¼1 n P j¼1

e xij zaIi for i ¼ 1; 2; /m

(1)

e xij zbIj for j ¼ 1; 2; /n

i¼1

0

Zw 

where

  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

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

188

D.K. Jana / Pacific Science Review A: Natural Science and Engineering 18 (2016) 178e189

 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

(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

D.K. Jana / Pacific Science Review A: Natural Science and Engineering 18 (2016) 178e189

189

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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