Theory of Approximation and Applications

Vol.12, No.1, (2018), 43-64

New Generalized Interval Valued

Intuitionistic Fuzzy Numbers

E. Baloui Jamkhaneh a,∗ A. saeidifar b

aDepartment of Statistics, Qaemshahr Branch, Islamic Azad University,Qaemshahr, Iran

bDepartment of Statistics, Arak Branch, Islamic Azad University, Arak, Iran

Received 12 March 2017; accepted 09 January 2018

Abstract

The aim of this paper is investigate the notion of a generalized interval valuedintuitionistic fuzzy number (GIV IFNB), which extends the interval valuedintuitionistic fuzzy number. Firstly, the concept of GIV IFNBs is introduced.Arithmetic operations and cut sets over GIV IFNBs are investigated. Then,the values and ambiguities of the membership degree and the non-membershipdegree and the value index and ambiguity index for GIV IFNBs are defined.Finally, we develop a value and ambiguity-based ranking method.

Key words: Generalized interval valued intuitionistic fuzzy sets, generalizedinterval valued intuitionistic fuzzy numbers, cut set, value index, ambiguityindex.

∗ Corresponding author’s E-mail: e baloui2008@yahoo.com

1 Introduction

Later on introduce fuzzy sets theory in Zadeh (1965), the concept offuzzy numbers and its arithmetic operations were first investigated byChang and Zadeh (1972) and others. The notion of fuzzy numbers wasintroduced by Dubois and Prade (1978) as a fuzzy subset of the real line.Subsequently, Atanassov (1986) introduced concept intuitionistic fuzzysets (IFSs) as a generalization of fuzzy sets. IFS can be used to deal withuncertainty by taking both degree of membership and degree of non-membership. Burillo et al. (1994) proposed the definition of intuitionis-tic fuzzy number. Mahapatra and Roy (2009) presented triangular intu-itionistic fuzzy number and used it for reliability evaluation. Wang andZhang (2009) defined the trapezoidal intuitionistic fuzzy number (TrIFN)and their operational laws. Also Mahapatra and Mahapatra (2010) de-fined trapezoidal intuitionistic fuzzy number and arithmetic operationsof TrIFN based on (α, β)-cut method. Parvathi (2012) introduced sym-metric trapezoidal intuitionistic fuzzy numbers (STrIFNs) and discussedtheir desirable properties and arithmetic operations based on (α, β)-cut.Atanassov and Gargov (1989) proposed the notion of the interval valuedintuitionistic fuzzy sets (IVIFSs) as combined IFS concept with intervalvalued fuzzy sets concept, which is characterized by membership func-tion and non-membership function whose values are interval rather thanexact numbers. They are very useful in the process of decision makingsince in many real word decision problems the values membership func-tion and non-membership function in an IFS are difficult to be expressedas an exact numbers. Yuan and Li (2009) and Adak and Bhowmik (2011)defined different types of cut sets on IVIFSs.Based on IVIFS, Xu (2007) defined the notion of interval valued intu-itionistic fuzzy number (IVIFN) and introduced some operations on IV-IFNs. Xu (2007) proposed score function and accuracy function to rankIVIFNs. Ye (2009) also proposed a novel accuracy function to rank IV-IFNs. Garg (2016) considered a new generalized improved score functionof IVIFSs and applications in expert systems.The literature review shows, notions of IVIFNs is very useful in mod-eling real life problems with imprecision or uncertainty and they havebeen applied to many different fields. For example, some recent appli-

44

cations of IVIFNs have been: multiple attribute group decision making(MAGDM) using elimination and choice translation reality (ELECTRE)method (Li et al. (2012); Veeramachaneni and Kandikonda, (2016)); ex-tension of the TOPSIS method to solve multiple attribute group decisionmaking problems (Park et al. (2011), Sudha et al. (2015)); interval val-ued intuitionistic fuzzy analytic hierarchy process (Abdullah and Najib,(2016)); The selection of technology forecasting method (Intepe et al.(2013)) and etc.Baloui Jamkhaneh and Nadarajah (2015) extended IFSs with the in-troduction of the concept of new generalized intuitionistic fuzzy sets(GIFSB) and introduced some operators over GIFSB. Shabani andBaloui Jamkhaneh (2014) introduced generalized intuitionistic fuzzy num-bers (GIFNB) base on GIFSB. Baloui Jamkhaneh (2016) study theconcepts of values and ambiguities of the degree of membership and thedegree of non-membership for GIFNB. Baloui Jamkhaneh (2015) consid-ered new generalized interval value intuitionistic fuzzy sets (GIV IFSB)and introduced some operators over GIV IFSB. The aim of this paper isto introduction the generalized interval valued intuitionistic fuzzy num-ber (GIV IFNB) based on generalization of the IVIFS related to BalouiJamkhaneh (2015) and to drive their specifications. The derived specifica-

tions include: i) (α1, α2)-cut, ii) (β1, β2)-cut, iii) (−→α ,←−β )-cut, iv) values of

the membership degree and the non-membership degree, v) ambiguitiesof the membership degree and the non-membership degree, vi) the valueindex, vii) the ambiguity index, viii) ranking function of GIV IFNB.In order to achieve this, the remainder of this paper is organized asfollows: In Section 2, we briefly introduce IFS and its generalizes. In Sec-tion 3 define new generalized interval valued intuitionistic fuzzy sets andarithmetic operations. In Section 4 introduce cut sets on GIV IFNB. InSection 5 define values and ambiguities of the membership degree andthe non-membership degree, the value index, ambiguity index and valueand ambiguity-based ranking method for GIV IFNBs are defined. Thepaper is concluded in Section 6.

45

2 Preliminaries

We some basic definitions of IFSs and IVIFSs are introduced to facilitatethe discussion.

Definition 2.1 (Atanassov,1986) An IFS A in X is defined as an objectof the form A = {〈x, µA (x) , νA(x)〉 : x ∈ X} where the functions µA :X → [0, 1] and νA: X → [0, 1] denotes the degree of membership andnon-membership functions of A, respectively and 0 ≤ µA (x)+νA (x) ≤ 1for each x ∈ X.

Definition 2.2 (Atanassov and Gargov, 1989) Let X be a non-emptyset. Interval valued intuitionistic fuzzy sets (IV IFS) A in X, is de-fined as an object of the form A = {〈x,MA (x) , NA(x)〉 | x ∈ X} wherethe functions MA (x) : X → [ I ] and NA(x) : X → [I], denote thedegree of membership and degree of non-membership of A respectively,where MA (x) = [MAL (x) ,MAU (x)], NA (x) = [NAL (x) , NAU (x)], 0 ≤MAU (x) +NAU (x) ≤ 1 for each x ∈ X.

Definition 2.3 (Baloui Jamkhaneh and Nadarajah (2015)) Let X be anon-empty set. Generalized intuitionistic fuzzy sets (GIFSB) A in X, isdefined as an object of the form A = {〈x, µA (x) , νA(x)〉 : x ∈ X} wherethe functions µA : X → [0, 1] and νA : X → [0, 1], denote the degree ofmembership and degree of non-membership functions of A respectively,and 0 ≤ µA(x)δ + vA(x)δ ≤ 1 for each x ∈ X and δ = n or 1

n, n =

1, 2, . . . , N .

Definition 2.4 Let [I] be the set of all closed subintervals of the interval[0, 1] and MA (x) = [MAL (x) ,MAU (x)] ∈ [I] and NA(x) = [NAL (x) ,NAU (x)] ∈ [I] then NA (x) ≤ MA (x) if and only if NAL (x) ≤MAL (x)and NAU (x) ≤MAU (x).

Definition 2.5 (Baloui Jamkhaneh(2015)) Let X be a non-empty set.Generalized interval valued intuitionistic fuzzy sets (GIV IFSBs) A inX, is defined as an object of the form A = {〈x,MA (x) , NA(x)〉 | x ∈ X}where the functions MA (x) : X → [I] and NA(x) : X → [I], denote thedegree of membership and degree of non-membership of A respectively,

46

and MA (x) = [MAL (x) ,MAU (x)], NA (x) = [NAL (x) , NAU (x)], where0 ≤ MAU (x)δ + NAU (x)δ ≤ 1, for each x ∈ X and δ = n or 1

n, n =

1, 2, . . . , N . The collection of all GIV IFSB(δ) is denoted by GIV IFSB(δ,X).

3 New Generalized Interval Valued Intuitionistic Fuzzy Num-bers

In this section, GIV IFNB and their operations are defined as follows.

Definition 3.1 In general, our generalized interval valued intuitionis-tic fuzzy number A can be described as any GIV IFSB (X) of the realline R whose membership function µA (x) = [µ

A(x) , µA (x)] and non-

membership function νA (x) = [νA (x) , νA (x)] are defined as follows:

µA

(x) =

((x−a)µ

b−a

) 1δ

, a ≤ x ≤ b

µ1δ , b ≤ x ≤ c((d−x)µ

d−c

) 1δ

, c ≤ x ≤ d

0, o.w.

, µA(x) =

((x−a)µb−a

) 1δ , a ≤ x ≤ b

µ1δ , b ≤ x ≤ c((d−x)µd−c

) 1δ , c ≤ x ≤ d

0, o.w.

,

νA(x) =

((b−x)+ν(x−a1)

b−a1

) 1δ , a1 ≤ x ≤ b

ν1δ , b ≤ x ≤ c((x−c)+ν(d1−x)

d1−c

) 1δ , c ≤ x ≤ d1

1, o.w.

, νA(x) =

((b−x)+ν(x−a1)

b−a1

) 1δ , a1 ≤ x ≤ b

ν1δ , b ≤ x ≤ c((x−c)+ν(d1−x)

d1−c

) 1δ , c ≤ x ≤ d1

1, o.w.

where a1 ≤ a ≤ b ≤ c ≤ d ≤ d1 and 0 ≤ µ ≤ µ ≤ 1, 0 ≤ ν ≤ ν ≤ 1, µ +ν ≤ 1. TheGIV IFNBA is denoted asA = (a1, a, b, c, d, d1, [µ, µ], [ν, ν], δ).

Definition 3.2 A GIV IFNB is said to be symmetric GIV IFNB if b−a = d− c and b− a1 = d1 − c.

Figure 1 shows membership and non-membership functions of A = (0.25,1, 2, 3, 4, 4.75, [0.54, 0.64], [0.26, 0.36], δ) with δ = 2 respectively. Figure 3shows membership and non-membership functions of A = (0.25, 1, 2, 3, 4,

47

4.75, [0.54, 0.64], [0.26, 0.36], δ) with δ = 1.

µA

(x) =

(0.54x− 0.54)1δ , 1 ≤ x ≤ 2

0.541δ , 2 ≤ x ≤ 3

(2.16− 0.54x)1δ , 3 ≤ x ≤ 4

0, o.w.

,

µA(x) =

(0.64x− 0.64)1δ , 1 ≤ x ≤ 2

0.641δ , 2 ≤ x ≤ 3

(2.56− 0.64x)1δ , 3 ≤ x ≤ 4

0, o.w.

,

νA(x) =

(1.1057143− 0.4228571x)1δ , 0.25 ≤ x ≤ 2

0.261δ , 2 ≤ x ≤ 3

(0.4228571x− 1.0085714)1δ , 3 ≤ x ≤ 4.75

1, o.w.

,

νA(x) =

(1.0914286− 0.3657143x)1δ , 0.25 ≤ x ≤ 2

0.361δ , 2 ≤ x ≤ 3

(0.3657143x− 0.7371429)1δ , 3 ≤ x ≤ 4.75

1, o.w.

Fig. 1. Membership and non-membership functions ofA = (0.25, 1, 2, 3, 4, 4.75, [0.54, 0.64], [0.26, 0.36])

48

Fig. 2. Membership and non-membership functions ofA = (0.25, 1, 2, 3, 4, 4.75, [0.54, 0.64], [0.26, 0.36])

Definition 3.3 Let A = (a′1, a1, b1, c1, d1, d′1, [µ1

, µ1], [ν1, ν1], δ) and B =(a′2, a2, b2, c2, d2, d

′2, [µ2

, µ2], [ν2, ν2], δ) be two GIV IFNBs, then,

A+B= (a′1 + a′2, a1 + a2, b1 + b2, c1 + c2, d1 + d2, d′1 + d′2

, [µ1

+ µ2− µ

1µ

2, µ1 + µ2 − µ1µ2], [ν1ν2, ν1ν2], δ),

kA= (ka′1, ka1, kb1, kc1, kd1, kd′1, [1− (1− µ

1)k, 1− (1− µ1)k]

, [νk1, νk1 ], δ), k = 2, 3, · · ·

kA= (kd′1, kd1, kc1, kb1, ka1, ka′1, [1− (1− µ

1)|k|, 1− (1− µ1)|k|]

, [ν|k|1 , ν

|k|1 ], δ), k = −2,−3, · · ·

−A = (−d′1,−d1,−c1,−b1,−a1,−a′1, [µ1, µ1], [ν1, ν1], δ),

A−B= (a′1 − d′2, a1 − d2, b1 − c2, c1 − b2, d1 − a2, d′1 − a′2

, [µ1

+ µ2− µ

1µ

2, µ1 + µ2 − µ1µ2], [ν1ν2, ν1ν2], δ),

A ·B= (a′1 · a′2, a1 · a2, b1 · b2, c1 · c2, d1 · d2, d′1 · d′2

49

, [µ1µ

2, µ1µ2], [ν1 + ν2 − ν1ν2, ν1 + ν2 − ν1ν2], δ),

Ak = (a′1k, ak1, b

k1, c

k1, d

k1, d′1k, [µk

1, µk1], [1− (1−ν1)k, 1− (1− ν1)k], δ), k > 0.

4 Cut Sets on GIV IFNB

Definition 4.1 A ~α- cut set (~α = (α1, α2)) of a GIV IFNBA is a crispsubset of R, which defined is as

A[~α, δ] ={〈x, µ

A(x) ≥ α1, µA(x) ≥ α2〉 : x ∈ X

}, 0 ≤ α1 ≤ µ

1δ , 0 ≤ α2 ≤ µ

1δ

According to the definition of membership function of GIV IFNB, it canbe shown that

A[~α, δ] = [L1(~α), U1(~α)] = [max(L1(α1), L1(α2)),min(U1(α1), U1(α2))],

L1(α1) = a+(b− a)αδ1

µ, L1(α2) = a+

(b− a)αδ2µ

,

U1(α1) = d− (d− c)αδ1µ

, U1(α2) = d− (d− c)αδ2µ

.

Theorem 4.1 Let A[~α, δ] be ~α- cut set of GIV IFNBA, then

i. If α2 ≥(µµ

) 1δ

α1, then A[~α, δ] = [L1(α2), U1(α2)],

ii. If α2 <(µµ

) 1δ

α1, then A[~α, δ] = [L1(α1), U1(α1)].

Proof. (i) If α2 ≥(µµ

) 1δ

α1 ⇒ αδ2

µ≥ αδ

1

µ⇒ a +

(b−a)αδ2

µ≥ a +

(b−a)αδ1

µ

⇒ L1(α2) ≥ L1(α1)

If α2 ≥(µµ

) 1δ

α1 ⇒ αδ2

µ≥ αδ

1

µ⇒ d − (d−c)αδ

2

µ≤ d − (d−c)αδ

1

µ⇒ U1(α2) ≤

U1(α1)Finally, we have

A[~α, δ] = [max(L1(α1), L1(α2)),min(U1(α1), U1(α2))] = [L1(α2), U1(α2)]

50

Proof (ii) is similar to (i).

Corollary 4.1 Let A[~α, δ] be ~α- cut set of GIV IFNBA, if α2 = α1,using Theorem 4.1-ii result is

A[~α, δ] = [L1(α1), U1(α1)].

Corollary 4.2 Let A[~α, δ] be ~α- cut set of GIV IFNBA, if α2 = µ1δ ,

α1 = µ1δ , using Definition 4.1 result is A[~α, δ] = [b, c].

Theorem 4.2 Let A[~α, δ] be ~α- cut set of GIV IFNBA, then

i. If 1 < δ1 ≤ δ2 then A[~α, δ1] ⊂ A[~α, δ2],ii. If δ1 ≤ δ2 < 1 then A[~α, δ2] ⊂ A[~α, δ1],

Proof. (i) Since for 1 ≤ δ, L1(α1) and L1(α2) are decreasing, also U1(α1)and U1(α2) are increasing, proof (i) is clear.(ii) Since for δ ≤ 1, L1(α1) and L1(α2) are increasing, also U1(α1) andU1(α2) are decreasing, proof (ii) is clear.

Remark 4.1 Let ~τ = (τ1, τ2) < ~γ = (γ1, γ2) (This means that τi <γi, i = 1, 2). Then A[~γ, δ] ⊆ A[~τ , δ].

Since L1(α1) and L1(α2) aspect to α1 and α2 are increasing, also U1(α1)and U1(α2) aspect to α1 and α2 are decreasing, proof is clear.

Definition 4.2 A ~β- cut set ( ~β = (β1, β2)) of a GIV IFNBA is a crispof R, which defined is as

A[ ~β, δ] = {〈x, νA(x) ≤ β1, νA(x) ≤ β2〉 : x ∈ X} , ν1δ ≤ β1 ≤ 1, ν

1δ ≤ β2 ≤ 1

According to the definition of non-membership function of GIV IFNB itcan be shown that

A[ ~β, δ] = [L2( ~β), U2( ~β)] = [max(L2(β1), L2(β2)),min(U2(β1), U2(β2))],

L2(β1) =b(1− β1

δ) + a1(β1δ − ν)

1− ν, L2(β2) =

b(1− β2δ) + a1(β2

δ − ν)

1− ν,

U2(β1) =c(1− β1

δ) + d1(β1δ − ν)

1− ν, U2(β2) =

c(1− β2δ) + d1(β2

δ − ν)

1− ν.

51

Theorem 4.3 Let A[ ~β, δ] be ~β- cut set of GIV IFNBA, then

i. Ifβδ1

1−ν−βδ2

1−ν ≥b−a1ν

(b−a1)(1−ν)− b−a1ν

(b−a1)(1−ν), then A[ ~β, δ] = [L2(β2), U2(β2)],

ii. Ifβδ1

1−ν−βδ2

1−ν <b−a1ν

(b−a1)(1−ν)− b−a1ν

(b−a1)(1−ν), then A[ ~β, δ] = [L2(β1), U2(β1)].

Proof. (i)βδ1

1−ν −βδ2

1−ν ≥b−a1ν

(b−a1)(1−ν)− b−a1ν

(b−a1)(1−ν)⇒ (a1−b)βδ

1+(b−a1ν)

1−ν −(a1−b)βδ

2+(b−a1ν)

1−ν ≤ 0,

⇒ L2(β1)− L2(β2) ≤ 0 ⇒ max(L2(β1), L2(β2)) = L2(β2),Since b−a1ν

(b−a1)(1−ν)− b−a1ν

(b−a1)(1−ν)= c−d1ν

(c−d1)(1−ν)− c−d1ν

(c−d1)(1−ν)then

βδ1

1−ν −βδ2

1−ν ≥c−d1ν

(c−d1)(1−ν)− c−d1ν

(c−d1)(1−ν)⇒ (d1−c)βδ

1+(c−d1ν)

1−ν − (d1−c)βδ2+(c−d1ν)

1−ν ≥0,

U2(β1)− U2(β2) ≥ 0⇒ min(U2(β1), U2(β2)) = U2(β2).

Proof is complete.

Proof (ii) is similar to (i).

Corollary 4.3 Let A[ ~β, δ] be ~β- cut set of GIV IFNBA, if β2 = β1,

using Definition 4.2 result is A[ ~β, δ] = [L2(β2), U2(β2)].

Corollary 4.4 Let A[ ~β, δ] be ~β- cut set of GIV IFNBA, if β2 = ν1δ ,

β1 = ν1δ , using Definition 4.2 result is A[ ~β, δ] = [b, c].

Theorem 4.4 Let A[ ~β, δ] be ~β- cut set of GIV IFNBA, then

i. If 1 ≤ δ1 ≤ δ2 then A[ ~β, δ2] ⊂ A[ ~β, δ1],

ii. If δ1 ≤ δ2 ≤ 1 then A[ ~β, δ1] ⊂ A[ ~β, δ2].

Proof. (i) Since for 1 ≤ δ, L2(β1) and L2(β2) are increasing, also U2(β1)and U2(β2) are decreasing, proof (i) is clear.(ii) Since for δ ≤ 1, L2(β1) and L2(β2) are decreasing, also U2(β1) andU2(β2) are increasing, proof (ii) is clear.

Remark 4.2 Let ~ξ = (ξ1, ξ2) < ~η = (η1, η2) (This means that ξi < ηi,

i = 1, 2). Then A[ ~ξ, δ] ⊆ A[ ~η, δ].

52

Since L1(β1) and L1(β2) aspect to β1 and β2 are decreasing, also U1(β1)and U1(β2) aspect to β1 and β2 are increasing, proof is clear.

Corollary 4.5 Let A = (a1, a, b, c, d, d1, [µ1, µ1], [ν1, ν1], δ) and

B = (a1, a, b, c, d, d1, [µ2, µ2], [ν2, ν2], δ) be two GIV IFNBs, then

i. If [µ1, µ1] ≤ [µ

2, µ2] then A[~α, δ] ⊂ B[~α, δ],

ii. If [ν2, ν2] ≤ [ν1, ν1] then A[ ~β, δ] ⊂ B[ ~β, δ].

Definition 4.3 Let αi, βi ∈ [0, 1] be fixed numbers such that 0 ≤ α1 ≤µ

1δ , 0 ≤ α2 ≤ µ

1δ , ν

1δ ≤ β1 ≤ 1, ν

1δ ≤ β2 ≤ 1, 0 ≤ αδ2 + βδ2 ≤ 1,

0 ≤ αδ1 + βδ1 ≤ 1, ~α = (α1, α2), ~β = (β1, β2).

A (~α, ~β)- cut set generated by a GIV IFNBA is defined by:

A[~α, ~β, δ] ={〈x, µ

A(x) ≥ α1, µA(x) ≥ α2, νA(x) ≤ β1, νA(x) ≤ β2〉 : x ∈ X

},

A[~α, ~β, δ] is defined as the crisp set of elements x which belong to A atleast to the degree α1 or α2 and which does not belong to A at most tothe degree β1 or β2. Therefore the (~α, ~β) -cut of a GIV IFNB is given by

A[~α, ~β, δ] ={x, x ∈ [L1(~α), U1(~α)] ∩ [L2( ~β), U2( ~β)]

}= [L(~α, ~β), U(~α, ~β)].

Corollary 4.6 Let A[~α, ~β, δ] be a (~α, ~β)- cut set generated by a GIV IFNBA,the using Theorem 4.1 and Theorem 4.3 result is:

Case I. If α2 ≥(µµ

) 1δ

α1 andβδ1

1−ν −βδ2

1−ν ≥b−a1ν

(b−a1)(1−ν)− b−a1ν

(b−a1)(1−ν)then

we have

L(~α, ~β) =

L1(α2), α2 ≥

(µ

1−ν ×b(1−βδ

2)+a1(βδ2−ν)−a(1−ν)

b−a

) 1δ

L2(β2), α2 <(

µ1−ν ×

b(1−βδ2)+a1(βδ

2−ν)−a(1−ν)

b−a

) 1δ,

and

U(~α, ~β) =

U1(α2), α2 ≥

(µ

1−ν ×c(1−βδ

2)+d1(βδ2−ν)−d(1−ν)

c−d

) 1δ

U2(β2), α2 <(

µ1−ν ×

c(1−βδ2)+d1(βδ

2−ν)−d(1−ν)

c−d

) 1δ.

53

In special case a1 = a, d1 = d

A[~α, ~β, δ] =

[L1(α2), U1(α2)], α2 ≥(

µ1−ν × (1− βδ2)

) 1δ

[L2(β2), U2(β2)], α2 <(

µ1−ν × (1− βδ2)

) 1δ.

Case II. If α2 <(µµ

) 1δ

α1 andβδ1

1−ν −βδ2

1−ν ≥b−a1ν

(b−a1)(1−ν)− b−a1ν

(b−a1)(1−ν)then

we have

L(~α, ~β) =

L1(α1), α1 ≥

(µ

1−ν ×b(1−βδ

2)+a1(βδ2−ν)−a(1−ν)

b−a

) 1δ

L2(β2), α1 <(

µ

1−ν ×b(1−βδ

2)+a1(βδ2−ν)−a(1−ν)

b−a

) 1δ,

and

U(~α, ~β) =

U1(α1), α1 ≥

(µ

1−ν ×c(1−βδ

2)+d1(βδ2−ν)−d(1−ν)

c−d

) 1δ

U2(β2), α1 <(

µ

1−ν ×c(1−βδ

2)+d1(βδ2−ν)−d(1−ν)

c−d

) 1δ.

In special case a1 = a, d1 = d

A[~α, ~β, δ] =

[L1(α1), U1(α1)], α1 ≥(

µ

1−ν × (1− βδ2)) 1

δ

[L2(β2), U2(β2)], α1 <(

µ

1−ν × (1− βδ2)) 1

δ.

Case III. If α2 ≥(µµ

) 1δ

α1 andβδ1

1−ν −βδ2

1−ν <b−a1ν

(b−a1)(1−ν)− b−a1ν

(b−a1)(1−ν)then

we have

L(~α, ~β) =

L1(α2), α2 ≥

(µ

1−ν ×b(1−βδ

1)+a1(βδ1−ν)−a(1−ν)

b−a

) 1δ

L2(β1), α2 <(

µ1−ν ×

b(1−βδ1)+a1(βδ

1−ν)−a(1−ν)

b−a

) 1δ,

54

and

U(~α, ~β) =

U1(α2), α2 ≥

(µ

1−ν ×c(1−βδ

1)+d1(βδ1−ν)−d(1−ν)

c−d

) 1δ

U2(β1), α2 <(

µ1−ν ×

c(1−βδ1)+d1(βδ

1−ν)−d(1−ν)

c−d

) 1δ.

In special case a1 = a, d1 = d

A[~α, ~β, δ] =

[L1(α2), U1(α2)], α2 ≥(

µ1−ν × (1− βδ1)

) 1δ

[L2(β1), U2(β1)], α2 <(

µ1−ν × (1− βδ1)

) 1δ.

Case IV. If α2 <(µµ

) 1δ

α1 andβδ1

1−ν −βδ2

1−ν <b−a1ν

(b−a1)(1−ν)− b−a1ν

(b−a1)(1−ν)then

we have

L(~α, ~β) =

L1(α1), α1 ≥

(µ

1−ν ×b(1−βδ

1)+a1(βδ1−ν)−a(1−ν)

b−a

) 1δ

L2(β1), α1 <(

µ

1−ν ×b(1−βδ

1)+a1(βδ1−ν)−a(1−ν)

b−a

) 1δ,

and

U(~α, ~β) =

U1(α1), α1 ≥

(µ

1−ν ×c(1−βδ

1)+d1(βδ1−ν)−d(1−ν)

c−d

) 1δ

U2(β1), α1 <(

µ

1−ν ×c(1−βδ

1)+d1(βδ1−ν)−d(1−ν)

c−d

) 1δ.

In special case a1 = a, d1 = d

A[~α, ~β, δ] =

[L1(α1), U1(α1)], α1 ≥(

µ

1−ν × (1− βδ1)) 1

δ

[L2(β1), U2(β1)], α1 <(

µ

1−ν × (1− βδ1)) 1

δ.

Example 4.1 Let A = (0.5, 1, 2, 3, 4, 4.5, [0.49, 0.64], [0.25, 0.36], 2) be aGIV IFNB. Then for 0 ≤ α1 ≤ 0.7, 0 ≤ α2 ≤ 0.8, 0.5 ≤ β1 ≤ 1,0.6 ≤ β2 ≤ 1, 0 ≤ α2

i + β2i ≤ 1, i = 1, 2, we get following ~α- cut and ~β-

cut set as

L1(α1) = 1+α2

1

0.49, L1(α2) = 1+

α22

0.64, U1(α1) = 4− α2

1

0.49, U1(α2) = 4− α2

2

0.64.

55

If α2 ≥ 1.1429α1 then A[~α, δ] =[1 +

α22

0.64, 4− α2

2

0.64

], otherwise A[~α, δ] =[

1 +α21

0.49, 4− α2

1

0.49

], for example α2 = 0.5 and α1 = 0.3 then A[α, δ] =

[1.3906, 3.6094].Also

L2(β1) = 2.5− 2β21 , L2(β2) = 2.8437− 2.3437β2

2 ,

U2(β1) = 2.5 + 2β21 , U2(β2) = 2.1562 + 2.3437β2

2 .

If 0.75β22 − 0.64β2

1 ≤ 0.11 then A[ ~β, δ] = [2.8437 − 2.3437β22 , 2.1562 +

2.3437β22 ] otherwise A[ ~β, δ] = [2.5−2β2

1 , 2.5+2β21 ], for example β1 = 0.6,

β2 = 0.7 then A[ ~β, δ] = [1.78, 3.22], therefore for α1 = 0.3, α2 = 0.5,

β1 = 0.6, β2 = 0.7, we have A[~α, ~β, δ] = [1.78, 3.22].

5 Indices of a GIV IFNB

Definition 5.1 Let A be a GIV IFNB. Then the values of the mem-bership function A and the non membership function A are defined asfollows:

Vµ(A, δ) =1

2

∫ µ1δ

0(L(α) + U(α))f(α)dα, f(α) =

2α

µ1δ

,

where L(α) = a+ (b−a)αδ

µ, U(α) = d− (d−c)αδ

µ, µ =

µ+µ

2,

Vν(A, δ) =1

2

∫ 1

ν1δ

(L(β) + U(β))f(β)dβ, f(β) =2(1− β)

1− ν 1δ

,

where L(β) = b(1−βδ)+a1(βδ−ν)1−ν , U(β) = c(1−βδ)+d1(βδ−ν)

1−ν , ν = ν+ν2

.

Corollary 5.1 Let A be a GIV IFNB, then using Definition 5.1

Vµ(A, δ) = µ1δ (a

2+d

2+

(b− a)− (d− c)δ + 2

),

56

Vν(A, δ) =b+ c

(1− ν)(1− ν 1δ )

1

2− 1

(δ + 1)(δ + 2)− ν

1δ +

νδ+1δ

δ + 1+ν

2δ

2− ν

δ+2δ

δ + 2

,+

a1 + d1

(1− ν)(1− ν 1δ )

1

(δ + 1)(δ + 2)− ν

2+δν

δ+1δ

δ + 1− δν

δ+2δ

2(δ + 1)

.

In this case Vµ(−A, δ) = −Vµ(A, δ) and Vν(−A, δ) = −Vν(A, δ).

Theorem 5.1 Let A = (a1, a, b, c, d, d1, [µ, µ], [ν, ν], δ), then

i. aµ1δ ≤ Vµ(A, δ) ≤ dµ

1δ ,

ii. (1− ν 1δ )a1 ≤ Vν(A, δ) ≤ (1− ν 1

δ )d1.

Proof. (i)

Vµ(A, δ) =1

2

∫ µ1δ

0(L(α) + U(α))f(α)dα,

=1

µ1δ

∫ µ1δ

0

(a+

(b− a)αδ

µ+ d− (d− c)αδ

µ

)αdα,

≥ 1

µ1δ

∫ µ1δ

02aαdα = aµ

1δ .

Vµ(A, δ) =1

2

∫ µ1δ

0(L(α) + U(α))f(α)dα,

=1

µ1δ

∫ µ1δ

0

(a+

(b− a)αδ

µ+ d− (d− c)αδ

µ

)αdα,

≤ 1

µ1δ

∫ µ1δ

02dαdα = dµ

1δ .

57

Proof is complete. (ii)

Vν(A, δ) = 1

1−ν1δ

∫ 1

ν1δ

(b(1−βδ)+a1(βδ−ν)

1−ν + c(1−βδ)+d1(βδ−ν)1−ν

)(1− β)dβ

= 1

1−ν1δ

∫ 1

ν1δ

((b−a1ν)−(b−a1)βδ

1−ν + (c−d1ν)−(c−d1)βδ

1−ν

)(1− β)dβ

≥ 1

1−ν1δ

∫ 1

ν1δ

2a1(1− β)dβ

= (1− ν 1δ )a1.

Vν(A, δ) =1

1− ν 1δ

∫ 1

ν1δ

(b(1−βδ)+a1(βδ−ν)

1− ν+c(1−βδ)+d1(βδ−ν)

1− ν

)(1− β)dβ,

=1

1− ν 1δ

∫ 1

ν1δ

((b−a1ν)−(b−a1)βδ

1− ν+

(c−d1ν)−(c−d1)βδ

1− ν

)(1− β)dβ,

≤ 1

1− ν 1δ

∫ 1

ν1δ

2a1(1− β)dβ = (1− ν1δ )d1.

Proof is complete.

Definition 5.2 Let A be a GIV IFNB. Then the ambiguity of the mem-bership function A and the non-membership function A are defined asfollows:

Gµ(A, δ) =∫ µ

1δ

0(U(α− L(α)))f(α)dα, f(α) =

2α

µ1δ

,

Gν(A, δ) =∫ 1

ν1δ

(U(β − L(β)))f(β)dβ, f(β) =2(1− β)

1− ν 1δ

.

Corollary 5.2 Let A be a GIV IFNB, then

58

Gµ(A, δ) =∫ µ

1δ

0(U(α)− L(α))f(α)dα,

=1

µ1δ

∫ µ1δ

0

(d− (d− c)αδ

µ− a− (b− a)αδ

µ

)2αdα,

=µ1δ

(d− a− 2(d− c) + 2(b− a)

δ + 2

).

And

Gν(A, δ) =∫ 1

ν1δ

(U(β)− L(β))f(β)dβ,

=2(−b+ c)

(1−ν)(1−ν 1δ )

1

2− 1

(δ+1)(δ+2)−ν

1δ +

νδ+1δ

δ+1+ν

2δ

2− ν

δ+2δ

δ+2

+

2(−a1 + d1)

(1− ν)(1− ν 1δ )

1

(δ+1)(δ+2)− ν

2+δν

δ+1δ

δ+1− δν

δ+2δ

2(δ+1)

.It can be easily shown that Gµ(A, δ) = Gµ(−A, δ) and Gν(A, δ) = Gν(−A, δ).

Theorem 5.2 Let A = (a1, a, b, c, d, d1, [µ, µ], [ν, ν], δ), then

i. µ1δ (c− b) ≤ Gµ(A, δ) ≤ µ

1δ (d− a),

ii. (1− ν 1δ )(c− b) ≤ Gν(A, δ) ≤ (1− ν 1

δ )(d1 − a1).

Proof. (i)

Gµ(A, δ) =∫ µ

1δ

0(U(α)− L(α))f(α)dα,

=∫ µ

1δ

0

(d− (d− c)αδ

µ− a− (b− a)αδ

µ

)f(α)dα,

≥ 1

µ1δ

∫ µ1δ

02(c− b)αdα = µ

1δ (c− b),

and

59

Gµ(A, δ) =∫ µ

1δ

0

(d− (d− c)αδ

µ− a− (b− a)αδ

µ

)f(α)dα,

≤ 1

µ1δ

∫ µ1δ

02(d− a)αdα = µ

1δ (d− a).

Proof is complete.(ii)

Gν(A, δ) =2

(1−ν 1δ )

∫ 1

ν1δ

(c(1−βδ)+d1(βδ−ν)

1− ν− b(1−β

δ)+a1(βδ−ν)

1− ν

)(1−β)dβ,

=2

(1−ν 1δ )

∫ 1

ν1δ

((c−d1ν)−(c−d1)βδ

1− ν− (b−a1ν)−(b−a1)βδ

1− ν

)(1−β)dβ,

≥ 2

(1− ν 1δ )

∫ 1

ν1δ

(c− b)(1− β)dβ = (1− ν1δ )(c− b).

Gν(A, δ) =2

(1−ν 1δ )

∫ 1

ν1δ

(c(1−βδ)+d1(βδ−ν)

1− ν− b(1−β

δ)+a1(βδ−ν)

1− ν

)(1−β)dβ,

=2

(1−ν 1δ )

∫ 1

ν1δ

((c−d1ν)−(c−d1)βδ

1− ν− (b−a1ν)−(b−a1)βδ

1− ν

)(1−β)dβ,

≤ 2

(1− ν 1δ )

∫ 1

ν1δ

(d1 − a1)(1− β)dβ = (1− ν1δ )(d1 − a1).

Proof is complete.

Definition 5.3 Let A = (a1, a, b, c, d, d1, [µ, µ], [ν, ν], δ). A value indexand an ambiguity index for the A are defined as follows:

V (A, δ) =Vµ(A, δ) + Vν(A, δ)

2, G(A, δ) =

Gν(A, δ) +Gµ(A, δ)

2.

60

Procedure for ranking GIV IFNB.Let A = (a

′1, a1, b1, c1, d1, d

′1, [µ1

, µ1], [ν1, ν1], δ)

and B = (a′2, a2, b2, c2, d2, d

′2, [µ2

, µ2], [ν2, ν2], δ) be two GV IIFNBs, thena ranking function is as follows

R(A, δ) = V (A, δ) +G(A, δ),

where

i. If R(A, δ) > R(B, δ), then A > B.ii. If R(A, δ) < R(B, δ), then A < B.

iii. If R(A, δ) = R(B, δ), then A = B.

Example 5.1 Let A = (0.5, 1, 2, 3, 4, 4.5, [0.49, 0.64], [0.25, 0.36], 2), µA =0.565, νA = 0.305, and B = (0.75, 1.25, 2, 3, 3.75, 4.25, [0.36, 0.49], [0.16, 0.25, 2]),µB = 0.425, νB = 0.205, thenVµ(A, δ) = 1.879, Vν(A, δ) = 0.9932, V (A, δ) = 1.4361, Vµ(B, δ) = 1.95,Vν(B, δ) = 1.3276, V (B, δ) = 1.6388, Gµ(A, δ) = 1.503, Gν(A, δ) =0.6327, G(A, δ) = 1.0662, Gµ(B, δ) = 1.141, Gν(B, δ) = 0.8451, G(B, δ) =0.99305, R(A, δ) = 2.5023, R(B, δ) = 2.63185. Since R(A, δ) < R(B, δ),therefore A < B.

6 Conclusions

We have introduced new generalized interval valued intuitionistic fuzzynumbers. We studied (α1, α2)-cut, (β1, β2)-cut and (~α, ~β)-cut ofGIV IFNB.Then, the values and ambiguities of the membership degree and thenon-membership degree and the value index and ambiguity index forGIV IFNBs are defined. They are used to define ranking function ofGIV IFNB.

61

References

[1] L. Abdullah, L. Najib, A new preference scale MCDM method based oninterval-valued intuitionistic fuzzy sets and the analytic hierarchy process,Soft Computing, 2016, 20(2), P.511–523.

[2] A. K. Adak, M. Bhowmik, Interval cut-set of interval-valued intuitionisticfuzzy sets, African Journal of Mathematics and Computer ScienceResearch, 2011, 4(4), P.192–200.

[3] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 1986,20, P.87–96.

[4] K. T. Atanassov, G. Gargov, Interval valued intuitionistic fuzzy sets, FuzzySets and Systems, 1989, 31(3), P.343–349.

[5] E. Baloui Jamkhaneh, S. Nadarajah, A New generalized intuitionisticfuzzy sets, Hacettepe Journal of Mathematics and Statistics, 2015, 44 (6),P.1537–1551.

[6] E. Baloui Jamkhaneh, New generalized interval value intuitionistic fuzzysets, Research and Communications in Mathematics and MathematicalSciences, 2015, 5(1), P.33–46.

[7] E. Baloui Jamkhaneh, A value and ambiguity-based ranking method ofgeneralized intuitionistic fuzzy numbers, Research and Communicationsin Mathematics and Mathematical Sciences, 2016, 6(2), P.89–103.

[8] P. Burillo, H. Bustince, V. Mohedano, Some definition of intuitionisticfuzzy number, Fuzzy based expert systems, fuzzy Bulgarian enthusiasts,September 28-30, 199), Sofia, Bulgaria.

[9] S. S. L. Chang, L.A. Zadeh, On fuzzy mapping and control, IEEETransaction on Systems, Man and Cybernetics, 1972, 2(1), P.30–34.

[10] D. Dubois, H. Prade, Operations on fuzzy numbers, International Journalof Systems Science, 1978, 9, P.613–626.

[11] H. Garg, A new generalized improved score function of interval valuedintuitionistic fuzzy sets and applications in expert systems, Applied SoftComputing, 2016, 38, P.988–999.

62

[12] G. Intepe, E. Bozdag, T. Koc, The selection of technology forecastingmethod using a multi-criteria interval-valued intuitionistic fuzzy groupdecision making approach, Computers and Industrial Engineering, 2013,65, P.277–285.

[13] J. Li, M. J. Lin, H. Chen, ELECTRE method based on interval valuedintuitionistic fuzzy number, Applied Mechanics and Materials, 2012, Vols.220-223, P.2308–2312.

[14] G. S. Mahapatra, T. K. Roy, Reliability evaluation using triangularintuitionistic fuzzy numbers arithmetic operations, Proceedings of WorldAcademy of Science, Engineering and Technology, Malaysia, 2009, 38,P.587–585.

[15] G. S. Mahapatra, B. S. Mahapatra, Intuitionistic fuzzy fault tree analysisusing intuitionistic fuzzy numbers, International Mathematical Forum,2010, 5(21), P.1015–1024.

[16] J. H. Park, I. Y. Park, Y. C. Kwun, X. Tan, Extension of the TOPSISmethod for decision making problems under interval-valued intuitionisticfuzzy environment, Applied Mathematical Modeling, 2011, 35, P.2544–2556.

[17] R. Parvathi, C. Malathi, Arithmetic operations on symmetric trapezoidalintuitionistic fuzzy numbers, International Journal of Soft Computing andEngineering, 2012, 02(2), P.268–273.

[18] A. Shabani, E. Baloui Jamkhaneh, A new generalized intuitionistic fuzzynumber, Journal of Fuzzy Set Valued Analysis, 2014, 4, P.1–10.

[19] S. Sudha, J. Rachel, I. Jeba, Crop production using interval-valuedintuitionistic fuzzy TOPSIS method, International Journal of EmergingResearch in Management and Technology, 2015, 4(11), P.435-466.

[20] S. Veeramachaneni, H. Kandikonda, An ELECTRE approach formulticriteria interval-valued intuitionistic trapezoidal fuzzy group decisionmaking problems, Advances in Fuzzy Systems, 2016, vol. 2016, ArticleID 1956303, 17 pages, doi:10.1155/2016/1956303.

[21] J. Q. Wang, Z. Zhang, Aggregation operators on intuitionistic trapezoidalfuzzy number and its application to multi-criteria decision makingproblems, Journal of Systems Engineering and Electronics, 2009, 20,P.321–326.

63

[22] J. Q. Wang, Z. Zhang, Multi-criteria decision making method withincomplete certain information based on intuitionistic fuzzy number,Control and Decision, 2009, 24, P.226–230.

[23] Z. S. Xu, Intuitionist fuzzy aggregation operators, IEEE Transactions onFuzzy Systems, 2007, 15(6), P.1179–1187.

[24] Z. S. Xu, Methods for aggregating interval valued intuitionistic fuzzyinformation and their application to decision making, Control andDecision, 2007, 22(2), P.215–219.

[25] J. Ye, Multicriteria fuzzy decision making method based on a novelaccuracy function under interval valued intuitionistic fuzzy environment,Expert Systems with Applications, 2009, 36, P.6899–6902.

[26] X. H. Yuan, H. X. Li, Cut sets on interval valued intuitionistic fuzzysets, Sixth International Conference on Fuzzy Systems and KnowledgeDiscovery, 2009, 6, P.167–171.

[27] L. A. Zadeh, Fuzzy sets, Information and Control, 1965, 8(3), P.338–356.

64