# A New Ranking Principle For Ordering Trapezoidal Intuitionistic Fuzzy Numbers

Authors: Lakshmana Gomathi Nayagam Velu; Jeevaraj Selvaraj; Dhanasekaran Ponnialagan
Journal: Complexity
 (2017)

---

## Abstract

Modelling real life (industrial) problems using intuitionistic fuzzy numbers is inevitable in the present scenario due to their efficiency in solving problems and their accuracy in the results. Particularly, trapezoidal intuitionistic fuzzy numbers (TrIFNs) are widely used in describing impreciseness and incompleteness of a data. Any intuitionistic fuzzy decision-making problem requires the ranking procedure for intuitionistic fuzzy numbers. Ranking trapezoidal intuitionistic fuzzy numbers play an important role in problems involving incomplete and uncertain information. The available intuitionistic fuzzy decision-making methods cannot perform well in all types of problems, due to the partial ordering on the set of intuitionistic fuzzy numbers. In this paper, a new total ordering on the class of TrIFNs using eight different score functions, namely, imprecise score, nonvague score, incomplete score, accuracy score, spread score, nonaccuracy score, left area score, and right area score, is achieved and our proposed method is validated using illustrative examples. Significance of our proposed method with familiar existing methods is discussed.


---

## Body

## 1. Introduction

Classical set theory cannot be the better choice for modelling problems involving qualitative or imprecise information. To model such problems, fuzzy number was introduced by Jain [1] and some operations on fuzzy numbers are defined in [2, 3]. Intuitionistic fuzzy numbers are comparatively better in modelling real life problems involving uncertainties and imprecise information. Particularly, trapezoidal intuitionistic fuzzy numbers are more effective in describing impreciseness and incompleteness of a data. To resolve the task of comparing trapezoidal intuitionistic fuzzy numbers, many authors have proposed different ranking methods but none of them yield a total order on the class of TrIFNs with finite number of scores. Different ranking methodologies on the class of intuitionistic fuzzy numbers are discussed in [4, 5].Nehi and Maleki [6] generalised the idea of natural ordering on real numbers to the triangular intuitionistic fuzzy numbers (TIFNs) by adopting a statistical view point. Nehi [7] compared TIFNs using lexicographic technique. Li [2] developed the idea of value and ambiguity of a triangular intuitionistic fuzzy number and introduced the new ranking method using the concept of the ratio of the value index to the ambiguity index. Ye [8] presented the new ranking method using expected value of a trapezoidal intuitionistic fuzzy number and solved the decision-making problem using weighted expected value of TrIFN. Dubey and Mehara [9] extended the concept of value and ambiguity to the slightly modified TIFN and proposed a new approach to solve intuitionistic fuzzy linear programing problem. Nehi [7] introduced the concept of characteristic values of membership and nonmembership functions of TrIFN and proposed a new ranking method for trapezoidal intuitionistic fuzzy numbers by using it. Zhang and Nan [10] developed a compromise ratio ranking method for fuzzy multiattribute decision-making (MADM) problem based on the concept that larger TIFN among other TIFNs will be closer to the maximum value index and it will be far away from the minimum ambiguity index simultaneously. Kumar and Kaur [11] proposed the ranking method for TrIFNs by modifying Nehi’s [10] method. Zeng et al. [12] introduced a new ranking method for TrIFNs by extending the concept value and ambiguity of TIFN defined in Li [2]. Wan and Dong [13] introduced the concept of lower and upper weighted possibility mean and possibility mean for a trapezoidal intuitionistic fuzzy numbers and proposed the new ranking method by use of it. Different ranking methods and their applications on multicriteria decision-making problem and other domains are studied in ([14–20]). Lakshmana Gomathi Nayagam et al. [19, 21, 22] have introduced a complete ranking procedure on the class of intuitionistic fuzzy numbers using countable number of parameter. In this paper, a new total ordering on the class of TrIFNs using finite number of score functions is achieved. Also the limitations and drawbacks of all the abovementioned methods are discussed and the efficiency of our proposed method is shown by comparing all existing methods.This paper is organised in the following manner. After introduction, some important definitions on intuitionistic fuzzy numbers are given in Section2. The different subclasses of TrIFNs are introduced and the new score functions on these subclasses are established in Section 3. In Section 4, a complete ranking on the class of trapezoidal intuitionistic fuzzy numbers by using score functions defined in Section 3 is explained. The ranking procedure is explained in detail with several examples and also our proposed method is compared with some other existing methods in the Section 5. Finally conclusions are given in Section 6.


## 2. Preliminaries

Here we give a brief review of some preliminaries.Definition 1 (Atanassov [23]).
LetX be a nonempty set. An intuitionistic fuzzy set (IFS) A in X is defined by A=x,μA(x),νA(x)∣x∈X, where μA(x):X→[0,1] and νA(x):X→[0,1],x∈X with the conditions 0≤μA(x)+νA(x)≤1,∀x∈X. The numbers μA(x),νA(x)∈[0,1] denote the degree of membership and nonmembership of x to lie in A, respectively. For each intuitionistic fuzzy subset A in X, πA(x)=1-μA(x)-νA(x) is called hesitancy degree of x to lie in A.Definition 2 (Burillo et al., [24]).
An intuitionistic fuzzy numberA=(μA,νA) in the set of real numbers R is defined as(1)μAx=fAxifa≤x≤b11ifb1≤x≤b2gAxifb2≤x≤c0Otherwise,νAx=hAxife≤x≤f10iff1≤x≤f2kAxiff2≤x≤g1Otherwise,where 0≤μA(x)+νA(x)≤1 and a,b1,b2,c,e,f1,f2,g∈R such that e≤a,f1≤b1≤b2≤f2,c≤g, and four functions fA,gA,hA,kA:R→[0,1] are the legs of membership function μA and nonmembership function νA. The functions fA and kA are nondecreasing continuous functions and the functions hA and gA are nonincreasing continuous functions.
An intuitionistic fuzzy number{(a,b1,b2,c),(e,f1,f2,g)} with (e,f1,f2,g)≤(a,b1,b2,c)c is shown in Figure 1.Figure 1
Intuitionistic fuzzy number.Definition 3.
A trapezoidal intuitionistic fuzzy numberA with parameters e≤a,f1≤b1≤b2≤f2,c≤g is denoted as A={(a,b1,b2,c),(e,f1,f2,g)} in the set of real numbers R is an intuitionistic fuzzy number whose membership function and nonmembership function are given as(2)μAx=x-a1a2-a1ifa1≤x≤a21ifa2≤x≤a3a4-xa4-a3ifa3≤x≤a40Otherwise,νAx=x-c2c1-c2ifc1≤x≤c20ifc2≤x≤c3x-c3c4-c3ifc3≤x≤c41Otherwise.If a2=a3 (and c2=c3) in a trapezoidal intuitionistic fuzzy number A, we have the triangular intuitionistic fuzzy numbers as special case of the trapezoidal intuitionistic fuzzy numbers.
A trapezoidal intuitionistic fuzzy numberA={(a,b1,b2,c),(e,f1,f2,g)} with f1≤b1, f2≥b2, e≤a, and g≥c is shown in Figure 2.Figure 2
Trapezoidal intuitionistic fuzzy number.We note that the condition(e,f1,f2,g)≤(a,b1,b2,c)c of the trapezoidal intuitionistic fuzzy number A={(a,b1,b2,c),(e,f1,f2,g)} whose membership and nonmembership fuzzy numbers of A are (a,b1,b2,c) and (e,f1,f2,g) implies f1≤b1, f2≥b2, e≤a, and g≥c on the legs of trapezoidal intuitionistic fuzzy number.Definition 4 (Atanassov and Gargov, [25]).
LetD[0,1] be the set of all closed subintervals of the interval [0,1]. An interval valued intuitionistic fuzzy set on a set X≠ϕ is an expression given by A=x,μA(x),νA(x):x∈X, where μA:X→D[0,1],νA:X→D[0,1] with the condition 0<supxμA(x)+supxνA(x)≤1.The intervalsμA(x) and νA(x) denote, respectively, the degree of belongingness and nonbelongingness of the element x to the set A. Thus for each x∈X, μA(x) and νA(x) are closed intervals whose lower and upper end points are, respectively, denoted by μAL(x), μAU(x) and νAL(x), μAU(x). We denote A=x,μALx,μAUx,νALx,νAUx:x∈X, where 0<μA(x)+νA(x)≤1.For each elementx∈X, we can compute the unknown degree (hesitance degree) of belongingness πA(x) to A as πA(x)=1-μA(x)-νA(x)=[1-μAU(x)-νAU(x),1-μAL(x)-νAL(x)]. An intuitionistic fuzzy interval number (IFIN) is denoted by A=(a,b,c,d) for convenience.Definition 5.
Letα,β∈(0,1]. An α,β-cut of a trapezoidal intuitionistic fuzzy number, denoted by AI(α,β), is defined as AI(α,β)=μAIα,νAIβ, where μAIα and νAIβ are α-cut and β-cut of membership and nonmembership trapezoidal fuzzy numbers.
Theα,β-cut of trapezoidal intuitionistic fuzzy number AI=(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′ is given by AIα,β=[μAILα,αμAIU],[νAILβ,νAIUβ] = (a1μ+αa2μ-a1μ,a4μ-αa4μ-a3μ, a1ν′+βa2ν′-a1ν′,a4ν′-βa4ν′-a3ν′), where μAILα and μAIUα represents the lower and upper end points of the α-cut of the membership function of AI and νAILβ and νAIUβ represent the lower and upper end points of the β-cut of the nonmembership function of AI, respectively.


## 3. Different Classes of Trapezoidal Intuitionistic Fuzzy Numbers

In this section the entire class of TrIFNs is partitioned into eight subclasses and further score functions using different concepts are defined in order to give total order on the entire class of TrIFNs and some theorems related to these concepts are established.In this paper,AI and BI always denote trapezoidal intuitionistic fuzzy number (TrIFN).A I = 〈 ( a 1 μ , a 2 μ , a 3 μ , a 4 μ ) , a 1 ν ′ , a 2 ν ′ , a 3 ν ′ , a 4 ν ′ 〉 with conditions that a1ν′≤a1μ and a2ν′≤a2μ, a3ν′≥a3μ and a4ν′≥a4μ and BI=(b1μ,b2μ,b3μ,b4μ),b1ν′,b2ν′,b3ν′,b4ν′ with the condition that b1ν′≤b1μ and b2ν′≤b2μ, b3ν′≥b3μ, and b4ν′≥b4μ unless otherwise specifically stated.


### 3.1. Imprecise Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, a new subclass of TrIFNs is introduced. The imprecise score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 6.
LetAI=(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′ and BI=〈b1μ,b2μ,b3μ,b4μ, b1ν′,b2ν′,b3ν′,b4ν′〉 be two TrIFNs.
A special subclassC1 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′. The imprecise relation on C1 is denoted as ⊑ and defined as follows: if AI,BI∈C1⊂TrIFN such that AI⊑BI then a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′.
IfAI⊏BI then one of the above inequalities becomes strict inequality.Note 1.
By Definition6, we note that any pair of members of C1 are related under ⊏.The score function which measures the preciseness is defined as follows.Definition 7.
LetAI=(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′ be a trapezoidal intuitionistic fuzzy number with a1μ,a2μ,a3μ,a4μ,a1ν′,a2ν′,a3ν′,a4ν′∈[0,1]. Then the imprecise score of a trapezoidal intuitionistic fuzzy number AI is defined by(3)J1AI=∫01μAILα-μAIUα4dα+∫01νAILβ-νAIUβ4dβ=∫01a1μ+αa2μ-a1μ4dα-∫01a4μ-αa4μ-a3μ4dα+∫01a1ν′+βa2ν′-a1ν′4dβ-∫01a4ν′-βa4ν′-a3ν′4dβ=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8.The proofs of the following propositions are immediate from the above definition and hence they are omitted.Proposition 8.
For any real numberr∈[0,1], J1(r)=0.Proposition 9.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J1(AI)=(a1μ+a2μ)-(a3μ+a4μ)/4.Proposition 10.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J1(AI)=(a1μ-a3μ)+a1ν′-a3ν′/4.Proposition 11.
LetAI=(a1μ,a2μ,a1ν′,a2ν′) be an interval valued intuitionistic fuzzy number. Then J1(AI)=(a1μ-a2μ)+a1ν′-a2ν′/2.Theorem 12.
LetAI and BI∈C1. If AI⊏BI then J1(AI)>J1(BI).Proof.
LetAI⊏BI. We claim that J1(AI)-J1(BI)>0.
By Definition6, we have(4)a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′.Now,(5)8J1AI-J1BI=a1μ-b1μ+a2μ-b2μ+b3μ-a3μ+b4μ-a4μ+a1ν′-b1ν′+a2ν′-b2ν′+b3ν′-a3ν′+b4ν′-a4ν′.From (4) it is very easy to see that all the terms in (5) are positive. Therefore their sum is also positive. From Definition 6, we know that at least one of the above inequalities in (4) becomes strict inequality and hence we get J1(AI)-J1(BI)>0.Theorem 13.
LetAI and BI∈C1 such that J1(AI)=J1(BI); then AI=BI.Proof.
LetJ1(AI)=J1(BI). We claim that AI=BI.
By Definition6, without loss of generality, we have(6)a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′.Now,(7)8J1AI-J1BI=a1μ-b1μ+a2μ-b2μ+b3μ-a3μ+b4μ-a4μ+a1ν′-b1ν′+a2ν′-b2ν′+b3ν′-a3ν′+b4ν′-a4ν′=0.Therefore from (6) and (7), it is clear that all the terms in (7) are positive and their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ, a2μ=b2μ, a3μ=b3μ, a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′, hence the proof.Note 2.
The imprecise score can be calculated to any TrIFN butJ1 gives total order in C1.Definition 14.
IfJ1(AI)>J1(BI)(J1(AI)<J1(BI)), then AI>BI(AI<BI).The following example explains the ranking procedure introduced in Definition14.Example 15.
LetAI=0.1,0.2,0.35,0.5, (0,0.15,0.45,0.6) and BI=〈(0.15,0.2,0.35,0.45), (0.1,0.1,0.5,0.55)〉∈C1. Now J1(AI)=-0.18125 and J1(BI)=-0.1625,J1(AI)<J1(BI).
HenceAI<BI.Example 16.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ+ϵ,a3μ-ϵ,a4μ-ϵ), a1ν′-ϵ,a2ν′-ϵ,a3ν′+ϵ,a4ν′+ϵ〉∈TrIFN but not in C1, where ϵ∈[0,1].
NowJ1(AI)=J1(BI)=(a1μ+a2μ)-(a3μ+a4μ)+(a1ν′+a2ν′)-a3ν′+a4ν′/8. But AI≠BI. Pictorial representation of this example is given in Figure 3.Figure 3
Pictorial representation of Example16.Example16 shows that J1 is not enough cover the entire class of TrIFNs. Therefore it is needed for us to define another score function that can cover some subclass of TrIFNs which cannot be covered by J1.


### 3.2. Nonvague Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, another subclass of TrIFNs is introduced using nonvague relation. The nonvague score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 17.
LetAI and BI be two TrIFNs.
A special subclassC2 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′. The nonvague relation on C2 is denoted as ⊒ and defined as follows: if AI,BI∈C2⊂TrIFN such that AI⊒BI then a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.
IfAI⊐BI then one of the above inequalities becomes strict inequality.Note 3.
By Definition17, we note that any pair of members of C2 are related under ⊐.The score function which measures the nonvagueness is defined as follows.Definition 18.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the nonvague score of a trapezoidal intuitionistic fuzzy number AI is defined by (8)J2AI=∫01μAILα-μAIUα4dα-∫01νAILβ-νAIUβ4dβ=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8.
The proofs of the following propositions are immediate from Definition18 and hence they are omitted.Proposition 19.
For any real numberr∈[0,1], J2(r)=0.Proposition 20.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J2(AI)=0.Proposition 21.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J2(AI)=(a1μ-a3μ)+-a1ν′+a3ν′/8.Proposition 22.
LetAI=a1μ,a2μ,a1ν′,a2ν′ be an interval valued intuitionistic fuzzy number. Then J2(AI)=(a1μ-a2μ)+-a1ν′+a2ν′/4.Theorem 23.
LetAI and BI∈C2. If AI⊐BI then J2(AI)>J2(BI).Proof.
Let us assumeAI⊐BI. We claim that J2(AI)-J2(BI)>0.
By Definition17, we have(9)a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.Now,(10)8J2AI-J2BI=a1μ-b1μ+a2μ-b2μ+b3μ-a3μ+b4μ-a4μ+b1ν′-a1ν′+b2ν′-a2ν′+a3ν′-b3ν′+a4ν′-b4ν′.From (9) it is very easy to see that all the terms in (10) are positive. Therefore their sum is also positive. From Definition 17, we know that at least one of the above inequalities in (9) becomes strict inequality and hence we get J2(AI)-J2(BI)>0.Theorem 24.
LetAI and BI∈C2. If J2(AI)=J2(BI) then AI=BI.Proof.
Let us assumeJ2(AI)=J2(BI). We claim that AI=BI.
By Definition17, without loss of generality, we have(11)a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.Now,(12)8J2AI-J2BI=a1μ-b1μ+a2μ-b2μ+b3μ-a3μ+b4μ-a4μ+b1ν′-a1ν′+b2ν′-a2ν′+a3ν′-b3ν′+a4ν′-b4ν′=0.Therefore from (11) and (12), it is clear that all the terms in (12) are positive; therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′. Hence AI=BI.Note 4.
The nonvague score can be calculated to any TrIFN. ButJ2 gives total order on C2.Definition 25.
IfJ2(AI)>J2(BI)(J2(AI)<J2(BI)), then AI>BI(AI<BI).Ranking relation defined above is explained in the following example.Example 26.
LetAI=(0.15,0.25,0.35,0.45), (0.1,0.2,0.4,0.5) and BI=〈(0.18,0.25,0.35,0.4), (0,0.15,0.45,0.55)〉∈C2. Now J2(AI)=0.025 and J2(BI)=0.06625,J2(BI)>J2(AI).
HenceBI>AI.Example27 shows the inefficiency of J1 in comparing any two arbitrary TrIFNs and the importance of defining new score function J2.Example 27.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ+ϵ,a3μ-ϵ,a4μ-ϵ), a1ν′-ϵ,a2ν′-ϵ,a3ν′+ϵ,a4ν′+ϵ〉∈TrIFN and ϵ∈[0,1].
Now(13)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8⟹AI=BI.But(14)J2AI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8+ϵ.Therefore J2(BI)>J2(AI); hence BI>AI.Example 28.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ+ϵ,a3μ+ϵ,a4μ+ϵ), (a1ν′+ϵ/2,a2ν′+ϵ/2,a3ν′+ϵ/2,a4ν′+ϵ/2)〉∈TrIFN and ϵ∈[0,1] but not in C2. Now(15)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8.But AI≠BI.
Pictorial representation of this example is given in Figure4.Figure 4
Pictorial representation of Example28.From Example28 it is easy to see that J1 and J2 are not enough to cover the entire class of TrIFNs. Therefore in the next subsection we are introducing a new score function which covers some more subclasses of TrIFNs that cannot be covered by J1 and J2.


### 3.3. Incomplete Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, another class of TrIFNs is introduced. The incomplete score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 29.
LetAI and BI be two TrIFNs.
A special subclassC3 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′. The incomplete relation on C3 is denoted as ⊆ and defined as follows: if AI,BI∈C3⊂TrIFN such that AI⊆BI then a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.
IfAI⊂BI then one of the above inequalities becomes strict inequality.Note 5.
By Definition29, we note that any pair of members of C3 are related under ⊂.The incomplete score function which measures the completeness is defined as follows.Definition 30.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the incomplete score of AI is defined by(16)J3AI=∫01-μAILα+μAIUα4dα+∫01νAILβ+νAIUβ4dβ=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8.The proofs of the following propositions are immediate from the above definition and hence they are omitted.Proposition 31.
For any real numberr∈[0,1], J3(r)=0.Proposition 32.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J3(AI)=0.Proposition 33.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then(17)J3AI=a1ν′+2a2ν′+a3ν′-a1μ+2a2μ+a3μ8.Proposition 34.
LetAI=(a1μ,a2μ,a1ν′,a2ν′) be an interval valued intuitionistic fuzzy number. Then(18)J3AI=a1ν′+a2ν′-a1μ+a2μ4.Theorem 35.
LetAI,BI∈C3. If AI⊂BI then J3(AI)>J3(BI).Proof.
Let us assume thatAI⊂BI. We claim that J3(AI)-J3(BI)>0.
By Definition29, we have(19)a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.Now,(20)8J3AI-J3BI=b1μ-a1μ+b2μ-a2μ+b3μ-a3μ+b4μ-a4μ+a1ν′-b1ν′+a2ν′-b2ν′+a3ν′-b3ν′+a4ν′-b4ν′.From (19) it is very easy to see that all the terms in (20) are positive. Therefore their sum is also positive. From Definition 29, we know that at least one of the above inequalities in (19) becomes strict inequality and hence we get J3(AI)-J3(BI)>0, hence the proof.Theorem 36.
LetAI,BI∈C3. If J3(AI)=J3(BI) then AI=BI.Proof.
Let us assumeJ3(AI)=J2(BI). We prove that AI=BI.
By Definition29, without loss of generality, we have(21)a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.Now,(22)8J3AI-J3BI=b1μ-a1μ+b2μ-a2μ+b3μ-a3μ+b4μ-a4μ+a1ν′-b1ν′+a2ν′-b2ν′+a3ν′-b3ν′+a4ν′-b4ν′=0.Therefore from (21) and (22), it is clear that all the terms in (22) are positive and therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′. Hence AI=BI.Note 6.
The incomplete score can be calculated to any TrIFN. ButJ3 gives total order on C3.Definition 37.
IfJ3(AI)>J3(BI)(J3(AI)<J3(BI)), then AI>BI(AI<BI).The following example is used to explain the ranking procedure defined in Definition37.Example 38.
LetAI=(0.15,0.3,0.4,0.45), (0.1,0.25,0.45,0.6) and BI=〈(0.2,0.4,0.45,0.55), (0,0.15,0.45,0.55)〉∈C3. Now J3(AI)=0.0125 and J3(BI)=-0.05625,J3(AI)>J3(BI).
HenceAI>BI.The inefficiency of the score functionsJ1 and J2 in discriminating any two arbitrary TrIFNs and the ability of J3 in comparing arbitrary TrIFNs is shown in the following example.Example 39.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ+ϵ,a3μ+ϵ,a4μ+ϵ), (a1ν′+ϵ/2,a2ν′+ϵ/2,a3ν′+ϵ/2,a4ν′+ϵ/2)〉∈TrIFN and ϵ∈[0,1].
Now(23)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8.But(24)J3AI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8+ϵ4⟹J3BI>J3AI.Hence BI>AI.The inefficiency of the score functionsJ1 to J3 in the task of comparing TrIFNs is explained in the following example.Example 40.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈a1μ+ϵ,a2μ+ϵ,a3μ+ϵ,a4μ+ϵ, a1ν′+ϵ,a2ν′+ϵ,a3ν′+ϵ,a4ν′+ϵ〉∈TrIFN and ϵ∈[0,1] but not in C3.
Now(25)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8.But AI≠BI.
Pictorial representation of this example is given in Figure5.Figure 5
Pictorial representation of Example40.Example40 shows that J1,J2, and J3 cannot be sufficient to cover the entire class of TrIFNs and the class of TrIFNs in the above example excite us to define new score function that can fill the subclass of TrIFNs which cannot be covered by J1, J2, and J3.


### 3.4. Accuracy Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, a new subclass of TrIFNs using accuracy relation is introduced and the accuracy score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 41.
LetAI and BI be two TrIFNs.
A special subclassC4 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′. The accuracy relation on C4 is denoted as ⪯ and defined as follows: if AI,BI∈C4⊂TrIFN such that AI⪯BI then a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′.
IfAI≺BI then one of the above inequalities becomes strict inequality.Note 7.
By Definition41, we note that any pair of members of C4 are related under ≺.The score function which measures the accuracy is defined as follows.Definition 42.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the accuracy score of AI is defined by(26)J4AI=∫01μAILα+μAIUα4dα+∫01νAILβ+νAIUβ4dβ=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8.The proofs of the following propositions are immediate from Definition42 and hence they are omitted.Proposition 43.
For any real numberr∈[0,1], J4(r)=r.Proposition 44.
For any trapezoidal fuzzy numberAI=(a1μ,a2μ,a3μ,a4μ), J4(AI)=(a1μ+a2μ+a3μ+a4μ)/4.Proposition 45.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J4(AI)=a1μ+2a2μ+a3μ+a1ν′+2a2ν′+a3ν′/8.Proposition 46.
LetAI=([a1μ,a2μ],[a1ν′,a2ν′]) be an interval valued intuitionistic fuzzy number. Then J4(AI)=(a1μ+a2μ)+a1ν′+a2ν′/4.Note 8.
The accuracy score can be calculated to any TrIFN. ButJ4 gives total order on C4 which is proved in Theorems 47 and 48.Theorem 47.
LetAI,BI∈C4. If AI≺BI then J4(AI)<J4(BI).Proof.
Let us assume thatAI≺BI. We claim that J4(BI)-J4(AI)>0.
By Definition41, we have(27)a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′.Now,(28)8J4BI-J4AI=b1μ-a1μ+b2μ-a2μ+b3μ-a3μ+b4μ-a4μ+b1ν′-a1ν′+b2ν′-a2ν′+b3ν′-a3ν′+b4ν′-a4ν′.From (27) it is very easy to see that all the terms in (28) are positive. Therefore their sum is also positive. From Definition 41, we know that at least one of the above inequalities in (27) becomes strict inequality and hence we get J4(BI)-J4(AI)>0.Theorem 48.
LetAI,BI∈C4 such that J4(AI)=J4(BI); then AI=BI.Proof.
Let us assumeJ4(AI)=J4(BI). We claim that AI=BI.
By Definition41, without loss of generality, we have(29)a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.Now,(30)8J4BI-J4AI=b1μ-a1μ+b2μ-a2μ+b3μ-a3μ+b4μ-a4μ+b1ν′-a1ν′+b2ν′-a2ν′+b3ν′-a3ν′+b4ν′-a4ν′=0.Therefore from (29) and (30), it is clear that all the terms in (30) are positive and therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′, a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′. Hence AI=BI.Definition 49.
IfJ4(AI)>J4(BI)(J4(AI)<J4(BI)) then AI>BI(AI<BI).Ranking procedure defined in Definition49 is illustrated in the following example.Example 50.
LetAI=(0.1,0.25,0.37,0.49), (0.1,0.2,0.4,0.6) and BI=〈(0.2,0.37,0.47,0.6), (0.15,0.32,0.52,0.7)〉∈C4. Now J4(AI)=0.31375 and J4(BI)=0.41625,J4(AI)<J4(BI).
HenceAI<BI.The relative importance of the score functionJ4 in discriminating two different TrIFNs when J1,J2, and J3 fail to discriminate them is explained by Example 51.Example 51.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈a1μ+ϵ,a2μ+ϵ,a3μ+ϵ,a4μ+ϵ, a1ν′+ϵ,a2ν′+ϵ,a3ν′+ϵ,a4ν′+ϵ〉∈TrIFN, and ϵ∈[0,1].
Now(31)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8.But(32)J4AI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8+ϵ⟹J4AI<J4BI.Hence AI<BI.Example 52.
LetAI=a1μ,a2μ,a3μ,a4μ,a1ν′,a2ν′,a3ν′,a4ν′ and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ+ϵ,a4μ-ϵ), (a1ν′+ϵ,a2ν′-ϵ,a3ν′+ϵ,a4ν′-ϵ)〉∈TrIFN and ϵ∈[0,1] but not in C4.
Now(33)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8.But AI≠BI.
Pictorial representation of this example is given in Figure6.Figure 6
Pictorial representation of Example52.Example52 shows that even J1,J2,J3, and J4 altogether are not enough to cover the entire class of TrIFNs; the class of TrIFNs in the above example excite us to define new score function that can cover the class of TrIFNs which cannot be covered by J1,J2,J3, and J4.


### 3.5. Spread Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, another subclass of TrIFNs is introduced. The spread score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 53.
LetAI and BI be two TrIFNs.
A special subclassC5 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′. The preference relation on C5 is denoted as ⊴ and defined as follows: if AI,BI∈C5⊂TrIFN such that AI⊴BI then a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′. If AI⊲BI then one of the above inequalities becomes strict inequality.Note 9.
By Definition53, we note that any pair of members of C5 are related under ⊲.The score function which measures the spread is defined as follows.Definition 54.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the spread score of AI is defined by(34)J5AI=a1μ+a3μ-a2μ+a4μ-a2ν′+a4ν′+a1ν′+a3ν′8.The proofs of the following propositions are immediate from Definition54 and hence they are omitted.Proposition 55.
For any real numberr∈[0,1], J5(r)=0.Proposition 56.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J5(AI)=(a1μ+a3μ)-a2μ+a4μ/4.Proposition 57.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J5(AI)=-a3μ-a1μ-a3ν′-a1ν′/8.Proposition 58.
LetAI=([a1μ,a2μ],[a1ν′,a2ν′]) be an interval valued intuitionistic fuzzy number. Then J5(AI)=0.Theorem 59.
LetAI,BI∈C5. If AI⊲BI then J5(AI)>J5(BI).Proof.
Let us assume thatAI⊲BI. We claim that J5(AI)-J5(BI)>0.
By Definition53, we have(35)a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′.Now,(36)8J5AI-J5BI=a1μ-b1μ+b2μ-a2μ+a3μ-b3μ+b4μ-a4μ+a1ν′-b1ν′+b2ν′-a2ν′+a3ν′-b3ν′+b4ν′-a4ν′.From (35) it is very easy to see that all the terms in (36) are positive. Therefore their sum is also positive. From Definition 53, we know that at least one of the above inequalities in (35) becomes strict inequality and hence we get J5(AI)-J5(BI)>0.Theorem 60.
LetAI,BI∈C5. If J5(AI)=J5(BI) then AI=BI.Proof.
Let us assumeJ5(AI)=J5(BI). We claim that AI=BI.
By Definition53, without loss of generality, we have(37)a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′.Now,(38)8J5AI-J5BI=a1μ-b1μ+b2μ-a2μ+a3μ-b3μ+b4μ-a4μ+a1ν′-b1ν′+b2ν′-a2ν′+a3ν′-b3ν′+b4ν′-a4ν′=0.Therefore from (37) and (38), it is clear that all the terms in (38) are positive and therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′. Hence AI=BI.Note 10.
The spread score can be calculated to any TrIFN. ButJ5 gives total order on C5 which is seen from Theorems 59 and 60.Definition 61.
IfJ5(AI)>J5(BI)(J5(AI)<J5(BI)) then AI>BI(AI<BI).The following example is used to explain the ranking procedure defined onC5.Example 62.
LetAI=(0.15,0.23,0.31,0.45), (0.1,0.16,0.35,0.5) and BI=〈(0.1,0.25,0.25,0.5), (0,0.2,0.3,0.6)〉∈C5. Now J5(AI)=-0.05375 and J5(BI)=-0.1125,J5(AI)>J5(BI).
HenceAI>BI.Example 63.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ+ϵ,a4μ-ϵ), (a1ν′+ϵ,a2ν′-ϵ,a3ν′+ϵ,a4ν′-ϵ)〉∈TrIFN and ϵ∈[0,1].
Now(39)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8.But(40)J5AI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8,J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8+ϵ⟹J5BI>J5AI.Hence BI>AI. In this example the importance of J5 in ranking arbitrary TrIFNs is shown.The inability ofJ5 in comparing any two TrIFNs is shown in the following example.Example 64.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ+ϵ,a4μ-ϵ), a1ν′-ϵ,a2ν′+ϵ,a3ν′-ϵ,a4ν′+ϵ〉∈TrIFN and ϵ∈[0,1] but not in C5.
Now(41)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J5AI=J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8.But AI≠BI.
Pictorial representation of this example is given in Figure7.Figure 7
Pictorial representation of Example64.Example64 shows that all the above defined scores are not enough to cover the entire class of TrIFNs; therefore we are introducing another score function in the next subsection.


### 3.6. Nonaccuracy Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, a new subclass of TrIFNs is introduced. The nonaccuracy score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 65.
LetAI and BI be two TrIFNs.
A special subclassC6 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′. The exact relation on C6 is denoted as ⊵ and defined as follows: if AI,BI∈C6⊂TrIFN such that AI⊵BI then a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′. If AI⊳BI then one of the above inequalities becomes strict inequality.Note 11.
By Definition65, we note that any pair of members of C6 are related under ⊳.The score function which measures the nonaccuracy is defined as follows.Definition 66.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the nonaccuracy score of AI is defined by(42)J6AI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8.The proofs of the following propositions are easy and hence they are omitted.Proposition 67.
For any real numberr∈[0,1], J6(r)=0.Proposition 68.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J6(AI)=0.Proposition 69.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J6(AI)=a1ν′-a3ν′-a1μ-a3μ/8.Proposition 70.
LetAI=([a1μ,a2μ],[a1ν′,a2ν′]) be an interval valued intuitionistic fuzzy number. Then J6(AI)=0.Note 12.
The nonaccuracy score can be calculated to any TrIFN. ButJ6 gives total order on C6 which is proved in Theorems 71 and 72.Theorem 71.
LetAI,BI∈C6. If AI⊳BI then J6(AI)<J6(BI).Proof.
Let us assume thatAI⊳BI. We claim that J6(BI)-J6(AI)>0.
By Definition65, we have(43)a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′and at least one of these inequalities becomes strict.
Now,(44)8J6BI-J6AI=a1μ-b1μ+b2μ-a2μ+a3μ-b3μ+b4μ-a4μ+b1ν′-a1ν′+a2ν′-b2ν′+b3ν′-a3ν′+a4ν′-b4ν′.From (43) it is very easy to see that all the terms in (44) are positive and therefore their sum is also positive. From Definition 65, we know that at least one of the above inequalities in (43) becomes strict inequality and hence we get J6(BI)-J6(AI)>0.Theorem 72.
LetAI,BI∈C6. If J6(AI)=J6(BI) then AI=BI.Proof.
Let us assumeJ6(AI)=J6(BI). We claim that AI=BI.
By Definition65, without loss of generality, we have(45)a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′.Now,(46)8J6BI-J6AI=a1μ-b1μ+b2μ-a2μ+a3μ-b3μ+b4μ-a4μ+b1ν′-a1ν′+a2ν′-b2ν′+b3ν′-a3ν′+a4ν′-b4ν′=0.Therefore from (45) and (46), it is clear that all the terms in (46) are positive and therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′. Hence AI=BI.Definition 73.
IfJ6(AI)>J6(BI)(J6(AI)<J6(BI)), then AI>BI(AI<BI).Ranking procedure introduced in Definition73 is explained in Example 74.Example 74.
LetAI=(0.23,0.27,0.34,0.51), (0.17,0.23,0.4,0.6) and BI=〈(0.2,0.3,0.3,0.55), (0.2,0.2,0.5,0.55)〉∈C6. Now J6(AI)=-0.00625 and J6(BI)=0.0375,J6(AI)<J6(BI).
HenceAI<BI.The efficiency ofJ6 in ranking TrIFNs is shown in the following example.Example 75.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ+ϵ,a4μ-ϵ), (a1ν′-ϵ,a2ν′+ϵ,a3ν′-ϵ,a4ν′+ϵ)〉∈TrIFN and ϵ∈[0,1].
Now(47)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J5AI=J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8.But(48)J6AI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8,J6BI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8+ϵ⟹J6AI<J6BI.Hence AI<BI.Example 76.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ-ϵ,a4μ+ϵ), (a1ν′-ϵ,a2ν′+ϵ,a3ν′+ϵ,a4ν′-ϵ)〉∈TrIFN  and  ϵ∈[0,1] but not in C6, where ϵ∈[0,1]. Now(49)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J5AI=J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8.But(50)J6AI=J6BI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8.But AI≠BI.
Pictorial representation of this example is given in Figure8.Figure 8
Pictorial representation of Example76.Example76 shows that the score functions J1 to J6 are not sufficient to cover the entire class of TrIFNs and hence the class of TrIFNs in the above example excite us to define new score function that can cover a subclass of TrIFNs which cannot be covered by J1 to J6.


### 3.7. Left Area of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, another subclass of TrIFNs is introduced and the left area score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 77.
LetAI and BI be two TrIFNs.
A special subclassC7 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′. The nonaccurate relation on C7 is denoted as ≪_ and defined as follows: if AI,BI∈C7⊂TrIFN such that AI≪_BI then a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′.
IfAI≪BI then one of the above inequalities becomes strict inequality.Note 13.
By Definition77, we note that any pair of members of C7 are related under ≪.The score function which measures the left area is defined as follows.Definition 78.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the left area of AI is defined by(51)J7AI=a3ν′-a1ν′+a2ν′-a4ν′-a3μ-a1μ-a2μ-a4μ8.The proofs of the following propositions are immediate application of Definition78 and hence they are omitted.Proposition 79.
For any real numberr∈[0,1], J7(r)=0.Proposition 80.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J7(AI)=0.Proposition 81.
LetAI=a1μ,a2μ,a3μ,a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J7(AI)=a1μ+a3μ-2a2μ+2a2ν′-a1ν′-a3ν′/8.Proposition 82.
LetAI=([a1μ,a2μ],[a1ν′,a2ν′]) be an interval valued intuitionistic fuzzy number. Then J7(AI)=0.Theorem 83.
LetAI,BI∈C7. If AI≪BI then J7(AI)<J7(BI).Proof.
Let us assume thatAI≪BI.
We claim thatJ7(BI)-J7(AI)>0.
By Definition77, we have(52)a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′.Now,(53)8J7BI-J7AI=b1μ-a1μ+a2μ-b2μ+a3μ-b3μ+b4μ-a4μ+a1ν′-b1ν′+b2ν′-a2ν′+b3ν′-a3ν′+a4ν′-b4ν′.From (52) it is very easy to see that all the terms in (53) are positive. Therefore their sum is also positive. From Definition 77, we know that at least one of the above inequalities in (52) becomes strict inequality and hence we get J7(BI)-J7(AI)>0, hence the proof.Theorem 84.
LetAI,BI∈C7. If J7(AI)=J7(BI) then AI=BI.Proof.
Let us assumeJ7(AI)=J7(BI). We claim that AI=BI.
By Definition77, without loss of generality, we have(54)a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′.Now,(55)8J7BI-J7AI=b1μ-a1μ+a2μ-b2μ+a3μ-b3μ+b4μ-a4μ+a1ν′-b1ν′+b2ν′-a2ν′+b3ν′-a3ν′+a4ν′-b4ν′=0.Therefore from (54) and (55), it is clear that all the terms in (55) are positive and therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′. Hence AI=BI.Note 14.
The left area score can be calculated to any TrIFN. ButJ7 gives total order on C7 which is seen from Theorems 83 and 84.Definition 85.
IfJ7(AI)>J7(BI)(J7(AI)<J7(BI)) then AI>BI(AI<BI).Ranking relation defined above is explained in the following example.Example 86.
LetAI=(0.17,0.23,0.34,0.49), (0.1,0.15,0.4,0.6) and BI=〈(0.2,0.2,0.3,0.55), (0,0.18,0.47,0.58)〉∈C7. Now J7(AI)=-0.0075 and J7(BI)=0.04,J7(BI)>J7(AI).
HenceBI>AI.The relative importance of the score functionJ7 is explained in the following example.Example 87.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ-ϵ,a4μ+ϵ), (a1ν′-ϵ,a2ν′+ϵ,a3ν′+ϵ,a4ν′-ϵ)〉∈TrIFN and ϵ∈[0,1].
Now(56)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J5AI=J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8,J6AI=J6BI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8.But(57)J7AI=a3ν′-a1ν′+a2ν′-a4ν′-a3μ-a1μ-a2μ-a4μ8,J7BI=a3ν′-a1ν′+a2ν′-a4ν′-a3μ-a1μ-a2μ-a4μ8-ϵ⟹J7AI>J7BI.Hence AI>BI.Example 88.
LetAI=a1μ,a2μ,a3μ,a4μ,a1ν′,a2ν′,a3ν′,a4ν′ and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ-ϵ,a4μ+ϵ), (a1ν′+ϵ,a2ν′-ϵ,a3ν′-ϵ,a4ν′+ϵ)〉∈TrIFN but not in C7, where ϵ∈[0,1].
Now(58)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J5AI=J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8,J6AI=J6BI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8,J7AI=J7BI=a3ν′-a1ν′+a2ν′-a4ν′-a3μ-a1μ-a2μ-a4μ8.But AI≠BI.
Pictorial representation of this example is given in Figure9.Figure 9
Pictorial representation of Example88.Example87 shows that all the above defined scores are not enough to cover the entire class of TrIFNs. Hence the new score function is introduced in the next subsection in order to give total ordering on the entire class of TrIFNs.


### 3.8. Right Area Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, a new subclass of TrIFNs is introduced and the right area score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 89.
LetAI and BI be two TrIFNs.
A special subclassC8 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′. The definite relation on C8 is denoted as ≫_ and defined as follows: if AI,BI∈C8⊂TrIFN such that AI≫_BI then a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′. If AI≫BI then one of the above inequalities becomes strict inequality.Note 15.
By Definition89, we note that any pair of members of C8 are related under ≫.The score function which measures the right area is defined as follows.Definition 90.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the right area score of AI is defined by(59)J8AI=a1μ-a3μ+a4μ-a2μ+a1ν′-a3ν′+a4ν′-a2ν′8.The proofs of the following propositions are immediate from Definition90 and hence they are omitted.Proposition 91.
For any real numberr∈[0,1], J8(r)=0.Proposition 92.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J8(AI)=(a1μ-a3μ)+(a4μ-a2μ)/4.Proposition 93.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J8(AI)=a1μ+a3μ-2a2μ+a1ν′+a3ν′-2a2ν′/8.Proposition 94.
LetAI=([a1μ,a2μ],[a1ν′,a2ν′]) be an interval valued intuitionistic fuzzy number. Then J8(AI)=0.Note 16.
The right area score can be calculated to any TrIFN. ButJ8 gives total order on C8 which is proved in the following theorems.Theorem 95.
LetAI,BI∈C8. If AI≫BI then J8(AI)<J8(BI).Proof.
Let us assume thatAI≫BI. We claim that J8(BI)-J8(AI)>0.
By Definition89, we have(60)a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′.Now,(61)8J8BI-J8AI=b1μ-a1μ+a2μ-b2μ+a3μ-b3μ+b4μ-a4μ+b1ν′-a1ν′+a2ν′-b2ν′+a3ν′-b3ν′+b4ν′-a4ν′.From (60) it is very easy to see that all the terms in (61) are positive. Therefore their sum is also positive. From Definition 89, we know that at least one of the above inequalities in (60) becomes strict inequality and hence we get J8(BI)-J8(AI)>0.Theorem 96.
LetAI=a1μ,a2μ,a3μ,a4μ,a1ν′,a2ν′,a3ν′,a4ν′ and BI=〈(b1μ,b2μ,b3μ,b4μ), b1ν′,b2ν′,b3ν′,b4ν′〉∈C8 such that J8(AI)=J8(BI); then AI=BI.Proof.
Let us assumeJ8(AI)=J8(BI). We claim that AI=BI.
By Definition89, without loss of generality, we have(62)a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′.Now,(63)8J1BI-J1AI=b1μ-a1μ+a2μ-b2μ+a3μ-b3μ+b4μ-a4μ+b1ν′-a1ν′+a2ν′-b2ν′+a3ν′-b3ν′+b4ν′-a4ν′=0.Therefore from (62) and (63), it is clear that all the terms in (63) are positive and therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′, hence the proof.Definition 97.
IfJ8(AI)>J8(BI)(J8(AI)<J8(BI)) then AI>BI(AI<BI).Ranking procedure introduced in Definition97 is demonstrated in Example 98.Example 98.
LetAI=(0.1,0.2,0.3,0.45), (0,0.15,0.35,0.5) and BI=〈(0.15,0.15,0.2,0.45), (0.1,0.1,0.3,0.45)〉∈C8. Now J8(AI)=0.00625 and J8(BI)=0.05,J8(BI)>J8(AI).
HenceBI>AI.The relative importance of the score functionJ8 in ranking arbitrary TrIFNs is explained mathematically in the following example.Example 99.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ-ϵ,a4μ+ϵ), (a1ν′+ϵ,a2ν′-ϵ,a3ν′-ϵ,a4ν′+ϵ)〉∈TrIFN but not in C7, where ϵ∈[0,1].
Now(64)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J5AI=J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8,J6AI=J6BI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8,J7AI=J7BI=a3ν′-a1ν′+a2ν′-a4ν′-a3μ-a1μ-a2μ-a4μ8.But(65)J8AI=a1μ-a3μ+a4μ-a2μ+a1ν′-a3ν′+a4ν′-a2ν′8,J8BI=a1μ-a3μ+a4μ-a2μ+a1ν′-a3ν′+a4ν′-a2ν′8-ϵ⟹J8AI>J8BI.Hence AI>BI.


## 3.1. Imprecise Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, a new subclass of TrIFNs is introduced. The imprecise score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 6.
LetAI=(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′ and BI=〈b1μ,b2μ,b3μ,b4μ, b1ν′,b2ν′,b3ν′,b4ν′〉 be two TrIFNs.
A special subclassC1 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′. The imprecise relation on C1 is denoted as ⊑ and defined as follows: if AI,BI∈C1⊂TrIFN such that AI⊑BI then a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′.
IfAI⊏BI then one of the above inequalities becomes strict inequality.Note 1.
By Definition6, we note that any pair of members of C1 are related under ⊏.The score function which measures the preciseness is defined as follows.Definition 7.
LetAI=(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′ be a trapezoidal intuitionistic fuzzy number with a1μ,a2μ,a3μ,a4μ,a1ν′,a2ν′,a3ν′,a4ν′∈[0,1]. Then the imprecise score of a trapezoidal intuitionistic fuzzy number AI is defined by(3)J1AI=∫01μAILα-μAIUα4dα+∫01νAILβ-νAIUβ4dβ=∫01a1μ+αa2μ-a1μ4dα-∫01a4μ-αa4μ-a3μ4dα+∫01a1ν′+βa2ν′-a1ν′4dβ-∫01a4ν′-βa4ν′-a3ν′4dβ=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8.The proofs of the following propositions are immediate from the above definition and hence they are omitted.Proposition 8.
For any real numberr∈[0,1], J1(r)=0.Proposition 9.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J1(AI)=(a1μ+a2μ)-(a3μ+a4μ)/4.Proposition 10.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J1(AI)=(a1μ-a3μ)+a1ν′-a3ν′/4.Proposition 11.
LetAI=(a1μ,a2μ,a1ν′,a2ν′) be an interval valued intuitionistic fuzzy number. Then J1(AI)=(a1μ-a2μ)+a1ν′-a2ν′/2.Theorem 12.
LetAI and BI∈C1. If AI⊏BI then J1(AI)>J1(BI).Proof.
LetAI⊏BI. We claim that J1(AI)-J1(BI)>0.
By Definition6, we have(4)a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′.Now,(5)8J1AI-J1BI=a1μ-b1μ+a2μ-b2μ+b3μ-a3μ+b4μ-a4μ+a1ν′-b1ν′+a2ν′-b2ν′+b3ν′-a3ν′+b4ν′-a4ν′.From (4) it is very easy to see that all the terms in (5) are positive. Therefore their sum is also positive. From Definition 6, we know that at least one of the above inequalities in (4) becomes strict inequality and hence we get J1(AI)-J1(BI)>0.Theorem 13.
LetAI and BI∈C1 such that J1(AI)=J1(BI); then AI=BI.Proof.
LetJ1(AI)=J1(BI). We claim that AI=BI.
By Definition6, without loss of generality, we have(6)a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′.Now,(7)8J1AI-J1BI=a1μ-b1μ+a2μ-b2μ+b3μ-a3μ+b4μ-a4μ+a1ν′-b1ν′+a2ν′-b2ν′+b3ν′-a3ν′+b4ν′-a4ν′=0.Therefore from (6) and (7), it is clear that all the terms in (7) are positive and their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ, a2μ=b2μ, a3μ=b3μ, a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′, hence the proof.Note 2.
The imprecise score can be calculated to any TrIFN butJ1 gives total order in C1.Definition 14.
IfJ1(AI)>J1(BI)(J1(AI)<J1(BI)), then AI>BI(AI<BI).The following example explains the ranking procedure introduced in Definition14.Example 15.
LetAI=0.1,0.2,0.35,0.5, (0,0.15,0.45,0.6) and BI=〈(0.15,0.2,0.35,0.45), (0.1,0.1,0.5,0.55)〉∈C1. Now J1(AI)=-0.18125 and J1(BI)=-0.1625,J1(AI)<J1(BI).
HenceAI<BI.Example 16.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ+ϵ,a3μ-ϵ,a4μ-ϵ), a1ν′-ϵ,a2ν′-ϵ,a3ν′+ϵ,a4ν′+ϵ〉∈TrIFN but not in C1, where ϵ∈[0,1].
NowJ1(AI)=J1(BI)=(a1μ+a2μ)-(a3μ+a4μ)+(a1ν′+a2ν′)-a3ν′+a4ν′/8. But AI≠BI. Pictorial representation of this example is given in Figure 3.Figure 3
Pictorial representation of Example16.Example16 shows that J1 is not enough cover the entire class of TrIFNs. Therefore it is needed for us to define another score function that can cover some subclass of TrIFNs which cannot be covered by J1.


## 3.2. Nonvague Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, another subclass of TrIFNs is introduced using nonvague relation. The nonvague score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 17.
LetAI and BI be two TrIFNs.
A special subclassC2 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′. The nonvague relation on C2 is denoted as ⊒ and defined as follows: if AI,BI∈C2⊂TrIFN such that AI⊒BI then a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.
IfAI⊐BI then one of the above inequalities becomes strict inequality.Note 3.
By Definition17, we note that any pair of members of C2 are related under ⊐.The score function which measures the nonvagueness is defined as follows.Definition 18.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the nonvague score of a trapezoidal intuitionistic fuzzy number AI is defined by (8)J2AI=∫01μAILα-μAIUα4dα-∫01νAILβ-νAIUβ4dβ=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8.
The proofs of the following propositions are immediate from Definition18 and hence they are omitted.Proposition 19.
For any real numberr∈[0,1], J2(r)=0.Proposition 20.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J2(AI)=0.Proposition 21.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J2(AI)=(a1μ-a3μ)+-a1ν′+a3ν′/8.Proposition 22.
LetAI=a1μ,a2μ,a1ν′,a2ν′ be an interval valued intuitionistic fuzzy number. Then J2(AI)=(a1μ-a2μ)+-a1ν′+a2ν′/4.Theorem 23.
LetAI and BI∈C2. If AI⊐BI then J2(AI)>J2(BI).Proof.
Let us assumeAI⊐BI. We claim that J2(AI)-J2(BI)>0.
By Definition17, we have(9)a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.Now,(10)8J2AI-J2BI=a1μ-b1μ+a2μ-b2μ+b3μ-a3μ+b4μ-a4μ+b1ν′-a1ν′+b2ν′-a2ν′+a3ν′-b3ν′+a4ν′-b4ν′.From (9) it is very easy to see that all the terms in (10) are positive. Therefore their sum is also positive. From Definition 17, we know that at least one of the above inequalities in (9) becomes strict inequality and hence we get J2(AI)-J2(BI)>0.Theorem 24.
LetAI and BI∈C2. If J2(AI)=J2(BI) then AI=BI.Proof.
Let us assumeJ2(AI)=J2(BI). We claim that AI=BI.
By Definition17, without loss of generality, we have(11)a1μ≥b1μ,a2μ≥b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.Now,(12)8J2AI-J2BI=a1μ-b1μ+a2μ-b2μ+b3μ-a3μ+b4μ-a4μ+b1ν′-a1ν′+b2ν′-a2ν′+a3ν′-b3ν′+a4ν′-b4ν′=0.Therefore from (11) and (12), it is clear that all the terms in (12) are positive; therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′. Hence AI=BI.Note 4.
The nonvague score can be calculated to any TrIFN. ButJ2 gives total order on C2.Definition 25.
IfJ2(AI)>J2(BI)(J2(AI)<J2(BI)), then AI>BI(AI<BI).Ranking relation defined above is explained in the following example.Example 26.
LetAI=(0.15,0.25,0.35,0.45), (0.1,0.2,0.4,0.5) and BI=〈(0.18,0.25,0.35,0.4), (0,0.15,0.45,0.55)〉∈C2. Now J2(AI)=0.025 and J2(BI)=0.06625,J2(BI)>J2(AI).
HenceBI>AI.Example27 shows the inefficiency of J1 in comparing any two arbitrary TrIFNs and the importance of defining new score function J2.Example 27.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ+ϵ,a3μ-ϵ,a4μ-ϵ), a1ν′-ϵ,a2ν′-ϵ,a3ν′+ϵ,a4ν′+ϵ〉∈TrIFN and ϵ∈[0,1].
Now(13)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8⟹AI=BI.But(14)J2AI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8+ϵ.Therefore J2(BI)>J2(AI); hence BI>AI.Example 28.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ+ϵ,a3μ+ϵ,a4μ+ϵ), (a1ν′+ϵ/2,a2ν′+ϵ/2,a3ν′+ϵ/2,a4ν′+ϵ/2)〉∈TrIFN and ϵ∈[0,1] but not in C2. Now(15)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8.But AI≠BI.
Pictorial representation of this example is given in Figure4.Figure 4
Pictorial representation of Example28.From Example28 it is easy to see that J1 and J2 are not enough to cover the entire class of TrIFNs. Therefore in the next subsection we are introducing a new score function which covers some more subclasses of TrIFNs that cannot be covered by J1 and J2.


## 3.3. Incomplete Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, another class of TrIFNs is introduced. The incomplete score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 29.
LetAI and BI be two TrIFNs.
A special subclassC3 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′. The incomplete relation on C3 is denoted as ⊆ and defined as follows: if AI,BI∈C3⊂TrIFN such that AI⊆BI then a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.
IfAI⊂BI then one of the above inequalities becomes strict inequality.Note 5.
By Definition29, we note that any pair of members of C3 are related under ⊂.The incomplete score function which measures the completeness is defined as follows.Definition 30.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the incomplete score of AI is defined by(16)J3AI=∫01-μAILα+μAIUα4dα+∫01νAILβ+νAIUβ4dβ=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8.The proofs of the following propositions are immediate from the above definition and hence they are omitted.Proposition 31.
For any real numberr∈[0,1], J3(r)=0.Proposition 32.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J3(AI)=0.Proposition 33.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then(17)J3AI=a1ν′+2a2ν′+a3ν′-a1μ+2a2μ+a3μ8.Proposition 34.
LetAI=(a1μ,a2μ,a1ν′,a2ν′) be an interval valued intuitionistic fuzzy number. Then(18)J3AI=a1ν′+a2ν′-a1μ+a2μ4.Theorem 35.
LetAI,BI∈C3. If AI⊂BI then J3(AI)>J3(BI).Proof.
Let us assume thatAI⊂BI. We claim that J3(AI)-J3(BI)>0.
By Definition29, we have(19)a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.Now,(20)8J3AI-J3BI=b1μ-a1μ+b2μ-a2μ+b3μ-a3μ+b4μ-a4μ+a1ν′-b1ν′+a2ν′-b2ν′+a3ν′-b3ν′+a4ν′-b4ν′.From (19) it is very easy to see that all the terms in (20) are positive. Therefore their sum is also positive. From Definition 29, we know that at least one of the above inequalities in (19) becomes strict inequality and hence we get J3(AI)-J3(BI)>0, hence the proof.Theorem 36.
LetAI,BI∈C3. If J3(AI)=J3(BI) then AI=BI.Proof.
Let us assumeJ3(AI)=J2(BI). We prove that AI=BI.
By Definition29, without loss of generality, we have(21)a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.Now,(22)8J3AI-J3BI=b1μ-a1μ+b2μ-a2μ+b3μ-a3μ+b4μ-a4μ+a1ν′-b1ν′+a2ν′-b2ν′+a3ν′-b3ν′+a4ν′-b4ν′=0.Therefore from (21) and (22), it is clear that all the terms in (22) are positive and therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′. Hence AI=BI.Note 6.
The incomplete score can be calculated to any TrIFN. ButJ3 gives total order on C3.Definition 37.
IfJ3(AI)>J3(BI)(J3(AI)<J3(BI)), then AI>BI(AI<BI).The following example is used to explain the ranking procedure defined in Definition37.Example 38.
LetAI=(0.15,0.3,0.4,0.45), (0.1,0.25,0.45,0.6) and BI=〈(0.2,0.4,0.45,0.55), (0,0.15,0.45,0.55)〉∈C3. Now J3(AI)=0.0125 and J3(BI)=-0.05625,J3(AI)>J3(BI).
HenceAI>BI.The inefficiency of the score functionsJ1 and J2 in discriminating any two arbitrary TrIFNs and the ability of J3 in comparing arbitrary TrIFNs is shown in the following example.Example 39.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ+ϵ,a3μ+ϵ,a4μ+ϵ), (a1ν′+ϵ/2,a2ν′+ϵ/2,a3ν′+ϵ/2,a4ν′+ϵ/2)〉∈TrIFN and ϵ∈[0,1].
Now(23)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8.But(24)J3AI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8+ϵ4⟹J3BI>J3AI.Hence BI>AI.The inefficiency of the score functionsJ1 to J3 in the task of comparing TrIFNs is explained in the following example.Example 40.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈a1μ+ϵ,a2μ+ϵ,a3μ+ϵ,a4μ+ϵ, a1ν′+ϵ,a2ν′+ϵ,a3ν′+ϵ,a4ν′+ϵ〉∈TrIFN and ϵ∈[0,1] but not in C3.
Now(25)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8.But AI≠BI.
Pictorial representation of this example is given in Figure5.Figure 5
Pictorial representation of Example40.Example40 shows that J1,J2, and J3 cannot be sufficient to cover the entire class of TrIFNs and the class of TrIFNs in the above example excite us to define new score function that can fill the subclass of TrIFNs which cannot be covered by J1, J2, and J3.


## 3.4. Accuracy Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, a new subclass of TrIFNs using accuracy relation is introduced and the accuracy score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 41.
LetAI and BI be two TrIFNs.
A special subclassC4 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′. The accuracy relation on C4 is denoted as ⪯ and defined as follows: if AI,BI∈C4⊂TrIFN such that AI⪯BI then a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′.
IfAI≺BI then one of the above inequalities becomes strict inequality.Note 7.
By Definition41, we note that any pair of members of C4 are related under ≺.The score function which measures the accuracy is defined as follows.Definition 42.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the accuracy score of AI is defined by(26)J4AI=∫01μAILα+μAIUα4dα+∫01νAILβ+νAIUβ4dβ=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8.The proofs of the following propositions are immediate from Definition42 and hence they are omitted.Proposition 43.
For any real numberr∈[0,1], J4(r)=r.Proposition 44.
For any trapezoidal fuzzy numberAI=(a1μ,a2μ,a3μ,a4μ), J4(AI)=(a1μ+a2μ+a3μ+a4μ)/4.Proposition 45.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J4(AI)=a1μ+2a2μ+a3μ+a1ν′+2a2ν′+a3ν′/8.Proposition 46.
LetAI=([a1μ,a2μ],[a1ν′,a2ν′]) be an interval valued intuitionistic fuzzy number. Then J4(AI)=(a1μ+a2μ)+a1ν′+a2ν′/4.Note 8.
The accuracy score can be calculated to any TrIFN. ButJ4 gives total order on C4 which is proved in Theorems 47 and 48.Theorem 47.
LetAI,BI∈C4. If AI≺BI then J4(AI)<J4(BI).Proof.
Let us assume thatAI≺BI. We claim that J4(BI)-J4(AI)>0.
By Definition41, we have(27)a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≤b4ν′.Now,(28)8J4BI-J4AI=b1μ-a1μ+b2μ-a2μ+b3μ-a3μ+b4μ-a4μ+b1ν′-a1ν′+b2ν′-a2ν′+b3ν′-a3ν′+b4ν′-a4ν′.From (27) it is very easy to see that all the terms in (28) are positive. Therefore their sum is also positive. From Definition 41, we know that at least one of the above inequalities in (27) becomes strict inequality and hence we get J4(BI)-J4(AI)>0.Theorem 48.
LetAI,BI∈C4 such that J4(AI)=J4(BI); then AI=BI.Proof.
Let us assumeJ4(AI)=J4(BI). We claim that AI=BI.
By Definition41, without loss of generality, we have(29)a1μ≤b1μ,a2μ≤b2μ,a3μ≤b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≥b4ν′.Now,(30)8J4BI-J4AI=b1μ-a1μ+b2μ-a2μ+b3μ-a3μ+b4μ-a4μ+b1ν′-a1ν′+b2ν′-a2ν′+b3ν′-a3ν′+b4ν′-a4ν′=0.Therefore from (29) and (30), it is clear that all the terms in (30) are positive and therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′, a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′. Hence AI=BI.Definition 49.
IfJ4(AI)>J4(BI)(J4(AI)<J4(BI)) then AI>BI(AI<BI).Ranking procedure defined in Definition49 is illustrated in the following example.Example 50.
LetAI=(0.1,0.25,0.37,0.49), (0.1,0.2,0.4,0.6) and BI=〈(0.2,0.37,0.47,0.6), (0.15,0.32,0.52,0.7)〉∈C4. Now J4(AI)=0.31375 and J4(BI)=0.41625,J4(AI)<J4(BI).
HenceAI<BI.The relative importance of the score functionJ4 in discriminating two different TrIFNs when J1,J2, and J3 fail to discriminate them is explained by Example 51.Example 51.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈a1μ+ϵ,a2μ+ϵ,a3μ+ϵ,a4μ+ϵ, a1ν′+ϵ,a2ν′+ϵ,a3ν′+ϵ,a4ν′+ϵ〉∈TrIFN, and ϵ∈[0,1].
Now(31)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8.But(32)J4AI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8+ϵ⟹J4AI<J4BI.Hence AI<BI.Example 52.
LetAI=a1μ,a2μ,a3μ,a4μ,a1ν′,a2ν′,a3ν′,a4ν′ and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ+ϵ,a4μ-ϵ), (a1ν′+ϵ,a2ν′-ϵ,a3ν′+ϵ,a4ν′-ϵ)〉∈TrIFN and ϵ∈[0,1] but not in C4.
Now(33)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8.But AI≠BI.
Pictorial representation of this example is given in Figure6.Figure 6
Pictorial representation of Example52.Example52 shows that even J1,J2,J3, and J4 altogether are not enough to cover the entire class of TrIFNs; the class of TrIFNs in the above example excite us to define new score function that can cover the class of TrIFNs which cannot be covered by J1,J2,J3, and J4.


## 3.5. Spread Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, another subclass of TrIFNs is introduced. The spread score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 53.
LetAI and BI be two TrIFNs.
A special subclassC5 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′. The preference relation on C5 is denoted as ⊴ and defined as follows: if AI,BI∈C5⊂TrIFN such that AI⊴BI then a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′. If AI⊲BI then one of the above inequalities becomes strict inequality.Note 9.
By Definition53, we note that any pair of members of C5 are related under ⊲.The score function which measures the spread is defined as follows.Definition 54.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the spread score of AI is defined by(34)J5AI=a1μ+a3μ-a2μ+a4μ-a2ν′+a4ν′+a1ν′+a3ν′8.The proofs of the following propositions are immediate from Definition54 and hence they are omitted.Proposition 55.
For any real numberr∈[0,1], J5(r)=0.Proposition 56.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J5(AI)=(a1μ+a3μ)-a2μ+a4μ/4.Proposition 57.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J5(AI)=-a3μ-a1μ-a3ν′-a1ν′/8.Proposition 58.
LetAI=([a1μ,a2μ],[a1ν′,a2ν′]) be an interval valued intuitionistic fuzzy number. Then J5(AI)=0.Theorem 59.
LetAI,BI∈C5. If AI⊲BI then J5(AI)>J5(BI).Proof.
Let us assume thatAI⊲BI. We claim that J5(AI)-J5(BI)>0.
By Definition53, we have(35)a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′.Now,(36)8J5AI-J5BI=a1μ-b1μ+b2μ-a2μ+a3μ-b3μ+b4μ-a4μ+a1ν′-b1ν′+b2ν′-a2ν′+a3ν′-b3ν′+b4ν′-a4ν′.From (35) it is very easy to see that all the terms in (36) are positive. Therefore their sum is also positive. From Definition 53, we know that at least one of the above inequalities in (35) becomes strict inequality and hence we get J5(AI)-J5(BI)>0.Theorem 60.
LetAI,BI∈C5. If J5(AI)=J5(BI) then AI=BI.Proof.
Let us assumeJ5(AI)=J5(BI). We claim that AI=BI.
By Definition53, without loss of generality, we have(37)a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′.Now,(38)8J5AI-J5BI=a1μ-b1μ+b2μ-a2μ+a3μ-b3μ+b4μ-a4μ+a1ν′-b1ν′+b2ν′-a2ν′+a3ν′-b3ν′+b4ν′-a4ν′=0.Therefore from (37) and (38), it is clear that all the terms in (38) are positive and therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′. Hence AI=BI.Note 10.
The spread score can be calculated to any TrIFN. ButJ5 gives total order on C5 which is seen from Theorems 59 and 60.Definition 61.
IfJ5(AI)>J5(BI)(J5(AI)<J5(BI)) then AI>BI(AI<BI).The following example is used to explain the ranking procedure defined onC5.Example 62.
LetAI=(0.15,0.23,0.31,0.45), (0.1,0.16,0.35,0.5) and BI=〈(0.1,0.25,0.25,0.5), (0,0.2,0.3,0.6)〉∈C5. Now J5(AI)=-0.05375 and J5(BI)=-0.1125,J5(AI)>J5(BI).
HenceAI>BI.Example 63.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ+ϵ,a4μ-ϵ), (a1ν′+ϵ,a2ν′-ϵ,a3ν′+ϵ,a4ν′-ϵ)〉∈TrIFN and ϵ∈[0,1].
Now(39)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8.But(40)J5AI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8,J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8+ϵ⟹J5BI>J5AI.Hence BI>AI. In this example the importance of J5 in ranking arbitrary TrIFNs is shown.The inability ofJ5 in comparing any two TrIFNs is shown in the following example.Example 64.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ+ϵ,a4μ-ϵ), a1ν′-ϵ,a2ν′+ϵ,a3ν′-ϵ,a4ν′+ϵ〉∈TrIFN and ϵ∈[0,1] but not in C5.
Now(41)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J5AI=J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8.But AI≠BI.
Pictorial representation of this example is given in Figure7.Figure 7
Pictorial representation of Example64.Example64 shows that all the above defined scores are not enough to cover the entire class of TrIFNs; therefore we are introducing another score function in the next subsection.


## 3.6. Nonaccuracy Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, a new subclass of TrIFNs is introduced. The nonaccuracy score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 65.
LetAI and BI be two TrIFNs.
A special subclassC6 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′. The exact relation on C6 is denoted as ⊵ and defined as follows: if AI,BI∈C6⊂TrIFN such that AI⊵BI then a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′. If AI⊳BI then one of the above inequalities becomes strict inequality.Note 11.
By Definition65, we note that any pair of members of C6 are related under ⊳.The score function which measures the nonaccuracy is defined as follows.Definition 66.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the nonaccuracy score of AI is defined by(42)J6AI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8.The proofs of the following propositions are easy and hence they are omitted.Proposition 67.
For any real numberr∈[0,1], J6(r)=0.Proposition 68.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J6(AI)=0.Proposition 69.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J6(AI)=a1ν′-a3ν′-a1μ-a3μ/8.Proposition 70.
LetAI=([a1μ,a2μ],[a1ν′,a2ν′]) be an interval valued intuitionistic fuzzy number. Then J6(AI)=0.Note 12.
The nonaccuracy score can be calculated to any TrIFN. ButJ6 gives total order on C6 which is proved in Theorems 71 and 72.Theorem 71.
LetAI,BI∈C6. If AI⊳BI then J6(AI)<J6(BI).Proof.
Let us assume thatAI⊳BI. We claim that J6(BI)-J6(AI)>0.
By Definition65, we have(43)a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′and at least one of these inequalities becomes strict.
Now,(44)8J6BI-J6AI=a1μ-b1μ+b2μ-a2μ+a3μ-b3μ+b4μ-a4μ+b1ν′-a1ν′+a2ν′-b2ν′+b3ν′-a3ν′+a4ν′-b4ν′.From (43) it is very easy to see that all the terms in (44) are positive and therefore their sum is also positive. From Definition 65, we know that at least one of the above inequalities in (43) becomes strict inequality and hence we get J6(BI)-J6(AI)>0.Theorem 72.
LetAI,BI∈C6. If J6(AI)=J6(BI) then AI=BI.Proof.
Let us assumeJ6(AI)=J6(BI). We claim that AI=BI.
By Definition65, without loss of generality, we have(45)a1μ≥b1μ,a2μ≤b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′.Now,(46)8J6BI-J6AI=a1μ-b1μ+b2μ-a2μ+a3μ-b3μ+b4μ-a4μ+b1ν′-a1ν′+a2ν′-b2ν′+b3ν′-a3ν′+a4ν′-b4ν′=0.Therefore from (45) and (46), it is clear that all the terms in (46) are positive and therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′. Hence AI=BI.Definition 73.
IfJ6(AI)>J6(BI)(J6(AI)<J6(BI)), then AI>BI(AI<BI).Ranking procedure introduced in Definition73 is explained in Example 74.Example 74.
LetAI=(0.23,0.27,0.34,0.51), (0.17,0.23,0.4,0.6) and BI=〈(0.2,0.3,0.3,0.55), (0.2,0.2,0.5,0.55)〉∈C6. Now J6(AI)=-0.00625 and J6(BI)=0.0375,J6(AI)<J6(BI).
HenceAI<BI.The efficiency ofJ6 in ranking TrIFNs is shown in the following example.Example 75.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ+ϵ,a4μ-ϵ), (a1ν′-ϵ,a2ν′+ϵ,a3ν′-ϵ,a4ν′+ϵ)〉∈TrIFN and ϵ∈[0,1].
Now(47)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J5AI=J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8.But(48)J6AI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8,J6BI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8+ϵ⟹J6AI<J6BI.Hence AI<BI.Example 76.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ-ϵ,a4μ+ϵ), (a1ν′-ϵ,a2ν′+ϵ,a3ν′+ϵ,a4ν′-ϵ)〉∈TrIFN  and  ϵ∈[0,1] but not in C6, where ϵ∈[0,1]. Now(49)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J5AI=J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8.But(50)J6AI=J6BI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8.But AI≠BI.
Pictorial representation of this example is given in Figure8.Figure 8
Pictorial representation of Example76.Example76 shows that the score functions J1 to J6 are not sufficient to cover the entire class of TrIFNs and hence the class of TrIFNs in the above example excite us to define new score function that can cover a subclass of TrIFNs which cannot be covered by J1 to J6.


## 3.7. Left Area of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, another subclass of TrIFNs is introduced and the left area score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 77.
LetAI and BI be two TrIFNs.
A special subclassC7 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′. The nonaccurate relation on C7 is denoted as ≪_ and defined as follows: if AI,BI∈C7⊂TrIFN such that AI≪_BI then a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′.
IfAI≪BI then one of the above inequalities becomes strict inequality.Note 13.
By Definition77, we note that any pair of members of C7 are related under ≪.The score function which measures the left area is defined as follows.Definition 78.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the left area of AI is defined by(51)J7AI=a3ν′-a1ν′+a2ν′-a4ν′-a3μ-a1μ-a2μ-a4μ8.The proofs of the following propositions are immediate application of Definition78 and hence they are omitted.Proposition 79.
For any real numberr∈[0,1], J7(r)=0.Proposition 80.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J7(AI)=0.Proposition 81.
LetAI=a1μ,a2μ,a3μ,a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J7(AI)=a1μ+a3μ-2a2μ+2a2ν′-a1ν′-a3ν′/8.Proposition 82.
LetAI=([a1μ,a2μ],[a1ν′,a2ν′]) be an interval valued intuitionistic fuzzy number. Then J7(AI)=0.Theorem 83.
LetAI,BI∈C7. If AI≪BI then J7(AI)<J7(BI).Proof.
Let us assume thatAI≪BI.
We claim thatJ7(BI)-J7(AI)>0.
By Definition77, we have(52)a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′.Now,(53)8J7BI-J7AI=b1μ-a1μ+a2μ-b2μ+a3μ-b3μ+b4μ-a4μ+a1ν′-b1ν′+b2ν′-a2ν′+b3ν′-a3ν′+a4ν′-b4ν′.From (52) it is very easy to see that all the terms in (53) are positive. Therefore their sum is also positive. From Definition 77, we know that at least one of the above inequalities in (52) becomes strict inequality and hence we get J7(BI)-J7(AI)>0, hence the proof.Theorem 84.
LetAI,BI∈C7. If J7(AI)=J7(BI) then AI=BI.Proof.
Let us assumeJ7(AI)=J7(BI). We claim that AI=BI.
By Definition77, without loss of generality, we have(54)a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≥b1ν′,a2ν′≤b2ν′,a3ν′≤b3ν′,a4ν′≥b4ν′.Now,(55)8J7BI-J7AI=b1μ-a1μ+a2μ-b2μ+a3μ-b3μ+b4μ-a4μ+a1ν′-b1ν′+b2ν′-a2ν′+b3ν′-a3ν′+a4ν′-b4ν′=0.Therefore from (54) and (55), it is clear that all the terms in (55) are positive and therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′. Hence AI=BI.Note 14.
The left area score can be calculated to any TrIFN. ButJ7 gives total order on C7 which is seen from Theorems 83 and 84.Definition 85.
IfJ7(AI)>J7(BI)(J7(AI)<J7(BI)) then AI>BI(AI<BI).Ranking relation defined above is explained in the following example.Example 86.
LetAI=(0.17,0.23,0.34,0.49), (0.1,0.15,0.4,0.6) and BI=〈(0.2,0.2,0.3,0.55), (0,0.18,0.47,0.58)〉∈C7. Now J7(AI)=-0.0075 and J7(BI)=0.04,J7(BI)>J7(AI).
HenceBI>AI.The relative importance of the score functionJ7 is explained in the following example.Example 87.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ-ϵ,a4μ+ϵ), (a1ν′-ϵ,a2ν′+ϵ,a3ν′+ϵ,a4ν′-ϵ)〉∈TrIFN and ϵ∈[0,1].
Now(56)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J5AI=J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8,J6AI=J6BI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8.But(57)J7AI=a3ν′-a1ν′+a2ν′-a4ν′-a3μ-a1μ-a2μ-a4μ8,J7BI=a3ν′-a1ν′+a2ν′-a4ν′-a3μ-a1μ-a2μ-a4μ8-ϵ⟹J7AI>J7BI.Hence AI>BI.Example 88.
LetAI=a1μ,a2μ,a3μ,a4μ,a1ν′,a2ν′,a3ν′,a4ν′ and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ-ϵ,a4μ+ϵ), (a1ν′+ϵ,a2ν′-ϵ,a3ν′-ϵ,a4ν′+ϵ)〉∈TrIFN but not in C7, where ϵ∈[0,1].
Now(58)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J5AI=J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8,J6AI=J6BI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8,J7AI=J7BI=a3ν′-a1ν′+a2ν′-a4ν′-a3μ-a1μ-a2μ-a4μ8.But AI≠BI.
Pictorial representation of this example is given in Figure9.Figure 9
Pictorial representation of Example88.Example87 shows that all the above defined scores are not enough to cover the entire class of TrIFNs. Hence the new score function is introduced in the next subsection in order to give total ordering on the entire class of TrIFNs.


## 3.8. Right Area Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, a new subclass of TrIFNs is introduced and the right area score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.Definition 89.
LetAI and BI be two TrIFNs.
A special subclassC8 of the set of TrIFNs consist of TrIFNs for which every pair of AI and BI are with a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′. The definite relation on C8 is denoted as ≫_ and defined as follows: if AI,BI∈C8⊂TrIFN such that AI≫_BI then a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ and a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′. If AI≫BI then one of the above inequalities becomes strict inequality.Note 15.
By Definition89, we note that any pair of members of C8 are related under ≫.The score function which measures the right area is defined as follows.Definition 90.
LetAI be a trapezoidal intuitionistic fuzzy number. Then the right area score of AI is defined by(59)J8AI=a1μ-a3μ+a4μ-a2μ+a1ν′-a3ν′+a4ν′-a2ν′8.The proofs of the following propositions are immediate from Definition90 and hence they are omitted.Proposition 91.
For any real numberr∈[0,1], J8(r)=0.Proposition 92.
IfAI=(a1μ,a2μ,a3μ,a4μ) is a trapezoidal fuzzy number, then J8(AI)=(a1μ-a3μ)+(a4μ-a2μ)/4.Proposition 93.
LetAI=(a1μ,a2μ,a3μ),a1ν′,a2ν′,a3ν′ be a triangular intuitionistic fuzzy number. Then J8(AI)=a1μ+a3μ-2a2μ+a1ν′+a3ν′-2a2ν′/8.Proposition 94.
LetAI=([a1μ,a2μ],[a1ν′,a2ν′]) be an interval valued intuitionistic fuzzy number. Then J8(AI)=0.Note 16.
The right area score can be calculated to any TrIFN. ButJ8 gives total order on C8 which is proved in the following theorems.Theorem 95.
LetAI,BI∈C8. If AI≫BI then J8(AI)<J8(BI).Proof.
Let us assume thatAI≫BI. We claim that J8(BI)-J8(AI)>0.
By Definition89, we have(60)a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′.Now,(61)8J8BI-J8AI=b1μ-a1μ+a2μ-b2μ+a3μ-b3μ+b4μ-a4μ+b1ν′-a1ν′+a2ν′-b2ν′+a3ν′-b3ν′+b4ν′-a4ν′.From (60) it is very easy to see that all the terms in (61) are positive. Therefore their sum is also positive. From Definition 89, we know that at least one of the above inequalities in (60) becomes strict inequality and hence we get J8(BI)-J8(AI)>0.Theorem 96.
LetAI=a1μ,a2μ,a3μ,a4μ,a1ν′,a2ν′,a3ν′,a4ν′ and BI=〈(b1μ,b2μ,b3μ,b4μ), b1ν′,b2ν′,b3ν′,b4ν′〉∈C8 such that J8(AI)=J8(BI); then AI=BI.Proof.
Let us assumeJ8(AI)=J8(BI). We claim that AI=BI.
By Definition89, without loss of generality, we have(62)a1μ≤b1μ,a2μ≥b2μ,a3μ≥b3μ,a4μ≤b4μ,a1ν′≤b1ν′,a2ν′≥b2ν′,a3ν′≥b3ν′,a4ν′≤b4ν′.Now,(63)8J1BI-J1AI=b1μ-a1μ+a2μ-b2μ+a3μ-b3μ+b4μ-a4μ+b1ν′-a1ν′+a2ν′-b2ν′+a3ν′-b3ν′+b4ν′-a4ν′=0.Therefore from (62) and (63), it is clear that all the terms in (63) are positive and therefore their sum gives zero only when each term is equal to zero. Hence a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′, hence the proof.Definition 97.
IfJ8(AI)>J8(BI)(J8(AI)<J8(BI)) then AI>BI(AI<BI).Ranking procedure introduced in Definition97 is demonstrated in Example 98.Example 98.
LetAI=(0.1,0.2,0.3,0.45), (0,0.15,0.35,0.5) and BI=〈(0.15,0.15,0.2,0.45), (0.1,0.1,0.3,0.45)〉∈C8. Now J8(AI)=0.00625 and J8(BI)=0.05,J8(BI)>J8(AI).
HenceBI>AI.The relative importance of the score functionJ8 in ranking arbitrary TrIFNs is explained mathematically in the following example.Example 99.
LetAI=〈(a1μ,a2μ,a3μ,a4μ),a1ν′,a2ν′,a3ν′,a4ν′〉 and BI=〈(a1μ+ϵ,a2μ-ϵ,a3μ-ϵ,a4μ+ϵ), (a1ν′+ϵ,a2ν′-ϵ,a3ν′-ϵ,a4ν′+ϵ)〉∈TrIFN but not in C7, where ϵ∈[0,1].
Now(64)J1AI=J1BI=a1μ+a2μ-a3μ+a4μ+a1ν′+a2ν′-a3ν′+a4ν′8,J2AI=J2BI=a1μ+a2μ-a3μ+a4μ-a1ν′+a2ν′+a3ν′+a4ν′8,J3AI=J3BI=a1ν′+a2ν′+a3ν′+a4ν′-a1μ+a2μ+a3μ+a4μ8,J4AI=J4BI=a1μ+a2μ+a3μ+a4μ+a1ν′+a2ν′+a3ν′+a4ν′8,J5AI=J5BI=a1μ+a3μ-a2μ+a4μ+a1ν′+a3ν′-a2ν′+a4ν′8,J6AI=J6BI=a2μ+a4μ-a1μ+a3μ+a1ν′+a3ν′-a2ν′+a4ν′8,J7AI=J7BI=a3ν′-a1ν′+a2ν′-a4ν′-a3μ-a1μ-a2μ-a4μ8.But(65)J8AI=a1μ-a3μ+a4μ-a2μ+a1ν′-a3ν′+a4ν′-a2ν′8,J8BI=a1μ-a3μ+a4μ-a2μ+a1ν′-a3ν′+a4ν′-a2ν′8-ϵ⟹J8AI>J8BI.Hence AI>BI.


## 4. A New Ranking Principle for Ordering Trapezoidal Intuitionistic Fuzzy Numbers

In this section a new ranking principle on trapezoidal intuitionistic fuzzy numbers is defined by using finite number of score functions defined in Sections3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, and 3.8 and we proved that the proposed ranking is a total order on the class of TrIFNs.Definition 100.
LetAI,BI∈TrIFN. An order relation < on the set of TrIFNs is defined by the following:IfJ1(AI)<J1(BI) then AI is smaller than BI, denoted by AI<BI.IfJ1(AI)=J1(BI) thenif J2(AI)<J2(BI) then AI is smaller than BI, denoted by AI<BI.IfJ1(AI)=J1(BI) and J2(AI)=J2(BI) then if J3(AI)<J3(BI) then AI is smaller than BI, denoted by AI<BI.IfJ1(AI)=J1(BI), J2(AI)=J2(BI), and J3(AI)=J3(BI) then if J4(AI)<J4(BI) then AI is smaller than BI, denoted by AI<BI.IfJ1(AI)=J1(BI), J2(AI)=J2(BI), J3(AI)=J3(BI), and J4(AI)=J4(BI) then if J5(AI)<J5(BI) then AI is smaller than BI, denoted by AI<BI.IfJ1(AI)=J1(BI), J2(AI)=J2(BI), J3(AI)=J3(BI), J4(AI)=J4(BI), and J5(AI)=J5(BI) then if J6(AI)<J6(BI) then AI is smaller than BI, denoted by AI<BI.IfJ1(AI)=J1(BI), J2(AI)=J2(BI), J3(AI)=J3(BI), J4(AI)=J4(BI), J5(AI)=J5(BI), and J6(AI)=J6(BI) then if J7(AI)<J7(BI) then AI is smaller than BI, denoted by AI<BI.IfJ1(AI)=J1(BI), J2(AI)=J2(BI), J3(AI)=J3(BI), J4(AI)=J4(BI), J5(AI)=J5(BI), J6(AI)=J6(BI), and J7(AI)=J7(BI) then if J8(AI)<J8(BI) then AI is smaller than BI, denoted by AI<BI.IfJ1(AI)=J1(BI), J2(AI)=J2(BI), J3(AI)=J3(BI), J4(AI)=J4(BI), J5(AI)=J5(BI), J6(AI)=J6(BI), J7(AI)=J7(BI), and J8(AI)=J8(BI) then AI=BI.The following theorem is proved to show the validity of Definition100.Theorem 101.
LetAI,BI∈TrIFN. If J1(AI)=J1(BI), J2(AI)=J2(BI), J3(AI)=J3(BI), J4(AI)=J4(BI), J5(AI)=J5(BI), J6(AI)=J6(BI), J7(AI)=J7(BI), and J8(AI)=J8(BI) then AI=BI. That is, a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ, and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′.Proof.
LetAI,BI∈TrIFN. Let us assume J1(AI)=J1(BI), J2(AI)=J2(BI), J3(AI)=J3(BI), J4(AI)=J4(BI), J5(AI)=J5(BI), J6(AI)=J6(BI), J7(AI)=J7(BI), and J8(AI)=J8(BI).
Claim  1 (AI=BI).(66)-J1AI+J4AI=-J1BI+J4BI⟹a3μ+a4μ+a3ν′+a4ν′=b3μ+b4μ+b3ν′+b4ν′,(67)J4AI+J1AI=J4BI+J1AI⟹a1μ+a2μ+a1ν′+a2ν′=b1μ+b2μ+b1ν′+b2ν′,(68)J2AI+J4AI=J2BI+J4BI⟹a1μ+a2μ+a3ν′+a4ν′=b1μ+b2μ+b3ν′+b4ν′,(69)-J3AI+J4AI=-J3BI+J4BI⟹a1μ+a2μ+a3ν+a4ν=b1μ+b2μ+b3ν+b4ν,(70)68-66⟹a1μ+a2μ-a3ν+a4ν=b1μ+b2μ-b3ν+b4ν,(71)69+70⟹a1μ+a2μ=b1μ+b2μ,(72)69-70⟹a3μ+a4μ=b3μ+b4μ.From (71), (66), and (67), we get(73)a1ν′+a2ν′=b1ν′+b2ν′,(74)a3ν′+a4ν′=b3ν′+b4ν′,(75)J4AI-J8AI=J4BI-J8BI⟹a2μ+a3μ+a2ν′+a3ν′=b2μ+b3μ+b2ν′+b3ν′,(76)J4AI-J7AI=J4BI-J7BI⟹a2μ+a3μ+a1ν′+a4ν′=b2μ+b3μ+b1ν′+b4ν′,(77)J4AI-J6AI=J4BI-J6BI⟹a1μ+a3μ+a2ν′+a4ν′=b1μ+b3μ+b2ν′+b4ν′,(78)J4AI-J5AI=J4BI-J5BI⟹a1μ+a3μ+a1ν′+a3ν′=b1μ+b3μ+b1ν′+b3ν′,(79)75-76⟹a2ν′-a1ν′+a3ν′-a4ν′=b2ν′-b1ν′+b3ν′-b4ν′,(80)77-78⟹a2ν′-a1ν′+a4ν′-a3ν′=b2ν′-b1ν′+b4ν′-b3ν′,(81)79+80⟹a2ν′-a1ν′=b2ν′-b1ν′,(82)80-79⟹a4ν′-a3ν′=b4ν′-b3ν′,(83)J2AI+J8AI=J2BI+J8BI⟹a1μ-a3μ+a4ν′-a2ν′=b1μ-b3μ+b4ν′-b2ν′,(84)-J6AI+J7AI=-J6BI+J7BI⟹a3μ-a4μ+a4ν′-a3ν′=b3μ-b4μ+a4ν′-b3ν′.From (73) and (74) and (81) and (82), we get(85)a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′.From (83), (84), and (85), we get(86)a1μ-a3μ=b1μ-b3μ,(87)a3μ-a4μ=b3μ-b4μ.(72) + (87) ⇒a3μ=b3μ and (72) − (87) ⇒a4=b4. From (86) a3μ=b3μ implies a1μ=b1μ. Put a1μ=b1μ in (71) and we get a2μ=b2μ. Therefore a1μ=b1μ,a2μ=b2μ,a3μ=b3μ,a4μ=b4μ and a1ν′=b1ν′,a2ν′=b2ν′,a3ν′=b3ν′,a4ν′=b4ν′, hence the proof.Theorem101 shows that the proposed ordering on TrIFNs satisfies Law of trichotomy and therefore the proposed ranking principle on TrIFNs gives total order on the class of TrIFNs.Remark 102.
The validation of the proposed ranking principle on the subclasses of TrIFNs is listed as follows:(1)
Accuracy score (J4) function is sufficient to rank the set of real numbers R totally, since J1,J2, and J3 scores become zero for any real number in [0,1].(2)
J1,J4,J5, and J8 give the total order on the class of trapezoidal fuzzy number.(3)
J1,J2,J3, and J4 together give the total order on the class of interval valued intuitionistic fuzzy number.(4)
J1,J2,J3,J4,J7, and J8 provide a complete ranking on the class of triangular intuitionistic fuzzy numbers.


## 5. Significance of the Proposed Method

In this section, significance of our proposed method is shown by comparing our proposed method with all other existing techniques. Table1 will show the shortcoming of existing methods and the advantage of our method. In Table 1, any trapezoidal intuitionistic fuzzy number is considered as normal.Table 1
Significance of proposed method.
Other existing methods Shortcomings of existing methods Numerical example Proposed method Li [2] R(A,λ)=a1+4a2+a323+a3-a1 A 1 = a 1 , a 2 , a 3 , 1,0,A2=a1-ϵ,a2+ϵ2,a3-ϵ,1,0 R(A1)=R(A2)⟹A1=A2 A 1 = 0,0.25,0.3 , 1,0,  A2=0.1,0.2,0.4,1,0 R(A1)=R(A2)=0.19697⟹A1=A2 J 1 ( A 1 ) = J 1 ( A 2 ) = - 0.075,  J2(A1)=J2(A2)=0 J3(A1)=J3(A2)=0   J4(A1)=0.2<J4(A2)=0.225  J4(A2)>J4(A1)⟹A2>A1  Zeng et al., [12] Vλ(A)=λVμ(A)+(1-λ)Vν(A) where VμA=a1+4a2+a3wA26,  Vν(A)=(a1+4a2+a3)1-uA26 Aλ(A)=λAμ(A)+(1-λ)Aν(A) where Aμ(A)=a3-a1wA26,Aν(A)=a3-a11-uA26 A 1 = a 1 , a 2 , a 3 , 1,0 A2=a1-ϵ,a2+ϵ2,a3-ϵ,1,0 VλA1=VλA2, Aλ(A1)=Aλ(A2)⟹A1=A2 A 1 = 0,0.25,0.3 , 1,0, A2=0.1,0.2,0.4,1,0 Vλ(A1)=Vλ(A2)=0.2167, AλA1=AλA2=0.05⟹A1=A2 J 1 ( A 1 ) = J 1 ( A 2 ) = - 0.075, J2(A1)=J2(A2)=0 J3(A1)=J3(A2)=0 J4(A1)=0.2<J4(A2)=0.225 J4A2>J4A1⟹A2>A1  Dubey and Mehara, [9] V(A,λ)=Vμ(A)+λVνA-VμA where VμA=a1+4a2+a3wA26, Vν(A)=(a1+4a2+a3)1-uA26 AA,λ=AνA-λAνA-AμA where AμA=a3-a1wA26,Aν(A)=(a3-a1)1-uA26 F(A,λ)=V(A,λ)-A(A,λ) A 1 = a 1 , a 2 , a 3 , 1,0 A2=a1-ϵ,a2+ϵ2,a3-ϵ,1,0 V(A1,λ)=V(A2,λ)  &  A(A1,λ)=A(A2,λ)  ⟹F(A1,λ)=F(A2,λ)⟹A1=A2 A 1 = 0,0.25,0.3 , 1,0, A2=0.1,0.2,0.4,1,0 V(A1,λ)=V(A2,λ)=0.2167, A(A1,λ)=A(A2,λ)=0.05 ⟹F(A1,λ)=F(A2,λ)=0.1667⟹A1=A2 J 1 ( A 1 ) = J 1 ( A 2 ) = - 0.075 J2(A1)=J2(A2)=0 J3(A1)=J3(A2)=0 J4(A1)=0.2<J4(A2)=0.225 J4(A2)>J4(A1)⟹A2>A1  Nehi, [7] Cμk(A)=a2+a32  +a1-a2+a4-a32(k+2),k∈R+ Cνk(A)=b1+b42 +b2-b1+b3-b42(k+2),k∈R+ A = a 1 , a 2 , a 3 , a 4 , b 1 , b 2 , b 3 , b 4 B=a1+ϵ,a2+ϵ,a3-ϵ,a4-ϵ,(b1+ϵ,b2+ϵ,b3-ϵ,b4-ϵ) Cμk(A)=Cμk(B), Cνk(A)=Cνk(B) ⟹A=B A = 0.2,0.3,0.5,0.6 , 0,0.1,0.6,0.7 B=0.3,0.4,0.4,0.5,0.1,0.2,0.5,0.6 Cμk(A)=Cμk(B)=0.4,Cνk(A)=Cνk(B)=0.35 ⟹A=B J 1 ( A ) = - 0.225 < J 1 ( B ) = - 0.125 ⟹A<B  Ye, [8] EV(A)=a1+a2+a3+a4+b1+b2+b3+b48 A 1 = a 1 , a 2 , a 3 , a 4 , b 1 , b 2 , b 3 , b 4, A2=a1+ϵ,a2+ϵ,a3+ϵ,a4+ϵ,b1-ϵ,b2-ϵ,b3-ϵ,b4-ϵ EV(A1)=EV(A2)⟹A1=A2 A 1 = 0.1,0.2,0.3,0.4 0.1,0.15,0.55,0.6, A2=0.2,0.3,0.4,0.5,0,0.05,0.45,0.5 EV(A1)=EV(A2)=0.3⟹A1=A2 J 1 ( A 1 ) = J 1 ( A 2 ) = - 0.1625,J2(A1)=J2(A2)=0.0625, J3(A1)=0.05,J3(A2)=-0.05 J3(A2)<J3(A1)⟹A2<A1  De and Das, [26] V(A¯)=Vμ(A¯)+λVνA¯-VμA¯ A(A1¯)=Aν(A¯)+λ(Aν(A¯)-Aμ(A¯)) where VμA¯=a1+2a2+2a3+a4wA¯6 Vν(A¯)=a1+2a2+2a3+a41-uA¯6 Aμ(A¯)=(a4-a1-2a2+2a3)(wA¯)6 Aν(A¯)=(a4-a1-2a2+2a3)(1-uA¯)6 A 1 ¯ = a 1 , a 2 , a 3 , a 4 , 1,0 A2¯=a1-ϵ,a2+ϵ2,a3+ϵ2,a4-ϵ,1,0 V(A1¯)=V(A2¯) and A(A1¯)=A(A2¯) ⟹A1¯=A2¯ A 1 ¯ = 0.20,0.35,0.40,0.55 , 1,0, A2¯=0.10,0.40,0.45,0.45,1,0 V(A1¯)=V(A2¯)=0.2333, A(A1¯)=A(A2¯)=0.0833⟹A1¯=A2¯ J 1 ( A 1 ¯ ) = J 1 ( A 2 ¯ ) = - 0.1 J2(A1¯)=J2(A2¯)=0 J3(A1¯)=J3(A2¯)=0 J4(A1¯)=0.375>J4(A2¯)=0.35 ⟹A1¯>A2¯  Kumar and Kaur, [11] Mμβ,kA=βa2+a32+1-βa1+a42 +1-βa2-a1+a3-a4k+12(k+2), Mνβ,kA=βa2′+a3′2+1-βa1′+a4′2 +1-βa2′-a1′+a3′-a4′k+12k+2, k∈R+  &  β∈[0,1] A = a 1 , a 2 , a 3 , a 4 , a 1 ′ , a 2 ′ , a 3 ′ , a 4 ′ B=a1-ϵ,a2-ϵ,a3+ϵ,a4+ϵ, a1′-ϵ,a2′-ϵ,a3′+ϵ,a4′+ϵ Mνβ,k(A)=Mνβ,k(B), Mνβ,k(A)=Mνβ,k(B) A=B A = 0.2,0.3,0.5,0.6 , 0,0.1,0.6,0.7 B=0.3,0.4,0.4,0.5,0.1,0.2,0.5,0.6 Mνβ,k(A)=Mνβ,k(B)=0.4,Mνβ,k(A)=Mνβ,k(B)=0.35 A=B J 1 ( A ) = - 0.225 , J 1 ( B ) = - 0.125 J1(A)<J1(B)⟹A<B  Wan and Dong, [13] mμ(A)=12m_μA+m¯μA mν(A)=12[m_ν(A)+m¯ν(A)] mμ(A)=16(a+2b+2c+d)wA mν(A)=16a+2b+2c+d1-uA A 1 = a 1 , a 2 , a 3 , a 4 , 1,0 A2=a1-ϵ,a2-ϵ,a3+ϵ,a4+ϵ,1,0 mμ(A1)=mν(A1)=mμ(A2)=mν(A2) ⟹A1=A2 A 1 = 0.40,0.45,0.50 ,  0.55 , 1,0,A2=0.30,0.35,0.60,0.65,1,0 mμ(A1)=mν(A1)=mμ(A2)=mν(A2)=0.475 ⟹A1=A2 J 1 ( A 1 ) = - 0.05 , J 1 ( A 2 ) = - 0.15 J1(A2)<J1(A1)⟹A2<A1That is,A=〈(a1,a2,a3,a4),(a1′,a2′,a3′,a4′),1,0〉. For example, let A=〈(0.2,0.3,0.5,0.6),(0,0.1,0.6,0.7)〉 and B=〈(0.3,0.4,0.4,0.5),(0.1,0.2,0.5,0.6)〉 be any two TrIFNs. Then by applying Nehi’s [7] approach, we get Cμk(A)=Cμk(B)=0.4 and Cνk(A)=Cνk(B)=0.35 and this implies that A and B are equal which is illogical. By applying the total order relation defined in Definition 100, we get J1(A)=0.225>J1(B)=0.125⇒A>B. Hence our proposed method ranks A as better.


## 6. Conclusion

In this paper, a complete ranking on the class of trapezoidal intuitionistic fuzzy numbers using eight different scores is proposed and compared with existing techniques using illustrative examples. A complete ranking on the class of TrIFNs yields the better results in MADM problems, fuzzy information systems, and artificial intelligence involving intuitionistic trapezoidal fuzzy information and hence our proposed method may be applied to control systems and other engineering fields. Our proposed method is conceptually easy to understand and operationally easy to use.



---

*Source: 3049041-2017-02-06.xml*