A New Ranking Principle For Ordering Trapezoidal Intuitionistic Fuzzy Numbers

Research Article

A New Ranking Principle For Ordering Trapezoidal

Intuitionistic Fuzzy Numbers

Lakshmana Gomathi Nayagam Velu, Jeevaraj Selvaraj, and Dhanasekaran Ponnialagan

Department of Mathematics, National Institute of Technology, Tiruchirappalli, India Correspondence should be addressed to Jeevaraj Selvaraj; alba.jeevi@gmail.com

Received 27 June 2016; Revised 4 November 2016; Accepted 13 November 2016; Published 6 February 2017 Academic Editor: Jose Egea

Copyright © 2017 Lakshmana Gomathi Nayagam Velu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 compar-ing 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

Volume 2017, Article ID 3049041, 24 pages https://doi.org/10.1155/2017/3049041

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 Section 2. 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, 𝜇_𝐴(𝑥), ]_𝐴(𝑥)⟩ | 𝑥 ∈ 𝑋}, where 𝜇_𝐴(𝑥) : 𝑋 → [0, 1] and ]_𝐴(𝑥) : 𝑋 → [0, 1], 𝑥 ∈ 𝑋 with the conditions 0 ≤ 𝜇_𝐴(𝑥) + ]_𝐴(𝑥) ≤ 1, ∀𝑥 ∈ 𝑋. The numbers 𝜇_𝐴(𝑥), ]_𝐴(𝑥) ∈ [0, 1] denote the degree of membership and nonmembership of𝑥 to lie in𝐴, respectively. For each intuitionistic fuzzy subset 𝐴 in𝑋, 𝜋_𝐴(𝑥) = 1 − 𝜇_𝐴(𝑥) − ]_𝐴(𝑥) is called hesitancy degree of 𝑥 to lie in 𝐴.

Definition 2 (Burillo et al., [24]). An intuitionistic fuzzy

number𝐴 = (𝜇_𝐴, ]_𝐴) in the set of real numbers 𝑅 is defined as 𝜇_𝐴(𝑥) = { { { { { { { { { { { { { { { 𝑓𝐴(𝑥) if 𝑎 ≤ 𝑥 ≤ 𝑏1 1 if 𝑏₁≤ 𝑥 ≤ 𝑏₂ 𝑔_𝐴(𝑥) if 𝑏2≤ 𝑥 ≤ 𝑐 0 Otherwise, ]_𝐴(𝑥) = { { { { { { { { { { { { { { { ℎ_𝐴(𝑥) if 𝑒 ≤ 𝑥 ≤ 𝑓₁ 0 if 𝑓₁≤ 𝑥 ≤ 𝑓₂ 𝑘_𝐴(𝑥) if 𝑓₂≤ 𝑥 ≤ 𝑔 1 Otherwise, (1) where0 ≤ 𝜇_𝐴(𝑥) + ]_𝐴(𝑥) ≤ 1 and 𝑎, 𝑏₁, 𝑏₂, 𝑐, 𝑒, 𝑓₁, 𝑓₂, 𝑔 ∈ R such that𝑒 ≤ 𝑎, 𝑓₁ ≤ 𝑏₁ ≤ 𝑏₂ ≤ 𝑓₂, 𝑐 ≤ 𝑔, and four functions 𝑓𝐴, 𝑔𝐴, ℎ𝐴, 𝑘𝐴 : R → [0, 1] are the legs of membership

(0, 1) (0, 0) (e, 1) (a, 1) (b1, 1) (b2, 1) (c, 1)(g, 1) 𝜇c A 𝜇A gA ℎA ]A fA kA (a ,0 ) (f1 ,0 ) (b1 ,0 ) (f2 , 0) (b2 ,0 ) (c ,0 )

Figure 1: Intuitionistic fuzzy number.

function𝜇_𝐴and nonmembership function]_𝐴. The functions 𝑓_𝐴and𝑘_𝐴are nondecreasing continuous functions and the functionsℎ_𝐴and𝑔_𝐴are nonincreasing continuous functions. An intuitionistic fuzzy number{(𝑎, 𝑏₁, 𝑏₂, 𝑐), (𝑒, 𝑓₁, 𝑓₂, 𝑔)} with(𝑒, 𝑓₁, 𝑓₂, 𝑔) ≤ (𝑎, 𝑏₁, 𝑏₂, 𝑐)𝑐is shown in Figure 1.

Definition 3. A trapezoidal intuitionistic fuzzy number 𝐴

with parameters𝑒 ≤ 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

𝜇𝐴(𝑥) = { { { { { { { { { { { { { { { { { 𝑥 − 𝑎₁ 𝑎₂− 𝑎₁ if 𝑎1≤ 𝑥 ≤ 𝑎2 1 if 𝑎₂≤ 𝑥 ≤ 𝑎₃ 𝑎₄− 𝑥 𝑎₄− 𝑎₃ if 𝑎3≤ 𝑥 ≤ 𝑎4 0 Otherwise, ]_𝐴(𝑥) = { { { { { { { { { { { { { { { { { 𝑥 − 𝑐2 𝑐₁− 𝑐₂ if 𝑐1≤ 𝑥 ≤ 𝑐2 0 if 𝑐₂≤ 𝑥 ≤ 𝑐₃ 𝑥 − 𝑐₃ 𝑐₄− 𝑐₃ if 𝑐3≤ 𝑥 ≤ 𝑐4 1 Otherwise. (2)

If𝑎₂ = 𝑎₃(and𝑐₂ = 𝑐₃) in a trapezoidal intuitionistic fuzzy number𝐴, we have the triangular intuitionistic fuzzy num-bers as special case of the trapezoidal intuitionistic fuzzy numbers.

A trapezoidal intuitionistic fuzzy number 𝐴 = {(𝑎, 𝑏₁, 𝑏₂, 𝑐), (𝑒, 𝑓₁, 𝑓₂, 𝑔)} with 𝑓₁ ≤ 𝑏₁,𝑓₂ ≥ 𝑏₂,𝑒 ≤ 𝑎, and 𝑔 ≥ 𝑐 is shown in Figure 2.

We note that the condition(𝑒, 𝑓₁, 𝑓₂, 𝑔) ≤ (𝑎, 𝑏₁, 𝑏₂, 𝑐)𝑐 of the trapezoidal intuitionistic fuzzy number𝐴 = {(𝑎, 𝑏₁, 𝑏₂, 𝑐), (𝑒, 𝑓₁, 𝑓₂, 𝑔)} whose membership and nonmembership fuzzy numbers of𝐴 are (𝑎, 𝑏₁, 𝑏₂, 𝑐) and (𝑒, 𝑓₁, 𝑓₂, 𝑔) implies

(0, 1) (0, 0) (e, 1) (a, 1) (b1, 1) (b2, 1) (c, 1) (g, 1) 𝜇cA 𝜇A ]A (a ,0 ) (f1 ,0 ) (b1 ,0 ) (f2 , 0) (b2 ,0 ) (c ,0 )

Figure 2: Trapezoidal intuitionistic fuzzy number.

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 set𝑋 ̸= 𝜙 is an expression given by𝐴 = {⟨𝑥, 𝜇_𝐴(𝑥), ]_𝐴(𝑥)⟩ : 𝑥 ∈ 𝑋}, where 𝜇_𝐴 : 𝑋 → 𝐷[0, 1], ]_𝐴: 𝑋 → 𝐷[0, 1] with the condition 0 < sup_𝑥𝜇_𝐴(𝑥)+ sup_𝑥]_𝐴(𝑥) ≤ 1.

The intervals 𝜇_𝐴(𝑥) and ]_𝐴(𝑥) denote, respectively, the degree of belongingness and nonbelongingness of the ele-ment𝑥 to the set 𝐴. Thus for each 𝑥 ∈ 𝑋, 𝜇_𝐴(𝑥) and ]_𝐴(𝑥) are closed intervals whose lower and upper end points are, respectively, denoted by𝜇_𝐴𝐿(𝑥), 𝜇_𝐴𝑈(𝑥) and ]_𝐴𝐿(𝑥), 𝜇_𝐴𝑈(𝑥). We denote𝐴 = {⟨𝑥, [𝜇_𝐴𝐿(𝑥), 𝜇_𝐴𝑈(𝑥)], []_𝐴𝐿(𝑥), ]_𝐴𝑈(𝑥)]⟩ : 𝑥 ∈ 𝑋}, where 0 < 𝜇_𝐴(𝑥) + ]_𝐴(𝑥) ≤ 1.

For each element𝑥 ∈ 𝑋, we can compute the unknown degree (hesitance degree) of belongingness 𝜋_𝐴(𝑥) to 𝐴 as 𝜋_𝐴(𝑥) = 1 − 𝜇_𝐴(𝑥) − ]_𝐴(𝑥) = [1 − 𝜇_𝐴𝑈(𝑥) − ]_𝐴𝑈(𝑥), 1 − 𝜇_𝐴𝐿(𝑥) − ]_𝐴𝐿(𝑥)]. An intuitionistic fuzzy interval number (IFIN) is denoted by𝐴 = ([𝑎, 𝑏], [𝑐, 𝑑]) for convenience.

Definition 5. Let𝛼, 𝛽∈ (0, 1]. An (𝛼, 𝛽)-cut of a trapezoidal

intuitionistic fuzzy number, denoted by(𝛼,𝛽)𝐴_𝐼, is defined as

(𝛼,𝛽)_𝐴 𝐼 = {(𝛼𝜇𝐴𝐼, 𝛽_] 𝐴𝐼)}, where 𝛼_𝜇 𝐴𝐼and 𝛽_]

𝐴𝐼 are𝛼-cut and 𝛽-cut of membership and nonmembership trapezoidal fuzzy numbers.

The(𝛼, 𝛽)-cut of trapezoidal intuitionistic fuzzy number 𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇), (𝑎󸀠 1], 𝑎󸀠2], 𝑎3]󸀠 , 𝑎4]󸀠 )⟩ is given by (𝛼,𝛽)_𝐴 𝐼 = {[𝛼𝜇𝐴_𝐼𝐿,𝛼𝜇𝐴_𝐼𝑈], [𝛽]𝐴_𝐼𝐿,𝛽]𝐴_𝐼𝑈]} = ([𝑎1𝜇 + 𝛼(𝑎2𝜇 − 𝑎_1𝜇), 𝑎_4𝜇− 𝛼(𝑎_4𝜇− 𝑎_3𝜇)], [𝑎󸀠 1]+ 𝛽 (𝑎2]󸀠 − 𝑎󸀠1]), 𝑎4]󸀠 − 𝛽 (𝑎4]󸀠 − 𝑎󸀠3])]), where𝛼𝜇_𝐴 𝐼𝐿 and 𝛼_𝜇

𝐴_𝐼𝑈 represents the lower and upper end points of the𝛼-cut of the membership function of 𝐴_𝐼 and

𝛽_]

𝐴_𝐼𝐿 and𝛽]𝐴_𝐼𝑈represent the lower and upper end points of the𝛽-cut of the nonmembership function of 𝐴_𝐼, 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, 𝐴_𝐼 and 𝐵_𝐼 always denote trapezoidal intuitionistic fuzzy number (TrIFN).

𝐴𝐼 = ⟨(𝑎1𝜇, 𝑎2𝜇, 𝑎3𝜇, 𝑎4𝜇), (𝑎1]󸀠, 𝑎2]󸀠 , 𝑎󸀠3], 𝑎4]󸀠 )⟩ with condi-tions that𝑎󸀠_1]≤ 𝑎_1𝜇and𝑎_2]󸀠 ≤ 𝑎_2𝜇,𝑎_3]󸀠 ≥ 𝑎_3𝜇and𝑎_4]󸀠 ≥ 𝑎_4𝜇and 𝐵_𝐼 = ⟨(𝑏_1𝜇, 𝑏_2𝜇, 𝑏_3𝜇, 𝑏_4𝜇), (𝑏󸀠

1], 𝑏2]󸀠, 𝑏3]󸀠 , 𝑏4]󸀠)⟩ with the condition that𝑏_1]󸀠 ≤ 𝑏_1𝜇and𝑏_2]󸀠 ≤ 𝑏_2𝜇,𝑏_3]󸀠 ≥ 𝑏_3𝜇, and𝑏_4]󸀠 ≥ 𝑏_4𝜇unless 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. Let𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇), (𝑎_1]󸀠 , 𝑎_2]󸀠 , 𝑎󸀠_3], 𝑎_4]󸀠 )⟩

and𝐵_𝐼= ⟨(𝑏_1𝜇, 𝑏_2𝜇, 𝑏_3𝜇, 𝑏_4𝜇), (𝑏_1]󸀠, 𝑏_2]󸀠 , 𝑏_3]󸀠, 𝑏_4]󸀠)⟩ be two TrIFNs. A special subclass𝐶₁of the set of TrIFNs consist of TrIFNs for which every pair of𝐴_𝐼and𝐵_𝐼are with𝑎_1𝜇 ≥ 𝑏_1𝜇, 𝑎_2𝜇 ≥ 𝑏_2𝜇, 𝑎_3𝜇 ≤ 𝑏_3𝜇, 𝑎_4𝜇 ≤ 𝑏_4𝜇 and 𝑎_1]󸀠 ≥ 𝑏_1]󸀠, 𝑎_2]󸀠 ≥ 𝑏_2]󸀠 , 𝑎_3]󸀠 ≤ 𝑏󸀠

3], 𝑎4]󸀠 ≤ 𝑏4]󸀠. The imprecise relation on𝐶1is denoted as⊑ and defined as follows: if𝐴_𝐼, 𝐵_𝐼 ∈ 𝐶₁ ⊂ TrIFN such that 𝐴_𝐼 ⊑ 𝐵_𝐼then𝑎_1𝜇 ≥ 𝑏_1𝜇, 𝑎_2𝜇 ≥ 𝑏_2𝜇, 𝑎_3𝜇 ≤ 𝑏_3𝜇, 𝑎_4𝜇 ≤ 𝑏_4𝜇and 𝑎󸀠

1]≥ 𝑏1]󸀠 , 𝑎󸀠2]≥ 𝑏2]󸀠 , 𝑎3]󸀠 ≤ 𝑏3]󸀠, 𝑎4]󸀠 ≤ 𝑏4]󸀠.

If𝐴_𝐼 ⊏ 𝐵_𝐼then one of the above inequalities becomes strict inequality.

Note 1. By Definition 6, we note that any pair of members of

𝐶₁are related under⊏.

The score function which measures the preciseness is defined as follows.

Definition 7. Let𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇), (𝑎_1]󸀠 , 𝑎_2]󸀠 , 𝑎󸀠_3], 𝑎_4]󸀠 )⟩

be a trapezoidal intuitionistic fuzzy number with 𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇, 𝑎󸀠

1], 𝑎󸀠2], 𝑎3]󸀠 , 𝑎󸀠4]∈ [0, 1]. Then the imprecise score of a trapezoidal intuitionistic fuzzy number𝐴_𝐼is defined by

𝐽1(𝐴𝐼) = ∫1 0 [𝛼_𝜇 𝐴_𝐼𝐿−𝛼𝜇𝐴_𝐼𝑈] 4 𝑑𝛼 + ∫ 1 0 [𝛽_] 𝐴_𝐼𝐿−𝛽]𝐴_𝐼𝑈] 4 𝑑𝛽 = ∫1 0 (𝑎1𝜇+ 𝛼 (𝑎2𝜇− 𝑎1𝜇)) 4 𝑑𝛼 − ∫1 0 (𝑎4𝜇− 𝛼 (𝑎4𝜇− 𝑎3𝜇)) 4 𝑑𝛼

+ ∫1 0 (𝑎1]󸀠 + 𝛽 (𝑎2]󸀠 − 𝑎1]󸀠 )) 4 𝑑𝛽 − ∫1 0 (𝑎󸀠 4]− 𝛽 (𝑎4]󸀠 − 𝑎3]󸀠 )) 4 𝑑𝛽 =(𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎2]󸀠) − (𝑎3]󸀠 + 𝑎󸀠4]) 8 . (3) The proofs of the following propositions are immediate from the above definition and hence they are omitted.

Proposition 8. For any real number 𝑟 ∈ [0, 1], 𝐽₁(𝑟) = 0.

Proposition 9. If 𝐴_𝐼 = (𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇) is a trapezoidal

fuzzy number, then𝐽₁(𝐴_𝐼) = ((𝑎_1𝜇+ 𝑎_2𝜇) − (𝑎_3𝜇+ 𝑎_4𝜇))/4.

Proposition 10. Let 𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇), (𝑎_1]󸀠 , 𝑎_2]󸀠 , 𝑎󸀠_3])⟩ be a

triangular intuitionistic fuzzy number. Then𝐽₁(𝐴_𝐼) = ((𝑎_1𝜇−

𝑎3𝜇) + (𝑎1]󸀠 − 𝑎3]󸀠 ))/4.

Proposition 11. Let 𝐴_𝐼= ([𝑎_1𝜇, 𝑎_2𝜇], [𝑎󸀠

1], 𝑎2]󸀠 ]) be an interval

valued intuitionistic fuzzy number. Then𝐽₁(𝐴_𝐼) = ((𝑎_1𝜇−𝑎_2𝜇)+

(𝑎󸀠

1]− 𝑎2]󸀠))/2.

Theorem 12. Let 𝐴_𝐼and𝐵_𝐼 ∈ 𝐶₁. If𝐴_𝐼⊏ 𝐵_𝐼then𝐽₁(𝐴_𝐼) >

𝐽₁(𝐵_𝐼).

Proof. Let𝐴_𝐼⊏ 𝐵_𝐼. We claim that𝐽₁(𝐴_𝐼) − 𝐽₁(𝐵_𝐼) > 0. By Definition 6, we have 𝑎1𝜇≥ 𝑏1𝜇, 𝑎_2𝜇≥ 𝑏_2𝜇, 𝑎_3𝜇≤ 𝑏_3𝜇, 𝑎_4𝜇≤ 𝑏_4𝜇, 𝑎󸀠_1]≥ 𝑏_1]󸀠 , 𝑎󸀠_2]≥ 𝑏_2]󸀠 , 𝑎󸀠_3]≤ 𝑏_3]󸀠 , 𝑎󸀠 4]≤ 𝑏4]󸀠 . (4) Now, 8 (𝐽₁(𝐴_𝐼) − 𝐽₁(𝐵_𝐼)) = [(𝑎_1𝜇− 𝑏_1𝜇) + (𝑎_2𝜇− 𝑏_2𝜇) + (𝑏_3𝜇− 𝑎_3𝜇) + (𝑏_4𝜇− 𝑎_4𝜇)] + [(𝑎_1]󸀠 − 𝑏_1]󸀠) + (𝑎_2]󸀠 − 𝑏_2]󸀠 ) + (𝑏_3]󸀠 − 𝑎󸀠_3]) + (𝑏_4]󸀠 − 𝑎󸀠_4])] . (5)

From (4) it is very easy to see that all the terms in (5) are pos-itive. Therefore their sum is also pospos-itive. From Definition 6, we know that at least one of the above inequalities in (4) becomes strict inequality and hence we get𝐽₁(𝐴_𝐼) − 𝐽₁(𝐵_𝐼) > 0.

Theorem 13. Let 𝐴_𝐼and𝐵_𝐼 ∈ 𝐶₁such that𝐽₁(𝐴_𝐼) = 𝐽₁(𝐵_𝐼);

then𝐴_𝐼= 𝐵_𝐼.

Proof. Let𝐽₁(𝐴_𝐼) = 𝐽₁(𝐵_𝐼). We claim that 𝐴_𝐼= 𝐵_𝐼. By Definition 6, without loss of generality, we have

𝑎_1𝜇≥ 𝑏_1𝜇, 𝑎2𝜇≥ 𝑏2𝜇, 𝑎_3𝜇≤ 𝑏_3𝜇, 𝑎_4𝜇≤ 𝑏_4𝜇, 𝑎󸀠_1]≥ 𝑏_1]󸀠 , 𝑎󸀠_2]≥ 𝑏_2]󸀠 , 𝑎󸀠_3]≤ 𝑏_3]󸀠 , 𝑎󸀠_4]≤ 𝑏_4]󸀠 . (6) Now, 8 (𝐽₁(𝐴_𝐼) − 𝐽₁(𝐵_𝐼)) = [(𝑎_1𝜇− 𝑏_1𝜇) + (𝑎_2𝜇− 𝑏_2𝜇) + (𝑏_3𝜇− 𝑎_3𝜇) + (𝑏_4𝜇− 𝑎_4𝜇)] + [(𝑎󸀠 1]− 𝑏1]󸀠) + (𝑎_2]󸀠 − 𝑏_2]󸀠 ) + (𝑏_3]󸀠 − 𝑎_3]󸀠) + (𝑏_4]󸀠 − 𝑎󸀠_4])] = 0. (7)

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. Hence𝑎_1𝜇 = 𝑏_1𝜇,𝑎_2𝜇 = 𝑏_2𝜇,𝑎_3𝜇 = 𝑏_3𝜇,𝑎_4𝜇 = 𝑏4𝜇and𝑎1]󸀠 = 𝑏1]󸀠 , 𝑎2]󸀠 = 𝑏2]󸀠, 𝑎3]󸀠 = 𝑏3]󸀠, 𝑎󸀠4] = 𝑏4]󸀠 , hence the proof.

Note 2. The imprecise score can be calculated to any TrIFN

but𝐽₁gives total order in𝐶₁.

Definition 14. If𝐽₁(𝐴_𝐼) > 𝐽₁(𝐵_𝐼)(𝐽₁(𝐴_𝐼) < 𝐽₁(𝐵_𝐼)), then 𝐴_𝐼>

𝐵_𝐼(𝐴_𝐼< 𝐵_𝐼).

The following example explains the ranking procedure introduced in Definition 14. Example 15. Let𝐴_𝐼= ⟨(0.1, 0.2, 0.35, 0.5), (0, 0.15, 0.45, 0.6)⟩ and 𝐵_𝐼 = ⟨(0.15, 0.2, 0.35, 0.45), (0.1, 0.1, 0.5, 0.55)⟩ ∈ 𝐶₁. Now 𝐽₁(𝐴_𝐼) = −0.18125 and 𝐽₁(𝐵_𝐼) = −0.1625, 𝐽₁(𝐴_𝐼) < 𝐽₁(𝐵_𝐼). Hence𝐴_𝐼< 𝐵_𝐼. Example 16. Let𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇), (𝑎_1]󸀠 , 𝑎_2]󸀠 , 𝑎󸀠_3], 𝑎_4]󸀠 )⟩ and𝐵_𝐼 = ⟨(𝑎_1𝜇+ 𝜖, 𝑎_2𝜇+ 𝜖, 𝑎_3𝜇− 𝜖, 𝑎_4𝜇− 𝜖), (𝑎󸀠_1]− 𝜖, 𝑎_2]󸀠 − 𝜖, 𝑎_3]󸀠 + 𝜖, 𝑎_4]󸀠 + 𝜖)⟩ ∈ TrIFN but not in 𝐶1, where𝜖 ∈ [0, 1].

Now𝐽₁(𝐴_𝐼) = 𝐽₁(𝐵_𝐼) = ((𝑎_1𝜇+ 𝑎_2𝜇) − (𝑎_3𝜇+ 𝑎_4𝜇) + (𝑎_1]󸀠 + 𝑎󸀠

2]) − (𝑎3]󸀠 + 𝑎4]󸀠 ))/8. But 𝐴𝐼 ̸= 𝐵𝐼. Pictorial representation of this example is given in Figure 3.

Example 16 shows that𝐽₁is not enough cover the entire class of TrIFNs. Therefore it is needed for us to define another

(0, 1) (0, 0) b󳰀 1] a󳰀1] b 󳰀 2] 󳰀 2_a] a2𝜇 b2𝜇 a1𝜇 b1𝜇 a3𝜇 b3𝜇 a4𝜇 b4𝜇 a󳰀 4] a 󳰀 3] b󳰀 4] b 󳰀 3] Figure 3: Pictorial representation of Example 16.

score function that can cover some subclass of TrIFNs which cannot be covered by𝐽₁.

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. Let𝐴_𝐼and𝐵_𝐼be two TrIFNs.

A special subclass𝐶₂of the set of TrIFNs consist of TrIFNs for which every pair of𝐴_𝐼and𝐵_𝐼are with𝑎_1𝜇 ≥ 𝑏_1𝜇, 𝑎_2𝜇 ≥ 𝑏2𝜇, 𝑎3𝜇 ≤ 𝑏3𝜇, 𝑎4𝜇 ≤ 𝑏4𝜇 and 𝑎1]󸀠 ≤ 𝑏1]󸀠, 𝑎2]󸀠 ≤ 𝑏2]󸀠, 𝑎3]󸀠 ≥ 𝑏󸀠

3], 𝑎4]󸀠 ≥ 𝑏4]󸀠. The nonvague relation on𝐶2is denoted as⊒ and defined as follows: if𝐴_𝐼, 𝐵_𝐼 ∈ 𝐶₂ ⊂ TrIFN such that 𝐴𝐼 ⊒ 𝐵𝐼then𝑎1𝜇 ≥ 𝑏1𝜇, 𝑎2𝜇 ≥ 𝑏2𝜇, 𝑎3𝜇 ≤ 𝑏3𝜇, 𝑎4𝜇 ≤ 𝑏4𝜇and 𝑎󸀠

1]≤ 𝑏1]󸀠 , 𝑎󸀠2]≤ 𝑏2]󸀠 , 𝑎3]󸀠 ≥ 𝑏3]󸀠, 𝑎4]󸀠 ≥ 𝑏4]󸀠.

If𝐴_𝐼 ⊐ 𝐵_𝐼then one of the above inequalities becomes strict inequality.

Note 3. By Definition 17, we note that any pair of members of

𝐶₂are related under⊐.

The score function which measures the nonvagueness is defined as follows.

Definition 18. Let 𝐴_𝐼 be a trapezoidal intuitionistic fuzzy number. Then the nonvague score of a trapezoidal intuition-istic fuzzy number𝐴_𝐼is defined by

𝐽2(𝐴𝐼) = ∫1 0 [𝛼_𝜇 𝐴_𝐼𝐿−𝛼𝜇𝐴_𝐼𝑈] 4 𝑑𝛼 − ∫ 1 0 [𝛽_] 𝐴_𝐼𝐿−𝛽]𝐴_𝐼𝑈] 4 𝑑𝛽 =(𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) − (𝑎 󸀠 1]+ 𝑎2]󸀠) + (𝑎3]󸀠 + 𝑎󸀠4]) 8 . (8)

The proofs of the following propositions are immediate from Definition 18 and hence they are omitted.

Proposition 19. For any real number 𝑟 ∈ [0, 1], 𝐽₂(𝑟) = 0.

Proposition 20. If 𝐴_𝐼 = (𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇) is a trapezoidal

fuzzy number, then𝐽₂(𝐴_𝐼) = 0.

Proposition 21. Let 𝐴_𝐼= ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇), (𝑎󸀠

1], 𝑎2]󸀠 , 𝑎󸀠3])⟩ be a

triangular intuitionistic fuzzy number. Then𝐽₂(𝐴_𝐼) = ((𝑎_1𝜇−

𝑎_3𝜇) + (−𝑎󸀠

1]+ 𝑎3]󸀠))/8.

Proposition 22. Let 𝐴_𝐼= ([𝑎_1𝜇, 𝑎_2𝜇], [𝑎󸀠

1], 𝑎2]󸀠 ]) be an interval

valued intuitionistic fuzzy number. Then𝐽₂(𝐴_𝐼) = ((𝑎_1𝜇−𝑎_2𝜇)+

(−𝑎󸀠_1]+ 𝑎󸀠_2]))/4.

Theorem 23. Let 𝐴_𝐼and𝐵_𝐼 ∈ 𝐶₂. If𝐴_𝐼⊐ 𝐵_𝐼then𝐽₂(𝐴_𝐼) >

𝐽₂(𝐵_𝐼).

Proof. Let us assume𝐴_𝐼⊐ 𝐵_𝐼. We claim that𝐽₂(𝐴_𝐼)−𝐽₂(𝐵_𝐼) > 0. By Definition 17, we have 𝑎_1𝜇≥ 𝑏_1𝜇, 𝑎_2𝜇≥ 𝑏_2𝜇, 𝑎3𝜇≤ 𝑏3𝜇, 𝑎_4𝜇≤ 𝑏_4𝜇, 𝑎󸀠_1]≤ 𝑏_1]󸀠 , 𝑎󸀠_2]≤ 𝑏_2]󸀠 , 𝑎󸀠_3]≥ 𝑏_3]󸀠 , 𝑎󸀠_4]≥ 𝑏_4]󸀠 . (9) Now, 8 (𝐽₂(𝐴_𝐼) − 𝐽₂(𝐵_𝐼)) = [(𝑎_1𝜇− 𝑏_1𝜇) + (𝑎_2𝜇− 𝑏_2𝜇) + (𝑏_3𝜇− 𝑎_3𝜇) + (𝑏_4𝜇− 𝑎_4𝜇)] + [(𝑏_1]󸀠 − 𝑎_1]󸀠 ) + (𝑏_2]󸀠 − 𝑎󸀠_2]) + (𝑎󸀠_3]− 𝑏_3]󸀠) + (𝑎󸀠_4]− 𝑏_4]󸀠)] . (10)

From (9) it is very easy to see that all the terms in (10) are pos-itive. Therefore their sum is also pospos-itive. From Definition 17, we know that at least one of the above inequalities in (9) becomes strict inequality and hence we get𝐽₂(𝐴_𝐼) − 𝐽₂(𝐵_𝐼) > 0.

Theorem 24. Let 𝐴_𝐼and𝐵_𝐼 ∈ 𝐶₂. If𝐽₂(𝐴_𝐼) = 𝐽₂(𝐵_𝐼) then

𝐴𝐼= 𝐵𝐼.

By Definition 17, without loss of generality, we have 𝑎_1𝜇≥ 𝑏_1𝜇, 𝑎2𝜇≥ 𝑏2𝜇, 𝑎_3𝜇≤ 𝑏_3𝜇, 𝑎_4𝜇≤ 𝑏_4𝜇, 𝑎󸀠_1]≤ 𝑏_1]󸀠 , 𝑎󸀠 2]≤ 𝑏2]󸀠 , 𝑎󸀠_3]≥ 𝑏_3]󸀠 , 𝑎󸀠_4]≥ 𝑏_4]󸀠 . (11) Now, 8 (𝐽₂(𝐴_𝐼) − 𝐽₂(𝐵_𝐼)) = [(𝑎_1𝜇− 𝑏_1𝜇) + (𝑎_2𝜇− 𝑏_2𝜇) + (𝑏_3𝜇− 𝑎_3𝜇) + (𝑏_4𝜇− 𝑎_4𝜇)] + [(𝑏_1]󸀠 − 𝑎_1]󸀠 ) + (𝑏_2]󸀠 − 𝑎󸀠_2]) + (𝑎_3]󸀠 − 𝑏_3]󸀠) + (𝑎󸀠_4]− 𝑏_4]󸀠)] = 0. (12)

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. Hence𝑎_1𝜇 = 𝑏_1𝜇, 𝑎_2𝜇 = 𝑏_2𝜇, 𝑎_3𝜇 = 𝑏_3𝜇, 𝑎_4𝜇 = 𝑏_4𝜇and𝑎_1]󸀠 = 𝑏_1]󸀠 , 𝑎󸀠_2] = 𝑏_2]󸀠, 𝑎_3]󸀠 = 𝑏_3]󸀠, 𝑎_4]󸀠 = 𝑏_4]󸀠 . Hence𝐴_𝐼= 𝐵_𝐼.

Note 4. The nonvague score can be calculated to any TrIFN.

But𝐽₂gives total order on𝐶₂.

Definition 25. If𝐽₂(𝐴_𝐼) > 𝐽₂(𝐵_𝐼)(𝐽₂(𝐴_𝐼) < 𝐽₂(𝐵_𝐼)), then 𝐴_𝐼>

𝐵_𝐼(𝐴_𝐼< 𝐵_𝐼).

Ranking relation defined above is explained in the follow-ing example. Example 26. Let𝐴_𝐼 = ⟨(0.15, 0.25, 0.35, 0.45), (0.1, 0.2, 0.4, 0.5)⟩ and 𝐵_𝐼 = ⟨(0.18, 0.25, 0.35, 0.4), (0, 0.15, 0.45, 0.55)⟩ ∈ 𝐶₂. Now𝐽₂(𝐴_𝐼) = 0.025 and 𝐽₂(𝐵_𝐼) = 0.06625, 𝐽₂(𝐵_𝐼) > 𝐽₂(𝐴_𝐼). Hence𝐵_𝐼> 𝐴_𝐼.

Example 27 shows the inefficiency of𝐽₁in comparing any two arbitrary TrIFNs and the importance of defining new score function𝐽₂. Example 27. Let𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇), (𝑎_1]󸀠 , 𝑎_2]󸀠 , 𝑎󸀠_3], 𝑎_4]󸀠 )⟩ and𝐵_𝐼 = ⟨(𝑎_1𝜇+ 𝜖, 𝑎_2𝜇+ 𝜖, 𝑎_3𝜇− 𝜖, 𝑎_4𝜇− 𝜖), (𝑎󸀠_1]− 𝜖, 𝑎_2]󸀠 − 𝜖, 𝑎󸀠 3]+ 𝜖, 𝑎4]󸀠 + 𝜖)⟩ ∈ TrIFN and 𝜖 ∈ [0, 1]. Now 𝐽1(𝐴𝐼) = 𝐽1(𝐵𝐼) =(𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎2]󸀠 ) − (𝑎󸀠3]+ 𝑎󸀠4]) 8 󳨐⇒ 𝐴𝐼= 𝐵𝐼. (13) (0, 1) (0, 0) b1󳰀] a1󳰀] b 󳰀 2] a 󳰀 2] a2𝜇 b2𝜇 a1𝜇 _b1𝜇 a3𝜇b3𝜇 a4𝜇 _b4𝜇 a4󳰀] a 󳰀 3] b4󳰀] b 󳰀 3] Figure 4: Pictorial representation of Example 28.

But 𝐽2(𝐴𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) − (𝑎 󸀠 1]+ 𝑎󸀠2]) + (𝑎󸀠3]+ 𝑎4]󸀠 ) 8 , 𝐽2(𝐵𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) − (𝑎 󸀠 1]+ 𝑎󸀠2]) + (𝑎󸀠3]+ 𝑎4]󸀠 ) 8 + 𝜖. (14) Therefore𝐽₂(𝐵_𝐼) > 𝐽₂(𝐴_𝐼); hence 𝐵_𝐼> 𝐴_𝐼. Example 28. Let𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇), (𝑎_1]󸀠 , 𝑎_2]󸀠 , 𝑎󸀠_3], 𝑎_4]󸀠 )⟩ and𝐵_𝐼 = ⟨(𝑎_1𝜇+ 𝜖, 𝑎_2𝜇+ 𝜖, 𝑎_3𝜇+ 𝜖, 𝑎_4𝜇+ 𝜖), (𝑎󸀠_1]+ 𝜖/2, 𝑎_2]󸀠 + 𝜖/2, 𝑎󸀠

3]+𝜖/2, 𝑎4]󸀠 +𝜖/2)⟩ ∈ TrIFN and 𝜖 ∈ [0, 1] but not in 𝐶2. Now 𝐽1(𝐴𝐼) = 𝐽1(𝐵𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎󸀠2]) − (𝑎󸀠3]+ 𝑎4]󸀠 ) 8 , 𝐽2(𝐴𝐼) = 𝐽2(𝐵𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) − (𝑎 󸀠 1]+ 𝑎󸀠2]) + (𝑎󸀠3]+ 𝑎4]󸀠 ) 8 . (15) But𝐴_𝐼 ̸= 𝐵_𝐼.

Pictorial representation of this example is given in Fig-ure 4.

From Example 28 it is easy to see that𝐽₁and𝐽₂are 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 by𝐽₁and𝐽₂.

3.3. Incomplete Score of a Trapezoidal Intuitionistic Fuzzy Number. In this subsection, another class of TrIFNs is

trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.

Definition 29. Let𝐴_𝐼and𝐵_𝐼be two TrIFNs.

A special subclass𝐶₃of the set of TrIFNs consist of TrIFNs for which every pair of𝐴_𝐼and𝐵_𝐼are with𝑎_1𝜇 ≤ 𝑏_1𝜇, 𝑎_2𝜇 ≤ 𝑏_2𝜇, 𝑎_3𝜇 ≤ 𝑏_3𝜇, 𝑎_4𝜇 ≤ 𝑏_4𝜇 and 𝑎_1]󸀠 ≥ 𝑏_1]󸀠, 𝑎_2]󸀠 ≥ 𝑏_2]󸀠, 𝑎_3]󸀠 ≥ 𝑏󸀠

3], 𝑎4]󸀠 ≥ 𝑏4]󸀠 . The incomplete relation on𝐶3 is denoted as ⊆ and defined as follows: if 𝐴_𝐼, 𝐵_𝐼 ∈ 𝐶₃ ⊂ TrIFN such that 𝐴_𝐼 ⊆ 𝐵_𝐼then𝑎_1𝜇 ≤ 𝑏_1𝜇, 𝑎_2𝜇 ≤ 𝑏_2𝜇, 𝑎_3𝜇 ≤ 𝑏_3𝜇, 𝑎_4𝜇 ≤ 𝑏_4𝜇and 𝑎󸀠

1]≥ 𝑏1]󸀠 , 𝑎󸀠2]≥ 𝑏2]󸀠 , 𝑎3]󸀠 ≥ 𝑏3]󸀠, 𝑎4]󸀠 ≥ 𝑏4]󸀠.

If𝐴_𝐼 ⊂ 𝐵_𝐼then one of the above inequalities becomes strict inequality.

Note 5. By Definition 29, we note that any pair of members

of𝐶₃are related under⊂.

The incomplete score function which measures the com-pleteness is defined as follows.

Definition 30. Let 𝐴_𝐼 be a trapezoidal intuitionistic fuzzy number. Then the incomplete score of𝐴_𝐼is defined by

𝐽3(𝐴𝐼) = ∫1 0 − [𝛼𝜇𝐴_𝐼𝐿 +𝛼𝜇𝐴_𝐼𝑈] 4 𝑑𝛼 + ∫1 0 [𝛽_] 𝐴_𝐼𝐿+𝛽]𝐴_𝐼𝑈] 4 𝑑𝛽 = (𝑎 󸀠 1]+ 𝑎2]󸀠 + 𝑎3]󸀠 + 𝑎4]󸀠 ) − (𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) 8 . (16)

The proofs of the following propositions are immediate from the above definition and hence they are omitted.

Proposition 31. For any real number 𝑟 ∈ [0, 1], 𝐽₃(𝑟) = 0.

Proposition 32. If 𝐴_𝐼 = (𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇) is a trapezoidal

fuzzy number, then𝐽₃(𝐴_𝐼) = 0.

Proposition 33. Let 𝐴_𝐼= ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇), (𝑎󸀠

1], 𝑎󸀠2], 𝑎3]󸀠)⟩ be a

triangular intuitionistic fuzzy number. Then

𝐽₃(𝐴_𝐼) = (𝑎 󸀠 1]+ 2𝑎󸀠2]+ 𝑎󸀠3]) − (𝑎1𝜇+ 2𝑎2𝜇+ 𝑎3𝜇) 8 . (17) Proposition 34. Let 𝐴_𝐼= ([𝑎_1𝜇, 𝑎_2𝜇], [𝑎󸀠 1], 𝑎󸀠2]]) be an interval

valued intuitionistic fuzzy number. Then

𝐽₃(𝐴_𝐼) = (𝑎 󸀠

1]+ 𝑎2]󸀠 ) − (𝑎1𝜇+ 𝑎2𝜇)

4 . (18)

Theorem 35. Let 𝐴_𝐼, 𝐵_𝐼 ∈ 𝐶₃. If 𝐴_𝐼 ⊂ 𝐵_𝐼then 𝐽₃(𝐴_𝐼) >

𝐽₃(𝐵_𝐼).

Proof. Let us assume that𝐴_𝐼 ⊂ 𝐵_𝐼. We claim that𝐽₃(𝐴_𝐼) − 𝐽3(𝐵𝐼) > 0. By Definition 29, we have 𝑎_1𝜇≤ 𝑏_1𝜇, 𝑎2𝜇≤ 𝑏2𝜇, 𝑎_3𝜇≤ 𝑏_3𝜇, 𝑎_4𝜇≤ 𝑏_4𝜇, 𝑎󸀠_1]≥ 𝑏_1]󸀠 , 𝑎󸀠_2]≥ 𝑏_2]󸀠 , 𝑎󸀠_3]≥ 𝑏_3]󸀠 , 𝑎󸀠_4]≥ 𝑏_4]󸀠 . (19) Now, 8 (𝐽₃(𝐴_𝐼) − 𝐽₃(𝐵_𝐼)) = [(𝑏_1𝜇− 𝑎_1𝜇) + (𝑏_2𝜇− 𝑎_2𝜇) + (𝑏_3𝜇− 𝑎_3𝜇) + (𝑏_4𝜇− 𝑎_4𝜇)] + [(𝑎󸀠_1]− 𝑏_1]󸀠) + (𝑎_2]󸀠 − 𝑏_2]󸀠) + (𝑎󸀠_3]− 𝑏_3]󸀠) + (𝑎󸀠_4]− 𝑏_4]󸀠)] . (20)

From (19) it is very easy to see that all the terms in (20) are positive. Therefore their sum is also positive. From Defini-tion 29, we know that at least one of the above inequalities in (19) becomes strict inequality and hence we get𝐽₃(𝐴_𝐼) − 𝐽₃(𝐵_𝐼) > 0, hence the proof.

Theorem 36. Let 𝐴_𝐼, 𝐵_𝐼 ∈ 𝐶₃. If𝐽₃(𝐴_𝐼) = 𝐽₃(𝐵_𝐼) then 𝐴_𝐼 =

𝐵_𝐼.

Proof. Let us assume𝐽₃(𝐴_𝐼) = 𝐽₂(𝐵_𝐼). We prove that 𝐴_𝐼= 𝐵_𝐼. By Definition 29, without loss of generality, we have

𝑎1𝜇≤ 𝑏1𝜇, 𝑎_2𝜇≤ 𝑏_2𝜇, 𝑎_3𝜇≤ 𝑏_3𝜇, 𝑎4𝜇≤ 𝑏4𝜇, 𝑎󸀠_1]≥ 𝑏_1]󸀠 , 𝑎󸀠_2]≥ 𝑏_2]󸀠 , 𝑎󸀠_3]≥ 𝑏_3]󸀠 , 𝑎󸀠_4]≥ 𝑏_4]󸀠 . (21) Now, 8 (𝐽₃(𝐴_𝐼) − 𝐽₃(𝐵_𝐼)) = [(𝑏_1𝜇− 𝑎_1𝜇) + (𝑏_2𝜇− 𝑎_2𝜇) + (𝑏3𝜇− 𝑎3𝜇) + (𝑏4𝜇− 𝑎4𝜇)] + [(𝑎󸀠1]− 𝑏1]󸀠) + (𝑎_2]󸀠 − 𝑏_2]󸀠) + (𝑎_3]󸀠 − 𝑏_3]󸀠) + (𝑎_4]󸀠 − 𝑏_4]󸀠)] = 0. (22)

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. Hence𝑎_1𝜇 = 𝑏_1𝜇, 𝑎_2𝜇 = 𝑏_2𝜇, 𝑎_3𝜇 = 𝑏_3𝜇, 𝑎_4𝜇 = 𝑏_4𝜇and𝑎_1]󸀠 = 𝑏_1]󸀠 , 𝑎󸀠_2] = 𝑏_2]󸀠, 𝑎_3]󸀠 = 𝑏_3]󸀠, 𝑎_4]󸀠 = 𝑏_4]󸀠 . Hence𝐴_𝐼= 𝐵_𝐼.

Note 6. The incomplete score can be calculated to any TrIFN.

But𝐽₃gives total order on𝐶₃.

Definition 37. If𝐽₃(𝐴_𝐼) > 𝐽₃(𝐵_𝐼)(𝐽₃(𝐴_𝐼) < 𝐽₃(𝐵_𝐼)), then 𝐴_𝐼>

𝐵_𝐼(𝐴_𝐼< 𝐵_𝐼).

The following example is used to explain the ranking procedure defined in Definition 37.

Example 38. Let𝐴_𝐼 = ⟨(0.15, 0.3, 0.4, 0.45), (0.1, 0.25, 0.45,

0.6)⟩ and 𝐵_𝐼 = ⟨(0.2, 0.4, 0.45, 0.55), (0, 0.15, 0.45, 0.55)⟩ ∈ 𝐶₃. Now𝐽₃(𝐴_𝐼) = 0.0125 and 𝐽₃(𝐵_𝐼) = −0.05625, 𝐽₃(𝐴_𝐼) > 𝐽₃(𝐵_𝐼).

Hence𝐴_𝐼> 𝐵_𝐼.

The inefficiency of the score functions 𝐽₁ and 𝐽₂ in discriminating any two arbitrary TrIFNs and the ability of 𝐽3in comparing arbitrary TrIFNs is shown in the following example. Example 39. Let𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇), (𝑎_1]󸀠 , 𝑎_2]󸀠 , 𝑎󸀠_3], 𝑎_4]󸀠 )⟩ and𝐵_𝐼= ⟨(𝑎_1𝜇+ 𝜖, 𝑎_2𝜇+ 𝜖, 𝑎_3𝜇+ 𝜖, 𝑎_4𝜇+ 𝜖), (𝑎󸀠_1]+ 𝜖/2, 𝑎_2]󸀠 + 𝜖/2, 𝑎󸀠 3]+ 𝜖/2, 𝑎󸀠4]+ 𝜖/2)⟩ ∈ TrIFN and 𝜖 ∈ [0, 1]. Now 𝐽1(𝐴𝐼) = 𝐽1(𝐵𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎󸀠2]) − (𝑎󸀠3]+ 𝑎4]󸀠) 8 , 𝐽2(𝐴𝐼) = 𝐽2(𝐵𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) − (𝑎 󸀠 1]+ 𝑎󸀠2]) + (𝑎󸀠3]+ 𝑎4]󸀠) 8 . (23) But 𝐽3(𝐴𝐼) =(𝑎 󸀠 1]+ 𝑎2]󸀠 + 𝑎3]󸀠 + 𝑎4]󸀠 ) − (𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) 8 , 𝐽3(𝐵𝐼) =(𝑎 󸀠 1]+ 𝑎2]󸀠 + 𝑎3]󸀠 + 𝑎4]󸀠 ) − (𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) 8 +𝜖₄ 󳨐⇒ 𝐽3(𝐵𝐼) > 𝐽3(𝐴𝐼) . (24) Hence𝐵_𝐼> 𝐴_𝐼.

The inefficiency of the score functions𝐽₁to𝐽₃in the task of comparing TrIFNs is explained in the following example.

(0, 1) (0, 0) b1󳰀] a1󳰀] b 󳰀 2] a 󳰀 2] a2𝜇 b2𝜇 a1𝜇 b1𝜇 a3𝜇 b3𝜇 a4𝜇 b4𝜇 a4󳰀] a 󳰀 3] b4󳰀] b 󳰀 3] Figure 5: Pictorial representation of Example 40.

Example 40. Let𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇), (𝑎_1]󸀠 , 𝑎_2]󸀠 , 𝑎󸀠_3], 𝑎_4]󸀠 )⟩

and𝐵_𝐼= ⟨(𝑎_1𝜇+𝜖, 𝑎_2𝜇+𝜖, 𝑎_3𝜇+𝜖, 𝑎_4𝜇+𝜖), (𝑎_1]󸀠 +𝜖, 𝑎󸀠_2]+𝜖, 𝑎_3]󸀠 + 𝜖, 𝑎󸀠

4]+ 𝜖)⟩ ∈ TrIFN and 𝜖 ∈ [0, 1] but not in 𝐶3. Now 𝐽₁(𝐴_𝐼) = 𝐽₁(𝐵_𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎2]󸀠) − (𝑎󸀠3]+ 𝑎4]󸀠) 8 , 𝐽₂(𝐴_𝐼) = 𝐽₂(𝐵_𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) − (𝑎 󸀠 1]+ 𝑎2]󸀠) + (𝑎󸀠3]+ 𝑎4]󸀠) 8 , 𝐽₃(𝐴_𝐼) = 𝐽₃(𝐵_𝐼) = (𝑎 󸀠 1]+ 𝑎2]󸀠 + 𝑎󸀠3]+ 𝑎4]󸀠 ) − (𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) 8 . (25) But𝐴_𝐼 ̸= 𝐵_𝐼.

Pictorial representation of this example is given in Fig-ure 5.

Example 40 shows that𝐽₁, 𝐽₂, and𝐽₃cannot 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 by𝐽₁, 𝐽₂, and𝐽₃.

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 func-tion on this class of trapezoidal intuifunc-tionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.

Definition 41. Let𝐴_𝐼and𝐵_𝐼be two TrIFNs.

A special subclass𝐶₄of the set of TrIFNs consist of TrIFNs for which every pair of𝐴_𝐼and𝐵_𝐼are with𝑎_1𝜇 ≤ 𝑏_1𝜇, 𝑎_2𝜇 ≤ 𝑏_2𝜇, 𝑎_3𝜇 ≤ 𝑏_3𝜇, 𝑎_4𝜇 ≤ 𝑏_4𝜇 and 𝑎_1]󸀠 ≤ 𝑏_1]󸀠, 𝑎_2]󸀠 ≤ 𝑏_2]󸀠, 𝑎_3]󸀠 ≤ 𝑏󸀠

3], 𝑎4]󸀠 ≤ 𝑏4]󸀠 . The accuracy relation on𝐶4is denoted as⪯ and defined as follows: if𝐴_𝐼, 𝐵_𝐼 ∈ 𝐶₄ ⊂ TrIFN such that 𝐴_𝐼 ⪯ 𝐵_𝐼then𝑎_1𝜇 ≤ 𝑏_1𝜇, 𝑎_2𝜇 ≤ 𝑏_2𝜇, 𝑎_3𝜇 ≤ 𝑏_3𝜇, 𝑎_4𝜇 ≤ 𝑏_4𝜇and 𝑎󸀠

1]≤ 𝑏1]󸀠 , 𝑎󸀠2]≤ 𝑏2]󸀠 , 𝑎3]󸀠 ≤ 𝑏3]󸀠, 𝑎4]󸀠 ≤ 𝑏4]󸀠.

If𝐴_𝐼 ≺ 𝐵_𝐼then one of the above inequalities becomes strict inequality.

Note 7. By Definition 41, we note that any pair of members of

𝐶₄are related under≺.

The score function which measures the accuracy is defined as follows.

Definition 42. Let 𝐴_𝐼 be a trapezoidal intuitionistic fuzzy number. Then the accuracy score of𝐴_𝐼is defined by

𝐽₄(𝐴_𝐼) = ∫1 0 [𝛼_𝜇 𝐴_𝐼𝐿 +𝛼𝜇𝐴_𝐼𝑈] 4 𝑑𝛼 + ∫ 1 0 [𝛽_] 𝐴_𝐼𝐿 +𝛽]𝐴_𝐼𝑈] 4 𝑑𝛽 = (𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎2]󸀠 + 𝑎󸀠3]+ 𝑎󸀠4]) 8 . (26)

The proofs of the following propositions are immediate from Definition 42 and hence they are omitted.

Proposition 43. For any real number 𝑟 ∈ [0, 1], 𝐽₄(𝑟) = 𝑟.

Proposition 44. For any trapezoidal fuzzy number 𝐴_𝐼 =

(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇), 𝐽₄(𝐴_𝐼) = (𝑎_1𝜇+ 𝑎_2𝜇+ 𝑎_3𝜇+ 𝑎_4𝜇)/4.

Proposition 45. Let 𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇), (𝑎󸀠

1], 𝑎󸀠2], 𝑎3]󸀠 )⟩ be a

triangular intuitionistic fuzzy number. Then𝐽₄(𝐴_𝐼) = ((𝑎_1𝜇+

2𝑎_2𝜇+ 𝑎_3𝜇) + (𝑎󸀠

1]+ 2𝑎2]󸀠 + 𝑎3]󸀠 ))/8.

Proposition 46. Let 𝐴_𝐼= ([𝑎_1𝜇, 𝑎_2𝜇], [𝑎󸀠_1], 𝑎_2]󸀠 ]) be an interval

valued intuitionistic fuzzy number. Then𝐽₄(𝐴_𝐼) = ((𝑎_1𝜇+𝑎_2𝜇)+

(𝑎_1]󸀠 + 𝑎_2]󸀠))/4.

Note 8. The accuracy score can be calculated to any TrIFN.

But𝐽₄gives total order on𝐶₄which is proved in Theorems 47 and 48.

Theorem 47. Let 𝐴𝐼, 𝐵𝐼 ∈ 𝐶4. If𝐴𝐼 ≺ 𝐵𝐼 then𝐽4(𝐴𝐼) <

𝐽₄(𝐵_𝐼).

Proof. Let us assume that𝐴_𝐼 ≺ 𝐵_𝐼. We claim that𝐽₄(𝐵_𝐼) − 𝐽₄(𝐴_𝐼) > 0. By Definition 41, we have 𝑎_1𝜇≤ 𝑏_1𝜇, 𝑎2𝜇≤ 𝑏2𝜇, 𝑎_3𝜇≤ 𝑏_3𝜇, 𝑎_4𝜇≤ 𝑏_4𝜇, 𝑎󸀠_1]≤ 𝑏_1]󸀠 , 𝑎󸀠_2]≤ 𝑏_2]󸀠 , 𝑎󸀠_3]≤ 𝑏_3]󸀠 , 𝑎󸀠_4]≤ 𝑏_4]󸀠 . (27) Now, 8 (𝐽₄(𝐵_𝐼) − 𝐽₄(𝐴_𝐼)) = [(𝑏_1𝜇− 𝑎_1𝜇) + (𝑏_2𝜇− 𝑎_2𝜇) + (𝑏_3𝜇− 𝑎_3𝜇) + (𝑏_4𝜇− 𝑎_4𝜇)] + [(𝑏_1]󸀠 − 𝑎_1]󸀠 ) + (𝑏󸀠 2]− 𝑎󸀠2]) + (𝑏3]󸀠 − 𝑎󸀠3]) + (𝑏4]󸀠 − 𝑎󸀠4])] . (28)

From (27) it is very easy to see that all the terms in (28) are positive. Therefore their sum is also positive. From Defini-tion 41, we know that at least one of the above inequalities in (27) becomes strict inequality and hence we get𝐽₄(𝐵_𝐼) − 𝐽₄(𝐴_𝐼) > 0.

Theorem 48. Let 𝐴_𝐼, 𝐵_𝐼∈ 𝐶₄such that𝐽₄(𝐴_𝐼) = 𝐽₄(𝐵_𝐼); then

𝐴_𝐼= 𝐵_𝐼.

Proof. Let us assume𝐽₄(𝐴_𝐼) = 𝐽₄(𝐵_𝐼). We claim that 𝐴_𝐼= 𝐵_𝐼. By Definition 41, without loss of generality, we have

𝑎_1𝜇≤ 𝑏_1𝜇, 𝑎2𝜇≤ 𝑏2𝜇, 𝑎_3𝜇≤ 𝑏_3𝜇, 𝑎_4𝜇≤ 𝑏_4𝜇, 𝑎󸀠_1]≥ 𝑏_1]󸀠 , 𝑎󸀠_2]≥ 𝑏_2]󸀠 , 𝑎󸀠_3]≥ 𝑏_3]󸀠 , 𝑎󸀠_4]≥ 𝑏_4]󸀠 . (29) Now, 8 (𝐽₄(𝐵_𝐼) − 𝐽₄(𝐴_𝐼)) = [(𝑏_1𝜇− 𝑎_1𝜇) + (𝑏_2𝜇− 𝑎_2𝜇) + (𝑏_3𝜇− 𝑎_3𝜇) + (𝑏_4𝜇− 𝑎_4𝜇)] + [(𝑏_1]󸀠 − 𝑎_1]󸀠 ) + (𝑏_2]󸀠 − 𝑎󸀠_2]) + (𝑏_3]󸀠 − 𝑎󸀠_3]) + (𝑏_4]󸀠 − 𝑎_4]󸀠 )] = 0. (30)

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. Hence𝑎_1𝜇 = 𝑏_1𝜇, 𝑎_2𝜇 = 𝑏_2𝜇, 𝑎_3𝜇 = 𝑏_3𝜇, 𝑎_4𝜇 = 𝑏_4𝜇and𝑎󸀠_1] = 𝑏_1]󸀠,𝑎_2]󸀠 = 𝑏_2]󸀠 , 𝑎󸀠_3] = 𝑏_3]󸀠 , 𝑎󸀠_4] = 𝑏_4]󸀠 . Hence𝐴_𝐼= 𝐵_𝐼.

Definition 49. If𝐽₄(𝐴_𝐼) > 𝐽₄(𝐵_𝐼)(𝐽₄(𝐴_𝐼) < 𝐽₄(𝐵_𝐼)) then 𝐴_𝐼>

𝐵_𝐼(𝐴_𝐼< 𝐵_𝐼).

Ranking procedure defined in Definition 49 is illustrated in the following example.

Example 50. Let 𝐴_𝐼 = ⟨(0.1, 0.25, 0.37, 0.49), (0.1, 0.2, 0.4,

0.6)⟩ and 𝐵_𝐼 = ⟨(0.2, 0.37, 0.47, 0.6), (0.15, 0.32, 0.52, 0.7)⟩ ∈ 𝐶₄. Now𝐽₄(𝐴_𝐼) = 0.31375 and 𝐽₄(𝐵_𝐼) = 0.41625, 𝐽₄(𝐴_𝐼) < 𝐽₄(𝐵_𝐼).

Hence𝐴_𝐼< 𝐵_𝐼.

The relative importance of the score function 𝐽₄ in discriminating two different TrIFNs when𝐽₁, 𝐽₂, and𝐽₃fail to discriminate them is explained by Example 51.

Example 51. Let𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇), (𝑎_1]󸀠 , 𝑎_2]󸀠 , 𝑎󸀠_3], 𝑎_4]󸀠 )⟩ and𝐵_𝐼 = ⟨(𝑎_1𝜇+ 𝜖, 𝑎_2𝜇+ 𝜖, 𝑎_3𝜇+ 𝜖, 𝑎_4𝜇+ 𝜖), (𝑎󸀠_1]+ 𝜖, 𝑎_2]󸀠 + 𝜖, 𝑎󸀠 3]+ 𝜖, 𝑎4]󸀠 + 𝜖)⟩ ∈ TrIFN, and 𝜖 ∈ [0, 1]. Now 𝐽1(𝐴𝐼) = 𝐽1(𝐵𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎2]󸀠 ) − (𝑎󸀠3]+ 𝑎4]󸀠 ) 8 , 𝐽2(𝐴𝐼) = 𝐽2(𝐵𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) − (𝑎 󸀠 1]+ 𝑎2]󸀠 ) + (𝑎󸀠3]+ 𝑎4]󸀠 ) 8 , 𝐽3(𝐴𝐼) = 𝐽3(𝐵𝐼) = (𝑎 󸀠 1]+ 𝑎2]󸀠 + 𝑎3]󸀠 + 𝑎4]󸀠 ) − (𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) 8 . (31) But 𝐽4(𝐴𝐼) =(𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎2]󸀠 + 𝑎3]󸀠 + 𝑎󸀠4]) 8 , 𝐽4(𝐵𝐼) =(𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎2]󸀠 + 𝑎3]󸀠 + 𝑎󸀠4]) 8 + 𝜖 󳨐⇒ 𝐽4(𝐴𝐼) < 𝐽4(𝐵𝐼) . (32) Hence𝐴_𝐼< 𝐵_𝐼. Example 52. Let𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇), (𝑎_1]󸀠 , 𝑎_2]󸀠 , 𝑎󸀠_3], 𝑎_4]󸀠 )⟩ and𝐵_𝐼 = ⟨(𝑎_1𝜇+ 𝜖, 𝑎_2𝜇− 𝜖, 𝑎_3𝜇+ 𝜖, 𝑎_4𝜇− 𝜖), (𝑎󸀠_1]+ 𝜖, 𝑎_2]󸀠 − 𝜖, 𝑎_3]󸀠 + 𝜖, 𝑎_4]󸀠 − 𝜖)⟩ ∈ TrIFN and 𝜖 ∈ [0, 1] but not in 𝐶4.

(0, 1) (0, 0) b 󳰀 1] a 󳰀 1] b 󳰀 2] 󳰀 2_a] a2𝜇 b2𝜇 a1𝜇 _b1𝜇 a3𝜇 b3𝜇 a4𝜇 b4𝜇 a 󳰀 4] a 󳰀 3] b 󳰀 4] b 󳰀 3] Figure 6: Pictorial representation of Example 52.

Now 𝐽₁(𝐴_𝐼) = 𝐽₁(𝐵_𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎2]󸀠) − (𝑎󸀠3]+ 𝑎4]󸀠) 8 , 𝐽₂(𝐴_𝐼) = 𝐽₂(𝐵_𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) − (𝑎 󸀠 1]+ 𝑎2]󸀠) + (𝑎󸀠3]+ 𝑎4]󸀠) 8 , 𝐽₃(𝐴_𝐼) = 𝐽₃(𝐵_𝐼) = (𝑎 󸀠 1]+ 𝑎2]󸀠 + 𝑎󸀠3]+ 𝑎4]󸀠 ) − (𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) 8 , 𝐽₄(𝐴_𝐼) = 𝐽₄(𝐵_𝐼) = (𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎2]󸀠 + 𝑎󸀠3]+ 𝑎󸀠4]) 8 . (33) But𝐴_𝐼 ̸= 𝐵_𝐼.

Pictorial representation of this example is given in Fig-ure 6.

Example 52 shows that even 𝐽₁, 𝐽₂, 𝐽₃, and𝐽₄ 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 by𝐽₁, 𝐽₂, 𝐽₃, and𝐽₄.

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 intu-itionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.

Definition 53. Let𝐴_𝐼and𝐵_𝐼be two TrIFNs.

A special subclass𝐶₅of the set of TrIFNs consist of TrIFNs for which every pair of𝐴_𝐼and𝐵_𝐼are with𝑎_1𝜇 ≥ 𝑏_1𝜇, 𝑎_2𝜇 ≤ 𝑏_2𝜇, 𝑎_3𝜇 ≥ 𝑏_3𝜇, 𝑎_4𝜇 ≤ 𝑏_4𝜇 and 𝑎_1]󸀠 ≥ 𝑏_1]󸀠, 𝑎_2]󸀠 ≤ 𝑏_2]󸀠 , 𝑎_3]󸀠 ≥ 𝑏_3]󸀠, 𝑎_4]󸀠 ≤ 𝑏_4]󸀠. The preference relation on𝐶₅ is denoted as

⊴ and defined as follows: if 𝐴_𝐼, 𝐵_𝐼 ∈ 𝐶₅ ⊂ TrIFN such that 𝐴_𝐼 ⊴ 𝐵_𝐼then𝑎_1𝜇 ≥ 𝑏_1𝜇, 𝑎_2𝜇 ≤ 𝑏_2𝜇, 𝑎_3𝜇 ≥ 𝑏_3𝜇, 𝑎_4𝜇 ≤ 𝑏_4𝜇and 𝑎󸀠

1]≥ 𝑏1]󸀠 , 𝑎󸀠2]≤ 𝑏2]󸀠 , 𝑎3]󸀠 ≥ 𝑏3]󸀠, 𝑎4]󸀠 ≤ 𝑏4]󸀠. If𝐴𝐼⊲ 𝐵𝐼then one of the above inequalities becomes strict inequality.

Note 9. By Definition 53, we note that any pair of members

of𝐶₅are related under⊲.

The score function which measures the spread is defined as follows.

Definition 54. Let 𝐴_𝐼 be a trapezoidal intuitionistic fuzzy number. Then the spread score of𝐴_𝐼is defined by

𝐽5(𝐴𝐼) = (𝑎1𝜇+ 𝑎3𝜇) − (𝑎2𝜇+ 𝑎4𝜇) − (𝑎 󸀠 2]+ 𝑎󸀠4]) + (𝑎󸀠1]+ 𝑎3]󸀠) 8 . (34)

The proofs of the following propositions are immediate from Definition 54 and hence they are omitted.

Proposition 55. For any real number 𝑟 ∈ [0, 1], 𝐽₅(𝑟) = 0.

Proposition 56. If 𝐴_𝐼 = (𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇) is a trapezoidal

fuzzy number, then𝐽₅(𝐴_𝐼) = ((𝑎_1𝜇+ 𝑎_3𝜇) − (𝑎_2𝜇+ 𝑎_4𝜇))/4.

Proposition 57. Let 𝐴_𝐼= ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇), (𝑎󸀠

1], 𝑎󸀠2], 𝑎3]󸀠 )⟩ be a

triangular intuitionistic fuzzy number. Then𝐽₅(𝐴_𝐼) = (−(𝑎_3𝜇−

𝑎_1𝜇) − (𝑎󸀠

3]− 𝑎1]󸀠 ))/8.

Proposition 58. Let 𝐴_𝐼= ([𝑎_1𝜇, 𝑎_2𝜇], [𝑎_1]󸀠 , 𝑎󸀠_2]]) be an interval

valued intuitionistic fuzzy number. Then𝐽₅(𝐴_𝐼) = 0.

Theorem 59. Let 𝐴_𝐼, 𝐵_𝐼 ∈ 𝐶₅. If𝐴_𝐼 ⊲ 𝐵_𝐼 then𝐽₅(𝐴_𝐼) >

𝐽₅(𝐵_𝐼).

Proof. Let us assume that𝐴_𝐼 ⊲ 𝐵_𝐼. We claim that𝐽₅(𝐴_𝐼) − 𝐽₅(𝐵_𝐼) > 0. By Definition 53, we have 𝑎_1𝜇≥ 𝑏_1𝜇, 𝑎_2𝜇≤ 𝑏_2𝜇, 𝑎_3𝜇≥ 𝑏_3𝜇, 𝑎_4𝜇≤ 𝑏_4𝜇, 𝑎󸀠_1]≥ 𝑏_1]󸀠 , 𝑎󸀠_2]≤ 𝑏_2]󸀠 , 𝑎󸀠_3]≥ 𝑏_3]󸀠 , 𝑎󸀠 4]≤ 𝑏4]󸀠 . (35) Now, 8 (𝐽5(𝐴𝐼) − 𝐽5(𝐵𝐼)) = [(𝑎1𝜇− 𝑏1𝜇) + (𝑏2𝜇− 𝑎2𝜇) + (𝑎_3𝜇− 𝑏_3𝜇) + (𝑏_4𝜇− 𝑎_4𝜇)] + [(𝑎󸀠_1]− 𝑏_1]󸀠) + (𝑏_2]󸀠 − 𝑎󸀠_2]) + (𝑎󸀠_3]− 𝑏_3]󸀠) + (𝑏_4]󸀠 − 𝑎_4]󸀠 )] . (36)

From (35) it is very easy to see that all the terms in (36) are positive. Therefore their sum is also positive. From Defini-tion 53, we know that at least one of the above inequalities in (35) becomes strict inequality and hence we get𝐽₅(𝐴_𝐼) − 𝐽₅(𝐵_𝐼) > 0.

Theorem 60. Let 𝐴_𝐼, 𝐵_𝐼 ∈ 𝐶₅. If𝐽₅(𝐴_𝐼) = 𝐽₅(𝐵_𝐼) then 𝐴_𝐼=

𝐵_𝐼.

Proof. Let us assume𝐽₅(𝐴_𝐼) = 𝐽₅(𝐵_𝐼). We claim that 𝐴_𝐼= 𝐵_𝐼. By Definition 53, without loss of generality, we have

𝑎_1𝜇≥ 𝑏_1𝜇, 𝑎2𝜇≤ 𝑏2𝜇, 𝑎_3𝜇≥ 𝑏_3𝜇, 𝑎_4𝜇≤ 𝑏_4𝜇, 𝑎󸀠_1]≥ 𝑏_1]󸀠 , 𝑎󸀠_2]≤ 𝑏_2]󸀠 , 𝑎󸀠_3]≥ 𝑏_3]󸀠 , 𝑎󸀠_4]≤ 𝑏_4]󸀠 . (37) Now, 8 (𝐽₅(𝐴_𝐼) − 𝐽₅(𝐵_𝐼)) = [(𝑎_1𝜇− 𝑏_1𝜇) + (𝑏_2𝜇− 𝑎_2𝜇) + (𝑎3𝜇− 𝑏3𝜇) + (𝑏4𝜇− 𝑎4𝜇)] + [(𝑎󸀠1]− 𝑏1]󸀠) + (𝑏_2]󸀠 − 𝑎󸀠_2]) + (𝑎_3]󸀠 − 𝑏_3]󸀠) + (𝑏_4]󸀠 − 𝑎󸀠_4])] = 0. (38)

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. Hence𝑎_1𝜇 = 𝑏_1𝜇, 𝑎_2𝜇 = 𝑏_2𝜇, 𝑎_3𝜇 = 𝑏_3𝜇, 𝑎_4𝜇 = 𝑏_4𝜇and𝑎_1]󸀠 = 𝑏_1]󸀠 , 𝑎󸀠_2] = 𝑏_2]󸀠, 𝑎_3]󸀠 = 𝑏_3]󸀠, 𝑎_4]󸀠 = 𝑏_4]󸀠 . Hence𝐴_𝐼= 𝐵_𝐼.

Note 10. The spread score can be calculated to any TrIFN. But

𝐽₅gives total order on𝐶₅which is seen from Theorems 59 and 60.

Definition 61. If𝐽₅(𝐴_𝐼) > 𝐽₅(𝐵_𝐼)(𝐽₅(𝐴_𝐼) < 𝐽₅(𝐵_𝐼)) then 𝐴_𝐼>

𝐵_𝐼(𝐴_𝐼< 𝐵_𝐼).

The following example is used to explain the ranking procedure defined on𝐶₅.

Example 62. Let 𝐴_𝐼 = ⟨(0.15, 0.23, 0.31, 0.45), (0.1, 0.16, 0.35, 0.5)⟩ and 𝐵_𝐼 = ⟨(0.1, 0.25, 0.25, 0.5), (0, 0.2, 0.3, 0.6)⟩ ∈ 𝐶₅. Now𝐽₅(𝐴_𝐼) = −0.05375 and 𝐽₅(𝐵_𝐼) = −0.1125, 𝐽₅(𝐴_𝐼) > 𝐽₅(𝐵_𝐼). Hence𝐴_𝐼> 𝐵_𝐼. Example 63. Let𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇), (𝑎_1]󸀠 , 𝑎_2]󸀠 , 𝑎󸀠_3], 𝑎_4]󸀠 )⟩ and𝐵_𝐼 = ⟨(𝑎_1𝜇+ 𝜖, 𝑎_2𝜇− 𝜖, 𝑎_3𝜇+ 𝜖, 𝑎_4𝜇− 𝜖), (𝑎󸀠_1]+ 𝜖, 𝑎_2]󸀠 − 𝜖, 𝑎󸀠 3]+ 𝜖, 𝑎4]󸀠 − 𝜖)⟩ ∈ TrIFN and 𝜖 ∈ [0, 1]. Now 𝐽1(𝐴𝐼) = 𝐽1(𝐵𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎󸀠2]) − (𝑎󸀠3]+ 𝑎4]󸀠) 8 , 𝐽₂(𝐴_𝐼) = 𝐽₂(𝐵_𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) − (𝑎 󸀠 1]+ 𝑎󸀠2]) + (𝑎󸀠3]+ 𝑎4]󸀠) 8 , 𝐽3(𝐴𝐼) = 𝐽3(𝐵𝐼) = (𝑎 󸀠 1]+ 𝑎2]󸀠 + 𝑎󸀠3]+ 𝑎4]󸀠 ) − (𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) 8 , 𝐽4(𝐴𝐼) = 𝐽4(𝐵𝐼) = (𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎2]󸀠 + 𝑎󸀠3]+ 𝑎󸀠4]) 8 . (39) But 𝐽5(𝐴𝐼) =(𝑎1𝜇+ 𝑎3𝜇) − (𝑎2𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎󸀠3]) − (𝑎2]󸀠 + 𝑎4]󸀠 ) 8 , 𝐽5(𝐵𝐼) =(𝑎1𝜇+ 𝑎3𝜇) − (𝑎2𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎󸀠3]) − (𝑎2]󸀠 + 𝑎4]󸀠 ) 8 + 𝜖 󳨐⇒ 𝐽5(𝐵𝐼) > 𝐽5(𝐴𝐼) . (40)

Hence 𝐵_𝐼 > 𝐴_𝐼. In this example the importance of 𝐽₅ in ranking arbitrary TrIFNs is shown.

The inability of𝐽₅in comparing any two TrIFNs is shown in the following example.

Example 64. Let𝐴_𝐼 = ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇), (𝑎_1]󸀠 , 𝑎_2]󸀠 , 𝑎󸀠_3], 𝑎_4]󸀠 )⟩

and𝐵_𝐼= ⟨(𝑎_1𝜇+𝜖, 𝑎_2𝜇−𝜖, 𝑎_3𝜇+𝜖, 𝑎_4𝜇−𝜖), (𝑎_1]󸀠 −𝜖, 𝑎󸀠_2]+𝜖, 𝑎_3]󸀠 − 𝜖, 𝑎_4]󸀠 + 𝜖)⟩ ∈ TrIFN and 𝜖 ∈ [0, 1] but not in 𝐶5.

(0, 1) (0, 0) b 󳰀 1] 󳰀 1_a] b 󳰀 2] a 󳰀 2] a2𝜇 b2𝜇 a1𝜇 _b1𝜇 a3𝜇 b3𝜇 a4𝜇 b4𝜇 a 󳰀 4] a 󳰀 3] b 󳰀 4] b 󳰀 3] Figure 7: Pictorial representation of Example 64.

Now 𝐽1(𝐴𝐼) = 𝐽1(𝐵𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎󸀠2]) − (𝑎󸀠3]+ 𝑎4]󸀠 ) 8 , 𝐽2(𝐴𝐼) = 𝐽2(𝐵𝐼) = (𝑎1𝜇+ 𝑎2𝜇) − (𝑎3𝜇+ 𝑎4𝜇) − (𝑎 󸀠 1]+ 𝑎󸀠2]) + (𝑎󸀠3]+ 𝑎4]󸀠 ) 8 , 𝐽3(𝐴𝐼) = 𝐽3(𝐵𝐼) = (𝑎 󸀠 1]+ 𝑎2]󸀠 + 𝑎3]󸀠 + 𝑎4]󸀠 ) − (𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) 8 , 𝐽4(𝐴𝐼) = 𝐽4(𝐵𝐼) = (𝑎1𝜇+ 𝑎2𝜇+ 𝑎3𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎󸀠2]+ 𝑎3]󸀠 + 𝑎4]󸀠) 8 , 𝐽5(𝐴𝐼) = 𝐽5(𝐵𝐼) = (𝑎1𝜇+ 𝑎3𝜇) − (𝑎2𝜇+ 𝑎4𝜇) + (𝑎 󸀠 1]+ 𝑎󸀠3]) − (𝑎󸀠2]+ 𝑎4]󸀠 ) 8 . (41) But𝐴_𝐼 ̸= 𝐵_𝐼.

Pictorial representation of this example is given in Fig-ure 7.

Example 64 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. Let𝐴_𝐼and𝐵_𝐼be two TrIFNs.

A special subclass𝐶₆of the set of TrIFNs consist of TrIFNs for which every pair of𝐴_𝐼and𝐵_𝐼are with𝑎_1𝜇 ≥ 𝑏_1𝜇, 𝑎_2𝜇 ≤ 𝑏_2𝜇, 𝑎_3𝜇 ≥ 𝑏_3𝜇, 𝑎_4𝜇 ≤ 𝑏_4𝜇 and 𝑎_1]󸀠 ≤ 𝑏_1]󸀠, 𝑎_2]󸀠 ≥ 𝑏_2]󸀠, 𝑎_3]󸀠 ≤ 𝑏󸀠

3], 𝑎4]󸀠 ≥ 𝑏4]󸀠. The exact relation on𝐶6is denoted as⊵ and defined as follows: if𝐴_𝐼, 𝐵_𝐼∈ 𝐶₆⊂ TrIFN such that 𝐴_𝐼⊵ 𝐵_𝐼 then𝑎_1𝜇 ≥ 𝑏_1𝜇, 𝑎_2𝜇 ≤ 𝑏_2𝜇, 𝑎_3𝜇 ≥ 𝑏_3𝜇, 𝑎_4𝜇 ≤ 𝑏_4𝜇and 𝑎󸀠_1] ≤ 𝑏󸀠

1], 𝑎2]󸀠 ≥ 𝑏2]󸀠, 𝑎3]󸀠 ≤ 𝑏3]󸀠, 𝑎4]󸀠 ≥ 𝑏4]󸀠 . If𝐴𝐼⊳ 𝐵𝐼then one of the above inequalities becomes strict inequality.

Note 11. By Definition 65, we note that any pair of members

of𝐶₆are related under⊳.

The score function which measures the nonaccuracy is defined as follows.

Definition 66. Let 𝐴_𝐼 be a trapezoidal intuitionistic fuzzy number. Then the nonaccuracy score of𝐴_𝐼is defined by

𝐽6(𝐴𝐼) = (𝑎2𝜇+ 𝑎4𝜇) − (𝑎1𝜇+ 𝑎3𝜇) + (𝑎 󸀠 1]+ 𝑎󸀠3]) − (𝑎󸀠2]+ 𝑎4]󸀠) 8 . (42)

The proofs of the following propositions are easy and hence they are omitted.

Proposition 67. For any real number 𝑟 ∈ [0, 1], 𝐽₆(𝑟) = 0.

Proposition 68. If 𝐴_𝐼 = (𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇, 𝑎_4𝜇) is a trapezoidal

fuzzy number, then𝐽₆(𝐴_𝐼) = 0.

Proposition 69. Let 𝐴_𝐼= ⟨(𝑎_1𝜇, 𝑎_2𝜇, 𝑎_3𝜇), (𝑎󸀠

1], 𝑎2]󸀠, 𝑎3]󸀠 )⟩ be a

triangular intuitionistic fuzzy number. Then𝐽₆(𝐴_𝐼) = ((𝑎_1]󸀠 −

𝑎󸀠

3]) − (𝑎1𝜇− 𝑎3𝜇))/8.

Proposition 70. Let 𝐴_𝐼= ([𝑎_1𝜇, 𝑎_2𝜇], [𝑎󸀠

1], 𝑎󸀠2]]) be an interval

valued intuitionistic fuzzy number. Then𝐽₆(𝐴_𝐼) = 0.

Note 12. The nonaccuracy score can be calculated to any

TrIFN. But 𝐽₆ gives total order on 𝐶₆ which is proved in Theorems 71 and 72.

Theorem 71. Let 𝐴_𝐼, 𝐵_𝐼 ∈ 𝐶₆. If𝐴_𝐼 ⊳ 𝐵_𝐼then𝐽₆(𝐴_𝐼) <

𝐽₆(𝐵_𝐼).

Proof. Let us assume that𝐴_𝐼 ⊳ 𝐵_𝐼. We claim that𝐽₆(𝐵_𝐼) − 𝐽₆(𝐴_𝐼) > 0. By Definition 65, we have 𝑎_1𝜇≥ 𝑏_1𝜇, 𝑎2𝜇≤ 𝑏2𝜇, 𝑎_3𝜇≥ 𝑏_3𝜇, 𝑎_4𝜇≤ 𝑏_4𝜇, 𝑎󸀠_1]≤ 𝑏_1]󸀠 , 𝑎󸀠_2]≥ 𝑏_2]󸀠 , 𝑎󸀠_3]≤ 𝑏_3]󸀠 , 𝑎󸀠_4]≥ 𝑏_4]󸀠 (43) and at least one of these inequalities becomes strict.

Now,

8 (𝐽₆(𝐵_𝐼) − 𝐽₆(𝐴_𝐼)) = [(𝑎_1𝜇− 𝑏_1𝜇) + (𝑏_2𝜇− 𝑎_2𝜇) + (𝑎_3𝜇− 𝑏_3𝜇) + (𝑏_4𝜇− 𝑎_4𝜇)] + [(𝑏_1]󸀠 − 𝑎_1]󸀠 ) + (𝑎󸀠_2]− 𝑏_2]󸀠) + (𝑏_3]󸀠 − 𝑎_3]󸀠 ) + (𝑎_4]󸀠 − 𝑏_4]󸀠 )] .

(44)

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 get𝐽₆(𝐵_𝐼) − 𝐽₆(𝐴_𝐼) > 0.

Theorem 72. Let 𝐴_𝐼, 𝐵_𝐼 ∈ 𝐶₆. If𝐽₆(𝐴_𝐼) = 𝐽₆(𝐵_𝐼) then 𝐴_𝐼 =

𝐵_𝐼.

Proof. Let us assume𝐽₆(𝐴_𝐼) = 𝐽₆(𝐵_𝐼). We claim that 𝐴_𝐼= 𝐵_𝐼. By Definition 65, without loss of generality, we have

𝑎_1𝜇≥ 𝑏_1𝜇, 𝑎2𝜇≤ 𝑏2𝜇, 𝑎_3𝜇≥ 𝑏_3𝜇, 𝑎_4𝜇≤ 𝑏_4𝜇, 𝑎󸀠_1]≤ 𝑏_1]󸀠 , 𝑎󸀠 2]≥ 𝑏2]󸀠 , 𝑎󸀠_3]≤ 𝑏_3]󸀠 , 𝑎󸀠_4]≥ 𝑏_4]󸀠 . (45) Now, 8 (𝐽₆(𝐵_𝐼) − 𝐽₆(𝐴_𝐼)) = [(𝑎_1𝜇− 𝑏_1𝜇) + (𝑏_2𝜇− 𝑎_2𝜇) + (𝑎3𝜇− 𝑏3𝜇) + (𝑏4𝜇− 𝑎4𝜇)] + [(𝑏1]󸀠 − 𝑎1]󸀠 ) + (𝑎󸀠_2]− 𝑏_2]󸀠) + (𝑏_3]󸀠 − 𝑎_3]󸀠 ) + (𝑎_4]󸀠 − 𝑏_4]󸀠 )] = 0. (46)

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. Hence𝑎_1𝜇 = 𝑏_1𝜇, 𝑎_2𝜇 = 𝑏_2𝜇, 𝑎_3𝜇 = 𝑏_3𝜇, 𝑎_4𝜇 = 𝑏_4𝜇and𝑎_1]󸀠 = 𝑏_1]󸀠 , 𝑎󸀠_2] = 𝑏_2]󸀠, 𝑎_3]󸀠 = 𝑏_3]󸀠, 𝑎_4]󸀠 = 𝑏_4]󸀠 . Hence𝐴_𝐼= 𝐵_𝐼.

Definition 73. If𝐽₆(𝐴_𝐼) > 𝐽₆(𝐵_𝐼)(𝐽₆(𝐴_𝐼) < 𝐽₆(𝐵_𝐼)), then 𝐴_𝐼>

𝐵_𝐼(𝐴_𝐼< 𝐵_𝐼).

Ranking procedure introduced in Definition 73 is explained in Example 74.