A complete ranking of incomplete trapezoidal information

Abstract

The intuitionistic fuzzy sets (IFSs) introduced by Atanassov are widely applied in all areas such as data analysis, artificial intelligence, decision support systems in modelling problems with incomplete and imprecise information due to their better accuracy. More precisely, trapezoidal intuitionistic fuzzy numbers (TraIFNs) are able to model incomplete and imprecise information based on qualitative in nature. Many researchers have proposed different ranking methods on TraIFNs, but none of them has covered the entire class of TraIFNs and also almost all the methods have disadvantage that they ranked two different IFNs as the same at some point of time. In this paper, a complete ranking on the class of TraIFNs using axiomatic set of total ordering on some special subclasses of TraIFN based on different score functions and an algorithm for making decisions from information system with incomplete trapezoidal information using proposed ordering, have been studied and also the significance of our proposed method over existing methods is shown by comparing the proposed method with existing methods.

Keywords

Trapezoidal intuitionistic fuzzy number membership non-membership vague and non-vague score functions absolute perfect integral complete score functions

1 Introduction

Zadeh [24] proposed fuzzy set theory to measure qualitative and sensitive information by membership functions. Intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets introduced by Atanassov [1] and Atanassov and Gargov [2] provide a substitute tool which measure incomplete and imprecise information in real world problems by membership function, non-membership function and hesitance degree.

In many applications especially in decision analysis, ranking of trapezoidal intuitionistic fuzzy numbers plays an important role. Wang and Zhang [20], Ye [22] introduced the concept of expected value for TraIFNs and proposed the new programming method for solving multi-criteria decision-making problem modelled by trapezoidal intuitionistic fuzzy number. Chen and Tan [4] introduced a score function and Hong and Choi [8] defined an accuracy function of an intuitionistic fuzzy value based on theory of vague sets. Lakshmana et al. [12] defined a method of intuitionistic fuzzy scoring to intuitionistic fuzzy numbers that generalizes Chen and Hwang’s scoring method for ranking of intuitionistic fuzzy numbers. Different ranking methods for fuzzy numbers have been studied in [6 , 18]. Geetha et al. [7] defined complete ranking of intuitionistic fuzzy interval numbers as defined in Lakshmana et al. [14] and studied intuitionistic fuzzy interval information system by using dominance degree based on the ranking of intuitionistic fuzzy interval numbers [7].

The imprecise, qualitative and incomplete information can be modelled better using trapezoidal intuitionistic fuzzy number than intuitionistic fuzzy interval number as the generalisation of intuitionistic fuzzy values and intuitionistic fuzzy intervals. A complete ranking on the class of trapezoidal intuitionistic fuzzy number is an open problem worldwide. The existing techniques for ranking of trapezoidal intuitionistic fuzzy information are problem dependent due to the partial ordering of TraIFNs. Our proposed ranking method on TraIFN will give better results over other existing methods, and this paper will give the better understanding over this TraIFNs. This type of IFNs are very much important in real life problems and this paper will give the significant impact in the literature. Modeling problems using this type of IFN will give better result.

This paper is organised as follows. Some basic definitions are briefly introduced in Section 2. InSection 3, axiomatic order relations on TraIFNs using membership, non-membership, vague, non-vague, absolute, perfect, integral score and complete score functions have been introduced and proved that they individually give total order on some special classes of TraIFNs. Further illustrative examples to validate and develop systematically all score functions have been studied in all subsections. Section 4 is devoted to define a complete ranking on the collection of TraIFNs using the axiomatic set of total ordering on some special subclasses of TraIFN defined in Section 3. In Section 5, the significance of our proposed method is studied by comparing our method with existing methods using illustrative examples and an algorithmic approach to identify the best alternative from an weighted trapezoidal intuitionistic fuzzy information system is developed using the proposed ranking principle and demonstrated using real life numerical problem. Finally conclusions are given in Section 6.

2 Preliminaries

Some basic definitions are given in this section.

Definition 2.0.1. (Atanassov [1]). Let X be a nonemp-ty set. An intuitionistic fuzzy set (IFS) A in X is defined by A = 〈 (x, μ_A (x) , ν_A (x)) : x ∈ X〉, where μ_A (x) : X → [0, 1] and ν_A (x) : X → [0, 1] , x ∈ X with the conditions 0 ≤ μ_A (x) + ν_A (x) ≤1, ∀ x ∈ X. The numbers μ_A (x) , ν_A (x) ∈ [0, 1] denote the degree of membership and non-membership of x to lie in A respectively. For each intuitionistic fuzzy subset A in X, π_A (x) =1 - μ_A (x) - ν_A (x) is called hesitancy degree of x to lie in A.

Definition 2.0.2. (Atanassov & Gargov, [2]). Let D [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, μ_A (x) , ν_A (x)) : x ∈ X〉 where μ_A : X → D [0, 1] , ν_A : X → D [0, 1] with the condition $0 < \sup_{x} μ_{A} (x) + \sup_{x} ν_{A} (x) \leq 1$ .

The intervals μ_A (x) and ν_A (x) denote, respectively, the degree of belongingness and non-belongingness of the element x to lie in the set A. Thus for each x ∈ X, μ_A (x) and ν_A (x) are closed intervals whose lower and upper end points are, respectively, denoted by μ_{A
_L} (x), μ_{A
_U} (x) and ν_{A
_L} (x), ν_{A
_U} (x). We denote A = 〈 (x, [μ_{A
_L} (x) , μ_{A
_U} (x)] , [ν_{A
_L} (x) , ν_{A
_U} (x)]) : x ∈ X〉 where 0 < μ_{A
_U} (x) + ν_{A
_U} (x) ≤1.

For each element x ∈ X, we can compute the unknown degree (hesitancy degree) of belongingness π_A (x) to A as π_A (x) =1 - μ_A (x) - ν_A (x) = [1 - μ_{A
_U} (x) - ν_{A
_U} (x) , 1 - μ_{A
_L} (x) - ν_{A
_L} (x)]. An intuitionistic fuzzy interval is denoted by A = ([a, b] , [c, d]) for convenience.

Definition 2.0.3. (Burillo et al. [3]). A trapezoidal intuitionistic fuzzy number A is defined by A = 〈 (μ_A, ν_A) ∣ x ∈ R〉, where μ_A and ν_A are trapezoidal fuzzy numbers with $ν_{A} (x) \leq μ_{A}^{c} (x)$ .

We note that the condition (a_1ν, a_2ν, a_3ν, a_4ν) ≤ (a_1μ, a_2μ, a_3μ, a_4μ) ^c on membership and non-membership of the trapezoidal intuitionistic fuzzynumber A = 〈 (a_1μ, a_2μ, a_3μ, a_4μ) , (a_1ν, a_2ν, a_3ν, a_4ν) 〉, impose either a_1ν ≥ a_3μ and a_2ν ≥ a_4μ or a_4ν ≤ a_2μ and a_3ν ≤ a_1μ on the foots of trapezoidal intuitionistic fuzzy number [13].

If a_2μ = a_3μ (and a_2ν = a_3ν) in a trapezoidal intuitionistic fuzzy number A, we have the triangular intuitionistic fuzzy number (TIFN).

Definition 2.0.4. (Lakshmana et al. [12]). The membership score of the triangular fuzzy number A = 〈 (a_1μ, a_2μ, a_3μ) 〉 is defined by $T (A) = \frac{1 + R (A) - L (A)}{2}$ , where $L (A) = \frac{1 - a_{1 μ}}{1 + a_{2 μ} - a_{1 μ}}$ and $R (A) = \frac{a_{3 μ}}{1 + a_{3 μ} - a_{2 μ}}$ .

Definition 2.0.5. (Lakshmana et al. [12]). The new membership score of the triangular fuzzy number A = 〈 (a_1μ, a_2μ, a_3μ) 〉 is defined by $NT (A) = \frac{1 + NL (A) - NR (A)}{2}$ , where $NL (A) = \frac{a_{1 μ}}{1 + a_{1 μ} - a_{2 μ}}$ and $NR (A) = \frac{1 - a_{3 μ}}{1 + a_{2 μ} - a_{3 μ}}$ .

Definition 2.0.6. (Lakshmana et al. [12]). The non-membership score of the triangular fuzzy number A = 〈 (a_1μ, a_2μ, a_3μ) 〉 is defined by NT_c (A) =1 - NT (A).

Definition 2.0.7. (Lakshmana et al. [12]). Let A = 〈 (a_1μ, a_2μ, a_3μ) , (a_1ν, a_2ν, a_3ν) 〉 be a triangular intuitionistic fuzzy number. If a_1ν ≥ a_2μ and a_2ν ≥ a_3μ, then the score of the intuitionistic fuzzy number A is defined by (T, NT_c), where T is the membership score of A which is obtained from (a_1μ, a_2μ, a_3μ) and NT_c is the nonmembership score of A obtained from (a_1ν, a_2ν, a_3ν).

3 Ranking principle for trapezoidal intuitionistic fuzzy numbers

In this section, an axiomatic relations on the subclasses of TraIFNs are introduced and some of their properties are studied using illustrative examples. In this paper, A and B always denote trapezoidal intuitionistic fuzzy numbers A = 〈 (a_1μ, a_2μ, a_3μ, a_4μ) , (a_1ν, a_2ν, a_3ν, a_4ν) 〉 withconditions that a_1ν ≥ a_3μ, a_2ν ≥ a_4μ or a_4ν ≤ a_2μ, a_3ν ≤ a_1μ and B = 〈 (b_1μ, b_2μ, b_3μ, b_4μ) , (b_1ν, b_2ν, b_3ν, b_4ν) 〉 with conditions that b_1ν ≥ b_3μ, b_2ν ≥ b_4μ or b_4ν ≤ b_2μ, b_3ν ≤ b_1μ unless or otherwisespecifically they are stated and a_1μ, a_2μ, a_3μ, a_4μ, a_1ν, a_2ν, a_3ν, a_4ν, b_1μ, b_2μ, b_3μ, b_4μ, b_1ν, b_2ν, b_3ν, b_4ν ∈ [0, 1].

Definition 3.0.1. Let A and B be two TraIFNs. A subset relation ‘⊆’ on TraIFNs is defined by A ⊆ B if a_1μ ≤ b_1μ, a_2μ ≤ b_2μ, a_3μ ≤ b_3μ, a_4μ ≤ b_4μ and a_1ν ≥ b_1ν, a_2ν ≥ b_2ν, a_3ν ≥ b_3ν, a_4ν ≥ b_4ν. If A ⊂ B then atleast one of the above inequalities become strict inequality. The collection C₁ of TraIFNs is a subclass of TraIFNs for which every pair of A, B ∈ C₁ either A ⊆ B or B ⊆ A.

Definition 3.0.2. Let A and B be two TraIFNs. A vague relation ‘ $\subseteq_{1}$ ’ on TraIFNs is defined by A ⊆ ₁B if a_1μ ≥ b_1μ, a_2μ ≥ b_2μ, a_3μ ≤ b_3μ, a_4μ ≤ b_4μ and a_1ν ≤ b_1ν, a_2ν ≤ b_2ν, a_3ν ≥ b_3ν, a_4ν ≥ b_4ν. If A ⊂ ₁B, then atleast one of the above inequalities become strict inequality. The collection C₂ of TraIFNs is a subclass of TraIFNs for which every pair of A, B ∈ C₂ either A ⊆ ₁B or B ⊆ ₁A.

Definition 3.0.3. Let A and B be two TraIFNs. An absolute relation ‘ $\subseteq_{2}$ ’ on TraIFNs is defined by A ⊆ ₂B if a_1μ ≥ b_1μ, a_2μ ≤ b_2μ, a_3μ ≥ b_3μ, a_4μ ≤ b_4μ and a_1ν ≥ b_1ν, a_2ν ≤ b_2ν, a_3ν ≥ b_3ν, a_4ν ≤ b_4ν. If A ⊂ ₂B, then atleast one of the above inequalities become strict inequality. The collection C₃ of TraIFNs is a subclass of TraIFNs for which every pair of A, B ∈ C₃ either A ⊆ ₂B or B ⊆ ₂A.

Definition 3.0.4. Let A and B be two TraIFNs. A perfect relation ‘ $\subseteq_{3}$ ’ on TraIFNs is defined by A ⊆ ₃B if a_1μ ≥ b_1μ, a_2μ ≤ b_2μ, a_3μ ≥ b_3μ, a_4μ ≤ b_4μ and a_1ν ≤ b_1ν, a_2ν ≥ b_2ν, a_3ν ≤ b_3ν, a_4ν ≥ b_4ν. If A ⊂ ₃B, then atleast one of the above inequalities become strict inequality. The collection C₄ of TraIFNs is a subclass of TraIFNs for which every pair of A, B ∈ C₄ either A ⊆ ₃B or B ⊆ ₃A.

Definition 3.0.5. Let A and B be two TraIFNs. An integral relation ‘ $\subseteq_{4}$ ’ on TraIFNs is defined by A ⊆ ₄B if a_1μ ≤ b_1μ, a_2μ ≥ b_2μ, a_3μ ≥ b_3μ, a_4μ ≤ b_4μ and a_1ν ≥ b_1ν, a_2ν ≤ b_2ν, a_3ν ≤ b_3ν, a_4ν ≥ b_4ν. If A ⊂ ₄B, then atleast one of the above inequalities become strict inequality. The collection C₅ of TraIFNs is a subclass of TraIFNs for which every pair of A, B ∈ C₅ either A ⊆ ₄B or B ⊆ ₄A.

Definition 3.0.6. Let A and B be two TraIFNs. A complete relation ‘ $\subseteq_{5}$ ’ on TraIFNs is defined by A ⊆ ₅B if a_1μ ≤ b_1μ, a_2μ ≥ b_2μ, a_3μ ≥ b_3μ, a_4μ ≤ b_4μ and a_1ν ≤ b_1ν, a_2ν ≥ b_2ν, a_3ν ≥ b_3ν, a_4ν ≤ b_4ν. If A ⊂ ₅B, then atleast one of the above inequalities become strict inequality. The collection C₆ of TraIFNs is a subclass of TraIFNs for which every pair of A, B ∈ C₆ either A ⊆ ₅B or B ⊆ ₅A.

3.1 Axioms of ranking principle in intuitionistic fuzzy set up

Diverse ranking approaches are available in the literature for comparing TraIFNs, but none of them yield a complete ordering on the class of TraIFNs. In this subsection, in order to make the set of TraIFNs as totally ordered set, some axioms on TraIFNs are introduced that need to be fulfilled by a ranking principles for TraIFNs to evade the illogicalities.

Different important axioms on TraIFNs are defined as follows.

The collection C₁ is totally ordered under membership score and non-membership score.

The collection C₂ is totally ordered under vague score and non-vague score.

The collection C₃ is totally ordered under absolute score.

The collection C₄ is totally ordered under perfect score.

The collection C₅ is totally ordered under integral score.

The collection C₆ is totally ordered under complete score.

3.2 Total order on C₁

In this subsection, the new membership and non-membership score functions on the set of TraIFNs are introduced and some of their properties are studied using illustrative examples. Further a relation <₁ on TraIFNs which is a total order on C₁ is defined.

Definition 3.2.1. Let A be a trapezoidal intuitionistic fuzzy number. Then the membership score function L of A is defined as $\begin{matrix} L (A) = \frac{2 (a_{1 μ} + a_{2 μ} + a_{3 μ} + a_{4 μ}) - 2 (a_{1 ν} + a_{2 ν} + a_{3 ν} + a_{4 ν})}{8} \\ + \frac{(a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) + (a_{3 μ} + a_{4 μ}) (a_{3 ν} + a_{4 ν})}{8} . \end{matrix}$

The proofs of following propositions are immediate from the above definition.

Proposition 3.2.1.IfA = ([a_1μ, a_2μ] , [a_1ν, a_2ν]) is an interval valued intuitionistic fuzzy number, then $L (A) = \frac{(a_{1 μ} + a_{2 μ}) - (a_{1 ν} + a_{2 ν}) + (a_{1 μ} a_{1 ν} + a_{2 μ} a_{2 ν})}{2} .$

Proposition 3.2.2. If A = 〈 (a_1μ, a_2μ, a_3μ) , (a_1ν, a_2ν, a_3ν) 〉 is a triangular intuitionistic fuzzy number (TIFN), then $\begin{matrix} L (A) = \frac{2 (a_{1 μ} + 2 a_{2 μ} + a_{3 μ}) - 2 (a_{1 ν} + 2 a_{2 ν} + a_{3 ν})}{8} \\ + \frac{(a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) + (a_{2 μ} + a_{3 μ}) (a_{2 ν} + a_{3 ν})}{8} . \end{matrix}$

Theorem 3.2.1.LetA, B ∈ C₁. IfA ⊂ B, thenL (A) < L (B).

Proof. Let A, B ∈ C₁ such that A ⊂ B. By Definition 3.0.1, we have a_1μ ≤ b_1μ, a_2μ ≤ b_2μ, a_3μ ≤ b_3μ, a_4μ ≤ b_4μ, and a_1ν ≥ b_1ν, a_2ν ≥ b_2ν, a_3ν ≥ b_3ν, a_4ν ≥ b_4ν. Now, 8 (L (B) - L (A)) =2 (b_1μ - a_1μ)+2 (b_2μ - a_2μ) + 2 (b_3μ - a_3μ) +2 (b_4μ - a_4μ) +2 (a_1ν - b_1ν) +2 (a_2ν - b_2ν) +2 (a_3ν - b_3ν) + 2 (a_4ν - b_4ν) + (b_1μ + b_2μ) (b_1ν + b_2ν) - (a_1μ + a_2μ) (a_1ν + a_2ν) + (b_3μ + b_4μ) (b_3ν + b_4ν) - (a_3μ + a_4μ) (a_3ν + a_4ν) =2 (b_1μ - a_1μ) +2 (b_2μ - a_2μ) + 2 (b_3μ - a_3μ) + 2 (b_4μ - a_4μ) + (2 - (a_1μ + a_2μ)) ((a_1ν - b_1ν) + (a_2ν - b_2ν)) + ((b_1μ - a_1μ) + (b_2μ - a_2μ)) (b_1ν + b_2ν) + (b_3ν + b_4ν) ((b_3μ - a_3μ) + (b_4μ - a_4μ)) + (2 - (a_3μ + a_4μ)) ((a_3ν - b_3ν) + (a_4ν - b_4ν)).

By Definition 3.0.1, all the above terms are positive and from the assumption, we know that at least one of the above inequalitites become strict inequality and hence we get L (B) - L (A) >0. □

Theorem 3.2.2.LetA, B ∈ C₁such thatL (A) = L (B) thenμ_A = μ_B.

Proof. Let A, B ∈ C₁ such that L (A) = L (B). By Definition 3.0.1, without loss of generality, we havea_1μ ≤ b_1μ, a_2μ ≤ b_2μ, a_3μ ≤ b_3μ, a_4μ ≤ b_4μ, and a_1ν ≥ b_1ν, a_2ν ≥ b_2ν, a_3ν ≥ b_3ν, a_4ν ≥ b_4ν. Now, 8 (L (B) - L (A)) = 2 (b_1μ - a_1μ) + 2 (b_2μ - a_2μ) +2 (b_3μ - a_3μ) + 2 (b_4μ - a_4μ) + (2 - (a_1μ + a_2μ)) ((a_1ν - b_1ν) + (a_2ν - b_2ν)) + ((b_1μ - a_1μ) + (b_2μ - a_2μ)) (b_1ν + b_2ν) + (b_3ν + b_4ν) ((b_3μ - a_3μ) + (b_4μ - a_4μ)) + (2 - (a_3μ + a_4μ)) ((a_3ν - b_3ν) + (a_4ν - b_4ν)) =0.

Since each term of the above sum is positive, we have each term is equal to 0. Hence a_1μ = b_1μ, a_2μ = b_2μ, a_3μ = b_3μ, a_4μ = b_4μ. □

Definition 3.2.2. Let A, B ∈ TraIFN. A relation < on TraIFNs is defined by A < B if L (A) < L (B).

Ranking relation defined in Definition 3.2.2 is demonstrated in the following example.

Example 3.2.1. Let A = 〈(0.3, 0.3, 0.5, 0.5) , (0.5, 0.6, 0.7, 0.8) 〉, B = 〈(0.3, 0.4, 0.4, 0.5) , (0.5, 0.6, 0.7, 0.8) 〉 ∈ C₁. Now L (A) =0.02 and L (B) =0.015. Hence A > B.

Generally, the equality of membership score function (L) for two different TraIFNs in C₁ does not imply their equality, but it guarentees their equality in membership functions (μ) as in the following example.

Example 3.2.2. Let A = 〈(0.5, 0.5, 0.5, 0.5) , (0.5, 0.51, 0.7, 0.89) 〉, B = 〈(0.5, 0.5, 0.5, 0.5) , (0.5, 0.52, 0.7, 0.88) 〉 ∈ C₁. Now L (A) = L (B) =0.175 and μ_A = μ_B. But A ≠ B.

The following examples are used to show the inefficiency of membership score function (L) in ranking arbitrary TraIFNs. So the relation < is not a total order relation on C₁.

Example 3.2.3. Let $A = 〈 (0.2, 0.4, 0.5, 0.5), (0.5, 0.6, 0.7, 0.9) 〉, B = 〈 (\sqrt{0.0075}, \sqrt{0.0075}, \sqrt{0.0075}, \sqrt{0.0075}), (\sqrt{0.0075}, \sqrt{0.0075}, \sqrt{0.0075}, \sqrt{0.0075}) 〉 \in TraIFN$ . Now L (A) = L (B) =0.0075. But A ≠ B.

Example 3.2.4. Let $A = 〈 (a_{1 μ}, a_{2 μ}, a_{3 μ}, a_{4 μ}), (a_{1 ν}, a_{2 ν}, a_{3 ν}, a_{4 ν}) 〉, B = 〈 (\sqrt{L (A)}, \sqrt{L (A)}, \sqrt{L (A)}, \sqrt{L (A)}), (\sqrt{L (A)}, \sqrt{L (A)}, \sqrt{L (A)}, \sqrt{L (A)}) 〉 \in TraIFN$ . Now $L (A) = L (B) = \frac{1}{8} (2 (a_{1 μ} + a_{2 μ} + a_{3 μ} + a_{4 μ}) - 2 (a_{1 ν} + a_{2 ν} + a_{3 ν} + a_{4 ν}) + (a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) + (a_{3 μ} + a_{4 μ}) (a_{3 ν} + a_{4 ν}))$ . But A ≠ B.

Example 3.2.4 shows that, the membership score function alone can not be sufficient to discriminate any two arbitrary TraIFNs. Therefore there is a need for us to define some more score functions to compare arbitrary TraIFNs.

The new score function which measures the non-membershipness is defined as follows.

Definition 3.2.3. Let A be any trapezoidal intuitionistic fuzzy number. Then the non-membership score function (LG) of A is defined as $\begin{matrix} LG (A) = \frac{- 2 (a_{1 μ} + a_{2 μ} + a_{3 μ} + a_{4 μ}) + 2 (a_{1 ν} + a_{2 ν} + a_{3 ν} + a_{4 ν})}{8} \\ + \frac{(a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) + (a_{3 μ} + a_{4 μ}) (a_{3 ν} + a_{4 ν})}{8} . \end{matrix}$ The proofs of following propositions are immediate from the above definition.

Proposition 3.2.3.For any TraIFNA, L (A) = LG (A^c) and L (A^c) = LG (A).

Proposition 3.2.4.IfA = ([a_1μ, a_2μ] , [a_1ν, a_2ν]) is an interval valued intuitionistic fuzzy number, then $LG (A) = \frac{- (a_{1 μ} + a_{2 μ}) + (a_{1 ν} + a_{2 ν}) + (a_{1 μ} a_{1 ν} + a_{2 μ} a_{2 ν})}{2} .$

Proposition 3.2.5.IfA = 〈 (a_1μ, a_2μ, a_3μ) , (a_1ν, a_2ν, a_3ν) 〉 ∈ TIFN, then $LG (A) = \frac{1}{8} (- 2 (a_{1 μ} + 2 a_{2 μ} + a_{3 μ}) + 2 (a_{1 ν} + 2 a_{2 ν} + a_{3 ν}) + (a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) + (a_{2 μ} + a_{3 μ}) (a_{2 ν} + a_{3 ν}))$ .

The proofs of the following two theorems are similar to Theorems 3.2.1 and 3.2.2.

Theorem 3.2.3.LetA, B ∈ C₁such thatA ⊂ B. ThenLG (A) > LG (B).

Theorem 3.2.4.LetA, B ∈ C₁. IfLG (A) = LG (B) thenν_A = ν_B.

Definition 3.2.4. Let A, B ∈ TraIFN. A <₁ relation on TraIFNs is defined by A < ₁B if L (A) < L (B) or if L (A) = L (B) and LG (A) > LG (B).

The following example is immediate application of the above definition.

Example 3.2.5. In Example 3.2.3, we have L (A) = L (B) =0.0075. But LG (A) =0.5575 > LG (B) =0.0075 ⇒ A < ₁B.

Note 3.2.1. Now L (A) and LG (A) scores can be calculated to any TraIFNs, but these two scores together gives total order <₁ on the collection of C₁ is seen from the following theorem whose proof is immediate application of Theorems 3.2.2 and 3.2.4. Hence <₁ satisfies Axiom A₁.

Theorem 3.2.5.LetAandB ∈C₁. IfL (A) = L (B) andLG (A) = LG (B) thenA = B.

Note 3.2.2. If the membership and non-membership score functions are equal for two trapezoidal intuitionistic fuzzy numbers in C₁, then they are the same. But the membership and nonmembership score functions are not enough to rank any two distinct TraIFNs, which can be seen from the following Example 3.2.6. In other words, <₁ is not a total order on TraIFNs.

Example 3.2.6. Let A = 〈(0.3, 0.3, 0.7, 0.7) , (0.9, 0.9, 0.9, 0.9) 〉 and B = 〈(0.1, 0.1, 0.9, 0.9) , (0.9, 0.9, 0.9, 0.9) 〉 ∈ TraIFN. Now L (A) = L (B) =0.05 and LG (A) = LG (B) =0.85 but A ≠ B. In this case some additional information is required to rank them.

3.3 Total order on C₂

In this subsection, the vague and the non-vague score functions are defined on TraIFNs and some of their properties are studied using illustrative examples. Further a total order <₂ is defined on the subclass C₂.

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

Definition 3.3.1. Let A be any TraIFN. Then the vague score function of A is defined as $\begin{matrix} P_{1} (A) = \frac{2 (a_{1 μ} + a_{2 μ} - a_{3 μ} - a_{4 μ}) + 2 (- a_{1 ν} - a_{2 ν} + a_{3 ν} + a_{4 ν})}{8} \\ + \frac{(a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) - (a_{3 μ} + a_{4 μ}) (a_{3 ν} + a_{4 ν})}{8} . \end{matrix}$ The proofs of following propositions are immediate from the above definition.

Proposition 3.3.1.IfA = ([a_1μ, a_2μ] , [a_1ν, a_2ν]) is an interval valued intuitionistic fuzzy number, then $\begin{matrix} P_{1} (A) = \frac{(a_{1 μ} - a_{2 μ}) + (- a_{1 ν} + a_{2 ν}) + (a_{1 μ} a_{1 ν} - a_{2 μ} a_{2 ν})}{2} . \end{matrix}$

Proposition 3.3.2.IfA = 〈 (a_1μ, a_2μ, a_3μ) , (a_1ν, a_2ν, a_3ν) 〉 ∈ TIFN, then $P_{1} (A) = \frac{1}{8} (2 (a_{1 μ} - a_{3 μ}) + 2 (- a_{1 ν} + a_{3 ν}) + (a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) - (a_{2 μ} + a_{3 μ}) (a_{2 ν} + a_{3 ν}))$ .

The proofs of the following two theorems are similar to Theorem 3.2.1 and 3.2.2.

Theorem 3.3.1.LetA, B ∈ C₂such thatA ⊂ ₁B. ThenP₁ (A) > P₁ (B).

Theorem 3.3.2.LetA, B ∈ C₂such thatP₁ (A) = P₁ (B). Thenμ_A = μ_B.

Definition 3.3.2. A relation < on TraIFNs is defined by A < B if P₁ (A) > P₁ (B).

Ranking relation introduced in Definition 3.3.2 is demonstrated in the Example 3.3.1. Further the above relation is significant in some TraIFN where membership score and nonmembership score fail to rank is seen as in the Example 3.3.2.

Example 3.3.1. Let A = 〈(0.5, 0.5, 0.5, 0.5) , (0.5, 0.86, 1, 1) 〉, B = 〈(0.5, 0.5, 0.5, 0.5) , (0.5, 0.94, 1, 1) 〉 ∈ TraIFN. Now P₁ (A) =0.08 and P₁ (B) =0.07 ⇒ P₁ (A) > P₁ (B). Hence A < B.

Example 3.3.2. Let A = 〈(0.3, 0.3, 0.5, 0.5) , (0.6, 0.6, 0.6, 0.6) 〉 , B = 〈(0.4, 0.4, 0.4, 0.4) , (0.6, 0.6, 0.6, 0.6) 〉 ∈ TraIFN. Now L (A) = L (B) =0.04, LG (A) = LG (B) = 0.44, P₁ (A) = -0.16 and P₁ (B) =0. Hence B < A.

The following example shows the inefficiency of the score functions L, LG, and P₁ in comparing arbitrary TraIFNs.

Example 3.3.3. Let A = 〈(0.3, 0.3, 0.4, 0.4) , (0.6, 0.6, 0.6, 0.6) 〉, B = 〈(0.256608438, 0.256608438, 0.42783361, 0.42783361) , (0.506608438, 0.506608438, 0.67783361, 0.67783361) 〉 ∈ TraIFN. Now L (A) = L (B) = -0.04, LG (A) = LG (B) = 0.46, P₁ (A) = P₁ (B) = -0.08. But A ≠ B.

Example 3.3.3 shows that, some more score functions are needed to cover the entire collection of TraIFNs. Therefore, a new non-vague score function on TraIFNs is introduced.

The score function which measures the non-vagueness is defined as follows.

Definition 3.3.3. Let A be a TraIFN.

Then the Non-Vague score function of A is defined as

$\begin{matrix} P_{2} (A) = \frac{2 (- a_{1 μ} - a_{2 μ} + a_{3 μ} + a_{4 μ}) + 2 (a_{1 ν} + a_{2 ν} - a_{3 ν} - a_{4 ν})}{8} \\ + \frac{(a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) - (a_{3 μ} + a_{4 μ}) (a_{3 ν} + a_{4 ν})}{8} . \end{matrix}$ The proofs of following propositions are immediate from the above definition.

Proposition 3.3.3.IfA = ([a_1μ, a_2μ] , [a_1ν, a_2ν]) is an interval valued intuitionistic fuzzy number, then $\begin{matrix} P_{2} (A) = \frac{(- a_{1 μ} + a_{2 μ}) + (a_{1 ν} - a_{2 ν}) + (a_{1 μ} a_{1 ν} - a_{2 μ} a_{2 ν})}{2} . \end{matrix}$

Proposition 3.3.4.IfA = 〈 (a_1μ, a_2μ, a_3μ) , (a_1ν, a_2ν, a_3ν) 〉 ∈ TIFN, then

$\begin{matrix} P_{2} (A) = \frac{2 (- a_{1 μ} + a_{3 μ}) + 2 (a_{1 ν} - a_{3 ν})}{8} \\ + \frac{(a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) - (a_{2 μ} + a_{3 μ}) (a_{2 ν} + a_{3 ν})}{8} . \end{matrix}$ The proofs of the following two theorems are similar to Theorems 3.2.1 and 3.2.2.

Theorem 3.3.3.LetA, B ∈ C₂. IfA ⊂ ₁BthenP₂ (A) < P₂ (B).

Theorem 3.3.4.LetA, B ∈ C₂such thatP₂ (A) = P₂ (B), thenν_A = ν_B.

Definition 3.3.4. A relation <₂ on TraIFNs is defined by A < ₂B if P₁ (A) > P₁ (B) or if P₁ (A) = P₁ (B) and P₂ (A) < P₂ (B).

Note 3.3.1. If the vague and non-vague score functions are equal for two trapezoidal intuitionistic fuzzy numbers in C₂, then they are the same. Hence <₂ is a total order on C₂ and it satisfies the Axiom A₂.

Ranking principle introduced in Definition 3.3.4 is illustrated in the following example and also the relative importance of P₂ when L, LG, and P₁ fail to rank arbitrary TraIFNs is shown.

Example 3.3.4. In Example 3.3.3, we get L (A) = L (B) = -0.04, LG (A) = LG (B) =0.46, P₁ (A) = P₁ (B) = -0.08, P₂ (A) =0.02. But P₂ (B) = -0.08.Hence B < ₂A.

Now the inefficiency of the score functions L, LG, P₁, and P₂ is seen in Example 3.3.5.

Example 3.3.5. Let A = 〈 (a_1μ, a_2μ, a_3μ, a_4μ) , (a_1ν, a_2ν, a_3ν, a_4ν) 〉 and B = 〈 (a_1μ + ɛ, a_2μ - ɛ, a_3μ + ɛ, a_4μ - ɛ) , (a_1ν + ɛ, a_2ν - ɛ, a_3ν + ɛ, a_4ν - ɛ) 〉 ∈ TraIFN and ɛ ∈ [0, 1]. Now

$\begin{matrix} L (A) = L (B) \\ = \frac{2 (a_{1 μ} + a_{2 μ}) + 2 (a_{3 μ} + a_{4 μ}) - 2 (a_{1 ν} + a_{2 ν}) - 2 (a_{3 ν} + a_{4 ν})}{8} \\ + \frac{(a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) + (a_{3 μ} + a_{4 μ}) (a_{3 ν} + a_{4 ν})}{8} \\ LG (A) = LG (B) \\ = \frac{- 2 (a_{1 μ} + a_{2 μ} + a_{3 μ} + a_{4 μ}) + 2 (a_{1 ν} + a_{2 ν})}{8} + \\ \frac{2 (a_{3 ν} + a_{4 ν}) + (a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) + (a_{3 μ} + a_{4 μ}) (a_{3 ν} + a_{4 ν})}{8}, \\ P_{1} (A) = P_{1} (B) \\ = \frac{2 (a_{1 μ} + a_{2 μ} - a_{3 μ} - a_{4 μ}) - 2 (a_{1 ν} + a_{2 ν})}{8} + \\ \frac{2 (a_{3 ν} + a_{4 ν}) + (a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) - (a_{3 μ} + a_{4 μ}) (a_{3 ν} + a_{4 ν})}{8} \end{matrix}$ and $\begin{matrix} P_{2} (A) = P_{2} (B) \\ = \frac{2 (- a_{1 μ} - a_{2 μ} + a_{3 μ} + a_{4 μ}) + 2 (a_{1 ν} + a_{2 ν})}{8} \\ - \frac{2 (a_{3 ν} + a_{4 ν}) + (a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) - (a_{3 μ} + a_{4 μ}) (a_{3 ν} + a_{4 ν})}{8} \Rightarrow \\ A = B, but A \neq B . \end{matrix}$ Equality in all the score functions defined above (L, LG, P₁, and P₂) for two different TraIFNs does not imply their equality in all the legs, but it gives some mathematical implications which is proved in the following theorem.

Theorem 3.3.5.LetA, B ∈ TraIFN. IfL (A) = L (B) , LG (A) = LG (B) , P₁ (A) = P₁ (B) and P₂ (A) = P₂ (B) thena_1μ + a_2μ = b_1μ + b_2μ, a_3μ + a_4μ = b_3μ + b_4μ, a_1ν + a_2ν = b_1ν + b_2ν, a_3ν + a_4ν = b_3ν + b_4ν.

Proof. Let A = 〈 (a_1μ, a_2μ, a_3μ, a_4μ) , (a_1ν, a_2ν, a_3ν, a_4ν) 〉, B = 〈 (b_1μ, b_2μ, b_3μ, b_4μ) , (b_1ν, b_2ν, b_3ν, b_4ν) 〉 be two TraIFNs.

Assuming L (A) + LG (A) = L (B) + LG (B) by Definitions 3.2.1 and 3.2.3.

⇒ (a_1μ + a_2μ) (a_1ν + a_2ν) + (a_3μ + a_4μ) (a_3ν + a_4ν) = (b_1μ + b_2μ) (b_1ν + b_2ν) + (b_3μ + b_4μ) (b_3ν + b_4ν) -- - (1)

Using Definitions 3.3.1 and 3.3.3 we have

P₁ (A) + P₂ (A) = P₁ (B) + P₂ (B) ⇒ (a_1μ + a_2μ) (a_1ν + a_2ν) - (a_3μ + a_4μ) (a_3ν + a_4ν) = (b_1μ + b_2μ) (b_1ν + b_2ν) - (b_3μ + b_4μ) (b_3ν + b_4ν) -- - (2)

Adding (1) and (2), we get (a_1μ + a_2μ) (a_1ν + a_2ν) = (b_1μ + b_2μ) (b_1ν + b_2ν) -- - (3)

Using (3) in (1), we get (a_3μ + a_4μ) (a_3ν + a_4ν) = (b_3μ + b_4μ) (b_3ν + b_4ν) -- - (4)

Using Definitions 3.2.1 and 3.3.1 we have L (A) + P₁ (A) = L (B) + P₁ (B) ⇒ (a_1μ + a_2μ) - (a_1ν + a_2ν) = (b_1μ + b_2μ) - (b_1ν + b_2ν) (Using (3)) - (5)

Using Definitions 3.2.1 and 3.3.3 we have L (A) + P₂ (A) = L (B) + P₂ (B) ⇒ (a_3μ + a_4μ) - (a_3ν + a_4ν) = (b_3μ + b_4μ) - (b_3ν + b_4ν) (Using (3)) -- (6)

From Equations (3) , (4) , (5) , (6), we get (a_1μ + a_2μ) = (b_1μ + b_2μ) ; (a_1ν + a_2ν) = (b_1ν + b_2ν) ; (a_3μ + a_4μ) = (b_3μ + b_4μ) ; (a_3ν + a_4ν) = (b_3ν + b_4ν). Hence the proof. □

All the score functions defined in the previous subsections are not enough to compare any two arbitra- ry TraIFNs which is seen from Example 3.3.5. It induce us to define some more new score functions on the class of TraIFNs. In the forthcoming sub-sections we will introduce some more score functions on the subclasses C₃, C₄, C₅ and C₆ of TraIFNs.

3.4 Absolute score of a trapezoidal intuitionistic fuzzy number

In this subsection, the absolute score function on the collection of TraIFNs is defined and its properties are studied using illustrative examples. Further a new relation on TraIFNs which is a total order on C₃ using absolute score is introduced.

The new absolute score function is defined as follows.

Definition 3.4.1. Let A be a TraIFN. Then the new absolute score of a TraIFN A is defined by

$\begin{matrix} D_{5} (A) = \frac{2 (a_{1 μ} - a_{2 μ} + a_{3 μ} - a_{4 μ}) + 2 (a_{1 ν} - a_{2 ν} + a_{3 ν} - a_{4 ν})}{8} \\ + \frac{(a_{1 μ} + a_{3 μ}) (a_{1 ν} + a_{3 ν}) - (a_{2 μ} + a_{4 μ}) (a_{2 ν} + a_{4 ν})}{8} . \end{matrix}$

The proofs of following propositions are immediate from the above definition.

Proposition 3.4.1.IfA = 〈 (a_1μ, a_2μ, a_3μ) , (a_1ν, a_2ν, a_3ν) 〉 ∈ TIFN, then

$\begin{matrix} D_{5} (A) = \frac{2 (a_{1 μ} - a_{3 μ}) + 2 (a_{1 ν} - a_{3 ν})}{8} \\ + \frac{(a_{1 μ} + a_{2 μ}) (a_{1 ν} + a_{2 ν}) - (a_{2 μ} + a_{3 μ}) (a_{2 ν} + a_{3 ν})}{8} . \end{matrix}$ Proposition 3.4.2.IfA = ([a_1μ, a_2μ] , [a_1ν, a_2ν]) is an interval valued intuitionistic fuzzy number, thenD₅ (A) =0.

Theorem 3.4.1.LetA, B ∈ C₃and IfA ⊂ ₂BthenD₅ (A) > D₅ (B).

Proof. Let A, B ∈ C₃ such that A ⊂ ₂B. By Definition 3.0.3, we have a_1μ ≥ b_1μ, a_2μ ≤ b_2μ, a_3μ ≥ b_3μ, a_4μ ≤ b_4μ and a_1ν ≥ b_1ν, a_2ν ≤ b_2ν, a_3ν ≥ b_3ν, a_4ν ≤ b_4ν . Now, 8 (D₅ (A) - D₅ (B)) =2 (a_1μ - b_1μ) + 2 (b_2μ - a_2μ) + 2 (a_3μ - b_3μ) + 2 (b_4μ - a_4μ) +2 (a_1ν - b_1ν) +2 (b_2ν - a_2ν) + 2 (a_3ν - b_3ν) +2 (b_4ν - a_4ν) + (a_1μ + a_3μ) (a_1ν + a_3ν) - (a_2μ + a_4μ) (a_2ν + a_4ν) - (b_1μ + b_3μ) (b_1ν + b_3ν) + (b_2μ + b_4μ) (b_2ν + b_4ν) ≥0 .

Since each term of the above sum is non-negative, we have D₅ (A) ≥ D₅ (B). From the assumption, we know that at least one of the above inequalities become strict inequality, we get D₅ (A) - D₅ (B) >0. Hence the proof. □

Theorem 3.4.2.LetA, B ∈ C₃such thatD₅ (A) = D₅ (B) thenA = B.

Proof. Let A, B ∈ C₃ and assume that D₅ (A) = D₅ (B). By Definition 3.0.3, without loss of generality, we have a_1μ ≥ b_1μ, a_2μ ≤ b_2μ, a_3μ ≥ b_3μ, a_4μ ≤ b_4μ and a_1ν ≥ b_1ν, a_2ν ≤ b_2ν, a_3ν ≥ b_3ν, a_4ν ≤ b_4ν . Now,8 (D₅ (A) - D₅ (B)) = 2 (a_1μ - b_1μ) +2 (b_2μ - a_2μ) +2 (a_3μ - b_3μ) +2 (b_4μ - a_4μ) + [2 (a_1ν - b_1ν) +2 (b_2ν - a_2ν) +2 (a_3ν - b_3ν) +2 (b_4ν - a_4ν)] + (a_1μ + a_3μ) (a_1ν + a_3ν) - (a_2μ + a_4μ) (a_2ν + a_4ν) - (b_1μ + b_3μ) (b_1ν + b_3ν) + (b_2μ + b_4μ) (b_2ν + b_4ν) =0 .

Since each term of the above sum is positive, we have each term is equal to 0. Hence a_1μ = b_1μ, a_2μ = b_2μ, a_3μ = b_3μ, a_4μ = b_4μ and a_1ν = b_1ν, a_2ν = b_2ν, a_3ν = b_3ν, a_4ν = b_4ν. Hence A = B. □

Definition 3.4.2. A relation <₃ on TraIFNs is defined by A < ₃B if D₅ (A) > D₅ (B).

Now the following examples are immediate application of the above definition.

Example 3.4.1. Let A = 〈(0.15, 0.23, 0.31, 0.45) , (0.1, 0.16, 0.35, 0.5) 〉, B = 〈(0.1, 0.25, 0.25, 0.5) , (0, 0.2, 0.3, 0.6) 〉 ∈ C₃. Now D₅ (A) = -0.05375 and D₅ (B) = -0.1125, D₅ (A) > D₅ (B). Hence A < ₃B.

Example 3.4.2. Let A = 〈(0.25, 0.25, 0.4, 0.5) , (0.5, 0.5, 0.5, 0.9) 〉, B = 〈(0.25, 0.25, 0.41, 0.49) , (0.5, 0.5, 0.5, 0.9) 〉 ∈ TraIFN. Now L (A) = L (B) = -0.03, LG (A) = LG (B) =0.47, P₁ (A) = P₁ (B) = -0.095, P₂ (A) = P₂ (B) = -0.095, D₅ (A) = -0.175 and D₅ (B) = -0.167, D₅ (A) < D₅ (B). Hence B < ₃A.

Note 3.4.1. The absolute score can be calculated to any TraIFN but D₅ gives total order <₃ on C₃. Hence the <₃ satisfies the Axiom A₃.

The inability of the score function D₅ with other score functons in ranking arbitrary TraIFNs is shown in the following example.

Example 3.4.3. Let A = 〈(0.04, 0.3, 0.38, 0.39) , (0.52, 0.69, 0.86, 0.99) 〉, B = 〈(0.0, 0.34, 0.38, 0.39) , (0.515264188, 0.694735812, 0.92, 0.93) ∈ TraIFN.Now L (A) = L (B) = -0.2580125, LG (A) = LG (B) =0.7169875, P₁ (A) = P₁ (B) = -0.074138, P₂ (A) = P₂ (B) = -0.179138 and D₅ (A) = D₅ (B) = -0.21495. But A ≠ B.

Example 3.4.3 shows that all the above defined scores are not enough to cover the entire collection of TraIFNs and therefore we are introducing another score function which measures the perfectness in the next subsection.

3.5 Perfect score of a trapezoidal intuitionistic fuzzy number

In this subsection, the perfect score function on TraIFNs is defined and some of its properties are studied using illustrative examples. Further a relation on TraIFNs which is a total order on C₄ is defined.

The perfect score function is defined as follows.

Definition 3.5.1 Let A be a TraIFN. Then the perfect score of a TraIFN A is defined by

$\begin{matrix} D_{6} (A) = \frac{2 (a_{1 μ} - a_{2 μ} + a_{3 μ} - a_{4 μ}) + 2 (- a_{1 ν} + a_{2 ν} - a_{3 ν} + a_{4 ν})}{8} \\ + \frac{(a_{1 μ} + a_{3 μ}) (a_{2 ν} + a_{4 ν}) - (a_{2 μ} + a_{4 μ}) (a_{1 ν} + a_{3 ν})}{8} . \end{matrix}$

The proofs of following propositions are immediate from the above definition.

Proposition 3.5.1.IfA = 〈 (a_1μ, a_2μ, a_3μ) , (a_1ν, a_2ν, a_3ν) 〉 ∈ TIFN, then

$\begin{matrix} D_{6} (A) = \frac{2 (a_{1 μ} - a_{3 μ}) - 2 a_{1 ν}}{8} \\ + \frac{2 a_{3 ν} + (a_{1 μ} + a_{2 μ}) (a_{2 ν} + a_{3 ν}) - (a_{2 μ} + a_{3 μ}) (a_{1 ν} + a_{2 ν})}{8} . \end{matrix}$ Proposition 3.5.2.IfA = ([a_1μ, a_2μ] , [a_1ν, a_2ν]) is an interval valued intuitionistic fuzzy number, thenD₆ (A) =0.

The proofs of the following two theorems are similar to Theorems 3.4.1 and 3.4.2.

Theorem 3.5.1.LetA, B ∈ C₄. IfA ⊂ ₃B. ThenD₆ (A) > D₆ (B).

Theorem 3.5.2.LetA, B ∈ C₄such thatD₆ (A) = D₆ (B). ThenA = B.

Definition 3.5.2.A relation <₄on TraIFNs is defined byA < ₄B if D₆ (A) < D₆ (B).

Note 3.5.1. The perfect score can be calculated to any TraIFN but D₆ with <₄ relation gives total order on C₄. Hence <₄ satisfies Axiom A₄.

Now the ranking principle introduced in Definition 3.3.2 is demonstrated in the following examples.

Example 3.5.1. In Example 3.4.3, we have L (A) = L (B) = -0.2580125, LG (A) = LG (B) =0.7169875, P₁ (A) = P₁ (B) = -0.0741375, P₂ (A) = P₂ (B) = -0.1791375, D₅ (A) = D₅ (B) = -0.21495, D₆ (A) = -0.023325. But D₆ (B) = -0.093925 and D₆ (A) > D₆ (B). Hence B < ₄A.

Example 3.5.2. Let A = 〈(0.25, 0.25, 0.45, 0.45) , (0.55, 0.75, 0.75, 0.95) 〉, B = 〈(0.25, 0.25, 0.45, 0.45) , (0.5, 0.8, 0.8, 0.9) 〉 ∈ TraIFN. Now L (A) = L (B) = -0.1275, LG (A) = LG (B) =0.6725, P₁ (A) = P₁ (B) = -0.11, P₂ (A) = P₂ (B) = -0.11, D₅ (A) = D₅ (B) = -0.135, D₆ (A) = D₆ (B) =0.135. But A ≠ B.

Example 3.5.2 shows that all the above scores are not enough to rank any two TraIFNs. Hence we need some more score functions to cover the entire collection of TraIFNs. We introduce an integral score function is as follows.

3.6 Integral score of a trapezoidal intuitionistic fuzzy number

In this subsection, the integral score function on TraIFNs is defined and its properties are studied using illustrative examples. Further a ranking relation <₅ which is a total order on C₅ is defined.

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

Definition 3.6.1. Let A be a trapezoidal intuitionistic fuzzy number. Then the integral score of a trapezoidal intuitionistic fuzzy number A is defined by

$\begin{matrix} D_{7} (A) = \frac{2 (- a_{1 μ} + a_{2 μ} + a_{3 μ} - a_{4 μ}) + 2 (a_{1 ν} - a_{2 ν} - a_{3 ν} + a_{4 ν})}{8} \\ + \frac{(a_{2 μ} + a_{3 μ}) (a_{1 ν} + a_{4 ν}) - (a_{1 μ} + a_{4 μ}) (a_{2 ν} + a_{3 ν})}{8} . \end{matrix}$

The proofs of following propositions are immediate from the above definition.

Proposition 3.6.1.IfA = 〈 (a_1μ, a_2μ, a_3μ) , (a_1ν, a_2ν, a_3ν) 〉 ∈ TIFN, then

$\begin{matrix} D_{7} (A) = \frac{(2 a_{2 μ} - a_{1 μ} - a_{3 μ})}{4} + \\ \frac{- (2 a_{2 ν} - a_{1 ν} - a_{3 ν}) - a_{2 ν} (a_{1 μ} + a_{3 μ}) + a_{2 μ} (a_{1 ν} + a_{3 ν})}{4} . \end{matrix}$

Proposition 3.6.2.IfA = ([a_1μ, a_2μ] , [a_1ν, a_2ν]) is an interval valued intuitionistic fuzzy number, thenD₇ (A) =0.

The proofs of the following two theorems are similar to Theorems 3.4.1 and 3.4.2.

Theorem 3.6.1.LetA, B ∈ C₅. IfA ⊂ ₄B, thenD₇ (A) > D₇ (B).

Theorem 3.6.2.LetA, B ∈ C₅such thatD₇ (A) = D₇ (B). ThenA = B.

Definition 3.6.2. A relation <₅ on TraIFNs is defined by A < ₅B if D₇ (A) > D₇ (B).

Note 3.6.1. The integral score can be calculated to any TraIFN but D₇ gives total order on C₅. Hence <₅ satisfies Axiom A₅.

Now the ranking principle defined above is demonstrated in the following examples.

Example 3.6.1. In Example 3.5.2, we get L (A) = L (B) = -0.1275, LG (A) = LG (B) = 0.6725, P₁ (A) = P₁ (B) = -0.11, P₂ (A) = P₂ (B) = -0.11, D₅ (A) = D₅ (B) = -0.135, D₆ (A) = D₆ (B) =0.135, D₇ (A) =0. But D₇ (B) = -0.0675 ⇒ D₇ (A) > D₇ (B). Hence A < ₅B.

Example 3.6.2. Let A = 〈(0.25, 0.25, 0.35, 0.55) , (0.55, 0.55, 0.8, 1.0) 〉, B = 〈(0.171739131, 0.32826087, 0.42826087, 0.47173913) , (0.45, 0.65, 0.9, 0.9) 〉 ∈ TraIFN. Now L (A) = L (B) = -0.10375, LG (A) = LG (B) = 0.64625, P₁ (A) = P₁ (B) = -0.05875, P₂ (A) = P₂ (B) = -0.20875, D₅ (A) = D₅ (B) = -0.15375, D₆ (A) = D₆ (B) = -0.01875, D₇ (A) = D₇ (B) = -0.01875. But A ≠ B.

Example 3.4.3 shows that all the above defined scores are not enough to cover the entire collection of TraIFNs. Hence the new score function which measures the completeness is defined in the next sub section.

3.7 Complete score of a trapezoidal intuitionistic fuzzy number

In this subsection, the complete score function on TraIFNs is defined and its properties are studied using illustrative examples. The score function which measures completeness is defined as follows.

Definition 3.7.1. Let A be a trapezoidal intuitionistic fuzzy number. Then the complete score of a trapezoidal intuitionistic fuzzy number A is defined by $\begin{matrix} D_{8} (A) = \frac{2 (- a_{1 μ} + a_{2 μ} + a_{3 μ} - a_{4 μ}) + 2 (- a_{1 ν} + a_{2 ν} + a_{3 ν} - a_{4 ν})}{8} \\ + \frac{(a_{2 μ} + a_{3 μ}) (a_{2 ν} + a_{3 ν}) - (a_{1 μ} + a_{4 μ}) (a_{1 ν} + a_{4 ν})}{8} . \end{matrix}$

The proofs of following propositions are immediate from the above definition.

Proposition 3.7.1.IfA = 〈 (a_1μ, a_2μ, a_3μ) , (a_1ν, a_2ν, a_3ν) 〉 ∈ TIFN, then

$\begin{matrix} D_{8} (A) = \frac{2 (2 a_{2 μ} - a_{1 μ} - a_{3 μ})}{8} \\ + \frac{2 (2 a_{2 ν} - a_{1 ν} - a_{3 ν}) - (a_{1 μ} + a_{3 μ}) (a_{1 ν} + a_{3 ν}) + 4 a_{2 μ} a_{2 ν}}{8} . \end{matrix}$

Proposition 3.7.2.IfA = ([a_1μ, a_2μ] , [a_1ν, a_2ν]) is an interval valued intuitionistic fuzzy number, thenD₈ (A) =0.

The proofs of the following two theorems are similar to Theorems 3.4.1 and 3.4.2.

Theorem 3.7.1.LetA, B ∈ C₆. IfA ⊂ ₅B. ThenD₈ (A) > D₈ (B).

Theorem 3.7.2.LetA, B ∈ C₆such thatD₈ (A) = D₈ (B). ThenA = B.

Definition 3.7.2. A relation <₆ is defined by B < ₆A if D₈ (A) > D₈ (B).

Now the ranking principle introduced in Definition 3.7.2 is explained in the following example. The efficiency of the score function D₈ in comparing any two arbitrary TraIFNs is shown in the following example.

Example 3.7.1. In Example 3.6.2, we have L (A) = L (B) = -0.10375, LG (A) = LG (B) = 0.64625, P₁ (A) = P₁ (B) = -0.05875, P₂ (A) = P₂ (B) = -0.20875, D₅ (A) = D₅ (B) = -0.15375, D₆ (A) = D₆ (B) = -0.01875, D₇ (A) = D₇ (B) = -0.01875, D₈ (A) = -0.15375. But D₈ (B) =0.11625. Hence A < ₆B.

Note 3.7.1. The complete score can be calculated to any TraIFN but D₈ gives total order on C₆. Hence the relation <₆ satifies Axiom A₆.

4 A Complete ranking of trapezoidal intuitionistic fuzzy numbers

In this section a new ranking principle on the class of trapezoidal intuitionistic fuzzy numbers is defined by using ranking principles defined in Sections 3.2, 3.3, 3.4, 3.5, 3.6 and 3.7 and proved that the proposed ranking is a complete ordering on TraIFNs.

Definition 4.0.1. Let A and B ∈ TraIFN

If A < ₁B then A is smaller than B, denoted by A < B.

If L (A) = L (B), LG (A) = LG (B) and A < ₂B then A is smaller than B, denoted by A < B.

If L (A) = L (B), LG (A) = LG (B), P₁ (A) = P₁ (B), P₂ (A) = P₂ (B) and A < ₃B then A is smaller than B, denoted by A < B.

If L (A) = L (B), LG (A) = LG (B), P₁ (A) = P₁ (B), P₂ (A) = P₂ (B), D₅ (A) = D₅ (B) and A < ₄B then A is smaller than B, denoted by A < B.

If L (A) = L (B), LG (A) = LG (B), P₁ (A) = P₁ (B), P₂ (A) = P₂ (B), D₅ (A) = D₅ (B), D₆ (A) = D₆ (B) and A < ₅B then A is smaller than B, denoted by A < B.

If L (A) = L (B), LG (A) = LG (B), P₁ (A) = P₁ (B), P₂ (A) = P₂ (B), D₅ (A) = D₅ (B), D₆ (A) = D₆ (B), D₇ (A) = D₇ (B) and A < ₆B then A is smaller than B, denoted by A < B.

If L (A) = L (B), LG (A) = LG (B), P₁ (A) = P₁ (B), P₂ (A) = P₂ (B), D₅ (A) = D₅ (B), D₆ (A) = D₆ (B) , D₇ (A) = D₇ (B) and D₈ (A) = D₈ (B) then A = B.

The following theorem is proved to show the validity of the Definition 4.0.1.

Theorem 4.1.LetA, B ∈ TraIFN. IfL (A) = L (B), LG (A) = LG (B), P₁ (A) = P₁ (B), P₂ (A) = P₂ (B), D₅ (A) = D₅ (B), D₆ (A) = D₆ (B) , D₇ (A) = D₇ (B) andD₈ (A) = D₈ (B), thenA = B. i.e.,a_1μ = b_1μ, a_2μ = b_2μ, a_3μ = b_3μ, a_4μ = b_4μ, a_1ν = b_1ν, a_2ν = b_2ν, a_3ν = b_3νanda_4ν = b_4ν.

Proof. If L (A) = L (B), LG (A) = LG (B), P₁ (A) = P₁ (B), P₂ (A) = P₂ (B), D₅ (A) = D₅ (B), D₆ (A) = D₆ (B) , D₇ (A) = D₇ (B) and D₈ (A) = D₈ (B).

Now we claim that A = B.

Using Theorem 3.5, we have a_1μ + a_2μ = b_1μ + b_2μ, a_3μ + a_4μ = b_3μ + b_4μ, a_1ν + a_2ν = b_1ν + b_2ν and a_3ν + a_4ν = b_3ν + b_4ν . -- - (1)

Since a_1μ + a_2μ = b_1μ + b_2μ and a_3μ + a_4μ = b_3μ + b_4μ, we get a_1μ + a_2μ + a_3μ + a_4μ = b_1μ + b_2μ + b_3μ + b_4μ -- - (2)

Also, a_1ν + a_2ν = b_1ν + b_2ν and a_3ν + a_4ν = b_3ν + b_4ν, we get a_1ν + a_2ν + a_3ν + a_4ν = b_1ν + b_2ν + b_3ν + b_4ν -- - (3)

Now, D₅ (A) + D₆ (A) = D₅ (B) + D₆ (B) ⇒ (a_1μ - a_2μ + a_3μ - a_4μ) (4 + a_1ν + a_2ν + a_3ν + a_4ν) = (b_1μ - b_2μ + b_3μ - b_4μ) (4 + b_1ν + b_2ν + b_3ν + b_4ν) -- - (4)

Also, D₇ (A) + D₈ (A) = D₇ (B) + D₈ (B) ⇒ (- a_1μ + a_2μ + a_3μ - a_4μ) (4 + a_1ν + a_2ν + a_3ν + a_4ν) = (- b_1μ + b_2μ + b_3μ - b_4μ) (4 + b_1ν + b_2ν + b_3ν + b_4ν) -- - (5)

Adding (4) and (5), we get (4 + a_1ν + a_2ν + a_3ν + a_4ν) (2a_3μ - 2a_4μ) = (4 + b_1ν + b_2ν + b_3ν + b_4ν) (2b_3μ - 2b_4μ) -- - (6)

Subtracting (5) from (4), we get (4 + a_1ν + a_2ν + a_3ν + a_4ν) (2a_1μ - 2a_2μ) = (4 + b_1ν + b_2ν + b_3ν + b_4ν) (2b_1μ - 2b_2μ) -- - (7)

Using (3) in (6) and (7), we get (2a_3μ - 2a_4μ) = (2b_3μ - 2b_4μ) and (2a_1μ - 2a_2μ) = (2b_1μ - 2b_2μ) -- - (8)

Using (1) in (8), we get a_1μ = b_1μ, a_2μ = b_2μ, a_3μ = b_3μ, a_4μ = b_4μ -- - (9)

Now, D₅ (A) - D₆ (A) = D₅ (B) - D₆ (B) ⇒ (a_1ν - a_2ν + a_3ν - a_4ν) (4 + a_1μ + a_2μ + a_3μ + a_4μ) = (b_1ν - b_2ν + b_3ν - b_4ν) (4 + b_1μ + b_2μ + b_3μ + b_4μ) -- - (10)

Also, D₇ (A) - D₈ (A) = D₇ (B) - D₈ (B) ⇒ (a_1ν - a_2ν - a_3ν + a_4ν) (4 + a_1μ + a_2μ + a_3μ + a_4μ) = (b_1ν - b_2ν - b_3ν + b_4ν) (4 + b_1μ + b_2μ + b_3μ + b_4μ) -- - (11)

Adding (10) and (11), we get (4 + a_1μ + a_2μ + a_3μ + a_4μ) (2a_1ν - 2a_2ν) = (4 + b_1μ + b_2μ + b_3μ + b_4μ) (2b_1ν - 2b_2ν) -- - (12)

Subtracting (11) from (10), we get (4 + a_1μ + a_2μ + a_3μ + a_4μ) (2a_3ν - 2a_4ν) = (4 + b_1μ + b_2μ + b_3μ + b_4μ) (2b_3ν - 2b_4ν) -- - (13)

Using (2) in (12) and (13), we get (2a_1ν - 2a_2ν) = (2b_1ν - 2b_2ν) and (2a_3ν - 2a_4ν) = (2b_3ν - 2b_4ν) -- - (14)

Using (1) in (14), we get a_1ν = b_1ν, a_2ν = b_2ν, a_3ν = b_3ν, a_4ν = b_4ν -- - (15)

From (9) and (15), we have a_1μ = b_1μ, a_2μ = b_2μ, a_3μ = b_3μ, a_4μ = b_4μ, a_1ν = b_1ν, a_2ν = b_2ν, a_3ν = b_3ν and a_4ν = b_4ν. Hence A = B. □

5 Significance of the proposed method

Many researchers have proposed different ranking methods on IFNs, but none of them has covered the entire class of IFNs and also almost all the methods have disadvantage that they ranked two different IFNs as the same at some point of time. In this paper a special type of IFNs which generalizes intuitionistic fuzzy values and intuitionistic fuzzy intervals is studied. Problems in different fields involving qualitative, quantitative and uncertain information can be modelled better using this type of IFNs when compared with usual IFNs. Our proposed ranking method on this type of IFN will give the better results over other existing methods, and this paper will give the better understanding over this new type of IFNs. This type of IFNs are very much important in real life problems and this paper will give the significant impact in the literature. Modeling problems using this type of IFN will give better result. In this section our proposed method is compared with different existing methods.

5.1 Comparison between our proposed method with the score function defined in Lakshmana et al.[12]:

In this subsection, our proposed method is compared with the total score function defined in Lakshmana et al. [12] with an illustrative example.

In the following example, Definitions 2.0.4 to 2.0.7 are demonstrated and also the illogicality of Lakshmana et al’s method is shown.

Example 5.1.1. Let M = 〈 (0, 0.2, 0.4) , (0.4, 0.45, . 5)〉 with (0, 0.2, 0.4) ^c ≤ (0.4, 0.45, 0.5) and 0.4 ≥ 0.2 & 0.45 ≥ 0.4 and N = 〈(0.25, 0.25, 0.25) , (0.4, 0.45, 0.5) 〉 with (0.25, 0.25, 0.25) ^c ≥ (0.4, 0.45, 0.5) and 0.4 ≥ 0.25 & 0.45 ≥ 0.25 be two triangular intuitionistic fuzzy numbers. Using Definition 2.0.4, we get L (M) =0.8333 and R (M) =0.3333 which implies T (M) =0.25. Applying Definition 2.0.5, 2.0.6 to M we get NL (M) =0.421053, NR (M) =0.526316, NT (M) =0.447368 and NT_c (M) =0.553. Therefore the total score of membership and nonmembership functions of M are T = 0.25 andNT_c = 0.553 respectively. Then the intuitionistic fuzzy score is represented by (T, NT_c) = (0.25, 0.553). Similarly using Definitions 2.0.4 to 2.0.7, we get the total score of membership and nonmembership functions of N are T = 0.25 and NT_c = 0.553 respectively.

Therefore from Definition 2.0.7, this method ranks M and N are equal but M & N are different triangular intuitionistic fuzzy numbers which is illogical. Applying our ranking principle introduced in Definition 4.0.1 to M and N, we get L (M) = -0.1575 and L (N) = -0.0875, i.e., L (M) < L (N). Hence our proposed method ranks N as better one.

5.2 Comparison of the proposed method with some existing methods

In this sub-section our proposed method is compared with some other existing methods using illustrative examples. The Table 1 shows that our proposed method is significant over the methods presented in [4 , 26].

For example, let A = ([0.1, 0.15] , [0.25, 0.35]) and B = ([0.05, 0.2] , [0.20, 0.40]) be two IVIFNs. Then by applying Jun Ye [10] approach, we get M (A) = M (B) = -0.45 which implies that A and B are equal which is illogical. By applying the total ordering < defined Definition 4.0.1, we have L (A) = -0.13625and L (B) = -0.13, i.e., L (A) < L (B). Hence A < B.

5.3 Trapezoidal Intuitionistic Fuzzy Information System (TraIFIS)

Information system (IS) is a decision model used to select the best alternative from the all the alternatives in hand under various attributes. The data collected from the experts may be incomplete or imprecise numerical quantities. To deal with such data the thoery of IFS provided by Atanassov [1] aids better. In information system, dominance relation rely on ranking of data, ranking of intuitionistic fuzzy numbers is inevitable. In this subsection a dominance relation is defined using proposed ranking method.

Definition 5.3.1. An Information System S = (U, AT, V, f) with V = ∪ _a∈ATV_a where V_a is a domainof attribute a is called trapezoidal intuitionisticfuzzy information system if V is a set of TraIFN.We denote f (x, a) ∈ V_a by f (x, a) = 〈 (a₁, a₂, a₃, a₄) , (a₁^′, a₂^′, a₃^′, a₄^′) 〉 where a_i and a_i^′ ∈ [0, 1].

The numerical illustration is given in Example 5.5.1.

Definition 5.3.2. An TraIFIS S = (U, AT, V, f) together with weights W = {w_a/a ∈ AT} is called Weighted Trapezoidal Intuitionistic Fuzzy Information System (WTraIFIS) and is denoted by S = (U, AT, V, f, W).

Definition 5.3.3. Let a ∈ AT be a criterion. Let x, y ∈ U. If f (x, a) > f (y, a) (as per Definition 4.0.1) then x > _ay which indicates that x is better than (outranks) y with respect to the criterion a. Also x = _ay means that x is equally good as y with respect to the criterion a, if f (x, a) = f (y, a).

Definition 5.3.4. Let S = (U, AT, V, f, W) be an WTraIFIS and A ⊆ AT. Let B_A (x, y) = {a ∈ A|x > _ay} and let C_A (x, y) ={ a ∈ A|x = _ay }. The weighted fuzzy dominance relation WR_A (x, y) : U × U → [0, 1] is defined by ${WR}_{A} (x, y) = \sum_{a \in B_{A} (x, y)} w_{a} + \frac{\sum_{a \in C_{A} (x, y)} w_{a}}{2}$ .

Definition 5.3.5. Let S = (U, AT, V, f, W) be an WTraIFIS and A ⊆ AT. The entire dominance degree of each object is defined as ${WR}_{A} (x_{i}) = \frac{1}{| U |} \sum_{j = 1}^{| U |} {WR}_{A} (x_{i}, y_{j})$

5.4 Algorithm for ranking of objects in WTraIFIS

Let S = (U, AT, V, f, W) be an WTraIFIS. The objects in U are ranked using following algorithm.

Algorithm: 5.4

1. Using Definition 4.0.1 find L and if neccessary LG, P₁, P₂, D₅, D₆, D₇, D₈ accordingly, to decide whether x_i > _a x_j or x_j > _a x_i or x_i = _a x_j for all a ∈ A (A ⊆ AT) and for all x_i, x_j ∈ U.

2. Enumerate B_A (x_i, x_j) using B_A (x_i, x_j) = {a ∈ A|x_i > _ax_j} and C_A (x_i, x_j) using C_A (x_i, x_j) ={ a ∈ A|x_i = _ax_j }.

3.Calculate the weighted fuzzy dominance relation using WR_A (x, y) : U × U → [0, 1] defined by ${WR}_{A} (x_{i}, x_{j}) = \sum_{a \in B_{A} (x_{i}, x_{j})} w_{a} + \frac{\sum_{a \in C_{A} (x_{i}, x_{j})} w_{a}}{2}$ .

4. Calculate the entire dominance degree of each object using ${WR}_{A} (x_{i}) = \frac{1}{| U |} \sum_{j = 1}^{| U |} {WR}_{A} (x_{i}, x_{j})$ .

5. The objects are ranked using entire dominance degree. The larger the value of WR_A (x_i), the better is the object.

5.5 Numerical Illustration

In this subsection, Algorithm 5.4 is illustrated by an Example 5.5.1.

Example 5.5.1. In this example we consider a selection problem of the best supplier for an automobile company from the available alternatives {x_i|i = 1 to 10} of pre evaluated 10 suppliers, based on WTraIFIS with attributes {a_j|j = 1 to 5} as product quality, relationship closeness, delivery performance, social responsibility and legal issue. An TraIFIS with U ={ x₁, x₂, . . . , x₁₀ }, AT ={ a₁, a₂, . . . , a₅ } is given in Table 2, and weights for the each attribute W_a is given by W ={ w_a|a ∈ AT } = { 0.3, 0.2, 0.15, 0.17, 0.18 }.

By step 1, the membership score function L (f (x_i, a_j)) using Definition 3.2.1, for all a_i ∈ AT and for all x_i ∈ U is found and tabulated in Table 3. If L (f (x_i, a_j)) = L (f (x_j, a_j)) for any alternatives x_i, x_j then LG and other necessary score functions (P₁, P₂, D₅, D₆, D₇ and D₈) are found wherever required. The bold letters are used in Tables 3 and 4 to represent the equality of scores.

The weighted fuzzy dominance relation using ${WR}_{A} (x, y) = \sum_{a \in B_{A} (x, y)} w_{a} + \frac{\sum_{a \in C_{A} (x, y)} w_{a}}{2}$ is calculated and is tabulated in Table 5. For example, B_A (x₁, x₂) ={ a₁, a₂, a₃ } and C_A (x₁, x₂) ={ } and hence WR_A (x₁, x₂) =0.3 + 0.2 + 0.15 = 0.65.

Now the entire dominance degree of each object using ${WR}_{A} (x_{i}) = \frac{1}{| U |} \sum_{j = 1}^{| U |} {WR}_{A} (x_{i}, y_{j})$ is found by Definiton 5.3.4. For example, ${WR}_{A} (x_{3}) = \frac{1}{10} \sum_{j = 1}^{10} {WR}_{A} (x_{3}, x_{j}) = 0.504$ . So by step 5, x₆ is selected as the best object from the weighted trapezoidal intuitionistic fuzzy information system is seen from Table 6.

6 Conclusions

In this paper, we have presented an axiomatic set of ranking method based on the set of membership, non membership, vague, non-vague, absolute, perfect, integral and complete score functions, which covers entire collection of TraIFNs and applied in TraIFIS. In the application point of view our proposed method is more applicable and more natural when it is compared with the existing methodology. The type of trapezoidal intuitionistic fuzzy number discussed in this paper is more natural in modeling problems with incompleteness. This opens a new area of research on the study of defining some more measuring functions in near future in all fields of data analysis, artificial intelligence, decision support systems.

Footnotes

Acknowledgments

Authors thank the anonymous referees for the valuable suggestions for improving the quality of the paper. The second author gratefully acknowledges the financial support provided by the Ministry of Human Resource Development (MHRD), New Delhi, India.

References

Atanassov

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems20 (1986), 87–96.

Atanassov

and Gargov

, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems31(3) (1989), 343–349.

Burillo

, Bustince

and Mohedano

, Some definitions of intuitionistic fuzzy number. First properties, Proc of the First Workshop on Fuzzy Based Expert Systems, ( Lakov

, Ed.), Sofia, 1994, 53–55.

Chen

S.M.

and Tan

J.M.

, Handling multi-criteria fuzzy decision-making problems based on vague set theory, Fuzzy Sets and Systems67 (1994), 163–172.

, Wu

and Lu

, Interval-valued intuitionistic fuzzy prioritized operators and their application in group decision making, Knowledge-Based Systems30 (2012), 57–66.

Eslamipoor

, Hosseini-nasab

and Sepehriar

, An improved ranking method for generalized fuzzy numbers based on Euclidian distance concept, Afrika Matematika26 (2015), 1291–1297.

Sivaraman

, Nayagam

V.L.G.

and Ponalagusamy

, A complete ranking of incomplete interval information, Expert Systems with Applications41 (2014), 1947–1954.

Hong

D.H.

and Choi

C.H.

, Multi-criteria fuzzy decision-making problems based on vague set theory, Fuzzy Sets and Systems114 (2000), 103–113.

Janizade-Haji

, Khademi Zare

, Eslamipoor

and Sepehriar

, A developed distance method for ranking generalized fuzzy numbers, Neural Computing and Applications25 (2014), 727–731.

10.

, Multi-criteria fuzzy decision making method based on a novel accuracy score function under interval valued intuitionistic fuzzy environment, Expert Systems with Applications36 (2009), 6899–6902.

11.

Klir

G.J.

and Yuan

, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall India, New Delhi, 1997.

12.

Nayagam

V.L.G.

, Venkateshwari

and Sivaraman

, Ranking of Intuitionistic Fuzzy Numbers, International Conference on Fuzzy Systems (FUZZ 2008), 2008, 1971–1974.

13.

Nayagam

V.L.G.

and Sivaraman

, Ranking of interval-valued intuitionistic fuzzy sets, Applied Soft Computing11 (2011), 3368–3372.

14.

Nayagam

V.L.G.

, Muralikrishnan

and Sivaraman

, Multi-criteria decision-making method based on interval-valued intuitionistic fuzzy sets, Expert Systems with Applications38 (2011), 1464–1467.

15.

Lin

L.G.

, Xu

L.Z.

and Wang

J.Y.

, Multi-criteria fusion decision-making method based on vague set, Computer Engineering31 (2005), 11–13. (in Chinese).

16.

Lin

, Yuan

X.H.

and Xia

Z.Q.

, Multicriteria fuzzy decision-making methods based on intuitionistic fuzzy sets, Journal of Computer and System Sciences73 (2007), 84–88.

17.

Liu

H.W.

, Vague set methods of multi-criteria fuzzy decision making, Systems Engineering - Theory & Practice24 (2004), 103–109(in Chinese).

18.

Urena

, Chiclana

and Herrera-Viedma

, Consistency based completion approaches of incomplete preference relations in uncertain decision contexts, Proceedings of 2015 IEEE International Conference on Fuzzy Systems, Istanbul, Turkey, 2015.

19.

Wang

, Zhang

and Liu

S.Y.

, A new score function for fuzzy MCDM based on vague set theory, International Journal of Computational Cognition4 (2006), 44–48.

20.

Wang

J.Q.

and Zhang

, Multi-criteria decision making method with incomplete certain information based on intuitionistic fuzzy number, Control and Decision24 (2009), 226–230(in Chinese).

21.

Z.S.

, Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making, Control and Decision22(2) (2007), 215–219.

22.

, Expected value method for intuitionistic trapezoidal fuzzy multicriteria decision-making problems, Expert Systems with Applications38 (2011), 11730–11734.

23.

, Using an improved measure function of vague sets for multicriteria fuzzy decision-making, Expert Systems with Applications37 (2010), 4706–4709.

24.

Zadeh

L.A.

, Fuzzy sets, Information and Control8 (1965), 338–356.

25.

Zhang

and Yu

, MADM method based on cross-entropy and extended TOPSIS with interval-valued intuitionistic fuzzy sets, Knowledge-Based Systems30 (2012), 115–120.

26.

Zhou

and Wu

Q.Z.

, Multicriteria fuzzy decision making method based on vague set, Mini-Micro Systems26 (2005), 1350–1353. (in Chinese).

A complete ranking of incomplete trapezoidal information

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Ranking principle for trapezoidal intuitionistic fuzzy numbers

3.1 Axioms of ranking principle in intuitionistic fuzzy set up

3.2 Total order on C1

3.3 Total order on C2

3.4 Absolute score of a trapezoidal intuitionistic fuzzy number

3.5 Perfect score of a trapezoidal intuitionistic fuzzy number

3.6 Integral score of a trapezoidal intuitionistic fuzzy number

3.7 Complete score of a trapezoidal intuitionistic fuzzy number

4 A Complete ranking of trapezoidal intuitionistic fuzzy numbers

5 Significance of the proposed method

5.1 Comparison between our proposed method with the score function defined in Lakshmana et al.[12]:

5.2 Comparison of the proposed method with some existing methods

5.3 Trapezoidal Intuitionistic Fuzzy Information System (TraIFIS)

5.4 Algorithm for ranking of objects in WTraIFIS

5.5 Numerical Illustration

6 Conclusions

Footnotes

Acknowledgments

References

3.2 Total order on C₁

3.3 Total order on C₂