Total ordering defined on the set of all intuitionistic fuzzy numbers

Abstract

L.A. Zadeh introduced the concept of fuzzy set theory as the generalisation of classical set theory in 1965 and further it has been generalised to intuitionistic fuzzy sets (IFSs) by Atanassov in 1983 to model information by the membership, non membership and hesitancy degree more accurately than the theory of fuzzy logic. The notions of intuitionistic fuzzy numbers in different contexts were studied in literature and applied in real life applications. Problems in different fields involving qualitative, quantitative and uncertain information can be modelled better using intutionistic fuzzy numbers introduced in [15] which generalises intuitionistic fuzzy values [1, 6, 15], interval valued intuitionistic fuzzy number (IVIFN) [9] than with usual IFNs [4, 10, 19]. Ranking of fuzzy numbers have started in early seventies in the last century and a complete ranking on the class of fuzzy numbers have achieved by W. Wang and Z. Wang only on 2014. A complete ranking on the class of IVIFNs, using axiomatic set of membership, non membership, vague and precise score functions has been introduced and studied by Geetha et al. [9]. In this paper, a total ordering on the class of IFNs [15] using double upper dense sequence in the interval [0, 1] which generalises the total ordering on fuzzy numbers (FNs) is proposed and illustrated with examples. Examples are given to show the proposed method on this type of IFN is better than existing methods and this paper will give the better understanding over this new type of IFNs.

Keywords

Double upper dense sequence total order relation intuitionistic fuzzy number interval valued intuitionistic fuzzy number trapezoidal intuitionistic fuzzy numbers (TrIFN)

1 Introduction

Information system (IS) is a decision model used to select the best alternative from 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 theory 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.

Many researchers have been working in the area of ranking of IFNs since last century. Different ranking methods for intuitionistic fuzzy values, interval valued intuitionistic fuzzy numbers have been studied in [3 , 26]. But till date, there exists no single method or combination of methods available to rank any two arbitrary IFNs. The difficulty of defining total ordering on the class of intuitionistic fuzzy numbers is that there is no effective tool to identify an arbitrarily given intuitionistic fuzzy number by finitely many real-valued parameters. In this work, by establising a new decomposition theorem for IFSs, any IFN can be identified by infinitely many but countable number of parameters. A new decomposition theorem for intuitionistic fuzzy sets is established by the use of an double upper dense sequence defined in [0, 1]. Actually there are many double upper dense sequences available in the interval [0, 1]. Since the choice of a double upper dense sequence is considered as the necessary reference systems for defining a complete ranking, infinitely many total orderings on the set of all IFNs can be well defined based on each choice of double upper dense sequence. After introduction, some necessary fundamental knowledge on ordering and intuitionistic fuzzy numbers, interval valued intuitionistic fuzzy numbers is introduced in Section 2. In Section 3, a new decomposition theorem for intuitionistic fuzzy sets is establised using double upper dense sequence defined in the interval [0, 1]. Section 4 is used to define total ordering on the set of all intuitionistic fuzzy numbers by using double upper dense sequence in the interval [0, 1]. Several examples are given in Section 5 to show how the total ordering on IFNs can be used for ranking better than some other existing methods. Application of our proposed method in solving intuitionistic fuzzy information system problem is shown in Section 5 by developing a new algorithm. Finally conclusions are given in Section 6.

1.1 Motivation

The capacity to handle dubious and uncertain data is more effectively done by stretching out intuitionistic fuzzy values to TrIFNs because the membership and non-membership degrees are better expressed as trapezoidal values rather than exact values. TrIFNs are generalisation of intuitionistic fuzzy values and IVIFNs. As a generalisation, the set of TrIFNs should contain the set of all intuitionistic fuzzy values and IVIFNs. But the existing defintion for TrIFNs [4 , 19] does not contain the set of intuitionistic fuzzy values which means the existing definition for TrIFN is not the real generalisation of intuitionistic fuzzy values. Hence the study about new structure for intuitionistic fuzzy number [15] is essential. In the application point of view our proposed method is more applicable and more natural when it is compared with the existing methodology. More precisely, the existing definition for Trapezoidal intuitionistic fuzzy numbers (TrIFN) present in the literature [4 , 19] is defined as A = 〈 (a, b, c, d) (e, f, g, h) 〉 with e ≤ a ≤ f ≤ b ≤ c ≤ g ≤ d ≤ h does not generalise even the intuitionistic fuzzy value of the kind A = (a, c) with a + c ≤ 1 and a < c. That is, if we write A = (a, c) in trapezoidal intuitionistic form, we get A = 〈 (a, a, a, a) (c, c, c, c) 〉 with a < c which contradicts the above definition for TrIFNs. So till today the real generalization of intuitionistic fuzzy values and interval valued intuitionistic fuzzy numbers have not been studied in detail. This problem motivate us to study this type of intuitionistic fuzzy numbers [15] and its ordering principles for ranking.

2 Preliminaries

Here we give a brief review of some preliminaries.

Definition 2.0.1. (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 where μ_A : X → D [0, 1] , ν_A : X → D [0, 1] with the condition $0 < \sup_{x} μ_{A} (x) + \sup_{x} ν_{A} (x) \leq 1$ .

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 (hesitance 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)]. We denote the set of all IVIFSs in X by IVIFS(X). An IVIF value is denoted by A = ([a, b] , [c, d]) for convenience.

Definition 2.0.2. (Lakshmana et al. [15]). An intuitionistic fuzzy set (IFS) A = (μ_A, ν_A) of R is said to be an intuitionistic fuzzy number if μ_A and ν_A are fuzzy numbers. Hence A = (μ_A, ν_A) denotes an intuitionistic fuzzy number if μ_A and ν_A are fuzzy numbers with ν_A ≤ μ_A^c, where μ_A^c denotes the complement of μ_A.

An intuitionistic fuzzy number

A = {(a, b₁, b₂, c) , (e, f₁, f₂, g)} with (e, f₁, f₂, g) ≤ (a, b₁, b₂, c) ^c is shown in Fig. 1.

Definition 2.0.3. (Lakshmana et al. [15]) 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 (e, f₁, f₂, g) ≤ (a, b₁, b₂, c) ^c of the trapezoidal intuitionistic fuzzy number A = {(a, b₁, b₂, c) , (e, f₁, f₂, g)} where (a, b₁, b₂, c) and (e, f₁, f₂, g) are membership and nonmembership fuzzy numbers of A with either e ≥ b₂ and f₁ ≥ c or f₂ ≤ a and g ≤ b₁ on the legs of trapezoidal intuitionistic fuzzy number.

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

Definition 2.0.4. (Lakshmana et al., [14]) For any IVIFN A = ([a, b] , [c, d]), the membership score function is defined as $L (A) = \frac{a + b - c - d + ac + bd}{2}$ .

Definition 2.0.5. (Geetha et al., [9]) For any IVIFN A = ([a, b] , [c, d]), the non-membership score function is defined as $LG (A) = \frac{- a - b + c + d + ac + bd}{2}$ .

Definition 2.0.6. (Geetha et al., [9]) For any IVIFN A = ([a, b] , [c, d]), the vague score function is defined as $P (A) = \frac{a - b - c + d + ac + bd}{2}$ .

Definition 2.0.7. (Geetha et al., [9]) For any IVIFN A = ([a, b] , [c, d]), the imprecise score function is defined as $IP (A) = \frac{- a + b - c + d - ac + bd}{2}$ .

Definition 2.0.8. Let X be a non-empty set. Any subset of the cartesian product X × X is called a relation, denoted by R, on X. We write aRb iff (a, b) ∈ R. A relation R is called a partial ordering on X if it is reflexive, antisymmetric, and transitive. A partial ordering R on X is called total ordering if either aRb or bRa for any a, b ∈ X. Two total orderings R₁ and R₂ are different iff there exist a, b ∈ X with a ≠ b such that aR₁b but bR₂a or aR₂b but bR₁a. For any given total ordered infinite set, there are infintely many ways to redefine a new total ordering on it. A relation R is called an equivalence relation if it is reflexive, symmetric and transitive.

3 Total ordering defined on intuitionistic fuzzy number

Geetha et al. [9] have achieved the total ordering on the set of IVIFNs using membership, nonmembership, vague and imprecise score functions. Let us recall the total ordering defined by Geetha et al. [9] on IVIFN.

Let A = ([a₁, b₁] , [c₁, d₁]) , B = ([a₂, b₂] , [c₂, d₂])∈IVIFN. The total ordering ≤ on IVIFN may be defined to one of the following criterion:

L (A) < L (B), or

L (A) = L (B) but - LG (A) < - LG (B), or

L (A) = L (B) and LG (A) = LG (B) butP (A) < P (B), or

L (A) = L (B) , LG (A) = LG (B) and P (A) = P (B) but - IP (A) ≤ - IP (B).

The above way of defining a total ordering is often referred to as lexicographic in literature [7].

In this section a new total ordering is defined in the set of all intuitionistic fuzzy numbers using the above ordering and a new decomposition theorem on intuitionistic fuzzy sets which is given in the following subsection.

3.1 Upper dense sequence in (0, 1]

The major difficulty in defining total ordering on the set of all fuzzy numbers is that there is no effective tool to identify an arbitrarily given fuzzy number by only finitely many real-valued parameters. To over come this difficulty, W. Wang, Z. Wang [21] has introduced the concept of upper dense sequence in the interval (0, 1], and defined the total ordering on the set of all fuzzy numbers using this sequence. This upper dense sequence gives values for α in the α-cut of FNs. But for IFNs, two sequences are needed to give values for α & β in the (α, β)-cut where α ∈ (0, 1] & β ∈ [0, 1). For convenience upper dense sequence of α is denoted by S₁ ={ α_i|i = 1, 2, . . . } and upper dense sequence of β is denoted by S₂ ={ β_i|i = 1, 2, . . . }. In this section, the upper dense sequence in (0, 1] is briefly reviewed.

Definition 3.1.1. (W. Wang, Z. Wang, [21]) A sequence S = { α_i|i = 1, 2, . . . } ⊂ (0, 1] is said to be upper dense in (0, 1] if, for every point x ∈ (0, 1] and ε > 0, there exists some α_i ∈ S such that α_i ∈ [x, x + ε) for some i. A sequence S ⊂ [0, 1) is said to be lower dense in [0, 1) if, for every point x ∈ [0, 1) and any ε > 0, there exists α_i ∈ S such that α_i ∈ (x - ε, x] for some i.

From definition 3.1.1 we note that, any upper dense sequence in (0, 1] is nothing but a dense sequence with real number 1 and it is also a lower dense sequence.

Definition 3.1.2. Let S₁ ={ α_i|i = 1, 2, . . . } and S₂ ={ β_i|i = 1, 2, . . . } be two upper dense sequences in (0, 1], then the double upper dense sequence is defined as S = (S₁, S₂) ={ (α_i, β_i) |i = 1, 2, . . . }.

Example for an upper dense sequence and double upper dense sequence in (0, 1] is given as follows.

Example 3.1.1. Let S₁ ={ α_i|i = 1, 2 . . . } be the set of all rational numbers in (0, 1], where α₁ = 1, $α_{2} = \frac{1}{2}, α_{3} = \frac{1}{3}, α_{4} = \frac{2}{3}, α_{5} = \frac{1}{4}, α_{6} = \frac{3}{4}, α_{7} = \frac{1}{5},$ $α_{8} = \frac{2}{5}, α_{9} = \frac{3}{5}, α_{10} = \frac{4}{5}, . . .$ .

Then sequence S₁ is an upper dense sequence in (0, 1]. If we allow a number to have multiple occurences in the sequence, the general members in upper dense sequence S_β ={ s_{β
_i}|i = 1, 2, . . . } can be expressed by $s_{β_{i}} = (\frac{i}{k} - \frac{k - 1}{2}), i = 1, 2, . . .$ where $k = ⌈ \sqrt{2 i + \frac{1}{4}} - \frac{1}{2} ⌉$ . That is, $s_{β_{1}} = 1, s_{β_{2}} = \frac{1}{2}, s_{β_{3}} = 1,$ $s_{β_{4}} = \frac{1}{3}, s_{β_{5}} = \frac{2}{3}, s_{β_{6}} = 1, s_{β_{7}} = \frac{1}{4}, s_{β_{8}} = \frac{2}{4}, s_{β_{9}} = \frac{3}{4}, . . .$ . In sequence S_β, for instance, s_{β
₃} is the same real number as s_{β
₁}.

Example 3.1.2. Consider S₁ as in example 1. Let S = (S₁, S₂) = { (α_i, β_i) |α_i = β_i, α_i ∈ S₁ } = {(1, 1) , (1/2, 1/2) , (1/3, 1/3) , (2/3, 2/3) , (1/4, 1/4) , (3/4, 3/4) , (1/5, 1/5) , . . .}. Clearly S is a double upper dense sequence in (0, 1].

In the forthcoming sections this double upper dense sequence will be very useful for defining total orderings on the set of IFNs.

3.2 Decomposition theorem for intuitionistic fuzzy number using upper dense sequence

In this section, a new decomposition theorem for IFSs is established using the double upper dense sequence defined in (0, 1]. Before establishing a new decomposition theorem for intuitionistic fuzzy sets, it is needed for us to define decompostion theorems for intuitionistic fuzzy sets using special α - β cuts.

Definition 3.2.1. Let X be a nonempty universal set and A = (μ_A, ν_A) be an intuitionistic fuzzy set of X with membership function μ_A and with a nonmembership function ν_A. Let α, β ∈ (0, 1]

The α - β cut, denoted by ^α - βA is defined by ^(α - β)A = ^αμ_A × ^βν_A where ^αμ_A = {x ∈ X|μ_A (x) ≥ α} and ^βν_A = {x ∈ X|ν_A (x) ≥ β}. Equivalently $^{(α - β)} A = μ_{A}^{- 1} ([α, 1]) \times ν_{A}^{- 1} ([β, 1])$ is a subset of ℘ (X) × ℘ (X).

The strong alpha beta cut, denoted by is defined by where ^α +μ_A = {x ∈ X|μ_A (x) > α} and X|ν_A (x) > β}. Equivalently $((α, 1]) \times ν_{A}^{- 1} ((β, 1])$ is a subset of ℘ (X) × ℘ (X).

The level set of A, denoted by L (A) is defined by L (A) = L_{μ
_A} × L_{ν
_A} where L_{μ
_A} = { α|μ_A (x) = α, x∈ X } and L_{ν
_A} ={ β|ν_A (x) = β, x ∈ X } which is a subset of ℘ ([0, 1]) × ℘ ([0, 1]).

Example 3.2.1. Let A = 〈(0.17, 0.3, 0.47, 0.56) , (0.05, 0.13, 0.16, 0.23) 〉 be a trapezezoidal intuitionistic fuzzy number. Then ^(α - β)A = ([0.17 + (0.3 - 0.17) α, 0.56 - (0.56 - 0.47) α] , [0.05 + (0.13-0.05) β, 0.23 - (0.23 - 0.16) β]) = ([0.17 + 0.13α, 0.56 - 0.09α] , [0.05 + 0.08β, 0.23 - 0.07β]) , ∀ α, β∈[0, 1]. (0.05 + 0.08β, 0.23 - 0.07β)) , ∀ α, β ∈ [0, 1]. L_{μ
_A} =({0.17 + (0.3 - 0.17) α, 0.56 - (0.56 - 0.47α} , {0.05+(0.13 - 0.05) β, 0.23 - (0.23 - 0.16) β)} = ({ 0.17 + 0.13α, 0.56 - 0.09α } , {0.05 + 0.08β, 0.23 - 0.07β}) , ∀α, β ∈ [0, 1].

One way of representing a fuzzy set is by special fuzzy sets on α-cuts and another way of representing a fuzzy set is by special fuzzy sets on Strong α-cuts. As a generalisaton of fuzzy sets, any intuitionistic fuzzy set can also be represented by the use of special intuitionistic fuzzy sets on α - β cuts and special intuitionistic fuzzy sets on strong α - β cuts.

Definition 3.2.2. The special intuitionistic fuzzy set _(α,β)A = (_αμ_A, _βν_A) is defined by its membership (_αμ_A) and non-membership function (_βν_A) as follows,

$_{α} μ_{A} (x) = {\begin{matrix} α, x \in^{α} μ_{A} \\ 0, otherwise \end{matrix}$

$_{β} ν_{A} (x) = {\begin{matrix} β, x \in^{β} ν_{A} \\ 0, otherwise \end{matrix}$ .

The following decomposition theorems will show the representation of an arbitrary IFS in terms of the special IFSs _(α,β)A.

Theorem 3.1. First Decomposition theorem of an IFS: LetXbe a non-empty set. For an intuitionistic fuzzy subsetA = (μ_A, ν_A) inX, A = (⋃ _α∈[0,1]_αμ_A, ⋃ _β∈[0,1]_βν_A) , where ⋃ is standard union.

Proof. Let x be an arbitrary element in X and let μ_A (x) = a & ν_A (x) = b.

Then $\begin{array}{l} ((\cup_{α \in {[0, 1]}_{α}} μ_{A}) (x), (\cup_{β \in [0, 1]}_{β} ν_{A}) (x)) = (S u p_{α \in [0, 1]}_{α} μ_{A} (x), S u p_{β \in [0, 1]}_{β} ν_{A} (x)) = (m a x [S u p_{α \in [0, a]}_{α} μ_{A} (x), S u p_{α \in (a, 1]}_{α} μ_{A} (x)], \\ m a x [S u p_{β \in [0, b]}_{β} ν_{A} (x), S u p_{β \in (b, 1]}_{β} ν_{A} (x)] \end{array}$

For each α ∈ [0, a], we have μ_A (x) = a ≥ α, therefore _αμ_A (x) = α. On the other hand, for each α ∈ (a, 1], we have μ_A (x) = a < α and _αμ_A (x) =0.

Similarly, for each β ∈ [0, b], we have ν_A (x) = b ≥ β, therefore _βν_A (x) = β. On the other hand, for each β ∈ (b, 1], we have ν_A (x) = b < β and _βν_A (x) =0. Therefore ((⋃ _α∈[0,1]_αμ_A) (x) , (⋃ _β∈[0,1]_βν_A) (x))=(Sup_α∈[0,a]α, Sup_β∈[0,b]β) = (a, b) = (μ_A (x) , ν_A (x)). Hence the theorem.

To illustrate the above theorem, let us consider a trapezoidal intuitionistic fuzzy number A as in Fig. 3.

For each α, β ∈ [0, 1], the α - β cut of A = {(a, b₁, b₂, c) , (e, f₁, f₂, g)} is given by ^(α - β)A =([a + (b₁ - a) α, c - (c - b₂) α] , [e + (f₁ - e) β, g - (g - f₂) β]) and the special intuitionistic fuzzy set _(α,β)A employed in definition 3.2.2 is defined by its membership (_αμ_A) and non-membership function (_βν_A) as follows

$_{α} μ_{A} (x) = {\begin{matrix} α, x \in [a + (b_{1} - a) α, c - (c - b_{2}) α] \\ 0, otherwise \end{matrix}$

$_{β} ν_{A} (x) = {\begin{matrix} β, x \in [e + (f_{1} - e) β, g - (g - f_{2}) β] \\ 0, otherwise \end{matrix}$

Examples of sets ^αμ_A, ^βν_A, _αμ_A and _βν_A for three values of α (namely α₁, α₂, α₃) and β (namely β₁, β₂, β₃) are shown in Fig. 3.

According to theorem 3.1, A is obtained by taking the standard fuzzy union of sets (_αμ_A, _βν_A) for all α, β ∈ [0, 1].

Theorem 3.2. Second Decomposition theorem of an IFS: LetXbe a non-empty set. For an intuitionistic fuzzy subsetAinX,

A = (⋃ _α∈[0,1]_α+μ_A, ⋃ _β∈[0,1]_β+ν_A), where ⋃ is standard union.

Proof. Let x be an arbitrary element in X and let μ_A (x) = a & ν_A (x) = b.

Then ((⋃ _α∈[0,1]_α+μ_A) (x) , (⋃ _β∈[0,1]_β+ν_A) (x)) = (Sup_{α∈[0,1]_α+μ_A (x) , Sup_{β∈[0,1]_β+ν_A (x)) = (max[Sup_{α∈[0,a)_α+μ_A (x) , Sup_{α∈[a,1]_α+μ_A (x)] , max[Sup_{β∈[0,b)_β+ν_A (x) , Sup_{β∈[b,1]_β+ν_A (x)]).}}}}}}

For each α ∈ [0, a), we have μ_A (x) = a > α, therefore _α +μ_A (x) = α. On the other hand, for each α ∈ [a, 1], we have μ_A (x) = a ≤ α and _α +μ_A (x) =0. Similarly, for each β ∈ [0, b), we have ν_A (x) = b > β, therefore _β +ν_A (x) = β. On the other hand, for each β ∈ [b, 1], we have ν_A (x) = b ≤ β and _β +ν_A (x) =0. Therefore

((⋃ _α∈[0,1]_α+μ_A) (x) , (⋃ _β∈[0,1]_β+ν_A) (x))=

(Sup_α∈[0,a)α, Sup_β∈[0,b)β)= (a, b) = (μ_A (x) , ν_A (x)). Hence the theorem.□

Theorem 3.3. Third Decomposition theorem of an IFS: LetXbe a non-empty set. For an intuitionistic fuzzy subsetAinX,

A = (⋃ _{α∈L_{μ
_A}}_αμ_A, ⋃ _{β∈L_{ν
_A}}_βν_A), where ⋃ is standard union.

Proof. The proof of this theorem is similar to the Theorem 3.1.

Regarding IFN as special intuitionistic fuzzy subset (IFS) of $R$ , these decomposition theorems are also available for IFNs. Since they identify an intuitionistic fuzzy number by uncountably infinite real valued parameters generally and therefore lexicography can not be used anymore, unfortunately none of them can be used to define a total ordering on the set of IFNs. Thus, establising a new decomposition theorem, which identifies any IFN by only countably many real valued parameters is essential.

Theorem 3.4. Fourth Decomposition theorem of an IFS:LetA = (μ_A, ν_A) be an intuitionistic fuzzy subset ofX, andS = (S₁, S₂) be a given double upper dense sequence in [0, 1].

ThenA = (⋃ _α∈S₁_αμ_A, ⋃ _β∈S₂_βν_A).

Proof. Let x be an arbitrary element in X and let μ_A (x) = a & ν_A (x) = b. Since S₁ ⊆ [0, 1] & S₂ ⊆ [0, 1], Thus we have ⋃_{α∈S₁_αμ_A ⊆ ⋃ _α∈[0,1]_αμ_A = μ_A and ⋃_β∈S₂_βν_A ⊆ ⋃ _β∈[0,1]_βν_A = ν_A.}

So (⋃ _α∈S₁_αμ_A, ⋃ _β∈S₂_βν_A) ⊆ A ........(1)

Now we have to show that μ_A (x) ≤ ⋃ _α∈S₁_αμ_A (x) and ν_A (x) ≤ ⋃ _β∈S₂_βν_A (x) to prove the theorem.

From second decomposition theorem we know that, for each x ∈ X, (μ_A (x) , ν_A (x)) = (Sup_α∈[0,1] (x) , Sup_{β∈[0,b)_β+ν_A (x)).}

For each α ∈ [0, a), since S₁ is upper dense in (0, 1]. We may find a real number K₁ ∈ S₁ such that K₁ ≥ α, which implies that K₁ ∈ [α, a) and . Thus taking the supremum with respect to α ∈ (0, μ_A (x)), we obtain μ_A (x) = Sup_{α∈(0,a)_α+μ_A (x) ≤ Sup_{K₁∈S₁_K₁μ_A (x) = Sup_{α∈S₁_αμ_A (x). i.e., μ_A (x) ≤ ⋃ _α∈S₁_αμ_A (x).}}}

Similarly for each β ∈ [0, b), since S₂ is upper dense in [0, 1]. We may find a real number K₂ ∈ S₂ such that K₂ ≥ β, which implies that K₂ ∈ [α, b) and . Thus taking the supremum with respect to β ∈ (0, ν_A (x)), we obtain ν_A (x) = Sup_{β∈(0,b)_β+ν_A (x) ≤ Sup_{K₂∈S₂_K₂ν_A (x) = Sup_{β∈S₂_βν_A (x).i.e., ν_A (x) ≤ ⋃ _β∈S₂_βν_A (x).}}}

Hence A (x) = (μ_A (x) , ν_A (x)) ⊆ (⋃ _α∈S₁_αμ_A (x) , ⋃ _β∈S₂_βν_A (x)). ......(2) (1) and (2) concludes the proof.□

4 Total ordering on the set of all intuitionistic fuzzy numbers

The new decomposition theorem established in Section 3 identifies an arbitrary intuitionistic fuzzy number by a countably many real-valued parameters. It provides us with a powerful tool for defining total order in the class of IFN, by extending lexicographic ranking relation defined in Geetha et al. [9].

Note 4.0.1. Let A be an intuitionistic fuzzy number. Let S = { (α_i, β_i) |i = 1, 2, … } ∈ [0, 1] be a double upper dense sequence. The (α_i - β_i)-cut of an IFN A at each α_i, β_i, i = 1, 2, . . ., is a combination of two closed intervals. Denote this intervals by ([a_i, b_i] , [c_i, d_i]), where [a_i, b_i] is the α-cut of the membership function of A and [c_i, d_i] is the β-cut of the non-membership function of A and

let $C_{4 i - 3} = \frac{a_{i} + b_{i} - c_{i} - d_{i} + a_{i} c_{i} + b_{i} d_{i}}{2}$ ,

$C_{4 i - 2} = \frac{a_{i} + b_{i} - c_{i} - d_{i} - a_{i} c_{i} - b_{i} d_{i}}{2}$ ,

$C_{4 i - 1} = \frac{a_{i} - b_{i} - c_{i} + d_{i} + a_{i} c_{i} + b_{i} d_{i}}{2}$ ,

$C_{4 i} = \frac{a_{i} - b_{i} + c_{i} - d_{i} + a_{i} c_{i} - b_{i} d_{i}}{2}, i = 1, 2, \dots$ .

By fourth decomposition theorem these countably many parameters {C_j|j = 1, 2, … } identify the intuitionistic fuzzy number. Using these parameters, we define a relation on the set of all intuitionistic fuzzy numbers as follows.

Definition 4.0.3. (Ranking Principle) Let A and B be any two IFNs. Consider any double upper dense sequence S ={ (α_i, β_i) |i = 1, 2, … } in [0, 1], using note 1, for each i = 1, 2 . . . we have C_4i-3, C_4i-2, C_4i-1, C_4i, which describe a sequences of C_j (A) and C_j (B). If A ≠ B, then there exists j such that C_j (A) ≠ C_j (B) and C_k (A) = C_k (B) for all positive integers k < j. The ’<’ relation on the set of all intuitionistic fuzzy numbers (IFN) is defined as,

A < B if there exists j such that C_j (A) < C_j (B) and C_k (A) = C_k (B) for all positive integers k < j.

The above ranking principle on IFNs is illustrated in the following examples.

Example 4.0.2. Let

A = 〈(0.20, 0.30, 0.50) , (0.35, 0.55, 0.65) 〉

B =〈 (0.17, 0.32, 0.58) , (0.37, 0.63, 0.73) 〉 and

C =〈 (0.25, 0.40, 0.70) , (0.45, 0.75, 0.85) 〉 be three triangular intuitionistic fuzzy numbers (TIFN).

The ordering < defined by using double upper dense sequence S given in example 3.1.2 and the way shown in definition 4.0.3 are now adapted. Thus we have , ,0.15α, 0.70 - 0.3α] , [0.45 + 0.3β, 0.85 - 0.1β]). For i = 1, (α₁, β₁) = (1, 1), C₁ (A) = -0.85, C₁ (B) = -0.1084, C₁ (C) = -0.05. i.e., C₁ (A) < C₁ (B) < C₁ (C). Hence A < B < C.

Example 4.0.3. Let

A =〈 (0.35, 0.35, 0.4, 0.6) , (0.1, 0.2, 0.3, 0.35) 〉

B =〈 (0.35, 0.35, 0.45, 0.55) , (0, 0.3, 0.3, 0.35) 〉 be two TrIFNs.

Thel ordering < defined by using double upper dense sequence S given in example 3.1.2 and the way shown in definition 4.0.3 are now adapted. We have , B = ([0.35, 0.55 - 0.1α] , [0.3β, 0.35 - 0.05β]). For i = 1, (α₁, β₁) = (1, 1), C₁ (A) =0.22 = C₁ (B), C₂ (A)= -0.03, C₂ (B) =0.02. Since C₂ (A) < C₂ (B). Hence A > B. From this example we come to know that C₁ alone can not rank any two given IFNs. With the help of C₂ we discriminate A and B.

The following example shows the importance of double upper dense sequence. In the previous examples, different IFNs are ranked by means of C₁ and C₂. But in general C₁ and C₂ alone need not be enough to rank the entire class of intuitionistic fuzzy number which is shown in example 4.0.4.

Example 4.0.4. Let

A =〈 0.3, 0.35, 0.4, 0.5) , (0.1, 0.2, 0.25, 0.3) 〉

B =〈 (0.35, 0.35, 0.4, 0.55) , (0, 0.2, 0.25, 0.35) 〉 be two TrIFNs.

The ordering < defined by using double upper dense sequence S given in example 3.1.2 and the way shown in definition 1 are now adapted. We have 0.35 - 0.1β]). For i = 1, (α₁, β₁) = (1, 1), we have C₁ (A) =0.235 = C₁ (B), C₂ (A) =0.065 = C₂ (B), C₃ (A) =0.085 = C₃ (B), C₄ (A) = -0.065 = C₄ (B). Now we have to find for i = 2, (α₂, β₂) = (1/2, 1/2), then C₅ (A) =0.26125 < C₅ (B) =0.30125. Since C₅ (A) < C₅ (B). Hence A < B.

Theorem 4.1.Relation < is a total ordering on the set of all intuitionistic fuzzy numbers.

Proof. Claim: < is total ordering on the set of IFNs.

To prove < is total ordering we need to show the following (1). < is Partial ordering on the set of IFNs

(2). Any two elements of the set of IFNs are comparable.

(1). < is partial ordering on the set of IFNs:

(i) < is refelxive:

It is very clear that the relation < is reflexive for any A.

(ii) < is antisymmetric:

claim: If A < B and B < A then A = B.

Suppose A ≠ B, then from the hypothesis A ≺ B and B ≺ A. From the definition 4.0.3, we can find j₁ such that C_{j
₁} (A) < C_{j
₁} (B) and C_j (A) = C_j (B) for all positive integers j < j₁. Similarly we are able to find j₂ such that C_{j
₂} (A) < C_{j
₂} (B) and C_j (A) = C_j (B) for all positive integers j < j₂. Then j₁ & j₂ must be the same, let it to be j₀. But C_{j
₀} (A) < C_{j
₀} (B), and C_{j
₀} (B) < C_{j
₀} (A) this contradicts our hypothesis. Therefore our assumption A ≠ B is wrong. Hence A = B.

(iii) < is transitive:

To prove (iii), we have to show that if A < B and B < C then A < C

Let A, B, C be three IFNs. Let us assume A < B and B < C.....(1)

Therefore from A < B, we can find a positive integer k₁ such that C_{k
₁} (A) < C_{k
₁} (B) and C_k (A) = C_k (B) for all positive integer k < k₁. Similarly from B < C, we can find a positive integer k₂ such that C_{k
₂} (B) < C_{k
₂} (C) and C_k (B) = C_k (C) for all positive integer k < k₂. Now taking j₀ = min (k₁, k₂), we have C_{k
₀} (A) < C_{k
₀} (C) and C_k (A) = C_k (C) for all positive integer k < k₀. i.e., A < C. Hence < is Transitive.

Therefore from (i), (ii), and (iii), we proved the relation < is Partial Ordering on the set of all IFNs. (2). Any two elements of the set of IFNs are comparable.

For any two IFNs A and B, they are either A = B, or A ≠ B. In the latter case, there are some integers j such that C_j (A) ≠ C_j (B). Let J ={ j|C_j (A) ≠ C_j (B) }. Then J is lower bounded 0 and therefore, according to the well ordering property, J has unique smallest element, denoted by j₀. Thus we have C_j (A) = C_j (B) for all positive integers j < j₀, and either C_{j
₀} (A) < C_{j
₀} (B) or C_{j
₀} (A) > C_{j
₀} (B), that is, either A ≺ B or B ≺ A in this case. So, for these two IFNs, either A < B or B < A. This means that partial ordering < is a total ordering on the set of all intuitionistic fuzzy numbers. Hence the proof.

Similar to the case of total orderings on the real line (- ∞ , + ∞) a the total ordering on the sets of special types of IFNs shown in Section 3, infinitely many different total orderings on the set of IFNs can be defined. A notable fact is that each of them is consistent with the natural ordering on the set of all real numbers. This can be regarded as a fundamental requirement for any practice of ordering method on the set of all IFNs.

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 at some point of time they ranked two different numbers as the same. In this paper a special type of IFNs which is shown in Fig. 1 which generalizes IFN more natural in real scenario is considered. 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. In this subsection our proposed method is compared with the total score function defined in Lakshmana et al. [15], which is explained here with an illustrative example.

5.1 Comparision between our proposed method with the score function defined in Lakshmana et al. [15]

In this subsection, our proposed method is compared with the total score function defined in Lakshmana et al. [15], which is explained here with an illustrative example.

Definition 5.1.1. The membership score of the triangular intuitionistic fuzzy number (TIFN) M = { (a, b, c) (e, f, g)} is defined by $T (M) = \frac{[1 + R (M) - L (M)]}{2}$ , where $L (M) = \frac{1 - a}{1 + b - a}$ and $R (M) = \frac{c}{1 + c - b}$ .

Definition 5.1.2. (Lakshmana et al. [15]) The new membership score of the triangular intuitionistic fuzzy number M ={ (a, b, c) (e, f, g) } is defined by $NT (M) = \frac{[1 + NL (M) - NR (M)]}{2}$ , where $NL (M) = \frac{e}{1 + e - f}$ and $NR (M) = \frac{1 - g}{1 + f - g}$ .

Definition 5.1.3. (Lakshmana et al. [15]) The nonmembership score of the triangular intuitionistic fuzzy number M ={ (a, b, c) (e, f, g) } is defined by NT_c (M) =1 - NT (M).

Definition 5.1.4. (Lakshmana et al. [15]) Let M =〈 (a, b, c) , (e, f, g) 〉 be an triangular intuitionistic fuzzy number. If e ≥ b and f ≥ c, then the score of the intuitionistic fuzzy number M is defined by (T, NT_c), where T is the membership score of M which is obtained from (a, b, c) and NT_c is the nonmembership score of M obtained from (e, f, g).

Definition 5.1.5 (Lakshmana et al. [15]) Ranking of Intuitionistic Fuzzy Numbers: Let M₁ =〈 (a₁, b₁, c₁) , (e₁, f₁, g₁) 〉 and M₂ = 〈 (a₂, b₂, c₂) , (e₂, f₂, g₂)〉 be two triangular intuitionistic fuzzy numbers with e_i ≥ b_i & f_i ≥ c_i. Then M₁ ≤ M₂ if the membership score of M₁≤ the membership score of M₂ and the nonmembership score of M₁≥ the nonmembership score of M₂.

In this example definition 5.1.1 to 5.1.5 are demonstrated and also the illogicality of Lakshmana et al.’s [15] 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 the two triangular intuitionistic fuzzy numbers. Using definition 5.1.1 we get L (M) =0.8333 and R (M) =0.3333 which implies T (M) =0.25. Applying definition 5.1.2, 5.1.3, 5.1.4 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 and NT_c = 0.553 respectively. Then the intuitionistic fuzzy score is represented by (T, NT_c) = (0.25, 0.553). Similarly using definitions 5.1.1 to 5.1.3, 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 5.1.5, this method ranks M and N are equal but M & N are different triangular intuitionistic fuzzy numbers.

The total ordering < defined by using double upper dense sequence D_(α,β) given in example 3.1.2 and the way shown in definition 4.0.3 are now adapted. For i = 1 and (α₁, β₁) = (1, 1), we have C₁ (M) = -0.16 and C₁ (N) = -0.0875.

i.e., C₁ (M) < C₁ (N). Hence our proposed method ranks N as better one.

5.2 Comparision of the proposed method with some existing methods

In this sub section significance of our proposed method over some existing methods are explained with examples. The table 1 shows that our proposed method is significant over the methods presented in [3 , 26], and it is supported by Geetha et al. [9].

For example, let A = ([0.2, 0.25] , [0.4, 0.45]) and B = ([0.15, 0.3] , [0.35, 0.5]) be two IVIFNs. Then by applying Lakshmana and Geetha [13] approach we get $LG (A) = LG (B) = \frac{0.45 + δ (0.70)}{2}$ this implies that A and B are equal which is illogical. By applying the total ordering < defined by using double upper dense sequence D_(α,β) given in example 3.1.2 and the way shown in definition 4.0.3 are now adapted. For i = 1 and (α₁, β₁) = (1, 1), we have C₁ (A) = -0.10375 and C₁ (B) = -0.09875, i.e., C₁ (A) < C₁ (B). Hence A < B which is supported by Geetha et al.’s approach. The double upper dense sequence S in example 3.1.2 is used as a necessary reference system for our proposed method in table 1 (i.e., for i = 1, (α₁, β₁) = (1, 1)).

5.3 Trapezoidal Intuitionistic Fuzzy Information System(TrIFIS)

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.

Definition 5.3.1. An Information System

S = (U, AT, V, f) with V = ∪ _a∈ATV_a where V_a is a domain of attribute a is called trapezoidal intuitionistic fuzzy information system if V is a set of TrIFN. We denote f (x, a) ∈ V_a by f (x, a) =〈 (a₁, a₂, a₃, a₄) , (a₁^′, a₂^′, a₃^′, a₄^′) 〉 where a_i, a_i^′ ∈ [0, 1]. The numerical illustration is given in example 5.5.1.

Definition 5.3.2. A TrIFIS, S = (U, AT, V, f) together with weights W = {w_a/a ∈ AT} is called Weighted Trapezoidal Intuitionistic Fuzzy Information System (WTrIFIS) 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 4.0.3) 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 a WTrIFIS 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 a WTrIFIS 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 WTrIFIS

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

Algorithm:5.3.

1. Using Definition 4.0.3 find $C_{j}^{'} s$ accordingly, to decide whether x_i > _a x_i 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 WTrIFIS with attributes {a_j|j = 1 to 5} as product quality, relation- ship closeness, delivery performance, social responsib- ility and legal issue.

An TrIFIS 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 }.

In the table 2, f (x_i, a_i) = (a₁, a₂, a₃, a₄) , (c₁, c₂, c₃, c₄) denotes the trapezoidal intuitionistic fuzzy numbers which include intuitionistic fuzzy values and interval valued intuitionstic fuzzy numbers and they stand for the evaluation of alternative x_i under the criteria a_i with acceptance of (a₁, a₂, a₃, a₄) and nonacceptance of (c₁, c₂, c₃, c₄). For example, f (x₁, a₁) denotes supplier x₁ is evaluated under the criteria ‘product quality’ (a₁) with “20% of acceptance and 40% of non acceptance” and f (x₅, a₃) denotes the supplier x₅ is evaluated under the criteria ‘delivery performance (a₃)’ with “around 30% to 35% of accepatance and around 45% to 50% of non acceptance”.

Step:1 For i = 1, (α, β) = (1, 1): By step 1, C₁ (f (x_i, a_j)) using defintion 4.0.3 and note 4.0.1, for all a_i ∈ AT and for all x_i ∈ U is found and tabulated in table 3. If C₁ (f (x_i, a_j)) = C₁ (f (x_j, a_j)) for any alternatives x_i, x_j then C₂ and other necessary score functions (C₃ and C₄) are found wherever required. The bold letters are used in table 3 to represent the equality of scores. From table 3 we observe that in many places $C_{j}^{'} s$ are not distiguishable for different IFNs.

Hence from table 3 we do not get the best alternative. Therefore the same procedure which is explained in step 1 is repeated for i = 2, (α, β) = (1/2, 1/2) and it is shown in table 4.

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, ${WR}_{A} (x_{1}) = \frac{1}{10} \sum_{j = 1}^{10} {WR}_{A} (x_{1}, x_{j}) = 0.469$ . So by step 5, x₈ is selected as the best object from the weighted trapezoidal intuitionistic fuzzy information system is seen from table 6.

6 Conclusion

In this paper, the total ordering on the set of all IFNs is achieved. The total orderings introduced and discussed in this paper are consistent with the natural ordering of real numbers. Actually this total ordering on IFNs generalises the total ordering on FNs defined by Wei Wang, Zhenyuan Wang [21]. Therefore this is an appropriate generalization of the total ordering on the set of all real numbers to the set of IFNs. This method can order intuitionistic fuzzy numbers, either alone or as a supplementary means with other ranking methods, and may be adopted in decision making with fuzzyinformation.

Footnotes

Acknowledgments

Authors thank the valuable suggestions from anonymous referees for the betterment of the quality of the paper.

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