A Novel Approach for Ranking Intuitionisitc Fuzzy Numbers and its Application to Decision Making
Meishe Liang Shijiazhuang Tiedao University Ju-Sheng Mi ( [email protected] ) Hebei Normal University https://orcid.org/0000-0002-3753-4490 Shaopu Zhang Shijiazhuang Tiedao University Chenxia Jin Hebei University of Science and Technology
Research Article
Keywords: Intuitionistic fuzzy number , Intuitionistic fuzzy set , Ideal measure , Multi-attribute decision making
Posted Date: July 6th, 2021
DOI: https://doi.org/10.21203/rs.3.rs-582033/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Soft Computing manuscript No. (will be inserted by the editor)
A novel approach for ranking intuitionisitc fuzzy numbers and its application to decision making
1,2 2 1 2,3 Meishe Liang · Jusheng Mi ∗ · Shaopu Zhang · Chenxia Jin
Received: date / Accepted: date
Abstract Ranking intuitionistic fuzzy numbers is an membership degree robustness, nonmembership degree important issue in practical application of intuitionis- robustness, and determinism. Numerical example is ap- tic fuzzy sets. For making a rational decision, people plied to illustrate the effectiveness and feasibility of this need to get an effective sorting over the set of intuition- method. Finally, using the presented approach, the op- istic fuzzy numbers. Many scholars rank intuitionistic timal alternative can be acquired in multi-attribute de- fuzzy numbers by defining different measures. These cision making problem. Comparison analysis shows that measures do not comprehensively consider the fuzzy se- the intuitionistic fuzzy value ordering method obtained mantics expressed by membership degree, nonmember- by the ideal measure is more effectiveness and simplic- ship degree and hesitancy degree of intuitionistic fuzzy ity than other existing methods. numbers. As a result, the ranking results are often coun- Keywords Intuitionistic fuzzy number Intuitionistic terintuitive, such as the indifference problems, the non- · fuzzy set Ideal measure Multi-attribute decision robustness problems, etc. In this paper, according to · · geometrical representation, a novel measure for intu- making itionistic fuzzy number is defined, which is called the ideal measure. After that a new sorting approach of intuitionistic fuzzy numbers is proposed. It is proved 1 Introduction that the intuitionistic fuzzy order obtained by the ideal measure satisfies the properties of weak admissibility, After the fuzzy set [31] was proposed by Zadeh in 1965, the mathematical research has been extended from the Meishe Liang precise problems to the fuzzy problems. The essence of E-mail: [email protected] fuzzy set is to extend the membership value µA(x) of Jusheng Mi element x for set A from 0, 1 to any value on closed E-mail: [email protected] { } interval [0, 1]. The value 1 µA(x) is used to express the − Shaopu Zhang degree to which x is not belong to the set A. As a di- E-mail: [email protected] rect extension of fuzzy set theory, intuitionistic fuzzy Chenxia Jin set (IFS) [2,3] was first introduced by Atanassov in E-mail: [email protected] 1986, in which membership degree, nonmembership de- 1Department of Mathematics and Physics, Shijiazhuang gree and hesitancy degree are used to characterize the Tiedao University, objective world in simultaneously. Therefore, it is more Shijiazhuang 050043, P.R.China delicate in describing the fuzziness of the real wrold 2School of Mathematical Science, Hebei Normal University, and more flexible in dealing with the uncertainty. At Shijiazhuang 050024, P.R.China present, the theoretical and practical application of IF- 3School of Economics and Management, Hebei University of S have been deeply and attentively studied and have Science and Technology, been widely used in many application fields [10,18,33], Shijiazhuang 050018, P.R.China such as medicine diagnosis [8,11], multi-attribute de- cision making [9,20,34,38], cluster analysis [5,19,36], 2 Meishe Liang et al. feature extraction [21–23], pattern recognition [24,25] the degree of uncertainty, which is obviously unreason- and other aspects. able. These methods mentioned above are important for ranking IFNs, and have been used in many practical With the in-depth study of IFS in the application problems. As we show in Section 3, the ranking results field, many pioneering works have been focused on com- by the mentioned approaches are often counterintuitive, paring and ranking intuitionistic fuzzy numbers (IFN- such as the indifference problems, the non-robustness s) [32]. A large number of scholars make great efforts problems, etc. It is necessary to present more effective in constructing different measures to rank IFSs (IFNs) and reasonable measure for sorting IFNs. and apply them to practical problems. Xu and Yager Based on the previous works, our purpose is to con- [30] firstly presented the rule to sort IFNs by two func- struct a new measure for sorting IFNs. This measure tions in 2006. They are score function and accuracy called ideal measure on the basis of Euclidean distance function. Comparing two IFNs, the larger the score and the geometrical representation of IFNs. Some im- function is, the higher order the IFN is. If the two IFNs portant properties are also addressed. have the same score function, the larger the accuracy The remainder of the paper is structured as follows: function is, the higher order the IFN is. This method In Section 2, we briefly review some preliminary notions is simple and easy to operate. However, if there is a s- and concepts, such as IFS, IFN, which will be used in light change of membership degree or nonmembership the following content. In Section 3, we introduce some degree, the ranking results may be completely oppo- exiting methods to rank IFNs in detail. In Section 4, site. Taking into account both the reliability informa- we introduce a novel approach for ranking IFNs. Mean- tion and the amount of information represented by an while, some desirable properties have been also checked. IFN, Szmidt [26] constructed the measure R to rank After that, illustrate examples are applied to show the IFNs. However, some counterintuitive ranking results reasonability of the proposed method. Section 5 discuss- will be produced by measure R. Based on the draw- es the optimal alternative selecting in multi-attribute back of the inadmissibility of IFNs order induced by decision making problem. In Section 6, we conclude this measure R, Guo [13] proposed a new measure Z to Q paper by an outlook for further research. rank IFNs by the view of the geometrical representa- tion of IFNs. The method can also set different param- eters to express different preference of decision makers. 2 Preliminaries Based on the similarity measure of IFNs, Zhang [35] proposed the measure l for sorting IFNs. In [27], Xing This section first briefly recalls some preliminary con- et al. pointed out that the order induced by measure cepts, such as intuitionistic fuzzy set and intuitionistic l could output unreasonable ordering results known as fuzzy number. the determinacy problem. Base on the Euclidean dis- Definition 1 [2,3] An IF set A on the universe X tance and the geometrical representation of IFNs, Xing is defined by A = µA(x),νA(x) : x X , in which {h i ∈ } et al. [27] introduced the measure P for sorting IFN- two mappings µA : X [0, 1] and νA : X [0, 1] → → s. This measurement takes into account the Euclidean represent the membership degree and nonmembership distance between the ideal positive IFN and the IFNs degree of the element x X to A. For any x X, ∈ ∈ of discussion. Xing et al. pointed out that the closer the 0 µA(x)+ νA(x) 1. We use 1 µA(x) νA(x) to ≤ ≤ − − IFN with the ideal positive IFN is, the more preference illustrate the hesitancy degree of x X to A (namely ∈ and the higher order of the IFN is. Although the rank πA(x)). Note that if for any x X, πA(x) = 0, the IFS ∈ of the IFNs induced by measure P possesses a lot of at- A degenerates into a fuzzy set. All IF subsets on X is tractive properties, this method does not consider the denoted by (X). IFS distance between the ideal negative IFN and the IFNs of Definition 2 [2,3] For two IFSs A = µA(x),νA(x) : {h i discussion. Huang et al. [16] indicated that the farther x X , B = µB(x),νB(x) : x X . The opera- ∈ } {h i ∈ } the distance between the IFN and the ideal negative tional law are defined as follows: IFN is, the more favorable and the higher order of the (1) A B if and only if µA(x) µB(x), νA(x) ⊆ ≤ ≥ corresponding IFN should be. In this view, the relative- νB(x), x X; ∀ ∈ distance-based favorable measure ME is defined to rank (2) A = B if and only if A B, B A; ⊆ ⊆ IFNs. For any two different IFNs, if the membership (3) A = νA(x),µA(x) : x X ; ∼ {h i ∈ } degree and the nonmembership degree have the same (4) A B = µA(x) µB(x),νA(x) νB(x) : x ∩ {h ∧ ∨ i ∈ value respectively, we have the same measure ME val- X ; } ue. However, the hesitation degree of the two IFNs are (5) A B = µA(x) µB(x),νA(x) νB(x) : x ∪ {h ∨ ∧ i ∈ obviously different, and the hesitation degree indicates X . } A novel ranking approach of IFNs 3
Definition 3 [28] Let A be an IFS on the universe order on IFNs is a linear order and refines the order X, for x X, A(x)= µA(x),νA(x) is called an IFN IFN , then it is an admissible order. However, the or- ∈ h i (or IF value). ders obtained by some measures, including the measure Definition 4 [29] Let A(x1), A(x2) be two IFNs. R, the measure l and so on are not admissible orders Some operational law are defined as follows: sometimes, since these measures are all single contin- (1) A(x ) A(x ) = µA(x )+ µA(x ) µA(x ) uous functions. In order to redefine the order of IFNs, 1 ⊕ 2 h 1 2 − 1 · µA(x ),νA(x ) νA(x ) ; Xing et al. [27] revised the definition of the admissible 2 1 · 2 i (2) A(x ) A(x ) = µA(x ) µA(x ),νA(x ) + order. 1 ⊗ 2 h 1 · 2 1 νA(x2) νA(x1) νA(x2) ; Definition 6 [27] Let (IFN, ) be a preordered set − · i λ λ (3) λA(x1)= 1 (1 µA(x1)) ,νA(x1) ; on the set of IFN. If the order satisfies the following λ h −λ − iλ (4) (A(x )) = µA(x ), 1 (1 νA(x )) . conditions: 1 h 1 − − 1 i These operations provide a useful way to aggregate (1) is a weak order on IFN; IFNs. (2) A(x ), A(x ) IFN, if A(x ) IFN A(x ), 1 2 ∈ 1 2 According to Definition 2, A∗(x) = 1, 0 is the then A(x1) A(x2). h i biggest IFN, then we denote A (x) as the ideal pos- Then the order is a weak admissible order. ∗ itive IFN [26], while A∗(x) = 0, 1 is called the In another word, if is a weak admissible order, ∼ h i ideal negative IFN. Let A(x ), A(x ) be two IFNs, if then is not only a weak order, but also refines the 1 2 A(x ) A(x ), then we note A(x ) IFN A(x ). The order IFN . 1 ⊆ 2 1 2 score function and the accuracy function are important Intuitively, the ordering of IFN should have certain measures and often applied to sort IFNs. stability. It should allow a small change to the member- Definition 5 [6,15] Let A(x) be an IFN, SA(x)= ship (and nonmembership) degree of the IFNs while the µA(x) νA(x), HA(x)= µA(x)+ νA(x) are commonly ordering remains unchanged. In other words, the order − referred to the score function (namely SA(x)) and the is both membership degree and nonmembership degree accuracy function (namely HA(x)), respectively. robust. From Definition 5, It is clear that the bigger the Definition 7 [27] Let be an order on IFN. A(x ), ≺ 1 value of SA(x), the higher the relative support of the A(x ) IFN are two IFNs, where µA(x ) µA(x ) = 0 2 ∈ 1 · 2 6 A(x); while the lager the value of HA(x), the more and πA(x ) πA(x ) = 0. The order is called mem- 1 · 2 6 ≺ information A(x) contains. bership degree robust if A(x ) A(x ) implies that 1 ≺ 2 there exists ε> 0 such that A′(x ) A(x ) (A(x ) 1 ≺ 2 1 ≺ A′(x )), where A′(x )= µA(x )+ε,νA(x ) (A′(x )= 2 1 h 1 1 i 2 3 The ranking principle of IFNs and Existing µA(x ) ε,νA(x ) ). h 2 − 2 i Methods Definition 8 [27] Let be an order on IFN. A(x ), ≺ 1 A(x2) IFN are two IFNs, where νA(x1) νA(x2) = 0 In this section, we will introduce some exiting methods ∈ · 6 and πA(x1) πA(x2) = 0. The order is called non- for ranking IFNs. In order to comprehend these meth- · 6 ≺ membership degree robust if A(x1) A(x2) implies ods, some terminologies related to sorting principle of ≺ that there exists ε > 0 such that A′(x1) A(x2) IFNs need to be introduced first. ≺ (A(x ) A′(x )), where A′(x ) = µA(x ),νA(x ) 1 ≺ 2 1 h 1 1 − ε (A′(x )= µA(x ),νA(x )+ ε ). i 2 h 2 2 i Intuitively, people are used to be vagueness averse 3.1 The ranking principle of IFNs when facing fuzzy information and incomplete informa- tion [32,37]. Thus, if two IFNs with the same value of A preorder is a kind of binary relationship on some score function, the more accurate of the IFN, the more particular nonempty set in which the order satisfies re- preferable it will be. Now, a definition is used to indi- flective and transitive [17]. Let be a preorder defined cate this fact. over set P , if is comparable, then (P, ) is called a Definition 9 [27] Let be an order on IFN. A(x ), weak order or total preorder [1]. In addition, if a weak ≺ 1 A(x ) IFN are two IFNs. The order is determi- order (P, ) satisfies antisymmetry [17], then (P, ) is 2 ∈ ≺ nate if SA(x )= SA(x ) and HA(x ) – Xu and Yager’ s rule [30]: Definition 12 The measure ZQ of A(x) is defined as Definition 10 Let A(x1), A(x2) IFN be two ∈ k 1 IFNs. ZQ(A(x)) = (1 πA(x))(µA(x)+ πA(x)), − 1+k 1+k (1) if S(A(x1)) < S(A(x2)), then A(x1) is smaller where k is used to express the preference of the decision than A(x2), denoted by A(x1) XY A(x2). ≺ maker. Moreover, k (0, 1) indicates the optimistic (2) if S(A(x1)) = S(A(x2)) and H(A(x1)) – Huang et al.’ s rule [16]: Proof (1) According to Definition 6, take the partial derivative of both sides of the measure I with respect Definition 15 The measure ME of A(x) is defined to µα, as 1 µα 1 µα 1 I (α)= ( + − + ) µ′ α √ 2 2 2 2 √ 2 √µα+(1 να) √(1 µα) +να 2 2 2 2 2 µA(x)+(1 νA(x)) (1 µA(x)) +νA(x) − − ME(A(x)) = − − . > 0. q 2 − q 2 It means that I(α) is monotone increasing with respect The order ME for sorting IFNs can be defined as to µα. follows: Then take the partial derivative of both sides of the Suppose that A(x1), A(x2) IFN are two IFNs. ∈ measure I with respect to να, A(x1) ME A(x2) if and only if ME(A(x1)) ME(A(x2)). ≤ 1 1 να να 1 I (α)= ( − + ) Similar to the order P , it can be proved that the ν′ α √ 2 2 2 2 √ 2 √µ +(1 να) √(1 µα) +ν 2 − α − − − α 1 1 1 να να order ME satisfies weak admissible and both member- = ( − ) √ √ 2 2 2 2 2 2 √µ +(1 να) √(1 µα) +ν ship robust and nonmembership robust. However, the − α − − − α 1 1 1 να να ( − ) √ √ 2 2 2 2 order ME is not determinate. Let us consider the fol- 2 2 √(1 να) +(1 να) √(1 µα) +ν ≤ − − − − − α 1 1 1 να να lowing example. For A(x) IFN, if µA(x)= νA(x) = ( − ) √2 √2 √2(1 ν ) 2 2 ∈ ≤ − α − √(1 µα) +να 0.5, then M (A(x)) 1, such as M ( 0.2, 0.2 ) = ν − − E E 1 ( α ) ≡ h i √ 2 2 2 √(1 µα) +ν ME( 0.3, 0.3 ) = ME( 0.5, 0.5 ) = 1. Intuitively, this ≤ − − α h i h i 0. result is obviously unreasonable. ≤ It is clear that I(α) is monotone decreasing with respect to να. 4 A novel approach for ranking IFNs Since α IFN, 0, 1 IFN 0,να IFN µα,να ∀ ∈ h i h i h i IFN µα, 0 IFN 1, 0 . According to monotony, I( 0, 1 ) h i h i h i In order to make a reasonable ordering, we should have I( 0,να ) I(α) I( µα, 0 ) I( 1, 0 ). According ≤ h i ≤ ≤ h i ≤ h i a deep understanding of the fuzzy semantics of mem- to Definition 17, I( 0, 1 )= 1 and I( 1, 0 ) = 1. Thus, h i − h i bership degree, nonmembership degree and hesitation α IFN, 1 I(α) 1. ∀ ∈ − ≤ ≤ degree in IFNs. In fact, three factors should be con- It can be seen from (1) that both (2) and (3) are sidered simultaneously while ranking IFNs. Firstly, the obvious. closer the distance between one IFN with the ideal pos- (4) Suppose α = µα,να ,β = µβ,νβ are two IFN- h i h i itive IFN is, the higher order of the IFN should be. Sec- s. If β IFN α, we need to verify that I(β) I(α). ≤ ondly, the farther the distance between one IFN with According to Definition 4, µβ µα and νβ να. That ≤ ≥ the ideal negative IFN is, the higher order of the IFN is to say µβ,νβ IFN µβ,να IFN µα,να . Since h i h i h i should be. Finally, the smaller the hesitancy degree is, I(α) is monotone increasing w.r.t µα and monotone de- the more information the IFN contains, and the higher creasing w.r.t να, respectively, we have I( µβ,νβ ) h i ≤ order of the IFN should be. Now we provide a list of I( µβ,να ) I( µα,να ). Thus, I(β) I(α). h i ≤ h i ≤ axioms that a “reasonable” measure of IFN should sat- In conclusion, I is an ideal measure of IFN. isfy for ranking IFNs. These axioms are summarized as follows. Lemma 1 Suppose α is an IFN. Then the ideal Definition 16 Denote I : IFN [ 1, 1], then I measure I of the α is also represented as → − 2 2 2 2 is called the ideal measure of IFN, if α,β IFN, it µ +(1 να) (1 µα) +ν π ∀ ∈ I(α)= α − − α α . satisfies the following properties: q 2 − q 2 − 2 (1) 1 I(α) 1; Proof According to Definition 1, the proof is obvious. − ≤ ≤ (2) I(α) = 1 if and only if α = 1, 0 ; h i Proposition 1 Suppose α is an IFN. If πα = 1 (3) I(α)= 1 if and only if α = 0, 1 ; − − h i µα να = 0, then we have I(α)= µα να. (4) β IFN α I(β) I(α). − − ⇒ ≤ According to Definition 16, we have a new measure Proof If πα = 1 µα να = 0, then µα = 1 να and − − − as following. να = 1 µα. Thus, − Definition 17 Suppose α = µα,να is an IFN. De- 2 2 2 2 µ +(1 να) (1 µα) +ν π h i I(α)= α − − α α fine the measure I of IFN as: q 2 − q 2 − 2 2 2 2 2 µ +(1 να) (1 µα) +ν α − − α = q 2 q 2 2 2 2 2 − µα+(1 να) (1 µα) +να 1 µα να 2 2 2 2 I α − − . µα+µα να+να ( )= q 2 q 2 − 2− = − − q 2 − q 2 = µα να. Theorem 1. I is the ideal measure of IFN. − 6 Meishe Liang et al. In fact, if πα = 1 µα να = 0, then the IFN α′ I β (and α I β′), where α′ = µα,να ε ,β′ = − − ≺ ≺ h − i α degenerates into a fuzzy number. I(α) = µα να = µβ,νβ + ε . Let − h i 2µα 1, the bigger membership degree of α is, the more − f(ε)= I(β) I(α′) favorable and the higher order α is, which is also a − 1 2 2 2 2 πβ standard way for sorting fuzzy numbers. = ( µ + (1 νβ) (1 µβ) + ν √2 q β − − q − β − √2 According to the value of the ideal measure I, the 2 2 2 2 πα′ µ ′ + (1 να′ ) + (1 µα′ ) + ν ′ + ) −p α − p − α √2 method for sorting IFNs can be interpreted as follows: 1 2 2 2 2 πβ = ( µβ + (1 νβ) (1 µβ) + νβ Definition 18 Suppose α = µα,να ,β = µβ,νβ √2 q − − q − − √2 h i h i 2 2 2 2 are two IFNs. β I α if and only if I(β) I(α). µα + (1 να + ε) + (1 µα) +(να ε) ≤ −pπα ε − p − − According to Definition 16, the order induced by I + + ) √2 √2 is weak order. In the following, we will verify whether Then the order for ranking IFNs induced by the ideal measure 1 1 1 να+ε να ε I satisfies the properties of weak admissibility, member- f (ε)= ( − − ) ′ √ √ 2 2 2 2 2 2 √µ +(1 να+ε) √(1 µα) +(να ε) − α − − − − ship degree robustness, nonmembership degree robust- 1 1 1 να+ε να ε < ( − − ) √ √ √ 2 2 2 2 2(1 να+ε) √(1 µα) +(να ε) ness, and determinacy. − − − − − < Theorem 2. The order induced by I is weak ad- 0 missibility. f(ε) is strictly monotone decreasing w.r.t parameter ε. Since f(0) = I(β) I(α) > 0, According to the Proof According to Definition 16 and Theorem 1, I is − a weak admissible order on IFN. function continuity properties, there exists ε > 0 such f(0) that f(ε)= > 0, which implies α′ I β. Similarly, 2 ≺ Theorem 3. The order induced by I is member- if α I β and β′ = µβ,νβ + ε , then there exists ε> 0 ≺ ≺ h i ship degree robust. such that α I β′. ≺ Proof Suppose that α = µα,να ,β = µβ,νβ are two Theorem 5. The order induced by I is determi- h i h i ≺ IFNs, where µα µβ = 0 and πα πβ = 0. If α I β, nate. · 6 · 6 ≺ we only need to prove that there exists ε> 0 such that Proof Suppose that α = µα,να ,β = µβ,νβ are two α′ I β (and α I β′), where α′ = µα + ε,να ,β′ = h i h i ≺ ≺ h i S β S α H β >H α µβ ε,νβ . Let IFNs, where ( )= ( ) and ( ) ( ). According h − i to Definition 5, we have µα να = µβ νβ and µβ +νβ > − − f(ε)= I(β) I(α′) µ + ν . In other words, µ µ = ν ν , µ <µ , − α α β α β α α β 1 2 2 2 2 πβ − − = ( µ + (1 νβ) (1 µβ) + ν να <νβ and πβ < πα. √2 q β − − q − β − √2 2 2 2 2 πα′ µ ′ + (1 να′ ) + (1 µα′ ) + ν ′ + ) −p α − p − α √2 (µα + να + µβ + νβ 2)(νβ να) < 0 1 2 2 2 2 πβ − − = ( µ + (1 νβ) (1 µβ) + ν (νβ + να)(νβ να) < (2 µα µβ)(νβ να) √2 q β − − q − β − √2 ⇔ − − − − 2 2 2 2 (νβ + να)(νβ να) < (µβ µα)(2 µα µβ) (µα + ε) + (1 να) + (1 µα ε) + να ⇔ − − − − −pπ − p − − (ν + ν )(ν ν ) < ((1 µ ) (1 µ ))((1 µ ) + α ε ) β α β α α β α √2 √2 ⇔ − − − − − − +(1 µβ)) 2 − 2 2 2 then νβ να < (1 µα) (1 µβ) ⇔ − 2 −2 − −2 2 1 1 µα+ε 1 µα ε (1 µβ) + νβ < (1 µα) + να f ′(ε)= − ( + + − − ) ⇔ − − √ √ 2 2 2 2 2 2 2 2 2 2 √(µα+ε) +(1 να) √(1 µα ε) +να (1 µβ ) +νβ (1 µα) +ν − − − q − < − α < 0 ⇔ 2 q 2 f(ε) is strictly monotone decreasing w.r.t parameter On the other hand, ε. Since f(0) = I(β) I(α) > 0, According to the 2 2 2 2 µβ +(1 νβ ) πβ µ +(1 να) π − q − α − + α function continuity properties, there exists ε > 0 such 2 − 2 − q 2 2 f(0) µ2 +(1 ν )2 2 2 f ε > α β β β 1 µβ νβ µα+(1 να) 1 µα να that ( )= 2 0, which implies ′ I . Similarly, = q − − − − + − − ≺ 2 − 2 − q 2 2 if α I β and β′ = µβ ε,νβ , then there exists ε> 0 1 2 2 2 2 ≺ h − i = ( µβ + (1 νβ) µα + (1 να) + √2(µβ µα)) such that α I β′. √2 q − − p − − ≺ 1 (µβ µα)(2 µα να µβ νβ ) = (√2(µβ µα) − − − − − ) √2 2 2 2 2 √µβ +(1 νβ ) +√µα+(1 να) Theorem 4. The order I is nonmembership de- − − − − µβ µα πα+πβ ≺ = − (√2 ) √ 2 2 2 2 gree robust. 2 µ +(1 νβ ) +√µ +(1 να) − √ β − α − µβ µα πα+πβ = − (√2 ) √2 2 2 2 2 2 2 Proof Suppose that α = µα,να ,β = µβ,νβ are two − √µβ +(µβ +πβ ) +√µα+(µα+πα) h i h i µβ µα πα+πβ IFNs, where να νβ = 0 and πα πβ = 0. If α I β, − (√2 ) · 6 · 6 ≺ ≥ √2 − πα+πβ we only need to prove that there exists ε> 0 such that 0 ≥ A novel ranking approach of IFNs 7 Thus, Table 1 Treatment evaluation using different measures µ2 +(1 ν )2 (1 µ )2+ν2 2 2 β β β β πβ µα+(1 να) IFNR Z l P M I − − A q q > − 1 0.490 0.400 0.500 0.461 0.000 −0.200 2 2 2 q 2 A − − − 2 0.250 0.500 0.500 0.500 0.000 0.000 2 2 A (1 µα) +ν 3 0.430 0.354 0.413 0.386 −0.205 −0.310 α πα A − 4 0.308 0.393 0.396 0.395 −0.210 −0.215 q 2 2 A 5 0.217 0.700 0.754 0.769 0.621 0.458 − A 6 0.150 0.700 0.700 0.700 0.400 0.400 From aforementioned process, we have α I β, which A 7 0.480 0.490 0.625 0.576 0.337 0.037 ≺ A concludes the proof. 8 0.604 0.418 0.586 0.500 0.237 −0.117 Now, a numerical example is applied to to demon- 5 IF multi-attribute decision making based on strate the reasonability of the ideal measure. ideal measure Example 1. Suppose that there are eight differ- ent treatments A , A , , A for one patient. The { 1 2 ··· 8} 5.1 Problem description treatments are evaluated by IFNs.Let A = 0.30, 0.30 , 1 h i A = 0.50, 0.50 , A = 0.39, 0.50 , A = 0.39, 0.60 , 2 h i 3 h i 4 h i Multi-attribute decision making (MADM) is widespread A5 = 0.673, 0.0 , A6 = 0.70, 0.30 , A7 = 0.40, 0.0 , in all areas of human social life, such as politics, e- h √2 i h i h i A8 = 1 , 0.0 , respectively. Specifically speaking, conomies, medical science, etc.. According to the infor- h − 2 i treatment A1 = 0.30, 0.30 influences 30% in a posi- mation contained by different attributes, decision mak- h i tive way for the symptoms, 30% in a negative way for ers (DMs) need to sort the alternatives of the discourse the symptoms, and 40% in a case of unknown. So do the or select the optimal alternative form the universe of other treatments. Different measures are used to pro- discourse. Here, we consider an evaluation example of cess these IFNs, the results are summarized in Table house selling taken from [20], which is described as fol- 1. lows. According to measure R, we have R(A6) Table 3 The aggregation information of each house H1 H2 H3 H4 H5 H6 AA (Hi) h0.766, 0.187i h0.749, 0.219i h0.722, 0.186i h0.748, 0.197i h0.713, 0.203i h0.722, 0.173i denote the aggregating of the decision-making informa- Table 6 The aggregation of the ideal measure 4 1 l H H H H H H tion as AA(Hi)= l=1 4 A (Hi). 1 2 3 4 5 6 ⊕ AI 0.545 0.509 0.467 0.492 0.458 0.449 Step 2 According to Definition 16, calculate the ide- al measure of the aggregation information I(AA(Hi)) for each alternative house. Note that the the sorting result we obtained above Step 3 Compare the ideal measure and rank the does not consider the decision makers’ preference to- alternative houses. wards different attributes. Suppose the main consider- Example 2 According to ranking method one, we ation of the DMs is good education resources. The DMs aggregate the decision-making information of alterna- insist that the location of the house is more important tive houses as shown in Table 3. than other factors. Or, if the decision maker puts the In terms of the computing the ideal measure of the dwelling quality in the first place, the property service aggregation information, the results is shown in Table will be more important than other factors. In this case, 4. decision makers can set different weights to different at- tributes according to the preference. Suppose that the weights to different attributes are ω = ω ,ω ,ω ,ω , Table 4 The ideal measure of the aggregation information { 1 2 3 4} in Ranking method 1, we set the aggregating of the H1 H2 H3 H4 H5 H6 AA 4 I( (Hi)) 0.555 0.514 0.486 0.522 0.465 0.492 decision-making information AA(Hi)= ωlAl(Hi). ⊕l=1 in Ranking method 2, we set the aggregating of the ideal measure of IFNs under different attributes AI(Hi)= Then we rank the alternative houses according to 4 Σl=lωlI(Al(Hi)). the ideal measure. The ranking result we get from Step Example 4 At this moment, two sets of weights are 3 is H1 >H4 >H2 >H6 >H3 >H5. fixed to different attribute. They are ω1 = 0.4, 0.2, 0.2, 0.2 { } Ranking method 2 and ω2 = 0.5, 0.1, 0.2, 0.2 . Similarly, we can obtain { } Step 1 According to Definition 16, calculate the ide- the ranking of the alternative houses, which are all sum- al measure of IFNs under different attributes I(Al(Hi)) marized as following. for each alternative house. By Ranking method 1: 1 Step 2 Aggregate the ideal measure of IFNs un- ω : H1 >H6 >H3 >H4 >H2 >H5, 2 der different attributes for different alternative houses. ω : H1 >H6 >H3 >H2 >H4 >H5. We denote the aggregating ideal measure as AI(Hi)= By Ranking method 2: 1 4 Al 1 4 Σl=lI( (Hi)). ω : H1 >H2 >H6 >H3 >H4 >H5, 2 Step 3 Compare the aggregation information and ω : H1 >H6 >H3 >H2 >H4 >H5. rank the alternative houses. Example 3 According to Definition 16, the ideal measure of IFNs under different attributes for each al- 5.3 Comparisons with related works ternative house are calculated as follows. In this subsection, comparison analysis is employed to illustrate the effectiveness of the proposed method. Table 5 The ideal measure of IFNs Example 5 According to literature [37], the main I(a1) I(a2) I(a3) I(a4) H1 0.600 0.579 0.395 0.604 MCDM ranking process can be explained as Max choice- H2 0.525 0.570 0.410 0.532 { H3 0.643 0.275 0.524 0.424 value - Min hesitation - Max score , in which dif- H4 0.524 0.654 0.181 0.610 H5 0.334 0.509 0.466 0.524 } { } { } H6 0.643 0.275 0.545 0.424 ferent pairs of level values are used to sort the alter- native houses. In detail, they are L(0.70, 0.30)-level, L(mid, mid)-level, L(top, bot)-level, L(top, top)-level, and In terms of the aggregating the ideal measure of L(bot, bot)-level. Now, we compute and compare the IFNs under different attributes, the results is shown in level soft set L(s,t) of each alternative house, the re- Table 6. sults are all summarized as we can see in Table 7. Then we rank the alternative houses according to Then, rank the alternative houses by the choice- the ideal measure. The ranking result we get from Step value. The ranking result of the alternative houses is 3 is H1 >H2 >H4 >H6 >H3 >H5. as following. A novel ranking approach of IFNs 9 Table 7 The choice-value of different level soft set L(s,t) different adjoint triples, the decision makers can obtain U L(0.70,0.30) L(mid,mid) L(top,bot) L(top,top) L(bot,bot) different results. H1 4 2 0 1 1 H2 3 2 0 0 0 For all i,j 4, τ : ai aj product: H3 3 2 1 2 1 ≤ × → H4 3 2 1 2 1 H6 >H3 >H1 >H4 >H5 >H2; H5 3 3 0 0 1 H6 2 2 2 2 2 If i,j 2, 3 , τ : ai aj L ukasiewicze; other- ∈ { } × → wise, τ : ai aj product: × → H6 >H1 >H3 >H5 >H4 >H2; L(0.70, 0.30) : H2 >H1 = H3 = H4 = H5 >H6; If i,j 1, 2 , τ : ai aj L ukasiewicze; other- L(mid, mid): H5 >H1 = H2 = H3 = H4 = H6; ∈ { } × → wise, τ : ai aj product): L(top, bot): H6 >H3 = H4 >H1 = H2 = H5; × → H6 >H3 >H1 >H4 >H2 >H5. L(top, top): H3 = H4 = H6 >H1 >H2 = H5; On the one hand, DMs need to consult relevant ex- L(bot, bot): H6 >H1 = H3 = H4 = H5 >H2. perts before making a decision. DMs may also represent We cannot rank the alternative houses in a com- explicit preferences among the attributes by associating pletely order only by the principle of the maximal choice- different family adjoint triples. However, this preference value. Hence, it is necessary to calculate and compare is hard to quantify. In other words, how to choose dif- the hesitation degree of the IFNs for all alternative ferent adjoint triples is difficult for DMs. At the same houses. The result is shown in Table 8. time, the calculation process is relatively complex. On the other hand, the IFNs of H2 is lager than H5 accord- Table 8 The hesitation degree of the alternative houses ing to attributes a1,a2 and a4. The IFN of a3 for H2 is nearly as the same as H5. Intuitively, H2 should be U H1 H2 H3 H4 H5 H6 hesitation 0.17 0.12 0.36 0.27 0.33 0.41 in front of H5 all the ranked results. However, we get H5 > H2 in the results of the first adjoint triple and the second adjoint triple above. As a matter of fact, our Next, we use the principle of Min hesitation on the { } approach gets the order H2 >H5, as desired. basis of the maximal choice-value principle, the ranking result of the alternative houses is as follows. L(0.70, 0.30) : H2 >H1 >H4 >H5 >H3 >H6; 6 Conclusion and further work L(mid, mid): H5 >H2 >H1 >H4 >H3 >H6; L(top, bot): H6 >H4 >H3 >H2 >H1 >H5; On the basis of analyzing the existing sorting methods L(top, top): H4 >H3 >H6 >H1 >H2 >H5; of IFNs, we define the ideal measure for sorting IFNs by L(bot, bot): H6 >H1 >H4 >H5 >H3 >H2. considering the fuzzy semantics expressed by the hesita- At this moment, we sort the alternative houses in a tion degree of IFNs and the Euclidean distance between strict order in all pairs of level values, and also get the IFNs with the ideal positive IFN and the ideal negative optimal object. IFN. Some desirable properties have also been strictly checked, such as weak admissibility, membership de- According to Table 2, since Al(H3) Al(H6) for ⊆ gree robustness, nonmembership degree robustness and each attribute, we should get H6 > H3 in all rank- determinism. After that, a numerical example is ap- ing results. However, we see that H3 > H6 in the re- sults of the level values L(0.70, 0.30), L(top, top) and plied to demonstrate the reasonability of the proposed L(mid, mid). By contrast, this case does not appear in measure. Moveover, we introduce the ranking method to acquire the optimal alternative in MADA problems. our method. In the other hand, the IFNs of H1 is bigger Comparison analysis are used to show the effectiveness than H2 with respect to attributes a1,a2 and a4. The and simplicity of the proposed method. In the following IFN of a3 for H1 is nearly as the same as H2. Intuitive- research, we will focus on how to apply ideal measure- ly, H1 should be in front of H2 all the ranked results. ment to practical problems in a wide range. However, we see that H2 >H1 in the results of the level values of L(0.70, 0.30),L(top, bot) and L(mid, mid). As a matter of fact, our approach gets the order H1 >H2, Acknowledgements This paper is supported by the Na- as desired. tional Natural Science Foundation of China (No. 62076088) and by the Natural Science Foundation of Hebei Province Example 6 According to literature [20], multi-adjoint (No. A2020208004). theory and Dempster-Shafer evidence theory are ap- plied to tackle the multi-attribute group decision mak- Authors’ contribution: ing problem. The main MADM ranking process can Mi J.S. directs and provides the conception of the work. be explained as Belief structure construction-Evidence Liang M.S. builds the measure, builds the applies, and combination-Belief internal comparison. On the basis of drafts the article. 10 Meishe Liang et al. Zhang S.P. and Jin C.X revise the article. 18. Jiao L, Yang H L, Li S G. Three-way decision based on decision-theoretic rough sets with single-valued neutrosoph- Compliance with Ethical Standards: ic information. International Journal of Machine Learning and Cybernetics, 2020, 11(15):657-665 We declare that we have no conflict of interest. This 19. Kuo R J, Lin T C, Zulvia F E, et al. A Hybrid Meta- article does not contain any studies with human par- heuristic and Kernel Intuitionistic Fuzzy c-means Algorith- ticipants or animals performed by any of the authors. m for Cluster Analysis. Applied Soft Computing, 2018, 67:299-308 Informed consent was obtained from all individual par- 20. Liang M S, Mi J S, Feng T, Xie B. Multi-adjoint based ticipants included in the study. group decision-making under an intuitionistic fuzzy infor- mation system. 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