# Induced Interval-Valued Intuitionistic Fuzzy Hybrid Aggregation Operators with TOPSIS Order-Inducing Variables

Authors: Jun-Ling Zhang; Xiao-Wen Qi
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Two induced aggregation operators with novelly designed TOPSIS order-inducing variables are proposed: Induced Interval-valued Intuitionistic Fuzzy Hybrid Averaging (I-IIFHA) operator and Induced Interval-valued Intuitionistic Fuzzy Hybrid Geometric (I-IIFHG) operator. The merit of two aggregation operators is that they can consider additional preference information of decision maker’s attitudinal characteristics besides argument-dependent information and argument-independent information. Some desirable properties of I-IIFHA and I-IIFHG are studied and theoretical analysis also shows that they can include a wide range of aggregation operators as special cases. Further, we extend these operators to form a novel group decision-making method for selecting the most desirable alternative in multiple attribute multi-interest group decision-making problems with attribute values and decision maker’s interest values taking the form of interval-valued intuitionistic fuzzy numbers, and application research to real estate purchase selection shows its practicality.


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## Body

## 1. Introduction

In many practical decision-making activities, the available knowledge of decision makers is vague or imprecise and cannot be assessed with numerical values, so Zadeh’s fuzzy set theory [1] and Atanassov’s intuitionistic fuzzy set (IFS) theory [2] are introduced to deal with uncertain decision environments. Over the last decades, the increasing researches have indicated that Atanassov’s IFS and interval-valued intuitionistic fuzzy set (IVIFS) [3] characterized by a membership function and nonmembership function are more suitable to deal with fuzziness and uncertainty than the generalization of fuzzy set developed by Zadeh. So recently, particular aggregation operators and corresponding decision-making methods have been studied and widely used for solving different decision-making problems under intuitionistic fuzzy or interval-valued intuitionistic fuzzy environments. As for the studies on single-person multiattribute decision-making (MADM) problems, Xu [4] investigated the methodology for solving multiattribute decision-making problems with intuitionistic fuzzy preference information aggregated by the intuitionistic fuzzy-weighted averaging (IFWA) operator, the intuitionistic fuzzy-ordered weighted averaging (IFOWA) operator, and the intuitionistic fuzzy hybrid averaging (IFHA) operator. Xu and Yager [5] proposed some geometric aggregation operators based on intuitionistic fuzzy sets, such as the intuitionistic fuzzy-weighted geometric (IFWG) operator, the intuitionistic fuzzy-ordered weighted geometric (IFOWG) operator and the intuitionistic fuzzy hybrid geometric (IFHG) operator and then applied them to multiple attribute decision-making. Xu and Xia [6] studied a method for multiattribute decision-making under IFS environment with induced generalized aggregation operators (I-GIFCOA and BS I-GIFCOA) designed by introducing Choquet integral and Dempster-Shafer evidence theory. Tan and Chen [7] considered interactions phenomena among the decision-making criteria under IFS environment and also introduced Choquet integral to design an intuitionistic fuzzy operator (I-IFC) for multicriteria decision-making, and so forth. With respect to the researches on dynamic multiattribute decision-making (DMADM) problems, Xu and Yager [8] developed two new dynamic intuitionistic fuzzy aggregation operators to solve the dynamic intuitionistic fuzzy multi-attribute decision-making problems. Wei [9] proposed the dynamic intuitionistic fuzzy-weighted geometric (DIFWG) operator and developed a procedure based on the DIFWG and IFWG operators to solve dynamic intuitionistic fuzzy multi-attribute decision-making (DIF-MADM) problems where all the decision information about attribute values takes the form of intuitionistic fuzzy numbers collected at different periods. For the researches on multiple attribute group decision-making (MAGDM) problems, Xu and Chen [10, 11] investigated decision-making methods for the IVIFS environment and defined several arithmetic aggregation operators and geometric aggregation operators, such as the interval-valued intuitionistic fuzzy-weighted averaging (IIFWA) operator, the interval-valued intuitionistic fuzzy-ordered weighted averaging (IIFOWA) operator, the interval-valued intuitionistic fuzzy hybrid averaging (IIFHA) operator, interval-valued intuitionistic fuzzy geometric (IIFG) operator, and interval-valued intuitionistic fuzzy-weighted geometric (IIFWG) operator. Wei [12] developed some induced geometric aggregation operators based on IFS and IVIFS, such as the induced intuitionistic fuzzy-ordered weighted geometric (I-IFOWG) operator and induced interval-valued intuitionistic fuzzy-ordered weighted geometric (I-IIFOWG) operator, and also extended them to solve the multi-attribute decision-making problems, and so forth.As can be seen from the researches mentioned, aggregation operators for fusion of decision preference information play a central role in solving different kinds of decision-making problems. In the existing literatures, different types of aggregation operators and their extensions can be found for fusing intuitionistic fuzzy and interval-valued intuitionistic fuzzy information, such as the weighted aggregation operators IFWA, IFWG, IIFWA, and IIFG, the ordered weighted aggregation operators IFOWA, IFOWG, IIFOWA, and IIFWG; the induced aggregation operators I-GIFCOA, I-GIFCOA, I-IFC, I-IFOWG, and I-IIFOWG; the hybrid aggregation operators IFHA, IFHG, and IIFHA. From the aggregation process of these representative operators, it can be seen that weighted aggregation operators or ordered weighted aggregation operators can manage to utilize argument-dependent (value weights) or argument-independent (position weights) decision preference information [4, 13–17]; hybrid operators combine the advantages of both ordered weighted aggregation and weighted aggregation operators and manage to consider both position weights and value weights [17–23]; induced operators [24–32] and generalized induced operators [6, 31, 33–38] integrating order-inducing variables are able to consider additional reordering criteria that do not depend on the values of the arguments. Obviously the combination of hybrid operators and induced operators can enable the aggregation process to utilize all the argument-dependent preference information, argument-independent preference information and additional preference criteria, and so as to select the most desirable alternative in accordance with preference of decision makers. Such as, Merigó and Casanovas [20] presented an induced generalized hybrid averaging (IGHA) operator under the exact numerical environment by combination of the induced aggregation operator and the hybrid aggregation operator. But there is scarcely any research on the induced hybrid aggregation operators for the intuitionistic fuzzy or interval-valued intuitionistic fuzzy decision environment as far as can be seen from existing literatures.In this paper, we investigate the induced hybrid operators with order-inducing variables not directly given by decision makers and their application to multiple attribute group decision-making with multi-interests under the interval-valued intuitionistic fuzzy decision environment. We present two induced interval-valued intuitionistic fuzzy hybrid aggregation operators: Induced Interval-valued Intuitionistic Fuzzy Hybrid Averaging (I-IIFHA) operator and Induced Interval-valued Intuitionistic Fuzzy Hybrid Geometric (I-IIFHG) operator. The two operators manage to use information more completely through two aspects: (1) hybridly including argument-dependent weighting vector and argument-independent weighting vector under interval-valued intuitionistic fuzzy environment, and (2) extending the IIFHA and IIFHG operators by including order-inducing variables ensured with a designed TOPSIS-based method in the aggregation process to use more attitudinal information in accordance with different decision maker’s interests.The first main advantage of this paper is that the proposed I-IIFHA and I-IIFHG operators can help decision-making process to consider preference information in a more complete way during aggregation than other ordinary aggregation operators mentioned previously. And obviously, if the order-inducing variables are the argument ordering, then the proposed induced operators can be reduced to other ordinary aggregation operators. So the I-IIFHA and I-IIFHG operators can include a wide range of interval-valued intuitionistic fuzzy aggregation operators as special cases, such as IIFHA, IIFOWA, IIFWA, IIFHG, IIFOWG and IIFWG operators. The other important contribution of this paper is the proposal of using the information related to various decision makers’ interests to deduce order-inducing criteria in induced interval-valued intuitionistic fuzzy hybrid aggregation. Recently, although many researches have been conducted on various induced operators, there are few researches on the methods to ensure order-inducing variables. Order-inducing variables as the reordering step can enable induced aggregation operators to consider additional decision rules that do not depend on the values of the arguments, such as attitudinal characteristic information of decision makers. In the literature [27, 39, 40], Merigó and Casanovas introduced some induced aggregation operators under exact number decision environment with a complex reordering process by using Euclidean or Hamming distance measures and the man-provided order-inducing variables according to the experts’ interests. But sometimes, when decision makers cannot provide their exact preference directly because of time pressure or limited expertise related to the problem domain, certain methods should be developed to ensure order-inducing variables. Marin et al. [25] proposed to use the specificity and fuzziness measures of the unbalanced linguistic terms to induce the order-inducing variables of the arguments. For the two situations with or without man-provided additional preference, Chiclana et al. [24] presented three induced aggregation operators under exact number environment: Importance IOWA, consistency IOWA, and preference IOWA operator; the former two operators are based on the reliability of the given arguments; the latter operator is based on a relative preference associated with each one of given arguments. In this paper, we also consider the situation where the decision makers provide their own interests, but the difference is that the decision makers express their attitudinal characteristics in the form of interval-valued intuitionistic fuzzy values which can represent the information in a more complete way, and meanwhile, they can provide both ideal solutions and negative solutions. To do this, in I-IIFHA and I-IIFHG we introduce the TOPSIS method to induce the order-inducing variables, which is called TOPSIS order-inducing variables. The developed TOPSIS order-inducing method especially can also be used to induce the order-inducing variables in the situation of more than one ideal solution or negative solution provided and in the situation of no ideal solution or negative solution provided. We can use the proposed induced hybrid aggregation operators with TOPSIS order-inducing variables to utilize preference information more completely under interval-valued intuitionistic fuzzy decision environment for solving multiple attribute group decision-making problems with multi-interests.The remainder of this paper is organized as follows. In Section2, we give a review of basic concepts related to intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets, and overview some aggregation operators. In Section 3, we propose induced interval-valued intuitionistic fuzzy hybrid averaging (I-IIFHA) operator and induced interval-valued intuitionistic fuzzy hybrid geometric (I-IIFHG) operator and analyze some desirable properties; then we introduce TOPSIS method to ensure order-inducing variables that reflect the complex attitudinal characteristic of the decision maker. In Section 4, based on the I-IIFHA and I-IIFHG operators, an approach is developed to solve the multiple attribute decision-making problems under the interval-valued intuitionistic fuzzy environment. In Section 5, an example is given to illustrate validity and practicality of the proposed methods. Finally, we conclude the paper.


## 2. Preliminaries

### 2.1. Related Definitions of IVIFSs

Some basic concepts on IVIFSs are introduced here in after to facilitate future discussions.In 1986, Atanassov [2] generalized the concept of fuzzy set and defined the concept of intuitionistic fuzzy set as follows.Definition 2.1 (see [2]).
A generalized fuzzy set called intuitionistic fuzzy set is shown as follows:(2.1)A={〈xi,μA(xi),νA(xi)〉∣xi∈X},
in which μA means a membership function, andνA means a nonmembership, with the condition 0≤μA(xi)+νA(xi)≤1, μA(xi),andνA(xi)∈[0,1], for all xi∈X. A=μA(xi)=νA(xi) especially; the given IFSA is degraded to an ordinary fuzzy set.In reality, it may not be easy to identify exact values for the membership and nonmembership degrees of an element to a set. In this case, a range of values may be a more appropriate measurement to accommodate the vagueness. As such, Atanassov and Gargov [3] introduce the notion of interval-valued intuitionistic fuzzy set (IVIFS).Definition 2.2 (see [3]).
An interval-valued intuitionistic fuzzy set (IVIFS)Ãin X is defined by
(2.2)Ã={〈xi,μ̃Ã(xi),ν̃Ã(xi)〉∣xi∈X}={〈xi,[μÃL(xi),μÃU(xi)],[νÃL(xi),νÃU(xi)]〉∣xi∈X},
where 0≤μÃL(xi)≤μÃU(xi)≤1,0≤νÃL(xi)≤νÃU(xi)≤1,0≤μÃU(xi)+νÃU(xi)≤1 for all xi∈X. Similarity, the intervals μ̃Ã(xi) and ν̃Ã(xi) denote the degree of membership and nonmembership of an element to a set.Then, we introduce some formulae to calculate the distance of IVIFSs.Definition 2.3.
Suppose that two interval-valued intuitionistic fuzzy sets (IVIFSs)Ã and B̃ in X are defined as
(2.3)Ã={〈xi,μ̃Ã(xi),ν̃Ã(xi)〉∣xi∈X}={〈xi,[μÃL(xi),μÃU(xi)],[νÃL(xi),νÃU(xi)]〉∣xi∈X},B̃={〈xi,μ̃B̃(xi),ν̃B̃(xi)〉∣xi∈X}={〈xi,[μB̃L(xi),μB̃U(xi)],[νB̃L(xi),νB̃U(xi)]〉∣xi∈X}.
The normalized Euclidean distance measure is
(2.4)D1(Ã,B̃)=16n∑i=1n[(μÃL(xi)-μB̃L(xi))2+(μÃU(xi)-μB̃U(xi))2+(νÃL(xi)-νB̃L(xi))2+B+H+K],
where ℬ denotes(νÃU(xi)-νB̃U(xi))2, ℋ denotes(πÃL(xi)-πB̃L(xi))2, and 𝒦denotes (πÃU(xi)-πB̃U(xi))2.
The normalized Hamming distance measure is(2.5)D2(Ã,B̃)=16n∑i=1n|μÃL(xi)-μB̃L(xi)|+|μÃU(xi)-μB̃U(xi)|+|νÃL(xi)-νB̃L(xi)|+|νÃU(xi)-νB̃U(xi)|+|πÃL(xi)-πB̃L(xi)|+|πÃU(xi)-πB̃U(xi)|.
The normalized Hausdorff distance measure is
(2.6)D3(Ã,B̃)=1n∑i=1nmax⁡{|μÃL(xi)-μB̃L(xi)|,|μÃU(xi)-μB̃U(xi)|,|νÃL(xi)-νB̃L(xi)|,|νÃU(xi)-νB̃U(xi)|,|πÃL(xi)-πB̃L(xi)|,|πÃU(xi)-πB̃U(xi)|}.In order to aggregate interval-valued intuitionistic fuzzy information, we introduce the following relations and operations given by Xu [10].Definition 2.4 (see [10]).
Letα̃=([a,b],[c,d]), α̃1=([a1,b1],[c1,d1]), α̃2=([a2,b2],[c2,d2]) be three IVIFNs; then(1)
α̃1⊕α̃2=([a1+a2-a1a2,b1+b2-b1b2],[c1c2,d1d2]);(2)
α̃1⊗α̃2=([a1a2,b1b2],[c1+c2-c1c2,d1+d2-d1d2]);(3)
λα̃=([1-(1-a)λ,1-(1-b)λ],[cλ,dλ]);(4)
α̃λ=([aλ,bλ],[1-(1-c)λ,1-(1-d)λ]).Definition 2.5 (see [10]).
Letα̃=([a,b],[c,d]), α̃1=([a1,b1],[c1,d1]), α̃2=([a2,b2],[c2,d2]) be three IVIFNs, λ,λ1,λ2≥0; then(1)
α̃1⊕α̃2=α̃2⊕α̃1;(2)
α̃1⊗α̃2=α̃2⊗α̃1;(3)
λ(α̃1⊕α̃2)=λα̃2⊕λα̃1;(4)
(α̃1⊗α̃2)λ=α̃2λ⊗α̃1λ;(5)
λ1α̃⊕λ2α̃=(λ1+λ2)α̃;(6)
α̃λ1⊗α̃λ2=(α̃)λ1+λ2.In order to rank alternatives, it is necessary to consider how to compare two IVIFNs. Xu [10] also devised an approach to compare two IVIFNs based on the concepts of score function and accuracy function.Definition 2.6 (see [10]).
For any three IVIFNsα̃=([μL,μU],[νL,νU]), α̃1=([μ1L,μ1U],[ν1L,ν1U]), α̃2=([μ2L,μ2U],[ν2L,ν2U]), score function defined as s(α̃)=(1/2)(μL+μU-νL-νU), and accuracy function defined as h(α̃)=(1/2)(μL+μU+νL+νU).Ifs(α̃1)<s(α̃2), then α̃1 is smaller than α̃2, denoted by α̃1<α̃2.Ifs(α̃1)=s(α̃2), then one has the following.Ifh(α̃1)<h(α̃2), then α̃1 is smaller than α̃2, denoted by α̃1<α̃2.Ifh(α̃1)=h(α̃2), then α̃1 and α̃2 represent the same information, denoted by α̃1=α̃2.


### 2.2. Some Interval-Valued Intuitionistic Fuzzy Aggregation Operators

We first introduce definitions of interval-valued intuitionistic fuzzy-weighted averaging (IIFWA) and interval-valued intuitionistic fuzzy-weighted geometric (IIFWG) operators.Definition 2.7 (see [10]).
Letα̃1,α̃2,…,α̃n be a collection of interval arguments, where α̃j=([aj,bj],[cj,dj]), and IIFWA operator of dimension n is a mapping IIFWA: Rn→R, that has an associated vector w=(w1,w2,…,wn)T such that ∑j=1nwj=1 and wj∈[0,1]; then
(2.7)IIFWAω(α̃1,α̃2,…,α̃n)=ω1α̃1⊕ω2α̃2⊕⋯⊕ωnα̃n=([1-∏j=1n(1-aj)ωj,1-∏j=1n(1-bj)ωj],[∏j=1ncjωj,∏j=1n∑djωj]).Definition 2.8 (see [10]).
Letα̃1,α̃2,…,α̃n be a collection of interval arguments, where α̃j=([aj,bj],[cj,dj]), and IIFWG operator of dimension n is a mapping IIFWG: Rn→R, that has an associated weight vector ω=(ω1,ω2,…,ωn)T such that ∑j=1nωj=1 and ωj∈[0,1]; then
(2.8)IIFWGω(α̃1,α̃2,…,α̃n)=α̃1ω1⊗α̃2ω2⊗⋯⊗α̃nωn=([∏j=1najωj,∏j=1nbjωj],[1-∏j=1n(1-cj)ωj,1-∏j=1n(1-dj)ωj]).Furthermore, consider the ordered positions of the interval-valued intuitionistic fuzzy values instead of weighting the interval-valued intuitionistic fuzzy values themselves. Xu and Chen [11] also proposed interval-valued intuitionistic fuzzy-ordered weighted averaging (IIFOWA) and interval-valued intuitionistic fuzzy-ordered weighted geometric (IIFOWG) operators.Definition 2.9 (see [11]).
Letα̃1,α̃2,…,α̃n be a collection of interval arguments, where α̃j=([aj,bj],[cj,dj]), and IIFOWA operator of dimension n is a mapping IIFOWA: Rn→R, that has an associated weight vector ω=(ω1,ω2,…,ωn)T such that ∑j=1nωj=1 and ωj∈[0,1]; then
(2.9)IIFOWAω(α̃1,α̃2,…,α̃n)=ω1α̃π(1)⊕ω2α̃π(2)⊕⋯⊕ωnα̃π(n)=([1-∏j=1n(1-aπ(j))ωj,1-∏j=1n(1-bπ(j))ωj],[∏j=1ncπ(j)ωj,∏j=1ndπ(j)ωj]),
where α̃π(1),α̃π(2),…,α̃π(n) is a permutation of α̃1,α̃2,…,α̃n, such that α̃π(j-1)≥α̃π(j) for all j.Definition 2.10 (see [11]).
Letα̃1,α̃2,…,α̃n be a collection of interval arguments, where α̃j=([aj,bj],[cj,dj]), and IIFOWG operator of dimension n is a mapping IIFOWG: Rn→R, that has an associated weight vector ω=(ω1,ω2,…,ωn)T such that ∑j=1nωj=1 and ωj∈[0,1]; then
(2.10)IIFOWGω(α̃1,α̃2,…,α̃n)=α̃π(1)ω1⊗α̃π(2)ω2⊗⋯⊗α̃π(n)ωn=([∏j=1naπ(j)ωj,∏j=1nbπ(j)ωj],[1-∏j=1n(1-cπ(j))ωj,1-∏j=1n(1-dπ(j))ωj]),
where α̃π(1),α̃π(2),…,α̃π(n) is a permutation of α̃1,α̃2,…,α̃n, such that α̃π(j-1)≥α̃π(j) for all j.Ordered weighted aggregation operators only consider the position weights of the interval-valued intuitionistic fuzzy values themselves. To overcome this limitation, Xu and Chen [11] provided definitions of interval-valued intuitionistic fuzzy hybrid averaging (IIFHA) operator and interval-valued intuitionistic fuzzy hybrid geometric (IIFHG) operator, which weights both the given interval-valued intuitionistic fuzzy value and its ordered position information.Definition 2.11 (see [11]).
An IIFHA operator of dimensionn is a mapping IIFHA: Rn→R that has an associated vector w=(w1,w2,…,wn)T such that ∑j=1nwj=1 and wj∈[0,1]; then
(2.11)IIFHAω,w(α̃1,α̃2,…,α̃n)=w1α̃̇π(1)⊕w2α̃̇π(2)⊕⋯⊕wnα̃̇π(n),
where α̃̇π(j) is the jth largest of the weighted intuitionistic fuzzy values α̃̇j=nωjα̃j, ω=(ω1,ω2,…,ωn)T is the weight vector α̃j(j=1,2,…,n) with ∑j=1nωj=1 and ωj∈[0,1], and n is the balancing coefficient.Definition 2.12 (see [11]).
An IIFHG operator of dimensionn is a mapping IIFHG: Rn→R that has an associated vector w=(w1,w2,…,wn)T such that ∑j=1nwj=1 and wj∈[0,1]; then
(2.12)IIFHGω,w(α̃1,α̃2,…,α̃n)=α̃̇π(1)w1⊗α̃̇π(2)w2⊗⋯⊗α̃̇π(n)wn,
where α̃̇π(j) is the jth largest of the weighted intuitionistic fuzzy values α̃̇j=α̃jnωj, ω=(ω1,ω2,…,ωn)T is the weight vector α̃j(j=1,2,…,n) with ∑j=1nωj=1 and ωj∈[0,1], and n is the balancing coefficient.The induced operator is an extension of the ordinary aggregation operators. The main difference is that the reordering step is not carried out with the arguments values. In this case, the reordering step is developed with order-inducing variables that reflect a more complex reordering process. Wei and Yi [29] introduced the induced interval-valued intuitionistic fuzzy-ordered weighted geometric (I-IIFOWG) operator, which is defined as follows.Definition 2.13 (see [29]).
An I-IIFOWG operator of dimensionn is a mapping I-IIFOWG: Rn→R that has an associated vector w=(w1,w2,…,wn)T such that ∑j=1nwj=1 and wj∈[0,1]; then
(2.13)I-IIFOWGw(〈ε1,α̃1〉,〈ε2,α̃2〉,…,〈εn,α̃n〉)=α̃π(1)w1⊕α̃π(2)w2⊕⋯⊕α̃π(n)wn=([∏j=1naπ(j)wj,∏j=1nbπ(j)wj],[1-∏j=1n(1-cπ(j))wj,1-∏j=1n(1-dπ(j))wj]),
where α̃π(j)=([aπ(j),bπ(j)],[cπ(j),dπ(j)]) is the α̃ value of the I-IIFOWG pair 〈εi,α̃i〉 having the jth largest εi(εi∈[0,1]), and εi in 〈εi,α̃i〉 is referred to as the order-inducing variable and α̃i=([ai,bi],[ci,di]) as the interval-valued intuitionistic fuzzy values.In the same way, we have the definition of interval-valued intuitionistic fuzzy-ordered weighted averaging (I-IIFOWA) operator.Definition 2.14.
An I-IIFOWA operator of dimensionn is a mapping I-IIFOWA: Rn→R, that has an associated vector with I-IIFOWA operator w=(w1,w2,…,wn)T such that ∑j=1nwj=1 and wj∈[0,1]; then
(2.14)I-IIFOWAw(〈ε1,α̃1〉,〈ε2,α̃2〉,…,〈εn,α̃n〉)=w1α̃π(1)⊕w2α̃π(2)⊕⋯⊕wnα̃π(n)=([1-∏j=1n(1-aπ(j))wj,1-∏j=1n(1-bπ(j))wj],[∏j=1ncπ(j)wj,∏j=1ndπ(j)wj]),
where α̃π(j)=([aπ(j),bπ(j)],[cπ(j),dπ(j)]) is the α̃ value of the I-IIFOWA pair 〈εi,α̃i〉 having the jth largest εi(εi∈[0,1]), and εi in 〈εi,α̃i〉 is referred to as the order-inducing variable and α̃i=([ai,bi],[ci,di])as the interval-valued intuitionistic fuzzy values.Xu [5] classified all these approaches for determining the weighting vector in aggregation operators into two categories: argument-independent weighting vector method and argument-dependent weighting vector method. The weights derived by the argument-independent approaches are associated with particular ordered positions of the aggregated arguments and have no connection with the aggregated arguments, while the argument-dependent approaches determine the weights based on the arguments.In this paper, the argument-weighting vectorω=(ω1,ω2,…,ωn)T associated with the inputs α̃j can be determined by the normal distribution method [13].Letα̃1,α̃2,…,α̃n be a collection of interval arguments, where α̃j=([aj,bj],[cj,dj]), the mean of these interval-valued intuitionistic fuzzy sets ũ=([aũ,bũ],[cũ,dũ]), where(2.15)aũ=1n∑j=1naj,bũ=1n∑j=1nbj,cũ=1n∑j=1ncj,dũ=1n∑j=1ndj,
and the variance of these interval-valued intuitionistic fuzzy arguments:(2.16)σ=1n∑j=1n(d(α̃j,ũ))2.
then we can have a normal distribution method for deriving the attribute weight associated with its value:(2.17)ωj=12πσe-d2(α̃j-ũ)/2σ2,j=1,2,…,n.
Consider that ωj∈[0,1] and ∑j=1nωj=1; we have normalized attribute weights:(2.18)ωj=(1/2πσ)e-d2(β̃j-ũ)/2σ2∑j=1n(1/2πσ)e-d2(β̃j-ũ)/2σ2,j=1,2,…,n.The position-weighting vectorw=(w1,w2,…,wn)T associated with the operator can be determined by RIM quantifiers [41]:(2.19)Q(r)=rα,α≥0,wi=Q(in)-Q(i-1n),fori=1,2,…,n.


## 2.1. Related Definitions of IVIFSs

Some basic concepts on IVIFSs are introduced here in after to facilitate future discussions.In 1986, Atanassov [2] generalized the concept of fuzzy set and defined the concept of intuitionistic fuzzy set as follows.Definition 2.1 (see [2]).
A generalized fuzzy set called intuitionistic fuzzy set is shown as follows:(2.1)A={〈xi,μA(xi),νA(xi)〉∣xi∈X},
in which μA means a membership function, andνA means a nonmembership, with the condition 0≤μA(xi)+νA(xi)≤1, μA(xi),andνA(xi)∈[0,1], for all xi∈X. A=μA(xi)=νA(xi) especially; the given IFSA is degraded to an ordinary fuzzy set.In reality, it may not be easy to identify exact values for the membership and nonmembership degrees of an element to a set. In this case, a range of values may be a more appropriate measurement to accommodate the vagueness. As such, Atanassov and Gargov [3] introduce the notion of interval-valued intuitionistic fuzzy set (IVIFS).Definition 2.2 (see [3]).
An interval-valued intuitionistic fuzzy set (IVIFS)Ãin X is defined by
(2.2)Ã={〈xi,μ̃Ã(xi),ν̃Ã(xi)〉∣xi∈X}={〈xi,[μÃL(xi),μÃU(xi)],[νÃL(xi),νÃU(xi)]〉∣xi∈X},
where 0≤μÃL(xi)≤μÃU(xi)≤1,0≤νÃL(xi)≤νÃU(xi)≤1,0≤μÃU(xi)+νÃU(xi)≤1 for all xi∈X. Similarity, the intervals μ̃Ã(xi) and ν̃Ã(xi) denote the degree of membership and nonmembership of an element to a set.Then, we introduce some formulae to calculate the distance of IVIFSs.Definition 2.3.
Suppose that two interval-valued intuitionistic fuzzy sets (IVIFSs)Ã and B̃ in X are defined as
(2.3)Ã={〈xi,μ̃Ã(xi),ν̃Ã(xi)〉∣xi∈X}={〈xi,[μÃL(xi),μÃU(xi)],[νÃL(xi),νÃU(xi)]〉∣xi∈X},B̃={〈xi,μ̃B̃(xi),ν̃B̃(xi)〉∣xi∈X}={〈xi,[μB̃L(xi),μB̃U(xi)],[νB̃L(xi),νB̃U(xi)]〉∣xi∈X}.
The normalized Euclidean distance measure is
(2.4)D1(Ã,B̃)=16n∑i=1n[(μÃL(xi)-μB̃L(xi))2+(μÃU(xi)-μB̃U(xi))2+(νÃL(xi)-νB̃L(xi))2+B+H+K],
where ℬ denotes(νÃU(xi)-νB̃U(xi))2, ℋ denotes(πÃL(xi)-πB̃L(xi))2, and 𝒦denotes (πÃU(xi)-πB̃U(xi))2.
The normalized Hamming distance measure is(2.5)D2(Ã,B̃)=16n∑i=1n|μÃL(xi)-μB̃L(xi)|+|μÃU(xi)-μB̃U(xi)|+|νÃL(xi)-νB̃L(xi)|+|νÃU(xi)-νB̃U(xi)|+|πÃL(xi)-πB̃L(xi)|+|πÃU(xi)-πB̃U(xi)|.
The normalized Hausdorff distance measure is
(2.6)D3(Ã,B̃)=1n∑i=1nmax⁡{|μÃL(xi)-μB̃L(xi)|,|μÃU(xi)-μB̃U(xi)|,|νÃL(xi)-νB̃L(xi)|,|νÃU(xi)-νB̃U(xi)|,|πÃL(xi)-πB̃L(xi)|,|πÃU(xi)-πB̃U(xi)|}.In order to aggregate interval-valued intuitionistic fuzzy information, we introduce the following relations and operations given by Xu [10].Definition 2.4 (see [10]).
Letα̃=([a,b],[c,d]), α̃1=([a1,b1],[c1,d1]), α̃2=([a2,b2],[c2,d2]) be three IVIFNs; then(1)
α̃1⊕α̃2=([a1+a2-a1a2,b1+b2-b1b2],[c1c2,d1d2]);(2)
α̃1⊗α̃2=([a1a2,b1b2],[c1+c2-c1c2,d1+d2-d1d2]);(3)
λα̃=([1-(1-a)λ,1-(1-b)λ],[cλ,dλ]);(4)
α̃λ=([aλ,bλ],[1-(1-c)λ,1-(1-d)λ]).Definition 2.5 (see [10]).
Letα̃=([a,b],[c,d]), α̃1=([a1,b1],[c1,d1]), α̃2=([a2,b2],[c2,d2]) be three IVIFNs, λ,λ1,λ2≥0; then(1)
α̃1⊕α̃2=α̃2⊕α̃1;(2)
α̃1⊗α̃2=α̃2⊗α̃1;(3)
λ(α̃1⊕α̃2)=λα̃2⊕λα̃1;(4)
(α̃1⊗α̃2)λ=α̃2λ⊗α̃1λ;(5)
λ1α̃⊕λ2α̃=(λ1+λ2)α̃;(6)
α̃λ1⊗α̃λ2=(α̃)λ1+λ2.In order to rank alternatives, it is necessary to consider how to compare two IVIFNs. Xu [10] also devised an approach to compare two IVIFNs based on the concepts of score function and accuracy function.Definition 2.6 (see [10]).
For any three IVIFNsα̃=([μL,μU],[νL,νU]), α̃1=([μ1L,μ1U],[ν1L,ν1U]), α̃2=([μ2L,μ2U],[ν2L,ν2U]), score function defined as s(α̃)=(1/2)(μL+μU-νL-νU), and accuracy function defined as h(α̃)=(1/2)(μL+μU+νL+νU).Ifs(α̃1)<s(α̃2), then α̃1 is smaller than α̃2, denoted by α̃1<α̃2.Ifs(α̃1)=s(α̃2), then one has the following.Ifh(α̃1)<h(α̃2), then α̃1 is smaller than α̃2, denoted by α̃1<α̃2.Ifh(α̃1)=h(α̃2), then α̃1 and α̃2 represent the same information, denoted by α̃1=α̃2.


## 2.2. Some Interval-Valued Intuitionistic Fuzzy Aggregation Operators

We first introduce definitions of interval-valued intuitionistic fuzzy-weighted averaging (IIFWA) and interval-valued intuitionistic fuzzy-weighted geometric (IIFWG) operators.Definition 2.7 (see [10]).
Letα̃1,α̃2,…,α̃n be a collection of interval arguments, where α̃j=([aj,bj],[cj,dj]), and IIFWA operator of dimension n is a mapping IIFWA: Rn→R, that has an associated vector w=(w1,w2,…,wn)T such that ∑j=1nwj=1 and wj∈[0,1]; then
(2.7)IIFWAω(α̃1,α̃2,…,α̃n)=ω1α̃1⊕ω2α̃2⊕⋯⊕ωnα̃n=([1-∏j=1n(1-aj)ωj,1-∏j=1n(1-bj)ωj],[∏j=1ncjωj,∏j=1n∑djωj]).Definition 2.8 (see [10]).
Letα̃1,α̃2,…,α̃n be a collection of interval arguments, where α̃j=([aj,bj],[cj,dj]), and IIFWG operator of dimension n is a mapping IIFWG: Rn→R, that has an associated weight vector ω=(ω1,ω2,…,ωn)T such that ∑j=1nωj=1 and ωj∈[0,1]; then
(2.8)IIFWGω(α̃1,α̃2,…,α̃n)=α̃1ω1⊗α̃2ω2⊗⋯⊗α̃nωn=([∏j=1najωj,∏j=1nbjωj],[1-∏j=1n(1-cj)ωj,1-∏j=1n(1-dj)ωj]).Furthermore, consider the ordered positions of the interval-valued intuitionistic fuzzy values instead of weighting the interval-valued intuitionistic fuzzy values themselves. Xu and Chen [11] also proposed interval-valued intuitionistic fuzzy-ordered weighted averaging (IIFOWA) and interval-valued intuitionistic fuzzy-ordered weighted geometric (IIFOWG) operators.Definition 2.9 (see [11]).
Letα̃1,α̃2,…,α̃n be a collection of interval arguments, where α̃j=([aj,bj],[cj,dj]), and IIFOWA operator of dimension n is a mapping IIFOWA: Rn→R, that has an associated weight vector ω=(ω1,ω2,…,ωn)T such that ∑j=1nωj=1 and ωj∈[0,1]; then
(2.9)IIFOWAω(α̃1,α̃2,…,α̃n)=ω1α̃π(1)⊕ω2α̃π(2)⊕⋯⊕ωnα̃π(n)=([1-∏j=1n(1-aπ(j))ωj,1-∏j=1n(1-bπ(j))ωj],[∏j=1ncπ(j)ωj,∏j=1ndπ(j)ωj]),
where α̃π(1),α̃π(2),…,α̃π(n) is a permutation of α̃1,α̃2,…,α̃n, such that α̃π(j-1)≥α̃π(j) for all j.Definition 2.10 (see [11]).
Letα̃1,α̃2,…,α̃n be a collection of interval arguments, where α̃j=([aj,bj],[cj,dj]), and IIFOWG operator of dimension n is a mapping IIFOWG: Rn→R, that has an associated weight vector ω=(ω1,ω2,…,ωn)T such that ∑j=1nωj=1 and ωj∈[0,1]; then
(2.10)IIFOWGω(α̃1,α̃2,…,α̃n)=α̃π(1)ω1⊗α̃π(2)ω2⊗⋯⊗α̃π(n)ωn=([∏j=1naπ(j)ωj,∏j=1nbπ(j)ωj],[1-∏j=1n(1-cπ(j))ωj,1-∏j=1n(1-dπ(j))ωj]),
where α̃π(1),α̃π(2),…,α̃π(n) is a permutation of α̃1,α̃2,…,α̃n, such that α̃π(j-1)≥α̃π(j) for all j.Ordered weighted aggregation operators only consider the position weights of the interval-valued intuitionistic fuzzy values themselves. To overcome this limitation, Xu and Chen [11] provided definitions of interval-valued intuitionistic fuzzy hybrid averaging (IIFHA) operator and interval-valued intuitionistic fuzzy hybrid geometric (IIFHG) operator, which weights both the given interval-valued intuitionistic fuzzy value and its ordered position information.Definition 2.11 (see [11]).
An IIFHA operator of dimensionn is a mapping IIFHA: Rn→R that has an associated vector w=(w1,w2,…,wn)T such that ∑j=1nwj=1 and wj∈[0,1]; then
(2.11)IIFHAω,w(α̃1,α̃2,…,α̃n)=w1α̃̇π(1)⊕w2α̃̇π(2)⊕⋯⊕wnα̃̇π(n),
where α̃̇π(j) is the jth largest of the weighted intuitionistic fuzzy values α̃̇j=nωjα̃j, ω=(ω1,ω2,…,ωn)T is the weight vector α̃j(j=1,2,…,n) with ∑j=1nωj=1 and ωj∈[0,1], and n is the balancing coefficient.Definition 2.12 (see [11]).
An IIFHG operator of dimensionn is a mapping IIFHG: Rn→R that has an associated vector w=(w1,w2,…,wn)T such that ∑j=1nwj=1 and wj∈[0,1]; then
(2.12)IIFHGω,w(α̃1,α̃2,…,α̃n)=α̃̇π(1)w1⊗α̃̇π(2)w2⊗⋯⊗α̃̇π(n)wn,
where α̃̇π(j) is the jth largest of the weighted intuitionistic fuzzy values α̃̇j=α̃jnωj, ω=(ω1,ω2,…,ωn)T is the weight vector α̃j(j=1,2,…,n) with ∑j=1nωj=1 and ωj∈[0,1], and n is the balancing coefficient.The induced operator is an extension of the ordinary aggregation operators. The main difference is that the reordering step is not carried out with the arguments values. In this case, the reordering step is developed with order-inducing variables that reflect a more complex reordering process. Wei and Yi [29] introduced the induced interval-valued intuitionistic fuzzy-ordered weighted geometric (I-IIFOWG) operator, which is defined as follows.Definition 2.13 (see [29]).
An I-IIFOWG operator of dimensionn is a mapping I-IIFOWG: Rn→R that has an associated vector w=(w1,w2,…,wn)T such that ∑j=1nwj=1 and wj∈[0,1]; then
(2.13)I-IIFOWGw(〈ε1,α̃1〉,〈ε2,α̃2〉,…,〈εn,α̃n〉)=α̃π(1)w1⊕α̃π(2)w2⊕⋯⊕α̃π(n)wn=([∏j=1naπ(j)wj,∏j=1nbπ(j)wj],[1-∏j=1n(1-cπ(j))wj,1-∏j=1n(1-dπ(j))wj]),
where α̃π(j)=([aπ(j),bπ(j)],[cπ(j),dπ(j)]) is the α̃ value of the I-IIFOWG pair 〈εi,α̃i〉 having the jth largest εi(εi∈[0,1]), and εi in 〈εi,α̃i〉 is referred to as the order-inducing variable and α̃i=([ai,bi],[ci,di]) as the interval-valued intuitionistic fuzzy values.In the same way, we have the definition of interval-valued intuitionistic fuzzy-ordered weighted averaging (I-IIFOWA) operator.Definition 2.14.
An I-IIFOWA operator of dimensionn is a mapping I-IIFOWA: Rn→R, that has an associated vector with I-IIFOWA operator w=(w1,w2,…,wn)T such that ∑j=1nwj=1 and wj∈[0,1]; then
(2.14)I-IIFOWAw(〈ε1,α̃1〉,〈ε2,α̃2〉,…,〈εn,α̃n〉)=w1α̃π(1)⊕w2α̃π(2)⊕⋯⊕wnα̃π(n)=([1-∏j=1n(1-aπ(j))wj,1-∏j=1n(1-bπ(j))wj],[∏j=1ncπ(j)wj,∏j=1ndπ(j)wj]),
where α̃π(j)=([aπ(j),bπ(j)],[cπ(j),dπ(j)]) is the α̃ value of the I-IIFOWA pair 〈εi,α̃i〉 having the jth largest εi(εi∈[0,1]), and εi in 〈εi,α̃i〉 is referred to as the order-inducing variable and α̃i=([ai,bi],[ci,di])as the interval-valued intuitionistic fuzzy values.Xu [5] classified all these approaches for determining the weighting vector in aggregation operators into two categories: argument-independent weighting vector method and argument-dependent weighting vector method. The weights derived by the argument-independent approaches are associated with particular ordered positions of the aggregated arguments and have no connection with the aggregated arguments, while the argument-dependent approaches determine the weights based on the arguments.In this paper, the argument-weighting vectorω=(ω1,ω2,…,ωn)T associated with the inputs α̃j can be determined by the normal distribution method [13].Letα̃1,α̃2,…,α̃n be a collection of interval arguments, where α̃j=([aj,bj],[cj,dj]), the mean of these interval-valued intuitionistic fuzzy sets ũ=([aũ,bũ],[cũ,dũ]), where(2.15)aũ=1n∑j=1naj,bũ=1n∑j=1nbj,cũ=1n∑j=1ncj,dũ=1n∑j=1ndj,
and the variance of these interval-valued intuitionistic fuzzy arguments:(2.16)σ=1n∑j=1n(d(α̃j,ũ))2.
then we can have a normal distribution method for deriving the attribute weight associated with its value:(2.17)ωj=12πσe-d2(α̃j-ũ)/2σ2,j=1,2,…,n.
Consider that ωj∈[0,1] and ∑j=1nωj=1; we have normalized attribute weights:(2.18)ωj=(1/2πσ)e-d2(β̃j-ũ)/2σ2∑j=1n(1/2πσ)e-d2(β̃j-ũ)/2σ2,j=1,2,…,n.The position-weighting vectorw=(w1,w2,…,wn)T associated with the operator can be determined by RIM quantifiers [41]:(2.19)Q(r)=rα,α≥0,wi=Q(in)-Q(i-1n),fori=1,2,…,n.


## 3. Induced Interval-Valued Intuitionistic Fuzzy Hybrid Aggregation Operators with TOPSIS Order-Inducing Variables

### 3.1. I-IIFHA and I-IIFHG Operators

Induced interval-valued intuitionistic fuzzy hybrid operators are the extensions of ordinary interval-valued intuitionistic fuzzy hybrid operators, and the difference resides in that they can consider more preference information during aggregation process by combination of order-inducing variables independent of arguments. Here, based on the IIFHA and IIFHG, we propose two more general formulations of the reordering process to form I-IIFHA and I-IIFHG operators, which can be defined as follows.Definition 3.1.
An I-IIFHA operator of dimensionn is a mapping I-IIFHA: Rn→R defined by an associated weighting vector w of dimension n such that the sum of the weights is 1 and wj∈[0,1], a set of order-inducing variables εj, according to the following formula:
(3.1)I-IIFHAω,w(〈ε1,α̃1〉,〈ε2,α̃2〉,…,〈εn,α̃n〉)=w1α̃̇π(1)⊕w2α̃̇π(2)⊕⋯⊕wnα̃̇π(n)=([1-∏j=1n(1-ã̇π(j))wj,1-∏j=1n(1-b̃̇π(j))wj],[∏j=1nc̃̇π(j)wj,∏j=1nd̃̇π(j)wj]),
where α̃̇i=([α̃̇i,b̃̇i],[c̃̇i,d̃̇i])=nωiα̃i=([1-(1-ãi)nωi,1-(1-b̃i)nωi],[c̃inωi,d̃inωi]),(α̃̇π(1),α̃̇π(2),…,α̃̇π(n)) is (α̃1,α̃2,…,α̃n) reordered in decreasing order of the order-inducing variables values of the εj, ω=(ω1,ω2,…,ωn)T is the weight vector α̃j(j=1,2,…,n) with ∑j=1nωj=1 and ωj∈[0,1], and n is the balancing coefficient.Definition 3.2.
An I-IIFHG operator of dimensionn is a mapping I-IIFHG: Rn→R defined by an associated weighting vector w of dimension n such that the sum of the weights is 1 and wj∈[0,1], a set of order-inducing variables εj, according to the following formula:
(3.2)I-IIFHGω,w(〈ε1,α̃1〉,〈ε2,α̃2〉,…,〈εn,α̃n〉)=∏j=1nα̃̇π(j)wj=([∏j=1nα̃̇π(j)wj,∏j=1nb̃̇π(j)wj],[1-∏j=1n(1-c̃̇π(j))wj,1-∏j=1n(1-d̃̇π(j))wj]),
where α̃̇i=([α̃̇i,b̃̇i],[c̃̇i,d̃̇i])=α̃inωi=([ainωi,binωi],[1-(1-ci)nωi,1-(1-di)nωi]), (α̃̇π(1),α̃̇π(2),…,α̃̇π(n)) is (α̃1,α̃2,…,α̃n) reordered in decreasing order of the order-inducing variables values of the εj, ω=(ω1,ω2,…,ωn)T is the weight vector α̃i(i=1,2,…,n) with ∑i=1nωi=1 and ωi∈[0,1], and n is the balancing coefficient.Remark 3.3.
The argument-weighting vectorω=(ω1,ω2,…,ωn)T associated with inputs (α̃1,α̃2,…,α̃n) can be determined by normal distribution method defined in (2.17) and (2.18); The position-weighting vector w=(w1,w2,…,wn)T associated with the operator can be determined by RIM quantifiers in (2.19).


### 3.2.  TOPSIS Order-Inducing Variables

Order-inducing variables as reordering step can enable induced aggregation operators to consider additional decision rules that do not depend on the values of the arguments. In this subsection, we propose a TOPSIS-based method to ensure order-inducing variables to be included in I-IIFHA and I-IIFHG operators; as such we can be able to deal with the complex decision-making problems in which the solution with highest score value is not the optimal one for the decision makers.For example, we have the temperature problem introduced by Marin et al. [25] under linguistic environment. If we are dealing with temperatures of the human body, the optimal term is adequate; if the temperature is higher, the results are not good; and on the other side, if the temperature is lower, the results are also not good. If we are dealing with temperatures at which we want to travel, different persons may be interested in a different ordering of the temperatures; a person may prefer to visit places with cold weather, whereas another person may prefer to warm weather. So proper order-inducing variables should be introduced to deal with the problems where the solution with highest values may not be the optimal one, because of additional decision criteria to influence the ordering of arguments, such as rules of nature and preference of person.In the induced operators mentioned previously, such as IOWA operator, ILOWA operator, and I-IIFOWA operator, the order-inducing variables have been researched under different environments of exact numbers, interval-valued numbers, or linguistic numbers. Here, we let the decision maker define his own preference by interval-valued intuitionistic fuzzy numbers; then a TOPSIS-based method is proposed as follows to derive the order-inducing variables, which is calledTOPSIS order-inducing variables.Step 1.
Let the decision maker define the interval-valued intuitionistic fuzzy ideal solution(s) and the interval-valued intuitionistic fuzzy negative ideal solution(s) corresponding to a collection of interval-valued intuitionistic interval arguments(α̃1,α̃2,…,α̃n):
(3.3)α̃j+=(α̃j1+,α̃j2+,…,α̃jp+),α̃j-=(α̃j1-,α̃j2-,…,α̃jq-),
where p(p≥0) is the number of ideal solution(s), and q(q≥0) is the number of negative solution(s).Step 2.
Calculate the distance(s) between the interval-valued intuitionistic interval arguments(α̃1,α̃2,…,α̃n) and ideal solution(s) and negative solution(s):
(3.4)dj+=max⁡l=1,2,…,pd(α̃j,α̃jl+),dj-=min⁡l=1,2,…,qd(α̃j,α̃jq-),
where d(α̃j,α̃jl+) and d(α̃j,α̃jl-) can be calculated by the distance measures defined in Definition 2.3. If there is more than one ideal solution, we choose the maximum distance one(s); if there exist more than one negative solution, we choose the minimum distance one(s).Step 3.
Calculate the coefficients(3.5)cj=dj-dj-+dj+,j=1,2,…,n.Step 4.
Derive the order-inducing variableuj by descending order of cj. The greater the value cj, the bigger the ordering-induced variable uj.Remark 3.4.
In decision-making problems, one decision maker has no interval-valued intuitionistic fuzzy ideal solution(s) or negative solution(s), and we assume thatdj+=1 and dj-=0 if the attribute is the benefit type and dj+=0 and dj-=1 if it is the cost type.


### 3.3. Properties

LetfA be the I-IIFHA operator, and let fG be the I-IIFHG operator; ω=(ω1,ω2,…,ωn)T is argument-weighting vector, and w=(w1,w2,…,wn)Tis position-weighting vector. One has two interval-valued intuitionistic fuzzy vectors:(3.6)A={〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉},B={〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉}.
Then we can get these theorems of I-IIFHA and I-IIFHG operator as follows.Theorem 3.5 (monotonicity).
The I-IIFHA and I-IIFHG operators are increasingly monotonous with respect to the argument values if the associated order-inducing variables remain unchanged.(3.7)fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≥fw,ωA(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉),fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≥fw,ωG(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉),ifα̃j≥β̃j,εα̃(j)=εβ̃(j)∀j.Proof.
Thus, one hasα̃j≥β̃j⇒α̃̇j=α̃jnωα̃(j)≥β̃̇j=β̃jnωβ̃(j). And εα̃(j)=εβ̃(j),
(3.8)fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≥fw,ωA(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉),fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≥fw,ωG(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉).Theorem 3.6 (commutativity).
The I-IIFHA and I-IIFHG operators are commutative.(3.9)fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=fw,ωA(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉),fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=fw,ωG(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉),
where (α̃1,α̃2,…,α̃n) is any permutation of the arguments (β̃1,β̃2,…,β̃n).Proof.
Since(α̃1,α̃2,…,α̃n) is a permutation of (β̃1,β̃2,…,β̃n), then we have α̃̇j=nωα̃(j)α̃j=β̃̇j=nωβ̃(j)β̃j and α̃̇j=α̃jnωα̃(j)=β̃̇j=β̃jnωβ̃(j) for all j. Meanwhile, (εα̃(1),εα̃(2),…,εα̃(n)) is equal to (εβ̃(1),εβ̃(2),…,εβ̃(n)) which is associated with the decision maker’s preference, so
(3.10)wα̃(1)α̃̇1⊕wα̃(2)α̃̇2⊕⋯⊕wα̃(n)α̃̇n=wβ̃(1)β̃̇1⊕wβ̃(2)β̃̇2⊕⋯⊕wβ̃(n)β̃̇n,α̃̇1wα̃(1)⊗α̃̇2wα̃(2)⊗⋯⊗α̃̇nwα̃(n)=β̃̇1wβ̃(1)⊗β̃̇2wβ̃(2)⊗⋯⊗β̃̇nwβ̃(n).
We get
(3.11)fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=fw,ωA(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉),fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=fw,ωG(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉).Theorem 3.7 (idempotency).
The I-IIFHA and I-IIFHG operators are idempotent. Ifα̃j=α̃ for all j, then
(3.12)fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=α̃,fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=α.̃Proof.
Sinceα̃j=α̃ for all j, we have (ω1,ω2,…,ωn)=(1/n,1/n,…,1/n), which is associated with the input arguments (α̃1,α̃2,…,α̃n); then
(3.13)fw,ωA(〈εα̃(1),nω1α̃1〉,〈εα̃(2),nω2α̃2〉,…,〈εα̃(n),nωnα̃n〉)=fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=w1α̃π(1)⊕w2α̃π(2)⊕⋯⊕wnα̃π(n)=(w1+w2+⋯+wn)α̃=α̃,fw,ωG(〈εα̃(1),α̃1nω1〉,〈εα̃(2),α̃2nω2〉,…,〈εα̃(n),α̃nnωn〉)=fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=α̃π(1)w1⊗α̃π(2)w2⊗⋯⊗α̃π(n)wn=α̃(w1+w2+⋯+wn)=α̃.Theorem 3.8 (bounded).
The I-IIFHA and I-IIFHG operators are bounded. That is, for any weighting vectorω and w:
(3.14)min⁡(α̃1,α̃2,…,α̃n)≤fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≤max⁡(α̃1,α̃2,…,α̃n),min⁡(α̃1,α̃2,…,α̃n)≤fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≤max⁡(α̃1,α̃2,…,α̃n).Proof.
By definitions of I-IIFHA and I-IIFHG operators and operations of IVIFNs, we can obtain that(3.15)min⁡{α̃i}≤min⁡{α̃π(i)}≤fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≤max⁡{α̃π(i)}≤max⁡{α̃i},min⁡{α̃i}≤min⁡{α̃π(i)}≤fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≤max⁡{α̃π(i)}≤max⁡{α̃i}.Theorem 3.9.
Ifω=(1/n,1/n,…,1/n), I-IIFHA and I-IIFHG operators reduce to I-IIFOWA and I-IIFOWG operators. If w=(1/n,1/n,…,1/n), they reduce to IIFWA and IIFWG operators. If there was no any positive or negative preference, (εα̃(1),εα̃(2),…,εα̃(n))=(εβ̃(1),εβ̃(2),…,εβ̃(n)), the IIFHA and IIFHG operators are the special cases of I-IIFHA and I-IIFHG operators.Proof.
Letω=(1/n,1/n,…,1/n); then
(3.16)I-IIFHA(〈εα̃(1),nω1α̃1〉,〈εα̃(2),nω2α̃2〉,…,〈εα̃(n),nωnα̃n〉)=w1α̃π(1)⊕w2α̃π(2)⋯⊕wnα̃π(n)=I-IIFOWAw(α̃1,α̃2,…,α̃n),I-IIFHG(〈εα̃(1),α̃1nω1〉,〈εα̃(2),α̃2nω2〉,…,〈εα̃(n),α̃nnωn〉)=α̃π(1)w1⊗α̃π(2)w2⊗⋯⊗α̃π(n)wn=I-IIFOWGw(α̃1,α̃2,…,α̃n).
Let w=(1/n,1/n,…,1/n); then
(3.17)I-IIFHA(〈εα̃(1),nω1α̃1〉,〈εα̃(2),nω2α̃2〉,…,〈εα̃(n),nωnα̃n〉)=ω1α̃1⊕ω2α̃2⊕⋯⊕ωnα̃n=IIFWAω(α̃1,α̃2,…,α̃n),I-IIFHG(〈εα̃(1),α̃1nω1〉,〈εα̃(2),α̃2nω2〉,…,〈εα̃(n),α̃nnωn〉)=α̃1ω1⊗α̃2ω2⊗⋯⊗α̃nωn=IIFWGω(α̃1,α̃2,…,α̃n).Theorem 3.10.
In the situation ofνjL=0 or νjL=0, interval-valued intuitionistic fuzzy geometric operators are better than interval-valued intuitionistic fuzzy averaging operators, while in μjL=0 or μjU=0, interval-valued intuitionistic fuzzy averaging operators are better than interval-valued intuitionistic fuzzy geometric operators.Proof.
Take the IIFWA and IIFWG operators, for example.
IfνjL=0 and νjL=0,(3.18)IIFWAω(α̃1,α̃2,…,α̃n)=ω1α̃1⊕ω2α̃2⊕⋯⊕ωnα̃n=([1-∏j=1n(1-μjL)ωj,1-∏j=1n(1-μjU)ωj],[∏j=1n(νjL)ωj,∏j=1n(νjU)ωj])=([1-∏j=1n(1-μjL)ωj,1-∏j=1n(1-μjU)ωj],[0,0]).
If μjL=0 and μjU=0,
(3.19)IIFWGω(α̃1,α̃2,…,α̃n)=α̃1ω1⊗α̃2ω2⊗⋯⊗α̃nωn=([∏j=1n(μjL)ωj,∏j=1n(μjU)ωj],[1-∏j=1n(1-νjL)ωj,1-∏j=1n(1-νjU)ωj])=([0,0],[1-∏j=1n(1-νjL)ωj,1-∏j=1n(1-νjU)ωj]).
Similarly, this rule is also suitable for IIFOWA, IIFOWG, IIFHA, IIFHG, I-IIFOWA, I-IIFOWG, I-IIFHA, I-IIFHG operators, and so forth.


## 3.1. I-IIFHA and I-IIFHG Operators

Induced interval-valued intuitionistic fuzzy hybrid operators are the extensions of ordinary interval-valued intuitionistic fuzzy hybrid operators, and the difference resides in that they can consider more preference information during aggregation process by combination of order-inducing variables independent of arguments. Here, based on the IIFHA and IIFHG, we propose two more general formulations of the reordering process to form I-IIFHA and I-IIFHG operators, which can be defined as follows.Definition 3.1.
An I-IIFHA operator of dimensionn is a mapping I-IIFHA: Rn→R defined by an associated weighting vector w of dimension n such that the sum of the weights is 1 and wj∈[0,1], a set of order-inducing variables εj, according to the following formula:
(3.1)I-IIFHAω,w(〈ε1,α̃1〉,〈ε2,α̃2〉,…,〈εn,α̃n〉)=w1α̃̇π(1)⊕w2α̃̇π(2)⊕⋯⊕wnα̃̇π(n)=([1-∏j=1n(1-ã̇π(j))wj,1-∏j=1n(1-b̃̇π(j))wj],[∏j=1nc̃̇π(j)wj,∏j=1nd̃̇π(j)wj]),
where α̃̇i=([α̃̇i,b̃̇i],[c̃̇i,d̃̇i])=nωiα̃i=([1-(1-ãi)nωi,1-(1-b̃i)nωi],[c̃inωi,d̃inωi]),(α̃̇π(1),α̃̇π(2),…,α̃̇π(n)) is (α̃1,α̃2,…,α̃n) reordered in decreasing order of the order-inducing variables values of the εj, ω=(ω1,ω2,…,ωn)T is the weight vector α̃j(j=1,2,…,n) with ∑j=1nωj=1 and ωj∈[0,1], and n is the balancing coefficient.Definition 3.2.
An I-IIFHG operator of dimensionn is a mapping I-IIFHG: Rn→R defined by an associated weighting vector w of dimension n such that the sum of the weights is 1 and wj∈[0,1], a set of order-inducing variables εj, according to the following formula:
(3.2)I-IIFHGω,w(〈ε1,α̃1〉,〈ε2,α̃2〉,…,〈εn,α̃n〉)=∏j=1nα̃̇π(j)wj=([∏j=1nα̃̇π(j)wj,∏j=1nb̃̇π(j)wj],[1-∏j=1n(1-c̃̇π(j))wj,1-∏j=1n(1-d̃̇π(j))wj]),
where α̃̇i=([α̃̇i,b̃̇i],[c̃̇i,d̃̇i])=α̃inωi=([ainωi,binωi],[1-(1-ci)nωi,1-(1-di)nωi]), (α̃̇π(1),α̃̇π(2),…,α̃̇π(n)) is (α̃1,α̃2,…,α̃n) reordered in decreasing order of the order-inducing variables values of the εj, ω=(ω1,ω2,…,ωn)T is the weight vector α̃i(i=1,2,…,n) with ∑i=1nωi=1 and ωi∈[0,1], and n is the balancing coefficient.Remark 3.3.
The argument-weighting vectorω=(ω1,ω2,…,ωn)T associated with inputs (α̃1,α̃2,…,α̃n) can be determined by normal distribution method defined in (2.17) and (2.18); The position-weighting vector w=(w1,w2,…,wn)T associated with the operator can be determined by RIM quantifiers in (2.19).


## 3.2.  TOPSIS Order-Inducing Variables

Order-inducing variables as reordering step can enable induced aggregation operators to consider additional decision rules that do not depend on the values of the arguments. In this subsection, we propose a TOPSIS-based method to ensure order-inducing variables to be included in I-IIFHA and I-IIFHG operators; as such we can be able to deal with the complex decision-making problems in which the solution with highest score value is not the optimal one for the decision makers.For example, we have the temperature problem introduced by Marin et al. [25] under linguistic environment. If we are dealing with temperatures of the human body, the optimal term is adequate; if the temperature is higher, the results are not good; and on the other side, if the temperature is lower, the results are also not good. If we are dealing with temperatures at which we want to travel, different persons may be interested in a different ordering of the temperatures; a person may prefer to visit places with cold weather, whereas another person may prefer to warm weather. So proper order-inducing variables should be introduced to deal with the problems where the solution with highest values may not be the optimal one, because of additional decision criteria to influence the ordering of arguments, such as rules of nature and preference of person.In the induced operators mentioned previously, such as IOWA operator, ILOWA operator, and I-IIFOWA operator, the order-inducing variables have been researched under different environments of exact numbers, interval-valued numbers, or linguistic numbers. Here, we let the decision maker define his own preference by interval-valued intuitionistic fuzzy numbers; then a TOPSIS-based method is proposed as follows to derive the order-inducing variables, which is calledTOPSIS order-inducing variables.Step 1.
Let the decision maker define the interval-valued intuitionistic fuzzy ideal solution(s) and the interval-valued intuitionistic fuzzy negative ideal solution(s) corresponding to a collection of interval-valued intuitionistic interval arguments(α̃1,α̃2,…,α̃n):
(3.3)α̃j+=(α̃j1+,α̃j2+,…,α̃jp+),α̃j-=(α̃j1-,α̃j2-,…,α̃jq-),
where p(p≥0) is the number of ideal solution(s), and q(q≥0) is the number of negative solution(s).Step 2.
Calculate the distance(s) between the interval-valued intuitionistic interval arguments(α̃1,α̃2,…,α̃n) and ideal solution(s) and negative solution(s):
(3.4)dj+=max⁡l=1,2,…,pd(α̃j,α̃jl+),dj-=min⁡l=1,2,…,qd(α̃j,α̃jq-),
where d(α̃j,α̃jl+) and d(α̃j,α̃jl-) can be calculated by the distance measures defined in Definition 2.3. If there is more than one ideal solution, we choose the maximum distance one(s); if there exist more than one negative solution, we choose the minimum distance one(s).Step 3.
Calculate the coefficients(3.5)cj=dj-dj-+dj+,j=1,2,…,n.Step 4.
Derive the order-inducing variableuj by descending order of cj. The greater the value cj, the bigger the ordering-induced variable uj.Remark 3.4.
In decision-making problems, one decision maker has no interval-valued intuitionistic fuzzy ideal solution(s) or negative solution(s), and we assume thatdj+=1 and dj-=0 if the attribute is the benefit type and dj+=0 and dj-=1 if it is the cost type.


## 3.3. Properties

LetfA be the I-IIFHA operator, and let fG be the I-IIFHG operator; ω=(ω1,ω2,…,ωn)T is argument-weighting vector, and w=(w1,w2,…,wn)Tis position-weighting vector. One has two interval-valued intuitionistic fuzzy vectors:(3.6)A={〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉},B={〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉}.
Then we can get these theorems of I-IIFHA and I-IIFHG operator as follows.Theorem 3.5 (monotonicity).
The I-IIFHA and I-IIFHG operators are increasingly monotonous with respect to the argument values if the associated order-inducing variables remain unchanged.(3.7)fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≥fw,ωA(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉),fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≥fw,ωG(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉),ifα̃j≥β̃j,εα̃(j)=εβ̃(j)∀j.Proof.
Thus, one hasα̃j≥β̃j⇒α̃̇j=α̃jnωα̃(j)≥β̃̇j=β̃jnωβ̃(j). And εα̃(j)=εβ̃(j),
(3.8)fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≥fw,ωA(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉),fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≥fw,ωG(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉).Theorem 3.6 (commutativity).
The I-IIFHA and I-IIFHG operators are commutative.(3.9)fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=fw,ωA(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉),fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=fw,ωG(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉),
where (α̃1,α̃2,…,α̃n) is any permutation of the arguments (β̃1,β̃2,…,β̃n).Proof.
Since(α̃1,α̃2,…,α̃n) is a permutation of (β̃1,β̃2,…,β̃n), then we have α̃̇j=nωα̃(j)α̃j=β̃̇j=nωβ̃(j)β̃j and α̃̇j=α̃jnωα̃(j)=β̃̇j=β̃jnωβ̃(j) for all j. Meanwhile, (εα̃(1),εα̃(2),…,εα̃(n)) is equal to (εβ̃(1),εβ̃(2),…,εβ̃(n)) which is associated with the decision maker’s preference, so
(3.10)wα̃(1)α̃̇1⊕wα̃(2)α̃̇2⊕⋯⊕wα̃(n)α̃̇n=wβ̃(1)β̃̇1⊕wβ̃(2)β̃̇2⊕⋯⊕wβ̃(n)β̃̇n,α̃̇1wα̃(1)⊗α̃̇2wα̃(2)⊗⋯⊗α̃̇nwα̃(n)=β̃̇1wβ̃(1)⊗β̃̇2wβ̃(2)⊗⋯⊗β̃̇nwβ̃(n).
We get
(3.11)fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=fw,ωA(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉),fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=fw,ωG(〈εβ̃(1),β̃1〉,〈εβ̃(2),β̃2〉,…,〈εβ̃(n),β̃n〉).Theorem 3.7 (idempotency).
The I-IIFHA and I-IIFHG operators are idempotent. Ifα̃j=α̃ for all j, then
(3.12)fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=α̃,fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=α.̃Proof.
Sinceα̃j=α̃ for all j, we have (ω1,ω2,…,ωn)=(1/n,1/n,…,1/n), which is associated with the input arguments (α̃1,α̃2,…,α̃n); then
(3.13)fw,ωA(〈εα̃(1),nω1α̃1〉,〈εα̃(2),nω2α̃2〉,…,〈εα̃(n),nωnα̃n〉)=fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=w1α̃π(1)⊕w2α̃π(2)⊕⋯⊕wnα̃π(n)=(w1+w2+⋯+wn)α̃=α̃,fw,ωG(〈εα̃(1),α̃1nω1〉,〈εα̃(2),α̃2nω2〉,…,〈εα̃(n),α̃nnωn〉)=fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)=α̃π(1)w1⊗α̃π(2)w2⊗⋯⊗α̃π(n)wn=α̃(w1+w2+⋯+wn)=α̃.Theorem 3.8 (bounded).
The I-IIFHA and I-IIFHG operators are bounded. That is, for any weighting vectorω and w:
(3.14)min⁡(α̃1,α̃2,…,α̃n)≤fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≤max⁡(α̃1,α̃2,…,α̃n),min⁡(α̃1,α̃2,…,α̃n)≤fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≤max⁡(α̃1,α̃2,…,α̃n).Proof.
By definitions of I-IIFHA and I-IIFHG operators and operations of IVIFNs, we can obtain that(3.15)min⁡{α̃i}≤min⁡{α̃π(i)}≤fw,ωA(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≤max⁡{α̃π(i)}≤max⁡{α̃i},min⁡{α̃i}≤min⁡{α̃π(i)}≤fw,ωG(〈εα̃(1),α̃1〉,〈εα̃(2),α̃2〉,…,〈εα̃(n),α̃n〉)≤max⁡{α̃π(i)}≤max⁡{α̃i}.Theorem 3.9.
Ifω=(1/n,1/n,…,1/n), I-IIFHA and I-IIFHG operators reduce to I-IIFOWA and I-IIFOWG operators. If w=(1/n,1/n,…,1/n), they reduce to IIFWA and IIFWG operators. If there was no any positive or negative preference, (εα̃(1),εα̃(2),…,εα̃(n))=(εβ̃(1),εβ̃(2),…,εβ̃(n)), the IIFHA and IIFHG operators are the special cases of I-IIFHA and I-IIFHG operators.Proof.
Letω=(1/n,1/n,…,1/n); then
(3.16)I-IIFHA(〈εα̃(1),nω1α̃1〉,〈εα̃(2),nω2α̃2〉,…,〈εα̃(n),nωnα̃n〉)=w1α̃π(1)⊕w2α̃π(2)⋯⊕wnα̃π(n)=I-IIFOWAw(α̃1,α̃2,…,α̃n),I-IIFHG(〈εα̃(1),α̃1nω1〉,〈εα̃(2),α̃2nω2〉,…,〈εα̃(n),α̃nnωn〉)=α̃π(1)w1⊗α̃π(2)w2⊗⋯⊗α̃π(n)wn=I-IIFOWGw(α̃1,α̃2,…,α̃n).
Let w=(1/n,1/n,…,1/n); then
(3.17)I-IIFHA(〈εα̃(1),nω1α̃1〉,〈εα̃(2),nω2α̃2〉,…,〈εα̃(n),nωnα̃n〉)=ω1α̃1⊕ω2α̃2⊕⋯⊕ωnα̃n=IIFWAω(α̃1,α̃2,…,α̃n),I-IIFHG(〈εα̃(1),α̃1nω1〉,〈εα̃(2),α̃2nω2〉,…,〈εα̃(n),α̃nnωn〉)=α̃1ω1⊗α̃2ω2⊗⋯⊗α̃nωn=IIFWGω(α̃1,α̃2,…,α̃n).Theorem 3.10.
In the situation ofνjL=0 or νjL=0, interval-valued intuitionistic fuzzy geometric operators are better than interval-valued intuitionistic fuzzy averaging operators, while in μjL=0 or μjU=0, interval-valued intuitionistic fuzzy averaging operators are better than interval-valued intuitionistic fuzzy geometric operators.Proof.
Take the IIFWA and IIFWG operators, for example.
IfνjL=0 and νjL=0,(3.18)IIFWAω(α̃1,α̃2,…,α̃n)=ω1α̃1⊕ω2α̃2⊕⋯⊕ωnα̃n=([1-∏j=1n(1-μjL)ωj,1-∏j=1n(1-μjU)ωj],[∏j=1n(νjL)ωj,∏j=1n(νjU)ωj])=([1-∏j=1n(1-μjL)ωj,1-∏j=1n(1-μjU)ωj],[0,0]).
If μjL=0 and μjU=0,
(3.19)IIFWGω(α̃1,α̃2,…,α̃n)=α̃1ω1⊗α̃2ω2⊗⋯⊗α̃nωn=([∏j=1n(μjL)ωj,∏j=1n(μjU)ωj],[1-∏j=1n(1-νjL)ωj,1-∏j=1n(1-νjU)ωj])=([0,0],[1-∏j=1n(1-νjL)ωj,1-∏j=1n(1-νjU)ωj]).
Similarly, this rule is also suitable for IIFOWA, IIFOWG, IIFHA, IIFHG, I-IIFOWA, I-IIFOWG, I-IIFHA, I-IIFHG operators, and so forth.


## 4. Application in Interval-Valued Intuitionistic Fuzzy Multiple-Attribute Group Decision-Making

For a multiple-attribute group decision-making problem, letX={x1,x2,…,xn} be a set of alternatives, and let G={g1,g2,…,gm} be a set of criteria. Let D={d1,d2,…,dt} be a set of decision makers, and let λ=(λ(1),λ(2),…,λ(t))T be the weighting vector of decision makers. In the following, we apply the I-IIFHA and I-IIFHG operators (whose position-weighting vector isw=(w1,w2,…,wm)T to multiple-attribute group decision-making based on interval-valued intuitionistic fuzzy information. The approach involves the following steps.Step 1.
Construct the interval-valued intuitionistic fuzzy evaluation matrix.R̃=(r̃ij)n×m=(μ̃ij,ν̃ij)n×m=([μijL,μijU],[νijL,νijU])n×m, where [μijL,μijU] indicates the degree at wich the alternative xi satisfies the attribute gj given by the decision maker, and [νijL,νijU] indicates the degree at which the alternative xi(i=1,2,…,n) does not satisfy the attribute gj(j=1,2,…,m).Step 2.
Obtain the interval-valued intuitionistic fuzzy positive-ideal vector and negative-ideal vector corresponding to each alternative’s attribute by every decision maker.
Let thekth decision maker provide the interval-valued intuitionistic fuzzy positive-ideal vector R̃(k)+ and the interval-valued intuitionistic fuzzy negative-ideal vector R̃(k)-(4.1)R̃(k)+=(r̃1(k)+,r̃2(k)+,…,r̃j(k)+,…,r̃m(k)+),R̃(k)-=(r̃1(k)-,r̃2(k)-,…,r̃j(k)-,…,r̃m(k)-),
where
(4.2)r̃j(k)+=(r̃j1(k)+,r̃j2(k)+,…,r̃jl(k)+,…,r̃jp(k)+),r̃j(k)-=(r̃j1(k)-,r̃j2(k)-,…,r̃jl(k)-,…,r̃jq(k)-).Step 3.
Calculate the argument-weighting vectorω=(ω1,ω2,…,ωn)T associated with the interval-valued intuitionistic fuzzy value r̃ij by normal distribution method.Step 4.
Construct aggregated weighted interval-valued intuitionistic fuzzy decision matrixR̃̇=(r̃̇ij)n×m:
(4.3)R̃̇=[g1g2⋯gmx1r̃̇11r̃̇12⋯r̃̇1mx2r̃̇21r̃̇22⋯r̃̇2m⋮⋮⋮⋮⋮xnr̃̇n1r̃̇n2⋯r̃̇nm]=(r̃̇ij)n×m,
where r̃̇ij=nωjr̃ij or r̃̇ij=r̃ijnωj, and ω=(ω1,ω2,…,ωn)T is the weighting vector associated with interval-valued intuitionistic fuzzy value r̃ij itself.Step 5.
Ensure the TOPSIS order-inducing variablesε by using of positive-ideal and negative-ideal information provided by each decision maker in Step 2.
First, we calculate the distance between the evaluation informationr̃ij and kth decision maker’s preference information.
(4.4)dij(k)+=max⁡l=1,2,…,pd(r̃ij,r̃jl(k)+),dij(k)-=min⁡l=1,2,…,qd(r̃ij,r̃jl(k)-)
where d(r̃ij,r̃j(k)+) and d(r̃ij,r̃j(k)-) can be calculated by the normalized Euclidean distance measure in Definition 2.3.
We can calculate the coefficients of each alternative’s attribute:(4.5)cij(k)=dij(k)-dij(k)-+dij(k)+,i=1,2,…,n,j=1,2,…,m,k=1,2,…,t,
and we can get coefficients matrix C(k)=(cij(k))n×m. By ascending order of cij(k), we get the ordering induced variables matrix ε(k)=(εij(k))n×m.Step 6.
Utilize the aggregated weighted interval-valued intuitionistic fuzzy decision matrixR̃̇=(r̃̇ij)n×m and I-IIFHA or I-IIFHG operator to derive the kth decision maker overall interval-valued intuitionistic fuzzy arguments R̃(k)=(r̃1(k),r̃2(k),…,r̃n(k)) corresponding to each alternative xi:
(4.6)r̃i(k)=([μi(k)L,μi(k)U],[νi(k)L,νi(k)U])=I-IIFHAω,w(〈εi1(k),r̃i1(k)〉,〈εi2(k),r̃i2(k)〉,…,〈εim(k),r̃im(k)〉)orr̃i(k)=([μi(k)L,μi(k)U],[νi(k)L,νi(k)U])=I-IIFHGω,w(〈εi1(k),r̃i1(k)〉,〈εi2(k),r̃i2(k)〉,…,〈εim(k),r̃im(k)〉).Step 7.
Utilize the individual overall interval-valued intuitionistic fuzzy arguments vectorr̃i(k) and IIFWA or IIFWG operator to derive the group overall interval-valued intuitionistic fuzzy argument vector r̃i:
(4.7)r̃i=([μiL,μiU],[νiL,νiU])=IIFWA(r̃i(1),r̃i(2),…,r̃i(t))orr̃i=([μiL,μiU],[νiL,νiU])=IIFWG(r̃i(1),r̃i(2),…,r̃i(t)).Step 8.
Calculate the scoress(r̃i)(i=1,2,…,n) and the accuracy values h(r̃i)(i=1,2,…,n) of the alternative xi(i=1,2,…,n), respectively.Step 9.
Utilize the scoress(r̃i)(i=1,2,…,n) and the accuracy values h(r̃i)(i=1,2,…,n) to rank all the alternatives xi(i=1,2,…,n) and then to select the optimal one(s).


## 5.  Applied Example

Here we take the real estate purchase for applied example. There are five candidate houses denoted byxi(i=1,2,3,4,5) that can be evaluated by the IVIFNs r̃ij with respect to the five attributes: (1)g1: size; (2)g2: price; (3)g3: structure; (4)g4: position; and (5)g5: environment, as listed in Table 1. In the real situation, the DMs may have his/her preference to each attribute. For example, someone may prefer the house close to suburb, while someone may prefer the position of urban area. DMs give their interval-valued intuitionistic fuzzy ideal values and negative ideal values corresponding to five attributes shown in Tables 2, 3, and 4. Suppose that the expert-weighting vector is λ=(0.3,0.3,0.4)T.Table 1
Decision matrixR̃ with IVIFNs.
g1g2g3g4g5x1([0.4,0.6],[0.3,0.4])([0.5,0.7],[0.0,0.2])([0.5,0.6],[0.2,0.4])([0.6,0.8],[0.1,0.2])([0.4,0.7],[0.2,0.3])x2([0.5,0.8],[0.1,0.2])([0.3,0.5],[0.2,0.3])([0.3,0.6],[0.2,0.4])([0.4,0.5],[0.2,0.4])([0.3,0.6],[0.2,0.3])x3([0.5,0.6],[0.0,0.1])([0.5,0.8],[0.1,0.2])([0.4,0.7],[0.2,0.3])([0.2,0.4],[0.2,0.3])([0.5,0.8],[0.0,0.2])x4([0.5,0.7],[0.1,0.3])([0.4,0.6],[0.0,0.1])([0.3,0.5],[0.2,0.4])([0.7,0.9],[0.0,0.1])([0.3,0.5],[0.2,0.2])x5([0.7,0.8],[0.0,0.1])([0.4,0.6],[0.0,0.2])([0.4,0.7],[0.2,0.3])([0.3,0.5],[0.1,0.3])([0.6,0.7],[0.1,0.2])Table 2
Preference decision matrixR̃(1) provided by d1.
d1g1g2g3g4g5R̃(1)+([0.5,0.5],[0.5,0.5])([1,1],[0,0])([0.5,0.5],[0.5,0.5])([1,1],[0,0])([1,1],[0,0])R̃(1)-([0,0],[1,1])([0,0],[1,1])([0,0],[1,1])([1,1],[0,0])([0,0],[1,1])Table 3
Preference decision matrixR̃(2) provided by d2.
d2g1g2g3g4g5R̃(2)+([1,1],[0,0])([1,1],[0,0])([0.3,0.4],[0.6,0.7])([0.5,0.5],[0.5,0.5])([1,1],[0,0])R̃(2)-([0,0],[1,1])([0,0],[1,1])([1,1],[0,0])([0,0],[1,1])([0,0],[1,1])Table 4
Preference decision matrixR̃(3) provided by d3.
d3g1g2g3g4g5R̃(3)+([1,1],[0,0])([1,1],[0,0])([0.8,0.9],[0.1,0.2])([0.5,0.5],[0.5,0.5])([1,1],[0,0])R̃(3)-([0,0],[1,1])([0,0],[1,1])([0,0],[1,1])([0,0],[1,1])([0,0],[1,1])Then we can get the alternative’s attribute coefficients matrixC(1), C(2), and C(3) as shown in Tables 5, 6, and 7 according to (4.4) and (4.5).Table 5
Alternative’s attribute coefficient matrixC(1).
d1g1g2g3g4g5x10.75130.67010.74460.73210.6182x20.69100.55190.67340.55940.5708x30.65580.69100.70490.51490.6865x40.70110.62750.66670.80000.5652x50.68550.62020.70490.56190.7101Table 6
Alternative’s attribute coefficient matrixC(2).
d2g1g2g3g4g5x10.56580.67010.57580.70610.6182x20.69100.55190.60730.70370.5708x30.65580.69100.54400.61800.6865x40.65690.62750.61760.68130.5652x50.78070.62020.54400.64100.7101Table 7
Alternative’s attribute coefficient matrixC(3).
d3g1g2g3g4g5x10.56580.67010.71130.70610.6182x20.69100.55190.63530.70370.5708x30.65580.69100.71120.61800.6865x40.65690.62750.60890.68130.5652x50.78070.62020.71120.64100.7101By the normal distribution method, we can determine argument-weighting vector associated with their own values:(5.1)ω=(0.2225,0.1911,0.2421,0.1516,0.1927)T.There existνxijL=0 and νxijU=0 in interval-valued intuitionistic fuzzy decision matrix R̃; we choose I-IIFHG and IIFWG operators to aggregate each decision information according to Theorem 3.10.We first get the weighted interval-valued intuitionistic fuzzy decision matrixR̃̇=(r̃̇ij)n×m as shown in Table 8.Table 8
Aggregate resultsR̃̇.
g1g2g3g4g5x1([0.3608,0.5665],[0.3275,0.4335])([0.5157,0.7112],[0.0,0.1920])([0.4321,0.5388],[0.2367,0.4612])([0.6790,0.8444],[0.0768,0.1556])([0.4136,0.7092],[0.1935,0.2908])x2([0.4625,0.7802],[0.1106,0.2198])([0.3165,0.5157],[0.1920,0.2888])([0.2328,0.5388],[0.2367,0.4612])([0.4993,0.5913],[0.1556,0.3210])([0.3135,0.6113],[0.1935,0.2908])x3([0.4625,0.5665],[0.0,0.1106])([0.5157,0.8080],[0.0958,0.1920])([0.3298,0.6494],[0.2367,0.3506])([0.2952,0.4993],[0.1556,0.2369])([0.5128,0.8065],[0.0,0.1935])x4([0.4625,0.6725],[0.1106,0.3275])([0.4166,0.6138],
[0.0,0.0958])([0.2328,0.4321],[0.2367,0.4612])([0.7631,0.9232],[0.0,0.0768])([0.3135,0.5128],[0.1935,0.1935])x5([0.6725,0.7802],[0.0,0.1106])([0.4166,0.6138],[0.0,0.1920])([0.3298,0.6494],[0.2367,0.3506])([0.4015,0.5913],[0.0768,0.2369])([0.6113,0.7092],[0.0965,0.1935])Step 6.
Suppose the position-weighting vectorw=(0.4472,0.1853,0.1421,0.1198,0.1056)T generalized by RIM quantifier. Utilize the aggregated weighted interval-valued intuitionistic fuzzy decision matrix R̃̇=(r̃̇ij)n×m and I-IIFHG operator to derive the kth decision maker overall interval-valued intuitionistic fuzzy arguments R̃(k)=(r̃1(k),r̃2(k),…,r̃n(k)) corresponding to each alternative xi(i=1,2,…,5).
(5.2)r̃1(1)=([0.5041,0.6482],[0.1449,0.2952]),r̃2(1)=([0.3410,0.5936],[0.1862,0.3348]),r̃3(1)=([0.4483,0.6866],[0.0704,0.1991]),r̃4(1)=([0.3709,0.5783],[0.1253,0.2871]),r̃5(1)=([0.4741,0.6772],[0.0806,0.2151]);r̃1(2)=([0.4583,0.6614],[0.1668,0.3184]),r̃2(2)=([0.3311,0.6091],[0.1840,0.3358]),r̃3(2)=([0.4246,0.6685],[0.1071,0.2202]),r̃4(2)=([0.4439,0.6469],[0.0898,0.1914]),r̃5(2)=([0.4741,0.6772],[0.0806,0.2151]);r̃1(3)=([0.4920,0.6994],[0.1344,0.2729]),r̃2(3)=([0.3311,0.6091],[0.1840,0.3358]),r̃3(3)=([0.4483,0.6866],[0.0704,0.1991]),r̃4(3)=([0.3858,0.5923],[0.1091,0.2616]),r̃5(3)=([0.4633,0.6618],[0.1073,0.2351]).Step 7.
Use the individual overall interval-valued intuitionistic fuzzy arguments vectorr̃i(k) and IIFWG operator to derive the group overall interval-valued intuitionistic fuzzy arguments vector r̃i(i=1,2,…,5):
(5.3)r̃1=([0.4829,0.6635],[0.1517,0.3002]),r̃2=([0.3350,0.6029],[0.1849,0.3354]),r̃3=([0.4387,0.6793],[0.0853,0.2076]),r̃4=([0.4017,0.6077],[0.1080,0.2450]),r̃5=([0.4719,0.6741],[0.0860,0.2191]).Step 8.
Calculate the scoress(r̃i)(i=1,2,…,5)of the collective overall intuitionistic fuzzy values by Definition 2.6.
(5.4)s(r̃1)=0.6945,s(r̃2)=0.4176,s(r̃3)=0.8251,s(r̃4)=0.6564,s(r̃5)=0.8409.Step 9.
Rank all the alternativesxi(i=1,2,…,5) in accordance with the scores s(r̃i)(i=1,2,…,5) of the collective overall intuitionistic fuzzy values: x2≺x4≺x1≺x3≺x5, and thus the most desirable alternative is x5.
In the previous analysis, the computation of illustrative example is conducted according to the proposed hybrid induced I-IIFHG operator. And we have also conducted computation according to other representative operators: two induced operators of I-IIFOWA and I-IIFOWG, which can consider the attitudinal characteristics of experts and the argument-independent preference information; two hybrid operators of IIFHA and IIFHG, which can consider the argument-dependent and argument-independent preference information; and the other proposed induced hybrid I-IIFHA operator. Comparative aggregated results and orderings are listed in Tables9 and 10.Table 9
Aggregated results by different operators.
I-IIFOWAI-IIFOWGIIFHAIIFHGI-IIFHAI-IIFHGx1([0.4979,0.6890],[0.0,0.2928])([0.4862,0.6724],[0.1716,0.3142])([0.4879,0.6802],
[0.0,0.2689])([0.5329,0.7249],[0.1311,0.2577])([0.4944,0.6719],[0.0,0.3230])([0.4829,0.6635],[0.1517,0.3002])x2([0.3930,0.6313],
[0.0,0.3116])([0.3796,0.5958],
[0.1747,0.3291])([0.4261,0.7182],
[0.1300,0.2510])([0.3945,0.6551],[0.1550,0.2866])([0.3870,0.6761],[0.1626,0.3057])([0.3350,0.6029],[0.1849,0.3354])x3([0.4394,0.7167],[0.0,0.2203])([0.4187,0.6827],[0.1212,0.2358])([0.4804,0.6919],[0.0,0.1463])([0.4519,0.7081],[0.0652,0.2057])([0.4634,0.7624],[0.0,0.2406])([0.4387,0.6793],[0.0853,0.2076])x4([0.5591,0.7852],[0.0,0.1572])([0.5008,0.7094],[0.0688,0.1940])([0.5075,0.7340],[0.0,0.1944])([0.5038,0.7039],[0.0685,0.1828])([0.4764,0.6653],[0.0,0.1843])([0.4017,0.6077],[0.1080,0.2450])x5([0.5676,0.7144],[0.0,0.1753])([0.5208,0.6922],[0.0721,0.1968])([0.6084,0.7492],[0.0,0.1465])([0.5382,0.7029],[0.0553,0.1817])([0.6104,0.7814],[0.0,0.1949])([0.4719,0.6741],[0.0860,0.2191])Table 10
Ordering of the candidate houses.
Different operatorsOrderingI-IIFOWA2≺1≺3≺5≺4I-IIFOWG2≺1≺3≺5≺4IIFHA2≺1≺3≺4≺5IIFHG2≺1≺3≺4≺5I-IIFHA2≺1≺4≺3≺5I-IIFHG2≺4≺1≺3≺5As can be seen from the comparative results in Table10, because the last four hybrid operators (IIFHA, IIFHG, I-IIFHA, and I-IIFHG) manage to give more consideration to the argument-dependent preference information than the first two induced operators (I-IIFOWA and I-IIFOWG), the 5th candidate alternative is emphasized to be the optimal one. And among the last four hybrid operators, because the last two induced hybrid operators (I-IIFHA and I-IIFHG) manage to give more consideration to the additional attitudinal characteristics of experts than the two hybrid operators (IIFHA and IIFHG), the 3rd alternative happens to meet the experts’ interests and is emphasized to be better than 1st, 2nd, and 4th alternatives. Also it is can be seen from Table 10, the optimal alternative is consistent with all the last four operators and the worst alternative is consistent with all the six comparative operators.Generally different aggregation operators for different decision environments may obtain different orderings of candidate solutions. The I-IIFHA and I-IIFHG operators can hybridly aggregate the argument-dependent preference information, argument-independent preference information, and the additional attitudinal characteristics of experts, and can be reduced to different ordinary operators under different decision environments. So the main advantage of the proposed I-IIFHA and I-IIFHG operator with TOPSIS order-inducing variables is that they can include a wide range of particular cases and so can consider a lot of possibilities with a more complete view of the different scenarios when solving multiple-attribute multi-interest group decision-making problems under interval-valued intuitionistic fuzzy decision environment.


## 6. Conclusion

We have presented a decision-making approach based on the proposed I-IIFHA and I-IIFHG operators. I-IIFHA and I-IIFHG operators are extensions of the IIFHA and IIFHG operators by using the TOPSIS order-inducing variables in the reordering process. The prominent main advantage of these operators is that they are able to consider more complex attitudinal characteristics in the decision process than other ordinary operators. We have studied some desirable properties of the I-IIFHA and I-IIFHG operators, such as commutativity, monotonicity, idempotency, and bounded. Theoretical analysis has shown that I-IIFHA and I-IIFHG can include a wide range of aggregation operators as special cases, such as IIFHA, IIFOWA, IIFWA, IIFHG, IIFOWG, and IIFWG operators. Furthermore, we have applied these operators to form an approach for the multiple attribute multi-interest group decision-making problems with interval-valued intuitionistic fuzzy information. In the future, we will computerize the proposed methods to facilitate the usage and continue working on the extension and application of developed operators to different environments and domains.



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