International Journal of Computational Intelligence Systems

<Previous Article In Issue
Download article (PDF)
Next Article In Issue>
Volume 12, Issue 2, 2019, Pages 580 - 596

A Novel Method Based on Fuzzy Tensor Technique for Interval-Valued Intuitionistic Fuzzy Decision-Making with High-Dimension Data

Authors
Shengyue Deng1, 2, Jianzhou Liu1, *, Jintao Tan2, Lixin Zhou1, 3
1Department of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China
2School of Science, Hunan University of Technology, Zhuzhou, Hunan 412008, China
3Faculty of Science, Guilin University of Aerospace Technology, Guilin, Guangxi 541004, China
*Corresponding author: Email: liujz@xtu.edu.cn
Corresponding Author
Shengyue Deng
Received 29 August 2018, Accepted 17 April 2019, Available Online 11 May 2019.
DOI
10.2991/ijcis.d.190424.001How to use a DOI?
Keywords
Fuzzy tensor; Interval-valued intuitionistic fuzzy tensor; Generalized interval-valued intuitionistic fuzzy weighted averaging (GIIFWA) operator; Generalized interval-valued intuitionistic fuzzy weighted geometric (GIIFWG) operator; Multiple attribute group decision-making; Dynamic multiple attribute group decision-making
Abstract

To solve the interval-valued intuitionistic fuzzy decision-making problems with high-dimension data, the fuzzy matrix is extended to the fuzzy tensor in this paper. Based on the constructed tensor definition, we propose the generalized interval-valued intuitionistic fuzzy weighted averaging (GIIFWA) and generalized interval-valued intuitionistic fuzzy weighted geometric (GIIFWG) operators. By exploring the properties of GIIFWA and GIIFWG operators, a new algorithm is presented to solve the interval-valued intuitionistic fuzzy multiple attribute group decision-making problem. Two typical application examples are also provided to demonstrate the efficiency and universal applicability of our proposed method.

Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

As an important branch of decision-making fields, the multiple attribute group decision-making has been paid a close attention in past decades. Normally, multiple attribute group decision-making problems is that multiple decision-makers select the optimal alternatives or ranking them from a set of feasible alternatives by the attribute weights and attribute values, for details refers to Xu and Cai [1]. However in some real applications such as Xu and Cai [1], Liu et al. [2], Wang et al. [3], Qin et al. [4], He [5], and Hashemi et al. [6], due to the undetermined decision-making environment, the multi-attribute group decision-making seems to be useless for decision-making. One alternative dealing with this difficulty is the fuzzy set, which was subsequently extended to intuitionistic fuzzy set by Atanassov [7] for applications in various decision-making areas, and Atanassov and Gargov [8] presented the concept and properties of interval-valued intuitionistic fuzzy set based on intuitionistic fuzzy set in 1989, which enriched intuitionistic fuzzy set theory. Especially in recent researches, multiple attribute group decision-making with incorporated interval-valued intuitionistic fuzzy sets has attracted great attentions and yielded plentiful results. For example, Xu [9] developed a method based on distance measure for group decision-making with interval-valued intuitionistic fuzzy matrices. Kabak and Ervural [10] devised a generic conceptual framework and a classification scheme for multiple attribute group decision-making methods. Yang et al. [11] proposed a new method based on dynamic intuitionistic normal fuzzy aggregation operators and VIKOR method with time sequence preference for the dynamic intuitionistic normal fuzzy multi-attribute decision-making problems. Liu [12] proposed the interval-valued intuitionistic fuzzy power Heronian aggregation operator and interval-valued intuitionistic fuzzy power weight Heronian aggregation operator for the multiple attribute group decision-making. Chen and Huang [13] proposed a new multi-attribute decision-making method by the interval-valued intuitionistic fuzzy weighted geometric average (IIFWGA) operator and the accuracy function of interval-valued intuitionistic fuzzy values. Wang and Chen [14] proposed an improved multiple attribute decision-making method by the score function SWC of interval-valued intuitionistic fuzzy values and the linear programming methodology. Qiu and Li [15] employed the plant growth simulation algorithm (PGSA) to calculate the optimal preferences of the entire expert group and proposed a new method to solve the multi-attribute group decision-making problem.

However the above mentioned models which are based on matrix frame meet with difficulties in processing higher dimension data and might lose their efficiency. To tackle this problem, we introduce a new developed tensor model which is a generalization of matrix. The concepts of higher-order tensor eigenvalues and eigenvectors were introduced by [16] and [17]. Subsequently, the theory and algorithms of some special tensors and the spectra of tensors with their various applications have attracted wide attention [18–31]. For example, Ding and Wei [18,19] investigated the solutions of some structured multi-linear systems whose coefficient tensor is M-tensor. Qi [20] proved two new spectral properties and a maximum property of the largest H-eigenvalue in a symmetric nonnegative tensor system. Ni et al. [21] obtained an upper bound of different US-eigenvalues and the count of US-eigenpairs corresponding to all nonzero eigenvalues in the symmetric tensors. Ng et al. [22] proposed an iterative method to calculate the largest eigenvalue of an irreducible nonnegative tensor. Rajesh Kannan et al. [23] gained some properties of strong H-tensors and (general) H-tensors. Based on the diagonal product dominance and S diagonal product dominance of tensor, Wang et al. [24] established some new implementable attribute which can be used for identifying nonsingular H-tensor. By studying the general product of two n-dimensional tensors A and ℬ with orders m⩾2 and k⩾1, Shao et al. [25,26] found that the product is a generalization of the usual matrix product and it satisfies the associative law. Bu et al. [27] gave some basic properties for the left (right) inverse, rank, and product of tensors. Pumplün [28] studied the tensor product of an associative and a nonassociative cyclic algebra. Giladi et al. [29] studied the volume ratio of the projective tensor products ℓpn⊗πℓqn⊗πℓrn with 1⩽p⩽q⩽r⩽∞ and obtained asymptotic formulas that are sharp in almost all cases. Gutiérrez García et al. [30] employed tensor products of complete lattices into fuzzy set theory. Hilberdink [31] studied operators having (infinite) matrix representations and gave such operators infinite tensor products over the primes. Moreover, we have defined the concept of fuzzy tensor and established the general form of the fuzzy synthetic evaluation model for solving multiple attribute group decision-making problems [32].

Based on the research results we have achieved [32], we will propose two new generalized aggregation operators based on interval-valued intuitionistic fuzzy tensor for solving the interval-valued intuitionistic fuzzy multiple attribute group decision-making problem. Specifically, we will first establish the generalized interval-valued intuitionistic fuzzy weighted averaging (GIIFWA) and generalized interval-valued intuitionistic fuzzy weighted geometric (GIIFWG) operators. Then some properties about those new generalized aggregation operators are developed and a new algorithm is presented for the corresponding decision-making problems. Indeed as shown in numerical experiments, the proposed interval-valued intuitionistic fuzzy tensor model does provide a new way for solving multiple attribute group decision-making problems with high-dimension data.

The whole paper is arranged as follows: In Section 2, we introduce some concepts and properties of the fuzzy tensor and interval-valued intuitionistic fuzzy aggregation. Section 3 is devoted to the derivation of the GIIFWA and GIIFWG operators by the product of tensor and vector, and gives some properties of two new generalized aggregation operators. In Section 4, we present an algorithm for solving the interval-valued intuitionistic fuzzy multiple attribution group decision-making problems. In Section 5, two different application examples are shown for illustrating the proposed approach. A conclusion is finally drawn in Section 6.

2. PRELIMINARIES

This section provides basic preliminaries about the fuzzy tensor, interval-valued intuitionistic fuzzy set, and interval-valued intuitionistic fuzzy information aggregation theory.

Let R be the real field and F and IVIF be the fuzzy set and interval-valued intuitionistic fuzzy set defined in universe R, respectively. The TRm,n, TFm,n, and TIVIFm,n denote the set of all mth-order n-dimension real tensors, fuzzy tensors, and interval-valued intuitionistic fuzzy tensors, respectively, and n=1,2,⋯,n. Fn and IVIFn denote the n-dimensional fuzzy vector in the F and n-dimensional interval-valued intuitionistic fuzzy vector in the IVIF, respectively.

Definition 2.1.

[8] Let X be a finite nonempty set. Then

A˜=〈x,μ˜A˜x,ν˜A˜x〉|x∈X
is called an interval-valued intuitionistic fuzzy set, where μ˜A˜x⊂0,1 and ν˜A˜x⊂0,1, x∈X, with the condition:
sup μ˜A˜x+sup ν˜A˜x⩽1,x∈X

Note: For convenience, the interval-valued intuitionistic fuzzy numbers (IVIFNs) [33] can be denoted as A˜=μA˜lx,μA˜ux,νA˜lx,νA˜ux in this paper, where

μA˜l,μA˜u⊂0,1,νA˜l,νA˜u⊂0,1,μA˜u+νA˜u⩽1.
and μA˜l,μA˜u and νA˜l,νA˜u represent the supported interval and opposed interval about an evaluation object, respectively.

Definition 2.2.

[33] Let α˜=μα˜l,μα˜u,να˜l,να˜u, α˜1=μα˜1l,μα˜1u,να˜1l,να˜1u and α˜2=μα˜2l,μα˜2u,να˜2l,να˜2u be IVIFNs. Then

  1. α˜¯=να˜l,να˜u,μα˜l,μα˜u, where α˜¯ is the complement of α˜.

  2. α˜1∧α˜2=min μα˜1l,μα˜2l,min μα˜1u,μα˜2u,  max να˜1l,να˜2l,max να˜1u,να˜2u;

  3. α˜1∨α˜2=max μα˜1l,μα˜2l,max μα˜1u,μα˜2u,  min να˜1l,να˜2l,minνα˜1u,να˜2u;

  4. α˜1+α˜2=μα˜1l+μα˜2l−μα˜1lμα˜2l,μα˜1u+μα˜2u−μα˜1uμα˜2u,να˜1lνα˜2l,να˜1uνα˜2u;.

  5. α˜1⋅α˜2=μα˜1lμα˜2l,μα˜1uμα˜2u,να˜1l+να˜2l−να˜1lνα˜2l,να˜1u+να˜2u−να˜1uνα˜2u.

  6. λα˜=1−1−μα˜lλ,1−1−μα˜uλ,να˜lλ,να˜uλ,λ>0;

  7. α˜λ=μα˜lλ,μα˜uλ,1−1−να˜lλ,1−1−να˜uλ,λ>0.

Definition 2.3.

[16] Let A∈TRm,n1×n2×⋯×nm, and its elements ai1i2⋯im∈R where i1∈n1,i2∈n2,⋯,im∈nm. Then A is called a mth-order tensor.

Note: According to the Definition 2.3, we know that the matrix is the 2nd-order tensor.

Definition 2.4.

[32] Let A˜∈TFm,n1×n2×⋯×nm, and its elements ai1i2⋯im∈0,1 where i1∈n1,i2∈n2,⋯,im∈nm, then A˜ is called a mth-order fuzzy tensor.

Definition 2.5.

Let A˜IVIF=ai1i2⋯imn1×n2×⋯×nm∈TIVIF m,n1 ×n2×⋯×nm, and its elements ai1i2⋯im= ui1i2⋯iml,μi1i2⋯imu,νi1i2⋯iml,νi1i2⋯imu where μi1i2⋯iml,μi1i2⋯imu⊂0,1, νi1i2⋯iml,νi1i2⋯imu⊂0,1 satisfy the condition

μi1i2⋯imu+νi1i2⋯imu⩽1,
and the interval μi1i2⋯iml,μi1i2⋯imu and νi1i2⋯iml,νi1i2⋯imu denote the supported interval and opposed interval about an evaluation object, respectively. Then A˜IVIF is called a mth-order interval-valued intuitionistic fuzzy tensor.

Definition 2.6.

[1] Let α˜ii=1,2,⋯,n be a collection of IVIFNs, and let IIFWA: FIVIFn→FIVIF. If

IIFWAωα˜1,α˜2,⋯,α˜n=ω1α˜1+ω2α˜2+⋯+ωnα˜n
where ω=ω1,ω2,⋯,ωnT is the weight vector of α˜ii=1,2,⋯,n, with ωi∈0,1 i=1,2,⋯,n, and ∑i=1nωi=1, then the function IIFWA is called an interval-valued intuitionistic fuzzy weighted averaging (IIFWA) operator.

Definition 2.7.

[1] Let IIFWG: FIVIFn→FIVIF. If

IIFWGωα˜1,α˜2,⋯,α˜n=α˜1ω1⋅α˜2ω2⋅⋯⋅α˜nωn
then the function IIFWG is called an interval-valued intuitionistic fuzzy weighted geometric (IIFWG) operator.

Definition 2.8.

[33] Let α˜=μα˜l,μα˜u,να˜l,να˜u be an IVIFN. Then we call

sα˜=12μα˜l−να˜l+μα˜u−να˜u
the score of α˜, where s is the score function of α˜, sα˜∈−1,1.

Definition 2.9.

[33] The accuracy function of an IVIFN α˜ is defined as

hα˜=12μα˜l+μα˜u+να˜l+να˜u
where hα˜∈0,1.

Definition 2.10.

[33] Let α˜1 and α˜2 be any two IVIFNs. Then

  1. If sα˜1<sα˜2, then α˜1<α˜2.

  2. If sα˜1=sα˜2, then

    1. If hα˜1<hα˜2, then α˜1<α˜2.

    2. If hα˜1>hα˜2, then α˜1>α˜2.

    3. If hα˜1=hα˜2, then α˜1∼α˜2.

Definition 2.11.

[16] Suppose that A=ai1i2⋯imn1×n2×⋯×nm∈TRm,n1×n2×⋯×nm is a mth-order tensor, and Xj=x1j,x2j,⋯,xnjT∈Rnj j∈m−1 is a nj-dimension vector, then the imth component of the vector A∘X1∘X2⋯∘Xm−1 in Rnm is defined as the following:

A∘X1∘X2⋯∘Xm−1im =∑i1=1n1⋯∑im−1=1nm−1ai1i2⋯imxi11xi22⋯xim−1m−1.

Definition 2.12.

[34] Let U and V be universes and FV be the set of all fuzzy sets in V (power set).

  • f:U→FV is a mapping

  • f is a fuzzy function iff

    μf(u)(v)=μR~(u,v),∀(u,v)∈U×V,
    where μR˜u,v is the membership function of a fuzzy relation.

Note: The mapping f in Definition 2.12 is also a fuzzy mapping.

Definition 2.13.

Let A˜IVIF=ai1i2⋯imn1×n2×⋯×nm∈ TIVIFm,n1×n2×⋯×nm, and let the function GIIFWA: FIVIFn2×⋯×nm→FIVIFn1. If

GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm =∑i2n2⋯∑imnmai1i2⋯im⋅xi22⋅⋯⋅ximm
where X2=x12,⋯,xi22,⋯,xn22T, ⋯, Xm=x1m,⋯,ximm,⋯,xnmmT are the weight vectors of a:i2:⋯:i2=1,2,⋯,n2,  ⋯, a:⋯:imim=1,2,⋯,nm, respectively, and ∑i2=1n2xi22=1, xi22⩾0;⋯; ∑im=1nmximm=1, ximm⩾0, then the function GIIFWA is called the GIIFWA operator.

Definition 2.14.

Suppose that the function GIIFWG: FIVIFn2×⋯×nm→FIVIFn1. If

GIIFWGA˜IVIF∘X2∘X3∘⋯∘Xm  =∏i2n2⋯∏imnmai1i2⋯imxi22⋅⋯⋅ximm
then the function GIIFWG is called the GIIFWG operator.

3. GENERALIZED INTERVAL-VALUED INTUITIONISTIC FUZZY AGGREGATION OPERATORS BASED ON FUZZY TENSOR TECHNIQUE

Since the interval-valued intuitionistic fuzzy information aggregation is helpful for dealing with fuzzy multiple attribute decision-making problem, we will first develop, in this section, the GIIFWA and GIIFWG operators by the product of the mth-order fuzzy tensor with vector. Then both the GIIFWA and the GIIFWG operators are proved to having properties of idempotency and boundedness, which lays a theoretical foundation for the algorithm to solve the fuzzy multiple attribute group decision-making problems in next section.

Theorem 3.1.

Let A˜IVIF=ai1i2⋯imn1×n2×⋯×nm ∈TIVIFm,n1×n2×⋯×nm be a mth-order interval-valued intuitionistic fuzzy tensor, and its elements ai1i2⋯im=μi1i2⋯iml,μi1i2⋯imu,νi1i2⋯iml,νi1i2⋯imu. Then the aggregated value by using Equation (1) is

GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm  =1−∏i2=1n2⋯∏im=1nm1−μi1i2⋯imlxi22⋯ximm,   1−∏i2=1n2⋯∏im=1nm1−μi1i2⋯imuxi22⋯ximm,   ∏i2=1n2⋯∏im=1nmνi1i2⋯imlxi22⋯ximm,   ∏i2=1n2⋯∏im=1nmνi1i2⋯imuxi22⋯ximm 
where X2=x12,⋯,xi22,⋯,xn22T,⋯, Xm=x1m,⋯,ximm,⋯,xnmmT are the weight vectors of a:i2:⋯:i2=1,2,⋯,n2,⋯, a:⋯:imim=1,2,⋯,nm, respectively, and ∑i2=1n2xi22=1, xi22⩾0;⋯ ; ∑im=1nmximm=1, ximm⩾0.

Proof.

We prove the Theorem 3.1 by using mathematical induction on n2,⋯,nm.

  1. When n2=⋯=nm=1, we have

    GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm=ai11⋯1x12⋯x1m=1−1−μi11⋯1lx12⋯x1m,1−1−μi11⋯1ux12⋯x1m, νi11⋯1lx12⋯x1m,νi11⋯1ux12⋯x1m.

  2. Let I1=2,3,⋯,m and I2=n2,n3,⋯,nm be indicator sets. When at least one element in the indicator set I2 add to “1,” then we consider the following cases:

    1. When j∈I1 and nj=2, then we have

      GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm  =∑ij=12ai11⋯ij⋯1x12⋯xijj⋯x1m  =1−∏ij=121−μi11⋯ij⋯1lx12⋯xijj⋯x1m,  1−∏ij=121−μi11⋯ij⋯1ux12⋯xijj⋯x1m,  ∏ij=12νi11⋯ij⋯1lx12⋯xijj⋯x1m,∏ij=12νi11⋯ij⋯1ux12⋯xijj⋯x1m.

    2. When j1,j2∈I1 j1≠j2 and nj1=nj2=2, then we have

      GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm  =∑ij1=12∑ij2=12ai11⋯ij1⋯ij2⋯1x12⋯xij1j1⋯xij2j2⋯x1m  =1−∏ij1=12∏ij2=121−μi11⋯ij1⋯ij2⋯1lx12⋯xij1j1⋯xij2j2⋯x1m,   1−∏ij1=12∏ij2=121−μi11⋯ij1⋯ij2⋯1ux12⋯xij1j1⋯xij2j2⋯x1m,   ∏ij1=12∏ij2=12νi11⋯ij1⋯ij2⋯1lx12⋯xij1j1⋯xij2j2⋯x1m,   ∏ij1=12∏ij2=12νi11⋯ij1⋯ij2⋯1ux12⋯xij1j1⋯xij2j2⋯x1m.  ⋯,

    3. When j1,j2,⋯,jl∈I1 j1≠j2≠⋯≠jl and nj1=nj2=⋯=njl=2, then we have

      GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm =∑ij1=12⋯∑ijl=12ai11⋯ij1⋯ijl⋯1x12⋯xij1j1⋯xijljl⋯x1m =1−∏ij1=12⋯∏ijl=121−μi11⋯ij1⋯ijl⋯1lx12⋯xij1j1⋯xijljl⋯x1m,  1−∏ij1=12⋯∏ijl=121−μi11⋯ij1⋯ijl⋯1ux12⋯xij1j1⋯xijljl⋯x1m,  ∏ij1=12⋯∏ijl=12νi11⋯ij1⋯ijl⋯1lx12⋯xij1j1⋯xijljl⋯x1m,  ∏ij1=12⋯∏ijl=12νi11⋯ij1⋯ijl⋯1ux12⋯xij1j1⋯xijljl⋯x1m.  ⋯,

    4. When all the elements in the indicator I2 add to “1,” that is, n2=⋯=nm=2, then we have

      GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm
       =∑i2=12⋯∑im=12ai1i2⋯imxi22⋯ximm =1−∏i2=12⋯∏im=121−μi1i2⋯imlxi22⋯ximm,  1−∏i2=12⋯∏im=121−μi1i2⋯imuxi22⋯ximm,  ∏i2=12⋯∏im=12νi1i2⋯imlxi22⋯ximm,  ∏i2=12⋯∏im=12νi1i2⋯imuxi22⋯ximm.

      Therefore, according to the above analysis, when at least one element in the indicator set I2 add to “1,” the Theorem 3.1 holds.

  3. Suppose that n2=K2,n3=K3,⋯,nm=Km, the Theorem 3.1 holds, that is,

    GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm =∑i2=1K2⋯∑im=1Kmai1i2⋯imxi22⋯ximm =1−∏i2=1K2⋯∏im=1Km1−μi1i2⋯imlxi22⋯ximm,  1−∏i2=1K2⋯∏im=1Km1−μi1i2⋯imuxi22⋯ximm,  ∏i2=1K2⋯∏im=1Kmνi1i2⋯imlxi22⋯ximm,  ∏i2=1K2⋯∏im=1Kmνi1i2⋯imuxi22⋯ximm.

    Let I3=K2,K3,⋯,Km be an indicator set. When at least one element in the indicator set I3 add to “1,” then we consider the following cases:

    1. When j∈I1 and nj=Kj+1, then we have

      GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm =∑i2=1K2⋯∑ij=1Kj+1⋯∑im=1Kmai1i2⋯ij⋯imx12⋯xijj⋯x1m =1−∏i2=1K2⋯∏ij=1Kj+1⋯∏im=1Km1−μi1i2⋯ij⋯imlxi22⋯xijj⋯ximm,  1−∏i2=1K2⋯∏ij=1Kj+1⋯∏im=1Km1−μi1i2⋯ij⋯imuxi22⋯xijj⋯ximm,  ∏i2=1K2⋯∏ij=1Kj+1⋯∏im=1Kmνi1i2⋯ij⋯imlxi22⋯xijj⋯ximm,  ∏i2=1K2⋯∏ij=1Kj+1⋯∏im=1Kmνi1i2⋯ij⋯imuxi22⋯xijj⋯ximm.

    2. When j1,j2∈I1 j1≠j2 and nj1=Kj1+1, nj2=Kj2+1, then we have

      GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm =∑i2=1K2⋯∑ij1=1Kj1+1⋯∑ij2=1Kj2+1⋯∑im=1Kmai1i2⋯ij1⋯ij2⋯imxi22  ⋯xij1j1⋯xij2j2⋯ximmGIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm =∑i2=1K2⋯∑ij1=1Kj1+1⋯∑ij2=1Kj2+1⋯∑im=1Kmai1i2⋯ij1⋯ij2⋯imxi22   ⋯xij1j1⋯xij2j2⋯ximm =1−∏i2=1K2⋯∏ij1=1Kj1+1⋯∏ij2=1Kj2+1⋯  ∏im=1Km1−μi1i2⋯ij1⋯ij2⋯imlxi22⋯xij1j1⋯xij2j2⋯ximm,  1−∏i2=1K2⋯∏ij1=1Kj1+1⋯∏ij2=1Kj2+1⋯  ∏im=1Km1−μi1i2⋯ij1⋯ij2⋯imuxi22⋯xij1j1⋯xij2j2⋯ximm,  ∏i2=1K2⋯∏ij1=1Kj1+1⋯∏ij2=1Kj2+1⋯  ∏im=1Kmνi1i2⋯ij1⋯ij2⋯imlxi22⋯xij1j1⋯xij2j2⋯ximm,  ∏i2=1K2⋯∏ij1=1Kj1+1⋯∏ij2=1Kj2+1⋯  ∏im=1Kmνi1i2⋯ij1⋯ij2⋯imuxi22⋯xij1j1⋯xij2j2⋯ximm.  ⋯,

    3. When j1,j2,⋯,jl∈I1 j1≠j2≠⋯≠jl and nj1=Kj1+1, nj2=Kj2+1, ⋯, njl=Kjl+1, then we have

      GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm =∑i2=1K2⋯∑ij1=1Kj1+1⋯∑ijl=1Kjl+1⋯∑im=1Kmai1i2⋯ij1⋯ij2⋯imxi22  ⋯xij1j1⋯xijljl⋯ximm =1−∏i2=1K2⋯∏ij1=1Kj1+1⋯∏ijl=1Kjl+1⋯  ∏im=1Km1−μi1i2⋯ij1⋯ijl⋯imlxi22⋯xij1j1⋯xijljl⋯ximm,  1−∏i2=1K2⋯∏ij1=1Kj1+1⋯∏ijl=1Kjl+1⋯  ∏im=1Km1−μi1i2⋯ij1⋯ijl⋯imuxi22⋯xij1j1⋯xijljl⋯ximm,  [∏i2=1K2⋯∏ij1=1Kj1+1⋯∏ij1=1Kj1+1⋯∏im=1Km  vi1i2⋯ij1⋯ijl⋯ijmlxi22⋯xj1j1⋯xjljl⋯xinm,  ∏i2=1K2⋯∏ij1=1Kj1+1⋯∏ij1=1Kj1+1⋯∏im=1Km  vi1i2⋯ij1⋯ijl⋯ijmuxi22⋯xj1j1⋯xjljl⋯ximm  ⋯,

    4. When all the elements in the indicator I3 add to “1,” that is, n2=K2+1, n3=K3+1, ⋯, nm=Km+1, then we have

      GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm =∑i2=1K2+1⋯∑im=1Km+1ai1i2⋯imxi22⋯ximm =1−∏i2=1K2+1⋯∏im=1Km+11−μi1i2⋯imlxi22⋯ximm,  1−∏i2=1K2+1⋯∏im=1Km+11−μi1i2⋯imuxi22⋯ximm,  ∏i2=1K2+1⋯∏im=1Km+1νi1i2⋯imlxi22⋯ximm,  ∏i2=1K2+1⋯∏im=1Km+1νi1i2⋯imuxi22⋯ximm.

      Therefore, for any n2, n3, ⋯, nm, the Theorem 3.1 holds from (1), (2), and (3). This completes the proof of Theorem 3.1.

Corollary 3.1.

[1] Let A˜IVIF∈TIVIF(2,n×m) be an interval-valued intuitionistic fuzzy matrix, and A˜IVIF=(aij)n×m where aij=μijl,μiju,νijl,νiju, then their aggregated value by using the GIIFWA operator is also an IVIFN and

GIIFWA(A˜IVIF∘X) =1−∏j=1m1−μijlxj,1−∏j=1m1−μijuxj,  ∏j=1mνijlxj,∏j=1mνijuxj
where X=(x1,⋯,xj,⋯,xm)T is the weight vector of a:j(j=1,2,⋯,m), with xj∈[0,1] and ∑j=1mxj=1.

Remark 3.1.

Clearly, the Theorem 3.1 is the extension of Corollary 3.1 which is the Theorem 2.3.1 in Xu [1].

Theorem 3.2.

Let A˜IVIF=ai1i2⋯imn1×n2×⋯×nm ∈TIVIFm,n1×n2×⋯×nm where its elements ai1i2⋯im=μi1i2⋯iml,μi1i2⋯imu],[νi1i2⋯iml,νi1i2⋯imu. Then the aggregated value by using Equation (2) is

GIIFWGA˜IVIF∘X2∘X3∘⋯∘Xm  =∏i2=1n2⋯∏im=1nmμi1i2⋯imlxi22⋯ximm,  ∏i2=1n2⋯∏im=1nmμi1i2⋯imuxi22⋯ximm,  1−∏i2=1n2⋯∏im=1nm1−νi1i2⋯imlxi22⋯ximm,  1−∏i2=1n2⋯∏im=1nm1−νi1i2⋯imuxi22⋯ximm

Proof.

The proof of the Theorem 3.2 is similar to the proof of Theorem 3.1.

Corollary 3.2.

[1] Suppose that A˜IVIF∈TIVIF(2, n×m) is an interval-valued intuitionistic fuzzy matrix, and A˜IVIF=aijn×m where aij=μijl,uiju,νijl,νiju, then their aggregated value by using the GIIFWG operator is also an IVIFN, and

GIIFWG(A˜IVIF∘X)=[∏j=1m(μijl)xj,∏j=1m(μiju)xj],[1−∏j=1m(1−vijl)xj,1−∏j=1m(1−viju)xj]
where X=x1,⋯,xj,⋯,xmT is the exponential weight vector of a:jj=1,2,⋯,m, with xj∈0,1 and ∑j=1mxj=1.

Remark 3.2.

The Theorem 3.2 is the general form of Corollary 3.2 which is the Theorem 2.3.2 in Xu [1].

Theorem 3.3.

The operational results in Theorems 3.1 and 3.2 are n1-dimension IVIF vectors.

Proof.

By the Theorems 3.1 and 3.2, we have

GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm =1−∏i2=1n2⋯∏im=1nm1−μi1i2⋯imlxi22⋯ximm,  1−∏i2=1n2⋯∏im=1nm1−μi1i2⋯imuxi22⋯ximm,  ∏i2=1n2⋯∏im=1nmνi1i2⋯imlxi22⋯ximm,  ∏i2=1n2⋯∏im=1nmνi1i2⋯imuxi22⋯ximm
and
GIIFWGA˜IVIF∘X2∘X3∘⋯∘Xm =∏i2=1n2⋯∏im=1nmμi1i2⋯imlxi22⋯ximm,  ∏i2=1n2⋯∏im=1nmμi1i2⋯imuxi22⋯ximm,  1−∏i2=1n2⋯∏im=1nm1−νi1i2⋯imlxi22⋯ximm,  1−∏i2=1n2⋯∏im=1nm1−νi1i2⋯imuxi22⋯ximm
and i1∈n1, then both GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm and GIIFWGA˜IVIF∘X2∘X3∘⋯∘Xm∈IVIFn1.

Therefore, the operational results in the Theorems 3.1 and 3.2 are n1-dimension IVIF vectors.

Theorem 3.4.

Let A˜IVIF=ai1i2⋯imn1×n2×⋯×nm ∈TIVIFm,n1×n2×⋯×nm be a mth-order interval-valued intuitionistic fuzzy tensor, and X2=x12,⋯,xi22,⋯,xn22T, ⋯, Xm=x1m,⋯,ximm,⋯,xnmmT are the weight vectors of a:i2:⋯:i2=1,2,⋯,n2, ⋯, a:⋯:im im=1,2,⋯,nm, respectively, that is, ∑i2=1n2xi22=1, xi22⩾0; ⋯; ∑im=1nmximm=1, ximm⩾0. Then GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm and GIIFWGA˜IVIF∘X2∘X3∘⋯∘Xm are fuzzy mappings.

Proof.

A˜IVIF∈TIVIFm,n1×n2×⋯×nm is a mth-order interval-valued intuitionistic fuzzy tensor.

According to the Definition 2.5, we have

A˜IVIF=μi1i2⋯iml,μi1i2⋯imu,νi1i2⋯iml,νi1i2⋯imun1×n2×⋯×nm
for arbitrary μi1i2⋯iml,μi1i2⋯imu⊂0,1, νi1i2⋯iml,νi1i2⋯imu⊂0,1 and μi1i2⋯imu+νi1i2⋯imu⩽1.

Owing to X2=x12,⋯,xi22,⋯,xn22T, ⋯, Xm=x1m,⋯,ximm,⋯,xnmmT are the weight vectors of a:i2:⋯: i2=1,2,⋯,n2,⋯, a:⋯:imim=1,2,⋯,nm, respectively, that is, ∀xi22∈0,1,⋯, ∀ximm∈0,1. Then we obtain X2∈0,1n2,⋯,Xm∈0,1nm.

On the basis of the Theorem 3.3, we get GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm∈IVIFn1 and GIIFWGA˜IVIF∘X2∘X3∘⋯∘Xm∈IVIFn1.

Thus GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm and GIIFWGA˜IVIF∘X2∘X3∘⋯∘Xm are fuzzy mappings from 0,1n2×n3×⋯×nm to IVIFn1 by the Definition 2.12.

Theorem 3.5.

Let A˜IVIF=(ai1i2⋯im)n1×n2×⋯×nm∈TIVIFm,n1×n2×⋯×nm be a mth-order interval-valued intuitionistic fuzzy tensor, where ai1i2⋯im=μi1i2⋯iml,μi1i2⋯imu,νi1i2⋯iml,νi1i2⋯imu. And X2=x12,⋯,xi22,⋯,xn22T,⋯, Xm=(x1m,⋯,ximm,⋯,xnmm)T are the weight vectors of a:i2:⋯:i2=1,2,⋯,n2,⋯, a:⋯:imim=1,2,⋯,nm, respectively, and ∑i2=1n2xi22=1, xi22⩾0;⋯; ∑im=1nmximm=1, ximm⩾0. Then we have the following properties of GIIFWA operator:

  1. (Idempotency). If all the elements of A˜IVIF are equal, that is, ai1i2⋯im=α, i1∈n1,i2∈n2,⋯,im∈nm, then

    GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm                   =α,α,⋯,αT∈IVIFn1

  2. (Boundedness). Let

    α−=mini1,i2,⋯,imμi1i2⋯iml,mini1,i2,⋯,imμi1i2⋯imu,               maxi1,i2,⋯,imνi1i2⋯iml,maxi1,i2,⋯,imνi1i2⋯imu,α+=maxi1,i2,⋯,imμi1i2⋯iml,maxi1,i2,⋯,imμi1i2⋯imu,               mini1,i2,⋯,imνi1i2⋯iml,mini1,i2,⋯,imνi1i2⋯imu
    and (α−,⋯,α−)T,(α+,⋯,α+)T∈IVIFn1. Then, for any X2,X3,⋯,Xm, we have
    α−,⋯,α−T⩽GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm⩽α+,⋯,α+T.

Proof.

  1. Let α=μl,μu,νl,νu. By the Theorem 3.1 and ai1i2⋯im=αi1∈n1,i2∈n2,⋯,im∈nm, we have the i1th component of GIIFWA operator and

    GIIFWAA˜IF∘X2∘X3∘⋯∘Xmi1 =1−∏i2=1n2⋯∏im=1nm1−μi1i2⋯imlxi22⋯ximm,  1−∏i2=1n2⋯∏im=1nm1−μi1i2⋯imuxi22⋯ximm,  ∏i2=1n2⋯∏im=1nmvi1i2⋯imlxi22⋯ximm  ∏i2=1n2⋯∏im=1nmvi1i2⋯imuxi22⋯ximmi1 =1−∏i2=1n2⋯∏im=1nm1−μlxi22⋯ximm,  1−∏i2=1n2⋯∏im=1nm1−μuxi22⋯ximm,  ∏i2=1n2⋯∏im=1nmvlxi22⋯ximm,  ∏i2=1n2⋯∏im=1nmvuxi22⋯ximmi1  =1−1−μl∑i2=1n2⋯∑im=1nmxi22⋯ximm,  1−1−μu∑i2=1n2⋯∑im=1nmxi22⋯ximm,  νl∑i2=1n2⋯∑im=1nmxi22⋯ximm,νu∑i2=1n2⋯∑im=1nmxi22⋯ximmi1  =1−1−μl,1−(1−μu),νl,νui1  =μl,μu,νl,νui1  =α
    and i1∈n1, then we get
    GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xm=α,α,⋯,αT∈IVIFn1.

  2. Since for any i1,i2,⋯,im, we have mini1,i2,⋯,imμi1i2⋯iml⩽μi1i2⋯iml⩽maxi1,i2,⋯,imμi1i2⋯iml, mini1,i2,⋯,imμi1i2⋯imu⩽μi1i2⋯imu⩽maxi1,i2,⋯,imμi1i2⋯imu, mini1,i2,⋯,imνi1i2⋯iml⩽νi1i2⋯iml⩽maxi1,i2,⋯,imνi1i2⋯iml and mini1,i2,⋯,imνi1i2⋯imu⩽νi1i2⋯imu⩽maxi1,i2,⋯,imνi1i2⋯imu.

    Then

    1−∏i2=1n2⋯∏im=1nm1−μi1i2⋯imlxi22⋯ximm ⩾1−∏i2=1n2⋯∏im=1nm1−mini1i2⋯imμi1i2⋯imlxi22⋯ximm =1−1−mini1i2⋯imμi1i2⋯iml∑i2=1n2⋯∑im=1nmxi22⋯ximm =mini1i2⋯imμi1i2⋯iml,
    1−∏i2=1n2⋯∏im=1nm1−μi1i2⋯imuxi22⋯ximm ⩾1−∏i2=1n2⋯∏im=1nm1−mini1i2⋯imμi1i2⋯imuxi22⋯ximm =1−1−mini1i2⋯imμi1i2⋯imu∑i2=1n2⋯∑im=1nmxi22⋯ximm =mini1i2⋯imμi1i2⋯imu,
    ∏i2=1n2⋯∏im=1nmνi1i2⋯imlxi22⋯ximm ⩾∏i2=1n2⋯∏im=1nmmini1i2⋯imνi1i2⋯imlxi22⋯ximm =mini1i2⋯imνi1i2⋯iml∑i2=1n2⋯∑im=1nmxi22⋯ximm =mini1i2⋯imνi1i2⋯iml
    and
    ∏i2=1n2⋯∏im=1nmνi1i2⋯imuxi22⋯ximm ⩾∏i2=1n2⋯∏im=1nmmini1i2⋯imνi1i2⋯imuxi22⋯ximm =mini1i2⋯imνi1i2⋯imu∑i2=1n2⋯∑im=1nmxi22⋯ximm =mini1i2⋯imνi1i2⋯imu.

    Similarly, we get

    1−∏i2=1n2⋯∏im=1nm1−μi1i2⋯imlxi22⋯ximm ⩽1−∏i2=1n2⋯∏im=1nm1−maxi1i2⋯imμi1i2⋯imlxi22⋯ximm =1−1−maxi1i2⋯imμi1i2⋯iml∑i2=1n2⋯∑im=1nmxi22⋯ximm =maxi1i2⋯imμi1i2⋯iml,
    1−∏i2=1n2⋯∏im=1nm1−μi1i2⋯imuxi22⋯ximm ⩽1−∏i2=1n2⋯∏im=1nm1−maxi1i2⋯imμi1i2⋯imuxi22⋯ximm =1−1−maxi1i2⋯imμi1i2⋯imu∑i2=1n2⋯∑im=1nmxi22⋯ximm =maxi1i2⋯imμi1i2⋯imu,
    ∏i2=1n2⋯∏im=1nmνi1i2⋯imlxi22⋯ximm ⩽∏i2=1n2⋯∏im=1nmmaxi1i2⋯imνi1i2⋯imlxi22⋯ximm =maxi1i2⋯imνi1i2⋯iml∑i2=1n2⋯∑im=1nmxi22⋯ximm =maxi1i2⋯imνi1i2⋯iml
    and
    ∏i2=1n2⋯∏im=1nmνi1i2⋯imuxi22⋯ximm ⩽∏i2=1n2⋯∏im=1nmmini1i2⋯imνi1i2⋯imuxi22⋯ximm =maxi1i2⋯imνi1i2⋯imu∑i2=1n2⋯∑im=1nmxi22⋯ximm =maxi1i2⋯imνi1i2⋯imu.

    Without loss of generality, for ∀i1∈n1, let

    GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xmi1=α
    where α=μl,μu,νl,νu. By the Definitions 2.8 and 2.10, we get
    sα=12μl−νl+μu−νu  ⩽12maxi1i2⋯imμi1i2⋯iml−mini1i2⋯imνi1i2⋯iml   +maxi1i2⋯imμi1i2⋯imu−mini1i2⋯imνi1i2⋯imu  =sα+
    and
    s(α)=12(μl−νl+μu−νu)⩾12mini1i2⋯im{μi1i2⋯iml}−maxi1i2⋯im{νi1i2⋯iml}+mini1i2⋯im{μi1i2⋯imu}−maxi1i2⋯im{νi1i2⋯imu}=s(α−).

Next, we will consider the following three cases:

  1. When sα<sα+ and sα>sα−, the conclusion (2) in Theorem 3.5 holds.

  2. When sα=sα+, we have α=α+, that is, μl=maxi1,i2,⋯,imμi1i2⋯iml, μu=maxi1,i2,⋯,imμi1i2⋯imu, νl=min i1,i2,⋯,imνi1i2⋯iml, and νu=mini1,i2,⋯,imνi1i2⋯imu.

    Hence, by the Definition 2.9, we get

    hα=12μl+μu+νl+νu  =12maxi1i2⋯imμi1i2⋯iml+maxi1i2⋯imμi1i2⋯imu   +mini1i2⋯imνi1i2⋯iml+mini1i2⋯imνi1i2⋯imu  =hα+.

    In this case, according to the Theorem 3.1 and Definition 2.10, we obtain GIIFWAA˜IF∘X2∘X3∘⋯∘Xmi1=α+. Due to the arbitrariness of i1, we get

    GIIFWAA˜IF∘X2∘X3∘⋯∘Xm=α+,⋯,α+∈IVIFn1.

  3. When sα=sα−, we have α=α−, that is, μl=mini1,i2,⋯,imμi1i2⋯iml, μu=mini1,i2,⋯,imμi1i2⋯imu, νl=maxi1,i2,⋯,imνi1i2⋯iml, and νu=maxi1,i2,⋯,imνi1i2⋯imu.

    Thus

    hα=12μl+μu+νl+νu  =12mini1i2⋯imμi1i2⋯iml+mini1i2⋯imμi1i2⋯imu   +maxi1i2⋯imνi1i2⋯iml+maxi1i2⋯imνi1i2⋯imu  =hα−.

In this case, on the basis of the Theorem 3.1 and Definition 2.10, we have GIIFWA(A˜IF∘X2∘X3∘⋯∘Xm)i1=α− for arbitrary i1∈[n1]. Then

GIIFWA(A˜IVIF∘X2∘X3∘⋯∘Xm)=(α−,⋯,α−)∈IVIFn1.

Therefore, based on cases (i), (ii), and (iii), we can see that the conclusion (2) in Theorem 3.5 holds.

This completes the proof of Theorem 3.5.

Theorem 3.6.

Let A˜IVIF=ai1i2⋯imn1×n2×⋯×nm ∈TIVIFm,n1×n2×⋯×nm be a mth-order interval-valued intuitionistic fuzzy tensor, where ai1i2⋯im=μi1i2⋯iml,μi1i2⋯imu,νi1i2⋯iml,νi1i2⋯imu. And X2=x12,⋯,xi22,⋯,xn22T, Xm=x1m,⋯,ximm,⋯,xnmmT,⋯ are the exponential weight vectors of a:i2:⋯:i2=1,2,⋯,n2,⋯, a:⋯:imim=1,2,⋯,nm, respectively, and ∑i2=1n2xi22=1, xi22⩾0; ⋯;∑im=1nmximm=1, ximm⩾0. Then we have the following properties of GIIFWG operator:

  1. (Idempotency). If all the elements of A˜IVIF∈TIVIFm,n1×n2×⋯×nm are equal, that is, ai1i2⋯im=α, i1∈n1,i2∈n2,⋯,im∈nm, then

    GIIFWGA˜IVIF∘X2∘X3∘⋯∘Xm=α,α,⋯,αT∈IVIFn1

  2. (Boundedness). For any X2,X3,⋯,Xm, we have α−,⋯,α−T⩽GIIFWGA˜IVIF∘X2∘X3∘⋯∘Xm⩽α+,⋯,α+T, where

    α−=mini1i2⋯imμi1i2⋯iml,mini1i2⋯imμi1i2⋯imu,    maxi1i2⋯imνi1i2⋯iml,maxi1i2⋯imνi1i2⋯imu,α+=maxi1i2⋯imμi1i2⋯iml,maxi1i2⋯imμi1i2⋯imu,    mini1i2⋯imνi1i2⋯iml,mini1i2⋯imνi1i2⋯imu

    and(α−,⋯,α−)T,(α+,⋯,α+)T∈IVIFn1.

Proof.

The proof of the Theorem 3.6 is similar to the proof of Theorem 3.5.

4. ALGORITHM

In this section, we will employ the generalized GIIFWA and GIIFWG operators to devise a new approach for solving the multiple attribute group decision-making problems with high-dimension data. The concrete steps of the algorithm are listed as follows:

Step 1. The interval-valued intuitionistic fuzzy decision matrices are transformed into interval-valued intuitionistic fuzzy tensor A˜IVIF;

Step 2. According to the Theorems 3.1 or 3.2, we utilize the GIIFWA operator:

c˜˙i1=GIIFWAA˜IVIF∘X2∘X3∘⋯∘Xmi1
or the GIIFWG operator:
c˜¨i1=GIIFWGA˜IVIF∘X2∘X3∘⋯∘Xmi1
to aggregate all the elements ai1i2⋯imi1∈n1,i2∈n2, ⋯,im∈nm of the interval-valued intuitionistic fuzzy tensor A˜IVIF and get the values c˜˙i1 (or c˜¨i1) corresponding to the alternatives Ai1i1∈n1;

Step 3. Calculate the sores sc˜˙i1 or sc˜¨i1 and the accuracy degrees hc˜˙i1 or hc˜¨i1 i1∈n1 by the Definitions 2.8 and 2.9.

Step 4. Rank the alternatives Ai1i1∈n1 by the Definition 2.10, and then obtain the best desirable alternative.

5. APPLICATION EXAMPLES AND DISCUSSION

5.1. Interval-Valued Intuitionistic Fuzzy Multiple Attribute Group Decision-Making

In this subsection, we apply the GIIFWA and GIIFWG operators to solving the interval-valued intuitionistic fuzzy multiple attribute group decision-making problem with the numerical example used in Qiu [15].

5.1.1. Numerical example

In this example, let us assume that someone intends to buy a car and consults a set of experts. The car supplier xi1i1=1,2,⋯,5 are evaluated by four decision-makers ei2i2=1,2,3,4, and each decision-maker evaluates the alternatives based on five different characteristics ci3i3=1,2,⋯,5. The interval-valued intuitionistic fuzzy decision matrix proposed by ei2i2=1,2,3,4 are listed in the Tables 1–4, and the weighted vector of the four experts is X2=0.3,0.2,0.3,0.2T, and the weighted vector of the five characteristics is X3=0.2,0.15,0.2,0.3,0.15T. Due to space limitations, the original interval-valued intuitionistic fuzzy decision matrices are omitted in this paper. For a detailed description, please see Qiu [15].

We now implement our algorithm to solve this problem.

C1 C2 C3 C4 C5
A1 ([0.3,0.4], [0.4,0.6]) ([0.5,0.6], [0.1,0.2]) ([0.6,0.7], [0.2,0.3]) ([0.7,0.8], [0.0,0.1]) ([0.6,0.7], [0.2,0.3])
A2 ([0.6,0.8], [0.1,0.2]) ([0.6,0.7], [0.2,0.3]) ([0.2,0.3], [0.4,0.6]) ([0.5,0.6], [0.1,0.3]) ([0.7,0.8], [0.0,0.2])
A3 ([0.5,0.8], [0.1,0.2]) ([0.7,0.8], [0.0,0.1]) ([0.5,0.5], [0.4,0.5]) ([0.2,0.3], [0.2,0.4]) ([0.4,0.6], [0.2,0.3])
A4 ([0.2,0.3], [0.4,0.5]) ([0.5,0.7], [0.1,0.3]) ([0.6,0.7], [0.1,0.2]) ([0.4,0.5], [0.1,0.3]) ([0.6,0.9], [0.0,0.1])
A5 ([0.6,0.8], [0.1,0.2]) ([0.3,0.5], [0.4,0.5]) ([0.4,0.6], [0.3,0.4]) ([0.6,0.8], [0.1,0.2]) ([0.5,0.6], [0.2,0.3])
Table 1

Interval-valued intuitionistic fuzzy decision matrix proposed by e1.

C1 C2 C3 C4 C5
A1 ([0.4,0.5], [0.3,0.4]) ([0.5,0.6], [0.1,0.2]) ([0.6,0.7], [0.2,0.3]) ([0.7,0.8], [0.1,0.2]) ([0.7,0.8], [0.0,0.2])
A2 ([0.6,0.8], [0.1,0.2]) ([0.5,0.6], [0.3,0.4]) ([0.4,0.5], [0.3,0.4]) ([0.4,0.6], [0.3,0.4]) ([0.4,0.7], [0.1,0.3])
A3 ([0.5,0.6], [0.3,0.4]) ([0.5,0.7], [0.1,0.2]) ([0.5,0.6], [0.3,0.4]) ([0.3,0.4], [0.2,0.5]) ([0.6,0.7], [0.2,0.3])
A4 ([0.5,0.6], [0.3,0.4]) ([0.7,0.8], [0.0,0.1]) ([0.4,0.5], [0.2,0.4]) ([0.5,0.7], [0.1,0.2]) ([0.5,0.7], [0.2,0.3])
A5 ([0.4,0.7], [0.2,0.3]) ([0.5,0.6], [0.2,0.4]) ([0.3,0.6], [0.3,0.4]) ([0.6,0.8], [0.1,0.2]) ([0.4,0.5], [0.2,0.3])
Table 2

Interval-valued intuitionistic fuzzy decision matrix proposed by e2.

C1 C2 C3 C4 C5
A1 ([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])
A2 ([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])
A3 ([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])
A4 ([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])
A5 ([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 3

Interval-valued intuitionistic fuzzy decision matrix proposed by e3.

C1 C2 C3 C4 C5
A1 ([0.3,0.4], [0.4,0.5]) ([0.8,0.9], [0.1,0.1]) ([0.7,0.8], [0.1,0.2]) ([0.4,0.5], [0.3,0.5]) ([0.2,0.4], [0.3,0.6])
A2 ([0.5,0.7], [0.1,0.3]) ([0.4,0.7], [0.2,0.3]) ([0.4,0.5], [0.2,0.2]) ([0.6,0.8], [0.1,0.2]) ([0.2,0.3], [0.0,0.1])
A3 ([0.2,0.4], [0.1,0.2]) ([0.4,0.5], [0.2,0.4]) ([0.5,0.8], [0.0,0.1]) ([0.4,0.6], [0.2,0.3]) ([0.5,0.6], [0.2,0.3])
A4 ([0.7,0.8], [0.0,0.2]) ([0.5,0.7], [0.1,0.2]) ([0.6,0.7], [0.1,0.3]) ([0.4,0.5], [0.1,0.2]) ([0.7,0.8], [0.1,0.2])
A5 ([0.5,0.6], [0.2,0.4]) ([0.5,0.8], [0.0,0.2]) ([0.4,0.7], [0.2,0.3]) ([0.3,0.6], [0.2,0.3]) ([0.7,0.8], [0.0,0.1])
Table 4

Interval-valued intuitionistic fuzzy decision matrix proposed by e4.

Step 1. If the interval-valued intuitionistic fuzzy tensor and the GIIFWA operator are employed for expressing data in Tables 1–4, then A˜IVIF=ai1i2i35×4×5∈TIVIF3,5×4×5, where its elements ai1i2i3=μi1i2i3l,μi1i2i3u,νi1i2i3l,νi1i2i3u, and ai1::i1=5 represent five suppliers, a:i2:i2=4 represent four experts and a::i3 i3=5 represent five different characteristics. The details are as follows:

a111=0.3,0.4,0.4,0.6,a112=0.5,0.6,0.1,0.2,
a113=0.6,0.7,0.2,0.3,a114=0.7,0.8,0.0,0.1,
a115=0.6,0.7,0.2,0.3,a121=0.4,0.5,0.3,0.4,
a122=0.5,0.6,0.1,0.2,a123=0.6,0.7,0.2,0.3,
a124=0.7,0.8,0.1,0.2,a125=0.7,0.8,0.0,0.2,
a131=0.4,0.6,0.3,0.4,a132=0.5,0.7,0.0,0.2,
a133=0.5,0.6,0.2,0.4,a134=0.6,0.8,0.1,0.2,
a135=0.4,0.7,0.2,0.3,a141=0.3,0.4,0.4,0.5,
a142=0.8,0.9,0.1,0.1,a143=0.7,0.8,0.1,0.2,
a144=0.4,0.5,0.3,0.5,a145=0.2,0.4,0.3,0.6,
a211=0.6,0.8,0.1,0.2,a212=0.6,0.7,0.2,0.3,
a213=0.2,0.3,0.4,0.6,a214=0.5,0.6,0.1,0.3,
a215=0.7,0.8,0.0,0.2,a221=0.6,0.8,0.1,0.2,
a222=0.5,0.6,0.3,0.4,a223=0.4,0.5,0.3,0.4,
a224=0.4,0.6,0.3,0.4,a225=0.4,0.7,0.1,0.3,
a231=0.5,0.8,0.1,0.2,a232=0.3,0.5,0.2,0.3,
a233=0.3,0.6,0.2,0.4,a234=0.4,0.5,0.2,0.4,
a235=0.3,0.6,0.2,0.3,a241=0.5,0.7,0.1,0.3,
a242=0.4,0.7,0.2,0.3,a243=0.4,0.5,0.2,0.2,
a244=0.6,0.8,0.1,0.2,a245=0.2,0.3,0.0,0.1,
a311=0.5,0.8,0.1,0.2,a312=0.7,0.8,0.0,0.1,
a313=0.5,0.5,0.4,0.5,a314=0.2,0.3,0.2,0.4,
a315=0.4,0.6,0.2,0.3,a321=0.5,0.6,0.3,0.4,
a322=0.5,0.7,0.1,0.2,a323=0.5,0.6,0.3,0.4,
a324=0.3,0.4,0.2,0.5,a325=0.6,0.7,0.2,0.3,
a331=0.5,0.6,0.0,0.1,a332=0.5,0.8,0.1,0.2,
a333=0.4,0.7,0.2,0.3,a334=0.2,0.4,0.2,0.3,
a335=0.5,0.8,0.0,0.2,a341=0.2,0.4,0.1,0.2,
a342=0.4,0.5,0.2,0.4,a343=0.5,0.8,0.0,0.1,
a344=0.4,0.6,0.2,0.3,a345=0.5,0.6,0.2,0.3,
a411=0.2,0.3,0.4,0.5,a412=0.5,0.7,0.1,0.3,
a413=0.6,0.7,0.1,0.2,a414=0.4,0.5,0.1,0.3,
a415=0.6,0.9,0.0,0.1,a421=0.5,0.6,0.3,0.4,
a422=0.7,0.8,0.0,0.1,a423=0.4,0.5,0.2,0.4,
a424=0.5,0.7,0.1,0.2,a425=0.5,0.7,0.2,0.3,
a431=0.5,0.7,0.1,0.3,a432=0.4,0.6,0.0,0.1,
a433=0.3,0.5,0.2,0.4,a434=0.7,0.9,0.0,0.1,
a435=0.3,0.5,0.2,0.2,a441=0.7,0.8,0.0,0.2,
a442=[0.5,0.7],[0.1,0.2],a443=0.6,0.7,0.1,0.3,
a444=0.4,0.5,0.1,0.2,a445=0.7,0.8,0.1,0.2,
a511=0.6,0.8,0.1,0.2,a512=0.3,0.5,0.4,0.5,
a513=0.4,0.6,0.3,0.4,a514=0.6,0.8,0.1,0.2,
a515=0.5,0.6,0.2,0.3,a521=0.4,0.7,0.2,0.3,
a522=0.5,0.6,0.2,0.4,a523=0.3,0.6,0.3,0.4,
a524=0.6,0.8,0.1,0.2,a525=0.4,0.5,0.2,0.3,
a531=0.7,0.8,0.0,0.1,a532=0.4,0.6,0.0,0.2,
a533=0.4,0.7,0.2,0.3,a534=0.3,0.5,0.1,0.3,
a535=0.6,0.7,0.1,0.2,a541=0.5,0.6,0.2,0.4,
a542=0.5,0.8,0.0,0.2,a543=0.4,0.7,0.2,0.3,
a544=0.3,0.6,0.2,0.3,a545=0.7,0.8,0.0,0.1.

Step 2. By the Theorem 3.1, and A˜IVIF∈TIVIF3,5×4×5, according to the experts weight X2 and the characteristics weight X3 in Qiu [15], we have

GIIFWG(A˜IVIF∘X2∘X3)  =1−∏i2=14∏i3=151−μi1i2i3lxi22xi33,   1−∏i2=14∏i3=151−μi1i2i3uxi22xi33,   ∏i2=14∏i3=15vi1i2i3lxi22xi33,∏i2=14∏i3=15vi1i2i3uxi22xi33  =(0.551,0.651,0.000,0.269,0.460,0.657,   0.000,0.290,0.431,0.570,0.000,0.264,   0.511,0.661,0.000,0.224,0.487,0.645,   0.000,0.255T.
that is, x1=0.551,0.651,0.000,0.267,
x2=0.460,0.657,0.000,0.290,
x3=0.431,0.570,0.000,0.264,
x4=0.511,0.661,0.000,0.224,
x5=0.487,0.645,0.000,0.255.

Step 3. To rank the IVIFNs xi1(i1=[5]), we calculate the scores s(xi1)(i1=[5]) by the Definition 2.8. s(x1)=0.466, s(x2)=0.414, s(x3)=0.369, s(x4)=0.474, s(x5)=0.438.

Step 4. By the scores s(xi1) result, the ranking order of all the alternatives is generated as x4≻x1≻x5≻x2≻x3. Therefore, the best car supplier is x4.

We can also replace the GIIFWA with the GIIFWG to resolve this problem. The difference starts from step 2.

Step 2'. By the Theorem 3.2, we have

GIIFWG(A˜IVIF∘X2∘X3) =∏i2=14∏i3=15μi1i2i3lxi22xi33,∏i2=14∏i3=15μi1i2i3uxi22xi33,  1−∏i2=14∏i3=151−νi1i2i3lxi22xi33,  1−∏i2=14∏i3=151−νi1i2i3uxi22xi33 =0.503,0.641,0.186,0.323,0.421,0.598,  0.178,0.349,0.387,0.560,0.171,0.307,  0.466,0.623,0.125,0.280,0.450,0.659,  0.158,0.282.

Then, we get

x1=0.503,0.641,0.186,0.323,
x2=0.421,0.598,0.178,0.349,
x3=0.387,0.560,0.171,0.307,
x4=0.466,0.623,0.125,0.280,
x5=0.450,0.659,0.158,0.282.

Step 3'. In order to rank the IVIFNs xi1(i1=[5]), we calculate the scores s(xi1)(i1=1,2,⋯,5) by the Definition 2.8, then we get s(x1)=0.318, s(x2)=0.246, s(x3)=0.234, s(x4)=0.342, s(x5)=0.334.

Step 4'. Then, by the scores s(xi) result, the ranking order of all the alternatives is generated as x4≻x5≻x1≻x2≻x3. Therefore, the optimal car supplier is x4.

5.1.2. Discussion

In this subsection, we try to explain the difference between our results with GIIFWA and GIIFWG operators and those in Qiu [15].

  1. The comparison of the results is shown in Table 5. By using the same data and weight information, the results calculated by GIIFWG operator are the same as the results in Qiu [15]. However, the results calculated by GIIFWA operator are slightly different from that in Qiu [15].

  2. The reason for the slightly different results calculated by the GIIFWA and GIIFWG operators is that the ranking result of GIIFWG operator is more accurate because zero-valued elements in expert preferences do not affect the calculation process; the GIIFWG operator ensure the reasonable of the alternative ranking in this numerical example.

  3. Compared with the method in Qiu [15], we get the same calculation result with the GIIFWG operator which is more simple in establishing and computing model.

Sort Function The Optimal
Method Results Values Preference order Car supplier
Qiu's [15] method x1=0.350,0.774,0.226,0.349 sx1=0.203 x4≻x5≻x1≻x2≻x3 x4
x2=0.423,0.692,0.171,0.227 sx2=0.181
x3=0.318,0.698,0.272,0.302 sx3=0.169
x4=0.259,0.740,0.191,0.200 sx4=0.235
x5=0.392,0.646,0.185,0.193 sx5=0.222
GIIFWA operator x1=0.551,0.650,0.000,0.269 sx1=0.466 x4≻x1≻x5≻x2≻x3 x4
x2=0.460,0.657,0.000,0.290 sx2=0.414
x3=([0.431,0.570],[0.000,0.264]) sx3=0.369
x4=([0.511,0.661],[0.000,0.224]) sx4=0.474
x5=([0.487,0.645],[0.000,0.255]) sx5=0.438
GIIFWG operator x1=0.503,0.641,0.186,0.323 sx1=0.318 x4≻x5≻x1≻x2≻x3 x4
x2=0.421,0.598,0.178,0.349 sx2=0.246
x3=0.387,0.560,0.171,0.307 sx3=0.234
x4=0.466,0.623,0.125,0.280 sx4=0.342
x5=0.450,0.659,0.158,0.282 sx5=0.334

GIIFWA, generalized interval-valued intuitionistic fuzzy weighted averaging; GIIFWG, generalized interval-valued intuitionistic fuzzy weighted geometric.

Table 5

The comparison among the results of the GIIFWA and GIIFWG operators in this paper and the results of Qiu.

5.2. Dynamic Interval-Valued Intuitionistic Fuzzy Multiple Attribute Group Decision-Making

In this subsection, we will use a practical example which is a slightly revised version of Case illustration in Xu and Yager [35] to illustrate the efficiency and universal applicability of the presented algorithm.

5.2.1. Practical example

Located in Central China and the middle reaches of the Changjiang (Yangtze) River, Hubei Province is distributed in a transitional belt where physical conditions and landscapes are on the transition from north to south and from east to west. Thus, Hubei Province is well known as a land of rice and fish since the region enjoys some of the favorable physical conditions, with a diversity of natural resources and the suitability for growing various crops. At the same time, however, there are also some restrictive factors for developing agriculture, such as a tight man-land relation between a constant degradation of natural resources and a growing population pressure on land resource reserve. Despite cherishing a burning desire to promote their standard of living, people living in the area are frustrated because they have no ability to enhance their power to accelerate economic development because of a dramatic decline in quantity and quality of natural resources and a deteriorating environment. Based on the distinctness and differences in environment and natural resources, Hubei Province can be roughly divided into seven agroecological regions: Y1-Wuhan-Ezhou-Huanggang; Y2-Northeast of Hubei; Y3-Southeast of Hubei; Y4-Jianghan region; Y5-North of Hubei; Y6-Northwest of Hubei; Y7-Southwest of Hubei. In order to prioritize these agroecological regions Yii=1,2,⋯,7 with respect to their comprehensive functions, a committee comprised of three experts Ell=1,2,3 has been set up to provide assessment information on Yii=1,2,⋯,7. The attributes which are considered here in assessment of Yii=1,2,⋯,7 are (1) G1 is ecological benefit, (2) G2 is economic benefit, and (3) G3 is social benefit. The committee evaluates the performance of agroecological regions Yii=1,2,⋯,7 in the years 2004−2006 according to the attributes Gjj=1,2,3, and constructs, respectively, the interval-valued intuitionistic fuzzy decision matrices Rtkll,k=1,2,3 (here, t1l denotes the year “2004,” t2l denotes the year “2005,” and t3l denotes the year “2006”) as listed in Tables 6–14. Let ω=1/6,2/6,3/6T be the weight vector of the years tklk=1,2,3, λ=0.5,0.2,0.3T be the weight vector of the experts El(l=1,2,3), and ξ=(0.3,0.4,0.3)T be the weight vector of the attributes Gjj=1,2,3.

G1 G2 G3
Y1 ([0.8,0.9], [0.0,0.1]) ([0.7,0.8], [0.1,0.2]) ([0.6,0.8], [0.0,0.2])
Y2 ([0.6,0.7], [0.2,0.3]) ([0.5,0.7], [0.2,0.3]) ([0.5,0.6], [0.2,0.3])
Y3 ([0.4,0.5], [0.2,0.4]) ([0.5,0.6], [0.2,0.3]) ([0.4,0.6], [0.1,0.2])
Y4 ([0.7,0.8], [0.1,0.2]) ([0.6,0.8], [0.0,0.1]) ([0.6,0.7], [0.1,0.2])
Y5 ([0.5,0.7], [0.1,0.3]) ([0.7,0.8], [0.1,0.2]) ([0.4,0.5], [0.2,0.4])
Y6 ([0.2,0.3], [0.5,0.6]) ([0.3,0.5], [0.4,0.5]) ([0.4,0.6], [0.3,0.4])
Y7 ([0.4,0.5], [0.3,0.4]) ([0.2,0.5], [0.3,0.5]) ([0.4,0.7], [0.2,0.3])
Table 6

Interval-valued intuitionistic fuzzy decision matrix Rt11.

G1 G2 G3
Y1 ([0.5,0.6], [0.2,0.3]) ([0.2,0.6], [0.1,0.2]) ([0.3,0.6], [0.2,0.3])
Y2 ([0.4,0.5], [0.1,0.3]) ([0.2,0.6], [0.1,0.4]) ([0.4,0.5], [0.3,0.5])
Y3 ([0.4,0.5], [0.2,0.3]) ([0.7,0.8], [0.1,0.2]) ([0.5,0.7], [0.2,0.3])
Y4 ([0.4,0.5], [0.2,0.3]) ([0.2,0.6], [0.1,0.3]) ([0.2,0.8], [0.1,0.2])
Y5 ([0.3,0.5], [0.2,0.3]) ([0.3,0.6], [0.1,0.3]) ([0.3,0.6], [0.1,0.2])
Y6 ([0.3,0.6], [0.2,0.3]) ([0.2,0.7], [0.1,0.2]) ([0.2,0.6], [0.1,0.4])
Y7 ([0.4,0.6], [0.2,0.3]) ([0.4,0.5], [0.1,0.2]) ([0.4,0.5], [0.2,0.3])
Table 7

Interval-valued intuitionistic fuzzy decision matrix Rt12.

G1 G2 G3
Y1 ([0.3,0.6], [0.1,0.3]) ([0.2,0.5], [0.2,0.5]) ([0.2,0.5], [0.3,0.4])
Y2 ([0.3,0.5], [0.2,0.5]) ([0.3,0.5], [0.3,0.4]) ([0.2,0.6], [0.2,0.3])
Y3 ([0.4,0.6], [0.1,0.3]) ([0.3,0.4], [0.2,0.3]) ([0.3,0.6], [0.1,0.2])
Y4 ([0.3,0.5], [0.1,0.3]) ([0.3,0.5], [0.2,0.3]) ([0.2,0.7], [0.1,0.2])
Y5 ([0.2,0.6], [0.1,0.2]) ([0.2,0.5], [0.1,0.4]) ([0.4,0.5], [0.2,0.3])
Y6 ([0.3,0.5], [0.2,0.3]) ([0.3,0.5], [0.1,0.2]) ([0.3,0.5], [0.2,0.4])
Y7 ([0.4,0.7], [0.1,0.3]) ([0.2,0.7], [0.2,0.3]) ([0.4,0.8], [0.1,0.2])
Table 8

Interval-valued intuitionistic fuzzy decision matrix Rt13.

G1 G2 G3
Y1 ([0.7,0.8], [0.1,0.2]) ([0.8,0.9], [0.0,0.1]) ([0.7,0.9], [0.0,0.1])
Y2 ([0.5,0.7], [0.1,0.2]) ([0.6,0.7], [0.1,0.3]) ([0.4,0.5], [0.2,0.4])
Y3 ([0.3,0.5], [0.1,0.3]) ([0.4,0.5], [0.1,0.3]) ([0.3,0.6], [0.3,0.4])
Y4 ([0.6,0.7], [0.1,0.2]) ([0.7,0.8], [0.1,0.2]) ([0.5,0.7], [0.1,0.3])
Y5 ([0.5,0.7], [0.2,0.3]) ([0.5,0.7], [0.1,0.3]) ([0.4,0.6], [0.2,0.3])
Y6 ([0.3,0.4], [0.4,0.6]) ([0.2,0.4], [0.5,0.6]) ([0.4,0.5], [0.4,0.5])
Y7 ([0.3,0.5], [0.3,0.5]) ([0.4,0.6], [0.3,0.4]) ([0.4,0.5], [0.2,0.4])
Table 9

Interval-valued intuitionistic fuzzy decision matrix Rt21.

G1 G2 G3
Y1 ([0.2,0.6], [0.3,0.4]) ([0.2,0.5], [0.3,0.4]) ([0.4,0.5], [0.2,0.5])
Y2 ([0.3,0.5], [0.2,0.3]) ([0.3,0.5], [0.1,0.2]) ([0.3,0.5], [0.2,0.4])
Y3 ([0.4,0.5], [0.1,0.2]) ([0.2,0.4], [0.2,0.3]) ([0.1,0.5], [0.2,0.3])
Y4 ([0.2,0.7], [0.1,0.3]) ([0.2,0.7], [0.1,0.3]) ([0.3,0.6], [0.2,0.4])
Y5 ([0.4,0.5], [0.2,0.3]) ([0.3,0.5], [0.1,0.2]) ([0.4,0.5], [0.1,0.3])
Y6 ([0.3,0.6], [0.2,0.3]) ([0.3,0.7], [0.1,0.2]) ([0.3,0.6], [0.1,0.4])
Y7 ([0.3,0.5], [0.1,0.2]) ([0.3,0.5], [0.2,0.4]) ([0.1,0.8], [0.1,0.2])
Table 10

Interval-valued intuitionistic fuzzy decision matrix Rt22.

G1 G2 G3
Y1 ([0.4,0.6], [0.1,0.3]) ([0.2,0.6], [0.1,0.2]) ([0.2,0.5], [0.2,0.4])
Y2 ([0.4,0.5], [0.3,0.5]) ([0.4,0.5], [0.1,0.2]) ([0.2,0.8], [0.1,0.2])
Y3 ([0.2,0.6], [0.2,0.3]) ([0.2,0.7], [0.1,0.2]) ([0.3,0.7], [0.2,0.3])
Y4 ([0.1,0.6], [0.2,0.3]) ([0.1,0.7], [0.2,0.3]) ([0.3,0.6], [0.1,0.4])
Y5 ([0.4,0.7], [0.1,0.2]) ([0.2,0.6], [0.2,0.3]) ([0.3,0.7], [0.1,0.2])
Y6 ([0.4,0.5], [0.2,0.3]) ([0.2,0.5], [0.1,0.2]) ([0.3,0.5], [0.2,0.3])
Y7 ([0.4,0.5], [0.2,0.1]) ([0.2,0.7], [0.1,0.3]) ([0.3,0.6], [0.1,0.2])
Table 11

Interval-valued intuitionistic fuzzy decision matrix Rt23.

G1 G2 G3
Y1 ([0.6,0.7], [0.1,0.3]) ([0.7,0.9], [0.0,0.1]) ([0.8,0.9], [0.0,0.1])
Y2 ([0.4,0.6], [0.1,0.2]) ([0.5,0.7], [0.1,0.2]) ([0.6,0.7], [0.1,0.3])
Y3 ([0.2,0.4], [0.2,0.3]) ([0.3,0.6], [0.2,0.3]) ([0.4,0.6], [0.2,0.4])
Y4 ([0.7,0.8], [0.0,0.1]) ([0.8,0.9], [0.0,0.1]) ([0.4,0.7], [0.2,0.3])
Y5 ([0.5,0.6], [0.2,0.3]) ([0.4,0.5], [0.1,0.2]) ([0.6,0.7], [0.2,0.3])
Y6 ([0.2,0.3], [0.5,0.6]) ([0.3,0.5], [0.3,0.4]) ([0.3,0.6], [0.2,0.4])
Y7 ([0.5,0.6], [0.3,0.4]) ([0.2,0.3], [0.4,0.5]) ([0.7,0.8], [0.1,0.2])
Table 12

Interval-valued intuitionistic fuzzy decision matrix Rt31.

G1 G2 G3
Y1 ([0.4,0.6], [0.2,0.3]) ([0.3,0.6], [0.1,0.3]) ([0.3,0.6], [0.2,0.4])
Y2 ([0.1,0.7], [0.2,0.3]) ([0.2,0.7], [0.1,0.2]) ([0.5,0.6], [0.1,0.3])
Y3 ([0.5,0.7], [0.2,0.3]) ([0.5,0.6], [0.1,0.3]) ([0.4,0.5], [0.1,0.2])
Y4 ([0.1,0.7], [0.2,0.3]) ([0.2,0.7], [0.1,0.3]) ([0.3,0.6], [0.1,0.2])
Y5 ([0.4,0.5], [0.1,0.3]) ([0.2,0.6], [0.1,0.4]) ([0.1,0.7], [0.2,0.3])
Y6 ([0.5,0.6], [0.1,0.3]) ([0.4,0.6], [0.2,0.4]) ([0.2,0.6], [0.1,0.3])
Y7 ([0.2,0.7], [0.1,0.2]) ([0.2,0.8], [0.1,0.2]) ([0.1,0.8], [0.1,0.2])
Table 13

Interval-valued intuitionistic fuzzy decision matrix Rt32.

G1 G2 G3
Y1 ([0.3,0.5], [0.2,0.4]) ([0.3,0.5], [0.1,0.2]) ([0.3,0.6], [0.2,0.3])
Y2 ([0.3,0.7], [0.2,0.3]) ([0.3,0.5], [0.1,0.4]) ([0.2,0.5], [0.2,0.4])
Y3 ([0.4,0.7], [0.2,0.3]) ([0.4,0.5], [0.1,0.3]) ([0.5,0.7], [0.1,0.2])
Y4 ([0.2,0.8], [0.1,0.2]) ([0.2,0.8], [0.1,0.2]) ([0.2,0.7], [0.1,0.2])
Y5 ([0.2,0.8], [0.1,0.2]) ([0.2,0.5], [0.1,0.3]) ([0.1,0.7], [0.2,0.3])
Y6 ([0.2,0.7], [0.1,0.3]) ([0.1,0.7], [0.2,0.3]) ([0.2,0.6], [0.3,0.4])
Y7 ([0.2,0.8], [0.1,0.2]) ([0.4,0.5], [0.2,0.3]) ([0.1,0.6], [0.2,0.4])
Table 14

Interval-valued intuitionistic fuzzy decision matrix Rt33.

Step 1. If the interval-valued intuitionistic fuzzy tensor and the GIIFWA operator are employed for expressing data in Tables 6–14, then A˜IVIF=ai1i2i3i47×3×3×3 ∈TIVIF(4,7×3×3×3), where its elements ai1i2i3i4=μi1i2i3i4l,μi1i2i3i4u,νi1i2i3i4l,νi1i2i3i4u, and ai1:::i1∈7 represent seven agroecological regions, a:i2::i2∈3 represent three years, a::i3:i3∈3 represent three experts, and a:::i4i4∈3 represent three attributes. The details are as follows:

a1111=0.8,0.9,0.0,0.1,a1112=0.7,0.8,0.1,0.2,
a1113=0.6,0.8,0.0,0.2,a1121=0.5,0.6,0.2,0.3,
a1122=0.2,0.6,0.1,0.2,a1123=0.3,0.6,0.2,0.3,
a1131=0.3,0.6,0.1,0.3,a1132=0.2,0.5,0.2,0.5,
a1133=0.2,0.5,0.3,0.4,a1211=0.7,0.8,0.1,0.2,
a1212=0.8,0.9,0.0,0.1,a1213=0.7,0.9,0.0,0.1,
a1221=0.2,0.6,0.3,0.4,a1222=0.2,0.5,0.3,0.4,
a1223=0.4,0.5,0.2,0.5,a1231=0.4,0.6,0.1,0.3,
a1232=0.2,0.6,0.1,0.2,a1233=0.2,0.5,0.2,0.4,
a1311=0.6,0.7,0.1,0.3,a1312=0.7,0.9,0.0,0.1,
a1313=0.8,0.9,0.0,0.1,a1321=0.4,0.6,0.2,0.3,
a1322=0.3,0.6,0.1,0.3,a1323=0.3,0.6,0.2,0.4,
a1331=0.3,0.5,0.2,0.4,a1332=0.3,0.5,0.1,0.2,
a1333=0.3,0.6,0.2,0.3,a2111=0.6,0.7,0.2,0.3,
a2112=0.5,0.7,0.2,0.3,a2113=0.5,0.6,0.2,0.3,
a2121=0.4,0.5,0.1,0.3,a2122=0.2,0.6,0.1,0.4,
a2123=0.4,0.5,0.3,0.5,a2131=0.3,0.5,0.2,0.5,
a2132=0.3,0.5,0.3,0.4,a2133=0.2,0.6,0.2,0.3,
a2211=0.5,0.7,0.1,0.2,a2212=0.6,0.7,0.1,0.3,
a2213=0.4,0.5,0.2,0.4,a2221=0.3,0.5,0.2,0.3,
a2222=0.3,0.5,0.1,0.2,a2223=0.3,0.5,0.2,0.4,
a2231=0.4,0.5,0.3,0.5,a2232=0.4,0.5,0.1,0.2,
a2233=0.2,0.8,0.1,0.2,a2311=0.4,0.6,0.1,0.2,
a2312=0.5,0.7,0.1,0.2,a2313=0.6,0.7,0.1,0.3,
a2321=0.1,0.7,0.2,0.3,a2322=0.2,0.7,0.1,0.2,
a2323=0.5,0.6,0.1,0.3,a2331=0.3,0.7,0.2,0.3,
a2332=0.3,0.5,0.1,0.4,a2333=0.2,0.5,0.2,0.4,
a3111=0.4,0.5,0.2,0.4,a3112=0.5,0.6,0.2,0.3,
a3113=0.4,0.6,0.1,0.2,a3121=0.4,0.5,0.2,0.3,
a3122=0.7,0.8,0.1,0.2,a3123=0.5,0.7,0.2,0.3,
a3131=0.4,0.6,0.1,0.3,a3132=0.3,0.4,0.2,0.3,
a3133=0.3,0.6,0.1,0.2,a3211=0.3,0.5,0.1,0.3,
a3212=0.4,0.5,0.1,0.3,a3213=0.3,0.6,0.3,0.4,
a3221=0.4,0.5,0.1,0.2,a3222=0.2,0.4,0.2,0.3,
a3223=0.1,0.5,0.2,0.3,a3231=0.2,0.6,0.2,0.3,
a3232=0.2,0.7,0.1,0.2,a3233=0.3,0.7,0.2,0.3,
a3311=0.2,0.4,0.2,0.3,a3312=0.3,0.6,0.2,0.3,
a3313=0.4,0.6,0.2,0.4,a3321=0.5,0.7,0.2,0.3,
a3322=0.5,0.6,0.1,0.3,a3323=0.4,0.5,0.1,0.2,
a3331=0.4,0.7,0.2,0.3,a3332=0.4,0.5,0.1,0.3,
a3333=0.5,0.7,0.1,0.2,a4111=0.7,0.8,0.1,0.2,
a4112=0.6,0.8,0.0,0.1,a4113=0.6,0.7,0.1,0.2,
a4121=0.4,0.5,0.2,0.3,a4122=0.2,0.6,0.1,0.3,
a4123=0.2,0.8,0.1,0.2,a4131=0.3,0.5,0.1,0.3,
a4132=0.3,0.5,0.2,0.3,a4133=0.2,0.7,0.1,0.2,
a4211=0.6,0.7,0.1,0.2,a4212=0.7,0.8,0.1,0.2,
a4213=0.5,0.7,0.1,0.3,a4221=0.2,0.7,0.1,0.3,
a4222=0.2,0.7,0.1,0.3,a4223=0.3,0.6,0.2,0.4,
a4231=0.1,0.6,0.2,0.3,a4232=0.1,0.7,0.2,0.3,
a4233=0.3,0.6,0.1,0.4,a4311=0.7,0.8,0.0,0.1,
a4312=0.8,0.9,0.0,0.1,a4313=0.4,0.7,0.2,0.3,
a4321=0.1,0.7,0.2,0.3,a4322=0.2,0.7,0.1,0.3,
a4323=0.3,0.6,0.1,0.2,a4331=0.2,0.8,0.1,0.2,
a4332=0.2,0.8,0.1,0.2,a4333=0.2,0.7,0.1,0.2,
a5111=0.5,0.7,0.1,0.3,a5112=0.7,0.8,0.1,0.2,
a5113=0.4,0.5,0.2,0.4,a5121=0.3,0.5,0.2,0.3,
a5122=0.3,0.6,0.1,0.3,a5123=0.3,0.6,0.1,0.2,
a5131=0.2,0.6,0.1,0.2,a5132=0.2,0.5,0.1,0.4,
a5133=0.4,0.5,0.2,0.3,a5211=0.5,0.7,0.2,0.3,
a5212=0.5,0.7,0.1,0.3,a5213=0.4,0.6,0.2,0.3,
a5221=0.4,0.5,0.2,0.3,a5222=0.3,0.5,0.1,0.2,
a5223=0.4,0.5,0.1,0.3,a5231=0.4,0.7,0.1,0.2,
a5232=0.2,0.6,0.2,0.3,a5233=0.3,0.7,0.1,0.2,
a5311=0.5,0.6,0.2,0.3,a5312=0.4,0.5,0.1,0.2,
a5313=0.6,0.7,0.2,0.3,a5321=0.4,0.5,0.1,0.3,
a5322=0.2,0.6,0.1,0.4,a5323=0.1,0.7,0.2,0.3,
a5331=0.2,0.8,0.1,0.2,a5332=0.2,0.5,0.1,0.3,
a5333=0.1,0.7,0.2,0.3,a6111=0.2,0.3,0.5,0.6,
a6112=0.3,0.5,0.4,0.5,a6113=0.4,0.6,0.3,0.4,
a6121=0.3,0.6,0.2,0.3,a6122=0.2,0.7,0.1,0.2,
a6123=0.2,0.6,0.1,0.4,a6131=0.3,0.5,0.2,0.3,
a6132=0.3,0.5,0.1,0.2,a6133=0.3,0.5,0.2,0.4,
a6211=0.3,0.4,0.4,0.6,a6212=0.2,0.4,0.5,0.6,
a6213=0.4,0.5,0.4,0.5,a6221=0.3,0.6,0.2,0.3,
a6222=0.3,0.7,0.1,0.2,a6223=0.3,0.6,0.1,0.4,
a6231=0.4,0.5,0.2,0.3,a6232=0.2,0.5,0.1,0.2,
a6233=0.3,0.5,0.2,0.3,a6311=0.2,0.3,0.5,0.6,
a6312=0.3,0.5,0.3,0.4,a6313=0.3,0.6,0.2,0.4,
a6321=0.5,0.6,0.1,0.3,a6322=0.4,0.6,0.2,0.4,
a6323=0.2,0.6,0.1,0.3,a6331=0.2,0.7,0.1,0.3,
a6332=0.1,0.7,0.2,0.3,a6333=0.2,0.6],0.3,0.4,
a7111=0.4,0.5,0.3,0.4,a7112=0.2,0.5,0.3,0.5,
a7113=0.4,0.7,0.2,0.3,a7121=0.4,0.6,0.2,0.3,
a7122=0.4,0.5,0.1,0.2,a7123=0.4,0.5,0.2,0.3,
a7131=0.4,0.7,0.1,0.3,a7132=0.2,0.7,0.2,0.3,
a7133=0.4,0.8,0.1,0.2,a7211=0.3,0.5,0.3,0.5,
a7212=0.4,0.6,0.3,0.4,a7213=0.4,0.5,0.2,0.4,
a7221=0.3,0.5,0.1,0.2,a7222=0.3,0.5,0.2,0.4,
a7223=0.1,0.8,0.1,0.2,a7231=0.4,0.5,0.2,0.1,
a7232=0.2,0.7,0.1,0.3,a7233=0.3,0.6,0.1,0.2,
a7311=0.5,0.6,0.3,0.4,a7312=0.2,0.3,0.4,0.5,
a7313=0.7,0.8,0.1,0.2,a7321=0.2,0.7,0.1,0.2,
a7322=0.2,0.8,0.1,0.2,a7323=0.1,0.8,0.1,0.2,
a7331=0.2,0.8,0.1,0.2,a7332=0.4,0.5,0.2,0.3,
a7333=0.1,0.6,0.2,0.4.

Step 2. By Theorem 3.1, A˜IVIF∈TIVIF4,7×3×3×3. Let X2=ω (the years weight), X3=λ (the decision-makers weight), and X4=ξ (the attributes weight), we have

GIIFWA(A˜IF∘X2∘X3∘X4)=1−∏i2=13∏i3=13∏i4=131−μi1i2i3i4lxi22xi33xi44, 1−∏i2=13∏i3=13∏i4=131−μi1i2i3i4uxi22xi33xi44, ∏i2=13∏i3=13∏i4=13νi1i2i3i4lxi22xi33xi44, ∏i2=13∏i3=13∏i4=13νi1i2i3i4uxi22xi33xi44=0.556,0.754,0.000,0.207,0.415,0.630, 0.134,0.283,0.363,0.581,0.153,0.290, 0.479,0.749,0.000,0.208,0.391,0.632, 0.135,0.273,0.279,0.542,0.233,0.384, 0.349,0.626,0.183,0.304T.

Then, we get

Y1=0.556,0.754,0.000,0.207,
Y2=0.415,0.630,0.134,0.283,
Y3=0.363,0.581,0.153,0.290,
Y4=0.479,0.749,0.000,0.208,
Y5=0.391,0.632,0.135,0.273,
Y6=0.279,0.542,0.233,0.384,
Y7=0.349,0.626,0.183,0.304.

Step 3. To rank the IVIFNs Yi1i1∈7, we calculate the scores sYi1i1∈7 by the Definition 2.8. Then, we have sY1=0.552, sY2=0.314, sY3=0.251, sY4=0.510, sY5=0.307, sY6=0.102, sY7=0.244.

Step 4. By the scores sYi1 result, the ranking order of all the alternatives is generated as Y1≻Y4≻Y2≻Y5≻Y3≻Y7≻Y6. Therefore, the agroecological region with the most comprehensive functions is Y1-Wuhan-Ezhou-Huanggang.

We can also replace the GIIFWA with the GIIFWG to resolve this problem. The difference starts from step 2.

Step 2'. By the Theorem 3.2, we have

GIIFWGA˜IF∘X2∘X3∘X4 =∏i2=13∏i3=13∏i4=13μi1i2i3i4lxi22xi33xi44,  ∏i2=13∏i3=13∏i4=13μi1i2i3i4uxi22xi33xi44,  1−∏i2=13∏i3=13∏i4=131−νi1i2i3i4lxi22xi33xi44,  1−∏i2=13∏i3=13∏i4=131−νi1i2i3i4uxi22xi33xi44 =0.444,0.684,0.107,0.249,0.369,0.610,   0.147,0.302,0.334,0.562,0.165,0.299,   0.346,0.724,0.104,0.231,0.332,0.610,   0.145,0.282,0.258,0.514,0.285,0.419,   0.290,0.575,0.214,0.338T

Then, we get

Y1=0.444,0.684,0.107,0.249,
Y2=0.369,0.610,0.147,0.302,
Y3=0.334,0.562,0.165,0.299,
Y4=0.346,0.724,0.104,0.231,
Y5=0.332,0.610,0.145,0.282,
Y6=0.258,0.514,0.285,0.419,
Y7=0.290,0.575,0.214,0.338.

Step 3'. To rank the IVIFNs Yi1i1∈7, we calculate the scores sYi1i1∈7 by the Definition 2.8, then we get sY1=0.386,sY2=0.266,sY3=0.216,sY4=0.367,sY5=0.258,sY6=0.034,sY7=0.156.

Step 4'. By the scores sYi1 result, the ranking order of all the alternatives is generated as Y1≻Y4≻Y2≻Y5≻Y3≻Y7≻Y6. Therefore, the agroecological region with the most comprehensive functions is also Y1-Wuhan-Ezhou-Huanggang.

5.2.2. Discussion

  1. The comparison of the results is shown in Table 15. By using the same data and weight information, we get the same results calculated by the GIIFWA and GIIFWG operators. That is, the agroecological region with the most comprehensive functions is Y1-Wuhan-Ezhou-Huanggang.

  2. The GIIFWA and GIIFWG operators proposed in this paper can effectively solve the dynamic multiple attribute group decision-making problem (four-dimensional data) through analyzing the above practical decision-making problem. Therefore, in order to solve the actual decision problem of high-dimensional data, the proposed methods have better adaptability. For example, it can effectively deal with multiple attribute group decision-making problem (three-dimensional data), dynamic multiple attribute group decision-making problem (four-dimensional data), and practical decision problems with higher dimension data.

Method Results Sort Function Values Preference Order The Agroecological Region with the Most Comprehensive Functions
GIIFWA Y1=0.556,0.754,0.000,0.207 sY1=0.552 Y1≻Y4≻Y2≻Y5≻Y3≻Y7≻Y6 Y1
operator Y2=0.415,0.630,0.134,0.283 sY2=0.314
Y3=0.363,0.581,0.153,0.290 sY3=0.251
Y4=0.479,0.749,0.000,0.208 sY4=0.510
Y5=0.391,0.632,0.135,0.273 sY5=0.307
Y6=0.279,0.542,0.233,0.384 sY6=0.102
Y7=0.349,0.626,0.183,0.304 sY7=0.244
GIIFWG Y1=0.444,0.684,0.107,0.249 sY1=0.386 Y1≻Y4≻Y2≻Y5≻Y3≻Y7≻Y6 Y1
operator Y2=0.369,0.610,0.147,0.302 sY2=0.266
Y3=0.334,0.562,0.165,0.299 sY3=0.216
Y4=0.346,0.724,0.104,0.231 sY4=0.367
Y5=0.332,0.610,0.145,0.282 sY5=0.258
Y6=0.258,0.514,0.285,0.419 sY6=0.034
Y7=0.290,0.575,0.214,0.338 sY7=0.156

GIIFWA, generalized interval-valued intuitionistic fuzzy weighted averaging; GIIFWG, generalized interval-valued intuitionistic fuzzy weighted geometric.

Table 15

The comparison between the results of the GIIFWA and GIIFWG operators in this paper.

6. CONCLUSION

As a generalization of fuzzy decision matrix, this paper has presented the concept of mth-order interval-valued intuitionistic fuzzy tensor and related properties. The GIIFWA and GIIFWG operators by the product of tensor with vector have been obtained and found effective to deal with the multiple attribute group decision-making and dynamic multiple attribute group decision-making problems in an interval-valued intuitionistic condition. Two typical examples have also been provided to demonstrate the efficiency and universal applicability of the proposed method.

ACKNOWLEDGMENT

The work was supported in part by the National Natural Science Foundation of China (Grant nos. 11571292 and 11747104), the General Project of Hunan Provincial Natural Science Foundation of China (2016JJ2043 and 2019JJ50125), the Postgraduate Innovation Project of Hunan Province of China (CX2017B263), Since the Guangxi Municipality Project for the Basic Ability Enhancement of Young and Middle-Aged Teachers (KY2016YB532) has been completed, the project is deleted in the final version.

REFERENCES

<Previous Article In Issue
Download article (PDF)
Next Article In Issue>
Journal
International Journal of Computational Intelligence Systems
Volume-Issue
12 - 2
Pages
580 - 596
Publication Date
2019/05/11
ISSN (Online)
1875-6883
ISSN (Print)
1875-6891
DOI
10.2991/ijcis.d.190424.001How to use a DOI?
Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

risenwbib
TY  - JOUR
AU  - Shengyue Deng
AU  - Jianzhou Liu
AU  - Jintao Tan
AU  - Lixin Zhou
PY  - 2019
DA  - 2019/05/11
TI  - A Novel Method Based on Fuzzy Tensor Technique for Interval-Valued Intuitionistic Fuzzy Decision-Making with High-Dimension Data
JO  - International Journal of Computational Intelligence Systems
SP  - 580
EP  - 596
VL  - 12
IS  - 2
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.190424.001
DO  - 10.2991/ijcis.d.190424.001
ID  - Deng2019
ER  -