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Article

Generalized Interval-Valued q-Rung Orthopair Hesitant Fuzzy Choquet Operators and Their Application

1
School of Management, Hebei University, Baoding 071002, China
2
College of Mathematics and Information Science, Hebei University, Baoding 071002, China
3
School of Big Data Science, Hebei Finance University, Baoding 071051, China
*
Author to whom correspondence should be addressed.
Symmetry 2023, 15(1), 127; https://doi.org/10.3390/sym15010127
Submission received: 14 November 2022 / Revised: 11 December 2022 / Accepted: 16 December 2022 / Published: 2 January 2023
(This article belongs to the Section Mathematics)

Abstract

:
Hesitant fuzzy evaluation strategy related to the interval-valued membership and nonmembership degrees should be an appropriate choice due to the lack of experience, ability and knowledge of some decision experts. In addition, it is important to reasonably model the interrelationship of these experts. In this work, firstly, the generalized interval-valued q-rung orthopair hesitant fuzzy sets (GIVqROHFSs) are defined, and some operational rules with respect to GIVqROF numbers are discussed. Secondly, two types of operators, which are denoted as GIVqROHFCA and GIVqROHFCGM, are developed. Thirdly, the desired properties and relationships of two operators are studied. Furthermore, a new multiple attributes group decision making (MAGDM) approach is proposed. Finally, three experiments are completed to illustrate the rationality of the developed method and the monotonicity of this approach concerning the parameter in the GIVqROHFCGM operator and the GIVqROHFCA operator which meets symmetrical characteristics, and shows the superiority and reliability of this new method in solving the GIVqROHF problems. The main advantages of this work include three points: (1) extending hesitant fuzzy sets to the interval-valued q-rung orthopair fuzzy case and proposing two types of aggregation operators for the GIVqROHF information; (2) considering the interaction among decision makers and among attributes in decision problems, and dealing with this interrelationship by fuzzy measure; (3) introducing the new decision method for the GIVqROHF environment and enriching the mathematical tools to solve multiple attributes decision-making problems.

1. Introduction

In many problems of practical decision making or human cognition, a single value may not be completely suitable to present values of the nonmembership and membership (NMMDs) [1,2]. In view of these restrictions, HFSs are useful to model the hesitant uncertainty in MAGDM problems [3,4], and the fuzzy filters theory of Sheffer stroke algebras can be used to power bridges between fuzzy and hesitant fuzzy structures [5,6]. The membership of the hesitant fuzzy set includes several hesitant values on the unit interval [0, 1], which have an extension on the basis of different fuzzy forms in many practical decision-making problems such as the interval case [7], intuitionistic case [8], and Pythagorean case [9]. For instance, Deli introduced the MAGDM approach based on the GTHF-numbers and Bonferroni mean operator [10]; Liao et al. extended the ORESTE method to the hesitant fuzzy linguistic environment for the MAGDM problems [11]; Mishra et al. integrated the Shapley measure and hesitant fuzzy sets theory to propose the COPRAS method for the MAGDM process [12]; Wan et al. considered the multiplicative consistency of hesitant fuzzy preference relations and presented a mathematical program on the basis of the new consistency index of HFPR [13]; Shen et al. utilized the binary connection number theory to the probabilistic hesitant fuzzy MAGDM problems [14], and Liu et al. studied the distance measures of the probabilistic hesitant fuzzy element for the MAGDM methods [15]; Meng et al. analysed the time-sequence situation of hesitant degrees of people and introduced the concept of time-sequential hesitant fuzzy set for the MAGDM problems [16]. Kinds of extension of HFSs have been proposed for some MAGDM problems. The interval-valued hesitant fuzzy sets (IVHFSs) permit that degrees of the membership take interval values on [0, 1] which describe the intuitionistic differences among decision experts [17]. The intuitionistic hesitant fuzzy sets (IHFSs) involve the possible membership and nonmembership values which decision makers provide for accessing an alternative [18]. The interval-valued intuitionistic hesitant fuzzy sets (IVIHFSs) are another generation of HFSs, which allow membership degrees on an element to contain several interval-valued intuitionistic fuzzy numbers [19]. Pythagorean hesitant fuzzy sets, expanding the scope of the NMMDs on IHFSs, can manage the situation where the 2-power sum of NMMDs on an element is not greater than the unit one, and it is also an extension of IHFSs [20,21].
In addition, for one thing, the interval-valued intuitionistic fuzzy sets (IVIFSs) were proposed [22]. For another thing, the interval-valued Pythagorean fuzzy sets (IVPFSs) were defined to extend Pythagorean fuzzy sets (PFSs) [23,24]. Moreover, the interval-valued q-rung orthopair fuzzy sets (IVqROFSs), generalizing IVIFSs and IVPFSs, are efficient to describe fuzzy information in the practical MAGDM problems [25,26]. As a result, some researchers have studied some MAGDM problems under fuzzy surroundings which were modeled based on the IVPFSs and IVqROFSs. For instance, Liu et al. developed the new distance measures with respect to the IVPFSs for the IVPFS-based MAGDM approach [27]; Garg defined two new exponential operational laws of the IVPFSs and extended these rules to interval numbers situation for dealing with the MAGDM problems [28]; Rahman et al. provided more general aggregation operators for the IVPFS multiple-attribute group decision-making method [29]; Mu et al. proposed a new comparison rule about the IVPFNs based on the new comparison functions for the MAGDM problems [30]; Li et al. considered the IVPFS decision-making method on the basis of the set pair analysis and fuzzy integral theory [31]; Zhao et al. presented the IVPFS TODIM method for the green supplier selection in terms of the cumulative prospect theory [32]; Peng et al. introduced a new distance measure about the IVPFSs for the IVPFS emergency decision-making process [33]; Gao et al. extended the Archimedean Muirhead mean operator to the IVqROFS environment for the MAGDM analysis [34]; Garg developed the Possibility degree measure in regard to two IVqROFSs and introduced the corresponding MAGDM method [35]; Garg et al. pooled the aggregation operators, AHP and TOPSIS, to study the concept of complex IVqROFSs for the IVqROFS decision process [36].
Compared to the above fuzzy method, the MAGDM approaches with grey system theory were developed on the basis of another mathematical perspective. On the other hand, the grey system theory can describe the uncertain phenomena with the explicit denotation and ambiguous connotation, and it has been the mathematical tool that models hesitation and uncertainty of the incomplete information. For instance, Turanoglu et al. ranked the grey decoration materials in the light of the AHP and grey correlation with TOPSIS [37]; Alkharabsheh et al. extended the classical AHP approach to grey values environment for ranking the supply quality criteria [38]; Esangbedo et al. proposed a new weighting method based on the point-allocation approach and the grey system theory and developed the weighted sum model with grey numbers [39]. However, the uncertain situation, which is the clear connotation and unclear extension in the MAGDM problems, can be addressed by the fuzzy theory or extensions of this mathematical tool. Furthermore, owing to some influencing factors such as the capacities of information processing, limited attention of decision makers and the lack of data or knowledge during the practical MAGDM process, they cannot present a single IVqROF number for MNMDs of a considered element, and may give several possible IVqROF numbers. For instance, in order to obtain a desired evaluation result, a decision corporation demands decision experts to provide the degree values with respect to an attribute, and supposing that three cases are included, that is, some experts present ( [ 0.8 , 0.95 ] , [ 0.45 , 0.7 ] ) , some provide ( [ 0.45 , 0.65 ] , [ 0.7 , 0.9 ] ) and the others assign ( [ 0.7 , 0.85 ] , [ 0.45 , 0.7 ] ) , where they cannot change their opinions. It can easily be obtained that the sum of the maximal value within the membership and non-membership interval is more than or equal to the unit one, and the ones of their different powers are yet not less than the one, that is 0.95 4 + 0.7 4 > 1 , 0.65 3 + 0.9 3 > 1 and 0.85 2 + 0.7 2 > 1 . It is shown that such a decision-making problem cannot be handled by the aforementioned MAGDM method, the fuzzy sets (FSs) theory [40] and extensions which involve the hesitant fuzzy sets (HFSs) theory [41], bipolar interval-valued neutrosophic sets (BIVNs) theory [42], dual hesitant fuzzy sets (DHFs) theory [43], interval-valued dual hesitant fuzzy sets (IVDHFs) theory [44], intuitionistic fuzzy sets theory [45], PFSs theory, interval-valued fuzzy sets (IVFSs) theory [46], IVIFSs theory, IVHFSs theory, IHFSs theory, IVIHFSs theory, PHFSs theory, IVPHFSs theory [47] and IVqROFSs theory [48]. Considering the concept of interval-valued q-rung dual fuzzy sets [49,50,51], the aforementioned hesitant fuzzy information may been modeled by an interval-valued q-rung dual fuzzy set (IVqRDHFS) { [ 0.8 , 0.95 ] , [ 0.45 , 0.65 ] , [ 0.7 , 0.85 ] } , { [ 0.45 , 0.7 ] ) , [ 0.7 , 0.9 ] } , However, it is easily shown that the hesitant fuzzy information ( [ 0.7 , 0.85 ] , [ 0.45 , 0.7 ] ) is omitted, thus this situation results in loss of the useful information. From another perspective, the IVqRDHFS { [ 0.8 , 0.95 ] , [ 0.45 , 0.65 ] , [ 0.7 , 0.85 ] } , { [ 0.45 , 0.7 ] ) , [ 0.7 , 0.9 ] } may be presented by several IVqROFNs as follows:
{ ( [ 0.8 , 0.95 ] , [ 0.45 , 0.7 ] ) , ( [ 0.8 , 0.95 ] , [ 0.7 , 0.9 ] ) , ( [ 0.45 , 0.65 ] , [ 0.45 , 0.7 ] ) , ( [ 0.45 , 0.65 ] , [ 0.7 , 0.9 ] ) , ( [ 0.7 , 0.85 ] } , [ 0.45 , 0.7 ] ) , ( [ 0.7 , 0.85 ] } , [ 0.7 , 0.9 ] ) }
According to this symbol, the hesitant fuzzy information in the above instance is contained, and the IVqRDHFS is justly a special case of a new type of hesitant fuzzy set which is characterized by several IVqROFSs. Therefore, the hesitant fuzzy sets should be extended to the IVqROFNs case. This paper takes the aim of the theory concerning the generalized interval-valued q-rung orthopair hesitant fuzzy sets (GIVqROHFSs) and the NMMDs set which may contain several possible IVqROFNs. In view of the previous instance, the NMMDs on an element can be described by the IVqROHFSs { ( [ 0.8 , 0.95 ] , [ 0.45 , 0.7 ] ) , ( [ 0.45 , 0.65 ] , [ 0.7 , 0.9 ] ) , ( [ 0.7 , 0.85 ] , [ 0.45 , 0.7 ] ) } . In spite of the fact that the attribute values may be a set of IVqROFNs in the practical application of many MAGDM problems, there is a matter of great concern to the aggregation approach which can deal with the GIVqROHF information. Nevertheless, the existing fusion methods have encountered a great difficulty under the GIVqROHF environment, and decision methods in the above-mentioned references only account for the restriction that all of the input arguments or decision attributes in decision problems are independent, although the practical decision-making process always involves interactive characteristics among the alternatives or attributes under the uncertain decision environment. As a key characteristic case, the Choquet integral, which is a symmetric aggregation operator, can merge the interaction among alternatives or attributes into the decision process, and it is an efficient mathematical tool to incorporate interacting attributes under uncertain circumstances [52,53]. In general, the discrete Choquet integral, which is applied in the decision-making process, linearly expresses the mean of all reordered elements in discourse domain, and it coincides with the weighted mean or ordered weighted averaging (OWA) operator. Moreover, the Choquet integral-based aggregation methods can cope with the decision situation in which the correlation with each other exists [54,55]. Specifically, In the decision methods based on the Choquet operator, where attributes or experts can be dependent, thus, the non-additive fuzzy measure is possible to model the interrelationship among attributes or experts by defining a weight on each subset of attributes set or expert set. Consequently, it is very necessary that the Choquet integral is used to overcome the restriction of independence assumption among alternatives or attributes in MAGDM problems [56].
Motivated by the advantages of the Choquet integral theory applied in MAGDM methods, and the GIVqROHFSs theory, which is proposed in this paper, we assign two types of aggregation operators based on Choquet integration to handle the GIVqROHF information, and properties concerning these operators are studied. Moreover, a new decision method is presented for the GIVqROHF information processing, which considers the interaction among inputs or attributes of an alternative. In addition, an application is performed to manifest the performance of this new method.
According to the above, we organize contents (see Figure 1). Section 2 focuses on the basic knowledge involving the HFSs, fuzzy measure, Choquet integral and IVqROFSs. The concept of the GIVqROHFSs is proposed and some operational rules of the GIVqROHFSs are introduced in Section 3. The GIVqROHFCA and GIVqROHFCGM operators are developed in Section 4, and some properties of these two operators are discussed. By the proposed operators, Section 5 demonstrates a new MAGDM approach for processing the GIVqROHF information. The contents of Section 6 include an illustrative example and two number experiences such as sensitivity analysis and compared experience. Some remarks are presented in Section 7.

2. Preliminaries

The hesitant fuzzy sets (HFSs), fuzzy measure, Choquet integral and IVqROFSs are introduced in the following:

2.1. HFSs

Definition 1 ([1]).
Provided that T is a nonempty set and ψ X ˜ ˜ t contains several real numbers on [0, 1] for any t T , a hesitant fuzzy set (HFS) is expressed by the following mathematical symbol:
X ˜ ˜ = t , ψ X ˜ ˜ t / t T
where ψ X ˜ ˜ t represents membership degree values of t T to X ˜ ˜ , and it is simply denoted the symbol ψ which is called hesitant fuzzy element (HFE) [1].
If ψ k and ψ l are two HFEs, several operations are introduced as follows and all of the operational results between HFEs are also HFEs [19]:
ψ k ψ l = t k + t l t k t l t k ψ k t l ψ l
ψ k ψ l = t k t l t k ψ k t l ψ l
k ψ l = { ( 1 t l ) k + 1 t l ψ l }
ψ l k = { ( t l ) k t l ψ l }
ψ l c = { 1 t l t l ψ l }
The ranking relationship is introduced [19].
Definition 2 ([19]).
Provided that ψ 1 and ψ 2 are HFEs, # ψ i indicates the cardinal number of ψ i and the score function of ψ i is denoted as v s ( ψ i ) = ( t ψ i t ) / # ψ i , therefore
(1) 
v s ( ψ 1 ) < v s ( ψ 2 ) ψ 1 < ψ 2 ; (2) v s ( ψ 1 ) = v s ( ψ 2 ) ψ 1 = ψ 2 .

2.2. λ -Fuzzy Measure and Choquet Integral

Definition 3 ([57]).
T is a universe of discourse, P ( T ) is the power of T and Y : P ( T ) [ 0 , 1 ] is a mapping with conditions.
(1) 
Y ( ϕ ) = 0 , Y ( T ) = 1 .
(2) 
T 1 , T 2 P ( T ) , T 1 T 2 Y ( T 1 ) Y ( T 2 ) .
(3) 
T 1 , T 2 P ( T ) , T 1 T 2 = ϕ , λ > 1 Y ( T 1 T 2 ) = Y ( T 1 ) + Y ( T 2 ) + λ Y ( T 1 ) Y ( T 2 ) .
We call Y : P ( T ) [ 0 , 1 ]   λ -fuzzy measure on P ( T ) .
When T = { t 1 , t 2 , , t n } is a finite space, a λ -fuzzy measure Y has the following equations:
Y ( T ) = ( 1 / λ ) i = 1 n [ 1 + λ Y ( { t i } ) ] 1 λ 0 i = 1 n Y ( { t i } ) λ = 0
Y ( T 1 ) = t i T 1 [ λ Y ( { t i } ) + 1 ] 1 / λ λ 0 t i T 1 Y ( { t i } ) λ = 0
1 + λ = i = 1 n [ 1 + λ Y ( { t i } ) ]
In MAGDM problems, Y ( { t i } ) can indicate the importance of each decision expert or attribute, and Y ( T 1 ) can describe the interaction among decision experts or attributes [46].
Definition 4 ([58]).
Supposing that the conditions hold.
(1) 
T = { t 1 , t 2 , , t n } and G : T R + is a positive real-valued function.
(2) 
{ κ ( 1 ) , κ ( 2 ) , , κ ( n ) } satisfies G ( t κ ( 1 ) ) G ( t κ ( 2 ) ) G ( t κ ( n ) ) with κ ( l ) { 1 , 2 , , n } .
(3) 
X κ ( m ) = t κ ( l ) / l m , m { 1 , 2 , , n } and X κ ( 0 ) = ϕ .
(4) 
Y is a λ -fuzzy measure on P ( T ) .
Choquet integral concerning λ -fuzzy measure Y is presented by:
( c ) G d M = l = 1 n G ( t κ ( l ) ) [ ( Y ( X κ ( l ) ) Y ( X κ ( l 1 ) ) ]
The Choquet integral is a significant tool of information fusion in in MAGDM processes [40,41,45,59].

2.3. IVqROFSs and IVq-RDHFSs

Definition 5 [25].
Suppose that three conditions hold:
(1) 
T is a nonempty set, q 1 is a real number and t T .
(2) 
For 0 M t M t + 1 , 0 Z t Z t + 1 and ( M t + ) q + ( Z t + ) q 1 .
(3) 
For M t = [ M t , M t + ] and Z t = [ Z t , Z t + ] .
An IVqROFS X ρ ˜ ˜ on T is structured:
X ρ ˜ ˜ = t , M t , Z t t T
where Z t and M t are, respectively, called the nonmembership and membership degree values of t T to X ρ ˜ ˜ , and the ordered pair ( M t , Z t ) in T is called IVqROFN [60], which is simply denoted ( M , Z ) with the conditions M = [ M , M + ] [ 0 , 1 ] , Z = [ Z , Z + ] [ 0 , 1 ] .
IVqROFNs have some laws with respect to operations.
Definition 6 [61].
Supposing that μ k = ( M μ k , Z μ k ) , k = 1 , 2 , μ = ( M μ , Z μ ) are three IVqROFNs, then operation laws are presented:
(1) 
μ k μ l = M μ k q + M μ l q M μ k q × M μ l q q , M μ k + q + M μ l + q M μ k + q × M μ l + q q , Z μ k × Z μ l , Z μ k + × Z μ l +
(2) 
μ k μ l = M μ k × M μ l , M μ k + × M μ l + , Z μ k q + Z μ l q Z μ k q × Z μ l q q , Z μ k + q + Z μ l + q Z μ k + q × Z μ l + q q
(3) 
m μ = M μ q + 1 m + 1 q , M μ + q + 1 m + 1 q , Z μ m , Z μ + m m ( 0 , + )
(4) 
μ m = M μ m , M μ + m , Z μ q + 1 m + 1 q , Z μ + q + 1 m + 1 q , m ( 0 , + )
(5) 
μ c = Z μ , Z μ + , M μ , M μ +
(6) 
μ k , μ l = M μ k , M μ l , M μ k + , M μ l + , Z μ k , Z μ l , Z μ k + , Z μ l +
(7) 
μ k , μ l = M μ k , M μ l , M μ k + , M μ l + , Z μ k , Z μ l , Z μ k + , Z μ l +
Theorem 7 [61].
From Definition 6, operational results of μ 1 μ 2 , μ 1 μ 2 , m μ , μ m , μ c , μ 1 μ 2 and μ 1 μ 2 are also IVqROFNs.
From the score and accuracy value with respect to IVqROFNs, an order relationship for them was presented.
Definition 8 [61].
Let μ 1 and μ 2 be IVqROFNs with μ k = [ M μ k , M μ k + ] , [ Z μ k , Z μ k + ] , then the score value v s ( μ k ) and the accuracy value v h ( μ k ) have the form:
v s ( μ k ) = M μ k + q + Z μ k + q M μ k q Z μ k q / 2
v h ( μ k ) = M μ k + q + Z μ k + q + M μ k q + Z μ k q / 2  
Consequently, the order relationship is presented as follows:
(1) v s ( μ 1 ) < v s ( μ 2 ) μ 1 < μ 2
(2) v s ( μ 1 ) = v s ( μ 2 ) , v h ( μ 1 ) < v h ( μ 2 ) μ 1 < μ 2
(3) v s ( μ 1 ) = v s ( μ 2 ) , v h ( μ 1 ) = v h ( μ 2 ) μ 1 = μ 2
(4) v s ( μ 1 ) = v s ( μ 2 ) , v h ( μ 1 ) > v h ( μ 2 ) μ 1 > μ 2
Definition 9 [49].
Suppose that there exist the conditions:
(1) 
T , T 1  and T 2  are three nonempty sets, q 1 is a real number.
(2) 
For any t T , t 1 T 1  and t 2 T 2 ,
0 M t 1 t M t 1 t + 1 ,   0 Z t 2 t Z t 2 t + 1   a n d   max t 1 T 1 M t 1 t + q + max t 2 T 2 Z t 1 t + q 1 .
(3) 
For any t T ,  M t = [ M t 1 t , M t 1 t + ] t 1 T t 1 and Z t = [ Z t 2 t , Z t 2 t + ] t 2 T t 2 .
An IVq-RDHFS X I V q D ˜ ˜ on T is characterized in the following symbol:
X I V q D ˜ ˜ = t , M t , Z t | t T
where Z t and M t are, respectively, called the nonmembership and membership degree values of t T to X I V q D ˜ ˜ , and the ordered pair M t , Z t in T is called IVq-RDHFE [Z11].

3. GIVqROHFSs

The GIVqROHFSs, the key feature of which is a set of the membership degree with several possible IVqROFNs, is defined. Some basic operational laws with respect to GIVqROHFSs are studied.
Definition 10.
Let t T = { t 1 , t 2 , , t n } ( n < + ) ,  [ M μ t , M μ t + ] [ 0 , 1 ] , [ Z μ t , Z μ t + ] [ 0 , 1 ]  and M μ t + q + Z μ t + q 1 , an GIVqROHFS on T is presented by the symbol:
X I V q ˜ ˜ = t , ψ X I V q ˜ ˜ t / t T
where ψ X I V q ˜ ˜ t consists of several IVqROFNs [ M μ t , M μ t + ] , [ Z μ t , Z μ t + ] μ ψ X I V q ˜ ˜ t  with condition M μ t + q + Z μ t + q 1 , which called an GIVqROHF element (GIVqROHFE), involving possible intervals of the MNMDs of t T to X I V q ˜ ˜ .
From Definition 10, there exist two considerations which need to be explained:
(1)
Provided that μ = ( [ M μ , M μ + ] , [ Z μ , Z μ + ] ) ψ X I V q ˜ ˜ t satisfies the condition
( M μ + ) + ( Z μ + ) 1   or   ( M μ + ) 2 + ( Z μ + ) 2 1 ,
ψ X I V q ˜ ˜ t , respectively, reduces to the IVIHFE. ψ X I V I ˜ ˜ t = [ M μ t , M μ t + ] , [ Z μ t , Z μ t + ] μ ψ X I V I ˜ ˜ t with the condition ( M μ + ) + ( Z μ + ) 1 [37] or the IVPHFE. ψ X I V P ˜ ˜ t = [ M μ t , M μ t + ] , [ Z μ t , Z μ t + ] μ ψ X I V P ˜ ˜ t with the condition ( M μ + ) 2 + ( Z μ + ) 2 1 [33], that is, the GIVqROHFS is an extension of the IVIHFS and IVPHFS, therefore it is also a generation of the HFS, IVHFS and IHFS.
(2)
Let [ M t 1 t , M t 1 t + ] t 1 T 1 , [ Z t 2 t , Z t 2 t + ] t 2 T 2 be an IVq-RDHFE, then one can obtain the GIVqROHFE [ M t 1 t , M t 1 t + ] , [ Z t 2 t , Z t 2 t + ] t 1 T 1 , t 2 T 2 which contains T 1 × T 2 pairs of intervals and does not make loss of information. On the contrary, when a GIVqROHFE [ M t 1 t , M t 1 t + ] , [ Z t 2 t , Z t 2 t + ] t 1 T 1 , t 2 T 2 is transformed into the IVq-RDHFE by the simple separation, due to the existence of the same membership degree interval and nonmembership degree interval in the interval pair set [ M t 1 t , M t 1 t + ] , [ Z t 2 t , Z t 2 t + ] t 1 T 1 , t 2 T t 2 , the following inequalities hold:
[ M t 1 t , M t 1 t + ] t 1 T 1 T 1 , [ Z t 2 t , Z t 2 t + ] t 2 T 2 T 2
Therefore, it is inevitable to lose the useful information, moreover, the GIVqROHFS is also the extension of the IVq-RDHFS according to Definition 9 and Definition 10.
Definition 11.
Let ψ k = ψ k ( t ) and ψ l = ψ l ( t ) be two GIVqROHFEs with m > 0 , some operations among GIVqROHFEs are defined as follows:
ψ k ψ l = M μ k q + 1 × M μ l q + 1 + 1 q , M μ k + q + 1 × M μ l + q + 1 + 1 q , Z μ k × Z μ l , Z μ k + × Z μ l + / μ k ψ k , μ l ψ l
ψ k ψ l = M μ k × M μ l , M μ k + × M μ l + , Z μ k q + 1 × Z μ l q + 1 + 1 q , Z μ k + q + 1 × Z μ l + q + 1 + 1 q / μ k ψ k , μ l ψ l
m ψ k = M μ k q + 1 m + 1 q , M μ k + q + 1 m + 1 q , Z μ k m , Z μ k + m / μ k ψ k ,   m ( 0 , + )
ψ k m = μ k m / μ k ψ k = M μ k m , M μ k + m , Z μ k q + 1 m + 1 q , Z μ k + q + 1 m + 1 q / μ k ψ k ,   m ( 0 , + )
ψ k ψ l = M μ k , M μ l , M μ k + , M μ l + , Z μ k , Z μ l , Z μ k + , Z μ l + / μ k ψ k , μ l ψ l
ψ k ψ l = M μ k , M μ l , M μ k + , M μ l + , Z μ k , Z μ l , Z μ k + , Z μ l + / μ k ψ k , μ l ψ l
ψ k c = Z μ k , Z μ k + , M μ k , M μ k + / μ k ψ k
Theorem 12.
If ψ k and ψ l are two GIVqROHFEs and m > 0 , then ψ k ψ l , ψ k m , m ψ k and ψ k ψ k are GIVqROHFEs.
Proof of Theorem 12.
Let ψ k = [ M μ k , M μ k + ] , [ Z μ k , Z μ k + ] / μ k ψ k ) , by Definition 10 and the definition of GIVqROFNs, we obtain [ M μ k , M μ k + ] , [ Z μ k , Z μ k + ] , [ M μ l , M μ l + ] , [ Z μ l , Z μ l + ] [ 0 , 1 ] , 1 M μ k + q Z μ k + q and 1 M μ l + q Z μ l + q .
(1) ψ k ψ l
0 M μ k q + 1 × M μ l q + 1 + 1 q M μ k + q + 1 × M μ l + q + 1 + 1 q 1
0 Z μ k × Z μ l Z μ k + × Z μ l + 1
M μ k + q + 1 × M μ l + q + 1 + 1 + Z μ k + q × Z μ l + q Z μ k + q × Z μ l + q + 1 + Z μ k + q × Z μ l + q = 1
(2) ψ k ψ l
0 M μ k × M μ l M μ k + × M μ l + 1
0 1 1 Z μ k q × 1 Z μ l q q 1 1 Z μ k + q × 1 Z μ l + q q 1
M μ k + q × M μ l + q + 1 1 Z μ k + q × 1 Z μ l + q M μ k + q × M μ l + q + 1 M μ k + q × M μ l + q = 1
(3) m ψ k ( m > 0 )
0 M μ k q + 1 m + 1 q M μ k + q + 1 m + 1 q 1 ,   0 Z μ k m Z μ k + m 1
M μ k + q + 1 m + 1 + Z μ k + m q Z μ k + q m + 1 + Z μ k + m q = 1
(4) ψ k m ( m > 0 )
0 M μ k m M μ k + m 1 ,   0 Z μ k q + 1 m + 1 q Z μ k + q + 1 m + 1 q 1
M μ k + m q Z μ k + q + 1 m + 1 M μ k + m q M μ k + q m + 1 = 1
Therefore, ψ k ψ l , ψ k m , m ψ k and ψ k ψ l are GIVqROHFEs. □
Theorem 13.
If ψ k and ψ l are two GIVqROHFEs and m > 0 , then GIVqROHFEs have properties.
ψ k ψ l = ψ l ψ k ,
( m l ) ψ k = m ( l ψ k )
m ψ k m ψ l = m ( ψ k ψ l )
ψ l ψ k = ψ k ψ l
ψ l ψ k = ψ k ψ l
ψ k m l = ( ψ k ) m l
( ψ k ) m + l = ψ k m ψ k l
m ψ k l ψ k = ( m + l ) ψ k
ψ k c ψ l c = ( ψ k ) ( ψ l ) c
ψ k c ψ l c = ( ψ k ) ( ψ l ) c
( ψ k ) ( ψ l ) c = ψ k c ψ l c
( ψ k ) ( ψ l ) c = ψ k c ψ l c
( ψ k c ) m = ( m ψ k ) c
m ( ψ k c ) = ( ψ k m ) c
For the proof of Theorem 13, refer to the Appendix A.
In order to rank the GIVqROHFEs, the comparison rules are presented as follows:
Definition 14.
Suppose that ψ k = [ M μ k , M μ k + ] , [ Z μ k , Z μ k + ] / μ k ψ k ( k = 1 , 2 ) are GIVqROHFEs. The score function of ψ k is defined:
V S ( ψ k ) = μ k ψ k v s ( μ k ) / # ψ k
where the cardinal number of ψ k is denoted as the symbol # ψ k .
The accuracy of ψ k is an equation:
V H ( ψ k ) = μ k ψ k v h ( μ k ) / # ψ k
ψ 1 and ψ 2 can be compared by the following order relation:
(1) V H ( ψ 1 ) < V H ( ψ 2 ) ψ 1 < ψ 2
(2) V S ( ψ 1 ) = V S ( ψ 2 ) , V H ( ψ 1 ) < V H ( ψ 2 ) ψ 1 < ψ 2
(3) V S ( ψ 1 ) = V S ( ψ 2 ) , V H ( ψ 1 ) = V H ( ψ 2 ) ψ 1 = ψ 2
(4) V S ( ψ 1 ) = V S ( ψ 2 ) , V H ( ψ 1 ) > V H ( ψ 2 ) ψ 1 > ψ 2
According to the above Definition 14, Theorem 15 can be proven.
Theorem 15.
If ψ = ( [ M μ , M μ + ] , [ Z μ , Z μ + ] ) / μ ψ is a GIVqROHFE and m > 0 , then the following inequalities hold:
(1) ψ ψ m , m ( 1 , + ) ; (2) ψ ψ m , m ( 0 , 1 ] ; (3) ψ m ψ , m ( 1 , + ) ; (4) ψ m ψ , m ( 0 , 1 ]
Proof of Theorem 15.
Since ψ = μ μ ψ = ( [ M μ , M μ + ] , [ Z μ , Z μ + ] ) / μ ψ is a GIVqROFE, we have:
M μ , M μ + , Z μ , Z μ + [ 0 , 1 ] ,   ( M μ + ) q + ( Z μ + ) q [ 0 , 1 ]
ψ m = μ m / μ ψ = ( M μ ) m , ( M μ + ) m , Z μ q + 1 m + 1 q , Z μ + q + 1 m + 1 q / μ ψ
When m > 1 , M μ q M μ m q , M μ + q M μ + m q , ( Z μ ) q ( Z μ ) q + 1 m + 1 and ( Z μ + ) q ( Z μ + ) q + 1 m + 1 . Thus, for any μ ψ , m > 1
v s ( μ ) = M μ q + M μ + q Z μ q Z μ + q / 2 1 2 M μ m q + M μ + m q 1 1 Z μ q m 1 1 Z μ + q m = v s ( μ m )
By the equation # ψ = # ψ m , V S ( ψ ) = μ ψ v s ( μ ) # ψ and V S ( ψ m ) = μ ψ v s ( μ m ) # ψ m , we have V S ( ψ ) V S ( ψ m ) . According to Definition 14, if V S ( ψ ) > V S ( ψ m ) ( m > 1 ) , then ψ > ψ m .
If V S ( ψ ) = μ ψ v s ( μ ) / # ψ = μ ψ v s ( μ m ) / # ψ m = V S ( ψ m ) , then we obtain the following results:
For any μ ψ , v s ( μ ) = M μ q + M μ + q Z μ q Z μ + q / 2 = M μ m q + M μ + m q + 1 Z μ q m 2 + 1 Z μ + q m / 2 = v s ( μ m )
From the conditions M μ q M μ m q , M μ + q M μ + m q , Z μ q 1 1 Z μ q m and Z μ + q 1 1 Z μ + q m , we have:
0 M μ q + M μ + q M μ m q M μ + m q = Z μ q + Z μ + q + 1 Z μ q m 2 + 1 Z μ + q m 0 ,   M μ q + M μ + q = M μ m q + M μ + m q ,   Z μ q + 1 Z μ q m + 1 Z μ + q m = 2 Z μ + q
Therefore, for any μ ψ ,
v h ( μ ) = W μ q + Z μ q + W μ + q + Z μ + q 2 = M μ m q + M μ + m q + 2 1 Z μ q m 1 Z μ + q m / 2 = v h ( μ m )
Consequently, V H ( ψ ) = μ ψ v h ( μ ) / # ψ = μ ψ v h ( μ m ) # ψ m = V H ( ψ m ) , i.e., ψ = ψ m , m ( 1 , + ) . □
From the order relationship of GIVqROHFEs, (1) ψ > ψ m , ( m > 1 ) is proven. By the similar inference, the other inequalities (2), (3) and (4) can be proven, which is omitted in this paper.

4. Aggregation Operators for the GIVqROHF Information

The GIVqROHFCA and GIVqROHFCGM Operators

Definition 16.
Let ψ k = ( [ M μ k , M μ k + ] , [ Z μ k , Z μ k + ] ) μ k ψ k be a GIVqROHFE with k 1 , 2 , , n , Y is a fuzzy measure on the power of { ψ 1 , , ψ k , , ψ n } , and a permutation { δ ( 1 ) , , δ ( k ) , , δ ( n ) } of { 1 , , k , , n } satisfies:
ψ δ ( 1 ) ψ δ ( 2 ) ψ δ ( n ) ,   λ > 0 , X δ ( 0 ) = ϕ , X δ ( i ) = { ψ δ ( j ) j i }
(1) 
A GIVqROHFCA operator is modeled.
G I V q R O H F C A λ ( ψ 1 , , ψ k , , ψ n ) = k = 1 n ψ δ ( k ) λ Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) 1 / λ
(2) 
A GIVqROHFCGM operator is a kind of following form.
G I V q R O H F C G M λ ( ψ 1 , , ψ k , , ψ n ) = 1 λ k = 1 n λ ψ δ ( k ) Y ( X δ ( k ) ) Y ( X δ ( k 1 ) )
From Definition 16, one can obtain a GIVqROHFS on the finite set 1 , 2 , , n which may present n attributes or n experts in multiple-attribute group decision making. Moreover, the GIVqROHF information, which needs generally to be considered in many practical problems, is characterized by this GIVqROHFS. On the basis of the above Definition 16, we discuss several cases of the GIVqROHFCA and GIVqROHFCGM operator:
(1)
If λ = 1 , the GIVqROHFCA and GIVqROHFCGM operator, respectively, reduce to the basic GIVqROHFCA operator abbreviated as IVqROHFCA and the basic GIVqROHFCGM operator with the symbol IVqROHFCGM.
I V q R O H F C A ( ψ 1 , , ψ k , , ψ n ) = k = 1 n ( ψ δ ( k ) Y ( X δ ( k ) ) ψ δ ( k ) Y ( X δ ( k 1 ) ) )
I V q R O H F C G M ( ψ 1 , , ψ k , , ψ n ) = k = 1 n ( ψ δ ( k ) ) ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) )
(2)
If k { 1 , 2 , , n } Y ( { ψ δ ( k ) } ) = Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) , the GIVqROHFCA operator is called the generalized IVqROHF weighted averaging (GIVqROHFWA) operator.
(3)
For a given k , Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) is an invariant constant w i for any permutation of { 1 , 2 , , n } , the GIVqROHFCA operator degenerates the generalized IVqROHF ordered weighted averaging (GIVqROHFOWA) operator. Especially, when Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) = 1 / n for any k , a GIVqROHFCA operator is a generalized IVqROHF mathematical averaging (GIVqROHFMA) operator.
Theorem 17.
Suppose that ψ k = [ M μ k , M μ k + ] , [ Z μ k , Z μ k + ] / μ k ψ k is a GIVqROHFE with k 1 , 2 , , n , then their aggregated values with respect to the GIVqROHFCA operator and the GIVqROHFCGM operator are both GIVqROHFEs which have the form:
G I V q R O H F C A λ ( ψ 1 , ψ 2 , , ψ n ) = k = 1 n ψ δ ( k ) λ Y ( X δ ( k ) ) ψ δ ( k ) λ Y ( X δ ( k 1 ) ) λ = k = 1 n M μ δ ( k ) q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ q , k = 1 n M μ δ ( k ) + q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ q , i = 1 n Z μ δ ( k ) q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + 1 q , i = 1 n Z μ δ ( k ) q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + 1 q μ δ ( k ) ψ δ ( k ) k 1 , 2 , , n
G I V q R O H F C G M λ ( ψ 1 , ψ 2 , , ψ n ) = 1 λ k = 1 n λ ψ δ ( k ) ( M ( T δ ( k ) ) M ( T δ ( k 1 ) ) ) = k = 1 n M μ δ ( k ) q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + 1 q , 1 1 k = 1 n M μ δ ( k ) + q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) λ q k = 1 n Z μ δ ( k ) q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ q , k = 1 n Z μ δ ( k ) + q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ q μ δ ( k ) ψ δ ( k ) k 1 , 2 , , n
For the proof of Theorem 17, refer to the Appendix A.
The GIVqROHFCA operator involves some properties such as idempotency, symmetry and boundedness.
Proposition 1 (Idempotency).
If ψ = [ M μ , M μ + ] , [ Z μ , Z μ + ] μ ψ is a GIVqROHFE then
G I V q R O H F C A λ ( ψ , ψ , , ψ ) = k = 1 n Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ψ λ 1 / λ = ψ
Proof of Proposition 1.
According to the operational properties of the GIVqROHFEs, Y ( X δ ( 0 ) ) = ϕ and M ( { ψ 1 , ψ 2 , , ψ n } ) = 1 ,
G I V q R O H F C A λ ( ψ , ψ , , ψ ) = k = 1 n ψ λ ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) 1 / λ = ψ λ i = 1 n ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) λ = ψ λ λ = ψ
 □
Proposition 2 (Symmetry).
If ψ k = ( [ M μ k , M μ k + ] , [ Z μ k , Z μ k + ] ) μ k ψ k is a GIVqROHFE with k 1 , 2 , , n , and a permutation { δ ( 1 ) , δ ( 2 ) , , δ ( n ) } of this sequence is provided, then
G I V q R O H F C A λ ( ψ 1 , ψ 2 , , ψ n ) = G I V q R O H F C A λ ( ψ δ ( 1 ) , ψ δ ( 2 ) , , ψ δ ( n ) )
Proof of Proposition 2.
Since the order of ψ 1 , , ψ k , ψ n is invariant, supposing that ψ δ ( 1 ) ψ δ ( k ) ψ δ ( n ) . Therefore,
G I V q R O H F C A λ ( ψ 1 , ψ 2 , , ψ n ) = k = 1 n ( Y ( X δ ( k ) ) ψ σ ( k ) λ Y ( X δ ( k 1 ) ) ψ σ ( k ) λ ) λ = G I V q R O H F C A λ ( ψ δ ( 1 ) , ψ δ ( 2 ) , , ψ δ ( n ) )
 □
Proposition 3 (Boundedness).
If n  GIVqROHFEs ψ k = [ M μ k , M μ k + ] , [ Z μ k , Z μ k + ] / μ k ψ k k = 1 , 2 , , n satisfy ψ δ ( 1 ) ψ δ ( k ) ψ δ ( n ) with any permutation { δ ( 1 ) , δ ( 2 ) , , δ ( n ) } concerning { 1 , 2 , , n } , and
ψ max = k = 1 n max j { M μ δ ( j ) } q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ q , k = 1 n max j { M μ δ ( j ) + } q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ q , k = 1 n ( min j { Z μ δ ( j ) } ) q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + 1 q , k = 1 n ( min j { Z μ δ ( j ) } ) q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + 1 q μ δ ( k ) ψ δ ( k ) k = 1 , 2 , , n ψ min = k = 1 n min j { M μ δ ( j ) } q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ q , k = 1 n min j { M μ δ ( j ) + } q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ q , k = 1 n ( max j { Z μ δ ( j ) } ) q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + 1 q , k = 1 n ( max j { Z μ δ ( j ) } ) q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + 1 q μ δ ( k ) ψ δ ( k ) k = 1 , 2 , , n
Then ψ min G I V q R O H F C A λ ( ψ 1 , , ψ k , , ψ n ) ψ max . □
Proof of Proposition 3.
By the method of mathematical analysis, when x ( 0 , 1 ) ,
d ( 1 x q λ ) ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) / d x = ( 1 x q λ ) ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) × q λ x q λ 1 × Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) < 0
d ( 1 ( 1 x q ) λ ) ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) / d x = ( 1 ( 1 x q ) λ ) ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) × ( λ ( 1 x q ) λ 1 ) × ( q x ) × ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) > 0
In other words, the function ( 1 x q λ ) ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) is decreasing on x , and 1 ( 1 x q ) λ ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) is increasing on x .
Thus,
k = 1 n ( max j { M μ δ ( j ) } ) q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ q k = 1 n ( M μ δ ( k ) ) q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ q
k = 1 n ( max j { M μ δ ( j ) + } ) q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ q k = 1 n ( M μ δ ( k ) + ) q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ q
1 1 k = 1 n 1 1 ( min j { Z μ δ ( j ) } ) q λ ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) λ q 1 1 k = 1 n 1 1 ( Z μ δ ( k ) ) q λ ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) λ q
1 1 k = 1 n 1 1 ( min j { Z μ δ ( j ) + } ) q λ ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) λ q 1 1 k = 1 n 1 1 ( Z μ δ ( k ) + ) q λ ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) λ q
On the basis of the inference method of Theorem 13, we obtain:
ψ min G I V q R O H F C A λ ( ψ 1 , ψ 2 , , ψ n ) ψ max
 □
Theorem 18.
If ψ k = [ M μ k , M μ k + ] , [ Z μ k , Z μ k + ] / μ k ψ k k { 1 , 2 , , n } are n  GIVqROHFEs and λ > 0 , then the GIVqROHFCA operator G I V q R O H F C A λ ( ψ 1 , ψ 2 , , ψ n ) is increasing with respect to λ , and the GIVqROHFGM operator G I V q R O H F C G M λ ( ψ 1 , ψ 2 , , ψ n ) is decreasing with respect to λ .
Proof of Theorem 18.
By Theorem 3.8 in [60], we obtain that the following formulas are increasing with respect to λ :
1 k = 1 n 1 M μ δ ( k ) q λ ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) λ q   1 k = 1 n 1 M μ δ ( k ) + q λ ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) λ q 1 k = 1 n 1 Z μ δ ( k ) q λ ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) λ q   1 k = 1 n 1 Z μ δ ( k ) + q λ ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) λ q
Moreover, the following formulas are decreasing with respect to λ :
k = 1 n ( M μ δ ( k ) ) q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + 1 q
k = 1 n ( M μ δ ( k ) + ) q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + 1 q
k = 1 n ( Z μ δ ( k ) ) q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + 1 q
k = 1 n ( Z μ δ ( k ) + ) q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + 1 q
Thus, according to Definition 14,
V S G I V q R O H F C A λ ψ 1 , , ψ k , , ψ n = # G I V q R O H F C A λ ψ 1 , , ψ k , , ψ n 1 2 μ δ ( k ) ψ δ ( k ) k = 1 , , n k = 1 n M μ δ ( k ) q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + k = 1 n Z μ δ ( k ) q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ 1 + # G I V q R O H F C A λ ψ 1 , , ψ k , , ψ n 1 2 μ δ ( k ) ψ δ ( k ) k = 1 , , n k = 1 n M μ δ ( k ) + q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + k = 1 n Z μ δ ( k ) + q λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ 1
The above equation is increasing with respect to λ , which illustrates that the GIVqROHFCA operator is increasing concerning λ .
Furthermore, V S G I V q R O H F C G M λ ψ 1 , , ψ k , , ψ n
= # G I V q R O H F C G M λ ψ 1 , , ψ k , , ψ n 1 2 μ δ ( k ) ψ δ ( k ) k = 1 , , n k = 1 n M μ δ ( k ) q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ k = 1 n Z μ δ ( k ) q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + 1 + # G I V q R O H F C G M λ ψ 1 , , ψ k , , ψ n 1 2 μ δ ( k ) ψ δ ( k ) , k = 1 , , n k = 1 n M μ δ ( k ) + q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ k = 1 n Z μ δ ( k ) + q + 1 λ + 1 ( Y ( X δ ( k ) ) Y ( X δ ( k 1 ) ) ) + 1 λ + 1
Therefore, it is decreasing on λ , which demonstrates that the GIVqROHFCGM operator is decreasing concerning λ . □
Theorem 19.
If ψ k k { 1 , 2 , , n } are n  GIVqROHFEs and λ > 0 , then the GIVqROHFCA, IVqROHFCA, GIVqROHFGM and GIVqROHFGM operators have order relations.
( 1 )   I V q R O H F C G M ( ψ 1 , , ψ k , , ψ n ) G I V q R O H F C A λ ( ψ 1 , , ψ k , , ψ n )
( 2 )   G I V q R O H F C G M λ ( ψ 1 , , ψ k , , ψ n ) G I V q R O H F C A λ ( ψ 1 , , ψ k , , ψ n )
( 3 )   G I V q R O H F C G M λ ( ψ 1 , , ψ k , , ψ n ) I V q R O H F C A ( ψ 1 , , ψ k , , ψ n )
( 4 )   I V q R O H F C G M ( ψ 1 , , ψ k , , ψ n ) I V q R O H F C A ( ψ 1 , , ψ k , , ψ n )
For the proof of Theorem 19, refer to the Appendix A.

5. A New Decision Method Based on GIVqROHFCA and GIVqROHFCGM Operators

In some decision problems with high uncertainty, due to the lack of experience, ability and knowledge, decision experts may hesitate to present their own evaluation regarding each alternative, and there does not exist sufficient information for the single-valued membership and nonmembership degrees; therefore, hesitant fuzzy evaluation strategy related to the interval-valued membership and nonmembership degrees should be a good choice. In addition, the experts are not independent, but interactive in decision making, consequently, it is important to reasonably model these interrelationships in efficient decision making. As a result, an approach is presented by using the proposed GIVqROHFCA and GIVqROHFCGM operators, which can handle some MAGDM problems under the GIVqROHF environment. In the whole framework model of this approach: (1) for each alternative, the hesitant evaluation of decision expert with respect to all attributes is defined as a GIVqROHF set on attribute set which can present the GIVqROHF information; (2) for each attribute, the hesitant assessment from all experts is modeled as a GIVqROHF set on expert set which may indicate the GIVqROHF uncertainty; (3) the basic operation unit of the GIVqROHFCA and GIVqROHFCGM operators is the GIVqROHFE, which is the basic composition unit of the aforementioned two types of GIVqROHFSs. For convenience, some symbols are denoted in Table 1.
Based on the above symbols, several steps are involved to describe the new method (see Figure 2).
  • Step 1. Obtaining the normalized decision-making matrix η m = ( η i j m ) n 1 × n 2 from the decision profile ζ m = ( ψ i j m ) n 1 × n 2 .
    η m = ( η i j m ) n 1 × n 2 = { ξ i j m ξ i j m η i j m } n 1 × n 2 = ( [ M ξ i j m , M ξ i j m + ] , [ Z ξ i j m , Z ξ i j m + ] ) ξ i j m η i j m n 1 × n 2
  • Step 2. Solving the following integer programming to determining the q-rung value q .
    min q s . t . 0 ( M ξ i j m + ) q + ( Z ξ u j m + ) q 1 q { 1 , 2 , } i { 1 , 2 , , n 1 } j { 1 , 2 , , n 2 } m { 1 , 2 , , l }
  • Step 3. Calculating values of the score function V S ( η i j m ) m = 1 , 2 , , l
    V S ( η i j m ) = ξ i j m η i j m v s ( ξ i j m ) / # η i j m = 1 2 ξ i j m η i j m [ M ξ i j m q + M ξ i j m + q Z ξ i j m q Z ξ i j m + q ] / # η i j m m = 1 , 2 , , l
    and the values of the accuracy function V H ( η i j m ) m = 1 , 2 , , l
    V H ( η i j m ) = ξ i j m η i j m v h ( ξ i j m ) / # η i j m = 1 2 ξ i j m η i j m [ M ξ i j m q + Z ξ i j m q + M ξ i j m + q + Z ξ i j m + q ] / # η i j m m = 1 , 2 , , l
  • Step 4. Determining the λ -fuzzy measure value g i of the decision expert D i i = 1 , 2 , , l , and solving the following equation to obtain the parameter value.
    γ 1 + 1 = i = 1 l [ 1 + γ 1 g i ]   or   γ 1 + 1 = i = 1 l [ 1 + γ 1 g i ]
    where γ 1 is determined by numerical calculation method.
  • Step 5. Ranking l GIVqROHFEs η i j 1 , η i j 2 , , η i j l such that η i j δ ( 1 ) η i j δ ( k ) η i j δ ( l ) , combining the ordered decision experts X δ ( p ) = { D δ ( 1 ) , , D δ ( k ) , , D δ ( p ) , } p { 1 , 2 , , l } and calculating λ -fuzzy measure value g ( X δ ( p ) ) , p = 1 , 2 , , l as the following formula:
    g ( X δ ( p ) ) = 1 γ 1 s = 1 p [ 1 + γ 1 g δ ( s ) ] 1 γ 1 0 s = 1 p g δ ( s ) γ 1 = 0
  • Step 6. Applying the GIVqROHFCA operator (16) Or the GIVqROHFCGM operator (17) in order to obtain the the collective GIVqROHF matrix η = ( η i j ) n 1 × n 2 with the following form:
    η i j = { ( [ M ξ i j , M ξ i j + ] , [ Z ξ i j , Z ξ i j + ] ) ξ i j η i j }
    η i j = G I V q R O H F C A λ 1 η i j 1 , η i j 2 , , η i j l = 1 s = 1 l ( 1 ( M ξ i j δ ( s ) ) q λ 1 ) ( g ( X δ ( s ) ) g ( X δ ( s 1 ) ) ) λ 1 q , 1 s = 1 l ( 1 ( M ξ i j δ ( s ) + ) q λ 1 ) ( g ( X δ ( s ) ) g ( X δ ( s 1 ) ) ) λ 1 q , 1 1 s = 1 l ( 1 ( 1 ( Z ξ i j δ ( s ) ) q ) λ 1 ) ( g ( X δ ( s ) ) g ( X δ ( s 1 ) ) ) λ 1 q , 1 1 s = 1 l ( 1 ( 1 ( Z ξ i j δ ( s ) + ) q ) λ 1 ) ( g ( X δ ( s ) ) g ( X δ ( s 1 ) ) ) λ 1 q ξ i j δ ( s ) η i j δ ( s ) s = 1 , 2 , , l
    η i j = G I V q R O H F C G M λ 2 η i j 1 , η i j 2 , , η i j l = 1 1 s = 1 l ( 1 ( 1 ( M ξ i j δ ( s ) ) q ) λ 2 ) ( g ( X δ ( s ) ) g ( X δ ( s 1 ) ) ) λ 2 q , 1 1 s = 1 l ( 1 ( 1 ( M ξ i j δ ( s ) + ) q ) λ 2 ) ( g ( X δ ( s ) ) g ( X δ ( s 1 ) ) ) λ 2 q , 1 s = 1 l 1 ( Z ξ i j δ ( s ) ) q λ ( g ( X δ ( s ) ) g ( X δ ( s 1 ) ) ) λ 2 q , 1 s = 1 l 1 ( Z ξ i j δ ( s ) + ) q λ ( g ( X δ ( s ) ) g ( X δ ( s 1 ) ) ) λ 2 q , ξ i j δ ( s ) η i j δ ( s ) s = 1 , 2 , , l
  • Step 7. Calculating the values of V S ( η i j ) to obtain score values of η i j j = 1 , 2 , , n 2
    V S ( η i j ) = 1 2 ξ i j η i j [ M ξ i j q Z ξ i j q + M ξ i j + q Z ξ i j + q ] # η i j
    and the values of the accuracy function V H ( η i j ) j = 1 , 2 , , n 2
    V H ( η i j ) = 1 2 ξ i j η i j [ W ξ i j q + Z ξ i j q + W ξ i j + q + Z ξ i j + q ] # η i j
  • Step 8. Provide the λ -fuzzy measure value ϑ i of the attribute C i i = 1 , 2 , , n 2 , and solving the equation as follows:
    γ 2 + 1 = i = 1 n 2 [ 1 + γ 2 ϑ i ]   or   γ 2 + 1 = i = 1 n 2 [ 1 + γ 2 ϑ i ]
  • Step 9. Ranking n GIVqROHFEs η i 1 , η i 2 , , η i n 2 such that:
    η i δ ( 1 ) η i δ ( k ) η i δ ( n 2 ) ,
    yielding the ordered attributes set X δ ( i ) = { C δ ( 1 ) ,   , C δ ( k ) , , C δ ( i ) , } i { 1 , 2 , , n 1 } and λ -fuzzy measure value ϑ ( X δ ( i ) ) , i = 1 , 2 , , n 1 as the following formula:
    ϑ ( X σ ( i ) ) = 1 γ 2 s = 1 i [ 1 + γ 2 ϑ δ ( s ) ] 1 γ 2 0 s = 1 i ϑ δ ( s ) γ 2 = 0
  • Step 10. Applying the GIVqROHFCA operator (16) or the GIVqROHFCGM operator (17) In order to derive the collective value η i = { ( [ M ξ i , M ξ i + ] , [ Z ξ i , Z ξ i + ] ) ξ i η i } of the alternative P i i = 1 , 2 , , n 1 .
    η i = G I V q R O H F C A λ 1 η i 1 , η i 2 , , η i n 2 = 1 s = 1 n 2 ( 1 ( M ξ i δ ( s ) ) q λ 1 ) ( ϑ ( X δ ( s ) ) ϑ ( X δ ( s 1 ) ) ) λ 1 q , 1 s = 1 n 2 ( 1 ( M ξ i δ ( s ) + ) q λ 1 ) ( ϑ ( X δ ( s ) ) ϑ ( X δ ( s 1 ) ) ) λ 1 q , 1 1 s = 1 n 2 ( 1 ( 1 ( Z ξ i δ ( s ) ) q ) λ 1 ) ( ϑ ( X δ ( s ) ) ϑ ( X δ ( s 1 ) ) ) λ 1 q , 1 1 s = 1 n 2 ( 1 ( 1 ( Z ξ i δ ( s ) + ) q ) λ 1 ) ( ϑ ( X δ ( s ) ) ϑ ( X δ ( s 1 ) ) ) λ 1 q ξ i δ ( s ) η i δ ( s ) s = 1 , 2 , , n 2
    η i = G I V q R O H F C G M λ 2 η i 1 , η i 2 , , η i n 2 = 1 1 s = 1 n 2