Complex Fuzzy Rough Aggregation Operators and their Applications in  for Multi-Criteria Group Decision-Making

Faiz Muhammad Khan; Naila Bibi; Saleem Abdullah; Azmat Ullah

doi:10.5391/IJFIS.2023.23.3.270

Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(3): 270-293

Published online September 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.3.270

Complex Fuzzy Rough Aggregation Operators and their Applications in for Multi-Criteria Group Decision-Making

Faiz Muhammad Khan¹, Naila Bibi^1,2, Saleem Abdullah³, and Azmat Ullah⁴

¹Department of Mathematics and Statistics, University of Swat, Khyber Pakhtunkhawa, Pakistan
²Government Girls Degree College, Swat, Pakistan
³Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan
⁴Department of Bio-Medical Sciences and Engineering, Graduate School of Sciences and Engineering, Koc University, Istanbul, Turkey

Correspondence to :
Naila Bibi (nailaazeemi963@gmail.com)

Received: March 11, 2023; Revised: July 31, 2023; Accepted: August 16, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

One of the notable advantages of the complex fuzzy set is its ability to incorporate not only satisfaction and dissatisfaction but also the absence of vague information in two-dimensional scenarios. By combining a fuzzy rough set with a complex fuzzy set, this study aims to provide a powerful and versatile tool for multi-criteria group decision-making (MCGDM) in complex and uncertain situations. This approach, based on EDAS (evaluation based on distance from average solution) method allows decision-makers to consider multiple criteria, account for uncertainty and vagueness, and make informed choices based on a wider range of factors. The main goal of this study is to introduce complex fuzzy (CF) rough averaging aggregation and geometric aggregation operators and embed these operators in EDAS to obtain remarkable results in MCGDM. Furthermore, we propose the CF rough weighted averaging (CFRWA), CF rough ordered weighted averaging (CFROWA), and CF rough hybrid averaging (CFRHA) aggregation operators. Additionally, we present the concepts of CF rough weighted geometric (CFRWG), CF rough ordered weighted geometric (CFROWG), and CF rough hybrid geometric (CFRHG) aggregation operators. A new score function is defined for the proposed method. The basic and useful aspects of the explored operators were discussed in detail. Next, a stepwise algorithm of the CFR-EDAS method is demonstrated to utilize the proposed approach. Moreover, a real-life numerical problem is presented for the developed model. Finally, a comparison of the explored method with various existing methods is discussed, demonstrating that the exploring model is more effective and advantageous than existing approaches.

Keywords: Complex fuzzy sets, Rough sets, Averaging and geometric operators, EDAS method, MCGDM

1. Introduction

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Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

Most information attributes are vague in this challenging technological era. This vague and uncertain information cannot be handled using classical set theory. This deficiency of classical set theory leads to fuzzy set theory, which was given by Zadeh [1]. In this competitive scenario, decision-making (DM) is more difficult when the information is imprecise. Traditionally, information on real-life problems is extensive in nature. It is becoming increasingly complicated owing to vague and imprecise information, which makes it difficult for a single decision-maker to make accurate decisions [2]. This breakthrough idea was a turning point in many fields, such as industrial control, human decision-making, and image processing.

Later, the traditional fuzzy set was generalized to complexity fuzzy sets (CFSs) by Ramot et al. [3, 4]. With this generalization, the range extends from the interval 0 ≤ y ≤ 1 to a unit circle in the complex plane. CFS has phase and amplitude terms, i.e., G = {(s, r_G (s) e_jγ^G⁽^s⁾) s ∈ Y}. Here, j=-1, r_G (s) ∈ [0, 1] represents the amplitude term and γ_G (s) ∈ [0, 2π] is the phase term. CFS can act as an ordinary fuzzy set by simply setting the phase term to zero. CFS is not only a simple extension of the traditional fuzzy set but also provides an intuitive extension for solving problems that are both complicated and unachievable. A CFS is an extended form of a fuzzy set, and its range extends from real to complex numbers with some imaginary qualities. CFS indicates a complex-valued membership grade that contains amplitude and phase terms. This implies that the CFS is more generalized than a classical fuzzy set. For example, if we are asked to obtain data with distance and direction simultaneously, then we may use CFS by specifying the distance and direction of the destination. Ramot et al. [4] initiated complex fuzzy (CF) operations and relations to deal with fuzzy information. Hu et al. [5, 6] presented the notion of approximate parallel and orthogonal relations of CFSs. Zhang et al. [7] introduced δ-equalities between CFSs. Bi et al. [8] presented two classes of entropy measures for CFSs. Hu and his colleagues [9, 10] and Alkouri and Salleh [11] developed several distance measures for CFSs. Tamir and Kandel [12] presented an axiomatic theory for complex fuzzy logic and classes. Liu et al. [13] measured the distance and cross-entropy on CFSs and their applications in decision-making. Dai [14] introduced a generalization of the rotational invariance in CF operations. Zahid et al. [15] introduced an ELECTRE-based method for group decision-making with complex spherical fuzzy information. Based on complex Pythagorean fuzzy information, Akram et al. [16] and Ma et al. [17] developed new approaches for multicriteria decision-making. Akram et al. [18] extended the VIKOR approach using a complex spherical fuzzy set for group decision-making.

However, with fuzzy sets, the decision-making process becomes simpler but still difficult in processes where multi-criteria attribute decision-making is required. This problem arises when only a single preference is required. Over the last few decades, aggregation operators have been introduced to address ambiguity. Different aggregation operators, such as average and geometric aggregation operators, are helpful tools for determining the best alternative in multicriteria group decision-making. Many authors have conducted appreciable research on the fundamentals of aggregation operators [19, 20]. The aggregation operator is a useful tool for combining multiple alternatives and selecting the best alternative. Essentially, aggregated information has remarkable value in multi-criteria group decision-making (MCGDM) for obtaining a concluding opinion. Wei and Lu [21] introduced Pythagorean fuzzy power aggregation operators in MCDM. Lui and Tang [22] proposed the neutrosophic fuzzy aggregation operator. Xu [23] proposed the notion of an intuitionistic fuzzy aggregation operator. To solve MCDM, Xu and Yager [24] introduced an ordered weighted aggregation (OWA) operator. In addition, Yager [25] proposed the notion of a generalized OWA operator. Bi et al. introduced the complex fuzzy arithmetic aggregation operator [23]. Over the past few decades, aggregation operators have utilized fuzzy information. Cholewa [26], Dubios and Koning [27] and Vanicek et al. [28] introduced aggregation operators and decision-making methods by using fuzzy averaging operators. To solve MCDM, using aggregation operators is not only useful in the field of fuzzy set theory but also has achieved remarkable results in more generalized forms of fuzzy sets, such as intuitionistic fuzzy sets [29], Pythagorean fuzzy sets [30], and q-rung orthopair fuzzy sets [31]. Seikh and Mandal [32] introduced the notion of intuitionistic fuzzy Dombi weighted averaging and geometric operators and utilized it in decision-making. Huang [33] used the concepts of the Hamacher t-norm and t-conorm to develop intuitionistic fuzzy Hamacher weighted averaging, ordered weighted averaging, and hybrid averaging operators and derived their important properties, which were investigated broadly.

Pawlak [34] is the innovator of rough set theory. Rough set theory is a new intelligent soft computing tool used for pattern recognition, attribute selection, conflicts between opinions, decision-making support, data mining, and discovering useful information in large datasets. It is an extension of classical set theory, which plays a vital role in intelligence systems characterized by imprecise and incomplete data. The main structure of a rough set depends on an approximation that can be induced in the upper and lower approximations. This theory soon evoked concern regarding the relationship between rough and fuzzy sets. Rough set theory [34] and fuzzy set theory [1] are the two main tools used to address information uncertainty. Dubois and Prade [35] were the pioneers among those investigating the fuzziness of rough sets. For rough complex fuzzy models, Sarwar et al. [36] defined the distance measure and δ-approximations. Using type-2 soft information, Sarwar and Akram [37] described certain hybrid rough models. A novel MCGDM approach based on rough soft approximations of graphs and hypergraphs was presented by Sarwar et al. [38].

Recently, complex fuzzy aggregation operators have been developed to aggregate complex fuzzy information. Fuzzy decision-making in a complex environment using a generalized aggregation operator was proposed by Merigo et al. [39]. Ramot et al. [3] used a vector aggregation operator for complex fuzzy information. Hu et al. [32] developed a power aggregation operator for complex fuzzy information. Ma et al. [40] investigated a product-sum aggregation operator and used it for multiple periodic factor predictions. Rani and Garg [41] studied power aggregation operators and ranking methods for complex fuzzy intuitionistic sets and their applications in decision-making. Garg and Rani [42] proposed generalized geometric aggregation operators based on t-norm operations for complex intuitionistic fuzzy sets and their uses in decision-making.

The method was proposed by Keshavarz Ghorabaee et al. [43], who solved decision-making problems using this method. The method plays a notable role in decision-making, particularly in situations where conflicts in criteria exist in MCGDM problems. Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) [44] and VIKOR [45] are the top MCDM methods. The TOPSIS method was introduced by Hwang et al. [46] and is based on the technique that the chosen alternative has the shortest distance from the positive ideal solution (PIS) and the longest distance from the negative ideal solution (NIS). In VIKOR, the alternatives are ranked, and the solution is the one closest to the ideal solution, while the method is based on calculating the best alternative from the list of possible options based on the positive distance from the average solution (PDAS) and negative distance from the average solution (NDAS), depending on the average solution (AS). PDAS and NDAS denote the differences between each solution and the AS. Therefore, the best one must have a larger PDAS value and a smaller NDAS value. Keshavarz Ghorabaee et al. [47] used the method on intuitionistic fuzzy information in supplier selection. Zhang et al. [48] used the method in MCGDM and developed a picture fuzzy weighted averaging and weighted geometric operator. Peng and Lui [49] proposed the neutrosophic soft decision approach using a similarity measure based on the method. Feng et al.[50] developed the method and applied it to hesitant fuzzy information. Li et al. [51] proposed the concept of a hybrid operator and its application in DM using the method. Liang [52] presented an extended form of the method in an intuitionistic fuzzy environment and its application in energy-saving projects. Kahraman et al. [53] used the method for site selection using intuitionistic fuzzy information. Illieva [54] introduced the concept of the method for MCGDM by using interval fuzzy information. Karasan and Kahraman [55, 56] proposed the method using interval-valued neutrosophic information. Stanujkic et al. [57] applied the notion of grey numbers to the method. The notion of a dynamic fuzzy approach was proposed by Keshavarz Ghorabaee et al. [58] for MCGDM based on the method. Stevic et al. [59] presented the method for the DM approach using fuzzy data. Keshavarz Ghorabaee et al. [60] proposed the concept of rank reversal and analyzed hybrid forms of the and TOPSIS methods.

1.1 Motivation of the Current Research

The primary motivations for this study are as follows:

• The first aspect that motivates this research is the limitations of existing fuzzy rough set and aggregation operator approaches in solving MCGDM problems. While these approaches have been useful, they have certain limitations in representing complex decision-making scenarios and dealing with uncertainty of a periodic nature. This motivates the exploration of complex fuzzy rough sets and complex fuzzy rough aggregation operators as potential solutions to overcome these limitations and enhance the decision-making process.
• The second aspect that motivates this research is the practical need for effective decision-making in daily-life MCGDM examples. The present difficult situations involve applying these concepts and methods to real-world scenarios and providing practical solutions. This motivates the research to not only propose new theoretical frameworks but also demonstrate their applicability and effectiveness in solving real-life decision-making problems.

This study makes four key contributions:

• This study proposes the concept of complex fuzzy rough sets as an extension to fuzzy rough sets. This introduces a two-dimensional approach to fuzzy rough sets, enabling a more comprehensive representation of complex decision-making problems.
• A CF rough aggregation is proposed as a method to handle the uncertainty of a periodic nature in MCGDM problems.
• Applying these aggregation operators within the EDASM, which is a well-established method in decision-making, further strengthens the decision-making process. The combination of sophisticated aggregation operators and established EDASM methodology is likely to produce remarkable results in MCGDM.
• The effectiveness of the proposed modified-EDAS method in determining the suitability of a site for fish culture and selecting the best location for a fish farm is demonstrated.

2. Basic Concepts

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Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

First, we recall some basic definitions, such as the complex fuzzy set, its basic operations, equivalence relation, and fuzzy rough set. These definitions provide a basis for the following sections.

Definition 1 ([3, 4])

Let G be a non-empty set. G is said to be a CFS on a universe of discourse Y and is defined as

G={(s,rG(s)ejγG(s))∣s∈Y}.

Here, j=-1, r_G(s) ∈ [0, 1] represents the amplitude term, and γ_G(s) ∈ [0, 2π] is the phase term. In addition, r_G(s) denotes the membership value of an element of the CFS G.

Definition 2

Let u = r_ue^jπϕ_u and v = r_ve^jπϕ_v be any two CFSs. Their average can be defined as

Avg=u+v2={ru+rv2}ejπ(φu+φv2).

Definition 3

Consider a universal setℳ, and let P ∈ℳ× ℳ be any relation. Then,

1. P is said to be reflexive if (u, u) ∈ P, ∀ u∈ℳ.
2. P is said to be symmetric if ∀u, v ∈ℳ, (u, v) ∈ P then (v, u) ∈ P.
3. P is said to be transitive if ∀u, v, x ∈ ℳ, (u, v) ∈ P and (v, x) ∈ P, then (u, x) ∈ P.

Definition 4

A relation is said to be an equivalence relation if it is reflexive, symmetric, and transitive.

Definition 5

Let U be a non-empty and finite universe of discourse and ℛ be a fuzzy equivalence relation defined on U × U. The pair (U,ℛ) is called a fuzzy approximation space. For any A ∈ F(U), the upper and lower approximations with respect to (U,ℛ) are denoted by ℛ_(A) and ℛ¯(A), respectively, and are two fuzzy sets defined as

R_(A)={(x,μR_(A)(x))∣x∈U},R¯(A)={(x,μR¯(A)(x))∣x∈U},

where

μR_(A)(x)=∧u∈U((1-μR(x,u))∨μA(u)), x∈U,μR_(A)(x)=∨u∈U(μR(x,u))∧μA(u)), x∈U.

The pair ℛ(A) = (ℛ_(A), ℛ¯(A)) is called the fuzzy rough set of A with respect to (U,ℛ).

3. Construction of Complex Fuzzy Rough Sets

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Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

In this section, we will present the notion of a complex fuzzy equivalence relation, score function, and a complex fuzzy rough set and its properties. We also solve related examples.

Definition 6

Let ℳ be a universal set and P ∈ CFS(ℳ× ℳ) be a CF equivalence relation. Then,

1. P is reflexive if μ_P(u, u) = 1, ∀u∈ℳ.
2. P is symmetric if for all (u, v) ∈ (ℳ×ℳ), μ_P(u, v) = μ_P(v, u).
3. P is transitive if for all (u, v)∈ℳ × ℳ, μ_P(u, v) ≥ ∨_x_∈_ℳ [μ_P(u, x) ∧ μ_P(x, v)]

Definition 7

Let ℳ be the universal set and P ∈ CF(ℳ× ℳ) be any complex fuzzy relation. Then, the pair (ℳ,P) is called a complex fuzzy approximation space (CFAS). Now, for any complex fuzzy set I ∈ CF(ℳ), the lower and upper approximations of I with respect to (ℳ,P) are defined as P(I)=<P_(I),P¯(I)>, where

P_(I)={<y,μP_(I)(y)ejπγ(x)> ∣y∈M},P¯(I)={<y,μP¯(I)(y)ejπψ(x)> ∣y∈M},

γ(x), ψ(x) ∈ [0, 1], and j=-1.

Also,

μP_(I)(y)=∧u∈M[(1-μP(y,u))∨μP(u)]×ejπ[(1-(γP(x),rP(x)))∨rP(x)],y∈M,μP¯(I)(y)=∨u∈M[μP(y,u)∧μP(u)]×ejπ[(ψP(x),rP(x))∧rP(x))],y∈M.

Example 1

Considerℳ = {y₁, y₂, y₃, y₄} and I = {(y₁, 0.4e^jπ^(0.3)), (y₂, 0.3e^jπ^(0.2)), (y₃, 0.6e^jπ^(0.4)), (y₄, 0.5e^jπ^(0.2))}, and let P be an equivalence class on CF(ℳ×ℳ) as Table 1.

By simple calculation, we can easily find the upper and lower approximations as follows:

P_(I)={y10.4ejπ(0.5),y20.3ejπ(0.7),y30.4ejπ(0.2),y40.4ejπ(0.2)},P¯(I)={y10.6ejπ(0.4),y20.4ejπ(0.2),y30.6ejπ(0.3),y40.6ejπ(0.4)}.

Definition 8

Let P1(I)=(P1_(I),P1¯(I)) and P2(I)=(P2_(I),P2¯(I)) be any two complex fuzzy rough sets. Then, we have the following operations:

1. P1(I)⊕P2(I)=(P1_(I)⊕P2_(I),P1¯(I)⊕P2¯(I)), where P1_(I)⊕P2_(I)=(μ_P1+μ_P2-μ_P1.μ_P2)×ejπ(r_1+r_2-r_1.r_2) and P1¯(I)⊕P2¯(I)=(μ¯P1+μ¯P2-μ¯P1.μ¯P2)ejπ(r_1+r_2-r_1.r_2).
2. P1(I)⊕P2(I)=(P1_(I)⊗P2_(I),P1¯(I)⊗P2¯(I)) where (P1_(I)⊗P2_(I),P1¯(I)⊗P2¯(I))=((μ_P1.μ_P2)×e(r_1.r_2),(μ¯P1.μ¯P2)e(r¯1.r¯2).
3. P1(I)⊕*P2(I)=(P1_(I)⊕P2_(I),P1¯(I)⊗P2¯(I)).
4. P1(I)⊗*P2(I)=(P1_(I)⊗P2_(I),P1¯(I)⊕P2¯(I)).
5. P1(I)∩P2(I)=(P1_(I)∩P2_(I),P1¯(I)∩P2¯(I)).
6. P1(I)∪P2(I)=(P1_(I)∪P2_(I),P1¯(I)∪P2¯(I)).
7. P1(I)⊆P2(I)⇒P1_(I)⊆P2_(I),P1¯(I)⊆P2¯(I).
8. λP1(I)=(λP1_(I),λP1¯(I)), where λP1_(I)=(1-(1-μ_P1(I))λ)ejπ(1-(1-r1)λ) and λP1¯(I)=(1-(1-μ_P1(I))λ)ejπ(1-(1-r1)λ).
9. P1(I)λ=(P1_(I)λ,P1¯(I)λ).
10. P1(I)C=(P1_(I)C,P1¯(I)C).

such that (P1_(I)C) and (P1¯(I)C) are the complements of the complex fuzzy rough approximation operator.
11. P1(I)=P2(I)⇔P1_(I)=P2_(I) and P1¯(I)=P2¯(I).

For the comparison of two complex fuzzy rough values (CFRVs), we use a score function. The smaller the score value of CFRVs, the more inferior that value is, and vice versa.

Definition 9

The score function for CFRV P(I)=(P_(I),P¯(I))=(μ_ejπr,μ¯ejπr′) is given as

S(P(I))=14(2+μ_+μ¯)ejπ14(2+r+r′).

4. Complex Fuzzy Rough Averaging Aggregation Operator

Go to

Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

This section presents complex aggregation operators by applying the idea of rough sets to obtain the aggregation concept of complex fuzzy rough weighted averaging (CFRWA), complex fuzzy rough order weighted averaging (CFROWA), and complex fuzzy rough hybrid averaging (CFRHA) operators. We will also discuss some basic properties of these operators.

4.1 Complex Fuzzy RoughWeighted Averaging Operator

Here, we discuss the concept of the CFRWA operator and its properties.

Definition 10

Consider the collection of CFRVs P(I1),P(I2),P(I3),…,P(In), where w = (w₁, w₂, w₃, ..., w_n)^T represents a weight vector such that ∑i=1nwi=1, 0 ≤ w_i ≤ 1. Then, the CFRWA operator is defined as

CFRWA(P(I1),P(I2),P(I3),…,P(In))=(⊕i=1nwiP_(Ii),⊕i=1nwiP¯(Ii)).

Based on this definition, the following Theorem is given for aggregated results.

Theorem 1

Consider the collection P(Ii)=(P_(Ii),P¯(Ii)), i = 1, 2, 3, ..., n of CFRVs with weighted vectors w = (w₁, w₂, w₃, ..., w_n)^T such that ∑i=1nwi=1, 0 ≤ w_i ≤ 1. Then, the CFRWA operator is determined as

CFRWA(P(I1),P(I2),P(I3),…,P(In))=(⊕i=1nwiP_(Ii),⊕i=1nwiP¯(Ii))=[(1-∏i=1n(1-μi_)wi) ejπ(1-∏i=1n(1-γi(x))wi),(1-∏i=1n(1-μi¯)wi) ejπ(1-∏i=1n(1-γi(x))wi)].

Proof

We will prove the desired result by mathematical induction. According to the defined operation, we have

P(I1)⊕P(I2)=[P_(I1)⊕P_(I2),P¯(I1)⊕P¯(I2)]=[{μ_1+μ_2-μ_1μ_2}ejπ(γ1+γ2-γ1γ2),{μ¯1+μ¯2-μ¯1μ¯2} ejπ(γ1+γ2-γ1γ2)],

and

λP(I1)=[λP_(I1),λP¯(I1)]=[(1-(1-μ_1)λ) ejπ(1-(1-γ1(x))λ),(1-(1-μ¯1)λ)ejπ(1-(1-γ1(x))λ)].

Let us consider n = 2. Then, we have

CFRWA(P(I1),P(I2))=(⊕i=12wiP_(Ii),⊕i=12wiP¯(Ii))=[(1-∏i=12(1-μi_)wi) ejπ ((1-∏i=12(1-γi(x))wi),(1-∏i=12(1-μi¯)wi) ejπ (1-∏i=12(1-γi(x))wi)].

Hence, this result holds for n = 2. Now, we consider that the result is true for n = k:

CFRWA(P(I1),P(I2),P(I3),…,P(Ik))=(⊕i=1kwiP_(Ii),⊕i=1kwiP¯(Ii))=[(1-∏i=1k(1-μi_)wi) ejπ ((1-∏i=1k(1-γi(x))wi),(1-∏i=1k(1-μi¯)wi) ejπ (1-∏i=1k(1-γi(x))wi)].

Now, we will prove the result is true for n = k + 1. We have

CFRWA(P(I1),P(I2),P(I3),…,P(Ik),P(Ik+1))=[(⊕i=1kwiP_(Ii)⊕wk+1P_(Ik+1)),(⊕i=1kwiP¯(Ii)⊕wk+1P¯(Ik+1))]=[(1-∏i=1k+1(1-μi_)wi) ejπ ((1-∏i=1k+1(1-γi(x))wi),(1-∏i=1k+1(1-μi¯)wi) ejπ (1-∏i=1k+1(1-γi(x))wi)]

Thus, the required result is true for n = k + 1, and the statement holds true for n ≥ 1.

From above observation, we see that P_(I) and P¯(I) are CFRVs. Therefore, according to Definition 10, ⊕i=1nwiP_(Ii) and ⊕i=1nwiP¯(Ii) are also CFRVs. Therefore, CFRWA(P(I1),P(I2),P(I3),…P(Ik)) is also a complex fuzzy rough set under CF approximation space (M,P).

Example 2

Consider

I⊆M={(y1_,0.5ejπ(0.3)),(y1¯,0.4ejπ(0.2)),(y2_,0.3ejπ(0.2)),(y2¯,0.4ejπ(0.32)),(y3_,0.6ejπ(0.4)),(y3¯,0.7ejπ(0.5))},

with weight vector w = (0.3, 0.2, 0.4)^T. Then, by Theorem 1, we have

CFRWA[P(I1),P(I2),P(I3)]=(⊕i=13wiP_(Ii),⊕i=13wiP¯(Ii))={[(1-(1-0.5)0.3(1-0.3)0.2(1-0.6)0.4)×ejπ(1-(1-0.3)0.3(1-0.2)0.2(1-0.4)0.4)],[(1-(1-0.4)0.3(1-0.4)0.2(1-0.7)0.4)×ejπ(1-(1-0.2)0.3(1-0.32)0.2(1-0.5)0.4)]}=(0.47ejπ(0.29),0.52ejπ(0.34)).

Some important characteristics of the CFRWA operator are given by Theorem 2.

Theorem 2

Consider the collection P(Ii)=(P_(Ii),P¯(Ii)), i = 1, 2, 3, ..., n, of CFRVs with weighted vectors w = (w₁, w₂, w₃, ..., w_n)^T such that ∑i=1nwi=1, 0 ≤ w_i ≤ 1. Then, the properties of the CFRWA operator are described as

i. Idempotency: If P(Ii)=L(K)∀i=1,2,3,…,n, where L(K)=(L_(K),L¯(K))=(h_ejπr,h¯ejπr), then, CFRWA(P(I1),P(I2),P(I3),…,P(In))=L(K).
ii. Boundedness: Let (P(I))−=(miniP_(Ii),maxiP¯(Ii)) and (P(I))+=(maxiP_(Ii),miniP¯(Ii)). Then, (P(I))−≤CFRWA(P(I1),P(I2),P(I3),…,P(In))≤P(I))+.
iii. Shift invariance: Consider CFRV B(K)=(B_(K),B¯(K)). Then, CFRWA(P(I1)⊕B(K),P(I2)⊕B(K),P(I3)⊕B(K),…,P(In)⊕B(K))=CFRWA(P(I1),P(I2),P(I3),…,P(In))⊕B(K).
iv. Monotonicity: Let F(Li)=(F_(Li),F¯(Li)), (i = 1, 2, 3, ..., n) be another collection of CFRVs such that F_(Li)≤P_(Ii) and F¯(Li)≤P¯(Ii). Then, CFRWA(F(L1),F(L2),F(L3),…,F(Ln))≤CFRWA(P(I1),P(I2),P(I3),…,P(In)).
v. Homogeneity: CFRWA(δP(I1),δP(I2),δP(I3),…,δP(In)) = δCFRWA(P(I1),P(I2),P(I3),…,P(In)) for any δ > 0.
vi. Commutativity: Let P′ (Ii)=(P′_(Ii),P′¯(Ii)), i = 1, 2, 3, ..., n, be any permutation of P(Ii)=(P_(Ii),P¯(Ii)). Then, CFRWA(P(I1),P(I2),P(I3),…,P(In))=CFRWA(P′(I1),P′(I2),P′(I3),…,P′(In)).

Proof

i. Idempotency: As

CFRWA(P(I1),P(I2),P(I3),…,P(In)=(⊕i=1nwiP_(Ii),⊕i=1nwiP¯(Ii))=[(1-∏i=1n(1-μi_)wi) ejπ ((1-∏i=1n(1-γi(x))wi),(1-∏i=1n(1-μi¯)wi) ejπ (1-∏i=1n(1-γi(x))wi)].

Moreover, for all i, P(Ii) = L(K), where L(K) = (L(K), L(K)) = (he^jπr, h̄e^jπr). Therefore,

CFRWA(P(I1),P(I2),P(I3),…,P(In)=[(1-∏i=1n(1-h_)wi) ejπ (1-∏i=1n(1-r)wi),(1-∏i=1n(1-h¯)wi) ejπ (1-∏i=1n(1-r)wi)]=[(1-∏i=1n(1-h_)) ejπ (1-∏i=1n(1-r)),(1-∏i=1n(1-h¯)) ejπ (1-∏i=1n(1-r))]=(h_ejπr,h¯ejπr)=(L_(K),L¯(K))=L(K).

ii. Boundedness: Let (P(I))−=mini(P_(Ii)) and (P(I))+=maxi(P¯(Ii)). Then, we will prove that

(P(I))-≤CFRWA(P(I1),P(I2),P(I3),…,P(In))≤(P(I))+.

As (P_(I))− = (min_i(μ_i)e^jπ^min^_i⁽^γ_i⁾, min_i(μ̄_i)e^jπ^min^_i⁽^γ_i⁾), (P_(I))+ = (max_i(μ_i)e^jπ^max^_i⁽^γ_i⁾, max_i(μ̄_i)e^jπ^max^_i⁽^γ_i⁾). For each i = 1, 2, 3, ..., n, we have

mini(μi_)ejπmini(γi)≤(μi_)ejπ(γi)≤maxi(μi_)ejπmaxi(γi)⇔{1-maxi(μi_)}ejπ(1-maxi(γi))≤{1-(μi_)}ejπ(1-γi) ≤ {1-mini(μi_)}ejπ(1-mini(γi))⇔∏i=1n{1-maxi(μi_)}wiejπ∏i=1n(1-maxi(γi))wi ≤∏i=1n{1-(μi_)}wiejπ∏i=1n(1-(γi))wi ≤∏i=1n{1-mini(μi_)}wiejπ∏i=1n(1-mini(γi))wi,⇔{1-maxi(μi_)}ejπ(1-maxi(γi)) ≤∏i=1n{1-(μi_)}wiejπ(1-(γi))wi ≤ {1-mini(μi_)}ejπ(1-mini(γi)),⇔(1-{1-mini(μi_)})ejπ(1-(1-mini(γi))) ≤ (1-∏i=1n{1-(μi_)}wi)ejπ(1-∏i=1n(1-γi)wi) ≤(1-{1-maxi(μi_)})ejπ(1-(1-maxi(γi))),⇒mini(μi_)ejπ(mini(γi)) ≤(1-∏i=1n{1-(μi_)}wi)ejπ(1-∏i=1n(1-(γi))wi) ≤maxi(μi_)ejπ(maxi(γi)).

Similarly, we can show that, for upper approximation,

mini(μi¯)ejπ(mini(γi))≤(1-∏i=1n{1-(μi¯)})wiejπ(1-∏i=1n(1-(γi)wi))≤maxi(μi¯)ejπ(maxi(γi)).

iii. Shift invariance: Let ℬ(K) = (ℬ(K), ℬ(K)) = (he^jπ⁽^v⁾, h̄e^jπ⁽^v⁾) be any CFRV and P(Ii)=(P_(Ii),P¯(Ii))=(μi_ejπ(γi),μi¯ejπ(γi)) be the collection of CFRVs. Thus,

P(I1)⊕ℬ(K)=(P_(I1)⊕ℬ_(K),P¯(I1)⊕ℬ¯(K))=[(1-∏i=1n(1-μi_)wi(1-h_)wi) ejπ (1-∏i=1n(1-γi)wi(1-v)wi),(1-∏i=1n(1-μi¯)wi(1-h¯)wi) ejπ (1-∏i=1n(1-γi)wi(1-v)wi)]=[(1-(1-h_)∏i=1n(1-μi_)wi) ejπ (1-(1-v)∏i=1n(1-γi)wi),(1-(1-h¯)∏i=1n(1-μi¯)wi) ejπ (1-(1-v)wi∏i=1n(1-γi)wi)]=[(1-∏i=1n(1-μi_)wi) ejπ (1-∏i=1n(1-γi) )wi)⊕h_,(1-∏i=1n(1-μi¯)wi) ejπ (1-∏i=1n(1-γi))wi)⊕h¯]=[(1-∏i=1n(1-μi_)wi) ejπ (1-∏i=1n(1-γi) )wi),(1-∏i=1n(1-μi¯)wi) ejπ (1-∏i=1n(1-γi))wi)⊕[h_,h¯]]=CFRWA(P(I1),P(I2),P(I3),…,P(In))⊕ℬ(K).

Thus, the proof is complete.

iv. Monotonicity: As F(Li)=(F_(Li),F¯(Li))=(hi_ejπ(vi),hi¯ejπ(vi)) and P(Ii)=(P_(Ii),P¯(Ii)) for (i = 1, 2, 3, ..., n), we have to show that F(L_i) ≤ (P_(Ii) and F(L_i) ≤ (P¯(Ii). Then,

(hi_)ejπ(vi)≤(μi_)ejπ(γi),(1-μi_)wiejπ(1-γi)wi≤(1-hi_)wiejπ(1-vi)wi,(∏i=1n(1-μi_)wi) ejπ(∏i=1n(1-γi)wi)≤(∏i=1n(1-hi_)wi) ejπ(∏i=1n(1-vi)wi),(1-∏i=1n(1-hi_)wi) ejπ(1-∏i=1n(1-vi)wi)≤(1-∏i=1n(1-μi_)wi) ejπ(1-∏i=1n(1-γi)wi).

Similarly, we can show that

(1-∏i=1n(1-hi¯)wi) ejπ(1-∏i=1n(1-vi)wi)≤(1-∏i=1n(1-μi¯)wi) ejπ(1-∏i=1n(1-γi)wi)

Hence, from above, F(L_i) ≤ (P_(Ii) and F(L_i) ≤ (P¯(Ii). Therefore,

CFRWA(F(L1),F(L2),F(L3),…,F(Ln))≤CFRWA(P(I1),P(I2),P(I3),…,P(In)).

v. Homogeneity: Take any real number δ > 0 and CFRV

P(Ii)=(P_(Ii),P¯(Ii)),δP(Ii)=(δP_(I1),δP¯(I1))=[1-(1-μ1_)δejπ(1-(1-γ1)δ),1-(1-μ1¯)δejπ(1-(1-γ1)δ)],CFRWA(δP(I1),δP(I2),δP(I3),…,δP(In))=[(1-∏i=1n((1-μi_)δ)wi) ejπ (1-∏i=1n((1-γi)δ)wi),(1-∏i=1n((1-μi¯)δ)wi) ejπ (1-∏i=1n((1-γi)δ)wi)]=[(1-∏i=1n((1-μi_)wi)δ) ejπ (1-∏i=1n((1-γi)wi)δ),(1-∏i=1n((1-μi¯)wi)δ) ejπ (1-∏i=1n((1-γi)wi)δ)]=δCFRWA(P(I1),P(I2),P(I3),…,P(In)).

This is the required proof.

vi. Commutativity: Consider the following:

CFRWA(P(I1),P(I2),P(I3),…,P(Ik))=(⊕i=1kwiP_(Ii),⊕i=1kwiP¯(Ii))=[(1-∏i=1k(1-μi_)wi) ejπ(1-∏i=1k(1-γi)wi),(1-∏i=1k(1-μi¯)wi) ejπ(1-∏i=1k(1-γi)wi)].

Because (P′(I1),P′(I2),P′(I3),…,P′(Ik)) is any permutation of (P(I1),P(I2),P(I3),…,P(In)), we have P(Ii)=P′(Ii), (i = 1, 2, 3, ..., n)

=[(1-∏i=1k(1-μi′_)wi) ejπ(1-∏i=1k(1-γi′)wi),(1-∏i=1k(1-μi′¯)wi) ejπ(1-∏i=1k(1-γi′)wi)]=(⊕i=1kwiP′_(Ii),⊕i=1kwiP′¯(Ii))=CFRWA(P′(I1),P′(I2),P′(I3),…,P′(Ik)).

4.2 Complex Fuzzy Rough Ordered Weighted Averaging Operator

In this subsection, we discuss the CFROWA operator in detail, along with its properties.

Definition 11

Consider the collection of CFRVs P(I1),P(I2),P(I3),…,P(In), where w = (w₁, w₂, w₃, ..., w_n)^T represents a weight vector such that ∑i=1nwi=1, 0 ≤ w_i ≤ 1. Then, the CFROWO operator is defined as CFROWA(P(I1),P(I2),P(I3),…,P(In))=(⊕i=1nwiPη_(Ii),⊕i=1nwiPη¯(Ii)).

Based on Definition 11, the following theorem is given for the CFROWA operator.

Theorem 3

Consider the collection P(Ii)=(P_(Ii),P¯(Ii)), i = 1, 2, 3, ..., n of CFRVs associated with weighted vectors w = (w₁, w₂, w₃, ..., w_n)^T such that ∑i=1nwi=1, 0 ≤ w_i ≤ 1. Then, the CFROWA operator is determined as

CFROWA(P(I1),P(I2),P(I3),…,P(In))=(⊕i=1nwiPη_(Ii),⊕i=1nwiPη¯(Ii))=[(1-∏i=1n(1-μηi_)wi) ejπ(1-∏i=1n(1-γρi(x))wi),(1-∏i=1n(1-μηi¯)wi) ejπ(1-∏i=1n(1-γρi(x))wi)],

where Pη(Ii)=(Pη_(Ii),Pη¯(Ii)) represents the largest value in the given collection of CFRVs.

Proof

The proof follows from Theorem 1.

Some important properties of the CFROWA operator are given by Theorem 4.

Theorem 4

Consider the collection P(Ii)=(P_(Ii),P¯(Ii)), i = 1, 2, 3, ..., n of CFRVs with weighted vectors w = (w₁, w₂, w₃, ..., w_n)^T such that ∑i=1nwi=1, 0 ≤ w_i ≤ 1. Then, the properties of the CFROWA operator are as follows:

i. Idempotency: If P(Ii) = L(K) ∀ i = 1, 2, 3, ..., n, where L(K) = (L(K), L(K)) = (he^jπr, h̄e^jπr), then

CFROWA(P(I1),P(I2),P(I3),…,P(In))=L(K).
ii. Boundedness: Let (P(I))−=(miniP_(Ii),maxiP¯(Ii)) and (P(I))+=(maxiP_(Ii),miniP¯(Ii)). Then,

(P(I))-≤CFROWA(P(I1),P(I2),P(I3),…,P(In))≤(P(I))+.
iii. Shift invariance: Consider CFRVs ℬ(K) = (ℬ(K), ℬ(K)). Then,

CFROWA(P(I1)⊕ℬ(K),P(I2)⊕ℬ(K),P(I3)⊕ℬ(K),…,P(In)⊕ℬ(K))=CFROWA(P(I1),P(I2),P(I3),…,P(In))⊕ℬ(K)
iv. Monotonicity: Let F(L_i) = (F(L_i), F(L_i)), (i = 1, 2, 3, ..., n) be another collection of CFRVs such that F(L_i) ≤ P_(Ii) and F(L_i) ≤ P¯(Ii). Then,

CFROWA(F(L1),F(L2),F(L3),.......,F(Ln))≤CFROWA(P(I1),P(I2),P(I3),…,P(In)).
v. Homogeneity:

CFROWA(δP(I1),δP(I2),δP(I3),…,δP(In))=δCFROWA(P(I1),P(I2),P(I3),…,P(In)),

for any δ > 0.
vi. Commutativity: Let P′(Ii)=(P′_(Ii),P′¯(Ii)), i = 1, 2, 3, ..., n be any permutation of P(Ii)=(P_(Ii),P¯(Ii)). Then,

CFROWA(P(I1),P(I2),P(I3),…,P(In))=CFROWA(P′(I1),P′(I2),P′(I3),…,P′(In)).

Proof

The proof follows from Theorem 2.

4.3 Complex Fuzzy Rough Hybrid Averaging Operator

In the following subsection, we discuss the CFRHA operator. The CFRHA operator weights both the values and the ordered positions of the CF argument simultaneously.

Definition 12

Consider the collection of CFRVs P(I1),P(I2),P(I3),…,P(In), where ν = (ν₁, ν₂, ν₃, ..., ν_n)^T represents the weight vector such that ∑i=1nνi=1, 0 ≤ ν_i ≤ 1. Let w = (w₁, w₂, w₃, ..., w_n)^T be the associated weight vector of the CFRV collection such that ∑i=1nwi=1, 0 ≤ w_i ≤ 1. Then, the CFRHA operator is defined as

CFRHA(P(I1),P(I2),P(I3),…,P(In))=⊕i=1nwiP¨η(Ii)=(⊕i=1nwiP¨η_(Ii),⊕i=1nwiP¨η¯(Ii)).

From the above definition, we can present the following theorem for the CFRHA operator.

Theorem 5

Then, the CFRHA operator is defined as

CFROWA(P(I1),P(I2),P(I3),…,P(In))=⊕i=1nwiP¨η(Ii)=(⊕i=1nwiP¨η_(Ii),⊕i=1nwiP¨η¯(Ii))=[(1-∏i=1n(1-μ¨ηi_)wi)ejπ((1-∏i=1n(1-γ¨ρi(x))wi),(1-∏i=1n(1-μ¨ηi¯)wi)ejπ((1-∏i=1n(1-γ¨ρi(x))wi)],

where P¨_η(I_i) = ζw_iP(Ii) = (ζw_iP_(Ii)), ζw_iP¯(Ii)) represents the largest value of permutation in the given collection of CFRVs, and ζ represents the balancing coefficient.

Proof

The proof follows from Theorem 1.

Remark

If ν=(1n,1n,1n,⋯,1n)T, then the CFRHA operator is reduced to the CFROWA operator.

Some important properties of the CFRHA operator are given by the following theorem.

Theorem 6

i. Idempotency: If P(Ii) = L(K) ∀ i = 1, 2, 3, ..., n, where L(K) = (L(K), L(K)) = (he^jπr, h̄e^jπr), then

CFRHA(P(I1),P(I2),P(I3),…,P(In))=L(K).
ii. Boundedness: Let (P(I))−=(miniP_(Ii),maxiP¯(Ii)) and (P(I))+=(maxiP_(Ii),miniP¯(Ii)). Then,

(P(I))-≤CFRHA(P(I1),P(I2),P(I3),…,P(In))≤(P(I))+.
iii. Shift invariance: Consider CFRVs ℬ(K) = (ℬ(K), ℬ(K)). Then,

CFRHA(P(I1)⊕ℬ(K),P(I2)⊕ℬ(K),P(I3)⊕ℬ(K),…,P(In)⊕ℬ(K))=CFRHA(P(I1),P(I2),P(I3),…,P(In))⊕ℬ(K).
iv. Monotonicity: Let F(L_i) = (F(L_i), F(L_i)), (i = 1, 2, 3, ..., n) be another collection of CFRVs such that F(L_i) ≤ P_(Ii) and F(L_i) ≤ P¯(Ii). Then,

CFRHA(F(L1),F(L2),F(L3),.......,F(Ln))≤CFRHA(P(I1),P(I2),P(I3),.......,P(In)).
v. Homogeneity:

CFRHA(δP(I1),δP(I2),δP(I3),…,δP(In))=δCFRHA(P(I1),P(I2),P(I3),…,P(In)),

for any δ > 0.
vi. Commutativity: Let P′(Ii)=(P′_(Ii),P′¯(Ii)), i = 1, 2, 3, ..., n, be any permutation of P(Ii)=(P_(Ii),P¯(Ii)). Then,

CFRHA(P(I1),P(I2),P(I3),…,P(In))=CFRHA(P′(I1),P′(I2),P′(I3),…,P′(In)).

Proof

The proof follows from Theorem 2.

5. Complex Fuzzy Rough Geometric Aggregation Operator

Go to

Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

Here, we introduce the concept of a complex fuzzy rough geometric aggregation operator using complex rough sets and a geometric operator. Subsequently, we consider the properties of this operator in detail.

5.1 Complex Fuzzy Rough Weighted Geometric Operator

This subsection presents a detailed study of the complex fuzzy rough geometric operator and its characteristics.

Definition 13

Consider the collection of CFRVs P(I1),P(I2),P(I3),…,P(In), where w = (w₁, w₂, w₃, ..., w_n)^T represents the weight vector such that ∑i=1nwi=1, 0 ≤ w_i ≤ 1. Then, the complex fuzzy rough weighted geometric (CFRWG) operator is defined as

CFRWG(P(I1),P(I2),P(I3),…,P(In))=(⊗i=1n(P_(I1))wi,⊗i=1n(P¯(I1))wi).

The aggregated result of the CFRWG operator is developed in the following theorem by using the above definition.

Theorem 7

Consider the collection P(Ii)=(P_(Ii),P¯(Ii)), i = 1, 2, 3, ..., n, of CFRVs with weighted vectors w = (w₁, w₂, w₃, ..., w_n)^T such that ∑i=1nwi=1, 0 ≤ w_i ≤ 1. Then, the CFRWG operator is determined as

CFRWG(P(I1),P(I2),P(I3),…,P(In))=(⊗i=1nwiP_(Ii),⊗i=1nwiP¯(Ii))=[(∏i=1n(μi_)wi)ejπ(∏i=1n(γi(x))wi),(∏i=1n(μi¯)wi)ejπ(∏i=1n(γi(x))wi)].

Proof

The proof follows directly from Theorem 1.

From the above observation, we see that P_(I) and P¯(I) are CFRVs. Therefore, by Definition 13, ⊗i=1nwiP_(Ii) and ⊗i=1nwiP¯(Ii) are also CFRVs, and CFRWG(P(I1),P(I2),P(I3),…P(Ik)) is a complex fuzzy rough set under CF approximation space (ℳ,P).

Theorem 8

i. Idempotency: If P(Ii) = L(K) ∀ i = 1, 2, 3, ..., n, where L(K) = (L(K), L(K)) = (he_jπr, h̄e_jπr), then

CFRWG(P(I1),P(I2),P(I3),…,P(In))=L(K).
ii. Boundedness: Let (P(I))−=(miniP_(Ii),maxiP¯(Ii)) and (P(I))+=(maxiP_(Ii),miniP¯(Ii)). Then,

(P(I))-≤CFRWG(P(I1),P(I2),P(I3),…,P(In))≤(P(I))+.
iii. Shift invariance: Consider CFRV ℬ(K) = (ℬ(K), ℬ(K)). Then,

CFRWG(P(I1)⊗ℬ(K),P(I2)⊗ℬ(K),P(I3)⊗ℬ(K),…,P(In)⊗ℬ(K))=CFRWG(P(I1),P(I2),P(I3),…,P(In))⊗ℬ(K).
iv. Monotonicity: Let F(L_i) = (F(L_i), F(L_i)), (i = 1, 2, 3, ..., n) be another collection of CFRVs such that F(L_i) ≤ P_(Ii) and F(L_i) ≤ P¯(Ii). Then,

CFRWG(F(L1),F(L2),F(L3),…,F(Ln))≤CFRWG(P(I1),P(I2),P(I3),…,P(In)).
v. Homogeneity:

CFRWG(δP(I1),δP(I2),δP(I3),…,δP(In))=δCFRWG(P(I1),P(I2),P(I3),…,P(In)),

for any δ > 0.
vi. Commutativity: Let P′(Ii)=(P′_(Ii),P′¯(Ii)), i = 1, 2, 3, ..., n, be any permutation of P(Ii)=(P_(Ii),P¯(Ii)). Then,

CFRWG(P(I1),P(I2),P(I3),…,P(In))=CFRWG(P′(I1),P′(I2),P′(I3),…,P′(In)).

Proof

The proof follows directly from Theorem 2.

5.2 Complex Fuzzy Rough OrderedWeight Geometric Operator

We now investigate the concept of complex fuzzy rough-ordered weighted geometric (CFROWG) aggregation operators and discuss some basic properties.

Definition 14

Consider the collection of complex fuzzy rough values P(I1),P(I2),P(I3),…,P(In), where w = (w₁, w₂, w₃, ..., w_n)^T represents the weight vector such that ∑i=1nwi=1, 0 ≤ w_i ≤ 1. Then, the CFROWG operator is defined as

CFROWG(P(I1),P(I2),P(I3),…,P(In))=(⊗i=1n(Pη_(Ii))wi,⊗i=1n(Pη¯(I1))wi).

Based on the above definition, we derive the aggregate result for the CFROWG operator in Theorem 9.

Theorem 9

Consider the collection P(Ii)=(P_(Ii),P¯(Ii)), i = 1, 2, 3, ..., n, of CFRVs associated with weighted vectors w = (w₁, w₂, w₃, ..., w_n)^T such that ∑i=1nwi=1, 0 ≤ w_i ≤ 1. Then, the CFROWG operator is determined as

CFRWOG(P(I1),P(I2),P(I3),…,P(In))=(⊗i=1nwiPη_(Ii),⊗i=1nwiPη¯(Ii))=[(∏i=1n(μηi_)wi) ejπ(∏i=1n(γρi(x))wi),(∏i=1n(μηi¯)wi) ejπ(∏i=1n(γρi(x))wi)],

where Pη(Ii)=(Pη_(Ii),Pη¯(Ii)) represents the largest value in the given collection of CFRVs.

Proof

The proof follows from Theorem 1.

Theorem 10

i. Idempotency: If P(Ii) = L(K) ∀ i = 1, 2, 3, ..., n, where L(K) = (L(K), L(K)) = (he^jπr, h̄e^jπr), then

CFROWG(P(I1),P(I2),P(I3),…,P(In))=L(K).
ii. Boundedness: Let (P(I))−=(miniP_(Ii),maxiP¯(Ii)) and (P(I))+=(maxiP_(Ii),miniP¯(Ii)). Then,

(P(I))-≤CFROWG(P(I1),P(I2),P(I3),…,P(In))≤(P(I))+.
iii. Shift invariance: Consider CFRV ℬ(K) = (ℬ(K), ℬ(K)). Then,

CFROWG(P(I1)⊗ℬ(K),P(I2)⊗ℬ(K),P(I3)⊗ℬ(K),…,P(In)⊗ℬ(K))=CFROWG(P(I1),P(I2),P(I3),…,P(In))⊗ℬ(K).
iv. Monotonicity: Let F(L_i) = (F(L_i), F(L_i)),(i = 1, 2, 3, ..., n) be another collection of CFRVs such that F(L_i) ≤ P_(Ii) and F(L_i) ≤ P¯(Ii). Then,

CFROWG(F(L1),F(L2),F(L3),…,F(Ln))≤CFROWG(P(I1),P(I2),P(I3),…,P(In)).
v. Homogeneity:

CFROWG(δP(I1),δP(I2),δP(I3),…,δP(In))=δCFROWG(P(I1),P(I2),P(I3),…,P(In)),

for any δ > 0.
vi. Commutativity: Let P′(Ii)=(P′_(Ii),P′¯(Ii)), i = 1, 2, 3, ..., n, be any permutation of P(Ii)=(P_(Ii),P¯(Ii)). Then,

CFROWG(P(I1),P(I2),P(I3),…,P(In))=CFROWG(P′(I1),P′(I2),P′(I3),…,P′(In)).

Proof

The proof follows from Theorem 2.

5.3 Complex Fuzzy Rough Hybrid Geometric Operator

In this subsection, we discuss the study of complex fuzzy rough hybrid geometric (CFRHG) operators that simultaneously weigh the value and position of complex fuzzy arguments in detail. In addition, we derive some fundamental results for the CFRHG operator.

Definition 15

Consider the collection of CFRVs P(I1),P(I2),P(I3),…,P(In), where ν = (ν₁, ν₂, ν₃, ..., ν_n)^T represents weight vector such that ∑i=1nνi=1, 0 ≤ ν_i ≤ 1. Let w = (w₁, w₂, w₃, ..., w_n)^T be the associated weight vector of the CFRV collection such that ∑i=1nwi=1, 0 ≤ w_i ≤ 1. Then, the CFRHG operator is defined as

CFRHG(P(I1),P(I2),P(I3),…,P(In))=⊗i=1n(P¨η(Ii))wi=(⊗i=1n(P¨η_(Ii))wi,⊗i=1n(P¨η¯(Ii))wi).

Consedring Definition 15, we demonstrate the aggregate result for the CFRHG operator in Theorem 11.

Theorem 11

CFRWOG(P(I1),P(I2),P(I3),…,P(In))=(⊗i=1nwiP¨η_(Ii),⊗i=1nwiP¨η¯(Ii))=[(∏i=1n(μ¨ηi_)wi) ejπ(∏i=1n(γ¨ρi(x))wi),(∏i=1n(μ¨ηi¯)wi) ejπ(∏i=1n(γ¨ρi(x))wi)],

where P¨_η(I_i) = ζw_iP(Ii) = (ζw_iP_(Ii), ζw_iP¯(Ii)) represents the largest value of permutation in the given collection of CFRVs, and ζ is the balancing coefficient.

Proof

The proof follows from Theorem 1.

Remark

If ν=(1n,1n,1n,…,1n)T, then the CFRHG operator is reduced to the CFROWG operator.

Some important properties of the CFRHG operator are provided in the following theorem.

Theorem 12

i. Idempotency: If P(Ii) = L(K) ∀ i = 1, 2, 3, ..., n, where L(K) = (L(K), L(K)) = (he^jπr, h̄e^jπr), then

CFRHG(P(I1),P(I2),P(I3),…,P(In))=L(K).
ii. Boundedness: Let (P(I))−=(mini(P_(Ii),maxi(P¯(Ii)) and (P(I))+=(maxi(P_(Ii),mini(P¯(Ii)). Then,

(P(I))-≤CFRHG(P(I1),P(I2),P(I3),…,P(In))≤(P(I))+.
iii. Shift invariance: Consider CFRV ℬ(K) = (ℬ(K), ℬ(K)). Then,

CFRHG(P(I1)⊗ℬ(K),P(I2)⊗ℬ(K),P(I3)⊗ℬ(K),…,P(In)⊗ℬ(K))=CFRHG(P(I1),P(I2),P(I3),…,P(In))⊗ℬ(K).
iv. Monotonicity: Let F(L_i) = (F(L_i), F(L_i)), (i = 1, 2, 3, ..., n) be another collection of CFRVs such that F(L_i) ≤ P_(Ii) and F(L_i) ≤ P¯(Ii). Then,

CFRHG(F(L1),F(L2),F(L3),…,F(Ln))≤CFRHG(P(I1),P(I2),P(I3),…,P(In)).
v. Homogeneity:

CFRHG(δP(I1),δP(I2),δP(I3),…,δP(In))=δCFRHG(P(I1),P(I2),P(I3),…,P(In)),

for any δ > 0.
vi. Commutativity: Let P′(Ii)=(P′_(Ii),P′¯(Ii)), i = 1, 2, 3, ..., n, be any permutation of P(Ii)=(P_(Ii),P¯(Ii)). Then,

CFRHG(P(I1),P(I2),P(I3),…,P(In))=CFRHG(P′(I1),P′(I2),P′(I3),…,P′(In)).

Proof

The proof follows from Theorem 2.

6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators

Go to

Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

Under present competitive circumstances, it is more challenging for a single expert to make decisions regarding the socioeconomic environment. Therefore, in real-life problems, a group of professional experts must form an opinion to obtain more reliable and satisfactory results. Therefore, MCGDM has a high capacity to evaluate multi-dispute criteria in all types of decision problems, achieving remarkable results. To address this situation, we use the method to solve MCGDM. The aim is to select the finest alternative from the given options based onthe average solution. By embedding CFRVs in the method, we introduce the idea of the method, where experts provide their opinions in the form of CFRVs. To establish the proposed approach, we take the following steps, where the data are in the form of complex fuzzy rough information:

Consider a set containing r alternatives such that M = {y₁, y₂, y₃, ..., y_r} and a set of n decision attributes, ℬ = {b₁, b₂, b₃, ..., b_n}. Let A = {A₁, A₂, A₃, ..., A_z} be the set of z professional decision-makers who will make their evaluation report for each alternative M = {y₁, y₂, y₃, ..., y_r} against the set of attributes ℬ = {b₁, b₂, b₃, ..., b_n}. Let w = {w₁, w₂, w₃, ..., w_n}^T be the weight vector for attribute b_i and ϕ = {ϕ₁, ϕ₂, ϕ₃, ..., ϕ_z}^T be the weight vector for decision-makers A_i (i = 1, 2, 3, ..., z) such that ∑i=1nwi=1,∑i=1nwi=1 and 0 ≤ w_i, ϕ_i ≤ 1. The standard algorithm for the hybrid version of the method in CF rough environment is discussed as follows:

Step 1: For the construction of the decision matrix, collect the evaluation information of professional decision-makers for each alternative y_i against their attribute b_i such that

H=[P(Iijl)]r×n,

where P(Iijl) denotes CFRV of alternative y_i against attribute b_i of expert β_k.

Step 2: Use the proposed approach for the aggregated decision matrix, which consists of the collective information of the decision matrix under the weight vector such that

H=[P(Iij)]r×n.

Step 3: Normalize the aggregated matrix H = [P(I_ij )]_r×n to Hn=[P(Iijn)]r×n, i.e.,

Hn=P(Iij)=((μij_eiπ(γ)),(μij¯eiπ(γ))).

Step 4: Calculate the average value by using the proposed approach for all alternatives under each attribute.

Avg=[Avg]1×n=[1r∑i=1rP(Iijn)]1×n=[(1-∏i=1r(1-μijn_)1reiπ(1-∏i=1r(1-γijn)1r)),(1-∏i=1r(1-μijn¯)1reiπ(1-∏i=1r(1-γijn)1r))]1×n.

Step 5: Based on the above , calculate the and through the following formulas:

PDASij=[PDASij](r×n)={max(0,[μijn)-Avgj])Avgj}ejπ{max(0,[γijn)-Avgj])Avgj},

and

NDASij=[NDASij](r×n)={max(0,[Avgj-μijn)])Avgj}ejπ{max(0,[Avgj-γijn)])Avgj}.

Step 6: Calculate the positive weight distance (S⁺) and negative weight distance (S⁻) by the following formulas:

S+=∑j=1nwiPDASij, S-=∑j=1nwiPDASij.

Step 7: Normalize the positive and negative weight distances by the formula

NS+yi=S+yimaxi(S+yi),NS-yi=1-S-yimaxi(S-yi).

Step 8: Calculate the appraisal score ( ) based on NS⁺y_i and NS⁻y_i using the following formula:

ASi=12(NS+yi+NS-yi).

Step 9: Rank all values in order depending on the value of . A larger value of is considered superior.

7. Illustrative Example

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Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

To demonstrate the effectiveness and dominance of the proposed model in MCGDM, we illustrate an example through which we determine the suitability of a site for fish culture, that is, to select the best location for a fish farm.

The construction of a fish farm requires careful planning and consideration, particularly regarding the crucial task of selecting an ideal site. Site location is important in determining the success of a fish farm and influences key factors such as water quality, temperature, and market accessibility. This article aims to guide the process of choosing the most suitable site for a fish farm, guaranteeing optimal conditions for growth and profitability. When selecting a site, it is crucial to consider the following criteria:

• Water quality plays a pivotal role in site selection for a fish farm, as it directly affects the health and well-being of the fish.
• Temperature is another critical factor to consider when selecting a site for a fish farm. Different species of fish thrive in specific temperature ranges, and it is vital to choose a site where the temperature remains within the preferred range of the fish species being cultivated.
• Hydrological characteristics, including factors such as water source, flow rate, and water depth, must be carefully evaluated during site selection.
• Soil characteristics also play a significant role in site selection. A suitable site should have well-drained soil with good water retention capacity.
• The proximity of a fish farm to potential markets holds great importance for efficient product distribution and reduced transportation costs.

By considering the above criteria during the site selection process, we can ensure that the fish farm operates under optimal conditions. Integrating these criteria into the site selection process will enable well-informed decisions and ultimately contribute to the success and profitability of the fish farming venture.

Consider that a pisciculture association wants to launch a project of four fish farms {y₁, y₂, y₃, y₄} at different sites in Pakistan, which will be further evaluated to achieve the most optimal fish farm. For the assessment of the fish farm, the association invites three professional experts β_k (k = 1, 2, 3) with weight vectors w = {0.21, 0.31, 0.23}^T. These experts will assess the four fish farms according to the following five criteria:

b₁ = Water quality
b₂ = Temperature
b₃ = Hydrological characteristics
b₄ = Soil characteristics
b₅ = Proximity to markets

The weight vector for these five criteria is given as v = {0.13, 0.17, 0.23, 0.24, 0.16}^T. Each β_k gives an assessment report for each y_i according to their corresponding criteria in the form of CFRVs. Next, we will use the proposed CFRWA operator to obtain the best fish farm site by making use of the above step-wise decision rules of the method.

Step 1: For the construction of the decision matrix, collect the evaluation information of professional decision-makers for each alternative y_i against their attribute b_i such that

H=[P(Iijl)]r×n,

where P(Iijl) denotes the CFRV of alternative y_i against attribute b_i by the expert β_k, as given in Tables 2–4.

Step 2: Use the proposed approach to aggregate the decision matrix consisting of the collective information of the decision matrix against the weight vector such that

H=[P(Iij)]r×n

The results are given in Table 5.

Step 3: The criteria are of the benefit type so must be normalized.

Step 4: Calculate the average value for each alternative under each criterion, as given in Table 6.

Step 5: Use in Table 6 to calculate the score values given in Tables 7 and 8, and use these score values to calculate the and , as given in Tables 9 and 10.

Step 6: Calculate S⁺y_i, S⁻y_i using the weight vector v = {0.13, 0.17, 0.23, 0.24, 0.16}^T and calculate |S⁺y_i|, |S⁻y_i|, as given in Tables 11 and 12.

Step 7: The normalized values of |S⁺y_i|, |S⁻y_i| are given as follows:

S+y1=1,S+y2=0.0420,S+y3=0.7853,S+y4=0.9945,S-y1=1,S-y2=0.8881,S-y3=0.9156,S-y4=0.0000.

Step 8: Calculate appraisal score using normalized positive and negative distance, which is given as

Ay1=0.9189,Ay2=0.4608,Ay3=0.5400,Ay4=0.5000.

Step 9: Depending on these calculations, rank all the values of the developed models based on the method, as shown in Table 13.

8. Comparative Study

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Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

To show the dominance of the proposed method, a comparative analysis was performed with some existing approaches (see [23, 24, 32, 33, 43, 44, 61, 62]). From Table 5 with weight vector v = {0.13, 0.17, 0.23, 0.24, 0.16}^T, the aggregation findings of the comparative study are given in Table 14.

• The theory of rough sets [35] is a powerful tool to navigate uncertainties in decision-making, which is an important skill. Applicable not just in the mathematics realm but in the real world too, it offers a perspective on fuzziness and rough information, opening new possibilities. With rough sets, we can effectively tackle uncertain scenarios and explore problems with imprecision, where solutions often vary. However, the introduction of phase values to complex fuzzy rough sets broadens the scope, enabling us to analyze how objects fluctuate and offering greater robustness. In comparison to rough fuzzy sets and other extensions in this domain, complex fuzzy rough sets exhibit strength by capturing variations, like a steady chain. This infusion of attributes brings new insights, empowering researchers and practitioners and making their studies more stable.
• Given uncertain data, CFAA operators [23], CFWG operators [24], and related extensions are practical. Complex fuzzy rough sets, however, offer a promising different approach. They effectively address uncertainty without additional functions or adjustments. By discussing uncertainty found in given data, complex fuzzy rough sets are effective, removing the need for subjective suppositions and making the results more aligned.
• It is clear from Table 14 that existing methods, including , IF-TOPSIS, IF-VIKOR, and IFGRA, as well as some aggregation operators,cannot be used to solve the illustrative example of Section 7 by using CF rough values. Although the methods introduced in [23, 24, 32, 33, 43, 44, 61, 62] have rough information, they cannot solve the proposed approach.

From the results in Table 14, we conclude that the existing approaches have a deficiency in complex fuzzy rough information, and these approaches are unable to solve and rank the illustrative example. Hence, the proposed approach is more reliable and useful than existing methods.

9. Conclusion

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Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

Within the realm of data imprecision, MCGDM prevails owing to its immense potential to handle uncertainties without failure. Aggregation seeks to merge diverse information from different resources to derive a precise conclusion. Complex fuzzy sets extend the domain of fuzzy sets; thus, their generality increases with the amplitude and phase terms. The phase term introduces a periodic and two-dimensional view, revealing previously unobserved phenomena and patterns. This study describes the concept and role of complex fuzzy rough sets, using complex fuzzy rough values for investigation, which was our primary goal. Aggregation operators such as CFRWA, CFROWA, and CFRHA were explored, and their essence was addressed. In addition, the concepts of the CFRWG, CFROWA, and CFRHG operators were examined. The characteristics of these operators were studied in detail to understand their strengths and limitations. A novel score function was presented to enhance the methodology, adding depth and effectiveness, which contribute to the strategy. The method was proposed alongside a stepwise algorithm, seeking optimal results. A real-life example was presented to demonstrate its practicality, addressing the optimal choice for constructing a fish farm. The stepwise algorithm of the method was employed to solve the problem in pisciculture. Finally, a comparative study was conducted with existing models to demonstrate the superiority of the proposed approach. In the realm of MCGDM, this method proves to be more effective and useful, handling imprecision in data with finesse and making fruitful decisions.

In future endeavors, we have the opportunity to expand the developed approach to encompass a range of diverse aggregation operators. These may include Einstein operations, Hamacher operations, Dombi operations, power aggregation operators, complex fuzzy-ordered weighted quadratic averaging operators, and Maclaurin symmetric mean operators with CFR information. Furthermore, we can explore the application of our proposed method in various domains, such as medical diagnosis.

Data Availability

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Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

The data used in this study were to support the findings of the proposed approach. Anyone can use it by simply citing this article.

Conflict of Interest

Go to

Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

No potential conflict of interest relevant to this article was reported.

Figures

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Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

Fig. 1.

Step-by-step diagram of modified EDAS method.

Fig. 2.

Ranking order of four different locations for a fish farm.

Tables

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Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

Table. 1.

Table 1. Equivalence relation based on CFAS.

P	y₁	y₂	y₃	y₄
y₁	1e^jπ⁰	0.4e^jπ^0.3	0.6e^jπ^0.4	0.7e^jπ^0.5
y₂	0.4e^jπ^0.3	1e^jπ⁰	0.5e^jπ^0.3	0.35e^jπ^0.2
y₃	0.6e^jπ^0.4	0.5e^jπ^0.3	1e^jπ⁰	0.8e^jπ^0.4
y₄	0.7e^jπ^0.5	0.35e^jπ^0.2	0.8e^jπ^0.4	1e^jπ⁰

Table. 2.

Table 2. CFR evaluation information by β₁.

	b₁	b₂	b₃	b₄	b₅
y₁	(0.6e^jπ^(0.3), 0.8e^jπ^(0.2))	(0.8e^jπ^(0.1), 0.7e^jπ^(0.4))	(0.9e^jπ^(0.5), 0.4e^jπ^(0.3))	(0.8e^jπ^(0.8), 0.7e^jπ^(0.5))	(0.4e^jπ^(0.4), 0.6e^jπ^(0.2))
y₂	(0.5e^jπ^(0.6), 0.9e^jπ^(0.6))	(0.6e^jπ^(0.3), 0.4e^jπ^(0.2))	(0.4e^jπ^(0.2), 0.8e^jπ^(0.3))	(0.6e^jπ^(0.3), 0.5e^jπ^(0.6))	(0.5e^jπ^(0.3), 0.5e^jπ^(0.4))
y₃	(0.6e^jπ^(0.4), 0.5e^jπ^(0.3))	(0.6e^jπ^(0.3), 0.5e^jπ^(0.3))	(0.6e^jπ^(0.5), 0.4e^jπ^(0.5))	(0.2e^jπ^(0.3), 0.9e^jπ^(0.2))	(0.8e^jπ^(0.2), 0.7e^jπ^(0.4))
y₄	(0.7e^jπ^(0.6), 0.4e^jπ^(0.9))	(0.4e^jπ^(0.9), 0.5e^jπ^(0.6))	(0.2e^jπ^(0.5), 0.5e^jπ^(0.8))	(0.3e^jπ^(0.7), 0.5e^jπ^(0.6))	(0.3e^jπ^(0.3), 0.5e^jπ^(0.8))

Table. 3.

Table 3. CFR evaluation information by β₂.

	b₁	b₂	b₃	b₄	b₅
y₁	(0.6e^jπ^(0.1), 0.3e^jπ^(0.1))	(0.8e^jπ^(0.3), 0.6e^jπ^(0.2))	(0.6e^jπ^(0.4), 0.7e^jπ^(0.5))	(0.5e^jπ^(0.8), 0.7e^jπ^(0.6))	(0.9e^jπ^(0.8), 0.8e^jπ^(0.8))
y₂	(0.4e^jπ^(0.2), 0.5e^jπ^(0.5))	(0.9e^jπ^(0.4), 0.4e^jπ^(0.8))	(0.4e^jπ^(0.6), 0.4e^jπ^(0.5))	(0.4e^jπ^(0.3), 0.2e^jπ^(0.4))	(0.8e^jπ^(0.9), 0.4e^jπ^(0.4))
y₃	(0.7e^jπ^(0.2), 0.2e^jπ^(0.3))	(0.4e^jπ^(0.4), 0.2e^jπ^(0.6))	(0.5e^jπ^(0.4), 0.5e^jπ^(0.6))	(0.2e^jπ^(0.4), 0.7e^jπ^(0.6))	(0.5e^jπ^(0.3), 0.3e^jπ^(0.4))
y₄	(0.5e^jπ^(0.2), 0.6e^jπ^(0.8))	(0.6e^jπ^(0.5), 0.4e^jπ^(0.3))	(0.3e^jπ^(0.5), 0.5e^jπ^(0.6))	(0.6e^jπ^(0.8), 0.4e^jπ^(0.6))	(0.6e^jπ^(0.6), 0.2e^jπ^(0.8))

Table. 4.

Table 4. CFR evaluation information by β₃.

	b₁	b₂	b₃	b₄	b₅
y₁	(0.7e^jπ^(0.3), 0.2e^jπ^(0.9))	(0.8e^jπ^(0.2), 0.6e^jπ^(0.4))	(0.7e^jπ^(0.3), 0.5e^jπ^(0.4))	(0.5e^jπ^(0.8), 0.4e^jπ^(0.3))	(0.8e^jπ^(0.6), 0.7e^jπ^(0.3))
y₂	(0.7e^jπ^(0.8), 0.3e^jπ^(0.7))	(0.5e^jπ^(0.3), 0.4e^jπ^(0.4))	(0.4e^jπ^(0.5), 0.3e^jπ^(0.5))	(0.7e^jπ^(0.3), 0.5e^jπ^(0.4))	(0.6e^jπ^(0.6), 0.4e^jπ^(0.3))
y₃	(0.4e^jπ^(0.9), 0.5e^jπ^(0.3))	(0.9e^jπ^(0.3), 0.9e^jπ^(0.4))	(0.3e^jπ^(0.6), 0.4e^jπ^(0.9))	(0.8e^jπ^(0.6), 0.3e^jπ^(0.3))	(0.6e^jπ^(0.4), 0.3e^jπ^(0.6))
y₄	(0.5e^jπ^(0.4), 0.7e^jπ^(0.5))	(0.5e^jπ^(0.9), 0.4e^jπ^(0.3))	(0.5e^jπ^(0.6), 0.1e^jπ^(0.8))	(0.3e^jπ^(0.4), 0.4e^jπ^(0.5))	(0.4e^jπ^(0.3), 0.5e^jπ^(0.8))

Table. 5.

Table 5. CFR aggregated decision matrix by CFRWA operator.

	b₁	b₂	b₃	b₄	b₅
y₁	(0.52e^jπ^(0.17), 0.39e^jπ^(0.46))	(0.70e^jπ^(0.17), 0.52e^jπ^(0.32))	(0.64e^jπ^(0.52), 0.47e^jπ^(0.33))	(0.50e^jπ^(0.70), 0.52e^jπ^(0.40))	(0.69e^jπ^(0.56), 0.62e^jπ^(0.47))
y₂	(0.44e^jπ^(0.47), 0.54e^jπ^(0.50))	(0.65e^jπ^(0.27), 0.31e^jπ^(0.48))	(0.31e^jπ^(0.31), 0.43e^jπ^(0.36))	(0.46e^jπ^(0.23), 0.31e^jπ^(0.37))	(0.57e^jπ^(0.63), 0.34e^jπ^(0.59))
y₃	(0.49e^jπ^(0.42), 0.31e^jπ^(0.33))	(0.58e^jπ^(0.27), 0.52e^jπ^(0.40))	(0.38e^jπ^(0.36), 0.35e^jπ^(0.62))	(0.38e^jπ^(0.36), 0.60e^jπ^(0.34))	(0.53e^jπ^(0.24), 0.35e^jπ^(0.38))
y₄	(0.46e^jπ^(0.32), 0.48e^jπ^(0.68))	(0.42e^jπ^(0.54), 0.34e^jπ^(0.32))	(0.27e^jπ^(0.64), 0.31e^jπ^(0.57))	(0.35e^jπ^(0.58), 0.34e^jπ^(0.47))	(0.37e^jπ^(0.36), 0.31e^jπ^(0.70))

Table. 6.

Table 6. Average solutions.

b₁	(0.4837e^jπ^(0.3791), 0.4404e^jπ^(0.4911))
b₂	(0.6038e^jπ^(0.3998), 0.4368e^jπ^(0.3653))
b₃	(0.4279e^jπ^(0.3877), 0.3998e^jπ^(0.5043))
b₄	(0.4326e^jπ^(0.5019), 0.4617e^jπ^(0.3978))
b₅	(0.5602e^jπ^(0.4690), 0.4243e^jπ^(0.4855))

Table. 7.

Table 7. Score values of average solutions.

b₁	0.7310e^jπ^(0.7175)
b₂	0.7601e^jπ^(0.6913)
b₃	0.7069e^jπ^(0.7230)
b₄	0.7236e^jπ^(0.7249)
b₅	0.7461e^jπ^(0.7386)

Table. 8.

Table 8. Score values of CFR aggregated decision matrix.

	b₁	b₂	b₃	b₄	b₅
y₁	0.7306e^jπ^(0.6572)	0.8068e^jπ^(0.6056)	0.7802e^jπ^(0.6636)	0.7585e^jπ^(0.7753)	0.8291e^jπ^(0.7561)
y₂	0.7455e^jπ^(0.7409)	0.7434e^jπ^(0.6888)	0.6893e^jπ^(0.6873)	0.69457e^jπ^(0.6521)	0.7296e^jπ^(0.7313)
y₃	0.7017e^jπ^(0.6852)	0.7775e^jπ^(0.6623)	0.6856e^jπ^(0.7547)	0.7484e^jπ^(0.6742)	0.7233e^jπ^(0.6548)
y₄	0.7383e^jπ^(0.7490)	0.6918e^jπ^(0.7566)	0.6477e^jπ^(0.7660)	0.6751e^jπ^(0.7628)	0.6727e^jπ^(0.7643)

Table. 9.

Table 9. Results of PDAS_ij.

	b₁	b₂	b₃	b₄	b₅
y₁	0.0000e^jπ^(0.0000)	0.0614e^jπ^(0.0000)	0.1036e^jπ^(0.0000)	0.0482e^jπ^(0.0695)	0.1111e^jπ^(0.0236)
y₂	0.0198e^jπ^(0.0325)	0.0000e^jπ^(0.0000)	0.0000e^jπ^(0.0000)	0.0000e^jπ^(0.0000)	0.0000e^jπ^(0.0000)
y₃	0.0000e^jπ^(0.0000)	0.0229e^jπ^(0.0000)	0.0000e^jπ^(0.0439)	0.0343e^jπ^(0.0000)	0.0000e^jπ^(0.0000)
y₄	0.0099e^jπ^(0.0438)	0.0987e^jπ^(0.0945)	0.0914e^jπ^(0.0595)	0.0718e^jπ^(0.0522)	0.1090e^jπ^(0.0348)

Table. 10.

Table 10. Results ofNDAS_ij.

	b₁	b₂	b₃	b₄	b₅
y₁	0.0005e^jπ^(0.0841)	0.0000e^jπ^(0.1238)	0.0000e^jπ^(0.0820)	0.0000e^jπ^(0.0000)	0.0000e^jπ^(0.0000)
y₂	0.0000e^jπ^(0.0000)	0.0219e^jπ^(0.0036)	0.0248e^jπ^(0.0493)	0.0401e^jπ^(0.1003)	0.0221e^jπ^(0.0098)
y₃	0.0400e^jπ^(0.0450)	0.0000e^jπ^(0.0418)	0.0300e^jπ^(0.0000)	0.0000e^jπ^(0.0699)	0.0305e^jπ^(0.1134)
y₄	0.9864e^jπ^(0.0000)	0.0898e^jπ^(0.0000)	0.0837e^jπ^(0.0575)	0.0670e^jπ^(0.0522)	0.0983e^jπ^(0.0000)

Table. 11.

Table 11. Results of S⁺, S⁻.

S⁺y₁ = 0.0636e^jπ^(0.0204)	S⁻y₁ = 0.0007e^jπ^(0.0508)
S⁺y₂ = 0.0025e^jπ^(0.0042)	S⁻y₂ = 0.0226e^jπ^(0.0376)
S⁺y₃ = 0.0121e^jπ^(0.0101)	S⁻y₃ = 0.0170e^jπ^(0.0479)
S⁺y₄ = 0.0738e^jπ^(0.0535)	S⁻y₄ = 0.1945e^jπ^(0.0257)

Table. 12.

Table 12. Modulus values of S⁺, S⁻.

\|S⁺y₁\| = 0.0617	\|S⁻y₁\| = 0.0000
\|S⁺y₂\| = 0.0024	\|S⁻y₂\| = 0.0217
\|S⁺y₃\| = 0.01212	\|S⁻y₃\| = 0.0164
\|S⁺y₄\| = 0.0737	\|S⁻y₄\| = 0.1945

Table. 13.

Table 13. Ranking order of the proposed model.

Investigated operators based on method	Appraisal modulus score values of alternatives				Ranking
Investigated operators based on method	y₁	y₂	y₃	y₄	Ranking
CFRWA	0.9189	0.4608	0.5400	0.5000	y₁> y₃> y₄> y₂
CFROWA	0.9870	0.4777	0.5738	0.5000	y₁> y₃> y₄> y₂
CFRHA	0.9905	0.4761	0.5747	0.5000	y₁> y₃> y₄> y₂
CFRWG	0.9315	0.2088	0.4989	0.4166	y₁> y₃> y₄> y₂
CFROWG	0.9301	0.4915	0.6133	0.6079	y₁> y₃> y₄> y₂
CFRHG	0.9509	0.4971	0.7774	0.6831	y₁> y₃> y₄> y₂

Table. 14.

Table 14. Comparative analysis of developed method with existing methods.

Investigated operators based on method	Appraisal modulus score values of alternatives				Ranking
Investigated operators based on method	y₁	y₂	y₃	y₄	Ranking
CFAA hybrid geometric		Inaccessible			×
CFWG [24]		Inaccessible			×
CFPA [32]		Inaccessible			×
BCFHA [33]		Inaccessible			×
IF-EDAS method [43]		Inaccessible			×
IF-TOPSIS method [44]		Inaccessible			×
IF-VIkOR method [61]		Inaccessible			×
IF-GIA method[62]		Inaccessible			×
CFRWA	0.9189	0.4608	0.5400	0.5000	y₁> y₃> y₄> y₂
CFROWA	0.9870	0.4777	0.5738	0.5000	y₁> y₃> y₄> y₂
CFRHA	0.9905	0.4761	0.5747	0.5000	y₁> y₃> y₄> y₂
CFRWG	0.9315	0.2088	0.4989	0.4166	y₁> y₃> y₄> y₂
CFROWG	0.9301	0.4915	0.6133	0.6079	y₁> y₃> y₄> y₂
CFRHG	0.9509	0.4971	0.7774	0.6831	y₁> y₃> y₄> y₂

References

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Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

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Biographies

Go to

Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability
Conflict of Interest
Figures
Tables
References
Biographies

Faiz Muhammad Khan is an associate professor of Mathematics in the Department of Mathematics and Statistics, University of Swat. He received his M.Phil. degree in Mathematics from Quaid-I-Azam University Islamabad in 2009, his Ph.D. degree from Universiti Teknologi Malaysia (UTM) in 2013 with International Doctoral Fellowship throughout the study. He also avail partial support from HEC Pakistan. He has completed post-doctoral in mathematics from Northwestern Polytechnical University Xi’an in 2019. He was Incharge of the Department of Mathematics and Statistics, University of Swat from March 2014 to May 2017. In his supervision, 9 M.Phil. scholars have completed their degrees. In co-supervision, two Ph.D. students from Malaysia and China successfully completed their degrees. Currently, 10 M.Phil. scholars and 2 Ph.D. scholars are working under his supervision. He is working on the fuzzy sets, soft sets in decision making problems using aggregation operators. Uncertainty are now adays the key problem faced in various fields like control theory, robotics, automata theory, computer science, economics, coding theory and finance. Using soft sets and various type of fuzzy sets, he deals with the uncertainty problems and recommend a reliable strategy. He has published more than 50 research publications in various reputed international journals with high impact factor. Various articles are under review. He has been awarded by one research project NRPU-6831 by Higher Education Commission (HEC) Pakistan for three years which has been already completed and the final report has been send to HEC.

Azmat Ullah is a Ph.D. student in the Biomedical Sciences and Engineering Department at Koc University. He holds a bachelor’s degree in Mechatronics Engineering from UET Peshawar and a master’s degree in Biomedical Engineering from NUST Islamabad. His research interests center around PINNs (Physics-Informed Neural Networks) and their applications in Cardiovascular Fluid Dynamics. Currently, he is focused on developing innovative algorithms based on PINNs for his doctoral research.

Naila Bibi serves as an assistant professor of Mathematics at Government Girls’ Degree College, Khwazakhela, Swat. She successfully obtained her M.Phil. degree from the University of Swat. Naila Bibi’s primary research focus revolves around fuzzy aggregation operators and their practical utilization within the realm of decision-making.

Saleem Abdullah serves as an associate professor of Mathematics at Abdul Wali Khan University Mardan. He holds his M.Phil. and Ph.D. degrees, both earned at Quaid-i-Azam University Islamabad. With a prolific academic career, he has authored over two hundred published articles. His research primarily centers around spherical fuzzy sets, fuzzy integral equations, fuzzy aggregation operators, and fuzzy decisionmaking.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(3): 270-293

Published online September 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.3.270

Complex Fuzzy Rough Aggregation Operators and their Applications in for Multi-Criteria Group Decision-Making

Faiz Muhammad Khan¹, Naila Bibi^1,2, Saleem Abdullah³, and Azmat Ullah⁴

Correspondence to:Naila Bibi (nailaazeemi963@gmail.com)

Received: March 11, 2023; Revised: July 31, 2023; Accepted: August 16, 2023

Abstract

Keywords: Complex fuzzy sets, Rough sets, Averaging and geometric operators, EDAS method, MCGDM

1. Introduction

Later, the traditional fuzzy set was generalized to complexity fuzzy sets (CFSs) by Ramot et al. [3, 4]. With this generalization, the range extends from the interval 0 ≤ y ≤ 1 to a unit circle in the complex plane. CFS has phase and amplitude terms, i.e., G = {(s, r_G (s) e_jγ^G⁽^s⁾) s ∈ Y}. Here, $j = \sqrt{- 1}$ , r_G (s) ∈ [0, 1] represents the amplitude term and γ_G (s) ∈ [0, 2π] is the phase term. CFS can act as an ordinary fuzzy set by simply setting the phase term to zero. CFS is not only a simple extension of the traditional fuzzy set but also provides an intuitive extension for solving problems that are both complicated and unachievable. A CFS is an extended form of a fuzzy set, and its range extends from real to complex numbers with some imaginary qualities. CFS indicates a complex-valued membership grade that contains amplitude and phase terms. This implies that the CFS is more generalized than a classical fuzzy set. For example, if we are asked to obtain data with distance and direction simultaneously, then we may use CFS by specifying the distance and direction of the destination. Ramot et al. [4] initiated complex fuzzy (CF) operations and relations to deal with fuzzy information. Hu et al. [5, 6] presented the notion of approximate parallel and orthogonal relations of CFSs. Zhang et al. [7] introduced δ-equalities between CFSs. Bi et al. [8] presented two classes of entropy measures for CFSs. Hu and his colleagues [9, 10] and Alkouri and Salleh [11] developed several distance measures for CFSs. Tamir and Kandel [12] presented an axiomatic theory for complex fuzzy logic and classes. Liu et al. [13] measured the distance and cross-entropy on CFSs and their applications in decision-making. Dai [14] introduced a generalization of the rotational invariance in CF operations. Zahid et al. [15] introduced an ELECTRE-based method for group decision-making with complex spherical fuzzy information. Based on complex Pythagorean fuzzy information, Akram et al. [16] and Ma et al. [17] developed new approaches for multicriteria decision-making. Akram et al. [18] extended the VIKOR approach using a complex spherical fuzzy set for group decision-making.

1.1 Motivation of the Current Research

The primary motivations for this study are as follows:

• The first aspect that motivates this research is the limitations of existing fuzzy rough set and aggregation operator approaches in solving MCGDM problems. While these approaches have been useful, they have certain limitations in representing complex decision-making scenarios and dealing with uncertainty of a periodic nature. This motivates the exploration of complex fuzzy rough sets and complex fuzzy rough aggregation operators as potential solutions to overcome these limitations and enhance the decision-making process.
• The second aspect that motivates this research is the practical need for effective decision-making in daily-life MCGDM examples. The present difficult situations involve applying these concepts and methods to real-world scenarios and providing practical solutions. This motivates the research to not only propose new theoretical frameworks but also demonstrate their applicability and effectiveness in solving real-life decision-making problems.

This study makes four key contributions:

• This study proposes the concept of complex fuzzy rough sets as an extension to fuzzy rough sets. This introduces a two-dimensional approach to fuzzy rough sets, enabling a more comprehensive representation of complex decision-making problems.
• A CF rough aggregation is proposed as a method to handle the uncertainty of a periodic nature in MCGDM problems.
• Applying these aggregation operators within the EDASM, which is a well-established method in decision-making, further strengthens the decision-making process. The combination of sophisticated aggregation operators and established EDASM methodology is likely to produce remarkable results in MCGDM.
• The effectiveness of the proposed modified-EDAS method in determining the suitability of a site for fish culture and selecting the best location for a fish farm is demonstrated.

2. Basic Concepts

First, we recall some basic definitions, such as the complex fuzzy set, its basic operations, equivalence relation, and fuzzy rough set. These definitions provide a basis for the following sections.

Definition 1 ([3, 4])

Let G be a non-empty set. G is said to be a CFS on a universe of discourse Y and is defined as

G = {(s, r_{G} (s) e^{j γ_{G} (s)}) ∣ s \in Y} .

Here, $j = \sqrt{- 1}$ , r_G(s) ∈ [0, 1] represents the amplitude term, and γ_G(s) ∈ [0, 2π] is the phase term. In addition, r_G(s) denotes the membership value of an element of the CFS G.

Definition 2

Let u = r_ue^jπϕ_u and v = r_ve^jπϕ_v be any two CFSs. Their average can be defined as

A v g = \frac{u + v}{2} = {\frac{r_{u} + r_{v}}{2}} e^{j π (\frac{φ_{u} + φ_{v}}{2})} .

Definition 3

Consider a universal setℳ, and let $P$ ∈ℳ× ℳ be any relation. Then,

1. $P$ is said to be reflexive if (u, u) ∈ $P$ , ∀ u∈ℳ.
2. $P$ is said to be symmetric if ∀u, v ∈ℳ, (u, v) ∈ $P$ then (v, u) ∈ $P$ .
3. $P$ is said to be transitive if ∀u, v, x ∈ ℳ, (u, v) ∈ $P$ and (v, x) ∈ $P$ , then (u, x) ∈ $P$ .

Definition 4

A relation is said to be an equivalence relation if it is reflexive, symmetric, and transitive.

Definition 5

Let U be a non-empty and finite universe of discourse and ℛ be a fuzzy equivalence relation defined on U × U. The pair (U,ℛ) is called a fuzzy approximation space. For any A ∈ F(U), the upper and lower approximations with respect to (U,ℛ) are denoted by $\underline{ℛ} (A)$ and $\bar{ℛ} (A)$ , respectively, and are two fuzzy sets defined as

\begin{array}{l} \underline{R} (A) = {(x, μ_{\underline{R} (A)} (x)) ∣ x \in U}, \\ \bar{R} (A) = {(x, μ_{\bar{R} (A)} (x)) ∣ x \in U}, \end{array}

where

\begin{array}{l} μ_{\underline{R} (A)} (x) = \land_{u \in U} ((1 - μ_{R} (x, u)) \lor μ_{A} (u)), x \in U, \\ μ_{\underline{R} (A)} (x) = \lor_{u \in U} (μ_{R} (x, u)) \land μ_{A} (u)), x \in U . \end{array}

The pair ℛ(A) = ( $\underline{ℛ} (A)$ , $\bar{ℛ} (A)$ ) is called the fuzzy rough set of A with respect to (U,ℛ).

3. Construction of Complex Fuzzy Rough Sets

In this section, we will present the notion of a complex fuzzy equivalence relation, score function, and a complex fuzzy rough set and its properties. We also solve related examples.

Definition 6

Let ℳ be a universal set and $P$ ∈ CFS(ℳ× ℳ) be a CF equivalence relation. Then,

1. $P$ is reflexive if μ_$P$(u, u) = 1, ∀u∈ℳ.
2. $P$ is symmetric if for all (u, v) ∈ (ℳ×ℳ), μ_$P$(u, v) = μ_$P$(v, u).
3. $P$ is transitive if for all (u, v)∈ℳ × ℳ, μ_$P$(u, v) ≥ ∨_x_∈_ℳ [μ_$P$(u, x) ∧ μ_$P$(x, v)]

Definition 7

Let ℳ be the universal set and $P$ ∈ CF(ℳ× ℳ) be any complex fuzzy relation. Then, the pair (ℳ, $P$ ) is called a complex fuzzy approximation space (CFAS). Now, for any complex fuzzy set I ∈ CF(ℳ), the lower and upper approximations of I with respect to (ℳ, $P$ ) are defined as $P (I) = < \underline{P} (I), \bar{P} (I) >$ , where

\begin{array}{l} \underline{P} (I) = {< y, μ_{\underline{P} (I)} (y) e^{j π γ (x)} > ∣ y \in M}, \\ \bar{P} (I) = {< y, μ_{\bar{P} (I)} (y) e^{j π ψ (x)} > ∣ y \in M}, \end{array}

γ(x), ψ(x) ∈ [0, 1], and $j = \sqrt{- 1}$ .

Also,

\begin{array}{l} μ_{\underline{P} (I)} (y) = \land_{u \in M} [(1 - μ_{P} (y, u)) \lor μ_{P} (u)] \times e^{j π [(1 - (γ_{P} (x), r_{P} (x))) \lor r_{P} (x)]}, y \in M, \\ μ_{\bar{P} (I)} (y) = \lor_{u \in M} [μ_{P} (y, u) \land μ_{P} (u)] \times e^{j π [(ψ_{P} (x), r_{P} (x)) \land r_{P} (x))]}, y \in M . \end{array}

Example 1

Considerℳ = {y₁, y₂, y₃, y₄} and I = {(y₁, 0.4e^jπ^(0.3)), (y₂, 0.3e^jπ^(0.2)), (y₃, 0.6e^jπ^(0.4)), (y₄, 0.5e^jπ^(0.2))}, and let $P$ be an equivalence class on CF(ℳ×ℳ) as Table 1.

By simple calculation, we can easily find the upper and lower approximations as follows:

\begin{array}{l} \underline{P} (I) = {\frac{y_{1}}{0.4 e^{j π (0.5)}}, \frac{y_{2}}{0.3 e^{j π (0.7)}}, \frac{y_{3}}{0.4 e^{j π (0.2)}}, \frac{y_{4}}{0.4 e^{j π (0.2)}}}, \\ \bar{P} (I) = {\frac{y_{1}}{0.6 e^{j π (0.4)}}, \frac{y_{2}}{0.4 e^{j π (0.2)}}, \frac{y_{3}}{0.6 e^{j π (0.3)}}, \frac{y_{4}}{0.6 e^{j π (0.4)}}} . \end{array}

Definition 8

Let $P_{1} (I) = (\underline{P_{1}} (I), \bar{P_{1}} (I))$ and $P_{2} (I) = (\underline{P_{2}} (I), \bar{P_{2}} (I))$ be any two complex fuzzy rough sets. Then, we have the following operations:

1. $P_{1} (I) \oplus P_{2} (I) = (\underline{P_{1}} (I) \oplus \underline{P_{2}} (I), \bar{P_{1}} (I) \oplus \bar{P_{2}} (I))$ , where $\underline{P_{1}} (I) \oplus \underline{P_{2}} (I) = ({\underline{μ}}_{P_{1}} + {\underline{μ}}_{P_{2}} - {\underline{μ}}_{P_{1}} . {\underline{μ}}_{P_{2}}) \times e^{j π ({\underline{r}}_{1} + {\underline{r}}_{2} - {\underline{r}}_{1} . {\underline{r}}_{2})}$ and $\bar{P_{1}} (I) \oplus \bar{P_{2}} (I) = ({\bar{μ}}_{P_{1}} + {\bar{μ}}_{P_{2}} - {\bar{μ}}_{P_{1}} . {\bar{μ}}_{P_{2}}) e^{j π ({\underline{r}}_{1} + {\underline{r}}_{2} - {\underline{r}}_{1} . {\underline{r}}_{2})}$ .
2. $P_{1} (I) \oplus P_{2} (I) = (\underline{P_{1}} (I) \otimes \underline{P_{2}} (I), \bar{P_{1}} (I) \otimes \bar{P_{2}} (I))$ where $(\underline{P_{1}} (I) \otimes \underline{P_{2}} (I), \bar{P_{1}} (I) \otimes \bar{P_{2}} (I)) = (({\underline{μ}}_{P_{1}} . {\underline{μ}}_{P_{2}}) \times e^{({\underline{r}}_{1} . {\underline{r}}_{2})}, ({\bar{μ}}_{P_{1}} . {\bar{μ}}_{P_{2}}) e^{({\bar{r}}_{1} . {\bar{r}}_{2})}$ .
3. $P_{1} (I) \oplus^{*} P_{2} (I) = (\underline{P_{1}} (I) \oplus \underline{P_{2}} (I), \bar{P_{1}} (I) \otimes \bar{P_{2}} (I))$ .
4. $P_{1} (I) \otimes^{*} P_{2} (I) = (\underline{P_{1}} (I) \otimes \underline{P_{2}} (I), \bar{P_{1}} (I) \oplus \bar{P_{2}} (I))$ .
5. $P_{1} (I) \cap P_{2} (I) = (\underline{P_{1}} (I) \cap \underline{P_{2}} (I), \bar{P_{1}} (I) \cap \bar{P_{2}} (I))$ .
6. $P_{1} (I) \cup P_{2} (I) = (\underline{P_{1}} (I) \cup \underline{P_{2}} (I), \bar{P_{1}} (I) \cup \bar{P_{2}} (I))$ .
7. $P_{1} (I) \subseteq P_{2} (I) \Rightarrow \underline{P_{1}} (I) \subseteq \underline{P_{2}} (I), \bar{P_{1}} (I) \subseteq \bar{P_{2}} (I)$ .
8. $λ P_{1} (I) = (λ \underline{P_{1}} (I), λ \bar{P_{1}} (I))$ , where $λ \underline{P_{1}} (I) = (1 - {(1 - {\underline{μ}}_{P_{1} (I)})}^{λ}) e^{j π (1 - {(1 - r_{1})}^{λ})}$ and $λ \bar{P_{1}} (I) = (1 - {(1 - {\underline{μ}}_{P_{1} (I)})}^{λ}) e^{j π (1 - {(1 - r_{1})}^{λ})}$ .
9. $P_{1} {(I)}^{λ} = (\underline{P_{1}} {(I)}^{λ}, \bar{P_{1}} {(I)}^{λ})$ .
10. $P_{1} {(I)}^{C} = (\underline{P_{1}} {(I)}^{C}, \bar{P_{1}} {(I)}^{C})$ .

such that ( $\underline{P_{1}} {(I)}^{C}$ ) and ( $\bar{P_{1}} {(I)}^{C}$ ) are the complements of the complex fuzzy rough approximation operator.
11. $P_{1} (I) = P_{2} (I) \Leftrightarrow$ $\underline{P_{1}} (I) = \underline{P_{2}} (I)$ and $\bar{P_{1}} (I) = \bar{P_{2}} (I)$ .

For the comparison of two complex fuzzy rough values (CFRVs), we use a score function. The smaller the score value of CFRVs, the more inferior that value is, and vice versa.

Definition 9

The score function for CFRV $P (I) = (\underline{P} (I), \bar{P} (I)) = (\underline{μ} e^{j π r}, \bar{μ} e^{j π r^{'}})$ is given as

S (P (I)) = \frac{1}{4} (2 + \underline{μ} + \bar{μ}) e^{j π \frac{1}{4} (2 + r + r^{'})} .

4. Complex Fuzzy Rough Averaging Aggregation Operator

4.1 Complex Fuzzy RoughWeighted Averaging Operator

Here, we discuss the concept of the CFRWA operator and its properties.

Definition 10

CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = (\oplus_{i = 1}^{n} w_{i} \underline{P} (I_{i}), \oplus_{i = 1}^{n} w_{i} \bar{P} (I_{i})) .

Based on this definition, the following Theorem is given for aggregated results.

Theorem 1

Consider the collection $P (I_{i}) = (\underline{P} (I_{i}), \bar{P} (I_{i}))$ , i = 1, 2, 3, ..., n of CFRVs with weighted vectors w = (w₁, w₂, w₃, ..., w_n)^T such that $\sum_{i = 1}^{n} w_{i} = 1$ , 0 ≤ w_i ≤ 1. Then, the CFRWA operator is determined as

\begin{array}{l} CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = (\oplus_{i = 1}^{n} w_{i} \underline{P} (I_{i}), \oplus_{i = 1}^{n} w_{i} \bar{P} (I_{i})) \\ = [(1 - \prod_{i = 1}^{n} {(1 - \underline{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - γ_{i} (x))}^{w_{i}})}, (1 - \prod_{i = 1}^{n} {(1 - \bar{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - γ_{i} (x))}^{w_{i}})}] . \end{array}

Proof

We will prove the desired result by mathematical induction. According to the defined operation, we have

\begin{array}{l} P (I_{1}) \oplus P (I_{2}) = [\underline{P} (I_{1}) \oplus \underline{P} (I_{2}), \bar{P} (I_{1}) \oplus \bar{P} (I_{2})] \\ = [{{\underline{μ}}_{1} + {\underline{μ}}_{2} - {\underline{μ}}_{1} {\underline{μ}}_{2}} e^{j π (γ_{1} + γ_{2} - γ_{1} γ_{2})}, {{\bar{μ}}_{1} + {\bar{μ}}_{2} - {\bar{μ}}_{1} {\bar{μ}}_{2}} e^{j π (γ_{1} + γ_{2} - γ_{1} γ_{2})}], \end{array}

and

\begin{array}{l} λ P (I_{1}) = [λ \underline{P} (I_{1}), λ \bar{P} (I_{1})] \\ = [(1 - {(1 - {\underline{μ}}_{1})}^{λ}) e^{j π (1 - {(1 - γ_{1} (x))}^{λ})}, (1 - {(1 - {\bar{μ}}_{1})}^{λ}) e^{j π (1 - {(1 - γ_{1} (x))}^{λ})}] . \end{array}

Let us consider n = 2. Then, we have

\begin{array}{l} CFRWA (P (I_{1}), P (I_{2})) = (\oplus_{i = 1}^{2} w_{i} \underline{P} (I_{i}), \oplus_{i = 1}^{2} w_{i} \bar{P} (I_{i})) \\ = [(1 - \prod_{i = 1}^{2} {(1 - \underline{μ_{i}})}^{w_{i}}) e^{j π ((1 - \prod_{i = 1}^{2} {(1 - γ_{i} (x))}^{w_{i}})}, (1 - \prod_{i = 1}^{2} {(1 - \bar{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{2} {(1 - γ_{i} (x))}^{w_{i}})}] . \end{array}

Hence, this result holds for n = 2. Now, we consider that the result is true for n = k:

\begin{array}{l} CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{k})) = (\oplus_{i = 1}^{k} w_{i} \underline{P} (I_{i}), \oplus_{i = 1}^{k} w_{i} \bar{P} (I_{i})) \\ = [(1 - \prod_{i = 1}^{k} {(1 - \underline{μ_{i}})}^{w_{i}}) e^{j π ((1 - \prod_{i = 1}^{k} {(1 - γ_{i} (x))}^{w_{i}})}, (1 - \prod_{i = 1}^{k} {(1 - \bar{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{k} {(1 - γ_{i} (x))}^{w_{i}})}] . \end{array}

Now, we will prove the result is true for n = k + 1. We have

\begin{array}{l} CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{k}), P (I_{k + 1})) = [(\oplus_{i = 1}^{k} w_{i} \underline{P} (I_{i}) \oplus w_{k + 1} \underline{P} (I_{k + 1})), (\oplus_{i = 1}^{k} w_{i} \bar{P} (I_{i}) \oplus w_{k + 1} \bar{P} (I_{k + 1}))] \\ = [(1 - \prod_{i = 1}^{k + 1} {(1 - \underline{μ_{i}})}^{w_{i}}) e^{j π ((1 - \prod_{i = 1}^{k + 1} {(1 - γ_{i} (x))}^{w_{i}})}, (1 - \prod_{i = 1}^{k + 1} {(1 - \bar{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{k + 1} {(1 - γ_{i} (x))}^{w_{i}})}] \end{array}

Thus, the required result is true for n = k + 1, and the statement holds true for n ≥ 1.

From above observation, we see that $\underline{P} (I)$ and $\bar{P} (I)$ are CFRVs. Therefore, according to Definition 10, $\oplus_{i = 1}^{n} w_{i} \underline{P} (I_{i})$ and $\oplus_{i = 1}^{n} w_{i} \bar{P} (I_{i})$ are also CFRVs. Therefore, CFRWA $(P (I_{1}), P (I_{2}), P (I_{3}), \dots P (I_{k}))$ is also a complex fuzzy rough set under CF approximation space $(M, P)$ .

Example 2

Consider

\begin{array}{l} I \subseteq M \\ = {(\underline{y_{1}}, 0.5 e^{j π (0.3)}), (\bar{y_{1}}, 0.4 e^{j π (0.2)}), (\underline{y_{2}}, 0.3 e^{j π (0.2)}), (\bar{y_{2}}, 0.4 e^{j π (0.32)}), (\underline{y_{3}}, 0.6 e^{j π (0.4)}), (\bar{y_{3}}, 0.7 e^{j π (0.5)})}, \end{array}

with weight vector w = (0.3, 0.2, 0.4)^T. Then, by Theorem 1, we have

\begin{array}{l} CFRWA [P (I_{1}), P (I_{2}), P (I_{3})] = (\oplus_{i = 1}^{3} w_{i} \underline{P} (I_{i}), \oplus_{i = 1}^{3} w_{i} \bar{P} (I_{i})) \\ = {[(1 - {(1 - 0.5)}^{0.3} {(1 - 0.3)}^{0.2} {(1 - 0.6)}^{0.4}) \times e^{j π (1 - {(1 - 0.3)}^{0.3} {(1 - 0.2)}^{0.2} {(1 - 0.4)}^{0.4})}], [(1 - {(1 - 0.4)}^{0.3} {(1 - 0.4)}^{0.2} {(1 - 0.7)}^{0.4}) \times e^{j π (1 - {(1 - 0.2)}^{0.3} {(1 - 0.32)}^{0.2} {(1 - 0.5)}^{0.4})}]} \\ = (0.47 e^{j π (0.29)}, 0.52 e^{j π (0.34)}) . \end{array}

Some important characteristics of the CFRWA operator are given by Theorem 2.

Theorem 2

Consider the collection $P (I_{i}) = (\underline{P} (I_{i}), \bar{P} (I_{i}))$ , i = 1, 2, 3, ..., n, of CFRVs with weighted vectors w = (w₁, w₂, w₃, ..., w_n)^T such that $\sum_{i = 1}^{n} w_{i} = 1$ , 0 ≤ w_i ≤ 1. Then, the properties of the CFRWA operator are described as

i. Idempotency: If $P (I_{i}) = L (K) \forall i = 1, 2, 3, \dots, n$ , where $L (K) = (\underline{L} (K), \bar{L} (K)) = (\underline{h} e^{j π r}, \bar{h} e^{j π r})$ , then, CFRWA $(P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = L (K)$ .
ii. Boundedness: Let ${(P (I))}^{-} = (\min_{i} \underline{P} (I_{i}), \max_{i} \bar{P} (I_{i})) and {(P (I))}^{+} = (\max_{i} \underline{P} (I_{i}), \min_{i} \bar{P} (I_{i})) .$ Then, ${(P (I))}^{-} \leq CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \leq P (I))^{+} .$
iii. Shift invariance: Consider $CFRV B (K) = (\underline{B} (K), \bar{B} (K)) .$ Then, $CFRWA (P (I_{1}) \oplus B (K), P (I_{2}) \oplus B (K), P (I_{3}) \oplus B (K), \dots, P (I_{n}) \oplus B (K)) = CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \oplus B (K) .$
iv. Monotonicity: Let $F (L_{i}) = (\underline{F} (L_{i}), \bar{F} (L_{i}))$ , (i = 1, 2, 3, ..., n) be another collection of CFRVs such that $\underline{F} (L_{i}) \leq \underline{P} (I_{i})$ and $\bar{F} (L_{i}) \leq \bar{P} (I_{i}) .$ Then, $CFRWA (F (L_{1}), F (L_{2}), F (L_{3}), \dots, F (L_{n})) \leq CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) .$
v. Homogeneity: CFRWA $(δ P (I_{1}), δ P (I_{2}), δ P (I_{3}), \dots, δ P (I_{n}))$ = δCFRWA $(P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n}))$ for any δ > 0.
vi. Commutativity: Let $P^{'} (I_{i}) = (\underline{P^{'}} (I_{i}), \bar{P^{'}} (I_{i}))$ , i = 1, 2, 3, ..., n, be any permutation of $P (I_{i}) = (\underline{P} (I_{i}), \bar{P} (I_{i})) .$ Then, $CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = CFRWA (P^{'} (I_{1}), P^{'} (I_{2}), P^{'} (I_{3}), \dots, P^{'} (I_{n})) .$

Proof

i. Idempotency: As

\begin{array}{l} CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n}) = (\oplus_{i = 1}^{n} w_{i} \underline{P} (I_{i}), \oplus_{i = 1}^{n} w_{i} \bar{P} (I_{i})) \\ = [(1 - \prod_{i = 1}^{n} {(1 - \underline{μ_{i}})}^{w_{i}}) e^{j π ((1 - \prod_{i = 1}^{n} {(1 - γ_{i} (x))}^{w_{i}})}, (1 - \prod_{i = 1}^{n} {(1 - \bar{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - γ_{i} (x))}^{w_{i}})}] . \end{array}

Moreover, for all i, $P (I_{i})$ = L(K), where L(K) = (L(K), L(K)) = (he^jπr, h̄e^jπr). Therefore,

\begin{array}{l} CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n}) \\ = [(1 - \prod_{i = 1}^{n} {(1 - \underline{h})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - r)}^{w_{i}})}, (1 - \prod_{i = 1}^{n} {(1 - \bar{h})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - r)}^{w_{i}})}] \\ = [(1 - \prod_{i = 1}^{n} (1 - \underline{h})) e^{j π (1 - \prod_{i = 1}^{n} (1 - r))}, (1 - \prod_{i = 1}^{n} (1 - \bar{h})) e^{j π (1 - \prod_{i = 1}^{n} (1 - r))}] \\ = (\underline{h} e^{j π r}, \bar{h} e^{j π r}) = (\underline{L} (K), \bar{L} (K)) \\ = L (K) . \end{array}

ii. Boundedness: Let ${(P (I))}^{-} = \min_{i} (\underline{P} (I_{i}))$ and ${(P (I))}^{+} = \max_{i} (\bar{P} (I_{i})) .$ Then, we will prove that

\begin{array}{l} {(P (I))}^{-} \leq CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ \leq {(P (I))}^{+} . \end{array}

As ${(\underline{P} (I))}^{-}$ = (min_i(μ_i)e^jπ^min^_i⁽^γ_i⁾, min_i(μ̄_i)e^jπ^min^_i⁽^γ_i⁾), ${(\underline{P} (I))}^{+}$ = (max_i(μ_i)e^jπ^max^_i⁽^γ_i⁾, max_i(μ̄_i)e^jπ^max^_i⁽^γ_i⁾). For each i = 1, 2, 3, ..., n, we have

\begin{array}{l} {min}_{i} (\underline{μ_{i}}) e^{j π {min}_{i} (γ_{i})} \leq (\underline{μ_{i}}) e^{j π (γ_{i})} \leq {max}_{i} (\underline{μ_{i}}) e^{j π {max}_{i} (γ_{i})} \\ \begin{array}{l} \Leftrightarrow & {1 - {max}_{i} (\underline{μ_{i}})} e^{j π (1 - {max}_{i} (γ_{i}))} \leq {1 - (\underline{μ_{i}})} e^{j π (1 - γ_{i})} \\ \leq {1 - {min}_{i} (\underline{μ_{i}})} e^{j π (1 - {min}_{i} (γ_{i}))} \\ \Leftrightarrow & \prod_{i = 1}^{n} {1 - {max}_{i} (\underline{μ_{i}})}^{w_{i}} e^{j π \prod_{i = 1}^{n} {(1 - {max}_{i} (γ_{i}))}^{w_{i}}} \\ \leq \prod_{i = 1}^{n} {1 - (\underline{μ_{i}})}^{w_{i}} e^{j π \prod_{i = 1}^{n} {(1 - (γ_{i}))}^{w_{i}}} \\ \leq \prod_{i = 1}^{n} {1 - {min}_{i} (\underline{μ_{i}})}^{w_{i}} e^{j π \prod_{i = 1}^{n} {(1 - {min}_{i} (γ_{i}))}^{w_{i}}}, \\ \Leftrightarrow & {1 - {max}_{i} (\underline{μ_{i}})} e^{j π (1 - {max}_{i} (γ_{i}))} \\ \leq \prod_{i = 1}^{n} {1 - (\underline{μ_{i}})}^{w_{i}} e^{j π {(1 - (γ_{i}))}^{w_{i}}} \\ \leq {1 - {min}_{i} (\underline{μ_{i}})} e^{j π (1 - {min}_{i} (γ_{i}))}, \\ \Leftrightarrow & (1 - {1 - {min}_{i} (\underline{μ_{i}})}) e^{j π (1 - (1 - {min}_{i} (γ_{i})))} \\ \leq (1 - \prod_{i = 1}^{n} {1 - (\underline{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - γ_{i})}^{w_{i}})} \\ \leq (1 - {1 - {max}_{i} (\underline{μ_{i}})}) e^{j π (1 - (1 - {max}_{i} (γ_{i})))}, \\ \Rightarrow & {min}_{i} (\underline{μ_{i}}) e^{j π ({min}_{i} (γ_{i}))} \\ \leq (1 - \prod_{i = 1}^{n} {1 - (\underline{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - (γ_{i}))}^{w_{i}})} \\ \leq {max}_{i} (\underline{μ_{i}}) e^{j π ({max}_{i} (γ_{i}))} . \end{array} \end{array}

Similarly, we can show that, for upper approximation,

\begin{array}{l} {min}_{i} (\bar{μ_{i}}) e^{j π ({min}_{i} (γ_{i}))} \leq {(1 - \prod_{i = 1}^{n} {1 - (\bar{μ_{i}})})}^{w_{i}} e^{j π (1 - \prod_{i = 1}^{n} (1 - {(γ_{i})}^{w_{i}}))} \\ \leq {max}_{i} (\bar{μ_{i}}) e^{j π ({max}_{i} (γ_{i}))} . \end{array}

iii. Shift invariance: Let ℬ(K) = (ℬ(K), ℬ(K)) = (he^jπ⁽^v⁾, h̄e^jπ⁽^v⁾) be any CFRV and $P (I_{i}) = (\underline{P} (I_{i}), \bar{P} (I_{i})) = (\underline{μ_{i}} e^{j π (γ_{i})}, \bar{μ_{i}} e^{j π (γ_{i})})$ be the collection of CFRVs. Thus,

\begin{array}{l} P (I_{1}) \oplus ℬ (K) = (\underline{P} (I_{1}) \oplus \underline{ℬ} (K), \bar{P} (I_{1}) \oplus \bar{ℬ} (K)) \\ = [(1 - \prod_{i = 1}^{n} {(1 - \underline{μ_{i}})}^{w_{i}} {(1 - \underline{h})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - γ_{i})}^{w_{i}} {(1 - v)}^{w_{i}})}, (1 - \prod_{i = 1}^{n} {(1 - \bar{μ_{i}})}^{w_{i}} {(1 - \bar{h})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - γ_{i})}^{w_{i}} {(1 - v)}^{w_{i}})}] \\ = [(1 - (1 - \underline{h}) \prod_{i = 1}^{n} {(1 - \underline{μ_{i}})}^{w_{i}}) e^{j π (1 - (1 - v) \prod_{i = 1}^{n} {(1 - γ_{i})}^{w_{i}})}, (1 - (1 - \bar{h}) \prod_{i = 1}^{n} {(1 - \bar{μ_{i}})}^{w_{i}}) e^{j π (1 - {(1 - v)}^{w_{i}} \prod_{i = 1}^{n} {(1 - γ_{i})}^{w_{i}})}] \\ = [(1 - \prod_{i = 1}^{n} {(1 - \underline{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} (1 - γ_{i}))^{w_{i}})} \oplus \underline{h}, (1 - \prod_{i = 1}^{n} {(1 - \bar{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - γ_{i}))}^{w_{i}})} \oplus \bar{h}] \\ = [(1 - \prod_{i = 1}^{n} {(1 - \underline{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} (1 - γ_{i}))^{w_{i}})}, (1 - \prod_{i = 1}^{n} {(1 - \bar{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - γ_{i}))}^{w_{i}})} \oplus [\underline{h}, \bar{h}]] \\ = CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \oplus ℬ (K) . \end{array}

Thus, the proof is complete.

iv. Monotonicity: As $F (L_{i}) = (\underline{F} (L_{i}), \bar{F} (L_{i})) = (\underline{h_{i}} e^{j π (v_{i})}, \bar{h_{i}} e^{j π (v_{i})})$ and $P (I_{i}) = (\underline{P} (I_{i}), \bar{P} (I_{i}))$ for (i = 1, 2, 3, ..., n), we have to show that F(L_i) ≤ $(\underline{P} (I_{i})$ and F(L_i) ≤ $(\bar{P} (I_{i}) .$ Then,

\begin{array}{l} (\underline{h_{i}}) e^{j π (v_{i})} \leq (\underline{μ_{i}}) e^{j π (γ_{i})}, \\ {(1 - \underline{μ_{i}})}^{w_{i}} e^{j π {(1 - γ_{i})}^{w_{i}}} \leq {(1 - \underline{h_{i}})}^{w_{i}} e^{j π {(1 - v_{i})}^{w_{i}}}, \\ (\prod_{i = 1}^{n} {(1 - \underline{μ_{i}})}^{w_{i}}) e^{j π (\prod_{i = 1}^{n} {(1 - γ_{i})}^{w_{i}})} \leq (\prod_{i = 1}^{n} {(1 - \underline{h_{i}})}^{w_{i}}) e^{j π (\prod_{i = 1}^{n} {(1 - v_{i})}^{w_{i}})}, \\ (1 - \prod_{i = 1}^{n} {(1 - \underline{h_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - v_{i})}^{w_{i}})} \leq (1 - \prod_{i = 1}^{n} {(1 - \underline{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - γ_{i})}^{w_{i}})} . \end{array}

Similarly, we can show that

(1 - \prod_{i = 1}^{n} {(1 - \bar{h_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - v_{i})}^{w_{i}})} \leq (1 - \prod_{i = 1}^{n} {(1 - \bar{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - γ_{i})}^{w_{i}})}

Hence, from above, F(L_i) ≤ $(\underline{P} (I_{i})$ and F(L_i) ≤ $(\bar{P} (I_{i}) .$ Therefore,

\begin{array}{l} CFRWA (F (L_{1}), F (L_{2}), F (L_{3}), \dots, F (L_{n})) \\ \leq CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) . \end{array}

v. Homogeneity: Take any real number δ > 0 and CFRV

\begin{array}{l} P (I_{i}) = (\underline{P} (I_{i}), \bar{P} (I_{i})), \\ δ P (I_{i}) = (δ \underline{P} (I_{1}), δ \bar{P} (I_{1})) \\ = [1 - {(1 - \underline{μ_{1}})}^{δ} e^{j π (1 - {(1 - γ_{1})}^{δ})}, 1 - {(1 - \bar{μ_{1}})}^{δ} e^{j π (1 - {(1 - γ_{1})}^{δ})}], \\ CFRWA (δ P (I_{1}), δ P (I_{2}), δ P (I_{3}), \dots, δ P (I_{n})) \\ = [(1 - \prod_{i = 1}^{n} {({(1 - \underline{μ_{i}})}^{δ})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {({(1 - γ_{i})}^{δ})}^{w_{i}})}, (1 - \prod_{i = 1}^{n} {({(1 - \bar{μ_{i}})}^{δ})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {({(1 - γ_{i})}^{δ})}^{w_{i}})}] \\ = [(1 - \prod_{i = 1}^{n} {({(1 - \underline{μ_{i}})}^{w_{i}})}^{δ}) e^{j π (1 - \prod_{i = 1}^{n} {({(1 - γ_{i})}^{w_{i}})}^{δ})}, (1 - \prod_{i = 1}^{n} {({(1 - \bar{μ_{i}})}^{w_{i}})}^{δ}) e^{j π (1 - \prod_{i = 1}^{n} {({(1 - γ_{i})}^{w_{i}})}^{δ})}] \\ = δ CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) . \end{array}

This is the required proof.

vi. Commutativity: Consider the following:

\begin{array}{l} CFRWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{k})) = (\oplus_{i = 1}^{k} w_{i} \underline{P} (I_{i}), \oplus_{i = 1}^{k} w_{i} \bar{P} (I_{i})) \\ = [(1 - \prod_{i = 1}^{k} {(1 - \underline{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{k} {(1 - γ_{i})}^{w_{i}})}, (1 - \prod_{i = 1}^{k} {(1 - \bar{μ_{i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{k} {(1 - γ_{i})}^{w_{i}})}] . \end{array}

Because $(P^{'} (I_{1}), P^{'} (I_{2}), P^{'} (I_{3}), \dots, P^{'} (I_{k}))$ is any permutation of $(P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n}))$ , we have $P (I_{i}) = P^{'} (I_{i})$ , (i = 1, 2, 3, ..., n)

\begin{array}{l} = [(1 - \prod_{i = 1}^{k} {(1 - \underline{μ_{i}^{'}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{k} {(1 - γ_{i}^{'})}^{w_{i}})}, (1 - \prod_{i = 1}^{k} {(1 - \bar{μ_{i}^{'}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{k} {(1 - γ_{i}^{'})}^{w_{i}})}] \\ = (\oplus_{i = 1}^{k} w_{i} \underline{P^{'}} (I_{i}), \oplus_{i = 1}^{k} w_{i} \bar{P^{'}} (I_{i})) \\ = CFRWA (P^{'} (I_{1}), P^{'} (I_{2}), P^{'} (I_{3}), \dots, P^{'} (I_{k})) . \end{array}

4.2 Complex Fuzzy Rough Ordered Weighted Averaging Operator

In this subsection, we discuss the CFROWA operator in detail, along with its properties.

Definition 11

Consider the collection of CFRVs $P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})$ , where w = (w₁, w₂, w₃, ..., w_n)^T represents a weight vector such that $\sum_{i = 1}^{n} w_{i} = 1$ , 0 ≤ w_i ≤ 1. Then, the CFROWO operator is defined as $CFROWA(P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = (\oplus_{i = 1}^{n} w_{i} \underline{P_{η}} (I_{i}), \oplus_{i = 1}^{n} w_{i} \bar{P_{η}} (I_{i}))$ .

Based on Definition 11, the following theorem is given for the CFROWA operator.

Theorem 3

Consider the collection $P (I_{i}) = (\underline{P} (I_{i}), \bar{P} (I_{i}))$ , i = 1, 2, 3, ..., n of CFRVs associated with weighted vectors w = (w₁, w₂, w₃, ..., w_n)^T such that $\sum_{i = 1}^{n} w_{i} = 1$ , 0 ≤ w_i ≤ 1. Then, the CFROWA operator is determined as

\begin{array}{l} CFROWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = (\oplus_{i = 1}^{n} w_{i} \underline{P_{η}} (I_{i}), \oplus_{i = 1}^{n} w_{i} \bar{P_{η}} (I_{i})) \\ = [(1 - \prod_{i = 1}^{n} {(1 - \underline{μ_{η i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - γ_{ρ i} (x))}^{w_{i}})}, (1 - \prod_{i = 1}^{n} {(1 - \bar{μ_{η i}})}^{w_{i}}) e^{j π (1 - \prod_{i = 1}^{n} {(1 - γ_{ρ i} (x))}^{w_{i}})}], \end{array}

where $P_{η} (I_{i}) = (\underline{P_{η}} (I_{i}), \bar{P_{η}} (I_{i}))$ represents the largest value in the given collection of CFRVs.

Proof

The proof follows from Theorem 1.

Some important properties of the CFROWA operator are given by Theorem 4.

Theorem 4

i. Idempotency: If $P (I_{i})$ = L(K) ∀ i = 1, 2, 3, ..., n, where L(K) = (L(K), L(K)) = (he^jπr, h̄e^jπr), then

$CFROWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = L (K) .$
ii. Boundedness: Let ${(P (I))}^{-} = (\min_{i} \underline{P} (I_{i}), \max_{i} \bar{P} (I_{i}))$ and ${(P (I))}^{+} = (\max_{i} \underline{P} (I_{i}), \min_{i} \bar{P} (I_{i})) .$ Then,

$\begin{array}{l} {(P (I))}^{-} \leq CFROWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ \leq {(P (I))}^{+} . \end{array}$
iii. Shift invariance: Consider CFRVs ℬ(K) = (ℬ(K), ℬ(K)). Then,

$\begin{array}{l} CFROWA (P (I_{1}) \oplus ℬ (K), P (I_{2}) \oplus ℬ (K), P (I_{3}) \oplus ℬ (K), \dots, P (I_{n}) \oplus ℬ (K)) \\ = CFROWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \oplus ℬ (K) \end{array}$
iv. Monotonicity: Let F(L_i) = (F(L_i), F(L_i)), (i = 1, 2, 3, ..., n) be another collection of CFRVs such that F(L_i) ≤ $\underline{P} (I_{i})$ and F(L_i) ≤ $\bar{P} (I_{i}) .$ Then,

$\begin{array}{l} CFROWA (F (L_{1}), F (L_{2}), F (L_{3}), ......., F (L_{n})) \\ \leq CFROWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) . \end{array}$
v. Homogeneity:

$\begin{array}{l} CFROWA (δ P (I_{1}), δ P (I_{2}), δ P (I_{3}), \dots, δ P (I_{n})) \\ = δ CFROWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})), \end{array}$

for any δ > 0.
vi. Commutativity: Let $P^{'} (I_{i}) = (\underline{P^{'}} (I_{i}), \bar{P^{'}} (I_{i}))$ , i = 1, 2, 3, ..., n be any permutation of $P (I_{i}) = (\underline{P} (I_{i}), \bar{P} (I_{i})) .$ Then,

\begin{array}{l} CFROWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ = CFROWA (P^{'} (I_{1}), P^{'} (I_{2}), P^{'} (I_{3}), \dots, P^{'} (I_{n})) . \end{array}

Proof

The proof follows from Theorem 2.

4.3 Complex Fuzzy Rough Hybrid Averaging Operator

In the following subsection, we discuss the CFRHA operator. The CFRHA operator weights both the values and the ordered positions of the CF argument simultaneously.

Definition 12

Consider the collection of CFRVs $P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})$ , where ν = (ν₁, ν₂, ν₃, ..., ν_n)^T represents the weight vector such that $\sum_{i = 1}^{n} ν_{i} = 1$ , 0 ≤ ν_i ≤ 1. Let w = (w₁, w₂, w₃, ..., w_n)^T be the associated weight vector of the CFRV collection such that $\sum_{i = 1}^{n} w_{i} = 1$ , 0 ≤ w_i ≤ 1. Then, the CFRHA operator is defined as

\begin{array}{l} CFRHA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ = \oplus_{i = 1}^{n} w_{i} {\ddot{P}}_{η} (I_{i}) = (\oplus_{i = 1}^{n} w_{i} \underline{{\ddot{P}}_{η}} (I_{i}), \oplus_{i = 1}^{n} w_{i} \bar{{\ddot{P}}_{η}} (I_{i})) . \end{array}

From the above definition, we can present the following theorem for the CFRHA operator.

Theorem 5

Then, the CFRHA operator is defined as

\begin{array}{l} CFROWA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = \oplus_{i = 1}^{n} w_{i} {\ddot{P}}_{η} (I_{i}) = (\oplus_{i = 1}^{n} w_{i} \underline{{\ddot{P}}_{η}} (I_{i}), \oplus_{i = 1}^{n} w_{i} \bar{{\ddot{P}}_{η}} (I_{i})) \\ = [(1 - \prod_{i = 1}^{n} {(1 - \underline{{\ddot{μ}}_{η i}})}^{w_{i}}) e^{j π ((1 - \prod_{i = 1}^{n} {(1 - {\ddot{γ}}_{ρ i} (x))}^{w_{i}})}, (1 - \prod_{i = 1}^{n} {(1 - \bar{{\ddot{μ}}_{η i}})}^{w_{i}}) e^{j π ((1 - \prod_{i = 1}^{n} {(1 - {\ddot{γ}}_{ρ i} (x))}^{w_{i}})}], \end{array}

where $\ddot{P}$ _η( $I$ _i) = ζw_i $P (I_{i})$ = (ζw_i $\underline{P} (I_{i})$ ), ζw_i $\bar{P} (I_{i})$ ) represents the largest value of permutation in the given collection of CFRVs, and ζ represents the balancing coefficient.

Proof

The proof follows from Theorem 1.

Remark

If $ν = {(\frac{1}{n}, \frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ , then the CFRHA operator is reduced to the CFROWA operator.

Some important properties of the CFRHA operator are given by the following theorem.

Theorem 6

i. Idempotency: If $P (I_{i})$ = L(K) ∀ i = 1, 2, 3, ..., n, where L(K) = (L(K), L(K)) = (he^jπr, h̄e^jπr), then

$CFRHA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = L (K) .$
ii. Boundedness: Let ${(P (I))}^{-} = (\min_{i} \underline{P} (I_{i}), \max_{i} \bar{P} (I_{i}))$ and ${(P (I))}^{+} = (\max_{i} \underline{P} (I_{i}), \min_{i} \bar{P} (I_{i})) .$ Then,

$\begin{array}{l} {(P (I))}^{-} \leq CFRHA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ \leq {(P (I))}^{+} . \end{array}$
iii. Shift invariance: Consider CFRVs ℬ(K) = (ℬ(K), ℬ(K)). Then,

$\begin{array}{l} CFRHA (P (I_{1}) \oplus ℬ (K), P (I_{2}) \oplus ℬ (K), P (I_{3}) \oplus ℬ (K), \dots, P (I_{n}) \oplus ℬ (K)) \\ = CFRHA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \oplus ℬ (K) . \end{array}$
iv. Monotonicity: Let F(L_i) = (F(L_i), F(L_i)), (i = 1, 2, 3, ..., n) be another collection of CFRVs such that F(L_i) ≤ $\underline{P} (I_{i})$ and F(L_i) ≤ $\bar{P} (I_{i}) .$ Then,

$\begin{array}{l} CFRHA (F (L_{1}), F (L_{2}), F (L_{3}), ......., F (L_{n})) \\ \leq CFRHA (P (I_{1}), P (I_{2}), P (I_{3}), ......., P (I_{n})) . \end{array}$
v. Homogeneity:

$\begin{array}{l} CFRHA (δ P (I_{1}), δ P (I_{2}), δ P (I_{3}), \dots, δ P (I_{n})) \\ = δ CFRHA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})), \end{array}$

for any δ > 0.
vi. Commutativity: Let $P^{'} (I_{i}) = (\underline{P^{'}} (I_{i}), \bar{P^{'}} (I_{i}))$ , i = 1, 2, 3, ..., n, be any permutation of $P (I_{i}) = (\underline{P} (I_{i}), \bar{P} (I_{i})) .$ Then,

$\begin{array}{l} CFRHA (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ = CFRHA (P^{'} (I_{1}), P^{'} (I_{2}), P^{'} (I_{3}), \dots, P^{'} (I_{n})) . \end{array}$

Proof

The proof follows from Theorem 2.

5. Complex Fuzzy Rough Geometric Aggregation Operator

5.1 Complex Fuzzy Rough Weighted Geometric Operator

This subsection presents a detailed study of the complex fuzzy rough geometric operator and its characteristics.

Definition 13

Consider the collection of CFRVs $P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})$ , where w = (w₁, w₂, w₃, ..., w_n)^T represents the weight vector such that $\sum_{i = 1}^{n} w_{i} = 1$ , 0 ≤ w_i ≤ 1. Then, the complex fuzzy rough weighted geometric (CFRWG) operator is defined as

\begin{array}{l} CFRWG(P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ = (\otimes_{i = 1}^{n} {(\underline{P} (I_{1}))}^{w_{i}}, \otimes_{i = 1}^{n} {(\bar{P} (I_{1}))}^{w_{i}}) . \end{array}

The aggregated result of the CFRWG operator is developed in the following theorem by using the above definition.

Theorem 7

\begin{array}{l} CFRWG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = (\otimes_{i = 1}^{n} w_{i} \underline{P} (I_{i}), \otimes_{i = 1}^{n} w_{i} \bar{P} (I_{i})) \\ = [(\prod_{i = 1}^{n} {(\underline{μ_{i}})}^{w_{i}}) e^{j π (\prod_{i = 1}^{n} {(γ_{i} (x))}^{w_{i}})}, (\prod_{i = 1}^{n} {(\bar{μ_{i}})}^{w_{i}}) e^{j π (\prod_{i = 1}^{n} {(γ_{i} (x))}^{w_{i}})}] . \end{array}

Proof

The proof follows directly from Theorem 1.

From the above observation, we see that $\underline{P} (I)$ and $\bar{P} (I)$ are CFRVs. Therefore, by Definition 13, $\otimes_{i = 1}^{n} w_{i} \underline{P} (I_{i})$ and $\otimes_{i = 1}^{n} w_{i} \bar{P} (I_{i})$ are also CFRVs, and CFRWG $(P (I_{1}), P (I_{2}), P (I_{3}), \dots P (I_{k}))$ is a complex fuzzy rough set under CF approximation space (ℳ, $P$ ).

Theorem 8

i. Idempotency: If $P (I_{i})$ = L(K) ∀ i = 1, 2, 3, ..., n, where L(K) = (L(K), L(K)) = (he_jπr, h̄e_jπr), then

$CFRWG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = L (K) .$
ii. Boundedness: Let ${(P (I))}^{-} = (\min_{i} \underline{P} (I_{i}), \max_{i} \bar{P} (I_{i}))$ and ${(P (I))}^{+} = (\max_{i} \underline{P} (I_{i}), \min_{i} \bar{P} (I_{i})) .$ Then,

$\begin{array}{l} {(P (I))}^{-} \leq CFRWG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ \leq {(P (I))}^{+} . \end{array}$
iii. Shift invariance: Consider CFRV ℬ(K) = (ℬ(K), ℬ(K)). Then,

$\begin{array}{l} CFRWG (P (I_{1}) \otimes ℬ (K), P (I_{2}) \otimes ℬ (K), P (I_{3}) \otimes ℬ (K), \dots, P (I_{n}) \otimes ℬ (K)) \\ = CFRWG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \otimes ℬ (K) . \end{array}$
iv. Monotonicity: Let F(L_i) = (F(L_i), F(L_i)), (i = 1, 2, 3, ..., n) be another collection of CFRVs such that F(L_i) ≤ $\underline{P} (I_{i})$ and F(L_i) ≤ $\bar{P} (I_{i}) .$ Then,

$\begin{array}{l} CFRWG (F (L_{1}), F (L_{2}), F (L_{3}), \dots, F (L_{n})) \\ \leq CFRWG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) . \end{array}$
v. Homogeneity:

$\begin{array}{l} CFRWG (δ P (I_{1}), δ P (I_{2}), δ P (I_{3}), \dots, δ P (I_{n})) \\ = δ CFRWG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})), \end{array}$

for any δ > 0.
vi. Commutativity: Let $P^{'} (I_{i}) = (\underline{P^{'}} (I_{i}), \bar{P^{'}} (I_{i}))$ , i = 1, 2, 3, ..., n, be any permutation of $P (I_{i}) = (\underline{P} (I_{i}), \bar{P} (I_{i})) .$ Then,

$\begin{array}{l} CFRWG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ = CFRWG (P^{'} (I_{1}), P^{'} (I_{2}), P^{'} (I_{3}), \dots, P^{'} (I_{n})) . \end{array}$

Proof

The proof follows directly from Theorem 2.

5.2 Complex Fuzzy Rough OrderedWeight Geometric Operator

We now investigate the concept of complex fuzzy rough-ordered weighted geometric (CFROWG) aggregation operators and discuss some basic properties.

Definition 14

Consider the collection of complex fuzzy rough values $P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})$ , where w = (w₁, w₂, w₃, ..., w_n)^T represents the weight vector such that $\sum_{i = 1}^{n} w_{i} = 1$ , 0 ≤ w_i ≤ 1. Then, the CFROWG operator is defined as

\begin{array}{l} CFROWG(P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ = (\otimes_{i = 1}^{n} {(\underline{P_{η}} (I_{i}))}^{w_{i}}, \otimes_{i = 1}^{n} {(\bar{P_{η}} (I_{1}))}^{w_{i}}) . \end{array}

Based on the above definition, we derive the aggregate result for the CFROWG operator in Theorem 9.

Theorem 9

Consider the collection $P (I_{i}) = (\underline{P} (I_{i}), \bar{P} (I_{i}))$ , i = 1, 2, 3, ..., n, of CFRVs associated with weighted vectors w = (w₁, w₂, w₃, ..., w_n)^T such that $\sum_{i = 1}^{n} w_{i} = 1$ , 0 ≤ w_i ≤ 1. Then, the CFROWG operator is determined as

\begin{array}{l} CFRWOG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = (\otimes_{i = 1}^{n} w_{i} \underline{P_{η}} (I_{i}), \otimes_{i = 1}^{n} w_{i} \bar{P_{η}} (I_{i})) \\ = [(\prod_{i = 1}^{n} {(\underline{μ_{η i}})}^{w_{i}}) e^{j π (\prod_{i = 1}^{n} {(γ_{ρ i} (x))}^{w_{i}})}, (\prod_{i = 1}^{n} {(\bar{μ_{η i}})}^{w_{i}}) e^{j π (\prod_{i = 1}^{n} {(γ_{ρ i} (x))}^{w_{i}})}], \end{array}

where $P_{η} (I_{i}) = (\underline{P_{η}} (I_{i}), \bar{P_{η}} (I_{i}))$ represents the largest value in the given collection of CFRVs.

Proof

The proof follows from Theorem 1.

Theorem 10

i. Idempotency: If $P (I_{i})$ = L(K) ∀ i = 1, 2, 3, ..., n, where L(K) = (L(K), L(K)) = (he^jπr, h̄e^jπr), then

$CFROWG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = L (K) .$
ii. Boundedness: Let ${(P (I))}^{-} = (\min_{i} \underline{P} (I_{i}), \max_{i} \bar{P} (I_{i}))$ and ${(P (I))}^{+} = (\max_{i} \underline{P} (I_{i}), \min_{i} \bar{P} (I_{i})) .$ Then,

$\begin{array}{l} {(P (I))}^{-} \\ \leq CFROWG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ \leq {(P (I))}^{+} . \end{array}$
iii. Shift invariance: Consider CFRV ℬ(K) = (ℬ(K), ℬ(K)). Then,

$\begin{array}{l} CFROWG (P (I_{1}) \otimes ℬ (K), P (I_{2}) \otimes ℬ (K), P (I_{3}) \otimes ℬ (K), \dots, P (I_{n}) \otimes ℬ (K)) \\ = CFROWG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \otimes ℬ (K) . \end{array}$
iv. Monotonicity: Let F(L_i) = (F(L_i), F(L_i)),(i = 1, 2, 3, ..., n) be another collection of CFRVs such that F(L_i) ≤ $\underline{P} (I_{i})$ and F(L_i) ≤ $\bar{P} (I_{i}) .$ Then,

$\begin{array}{l} CFROWG (F (L_{1}), F (L_{2}), F (L_{3}), \dots, F (L_{n})) \\ \leq CFROWG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) . \end{array}$
v. Homogeneity:

$\begin{array}{l} CFROWG (δ P (I_{1}), δ P (I_{2}), δ P (I_{3}), \dots, δ P (I_{n})) \\ = δ CFROWG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})), \end{array}$

for any δ > 0.
vi. Commutativity: Let $P^{'} (I_{i}) = (\underline{P^{'}} (I_{i}), \bar{P^{'}} (I_{i}))$ , i = 1, 2, 3, ..., n, be any permutation of $P (I_{i}) = (\underline{P} (I_{i}), \bar{P} (I_{i})) .$ Then,

$\begin{array}{l} CFROWG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ = CFROWG (P^{'} (I_{1}), P^{'} (I_{2}), P^{'} (I_{3}), \dots, P^{'} (I_{n})) . \end{array}$

Proof

The proof follows from Theorem 2.

5.3 Complex Fuzzy Rough Hybrid Geometric Operator

Definition 15

Consider the collection of CFRVs $P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})$ , where ν = (ν₁, ν₂, ν₃, ..., ν_n)^T represents weight vector such that $\sum_{i = 1}^{n} ν_{i} = 1$ , 0 ≤ ν_i ≤ 1. Let w = (w₁, w₂, w₃, ..., w_n)^T be the associated weight vector of the CFRV collection such that $\sum_{i = 1}^{n} w_{i} = 1$ , 0 ≤ w_i ≤ 1. Then, the CFRHG operator is defined as

CFRHG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = \otimes_{i = 1}^{n} {({\ddot{P}}_{η} (I_{i}))}^{w_{i}} = (\otimes_{i = 1}^{n} {(\underline{{\ddot{P}}_{η}} (I_{i}))}^{w_{i}}, \otimes_{i = 1}^{n} {(\bar{{\ddot{P}}_{η}} (I_{i}))}^{w_{i}}) .

Consedring Definition 15, we demonstrate the aggregate result for the CFRHG operator in Theorem 11.

Theorem 11

\begin{array}{l} CFRWOG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = (\otimes_{i = 1}^{n} w_{i} \underline{{\ddot{P}}_{η}} (I_{i}), \otimes_{i = 1}^{n} w_{i} \bar{{\ddot{P}}_{η}} (I_{i})) \\ = [(\prod_{i = 1}^{n} {(\underline{{\ddot{μ}}_{η i}})}^{w_{i}}) e^{j π (\prod_{i = 1}^{n} {({\ddot{γ}}_{ρ i} (x))}^{w_{i}})}, (\prod_{i = 1}^{n} {(\bar{{\ddot{μ}}_{η i}})}^{w_{i}}) e^{j π (\prod_{i = 1}^{n} {({\ddot{γ}}_{ρ i} (x))}^{w_{i}})}], \end{array}

where $\ddot{P}$ _η( $I$ _i) = ζw_i $P (I_{i})$ = (ζw_i $\underline{P} (I_{i})$ , ζw_i $\bar{P} (I_{i})$ ) represents the largest value of permutation in the given collection of CFRVs, and ζ is the balancing coefficient.

Proof

The proof follows from Theorem 1.

Remark

If $ν = {(\frac{1}{n}, \frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ , then the CFRHG operator is reduced to the CFROWG operator.

Some important properties of the CFRHG operator are provided in the following theorem.

Theorem 12

i. Idempotency: If $P (I_{i})$ = L(K) ∀ i = 1, 2, 3, ..., n, where L(K) = (L(K), L(K)) = (he^jπr, h̄e^jπr), then

$CFRHG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) = L (K) .$
ii. Boundedness: Let ${(P (I))}^{-} = (\min_{i} (\underline{P} (I_{i}), \max_{i} (\bar{P} (I_{i}))$ and ${(P (I))}^{+} = (\max_{i} (\underline{P} (I_{i}), \min_{i} (\bar{P} (I_{i})) .$ Then,

$\begin{array}{l} {(P (I))}^{-} \leq CFRHG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ \leq {(P (I))}^{+} . \end{array}$
iii. Shift invariance: Consider CFRV ℬ(K) = (ℬ(K), ℬ(K)). Then,

$\begin{array}{l} CFRHG (P (I_{1}) \otimes ℬ (K), P (I_{2}) \otimes ℬ (K), P (I_{3}) \otimes ℬ (K), \dots, P (I_{n}) \otimes ℬ (K)) \\ = CFRHG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \otimes ℬ (K) . \end{array}$
iv. Monotonicity: Let F(L_i) = (F(L_i), F(L_i)), (i = 1, 2, 3, ..., n) be another collection of CFRVs such that F(L_i) ≤ $\underline{P} (I_{i})$ and F(L_i) ≤ $\bar{P} (I_{i}) .$ Then,

$\begin{array}{l} CFRHG (F (L_{1}), F (L_{2}), F (L_{3}), \dots, F (L_{n})) \\ \leq CFRHG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) . \end{array}$
v. Homogeneity:

$\begin{array}{l} CFRHG (δ P (I_{1}), δ P (I_{2}), δ P (I_{3}), \dots, δ P (I_{n})) \\ = δ CFRHG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})), \end{array}$

for any δ > 0.
vi. Commutativity: Let $P^{'} (I_{i}) = (\underline{P^{'}} (I_{i}), \bar{P^{'}} (I_{i}))$ , i = 1, 2, 3, ..., n, be any permutation of $P (I_{i}) = (\underline{P} (I_{i}), \bar{P} (I_{i})) .$ Then,

$\begin{array}{l} CFRHG (P (I_{1}), P (I_{2}), P (I_{3}), \dots, P (I_{n})) \\ = CFRHG (P^{'} (I_{1}), P^{'} (I_{2}), P^{'} (I_{3}), \dots, P^{'} (I_{n})) . \end{array}$

Proof

The proof follows from Theorem 2.

6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators

Consider a set containing r alternatives such that M = {y₁, y₂, y₃, ..., y_r} and a set of n decision attributes, ℬ = {b₁, b₂, b₃, ..., b_n}. Let A = {A₁, A₂, A₃, ..., A_z} be the set of z professional decision-makers who will make their evaluation report for each alternative M = {y₁, y₂, y₃, ..., y_r} against the set of attributes ℬ = {b₁, b₂, b₃, ..., b_n}. Let w = {w₁, w₂, w₃, ..., w_n}^T be the weight vector for attribute b_i and ϕ = {ϕ₁, ϕ₂, ϕ₃, ..., ϕ_z}^T be the weight vector for decision-makers A_i (i = 1, 2, 3, ..., z) such that $\sum_{i = 1}^{n} w_{i} = 1$ , $\sum_{i = 1}^{n} w_{i} = 1$ and 0 ≤ w_i, ϕ_i ≤ 1. The standard algorithm for the hybrid version of the method in CF rough environment is discussed as follows:

Step 1: For the construction of the decision matrix, collect the evaluation information of professional decision-makers for each alternative y_i against their attribute b_i such that

H = {[P (I_{i j}^{l})]}_{r \times n},

where $P (I_{i j}^{l})$ denotes CFRV of alternative y_i against attribute b_i of expert β_k.

Step 2: Use the proposed approach for the aggregated decision matrix, which consists of the collective information of the decision matrix under the weight vector such that

H = {[P (I_{i j})]}_{r \times n} .

Step 3: Normalize the aggregated matrix H = [P(I_ij )]_r×n to $H^{n} = {[P (I_{i j}^{n})]}_{r \times n}$ , i.e.,

H^{n} = P (I_{i j}) = ((\underline{μ_{i j}} e^{i π (γ)}), (\bar{μ_{i j}} e^{i π (γ)})) .

Step 4: Calculate the average value by using the proposed approach for all alternatives under each attribute.

\begin{array}{l} A v g = {[A v g]}_{1 \times n} = {[\frac{1}{r} \sum_{i = 1}^{r} P (I_{i j}^{n})]}_{1 \times n} \\ = {[(1 - \prod_{i = 1}^{r} {(1 - \underline{μ_{i j}^{n}})}^{\frac{1}{r}} e^{i π (1 - \prod_{i = 1}^{r} {(1 - γ_{i j}^{n})}^{\frac{1}{r}})}), (1 - \prod_{i = 1}^{r} {(1 - \bar{μ_{i j}^{n}})}^{\frac{1}{r}} e^{i π (1 - \prod_{i = 1}^{r} {(1 - γ_{i j}^{n})}^{\frac{1}{r}})})]}_{1 \times n} . \end{array}

Step 5: Based on the above , calculate the and through the following formulas:

\begin{array}{l} P D A S_{i j} \\ = {[P D A S_{i j}]}_{(r \times n)} \\ = {\frac{m a x (0, [μ_{i j}^{n}) - A v g_{j}])}{A v g_{j}}} e^{j π {\frac{m a x (0, [γ_{i j}^{n}) - A v g_{j}])}{A v g_{j}}}}, \end{array}

and

\begin{array}{l} N D A S_{i j} \\ = {[N D A S_{i j}]}_{(r \times n)} \\ = {\frac{m a x (0, [A v g_{j} - μ_{i j}^{n})])}{A v g_{j}}} e^{j π {\frac{m a x (0, [A v g_{j} - γ_{i j}^{n})])}{A v g_{j}}}} . \end{array}

Step 6: Calculate the positive weight distance (S⁺) and negative weight distance (S⁻) by the following formulas:

S^{+} = \sum_{j = 1}^{n} w_{i} P D A S_{i j}, S^{-} = \sum_{j = 1}^{n} w_{i} P D A S_{i j} .

Step 7: Normalize the positive and negative weight distances by the formula

N S^{+} y_{i} = \frac{S^{+} y_{i}}{m a x_{i} (S^{+} y_{i})}, N S^{-} y_{i} = 1 - \frac{S^{-} y_{i}}{m a x_{i} (S^{-} y_{i})} .

Step 8: Calculate the appraisal score ( ) based on NS⁺y_i and NS⁻y_i using the following formula:

A S_{i} = \frac{1}{2} (N S^{+} y_{i} + N S^{-} y_{i}) .

Step 9: Rank all values in order depending on the value of . A larger value of is considered superior.

7. Illustrative Example

• Water quality plays a pivotal role in site selection for a fish farm, as it directly affects the health and well-being of the fish.
• Temperature is another critical factor to consider when selecting a site for a fish farm. Different species of fish thrive in specific temperature ranges, and it is vital to choose a site where the temperature remains within the preferred range of the fish species being cultivated.
• Hydrological characteristics, including factors such as water source, flow rate, and water depth, must be carefully evaluated during site selection.
• Soil characteristics also play a significant role in site selection. A suitable site should have well-drained soil with good water retention capacity.
• The proximity of a fish farm to potential markets holds great importance for efficient product distribution and reduced transportation costs.

b₁ = Water quality
b₂ = Temperature
b₃ = Hydrological characteristics
b₄ = Soil characteristics
b₅ = Proximity to markets

Step 1: For the construction of the decision matrix, collect the evaluation information of professional decision-makers for each alternative y_i against their attribute b_i such that

H = {[P (I_{i j}^{l})]}_{r \times n},

where $P (I_{i j}^{l})$ denotes the CFRV of alternative y_i against attribute b_i by the expert β_k, as given in Tables 2–4.

Step 2: Use the proposed approach to aggregate the decision matrix consisting of the collective information of the decision matrix against the weight vector such that

H = {[P (I_{i j})]}_{r \times n}

The results are given in Table 5.

Step 3: The criteria are of the benefit type so must be normalized.

Step 4: Calculate the average value for each alternative under each criterion, as given in Table 6.

Step 5: Use in Table 6 to calculate the score values given in Tables 7 and 8, and use these score values to calculate the and , as given in Tables 9 and 10.

Step 6: Calculate S⁺y_i, S⁻y_i using the weight vector v = {0.13, 0.17, 0.23, 0.24, 0.16}^T and calculate |S⁺y_i|, |S⁻y_i|, as given in Tables 11 and 12.

Step 7: The normalized values of |S⁺y_i|, |S⁻y_i| are given as follows:

\begin{array}{l} S^{+} y_{1} = 1, & S^{+} y_{2} = 0.0420, \\ S^{+} y_{3} = 0.7853, & S^{+} y_{4} = 0.9945, \\ S^{-} y_{1} = 1, & S^{-} y_{2} = 0.8881, \\ S^{-} y_{3} = 0.9156, & S^{-} y_{4} = 0.0000. \end{array}

Step 8: Calculate appraisal score using normalized positive and negative distance, which is given as

\begin{array}{l} A y_{1} = 0.9189, & A y_{2} = 0.4608, \\ A y_{3} = 0.5400, & A y_{4} = 0.5000. \end{array}

Step 9: Depending on these calculations, rank all the values of the developed models based on the method, as shown in Table 13.

8. Comparative Study

• The theory of rough sets [35] is a powerful tool to navigate uncertainties in decision-making, which is an important skill. Applicable not just in the mathematics realm but in the real world too, it offers a perspective on fuzziness and rough information, opening new possibilities. With rough sets, we can effectively tackle uncertain scenarios and explore problems with imprecision, where solutions often vary. However, the introduction of phase values to complex fuzzy rough sets broadens the scope, enabling us to analyze how objects fluctuate and offering greater robustness. In comparison to rough fuzzy sets and other extensions in this domain, complex fuzzy rough sets exhibit strength by capturing variations, like a steady chain. This infusion of attributes brings new insights, empowering researchers and practitioners and making their studies more stable.
• Given uncertain data, CFAA operators [23], CFWG operators [24], and related extensions are practical. Complex fuzzy rough sets, however, offer a promising different approach. They effectively address uncertainty without additional functions or adjustments. By discussing uncertainty found in given data, complex fuzzy rough sets are effective, removing the need for subjective suppositions and making the results more aligned.
• It is clear from Table 14 that existing methods, including , IF-TOPSIS, IF-VIKOR, and IFGRA, as well as some aggregation operators,cannot be used to solve the illustrative example of Section 7 by using CF rough values. Although the methods introduced in [23, 24, 32, 33, 43, 44, 61, 62] have rough information, they cannot solve the proposed approach.

9. Conclusion

Data Availability

The data used in this study were to support the findings of the proposed approach. Anyone can use it by simply citing this article.

Abstract
1. Introduction
2. Basic Concepts
3. Construction of Complex Fuzzy Rough Sets
4. Complex Fuzzy Rough Averaging Aggregation Operator
5. Complex Fuzzy Rough Geometric Aggregation Operator
6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators
7. Illustrative Example
8. Comparative Study
9. Conclusion
Data Availability

Fig 1.

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Figure 1.

Step-by-step diagram of modified EDAS method.

The International Journal o

\|S⁺y₁\| = 0.0617	\|S⁻y₁\| = 0.0000
\|S⁺y₂\| = 0.0024	\|S⁻y₂\| = 0.0217
\|S⁺y₃\| = 0.01212	\|S⁻y₃\| = 0.0164
\|S⁺y₄\| = 0.0737	\|S⁻y₄\| = 0.1945

International Journalof Fuzzy Logic andIntelligent Systems

Original Article

Complex Fuzzy Rough Aggregation Operators and their Applications in for Multi-Criteria Group Decision-Making

Abstract

1. Introduction

1.1 Motivation of the Current Research

2. Basic Concepts

Definition 1 ([3, 4])

Definition 2

Definition 3

Definition 4

Definition 5

3. Construction of Complex Fuzzy Rough Sets

Definition 6

Definition 7

Example 1

Definition 8

Definition 9

4. Complex Fuzzy Rough Averaging Aggregation Operator

4.1 Complex Fuzzy RoughWeighted Averaging Operator

4.2 Complex Fuzzy Rough Ordered Weighted Averaging Operator

4.3 Complex Fuzzy Rough Hybrid Averaging Operator

5. Complex Fuzzy Rough Geometric Aggregation Operator

5.1 Complex Fuzzy Rough Weighted Geometric Operator

5.2 Complex Fuzzy Rough OrderedWeight Geometric Operator

5.3 Complex Fuzzy Rough Hybrid Geometric Operator

6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators

7. Illustrative Example

8. Comparative Study

9. Conclusion

Data Availability

Conflict of Interest

Figures

Tables

References

Biographies

Article

Original Article

Complex Fuzzy Rough Aggregation Operators and their Applications in for Multi-Criteria Group Decision-Making

Abstract

1. Introduction

1.1 Motivation of the Current Research

2. Basic Concepts

Definition 1 ([3, 4])

Definition 2

Definition 3

Definition 4

Definition 5

3. Construction of Complex Fuzzy Rough Sets

Definition 6

Definition 7

Example 1

Definition 8

Definition 9

4. Complex Fuzzy Rough Averaging Aggregation Operator

4.1 Complex Fuzzy RoughWeighted Averaging Operator

4.2 Complex Fuzzy Rough Ordered Weighted Averaging Operator

4.3 Complex Fuzzy Rough Hybrid Averaging Operator

5. Complex Fuzzy Rough Geometric Aggregation Operator

5.1 Complex Fuzzy Rough Weighted Geometric Operator

5.2 Complex Fuzzy Rough OrderedWeight Geometric Operator

5.3 Complex Fuzzy Rough Hybrid Geometric Operator

6. Use of CF Information in Method for MCGDM Based on Complex Rough Aggregation Operators

7. Illustrative Example

8. Comparative Study

9. Conclusion

Data Availability

Fig 1.

International Journal
of Fuzzy Logic and
Intelligent Systems