# Intuitionistic Fuzzy Dombi Hybrid Decision-Making Method and Their Applications to Enterprise Financial Performance Evaluation

Authors: Chiranjibe Jana; G. Muhiuddin; Madhumangal Pal; D. Al-Kadi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In the era of the knowledge-based economy, the active branch of information technology plays a crucial role. The enterprise administration covers efficient changes, and it has been entered in the age of reasonable management argument. The standard enterprise financial review evaluation centers on the importance of bondholders. The investor takes operational data as an issue and pays surveillance to the study of material attraction and the result. Otherwise, it is not intelligent to adjust in a modern marketplace period. Therefore, enterprise financial directing the interests of shareholders and business policies that are taking into account stakeholders’ needs is continually investigated in the future lively competition. In that view, accumulating data is an essential research tool to draw the researchers’ recent attention during the knowledge investigation. In this research, multiple attribute decision-making (MADM) approach has been proposed for the enterprise financial performance evaluation. To this view, financial performance evaluation has been done with intuitionistic fuzzy arguments. We apply new Dombi hybrid operators such as the intuitionistic fuzzy Dombi hybrid average (IFDHWA) operator and intuitionistic fuzzy Dombi hybrid geometric (IFDHWG) operator. These operators have a good advantage of adaptability in the working parameter. Finally, a realistic instance for enterprise financial performance is reported following comment on the benefit or utility of the recommended output.


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## Body

## 1. Introduction

Financial control is an integral part of administration examination which shows how companies exploit proper approaches to generate and keep up reasonable dominance. Nowadays, the study on contest has enlarged exponentially. Mintzberg et al. [1] condemned excessively explanatory orientation, upper administration slant, and an absence of thoughtfulness regarding taking movement and learning and disregarding the components that prompt those formations of methodologies. Shrivastava [2] focused on his examination of hierarchical learning forms; this can offer bits of knowledge into these apparent disadvantages. The hierarchical information is justified as the groundwork for gaining defended aggressive capabilities and a resolution alternative of the exhibition of firm performance [3]. Moreover, a few studies set up evidence of the beneficial connection between policy-making learning and obstinate execution. For example, the learning intention has an immediate impact on firm production [4]. Ussahawanitchakit [5] executed advanced model research on the economy and obtained related issues.The enterprise financial performance appraisal is not only a growth of the market economy at a particular time. But, there also a precise approach to supervising the enterprises for a nation with the contemporary market economy. It is the direction to manage successfully with modern enterprise management. The enterprise management team must learn about the foreign market economy, which can supervise its performance. The methods of contemporary enterprise management systems change rapidly and face the challenges of a rapidly developing economy. Also, a policy of offering judgment in which our business community plays an especial operation improving the management level, sharpening the business competitiveness, and further increasing the industrial growth quality is shown. Numerous MCDM or MADM approaches (review from business management, financial, and strategic management) have been considered using aggregation operators (AOs) in probabilistic notion [6–8]. The ordered weighted averaging induces geometric operators and then utilizes these to construct strategic decision theory [9]. The management study [10] based on game theory has been developed in the IFS environment. The survey of aggregation under the IFS environment is a vital research machine in decision theory. In the next part, we shortly overview some related decision-making-based problems. It is hard to take absolute values of attributes because of complexity level to be higher in the field of decision science. Zadeh [11] started the hypothesis of fuzzy sets (FSs) new mathematical knowledge of handling MADM issues efficiently and multiattribute group decision-making (MAGDM) problems.However, FS cannot handle complex fuzzy information because it expresses only membership degree. In that situation, Atanassov [12] defined IFSs, which easily handle complex FS information. IFS addressed an object characterized by membership degree (MD) and nonmembership degree (NMD). In the preceding several decades, researchers have more thought about IFSs and interval-valued intuitionistic fuzzy sets (IVIFSs) because these thoughts have been connected to many practical results, such as medical science, decision-making, pattern recognition, and clustering. Recently, many tremendous works have been developed in the IFS environment, such as De et al. [13] defined operation on IFSs, Szmidt and Kacprzyk [14] measured the similarity between IFSs, and Guo and Song [15] proposed entropy between IFSs. It is generally seen that the theory of IFSs is used to manage the MAGDM issues and clustering algorithms for economic risk evaluation utilizing MCDM procedures [16], Li et al. [17] introduced an MADM method using Hassdrof’s distance measure generalized fuzzy numbers, Garg [18] proposed a generalized improved score function of IVIFSs and applied it in expert systems, and Chen and Chiou [19] solved MADM problems based on IVIFSs using PSO procedures and evidential logic methodology. Kumar and Garg [20] utilized the technique for order of preference by similarity to ideal solution (TOPSIS) process based on set pair evaluation relationship under the IVIFSs environment. Li and Pen [21] pointed out the MADM method using the amount and reliability of IFS information. Lourenzutti and Krohling [22] studied TODIM (an acronym in Portuguese for interactive multicriteria decision making) problems based on IFS random methodology.Nowadays, information AOs are a major research topic in the multiattribute group decision-making (MAGDM) environment and become a concentration of the researchers to developed critical works [23–30]. Some traditional results [31–38] have been developed based on aggregation operators. At present, some research studies have been made using extended AOs; for instance, Zhang et al. [39] delivered power AOs on IFSs and Liu and Yu [40] used density operators for IFSs or IVIFSs, respectively. Wu and Su [41] introduced prioritized AOs for IFSs that bear in mind priorities of characteristics using precedence weights. In [42, 43], a few AOs based on algebraic operational laws for IFSs are suggested, which is an exceptional case of triangular norms. The Archimedean triangular norms are the Hamacher norms, Algebraic norms, Einstein norms, Frank norms, and Dombi norms. Liu [44] utilized Hamacher AOs on IVIFSs and developed a MAGDM model. Zhang [45] proposed Frank AOs for IVIFSs and their applications to MAGDM procedures. Einstein hybrid AOs [46] for IFSs are applied to the MADM approach. Yu [47] introduced Choquet AOs based on Einstein norms for IFS. Dombi triangular norms are general triangular norms, which can deal with the data collection process more adaptable by a parameter. Dombi [48] provided Dombi norms which have good operational flexibility to find the results. For this advantage, Bonferroni operations to IFSs are applied it to develop a MAGDM [49] problem. Chen and Ye [50] referred MADM problem using Dombi AOs in the SVN environment. Some research studies [51, 52] are developed based on Dombi norms in the different uncertain fuzzy environment, and hybrid aggregation operators [53, 54] in intuitionistic ambiguous environment have been motivated to study the proposed work. They are gripping in mind that the IFS has a powerful technique to model the vague and imprecise knowledge that appears in real-world problems. The decision-making problems in a complex fuzzy environment under Dombi operations present sufficient motivation to improve our present paper. The central object of this paper is to develop an MADM approach based on intuitionistic Dombi hybrid operators. The objectives of this paper are as follows:A new approach is developed in connecting with intuitionistic fuzzy Dombi hybrid operatorsThe proposed operators are utilized for IFMADM approachAn illustrative example is demonstrated by a numerical exampleSuperiority of the proposed method is verified numerically.The organization of the paper is sorted as follows: in the next section, some basic ideas of IFNs are briefly reviewed, and their operational laws and some operators are presented. Intuitionistic fuzzy Dombi weighted averaging, order weighted averaging, and hybrid weighted averaging operator are defined in Section3. In Section 4, intuitionistic fuzzy Dombi hybrid weighted geometric operator, order weighted geometric operator, and hybrid weighted geometric operators are mentioned. In Section 5, we have applied IFDHWA and IFDHWG operators to construct MADM problems. An illustrative example is given for the preference of the best choice of the alternatives in Section 6. Section 7 incorporates assured comparative discussions with different broadly used MADM procedures. In Section 8, some comments are given to the paper.


## 2. Preliminaries

In this section, we recall a few fundamental ideas associated with IFSs over the universe of discourseX.Definition 1 (see [55]).
An IFS over the fixed setX is interpreted as(1)I˜=x,μx,νx|x∈X,where μx:X⟶0,1 follows MD and νx:X⟶0,1 indicates NMD for an element x∈X to an IFS. πx=1−μx−νx is the depicted degree of indeterminacy for an element x to the set I˜. μ,ν is denoted as intuitionistic fuzzy elements (IFEs) or intuitionistic fuzzy values (IFVs).Reference [32] introduced the IFWA operator, IFOWA operator, and IFHWA operator as follows.for which σ1,σ2,…,ση is a permutation of 1,2,…,η, where I˜σϕ−1≥I˜σϕ for every ϕ=1,2,…,η.where Ω˜σϕ is the ϕ-th largest weighted IFV Ω˜ϕ=ηψϕΩ˜ϕ,ϕ=1,2,…,η and ψ=ψ1,ψ2,…,ψηT be the weight vector of Ω˜ϕ with ψϕ>0 and ∑ϕ=1ηψϕ=1.Definition 2 (see [32]).
LetΩ˜ϕ=μϕ,νϕϕ=1,2,…,η be a group of IFEs. The IFWA operator of dimension η is a function I˜FEη⟶I˜FE with weighting vector ψ=ψ1,ψ2,…,ψηT and react as ψ>0 and ∑t=1ηψϕ=1; then,(2)IFWAψΩ˜1,Ω˜2,…,Ω˜η=⊕ϕ=1ηψϕΩ˜ϕ=1−∏ϕ=1η1−μϕψϕ,∏ϕ=1ηνϕψϕ.Definition 3 (see [32]).
LetΩ˜ϕ=μϕ,νϕϕ=1,2,…,η be a group of IFEs. An IFOWA operator is a function with dimension η defined as IFEη⟶IFE with acting weight vector ψ=ψ1,ψ2,…,ψηT such that ψ>0 and ∑ϕ=1ηψϕ=1. Furthermore,(3)IFOWAψΩ˜1,Ω˜2,…,Ω˜η⊕ϕ=1ηψϕΩ˜σϕ=1−∏ϕ=1η1−μσϕψϕ,∏ϕ=1ηνσϕψϕ,Definition 4 (see [32]).
LetΩ˜ϕ=μϕ,νϕϕ=1,2,…,η be a group of IFEs. A function IFHWA of dimension η defined as I˜FEη⟶I˜FE is weight vector acting ψ=ψ1,ψ2,…,ψηT with ψ>0 and ∑ϕ=1ηψϕ=1. Furthermore,(4)IFHWAψΩ˜1,Ω˜2,…,Ω˜η=⊕ϕ=1ηψϕΩ˜σϕ=1−∏ϕ=1η1−μσϕψϕ,∏ϕ=1ηνσϕψϕ,Xu [33] also used weighted geometric operators such as IFWG, IFOWG, and IFHWG operators.We defined score and accuracy functions [54] as follows.Definition 5 (see [36]).
LetU=μU,νU be an IFEs. Then, score U˜ for IFEs is computed as follows:(5)ΛU=1+μU−νU2,ΛU∈0,1,and accuracy function Φ for IFEs is evaluated as follows:(6)ΦU=μU+νU2,ΦU∈0,1.On the basis ofΛU and ΦU, we used order relation between two IFEs U=〈μUx,νUx〉 and V=〈μVx,νVx〉 as follows.Definition 6 (see [36]).
LetU and V be any two IFEs. Then,(i)
IfΛU<ΛV, then U≺V(ii)
IfΛU>ΛV, then U≻V(iii)
IfΛU=ΛV, then(1)
IfΦU<ΦV, then U≺V(2)
IfΦU>ΦV, then U≻V(3)
IfΦU=ΦV, then U∼VIn the following section, we defined Dombi operator [48].


### 2.1. Dombi Operations on IFEs

Dombi triangular norms and conorms are defined as follows.Definition 7 (see [48]).
Let us takep and q as any two real numbers. Then,(7)Domp,q=11+1−p/pϱ+1−q/qϱ1/ϱ,Dom∗p,q=1−11+p/1−pϱ+q/1−qϱ1/ϱ,where ϱ≥1 and p,q∈0,1×0,1.In view of Dom-norms and Dom-conorms, we explained Dombi operations with respect to IFEs.LetΩ˜1 and Ω˜2 be two IFEs and λ>0, then Dom-product and Dom-sum of Ω˜1 and Ω˜2 are, respectively, denoted as Ω˜1⊗Ω˜2 and Ω˜1⊕Ω˜2 given as follows:(8)iΩ˜1⊕Ω˜2=1−11+μ1/1−μ1ϱ+μ2/1−μ2ϱ1/ϱ,11+1−ν1/ν1ϱ+1−ν2/ν2ϱ1/ϱ,(9)iiΩ˜˜1⊗Ω˜2=11+1−μ1/μ1ϱ+1−μ2/μ2ϱ1/ϱ,1−11+ν1/1−ν1ϱ+ν2/1−ν2ϱ1/ϱ,(10)iiiλ⋅Ω˜1=1−11+λμ1/1−μ1ϱ1/ϱ,11+λ1−ν1/ν1ϱ1/ϱ,(11)ivΩ˜11λ=11+λ1−μ1/μ1ϱ1/ϱ,1−11+λν1/1−ν1ϱ1/ϱ.


## 2.1. Dombi Operations on IFEs

Dombi triangular norms and conorms are defined as follows.Definition 7 (see [48]).
Let us takep and q as any two real numbers. Then,(7)Domp,q=11+1−p/pϱ+1−q/qϱ1/ϱ,Dom∗p,q=1−11+p/1−pϱ+q/1−qϱ1/ϱ,where ϱ≥1 and p,q∈0,1×0,1.In view of Dom-norms and Dom-conorms, we explained Dombi operations with respect to IFEs.LetΩ˜1 and Ω˜2 be two IFEs and λ>0, then Dom-product and Dom-sum of Ω˜1 and Ω˜2 are, respectively, denoted as Ω˜1⊗Ω˜2 and Ω˜1⊕Ω˜2 given as follows:(8)iΩ˜1⊕Ω˜2=1−11+μ1/1−μ1ϱ+μ2/1−μ2ϱ1/ϱ,11+1−ν1/ν1ϱ+1−ν2/ν2ϱ1/ϱ,(9)iiΩ˜˜1⊗Ω˜2=11+1−μ1/μ1ϱ+1−μ2/μ2ϱ1/ϱ,1−11+ν1/1−ν1ϱ+ν2/1−ν2ϱ1/ϱ,(10)iiiλ⋅Ω˜1=1−11+λμ1/1−μ1ϱ1/ϱ,11+λ1−ν1/ν1ϱ1/ϱ,(11)ivΩ˜11λ=11+λ1−μ1/μ1ϱ1/ϱ,1−11+λν1/1−ν1ϱ1/ϱ.


## 3. Intuitionistic Fuzzy Dombi Aggregation Operator

To this section, we propose Dombi arithmetic AOs with IFEs such as intuitionistic fuzzy Dombi weighted averaging (IFDWA) operator, intuitionistic fuzzy Dombi ordered weighted averaging (IFDOWA) operator, and intuitionistic fuzzy Dombi hybrid weighted averaging (IFDHWA) operator.


### 3.1. Intuitionistic Fuzzy Dombi Hybrid Averaging Operator

In this section, we introduce IFDHWA operator.Definition 8 (see [55]).
LetΩ˜ϕ=μϕ,νϕϕ=1,2,…,η be a group of IFEs. Then, IFDWA operator is a function IFEη⟶IFE such that(12)IFDWAψΩ˜1,Ω˜2,…,Ω˜η=⊕ϕ=1ηψϕΩ˜ϕ=1−11+∑ϕ=1ηψϕμϕ/1−μϕϱ1/ϱ,11+∑ϕ=1ηψϕ1−νϕ/νϕϱ1/ϱ,where ψ=ψ1,ψ2,…,ψηT be the weight vector of Ω˜ϕϕ=1,2,…,η with ψϕ>0 and ∑ϕ=1ηψϕ=1.Now, we introduce IFDOWA operator.Definition 9 (see [36]).
LetΩ˜ϕ=μϕ,νϕt=1,2,…,n be a number of IFEs. The IFDOWA operator of dimension η is a function IFDOWA:IFEη⟶IFE with related vector ψ=ψ1,ψ2,…,ψηT such that ψt>0 and ∑ϕ=1ηψϕ=1. Therefore,(13)IFDOWAψΩ˜1,Ω˜2,…,Ω˜η=⊕ϕ=1ηψϕΩ˜ϕσ=1−11+∑ϕ=1ηψϕμσϕ/1−μσϕϱ1/ϱ,11+∑ϕ=1ηψϕ1−νσϕ/νσϕϱ1/ϱ,where σ1,σ2,…,ση are the permutation of ϕ=1,2,…,η, for which Ω˜σϕ−1≥Ω˜σϕ for all ϕ=1,2,…,η.In Definitions8 and 9, the IFDWA considered the weights of the IFVs whereas the IFDOWA operator used the weights in their ordered locations of IFVs instead of IFVs weighting themselves. Thus, weighting for IFDWA and IFDOWA is used in different aspects. However, they are produced one time only. To overcome this happening, we focus on IFDHA operator.Definition 10.
The IFDHWA operator of dimensionη is a function IFDHWA:IFEη⟶IFE, with related weight vector W=w1,w2,…,wη such that wϕ>0 and ∑ϕ=1ηwϕ=1. Therefore, IFDHWA operator can be evaluated as(14)IFDHWAψΩ˜1,Ω˜2,…,Ω˜η=⊕t=1ηψtΩ˜⋅σϕ=1−11+∑ϕ=1ηwϕμ˙σϕ/1−μ˙σϕϱ1/ϱ,11+∑ϕ=1ηwϕ1−ν˙σϕ/ν˙σϕϱ1/ϱ,where Ω˜⋅σϕ is the ϕth biggest weighted IFVs Ω˜⋅ϕΩ˜⋅ϕ=ηψϕΩϕ,ϕ=1,2,…,η, and weighting vector be ψ=ψ1,ψ2,…,ψηT of Ω˜⋅ϕ with ψϕ>0 and ∑ϕ=1ηψϕ=1, where η is the equipoise coefficient.Example 1.
There are four IFEsΩ˜1=0.5,0.3, Ω˜2=0.6,0.3, Ω˜3=0.7,0.3, and Ω˜4=0.2,0.4 and ψ=0.20,0.30,30,0.20T is the weight vector of these four IFEs and W=0.2,0.1,0.3,0.4T is the associated weight vector. Then, by Definition 10, for aggregated of IFEs for κ=3, by the way,(15)Ω˜⋅1=1−11+4×0.20×0.5/1−0.531/3,11+4×0.20×1−0.3/0.331/3=0.4814,0.3158,Ω˜⋅2=1−11+4×0.30×0.6/1−0.631/3,11+4×0.30×1−0.3/0.331/3=0.6145,0.2874,Ω˜⋅3=1−11+4×0.30×0.7/1−0.731/3,11+4×0.30×1−0.3/0.331/3=0.7126,0.2874,Ω˜⋅4=1−11+4×0.20×0.2/1−0.231/3,11+4×0.20×1−0.4/0.431/3=0.1884,0.4180.
Scores ofΩ˜tt=1,2,3,4 are calculated as follows:(16)ΛΩ˜⋅1=1+0.4814−0.31582=0.5828,ΛΩ˜⋅2=1+0.6145−0.28742=0.6636,ΛΩ˜⋅3=1+0.7126−0.28742=0.7126,ΛΩ˜⋅4=1+0.1884−0.41802=0.3852.
Since(17)ΛΩ˜⋅3>ΛΩ˜⋅2>ΛΩ˜⋅1>ΛΩ˜⋅4,then(18)Ω˜⋅σ1=Ω˜⋅3=0.7126,0.2874,Ω˜⋅σ2=Ω˜⋅2=0.6145,0.2874,Ω˜⋅σ3=Ω˜⋅1=0.4814,0.3158,Ω˜⋅σ4=Ω˜⋅4=0.1884,0.4180.
Therefore, aggregated values of IFEs forϱ=3 by IFDHWA operator are as follows:(19)IFDHWAωΩ˜1,Ω˜2,…,Ω˜4=⊕q=14ωqΩ˜⋅σq=1−11+∑ϕ=14wϕμ˙σϕ/1−μ˙σϕ31/3,11+∑ϕ=14wϕ1−ν˙σϕ/ν˙σϕ31/3=1−11+0.2×0.7126/1−0.71263+0.1×0.6145/1−0.61453+0.3×0.4814/1−0.48143+0.4×0.1884/1−0.188431/311+0.2×1−0.2874/0.28743+0.1×1−0.2874/0.28743+0.3×1−0.3158/0.31583+0.4×1−0.4180/0.418031/3=0.6073,0.3271.


## 3.1. Intuitionistic Fuzzy Dombi Hybrid Averaging Operator

In this section, we introduce IFDHWA operator.Definition 8 (see [55]).
LetΩ˜ϕ=μϕ,νϕϕ=1,2,…,η be a group of IFEs. Then, IFDWA operator is a function IFEη⟶IFE such that(12)IFDWAψΩ˜1,Ω˜2,…,Ω˜η=⊕ϕ=1ηψϕΩ˜ϕ=1−11+∑ϕ=1ηψϕμϕ/1−μϕϱ1/ϱ,11+∑ϕ=1ηψϕ1−νϕ/νϕϱ1/ϱ,where ψ=ψ1,ψ2,…,ψηT be the weight vector of Ω˜ϕϕ=1,2,…,η with ψϕ>0 and ∑ϕ=1ηψϕ=1.Now, we introduce IFDOWA operator.Definition 9 (see [36]).
LetΩ˜ϕ=μϕ,νϕt=1,2,…,n be a number of IFEs. The IFDOWA operator of dimension η is a function IFDOWA:IFEη⟶IFE with related vector ψ=ψ1,ψ2,…,ψηT such that ψt>0 and ∑ϕ=1ηψϕ=1. Therefore,(13)IFDOWAψΩ˜1,Ω˜2,…,Ω˜η=⊕ϕ=1ηψϕΩ˜ϕσ=1−11+∑ϕ=1ηψϕμσϕ/1−μσϕϱ1/ϱ,11+∑ϕ=1ηψϕ1−νσϕ/νσϕϱ1/ϱ,where σ1,σ2,…,ση are the permutation of ϕ=1,2,…,η, for which Ω˜σϕ−1≥Ω˜σϕ for all ϕ=1,2,…,η.In Definitions8 and 9, the IFDWA considered the weights of the IFVs whereas the IFDOWA operator used the weights in their ordered locations of IFVs instead of IFVs weighting themselves. Thus, weighting for IFDWA and IFDOWA is used in different aspects. However, they are produced one time only. To overcome this happening, we focus on IFDHA operator.Definition 10.
The IFDHWA operator of dimensionη is a function IFDHWA:IFEη⟶IFE, with related weight vector W=w1,w2,…,wη such that wϕ>0 and ∑ϕ=1ηwϕ=1. Therefore, IFDHWA operator can be evaluated as(14)IFDHWAψΩ˜1,Ω˜2,…,Ω˜η=⊕t=1ηψtΩ˜⋅σϕ=1−11+∑ϕ=1ηwϕμ˙σϕ/1−μ˙σϕϱ1/ϱ,11+∑ϕ=1ηwϕ1−ν˙σϕ/ν˙σϕϱ1/ϱ,where Ω˜⋅σϕ is the ϕth biggest weighted IFVs Ω˜⋅ϕΩ˜⋅ϕ=ηψϕΩϕ,ϕ=1,2,…,η, and weighting vector be ψ=ψ1,ψ2,…,ψηT of Ω˜⋅ϕ with ψϕ>0 and ∑ϕ=1ηψϕ=1, where η is the equipoise coefficient.Example 1.
There are four IFEsΩ˜1=0.5,0.3, Ω˜2=0.6,0.3, Ω˜3=0.7,0.3, and Ω˜4=0.2,0.4 and ψ=0.20,0.30,30,0.20T is the weight vector of these four IFEs and W=0.2,0.1,0.3,0.4T is the associated weight vector. Then, by Definition 10, for aggregated of IFEs for κ=3, by the way,(15)Ω˜⋅1=1−11+4×0.20×0.5/1−0.531/3,11+4×0.20×1−0.3/0.331/3=0.4814,0.3158,Ω˜⋅2=1−11+4×0.30×0.6/1−0.631/3,11+4×0.30×1−0.3/0.331/3=0.6145,0.2874,Ω˜⋅3=1−11+4×0.30×0.7/1−0.731/3,11+4×0.30×1−0.3/0.331/3=0.7126,0.2874,Ω˜⋅4=1−11+4×0.20×0.2/1−0.231/3,11+4×0.20×1−0.4/0.431/3=0.1884,0.4180.
Scores ofΩ˜tt=1,2,3,4 are calculated as follows:(16)ΛΩ˜⋅1=1+0.4814−0.31582=0.5828,ΛΩ˜⋅2=1+0.6145−0.28742=0.6636,ΛΩ˜⋅3=1+0.7126−0.28742=0.7126,ΛΩ˜⋅4=1+0.1884−0.41802=0.3852.
Since(17)ΛΩ˜⋅3>ΛΩ˜⋅2>ΛΩ˜⋅1>ΛΩ˜⋅4,then(18)Ω˜⋅σ1=Ω˜⋅3=0.7126,0.2874,Ω˜⋅σ2=Ω˜⋅2=0.6145,0.2874,Ω˜⋅σ3=Ω˜⋅1=0.4814,0.3158,Ω˜⋅σ4=Ω˜⋅4=0.1884,0.4180.
Therefore, aggregated values of IFEs forϱ=3 by IFDHWA operator are as follows:(19)IFDHWAωΩ˜1,Ω˜2,…,Ω˜4=⊕q=14ωqΩ˜⋅σq=1−11+∑ϕ=14wϕμ˙σϕ/1−μ˙σϕ31/3,11+∑ϕ=14wϕ1−ν˙σϕ/ν˙σϕ31/3=1−11+0.2×0.7126/1−0.71263+0.1×0.6145/1−0.61453+0.3×0.4814/1−0.48143+0.4×0.1884/1−0.188431/311+0.2×1−0.2874/0.28743+0.1×1−0.2874/0.28743+0.3×1−0.3158/0.31583+0.4×1−0.4180/0.418031/3=0.6073,0.3271.


## 4. Intuitionistic Fuzzy Dombi Geometric Aggregation Operators

To this section, we propose Dombi geometric AOs with IFEs such as intuitionistic fuzzy Dombi weighted geometric (IFDWG) operator, intuitionistic fuzzy Dombi ordered weighted geometric (IFDOWG) operator, and intuitionistic fuzzy Dombi hybrid weighted geometric (IFDHWG) operator.


### 4.1. Intuitionistic Fuzzy Dombi Hybrid Geometric Operator

In this section, we introduce IFDHWG operator.Definition 11 (see [36]).
Assume thatΩ˜ϕ=μϕ,νϕϕ=1,2,…,η is a group of IFEs. Then, IFDWG operator is a function IFEη⟶IFE such that(20)IFDWGψΩ˜1,Ω˜2,…,Ω˜η=⊗ϕ=1ηΩ˜ϕψϕ=11+∑ϕ=1ηψϕ1−μϕ/μϕϱ1/ϱ,1−11+∑ϕ=1ηψϕνϕ/1−νϕϱ1/ϱ,where ψ=ψ1,ψ2,…,ψηT be the weight vector of Ω˜ϕϕ=1,2,…,η such that ψϕ>0 and ∑ϕ=1ηψϕ=1.Now, we introduce IFDOWG operator.Definition 12 (see [36]).
LetΩ˜ϕ=μϕ,νϕϕ=1,2,…,η be a group of IFEs. An IFDOWG operator of dimension η is a function IFDOWG:IFEη⟶IFE with related weight vector w=w1,w2,…,wηT such that wϕ>0 and ∑ϕ=1ηwϕ=1. Therefore,(21)IFDOWGwΩ˜1,Ω˜2,…,Ω˜η=⊗ϕ=1ηΩ˜ϕσwϕ=11+∑ϕ=1ηwϕ1−μσϕ/μσϕϱ1/ϱ,1−11+∑ϕ=1ηwϕνσϕ/1−νσϕϱ1/ϱ,where σ1,σ2,…,ση are the permutation of ϕ=1,2,…,η, for which Ω˜σϕ−1≥Ω˜σϕ for all ϕ=1,2,…,η.In Definitions11 and 12, the IFDWG considered the weights of the IFVs, however, in IFDOWG weights considered the ordered ground of the IFVs in view of weights of the IFVs among. Thus, the weighting for IFDWG and IFDOWG is utilized in various ways. However, they are produced only one time. To overcome this case, we introduce IFDHWG operator.Definition 13.
An IFDHWG operator of dimensionη is a function IFDHWG:IFEη⟶IFE, with related weight vector w=w1,w2,…,wη such that wϕ>0 and ∑ϕ=1ηwϕ=1. Therefore, IFDHWG operator can be evaluated as(22)IFDHWGψ,wΩ˜1,Ω˜2,…,Ω˜η=⊗ϕ=1ηΩ˜⋅σϕwϕ=11+∑ϕ=1ηwϕ1−μ˙σϕ/μ˙σϕϱ1/ϱ,1−11+∑ϕ=1ηwϕν˙σϕ/1−ν˙σϕϱ1/ϱ,where Ω˜⋅σϱ is the ϕth largest weighted IFVs Ω˜⋅ϕ=ηψϕIϕ,ϕ=1,2,…,η and weight vector be ψ=ψ1,ψ2,…,ψηT of I˜⋅ϕ with ψϕ>0 and ∑ϕ=1ηψϕ=1, where η is the equipoise coefficient.Example 2.
There are four IFEsΩ˜1=0.5,0.3, Ω˜2=0.6,0.3, Ω˜3=0.7,0.3, and Ω˜4=0.2,0.4, and ψ=0.20,0.30,30,0.20T is the weight vector of these four IFEs and W=0.2,0.1,0.3,0.4T is the associated weight vector. Then, by Definition 13, for aggregated of IFEs for κ=3, by the way,(23)Ω˜⋅1=11+4×0.20×1−0.5/0.531/3,1−11+4×0.20×0.3/1−0.331/3=0.5186,0.2846,Ω˜⋅2=11+4×0.30×1−0.6/0.631/3,1−11+4×0.30×0.3/1−0.331/3=0.5853,0.3129,Ω˜⋅3=11+4×0.30×1−0.7/0.731/3,1−11+4×0.30×0.3/1−0.331/3=0.6871,0.3129,Ω˜⋅4=11+4×0.20×1−0.2/0.231/3,1−11+4×0.20×0.4/1−0.431/3=0.2122,0.3823.
Scores ofΩ˜ϕϕ=1,2,3,4 are calculated as follows:(24)ΛΩ˜⋅1=1+0.5186−0.28462=0.6170,ΛI˜⋅2=1+0.5853−0.31292=0.6362,ΛΩ˜⋅3=1+0.6871−0.31292=0.6871,ΛI˜⋅4=1+0.2122−0.38232=0.4150.
Since(25)ΛΩ˜⋅3>EΩ˜⋅2>ΛΩ˜⋅1>ΛΩ˜⋅4,then(26)Ω˜⋅σ1=Ω˜⋅3=0.6871,0.3129,Ω˜⋅σ2=Ω˜⋅2=0.5853,0.3129,Ω˜⋅σ3=Ω˜⋅1=0.5186,0.2846,Ω˜⋅σ4=Ω˜⋅4=0.2122,0.3823.
Therefore, aggregated values of IFEsϱ=3 by IFDHWG operator are as follows: (27)IFDHWGωΩ˜1,Ω˜2,…,Ω˜4=⊕ϕ=14ωqΩ˜⋅σq=11+∑ϕ=14wϕ1−μ˙σϕ/μ˙σϕ31/3,1−11+∑ϕ=14wϕν˙σϕ/1−ν˙σϕ31/3=1−11+0.2×1−0.6871/0.68713+0.1×1−0.5853/0.58533+0.3×1−0.5186/0.51863+0.4×31/311+0.2×0.3129/1−0.31293+0.1×0.3129/1−0.31293+0.3×0.2846/1−0.28463+0.4×0.3823/1−0.382331/3=0.3582,0.3429.


## 4.1. Intuitionistic Fuzzy Dombi Hybrid Geometric Operator

In this section, we introduce IFDHWG operator.Definition 11 (see [36]).
Assume thatΩ˜ϕ=μϕ,νϕϕ=1,2,…,η is a group of IFEs. Then, IFDWG operator is a function IFEη⟶IFE such that(20)IFDWGψΩ˜1,Ω˜2,…,Ω˜η=⊗ϕ=1ηΩ˜ϕψϕ=11+∑ϕ=1ηψϕ1−μϕ/μϕϱ1/ϱ,1−11+∑ϕ=1ηψϕνϕ/1−νϕϱ1/ϱ,where ψ=ψ1,ψ2,…,ψηT be the weight vector of Ω˜ϕϕ=1,2,…,η such that ψϕ>0 and ∑ϕ=1ηψϕ=1.Now, we introduce IFDOWG operator.Definition 12 (see [36]).
LetΩ˜ϕ=μϕ,νϕϕ=1,2,…,η be a group of IFEs. An IFDOWG operator of dimension η is a function IFDOWG:IFEη⟶IFE with related weight vector w=w1,w2,…,wηT such that wϕ>0 and ∑ϕ=1ηwϕ=1. Therefore,(21)IFDOWGwΩ˜1,Ω˜2,…,Ω˜η=⊗ϕ=1ηΩ˜ϕσwϕ=11+∑ϕ=1ηwϕ1−μσϕ/μσϕϱ1/ϱ,1−11+∑ϕ=1ηwϕνσϕ/1−νσϕϱ1/ϱ,where σ1,σ2,…,ση are the permutation of ϕ=1,2,…,η, for which Ω˜σϕ−1≥Ω˜σϕ for all ϕ=1,2,…,η.In Definitions11 and 12, the IFDWG considered the weights of the IFVs, however, in IFDOWG weights considered the ordered ground of the IFVs in view of weights of the IFVs among. Thus, the weighting for IFDWG and IFDOWG is utilized in various ways. However, they are produced only one time. To overcome this case, we introduce IFDHWG operator.Definition 13.
An IFDHWG operator of dimensionη is a function IFDHWG:IFEη⟶IFE, with related weight vector w=w1,w2,…,wη such that wϕ>0 and ∑ϕ=1ηwϕ=1. Therefore, IFDHWG operator can be evaluated as(22)IFDHWGψ,wΩ˜1,Ω˜2,…,Ω˜η=⊗ϕ=1ηΩ˜⋅σϕwϕ=11+∑ϕ=1ηwϕ1−μ˙σϕ/μ˙σϕϱ1/ϱ,1−11+∑ϕ=1ηwϕν˙σϕ/1−ν˙σϕϱ1/ϱ,where Ω˜⋅σϱ is the ϕth largest weighted IFVs Ω˜⋅ϕ=ηψϕIϕ,ϕ=1,2,…,η and weight vector be ψ=ψ1,ψ2,…,ψηT of I˜⋅ϕ with ψϕ>0 and ∑ϕ=1ηψϕ=1, where η is the equipoise coefficient.Example 2.
There are four IFEsΩ˜1=0.5,0.3, Ω˜2=0.6,0.3, Ω˜3=0.7,0.3, and Ω˜4=0.2,0.4, and ψ=0.20,0.30,30,0.20T is the weight vector of these four IFEs and W=0.2,0.1,0.3,0.4T is the associated weight vector. Then, by Definition 13, for aggregated of IFEs for κ=3, by the way,(23)Ω˜⋅1=11+4×0.20×1−0.5/0.531/3,1−11+4×0.20×0.3/1−0.331/3=0.5186,0.2846,Ω˜⋅2=11+4×0.30×1−0.6/0.631/3,1−11+4×0.30×0.3/1−0.331/3=0.5853,0.3129,Ω˜⋅3=11+4×0.30×1−0.7/0.731/3,1−11+4×0.30×0.3/1−0.331/3=0.6871,0.3129,Ω˜⋅4=11+4×0.20×1−0.2/0.231/3,1−11+4×0.20×0.4/1−0.431/3=0.2122,0.3823.
Scores ofΩ˜ϕϕ=1,2,3,4 are calculated as follows:(24)ΛΩ˜⋅1=1+0.5186−0.28462=0.6170,ΛI˜⋅2=1+0.5853−0.31292=0.6362,ΛΩ˜⋅3=1+0.6871−0.31292=0.6871,ΛI˜⋅4=1+0.2122−0.38232=0.4150.
Since(25)ΛΩ˜⋅3>EΩ˜⋅2>ΛΩ˜⋅1>ΛΩ˜⋅4,then(26)Ω˜⋅σ1=Ω˜⋅3=0.6871,0.3129,Ω˜⋅σ2=Ω˜⋅2=0.5853,0.3129,Ω˜⋅σ3=Ω˜⋅1=0.5186,0.2846,Ω˜⋅σ4=Ω˜⋅4=0.2122,0.3823.
Therefore, aggregated values of IFEsϱ=3 by IFDHWG operator are as follows: (27)IFDHWGωΩ˜1,Ω˜2,…,Ω˜4=⊕ϕ=14ωqΩ˜⋅σq=11+∑ϕ=14wϕ1−μ˙σϕ/μ˙σϕ31/3,1−11+∑ϕ=14wϕν˙σϕ/1−ν˙σϕ31/3=1−11+0.2×1−0.6871/0.68713+0.1×1−0.5853/0.58533+0.3×1−0.5186/0.51863+0.4×31/311+0.2×0.3129/1−0.31293+0.1×0.3129/1−0.31293+0.3×0.2846/1−0.28463+0.4×0.3823/1−0.382331/3=0.3582,0.3429.


## 5. MADM Model Based on Hybrid Operators

For the section, we form an MADM approach applying IFS AOs in which the weight of the attributes is in real numbers, and one can apply different existing methods for the evaluation of the attribute’s weight vector. Here, MADM problem is used to develop for enterprise financial performance evaluation under intuitionistic fuzzy information. Let a group of alternative beQ=Q1,Q2,…,Qξ and attributes be G=G1,G2,…,Gη. Let a set of weight vector be ψ=ψ1,ψ2,…,ψηGϕϕ=1,2,…,η of Qϕ which are real-values where ψϕ>0 and ∑ϕ=1ηψϕ=1. Suppose the decision matrix D˜=μξϕ,νξϕδ×η where μξϕ is the MD considered for the alternative Qϕ satified under the attribute Gϕ proposed by the DMs, and νξϕ implied NMD for the alternative Qξ does not comfort Gϕ stated by the DMs, where μξϕ⊂0,1 and νξϕ⊂0,1 such that 0≤μξϕ+νξϕ≤1, ξ=1,2,…,δ and ϕ=1,2,…,η.In the following algorithm, IFDHWA and IFDHWG operators are applied to solve an MADM problem:Step 1. We operate the decision said matrix D and apply the IFDHWA operator as follows:(28)βξ=IFDHWAβξ1,βξ2,…,βδη=⊕ϕ=1ηwϕβξϕ=1−11+∑ϕ=1ηwϕμ˙σϕ/1−μ˙σϕϱ1/ϱ,11+∑ϕ=1ηwϕ1−ν˙σϕ/ν˙σϕϱ1/ϱ,(29)orβξ=IFDHWGβξ1,βξ2,…,βδη=⊕ϕ=1ηβξϕwϕ=11+∑ϕ=1ηwϕ1−μ˙σϕ/μ˙σϕϱ1/ϱ,1−11+∑ϕ=1ηwϕν˙σϕ/1−ν˙σϕϱ1/ϱ,to achieve the overall decision valuesβξξ=1,2,…,δ of the alternative Qϕ.Step 2. Measure the score Λβξξ=1,2,…,δ approaching overall IFVs βξξ=1,2,…,δ in established to list all the alternatives Qξξ=1,2,…,δ to get favorable Qξ. If there is no divergence among score functions Λβξ and Λβϕ, then we proceed to compute accuracy Φβξ and Φβϕ based on overall IFVs of βξ and βϕ and rank the alternatives Qξ depending on accuracy degrees of Φβξ and Φβϕ.Step 3. List all Qξξ=1,2,…,δ to get the leading one(s) conforming with Λβξξ=1,2,…,δ.Step 4. Stop.


## 6. Case Study

The long-time steady increase of enterprise hindered due to specific problems: improvement of production, sounding environmental impairment, reduced class production, loss of sources, and reduced security of the benefits. As a result, its employees lose interest in spending their wealth, and they move to special-purpose funding in the organization and support the funding opportunity. Thus, the enterprise’s growth and survival rely upon its capacity to manage the relationship among different investors. The necessary management expert’s steadily understood that it is small-minded conduct for enterprises if they want to achieve the aim of stockholder value in manufacturing procedure, regardless of the importance of different stakeholder situations. Finally, they reach the strategic control purpose of an enterprise. In this article, we will present a project to select the best enterprise alternative(s). Here, we investigate the financial performance of five possible enterprise for their rankingQϕϕ=1,2,3,4,5. An organization invests its money in an enterprise with the enterprise financial shows and to maximize the demanded profit. In that direction, it is required to measure the enterprise financial performance of five possible enterprises to select attractive ones. The funding enterprise makes a decision depending on the consequent four attributes as follows: G1: dept paying ability; G2: management capability; G3: earning capacity; and G4: development and proper application. Enterprise financial performance of five possible enterprises Qϕϕ=1,2,…,5 is evaluated using IFS data by the DMs under the IFDHWA and IFDHWG operators in which triangular Dombi norms are used which have a good advantage of flexibility for the operation of working parameter ϱ based on Gϕ and their weight vector ψ=0.2,0.1,0.3,0.4T, for resulting matrix D˜=βξϕ5×4 given in Table 1, where βξϕ is IFEs.Table 1
Intuitionistic fuzzy elements.
Q1Q2Q3Q4Q5G1(0.7, 0.3)(0.6, 0.2)(0.4, 0.3)(0.6, 0.1)(0.6, 0.4)G2(0.5, 0.2)(0.7, 0.2)(0.6, 0.2)(0.5, 0.2)(0.5, 0.1)G3(0.6, 0.2)(0.5, 0.1)(0.6, 0.4)(0.8, 0.1)(0.6, 0.2)G4(0.8, 0.2)(0.4, 0.6)(0.6, 0.3)(0.5, 0.3)(0.4, 0.1)In order to find most desirable enterpriseQϕϕ=1,2,…,η, we utilize the IFDHA operator to develop an MADM theory with intuitionistic fuzzy data, which can be evaluated as follows:Step 1. Utilizing the decision data stated in matrix D˜, I˜⋅ξϕ=ηψI˜ξϕ, we get(30)Ω˜⋅11=0.6512,0.3488,Ω˜⋅12=0.2857,0.3846,Ω˜⋅13=0.6429,0.1724,Ω˜⋅14=0.8649,0.1351,Ω˜⋅21=0.5455,0.2381,Ω˜⋅22=0.4828,0.3846,Ω˜˙23=0.5455,0.0847Ω˜⋅24=0.5161,0.4839,Ω˜⋅31=0.3478,0.3488,Ω˜⋅32=0.3750,0.3846,Ω˜⋅33=0.6429,0.3571,Ω˜⋅34=0.7059,0.2113,Ω˜⋅41=0.5455,0.1220,Ω˜⋅42=0.2857,0.3846,Ω˜⋅43=0.8276,0.0847,Ω˜⋅44=0.6154,0.2113,Ω˜⋅51=0.5455,0.4545,Ω˜⋅52=0.2857,0.2174,Ω˜⋅53=0.6429,0.1724,Ω˜⋅54=0.5161,0.0649.Then, by score function, we obtained(31)Ω˜⋅σ11=0.8649,0.1351,Ω˜⋅σ12=0.6429,0.1724,Ω˜⋅σ13=0.6512,0.3488,Ω˜⋅σ14=0.2857,0.3846,Ω˜⋅σ21=0.5455,0.0847,Ω˜⋅σ22=0.5455,0.2381,Ω˜⋅σ23=0.4828,0.3846,Ω˜⋅σ24=0.5161,0.4839,Ω˜⋅σ31=0.7059,0.2113,Ω˜⋅σ32=0.6429,0.3571,Ω˜⋅σ33=0.3478,0.3488,Ω˜⋅σ34=0.3750,0.3846,Ω˜⋅σ41=0.8276,0.0847,Ω˜⋅σ42=0.5455,0.1220,Ω˜⋅σ43=0.6154,0.2113,Ω˜⋅σ44=0.2857,0.3846,Ω˜⋅σ51=0.6429,0.1724,Ω˜⋅σ52=0.5161,0.0649,Ω˜⋅σ53=0.5455,0.4545,Ω˜⋅σ54=0.2857,0.2174.Step 2. Utilizing the D˜ using the IFDHWA operator and applying weighted vector w=0.20,0.30,0.30,0.20T, then obtain overall decision assessments βξ for Qξξ=1,2,…,5 as follows: β1=0.7110,0.2174, β2=0.5223,0.2077, β3=0.5652,0.3158, β4=0.6528,0.1479, and β5=0.5283,0.1358.Step 3. Compute the score of Λβξξ=1,2,…,5 of each Qξβξξ=1,2,…,5 as follows:(32)Λβ1=0.7468,Λβ2=0.6573,Λβ3=0.6247,Λβ4=0.7525,Λβ5=0.6963.Step 4. Rank Qξξ=1,2,…,5 in accordance with the value of Λβξ as follows: Q4≻Q1≻Q5≻Q2≻Q3.Step 5. Thus, Q4 is finalized as the favorable enterprise.If IFDWG operator is applied, proceed similarly as above.Step 1. Utilizing the decision matrix D˜, I˜⋅ξϕ=ηψI˜ξϕ, we get(33)Ω˜⋅11=0.7447,0.2553,Ω˜⋅12=0.7143,0.0909,Ω˜⋅13=0.5556,0.2308,Ω˜⋅14=0.7143,0.2857,Ω˜⋅21=0.6522,0.1667,Ω˜⋅22=0.8537,0.0909,Ω˜⋅23=0.4545,0.1176,Ω˜⋅24=0.2941,0.7059,Ω˜⋅31=0.4545,0.2553,Ω˜⋅32=0.7895,0.0909,Ω˜⋅33=0.5556,0.4444,Ω˜⋅34=0.4839,0.4068,Ω˜⋅41=0.6523,0.0816,Ω˜⋅42=0.7143,0.0909,Ω˜⋅43=0.7692,0.1176,Ω˜⋅44=0.3846,0.4068,Ω˜⋅51=0.6522,0.3478,Ω˜⋅52=0.7143,0.0426,Ω˜⋅53=0.5556,0.2308,Ω˜⋅54=0.3846,0.1509.Then, by score function, we obtained(34)Ω˜⋅σ11=0.7143,0.0909,Ω˜⋅σ12=0.7447,0.2553,Ω˜⋅σ13=0.7143,0.2857,Ω˜⋅σ14=0.5556,0.2308,Ω˜⋅σ21=0.8537,0.0909,Ω˜⋅σ22=0.6522,0.1667,Ω˜⋅σ23=0.4545,0.1176,Ω˜⋅σ24=0.2941,0.7059,Ω˜⋅σ31=0.7895,0.0909,Ω˜⋅σ32=0.4545,0.2553,Ω˜⋅σ33=0.5556,0.4444,Ω˜⋅σ34=0.4839,0.4068,Ω˜⋅σ41=0.7692,0.1176,Ω˜⋅σ42=0.7143,0.0909,Ω˜⋅σ43=0.6523,0.0816,Ω˜⋅σ44=0.3846,0.4068,Ω˜⋅σ51=0.7143,0.0426,Ω˜⋅σ52=0.5556,0.2308,Ω˜⋅σ53=0.6522,0.3478,Ω˜⋅σ54=0.3846,0.1509.Step 2. Utilizing matrix D˜ using the IFDHWAG operator and applying weighted vector w=0.20,0.30,0.30,0.20T, we get overall decision costs βϕ of Qϕϕ=1,2,…,5 as follows:(35)β1=0.6836,0.2324,β2=0.4915,0.3750,β3=0.5357,0.3333,β4=0.6024,0.1807,β5=0.5556,0.2274.Step 3. Calculate the value of Λβϕϕ=1,2,…,5 of each βϕϕ=1,2,…,5 as follows:(36)Λβ1=0.7256,Λβ2=0.5583,Λβ3=0.6012,Λβ4=0.7109,Λβ5=0.6641.Step 4. Now, enterprise ranking Qϕξ=1,2,…,5 as per score values Λβϕ is Q1≻Q4≻Q5≻Q3≻Q2.Step 5. Thus, Q1 is selected as attractive enterprise.


## 7. Comparative Analysis

A comparative study was directed to approve the aftereffects of the proposed technique with those of the other approach. The relative outcomes appear in Tables2 and 3. Therefore, with regard to the same MADM problem for assessing enterprise financial performance, this paper contrasted the outcomes acquired with those of the Xu method, the Liao and Xu method, the Zhao and Wei method, and the Huang methods inside the intuitionistic fuzzy surroundings. Table 4 gives an aggregate value of overall evaluation by IFDHWA, IFDHWG, and other methods. Table 2 gives a synopsis of solution consequences for the choice issue of enterprise financial performances.Table 2
Comparison analysis with some of the existing method.
MethodsΛβ1Λβ2Λβ3Λβ4Λβ5OrderingXu [32] method by IFWA0.47040.19830.22800.48190.3135Q4≻Q1≻Q5≻Q3≻Q2Liao and Xu [53] by IFHWA0.71920.61750.61510.69370.6363Q1≻Q4≻Q5≻Q3≻Q2Zhao and Wei [46] by IFEHWA0.47180.23620.23560.46470.3518Q1≻Q4≻Q5≻Q2≻Q3Huang [54] by IFHHWA0.68190.50290.54060.62060.5331Q1≻Q4≻Q3≻Q5≻Q2Proposed method by IFDHWA0.74680.65730.62470.75250.6963Q4≻Q1≻Q5≻Q2≻Q3Proposed method by IFDHWG0.72560.55830.60120.71090.6641Q1≻Q4≻Q5≻Q3≻Q2Table 3
Characteristic comparisons of different methods.
MethodsFuzzy information easierAggregation flexibilityXu [32] method by IFWAYesNoLiao and Xu [53] by IFHWAYesNoZhao and Wei [46] by IFEHWAYesNoHuang [54] by IFHHWAYesNoProposed methodYesYesTable 4
Aggregate value of overall evaluation by IFDHWA, IFDHWG, and other methods.
Methodsβ1β2β3β4β5Reference [32] method by IFWA0.6956,0.22520.5177,0.31940.5587,0.33070.6592,0.17730.5231,0.2096Liao and Xu [53] by IFHWA0.6956,0.22040.5177,0.22150.5749,0.32080.6291,0.16000.5263,0.1713Zhao and Wei [46] by IFEHWA0.6927,0.22090.5160,0.27980.5570,0.32140.6278,0.16310.5337,0.1819Huang [54] by IFHHWA0.6915,0.00960.5163,0.01340.5559,0.01530.6276,0.00700.5409,0.0078Proposed method by IFDHWA0.7110,0.21740.5223,0.20770.5652,0.31580.6528,0.14790.5283,0.1358Proposed method by IFDHWG0.6836,0.23240.4915,0.37500.5357,0.33330.6024,0.18070.5556,0.2274As indicated in Table2, the Xu method and the proposed method by IFDHWA yielded the same complete-preference ranking result of the financial performance of five possible enterprises: Q4≻Q1≻Q5≻Q2≻Q3. The distinction exists within the preference relation between Q2 and Q3. The comparative outcomes exhibit that the recommended method by IFDHWA is powerful and legitimate for addressing the interpretative issue within the intuitionistic fuzzy surroundings. By using the Liao and Xu method, we get the complete-preference ranking result of the alternatives as Q1≻Q4≻Q5≻Q3≻Q2, which is the same as our obtained result by IFDHWG operator. Again, by applying the Zhao and Wei method and the Huang method, we get the complete-preference ranking outcome of the alternatives as Q1≻Q4≻Q5≻Q2≻Q3 and Q1≻Q4≻Q3≻Q5≻Q2, respectively. The difference exists in the preference relation between Q2, Q3, and Q5. For this proposed approach, we apply intuitionistic fuzzy Dombi hybrid averaging and geometric operators which have an advantage of flexibility for the operation of ϱ; on the other hand, the existing operators IFWA [32], IFHWA [53], IFEHWA [46], and IFHHWA [54] have no such advantages provided in Table 3. Thus, our proposed methods are more general and more flexible than some existing methods to control intuitionistic fuzzy MADM problems.


## 8. Conclusions

Enterprises are an important key factor of employees, stockholders, customer, creditor, government, and other stakeholder. In the financial performance of enterprises, double characters should be considered: economic and society, so we should consider all stakeholder’s benefit in performance of enterprise evaluating time. We set up a performance evaluating system on basis of stakeholder benefits. In this article, we have studied MADM problems for enterprise financial performance with intuitionistic fuzzy data. We employ IFDHWA and IFDHWG operators to develop MADM approach for evaluating enterprises financial performance. In these operators, we use triangular Dombi norms which have a good advantage of flexibility for the operation of working parameterϱ. For this proposed approach, we apply intuitionistic fuzzy Dombi hybrid averaging and geometric operators which have an advantage of flexibility for the operation of ϱ; on the other hand, the existing operators IFWA [32], IFHWA [53], IFEHWA [46], and IFHHWA [54] have no such advantages. In future, proposed approach can be applied in (1) green supplier selection [56], (2) economic risk evaluation [8], (3) hybrid rough approach study [57], and other domains under ambiguous environments.



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*Source: 3218133-2021-07-20.xml*