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

Jana, Chiranjibe; Muhiuddin, G.; Pal, Madhumangal; Al-Kadi, D.

doi:https://doi.org/10.1155/2021/3218133

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Volume 2021 | Article ID 3218133 | https://doi.org/10.1155/2021/3218133

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

Chiranjibe Jana,¹G. Muhiuddin,²Madhumangal Pal,¹and D. Al-Kadi³

Academic Editor: Ewa Rak

Received29 May 2021

Accepted02 Jul 2021

Published20 Jul 2021

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.

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 operators The proposed operators are utilized for IFMADM approach An illustrative example is demonstrated by a numerical example Superiority 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 Section 3. 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 discourse .

Definition 1 (see [55]). An IFS over the fixed set is interpreted aswhere follows MD and indicates NMD for an element to an IFS. is the depicted degree of indeterminacy for an element to the set . 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 is a permutation of , where for every .where is the -th largest weighted IFV and be the weight vector of with and .

Definition 2 (see [32]). Let be a group of IFEs. The IFWA operator of dimension is a function with weighting vector and react as and ; then,

Definition 3 (see [32]). Let be a group of IFEs. An IFOWA operator is a function with dimension defined as with acting weight vector such that and . Furthermore,

Definition 4 (see [32]). Let be a group of IFEs. A function IFHWA of dimension defined as is weight vector acting with and . Furthermore,

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]). Let be an IFEs. Then, score for IFEs is computed as follows:and accuracy function for IFEs is evaluated as follows:

On the basis of and , we used order relation between two IFEs and as follows.

Definition 6 (see [36]). Let and be any two IFEs. Then,(i)If , then (ii)If , then (iii)If , then(1)If , then (2)If , then (3)If , then

In 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 take and as any two real numbers. Then,where and .

In view of Dom-norms and Dom-conorms, we explained Dombi operations with respect to IFEs.

Let and be two IFEs and , then Dom-product and Dom-sum of and are, respectively, denoted as and given as follows:

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]). Let be a group of IFEs. Then, IFDWA operator is a function such thatwhere be the weight vector of with and .

Now, we introduce IFDOWA operator.

Definition 9 (see [36]). Let be a number of IFEs. The IFDOWA operator of dimension is a function with related vector such that and . Therefore,where are the permutation of , for which for all .

In Definitions 8 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 dimension is a function , with related weight vector such that and . Therefore, IFDHWA operator can be evaluated aswhere is the biggest weighted IFVs , and weighting vector be of with and , where is the equipoise coefficient.

Example 1. There are four IFEs , , , and and is the weight vector of these four IFEs and is the associated weight vector. Then, by Definition 10, for aggregated of IFEs for , by the way,Scores of are calculated as follows:SincethenTherefore, aggregated values of IFEs for by IFDHWA operator are as follows:

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 that is a group of IFEs. Then, IFDWG operator is a function such thatwhere be the weight vector of such that and .

Now, we introduce IFDOWG operator.

Definition 12 (see [36]). Let be a group of IFEs. An IFDOWG operator of dimension is a function with related weight vector such that and . Therefore,where are the permutation of , for which for all .

In Definitions 11 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 dimension is a function , with related weight vector such that and . Therefore, IFDHWG operator can be evaluated aswhere is the largest weighted IFVs and weight vector be of with and , where is the equipoise coefficient.

Example 2. There are four IFEs , , , and , and is the weight vector of these four IFEs and is the associated weight vector. Then, by Definition 13, for aggregated of IFEs for , by the way,Scores of are calculated as follows:SincethenTherefore, aggregated values of IFEs by IFDHWG operator are as follows:

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 be and attributes be . Let a set of weight vector be of which are real-values where and . Suppose the decision matrix where is the MD considered for the alternative satified under the attribute proposed by the DMs, and implied NMD for the alternative does not comfort stated by the DMs, where and such that , and .

In the following algorithm, IFDHWA and IFDHWG operators are applied to solve an MADM problem: Step 1. We operate the decision said matrix and apply the IFDHWA operator as follows: to achieve the overall decision values of the alternative . Step 2. Measure the score approaching overall IFVs in established to list all the alternatives to get favorable . If there is no divergence among score functions and , then we proceed to compute accuracy and based on overall IFVs of and and rank the alternatives depending on accuracy degrees of and . Step 3. List all to get the leading one(s) conforming with . 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 ranking . 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: : dept paying ability; : management capability; : earning capacity; and : development and proper application. Enterprise financial performance of five possible enterprises 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 based on and their weight vector , for resulting matrix given in Table 1, where is IFEs.

In order to find most desirable enterprise , 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 , , we get Then, by score function, we obtained Step 2. Utilizing the using the IFDHWA operator and applying weighted vector , then obtain overall decision assessments for as follows: , , , , and . Step 3. Compute the score of of each as follows: Step 4. Rank in accordance with the value of as follows: . Step 5. Thus, is finalized as the favorable enterprise.

If IFDWG operator is applied, proceed similarly as above. Step 1. Utilizing the decision matrix , , we get Then, by score function, we obtained Step 2. Utilizing matrix using the IFDHWAG operator and applying weighted vector , we get overall decision costs of as follows: Step 3. Calculate the value of of each as follows: Step 4. Now, enterprise ranking as per score values is . Step 5. Thus, 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 Tables 2 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.

As indicated in Table 2, the Xu method and the proposed method by IFDHWA yielded the same complete-preference ranking result of the financial performance of five possible enterprises: . The distinction exists within the preference relation between and . 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 , 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 and , respectively. The difference exists in the preference relation between , , and . For this proposed approach, we apply intuitionistic fuzzy Dombi hybrid averaging and geometric operators which have an advantage of flexibility for the operation of ; 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 parameter . For this proposed approach, we apply intuitionistic fuzzy Dombi hybrid averaging and geometric operators which have an advantage of flexibility for the operation of ; 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.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Taif University Researchers Supporting Project (TURSP-2020/246), Taif University, Taif, Saudi Arabia.

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Copyright © 2021 Chiranjibe Jana et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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