Decision Support System Based on CoCoSo Method with the Picture Fuzzy Information

Qiyas, Muhammad; Naeem, Muhammad; Khan, Saifullah; Abdullah, Saleem; Botmart, Thongchi; Shah, Tajammul

doi:https://doi.org/10.1155/2022/1476233

Journal of Mathematics

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 1476233 | https://doi.org/10.1155/2022/1476233

Decision Support System Based on CoCoSo Method with the Picture Fuzzy Information

Muhammad Qiyas,¹Muhammad Naeem,²Saifullah Khan,¹Saleem Abdullah,¹Thongchi Botmart,³and Tajammul Shah¹

Academic Editor: Ghous Ali

Received16 Feb 2022

Revised29 Mar 2022

Accepted31 May 2022

Published08 Jul 2022

Abstract

Coronavirus disease 2019 (COVID-19) is currently threatening the entire world, and a novel coronavirus is a virus from the corona family that has spread a new infection. The number of instances of this disease is increasing at an exponential, but there are now commercially accessible COVID-19 vaccines. The weak symptoms of COVID-19 disease, on the other hand, are treated with a variety of antiviral treatments. It is still choosing the optimal antiviral medicine to manage COVID-19s. It is a challenging and difficult alternative to reduce the risk of infection. In this study, an improved combined compromise solution (CoCoSo) method is proposed to identify the ranking of alternatives. The introduction of a logarithmic picture fuzzy set is a more effective technique for representing variance, represented by three memberships (positive, neutral, and negative membership) degrees. This work introduces a fresh logarithmic picture fuzzy score function, to deal with the problem of comparison. The CoCoSo method-based logarithmic picture fuzzy decision-making algorithm is given. To achieve so, a new divergence measure for the logarithmic picture fuzzy number is introduced. To demonstrate the viability and efficacy of the established approach in real-world applications, a case study of COVID-19 disease drug selection is discussed.

1. Introduction

1.1. Background

In December 2019, the coronavirus was identified in Chinese seafood [1], and the chicken market has entered almost every country. Life is being turned upside down, and the world economy is being thrown off. The virus has infected over 1.6 million people and sickened over 76 million people in the last year. The condition has been declared a global pandemic by the World Health Organization.

In early December 2020, nations began a race to obtain vaccines. With a few Western countries leading the charge to control the virus, they began giving vaccinations to their most vulnerable populations. Here is how the pandemic has progressed over the previous 12 months. On December 31, the administration of Wuhan, China, claimed that health authorities [2] were treating thousands of cases. In [1], researchers in China found a new virus a few days later that had infected thousands of individuals across Asia. At the time, there was no confirmation that the people easily transmitted the virus. China’s health officials said they were keeping an eye on it to make sure it did not turn into anything more profound [3]. Hesitant fuzzy sets for medicine selection to treat mild symptoms of coronavirus disease will be available in September 2020 [3] is currently spreading. An improved fuzzy decision-making structure is defined for the treatment of mild symptoms of coronavirus virus using hesitant fuzzy sets 2019 [4]. The virus, biology, and objective proof prevention and therapy options for COVID-19 have all advanced [5]. Since the COVID-19 pandemic began in early December 2019, China has documented confirmed cases and 3242 deaths in March . The Chinese government has helped contribute to the fight against this disease, resulting in a significant improvement in the situation. COVID-19 was recognized in December 2019, but it is still in a precarious evaluation stage due to a lack of data and medical technology, specifically clinical trials. Previously, it was difficult to explicitly relate available disease data to existing statistical methods and determine the effectiveness of the ongoing medical response. To avoid further crisis development, physicians, medical researchers, or sections should be included to adopt tests, tactics, or picking an option appropriate strategy for the treatment process. The division in charge of implementing strategy must make fast and accurate decision-making. Individuals are usually logically bound in place by entirely rational decisions in this case. As a result, finding appropriate DM models that consider people’s actions are critical in providing people with an efficient way of responding to an emergency. Taking care of ambiguous is unclear values in practical situations has always been a challenge.

1.1.1. China Reported Its First Death

The first confirmed case is on January 11, and Chines official media reported a fatality from an infection induced by the virus, which had afflicted thousands of individuals. The deceased 61-year-old guy was a regular visitor to Wuhan’s market. His death comes days before each of China’s most important holidays when hundreds of millions of people travel throughout the globe.

1.1.2. Cases in Several Countries

According to the World Health Organization’s first situation study, Thailand, South Korea, and Japan were among the first countries to experience the disease outside of mainland China. The first confirmed case in the United States happened the next day in Washington State when a 30-year-old guy began to have symptoms after returning from a vacation to Wuhan.

1.1.3. The First COVID-19 Death Outside China

According to officials, a 44-year-old man died in the Philippines after getting the disease; the first death verified outside of China. By this moment, almost 359 individuals had died.

1.1.4. The Virus Was Given the Name of the Disease

The World Health Organization suggested Covid-19 as an official name for the virus outbreak; coronavirus disease 2019 is the abbreviation for coronavirus disease. Given the intention of avoiding stigma, there is no reference to any of the people, places, or animals related to the coronavirus in the name.

1.1.5. COVID-19 Pandemic in Pakistan

In Pakistan, coronavirus is thought to be spreading. On February , the first case of coronavirus was recorded in Karachi, Pak, which has a population of 204.65 million people 3.4. Since the virus has spread to different parts of the country, it has now spread to record levels. Pakistan will be ready in 45 days on April had 4601 confirmed COVID-19 cases; seven hundred and thirty patients had recovered, but seventy had died.

This brief correspondence aims to raise awareness about the country’s coronavirus outbreak. It will help to highlight the current in a word, this is the problem, and steps are recorded by Pakistan’s department of health to reduce the possibility of connection.

1.1.6. March

Pakistan had administered vaccine doses by March . Nadhim Zahawi announced on March , that Pakistan would receive 16 million vaccine doses for COVID-19 from the United Kingdom. On March 16, Pakistan received 500000 Sinopharm vaccination pills as a gift from China.

1.2. Literature Review

Multiattribute decision-making (MADM) problems are faced in everyday life. In MADM problems, there are more than one alternative and criteria. The MADM problems are mainly used in some practical decision-making problems which are Web 2.0 [6], social network [7], and trust recommendation system [8, 9]. The decision-making problems are in the form of two ways, the first one is MADM problems and the second one is a multicriteria group decision-making problem. On DMs problems, one cannot decide directly. As a result, it becomes difficult for decision-makers to express attribute values numerically. Many theoretical researches on MADM problems have been published in recent years. There are some problems that are not solved by a classical set, and to solve this type of problem, Zadeh [10] introduced the fuzzy set (FS) which contained one degree called a positive degree that belongs to the close interval zero and one. Intuitionistic fuzzy sets (IFSs) a hybridization of FS presented by Atanassov [11] are one of the most effective modifications of FS and are applied to handle many DM problems. Each element in this collection is expressed in terms of positive degree and negative degree in such a way that their sum is less than or equal to zero and one. IFSs have been successfully implemented by numerous researchers in various areas for handling decision-making issues. For example, weighted averaging and geometric aggregation operators were first presented by Xu [12], and averaging operator, ordered weighted geometric aggregation operator, and the hybrid geometric aggregation operator were discussed by Xu and Chen [13] and Wei and Wang [14], respectively. IFS has been used by a lot of researchers in various fields. One of the most crucial concepts of neutral grade (NuG) is missing from the IFS theory.

In a different scenario, professional experts are limited in their options within the IFS range. To address this shortcoming, Yager [15] looked into the Pythagorean fuzzy set (PyFS), a powerful paradigm in which the square sum of MG and NMG must lie between the real numbers. By giving the decision-maker more space, PyFS relaxes and broadens the boundary range. Using Pythagorean fuzzy information, Yager [16] invented the geometric and averaging aggregation operation. Peng and Yang [17] were the first to introduce the concept of subtraction and division operators and to demonstrate some of their fundamental properties. Pythagorean fuzzy Choquet integral average and Pythagorean fuzzy Choquet integral geometric operators were investigated by Peng and Yang [18]. Garg [19, 20] presented the basic characteristics of Pythagorean fuzzy Einstein averaging and Pythagorean fuzzy Einstein geometric operators. Garg [21] looked into the basic properties of confidence Pythagorean fuzzy weighted and ordered weighted averaging operators. Wei and Lu [22] were the first to introduce the concepts of Pythagorean fuzzy power averaging and geometric operators, as well as their desirable characteristics. Wei [23] used Pythagorean fuzzy information to present some interaction averaging and geometric operators. Wu and Wei [24] introduced the concept of Hamacher operations for Pythagorean fuzzy averaging and geometric operators. Decision-makers in the PyF environment are constrained by their boundary limitations and are unable to freely provide their preferred values. Due to these limitations, PyFS is unable to effectively handle some critical information. Khan et al. [25] proposed Pythagorean fuzzy Dombi AOs and their application in decision support system.

Yager [26] recently introduced the q-rung orthopair fuzzy (q-ROF) set (q-ROFS), a new concept from which the most significant generalization of PFS emerged. In q-ROFS, the sum of MG and NMG power must be confined to the unit interval, and as the rung increases, the range of orthopair satisfies the boundary restriction that is required. Because IFS and PFS are special cases of q-ROFS, the concept of q-ROFS is more useful and powerful than IFS and PFS. Yager and Alajlan [27] proposed the basic properties of q-ROFS, which have been used in knowledge representation. Hussain et al. [28] proposed a different perspective on q-ROFS based on the concept of orbits. The concepts of q-ROF weighted averaging (q-ROFWA) and q-ROF weighted geometric averaging (q-ROFWA) were proposed by Liu and Wang [29] (q-ROFWG). Liu et al. [30] presented a combined study of Bonferroni mean (BM) operators with q-ROFS in order to investigate q-ROF Bonferroni mean operators as well as q-ROF geometric BM operators with desirable properties. Jana et al. [31] were the first to introduce the q-ROF Dombi averaging and geometric aggregation operators, which have fundamentally desirable properties. Wang et al. [32] investigated the concept of combining Muirhead means (MM) operators with q-ROFS to create q-ROF Muirhead means operators, which are new aggregation operators. Joshi and Gegov [33] proposed some aggregation operators based on the confidence level of experts in the original information in a q-ROF environment, such as confidence q-ROFWA (Cq-ROFWA) and confidence q-ROFWG (Cq-ROFWG), Cq-ROFOWA, and Cq-ROFOWG. Yang et al. [34] proposed q-RO normal fuzzy sets and defined their operational laws and score function. They also created q-RONFWA and q-RONFOWG, which are aggregation operators for the same concept. Hussain et al. [35] also proposed and discussed the hesitant q-ROFWA and hesitant q-ROFWG operators, as well as their desirable properties. Hussain et al. [36] defined the generalized and group generalized averaging operation based on q-ROF information. Wang et al. [37] defined the decision-making based on q-rung orthopair fuzzy soft rough sets.

Picture fuzzy sets (PFSs), which were first presented by Cuong and Kreinovich [38], have shown to be an effective strategy for describing the fuzziness of MCDM issues when confronted with multiple types of responses: off, obstruct, no, and refusal. There are three memberships; positive, neutral, and negative membership, where the number is equal to or less than one. When the neutral membership for the relevant domain is equal to 0, the PFS is reduced to IFS [39]. Garg [40] proposed some AOs based on PFS and discussed their applications to MCDM. Joshi and Kumar [41] developed an approach to MCDM problems using dice similarity measures for PFSs. Ashraf et al. [42] defined some approaches for MCGDM problems using picture fuzzy information. Jana and Pal [43] developed the assessment of enterprise performance based on picture fuzzy Hamacher AOs. Ju et al. [44] studied the site selection of electric vehicle charging stations based on the extended GRP method with picture fuzzy information. Khan et al. [45] introduced some new picture fuzzy AOs based on Einstein’s operations. As a result, PFSs are better suited to dealing with uncertain data. Because of this benefit, it quickly becomes a major topic among aggregation operators. Liu et al. [46] give some models for MADM with picture fuzzy information. Luqman et al. [47] proposed a granulation of hypernetwork models under the q-rung picture fuzzy environment. Liu et al. [48] developed a specific type of q-rung picture fuzzy Yager AOs for decision-making. Akram et al. [49] defined a DM model under complex picture fuzzy Hamacher aggregation operators. Ashraf et al. [50] defined a decision support model for the patient admission scheduling problem based on picture fuzzy aggregation information and TOPSIS methodology. As we know that PFSs are better suited to dealing with unclear data. Given this benefit, it quickly becomes a hot topic, including aggregation operators (Wang et al. [51–53]). Qiyas et al. [54] defined triangular picture fuzzy linguistic induced ordered weighted AOs and its application on decision-making problems. In [55], picture fuzzy sets are used to deal with the overall uncertainties; several aggregation operators depending on the algebraic t-norm [56] and t-conorm have been created. Logarithmic operational rules are valuable mathematical processes for gathering complex and incorrect data. Akram et al. [57] proposed a hybrid decision-making analysis under complex q-rung picture fuzzy Einstein averaging operators. Qiyas et al. [58] defined the concept of Yager operators using picture fuzzy set information and discussed its application to emergency program selection. Akram et al. [59] defined the MADM approach using the q-rung picture fuzzy information. Zeng et al. [60] proposed a social network MCDM approach for evaluating unmanned ground delivery vehicles under the Pythagorean fuzzy environment. In some cases, the concept of neutrality degree can be seen when we are faced with human viewpoints containing multiple types of responses such as yes, no, and refuse. Khoshaim et al. [61] developed an approach for supplier selection problems based on picture cubic fuzzy aggregation operators. Abdullah et al. [62] defined a novel approach based on sine trigonometric picture fuzzy AOs and their application in decision support systems. Khan et al. [63] proposed an analysis of robot selection based on 2-tuple picture fuzzy linguistic aggregation operators.

1.3. CoCoSo Methods

The combined compromise solution (CoCoSo) was initially proposed by Yazdani [64]. In the decision-making method, three aggregation methods were used by combining the actual weight in addition (SAW) and a product that is exponentially weighted (EWP) models, which we can see as a workable compromise. The CoCoSo approach has been successfully used in the selection of logistic providers [65]. During the decision-making process, we are going to have some real-world problems. When dealing with complex preference data, the CoCoSo approach has been used in several ambiguous conditions to deal with such circumstances. Karasan and Bolturk [66] defined that, in the interval, the CoCoSo technique was extended. Wen et al. [67] defined that, in a hesitant fuzzy language context, the CoCoSo technique was tested. It integrates the SAW [68], WASPAS (weighted aggregated sum product assessment) [69], and MEW (multiplicative exponential weighting) methods [70] with aggregation strategies. By this method, decision-makers can obtain a multifaceted compromise solution, which is consistent with the solution obtained by other MCDM methods, such as the VIKOR [71] and MOORA (multiobjective optimization on the basis of ratio analysis) [72] methods.

1.4. Objective

(1)The CoCoSo method is used to develop a novel logarithmic picture fuzzy set decision-making method that can achieve the best alternative(2)The combined weight method uses CRITIC (criteria importance through intercriteria correlation) and the linear weighted comprehensive method to consider subjective and objective data simultaneously(3)A logarithmic picture fuzzy CoCoSo method is defined and solved the problem of choosing the optimal antiviral medicine to manage COVID-19(4)The validity and superiority of the established measures are also examined through detailed comparisons between proposed and existing measures

In Section 2, we are to discuss some basic concepts of LPFSs. Section 3 explores the novel score function with its properties. Section 4 introduces a new logarithmic picture fuzzy MCDM method based on the CoCoSo method with CRITIC, and the sensitivity analysis is shown. Section 5 gives a comparison with some existing MCDM methods employing a numerical example to state the feasibility of the developed method. Section 5 is the comparison and Section 6 is the conclusion.

2. Preliminaries

Now, we will go over some basic notion of logarithmic picture fuzzy numbers.

Definition 1 (see [38]). If we have a fixed set , thenis said to be PFS, and are the positive, natural, and negative membership degree of the element in , respectively. satisfying the following condition for all ;for all , and the refusal degree is defined as .

Definition 2 (see [40]). Let be nonempty fixed set and be the PFS. Then, the logarithmic PFNs is defined aswhere are the positive, neutral, and negative membership degrees of the function in , respectively, with condition that . Then, the positive membership is defined as, and the neutral membership is defined as, and the negative membership is defined as, and the refusal degree is defined as

Definition 3. (see [40]). Let , be two PFNs. Then, the comparison rules are defined.(1)If , then (2)If and , then (3)If and , then

Definition 4. (see [40]). Let are three PFNs. Then,(1)(2)(3)(4)

2.1. The Novel Logarithmic Picture Fuzzy Score Function

Now, in this section, we are going to discuss some existing score function and accuracy function.

2.1.1. Existing Logarithmic Picture Fuzzy Score Function

Some existing score functions are displayed in a Table 1.

2.1.2. Existing Logarithmic Picture Fuzzy Accuracy Function

There is some existing accuracy function which is given in Table 2.

2.2. New Score Function

In this section, we have to discuss some novel score functions for logarithmic PFN. This section’s major goal is to suggest a new score function as an effective score function. It is the positive degree, neutral, negative, and refusal degree that should be accepted. According to the expert, PFN entails a more enormous positive, a lower neutral, a lower negative, and a smaller refusal. Thus, we must demonstrate that the supplied score is functional.

Definition 5. Let , be logarithmic picture fuzzy set. Then, the corresponding score function can be defined as

Theorem 1. For -PFN, we have , , as increases, monotonically increases and or increases, monotonically decreases.

Proof. Using Definition 5, we have first partial derivative that corresponds with respect to ,Similarly, we are to calculate first partial derivative of in relation to and Hence, we proved that.

Theorem 2. For -PFN, the corresponding score function are the following relation.(1)(2), iff (3), iff

Proof. Now, using Theorem 1, we are to consider , or , can present the minimum-value or maximum-value when or , and in other word, or , hence proved.

3. New Logarithmic Picture Fuzzy MCDM Method

Consider that be a series of alternatives be a set of distinct criteria, and weight vector with . Assume that the alternative evaluation in terms of criteria is represented by the picture fuzzy (LPF) matrix , which is shown in Table 3.

Let , and

3.1. Logarithmic Picture Fuzzy MCDM Approach Based on CoCoSo Method

Yazdani et al. [64] present the CoCoSo (combined compromise solution) method, which is a novel and effective MCDM method. The proposed method is based on an integrated exponentially weighted product (EWP) and simple additive weighting (SAW) model, which can be used to create a collection of compromise solutions. We propose a log-PFS-CoCoSo approach to solve the MCDM problem.

The log-PFS-CoCoSo method, in general, consists of the steps listed as follows: Step 1. Obtain the decision matrix Step 2. By using equation (12), normalize the picture fuzzy matrix of Step 3. Determine the score function’s values of using Step 4. Calculate the for matrix of a score function using Step 5. Obtain the objective weight using Step 6. Evaluate the combined weight using Step 7. Evaluate the comparability sequences of the weight for each alternative as Step 8. For each alternative, compute the comparability sequence for the power weight as Step 9. Using the aggregation method given in equations (18)–(20) Step 10. Calculate the score value using Step 11. Alternatives are rated in order of decreasing assessed score values .

4. Example

According to a study released in February , there have been estimated COVID-19 cases have been recorded all over the world, resulting in deaths. A total of people have been rescued. According to research, a large number of people with COVID-19 will show the symptoms and indications listed below; flu , vomiting , weakness , alcoholism , nausea , phlegm development , and dysphagia . Here, we prefer five alternatives to medications for COVID-19 patients that must be kept under control, in particular, LVPV/RTV-IFNb , favipiravir , LPV/RTV , danger , and clobazam . Also, we prefer seven signs as alternatives anorexia , cough , fatigue , fever , myalgia , shortness of breath , sputum production . Step 1. The decision matrix is given in Table 4 Step 2. Since all the attributes are benefit types, there is no need to convert Step 3. Using equation (8), to obtain the score function’s values of are given in Table 5 Step 4. Using equation (13), to obtain for a matrix of a score function Step 5. Using equation (14), to obtain the combined weight Step 6. Determine the entire weighted comparability sequence as Step 7. Determine the total number of power-weighted comparable sequences as Step 8. Obtained the three aggregation strategies , and as Step 9. Obtained the corresponding assessment value as Step 10. Give raking according to the score values as

5. Comparison

This section contains the following information that provides the contrast between the existing method and the proposed method. The existing aggregation operators were introduced in Peng and Luo [74] based on the CoCoSo method with picture fuzzy information; Ashraf et al. [42] based on picture fuzzy information; Garg [40] based on picture fuzzy aggregation operators; Jana and Pal [43] based on picture fuzzy Hamacher aggregation operators; Joshi and Kumar [41] based on dice similarity measures for picture fuzzy sets; Ju et al. [44] based on the extended GRP method under picture fuzzy information; Khan et al. [45] based on picture fuzzy Einstein aggregation operators to handle the logarithmic picture fuzzy information using the algebraic operation laws. Now, we compared our proposed method logarithmic picture fuzzy set to other existing PFS aggregation operators. The best opportunity is obtained by using logarithmic picture fuzzy Einstein aggregation operators. As a result, this research suggested novel idea logarithmic Einstein aggregation operators to aggregate the picture fuzzy data. Now, we have been using the existing aggregation operators on the same data. So, we have the same result as in the existing methods. Now, the proposed methods are more effectively and efficiently. As a result, the proposed method is the best solution for MCDM problem. The ranking of the proposed method and existing method is given in Table 6.

As a result of the above analysis, it is concluded that it is the most suitable and desirable alternative . The higher the score value, the more optimistic it is; the lower the score value, the more pessimistic it is. According to the comparisons and analysis above, the methods proposed in this paper for aggregating picture fuzzy information based on the logarithmic PFWA and logarithmic PFWG operators are better than the existing other methods. As a result, they are better suited to solving the MADM problem.

6. Conclusion

In all areas of decision-making, MCDM has a high potential and discipline process for improving and evaluating multiple conflicting criteria. Experts evaluate each and every character of an alternative before making a decision to make an intelligent decision. Experts must carefully prepare and evaluate each and every character for an alternative in order to make an intelligent and successful decision. Once they have collected all of the data and information they require, they can make a good decision. The following are the most important contributions.(1)Give an original solution to the COVID-19 disease symptom drug selection problem using fuzzy logarithmic picture numbers.(2)It presents a new logarithmic picture fuzzy scoring function that considers refusal data and reduces information distortion. It has much experience when it comes to separating PFNs.(3)The CoCoSo-based innovative logarithmic picture fuzzy MCDM method is presented. Furthermore, the efficacy and utility of this methodology have been established by evaluating medications for COVID-19 individuals with mild symptoms.

The remarkable CoCoSo approach can expand this work to other generalized theories of fuzzy information in the future such as cubic fuzzy sets, Hamacher operators, 2-tuple picture fuzzy linguistic set, T-spherical fuzzy sets, Dombi operators, power mean aggregation operators, Bonferroni mean operators, and Heronian mean operators.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors participated in every stage of the research, and all authors read and approved the final manuscript.

Acknowledgments

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work under grant code 22UQU4310396DSR13.

References

S. Ashraf, S. Abdullah, M. Aslam, M. Qiyas, and M. A. Kutbi, “Spherical fuzzy sets and its representation of spherical fuzzy t-norms and t-conorms,” Journal of Intelligent and Fuzzy Systems, vol. 36, no. 6, pp. 6089–6102, 2019.
View at: Publisher Site | Google Scholar
I. Cromwell, S. J. Peacock, and C. Mitton, ““Real-world” health care priority setting using explicit decision criteria: a systematic review of the literature,” BMC Health Services Research, vol. 15, no. 1, p. 164, 2015.
View at: Publisher Site | Google Scholar
M. A. Martinez, “Compounds with therapeutic potential against novel respiratory 2019 coronavirus,” Antimicrobial Agents and Chemotherapy, vol. 64, no. 5, 2020.
View at: Publisher Site | Google Scholar
A. K. Azkur, M. Akdis, D. Azkur et al., “Immune response to SARS‐CoV‐2 and mechanisms of immunopathological changes in COVID‐19,” Allergy, vol. 75, no. 7, pp. 1564–1581, 2020.
View at: Publisher Site | Google Scholar
G. Zhou, S. Chen, and Z. Chen, “Advances in COVID-19: the virus, the pathogenesis, and evidence-based control and therapeutic strategies,” Frontiers of Medicine, vol. 14, no. 2, pp. 117–125, 2020.
View at: Publisher Site | Google Scholar
S. Alonso, I. J. Pérez, F. J. Cabrerizo, and E. Herrera-Viedma, “A linguistic consensus model for web 2.0 communities,” Applied Soft Computing, vol. 13, no. 1, pp. 149–157, 2013.
View at: Publisher Site | Google Scholar
J. Wu, F. Chiclana, and E. Herrera-Viedma, “Trust based consensus model for social network in an incomplete linguistic information context,” Applied Soft Computing, vol. 35, pp. 827–839, 2015.
View at: Publisher Site | Google Scholar
C. Martinez-Cruz, C. Porcel, J. Bernabé-Moreno, and E. Herrera-Viedma, “A model to represent users trust in recommender systems using ontologies and fuzzy linguistic modeling,” Information Sciences, vol. 311, pp. 102–118, 2015.
View at: Publisher Site | Google Scholar
I. J. Pérez, F. J. Cabrerizo, S. Alonso, and E. Herrera-Viedma, “A new consensus model for group decision making problems with non-homogeneous experts,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 44, no. 4, pp. 494–498, 2014.
View at: Publisher Site | Google Scholar
L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986.
View at: Publisher Site | Google Scholar
Z. Xu, “Intuitionistic fuzzy aggregation operators,” IEEE Transactions on Fuzzy Systems, vol. 15, no. 6, pp. 1179–1187, 2007.
View at: Publisher Site | Google Scholar
Z. Xu and J. Chen, “On geometric aggregation over interval-valued intuitionistic fuzzy information,” in Proceedings of the 4th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2007), vol. 2, pp. 466–471, Harbin, China, August 2007.
View at: Publisher Site | Google Scholar
G. Wei and X. Wang, “Some geometric aggregation operators based on interval-valued intuitionistic fuzzy sets and their application to group decision making,” in Proceedings of the 2007 International Conference on Computational Intelligence and Security (CIS 2007), Harbin, China, December 2007.
View at: Publisher Site | Google Scholar
R. R. Yager, “Pythagorean fuzzy subsets,” in Proceedings of the 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), Edmonton, Canada, June 2013.
View at: Google Scholar
R. R. Yager, “Pythagorean membership grades in multicriteria decision making,” IEEE Transactions on Fuzzy Systems, vol. 22, no. 4, pp. 958–965, 2014.
View at: Publisher Site | Google Scholar
X. Peng and Y. Yang, “Some results for Pythagorean fuzzy sets,” International Journal of Intelligent Systems, vol. 30, no. 11, pp. 1133–1160, 2015.
View at: Publisher Site | Google Scholar
X. Peng and Y. Yang, “Pythagorean fuzzy Choquet integral based MABAC method for multiple attribute group decision making,” International Journal of Intelligent Systems, vol. 31, no. 10, pp. 989–1020, 2016.
View at: Publisher Site | Google Scholar
H. Garg, “A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making,” International Journal of Intelligent Systems, vol. 31, no. 9, pp. 886–920, 2016.
View at: Publisher Site | Google Scholar
H. Garg, “Generalized pythagorean fuzzy geometric aggregation operators using einsteint-norm andt-conorm for multicriteria decision-making process,” International Journal of Intelligent Systems, vol. 32, no. 6, pp. 597–630, 2017.
View at: Publisher Site | Google Scholar
H. Garg, “Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process,” Computational & Mathematical Organization Theory, vol. 23, no. 4, pp. 546–571, 2017.
View at: Publisher Site | Google Scholar
G. Wei and M. Lu, “Pythagorean fuzzy power aggregation operators in multiple attribute decision making,” International Journal of Intelligent Systems, vol. 33, no. 1, pp. 169–186, 2018.
View at: Publisher Site | Google Scholar
G. Wei, “Pythagorean fuzzy interaction aggregation operators and their application to multiple attribute decision making,” Journal of Intelligent and Fuzzy Systems, vol. 33, no. 4, pp. 2119–2132, 2017.
View at: Publisher Site | Google Scholar
S. J. Wu and G. W. Wei, “Pythagorean fuzzy Hamacher aggregation operators and their application to multiple attribute decision making,” International Journal of Knowledge-Based and Intelligent Engineering Systems, vol. 21, no. 3, pp. 189–201, 2017.
View at: Publisher Site | Google Scholar
A. Khan, S. Ashraf, S. Abdullah, M. Qiyas, J. Luo, and S. Khan, “Pythagorean fuzzy Dombi aggregation operators and their application in decision support system,” Symmetry, vol. 11, no. 3, p. 383, 2019.
View at: Publisher Site | Google Scholar
R. R. Yager, “Generalized orthopair fuzzy sets,” IEEE Transactions on Fuzzy Systems, vol. 25, no. 5, pp. 1222–1230, 2017.
View at: Publisher Site | Google Scholar
R. R. Yager and N. Alajlan, “Approximate reasoning with generalized orthopair fuzzy sets,” Information Fusion, vol. 38, pp. 65–73, 2017.
View at: Publisher Site | Google Scholar
A. Hussain, M. Irfan Ali, and T. Mahmood, “Covering based q-rung orthopair fuzzy rough set model hybrid with TOPSIS for multi-attribute decision making,” Journal of Intelligent and Fuzzy Systems, vol. 37, no. 1, pp. 981–993, 2019.
View at: Publisher Site | Google Scholar
P. Liu and P. Wang, “Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making,” International Journal of Intelligent Systems, vol. 33, no. 2, pp. 259–280, 2018.
View at: Publisher Site | Google Scholar
P. Liu and J. Liu, “Some q-rung orthopai fuzzy Bonferroni mean operators and their application to multi-attribute group decision making,” International Journal of Intelligent Systems, vol. 33, no. 2, pp. 315–347, 2018.
View at: Publisher Site | Google Scholar
C. Jana, G. Muhiuddin, and M. Pal, “Some Dombi aggregation of Q‐rung orthopair fuzzy numbers in multiple‐attribute decision making,” International Journal of Intelligent Systems, vol. 34, no. 12, pp. 3220–3240, 2019.
View at: Publisher Site | Google Scholar
J. Wang, R. Zhang, X. Zhu, Z. Zhou, X. Shang, and W. Li, “Some q-rung orthopair fuzzy Muirhead means with their application to multi-attribute group decision making,” Journal of Intelligent and Fuzzy Systems, vol. 36, no. 2, pp. 1599–1614, 2019.
View at: Publisher Site | Google Scholar
B. P. Joshi and A. Gegov, “Confidence levels q‐rung orthopair fuzzy aggregation operators and its applications to MCDM problems,” International Journal of Intelligent Systems, vol. 35, no. 1, pp. 125–149, 2020.
View at: Publisher Site | Google Scholar
Z. Yang, X. Li, Z. Cao, and J. Li, “Q-rung orthopair normal fuzzy aggregation operators and their application in multi-attribute decision-making,” Mathematics, vol. 7, no. 12, p. 1142, 2019.
View at: Publisher Site | Google Scholar
A. Hussain, M. I. Ali, and T. Mahmood, “Hesitant q-rung orthopair fuzzy aggregation operators with their applications in multi-criteria decision making,” Iranian Journal of Fuzzy Systems, vol. 17, no. 3, pp. 117–134, 2020.
View at: Google Scholar
A. Hussain, M. I. Ali, T. Mahmood, and M. Munir, “Group-based generalized q-rung orthopair average aggregation operators and their applications in multi-criteria decision making,” Complex & Intelligent Systems, vol. 7, no. 1, pp. 123–144, 2021.
View at: Publisher Site | Google Scholar
Y. Wang, A. Hussain, T. Mahmood, M. I. Ali, H. Wu, and Y. Jin, “Decision-making based on q-rung orthopair fuzzy soft rough sets,” Mathematical Problems in Engineering, vol. 2020, Article ID 6671001, 21 pages, 2020.
View at: Publisher Site | Google Scholar
B. C. Cuong and V. Kreinovich, “Picture fuzzy sets-a new concept for computational intelligence problems,” in Proceedings of the 2013 3rd World Congress on Information and Communication Technologies (WICT 2013), Hanoi, Vietnam, December 2013.
View at: Publisher Site | Google Scholar
K. T. Atanassov, Intuitionistic Fuzzy Sets, Physica-Verlag HD, Heidelberg, Germany, 1999.
H. Garg, “Some picture fuzzy aggregation operators and their applications to multicriteria decision-making,” Arabian Journal for Science and Engineering, vol. 42, no. 12, pp. 5275–5290, 2017.
View at: Publisher Site | Google Scholar
D. Joshi and S. Kumar, An Approach to Multi-Criteria Decision Making Problems Using Dice Similarity Measure for Picture Fuzzy Sets, Mathematics and Computing. ICMC 2018. Communications in Computer and Information Science, Springer, Singapore, 2018.
S. Ashraf, T. Mahmood, S. Abdullah, and Q. Khan, “Different approaches to multi-criteria group decision making problems for picture fuzzy environment,” Bulletin of the Brazilian Mathematical Society, New Series, vol. 50, no. 2, pp. 373–397, 2019.
View at: Publisher Site | Google Scholar
C. Jana and M. Pal, “Assessment of enterprise performance based on picture fuzzy Hamacher aggregation operators,” Symmetry, vol. 11, no. 1, p. 75, 2019.
View at: Publisher Site | Google Scholar
Y. Ju, D. Ju, E. D. Santibanez Gonzalez, M. Giannakis, and A. Wang, “Study of site selection of electric vehicle charging station based on extended GRP method under picture fuzzy environment,” Computers & Industrial Engineering, vol. 135, pp. 1271–1285, 2019.
View at: Publisher Site | Google Scholar
S. Khan, S. Abdullah, and S. Ashraf, “Picture fuzzy aggregation information based on Einstein operations and their application in decision making,” Mathematical Sciences, vol. 13, no. 3, pp. 213–229, 2019.
View at: Publisher Site | Google Scholar
H. Liu, H. Wang, Y. Yuan, and C. Zhang, “Models for multiple attribute decision making with picture fuzzy information,” Journal of Intelligent and Fuzzy Systems, vol. 37, no. 2, pp. 1973–1980, 2019.
View at: Publisher Site | Google Scholar
A. Luqman, M. Akram, and A. N. A. Koam, “Granulation of hypernetwork models under the q-rung picture fuzzy environment,” Mathematics, vol. 7, no. 6, p. 496, 2019.
View at: Publisher Site | Google Scholar
P. Liu, G. Shahzadi, and M. Akram, “Specific types of q-rung picture fuzzy Yager aggregation operators for decision-making,” International Journal of Computational Intelligence Systems, vol. 13, no. 1, p. 1072, 2020.
View at: Publisher Site | Google Scholar
M. Akram, A. Bashir, and H. Garg, “Decision-making model under complex picture fuzzy Hamacher aggregation operators,” Computational and Applied Mathematics, vol. 39, no. 3, pp. 1–38, 2020.
View at: Google Scholar
S. Ashraf, N. Rehman, N. Rehman et al., “Decision support model for the patient admission scheduling problem based on picture fuzzy aggregation information and TOPSIS methodology,” Mathematical Biosciences and Engineering, vol. 19, no. 3, pp. 3147–3176, 2022.
View at: Publisher Site | Google Scholar
C. Wang, X. Zhou, H. Tu, and S. Tao, “Some geometric aggregation operators based on picture fuzzy sets and their application in multiple attribute decision making,” Italian Journal of Pure and Applied Mathematics, vol. 37, pp. 477–492, 2017.
View at: Google Scholar
R. Wang, J. Wang, H. Gao, and G. Wei, “Methods for MADM with picture fuzzy muirhead mean operators and their application for evaluating the financial investment risk,” Symmetry, vol. 11, no. 1, p. 6, 2018.
View at: Publisher Site | Google Scholar
H. Wang, S. Lu, and J. Zhao, “Aggregating multiple types of complex data in stock market prediction: a model-independent framework,” Knowledge-Based Systems, vol. 164, pp. 193–204, 2019.
View at: Google Scholar
M. Qiyas, S. Abdullah, S. Ashraf, S. Khan, and A. Khan, “Triangular picture fuzzy linguistic induced ordered weighted aggregation operators and its application on decision making problems,” Mathematical Foundations of Computing, vol. 2, no. 3, pp. 183–201, 2019.
View at: Publisher Site | Google Scholar
G. Wei, “Picture fuzzy aggregation operators and their application to multiple attribute decision making,” Journal of Intelligent and Fuzzy Systems, vol. 33, no. 2, pp. 713–724, 2017.
View at: Publisher Site | Google Scholar
G. Wei, “Some cosine similarity measures for picture fuzzy sets and their applications to strategic decision making,” Informatica, vol. 28, no. 3, pp. 547–564, 2017.
View at: Publisher Site | Google Scholar
M. Akram, A. Bashir, and S. A. Edalatpanah, “A hybrid decision-making analysis under complex q-rung picture fuzzy Einstein averaging operators,” Computational and Applied Mathematics, vol. 40, no. 8, p. 305, 2021.
View at: Publisher Site | Google Scholar
M. Qiyas, M. A. Khan, S. Khan, and S. Abdullah, “Concept of Yager operators with the picture fuzzy set environment and its application to emergency program selection,” International Journal of Intelligent Computing and Cybernetics, vol. 13, no. 4, pp. 455–483, 2020.
View at: Publisher Site | Google Scholar
M. Akram, G. Shahzadi, and J. C. R. Alcantud, “Multi-attribute decision-making with q-rung picture fuzzy information,” Granular Computing, vol. 7, no. 1, pp. 197–215, 2022.
View at: Publisher Site | Google Scholar
S. Zeng, N. Zhang, C. Zhang, W. Su, and L. A. Carlos, “Social network multiple-criteria decision-making approach for evaluating unmanned ground delivery vehicles under the Pythagorean fuzzy environment,” Technological Forecasting and Social Change, vol. 175, Article ID 121414, 2022.
View at: Publisher Site | Google Scholar
A. B. Khoshaim, M. Qiyas, S. Abdullah, and M. Naeem, “An approach for supplier selection problem based on picture cubic fuzzy aggregation operators,” Journal of Intelligent and Fuzzy Systems, vol. 40, no. 5, pp. 10145–10162, 2021.
View at: Publisher Site | Google Scholar
S. Abdullah, S. Khan, M. Qiyas, and R. Chinram, “A novel approach based on sine trigonometric picture fuzzy aggregation operators and their application in decision support system,” Journal of Mathematics, vol. 2021, Article ID 8819517, 19 pages, 2021.
View at: Publisher Site | Google Scholar
A. A. Khan, M. Qiyas, S. Abdullah, J. Luo, and M. Bano, “Analysis of robot selection based on 2-tuple picture fuzzy linguistic aggregation operators,” Mathematics, vol. 7, no. 10, p. 1000, 2019.
View at: Publisher Site | Google Scholar
M. Yazdani, P. Zarate, E. Kazimieras Zavadskas, and Z. Turskis, “A combined compromise solution (CoCoSo) method for multi-criteria decision-making problems,” Management Decision, vol. 57, no. 9, pp. 2501–2519, 2019.
View at: Publisher Site | Google Scholar
F. Ecer, D. Pamucar, S. Hashemkhani Zolfani, and M. Keshavarz Eshkalag, “Sustainability assessment of OPEC countries: application of a multiple attribute decision making tool,” Journal of Cleaner Production, vol. 241, Article ID 118324, 2019.
View at: Publisher Site | Google Scholar
A. Karasan and E. Bolturk, “Solid waste disposal site selection by using neutrosophic combined compromise solution method,” in Proceedings of the 2019 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology (EUSFLAT 2019), Birmingham, UK, September 2019.
View at: Publisher Site | Google Scholar
Z. Wen, H. Liao, A. Mardani, and A. Al-Barakati, A Hesitant Fuzzy Linguistic Combined Compromise Solution Method for Multiple Criteria Decision Making, Springer, Berlin, Germany, 2019.
A. Afshari, M. Mojahed, and R. M. Yusuff, “Simple additive weighting approach to personnel selection problem,” International Journal of Innovation, Management and Technology, vol. 1, no. 5, p. 511, 2010.
View at: Google Scholar
E. K. Zavadskas, Z. Turskis, J. Antucheviciene, and A. Zakarevicius, “Optimization of weighted aggregated sum product assessment,” Electronics and Electrical Engineering, vol. 122, no. 6, pp. 3–6, 2012.
View at: Publisher Site | Google Scholar
S. H. Zanakis, A. Solomon, N. Wishart, and S. Dublish, “Multi-attribute decision making: a simulation comparison of select methods,” European Journal of Operational Research, vol. 107, no. 3, pp. 507–529, 1998.
View at: Publisher Site | Google Scholar
S. Opricovic, “Multicriteria optimization of civil engineering systems,” Faculty of Civil Engineering, vol. 2, no. 1, pp. 5–21, 1998.
View at: Google Scholar
W. K. Brauers and E. K. Zavadskas, “The MOORA method and its application to privatization in a transition economy,” Control and Cybernetics, vol. 35, no. 2, pp. 445–469, 2006.
View at: Google Scholar
S. Khan, S. Abdullah, L. Abdullah, and S. Ashraf, “Logarithmic aggregation operators of picture fuzzy numbers for multi-attribute decision making problems,” Mathematics, vol. 7, no. 7, p. 608, 2019.
View at: Publisher Site | Google Scholar
X. Peng and Z. Luo, “Decision-making model for China’s stock market bubble warning: the CoCoSo with picture fuzzy information,” Artificial Intelligence Review, vol. 54, no. 8, pp. 5675–5697, 2021.
View at: Publisher Site | Google Scholar
A. R. Mishra, P. Rani, R. Krishankumar, K. S. Ravichandran, and S. Kar, “An extended fuzzy decision-making framework using hesitant fuzzy sets for the drug selection to treat the mild symptoms of coronavirus disease 2019 (COVID-19),” Applied Soft Computing, vol. 103, Article ID 107155, 2021.
View at: Publisher Site | Google Scholar
Z. Xu and R. R. Yager, “Some geometric aggregation operators based on intuitionistic fuzzy sets,” International Journal of General Systems, vol. 35, no. 4, pp. 417–433, 2006.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Muhammad Qiyas 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.

PDF Download Citation

Download other formats

Order printed copies

Views

532

Downloads

475

Citations