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

Authors: Muhammad Qiyas; Muhammad Naeem; Saifullah Khan; Saleem Abdullah; Thongchi Botmart; Tajammul Shah
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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 81174 confirmed cases and 3242 deaths in March 19,2020. 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 February26,2020, 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 10,2020 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. March2021

Pakistan had administered197,000 vaccine doses by March 5,2021. Nadhim Zahawi announced on March 6,2021, 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 measuresIn Section2, 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.


## 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 81174 confirmed cases and 3242 deaths in March 19,2020. 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 February26,2020, 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 10,2020 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. March2021

Pakistan had administered197,000 vaccine doses by March 5,2021. Nadhim Zahawi announced on March 6,2021, 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.


## 1.1.1. 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 February26,2020, 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 10,2020 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. March2021

Pakistan had administered197,000 vaccine doses by March 5,2021. Nadhim Zahawi announced on March 6,2021, 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 measuresIn Section2, 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 setS^≠ϕ, then(1)β=Ωβs^,γβs^,δβs^|s^∈S^,
is said to be PFS, andΩβs^,γβs^,δβs^∈0,1 are the positive, natural, and negative membership degree of the element s^∈S^ in β, respectively. Ωβs^,γβs^,δβs^ satisfying the following condition for all s^∈S^;(2)0≤Ωβs^+γβs^+δβs^≤1,for all s^∈S^, and the refusal degree is defined as Ξβ=1−Ωβs^−γβs^−δβs^.Definition 2 (see [40]).
LetS^ be nonempty fixed set and β=Ωβs^,ϑ˜βs^,ϒ˜βs^|s^∈S^ be the PFS. Then, the logarithmic PFNs is defined as(3)logτβ=1−logτΩβs^,logτ1−ϑ˜βs^,logτ1−ϒ˜βs^|s^∈S^0<τ<β≤1;τ≠1,where Ωβs^,ϑ˜βs^,ϒ˜βs^ are the positive, neutral, and negative membership degrees of the function s^∈S^ in β, respectively, with condition that 0≤Ωβs^,ϑ˜βs^,Υ˜βs^≤1. Then, the positive membership is defined as(4)1−logτΩβs^:S^⟶0,1,where1−logτΩβs^∈0,1,∀s^∈S^, and the neutral membership is defined as(5)logτ1−ϑ˜βs^:S^⟶0,1,wherelogτ1−ϑ˜βs^∈0,1,∀s^∈S^, and the negative membership is defined as(6)logτ1−ϒ˜βs^:S^⟶0,1,wherelogτ1−ϒ˜βs^∈0,1,∀s^∈S^, and the refusal degree is defined as(7)1−1−logτΩβs^+logτ1−ϑ˜βs^+logτ1−ϒ˜βs^∈0,1,∀s^∈S^.Definition 3.
(see [40]). Let logτβ1=1−logτΩ1s^,logτ1−ϑ˜1s^,logτ1−ϒ˜1s^, logτβ2=1−logτΩ2s^,logτ1−ϑ˜2s^,logτ1−ϒ˜2s^ be two PFNs. Then, the comparison rules are defined.(1)
IfClogτβ1>Clogτβ2, then logτβ1>logτβ2(2)
IfClogτβ1=Clogτβ2 and Alogτβ1>Alogτβ2, then logτβ1>logτβ2(3)
IfClogτβ1=Clogτβ2 and Alogτβ1=Alogτβ2, then logτβ1=logτβ2Definition 4.
(see [40]). Let logτβ=1−logτΩβs^,logτ1−ϑ˜s^,logτ1−ϒ˜s^,logτβ1=1−logτΩ1s^,logτ1−ϑ˜1s^,logτ1−ϒ˜1s^,logτβ2=1−logτΩ2s^,logτ1−ϑ˜2s^,logτ1−ϒ˜2s^ are three PFNs. Then,(1)
logτβc=logτ1−ϒ˜s^,logτ1−ϑ˜s^,1−logτΩs^(2)
logτβ1∧logτβ2=min1−logτΩ1s^,1−logτΩ2s^maxlogτ1−ϑ˜1s^,logτ1−ϑ˜2s^maxlogτ1−ϒ˜1s^,logτ1−ϒ˜2s^(3)
logτβ1∨logτβ2=max1−logτΩ1s^,1−logτΩ2s^minlogτ1−ϑ˜1s^,logτ1−ϑ˜2s^minlogτ1−ϒ˜1s^,logτ1−ϒ˜2s^(4)
logτβ1=logτβ2,if1−logτΩβ1s^=1−logτΩβ2s^,logτ1−ϒ˜1s^=logτ1−ϒ˜2s^,logτ1−ϑ˜1s^=logτ1−ϑ˜2s^


### 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 Table1.Table 1
Some score function of the existing methods.
Authors
Score function
Khan et al. [73]
C
log
τ
β
1
=
1
−
log
τ
Ω
1
s
^
−
log
τ
1
−
ϑ
˜
1
s
^
−
log
τ
1
−
ϒ
˜
1
s
^
,
C
log
τ
β
1
∈
−
1,1
Khan et al. [73]
S
log
τ
β
1
=
1
−
log
τ
Ω
1
s
^
2
−
log
τ
1
−
ϑ
˜
1
s
^
2
−
log
τ
1
−
ϒ
˜
1
s
^
2
Peng and Luo [74]
S
PH
p
=
Ω
+
1
/
2
ϑ
˜
−
ϒ
˜


#### 2.1.2. Existing Logarithmic Picture Fuzzy Accuracy Function

There is some existing accuracy function which is given in Table2.Table 2
Some accuracy function of the existing methods.
Authors
Accuracy function
Khan et al. [73]
C
log
τ
β
1
=
1
−
log
τ
Ω
1
s
^
+
log
τ
1
−
ϑ
˜
1
s
^
+
log
τ
1
−
ϒ
˜
1
s
^
,
C
log
τ
β
1
∈
0,1
Khan et al. [73]
A
log
τ
β
1
=
1
−
log
τ
Ω
1
s
^
2
+
log
τ
1
−
ϑ
˜
1
s
^
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.
Letlogτβ=1−logτΩβs^,logτ1−ϑ˜βs^,logτ1−ϒ˜βs^|s^∈S^, 0<τ<β≤1;τ≠1 be logarithmic picture fuzzy set. Then, the corresponding score function can be defined as(8)Sscologτβ=1−logτΩβ−logτ1−γβ+1+e1−logτΩβ−logτ1−ϑ˜βs^−logτ1−Υ˜βs^2−1−logτΩβ−logτ1−ϑ˜βs^−logτ1−ϒ˜βs^,Sscologτβ∈1e,2+e.Theorem 1.
Forlog-PFN, we have logτβ=1−logτΩβs^,logτ1−ϑ˜βs^,logτ1−ϒ˜βs^|s^∈S^, 0<τ<β≤1;τ≠1, as 1−logτΩβs^ increases, Sscologτβ monotonically increases and logτ1−ϑ˜βs^ or logτ1−ϒ˜βs^ increases, Sscologτβ monotonically decreases.Proof.
Using Definition5, we have first partial derivative that corresponds Sscoβ with respect to 1−logτΩβs^,(9)∂Sscologτβ∂1−logτΩβs^=1+e1−logτΩβ−logτ1−γβ−logτ1−δβ3−1−logτΩβ−logτ1−γβ−logτ1−δβ2−1−logτΩβ−logτ1−γβ−logτ1−δβ2≥0.
Similarly, we are to calculate first partial derivative ofSscoβ in relation to logτ1−ϒ˜βs^ and logτ1−ϑ˜βs^(10)∂Sscoβ∂logτ1−ϒ˜βs^=−1+e1−logτΩβ−logτ1−γβ−logτ1−δβ1−logτΩβ+logτ1−γβ+logτ1−δβ−12−1−logτΩβ−logτ1−γβ−logτ1−δβ2≤0,(11)∂Sscoβ∂logτ1−ϑ˜βs^=e1−logτΩβ−logτ1−γβ−logτ1−δβ1−logτΩβ+logτ1−γβ+logτ1−δβ−12−1−logτΩβ−logτ1−γβ−logτ1−δβ2≤0.
Hence, we proved that.Theorem 2.
Forlog-PFN, logτβ the corresponding score function Sscologτβ are the following relation.(1)
e−1Sscologτβ≤2+e(2)
Sscologτβ=2+e, iff logτβ=1,0,0(3)
Sscologτβ=e−1, iff logτβ=0,0,1Proof.
Now, using Theorem1, we are to consider 1−logτΩβs^, logτ1−ϑ˜βs^ or logτ1−ϒ˜βs^, Sscologτβ can present the minimum-value or maximum-value when logτβ=0,0,1 or logτβ=1,0,0, and in other word, Sscologτβ=e−1 or Sscologτβ=2+e, hence proved.


## 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 Table1.Table 1
Some score function of the existing methods.
Authors
Score function
Khan et al. [73]
C
log
τ
β
1
=
1
−
log
τ
Ω
1
s
^
−
log
τ
1
−
ϑ
˜
1
s
^
−
log
τ
1
−
ϒ
˜
1
s
^
,
C
log
τ
β
1
∈
−
1,1
Khan et al. [73]
S
log
τ
β
1
=
1
−
log
τ
Ω
1
s
^
2
−
log
τ
1
−
ϑ
˜
1
s
^
2
−
log
τ
1
−
ϒ
˜
1
s
^
2
Peng and Luo [74]
S
PH
p
=
Ω
+
1
/
2
ϑ
˜
−
ϒ
˜


### 2.1.2. Existing Logarithmic Picture Fuzzy Accuracy Function

There is some existing accuracy function which is given in Table2.Table 2
Some accuracy function of the existing methods.
Authors
Accuracy function
Khan et al. [73]
C
log
τ
β
1
=
1
−
log
τ
Ω
1
s
^
+
log
τ
1
−
ϑ
˜
1
s
^
+
log
τ
1
−
ϒ
˜
1
s
^
,
C
log
τ
β
1
∈
0,1
Khan et al. [73]
A
log
τ
β
1
=
1
−
log
τ
Ω
1
s
^
2
+
log
τ
1
−
ϑ
˜
1
s
^
2


## 2.1.1. Existing Logarithmic Picture Fuzzy Score Function

Some existing score functions are displayed in a Table1.Table 1
Some score function of the existing methods.
Authors
Score function
Khan et al. [73]
C
log
τ
β
1
=
1
−
log
τ
Ω
1
s
^
−
log
τ
1
−
ϑ
˜
1
s
^
−
log
τ
1
−
ϒ
˜
1
s
^
,
C
log
τ
β
1
∈
−
1,1
Khan et al. [73]
S
log
τ
β
1
=
1
−
log
τ
Ω
1
s
^
2
−
log
τ
1
−
ϑ
˜
1
s
^
2
−
log
τ
1
−
ϒ
˜
1
s
^
2
Peng and Luo [74]
S
PH
p
=
Ω
+
1
/
2
ϑ
˜
−
ϒ
˜


## 2.1.2. Existing Logarithmic Picture Fuzzy Accuracy Function

There is some existing accuracy function which is given in Table2.Table 2
Some accuracy function of the existing methods.
Authors
Accuracy function
Khan et al. [73]
C
log
τ
β
1
=
1
−
log
τ
Ω
1
s
^
+
log
τ
1
−
ϑ
˜
1
s
^
+
log
τ
1
−
ϒ
˜
1
s
^
,
C
log
τ
β
1
∈
0,1
Khan et al. [73]
A
log
τ
β
1
=
1
−
log
τ
Ω
1
s
^
2
+
log
τ
1
−
ϑ
˜
1
s
^
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.
Letlogτβ=1−logτΩβs^,logτ1−ϑ˜βs^,logτ1−ϒ˜βs^|s^∈S^, 0<τ<β≤1;τ≠1 be logarithmic picture fuzzy set. Then, the corresponding score function can be defined as(8)Sscologτβ=1−logτΩβ−logτ1−γβ+1+e1−logτΩβ−logτ1−ϑ˜βs^−logτ1−Υ˜βs^2−1−logτΩβ−logτ1−ϑ˜βs^−logτ1−ϒ˜βs^,Sscologτβ∈1e,2+e.Theorem 1.
Forlog-PFN, we have logτβ=1−logτΩβs^,logτ1−ϑ˜βs^,logτ1−ϒ˜βs^|s^∈S^, 0<τ<β≤1;τ≠1, as 1−logτΩβs^ increases, Sscologτβ monotonically increases and logτ1−ϑ˜βs^ or logτ1−ϒ˜βs^ increases, Sscologτβ monotonically decreases.Proof.
Using Definition5, we have first partial derivative that corresponds Sscoβ with respect to 1−logτΩβs^,(9)∂Sscologτβ∂1−logτΩβs^=1+e1−logτΩβ−logτ1−γβ−logτ1−δβ3−1−logτΩβ−logτ1−γβ−logτ1−δβ2−1−logτΩβ−logτ1−γβ−logτ1−δβ2≥0.
Similarly, we are to calculate first partial derivative ofSscoβ in relation to logτ1−ϒ˜βs^ and logτ1−ϑ˜βs^(10)∂Sscoβ∂logτ1−ϒ˜βs^=−1+e1−logτΩβ−logτ1−γβ−logτ1−δβ1−logτΩβ+logτ1−γβ+logτ1−δβ−12−1−logτΩβ−logτ1−γβ−logτ1−δβ2≤0,(11)∂Sscoβ∂logτ1−ϑ˜βs^=e1−logτΩβ−logτ1−γβ−logτ1−δβ1−logτΩβ+logτ1−γβ+logτ1−δβ−12−1−logτΩβ−logτ1−γβ−logτ1−δβ2≤0.
Hence, we proved that.Theorem 2.
Forlog-PFN, logτβ the corresponding score function Sscologτβ are the following relation.(1)
e−1Sscologτβ≤2+e(2)
Sscologτβ=2+e, iff logτβ=1,0,0(3)
Sscologτβ=e−1, iff logτβ=0,0,1Proof.
Now, using Theorem1, we are to consider 1−logτΩβs^, logτ1−ϑ˜βs^ or logτ1−ϒ˜βs^, Sscologτβ can present the minimum-value or maximum-value when logτβ=0,0,1 or logτβ=1,0,0, and in other word, Sscologτβ=e−1 or Sscologτβ=2+e, hence proved.


## 3. New Logarithmic Picture Fuzzy MCDM Method

Consider thatS=S1,S2,…,Sn be a series of alternatives M=M1,M2,…,Mn be a set of distinct criteria, and ω=ω1,ω2,…,ωn weight vector with ωj∈0,1,∑j=1nωj=1. Assume that the alternative evaluation Si in terms of criteria Mj is represented by the picture fuzzy (LPF) matrix P=pijm×n=1−logτΩβs^ij,logτ1−ϑ˜βs^ij,logτ1−ϒ˜βs^ijm×n, which is shown in Table 3.Table 3
The picture fuzzy MCDM matrix.
M
1
M
2
⋯
M
n
S
1
a
11
,
b
11
,
c
11
a
12
,
b
12
,
c
12
⋯
a
1
n
,
b
1
n
,
c
1
n
S
2
a
21
,
b
21
,
c
21
a
22
,
b
22
,
c
22
⋯
a
2
n
,
b
2
n
,
c
2
n
⋮
⋮
⋮
⋱
⋮
S
n
a
m
1
,
b
m
1
,
c
m
1
a
m
2
,
b
m
2
,
c
m
2
⋯
a
m
n
,
b
m
n
,
c
m
nLeta=1−logτΩβs^,b=logτ1−ϑ˜βs^, and c=logτ1−ϒ˜βs^


### 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 matrixP=pijm×nStep 2. By using equation (12), normalize the picture fuzzy matrix P=pijm×n of P′=Pij′m×n(12)Pij′=aij,bij,cij,if benefit type criteria,cij,bij,aij,if cost type criteria.Step 3. Determine the score function’s valuesR=rijm×n of P′=Pij′m×n usingStep 4. Calculate theEj for matrix R of a score function using(13)Ej=11−αlog∑i=1mrijα.Step 5. Obtain the objective weightωj using(14)ωj=1−Ej∑j=1n1−Ej.Step 6. Evaluate the combined weightϖj using(15)ϖj=wjϖj∑j=1nwjϖj.Step 7. Evaluate the comparability sequences of the weight for each alternativeSi as(16)Si=∑j=1nωjrij′.Step 8. For each alternative, compute the comparability sequence for the power weightPi as(17)Pi=∑j=1nrij′Wj.Step 9. Using the aggregation method given in equations (18)–(20)(18)Tia=Pi+Si∑i=1mPi+Si,(19)Tib=SiminiSi+PiminiPi,(20)Tic=λSi+1−λPiλmaxiSi+1−λmaxiPi,0≤λ≤1.Step 10. Calculate the score valueTi using(21)Ti=TiaTibTic1/3+Tia+Tib+Tic3.Step 11. Alternatives are rated in order of decreasing assessed score valuesTii=1,…,m.


## 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 matrixP=pijm×nStep 2. By using equation (12), normalize the picture fuzzy matrix P=pijm×n of P′=Pij′m×n(12)Pij′=aij,bij,cij,if benefit type criteria,cij,bij,aij,if cost type criteria.Step 3. Determine the score function’s valuesR=rijm×n of P′=Pij′m×n usingStep 4. Calculate theEj for matrix R of a score function using(13)Ej=11−αlog∑i=1mrijα.Step 5. Obtain the objective weightωj using(14)ωj=1−Ej∑j=1n1−Ej.Step 6. Evaluate the combined weightϖj using(15)ϖj=wjϖj∑j=1nwjϖj.Step 7. Evaluate the comparability sequences of the weight for each alternativeSi as(16)Si=∑j=1nωjrij′.Step 8. For each alternative, compute the comparability sequence for the power weightPi as(17)Pi=∑j=1nrij′Wj.Step 9. Using the aggregation method given in equations (18)–(20)(18)Tia=Pi+Si∑i=1mPi+Si,(19)Tib=SiminiSi+PiminiPi,(20)Tic=λSi+1−λPiλmaxiSi+1−λmaxiPi,0≤λ≤1.Step 10. Calculate the score valueTi using(21)Ti=TiaTibTic1/3+Tia+Tib+Tic3.Step 11. Alternatives are rated in order of decreasing assessed score valuesTii=1,…,m.


## 4. Example

According to a study released in February10,2021, there have been estimated 62,570,316 COVID-19 cases have been recorded all over the world, resulting in 1,466,426 deaths. A total of 44,671,725 people have been rescued. According to research, a large number of people with COVID-19 will show the symptoms and indications listed below; flu 83%–99%, vomiting 58%–81%, weakness 45%–70%, alcoholism 41%–83%, nausea 31%–40%, phlegm development 27%–33%, and dysphagia 12%–35%. Here, we prefer five alternatives to medications for COVID-19 patients that must be kept under control, in particular, LVPV/RTV-IFNb M1, favipiravir M2, LPV/RTV M3, danger M4, and clobazam M5. Also, we prefer seven signs as alternatives anorexia S1, cough S2, fatigue S3, fever S4, myalgia S5, shortness of breath S6, sputum production S7.Step 1. The decision matrixP=pij5×7 is given in Table 4Step 2. Since all the attributes are benefit types, there is no need to convertStep 3. Using equation (8), to obtain the score function’s values R=rijm×n of P′=Pij′m×n are given in Table 5Step 4. Using equation (13), to obtain Ej for a matrix R of a score function(22)E1=0.72835,E2=0.29797,E3=0.335161,E4=0.11675,E5=0.63075.Step 5. Using equation (14), to obtain the combined weight ωjj=1,…,5(23)ω1=0.372643,ω2=0.144633,ω3=0.161038,ω4=0.0134843,ω5=0.308199.Step 6. Determine the entire weighted comparability sequencesii=1,…,7 as(24)s1=2.646600,s2=2.740795,s3=2.842651,s4=2.753345,s5=2.962960,s6=2.684800,s7=2.656870.Step 7. Determine the total number of power-weighted comparable sequencesPii=1,…,7 as(25)P1=6.10763,P2=6.11257,P3=6.13080,P4=6.15539,P5=6.21459,P6=6.11583,P7=6.11832.Step 8. Obtained the three aggregation strategiesTia,Tib, and Tici=1,…,7 as(26)T1a=0.140645,T2a=0.142238,T3a=0.144167,T4a=0.143127,T5a=0.147446,T6a=0.141391,T7a=0.140982,T1b=2.0000,T2b=2.0364,T3b=2.0778,T4b=2.0481,T5b=2.13704,T6b=2.01577,T7b=2.00563,T1c=0.97828,T2c=0.98063,T3c=0.98515,T4c=0.98739,T5c=1.0000,T6c=0.98018,T7c=0.98009.Step 9. Obtained the corresponding assessment valueTii=1,…,7 as(27)T1=1.69008,T2=1.71044,T3=1.73484,T4=1.72105,T5=1.77531,T6=1.69950,T7=1.69420.Step 10. Give raking according to the score values as(28)S5≻S3≻S4≻S2≻S6≻S7≻S1.Table 4
Picture fuzzy information given by experts.
M
1
M
2
M
3
M
4
M
5
S
1
0.4
,
0.1
,
0.2
0.3
,
0.1
,
0.1
0.4
,
0.1
,
0.3
0.5
,
0.1
,
0.2
0.34
,
0.4
,
0.21
S
2
0.7
,
0.17
,
0.01
0.2
,
0.1
,
0.1
0.4
,
0.05
,
0.3
0.6
,
0.02
,
0.3
0.2
,
0.7
,
0.1
S
3
0.3
,
0.2
,
0.3
0.5
,
0.2
,
0.09
0.9
,
0.03
,
0.01
0.7
,
0.01
,
0.1
0.3
,
0.06
,
0.2
S
4
0.4
,
0.21
,
0.29
0.53
,
0.2
,
0.2
0.4
,
0.06
,
0.2
0.4
,
0.5
,
0.03
0.2
,
0.7
,
0.03
S
5
0.5
,
0.2
,
0.2
0.4
,
0.3
,
0.1
0.8
,
0.09
,
0.02
0.6
,
0.03
,
0.01
0.5
,
0.1
,
0.2
S
6
0.6
,
0.1
,
0.2
0.3
,
0.6
,
0.1
0.2
,
0.7
,
0.03
0.5
,
0.06
,
0.2
0.1
,
0.04
,
0.03
S
7
0.3
,
0.01
,
0.1
0.2
,
0.09
,
0.1
0.3
,
0.07
,
0.2
0.4
,
0.07
,
0.02
0.5
,
0.05
,
0.2Table 5
Score values of the given matrix.
M
1
M
2
M
3
M
4
M
5
S
1
2.7663
2.4587
2.6949
3.1078
2.5042
S
2
3.8460
2.0223
2.6999
3.3703
1.9322
S
3
1.42077
2.3077
3.1647
4.4311
3.8033
S
4
2.3977
2.6916
3.1980
2.7711
2.8192
S
5
1.9676
2.8040
4.1358
3.5499
3.1075
S
6
3.4304
2.3893
1.9676
3.1116
2.4790
S
7
2.4719
2.0236
2.3963
2.8847
3.1127


## 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 S5 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 S5 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.Table 6
Ranking of the proposed and existing methods.
Method
Ranking
Proposed method
S
5
≻
S
3
≻
S
4
≻
S
2
≻
S
6
≻
S
7
≻
S
1
Peng and Luo [74]
S
5
≻
S
4
≻
S
3
≻
S
1
≻
S
7
≻
S
2
≻
S
6
Ashraf et al. [42]
S
5
≻
S
4
≻
S
3
≻
S
1
≻
S
2
≻
S
7
≻
S
6
Garg [40]
S
5
≻
S
2
≻
S
4
≻
S
3
≻
S
7
≻
S
1
≻
S
7
Jana and Pal [43]
S
5
≻
S
1
≻
S
6
≻
S
3
≻
S
2
≻
S
4
≻
S
7
Joshi and Kumar [41]
S
5
≻
S
6
≻
S
2
≻
S
7
≻
S
3
≻
S
1
≻
S
4
Ju et al. [44]
S
5
≻
S
3
≻
S
7
≻
S
4
≻
S
2
≻
S
6
≻
S
1
Khan et al. [45]
S
5
≻
S
4
≻
S
3
≻
S
2
≻
S
1
≻
S
7
≻
S
6As a result of the above analysis, it is concluded that it is the most suitable and desirable alternativeS5. 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.



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*Source: 1476233-2022-07-08.xml*