key: cord-0065230-8cicl04g
authors: Rehman, Noor; Ali, Abbas; Liu, Peide; Hila, Kostaq
title: A comprehensive study of upward fuzzy preference relation based fuzzy rough set models: Properties and applications in treatment of coronavirus disease
date: 2021-06-03
journal: nan
DOI: 10.1002/int.22433
sha: e58e791a9f512f2332065f25c757661e87ca7a5c
doc_id: 65230
cord_uid: 8cicl04g

In this paper, we first introduce a new type of rough sets called [Formula: see text] ‐upward fuzzified preference rodownward fuzzy preferenceugh sets using upward fuzy preference relation. Thereafter on the basis of [Formula: see text] ‐upward fuzzified preference rough sets, we propose approximate precision, rough degree, approximate quality and their mutual relationships. Furthermore, we presented the idea of new types of fuzzy upward [Formula: see text] ‐coverings, fuzzy upward [Formula: see text] ‐neighborhoods and fuzzy upward complement [Formula: see text] ‐neighborhoods and some relavent properties are discussed. Hereby, we formulate a new type of upward lower and upward upper approximations by applying an upward [Formula: see text] ‐neighborhoods. After employing the upward [Formula: see text] ‐neighborhoods based upward rough set approach to it any times, we can only get the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Subsequently, we presented the idea to combine the fuzzy implicator and [Formula: see text] ‐norm to introduce multigranulation [Formula: see text] ‐fuzzy upward rough set applying fuzzy upward [Formula: see text] ‐covering and some relative properties are discussed. Finally we presented a new technique for the selection of medicine for treatment of coronavirus disease (COVID‐19) using multigranulation [Formula: see text] ‐fuzzy upward rough sets.

Multiattribute group decision-making problems has always been an important direction of modern decision making sciences. Based on multiattribute group decision making systems with different natures, researchers not only developed a wealth of decision-making scheme, but also solved various practical problems such as Kreyea et al. 1 developed a new approach of group decision making problem to manage their application in logistics, Mou et al. 2 introduced group decision-making technique based on graph approach under the intuitionistic fuzzy (fuzzy preference relation) information and applied to energy related problem. Ishizaka and Nemery 3 introduced a new approach to multicriteria decision analysis and discussed their application in safety management. Alcantud 4 developed a group decision making technique to handle a problem related to facility location. Inan et al. 5 initiated a multiattribute group decision approach for the comparison of firms occupational health and safety management. Ishizaka and Nemery 6 put forward a new idea of decision making technique to solve a problems related to supplier selection. Aldape-Perez et al. 7 defined a novel approach to perform pattern classification tasks for medical decision support systems. Arsene et al. 8 applied an expert system under the framework of software agents for medicine diagnosis. Azar and Metwally 9 presented a decision tree classifier for automated medical diagnosis problem, Esfandiari et al. 10 presented data mining application in medicine. The whole of the world is engulfed with the spread of COVID-19 and it is very painstakingly difficult for the denizens of the world to live a peaceful life. The epidemic is viral and the attack is so severe that the World Health Organization (WHO) is compelled to announce a global emergency. In the last quarter of 2019, some cases of the disease were reported in Wuhan city China, which after the diagnosis was found as coronavirus . In the wake of this incident, the virus is circulated in the entire world and became the sole cause of the demise of thousands and thousands of people in the whole of the world. The word "coronavirus" is derived from the Latin word "corona" which means a "crown i.e., a circle of light or nimbus." The virus promptly affects the lungs. It has similar symptoms as those of influenza and pneumonia. In the very outset, it was found that the people who worked or shopped at the seafood market in Wuhan became the victims of this virus. After that it pervaded universally through import, export, travelling, and social contacting with infected people. Several researchers investigated and developed different methods to address the problem. In decision making, there have been a lot of uncertainties, imprecise, and vague information, whose representation and management are always the central issues. Health professionals and healthcare administrators are working to reduce clinical and maintenance costs for the prevention and management of disease. It is concluded that the coronavirus is a burning issue and needs mathematical formulation/technique for selection of medicine for treatment of the disease.

The main objective of this study is to establish decision making with the help of fuzzy rough set theory. We developed a technique for the selection of medicine to treat the coronavirus disease (COVID-19) using fuzzy rough sets.

The theory of rough sets introduced by Pawlak 11 provides a new mathematical approach to extend classical set theory. In rough set theory, creating a pair of approximation operators called lower and upper approximation operators is important. The classical approximation operators are described with the help of an equivalent relation in the universe. Rough set methodology is the fundamental method of solving uncertain knowledge and their application in various fields such as, the field of expert systems, pattern recognition, image processing, decision analysis, artificial intelligence, and so on. Accordingly, to enhance the utilization rate of information in information systems, several authors have extended the definition of rough set approximation through applying general relations, such as Bonikowski introduced algebraic structures of rough sets, 12 Liu and Zhu 13 further generalized the work of Bonikowski by introducing algebraic structures of generalized rough sets, Slowinski and Stefanowski 14 discussed the solution of medical information systems problems based on rough fuzzy hybridized structure. Dubois and Prade 15 combined the notion of rough sets and fuzzy sets to form fuzzy rough sets and rough fuzzy sets. Then the upper and lower approximation operators of approximate space in fuzzy environment are widely used. 16 After the formation of rough set, 11 number of generalizations have been presented in terms of various demands. Dubois and Prade 15 initiated the idea of rough set theory based on set valued mapping. Zhu 17 integrated the idea of generalized rough sets using general relation. Yao and Yao 18 initiated the idea of rough sets based on covering approach and their applications. Zhu 19 presented the idea of covering rough sets based on topological properties and their applications. Zhu and Wang 20 studied covering rough sets based on reduction of attributes and their applications. Further, Zhu and Wang discussed three different types of rough sets based on covering and their mutual relationships. 21 Based on topological approach, Zhao 22 introduced various types of rough sets based on covering and their applications. Deng and Yao 23 studied fuzzy environment based three way approximation with decision theoretic fuzzy rough sets and applied to group decision making problems. Sun et al. 24 integrated the idea of rough fuzzy sets based on decision theoretic approach and applied to multiattribute group decision-making problems. Ziarko 25 introduced another rough set model called variable precision rough sets model, which is the generalized form of Pawlak's rough set model. Yao 26 initiated three way decisions analysis with the help of probabilistic rough sets and discussed their applications in multiattribute group decision making problem. Greco et al. [27] [28] [29] initiated rough approximations based on dominance relations and their applications in various multicriteria group decision making problems. Qian et al. 30, 31 generalized the Pawlak's single granulation rough set model to a multigranulation rough set model for finding two terminologies called optimistic/pessimistic multigranulation rough set models and disclosed their applications in decision making process. Qian et al. 32 further extended multigranulation methodology to decision theoretic rough sets and applied them to multicriteria group decision making problem. Lin et al. 33 initiated covering based multigranulation rough sets and applied them in decision making problems. Ali et al. 34 REHMAN ET AL. | 3 originated multigranulation rough sets approaches based on dominance relations and their application in labor management negotiations in conflict analysis problems. Rehman et al. 35 applied soft preference relation for the construction of soft optimistic/soft pessimistic multigranulation rough sets and presented their application in conflict problems. Different researchers have confabulated the applications of rough sets in medical sciences such as, Cheng and Liu 36 argued wavelet packet based rough set technique for Identify brain disease.

The original definition of a fuzzy covering is given in Reference [37] . Let q q q = { , , } 

denotes the efficiency of the medicine q j for the test i  . Ma defined fuzzy β-covering which is the generalized form of fuzzy covering to replace 1 by a parameter β (0, 1] ∈ . 38 Subsequently, Ma 38 presented two new types of rough set models based on fuzzy covering by applying the concept of fuzzy β-neighborhood. Further, Ma 38 defined two types of rough set models based on fuzzy covering and presented their applications in fuzzy lattice theory. Yang and Hu 39 initiated various types of rough set models using fuzzy covering approach and applied them to medical diagnosis problems. Zhan et al. 40 combined the fuzzy implicator and t-norm to introduce multigranulation ( , )   -fuzzy rough set models based on covering and further studied their application in assessment of appointing a system analysis engineer. Based on fuzzy implicator and t-norm, Jiang et al. 41 presented variable precision ( , )   -fuzzy rough sets using covering methodology and presented their application to supplier selection problems. Zhang et al. 42 integrated the idea of fuzzy rough sets applying fuzzy soft covering methodology and discussed their applications to select athletes for table tennis team. Zhang and Zhan, 43 introduced a new hybridized structure called fuzzy rough sets using fuzzy soft β-covering model and its application in decision making problem.

There are some limitation, for instance, the above example is fuzzy β -covering model for β (0, 0.4] ∈ . If the required critical value β = 0.5, then how is it possible to make β-covering model for β (0, 0.5] ? ∈ Hu et al. 44 adopted the well-known logis transfer function to compute the fuzzy preference degree of the feasible alternatives. Pan et al. 45 pointed out that the transfer fuzzy preference degree of Hu et al. is not additive consistent and suggested another transfer function. The motive of this paper is first to point out that the transfer function for computing the fuzzy preference degree of Pan et al. 45 for the construction of upward/downward fuzzy preference relations are not additive consistent. The appropriate counterexample is given and their modified versions are presented. Furthermore, we construct upward consistency matrices of experts which satisfy the upward additive consistency and the upward order consistency simultaneously. Subsequently, we introduced a new type of rough sets called α-upward fuzzified preference rough sets using upward fuzzy preference relation. On the basis of α-upward fuzzified preference rough sets, we introduced approximate precision, rough degree, approximate quality and their mutual relationships. Furthermore, we presented the idea of new types of fuzzy upward β-coverings, fuzzy upward β-neighborhoods and fuzzy upward complement β-neighborhoods and related properties are discussed. Hereby, we propose a new type of upward lower and upward upper approximations by employing an upward β-neighborhoods. It is worth mentioning by applying an upward β-neighborhoods based upward rough set approach to it any times, we can only get the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Afterwords, we presented the idea to combine the fuzzy implicator and t-norm to introduce multigranulation ( , )   -fuzzy upward rough set applying fuzzy upward β-covering and some related properties are discussed. Finally we presented a new technique for the selection of medicine to treat the coronavirus disease (COVID-19) using multigranulation ( , )   -fuzzy upward rough sets.

The remainder of this manuscript is as follows: Section 2 recalls preliminary notions concerning fuzzy preference relation, fuzzy additive consistency and logis sigmoid transfer function. In Section 3, we construct upward/downward fuzzy preference relations which are additive consistent. Furthermore, we construct upward consistency matrices of experts which satisfy the upward additive consistency and the upward order consistency. Section 4, introduce a new type of roughness called α-upward fuzzified preference rough sets using upward fuzzy preference relation. A new type of upward lower and upward upper approximations by applying an upward β-neighborhoods and after employing an upward β-neighborhoods based upward rough sets approach to it any times, we get the six different sets at most are discussed in Section 5. In Section 6, we presented the idea to combine the fuzzy implicator and t-norm to introduce optimistic/pessimistic multigranulation ( , )   -fuzzy upward rough sets using fuzzy upward β-covering approach. Section 7, confabulated the algorithm to handle the uncertainty problems using multigranulation ( , )   -fuzzy upward rough sets. Section 8 highlights the applications of the proposed model in prescription of medicine for treatment of coronavirus disease . In Section 9, we focus our attention on comparison of various models with the proposed technique. The paper is concluded in Section 10.

In this section, some basic notations of fuzzy preference relation, fuzzy additive consistency and logis sigmoid transfer function have been discussed. 

· · · · · · · · · · · · · · · · · · · · · · · · · · · ij n n n n n n n n nn 

. Especially, f = 0.5 ij means that the behavior of q i and q j are same in fuzzy information system; f > 0.5 ij shows that the behavior of q i is better than the behavior of q f ; = 1 j ij means that the behavior of q i is absolutely better than the behavior of q j ; the f < 0.5 ij depicts that the behavior of q j is better than the behavior of q f ; = 0 i ij means that the behavior of q j is absolutely better than the behavior of q i .

In the above definition, the fuzzy preference relation is considered, f ij merely presents that the degree of preference of feasible alternative q i is before the feasible alternative q j . However, in practical applications, we need to show the degree of q i is poorer than q j . To satisfy both the cases, we call the fuzzy preference relation as an upward fuzzy preference relation and the other downward fuzzy preference relation. The upward fuzzy preference relation is denoted as

and downward fuzzy preference relation as

. For downward fuzzy preference relation, f = 0.5 ij ⇓ means that the behavior of q i and q j are same in fuzzy

depicts that the behavior of q i is poorer than the behavior of q j ; f = 1 ij ⇓ means the behavior of q i is absolutely poorer than the behavior of q f ; < 0.5 j ij ⇓ shows that the behavior of q j is poorer than the behavior of q f ; = 0 i ij ⇓ means that behavior of q j is absolutely poor than the behavior of q i .

Hu et al. 44 used the well-known logis sigmoid transfer function e 1 1 + k g q i c g q j c ( ( , )+ ( , )) for the construction of fuzzy preference degree of the feasible alternative q i to the feasible alternative q j as

where k is a positive constant. Pan et al. 45 showed that the fuzzy preference degree based on logis sigmoid transfer function is not additive consistent and they suggested another transfer function to compute the fuzzy preference degree. The fuzzy preference degree of q i to q j is given by Based on the initial record (birth information, weight, age, etc. of the babies), of the babies birth's taken from various hospitals, we have the following example. Let A q j = { : = 1, 2, …, 9} j be the set of (feasible alternatives) babies born in a same minute in the world, C c c c = { , , } 1 2 3 be the set criteria, where c 1 , shows the weight of the babies, c 2 , represent the ages of babies in the same minute and c 3 shows baby birth on normal delivery. In this study, the value of information function g q c ( , ) j i is belonging to [0, 1] and g q c ( , ) j i describe the fuzzy evaluation of q j on criterion c i . The information system is given in the following Table 1 .

Based on criterion c 1 and using Equation (1) , to calculate the upward fuzzy preference degree of q i i ( = 1, 2, …, 9) to q j j ( = 1, 2, …, 9) , one can acquire 

Based on criterion c 1 and using Equation (5) , to calculate the upward fuzzy preference degree of q i i ( = 1, 2, …, 9) to q j j ( = 1, 2, …, 9) , one can acquire 

Further based on criterion c 2 and using Equation (5) , to calculate the upward fuzzy preference degree of q i i ( = 1, 2, …, 9) to q j j ( = 1, 2, …, 9) , one can get q q ( , ) = 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000

Finally based on criterion c 3 and using Equation (5) , to calculate the upward fuzzy preference degree of q i i ( = 1, 2, …, 9) to q j j ( = 1, 2, …, 9) , one can acquire q q ( , ) = 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 0.50000 In real world problems, the information system depends on the behavior of the decision maker(s), so such type of attributes (criteria) like c 2 and/or c 3 may or may not exist. If attributes (criteria) like c 2 and/or c 3 are exist, then we do not have any technique to handle the situation. Existence of criterion like c 2 or c 3 in a fuzzy information system has its own importance. Regarding the reducing a criterion from the information system by a decision maker using his/her own technique arises the question that whether the technique works on this criterion or not? If c 2 or c 3 to be reduced from the information system to apply Pan et al. Further we prove that our constructed upward fuzzy preference relation is upward fuzzy additive consistent we can get 

ij jk g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c g q c

be the upward fuzzy preference relation which satisfy the upward additive consistency condition. Then based on  ⇑ , we can derive the upward additive consistency matrix for  ⇑ which satisfies the additive consistency, where

satisfies the upward additive consistency condition and the upward order consistency condition as follows:

(

Proof. The proof is straightforward. □

Preference relations are very useful in expressing decision maker's preference information in ordinal decision problems. Fuzzy preference relation is first proposed by Orlovsky (1978) to represent an expert's opinion about a set of alternatives. The fuzzy preference relation not only can reflect that one alternative is before another alternative, but also can show the preference degree. The Pawlak's rough set model and fuzzy rough set are not able to receive and extract the information of ordinal structure and cannot be used to analyze the information with preference relations. Pawlak discussed this problem in Reference [47] . Greco et al. proposed a novel rough set model for preference analysis and constructed dominance relation based on the decision preference. [27] [28] [29] Hu et al. proposed a type of fuzzy preference relation rough sets model in Reference [44] . Hu et al. 44 adopted the well-known logis transfer function to compute the fuzzy preference degree of the feasible alternatives for the construction of fuzzy preference relation and proposed a new type of fuzzy preference relation rough set model. Pan et al. 45 pointed out that the transfer fuzzy preference degree of Hu et al. is not additive consistent and they suggested another transfer function to modify the fuzzy preference relation rough set model of Hu et al. As mentioned earlier that the transfer function for computing the fuzzy preference degree in Pan et al. 45 for the construction of upward/downward fuzzy preference relations are not additive consistent. Less effort has been made to explore the structure of these fuzzy preference relation rough sets using fuzzy preference relations. In current literature the researchers utilized the idea of fuzzy preference relation to find the fuzzy approximations. However the scholars were unable/incapable to find the crisp approximations with the help of fuzzy preference relations. Naturally question arises that whether we can find the crisp approximations with the help of fuzzy preference relations? The affirmative answer to this question has led the present authors to the introduction of α-upward fuzzified preference rough sets. Moreover, the approximation defined based on α-upward fuzzified preference rough sets play a bridging role between fuzzy preference relation and crisp set. Furthermore, the approximation defined based on α-upward fuzzified preference rough sets are useful in different uncertainties such as approximate precision, rough degree, and approximate quality and their mutual relationships. Similarly the very same concept can be applied for linguistic/ordinal information systems. 

where "+" means the union operation. The upward fuzzy preference relation and downward fuzzy preference relation from a family of fuzzy information granules from the universe which composes the upward fuzzy preference granular structure and downward fuzzy preference granular structure given by

where  is an arbitrary universe and ( )   ⇑ an upward additive fuzzy preference granular structure. For any α [0.5, 1) ∈ , the upward fuzzified preference lower and upper approximations for a given set 1   ⊆ are defined as

is referred as an α-upward fuzzified preference rough set. The positive, negative and boundary regions of 1   ⊆ for any α [0.5, 1) ∈ are defined and denoted as:

Proof.

Proof.

(i) and (ii) straightforward.

On the other hand, for any q (( ( ) ) )

Similarly we can get (

The proof process of (vi) and (vii) is similar to the proof of (iv).

(viii) By using (iv), we can write ( A ( ) ) ( A ( ) )

The proof process is similar to the proof of (viii). □ Definition 5. Let ( , ( ))    ⇑ be an upward fuzzy preference approximation space and

where 1  is a nonempty subset of  and |⋅| denotes the cardinality of a set. Let

The following theorem describes the relationship of the rough degree μ ( )

for the union and intersection of subsets 1  and 1  of the universe  .

Theorem 5. Let ( , ( ))    ⇑ be an upward fuzzy preference approximation space and α [0.5, 1) ∈ . Then the approximate precision and rough degree of the subsets , ,

  ∩ of the universe  satisfy the following relations.

The following theorem describes the relationship of the rough degree μ ( )

for the union and intersection of subsets 1  and 1  of the universe  .

Theorem 6. Let ( , ( ))    ⇑ be an upward fuzzy preference approximation space and α [0.5, 1) ∈ . Then the rough degree and approximate quality for all subsets , ,

of the universe  satisfy the following relation:

Proof. The proof is straightforward. □

The following theorem highlights the relationship between approximate precision and approximate quality for the union and intersection of two sets. Theorem 7. Let ( , ( ))    ⇑ be an upward fuzzy preference approximation space and α [0.5, 1) ∈ . Then the approximate quality and approximate precision for all subsets , 1 1   , 1 1   ∪ and 1 1   ∩ of the universe  satisfy the following relation:

Proof. The proof is straightforward. □

In this section, we first find upward β-neighborhood in the fuzzy upward covering approximation space and then present upward rough sets and discussed some of their properties.

Definition 7. Let  be an arbitrary universal set and ( )   ⇑ be an upward additive fuzzy preference granular structure. Then for each β (0, 1] ∈ , ( )   ⇑ is a fuzzy upward β-

The pair ( , ( ))    ⇑ is called fuzzy upward covering approximation space. 

Proof. Let q  ∈ . Then it follows that

Proof. For

Let ( , ( ))    ⇑ be a fuzzy upward covering approximation space, where ( )   ⇑ is a fuzzy upward β-covering of  for some β (0, 1] ∈ . Then for each q  ∈ , we define the upward β-neighborhood N q β ⇑ of q as: 

Otherwise, the pair (( A ( )) , ( ( )) )

Theorem 8. Let ( , ( ))    ⇑ be a fuzzy upward covering approximation space, where ( )   ⇑ is a fuzzy upward β-covering of  for some β (0, 1] ∈ . Then

In classical rough set theory, if the lower approximation R ( ) 1  and upper approximation Also N q q q q q q q q q N q q q q q q N q q N q q q q q q q q N q q q q q q q q N q q N q q q q q q N q q q q q q N q q q q q q .

From the above example we see that ( A ( )) 

( A ( ( ))) , ( ( A ( ( )))) .

Theorem 9. Let ( , ( ))    ⇑ be a fuzzy upward covering approximation space, where ( )   ⇑ is a fuzzy upward β-covering of  for some β (0, 1] ∈ . For any crisp subset 1  of  , the following properties hold:

Proof.

(i) As we know that ( A ( ))

⇑ . This implies that ( ( A ( ( A ( )))))

Therefore ( ( A ( ( A ( ))))) = ( ( A ( )))

Moreover, it follows from that ( A ( ( ))) ( ( A ( ( ))))

Therefore ( A ( ( A ( ( ))))) = ( A ( ( )))

The following theorem gives the relationship between the aforesaid six sets. 

Qian et al. generalized the Pawlak's single granulation rough set model to a multigranulation rough set model for finding two terminologies called optimistic/pessimistic multigranulation rough set models and discussed their applications in decision making process. 30, 31 Qian et al. 32 further extended multigranulation methodology to decision theoretic rough sets and applied them to multicriteria group decision making problem. In this section, we presented the idea to combine the fuzzy implicator and t-norm to introduce optimistic/pessimistic multigranulation ( , )   -fuzzy upward rough sets using fuzzy upward β-covering approach and some relative properties are discussed. 

Definition 13. Let ( , ( ))    ⇑ be a fuzzy upward covering approximation space and

for all t l = 1, 2, …, . Assume that 

  -fuzzy upward rough set of the fuzzy subset μ if ( ) 

is the optimistic multigranulation fuzzy upward rough set of μ.

, then the above expression become: 

is the fuzzy upward rough set of μ.

, then the above expression become as follows:

This means that (

is the fuzzy rough set of μ as proposed by Ma. 38 Remark 1. Let  be an implicator 1  and 2  be t-norms and  be a t-conorm. Then the following hold:

and  is right monotonic, then the inequality will be equality. C ( ) 2 q y z q y q z ( , ( , )) ( ( , ), ( , ))

 , then the inequality will be equality.

and  is right monotonic, then the inequality will be equality.

Theorem 11. Let ( , ( ))    ⇑ be a fuzzy upward covering approximation space and

, are fuzzy sets in  , then the following hold:

(vii) If 1  and  satisfy the weak distributivity laws, then ( )

Definition 14. Let ( , ( ))    ⇑ be a fuzzy upward covering approximation space and 

. Otherwise μ is called definable on multigranulation ( , )   -fuzzy upward approximation space.

(i) If  and  are the Kleene-Dienes implicator KD  and standard min operator M  based on S  and M  , respectively, then from the above definition it follows that

becomes the pessimistic multigranulation fuzzy upward rough set of μ. 

becomes the pessimistic multigranulation fuzzy rough set of μ as proposed by Zhan et al. 40 Theorem 12. Let ( , ( ))    ⇑ be a fuzzy upward covering approximation space and

for all t l = 1, 2, …, . Assume that q β c t  ⇑ is a fuzzy upward β-neighborhood of q in  induced by c t l , = 1, 2, …, t . If μ and λ are fuzzy sets in  , then the following hold:

(v) If 1  and 2  satisfy the condition C ( ) 2 , then ( )

(vii) If 1  and  satisfy the weak distributivity laws, then ( )

Definition 15. Let ( , ( ))    ⇑ be a fuzzy upward covering approximation space and 

  -fuzzy upward rough set of the fuzzy set μ if ( )

. Otherwise μ is called a definable on multigranulation ( , )   -fuzzy upward approximation space.

Theorem 13. Let ( , ( ))    ⇑ be a fuzzy upward covering approximation space and

, for all t l = 1, 2, …, . Assume that the

, are fuzzy sets in  , then the following hold:

(vii) If 1  and  satisfy the weak distributive laws, then ( )

Definition 16. Let ( , ( ))    ⇑ be a fuzzy upward covering approximation space and

(v) If 1  and 2  satisfy the condition C ( ) 2 , then ( )

(vii) If 1  and  satisfy the weak distributive laws, then ( )

Over the years, numerous decision making procedures have been introduced in the literature, of which technique for order preference by similarity to ideal solution (TOPSIS) is one of the extensively and efficiently used famous methods. Hwang and Yoon 49 presented the TOPSIS to deal multiattribute decision making problems. According to which the alternative is the smallest distance from the positive ideal solution and the furthest distance from the negative ideal solution in decision making problems is the best alternative. Some advantages of fuzzy TOPSIS are:

(i) When the fuzzy analytic hierarchy process (AHP) and fuzzy TOPSIS methods are compared with respect to the amount of computations, fuzzy AHP requires more complex computations than fuzzy TOPSIS. is the collection of different attributes/tests. The unknown weight vector of l attributes is denoted by W ω ω ω = ( , , …, )

. Let E be a finite set of the domain for the information function g q c ( ,

Here we present the fuzzy information system W E ( , C, , )  .

To find the most suitable medicine among the given ones, we initiate an algorithm based on the proposed multigranulation ( , )   -fuzzy upward rough sets applying fuzzy upward β-covering approach and their corresponding steps are compiled as follows:

Input: Given fuzzy information system W E ( , C, , )  ; REHMAN ET AL.

Step 1: Using the proposed transfer function for the construction of c t  ⇑ , where t m = 1, 2, …, ;

Step 2: Compute q β c j t  ⇑ of q j with respect to c t ;

Step 3: Apply the principle of fuzzy TOPSIS method to compute the individual's best and worst fuzzy decision making objects c 1

and { } q i n = min ( ): = 1, 2, …, ,

where t l = 1, 2, …, .

Step 4: Integrate the following approximations (

Step 5: Find the ranking function δ q

Step 6: Constructed the optimal index function δ q (10) and determine the weight vectors of every attributes/tests according to δ q ( ) k j by

Step 7: Rank the alternatives/medicines by the value of the overall ranking function δ q ( ) j and make the decision. Output: A ranking result of all the alternatives/medicines.

Begin for i = 1 to n and t = 1 to l do compute c t  ⇑ , where t = 1 to l//according to Equation (5) end for j = 1 to n and t = 1 to l do compute q β c j t  ⇑ of q j with respect to c t //according to Definition 8 28 | end for i = 1 to n j , = 1 to n and t = 1 to l do compute c 1 + t  and c 1 − t  //according to Equation (7) and Equation (8) . For a critical value β suppose that for each medicine q i  ∈ , there is at least one symptoms y p  ∈ such that the efficacy value of the medicine q j for the symptom y i is not less than β, and ( ) c   ⇑ is a fuzzy upward β-covering of  . Then the fuzzy upward β-neighborhood q β c j  ⇑ of q j with respect to c is a fuzzy set given by

which denotes the minimum value among all the efficacy values of each medicine q k for treating the symptoms. If a fuzzy set μ denotes the ability of all medicines in  to cure the coronavirus disease A, since the inaccuracy of μ, then we can take it approximate evaluation according to the lower and upper approximation of μ. Let q i = { : = 1, 2, …, 9} i  be the set of medicines and c i be the criteria. Then the evaluation of  by the c, is given in Table 2 . 

Based on criterion c 1 and using Equation (5), to compute the fuzzy preference degree of q i i ( = 1, 2, …, 9) to q j j ( = 1, 2, …, 9), one can derive 

Based on criterion c 2 and using Equation (5), to compute the fuzzy preference degree of q i i ( = 1, 2, …, 9) to q j j ( = 1, 2, …, 9), one can acquire 

Based on criterion c 3 and using Equation (5), to compute the fuzzy preference degree of q i i ( = 1, 2, …, 9) to q j j ( = 1, 2, …, 9), one can derive The weight of every attributes are as follows, respectively ω ω ω = 0.30279, = 0.32236, = 0.37485. 1 2 3 Now, we are ready to apply optimal index formula for alternative q j , Finally, we can see that the ranking of the nine alternatives is q q q q q q q q q . This ranking shows that the medicine q 8 is the most important for the treatment of the coronavirus disease A.

A comparative analysis among the methods of Ma 38 and Yang and Hu 39 with our proposed method is discussed in this section. On one hand, in light of the numerical example of the previous section, we compare the methods of Ma, Yang, and Hu, with our proposed method. On the other hand, for the drawbacks of the above mentioned methods that cannot make a decision in some situations for example when β = 0.5, we find that proposed method can make up for this defect.

In the study of multiple attributes decision making problems with fuzzy information, there are many decision making methods based on a fuzzy binary relation. However, not all multiple attributes decision making problems can be characterized by a fuzzy binary relation. For this reason, we set methods to solve multiple attributes decision making problems with fuzzy information based on the optimistic multigranulation ( , )   -fuzzy upward rough set based on fuzzy upward β-covering. Furthermore, by comparative analysis, we find that our proposed method is more widely used than the above mentioned methods based on a fuzzy binary relation.

(i) The proposed transfer function is suitable for computing the fuzzy preference degree of Pan et al. 45 for the construction of upward/downward fuzzy preference relations. (ii) Another novelty of the proposed method is that it can be applied to form fuzzy preference relation from simple fuzzy set rather to be a preassumed one. Further the very same method can be applied to construct a fuzzy β-coverings called fuzzy upward β-coverings. (iii) The proposed method more suitable to construct the upward consistency matrices of experts which satisfy the upward additive consistency and the upward order consistency based on upward fuzzy preference relations. (iv) The advantage of the proposed model is that, it can be used for ranking of the feasible alternatives with auto adjustable weights. (v) The present work is more technically advanced as compared with other existing techniques. As stated in Definitions 7, 12, 13, and 14 other existing literature/techniques become the particular cases of the present technique.

Approaches to displaying information to assist decisions under uncertainty

A graph based group decision making approach with intuitionistic fuzzy preference relations

Multi-Criteria Decision Analysis: Methods and Software

Revealed indifference and models of choice behavior

A multiple attribute decision model to compare the the firms occupational health and safety management perspectives

A multi-criteria group decision framework for partner grouping when sharing facilities

An associative memory approach to medical decision support systems

Expert system for medicine diagnosis using software agents

Decision tree classifiers for automated medical diagnosis

Knowledge discovery in medicine: current issue and future trend

Rough sets

Algebraic structures of rough sets

The algebraic structures of generalized rough set theory

Medical information systems problems with analysis and way of solution

Rough fuzzy sets and fuzzy rough sets

Attribute reduction in ordered decision tables via evidence theory

Generalized rough sets based on relations

Covering based rough sets approximations

Topological approaches to covering rough sets

Reduction and axiomization of covering generalized rough sets

On three types of covering-based rough sets

On some types of covering rough sets from topological points of view

Decision-theoretic rough fuzzy set model and application

Decision-theoretic three-way approximations of fuzzy sets

Variable precision rough sets model

Three-way decisions with probabilistic rough sets

Rough approximation of a preference relation by dominance relations

Rough sets theory for multicriteria decision analysis

Rough sets methodology for sorting problems in presence of multiple attributes and criteria

MGRS: a multi-granulation rough set

Pessimistic rough set based decisions: a multigranulation fusion strategy

Multi-granulation decision theoretic rough sets

Multi-granulation rough sets: from partition to covering

New types of dominance based multigranulation rough sets and their applications in conflict analysis problems

SDMGRS: soft dominance based multi granulation rough sets and their applications in conflict analysis problems

Identifying degenerative brain disease using rough set classifier based on Wavelet packet method

Generalized fuzzy rough approximation operators based on fuzzy coverings

Two fuzzy covering rough set models and their generalizations over fuzzy lattices

On some types of fuzzy covering-based rough sets

Covering based multigranulation ( , )

  -fuzzy rough set models and applications in multi-attribute group decision-making

Covering based variable precision

  -fuzzy rough sets with applications to multi-attribute decision-making

Novel classes of fuzzy soft coverings based fuzzy rough sets with applications to multi-criteria fuzzy group decision making

Fuzzy soft β-covering based fuzzy rough sets and corresponding decision-making applications

Fuzzy preference based rough sets

Multi-granulation fuzzy preference relation rough set for ordinal decision system

Some issues on consistency of fuzzy preference relations

Rough set approach to multi-attribute decision analysis

A comparative study of fuzzy rough sets. Fuzzy Sets and Systems

Multiple Attribute Decision Making Methods Application

TOPSIS method based on a fuzzy covering approximation space: an application to biological nano-materials selection

A comprehensive study of upward fuzzy preference relation based fuzzy rough set models: Properties and applications in treatment of coronavirus disease

Then for all fuzzy set μ in the universe  , define the pessimistic multigranulation ( , )  fuzzy upward lower approximation ( )and the pessimistic multigranulation. Otherwise μ is called definable on multigranulation ( , )   -fuzzy upward approximation space.Theorem 14. Let ( , ( ))    ⇑ be a fuzzy upward covering approximation space and, for all t l = 1, 2, …, . Assume that theIf μ and λ are fuzzy sets in  , the following hold: . The upward fuzzy preference classes i c3   ⇑ are given by:In this study, we have discussed a new type of rough sets called α-upward fuzzified preference rough sets using upward fuzzy preference relation. On the basis of α-upward fuzzified preference rough sets, we introduced approximate precision, rough degree, approximate quality, and their mutual relationships. Furthermore, we presented the idea of new types of fuzzy upward β-coverings, fuzzy upward β-neighborhoods and fuzzy upward complement βneighborhoods and some relative properties are discussed. We further proposed a new type of upward lower and upward upper approximations by applying an upward β-neighborhoods.After employing an upward β-neighborhoods based upward rough set approach. We observed that every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Subsequently, we presented the idea to combined the fuzzy implicator and t-norm to introduce multigranulation ( , )   -fuzzy upward rough set applying fuzzy upward β-covering and some relative properties are discussed. Finally we presented a new technique/algorithm for the selection of medicine using multigranulation ( , )   -fuzzy upward rough set.