Abstract
Fuzzy soft set is the most powerful and effective extension of soft sets which deals with parameterized values of the alternative. It is an extended model of soft set and a new mathematical tool that has great advantages in dealing with uncertain information and is proposed by combining soft sets and fuzzy sets. Many fuzzy decision making algorithms based on fuzzy soft sets were given. However, these do not consider the external effective on the decision it depends on the parameters without considering any external effective. In order to solve these problems, in this paper, we introduce the concept of effective fuzzy soft set and its operation and study some of its properties. We also give an application of this concept in decision making (DM) problem. Finally, we give an application of this theory to medical diagnosis (MD) and exhibit the technique with a hypothetical case study.
1. Introduction
Currently, many researchers try to find suitable solutions to some uncertainty in mathematics that the classical methods cannot solve it since the classical methods cannot solve all the uncertainty decision making problems in economy, engineering, medicine, and others. Fuzzy set is one of these solutions defined by Zadeh as new mathematical tool [1] which was published in 1965. Molodtsove [2] defined one of the most important solutions as a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects which is a soft set theory where Maji et al. [3] went deep into the study of this theory through defining some operations such as AND, OR, union, and intersection. Then, Maji and Roy [4] apply a soft sets theory to find a solution for some decision making problems using rough mathematics. In 2008, Majumdar and Samanta [5] proposed some similarity measures between soft sets and gave an application of these similarities to solve DM problems, and then Kharal and Ahmad [6] defined new definitions and applications on the similarity measure between soft sets. In 2001, as a combination between fuzzy and soft sets, Maji et al. [7] proposed a theory of fuzzy soft set and many researchers have studied this concept and its properties with applications such as Roy and Maji [8], in 2007, and Feng et al. [9]. Chaudhuri and De [10] gave an attention on soft relation and fuzzy soft relation and they used these concepts to solve some of the decision making problems. On the other hand, Majumdar and Samanta [11] in 2010 presented generalised fuzzy soft sets theory and studied some of its properties and their applications in DM problems and MD problems. Çaǧman et al. [12] proposed a new DM method by using a theory of fuzzy parameterized fuzzy soft sets and they also in 2011 [13] proposed a DM method based on fuzzy parameterized-soft sets theory. Alkhazaleh et al. [14] and as a combination between soft multiset [15] and fuzzy set defined the concept of fuzzy soft multiset with its application in DM. They also generalised the concepts of fuzzy parameterized fuzzy soft set (FPFSS) to fuzzy parameterized interval-valued fuzzy soft set (FPIVFSS) [16] and FPFSS to possibility fuzzy soft set [17] and gave some applications in DM and MD problems. One of the most important concepts related to soft set is soft expert set defined by Alkhazaleh and Salleh [18] which is generalised later by the same authors to fuzzy soft expert set theory [19] and they also presented the applications of these two theories in DM and MD problems. Certainty and coverage of a parameter defined by Renukadevi and Sangeetha in 2020 [20] as a new concept are related to the soft set and they presented a new approach using the certainty of a parameter to solve a DM problem over the soft universe. Debnath in his paper in 2021 [21] proposed a fuzzy hypersoft set as a combination between fuzzy set and hypersoft set. The adaptability of this hypothesis is to handle the parameterized issues of instability as more as compared to fuzzy soft set. In 2021, Phaengtan et. al. [22] defined partial averages of fuzzy soft sets and presented a new algorithm for solving some DM problems based on partial averages. Furthermore, they also showed that this algorithm is practical for solving DM problems. Močkoř and Hurtík in 2021 [23] defined fuzzy soft relations and introduced the fuzzy soft approximation of fuzzy soft sets related to this relation. They also used fuzzy soft approximations in selective color segmentation problem, where authoritative and fully automated methods do not yet exist. They proposed three novel hybrid models and presented some properties of these models and defined multi-soft rough approximation operators in terms of multi-soft relations. For more information, see [24, 25].
When studying soft sets and their applications, all researchers deal with the parameters and the universal set, ignoring external effective that may affect their decisions. In this research, we will study for the first time the extent of the effect of external effectiveness on soft sets and on the outcome of decisions issued by these sets. Firstly, we define the concept of effective fuzzy soft set (EFRSS) and some definitions related to this concept are given and we use these definitions to solve DM problems by giving a new algorithm. A medical diagnosis method (MD) is established for EFSS setting using similarity measures. Lastly, a numerical example is given to demonstrate the possible application of similarity measures in (MD).
2. Preliminary
In this section, we review some definitions relevant to this work. The soft set defined by Molodtsov can be expressed as the following: let be a set of universe, be a set of parameters, and denote the power set of and .
Definition 1. (see [2]). A pair is called a soft set over , where is a mapping
Reference [7] generalised soft set of [2] to fuzzy soft set. In this section, we will review some definitions and properties related to fuzzy soft set theory, which we will use in our work. The following definitions and propositions are due to [7].
Definition 2. Let be an initial universal set, be a set of parameters, and be the power set of fuzzy set of . Let and is a pair called a fuzzy soft set over where is a mapping given by
Definition 3. The union of two fuzzy soft sets and over a common universe is the fuzzy soft set where , and ,where s is any s-norm.
Definition 4. The intersection of two fuzzy soft sets and over a common universe is the fuzzy soft set where , and ,where t is any t-norm.
3. Effective Fuzzy Soft Set (EFSS)
In this section, we generalise the concept of fuzzy soft sets as introduced by Maji et al. [7]. In our generalisation of fuzzy soft set, an effective set is applied with the parameterization of fuzzy sets while defining a fuzzy soft set. Let us start with definitions of effective set and effect scale, respectively, which are introduced for the first time.
Definition 5. An effective set is a fuzzy set in a universe of discourse where is a function . is the set of effective parameters that may change the membership values by making positive effect (or no effect) on values of memberships after applying it and defined as follows:
Definition 6. Let be an initial universal set, be a set of parameters, be a set of effective parameters, and be the effective set over . Let denote all fuzzy subsets of ; a pair is called an effective fuzzy soft set (EFSS in short) over where is a mapping given bydefined as follows:where ,
Example 1. Let be a set of universe. Let be a set of parameters and let be a set of effective parameters. Suppose that the effective set over for all is given by expert as follows:Let the fuzzy soft set be defined as follows:After applying Definition 6 on , we obtain the following:Then, we have .
By using the same method, we get the following EFSS:
Definition 7. The union and intersection of two effective sets and over the set of effective parameters is the effective set and , respectively, where s is any s-norm and t is any t-norm.
Definition 8. The complement of effective set over the set of effective parameters is the effective set where c is any fuzzy complement.
Definition 9. The of the EFSS is the EFSS , where is any fuzzy complement of .
Here, we keep the fuzzy soft set as is and find the fuzzy complement of the effective set , and then we apply equation (6) to get a new EFSS.
Definition 10. The of the EFSS is the EFSS , where is the fuzzy soft complement of .
Here, we find the fuzzy soft complement of which is and keep the effective set as is, then we apply equation (6) to get a new EFSS.
Definition 11. The of the EFSS is the EFSS , where is the fuzzy soft complement of and is any fuzzy complement of .
Here, we find the fuzzy soft complement of which is and the fuzzy complement of the effective set which is , then we apply Definition 6 to get a new EFSS.
Example 2. Consider Example 1. LetBy using the basic fuzzy complement of effective set , we get the following effective set :Also, letbe the fuzzy soft set, and by using the fuzzy soft complement over , we have the following fuzzy soft set:By using Definitions 9, 10, and 11 with applying Definition 6, we obtain the following , , and , respectively:
Definition 12. The union of two EFSSs and over a common universe is the EFSS where and ,where s is any s-norm and is the fuzzy soft union between and .
The idea of this union is to create a new effective set resulting from the union of and and then apply this set to the fuzzy soft set , resulting from the union of and by using Definition 6.
Example 3. Consider Example 1. Letbe any two effective sets given by two different experts. Also, letbe a fuzzy soft set over , and letbe a fuzzy soft set over .
By using the basic fuzzy union (max), we have the following effective set:Also, by using the basic fuzzy soft union, we have the following fuzzy soft set (in this example ):Then, by using Definitions 12 and 6, we obtain the following EFSS :
Definition 13. The intersection of two EFSSs and over a common universe is the EFSS where and ,where t is any t-norm; is the fuzzy soft intersection between and .
The idea of this intersection is to create a new effective set resulting from the intersection of and and then apply this set to the fuzzy soft set resulting from the intersection of and by using Definition 6.
Example 4. Consider Example 3.
By using the basic fuzzy intersection (min), we have the following effective set:Also, by using the basic fuzzy soft intersection, we have the following fuzzy soft set (in this example ):Then, by using Definitions 13 and 6, we obtain the following EFSS :
4. An Effective Fuzzy Soft Set Theoretic Approach to Decision Making Problems
In this section, we give an effective fuzzy soft set theoretic approach to get a solution of decision making problem.
Definition 14. (see [8]). A comparison table is a square table with an equal number of rows and columns, both labelled by the object names of the universe, and the entries are , given by the number of parameters for which the membership value of exceeds or is equal to the membership value of .
4.1. Algorithm
Here, we give an algorithm as a modification of the algorithm given by Maji and Roy [8]. Then, we will compare between the original algorithm and the modified algorithm. Firstly, we present the algorithm given by Maji and Roy as follows: let , , and be three fuzzy soft sets.(1)Input , , and .(2)Input the set of parameters as observed by the observer.(3)Compute the corresponding resultant fuzzy soft set from , , and and place it in tabular form.(4)Construct the comparison table of the fuzzy soft set and compute and for .(5)Compute the score of .(6)The decision is if .(7)If has more than one value, then any one of may be chosen.
Then, we give our algorithm as follows:(1)Input the fuzzy soft sets and .(2)Input the effective sets of parameters .(3)Input the effective sets and over for the fuzzy soft sets and , respectively.(4)Compute the corresponding resultant EFSS and .(5)Compute the corresponding resultant EFSS from the EFFSs and and place it in tabular form.(6)Construct the comparison table of the EFSS and compute and for .(7)Compute the score of .(8)The decision is if .
4.2. Application in a Decision Making Problem
Let be a set of cars with the same model. This type of car is manufactured in four countries, and one of these factories is the main factory. Let be a set of parameters where safety, affordable, maintenance, comfortable, performance, and reliable. Let be a set of effective parameters where all of its parts are made in the original factory, it reassembled at the original factory, never worked under car Apps, and it was not owned by more than one person. Let the effective set over given by experts be as follows:
Also, letbe a fuzzy soft set over , and letbe a fuzzy soft set over .
Now, let the parameter set be as observed by the observer. To using Maji and Roy algorithm, we firstly find which is the union of and as follows:
The tabular representation of resultant fuzzy soft set will be, as in Table 1.
The comparison table of the above resultant fuzzy soft set is in Table 2.
Now, we compute the row-sum, column-sum, and the score for each as shown in Table 3:
It is clear that our decision is to select Car 3 since the maximum score is 28, scored by . To use our algorithm, firstly, we use the basic fuzzy union (max) to get the new effective set from and as follows:
Then, by using the FSS as above, Definitions 12 and 6, we obtain the following EFSS :
The tabular representation of resultant EFSS will be as in Table 4.
The comparison table of the above resultant effective fuzzy soft set is in Table 5.
Now, we compute the row-sum, column-sum, and the score for each as shown in Table 6.
It is clear that our decision is to select Car 6 since the maximum score is 17, scored by , and by comparing with Maji and Roy algorithm, we conclude that the effective set made the change in decision from Car 3 to Car 6.
5. Application of EFSS in Medical Diagnosis
There are many applications and theories that seek to facilitate the process of medical diagnosis, but each of these applications and theories take into account the symptoms that appear on the patient without looking at external effects that can change the diagnosis completely. In this section, we will try for the first time to find the closest diagnosis of the disease, depending on the symptoms and external effects.
Assume that be a set of 4 patients in the hospital. The hospital diagnostic expert identified the following symptoms to find out what patients were suffering from:where , , , , , , , , , , , , , , , , , , , and. Also, let be a set of diseases such that COVID-19, , , and .
Suppose be a set of effective parameters, where in the past two weeks, he visited a country in Europe, America, Africa, or East Asia, he close contacted (less than 6 feet) with anyone who is suffering from COVID-19, he works in medical centers, he is working in large gatherings or using public transportation daily, he is eating his food in the restaurant or eating fast food, he was in an area with stagnant water, especially at dawn and dusk, he used to sleep without a cover or mosquito net, eating food that is raw or undercooked, and eating foods and beverages purchased from street vendors. After talking with patients, we found out the patients daily activities and lives, as in Table 7.
The relation between the above parameters and the given diseases is presented in Table 8.
By using the expertise of the medical team and the information in Tables 7 and 8, we construct the effective sets for each patient with respect to the given diseases as the representation, Tables 9–12.
Now, suppose the tabular representation of (patient symptom) given in Tables 13 and 14.
Also, the tabular representation of (model symptom) is given in Tables 15 and 16.
w, we construct the EFSSs using Definition 6 and Tables 13 and 14, as given in Tables 17 and 24.
Finally, we compute the score table by finding the similarity between each row in Tables 17–24 with each row in Tables 15 and 16 and find the maximum value for each patient and the diseases related to these values. We use the following formula to find the similarity:
This step can be as follows:
By similar calculations, consequently, we get the score table, as in Table 25.
It is clear from Table 25 that the first patient suffers from dengue fever, the second patient suffers from typhoid, the third patient suffers from malaria, and the fourth patient suffers from COVID-19
6. Conclusion
As a new tool dealing with uncertainty, we have introduced the effective fuzzy soft set theory which is more efficient and useful and studied some of its properties. We also defined basic operations on effective fuzzy soft sets, such as complement, union, and intersection. The theory has been applied to solve DM and MD problems. In future work, researchers can generalise this concept to interval-valued effective fuzzy soft set and they also can develop it to effective fuzzy soft expert set to give it more efficiency.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
The authors would like to acknowledge the financial support received from Jadara University, Scientific Research Incentives for 2022, Grant No. 3.