# Effective Fuzzy Soft Set Theory and Its Applications

Authors: Shawkat Alkhazaleh
Journal: Applied Computational Intelligence and Soft Computing
 (2022)

---

## 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.


---

## Body

## 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-Q,N−soft rough approximation operators in terms of multi-Q,N−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: letU be a set of universe, E be a set of parameters, and PU denote the power set of U and A⊆E.Definition 1.
(see [2]). A pair F,A is called a soft set over U, where F is a mapping(1)F:A⟶PU.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.
LetU be an initial universal set, E be a set of parameters, and IU be the power set of fuzzy set of U. Let A⊆E and F,E is a pair called a fuzzy soft set over U where F is a mapping given by(2)F:A⟶IU.Definition 3.
The union of two fuzzy soft setsF,A and G,B over a common universe U is the fuzzy soft set H,C where C=A∪B, and ∀ε∈C,(3)Hε=Fε,ifε∈A−B,Gε,ifε∈B−A,Fε,Gε,ifε∈A∩B,where s is any s-norm.Definition 4.
The intersection of two fuzzy soft setsF,A and G,B over a common universe U is the fuzzy soft set H,C where C=A∪B, and ∀ε∈C,(4)Hε=Fε,ifε∈A−B,Gε,ifε∈B−A,tFε,Gε,ifε∈A∩B,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 setΛ in a universe of discourse A where Λ is a function Λ:A⟶0,1. A 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:(5)Λ=a,δΛa:a∈A.Definition 6.
LetU be an initial universal set, E be a set of parameters, A be a set of effective parameters, and Λ be the effective set over A. Let IU denote all fuzzy subsets of U; a pair F,EΛ is called an effective fuzzy soft set (EFSS in short) over U where F is a mapping given by(6)F:E⟶IU,defined as follows:(7)FeiΛ=xjμUxjΛ:xj∈U,ei∈E,where ∀ak∈A,(8)μUxjΛ=μUxj+1−μUxj∑kδΛxjakA,ifμUxj∈0,1,μUxj,O.W.Example 1.
LetU=x1,x2,x3 be a set of universe. Let E=e1,e2,e3,\\e4,e5,e6 be a set of parameters and let A=a1,a2,a3,a4 be a set of effective parameters. Suppose that the effective set over A for all x1,x2,x3 is given by expert as follows:(9)Λx1=a10.8,a21,a30,a40.2,Λx2=a10.4,a20.5,a30,a40.8,Λx3=a11,a21,a30.7,a40.6.
Let the fuzzy soft setF be defined as follows:(10)Fe1=x10.3,x20.7,x30.5,Fe2=x10.5,x20.6,x30.6,Fe3=x10.7,x20.6,x30.5,Fe4=x10.8,x20.4,x30.4,Fe5=x10.6,x20.9,x30.3,Fe6=x10.2,x20.4,x30.7.
After applying Definition6 on F, we obtain the following:(11)FΛe1=x10.3+0.70.8+1+0+0.2/4,x20.7+0.34+0.5+0+0.8/4,x30.5+0.51+1+0.7+0.6/4.
Then, we haveFΛe1=x1/0.65,x2/0.83,x3/0.91.
By using the same method, we get the following EFSS:(12)FΛ,E=e1,x10.65,x20.83,x30.91,e2,x10.75,x20.77,x30.93,e3,x10.85,x20.77,x30.91,e4,x10.9,x20.66,x30.9,e5,x10.8,x20.94,x30.88,e6,x10.6,x20.66,x30.95.Definition 7.
The union and intersection of two effective setsΛ′ and Λ′′ over the set of effective parameters A is the effective set Λs and Λt, respectively, where s is any s-norm and t is any t-norm.Definition 8.
The complement of effective setΛ over the set of effective parameters A is the effective set Λc where c is any fuzzy complement.Definition 9.
TheΛcomplement of the EFSS FΛ,E is the EFSS FΛc,E, where Λc is any fuzzy complement of Λ.Here, we keep the fuzzy soft setF as is and find the fuzzy complement of the effective set Λ, and then we apply equation (6) to get a new EFSS.Definition 10.
TheSoftcomplement of the EFSS FΛ,E is the EFSS FΛc,E, where Fc is the fuzzy soft complement of F.Here, we find the fuzzy soft complement ofF which is Fc and keep the effective set Λ as is, then we apply equation (6) to get a new EFSS.Definition 11.
TheTotalcomplement of the EFSS FΛ,E is the EFSS FΛcc,E, where Fc is the fuzzy soft complement of F and Λc is any fuzzy complement of Λ.Here, we find the fuzzy soft complement ofF which is Fc and the fuzzy complement of the effective set Λ which is Λc, then we apply Definition 6 to get a new EFSS.Example 2.
Consider Example1. Let(13)Λx1=a10.8,a21,a30,a40.2,Λx2=a10.4,a20.5,a30,a40.8,Λx3=a11,a21,a30.7,a40.6.
By using the basic fuzzy complement of effective setΛ, we get the following effective set Λc:(14)Λcx1=a10.2,a20,a31,a40.8,Λcx2=a10.6,a20.5,a31,a40.2,Λcx3=a10,a20,a30.3,a40.4.
Also, let(15)Fe1=x10.3,x20.7,x30.5,Fe2=x10.5,x20.6,x30.6,Fe3=x10.7,x20.6,x30.5,Fe4=x10.8,x20.4,x30.4,Fe5=x10.6,x20.9,x30.3,Fe6=x10.2,x20.4,x30.7,be the fuzzy soft set, and by using the fuzzy soft complement over F, we have the following fuzzy soft set:(16)Fce1=x10.7,x20.3,x30.5,Fce2=x10.5,x20.4,x30.4,Fce3=x10.3,x20.6,x30.5,Fce4=x10.2,x20.6,x30.6,Fce5=x10.6,x20.9,x30.3,Fce6=x10.2,x20.6,x30.6.
By using Definitions9, 10, and 11 with applying Definition 6, we obtain the following Totalcomplement, Λcomplement, and Softcomplement, respectively:(17)FΛcc,E=e1,x10.85,x20.7,x30.59,e2,x10.75,x20.75,x30.5,e3,x10.65,x20.63,x30.59,e4,x10.6,x20.83,x30.67,e5,x10.8,x20.95,x30.43,e6,x10.6,x20.83,x30.47,FΛc,E=e1,x10.65,x20.87,x30.59,e2,x10.75,x20.83,x30.67,e3,x10.85,x20.83,x30.59,e4,x10.9,x20.75,x30.5,e5,x10.8,x20.96,x30.43,e6,x10.6,x20.75,x30.75.Definition 12.
The union of two EFSSsFΛ′,E1 and GΛ′′,E2 over a common universe U is the EFSS HΛs,E where E=E1∪E2 and ∀υ∈E,(18)HΛsυ=FΛsυ,ifυ∈E1−E2,GΛsυ,ifυ∈E2−E1,F∪GΛsυ,ifυ∈E1∩E2,where s is any s-norm and H is the fuzzy soft union between F and G.The idea of this union is to create a new effective setΛs resulting from the union of Λ′ and Λ′′ and then apply this set to the fuzzy soft set H, resulting from the union of F and G by using Definition 6.Example 3.
Consider Example1. Let(19)Λ′x1=a10.8,a21,a30,a40.2,Λ′x2=a10.4,a20.5,a30,a40.8,Λ′x3=a11,a21,a30.7,a40.6,Λ″x1=a10.7,a20.8,a30.2,a40.2,Λ″x2=a10.4,a20.6,a30.1,a40.8,Λ″x3=a10.9,a20.8,a30.8,a40.7,be any two effective sets given by two different experts. Also, let(20)F,E1=e1,x10.8,x20.6,x30.9,e3,x10.5,x20.7,x30.8,e4,x10.6,x20.7,x30.8,e6,x10.6,x20.6,x30.9,be a fuzzy soft set over E1⊂E, and let(21)G,E2=e1,x10.7,x20.7,x30.8,e2,x10.8,x20.6,x30.4,e3,x10.5,x20.7,x30.8,e4,x10.7,x20.6,x30.9,e5,x10.4,x20.5,x30.7,be a fuzzy soft set over E2⊂E.
By using the basic fuzzy union (max), we have the following effective set:(22)Λsx1=a10.8,a21,a30.2,a40.2,Λsx2=a10.4,a20.6,a30.1,a40.8,Λsx3=a11,a21,a30.8,a40.7.
Also, by using the basic fuzzy soft union, we have the following fuzzy soft setH,E (in this example E=E1∪E2):(23)H,E=e1,x10.8,x20.7,x30.9,e2,x10.8,x20.6,x30.4,e3,x10.5,x20.7,x30.8,e4,x10.7,x20.7,x30.9,e5,x10.4,x20.5,x30.7,e6,x10.6,x20.6,x30.9.
Then, by using Definitions12 and 6, we obtain the following EFSS HΛs,E:(24)HΛs,E=e1,x10.99,x20.9,x30.99,e2,x10.99,x20.88,x30.93,e3,x10.78,x20.91,x30.98,e4,x10.87,x20.9,x30.99,e5,x10.77,x20.85,x30.96,e6,x10.82,x20.88,x30.99.Definition 13.
The intersection of two EFSSsFΛ′,E1 and GΛ′′,E2 over a common universe U is the EFSS KΛt,E where E=E1∪E2 and ∀υ∈E,(25)KΛsυ=FΛtυ,ifυ∈E1−E2,GΛtυ,ifυ∈E2−E1,F∩GΛtυ,ifυ∈E1∩E2,where t is any t-norm; K is the fuzzy soft intersection between F and G.The idea of this intersection is to create a new effective setΛt resulting from the intersection of Λ′ and Λ′′ and then apply this set to the fuzzy soft set K resulting from the intersection of F and G by using Definition 6.Example 4.
Consider Example3.
By using the basic fuzzy intersection (min), we have the following effective set:(26)Λtx1=a10.7,a20.8,a30,a40.2,Λtx2=a10.4,a20.5,a30,a40.8,Λtx3=a10.9,a20.8,a30.7,a40.6.
Also, by using the basic fuzzy soft intersection, we have the following fuzzy soft setH,E (in this example E=E1∪E2):(27)K,E=e1,x10.7,x20.6,x30.8,e2,x10.8,x20.6,x30.4,e3,x10.5,x20.7,x30.8,e4,x10.6,x20.6,x30.8,e5,x10.4,x20.5,x30.7,e6,x10.6,x20.6,x30.9.
Then, by using Definitions13 and 6, we obtain the following EFSS KΛt,E:(28)KΛt,E=e1,x10.83,x20.77,x30.95,e2,x10.87,x20.77,x30.85,e3,x10.72,x20.83,x30.95,e4,x10.77,x20.77,x30.95,e5,x10.66,x20.72,x30.93,e6,x10.77,x20.77,x30.98.


## 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 o1,o2,o3,…,on of the universe, and the entries are cij,i,j=1,2,…,n, given by cij= the number of parameters for which the membership value of oi exceeds or is equal to the membership value of oj.


### 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 F,A, G,B, and H,C be three fuzzy soft sets.(1)
InputF,A, G,B, and H,C.(2)
Input the set of parametersP as observed by the observer.(3)
Compute the corresponding resultant fuzzy soft setS,P from F,A, G,B, and H,C and place it in tabular form.(4)
Construct the comparison table of the fuzzy soft setS,P and compute ri and ti for oi,∀i.(5)
Compute the score ofoi,∀i.(6)
The decision isSk if Sk=maxiSi.(7)
Ifk has more than one value, then any one of ok may be chosen.Then, we give our algorithm as follows:(1)
Input the fuzzy soft setsF,A and G,B.(2)
Input the effective sets of parametersA.(3)
Input the effective setsΛF and ΛG over A for the fuzzy soft sets F,A and G,B, respectively.(4)
Compute the corresponding resultant EFSSFΛF and GΛG.(5)
Compute the corresponding resultant EFSSHΛ from the EFFSs FΛF and GΛG and place it in tabular form.(6)
Construct the comparison table of the EFSSHΛ and compute ri and ti for oi,∀i.(7)
Compute the score ofoi,∀i.(8)
The decision isSk if Sk=maxiSi.


### 4.2. Application in a Decision Making Problem

LetU=x1,x2,x3,x4,x5,x6 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 E=e1,e2,e3,\\e4,e5,e6 be a set of parameters where e1= safety, e2= affordable, e3= maintenance, e4= comfortable, e5= performance, and e6= reliable. Let A=a1,a2,a3,a4 be a set of effective parameters where a1= all of its parts are made in the original factory, a2= it reassembled at the original factory, a3= never worked under car Apps, and a4= it was not owned by more than one person. Let the effective set over A,∀xi∈U given by experts be as follows:(29)Λ′x1=a10.8,a21,a30,a40.2,Λ′x2=a10.6,a21,a30.5,a40.4,Λ′x3=a10.2,a20.3,a30,a40.3,Λ′x4=a10.5,a20.4,a30.3,a40.4,Λ′x5=a10.7,a20.7,a30.5,a40.8,Λ′x6=a11,a21,a30.7,a40.6,Λ″x1=a10.6,a21,a30.1,a40.3,Λ″x2=a10.5,a21,a30.7,a40.5,Λ″x3=a10.3,a20.4,a30,a40.4,Λ″x4=a10.6,a20.6,a30.5,a40.5,Λ″x5=a10.7,a20.8,a30.4,a40.6,Λ″x6=a10.9,a20.8,a30.7,a40.6.Also, let(30)F,E1=e1,x10.8,x20.6,x30.9,x40.4,x50.8,x60.3,e3,x10.5,x20.7,x30.8,x40.6,x50.7,x60.8,e4,x10.6,x20.7,x30.8,x40.6,x50.2,x60.3,,e6,x10.6,x20.6,x30.9,x40.7,x50.6,x60.5,be a fuzzy soft set over E1⊂E, and let(31)G,E2=e1,x10.6,x20.5,x30.8,x40.3,x50.7,x60.4,e2,x10.4,x20.5,x30.7,x40.4,x50.5,x60.5,e3,x10.6,x20.8,x30.9,x40.6,x50.6,x60.7,e4,x10.6,x20.9,x30.7,x40.7,x50.4,x60.4,,e5,x10.8,x20.8,x30.9,x40.7,x50.5,x60.6,be a fuzzy soft set over E2⊂E.Now, let the parameter setP=E be as observed by the observer. To using Maji and Roy algorithm, we firstly find H,E which is the union of F,E1 and G,E2 as follows:(32)H,E=e1,x10.8,x20.6,x30.9,x40.4,x50.8,x60.4,e2,x10.4,x20.5,x30.7,x40.4,x50.5,x60.5,e3,x10.6,x20.8,x30.9,x40.6,x50.7,x60.8,e4,x10.6,x20.9,x30.8,x40.7,x50.4,x60.4,,e5,x10.8,x20.8,x30.9,x40.7,x50.5,x60.6,e6,x10.6,x20.6,x30.9,x40.7,x50.6,x60.5.The tabular representation of resultant fuzzy soft setH,E will be, as in Table 1.Table 1
Tabular representation ofH,E.
Ue1e2e3e4e5e6x10.80.40.60.60.80.6x20.60.50.80.90.80.6x30.90.70.90.80.90.9x40.40.40.60.70.70.7x50.80.50.70.40.50.6x60.40.50.80.40.60.5The comparison table of the above resultant fuzzy soft set is in Table2.Table 2
Comparison table ofH,E.
Ux1x2x3x4x5x6x1630434x2561556x3656666x4410634x5430364x6220346Now, we compute the row-sum, column-sum, and the score for eachoi as shown in Table 3:Table 3
oi scores.
Row-sumriColumn-sumtiScoreSix12027−7x228208x335728x41827−9x52027−7x61730−13It is clear that our decision is to select Car 3 since the maximum score is 28, scored byx3. To use our algorithm, firstly, we use the basic fuzzy union (max) to get the new effective set Λs from Λ′ and Λ′′ as follows:(33)Λsx1=a10.8,a21,a30.1,a40.3,Λsx2=a10.6,a21,a30.7,a40.5,Λsx3=a10.3,a20.4,a30,a40.4,Λsx4=a10.6,a20.6,a30.5,a40.5,Λsx5=a10.7,a20.8,a30.5,a40.8,Λsx6=a11,a21,a30.7,a40.6.Then, by using the FSSH,E as above, Definitions 12 and 6, we obtain the following EFSS HΛs,E:(34)H,E=e1,x10.8,x20.6,x30.9,x40.4,x50.8,x60.4,e2,x10.4,x20.5,x30.7,x40.4,x50.5,x60.5,e3,x10.6,x20.8,x30.9,x40.6,x50.7,x60.8,e4,x10.6,x20.9,x30.8,x40.7,x50.4,x60.4,,e5,x10.8,x20.8,x30.9,x40.7,x50.5,x60.6,e6,x10.6,x20.6,x30.9,x40.7,x50.6,x60.5.The tabular representation of resultant EFSSHΛs,E will be as in Table 4.Table 4
Tabular representation ofHΛs,E.
Ue1e2e3e4e5e6x10.910.730.820.820.910.82x20.880.850.940.970.940.88x30.930.780.930.860.930.93x40.730.730.820.870.820.87x50.940.850.910.820.850.88x60.90.920.970.90.930.92The comparison table of the above resultant effective fuzzy soft set is in Table5.Table 5
Comparison table ofHΛs,E.
Ux1x2x3x4x5x6x1610421x2564651x3626543x4401610x5532561x6544656Now, we compute the row-sum, column-sum, and the score for eachoi as shown in Table 6.Table 6
oi scores of HΛs,E.
Row-sumriColumn-sumtiScoreSix11431−17x2271611x326179x41232−20x52223−1x6301317It is clear that our decision is to select Car 6 since the maximum score is 17, scored byx6, and by comparing with Maji and Roy algorithm, we conclude that the effective set Λs made the change in decision from Car 3 to Car 6.


## 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 F,A, G,B, and H,C be three fuzzy soft sets.(1)
InputF,A, G,B, and H,C.(2)
Input the set of parametersP as observed by the observer.(3)
Compute the corresponding resultant fuzzy soft setS,P from F,A, G,B, and H,C and place it in tabular form.(4)
Construct the comparison table of the fuzzy soft setS,P and compute ri and ti for oi,∀i.(5)
Compute the score ofoi,∀i.(6)
The decision isSk if Sk=maxiSi.(7)
Ifk has more than one value, then any one of ok may be chosen.Then, we give our algorithm as follows:(1)
Input the fuzzy soft setsF,A and G,B.(2)
Input the effective sets of parametersA.(3)
Input the effective setsΛF and ΛG over A for the fuzzy soft sets F,A and G,B, respectively.(4)
Compute the corresponding resultant EFSSFΛF and GΛG.(5)
Compute the corresponding resultant EFSSHΛ from the EFFSs FΛF and GΛG and place it in tabular form.(6)
Construct the comparison table of the EFSSHΛ and compute ri and ti for oi,∀i.(7)
Compute the score ofoi,∀i.(8)
The decision isSk if Sk=maxiSi.


## 4.2. Application in a Decision Making Problem

LetU=x1,x2,x3,x4,x5,x6 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 E=e1,e2,e3,\\e4,e5,e6 be a set of parameters where e1= safety, e2= affordable, e3= maintenance, e4= comfortable, e5= performance, and e6= reliable. Let A=a1,a2,a3,a4 be a set of effective parameters where a1= all of its parts are made in the original factory, a2= it reassembled at the original factory, a3= never worked under car Apps, and a4= it was not owned by more than one person. Let the effective set over A,∀xi∈U given by experts be as follows:(29)Λ′x1=a10.8,a21,a30,a40.2,Λ′x2=a10.6,a21,a30.5,a40.4,Λ′x3=a10.2,a20.3,a30,a40.3,Λ′x4=a10.5,a20.4,a30.3,a40.4,Λ′x5=a10.7,a20.7,a30.5,a40.8,Λ′x6=a11,a21,a30.7,a40.6,Λ″x1=a10.6,a21,a30.1,a40.3,Λ″x2=a10.5,a21,a30.7,a40.5,Λ″x3=a10.3,a20.4,a30,a40.4,Λ″x4=a10.6,a20.6,a30.5,a40.5,Λ″x5=a10.7,a20.8,a30.4,a40.6,Λ″x6=a10.9,a20.8,a30.7,a40.6.Also, let(30)F,E1=e1,x10.8,x20.6,x30.9,x40.4,x50.8,x60.3,e3,x10.5,x20.7,x30.8,x40.6,x50.7,x60.8,e4,x10.6,x20.7,x30.8,x40.6,x50.2,x60.3,,e6,x10.6,x20.6,x30.9,x40.7,x50.6,x60.5,be a fuzzy soft set over E1⊂E, and let(31)G,E2=e1,x10.6,x20.5,x30.8,x40.3,x50.7,x60.4,e2,x10.4,x20.5,x30.7,x40.4,x50.5,x60.5,e3,x10.6,x20.8,x30.9,x40.6,x50.6,x60.7,e4,x10.6,x20.9,x30.7,x40.7,x50.4,x60.4,,e5,x10.8,x20.8,x30.9,x40.7,x50.5,x60.6,be a fuzzy soft set over E2⊂E.Now, let the parameter setP=E be as observed by the observer. To using Maji and Roy algorithm, we firstly find H,E which is the union of F,E1 and G,E2 as follows:(32)H,E=e1,x10.8,x20.6,x30.9,x40.4,x50.8,x60.4,e2,x10.4,x20.5,x30.7,x40.4,x50.5,x60.5,e3,x10.6,x20.8,x30.9,x40.6,x50.7,x60.8,e4,x10.6,x20.9,x30.8,x40.7,x50.4,x60.4,,e5,x10.8,x20.8,x30.9,x40.7,x50.5,x60.6,e6,x10.6,x20.6,x30.9,x40.7,x50.6,x60.5.The tabular representation of resultant fuzzy soft setH,E will be, as in Table 1.Table 1
Tabular representation ofH,E.
Ue1e2e3e4e5e6x10.80.40.60.60.80.6x20.60.50.80.90.80.6x30.90.70.90.80.90.9x40.40.40.60.70.70.7x50.80.50.70.40.50.6x60.40.50.80.40.60.5The comparison table of the above resultant fuzzy soft set is in Table2.Table 2
Comparison table ofH,E.
Ux1x2x3x4x5x6x1630434x2561556x3656666x4410634x5430364x6220346Now, we compute the row-sum, column-sum, and the score for eachoi as shown in Table 3:Table 3
oi scores.
Row-sumriColumn-sumtiScoreSix12027−7x228208x335728x41827−9x52027−7x61730−13It is clear that our decision is to select Car 3 since the maximum score is 28, scored byx3. To use our algorithm, firstly, we use the basic fuzzy union (max) to get the new effective set Λs from Λ′ and Λ′′ as follows:(33)Λsx1=a10.8,a21,a30.1,a40.3,Λsx2=a10.6,a21,a30.7,a40.5,Λsx3=a10.3,a20.4,a30,a40.4,Λsx4=a10.6,a20.6,a30.5,a40.5,Λsx5=a10.7,a20.8,a30.5,a40.8,Λsx6=a11,a21,a30.7,a40.6.Then, by using the FSSH,E as above, Definitions 12 and 6, we obtain the following EFSS HΛs,E:(34)H,E=e1,x10.8,x20.6,x30.9,x40.4,x50.8,x60.4,e2,x10.4,x20.5,x30.7,x40.4,x50.5,x60.5,e3,x10.6,x20.8,x30.9,x40.6,x50.7,x60.8,e4,x10.6,x20.9,x30.8,x40.7,x50.4,x60.4,,e5,x10.8,x20.8,x30.9,x40.7,x50.5,x60.6,e6,x10.6,x20.6,x30.9,x40.7,x50.6,x60.5.The tabular representation of resultant EFSSHΛs,E will be as in Table 4.Table 4
Tabular representation ofHΛs,E.
Ue1e2e3e4e5e6x10.910.730.820.820.910.82x20.880.850.940.970.940.88x30.930.780.930.860.930.93x40.730.730.820.870.820.87x50.940.850.910.820.850.88x60.90.920.970.90.930.92The comparison table of the above resultant effective fuzzy soft set is in Table5.Table 5
Comparison table ofHΛs,E.
Ux1x2x3x4x5x6x1610421x2564651x3626543x4401610x5532561x6544656Now, we compute the row-sum, column-sum, and the score for eachoi as shown in Table 6.Table 6
oi scores of HΛs,E.
Row-sumriColumn-sumtiScoreSix11431−17x2271611x326179x41232−20x52223−1x6301317It is clear that our decision is to select Car 6 since the maximum score is 17, scored byx6, and by comparing with Maji and Roy algorithm, we conclude that the effective set Λs 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 thatP=p1,p2,p3,p4 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:(35)S=s1,s2,s3,s4,s5,s6,s7,s8,s9,s10,s11,s12,s13,s14,s15,s16,s17,s18,s19,s20,where s1=fever, s2=dry cough, s3=loose motion, s4=shortness of breath or difficulty breathing, s5=headache, s6=tiredness, s7=aches, s8=runny nose, s9=sore throat, s10=severe pneumonia, s11=rash, e12=diarrhoea, s13=bone and joint pain, s14=nausea, s15=vomiting, s16=pain behind the eyes, s17=chills, s18=sweating, s19=abdominal pain, ands20=swollen glands. Also, let D=d1,d2,d3,d4 be a set of diseases such that d1= COVID-19, d2=dengue fever, d3=malaria, and d4=typhoid.SupposeA=a1,a2,a3,a4,a5,a6,a7,a8 be a set of effective parameters, where a1= in the past two weeks, he visited a country in Europe, America, Africa, or East Asia, a2= he close contacted (less than 6 feet) with anyone who is suffering from COVID-19, a3= he works in medical centers, a4= he is working in large gatherings or using public transportation daily, a5= he is eating his food in the restaurant or eating fast food, a6= he was in an area with stagnant water, especially at dawn and dusk, a7= he used to sleep without a cover or mosquito net, a8= eating food that is raw or undercooked, and a9= 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.Table 7
Patients daily activities and lives.
Pa1a2a3a4a5a6a7a8a9p1YesYesNoYesYesNoNoNoYesp2NoNoYesNoYesNoYesYesYesp3YesNoNoNoNoYesYesNoNop4NoNoNoYesYesNoNoYesNoThe relation between the above parameters and the given diseases is presented in Table8.Table 8
Parameters-diseases relation.
Da1a2a3a4a5a6a7a8a9Ajd1YesYesYesYesNoNoNoNoNo4d2YesNoNoNoNoYesYesNoNo3d3YesNoNoNoNoYesYesNoNo3d4YesNoNoNoYesNoNoYesYes4By using the expertise of the medical team and the information in Tables7 and 8, we construct the effective sets Λdjpi for each patient with respect to the given diseases as the representation, Tables 9–12.Table 9
Tabular representation ofΛd1pi.
pia1a2a3a4a5a6a7a8a9Sump1d11101000003p2d10010000001p3d11000000001p4d10000100001Table 10
Tabular representation ofΛd2pi.
pia1a2a3a4a5a6a7a8a9Sump1d21000000001p2d20000001001p3d21000011003p4d20000000000Table 11
Tabular representation ofΛd3pi.
pia1a2a3a4a5a6a7a8a9Sump1d31000000001p2d30000000101p3d31000011003p4d30000000000Table 12
Tabular representation ofΛd4pi.
pia1a2a3a4a5a6a7a8a9Sump1d41000100013p2d40000100113p3d41000000001p4d40000100102Now, suppose the tabular representation ofF,S (patient symptom) given in Tables 13 and 14.Table 13
Tabular representation ofF,S part 1.
Ps1s2s3s4s5s6s7s8s9s10p10.20.10.10.70.80.4000.10p20.70.50.100.80.90.60.20.30p30.80.70.200.60.10.700.20p40.60.70.70.90.50.10.60.60.40.9Table 14
Tabular representation ofF,S part 2.
Ps11s12s13s14s15s16s17s18s19s20p10.70.20.90.70.60.900.10.70.7p20.70.710.8000.60.40.60p30.100.910.700.60.60.40p40.10.60.10.200.20000Also, the tabular representation ofG,S (model symptom) is given in Tables 15 and 16.Table 15
Tabular representation ofG,S (Part 1).
Ds1s2s3s4s5s6s7s8s9s10d111110.500.50.50.51d20000.510.50000d310.500101000d410.500111000Table 16
Tabular representation ofG,S (Part 2).
Ds11s12s13s14s15s16s17s18s19s20d100.500000000d2101111000.51d300111010.50.50d40.5111000.50.510w, we construct the EFSSs using Definition6 and Tables 13 and 14, as given in Tables 17 and 24.Table 17
Tabular representation ofFΛd1,S part 1.
Ps1s2s3s4s5s6s7s8s9s10p10.80.760.760.930.950.85000.760p20.760.630.3300.850.930.70.40.480p30.850.760.400.70.330.7600.40p40.70.760.760.930.630.330.70.70.550.93Table 18
Tabular representation ofFΛd1,S (part 2).
Ps11s12s13s14s15s16s17s18s19s20p10.930.80.980.930.90.9800.760.930.93p20.760.7610.85000.70.550.70p30.3300.9310.7600.70.70.550p40.330.70.330.400.40000Table 19
Tabular representation ofFΛd2,S (Part 1).
Ps1s2s3s4s5s6s7s8s9s10p10.470.40.40.80.870.6000.40p20.80.670.400.870.930.730.470.530p31110111010p40.60.70.70.90.50.10.60.60.40.9Table 20
Tabular representation ofFΛd2,S (Part 2).
Ps11s12s13s14s15s16s17s18s19s20p10.80.470.930.80.730.9300.40.80.8p21111001110p30.100.910.700.60.60.40p40.10.60.10.200.20000Table 21
Tabular representation ofFΛd3,S (Part 1).
Ps1s2s3s4s5s6s7s8s9s10p10.470.40.40.80.870.6000.40p20.80.670.400.870.930.730.470.530p31110111010p40.60.70.70.90.50.10.60.60.40.9Table 22
Tabular representation ofFΛd3,S (Part 2).
Ps11s12s13s14s15s16s17s18s19s20p10.80.470.930.80.730.9300.40.80.8p21111001110p30.100.910.700.60.60.40p40.10.60.10.200.20000Table 23
Tabular representation ofFΛd4,S (Part 1).
Ps1s2s3s4s5s6s7s8s9s10p10.80.760.760.930.950.85000.760p20.930.880.7600.950.980.90.80.830p30.850.780.400.70.330.7800.40p40.80.850.850.950.750.550.80.80.80.95Table 24
Tabular representation ofFΛd4,S (Part 2).
Ps11s12s13s14s15s16s17s18s19s20p10.70.20.90.70.60.900.10.70.7p20.930.9310.95000.90.850.90p30.3300.9310.7800.70.70.550p40.550.80.550.600.60000Finally, we compute the score table by finding the similarity between each row in Tables17–24 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:(36)Spi,dj=∑l20minFdjpisl,Gdjsl∑l20minFdjpisj,Gdksj.This step can be as follows:(37)Sp1,d1=min0.8,1+min0.76,1+⋯+min0.93,0+min0.93,0max0.8,1+max0.76,1+⋯+max0.93,0+max0.93,0=4.7516.7=0.28.By similar calculations, consequently, we get the score table, as in Table25.Table 25
Spi,dj.
d1d2d3d4maxp10.280.630.360.40.63p20.30.30.490.680.68p30.260.30.620.570.62p40.650.170.260.380.65It is clear from Table25 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.



---

*Source: 6469745-2022-04-15.xml*