# An Efficient Model for the Approximation of Intuitionistic Fuzzy Sets in terms of Soft Relations with Applications in Decision Making

Authors: Muhammad Zishan Anwar; Shahida Bashir; Muhammad Shabir
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The basic notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation. This paper proposes an intuitionistic fuzzy rough set (IFRS) model which is a combination of intuitionistic fuzzy set (IFS) and rough set. We approximate an IFS by using soft binary relations instead of fixed binary relations. By using this technique, we get two pairs of intuitionistic fuzzy (IF) soft sets, called the upper approximation and lower approximation with respect to foresets and aftersets. Properties of newly defined rough set model (IFRS) are studied. Similarity relations between IFS with respect to this rough set model (IFRS) are also studied. Finally, an algorithm is constructed depending on these approximations of IFSs and score function for decision-making problems, although a method of decision-making algorithm has been introduced for fuzzy sets already. But, this new IFRS model is more accurate to solve the problem because IFS has degree of nonmembership and degree of hesitant.


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## Body

## 1. Introduction

To control uncertainty usually, probability theory is deliberated as an applicable tool, but for its practical work, a randomly stable system must be a very basic requirement. To establish such kind of system, a lot of time is needed. In today’s speedy life, as everyone has shortage of time, in an unreliable environment, researchers have introduced many updated methods and techniques to solve uncertainties. Rough set (RS), fuzzy set (FS), and soft set (SS) are expressive methods to control uncertainty, vagueness, and incompleteness in the information systems. The abovementioned sets have their own operations and properties. These sets are much applicable in real life, computer science, and artificial intelligence. We are encountered by different types of real-world problems everyday which have uncertainty and vagueness rather than preciseness. Precise and complete reasoning would not be possible if our information data are inexact, vague, and incomplete. Recently, the gap between traditional mathematics with precise concepts and the world full of uncertainty become much smaller than earlier. In different fields, nature of vagueness can be different. Researchers are very active and interested to study many newly defined theories to solve this problem [1].Fuzzy set (FS) theory was introduced by Zadeh in 1965 [2] which is a very revolutionary attempt to deal with uncertainty. The FS theory is a generalization of classical set theory. It has greater richness in application than the classical set theory. It has ability to translate human linguistic terms mathematically. Although FS has the membership degree, but often the nonmembership degree is required also to handle critical situations in real-world problems. Atanassov presented IFS in 1986 [3, 4] which is the generalization of FS. The IFS describes the fuzzy characteristic of things more comprehensively than FS and thus is a powerful and successful tool to express fuzzy information of real-world problems. Elements of IFS are written in the form of ordered pairs and these ordered pairs are said to be intuitionistic fuzzy numbers or intuitionistic fuzzy values. Each intuitionistic fuzzy value is characterized by a membership degree, a nonmembership degree, and a hesitant degree. The sum of these three degrees is equal to 1. The IFSs are important in fuzzy mathematics due to its wide applications in real life, such as in pattern recognition, career determination, medical diagnosis, electoral system, and machine learning [5, 6].In 1982, rough set (RS) theory introduced by Pawlak [7] is one of the untraditional methods to control uncertainty. A subset distinguished by lower approximation and upper approximation is known as RS. Pawlak used equivalence relations to prepare approximations in a set [7, 8]. However, the equivalence relations in RS seem to be very restrictive that may limit the scope of RS model. For instance, a frequent and significant problem in the medical field is the stomach pain in the children, which can be expected to some reasons and it is a demanding job to diagnose the reason correctly. The RS theory can help the doctors to diagnose the correct reason by discharge comments. Also, it has a broad scale of applications in image processing, knowledge finding, recognition of optical characters, and pattern recognition and to recognize various facial expressions in artificial intelligence, in data clustering, in decision-making problems with precised accuracy, and in business and finance because of their capacity to find the rule induction and knowledge. Figure 1 shows the graphical representation of RS with lower approximation and upper approximation. Upper approximation is a set which has elements having possible belonging with the target set and lower approximation set has objects having positive belonging with the target set [9–11].Figure 1
Graphical representation of rough set.Figure1 shows that the target set is in red line circle, yellow box is the lower approximated set, blue box is the upper approximated set, and green box is the universal set.In 1999, Molodtsov [12, 13] introduced the key notion of SS to deal with uncertainty. This new technique is free from the problems related with existing techniques of uncertainty. An appropriate number of parameters is available in this theory which makes it possible. The SS theory has a wide variety of applications in many fields, such as operational research, the smoothness of functions, Riemann integration, and game theory. Moreover, the SSs have a rich number of operations which are very helpful to deal with uncertainty in different types of situations. The concept of parametric reduction in SSs has been studied by many authors [14, 15]. Ali et al. [16] initiated some new operations in SS theory. Abbas et al. [17] initiated various generalized operations in SS theory by applying many relaxed conditions on parameters. Applications of SSs can be found in [14, 18–26].Many extensions of SSs have been presented such as probabilistic SS theory, bijective SS theory, fuzzy bipolar SS theory, and intuitionistic fuzzy soft set (IFSS) theory. Akram et al. [27] introduced three hybrid models, namely, N-soft rough IF sets, IF N-soft sets, and IF N-soft RSs with real-life applications of decision-making algorithm. Alcantud et al. [28] presented covering-based fuzzy RS model by t-norm and fuzzy logical implicator. This fuzzy RS model is useful to characterize the covering-based optimistic and pessimistic multigranulation. They also presented two kinds of decision-making methods to analyze this model theoretically. Alcantud et al. [29] presented a tool which aggregates infinite chains of IF sets over time. They presented IF sets along an indefinitely long number of periods by using score and accuracy degrees of temporal IF elements.A parameterized collection of ordinary binary relations is called a soft binary relation on a universe and this is generalization of binary relations. In RS theory, rough approximations just address single binary relations but rough approximations by using soft binary relations can deal with different binary relations. The idea of soft relation overU is given by Feng et al. [30] in 2013. Babitha and Sunil [31] presented some results on soft set relations. Many authors have generalized the notion of Pawlak RS model by using dominance relations, covering relations, similarity relations, tolerance relations, fuzzy relations, neighbourhood relations, and other indiscernibility relations, see [28, 32–42].


### 1.1. Related Works

Feng et al. [1] presented a hybrid model of SSs which is rough approximation of SS. They used an SS instead of an equivalence relation to granulate the universe. In the result, soft approximation space and soft RSs have been introduced as a deviation of rough approximation space. Furthermore, they also extended Dubois and Prade’s RSs by approximating a FS in a soft approximation space and called soft rough FSs. Feng et al. have done a lot of work by combining SSs, RSs, and FSs and defined new models [1, 43, 44]. Ali and Shabir [45] studied fuzzy SSs. Roy and Maji [46] initiated the study of fuzzy SSs. In 2020, Bashir et al. introduced a model of rough fuzzy ternary semigroups based on three-dimensional congruence relation [47]. Many authors introduced RS approximations in IFSs [48, 49]. Kanwal and Shabir approximated the ideals and fuzzy sets in semigroups based on soft relations [50, 51]. Later in 2020, Shabir and Kanwal used soft relations to define lower and upper approximations of a set in [51]. We have generalized the concept in [51] in terms of IFS by introducing nonmembership degree. Our model IFRS gives approximations corresponding to every attribute or parameter. In this way, we get more accurate results by reducing errors than all previous ones.


### 1.2. Connection of IFRS Model with Rough Sets

The IF set theory deals mainly with vagueness, while the RS theory deals with incompleteness. The study of the combination of these two theories is useful to deal with impreciseness. It means that the rough IF sets are useful to deal with both vagueness and incompleteness. Recently, RS approximations have been discussed in IF environment. In the result, IF rough sets, rough IF sets, and generalized IF rough sets have been presented. Zhou et al. studied different relation-based IF approximation operators in the axiomatic and constructive approaches [52].


### 1.3. Innovative Contribution

Some researchers have words that one theory is better than other theory to deal with inexact data. Majority of researchers admitted that RSs and FSs are very closely related with each other, but distinction is there that they model different types of vagueness. The RS is a coarsely described crisp set, whereas the FS is viewed as a class with blunt boundaries. Since the FS has only membership degree but the IF set has also degree of nonmembership which is more useful in medical science. To diagnose a decease, IF environment is better than fuzzy environment due to the presence of nonmembership degree. In 2020, Shabir et al. [51] presented a model of RS which is combination of SS and FS. They approximated FS in terms of soft binary relations. In our paper, we considered an IF set instead of a FS and an IF set has more accurate results than a FS in medical science. The IFS with other algebraic structures generalizes hybrid models which are very useful in medical science, computer science, and other fields. Samanta and Mondal [53] presented the IF rough set A,B which is generalized IFS in terms of fuzzy rough sets A and B. On the other hand, an IF rough set A,B presented by Chakrabarty et al. [48] is the generalization of fuzzy rough set in terms of IFSs A and B. Zhou [54] proposed IF rough sets induced by IF approximation spaces and discussed their properties. Recently, IF set has combined with rough set approximations and resulting sets are called IF rough sets and rough IF and generalized IF rough sets. In axiomatic and constructive approach, Zhou et al. presented a useful framework and studied different IF rough approximation operators by using a special type of IF triangular norm min. Zhou (2014) presented IF soft rough sets and soft rough If sets. These newly presented models were very useful as new approaches for decision-making problems. By integrating IF sets with SSs, Maji et al. presented IF soft sets. Jiang et al. discussed an approach of IF soft sets-based decision making and they also presented interval-valued IF soft set model. IF soft set is an important combination of IF sets and SSs. It makes more accurate and realistic descriptions of materialistic world. Zhang presented a useful model combining IF soft sets with RSs [1, 52, 55, 56].


### 1.4. Motivation

In the present paper, we extend the concept given by Shabir et al. [51] in terms of FSs. We use IFSs instead of FSs which is more valuable to manage uncertainty in many scientific fields, such as medical diagnosis and pattern recognition. Our proposed model based on soft relations is very useful due to importance of IFS in real-life situations [5, 6]. In our research, we propose a decision-making algorithm by using our model based on soft relations. Then, we present an example to illustrate the validity of our proposed decision-making algorithm. Our IFRS model is the combination of RS, IFS, and SS which is helpful to control impreciseness and uncertainty.


### 1.5. Organization of the Paper

The pattern of this paper is as follows. In Section2, some foundational concepts are identified with FSs, IFS, RSs, and SSs and soft binary relations are described. In Section 3, we presented IFRS model based on soft relations and discussed some properties. Soft similarity relations have been examined in Section 4. In Section 5, we gave an approach to a decision-making problem based on an IFS. Moreover, an example is presented to illustrate this decision-making algorithm in Section 6.


## 1.1. Related Works

Feng et al. [1] presented a hybrid model of SSs which is rough approximation of SS. They used an SS instead of an equivalence relation to granulate the universe. In the result, soft approximation space and soft RSs have been introduced as a deviation of rough approximation space. Furthermore, they also extended Dubois and Prade’s RSs by approximating a FS in a soft approximation space and called soft rough FSs. Feng et al. have done a lot of work by combining SSs, RSs, and FSs and defined new models [1, 43, 44]. Ali and Shabir [45] studied fuzzy SSs. Roy and Maji [46] initiated the study of fuzzy SSs. In 2020, Bashir et al. introduced a model of rough fuzzy ternary semigroups based on three-dimensional congruence relation [47]. Many authors introduced RS approximations in IFSs [48, 49]. Kanwal and Shabir approximated the ideals and fuzzy sets in semigroups based on soft relations [50, 51]. Later in 2020, Shabir and Kanwal used soft relations to define lower and upper approximations of a set in [51]. We have generalized the concept in [51] in terms of IFS by introducing nonmembership degree. Our model IFRS gives approximations corresponding to every attribute or parameter. In this way, we get more accurate results by reducing errors than all previous ones.


## 1.2. Connection of IFRS Model with Rough Sets

The IF set theory deals mainly with vagueness, while the RS theory deals with incompleteness. The study of the combination of these two theories is useful to deal with impreciseness. It means that the rough IF sets are useful to deal with both vagueness and incompleteness. Recently, RS approximations have been discussed in IF environment. In the result, IF rough sets, rough IF sets, and generalized IF rough sets have been presented. Zhou et al. studied different relation-based IF approximation operators in the axiomatic and constructive approaches [52].


## 1.3. Innovative Contribution

Some researchers have words that one theory is better than other theory to deal with inexact data. Majority of researchers admitted that RSs and FSs are very closely related with each other, but distinction is there that they model different types of vagueness. The RS is a coarsely described crisp set, whereas the FS is viewed as a class with blunt boundaries. Since the FS has only membership degree but the IF set has also degree of nonmembership which is more useful in medical science. To diagnose a decease, IF environment is better than fuzzy environment due to the presence of nonmembership degree. In 2020, Shabir et al. [51] presented a model of RS which is combination of SS and FS. They approximated FS in terms of soft binary relations. In our paper, we considered an IF set instead of a FS and an IF set has more accurate results than a FS in medical science. The IFS with other algebraic structures generalizes hybrid models which are very useful in medical science, computer science, and other fields. Samanta and Mondal [53] presented the IF rough set A,B which is generalized IFS in terms of fuzzy rough sets A and B. On the other hand, an IF rough set A,B presented by Chakrabarty et al. [48] is the generalization of fuzzy rough set in terms of IFSs A and B. Zhou [54] proposed IF rough sets induced by IF approximation spaces and discussed their properties. Recently, IF set has combined with rough set approximations and resulting sets are called IF rough sets and rough IF and generalized IF rough sets. In axiomatic and constructive approach, Zhou et al. presented a useful framework and studied different IF rough approximation operators by using a special type of IF triangular norm min. Zhou (2014) presented IF soft rough sets and soft rough If sets. These newly presented models were very useful as new approaches for decision-making problems. By integrating IF sets with SSs, Maji et al. presented IF soft sets. Jiang et al. discussed an approach of IF soft sets-based decision making and they also presented interval-valued IF soft set model. IF soft set is an important combination of IF sets and SSs. It makes more accurate and realistic descriptions of materialistic world. Zhang presented a useful model combining IF soft sets with RSs [1, 52, 55, 56].


## 1.4. Motivation

In the present paper, we extend the concept given by Shabir et al. [51] in terms of FSs. We use IFSs instead of FSs which is more valuable to manage uncertainty in many scientific fields, such as medical diagnosis and pattern recognition. Our proposed model based on soft relations is very useful due to importance of IFS in real-life situations [5, 6]. In our research, we propose a decision-making algorithm by using our model based on soft relations. Then, we present an example to illustrate the validity of our proposed decision-making algorithm. Our IFRS model is the combination of RS, IFS, and SS which is helpful to control impreciseness and uncertainty.


## 1.5. Organization of the Paper

The pattern of this paper is as follows. In Section2, some foundational concepts are identified with FSs, IFS, RSs, and SSs and soft binary relations are described. In Section 3, we presented IFRS model based on soft relations and discussed some properties. Soft similarity relations have been examined in Section 4. In Section 5, we gave an approach to a decision-making problem based on an IFS. Moreover, an example is presented to illustrate this decision-making algorithm in Section 6.


## 2. Preliminaries and Basic Concepts

In this section, some basic notions about binary relations, IFS, soft sets, and intuitionistic fuzzy soft sets are given. Throughout this paper,U1 and U2 represent two nonempty finite sets unless stated otherwise.A binary relationℛ from U1 to U2 is a subset of U1×U2 and a subset of U×U is called a binary relation on U. If ℛ is a binary relation on U, then ℛ is said to be reflexive if u,u∈ℛ for all u∈U, symmetric if u,v∈ℛ implies v,u∈ℛ for all u,v∈U, and transitive if u,v∈ℛ and v,w∈ℛ imply u,w∈ℛ for all u,v,w∈U. If a binary relation ℛ is reflexive, symmetric, and transitive, then it is called an equivalence relation. An equivalence relation partitions the set into disjoint classes.LetU be a nonempty universe. An IF set M in the universe U is an object having the form M=x,μMx,γMx:x∈U, where μM:U⟶0,1 and γM:U⟶0,1, satisfying 0≤μMx+γMx≤1 for all x∈U. The values μMx and γMx are called degree of membership and degree of nonmembership of x∈U to M, respectively. The number πMx=1−μMx−γMx is called the degree of hesitancy of x∈U to M. The collection of all IFSs in U is denoted by IFU. In the remaining paper, we shall write an IFS by M=μM,γM instead of M=x,μMx,γMx:x∈U. Let M=μM,γM and N=μN,γN be two IFSs in U. Then, M⊆N if and only if μMx≤μNx and γNx≤γMx for all x∈U. Two IFSs M and N are said to be equal if and only if M⊆N and N⊆M. The union and intersection of two IFSs M and N in U are denoted and defined by M∩N=μM∩μN,γM∪γN and M∪N=μM∪μN,γM∩γN, where μM∩μNx=infμMx,μNx, γM∪γNx=supγMx,γNx, μM∪μNx=supμMx,μNx, γM∩γNx=infγMx,γNx.Next, we define two special types of (IFSs) as follows:The IF universe setU=1U=1,0 and IF empty set ∅=0U=0,1, where 1x=1 and 0x=0 for all x∈U. The complement of an IFS M=μ,γ is denoted and defined as Mc=γ,μ [3].For a fixedx∈U, the pair μMx,γMx is called IF value or IF number. In order to define the order between two IFNs, Chen and Tan [57] presented the score function as Sx=μMx−γMx and Hong and Choi [58] defined the accuracy function as Hx=μMx+γMx, where x is an IFV. Xu [59,60] combined the accuracy and score functions and formed the order relations between any pair of IFVs x,y as follows:(i)
IfSx>Sy, then x>y;(ii)
IfSx=Sy, then(a)
IfHx=Hy, then x=y;(b)
IfHx<Hy, then x<y.A pairF,A is called a soft set over U if F is a mapping given by F:A⟶PU, where A is a subset of E (the set of parameters) and PU is the power set of U. Thus, Fe is a subset of U for all e∈A. Hence, a soft set over U is a parametrized collection of subsets of U. A pair F,A is called an intuitionistic fuzzy soft set over U if F is a mapping given by F:A⟶IFU and A is a subset of E (the set of parameters). Thus, Fe is an IF set in U for all e∈A. Hence, an IF soft set over U is a parametrized collection of IF sets in U. For two IF soft sets F,A and G,B over a common universe U, we say that F,A is an IF soft subset of G,B if 1A⊆B and 2Fe is an IF subset of Ge for all e∈A. Two IF soft sets F,A and G,B over a common universe U are said to be IF soft equal ifF,A is an IF soft subset of G,B and G,B is an IF soft subset of F,A. The union of two IF soft sets F,A and G,A over the common universe U is the IF soft set H,A, where He=Fe∪Ge for all e∈A. The intersection of two IF soft sets F,A and G,A over the common universe U is the IF soft set K,A, where Ke=Fe∩Ge for all e∈A [12, 61, 62].An IF soft set can be represented by a table, which is shown in the following example.Example 1.
LetU=x,y,z,s,t, A=e1,e2. Consider an IF soft set F,A over U defined by Fe1x=0.3,0.4,Fe1y=0.4,0.3,Fe1z=0.4,0.2, Fe1s=0.8,0.1,Fe1t=0.2,0.6 and Fe2x=0.9,0,Fe2y=0.5,0.5,Fe2z=0.4,0.5, Fe2s=0.3,0.7,Fe2t=0.6,0.3.
The above intuitionistic fuzzy soft set can be represented as in Table1.Table 1
Representation of IF soft set.
Ue1e2x0.3,0.40.9,0y0.4,0.30.5,0.5z0.4,0.20.4,0.5s0.8,0.10.3,0.7t0.2,0.60.6,0.3


## 3. Approximations of an IFS by Soft Binary Relation

In this section, we consider soft binary relation fromU1 to U2 and approximate an IFS of U2 by using aftersets and get two IF soft sets of U1. Similarly, we approximate an IFS of U1 by using foresets and get two IF soft sets of U2. We also study some properties of these approximations.Definition 1.
(see [30]). A soft binary relation σ,A from U1 to U2 is a soft set over U1×U2, that is, σ:A⟶PU1×U2, where A is a subset of the set of parameters E.
Of course,σ,A is a parameterized collection of binary relations from U1 to U2. That is, for each e∈A, we have a binary relation σe from U1 to U2.Definition 2.
Letσ,A be a soft binary relation from U1 to U2 and M=μM,γM be an IFS in U2. Then, we define lower approximation σ¯M=σ¯μM,σ¯γM and upper approximation σ¯M=σ¯μM,σ¯γM of M=μM,γM with respect to aftersets as follows:(1)σ¯μMeu1=∧a∈u1σeμMa,ifu1σe≠∅,1,ifu1σe=∅,σ¯γMeu1=∨a∈u1σeγMa,ifu1σe≠∅,0,ifu1σe=∅,σ¯μMeu1=∨a∈u1σeμMa,ifu1σe≠∅,0,u1σe=∅,σ¯γMeu1=∧a∈u1σeγMa,ifu1σe≠∅,1,ifu1σe=∅,where u1σe=a∈U2:u1,a∈σe and is called the afterset of u1 for u1∈U1 and e∈A.(i)
σ¯μMeu1 indicates the degree to which u1 definitely have the property e.(ii)
σ¯γMeu1 indicates the degree to which u1 probably do not have the property e.(iii)
σ¯μMeu1 indicates the degree to which u1 probably have the property e.(iv)
σ¯γMeu1 indicates the degree to which u1 definitely do not have the property e.
In Definition2, soft binary relation from U1 to U2 is given and IFS in U2 can be approximated as lower and upper approximations with respect to the aftersets. The resulting sets are two pairs of IF soft sets.Definition 3.
Letσ,A be a soft binary relation from U1 to U2 and M=μM,γM be an IFS in U1. Then, we define lower approximation σ¯M=μMσ¯,γMσ¯ and upper approximation Mσ¯=μMσ¯,γMσ¯ of M=μM,γM with respect to foresets as follows:(2)μMσ¯eu2=∧a∈σeu2μMa,ifσeu2≠∅,1,ifσeu2=∅,σ¯γMeu2=∨a∈σeu2γMa,ifσeu2≠∅,0,ifσeu2=∅,σ¯μMeu2=∨a∈σeu2μMaifσeu2≠∅,0ifσeu2=∅,σ¯γMeu2=∧a∈σeu2γMa,ifσeu2≠∅,1,ifσeu2=∅,where σeu2=a∈U1:a,u2∈σe and is called the foreset ofu2 for u2∈U2 and e∈A. Of course, σ¯M:A⟶IFU1, σ¯M:A⟶IFU1 and Mσ¯:A⟶IFU2, σ¯M:A⟶IFU2. The following example explains these concepts.
In Definition3, soft binary relation from U1 to U2 is given and IFS in U1 can be approximated as lower and upper approximations with respect to the foresets. The resulting sets are two pairs of IF soft sets.Example 2.
Suppose thatMr. X wants to buy a shirt for his own use. Let U1=the set of all shirts designs=d1,d2,d3,d4,d5,d6 and U2=the colors of all designs=c1,c2,c3,c4 and the set of attributes be A=e1,e2,e3=the set of stores near his house.
Defineσ:A⟶PU1×U2 by(3)σe1=d1,c1,d1,c2,d1,c3,d2,c2,d2,c4,d4,c2,d4,c3,d5,c3,d5,c4,d6,c1,σe2=d1,c3,d2,c3,d4,c1,d5,c1,d6,c2,d6,c3,σe3=d2,c4,d3,c1,d3,c3,d5,c3,d5,c4,which represents the relation between designs and colors available on store ei for 1≤i≤3. Then,(4)d1σe1=c1,c2,c3,d2σe1=c2,c4,d3σe1=∅,d4σe1=c2,c3,d5σe1=c3,c4,d6σe1=c1d1σe2=c3,d2σe2=c3,d3σe2=∅,d4σe2=c1,d5σe2=c1,d6σe2=c2,c3d1σe3=∅,d2σe3=c4,d3σe3=c1,c3,d4σe3=∅,d5σe3=c3,c4,d6σe3=∅,where diσej represents the color of the design di available on the store ej.
Also,(5)σe1c1=d1,d6,σe1c2=d1,d2,d4,σe1c3=d1,d4,d5,σe1c4=d2,d5,σe2c1=d4,d5,σe2c2=d6,σe2c3=d1,d2,d6,σe2c4=∅,σe3c1=d3,σe3c2=∅,σe3c3d3,d5,σe3c4=d2,d5,where σejci represents the design of the color ci available on the store ej.(6)DefineM=μM,γM:U2⟶0,1which represents the preference of thecolors given by Mr.Xsuch thatμMc1=0.9,μMc2=0.8,μMc3=0.4,μMc4=0γMc1=0.0,γMc2=0.2,γMc3=0.5,γMc4=0.8.DefineN=μN,γN:U1⟶0,1which represents the preference of thedesigns given by Mr.Xsuch thatμNd1=1,μNd2=0.7,μNd3=0.5,μNd4=0.1,μNd5=0,μNd6=0.4γNd1=0,γNd2=0.2,γNd3=0.5,γNd4=0.7,γNd5=1,γNd6=0.5.
Therefore, the lower and upper approximations (with respect to the aftersets as well as with respect to the foresets) are(7)σ¯M=σ¯μM,σ¯γMgiven in Table2,σ¯M=σ¯μM,σ¯γMgiven in Table3,and(8)Nσ¯=σ¯μN,σ¯γNgiven in Table4Nσ¯=μNσ¯,γNσ¯given in Table5.
Table2 shows the lower approximation of IFS M with respect to the aftersets by using Definition 2. Table 3 shows the upper approximation of IFS M with respect to the aftersets by using Definition 2. Table 4 shows the lower approximation of IFS N with respect to the foresets by using Definition 3. Table 5 shows the upper approximation of IFS N with respect to the foresets by using Definition 3.Table 2
Lower approximation of intuitionistic fuzzy setM.
σ¯Me1e2e3d10.4,0.50.4,0.51,0d20,0.80.4,0.50,0.8d31,01,00.4,0.5d40.4,0.50.9,01,0d50,0.80.9,00,0.8d60.9,00.4,0.51,0Table 3
Upper approximation of intuitionistic fuzzy setM.
σ¯Me1e2e3d10.9,00.4,0.50,1d20.8,0.20.4,0.50,0.8d30,10,10.9,0d40.8,0.20.9,00,1d50.4,0.50.9,00.4,0.5d60.9,00.8,0.20,1Table 4
Lower approximation of intuitionistic fuzzy setN.
σ¯Ne1e2e3c10.4,0.50,10.5,0.5c20.1,0.70.4,0.51,0c30,10.4,0.50,1c40,11,00,1Table 5
Upper approximation of intuitionistic fuzzy setN.
Nσ¯e1e2e3c11,00.1,0.70.5,0.5c21,00.4,0.50,1c31,01,00.5,0.5c40.7,0.20,10.7,0.2Theorem 1.
Letσ,A be a soft binary relation from U1 to U2, that is, σ:A⟶PU1×U2. For any IFSs, M=μM,γM, N=μN,γN, and P=μP,γP of U2, the following are true:(1)
IfN⊆P, thenσ¯N⊆σ¯P;(2)
IfN⊆P, thenσ¯N⊆σ¯P;(3)
σ¯N∩σ¯P=σ¯N∩P;(4)
σ¯N∩σ¯P⊇σ¯N∩P;(5)
σ¯N∪σ¯P⊆σ¯N∪P;(6)
σ¯N∪σ¯P=σ¯N∪P;(7)
σ¯1U2=1U1ifu1σe≠∅;(8)
σ¯1U2=1U1ifu1σe≠∅;(9)
σ¯M=σ¯Mccifu1σe≠∅;(10)
σ¯M=σ¯Mccifu1σe≠∅;(11)
σ¯0U2=0U1=σ¯0U2ifu1σe≠∅.Proof.
(1)
Letu1∈U1. If u1σe=∅, then σ¯μNeu1=1=σ¯μPeu1 and σ¯γNeu1=0=σ¯γPeu1. If u1σe≠∅, then σ¯μNeu1=∧a∈u1σeμNa≤∧a∈u1σeμPa because μNa≤μPa=σ¯μPeu1.Thus,σ¯μNeu1≤σ¯μPeu1.Also,σ¯γNeu1=∨a∈u1σeγNa≥∨a∈u1σeγPa because γNa≥γPa=σ¯γPeu1.Thus,σ¯γNeu1≥σ¯γPeu1. Hence, σ¯N⊆σ¯P.(2)
Letu1∈U1. If u1σe=∅, then σ¯μNeu1=0=σ¯μPeu1 and σ¯γNeu1=1=σ¯γPeu1. If u1σe≠∅, thenσ¯μNeu1=∨a∈u1σeμNa≤∨a∈u1σeμPa because μNa≤μPa=σ¯μPeu1.Thus,σ¯μNeu1≤σ¯μPeu1.Also,σ¯γNeu1=∧a∈u1σeγNa≥∧a∈u1σeγPa because γNa≥γPa=σ¯γPeu1.Thus,σ¯γNeu1≥σ¯γPeu1. Hence, σ¯N⊆σ¯P.(3)
Letu1∈U1. If u1σe=∅, then σ¯μN∩Peu1=1=σ¯μNeu1∩σ¯μPeu1 and σ¯γN∩Peu1=0=σ¯γNeu1∪σ¯γPeu1. If u1σe≠∅, thenσ¯μN∩σ¯μPeu1=σ¯μNeu1∧σ¯μPeu1=∧a∈u1σeμNa∧∧a∈u1σeμPa=∧a∈u1σeμNa∧μPa=∧a∈u1σeμN∧μPa=∧a∈u1σeμN∩Pa=σ¯μN∩Peu1.Also,σ¯γN∪σ¯γPeu1=σ¯γNeu1∨σ¯γPeu1=∨a∈u1σeγNa∨∨a∈u1σeγPa=∨a∈u1σeγNa∨γPa=∨a∈u1σeγN∨γPa=∨a∈u1σeγN∩Pa=σ¯γN∩Peu1.This shows thatσ¯N∩σ¯P=σ¯N∩P.(4)
SinceN∩P⊆N and N∩P⊆P, we have from part (2) σ¯N∩P⊆σ¯N and σ¯N∩P⊆σ¯P. Thus, σ¯N∩P⊆σ¯N∩σ¯P.(5)
SinceN∪P⊇N and N∪P⊇P, we have from part (1) σ¯N∪P⊇σ¯N and σ¯N∪P⊇σ¯P. Thus, σ¯N∪P⊇σ¯N∪σ¯P.(6)
Letu1∈U1. If u1σe=∅, then σ¯μN∪Peu1=0=σ¯μNeu1∪σ¯μPeu1 and σ¯γN∪Peu1=1=σ¯γNeu1∩σ¯γPeu1. If u1σe≠∅, then σ¯μN∪σ¯μPeu1=σ¯μNeu1∨σ¯μPeu1=∨a∈u1σeμNa∨∨a∈u1σeμPa=∨a∈u1σeμNa∨μPa=∨a∈u1σeμN∨μPa=∨a∈u1σeμN∪Pa=σ¯μN∪Peu1.Also,σ¯γN∩σ¯γPeu1=σ¯γNeu1∧σ¯γPeu1=∧a∈u1σeγNa∧∧a∈u1σeγPa=∧a∈u1σeγNa∧γPa=∧a∈u1σeγN∧γPa=∧a∈u1σeγN∪Pa=σ¯γN∪Peu1.This shows thatσ¯N∪σ¯P=σ¯N∪P.(7)
Considerσ¯1eu1=∧a∈u1σe1a=∧a∈u1σe1=1, because u1σe≠∅andσ¯0eu1=∨a∈u1σe0a=∨a∈u1σe0=0, because u1σe≠∅.Thus,σ¯1U2=1U1.(8)
The proof is similar to the proof of part (7).(9)
LetM=<μM,γM> be an IFS on U2. Then, Mc=<μMc,γMc>=<γM,μM>, that is, μMc=γM and γMc=μM. Now, σ¯Mc=σ¯μMc,σ¯γMc=σ¯γM,σ¯μM. Thus, σ¯μMceu1=∨a∈u1σeμMca=∨a∈u1σeγMa=σ¯γMeu1 and σ¯γMceu1=∧a∈u1σeγMca=∧a∈u1σeμMa=σ¯μMeu1. Hence, σ¯Mc=σ¯μMc,σ¯γMc=σ¯γM,σ¯μM=σ¯μM,σ¯γMc, that is, σ¯Mcc=σ¯M.(10)
follows from part (9).(11)
Straightforward.
Theorem1 describes the properties of newly defined IFRS model based on soft relations. It shows that if an IFS N is the subset of IFS P, then the lower approximation of N is also a subset of the lower approximation of P, and if an IFS N is subset of IFS P, then the upper approximation of N is also a subset of the upper approximation of P. Similarly, the empirical relations among the operations union, intersection, and complement have been described.Theorem 2.
Letσ,A be a soft binary relation from U1 to U2, that is, σ:A⟶PU1×U2. For any IFSs, M=μM,γM, N=μN,γN, and P=μP,γP of U1, the following are true:(1)
IfN⊆P, thenNσ¯⊆Pσ¯;(2)
IfN⊆P, thenNσ¯⊆Pσ¯;(3)
Nσ¯∩Pσ¯=N∩Pσ¯;(4)
Nσ¯∩Pσ¯⊇N∩Pσ¯;(5)
Nσ¯∪Pσ¯⊆N∪Pσ¯;(6)
Nσ¯∪Pσ¯=N∪Pσ¯;(7)
σ¯1U1=1U2ifu1σe≠∅;(8)
σ¯1U1=1U2ifu1σe≠∅;(9)
Mσ¯=Mcσ¯cifu1σe≠∅;(10)
Mσ¯=Mcσ¯cifu1σe≠∅;(11)
0U1σ¯=0U2=0U1σ¯.Proof.
The proof is similar to the proof of Theorem1.
The following example shows that equality does not hold in (4) and (5) assertions of above theorems in general.Example 3.
Consider Example 2. LetP=μP,γP:U2⟶0,1 (given in Table 6).
Table6 simply shows the degree of membership and degree of nonmembership of IFS P.
Then,M∪P=μM∪P,γM∪P=μM∪μP,γM∩γP (given in Table 7) and M∩P=μM∩P,γM∩P=μM∩μP,γM∪γP (given in Table 7).
Table7 shows the calculations of union and intersection of two IFSs M and P, respectively.
Now,σ¯P=σ¯μP,σ¯γP (given in Table 8) and σ¯M∪P=σ¯μM∪P,σ¯γM∪P (given in Table 9).
Now,σ¯M∪σ¯P (given in Table 10).
In Table10, we calculated σ¯M∪σ¯P. Tables 9 and 10 show that σ¯M∪σ¯P≠σ¯M∪P.
Now,σ¯P=σ¯μP,σ¯γP (given in Table 11).
In Table11, we calculated upper approximation of P.
Now,σ¯M∩P=σ¯μM∩P,σ¯γM∩P (given in Table 12).
Now,σ¯M∩σ¯P (given in Table 13).
In Table12, we calculated upper approximation of M∩P. In Table 13, we calculated the intersection of upper approximations of M and P. Tables 12 and 13 show that σ¯M∩σ¯P≠σ¯M∩P.Table 6
Intuitionistic fuzzy setP.
c1c2c3c4μP0.10.60.51γP0.90.10.50Table 7
Intuitionistic fuzzy setsM∪P, M∩P.
c1c2c3c4M∪P0.9,00.8,0.10.5,0.51,0M∩P0.1,0.90.6,0.20.4,0.50,0.8Table 8
Lower approximation ofP.
σ¯Pe1e2e3d10.1,0.90.5,0.51,0d20.6,0.10.5,0.51,0d31,01,00.1,0.9d40.5,0.50.1,0.91,0d50.5,0.50.1,0.90.5,0.5d60.1,0.90.5,0.51,0Table 9
Lower approximation ofM∪P.
σ¯M∪Pe1e2e3d10.5,0.50.5,0.51,0d20.8,0.10.5,0.51,0d31,01,00.5,0.5d40.5,0.50.9,01,0d50.5,0.50.9,00.5,0.5d60.9,00.5,0.51,0Table 10
Union of lower approximations ofM and P.
σ¯M∪σ¯Pe1e2e3d10.4,0.50.5,0.51,0d20.6,0.10.5,0.51,0d31,01,00.4,0.5d40.5,0.50.9,01,0d50.5,0.50.9,00.5,0.5d60.9,00.5,0.51,0Table 11
Upper approximation ofP.
σ¯Pe1e2e3d10.6,0.10.5,0.50,1d21,00.5,0.51,0d30,10,10.5,0.5d40.6,0.10.1,0.90,1d51,00.1,0.91,0d60.1,0.90.6,0.10,1Table 12
Upper approximation ofM∩P.
σ¯M∩Pe1e2e3d10.6,0.20.4,0.50,1d20.6,0.20.4,0.50,0.8d30,10,10.4,0.5d40.6,0.20.1,0.90,1d50.4,0.50.1,0.90.4,0.5d60.1,0.90.6,0.20,1Table 13
Intersection of upper approximations ofM and P.
σ¯M∩σ¯Pe1e2e3d10.6,0.10.4,0.50,1d20.8,0.20.4,0.50,0.8d30,10,10.5,0.5d40.6,0.20.1,0.90,1d50.4,0.50.1,0.90.4,0.5d60.1,0.90.6,0.20,1Theorem 3.
Letσ1,A and σ2,A be two soft binary relations from U1 to U2, such that σ1,A⊆σ2,A, that is, σ1e⊆σ2e for all e∈A. Then, for any IFS M=μM,γM of U2, the following are true:(i)
σ2¯M⊆σ1¯M;(ii)
σ¯1M⊆σ¯2M.Proof.
(1)
Letu1∈U1. If u1σ1e=∅, then σ1¯μMeu1=1≥σ2¯μMeu1 and σ1¯γMeu1=0≤σ2¯γMeu1. If u1σ1e≠∅, then u1σ2e≠∅, we haveσ1¯μMeu1=∧a∈u1σ1eμMa≥∧a∈u1σ2eμMa because u1σ1e⊆u1σ2e=σ2¯μMeu1.Also,σ1¯γMeu1=∨a∈u1σ1eγMa≤∨a∈u1σ2eγMa because u1σ1e⊆u1σ2e=σ2¯γMeu1.Hence,σ2¯M⊆σ1¯M.(2)
Letu1∈U1. If u1σ1e=∅, then σ1¯μMeu1=0≤σ2¯μMeu1 and σ¯1γMeu1=1≥σ2¯γMeu1. If u1σ1e≠∅, then u1σ2e≠∅, and we haveσ1¯μMeu1=∨a∈u1σ1eμMa≤∨a∈u1σ2eμMa because u1σ1e⊆u1σ2e=σ2¯μMeu1.Also,σ1¯γMeu1=∧a∈u1σ1eγMa≥∧a∈u1σ2eγMa because u1σ1e⊆u1σ2e=σ2¯γMeu1.Hence,σ1¯M⊆σ2¯M.
Theorem3 shows that if any soft relation σ1,A⊆σ2,A, then for any IFS M in U2, the lower approximation associated with σ2,A is a subset of σ1,A. Similarly, if any soft relation σ1,A⊆σ2,A, then for any IFS M in U2, the upper approximation associated with σ1,A is a subset of σ2,A.Theorem 4.
Letσ1,A and σ2,A be two soft binary relations from U1 to U2, such that σ1,A⊆σ2,A, that is, σ1e⊆σ2e for all e∈A. Then, for any IFS M=μM,γM of U1, the following are true:(1)
Mσ2¯⊆Mσ1¯(2)
σ¯M1⊆σ¯M2Proof.
(1)
Letu2∈U2. If σ1eu2=∅, then μMσ1¯eu2=1≥μMσ2¯eu2 and σ1¯γMeu2=0≤σ2¯γMeu2. If σ1eu2≠∅, then σ2eu2≠∅, we haveσ1¯μMeu2=∧a∈σ1eu2μMa≥∧a∈σ2eu2μMa because σ1eu2⊆σ2eu2=σ2¯μMeu2.Also,σ1¯γMeu2=∨a∈σ1eu2γMa≤∨a∈σ2eu2γMa because σ1eu2⊆σ2eu2=σ2¯γMeu2.Hence,σ2¯M⊆σ1¯M.(2)
Letu2∈U2. If σ1eu2=∅, then σ1¯μMeu2=0≤σ2¯μMeu2 and σ1¯γMeu2=1≥σ2¯γMeu2. If σ1eu2≠∅, then σ2eu2≠∅, and we haveσ1¯μMeu2=∨a∈σ1eu2μMa≤∨a∈σ2eu2μMa because σ1eu2⊆σ2eu2=σ2¯μMeu2.Also,σ1¯γMeu2=∧a∈σ1eu2γMa≥∧a∈σ2eu2γMa because σ1eu2⊆σ2eu2=γMσ2¯eu2.Hence,Mσ1¯⊆Mσ2¯.Theorem 5.
Letσ1,A and σ2,A be two soft binary relations from U1 to U2. Then, for any IFS M=μM,γM of U2, the following are true:(1)
σ1¯M⊆σ1∩σ2¯M;(2)
σ2¯M⊆σ1∩σ2¯M;(3)
σ1∩σ2¯M⊆σ¯1M;(4)
σ1∩σ2¯M⊆σ¯2M.Proof.
(1)
Asσ1∩σ2⊆σ1, therefore from Theorem 3 part (1), σ1¯M⊆σ1∩σ2¯M.(2) Asσ1∩σ2⊆σ2, therefore from Theorem 3 part (1), σ2¯M⊆σ1∩σ2¯M.(3) Asσ1∩σ2⊆σ1, therefore from Theorem 3 part (2), σ1∩σ2¯M⊆σ¯1M.(4) Asσ1∩σ2⊆σ2, therefore from Theorem 3 part (2), σ1∩σ2¯M⊆σ¯2M.Theorem 6.
Letσ1,A and σ2,A be two soft binary relations from U1 to U2. Then, for any IFS M=μM,γM of U1, the following are true:(1)
σ1¯M⊆Mσ1∩σ2¯;(2)
Mσ2¯⊆Mσ1∩σ2¯;(3)
Mσ1∩σ2¯⊆Mσ¯1;(4)
Mσ1∩σ2¯⊆Mσ¯2.Proof.
(1)
Asσ1∩σ2⊆σ1, therefore from Theorem 4 part (1), Mσ1¯⊆Mσ1∩σ2¯.(2)
Asσ1∩σ2⊆σ2, therefore from Theorem 4 part (1), Mσ2¯⊆Mσ1∩σ2¯.(3)
Asσ1∩σ2⊆σ1, therefore from Theorem 4 part (2), Mσ1∩σ2¯⊆Mσ¯1.(4)
Asσ1∩σ2⊆σ2, therefore from Theorem 4 part (2), Mσ1∩σ2¯⊆Mσ¯2.
In Theorems5 and 6, some empirical relations have been discussed about union and intersection of two soft relations σ1,A and σ2,A with respect to the aftersets and with respect to the foresets, respectively.Definition 4.
Ifσ,A is a soft set over U×U, then σ,A is called a soft binary relation on U.
In fact,σ,A is a parameterized collection of binary relations on U. That is, for each parameter e∈A, we have a binary relation σe on U. A soft binary relation σ,A on U is said to be soft reflexive relation on U if σe is a reflexive relation on U for all e∈A. If σ,A is a soft reflexive binary relation on U, then uσeresp.σeu is nonempty and u∈uσeresp.u∈σeu. It is not necessary that uσe=σeu. A soft binary relation σ,A on U is said to be soft symmetric relation on U if σe is a symmetric relation on U for all e∈A. A soft binary relation σ,A on U is said to be soft transitive relation on U if σe is a transitive relation on U for all e∈A.
A soft binary relationσ,A over U is soft equivalence relation over U if it is soft reflexive, soft symmetric, and soft transitive relation over U. A soft binary relation σ,A over U is a soft equivalence relation over U if σe for all e∈A is an equivalence relation over U. In this case, uσe=σeu and uσe:u∈U is a partition of U. Also, in this case, σ¯Me=σ¯eM and σ¯Me=σ¯eM, for any IFS M=μM,γM of U.
The approximation operators have additional properties with respect to soft reflexive binary relation as follows.Theorem 7.
Letσ,A be a soft reflexive binary relation on U. Then, for any IFS M=μM,γM of U, the following are true:(1)
σ¯μMe≤μMfor alle∈A;(2)
μM≤σ¯μMefor alle∈A;(3)
σ¯μMe≤σ¯μMefor alle∈A;(4)
σ¯γMe≥γMfor alle∈A;(5)
γM≥σ¯γMefor alle∈A;(6)
σ¯γMe≥σ¯γMefor alle∈A.Proof.
(1)
Letu∈U. Then,σ¯μMeu=∧a∈uσ1eμMa≤μMu because u∈uσe.(2)
Hence,σ¯μMe≤μM.Letu∈U. Then,σ¯μMeu=∨a∈uσ1eμMa≥μMu because u∈uσeHence,μM≤σ¯μMe.(3)
It follows from part (1) and part (2).(4)
Letu∈U. Then,σ¯γMeu=∨a∈uσ1eγMa≥γMu because u∈uσeHence,σ¯γMe≥γM.(5)
Letu∈U. Then,σ¯γMeu=∧a∈uσ1eγMa≤γMu because u∈uσeHence,γM≥σ¯γMe.It follows from part (4) and part (5).
Theorem7 shows the empirical relations between IFS M and a soft reflexive relation σ,A.Theorem 8.
Letσ,A be a soft reflexive binary relation on U. Then, for any IFS M=μM,γM of U, the following are true:(1)
μMσ¯e≤μM for all e∈A;(2)
μM≤μMσ¯e for all e∈A;(3)
μMσ¯e≤μMσ¯e for all e∈A;(4)
γMσ¯e≥γM for all e∈A;(5)
γM≥γMσ¯e for all e∈A;(6)
γMσ¯e≥γMσ¯e for all e∈A.Proof.
(1)
Letu∈U. Then,σ¯μMeu=∧a∈σ1euμMa≤μMu because u∈σeu.Hence,σ¯μMe≤μM.(2)
Letu∈U. Then,σ¯μMeu=∨a∈σ1euμMa≥μMu because u∈σeuHence,μM≤μMσ¯e.(3)
It follows from part (1) and part (2).(4)
Letu∈U. Then,σ¯γMeu=∨a∈σ1euγMa≥γMu because u∈σeuHence,γMσ¯e≥γM.(5)
Letu∈U. Then,γMσ¯eu=∧a∈σ1euγMa≤γMu because u∈σeu(i)
Hence,γM≥γMσ¯e.(6)
It follows from part (4) and part (5).


## 4. Similarity Relations

In this section, we define some relations between IFS ofU2 with the help of a soft relation from U1 to U2. We say that two intuitionistic fuzzy sets in U2 are related if the lower (upper) approximations in U1 are equal. Similarly, we define relations between intuitionistic fuzzy sets of U1.Definition 5.
Letσ,A be a soft binary relation from U1 to U2. Then, for any IFS N=μN,γN and P=μP,γP of U2, we define
N≃AP if and only if σ¯N=σ¯PN≂AP if and only if σ¯N=σ¯PN≈AP if and only if σ¯N=σ¯P and σ¯N=σ¯P.Definition 6.
Letσ,A be a soft binary relation from U1 to U2. Then, for any IFS N=μN,γN and P=μP,γP of U1, we define
N≃σP if and only if σ¯N=σ¯PN≂σP if and only if σ¯N=σ¯PN≈σP if and only if Nσ¯=σ¯P and σ¯N=σ¯P.
These binary relations may be called the lower similarity relation, upper similarity relation, and similarity relation, respectively.
Definitions5 and 6 show that if an IFS N has upper(lower) similarity relation with an IFS P, then its associated lower(upper) approximation has also upper(lower) similarity relation.Proposition 1.
The relations≃A, ≂A, and ≈A are equivalence relations on IFU2.Proof.
≃Ais reflexive: let N be an IFS of U2. Since σ¯N=σ¯N, so we have N≃N. ≃Ais symmetric: let N and P be IFSs of U2 such that N≃P; this implies σ¯N=σ¯P, so σ¯P=σ¯N; this implies P≃N.
≃Ais transitive: let N,P and Q be IFSs of U2 such that N≃P and P≃Q; this implies σ¯N=σ¯P and σ¯P=σ¯Q, so σ¯N=σ¯Q; this implies N≃Q.
Thus,≃Ais an equivalence relation on IFU2.
Similarly,≂Aand ≈Aare equivalence relations on IFU2.Proposition 2.
The relations≃σ, ≂σ, and ≈σ are equivalence relations on IFU1.Proof.
≂σis reflexive: let N be an IFS of U1. Since Nσ¯=Nσ¯, so we have N≃N.
≂σis symmetric: let N and P be IFSs of U1 such that N≃P; this implies Nσ¯=Pσ¯, so Pσ¯=Nσ¯; this implies P≃N.
≂σis transitive: let N,P, and Q be IFSs of U1 such that N≃P and P≃Q; this implies Nσ¯=Pσ¯ and Pσ¯=Qσ¯, so Nσ¯=Qσ¯; this implies N≃Q.
Thus,≂σis an equivalence relation on IFU1.
Similarly,≂σand ≈σare equivalence relations on IFU1.Theorem 9.
Letσ,A be a soft binary relation from U1 to U2. Let N,P,Q, and T be IFSs of U2. Then, the following are true:(1)
N≂APif and only ifN≂AN∪P≂AP;(2)
N≂APandQ≂ATimply thatN∪Q≂AP∪T;(3)
N⊆PandP≂A0U2imply thatN≂A0;(4)
N∪P≂A0U2if and only ifN≂A0U2andP≂A0U2;(5)
N⊆PandN≂A1U2imply thatP≂A1;(6)
IfN∩P≂A1U2, thenN≂A1U2andP≂A1U2.Proof.
(1)
LetN≂AP. Then, σ¯N=σ¯P. By Theorem 1, we get σ¯N∪P=σ¯N∪σ¯P=σ¯N=σ¯P. This implies that N≂AN∪P≂AP. Conversely, it holds due to transitive property of relation ≂A.(2)
LetN≂AP and Q≂AT. Then, σ¯N=σ¯P and σ¯Q=σ¯T.By Theorem1, we get σ¯N∪Q=σ¯N∪σ¯Q=σ¯P∪σ¯T=σ¯P∪T. This implies that N∪Q≂AP∪T.(3)
LetN⊆P and P≂A0U2. Then, σ¯P=σ¯0U2.Also, by Theorem1, N⊆P implies that σ¯N⊆σ¯P=σ¯0U2. But σ¯0U2⊆σ¯N. Thus, σ¯N=σ¯0U2. This implies that N≂A0.(4)
IfN≂A0U2 and P≂A0U2, then σ¯N=σ¯0U2 and σ¯P=σ¯0U2. Now, by Theorem 1, we have σ¯N∪P=σ¯N∪σ¯P=σ¯0U2∪σ¯0U2=σ¯0U2. This implies that N∪P≂A0U2. Conversely, if N∪P≂A0U2, then by part (3), we have N≂A0U2 and P≂A0U2.(5)
SupposeN≂A1U2. Then, σ¯N=σ¯1U2. AsN⊆P, we have σ¯P⊇σ¯N=σ¯1U2. On the other hand, P⊆1U2, so we have σ¯P⊆σ¯1U2. This implies that σ¯P=σ¯1U2, that is, P≂A1U2.(6)
It follows from (5).
Theorem9 shows some lower similarity relations of union and intersection of IFSs N,P,Q, and T in U2 with respect to the aftersets.Theorem 10.
Letσ,A be a soft binary relation from U1 to U2. Let N,P,Q, and T be IFSs of U1. Then, the following are true:(1)
N≂σPif and only ifN≂σN∪P≂σP;(2)
N≂σPandQ≂σTimply thatN∪Q≂σP∪T;(3)
N⊆PandP≂σ0U1imply thatN≂σ0U1;(4)
N∪P≂σ0U1if and only ifN≂σ0U1andP≂σ0U1;(5)
N⊆PandN≂σ1U1imply thatP≂σ1U1;(6)
IfN∩P≂σ1U1, thenN≂σ1U1andP≂σ1U1.Proof.
(1)
LetN≂σP. Then, Nσ¯=Pσ¯. By Theorem 2, we get N∪Pσ¯=Nσ¯∪Pσ¯=Nσ¯=Pσ¯. This implies that N≂σN∪P≂σP. Converse holds by the transitivity of the relation ≂σ.(2)
LetN≂σP and Q≂σT. Then, Nσ¯=Pσ¯ and Qσ¯=Tσ¯.By Theorem2, we get N∪Qσ¯=Nσ¯∪Qσ¯=Pσ¯∪Tσ¯=P∪Tσ¯. This implies that N∪Q≂σP∪T.(3)
LetN⊆P and P≂σ0U1. Then, Pσ¯=0U1σ¯.Also, by Theorem2, N⊆P implies that Nσ¯⊆Pσ¯=0U1σ¯. But, 0U1σ¯⊆Nσ¯. Thus, Nσ¯=0U1σ¯. This implies that N≂σ0.(4)
IfN≂σ0U1 and P≂σ0U1, then Nσ¯=0U1σ¯ and Pσ¯=0U1σ¯. Now, by Theorem 2, we have N∪Pσ¯=Nσ¯∪Pσ¯=0U1σ¯∪0U1σ¯=0U1σ¯. This implies that N∪P≂σ0U1. Conversely, if N∪P≂σ0U1, then by part (3), we have N≂σ0U1 and P≂σ0U1.(5)
SupposeN≂σ1U1. Then, Nσ¯=1U1σ¯. AsN⊆P, we have Pσ¯⊇Nσ¯=1U1σ¯. On the other hand, P⊆1U1, so we have Pσ¯⊆1U1σ¯. This implies that Pσ¯=1U1σ¯, that is, P≂σ1U1.(6)
It follows from (5).
Theorem10 shows some lower similarity relations of union and intersection of IFSs N,P,Q, and T in U1 with respect to the foresets.Theorem 11.
Letσ,A be a soft binary relation from U1 to U2. Let N,P,Q, and T be IFSs of U2. Then, the following are true:(1)
N≃APif and only ifN≃AN∩P≃AP;(2)
N≃APandQ≃ATimply thatN∩Q≃AP∩T;(3)
N⊆PandP≃A0U2imply thatN≃A0U2;(4)
N∩P≃A0U2if and only ifN≃A0U2andP≃A0U2;(5)
N⊆PandN≃A1U2imply thatP≃A1U2;(6)
IfN∩P≃A1U2, thenN≃A1U2andP≃A1U2.Proof.
(1)
LetN≃AP. Then, σ¯N=σ¯P. By Theorem 1, we get σ¯N∩P=σ¯N∩σ¯P=σ¯N=σ¯P. This implies that N≃AN∩P≃AP. Converse holds by the transitivity of the relation ≃A.(2)
LetN≃AP and Q≃AT. Then, σ¯N=σ¯P and σ¯Q=σ¯T.By Theorem1, we get σ¯N∩Q=σ¯N∩σ¯Q=σ¯P∩σ¯T=σ¯P∩T. This implies that N∩Q≃AP∩T.(3)
LetN⊆P and P≃A0U2. Then, σ¯P=σ¯0U2.Also, by Theorem1, N⊆P implies that σ¯N⊆σ¯P=σ¯0U2. But, σ¯0U2⊆σ¯N. Thus, σ¯N=σ¯0U2. This implies that N≃A0U2.(4)
IfN≃A0U2 and P≃A0U2, then σ¯N=σ¯0U2 and σ¯P=σ¯0U2. Now, by Theorem 1, we haveσ¯N∩P=σ¯N∩σ¯P=σ¯0U2∩σ¯0U2=σ¯0U2, so σ¯N∩P=σ¯0U2 This implies that N∩P≃A0U2. Conversely, if N∩P≃A0U2, then by part (3), we have N≃A0U2 and P≃A0U2.(5)
SupposeN≃A1U2. Then, σ¯N=σ¯1U2. As N⊆P, we have σ¯P⊇σ¯N=σ¯1U2. On the other hand, P⊆1U2, so we have σ¯P⊆σ¯1U2. This implies that σ¯P=σ¯1U2, that is, P≃A1U2.(6)
It follows from (5).
Theorem11 shows some upper similarity relations of union and intersection of IFSs N,P,Q, and T in U2 with respect to the aftersets.Theorem 12.
Letσ,A be a soft binary relation from U1 to U2. Let N,P,Q, and T be IFSs of U1. Then, the following are true:(1)
N≃σPif and only ifN≃σN∩P≃σP;(2)
N≃σPandQ≃σTimply thatN∩Q≃σP∩T;(3)
N⊆PandP≃σ0U1imply thatN≃σ0U1;(4)
N∩P≃σ0U1if and only ifN≃σ0U1andP≃σ0U1;(5)
N⊆PandN≃σ1U1imply thatP≃σ1U1;(6)
IfN∩P≃σ1U1, thenN≃σ1U1andP≃σ1U1.Proof.
(1)
LetN≃σP. Then, Nσ¯=Pσ¯. By Theorem 2, we get σ¯N∩P=σ¯N∩σ¯P=σ¯N=σ¯P. This implies that N≃σN∩P≃σP. Converse holds by the transitivity of the relation ≃σ.(2)
LetN≃σP and Q≃σT. Then, Nσ¯=Pσ¯ and Qσ¯=Tσ¯.By Theorem2, we get N∩Qσ¯=Nσ¯∩Qσ¯=Pσ¯∩Tσ¯=P∩Tσ¯. This implies that N∩Q≃σP∩T.(3)
LetN⊆P and P≃σ0U1. Then, Pσ¯=0U1σ¯.Also, by Theorem2, N⊆P implies that Nσ¯⊆Pσ¯=0U1σ¯. But, 0U1σ¯⊆Nσ¯. Thus, Nσ¯=0U1σ¯. This implies that N≃σ0U1.(4)
IfN≃σ0U1 and P≃σ0U1, then σ¯N=0U1σ¯ and Pσ¯=0U1σ¯. Now, by Theorem 2, we haveN∩Pσ¯=Nσ¯∩Pσ¯=0U1σ¯0U1∩0U1σ¯=0U1σ¯, so N∩Pσ¯=0U1σ¯. This implies that N∩P≃σ0U1. Conversely, if N∩P≃σ0U1, then by part (3), we have N≃σ0U1 and P≃σ0U1.(5)
SupposeN≃σ1U1. Then, Nσ¯=1U1σ¯. As N⊆P, we have Pσ¯⊇Nσ¯=1U1σ¯. On the other hand, P⊆1U1, so we have Pσ¯⊆1U1σ¯. This implies that Pσ¯=1U1σ¯, that is, P≃σ1U1.(6)
It follows from (5).Theorem 13.
Letσ,A be a soft binary relation from U1 to U2. Let N,P,Q, and T be IFSs of U2. Then, the following are true:(1)
N⊆PandP≈A0U2imply thatN≈A0U2;(2)
N⊆PandN≈A1U2imply thatP≈A1U2;(3)
N∪P≈A0U2, thenN≈A0U2andP≈A0U2;(4)
N∩P≈A1U2, thenN≈A1U2andP≈A1U2;(5)
N≈APif and only ifN≂AN∪P≂APandN≃AN∩P≃AP.Proof.
(1)
SupposeP≈A0U2, this implies that σ¯N=σ¯0U2 and σ¯P=σ¯0U2. As N⊆P, we have σ¯N⊆σ¯P=σ¯0U2 and σ¯N⊆σ¯P=σ¯0U2. On the other hand, N⊇0U2; this implies that σ¯N⊇σ¯0U2 and σ¯N⊇σ¯0U2. So,σ¯N=σ¯0U2 and σ¯N=σ¯0U2, thus N≈A0U2.(2)
SupposeN≈A1U2, this implies that σ¯N=σ¯1U2 and σ¯P=σ¯1U2. As N⊆P, we have σ¯P⊇σ¯N=σ¯1U2 and σ¯P⊇σ¯N=σ¯1U2. On the other hand, P⊆1U2; this implies that σ¯P⊆σ¯1U2 and σ¯P⊆σ¯1U2. So,σ¯P=σ¯1U2 and σ¯P=σ¯1U2, thus P≈A1U2.(3)
It follows from part (1).(4)
It follows from part (2).(5)
LetN≈AP. Then, σ¯N=σ¯P and σ¯P=σ¯N. By Theorem 1, we get σ¯N∩P=σ¯N∩σ¯P=σ¯N=σ¯P and σ¯N∪P=σ¯N∪σ¯P=σ¯N=σ¯P. This implies that N≃AN∩P≃AP and N≂AN∪P≂AP.Converse holds by the transitivity of the relation≈A.
Theorem13 shows some similarity relations of union and intersection of IFSs N,P,Q, and T in U2 with respect to the aftersets.Theorem 14.
Letσ,A be a soft binary relation from U1 to U2. Let N,P,Q, and T be IFSs of U1. Then, the following are true:(1)
N⊆PandP≈σ0U1imply thatN≈σ0U1;(2)
N⊆PandN≈σ1U1imply thatP≈σ1U1;(3)
N∪P≈σ0U1, thenN≈σ0U1andP≈σ0U1;(4)
N∩P≈σ1U1, thenN≈σ1U1andP≈σ1U1;(5)
N≈σPif and only ifN≂σN∪P≂σPandN≃σN∩P≃σP.Proof.
(1)
SupposeP≈0σU1, this implies that σ¯N=σ¯0U1 and σ¯P=σ¯0U1. As N⊆P, we have σ¯N⊆σ¯P=σ¯0U1 and σ¯N⊆σ¯P=σ¯0U1. On the other hand, N⊇0U1; this implies that σ¯⊇σ¯0U1 and σ¯N⊇σ¯0U1. So,σ¯N=σ¯0U1 and σ¯N=σ¯0U1, thus N≈σ0U1.(2)
SupposeN≈σ1U1, this implies that σ¯N=σ¯1U1 and Pσ¯=σ¯1U1. As N⊆P, we have σ¯P⊇σ¯N=σ¯1U1 and σ¯P⊇σ¯N=σ¯1U1. On the other hand, P⊆1U1; this implies that σ¯P⊆σ¯1U1 and σ¯P⊆σ¯1U1. So,σ¯P=σ¯1U1 and Pσ¯=1U1σ¯, thus P≈σ1U1.(3)
It follows from part (1).(4)
It follows from part (2).(5)
LetN≈σP. Then, Nσ¯=Pσ¯ and Pσ¯=Nσ¯. By Theorem 2, we get σ¯N∩P=σ¯N∩σ¯P=σ¯N=σ¯P and σ¯N∪P=σ¯N∪σ¯P=σ¯N=σ¯P. This implies that N≃σN∩P≃σP and N≂σN∪P≂σP.Converse holds by the transitivity of the relation≈σ.


## 5. Application in Decision-Making Problem

A major area of study in all kinds of data analysis is decision making. Many experts and researchers introduced many methods to find a wise decision. RS theory [7], SS theory [12], and IFS theory [3] are the theories which are mostly used in the decision-making problems. In the above sections, we develop a rough set model using soft binary relations. We used soft binary relation to approximate an IFS. We used score function defined by Chen and Tan [57] and accuracy function defined by Hong and Choi [58] to define order between objects. Now, we present an algorithm for the approach to a decision-making problem and this problem is depended on IF soft rough set theory based on soft binary relations. This algorithm extends the already existing approach which is described by Kanwal and Shabir [51]. For our new approach, data information is only needed which is provided by the decision-making problem and no need of any additional information by any supplementary ways. So, the decision results can be avoided by the effect of subjective information. Therefore, the outcomes could avoid the inconsistent results for the same problem and could be better objective. The decision Algorithm 1 is as follows:Algorithm 1:Procedural steps for better decision with the help of score function.
(1)
Compute the upper IF soft set approximationσ¯M and lower IF soft set approximation σ¯M of an IF set M=μM,γM with respect to the aftersets;(2)
Compute the score values for each of the entries of theσ¯M and σ¯M and denote them by S¯ijxi,ej and S¯ijxi,ej for all i,j;(3)
Compute the aggregated scoreS¯xi=∑j=1nS¯ijxi,ej and S¯xi=∑j=1nS¯ijxi,ej;(4)
ComputeSxi=S¯xi+S¯xi;(5)
The best decision isxk=maxiSxi;(6)
Ifk has more than one value, say k1,k2, then we calculate the accuracy values H¯ijxi,ej and H¯ijxi,ej for only those xk for which Sxk are equal;(7)
ComputeHxk=∑j=1nH¯kjxk,ej+∑j=1nH¯kjxk,ej for k=k1,k2;(8)
IfHxk1>Hxk2, then we select x;(9)
IfHxk1=Hxk2, then select any one of xk1 and xk2.Now, we show this approach step by step to decision making which is proposed by using the following example. The following example discusses algorithm to make wise decision for the selection of a car.Example 4.
Suppose a personMr. X wants to select a car out of available models. Let U1=the set of all modelsavailablein range=m1,m2,m3,m4,m5,m6 and U2=the colors of all models=c1,c2,c3,c4 and the set of attributes be A=e1,e2,e3=the set of brands=e1=Suzuki,e2=Toyota,e3=Honda.
Defineσ:A⟶PU1×U2 by(9)σe1=m1,c1,m1,c2,m1,c3,m2,c2,m2,c4,m4,c2,m4,c3,m5,c3,m5,c4,m6,c1,σe2=m1,c3,m2,c3,m4,c1,m5,c1,m6,c2,m6,c3,σe3=m3,c3,m3,c1,m2,c4,m5,c3,m5,c4,which represents the relation between models and colors available in brand ei for 1≤i≤3. Then,(10)m1σe1=c1,c2,c3,m2σe1=c2,c4,m3σe1=∅,m4σe1=c2,c3,m5σe1=c3,c4,m6σe1=c1,m1σe2=c3,m2σe2=c3,m3σe2=∅,m4σe2=c1,m5σe2=c1,m6σe2=c2,c3,m1σe3=∅,m2σe3=c4,m3σe3=c1,c3,m4σe3=∅,m5σe3=c3,c4,m6σe3=∅,where miσej represents the color of the model mi available in the brand ej.
Also,(11)σe1c1=m1,m6,σe1c2=m1,m2,m4,σe1c3=m1,m4,m5,σe1c4=m2,m5,σe2c1=m4,m5,σe2c2=m6,σe2c3=m1,m2,σe2c4=∅,σe3c1=m3,σe3c2=∅,σe3c3=m3,m5,σe3c4=m2,m5,where σejci represents the model of the color ci available in the brand ej.(12)DefineM=μM,γM:U2⟶0,1,which represents the preference of thecolors given by Mr.Xsuch thatμMc1=0.9,μMc2=0.8,μMc3=0.4,μMc4=0γMc1=0.0,γMc2=0.2,γMc3=0.5,γMc4=0.8.DefineN=μN,γN:U1⟶0,1,which represents the preference of themodel given by Mr.Xsuch thatμNm1=1,μNm2=0.7,μNm3=0.5,μNm4=0.1,μNm5=0,μNm6=0.4γNm1=0,γNm2=0.2,γNm3=0.5,γNm4=0.7,γNm5=1,γNm6=0.5.
Therefore, the lower and upper approximations (with respect to the aftersets as well as with respect to the foresets) are as follows (Tables14 and 15, respectively):(13)σ¯M=σ¯μM,σ¯γM,σ¯M=σ¯μM,σ¯γM.
The values of score function for car models are given in Table16.
Table16 shows that Sm4=Sm6, so we calculate accuracy values for m4 and m6.
Hence, the values of accuracy function are given in Table17.
Table17 shows that Hm4=Hm6, so we can select any one, m4or m6.
Now,σ¯N=μNσ¯,γNσ¯ (given in Table 18) and σ¯N=μNσ¯,γNσ¯ (given in Table 19).
The values of score function for colors of cars are given in Table20.
Table20 shows that Sc3=0.5 is maximum, so he will select color c4.
The flow chart of our decision-making algorithm is given in Figure2.
In Figure2, the flow chart shows that if a person wants to buy a car with his favourite design and color, then this algorithm helps him to make best and suitable decision according to his choice.Table 14
Lower approximation ofM.
σ¯μM,σ¯γMm1m2m3m4m5m6σ¯μMe10.4010.400.9σ¯μMe20.40.410.90.90.4σ¯μMe3100.4101σ¯γMe10.50.800.50.80σ¯γMe20.50.50000.5σ¯γMe300.80.500.80Table 15
Upper approximation ofM.
σ¯μM,σ¯γMm1m2m3m4m5m6σ¯μMe10.90.800.80.40.9σ¯μMe20.40.400.90.90.8σ¯μMe3000.900.40σ¯γMe100.210.20.50σ¯γMe20.50.51000.2σ¯γMe310.8010.51Table 16
Values of score function for car models.
S¯ije1S¯ije2S¯ije3S¯ije1S¯ije2S¯ije3S¯xiS¯xiSxim1−0.1−0.110.9−0.1−10.8−0.20.6m2−0.8−0.1−0.80.6−0.1−0.8−1.7−0.3−2m311−0.1−1−10.91.9−1.10.8m4−0.10.910.60.9−11.80.52.3m5−0.80.9−0.8−0.10.90.3−0.71.10.4m60.9−0.110.90.6−11.80.52.3Table 17
Values of accuracy function.
H¯ije1H¯ije2H¯ije3H¯ije1H¯ije2H¯ije3Hm40.90.9110.915.7m60.90.910.9115.7Table 18
Lower approximation ofN.
μNσ¯,γNσ¯c1c2c3c4σ¯μNe10.40.100σ¯μNe200.40.71σ¯μNe30.5100σ¯γNe10.50.711σ¯γNe210.50.20σ¯γNe30.5011Table 19
Upper approximation ofN.
μNσ¯,γNσ¯c1c2c3c4σ¯μNe11110.7σ¯μNe20.10.410σ¯μNe30.500.50.7σ¯γNe10000.2σ¯γNe20.70.501σ¯γNe30.510.50.2Table 20
Values of score function for colors of cars.
S¯ije1S¯ije2S¯ije3S¯ije1S¯ije2S¯ije3S¯xiS¯xiSxic1−0.1−101−0.60−1.10.4−0.7c2−0.6−0.111−0.1−10.3−0.10.2c3−10.5−1110−1.520.5c4−11−10.5−10.5−10−1Figure 2
Flow chart of decision-making algorithm of IFRS proposed model.


## 6. Comparison

First, we review existing approaches to intuitionistic fuzzy rough set (IFRS) model-based decision making and then finally we show that our newly proposed IFRS model is very useful than other existing theories. Since the combination of IFS with SS and RS is very helpful to deal with uncertainty and impreciseness, Maji et al. [24] presented a useful model of RS and SS. Chen et al. [20] used SS parameterization reduction and improved SSs-based decision making in [24]. Cagman et al. [63] presented a uni-int decision-making method by using redefined operations of soft sets. But all the above work in decision making is about only crisp soft set. Then, Roy et al. [46] solved recognition problems by using their newly proposed algorithm of fuzzy SSs [64]. Later, Kong et al. [65] modified Roy et al.’ algorithm and proved that their algorithm was not able to obtain optimal choice generally. Feng et al. [21] also worked on fuzzy soft set-based algorithm. Later, Jiang et al. [66] discussed intuitionistic fuzzy soft sets with an adjusted approach.


### 6.1. Maji and Roy’s Method and Its Limitation

Maji et al. used concept of knowledge reduction in RSs with SSs to solve decision-making problems. This method consists of two steps. In first step, find one reduct soft set of the original SS based on the knowledge reduction of RSs, and then calculate the choice values of all elements and select the element with the maximum value as the optimum alternative. Chen et al. [20] claimed that soft set reduction in [24] has incorrect results in Step 1.


### 6.2. Cagman’s Method and Its Limitation

Cagman et al. [63] proposed a soft max-min decision-making method. Optimum alternatives are selected from the alternatives set by this method. In this method, the noting point is that this method has its constitutive limitation. An algorithm of this method gets an empty optimum set.Dubious and Prade presented rough FSs [35] and Feng et al. [1] extended their model in terms of SSs. Feng et al. [1] approximated a FS in a soft approximation space and extended a concept called soft rough FSs. The FSs and IF relations-based models are useful in many fields, but we used soft binary relations in our proposed model. Soft binary relation is the generalization of a binary relation and soft binary relation is the parameterized collection of ordinary binary relations. In soft binary relation, we can use different parameters according to the nature of problem. That is why our proposed model is more useful to manage uncertainty in different types of problems.


### 6.3. Advantages of Our Proposed IFRS Model

(1)
In our IFRS model based on soft binary relations, we also get information about what candidate is optimum alternative and what candidate should not be optimum alternative, whereas other existing theories only get optimum alternatives. That is why our newly proposed IFRS model is more precise and flexible for decision-making problems.(2)
Our proposed model also gives a solution IFRS-based group decision making, whereas other existing approaches have no directions to discuss the intuitionistic fuzzy set group decision making.(3)
This IFRS model based on soft relations can be applied to solve decision-making problems involving intuitionistic fuzzy sets in real life [56].In 2012, Zhang [56] proposed a RS model based on ordinary binary relation induced by an IF relation over two universes and presented a decision-making algorithm based on RS model with IFSs. In 2020, Shabir et al. [51] proposed a RS model of FSs based on soft relations and presented decision-making algorithm. In comparison of these models, we proposed a RS model of IFSs based on soft relations and presented a decision-making algorithm based on IFRS which is a better technique to manage uncertainty and impreciseness. We used an IFS instead of a crisp set or a FS in our proposed model due to its importance in scientific fields and decision making, such as medical diagnosis, career determination, pattern recognition, and electoral system. An IFS has degree of membership and also degree of nonmembership which is helpful to make better decision in real-life problems.


## 6.1. Maji and Roy’s Method and Its Limitation

Maji et al. used concept of knowledge reduction in RSs with SSs to solve decision-making problems. This method consists of two steps. In first step, find one reduct soft set of the original SS based on the knowledge reduction of RSs, and then calculate the choice values of all elements and select the element with the maximum value as the optimum alternative. Chen et al. [20] claimed that soft set reduction in [24] has incorrect results in Step 1.


## 6.2. Cagman’s Method and Its Limitation

Cagman et al. [63] proposed a soft max-min decision-making method. Optimum alternatives are selected from the alternatives set by this method. In this method, the noting point is that this method has its constitutive limitation. An algorithm of this method gets an empty optimum set.Dubious and Prade presented rough FSs [35] and Feng et al. [1] extended their model in terms of SSs. Feng et al. [1] approximated a FS in a soft approximation space and extended a concept called soft rough FSs. The FSs and IF relations-based models are useful in many fields, but we used soft binary relations in our proposed model. Soft binary relation is the generalization of a binary relation and soft binary relation is the parameterized collection of ordinary binary relations. In soft binary relation, we can use different parameters according to the nature of problem. That is why our proposed model is more useful to manage uncertainty in different types of problems.


## 6.3. Advantages of Our Proposed IFRS Model

(1)
In our IFRS model based on soft binary relations, we also get information about what candidate is optimum alternative and what candidate should not be optimum alternative, whereas other existing theories only get optimum alternatives. That is why our newly proposed IFRS model is more precise and flexible for decision-making problems.(2)
Our proposed model also gives a solution IFRS-based group decision making, whereas other existing approaches have no directions to discuss the intuitionistic fuzzy set group decision making.(3)
This IFRS model based on soft relations can be applied to solve decision-making problems involving intuitionistic fuzzy sets in real life [56].In 2012, Zhang [56] proposed a RS model based on ordinary binary relation induced by an IF relation over two universes and presented a decision-making algorithm based on RS model with IFSs. In 2020, Shabir et al. [51] proposed a RS model of FSs based on soft relations and presented decision-making algorithm. In comparison of these models, we proposed a RS model of IFSs based on soft relations and presented a decision-making algorithm based on IFRS which is a better technique to manage uncertainty and impreciseness. We used an IFS instead of a crisp set or a FS in our proposed model due to its importance in scientific fields and decision making, such as medical diagnosis, career determination, pattern recognition, and electoral system. An IFS has degree of membership and also degree of nonmembership which is helpful to make better decision in real-life problems.


## 7. Conclusion

Since the combination of IFS with SS and RS is very helpful to deal with uncertainty, Maji et al. presented a useful model of RS and SS. Chen et al. used SS parameterization reduction and improved SSs-based decision making. Cagman et al. presented a uni-int decision-making method by using redefined operations of SSs. But, all the above work in decision making is about only crisp SS. Then, Roy et al. solved recognition problems by using their newly proposed algorithm of fuzzy SSs. Later, Kong et al. modified Roy et al.’s algorithm and proved that their algorithm was not able to obtain optimal choice generally. Feng et al. also worked on fuzzy SS-based algorithm. Later, Jiang et al. discussed IF soft sets with an adjusted approach. In our paper, we have given a generalization of [51] and we have approximated an IFS by soft binary relations. We used foresets and aftersets to approximate IFS. In this way, we get two pairs of intuitionistic fuzzy soft sets, called the lower approximation and upper approximation. Properties of these approximations are studied. Similarity relations between IFS with respect to this rough set model are also studied. Finally, we developed an algorithm for intuitionistic fuzzy rough sets (IFRS) based on decision making and an example is provided to illustrate the developed algorithm. Further study can be performed to investigate the roughness in interval-valued IFS and multigranulation roughness of IFS by using soft relations.



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