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.
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 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 over 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 which is generalized IFS in terms of fuzzy rough sets and . On the other hand, an IF rough set presented by Chakrabarty et al. [48] is the generalization of fuzzy rough set in terms of IFSs and . 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 Section 2, 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, and represent two nonempty finite sets unless stated otherwise.
A binary relation from to is a subset of and a subset of is called a binary relation on . If is a binary relation on , then is said to be reflexive if for all , symmetric if implies for all , and transitive if and imply for all . If a binary relation is reflexive, symmetric, and transitive, then it is called an equivalence relation. An equivalence relation partitions the set into disjoint classes.
Let be a nonempty universe. An IF set in the universe is an object having the form , where and , satisfying for all . The values and are called degree of membership and degree of nonmembership of to , respectively. The number is called the degree of hesitancy of to . The collection of all IFSs in is denoted by . In the remaining paper, we shall write an IFS by instead of . Let and be two IFSs in . Then, if and only if and for all . Two IFSs and are said to be equal if and only if and . The union and intersection of two IFSs and in are denoted and defined by and , where , , , .
Next, we define two special types of (IFSs) as follows:
The IF universe set and IF empty set , where and for all . The complement of an IFS is denoted and defined as [3].
For a fixed , the pair 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 and Hong and Choi [58] defined the accuracy function as , where is an IFV. Xu [59,60] combined the accuracy and score functions and formed the order relations between any pair of IFVs as follows:(i)If , then ;(ii)If , then(a)If , then ;(b)If , then .
A pair is called a soft set over if is a mapping given by , where is a subset of (the set of parameters) and is the power set of . Thus, is a subset of for all . Hence, a soft set over is a parametrized collection of subsets of . A pair is called an intuitionistic fuzzy soft set over if is a mapping given by and is a subset of (the set of parameters). Thus, is an IF set in for all . Hence, an IF soft set over is a parametrized collection of IF sets in . For two IF soft sets and over a common universe , we say that is an IF soft subset of if and is an IF subset of for all . Two IF soft sets and over a common universe are said to be IF soft equal if is an IF soft subset of and is an IF soft subset of . The union of two IF soft sets and over the common universe is the IF soft set , where for all . The intersection of two IF soft sets and over the common universe is the IF soft set , where for all [12, 61, 62].
An IF soft set can be represented by a table, which is shown in the following example.
Example 1. Let , . Consider an IF soft set over defined by , and , .
The above intuitionistic fuzzy soft set can be represented as in Table 1.
3. Approximations of an IFS by Soft Binary Relation
In this section, we consider soft binary relation from to and approximate an IFS of by using aftersets and get two IF soft sets of . Similarly, we approximate an IFS of by using foresets and get two IF soft sets of . We also study some properties of these approximations.
Definition 1. (see [30]). A soft binary relation from to is a soft set over , that is, , where is a subset of the set of parameters .
Of course, is a parameterized collection of binary relations from to . That is, for each , we have a binary relation from to .
Definition 2. Let be a soft binary relation from to and be an IFS in . Then, we define lower approximation and upper approximation of with respect to aftersets as follows:where and is called the afterset of for and .(i) indicates the degree to which definitely have the property .(ii) indicates the degree to which probably do not have the property .(iii) indicates the degree to which probably have the property .(iv) indicates the degree to which definitely do not have the property .In Definition 2, soft binary relation from to is given and IFS in 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. Let be a soft binary relation from to and be an IFS in . Then, we define lower approximation and upper approximation of with respect to foresets as follows:where and is called the foreset of for and . Of course, , and , . The following example explains these concepts.
In Definition 3, soft binary relation from to is given and IFS in 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 that . wants to buy a shirt for his own use. Let and and the set of attributes be .
Define bywhich represents the relation between designs and colors available on store for . Then,where represents the color of the design available on the store .
Also,where represents the design of the color available on the store .Therefore, the lower and upper approximations (with respect to the aftersets as well as with respect to the foresets) areandTable 2 shows the lower approximation of IFS with respect to the aftersets by using Definition 2. Table 3 shows the upper approximation of IFS with respect to the aftersets by using Definition 2. Table 4 shows the lower approximation of IFS with respect to the foresets by using Definition 3. Table 5 shows the upper approximation of IFS with respect to the foresets by using Definition 3.
Theorem 1. Let be a soft binary relation from to , that is, . For any IFSs, , , and of , the following are true:(1)If, then;(2)If, then;(3);(4);(5);(6);(7)if;(8)if;(9)if;(10)if;(11)if.
Proof. (1)Let . If , then and . If , then because . Thus, . Also, because . Thus, . Hence, .(2)Let . If , then and . If , then because . Thus, . Also, because . Thus, . Hence, .(3)Let . If , then and . If , then . Also, . This shows that .(4)Since and , we have from part (2) and . Thus, .(5)Since and , we have from part (1) and . Thus, .(6)Let . If , then and . If , then . Also, . This shows that .(7)Consider , because and , because . Thus, .(8)The proof is similar to the proof of part (7).(9)Let be an IFS on . Then, , that is, and . Now, . Thus, and . Hence, , that is, .(10)follows from part (9).(11)Straightforward.Theorem 1 describes the properties of newly defined IFRS model based on soft relations. It shows that if an IFS is the subset of IFS , then the lower approximation of is also a subset of the lower approximation of and if an IFS is subset of IFS , then the upper approximation of is also a subset of the upper approximation of . Similarly, the empirical relations among the operations union, intersection, and complement have been described.
Theorem 2. Let be a soft binary relation from to , that is, . For any IFSs, , , and of , the following are true:(1)If, then;(2)If, then;(3);(4);(5);(6);(7)if;(8)if;(9)if;(10)if;(11).
Proof. The proof is similar to the proof of Theorem 1.
The following example shows that equality does not hold in (4) and (5) assertions of above theorems in general.
Example 3. Consider Example 2. Let (given in Table 6).
Table 6 simply shows the degree of membership and degree of nonmembership of IFS .
Then, (given in Table 7) and (given in Table 7).
Table 7 shows the calculations of union and intersection of two IFSs and , respectively.
Now, (given in Table 8) and (given in Table 9).
Now, (given in Table 10).
In Table 10, we calculated . Tables 9 and 10 show that .
Now, (given in Table 11).
In Table 11, we calculated upper approximation of .
Now, (given in Table 12).
Now, (given in Table 13).
In Table 12, we calculated upper approximation of . In Table 13, we calculated the intersection of upper approximations of and . Tables 12 and 13 show that .
Theorem 3. Let and be two soft binary relations from to , such that , that is, for all . Then, for any IFS of , the following are true:(i);(ii).
Proof. (1)Let . If , then and . If , then , we have because . Also, because . Hence, .(2)Let . If , then and . If , then , and we have because . Also, because . Hence, .Theorem 3 shows that if any soft relation , then for any IFS in , the lower approximation associated with is a subset of . Similarly, if any soft relation , then for any IFS in , the upper approximation associated with is a subset of .
Theorem 4. Let and be two soft binary relations from to , such that , that is, for all . Then, for any IFS of , the following are true:(1)(2)
Proof. (1)Let . If , then and . If , then , we have because . Also, because . Hence, .(2)Let . If , then and . If , then , and we have because . Also, because . Hence, .
Theorem 5. Let and be two soft binary relations from to . Then, for any IFS of , the following are true:(1);(2);(3);(4).
Proof. (1)As , therefore from Theorem 3 part (1), . (2) As , therefore from Theorem 3 part (1), . (3) As , therefore from Theorem 3 part (2), . (4) As , therefore from Theorem 3 part (2), .
Theorem 6. Let and be two soft binary relations from to . Then, for any IFS of , the following are true:(1);(2);(3);(4).
Proof. (1)As , therefore from Theorem 4 part (1), .(2)As , therefore from Theorem 4 part (1), .(3)As , therefore from Theorem 4 part (2), .(4)As , therefore from Theorem 4 part (2), .In Theorems 5 and 6, some empirical relations have been discussed about union and intersection of two soft relations and with respect to the aftersets and with respect to the foresets, respectively.
Definition 4. If is a soft set over , then is called a soft binary relation on .
In fact, is a parameterized collection of binary relations on . That is, for each parameter , we have a binary relation on . A soft binary relation on is said to be soft reflexive relation on if is a reflexive relation on for all . If is a soft reflexive binary relation on , then is nonempty and . It is not necessary that . A soft binary relation on is said to be soft symmetric relation on if is a symmetric relation on for all . A soft binary relation on is said to be soft transitive relation on if is a transitive relation on for all .
A soft binary relation over is soft equivalence relation over if it is soft reflexive, soft symmetric, and soft transitive relation over . A soft binary relation over is a soft equivalence relation over if for all is an equivalence relation over . In this case, and is a partition of . Also, in this case, and , for any IFS of .
The approximation operators have additional properties with respect to soft reflexive binary relation as follows.
Theorem 7. Let be a soft reflexive binary relation on . Then, for any IFS of , the following are true:(1)for all;(2)for all;(3)for all;(4)for all;(5)for all;(6)for all.
Proof. (1)Let . Then, because .(2)Hence, . Let . Then, because Hence, .(3)It follows from part (1) and part (2).(4)Let . Then, because Hence, .(5)Let . Then, because Hence, . It follows from part (4) and part (5).Theorem 7 shows the empirical relations between IFS and a soft reflexive relation .
Theorem 8. Let be a soft reflexive binary relation on . Then, for any IFS of , the following are true:(1) for all ;(2) for all ;(3) for all ;(4) for all ;(5) for all ;(6) for all .
Proof. (1)Let . Then, because Hence, .(2)Let . Then, because Hence, .(3)It follows from part (1) and part (2).(4)Let . Then, because Hence, .(5)Let . Then, because (i)Hence, .(6)It follows from part (4) and part (5).
4. Similarity Relations
In this section, we define some relations between IFS of with the help of a soft relation from to . We say that two intuitionistic fuzzy sets in are related if the lower (upper) approximations in are equal. Similarly, we define relations between intuitionistic fuzzy sets of .
Definition 5. Let be a soft binary relation from to . Then, for any IFS and of , we define
if and only if if and only if if and only if and .
Definition 6. Let be a soft binary relation from to . Then, for any IFS and of , we define
if and only if if and only if if and only if and .
These binary relations may be called the lower similarity relation, upper similarity relation, and similarity relation, respectively.
Definitions 5 and 6 show that if an IFS has upper(lower) similarity relation with an IFS , then its associated lower(upper) approximation has also upper(lower) similarity relation.
Proposition 1. The relations , , and are equivalence relations on .
Proof. is reflexive: let be an IFS of . Since , so we have . is symmetric: let and be IFSs of such that ; this implies , so ; this implies .
is transitive: let and be IFSs of such that and ; this implies and , so ; this implies .
Thus, is an equivalence relation on .
Similarly, and are equivalence relations on .
Proposition 2. The relations , , and are equivalence relations on .
Proof. is reflexive: let be an IFS of . Since , so we have .
is symmetric: let and be IFSs of such that ; this implies , so ; this implies .
is transitive: let , and be IFSs of such that and ; this implies and , so ; this implies .
Thus, is an equivalence relation on .
Similarly, and are equivalence relations on .
Theorem 9. Let be a soft binary relation from to . Let , and be IFSs of . Then, the following are true:(1)if and only if;(2)andimply that;(3)andimply that;(4)if and only ifand;(5)andimply that;(6)If, thenand.
Proof. (1)Let . Then, . By Theorem 1, we get . This implies that . Conversely, it holds due to transitive property of relation .(2)Let and . Then, and . By Theorem 1, we get . This implies that .(3)Let and . Then, . Also, by Theorem 1, implies that . But . Thus, . This implies that .(4)If and , then and . Now, by Theorem 1, we have . This implies that . Conversely, if , then by part (3), we have and .(5)Suppose . Then, . As , we have . On the other hand, , so we have . This implies that , that is, .(6)It follows from (5).Theorem 9 shows some lower similarity relations of union and intersection of IFSs , and in with respect to the aftersets.
Theorem 10. Let be a soft binary relation from to . Let , and be IFSs of . Then, the following are true:(1)if and only if;(2)andimply that;(3)andimply that;(4)if and only ifand;(5)andimply that;(6)If, thenand.
Proof. (1)Let . Then, . By Theorem 2, we get . This implies that . Converse holds by the transitivity of the relation .(2)Let and . Then, and . By Theorem 2, we get . This implies that .(3)Let and . Then, . Also, by Theorem 2, implies that . But, . Thus, . This implies that .(4)If and , then and . Now, by Theorem 2, we have . This implies that . Conversely, if , then by part (3), we have and .(5)Suppose . Then, . As , we have . On the other hand, , so we have . This implies that , that is, .(6)It follows from (5).Theorem 10 shows some lower similarity relations of union and intersection of IFSs , and in with respect to the foresets.
Theorem 11. Let be a soft binary relation from to . Let , and be IFSs of . Then, the following are true:(1)if and only if;(2)andimply that;(3)andimply that;(4)if and only ifand;(5)andimply that;(6)If, thenand.
Proof. (1)Let . Then, . By Theorem 1, we get . This implies that . Converse holds by the transitivity of the relation .(2)Let and . Then, and . By Theorem 1, we get . This implies that .(3)Let and . Then, . Also, by Theorem 1, implies that . But, . Thus, . This implies that .(4)If and , then and . Now, by Theorem 1, we have, so This implies that . Conversely, if , then by part (3), we have and .(5)Suppose . Then, . As , we have . On the other hand, , so we have . This implies that , that is, .(6)It follows from (5).Theorem 11 shows some upper similarity relations of union and intersection of IFSs , and in with respect to the aftersets.
Theorem 12. Let be a soft binary relation from to . Let , and be IFSs of . Then, the following are true:(1)if and only if;(2)andimply that;(3)andimply that;(4)if and only ifand;(5)andimply that;(6)If, thenand.
Proof. (1)Let . Then, . By Theorem 2, we get . This implies that . Converse holds by the transitivity of the relation .(2)Let and . Then, and . By Theorem 2, we get . This implies that .(3)Let and . Then, . Also, by Theorem 2, implies that . But, . Thus, . This implies that .(4)If and , then and . Now, by Theorem 2, we have, so . This implies that . Conversely, if , then by part (3), we have and .(5)Suppose . Then, . As , we have . On the other hand, , so we have . This implies that , that is, .(6)It follows from (5).
Theorem 13. Let be a soft binary relation from to . Let , and be IFSs of . Then, the following are true:(1)andimply that;(2)andimply that;(3), thenand;(4), thenand;(5)if and only ifand.
Proof. (1)Suppose , this implies that and . As , we have and . On the other hand, ; this implies that and . So, and , thus .(2)Suppose , this implies that and . As , we have and . On the other hand, ; this implies that and . So, and , thus .(3)It follows from part (1).(4)It follows from part (2).(5)Let . Then, and . By Theorem 1, we get and . This implies that and . Converse holds by the transitivity of the relation .Theorem 13 shows some similarity relations of union and intersection of IFSs , and in with respect to the aftersets.
Theorem 14. Let be a soft binary relation from to . Let , and be IFSs of . Then, the following are true:(1)andimply that;(2)andimply that;(3), thenand;(4), thenand;(5)if and only ifand.
Proof. (1)Suppose , this implies that and . As , we have and . On the other hand, ; this implies that and . So, and , thus .(2)Suppose , this implies that and . As , we have and . On the other hand, ; this implies that and . So, and , thus .(3)It follows from part (1).(4)It follows from part (2).(5)Let . Then, and . By Theorem 2, we get and . This implies that and . Converse holds by the transitivity of the relation .
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:
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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 person . wants to select a car out of available models. Let and and the set of attributes be .
Define bywhich represents the relation between models and colors available in brand for . Then,where represents the color of the model available in the brand .
Also,where represents the model of the color available in the brand .Therefore, the lower and upper approximations (with respect to the aftersets as well as with respect to the foresets) are as follows (Tables 14 and 15, respectively):The values of score function for car models are given in Table 16.
Table 16 shows that , so we calculate accuracy values for and .
Hence, the values of accuracy function are given in Table 17.
Table 17 shows that , so we can select any one, or .
Now, (given in Table 18) and (given in Table 19).
The values of score function for colors of cars are given in Table 20.
Table 20 shows that is maximum, so he will select color .
The flow chart of our decision-making algorithm is given in Figure 2.
In Figure 2, 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.
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.
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.
Data Availability
No data were used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.