1. Introduction
In real life, the limitation of precise research is progressively being recognized in various fields such as economics, social sciences, medical sciences, computer sciences, physical sciences, environmental sciences, management sciences, and engineering. It is familiar that the real world is full of vagueness, imprecision, and uncertainty, so research on these areas is of great significance. The solutions to such problems engaged the use of mathematical principles on the basis of imprecision and uncertainty. This article expands the scope of applications of one of the theories that can be used to deal with these attributes or characteristics, namely soft set theory.
In this unrealistic environment, there are many problems related to uncertainty [
1,
2,
3,
4]. However, the maximum mathematical tools that are in existence are crisp [
5]. Numerous theories have been introduced to explore uncertainty in an efficient way. For instance, Bayesian network [
6], evidence theory [
7,
8,
9], fuzzy set theory [
10,
11], intuitionistic fuzzy set (IFS) [
12,
13], and gray prediction model [
14]. Meanwhile, numerous properties of these theories have also been studied broadly [
14]. In [
15], Molodtsov indicates that there is a difficulty in the fuzzy set and intuitionistic fuzzy set theory, that is, the level of the membership defined by the individual regarded depends on the knowledge received by the individual, in consequence, vulnerable to subjective factors. Additionally, different attributes in one problem need to be thought about in an integrated manner. A soft set computing model was developed by Molodtsov [
15] to address these restrictions. A problem considering multiple attributes is a virtue of the soft set theory and it has very good potential to solve problems and plays a very significant role in various fields [
16,
17]. Therefore, for soft set theory, many researchers are introducing methods and operations for it. For instance, the fuzzy soft set theory is an extension of the soft set proposed by Xu [
18].
By accumulating soft set theory with the fuzzy set theory, Maji [
19] introduced the notation of FSS (fuzzy soft set), and this theory was used to address decision-making problems. Generalized fuzzy soft sets were proposed by Majumdar and Samanta [
20]; their properties were studied and used to solve problems of uncertainty. Maji [
21,
22] also introduced intuitionistic fuzzy soft sets by integrating IFS with the soft set. Dinda [
23] introduced the generalized intuitionistic fuzzy soft sets, belief interval-valued soft sets [
24], generalized belief interval-valued soft sets [
25], interval-valued intuitionistic fuzzy soft sets [
26], interval-valued picture fuzzy soft sets [
27], interval-valued neutrosophic soft sets [
28], and generalized picture fuzzy soft sets [
29]. Further, there are many extension models of the soft set theory rapidly developed; for instance, possibility fuzzy soft set [
30], possibility m-polar fuzzy soft sets [
31], possibility Pythagorean fuzzy soft sets [
32], possibility neutrosophic soft sets [
33], possibility multi-fuzzy soft sets [
34], and possibility belief interval-valued soft sets [
35].
The belief theory was proposed by Dempster and Shafer [
35,
36]. This theory has been applied in various fields. For instance, uncertainty modeling [
37], uncertainty reasoning [
14,
38,
39], decision-making [
40,
41], information fusion [
42,
43], and other fields [
44]. Fatimah [
45] extended the soft set model under a non-binary evaluation environment and introduced the concept of N-soft set (NSS) and explained the significance of ordered grades in the practical problems. Furthermore, they also developed decision-making procedures for the N-soft set. Later on, Akram [
46] proposed a novel hybrid model known as hesitant N-soft set (HNSS) by accumulating hesitancy and N-soft set. Meanwhile, in [
47], they also introduced the concept of fuzzy N-soft set (FNSS) by accumulating a fuzzy set with an N-soft set. Many problems related to decision-making are discussed by using different kinds of environments in [
48,
49,
50,
51,
52,
53,
54,
55,
56,
57,
58]. The developed model gives a more flexible decision-making method for dealing with uncertainties referring to which specific level is allocated to objects in the parameterizations by attributes.
In this article, we present the concept of a possibility belief interval-valued N-soft set, which can be viewable as a possibility belief interval-valued N-soft model. In
Section 2, we review the basic idea concerning the Dempster–Shafer theory and in addition, soft set, belief interval-valued soft set (BIVSS) and N-soft set are briefly reminded of with examples. In
Section 3, we propose the model of the belief interval-valued N-soft set (BIVNSS). In
Section 4, we discuss some algebraic operations (for instance, restricted intersection, restricted union, extended intersection, extended union, complement, top complement, bottom complement, max-AND, and min-OR) on the belief interval-valued N-soft set and many fundamental properties of these operations are introduced. In
Section 5, we proposed the model of possibility belief interval-valued N-soft set (PBIVNSS). In
Section 6, we introduce many algebraic operations (for example, restricted intersection, restricted union, extended intersection, extended union, complement, top complement, bottom complement, max-AND, and min-OR) on possibility belief interval-valued N-soft set, and various fundamental properties on these operations are also discussed. In
Section 7, we develop algorithms on max-AND and min-OR operations of possibility belief interval-valued N-soft sets for decision making. Then in
Section 8, we present the applications on decision-making problems that yield the optimum solution. While in
Section 9, we conclude the article.
2. Preliminaries
In this section, a short review of basic definitions and relevant theories are given, which we used to develop the methods introduced in this paper. There are several problems related to uncertainty in this real-life [
59,
60,
61]. The Dempster–Shafer theory has been broadly used in dealing with the uncertain problems [
62,
63]. The Dempster–Shafer theory is a generalized scheme for demonstrating uncertainty. Dempster proposed a belief measure theory that developed lower and upper probabilities of a system while Shafer provided a thorough belief function explanation.
Definition 1. Let be a finite set of frame of discernment (hypotheses), be the set of all subsets of and . The belief structure of Dempster-Shafer is associated with a mapping such thatis the basic probability assignment function, where indicates the belief values of . For which subsets of mapping allot non-zero values are known as focal elements [24]. Basic probability assignment has various operations for instance divergence [64], entropy function [65,66,67], and others [68]. Definition 2 ([
35]).
The measure of belief function associated with is determined as a mapping such that for any subset of , Definition 3 ([
35]).
The measure of plausibility function associated with is determined as a mapping such that for any subset of , Obviously,
. The interval
is called belief interval (BI) [
69].
Definition 4. Let U be the non-empty universal set of objects and E be the set of attributes, for any non-empty set . A pair is called soft set over U if there is a mapping where denotes the set of all subsets of U.
Thus, the soft set is a parametric family of the subsets of a universal set. For each , we can interpret as a subset of universal set U. We can also consider as a mapping and then equivalent to , otherwise [45]. Molodtsov considered many examples in [15] to illustrate the soft set. Definition 5. Let U be the non-empty universal set of objects. Let denote the collection of all belief interval-valued subsets of U and E be the set of attributes, for any non-empty set . A pair is called a belief interval-valued soft set over U (in short ) if there is a mapping.
It is represented as:where , and for all . Example 1. Let be the set of universe, be the set of attributes, and such that . Then over U is: Definition 6. Let U be the non-empty universal set of objects. Let denote the set of all subsets of U and let be a set of ordered grades where and E are the set of attributes, for any non-empty set . A triple is called N-soft set over U if there is a mapping , with the property that for each there exists a unique such that , where is the collection of all soft sets over [45]. Example 2. Let be the set of students, be the set of attributes evaluations of students by skills, and such that
and let be the set of grade evaluation. Then, is the 6-soft set as follows:
It can also be represented in tabular form as follows: | | | |
| 4 | 2 | 5 |
| 2 | 1 | 3 |
| 3 | 5 | 0 |
For illustration, the above table is of a 6-soft set
established on communication skills, collaboration skills, and critical thinking of the students. Where in the top left cell 4 is the ordered grade
of the student
with respect to
= communication skills. Similarly, in the bottom right cell, 0 is the ordered grade
of the student
with respect to
= critical thinking. Here, 0 is the lowest grade; it does not mean that there is no evaluation or incomplete information. There are many examples to illustrate the N-soft set in [
45].
4. Operations on
In this section, we discussed some algebraic operations on belief interval-valued N-soft set and their properties.
Definition 8. Let U be the non-empty universal set of objects. Given that and are two on U, their restricted intersection is defined as:where and where, If with
Definition 9. Let U be the non-empty universal set of objects. Given that and are two on U, their restricted union is defined as:where and ,where, If with
Definition 10. Let U be the non-empty universal set of objects. Given that and are two on U, their extended intersection is defined as:whereand Definition 11. Let U be the non-empty universal set of objects. Given that and are two on U, their extended union is defined as:where and
,
Example 4. Let be the set of Covid-19 patients, be the set of attributes and such that . The are defined as follows: Then their restricted intersection is:their restricted union is:their extended intersection is:their extended union is: Definition 12. Let be a on a non-empty universe U. Then a weak belief interval-valued complement is denoted by where and is defined as:where, . Definition 13. For any on U. The bottom weak belief interval-valued complement of is defined as: Definition 14. For any on U. The top weak belief interval-valued complement of is defined as: Example 5. Consider as described in Example 3 then its weak belief interval-valued complement is:its bottom weak belief interval valued complement is:its top weak belief interval-valued complement is: Definition 15. Soft max-AND operation of two and (where and defined as:with and . Definition 16. Soft min-OR operation of two and (where and defined as:where, where,with and . Example 6. Consider and as described in Example 4, then their soft max-AND is:
their soft min-OR is:
Proposition 1. Given that and are any three on U, then the commutative and associative properties are held:
- (1)
- (2)
- (3)
- (4)
- (5)
,
- (6)
,
- (7)
,
- (8)
,
- (9)
,
- (10)
,
- (11)
,
- (12)
.
Proof. (1)–(8) follows from the definition. (9) Let
and
(where
and
) be two
. By the definition of extended intersection we have
such that,
where
with
and
,
As,
Suppose that
such that
, where
with
,
and
,
Again, let
and
(where
and
) be two
. By the definition of extended intersection we have
where
such that,
where
with
and
As,
. Suppose that
such that
where
with
,
and
,
Then . Hence, (9) is held, (10)-(12) are similar to (9). □
Proposition 2. Given that and are any two on U, then the following results hold:
- (1)
,
- (2)
,
- (3)
,
- (4)
,
- (5)
.
Proof. (1) is straight-forward. We start from (2), let
and
(where
) be two
on
U. Then by the definition of weak belief interval-valued complement and extended union we have,
where
with
and
,
Again by the definition of extended intersection we have
with
and
,
Thus, by the definition of weak complement we have,
Then, . Hence, (2) holds, and (3)–(5) are similar to (2). □
6. Operations on
In this section, we discussed some algebraic operations on a possibility belief interval valued N-soft set and their fundamental properties.
Definition 18. Let U be the non-empty universal set of objects. Given that and are two on U, their restricted intersection is defined as:where and ,where,and If and with
Definition 19. Let U be the non-empty universal set of objects. Given that and are two on U, their restricted union is defined as:where ,where,and If and with
Definition 20. Let U be the non-empty universal set of objects. Given that and are two on U, their extended intersection is defined as:where and Definition 21. Let U be the non-empty universal set of objects. Given that and are two on U, their extended union is defined as:where and Example 8. Let and such that . Then are defined as follows:
Then their restricted intersection is:their restricted union is:their extended intersection is
their extended union is:
Definition 22. Let be a on a non-empty universe U. Then a weak possibility belief interval-valued complement is denoted by where and is defined as: Definition 23. For any on U. The bottom weak possibility belief interval-valued complement of is defined as:
Definition 24. For any on U. The top weak possibility belief interval-valued complement of is defined as: Example 9. Consider as described in Example 7, then its weak possibility belief interval-valued complement is:
its bottom weak possibility belief interval-valued complement is:
its top weak possibility belief interval-valued complement is:
Definition 25. Soft max-AND operation of two and (where and defined as:where where,andwith and . Definition 26. Soft min-OR operation of two and (where and defined as: