Belief and Possibility Belief Interval-Valued N-Soft Set and Their Applications in Multi-Attribute Decision-Making Problems

In this research article, we motivate and introduce the concept of possibility belief interval-valued N-soft sets. It has a great significance for enhancing the performance of decision-making procedures in many theories of uncertainty. The N-soft set theory is arising as an effective mathematical tool for dealing with precision and uncertainties more than the soft set theory. In this regard, we extend the concept of belief interval-valued soft set to possibility belief interval-valued N-soft set (by accumulating possibility and belief interval with N-soft set), and we also explain its practical calculations. To this objective, we defined related theoretical notions, for example, belief interval-valued N-soft set, possibility belief interval-valued N-soft set, their algebraic operations, and examined some of their fundamental properties. Furthermore, we developed two algorithms by using max-AND and min-OR operations of possibility belief interval-valued N-soft set for decision-making problems and also justify its applicability with numerical examples.


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

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 Y be a finite set of frame of discernment (hypotheses), 2 Y be the set of all subsets of Y andŶ ⊆ Y. The belief structure of Dempster-Shafer is associated with a mapping E : 2 Y → [0, 1] such that is the basic probability assignment function, where E (Ŷ ) indicates the belief values of Y. For which subsets of Y 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 E is determined as a mapping Bel : 2 Y → [0, 1] such that for any subsetẐ of Y, Definition 3 ([35]). The measure of plausibility function associated with E is determined as a mapping Pl : 2 Y → [0, 1] such that for any subsetẐ of Y, Obviously, Bel(Ẑ ) ≤ Pl(Ẑ ). The interval [Bel(Ẑ ), Pl(Ẑ )] 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 K ⊆ E. A pair (A, K) is called soft set over U if there is a mapping A : K → 2 U where 2 U 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 k j ∈ K, we can interpret A(k j ) as a subset of universal set U. We can also consider A(k j ) as a mapping A(k j ) : U → {0, 1} and then A(k j )(u i ) = 1 equivalent to u i ∈ A(k j ), otherwise A(k j )(u i ) = 0 [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 BI U denote the collection of all belief interval-valued subsets of U and E be the set of attributes, for any non-empty set K ⊆ E. A pair (B, K) is called a belief interval-valued soft set over U (in short BIVSS) if there is a mapping B : K → BI U . It is represented as: where BI B(k j ) (u i ) = [Bel B(k j ) (u i ), Pl B(k j ) (u i )], Bel B(k j ) (u i ) ∈ [0, 1], Pl B(k j ) (u i ) ∈ [0, 1], and 0 ≤ Bel B(k j ) (u i ) ≤ Pl B(k j ) (u i ) ≤ 1 for all u i ∈ U. Example 1. Let U = {u 1 , u 2 , u 3 , u 4 , u 5 } be the set of universe, E = {k 1 , k 2 , k 3 , k 4 } be the set of attributes, and K ⊆ E such that K = {k 1 , k 2 }. Then BIVSS over U is: Definition 6. Let U be the non-empty universal set of objects. Let 2 U denote the set of all subsets of U and let R = {0, 1, 2, . . ., N − 1} be a set of ordered grades where N ∈ {2, 3, 4, . . .} and E are the set of attributes, for any non-empty set K ⊆ E. A triple (C, K, N) is called N-soft set over U if there is a mapping C : K → 2 U×R , with the property that for each k j ∈ K there exists a unique (u i , r ij ) ∈ (U × R) such that (u i , r ij ) ∈ C(k j ), k j ∈ K, u i ∈ U and r ij ∈ R, where 2 U×R is the collection of all soft sets over U × R [45].
Example 2. Let U = {u 1 , u 2 , u 3 } be the set of students, E = {k 1 , k 2 , k 3 , k 4 , k 5 } be the set of attributes evaluations of students by skills, and K ⊆ E such that K = {k 1 = communication skills, k 3 = collaboration skills, k 5 = critical thinking} and let R = {0, 1, 2, 3, 4, 5} be the set of grade evaluation. Then, (C, K, 6) is the 6-soft set as follows: C(k 1 ) = (u 1 , 4), (u 2 , 2), (u 3 , 3) , C(k 3 ) = (u 1 , 2), (u 2 , 1), (u 3 , 5) , and C(k 5 ) = (u 1 , 5), (u 2 , 3), (u 3 , 0) . It can also be represented in tabular form as follows: For illustration, the above table is of a 6-soft set (C, K, 6) established on communication skills, collaboration skills, and critical thinking of the students. Where in the top left cell 4 is the ordered grade (r 11 ) of the student u 1 with respect to k 1 = communication skills. Similarly, in the bottom right cell, 0 is the ordered grade (r 35 ) of the student u 3 with respect to k 5 = 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].

Belief Interval-Valued N-Soft Set (BIV NSS)
In this section, we derive some basic concepts of a new extended model of a belief interval-valued N-soft set with examples from real practice.

Definition 7.
Let U be the non-empty universal set of objects and E be the set of attributes, for any non-empty set K ⊆ E and let BI U denote the collection of all belief interval-valued subsets of U and R = {0, 1, 2, . . ., N − 1} be a set of ordered grades where N ∈ {2, 3, 4, . . .}. A triple (A, K, N) is called a belief interval valued N-soft set over U if there is a mapping A : K → BI U×R , where BI U×R is the collection of all belief interval-valued soft sets over U × R. It is represented as: where, and Example 3. Let U = {u 1 , u 2 , u 3 } be the universe of gardens, R = {0, 1, 2, 3, 4} be the set of grade evaluation, E = {k 1 = Rose, k 2 = Tulip, k 3 = Jasmine, k 4 = Da f f odils} be the set of attributes (evaluation of gardens by flowers), and K ⊆ E such that K = {k 1 , k 3 , k 4 }. Thus, (A, K, 5) is the belief interval of 5−soft set as follow:

Operations on BIV NSS
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 (B, K, M) and (C, L, N) are two BIV NSS on U, their restricted intersection is defined as: Definition 9. Let U be the non-empty universal set of objects. Given that (B, K, M) and (C, L, N) are two BIV NSS on U, their restricted union is defined as: where E = B ∪ R C, T = K ∩ L and P = max(M, N), i.e., ∀t j ∈ T and u i = U, where, where F = B ∩ E C, S = K ∪ L and P = max(M, N), i.e., ∀s j ∈ S, u i = U with s 1 j ∈ K, and s 2 j ∈ L, where H = B ∪ E C, S = K ∪ L and P = max(M, N), i.e., ∀s j ∈ S, u i = U with s 1 j ∈ K and s 2 j ∈ L, Example 4. Let U = {u 1 , u 2 , u 3 } be the set of Covid-19 patients, E = {k 1 = tiredness, k 2 = skin rashes, k 3 = dry cough, k 4 = shortness o f breath} be the set of attributes and K, L ⊆ E such that K = {k 1 , k 3 , k 4 }, L = {k 1 , k 3 }. The BIV NSS are defined as follows: Then their restricted intersection is: their restricted union is: their extended intersection is: their extended union is: Definition 12. Let (A, K, N) be a BIV NSS on a non-empty universe U. Then a weak belief interval-valued complement is denoted by(A c , K, N) where A c (k j ) ∩ A(k j ) = Φ; ∀k j ∈ K and A c (k j ) is defined as: , if r ij = 0.
Example 5. Consider (A, K, 5) as described in Example 3 then its weak belief interval-valued complement is: its bottom weak belief interval valued complement is: Entropy 2021, 23, 1498 8 of 37 its top weak belief interval-valued complement is: Definition 15. Soft max-AND operation of two BIV NSS (B, K, M) and (C, L, N) (where B : K → BI U×R and C : L → BI U×R ) defined as:

Definition 16.
Soft min-OR operation of two BIV NSS (B, K, M) and (C, L, N) (where B : K → BI U×R and C : L → BI U×R ) defined as: where, Example 6. Consider (B, K, 5) and (C, L, 4) as described in Example 4, then their soft max-AND is: their soft min-OR is: Proof.

Proposition 2.
Given that (B, K, M) and (C, L, N) are any two BIV NSS on U, then the following results hold:

Proof.
(1) is straight-forward. We start from (2), let (B, K, M) and (C, L, N) (where B : K → BI U×R C : L → BI U×R ) be two BIV NSS on U. Then by the definition of weak belief interval-valued complement and extended union we have, Again by the definition of extended intersection we have (F, S, Thus, by the definition of weak complement we have, Then, J(s j ) = F c (s j ); ∀s j ∈ S. Hence, (2) holds, and (3)-(5) are similar to (2).

Possibility Belief Interval-Valued N-Soft Set (PBIV NSS)
In this section, we defined the notion of possibility belief interval-valued N-soft set.

Definition 17.
Assume that BI U is the set of all belief interval-valued subsets of U and E is the set of attributes, for any non-empty set K ⊆ E. The pair (U, K) is called a soft universe, and R = {0, 1, 2, . . ., N − 1} is a set of ordered grades where N ∈ {2, 3, 4, . . .}. Let B : K → BI U×R , and b is a belief interval-valued subsets of K, i.e., b : K → BI U×R where BI U×R is the collection of all belief interval-valued soft sets over U × R. A triple (B b , K, N) is called a possibility belief interval-valued N-soft set over (U, K) if there is a mapping B b : K → BI U×R × BI U×R . It is represented as:

Operations on PBIV NSS
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 (B b , K, M) and (Y y , L, N) are two PBIV NSS on U, their restricted intersection is defined as: (M, N), i.e., ∀t j ∈ T and u i = U, where, and BI e(t j ) (u i , r ij ) = Bel e(t j ) (u i , r ij ), Pl e(t j ) (u i , r ij ) . where, and Definition 20. Let U be the non-empty universal set of objects. Given that (B b , K, M) and (Y y , L, N) are two PBIV NSS on U, their extended intersection is defined as: N), i.e., ∀s j ∈ S, u i = U with s 1 j ∈ K and s 2 j ∈ L, where H = B ∪ E Y, h = b ∪ E y, S = K ∪ L and P = max(M, N), i.e., ∀s j ∈ S, u i = U, u i = U with s 1 j ∈ K and s 2 j ∈ L, Then PBIV NSS are defined as follows: Then their restricted intersection is: their extended union is: Example 9. Consider (B b , K, 7) as described in Example 7, then its weak possibility belief intervalvalued complement is: its bottom weak possibility belief interval-valued complement is: where G g : K × L → BI U×R × BI U×R ; ∀(k s , l t ) ∈ (K × L), s , t ∈ Λ, and P = max(M, N), t ) ), where, and Bel g(k s ,l t ) (u i , r i(s ,t ) ), Pl g(k s ,l t ) (u i , r i(s ,t ) ) = where, and Bel q(k s ,l t ) (u i , r i(s ,t ) ), Pl q(k s ,l t ) (u i , r i(s ,t ) ) = and (Y y , L, 6) as described in Example 8, then their soft max-AND is: their soft min-OR is:   , N) ; ∀s j ∈ S with s 1 j ∈ K and s 2 j ∈ L, Then, G g (w j ) = I i (w j ), ∀w j ∈ W. Hence, (9) is hwld and (10)- (12) are similar to (9).

Proposition 4.
Given that (B b , K, M) and (Y y , L, N) are any two PBIV NSS on U, then the following results hold: Again by the definition of extended intersection we have (F f , S, P) = (B b ∩ E Y y ), S = K ∪ L, P = max(M, N); ∀s j ∈ S with s 1 j ∈ K and s 2 j ∈ L, Thus, by the definition of weak possibility belief interval-valued complement we have, Then, J j (s j ) = F c f c (s j ); ∀s j ∈ S. Hence, (2) holds. Remaining (3)-(5) are similar to (2).
In Figures 1 and 2, we give the flow charts of Algorithms 1 and 2 respectively.

Algorithms
In this section, we will present the algorithms on soft max-AND and soft min-OR operations on two possibility belief interval valued N-soft sets for decision making.

Algorithm 1 Soft max-AND operations
Step 1: We have two PBIV NSS (B b , K, M) and (Y y , L, N) (where B b : K → BI U×R × BI U×R and Y y : L → BI U×R × BI U×R ) on universal set U = {u 1 , u 2 , . . ., u l }.
Step 3: Evaluate the choice value C(k s , l t )(u i ); ∀ u i ∈ U, (k s , l t ) ∈ (K × L), s , t ∈ Λ defined as: where the interval of choice value is: Step 4: Evaluate the score S(k s , l t )(u i ); where, α 1 I C(k s ,l t ) (u i , r i(s ,t ) ) − α 1 I C(k s ,l t ) (t q , r q(s ,t ) )+ α 2 I C(k s ,l t ) (u i , r i(s ,t ) ) − α 2 I C(k s ,l t ) (t q , r q(s ,t ) ) .
where α m : [0, 1] → [0, 1] is the mth projection mapping such that α 1 is the lower membership value and α 2 is the upper membership value of choice interval and S 2 (k s , l t )(u i ) = l ∑ q=1 max r i(s ,t ) , r q(s ,t ) .
Step 5: For each S(u l ) it's weighted value is: S(k s , l t )(u l ).

Algorithm 2 Soft min-OR operations
Step 1: Let we have two PBIV NSS (B b , K, M) and (Y y , L, N) (where B b : K → BI U×R × BI U×R and Y y : L → BI U×R × BI U×R ) on universal set U = {u 1 , u 2 , . . ., u l }.
Step 3: Evaluate the choice value C(k s , l t )(u i ); ∀ u i ∈ U, (k s , l t ) ∈ (K × L), s , t ∈ Λ defined as: where the interval of choice value is: Step 4: Evaluate the score S(k s , l t )(u i ); where, where α m : [0, 1] → [0, 1] is the mth projection mapping such that α 1 is the lower membership value and α 2 is the upper membership value of choice interval and S 2 (k s , l t )(u i ) = l ∑ q=1 max r i(s ,t ) , r q(s ,t ) .
Step 5: For each S(u l ) it's weighted value is: S(k s , l t )(u l ).

Applications
In this section, we give the application of our proposed sets. Firstly, we will evaluate the soft max-AND operation (G g , K × L, 7) = (B b , K, 4) ∧ (Y y , L, 7) by using step 2 of Algorithm 1: The Score S(k s , k t )(u i ) The Score S(k 1 , k 4 )(u 1 ) −52.14 S(k 3 , k 4 )(u 1 ) −10.34  Here the optimal decision by using step 6 of Algorithm 1 is: Thus, u 5 is the best choice. Hence Mr. H will buy the u 5 smartphone.
In Figures 3 and 4, we give the graphical behavior of score values of Examples 11 and 12 by means of Algorithms 1 and 2 respectively.   Now we will evaluate the Score S(k s , l t )(u i ); ∀ u i ∈ U, (k s , l t ) ∈ (K × L), s = 1, 3 and t = 4, 5, 6 by using step 4 of Algorithm 2. Here the optimal decision by using step 6 of Algorithm 2 is: X = arg max {S(u 1 ), S(u 2 ), S(u 3 ), S(u 4 ), S(u 5 ), S(u 6 )}.
Thus, u 2 is the best choice. Hence Mr. H will buy the u 2 smartphone.
In Figure 5, we observe that the following relations between the score values of Examples 11 and 12 by means of Algorithms 1 and 2, respectively. Here the optimal decision by using step 6 of Algorithm 1 is: X = arg max {S(u 1 ), S(u 2 ), S(u 3 ), S(u 4 ), S(u 5 ), S(u 6 )}.
Thus, u 2 is the best choice. Hence Mr. X will select u 2 person as a personal secretary.
In Figures 6 and 7, we give the graphical behavior of score values of Examples 13 and 14 by means of Algorithms 1 and 2 respectively.

Conclusions
The main intent of this work is to present a possibility belief interval-valued N-soft set by consolidating a belief interval value and possibility with N-soft set and its applications for solving the complicated decision-making problems in various fields of life. There are great applications for the belief interval in many different fields of life, while the other tool N-soft set theory is arising as a prosperous mathematical approach for manipulating ambivalent information. First, we defined basic theory and definitions of important sets in a very clear way. Then we discussed the BIVNSS, various algebraic operations, and their fundamental properties. Then we defined PBIVNSS, its algebraic operations, its elemental properties, and also its applications for decision-making problems. We have also provided algorithms for these decision-making methods and showed how decision-making methods are applied successfully in the problems of real life. In further work, this idea can be seen in many other algebraic expressions and topological structures.