Hybrid multi-attribute decision-making model based on ( m,N )-soft rough sets

Abstract

The theory of soft sets is a powerful mathematical tool to handle the vagueness in data, while the m-polar fuzzy (mF, for short) sets have ability to deal with uncertainty as well as multi-polarity of the data in many situations. In this research article we present two novel hybrid models for soft computing, namely, m-polar fuzzy N-soft sets ((m, N)-soft sets, for short) and m-polar fuzzy N-soft rough sets ((m, N)-soft rough sets, for short). We define related concepts and investigate some of their fundamental properties. We also present applications of these hybrid models to multi-attribute decision-making.

Keywords

soft set rough set decision-making

1 Introduction

Chen et al. [17] generalized bipolar fuzzy sets [44] and proposed a strong idea of mF sets. The idea behind this concept is that “multipolar information", exists because in various real life decision-making problems data sometimes produced by m sources (m ≥ 2). Akram [3] proposed various new concepts on mF graphs. The theory of rough sets [39] is a successful mathematical tool for handling imprecise and uncertain information. There are several real life problems in various fields, including physical sciences, life sciences and social sciences contain uncertainties. Many classical mathematical theories including Zadeh’s fuzzy set theory [42] and Pawlak’s rough set theory [40] are very useful tools for handling the different types of uncertainties. Molodtsov [38] pointed out the difficulties of these theories. One reason behind the difficulties of these theories is the inadequacy of the parametrization tool. To overcome these drawbacks, he [38] proposed the idea of soft sets. The theory of soft sets is playing a very important role in many fields including, data analysis [48] and decision-making [27 , 41]. Some algebraic operations on soft set theory were discussed by Maji et al. [36], see also Ali et al. [13]. Based on the idea of knowledge of reduction in rough set theory, Maji et al. [35] proposed the first real life application of soft sets in decision-making problems. A branch of this literature that has produced many fruitful approaches is parameter reduction (for example, Ali [15], Danjuma et al. [20], Danjuma et al. [21] and Zhan and Alcantud [43] are updated surveys). Maji et al. [37] introduced a new hybrid model fuzzy soft sets by the combination of fuzzy sets and soft sets. Alcantud [8 –10] introduce a new approach to the decision-making problems of fuzzy soft sets (see also Alcantud and Mathew [12], Alcantud et al. [11], Liu et al. [32]). Feng et al. [25] combined soft sets, fuzzy sets and rough sets, see also Ali [14]. Moreover, Feng et al. [26] combined soft sets and rough sets, which were applied to multi-criteria group decision-making in Feng [24]. Fatimah et al. [23] proposed new algorithms for decision-making based on the probabilistic soft set theory.

An inspection of the existing hybrid soft set models readily shows that the researchers attracted by soft sets and their hybrid models typically worked on either a binary framework of evaluations (either 0 or 1) or else, real numbers between 0 and 1 (see [34, 48]). However examples abound that daily real situations contain data with a non-binary and discrete structure, which therefore do not belong to those formats. For instance in the realm of the aggregation of social opinions, Alcantud and Laruelle [7] specified the characteristics of ternary voting systems and they gave examples where ternary opinions must be aggregated. Examples that are closer to our daily experience are the non-binary estimations that we often find in rating or ranking positions. Rankings of tourist resorts, films, or electronic devices, often take the form of number of dots and stars (like ‘one big dot,’ ‘one star,’ ‘two stars,’ ‘three stars’), which can also be seen in the form of natural numbers (like ‘0’ for one big circle, ‘1’ for one star, ‘2’ for two stars, ‘3’ for three stars). Furthermore, Herawan and Deris [28] developed n binary-valued information system in soft sets where every parameter has its own ranking, when compared to the rating order described in Chen et al. [18]. Instead of ratings as estimations, Ali et al. [16] designed rating systems among the elements of soft sets parameters. Motivated by these practical considerations, Fatimah et al. [22] introduced the notion of N-soft sets as a generalization of soft sets with a multinary nature. This model has soon be combined with other features like hesitancy and fuzziness. Indeed, Akram et al. [1] developed a new powerful hybrid model for group decision-making, namely, hesitant N-soft sets. Akram et al. [2] combined fuzzy sets with N-soft sets, and proposed another useful decision-making model called fuzzy N-soft sets. Multi-attribute decision-making (MADM) is an important part of modern decision science. It assumes that there exists a set of alternatives with multiple attributes which a decision maker should evaluate and analyze. The aim of MADM is to find the most desirable alternative or rank the feasible alternatives for supporting decision makings. It is used in various fields, including location choice of manufactory, investment decision-making, selection of person with ability. It also has been extensively applied to many aspects such as economics, management, engineering, and technology [4 , 45–47].

In this article, we introduce two novel MADM models, (m, N)-soft sets and (m, N)-soft rough sets. We investigate their fundamental properties. We also introduce some helpful related notions and operations. Furthermore, we present respective applications of these models to decision-making situations. Our solutions are given by algorithms whose implementation and feasibility are proven by real examples.

The outline of the paper is as follows. Section 2 defines (m, N)-soft sets and basic operations on them. Some relationships with existing models are put forward too. Section 3 defines (m, N)-soft rough sets and basic operations on them. The decision-making mechanisms that operate on these models are described in Section 4. We also apply them to real examples that are fully described in order to prove their implementability and feasibility. Section 5 studies the advantages and disadvantages of our proposed models. We conclude in Section 6.

2 Construction of (m, N)-soft sets

We proceed to introduce the novel concept of (m, N)-soft sets, which needs some previous definitions inclusive of the new notion of sub-mF set.

Definition 2.1. [17] An mF set on a nonempty universe set V is a mapping $A (v) = (p_{1} \circ A (v), p_{2} \circ A (v), \dots, p_{m} \circ A (v)) : V \to {[0, 1]}^{m}$ , where the i-th projection mapping is defined as p_i ∘ A : [0, 1] ^m → [0, 1]. The collection of all mF sets over V is denoted by $M (V)$ . 0 = (0, 0, ⋯ , 0) is the smallest element in [0, 1] ^m and 1 = (1, 1, ⋯ , 1) is the biggest element in [0, 1] ^m.

Definition 2.2. [22] Let V be a universe, Z a set of attributes and C ⊆ Z. Let $M = {0, 1, \dots, N - 1}$ be a set of ordered grades where $N \in {2, 3, \dots}$ . A triple (F, C, N) is called an N-soft set over V if F is a mapping from C to 2^V×M, with the property that for each c_k ∈ C and v ∈ V there exists a unique (v, r_{c
_k}) ∈ V × M such that (v, r_{c
_k}) ∈ F (c_k) , v ∈ V, r_{c
_k} ∈ M.

Definition 2.3.3 [6] Let V be a universal set, Z a set of parameters and C ⊆ Z. Define a mapping ξ from C to $M (V)$ . Then, a pair (ξ, C) is called an mF soft set on V, which is defined by $\begin{matrix} (ξ, C) = {〈 v, (p_{1} \circ C_{c_{k}} (v), \dots, p_{m} \circ C_{c_{k}} (v) 〉 | \\ v \in V, c_{k} \in C} . \end{matrix}$

Definition 2.4. Let V be a universal set. Any mF set on a subset V′ of V is called sub-mF set on V. The family of all sub-mF sets over V is denoted by $M^{'} (V)$ .

We now define an m-polar fuzzy N-soft set or (m, N)-soft set.

Definition 2.5. An (m, N)-soft set is a triple (f, D, m) where D = (F, C, N) is an N-soft set on V and f maps any attribute in C with an mF set A on F (c_k), which is a convenient subset of V × M and c_k ∈ C. Therefore, the domain of f is of course C, and its codomain is $M^{'} (V \times M)$ , the family of all sub-mF sets over V × M.

Example 2.6. In Australia star rating for hotels is awarded by AAA (Accommodation Association of Australia) Tourism and its range is from 1 to 5 star. AAA Tourism is represented by the states automotive clubs, including RACV, RAC, RAA and RACT. AAA inspectors collectively visit hotels and inspect the facilities and levels of comfort and luxury, allowing them to identify emerging trends in the industry.

Let V = {v₁, v₂, v₃, v₄, v₅, v₆} be a universe of six hotels in Australia and let C = {c₁, c₂, c₃, c₄} ⊆ Z be the set of attributes, which gives grades to hotels based on location, services and parking area. we can obtain a (3, 6)-soft set from Table 1, where

Five star represents ‘luxury’,

Four star represents ‘excellent’,

Three star represents ‘very good’,

Two star represents ‘good’,

One star represents ‘regular’,

Circle represents ‘bad’.

Table 1
Information extracted from the related data

V c ₁ c ₂ c ₃ c ₄

v ₁ ★★★ ★ ★★ ★★★

v ₂ ★★★ ★★★ ★ ★★★ ★★

v ₃ ★ ★★★ ★★ ★ ★★★ ★

v ₄ ★★ ∘ ★★★ ★ ★★★

v ₅ ★★★ ★★ ★★ ★★★ ★★ ★★★

v ₆ ★★★ ★ ★★★ ★★★ ★ ★★★ ★

V	c ₁	c ₂	c ₃	c ₄
v ₁	★★★	★	★★	★★★
v ₂	★★★	★★★ ★	★★★	★★
v ₃	★	★★★ ★★	★	★★★ ★
v ₄	★★	∘	★★★ ★	★★★
v ₅	★★★ ★★	★★	★★★ ★★	★★★
v ₆	★★★ ★	★★★	★★★ ★	★★★ ★

The set of ordered grades M = {0, 1, 2, 3, 4, 5} can be easily associated with stars as follows:

0 stand for ‘∘’,

1 stand for ‘★’,

2 stand for ‘★★’,

3 stand for ‘★★★’,

4 stand for ‘★★★ ★’,

5 stand for ‘★★★ ★★’.

From Definition 2.2, tabular form of a 6-soft set is given by Table 2.

Table 2

Tabular representation of 6-soft set (F, C, 6)

(F, C, 6)	c ₁	c ₂	c ₃	c ₄
v ₁	3	1	2	3
v ₂	3	4	3	2
v ₃	1	5	1	4
v ₄	2	0	4	3
v ₅	5	2	5	3
v ₆	4	3	4	4

Therefore, by Definition 2.5 a (3, 6)-soft set (f, D, 3) is defined as follows: $(f, D, 3) =$ ${\begin{matrix} f (c_{1}) = {〈 (v_{1}, 3), 0.3, 0.5, 0.7 〉, 〈 (v_{2}, 3), 0.1, 0.6, 0.8 〉, \\ 〈 (v_{3}, 1), 0.4, 0.3, 0.5 〉, 〈 (v_{4}, 2), 0.2, 0.7, 0.5 〉, \\ 〈 (v_{5}, 5), 0.9, 0.5, 0.7 〉, 〈 (v_{6}, 4), 0.4, 0.8, 0.7 〉}, \\ f (c_{2}) = {〈 (v_{1}, 1), 0.4, 0.2, 0.6 〉, 〈 (v_{2}, 4), 0.3, 0.4, 0.8 〉, \\ 〈 (v_{3}, 5), 0.5, 0.6, 0.8 〉, 〈 (v_{4}, 0), 0.4, 0.3, 0.4 〉, \\ 〈 (v_{5}, 2), 0.2, 0.6, 0.5 〉, 〈 (v_{6}, 3), 0.4, 0.1, 0.9 〉}, \\ f (c_{3}) = {〈 (v_{1}, 2), 0.7, 0.5, 0.3 〉, 〈 (v_{2}, 3), 0.9, 0.4, 0.6 〉, \\ 〈 (v_{3}, 1), 0.1, 0.5, 0.8 〉, 〈 (v_{4}, 4), 0.8, 0.7, 0.7 〉, \\ 〈 (v_{5}, 5), 1.0, 1.0, 1.0 〉, 〈 (v_{6}, 4), 0.4, 0.7, 0.9 〉}, \\ f (c_{4}) = {〈 (v_{1}, 3), 0.4, 0.6, 0.9 〉, 〈 (v_{2}, 2), 0.2, 0.4, 0.8 〉, \\ 〈 (v_{3}, 4), 0.7, 0.6, 0.9 〉, 〈 (v_{4}, 3), 0.6, 0.8, 0.1 〉, \\ 〈 (v_{5}, 3), 0.4, 0.5, 0.7 〉, 〈 (v_{6}, 4), 0.5, 0.3, 0.9 〉} . \end{matrix}}$

The (3, 6)-soft set (f, D, 3) can be represented by Table 3.

Table 3

Tabular representation of (3, 6)-soft set (f, D, 3)

(f, D, 3)	c ₁	c ₂	c ₃	c ₄
v ₁	〈3, (0.3, 0.5, 0.7) 〉	〈1, (0.4, 0.2, 0.6) 〉	〈2, (0.7, 0.5, 0.3) 〉	〈3, (0.4, 0.6, 0.9) 〉
v ₂	〈3, (0.1, 0.6, 0.8) 〉	〈4, (0.3, 0.4, 0.8) 〉	〈3, (0.9, 0.4, 0.6) 〉	〈2, (0.2, 0.4, 0.8) 〉
v ₃	〈1, (0.4, 0.3, 0.5) 〉	〈5, (0.5, 0.6, 0.8) 〉	〈1, (0.1, 0.5, 0.8) 〉	〈4, (0.7, 0.6, 0.9) 〉
v ₄	〈2, (0.2, 0.7, 0.5) 〉	〈0, (0.4, 0.3, 0.4) 〉	〈4, (0.8, 0.7, 0.7) 〉	〈3, (0.6, 0.8, 0.1) 〉
v ₅	〈5, (0.9, 0.5, 0.7) 〉	〈2, (0.2, 0.6, 0.5) 〉	〈5, (1.0, 1.0, 1.0) 〉	〈3, (0.4, 0.5, 0.7) 〉
v ₆	〈4, (0.4, 0.8, 0.7) 〉	〈3, (0.4, 0.1, 0.9) 〉	〈4, (0.4, 0.7, 0.9) 〉	〈4, (0.5, 0.3, 0.9) 〉

Thus, (f, D, 3) is a (3, 6)-soft set based on location, services and parking area of the hotels. For example, in the Table 3, 〈3, (0.3, 0.5, 0.7) 〉 means that the hotel v₁ is 30% suitable with the location, 50% suitable with the services and 70% suitable with parking area, where 3 is the evaluation grade of v₁ according to attribute c₁. Note that the criterion of assigning grades corresponding to each parameter based upon the membership values of the poles, for example, according to parameter c₁ grades are assigned based on average of membership values of all three poles for each object v_i, i = 1, 2, …, 6 . And the criterion of assigning grades may vary from one parameter to other.

Remark.

An (m, 2)-soft set can be naturally associated with an mF soft set. We identify an (m, 2)-soft set $f : C \to M^{'} (V \times {0, 1})$ with an mF soft set (ξ, C) by Definition 2.3.

If we choose m = 1, then a fuzzy 2-soft set $f : C \to M^{'} (V \times {0, 1})$ can be identified with a fuzzy soft set $G : C \to F (V)$ defined by $G (c_{k}) = {(v, μ (v)) | 〈 (v, 1), μ (v) 〉 \in f (c_{k}), c_{k} \in C},$ where $F (V)$ is the family of all fuzzy subsets of V.

Any (m, N)-soft set over a universe V can be identified as an (m, N + 1)-soft set. For example, from Table 3, the (3, 6)-soft set (f, D, 3) in Example 2.6 can be identified as (3, 7)-soft set over V. In (3, 7)-soft set, let us assume that we have a 6 grade, which is not required.

Grade 0 ∈ M in Definition 2.2 represents the lowest score. It does not mean that there is incomplete information or no assessment.

Example 2.7. Let V = {s₁, s₂, s₃} be a set of subjects and let C = {c₁, c₂, c₃} be a set of students, candidate c₁ selects subject s₁, candidate c₂ selects subject s₃, candidate c₃ selects subject s₃. Let M = {0, 1}. Therefore, a fuzzy 2-soft set is given as follows: $(f, D, 1) = {\begin{matrix} f (c_{1}) = {〈 (s_{1}, 1), 0.7 〉, 〈 (s_{2}, 0), 0.1 〉, \\ 〈 (s_{3}, 0), 0.3 〉}, \\ f (c_{2}) = {〈 (s_{1}, 0), 0.2 〉, 〈 (s_{2}, 0), 0.3 〉, \\ 〈 (s_{3}, 1), 0.9 〉}, \\ f (c_{3}) = {〈 (s_{1}, 0), 0.1 〉, 〈 (s_{2}, 0), 0.0 〉, \\ 〈 (s_{3}, 1), 0.6 〉} . \end{matrix}}$ This fuzzy 2-soft set (f, D, 1) can be identified with fuzzy soft set (ξ, C), which is defined by $(ξ, C) = {\begin{matrix} ξ (c_{1}) = {(s_{1}, 0.7)}, \\ ξ (c_{2}) = {(s_{3}, 0.9)}, \\ ξ (c_{3}) = {(s_{3}, 0.6)} . \end{matrix}}$

Definition 2.8.An (m, N)-soft set (f, D, m) is said to be efficient where D = (F, C, N) is an N-soft set, if f (c_k) = 〈 (v_j, N - 1) , 1〉 for some c_k ∈ C, v_j ∈ V.

Example 2.9. The (3, 6)-soft set defined in Example 2.6 is efficient.

Definition 2.10. Let (f, D, m) be an efficient (m, N)-soft set, where D = (F, C, N) is an N-soft set and f maps any attribute in C with an mF set A on F (c_k). Then its minimized efficient (m, N)-soft set on V is denoted by (f_l, D₁, m) where D₁ = (F_l, C, L) and is defined as $L = max_{j, k} F_{l} (c_{k}) (v_{j}) + 1$ , $(p_{1} \circ A (v_{j}, r_{c_{k}}), \dots, p_{m} \circ A (v_{j}, r_{c_{k}})) = 1$ for some (v_j, r_{c
_k}), F_l (c_k) (v_j) = F (c_k) (v_j) , ∀ c_k ∈ C, v_j ∈ V .

Proposition 2.11. Every efficient (m, N)-soft set corresponds with its minimized efficient (m, N)-soft set.

Proof. Its proof follows directly from the Definitions 2.8 and 2.10.□

Definition 2.12. Let (f₁, D₁, m) and (f₂, D₂, m) be two (m, N)-soft sets on a universe V where D₁ = (F₁, R, N₁) , D₂ = (F₂, S, N₂) are N-soft sets. Then (f₁, D₁, m) and (f₂, D₂, m) are said to be equal if and only if f₁ = f₂, D₁ = D₂.

Definition 2.13. Let (f, D, m) be an (m, N)-soft set on V, where D = (F, C, N) is an N-soft set. Then its weak complement is denoted by (f^c, D₁, m) where D₁ = (F^c, C, N) is an arbitrary N-soft set and f^c maps any attribute in C with an mF set A on F^c (c_k), F^c (c_k) ∩ F (c_k) = ∅ , for all c_k ∈ C and f^c (c_k) is given by $\begin{matrix} f^{c} (c_{k}) = {〈 ( & v_{j}, r_{c_{k}}), 1 - (p_{1} \circ A (v_{j}, r_{c_{k}}), \dots, \\ p_{m} \circ A (v_{j}, r_{c_{k}})) 〉 | (v_{j}, r_{c_{k}}) \in V \times M} . \end{matrix}$

Example 2.14. Reconsider Example 2.6, the weak complement of the (3, 6)-soft set (f, D, 3) in Example 2.6 is denoted by (f^c, D₁, 3) and is defined in Table 4.

Table 4

The weak complement of (3, 6)-soft set (f, D, 3)

(f^c, D₁, 3)	c ₁	c ₂	c ₃	c ₄
v ₁	〈4, (0.7, 0.5, 0.3) 〉	〈4, (0.6, 0.8, 0.4) 〉	〈3, (0.3, 0.5, 0.7) 〉	〈2, (0.6, 0.4, 0.1) 〉
v ₂	〈4, (0.9, 0.4, 0.2) 〉	〈2, (0.7, 0.6, 0.2) 〉	〈2, (0.1, 0.6, 0.4) 〉	〈1, (0.8, 0.6, 0.2) 〉
v ₃	〈5, (0.6, 0.7, 0.5) 〉	〈3, (0.5, 0.4, 0.2) 〉	〈4, (0.9, 0.5, 0.2) 〉	〈1, (0.3, 0.4, 0.1) 〉
v ₄	〈4, (0.8, 0.3, 0.5) 〉	〈5, (0.6, 0.7, 0.6) 〉	〈3, (0.2, 0.3, 0.3) 〉	〈2, (0.4, 0.2, 0.9) 〉
v ₅	〈1, (0.1, 0.5, 0.3) 〉	〈4, (0.8, 0.4, 0.5) 〉	〈0, (0.0, 0.0, 0.0) 〉	〈2, (0.6, 0.5, 0.3) 〉
v ₆	〈2, (0.6, 0.2, 0.3) 〉	〈2, (0.6, 0.9, 0.1) 〉	〈3, (0.6, 0.3, 0.1) 〉	〈3, (0.5, 0.7, 0.1) 〉

Definition 2.15. For an (m, N)-soft set (f, D, m) where D = (F, C, N) is an N-soft set and f maps any attribute c_k ∈ C with an mF set A on F (c_k), the top weak complement (f^>, D, m) of (f, D, m) defined by $(f^{>}, D, m) = {\begin{matrix} f (c_{k}) = 〈 (v_{j}, N - 1), 1 - (p_{1} \circ A (v_{j}, \\ r_{c_{k}}), \dots, p_{m} \circ A (v_{j}, r_{c_{k}})) 〉, \\ if r_{c}^{k} < N - 1, \\ f (c_{k}) = 〈 (v_{j}, 0), 1 - (p_{1} \circ A (v_{j}, r_{c_{k}}), \\ \dots, p_{m} \circ A (v_{j}, r_{c_{k}})) 〉, \\ if r_{c}^{k} = N - 1 . \end{matrix}$

Example 2.16. Reconsider Example 2.6, the top weak complement of the (3, 6)-soft set (f, D, 3) in Example 2.6 denoted by (f^>, D, 3) and is defined in Table 5.

Table 5

The top weak complement of (3, 6)-soft set (f, D, 3)

(f^>, D, 3)	c ₁	c ₂	c ₃	c ₄
v ₁	〈5, (0.7, 0.5, 0.3) 〉	〈5, (0.6, 0.8, 0.4) 〉	〈5, (0.3, 0.5, 0.7) 〉	〈5, (0.6, 0.4, 0.1) 〉
v ₂	〈5, (0.9, 0.4, 0.2) 〉	〈5, (0.7, 0.6, 0.2) 〉	〈5, (0.1, 0.6, 0.4) 〉	〈5, (0.8, 0.6, 0.2) 〉
v ₃	〈5, (0.6, 0.7, 0.5) 〉	〈0, (0.5, 0.4, 0.2) 〉	〈5, (0.9, 0.5, 0.2) 〉	〈5, (0.3, 0.4, 0.1) 〉
v ₄	〈5, (0.8, 0.3, 0.5) 〉	〈5, (0.6, 0.7, 0.6) 〉	〈5, (0.2, 0.3, 0.3) 〉	〈5, (0.4, 0.2, 0.9) 〉
v ₅	〈0, (0.1, 0.5, 0.3) 〉	〈5, (0.8, 0.4, 0.5) 〉	〈0, (0.0, 0.0, 0.0) 〉	〈5, (0.6, 0.5, 0.3) 〉
v ₆	〈5, (0.6, 0.2, 0.3) 〉	〈5, (0.6, 0.9, 0.1) 〉	〈5, (0.6, 0.3, 0.1) 〉	〈5, (0.5, 0.7, 0.1) 〉

Definition 2.17. For an (m, N)-soft set (f, D, m) where D = (F, C, N) is an N-soft set and f maps any attribute c_k ∈ C with an mF set A on F (c_k), the bottom weak complement (f^<, D, m) of (f, D, m) is defined as follows: $(f^{<}, D, m) = {\begin{matrix} f (c_{k}) = 〈 (v_{j}, 0), 1 - (p_{1} \circ A (v_{j}, r_{c_{k}}), \\ \dots, p_{m} \circ A (v_{j}, r_{c_{k}})) 〉, \\ if r_{c}^{k} > 0, \\ f (c_{k}) = 〈 (v_{j}, N - 1), 1 - (p_{1} \circ A (v_{j}, \\ r_{c_{k}}), \dots, p_{m} \circ A (v_{j}, r_{c_{k}})) 〉, \\ if r_{c}^{k} = 0 . \end{matrix}$

Example 2.18. Reconsider Example 2.6, the bottom weak complement of the (3, 6)-soft set (f, D, 3) in Example 2.6 denoted by (f^<, D, 3) and is defined in Table 6.

Table 6

The bottom weak complement of (3, 6)-soft set (f, D, 3)

(f^<, D, 3)	c ₁	c ₂	c ₃	c ₄
v ₁	〈0, (0.7, 0.5, 0.3) 〉	〈0, (0.6, 0.8, 0.4) 〉	〈0, (0.3, 0.5, 0.7) 〉	〈0, (0.6, 0.4, 0.1) 〉
v ₂	〈0, (0.9, 0.4, 0.2) 〉	〈0, (0.7, 0.6, 0.2) 〉	〈0, (0.1, 0.6, 0.4) 〉	〈0, (0.8, 0.6, 0.2) 〉
v ₃	〈0, (0.6, 0.7, 0.5) 〉	〈0, (0.5, 0.4, 0.2) 〉	〈0, (0.9, 0.5, 0.2) 〉	〈0, (0.3, 0.4, 0.1) 〉
v ₄	〈0, (0.8, 0.3, 0.5) 〉	〈5, (0.6, 0.7, 0.6) 〉	〈0, (0.2, 0.3, 0.3) 〉	〈0, (0.4, 0.2, 0.9) 〉
v ₅	〈0, (0.1, 0.5, 0.3) 〉	〈0, (0.8, 0.4, 0.5) 〉	〈0, (0.0, 0.0, 0.0) 〉	〈0, (0.6, 0.5, 0.3) 〉
v ₆	〈0, (0.6, 0.2, 0.3) 〉	〈0, (0.6, 0.9, 0.1) 〉	〈0, (0.6, 0.3, 0.1) 〉	〈0, (0.5, 0.7, 0.1) 〉

Definition 2.19. Let V be a universal set of objects and let (f₁, D₁, m) and (f₂, D₂, m) be two (m, N)-soft sets where D₁ = (F₁, R, N₁) , D₂ = (F₂, S, N₂) are N-soft sets on V. Then their restricted intersection is denoted by (f₁, D₁, m) ∩ _R (f₂, D₂, m) and is defined as $(g, D_{1} \cap_{R} D_{2}, m)$ where $D_{1} \cap_{R} D_{2} = (G, R \cap S, min (N_{1}, N_{2}))$ for all c_k ∈ R ∩ S, v_j ∈ V, $〈 (v_{j}, r_{c_{k}}), z 〉 \in g (c_{k}) \Leftrightarrow r_{c_{k}} = min (r_{c_{k}}^{1}, r_{c_{k}}^{2}), z = min (p_{i} \circ A (v_{j}, r_{c_{k}}^{1}), p_{i} \circ B (v_{j}, r_{c_{k}}^{2}))$ , if $〈 (v_{j}, r_{c_{k}}^{1}), p_{i} \circ A (v_{j}, r_{c_{k}}^{1}) 〉 \in f_{1} (c_{k})$ and $〈 (v_{j}, r_{c_{k}}^{2}), p_{i} \circ B (v_{j}, r_{c_{k}}^{2}) 〉 \in f_{2} (c_{k})$ , for all 1 ≤ i ≤ m, A and B are mF sets on F₁ (c_k) and F₂ (c_k), respectively.

Example 2.20. Suppose that (f₁, D₁, 3) and (f₂, D₂, 3) are the (3, 5)-soft set and (3, 4)-soft set defined by Tables 7 and 8, respectively, where D₁ = (F₁, R, 5) , D₂ = (F₂, S, 4) are N-soft sets on V. Then their restricted intersection $(f_{1}, D_{1}, 3) \cap_{R} (f_{2}, D_{2}, 3) = (g, D_{1} \cap_{R} D_{2}, 3)$ is defined by Table 9.

Table 7

Tabular representation of (3, 5)-soft set (f₁, D₁, 3)

(f₁, D₁, 3)	c ₁	c ₂	c ₃
v ₁	〈3, (0.7, 0.5, 0.3) 〉	〈4, (0.6, 0.8, 0.5) 〉	〈3, (0.3, 0.5, 0.7) 〉
v ₂	〈3, (0.9, 0.4, 0.2) 〉	〈2, (0.7, 0.6, 0.2) 〉	〈2, (0.4, 0.6, 0.4) 〉
v ₃	〈4, (0.6, 0.7, 0.5) 〉	〈1, (0.5, 0.4, 0.2) 〉	〈4, (0.9, 0.5, 1.0) 〉
v ₄	〈4, (0.8, 0.5, 0.5) 〉	〈4, (0.6, 0.7, 0.6) 〉	〈0, (0.2, 0.3, 0.3) 〉
v ₅	〈1, (0.1, 0.5, 0.3) 〉	〈3, (0.8, 0.4, 0.5) 〉	〈0, (0.0, 0.0, 0.0) 〉
v ₆	〈2, (0.6, 0.4, 0.3) 〉	〈2, (0.5, 0.9, 0.1) 〉	〈1, (0.6, 0.3, 0.1) 〉

Table 8

Tabular representation of (3, 4)-soft set (f₂, D₂, 3)

(f₂, D₂, 3)	c ₁	c ₃	e
v ₁	〈2, (0.8, 0.7, 0.6) 〉	〈3, (0.9, 0.7, 0.6) 〉	〈2, (0.4, 0.8, 0.5) 〉
v ₂	〈1, (0.6, 0.5, 0.4) 〉	〈3, (0.8, 0.5, 0.9) 〉	〈3, (0.7, 0.8, 0.6) 〉
v ₃	〈3, (0.8, 0.8, 0.8) 〉	〈2, (0.6, 0.3, 0.4) 〉	〈3, (0.8, 0.9, 0.5) 〉
v ₄	〈2, (0.4, 0.9, 0.8) 〉	〈3, (0.8, 0.6, 0.7) 〉	〈1, (0.6, 0.5, 0.2) 〉
v ₅	〈0, (0.1, 0.2, 0.1) 〉	〈3, (0.7, 0.6, 0.8) 〉	〈1, (0.6, 0.3, 0.4) 〉
v ₆	〈3, (0.8, 0.6, 0.9) 〉	〈2, (0.4, 0.6, 0.5) 〉	〈2, (0.7, 0.6, 0.2) 〉

Table 9

Tabular representation of the restricted intersection (f₁, D₁, 3) ∩ _R (f₂, D₂, 3)

$(g, D_{1} \cap_{R} D_{2}, 3)$	c ₁	c ₃
v ₁	〈2, (0.7, 0.5, 0.3) 〉	〈3, (0.3, 0.5, 0.6) 〉
v ₂	〈1, (0.6, 0.4, 0.2) 〉	〈2, (0.4, 0.5, 0.4) 〉
v ₃	〈3, (0.6, 0.7, 0.5) 〉	〈2, (0.6, 0.3, 0.4) 〉
v ₄	〈2, (0.4, 0.5, 0.5) 〉	〈0, (0.2, 0.3, 0.3) 〉
v ₅	〈0, (0.1, 0.2, 0.1) 〉	〈0, (0.0, 0.0, 0.0) 〉
v ₆	〈2, (0.6, 0.4, 0.3) 〉	〈1, (0.4, 0.3, 0.1) 〉

Definition 2.21. Let V be a universal set and let (f₁, D₁, m) and (f₂, D₂, m) be two (m, N)-soft sets where D₁ = (F₁, R, N₁) , D₂ = (F₂, S, N₂) are N-soft sets on V. Then their extended intersection is denoted by (f₁, D₁, m) ∩ _E (f₂, D₂, m) and is defined as $(h, D_{1} \cap_{E} D_{2}, m)$ , where $D_{1} \cap_{E} D_{2} = (H, R \cup S, max (N_{1}, N_{2}))$ , and h (c_k) is given by $h (c_{k}) = {\begin{matrix} f_{1} (c_{k}), if c_{k} \in R - S, \\ f_{2} (c_{k}), if c_{k} \in S - R, \\ 〈 (v_{j}, r_{c_{k}}), z 〉 such that r_{c_{k}} = min (r_{c_{k}}^{1}, r_{c_{k}}^{2}), \\ z = min (p_{i} \circ A (v_{j}, r_{c_{k}}^{1}), p_{i} \circ B (v_{j}, r_{c_{k}}^{2})) \\ where 〈 (v_{j}, r_{c_{k}}^{1}), p_{i} \circ A (v_{j}, r_{c_{k}}^{1}) 〉 \in f_{1} (c_{k}) \\ and 〈 (v_{j}, r_{c_{k}}^{2}), p_{i} \circ B (v_{j}, r_{c_{k}}^{2}) 〉 \in f_{2} (c_{k}), \\ \forall 1 \leq i \leq m, A and B are mF sets on \\ F_{1} (c_{k}) and F_{2} (c_{k}), respectively . \end{matrix}$

Example 2.22. Suppose that (f₁, D₁, 3) and (f₂, D₂, 3) are the (3, 5)-soft set and (3, 4)-soft set represented by Tables 7 and 8, respectively, where D₁ = (F₁, R, 5) , D₂ = (F₂, S, 4) are 5-soft set and 4-soft set on V, respectively. Then their extended intersection $(f_{1}, D_{1}, 3) \cap_{E} (f_{2}, D_{2}, 3) = (h, D_{1} \cap_{E} D_{2}, 3)$ is defined by Table 10.

Table 10

Tabular representation of extended intersection (f₁, D₁, 3) ∩ _E (f₂, D₂, 3)

$(h, D_{1} \cap_{E} D_{2}, 3)$	c ₁	c ₂	c ₃	e
v ₁	〈2, (0.7, 0.5, 0.3) 〉	〈4, (0.6, 0.8, 0.5) 〉	〈3, (0.3, 0.5, 0.6) 〉	〈2, (0.4, 0.8, 0.5) 〉
v ₂	〈1, (0.6, 0.4, 0.2) 〉	〈2, (0.7, 0.6, 0.2) 〉	〈2, (0.4, 0.5, 0.4) 〉	〈3, (0.7, 0.8, 0.6) 〉
v ₃	〈3, (0.6, 0.7, 0.5) 〉	〈1, (0.5, 0.4, 0.2) 〉	〈2, (0.6, 0.3, 0.4) 〉	〈3, (0.8, 0.9, 0.5) 〉
v ₄	〈2, (0.4, 0.5, 0.5) 〉	〈4, (0.6, 0.7, 0.6) 〉	〈0, (0.2, 0.3, 0.3) 〉	〈1, (0.6, 0.5, 0.2) 〉
v ₅	〈0, (0.1, 0.2, 0.1) 〉	〈3, (0.8, 0.4, 0.5) 〉	〈0, (0.0, 0.0, 0.0) 〉	〈1, (0.6, 0.3, 0.4) 〉
v ₆	〈2, (0.6, 0.4, 0.3) 〉	〈2, (0.5, 0.9, 0.1) 〉	〈1, (0.4, 0.3, 0.1) 〉	〈2, (0.7, 0.6, 0.2) 〉

Definition 2.23. Let V be a universe and let (f₁, D₁, m) and (f₂, D₂, m) be two (m, N)-soft sets over V where D₁ = (F₁, R, N₁) , D₂ = (F₂, S, N₂) are N-soft sets on V. Then their restricted union is denoted by (f₁, D₁, m) ∪ _R (f₂, D₂, m) and is defined as $(w, D_{1} \cup_{R} D_{2}, m)$ where $D_{1} \cup_{R} D_{2} = (W, R \cap S, max (N_{1}, N_{2}))$ for all c_k ∈ R ∩ S, v_j ∈ V, $〈 (v_{j}, r_{c_{k}}), z 〉 \in w (c_{k}) \Leftrightarrow r_{c_{k}} = max (r_{c_{k}}^{1}, r_{c_{k}}^{2}), z = max (p_{i} \circ A (v_{j}, r_{c_{k}}^{1}), p_{i} \circ B (v_{j}, r_{c_{k}}^{2}))$ if $〈 (v_{j}, r_{c_{k}}^{1}), p_{i} \circ A (v_{j}, r_{c_{k}}^{1}) 〉 \in f_{1} (c_{k})$ and $〈 (v_{j}, r_{c_{k}}^{2}), p_{i} \circ B (v_{j}, r_{c_{k}}^{2}) 〉 \in f_{2} (c_{k})$ , for all 1 ≤ i ≤ m, A and B are mF sets on F₁ (c_k) and F₂ (c_k), respectively.

Example 2.24. Suppose that (f₁, D₁, 3) and (f₂, D₂, 3) are the (3, 5)-soft set and (3, 4)-soft set represented by Tables 7 and 8, respectively. Then their restricted union $(f_{1}, D_{1}, 3) \cup_{R} (f_{2}, D_{2}, 3) = (w, D_{1} \cup_{R} D_{2}, 3)$ is defined by Table 11.

Table 11

Tabular representation of the restricted union (f₁, D₁, 3) ∪ _R (f₂, D₂, 3)

$(w, D_{1} \cup_{R} D_{2}, 3)$	c ₁	c ₃
v ₁	〈3, (0.8, 0.7, 0.6) 〉	〈3, (0.9, 0.7, 0.7) 〉
v ₂	〈3, (0.9, 0.5, 0.4) 〉	〈3, (0.8, 0.6, 0.9) 〉
v ₃	〈4, (0.8, 0.8, 0.8) 〉	〈4, (0.9, 0.5, 1.0) 〉
v ₄	〈4, (0.8, 0.9, 0.8) 〉	〈3, (0.8, 0.6, 0.7) 〉
v ₅	〈1, (0.1, 0.5, 0.3) 〉	〈3, (0.7, 0.6, 0.8) 〉
v ₆	〈3, (0.8, 0.6, 0.9) 〉	〈2, (0.6, 0.6, 0.5) 〉

Definition 2.25. Let V be a universal set and let (f₁, D₁, m) and (f₂, D₂, m) be two (m, N)-soft sets on V where D₁ = (F₁, R, N₁) , D₂ = (F₂, S, N₂) are N-soft sets on V. Then their extended union is denoted by (f₁, D₁, m) ∪ _E (f₂, D₂, m) and is defined as $(y, D_{1} \cup_{E} D_{2}, m)$ , where $D_{1} \cup_{E} D_{2} = (Y, R \cup S, max (N_{1}, N_{2}))$ and y (c_k) is given by $y (c_{k}) = {\begin{matrix} f_{1} (c_{k}), if c_{k} \in R - S, \\ f_{2} (c_{k}), if c_{k} \in S - R, \\ 〈 (v_{j}, r_{c_{k}}), z 〉 such that r_{c_{k}} = max (r_{c_{k}}^{1}, r_{c_{k}}^{2}), \\ z = max (p_{i} \circ A (v_{j}, r_{c_{k}}^{1}), p_{i} \circ B (v_{j}, r_{c_{k}}^{2})) \\ where 〈 (v_{j}, r_{c_{k}}^{1}), p_{i} \circ A (v_{j}, r_{c_{k}}^{1}) 〉 \in f_{1} (c_{k}) \\ and 〈 (v_{j}, r_{c_{k}}^{2}), p_{i} \circ B (v_{j}, r_{c_{k}}^{2}) 〉 \in f_{2} (c_{k}), \\ \forall 1 \leq i \leq m, A and B are mF sets on \\ F_{1} (c_{k}) and F_{2} (c_{k}), respectively . \end{matrix}$

Example 2.26. Suppose that (f₁, D₁, 3) and (f₂, D₂, 3) are the (3, 5)-soft set and (3, 4)-soft set represented by Tables 7 and 8, respectively. Then their extended union $(f_{1}, D_{1}, 3) \cup_{E} (f_{2}, D_{2}, 3) = (y, D_{1} \cup_{E} D_{2}, 3)$ is defined by Table 12.

Table 12

Tabular representation of extended union (f₁, C, 3) ∪ _E (f₂, D, 3)

$(y, D_{1} \cup_{E} D_{2}, 3)$	c ₁	c ₂	c ₃	e
v ₁	〈3, (0.8, 0.7, 0.6) 〉	〈4, (0.6, 0.8, 0.5) 〉	〈3, (0.9, 0.7, 0.7) 〉	〈2, (0.4, 0.8, 0.5) 〉
v ₂	〈3, (0.9, 0.5, 0.4) 〉	〈2, (0.7, 0.6, 0.2) 〉	〈3, (0.8, 0.6, 0.9) 〉	〈3, (0.7, 0.8, 0.6) 〉
v ₃	〈4, (0.8, 0.8, 0.8) 〉	〈1, (0.5, 0.4, 0.2) 〉	〈4, (0.9, 0.5, 1.0) 〉	〈3, (0.8, 0.9, 0.5) 〉
v ₄	〈4, (0.8, 0.9, 0.8) 〉	〈4, (0.6, 0.7, 0.6) 〉	〈3, (0.8, 0.6, 0.7) 〉	〈1, (0.6, 0.5, 0.2) 〉
v ₅	〈1, (0.1, 0.5, 0.3) 〉	〈3, (0.8, 0.4, 0.5) 〉	〈3, (0.7, 0.6, 0.8) 〉	〈1, (0.6, 0.3, 0.4) 〉
v ₆	〈3, (0.8, 0.6, 0.9) 〉	〈2, (0.5, 0.9, 0.1) 〉	〈2, (0.6, 0.6, 0.5) 〉	〈2, (0.7, 0.6, 0.2) 〉

Definition 2.27. Let V be a universe, (f, D, m) an (m, N)-soft set where D = (F, C, N) is an N-soft set on V. Let 0 < S < N be a threshold. An mF soft set related with (f, D, m) and S, denoted by (f^S, C), is given as follows: $(f^{S}, C) = {\begin{matrix} f^{S} (c_{k}) = (v_{j}, 1), if r_{c_{k}} \geq S, \\ f^{S} (c_{k}) = (v_{j}, 0), otherwise . \end{matrix}$

Specifically, (f¹, C) is said to be bottom mF soft set associated with (f, D, m) and (f^N-1, C) is said to be top mF soft set.

Example 2.28. Consider the (3, 5)-soft set (f₁, D₁, 3) where D₁ = (F₁, R, 5) is a 5-soft set, represented by Table 7. From Definition 2.27, we have 0 < S < 5. The possible mF soft sets associated with feasible thresholds and (f₁, D₁, 3) are given by $(f_{1}^{1}, R) = {\begin{matrix} f_{1}^{1} (c_{1}) = {(v_{1}, α), (v_{2}, α), (v_{3}, α), \\ (v_{4}, α), (v_{5}, α), (v_{6}, α)}, \\ f_{1}^{1} (c_{2}) = {(v_{1}, α), (v_{2}, α), (v_{3}, α), \\ (v_{4}, α), (v_{5}, α), (v_{6}, α)}, \\ f_{1}^{1} (c_{3}) = {(v_{1}, α), (v_{2}, α), (v_{3}, α), \\ (v_{4}, β), (v_{5}, β), (v_{6}, α)} . \end{matrix}}$

$(f_{1}^{2}, R) = {\begin{matrix} f_{1}^{2} (c_{1}) = {(v_{1}, α), (v_{2}, α), (v_{3}, α), \\ (v_{4}, α), (v_{5}, β), (v_{6}, α)}, \\ f_{1}^{2} (c_{2}) = {(v_{1}, α), (v_{2}, α), (v_{3}, α), \\ (v_{4}, α), (v_{5}, α), (v_{6}, α)}, \\ f_{1}^{2} (c_{3}) = {(v_{1}, β), (v_{2}, β), (v_{3}, α), \\ (v_{4}, β), (v_{5}, β), (v_{6}, β)} . \end{matrix}}$

$(f_{1}^{3}, R) = {\begin{matrix} f_{1}^{3} (c_{1}) = {(v_{1}, α), (v_{2}, α), (v_{3}, α), \\ (v_{4}, β), (v_{5}, β), (v_{6}, β)}, \\ f_{1}^{3} (c_{2}) = {(v_{1}, α), (v_{2}, β), (v_{3}, β), \\ (v_{4}, α), (v_{5}, α), (v_{6}, α)}, \\ f_{1}^{3} (c_{3}) = {(v_{1}, β), (v_{2}, β), (v_{3}, α), \\ (v_{4}, β), (v_{5}, β), (v_{6}, β)} . \end{matrix}}$

$(f_{1}^{4}, R) = {\begin{matrix} f_{1}^{4} (c_{1}) = {(v_{1}, α), (v_{2}, β), (v_{3}, α), \\ (v_{4}, β), (v_{5}, β), (v_{6}, β)}, \\ f_{1}^{4} (c_{2}) = {(v_{1}, β), (v_{2}, β), (v_{3}, β), \\ (v_{4}, α), (v_{5}, β), (v_{6}, β)}, \\ f_{1}^{4} (c_{3}) = {(v_{1}, β), (v_{2}, β), (v_{3}, α), \\ (v_{4}, β), (v_{5}, β), (v_{6}, β)} . \end{matrix}}$ where α = (1.0, 1.0, 1.0) , β = (0.0, 0.0, 0.0).

Definition 2.29. An (m, N)-soft set over V is said to be a null (m, N)-soft set, denoted by (o, C, m), if for all c_k ∈ C, (O, C, N) is the N-soft set on V with the null mF set $\bar{O}$ of V where $\bar{O} (v) = 0$ for all v ∈ V. An (m, N)-soft set on V is called a whole (m, N)-soft set, denoted by (w, C, m), if for all c_k ∈ C, (W, C, N) is the N-soft set on V with the whole mF set $\bar{I}$ of V where $\bar{I} (v) = 1$ for all v ∈ V.

3 Construction of (m, N)-soft rough sets

In this section we introduce the novel concept of m-polar fuzzy N-soft rough sets or (m, N)-soft rough sets, which relies on some previous new definitions that we proceed to state and exemplify.

Definition 3.1. Let (f, D, m) be an (m, N)-soft set on V, where D = (F, C, N) is an N-soft set. Then, an (m, N)-soft subset λ of (V × M) × Z is called an (m, N)-soft relation over (V × M) × Z, which is defined as $\begin{matrix} λ = { & 〈 ((v_{j}, r_{jk}), z_{k}), p_{i} \circ λ ((v_{j}, r_{jk}), z_{k}) 〉 | (v_{j}, r_{jk}) \\ \in V \times M, z_{k} \in Z, 1 \leq i \leq m}, \end{matrix}$ where λ : (V × M) × Z → [0, 1] ^m.

If V = {v₁, v₂, ⋯ , v_j}, Z = {z₁, z₂, ⋯ , z_k}, $M = {r_{jk} | j, k = 1, 2, \dots}$ , then tabular representation of an (m, N)-soft relation λ over (V × M) × Z can be given as follows: $\begin{matrix} λ & z_{1} & \dots & z_{k} \\ v_{1} & 〈 r_{11}, p_{i} \circ λ ((v_{1}, r_{11}), z_{1}) 〉 & \dots & 〈 r_{1 k}, p_{i} \circ λ ((v_{1}, r_{1 k}), z_{k}) 〉 \\ v_{2} & 〈 r_{21}, p_{i} \circ λ ((v_{2}, r_{21}), z_{1}) 〉 & \dots & 〈 r_{2 k}, p_{i} \circ λ ((v_{2}, r_{2 k}), z_{k}) 〉 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ v_{j} & 〈 r_{j 1}, p_{i} \circ λ ((v_{j}, r_{j 1}), z_{1}) 〉 & \dots & 〈 r_{jk}, p_{i} \circ λ ((v_{j}, r_{jk}), z_{k}) 〉 \end{matrix}$

Example 3.2. Let V = {v₁, v₂, v₃} be a universe, Z = {z₁, z₂, z₃} a set of attributes and M = {0, 1, 2, 3, 4}. A (3, 5)-soft relation λ : V × M → Z is given as follows: $\begin{matrix} λ & z_{1} & z_{2} & z_{3} \\ v_{1} & 〈 1, (0.6, 0.3, 0.1) 〉 & 〈 3, (0.4, 0.7, 0.6) 〉 & 〈 2, (0.4, 0.6, 0.2) 〉 \\ v_{2} & 〈 1, (0.5, 0.3, 0.2) 〉 & 〈 2, (0.5, 0.2, 0.8) 〉 & 〈 4, (0.6, 0.9, 0.6) 〉 \\ v_{3} & 〈 0, (0.3, 0.2, 0.1) 〉 & 〈 2, (0.3, 0.4, 0.8) 〉 & 〈 3, (0.7, 0.3, 0.5) 〉 \end{matrix}$

Now we define (m, N)-soft rough sets.

Definition 3.3. Let V be a universal set, Z a set of attributes. For an arbitrary (m, N)-soft relation λ on (V × M) × Z, a tuple (V, Z, N, λ) is called an (m, N)-soft approximation space. For each $P \in M^{'} (Z)$ , we define the lower N-soft approximations of P $({\underline{λ}}_{N} (P))$ and upper N-soft approximations of P $({\bar{λ}}^{N} (P))$ , respectively, as follows: $\begin{matrix} {\underline{λ}}_{N} (P) = {〈 & (v_{j}, ⋀_{j, k} r_{jk}), P_{\underline{λ}} (v_{j}, ⋀_{j, k} r_{jk}) 〉 | v_{j} \in V, \\ r_{jk} \in M}, \\ {\bar{λ}}^{N} (P) = {〈 & (v_{j}, ⋁_{j, k} r_{jk}), P_{\bar{λ}} (v_{j}, ⋁_{j, k} r_{jk}) 〉 | v_{j} \in V, \\ r_{jk} \in M}, \end{matrix}$ where $\begin{matrix} P_{\underline{λ}} (v_{j}, ⋀_{j, k} r_{jk}) = inf_{z_{k} \in T} [( & 1 - p_{i} \circ P_{λ} ((v_{j}, r_{jk}), z_{k})) \lor \\ p_{i} \circ P (z_{k})], \\ P_{\bar{λ}} (v_{j}, ⋁_{j, k} r_{jk}) = sup_{z_{k} \in T} [ & p_{i} \circ P_{λ} ((v_{j}, r_{jk}), z_{k}) \land p_{i} \circ \\ P (z_{k})] . \end{matrix}$ for all 1 ≤ i ≤ m. The pair $({\underline{λ}}_{N} (P), {\bar{λ}}^{N} (P))$ is referred to as an (m, N)-soft rough set of P with respect to (V, Z, N, λ), and ${\underline{λ}}_{N}, {\bar{λ}}^{N} : M^{'} (Z) \to M^{'} (V \times M)$ are referred to as lower and upper (m, N)-soft rough approximation operators.

Example 3.4. Let V = {v₁, v₂, v₃, v₄} be a universe of TV shows and let Z = {z₁, z₂, z₃} be a set of parameters and M = {0, 1, 2, 3, 4}. Consider a (3, 5)-soft relation λ : V × M → Z is given by $\begin{matrix} λ & z_{1} & z_{2} & z_{3} \\ v_{1} & 〈 2, (0.6, 0.3, 0.1) 〉 & 〈 3, (0.4, 0.7, 0.6) 〉 & 〈 2, (0.4, 0.6, 0.2) 〉 \\ v_{2} & 〈 2, (0.5, 0.3, 0.2) 〉 & 〈 3, (0.5, 0.2, 0.8) 〉 & 〈 4, (0.6, 0.9, 0.6) 〉 \\ v_{3} & 〈 1, (0.3, 0.2, 0.1) 〉 & 〈 3, (0.3, 0.4, 0.8) 〉 & 〈 3, (0.7, 0.3, 0.5) 〉 \\ v_{4} & 〈 2, (0.4, 0.3, 0.6) 〉 & 〈 2, (0.5, 0.1, 0.4) 〉 & 〈 0, (0.3, 0.1, 0.0) 〉 \end{matrix}$ Suppose that P is a 3-polar fuzzy subset of Z, given by $\begin{matrix} P = { & (z_{1}, 0.3, 0.1, 0.7), (z_{2}, 0.3, 0.6, 0.4), \\ (z_{3}, 0.5, 0.6, 0.1)} . \end{matrix}$ From Definition 3.3, the lower and upper N-soft approximations are given by

$P_{\underline{λ}} (v_{1}, 2) = (0.4, 0.6, 0.4), P_{\bar{λ}} (v_{1}, 3) = (0.4, 0.6, 0.4),$ $P_{\underline{λ}} (v_{2}, 2) = (0.5, 0.6, 0.4), P_{\bar{λ}} (v_{2}, 4) = (0.5, 0.6, 0.4),$ $P_{\underline{λ}} (v_{3}, 1) = (0.5, 0.6, 0.5), P_{\bar{λ}} (v_{3}, 3) = (0.5, 0.4, 0.4),$ $P_{\underline{λ}} (v_{4}, 0) = (0.5, 0.7, 0.6), P_{\bar{λ}} (v_{4}, 2) = (0.3, 0.1, 0.6),$

Thus, $\begin{matrix} {\underline{λ}}_{N} (P) = { & 〈 (v_{1}, 2), 0.4, 0.6, 0.4 〉, 〈 (v_{2}, 2), 0.5, 0.6, 0.4 〉, \\ 〈 (v_{3}, 1), 0.5, 0.6, 0.4 〉, 〈 (v_{4}, 0), 0.5, 0.7, 0.6 〉}, \\ {\bar{λ}}^{N} (P) = { & 〈 (v_{1}, 3), 0.4, 0.6, 0.4 〉, 〈 (v_{2}, 4), 0.5, 0.6, 0.4 〉, \\ 〈 (v_{3}, 3), 0.5, 0.4, 0.4 〉, 〈 (v_{4}, 2), 0.3, 0.1, 0.6 〉} . \end{matrix}$ Hence, the pair $({\underline{λ}}_{N} (P), {\bar{λ}}^{N} (P))$ is referred to as a (3, 5)-soft rough set.

Definition 3.5. Let (V, Z, N, λ) be an (m, N)-soft approximation space. For any $P \in M^{'} (Z)$ , the complement of (m, N)-soft rough set $({\underline{λ}}_{N} (P), {\bar{λ}}^{N} (P))$ of P about (V, Z, N, λ), denoted by $(\sim {\underline{λ}}_{N} (P), \sim {\bar{λ}}^{N} (P))$ , is given by $\begin{matrix} \sim {\underline{λ}}_{N} (P) = {〈 & (v_{j}, N - ⋀_{j, k} r_{jk}), \neg P_{\underline{λ}} (v_{j}, N - ⋀_{j, k} r_{jk}) 〉 | \\ v_{j} \in V, r_{jk} \in M}, \\ \sim {\bar{λ}}^{N} (P) = {〈 & (v_{j}, N - ⋁_{j, k} r_{jk}), \neg P_{\bar{λ}} (v_{j}, N - ⋁_{j, k} r_{jk}) 〉 | \\ v_{j} \in V, r_{jk} \in M}, \end{matrix}$ where $\begin{matrix} \neg P_{\underline{λ}} (v_{j}, N - ⋀_{j, k} r_{jk}) = sup_{z_{k} \in T} [ & p_{i} \circ P_{λ} ((v_{j}, r_{jk}), z_{k}) \land \\ (1 - p_{i} \circ P (z_{k}))], \\ \neg P_{\bar{λ}} (v_{j}, N - ⋁_{j, k} r_{jk}) = inf_{z_{k} \in T} [ & (1 - p_{i} \circ P_{λ} ((v_{j}, r_{jk}), \\ z_{k})) \lor (1 - p_{i} \circ P (z_{k}))] . \end{matrix}$ Example 3.6. Let V = {v₁, v₂, v₃, v₄} be a universe and let Z = {z₁, z₂, z₃} be a set of parameters and M = {0, 1, 2, 3, 4}. Consider a (3, 5)-soft relation λ : (V × M) × Z → [0, 1] ^m is defined as follows: $\begin{matrix} λ & z_{1} & z_{2} & z_{3} \\ v_{1} & 〈 2, (0.6, 0.3, 0.1) 〉 & 〈 3, (0.4, 0.7, 0.6) 〉 & 〈 2, (0.4, 0.6, 0.2) 〉 \\ v_{2} & 〈 2, (0.5, 0.3, 0.2) 〉 & 〈 3, (0.5, 0.2, 0.8) 〉 & 〈 4, (0.6, 0.9, 0.6) 〉 \\ v_{3} & 〈 1, (0.3, 0.2, 0.1) 〉 & 〈 3, (0.3, 0.4, 0.8) 〉 & 〈 3, (0.7, 0.3, 0.5) 〉 \\ v_{4} & 〈 2, (0.4, 0.3, 0.6) 〉 & 〈 2, (0.5, 0.1, 0.4) 〉 & 〈 0, (0.3, 0.1, 0.0) 〉 \end{matrix}$ Suppose P is a 3-polar fuzzy subset of Z defined as follows: $\begin{matrix} P = {(z_{1}, 0.5, 0.2, 0.7), (z_{2}, 0.3, 0.5, 0.6), \\ (z_{3}, 0.9, 0.1, 0.4)} . \end{matrix}$

By Definition 3.5, we have

$\neg P_{\underline{λ}} (v_{1}, 3) = (0.5, 0.6, 0.4), \neg P_{\bar{λ}} (v_{1}, 2) = (0.5, 0.5, 0.4),$

$\neg P_{\underline{λ}} (v_{2}, 3) = (0.5, 0.9, 0.6), \neg P_{\bar{λ}} (v_{2}, 1) = (0.4, 0.8, 0.4),$

$\neg P_{\underline{λ}} (v_{3}, 4) = (0.3, 0.4, 0.5), \neg P_{\bar{λ}} (v_{3}, 2) = (0.3, 0.6, 0.4),$

$\neg P_{\underline{λ}} (v_{4}, 5) = (0.5, 0.3, 0.4), \neg P_{\bar{λ}} (v_{4}, 3) = (0.6, 0.8, 0.4),$

Thus, $\begin{matrix} \sim {\underline{λ}}_{N} (P) = { & 〈 (v_{1}, 3), 0.5, 0.6, 0.4 〉, 〈 (v_{2}, 3), 0.5, 0.9, 0.6 〉, \\ 〈 (v_{3}, 4), 0.3, 0.4, 0.5 〉, 〈 (v_{4}, 5), 0.5, 0.3, 0.4 〉}, \\ \sim {\bar{λ}}^{N} (P) = { & 〈 (v_{1}, 2), 0.5, 0.5, 0.4 〉, 〈 (v_{2}, 1), 0.4, 0.8, 0.4 〉, \\ 〈 (v_{3}, 2), 0.3, 0.6, 0.4 〉, 〈 (v_{4}, 3), 0.6, 0.8, 0.4 〉} . \end{matrix}$ Hence, the pair $(\sim {\underline{λ}}_{N} (P), \sim {\bar{λ}}^{N} (P))$ is referred to as a complement of (3, 5)-soft rough set $({\underline{λ}}_{N} (P), {\bar{λ}}^{N} (P))$ .

Theorem 3.7. Let (V, Z, N, λ) be an (m, N)-soft approximation space. Then the lower (m, N)-soft rough approximation operators ${\underline{λ}}_{N}$ and the upper (m, N)-soft rough approximation operators ${\bar{λ}}^{N}$ , satisfy the following properties for all $P, Q \in M^{'} (Z)$ .

${\underline{λ}}_{N} (P) = \sim {\bar{λ}}^{N} (\sim P),$

$P \subseteq Q \Rightarrow {\underline{λ}}_{N} (P) \subseteq {\underline{λ}}_{N} (Q),$

${\underline{λ}}_{N} (P \cup Q) \supseteq {\underline{λ}}_{N} (P) \cup {\underline{λ}}_{N} (Q),$

${\underline{λ}}_{N} (P \cap Q) = {\underline{λ}}_{N} (P) \cap {\underline{λ}}_{N} (Q),$

${\bar{λ}}^{N} (P) = \sim {\underline{λ}}_{N} (\sim P),$

$P \subseteq Q \Rightarrow {\bar{λ}}^{N} (P) \subseteq {\bar{λ}}^{N} (Q),$

${\bar{λ}}^{N} (P \cup Q) = {\bar{λ}}^{N} (P) \cup {\bar{λ}}^{N} (Q),$

${\bar{λ}}^{N} (P \cap Q) \subseteq {\bar{λ}}^{N} (P) \cap {\bar{λ}}^{N} (Q),$

where ∼P denotes the complement of P.

Proof. Its proof follows directly by Definitions 3.3 and 3.5.□

Proposition 3.8. Let (V, Z, N, λ) be an (m, N)-soft approximation space and $P, Q \in M^{'} (Z)$ . Then the lower and upper N-soft approximations of P and Q satisfies the following properties:

$\sim ({\underline{λ}}_{N} (P) \cup {\underline{λ}}_{N} (Q)) = {\bar{λ}}_{N} (\sim P) \cap {\bar{λ}}_{N} (\sim Q),$

$\sim ({\underline{λ}}_{N} (P) \cup {\bar{λ}}_{N} (Q)) = {\bar{λ}}_{N} (\sim P) \cap {\underline{λ}}_{N} (\sim Q),$

$\sim ({\bar{λ}}_{N} (P) \cup {\underline{λ}}_{N} (Q)) = {\underline{λ}}_{N} (\sim P) \cap {\bar{λ}}_{N} (\sim Q),$

$\sim ({\bar{λ}}_{N} (P) \cup {\bar{λ}}_{N} (Q)) = {\underline{λ}}_{N} (\sim P) \cap {\underline{λ}}_{N} (\sim Q),$

$\sim ({\underline{λ}}_{N} (P) \cap {\underline{λ}}_{N} (Q)) = {\bar{λ}}_{N} (\sim P) \cup {\bar{λ}}_{N} (\sim Q),$

$\sim ({\underline{λ}}_{N} (P) \cap {\bar{λ}}_{N} (Q)) = {\bar{λ}}_{N} (\sim P) \cup {\underline{λ}}_{N} (\sim Q),$

$\sim ({\bar{λ}}_{N} (P) \cap {\underline{λ}}_{N} (Q)) = {\underline{λ}}_{N} (\sim P) \cup {\bar{λ}}_{N} (\sim Q),$

$\sim ({\bar{λ}}_{N} (P) \cap {\bar{λ}}_{N} (Q)) = {\underline{λ}}_{N} (\sim P) \cup {\underline{λ}}_{N} (\sim Q) .$

Proof. Its proof follows directly from the Definitions 3.3 and 3.5.□

Definition 3.9. [41] Let $V = {v_{1}, v_{2}, v_{3}, \dots, v_{n}}$ be a universe of objects. A comparison table is a square table containing equal number of rows and columns labeled by the objects of the universe and the entries c_jk where c_jk = the number of parameters for which the membership value of v_j exceeds or equals to the value of v_k.

Definition 3.10. Let V be a universe, Z a set of attributes and $P \in M^{'} (Z)$ . Then, the ring sum operation about the lower N-soft approximation ${\underline{λ}}_{N} (P)$ and the upper N-soft approximation ${\bar{λ}}^{N} (P)$ is defined by $\begin{matrix} {\underline{λ}}_{N} (P) \oplus {\bar{λ}}^{N} (P) = {〈 & (v, \underline{r_{z}} + \bar{r^{z}} - \underline{r_{z}} . \bar{r^{z}}), p_{i} \circ P_{\underline{λ}} (v) + \\ p_{i} \circ P_{\bar{λ}} (v) - p_{i} \circ P_{\underline{λ}} (v) \times p_{i} \circ \\ P_{\bar{λ}} (v) 〉 | 〈 (v, \underline{r_{z}}), p_{i} \circ P_{\underline{λ}} (v) 〉 \in \\ {\underline{λ}}_{N} (P), 〈 (v, \bar{r^{z}}), p_{i} \circ P_{\bar{λ}} (v) 〉 \in \\ {\bar{λ}}^{N} (P), v \in V, 1 \leq i \leq m} . \end{matrix}$

Example 3.11. Reconsider Example 3.4, let V = {v₁, v₂, v₃, v₄} be a universe of TV shows and let Z = {z₁, z₂, z₃} be a set of parameters and M = {0, 1, 2, 3, 4}. Consider a (3, 5)-soft relation λ : V × M → Z is given by $\begin{matrix} λ & z_{1} & z_{2} & z_{3} \\ v_{1} & 〈 2, (0.6, 0.3, 0.1) 〉 & 〈 3, (0.4, 0.7, 0.6) 〉 & 〈 2, (0.4, 0.6, 0.2) 〉 \\ v_{2} & 〈 2, (0.5, 0.3, 0.2) 〉 & 〈 3, (0.5, 0.2, 0.8) 〉 & 〈 4, (0.6, 0.9, 0.6) 〉 \\ v_{3} & 〈 1, (0.3, 0.2, 0.1) 〉 & 〈 3, (0.3, 0.4, 0.8) 〉 & 〈 3, (0.7, 0.3, 0.5) 〉 \\ v_{4} & 〈 2, (0.4, 0.3, 0.6) 〉 & 〈 2, (0.5, 0.1, 0.4) 〉 & 〈 0, (0.3, 0.1, 0.0) 〉 \end{matrix}$ Suppose that P is a 3-polar fuzzy subset of Z, given by $\begin{matrix} P = { & (z_{1}, 0.3, 0.1, 0.7), (z_{2}, 0.3, 0.6, 0.4), \\ (z_{3}, 0.5, 0.6, 0.1)} . \end{matrix}$ Using Definition 3.3, the lower and upper N-soft approximations of P are given by $\begin{matrix} {\underline{λ}}_{N} (P) = { & 〈 (v_{1}, 2), 0.4, 0.6, 0.4 〉, 〈 (v_{2}, 2), 0.5, 0.6, 0.4 〉, \\ 〈 (v_{3}, 1), 0.5, 0.6, 0.4 〉, 〈 (v_{4}, 0), 0.5, 0.7, 0.6 〉}, \\ {\bar{λ}}^{N} (P) = { & 〈 (v_{1}, 3), 0.4, 0.6, 0.4 〉, 〈 (v_{2}, 4), 0.5, 0.6, 0.4 〉, \\ 〈 (v_{3}, 3), 0.5, 0.4, 0.4 〉, 〈 (v_{4}, 2), 0.3, 0.1, 0.6 〉} . \end{matrix}$ From Definition 3.10, the ring sum operation about the lower N-soft approximation ${\underline{λ}}_{N} (P)$ and the upper N-soft approximation ${\bar{λ}}^{N} (P)$ is given as follows: $\begin{matrix} {\underline{λ}}_{N} (P) \oplus {\bar{λ}}_{N} (P) = { & 〈 (v_{1}, - 1), 0.64, 0.84, 0.64 〉, \\ 〈 (v_{2}, - 2), 0.75, 0.84, 0.64 〉, \\ 〈 (v_{3}, 1), 0.75, 0.76, 0.64 〉, \\ 〈 (v_{4}, 2), 0.65, 0.1, 0.84) 〉} . \end{matrix}$

4 Applications

This section explains the decision making mechanisms that operate on the models that we have described in sections 2 and 3. Therefore, we define respective algorithms for problems that are characterized by (m, N)-soft sets and (m, N)-soft rough sets. In order to prove their implementability and feasibility we apply them to real situations that are fully developed.

4.1 Selection of a restaurant

Stars are used by reviewers for ranking things, including hotels, restaurants, TV shows and resorts. Buying a suitable restaurant is a difficult task due to the variation in ratings and reviews on one website to another for the same restaurant. The rating of a restaurant depend on different parameters and reviewers. Suppose that a businessman (Mr. Ali) wants to purchase a suitable restaurant from the alternatives v₁, v₂, v₃, v₄. Let V = {v₁, v₂, v₃, v₄} be a universe of four restaurants and let C = {c₁, c₂, c₃} ⊆ Z be a set of attributes, which gives grades to restaurants based on location, meal options, services and parking area. We can obtain a (4, 6)-soft set from Table 13, where

Table 13
Information obtained from the related data

V c ₁ c ₂ c ₃

v ₁ ★★★ ★★ ★★

v ₂ ★★★ ★★★ ★★ ★★

v ₃ ★★★ ★★ ★★★ ★★★ ★

v ₄ ★★★ ★ ★★★ ★★★

V	c ₁	c ₂	c ₃
v ₁	★★★	★★	★★
v ₂	★★★	★★★ ★★	★★
v ₃	★★★ ★★	★★★	★★★ ★
v ₄	★★★ ★	★★★	★★★

Five star represents ‘luxury’,

Four star represents ‘excellent’,

Three star represents ‘very good’,

Two star represents ‘good’.

The set of ordered grades M = {0, 1, 2, 3, 4, 5} can be easily associated with stars as follows:

2 stand for ‘★★’,

3 stand for ‘★★★’,

4 stand for ‘★★★ ★’,

5 stand for ‘★★★ ★★’.

From Definition 2.2, tabular form of a 6-soft set is given by Table 14.

Table 14

Tabular representation of 6-soft set (F, C, 6)

(F, C, 6)	c ₁	c ₂	c ₃
v ₁	3	2	2
v ₂	3	5	2
v ₃	5	3	4
v ₄	4	3	3

Therefore, by Definition 2.5, a (4, 6)-soft set (f, D, 4) is defined by Table 15.

Table 15

Tabular representation of (4, 6)-soft set (f, D, 4)

(f, D, 4)	c ₁	c ₂	c ₃
v ₁	〈3, (0.5, 0.4, 0.7, 0.6) 〉	〈2, (0.6, 0.5, 0.7, 0.6) 〉	〈2, (0.5, 0.8, 0.6, 0.9) 〉
v ₂	〈3, (0.5, 0.6, 0.4, 0.7) 〉	〈5, (0.8, 0.9, 0.7, 0.9) 〉	〈2, (0.6, 0.8, 0.7, 0.7) 〉
v ₃	〈5, (0.9, 0.8, 0.7, 0.9) 〉	〈3, (0.3, 0.7, 0.6, 0.9) 〉	〈4, (0.9, 0.8, 0.9, 0.7) 〉
v ₄	〈4, (0.9, 0.6, 0.7, 0.8) 〉	〈3, (0.8, 0.5, 0.8, 0.4) 〉	〈3, (0.7, 0.6, 0.8, 0.9) 〉

Thus, (f, D, 4) is a (4, 6)-soft set based location, meal option, services and parking area of the restaurants. For example, in the Table 15, 〈3, (0.5, 0.4, 0.7, 0.6) means that the restaurant v₁ is 50% suitable with the location, 40% suitable with the meal options, 70% suitable with services and 60% suitable with the parking area, and 3 is the evaluation grade of v₁ according to attribute c₁. The tabular representation of the first pole is given by Table 16.

Table 16

Tabular representation of 1st pole

p ₁	c ₁	c ₂	c ₃
v ₁	0.5	0.6	0.5
v ₂	0.5	0.8	0.6
v ₃	0.9	0.3	0.9
v ₄	0.9	0.8	0.7

Now we will make the comparison table for the 1st pole, see Table 17.

The membership score for each restaurant with sum of grades $(\sum_{i = 1}^{3} r_{c_{i}})$ is obtained by subtracting the column sum from the row sum of Table 17, which is displayed in Table 18.

Table 17

Comparison table for the 1st pole

.	v ₁	v ₂	v ₃	v ₄
v ₁	3	1	1	0
v ₂	3	3	1	1
v ₃	2	2	3	2
v ₄	3	3	2	3

Table 18

Membership score table of the 1st pole

.	Grades sum	Row sum	Column sum	V₁ = e - f
	$(\sum_{i = 1}^{3} r_{c_{i}})$	(e)	(f)
v ₁	7	5	11	-6
v ₂	10	8	9	-1
v ₃	12	9	7	2
v ₄	10	11	6	5

Similarly, we represent the remaining three poles in tabular form and calculate the membership score with the help of comparison tables for each pole, respectively, see Tables 19 –27.

Table 19

Tabular representation of the 2nd pole

p ₂	c ₁	c ₂	c ₃
v ₁	0.4	0.5	0.8
v ₂	0.6	0.9	0.8
v ₃	0.8	0.7	0.8
v ₄	0.6	0.5	0.6

Table 20

Comparison table for the 2nd pole

.	v ₁	v ₂	v ₃	v ₄
v ₁	3	1	1	2
v ₂	3	3	2	3
v ₃	3	2	3	3
v ₄	2	1	0	3

Table 21

Membership score table of the 2nd pole

.	Grades sum	Row sum	Column sum	V₂ = g - h
	$(\sum_{i = 1}^{3} r_{c_{i}})$	(g)	(h)
v ₁	7	7	11	-4
v ₂	10	11	7	4
v ₃	12	11	6	5
v ₄	10	6	11	-5

Table 22

Tabular representation of the 3rd pole

p ₃	c ₁	c ₂	c ₃
v ₁	0.7	0.7	0.6
v ₂	0.4	0.7	0.7
v ₃	0.7	0.6	0.9
v ₄	0.7	0.8	0.8

Table 23

Comparison table for the 3rd pole

.	v ₁	v ₂	v ₃	v ₄
v ₁	3	2	2	1
v ₂	2	3	1	0
v ₃	2	1	3	2
v ₄	3	3	2	3

Table 24

Membership score table of the 3rd pole

.	Grades sum	Row sum	Column sum	V₃ = c - d
	$(\sum_{i = 1}^{3} r_{c_{i}})$	(c)	(d)
v ₁	7	8	10	-2
v ₂	10	6	9	-3
v ₃	12	8	8	0
v ₄	10	11	6	5

Table 25

Tabular representation of the 4th pole

p ₄	c ₁	c ₂	c ₃
v ₁	0.6	0.6	0.9
v ₂	0.7	0.9	0.7
v ₃	0.9	0.9	0.7
v ₄	0.8	0.4	0.9

Table 26

Comparison table for the 4th pole

.	v ₁	v ₂	v ₃	v ₄
v ₁	3	1	1	2
v ₂	2	3	2	1
v ₃	2	3	3	2
v ₄	2	2	1	3

Table 27

Membership score table of the 4th pole

.	Grades sum	Row sum	Column sum	V₄ = n - o
	$(\sum_{i = 1}^{3} r_{c_{i}})$	(n)	(o)
v ₁	7	7	9	-2
v ₂	10	8	9	-1
v ₃	12	10	7	3
v ₄	10	8	8	0

The final score for each restaurant is obtained by adding the membership scores V₁, V₂, V₃, V₄ of all alternatives as shown by Table 28.

Table 28

Final score table with grades

.	Grades sum	V ₁	V ₂	V ₃	V ₄	Final Score
	$(\sum_{i = 1}^{3} r_{c_{i}})$					( $\sum_{i = 1}^{4} V_{i}$ )
v ₁	7	-6	-4	-2	-2	-14
v ₂	10	-1	4	-3	-1	-1
v ₃	12	2	5	0	3	10
v ₄	10	5	-5	5	0	5

It is clear from the above calculations that the maximum score is 10 by v₃ with maximum grades. Therefore, Mr. Ali will select the restaurant v₃ to buy. We give the following Algorithm 1 for the selection of a suitable restaurant.

Algorithm 1

Selection of an alternative in an (m, N)-soft set

1. Input V as a universe with n elements.

2. Input C ⊆ Z as a set of attributes.

3. Consider an (m, N)-soft set in tabular form.

4. Determine the comparison table for all 3-poles.

5. Compute the information score for all poles.

6. Compute the final score by adding the scores of all poles.

7. Find the biggest score with maximum grades, if it occurs in

kth row, then we will choose option v_k as output, 1 ≤ k ≤ n.

If the set of optimal choices in the last step of Algorithm 1 contain more than one values, that is, v_a = v_b, where 1 ≤ a ≠ b ≤ n, then any one of them can be chosen. For clarification we explain that: step 4 produces Tables 17, 20, 23 and 26; step 5 produces Tables 18, 21, 24 and 27; and Table 28 is the result of step 6.

Next we apply Algorithm 1 to a different real situation.

4.2 Selection of a hotel

Choosing a suitable hotel on a tour to stay is a very difficult task due to different star rating for the same hotel on different websites based on different attributes, for example, star rating of the hotel “The Nishat Hotel” is different on various websites including, www.agoda.com and www.booking.com. Suppose that a tourist (Mr. Nabeel) wants to select a suitable hotel from the alternatives q₁, q₂, q₃, q₄. Let V = {q₁, q₂, q₃, q₄} be a universe of four hotels and let C = {c₁, c₂, c₃} ⊆ Z be a set of attributes, which gives grades to hotels based on location, price and services. We can obtain a (3, 6)-soft set from Table 29, where

Five star represents ‘luxury’,

Four star represents ‘excellent’,

Three star represents ‘very good’,

Two star represents ‘good’,

One star represents ‘regular’.

Table 29
Information obtained from the related data

V c ₁ c ₂ c ₃

q ₁ ★★★ ★ ★★★ ★

q ₂ ★★★ ★ ★ ★★

q ₃ ★★★ ★★ ★★★ ★ ★★★

q ₄ ★★★ ★ ★★ ★★

V	c ₁	c ₂	c ₃
q ₁	★★★ ★	★★★	★
q ₂	★★★ ★	★	★★
q ₃	★★★ ★★	★★★ ★	★★★
q ₄	★★★ ★	★★	★★

We can easily identify ordered grades M = {0, 1, 2, 3, 4, 5} with stars as follows:

1 stand for ‘★’,

2 stand for ‘★★’,

3 stand for ‘★★★’,

4 stand for ‘★★★ ★’,

5 stand for ‘★★★ ★★’.

From Definition 2.2, tabular form of a 6-soft set is given by Table 30.

Table 30

Tabular representation of 6-soft set (F, C, 6)

(F, C, 6)	c ₁	c ₂	c ₃
q ₁	4	3	1
q ₂	4	1	2
q ₃	5	4	3
q ₄	4	2	2

Therefore, by Definition 2.5 a (3, 6)-soft set (f, D, 3) is defined by Table 31.

Table 31

Tabular representation of (3, 6)-soft set (f, D, 3)

(f, D, 3)	c ₁	c ₂	c ₃
q ₁	〈4, (0.7, 0.6, 0.5) 〉	〈3, (0.6, 0.7, 0.6) 〉	〈1, (0.5, 0.7, 0.6) 〉
q ₂	〈4, (0.5, 0.6, 0.7) 〉	〈1, (0.7, 0.4, 0.5) 〉	〈2, (0.7, 0.6, 0.7) 〉
q ₃	〈5, (0.8, 0.8, 0.7) 〉	〈4, (0.7, 0.7, 0.8) 〉	〈3, (0.9, 0.7, 0.7) 〉
q ₄	〈4, (0.7, 0.5, 0.6) 〉	〈2, (0.5, 0.6, 0.7) 〉	〈2, (0.7, 0.6, 0.7) 〉

Thus, (f, D, 3) is a (3, 6)-soft set based on location, price and services of the hotels. For example, in the Table 31, 〈4, (0.7, 0.6, 0.5) means that the hotel q₁ is 70% suitable with the location, 60% suitable with the price and 50% suitable with services, where 4 is the evaluation grade of q₁ according to attribute c₁. The tabular form of the 1st pole is given by the Table 32.

Table 32

Tabular representation of the 1st pole

p ₁	c ₁	c ₂	c ₃
q ₁	0.7	0.6	0.5
q ₂	0.5	0.7	0.7
q ₃	0.8	0.7	0.9
q ₄	0.7	0.5	0.7

Now we will make the comparison table for the 1st pole given by Table 33.

Table 33

Comparison Table for the 1st pole

.	q ₁	q ₂	q ₃	q ₄
q ₁	3	1	0	2
q ₂	2	3	1	2
q ₃	3	3	3	3
q ₄	2	2	0	3

The membership score for each hotel with sum of grades $(\sum_{i = 1}^{3} r_{c_{i}})$ is obtained by subtracting the column sum from the row sum of Table 33, which is given in Table 34.

Table 34

Membership score Table of the 1st pole

.	Grades sum	Row sum	Column sum	V₁ = c - d
	$(\sum_{i = 1}^{3} r_{c_{i}})$	(c)	(d)
q ₁	8	6	10	-4
q ₂	7	8	9	-1
q ₃	12	12	4	8
q ₄	8	7	10	-3

Similarly, we represent the remaining two poles in tabular form and calculate the membership score with the help of comparison tables for each pole, respectively, see Tables 35 –40.

Table 35

Tabular representation of 2nd pole

p ₂	c ₁	c ₂	c ₃
q ₁	0.6	0.7	0.7
q ₂	0.6	0.4	0.6
q ₃	0.8	0.7	0.7
q ₄	0.5	0.6	0.6

Table 36

Comparison Table for the 2nd pole

.	q ₁	q ₂	q ₃	q ₄
q ₁	3	3	2	3
q ₂	1	3	0	3
q ₃	3	3	3	3
q ₄	0	2	0	3

Table 37

Membership score Table of the 2nd pole

.	Grades sum	Row sum	Column sum	V₂ = e - f
	$(\sum_{i = 1}^{3} r_{c_{i}})$	(e)	(f)
q ₁	8	11	7	4
q ₂	7	6	11	-5
q ₃	12	12	5	7
q ₄	8	5	11	-6

Table 38

Tabular representation of the 3rd pole

p ₃	c ₁	c ₂	c ₃
q ₁	0.5	0.6	0.6
q ₂	0.7	0.5	0.7
q ₃	0.7	0.8	0.7
q ₄	0.6	0.7	0.7

Table 39

Comparison Table for the 3rd pole

.	q ₁	q ₂	q ₃	q ₄
q ₁	3	1	0	0
q ₂	2	3	2	2
q ₃	3	3	3	3
q ₄	3	2	1	3

Table 40

Membership score Table of the 3rd pole

.	Grades sum	Row sum	Column sum	V₃ = g - h
	$(\sum_{i = 1}^{3} r_{c_{i}})$	(g)	(h)
q ₁	8	4	11	–7
q ₂	7	9	9	0
q ₃	12	12	6	6
q ₄	8	9	8	1

The final score for each hotel with grades is obtained by adding the membership scores V₁, V₂, V₃ of all the alternatives.

It is clear from the Table 41, the maximum grade is 12 and maximum score is 21 scored by q₃. Therefore, Mr. Nabeel will select the hotel q₃ to stay.

Table 41

Final Score Table with grades

.	Grades sum	V ₁	V ₂	V ₃	Final Score
	$(\sum_{i = 1}^{3} r_{c_{i}})$				( $\sum_{i = 1}^{3} V_{i}$ )
q ₁	8	-4	4	-7	-7
q ₂	7	-1	-5	0	-6
q ₃	12	8	7	6	21
q ₄	8	-3	-6	1	-8

4.3 Selection of a resort

Stars are a very subjective thing. It’s usually referring to the service and amenities. Choosing a resort is very difficult task due to different star ratings for the same resort on different websites, for example, the resort named as“Komandoo Maldives Resort” has different star ratings on different websites including, www.expedia.com and www.tripadvisor.com. Suppose that a family plan to spend their vacations in a resort in Maldives from the alternatives y₁, y₂, y₃, y₄. They want to select the best resort to spend vacations with comfort. It is difficult for family members to agree at one option. The natural environment, cost and social environment are the leading parameters for selecting a resort.

Let V = {y₁, y₂, y₃, y₄} be a set of resorts and let Z = {z₁, z₂} be a set of attributes which gives grades to these resorts based on natural environment, social environment and cost.

We can obtain a (3, 6)-soft relation from Table 42, where

Table 42
Information obtained from the related data

V z ₁ z ₂

y ₁ ★★★ ★★ ★★★

y ₂ ★★★ ★ ★★★ ★

y ₃ ★★★ ★ ★★★ ★★

y ₄ ★★★ ★ ★★★ ★★

V	z ₁	z ₂
y ₁	★★★ ★★	★★★
y ₂	★★★ ★	★★★ ★
y ₃	★★★ ★	★★★ ★★
y ₄	★★★ ★	★★★ ★★

Five star represents ‘luxury’,

Four star represents ‘excellent’,

Three star represents ‘very good’.

The set of ordered grades M = {0, 1, 2, 3, 4, 5} can be easily associated with stars as follows:

3 stand for ‘★★★’,

4 stand for ‘★★★ ★’,

5 stand for ‘★★★ ★★’.

Assume that they explain the “attractiveness of the resort” by composing a (3, 6)-soft relation λ over (V × M) × Z, which is given by $\begin{matrix} λ & z_{1} & z_{2} \\ y_{1} & 〈 5, (0.9, 0.8, 0.8) 〉 & 〈 3, (0.6, 0.6, 0.6) 〉 \\ y_{2} & 〈 4, (0.7, 0.8, 0.7) 〉 & 〈 4, (0.6, 0.7, 0.6) 〉 \\ y_{3} & 〈 4, (0.8, 0.6, 0.8) 〉 & 〈 5, (0.8, 0.9, 0.5) 〉 \\ y_{4} & 〈 4, (0.7, 0.8, 0.7) 〉 & 〈 5, (0.7, 0.8, 0.7) 〉 \end{matrix}$ Thus λ : (V × M) × Z → [0, 1] ^m is a (3, 6)-soft relation based on natural environment, social environment and cost of the resorts. For example, 〈 ((y₁, 5) , z₁) , 0.9, 0.8, 0.8〉 means that the resort y₁ has grade 5 include, 90% natural environment, 80% social environment and 80% cost. Now suppose that they provide the optimal decision object P as follows: $P = {(z_{1}, 0.9, 0.8, 0.7), (z_{2}, 0.7, 0.6, 0.8)} .$ From Definition 3.3,

$P_{\underline{λ}} (y_{1}, 3) = (0.7, 0.6, 0.7), P_{\bar{λ}} (y_{1}, 5) = (0.9, 0.8, 0.7),$

$P_{\underline{λ}} (y_{2}, 4) = (0.7, 0.6, 0.7), P_{\bar{λ}} (y_{2}, 4) = (0.7, 0.8, 0.7),$

$P_{\underline{λ}} (y_{3}, 4) = (0.7, 0.8, 0.5), P_{\bar{λ}} (y_{3}, 5) = (0.8, 0.6, 0.7),$

$P_{\underline{λ}} (y_{4}, 4) = (0.7, 0.6, 0.7), P_{\bar{λ}} (y_{4}, 5) = (0.7, 0.6, 0.7) .$

By Definition 3.3, the operators ${\underline{λ}}_{N} (P)$ and ${\bar{λ}}^{N} (P)$ , respectively, are given as follows: $\begin{matrix} {\underline{λ}}_{N} (P) = { & 〈 (y_{1}, 3), 0.7, 0.6, 0.7 〉, 〈 (y_{2}, 4), 0.7, 0.6, 0.7 〉, \\ 〈 (y_{3}, 4), 0.7, 0.8, 0.5 〉, 〈 (y_{4}, 4), 0.7, 0.6, 0.7 〉}, \\ {\bar{λ}}^{N} (P) = { & 〈 (y_{1}, 5), 0.9, 0.8, 0.7 〉, 〈 (y_{2}, 4), 0.7, 0.8, 0.7 〉, \\ 〈 (y_{3}, 5), 0.8, 0.6, 0.7 〉, 〈 (y_{4}, 5), 0.7, 0.6, 0.7 〉} . \end{matrix}$ By Definition 3.10, $\begin{matrix} {\underline{λ}}_{N} (P) \oplus {\bar{λ}}^{N} (P) = { & 〈 (y_{1}, - 7), 0.97, 0.92, 0.91 〉, \\ 〈 (y_{2}, - 8), 0.91, 0.92, 0.91, \\ 〈 (y_{3}, - 11), 0.94, 0.92, 0.85 〉, \\ 〈 (y_{4}, - 11), 0.91, 0.84, 0.91 〉} . \end{matrix}$ Thus, y₁ is the resort containing optimal values in the set ${\underline{λ}}_{N} (P) \oplus {\bar{λ}}^{N} (P)$ . Therefore, they will select the resort y₁.

The following Algorithm 2 is organized for the selection of a suitable resort.

Algorithm 2

Selection of an alternative in an (m, N)-soft rough set

1. Input V as a universe with n objects.

2. Input Z as a set of attributes.

3. Consider an (m, N)-soft relation λ on (V × M) × Z.

4. Input

P \in M^{'} (Z)

as an optimal decision according to needs of

decision makers.

5. Determine

{\underline{λ}}_{N} (P)

and

{\bar{λ}}^{N} (P)

by Definition 3.3.

6. Compute the choice set

Z = {\underline{λ}}_{N} (P) \oplus {\bar{λ}}^{N} (P)

Definition 3.10.

7. Find k for which p_i ∘ Z (y_k, r) ≥ Q, where

Q = ⋁ _1≤k≤np_i ∘ Z (y_k, r), then the optimal decision will be y_k,

which can be chosen as output.

If the set of optimal choices in the last step of Algorithm 2 contain more than one values, that is, y_a = y_b, where 1 ≤ a ≠ b ≤ n, change the mF subset P and repeat the Algorithm 2 until the optimal decision is only one.

We now apply Algorithm 2 to another real situation.

4.4 Selection of a laptop

The selection of a laptop is a difficult task due to the variation in ratings and reviews on one website to another for the same laptop, for example, star rating of the laptop “Lenovo ThinkPad X1 Carbon 2017” is different on various websites including, www.cnet.com and www.computershopper.com. Since every person has different requirements about a laptop such as, design, technology, price etc. Suppose a computer programmer (Mr. Raheel) wishes to purchase a most suitable laptop from the alternatives r₁, r₂, r₃, r₄. The design, technology and price are the leading parameters for selecting a laptop. Let V = {r₁, r₂, r₃, r₄} be a universe of laptops, Z = {z₁, z₂, z₃} a set of attributes, which gives grades to these laptops based on design, technology and price.

We can obtain a (3, 6)-soft relation from Table 43, where

Five star represents ‘luxury’,

Four star represents ‘excellent’,

Three star represents ‘very good’,

Two star represents ‘good’,

One star represents ‘regular’,

Table 43
Information extracted from the related data

V z ₁ z ₂ z ₃

r ₁ ★ ★★★ ★★

r ₂ ★★★ ★ ★★★ ★★

r ₃ ★★★ ★★ ★★★ ★ ★★

r ₄ ★★★ ★★★ ★★★

V	z ₁	z ₂	z ₃
r ₁	★	★★★	★★
r ₂	★★★ ★	★★★	★★
r ₃	★★★ ★★	★★★ ★	★★
r ₄	★★★	★★★	★★★

We can easily identify ordered grades M = {0, 1,2, 3, 4, 5} with stars as follows:

1 stand for ‘★’,

2 stand for ‘★★’,

3 stand for ‘★★★’,

4 stand for ‘★★★ ★’,

5 stand for ‘★★★ ★★’.

Assume that Raheel describes the “attractiveness of the laptop” by a (3, 6)-soft relation λ : (V × M) × Z → [0, 1] ^m, given as follows: $\begin{matrix} λ & z_{1} & z_{2} & z_{3} \\ r_{1} & 〈 1, (0.2, 0.6, 0.1) 〉 & 〈 3, (0.3, 0.4, 0.7) 〉 & 〈 2, (0.7, 0.3, 0.2) 〉 \\ r_{2} & 〈 4, (0.4, 0.5, 0.7) 〉 & 〈 3, (0.4, 0.5, 0.5) 〉 & 〈 2, (0.7, 0.4, 0.1) 〉 \\ r_{3} & 〈 5, (0.7, 0.8, 0.3) 〉 & 〈 4, (0.8, 0.9, 0.4) 〉 & 〈 2, (0.6, 0.2, 0.6) 〉 \\ r_{4} & 〈 3, (0.5, 0.6, 0.4) 〉 & 〈 3, (0.6, 0.7, 0.1) 〉 & 〈 3, (0.8, 0.5, 0.3) 〉 \end{matrix}$ Thus λ over (V × M) × Z is the (3, 6)-soft relation based on design, technology and price of the laptops. For example, ((r₁, 1) , z₁) , 0.2, 0.6, 0.1) means that the laptop r₁ contains suitability, 20% for design, 60% for technology and 10% for price. We now suppose that Raheel provides the optimal decision object P as follows: $\begin{matrix} P = { & (z_{1}, 0.5, 0.6, 0.7), (z_{2}, 0.7, 0.6, 0.9), \\ (z_{3}, 0.9, 0.6, 0.8)} . \end{matrix}$ From Definition 3.3,

$P_{\underline{λ}} (r_{1}, 1) = (0.7, 0.6, 0.8), P_{\bar{λ}} (r_{1}, 3) = (0.7, 0.6, 0.7),$

$P_{\underline{λ}} (r_{2}, 2) = (0.6, 0.6, 0.7), P_{\bar{λ}} (r_{2}, 4) = (0.7, 0.5, 0.7),$

$P_{\underline{λ}} (r_{3}, 2) = (0.5, 0.6, 0.7), P_{\bar{λ}} (r_{3}, 5) = (0.7, 0.6, 0.6),$

$P_{\underline{λ}} (r_{4}, 3) = (0.5, 0.6, 0.7), P_{\bar{λ}} (r_{4}, 3) = (0.8, 0.6, 0.4) .$

Now (3, 6)-soft rough approximation operators ${\underline{λ}}_{N} (P), {\bar{λ}}^{N} (P)$ , respectively, are given by $\begin{matrix} {\underline{λ}}_{N} (P) = { & 〈 (r_{1}, 1), 0.7, 0.6, 0.8 〉, 〈 (r_{2}, 2), 0.6, 0.6, 0.7 〉, \\ 〈 (r_{3}, 2), 0.5, 0.6, 0.7 〉, 〈 (r_{4}, 3), 0.5, 0.6, 0.7 〉}, \\ {\bar{λ}}^{N} (P) = { & 〈 (r_{1}, 3), 0.7, 0.6, 0.7 〉, 〈 (r_{2}, 4), 0.7, 0.5, 0.7 〉, \\ 〈 (r_{3}, 5), 0.7, 0.6, 0.6 〉, 〈 (r_{4}, 3), 0.8, 0.6, 0.4 〉} . \end{matrix}$ From Definition 3.10, $\begin{matrix} {\underline{λ}}_{N} (P) \oplus {\bar{λ}}_{N} (P) = { & 〈 (r_{1}, 1), 0.91, 0.84, 0.94 〉, \\ 〈 (r_{2}, - 2), 0.88, 0.8, 0.91 〉, \\ 〈 (r_{3}, - 3), 0.85, 0.84, 0.88 〉, \\ 〈 (r_{4}, - 3), 0.9, 0.84, 0.82) 〉} . \end{matrix}$ Therefore, Raheel will choose the laptop r₁ to buy because r₁ contains the optimal values in the set ${\underline{λ}}_{N} (P) \oplus {\bar{λ}}_{N} (P)$ .

5 Discussion

Like soft set theory in the recent past, the theory of N-soft sets is now emerging as a powerful mathematical tool to handle the uncertainty of situations with either binary or multinary data in discrete form. When N = 2, an N-soft set degenerates into a soft set. Thus N-soft set theory has a wider range of practical applications as compared to soft set theory. Recently, Akram et al. [2] developed a new decision-making hybrid model called fuzzy N-soft sets and illustrated it through real life applications to decision-making. The fuzzy N-soft set model adds fuzziness on top of the requirements of N-soft sets. But there are many situations where data come from multiple poles (agents). For example, weighted games; a company that decides to manufacture an item or product; a country that elects its leader; or a group of friends that plan to visit a country. Our proposed models, namely, (m, N)-soft sets and (m, N)-soft rough sets, enable us to deal with such type of data more precisely than existing models in the literature (e.g., mF soft sets [6] and the aforementioned models). Since an (m, N)-soft set is a generalization of both fuzzy N-soft sets [2] and mF soft sets [6] (v., Remark 2), the applicability of the (m, N)-soft set model is wider than the applicability of fuzzy N-soft set and mF soft set models. Further, (m, N)-soft rough sets describe the rough approximations of (m, N)-soft sets. In an (m, N)-soft rough set model, the decision is based on an (m, N)-soft relation and an optimal decision object. But in an (m, N)-soft set model, the optimal decision is totally based on a comparison table approach. Therefore it does not seem feasible to contrast these models that are intended to capture alternative forms of inputs.

6 Conclusion

The notion N-soft sets is a practical generalization of soft sets. As in the case of soft sets, there are many potential applications of N-soft sets and in fact already Fatimah et al. [22] have provided a few real examples. This paper introduces two hybrid models that benefit from the hybridization of N-soft sets with other successful models, namely, (m, N)-soft sets and (m, N)-soft rough sets. An (m, N)-soft set emerges from the combination of mF sets and N-soft sets. It allows to conceive of (m, N)-soft relations, which in turn permit to define (m, N)-soft approximation spaces and ultimately yield (m, N)-soft rough sets. We have given a few natural operations on these new objects that will encourage further research. We have also shown that natural decision-making procedures can give raise to optimal solutions in each of these settings. All in all, we have settled the ground for future deeper analysis of these general models.

Conflict of interest

The authors declare that they have no conflict of interest regarding the publication of this research article.

Footnotes

Acknowledgments

The authors are highly thankful to the Associate Editor and the three anonymous referees for their valuable comments and suggestions.

References

Akram

, Adeel

and Alcantud

J.C.R.

, Group decisionmaking methods based on hesitant N– soft sets, Expert Systems with Application 115 (2019), 95–105.

Akram

, Adeel

and Alcantud

J.C.R.

, Fuzzy N-soft sets: A novel model with applications, Journal of Intelligent and Fuzzy Systems 35(4) (2018), 4757–4771.

Akram

, m-Polar fuzzy graphs-theory, methods & applications, Studies in Fuzziness and Soft Computing 371 (2019), 1–284. Springer.

Akram

and Adeel

, Novel hybrid decision-making methods based on mF rough information, Granular Computing (2018). doi: 10.1007/s41066-018-00142-6.

Akram

, Ali

and Alshehri

N.O.

, A new multiattribute decision-making method based on m-polar fuzzy soft rough sets, Symmetry 9(11) (2017). 271. https://doi.org/10.3390/sym9110271.

Akram

, Ali

, Waseem

and Davvaz

, Decisionmaking methods based on hybrid mF models, Journal of Intelligent and Fuzzy Systems 35(3) (2018), 3387–3403.

Alcantud

J.C.R.

and Laruelle

, Dis & approval voting: A characterization, Social Choice and Welfare 43(1) (2014), 1–10.

Alcantud

J.C.R.

, A novel algorithm for fuzzy soft set based decision making from multiobserver input parameter data set, Information Fusion 29 (2016), 142–148.

Alcantud

J.C.R.

, Fuzzy soft set based decision making: A novel alternative approach, IFSA-EUSFLAT, 2015.

10.

Alcantud

J.C.R.

, Fuzzy soft set decision making algorithms: Some clarifications and reinterpretations, In Conference of the Spanish Association for Artificial Intelligence, 2016, pp. 479–488. Springer International Publishing.

11.

Alcantud

J.C.R.

, Cruz-Rambaud

and Muñoz

M.J.

, Torrecillas, Valuation fuzzy soft sets: A flexible fuzzy soft set based decision making procedure for the valuation of assets, Symmetry 9 (2017), 253. doi: 10.3390/sym9110253.

12.

Alcantud

J.C.R.

and Mathew

T.J.

, Separable fuzzy soft sets and decision making with positive and negative attributes, Applied Soft Computing 59 (2017), 586–595.

13.

Ali

M.I.

, Feng

, Liu

X.Y.

, Min

W.K.

and Shabir

, On some new operations in soft set theory, Computers and Mathematics with Applications 57(9) (2009), 1547–1553.

14.

Ali

M.I.

, A note on soft sets, rough soft sets and fuzzy soft sets, Applied Soft Computing 11(4) (2011), 3329–3332.

15.

Ali

M.I.

, Another view on reduction of parameters in soft sets, Applied Soft Computing 12(6) (2012), 1814–1821.

16.

Ali

M.I.

, Mahmood

, Rehman

M.M.U.

and Aslam

M.F.

, On lattice ordered soft sets, Applied Soft Computing 36 (2015), 499–505.

17.

Chen

, Li

, Ma

and Wang

, m-polar fuzzy sets: An extension of bipolar fuzzy sets, The Scientific World Journal 2014 (2014), 8. Article ID 416530.

18.

Chen

, Liu

, Wang

and Augusto

J.C.

, Ordering based decision-making a survey, Information Fusion 14(4) (2013), 521–531.

19.

Dong

, Liu

, Liang

, Chiclana

and Herrera-Viedma

, Strategic weight manipulation in multiple attribute decision making, Omega 75 (2018), 154–164.

20.

Danjuma

, Ismail

M.A.

and Herawan

, An alternative approach to normal parameter reduction algorithm for soft set theory, IEEE Access 5 (2017), 4732–4746.

21.

Danjuma

, Herawan

, Ismail

M.A.

, Chiroma

, Abubakar

A.I.

and Zeki

A.M.

, A review on soft set-based parameter reduction and decision making, IEEE Access 5 (2017), 4671–4689.

22.

Fatimah

, Rosadi

, Hakim

R.B.F.

and Alcantud

J.C.R.

, N-soft sets and their decision making algorithms, Soft Computing 22(12) (2018), 3829–3842.

23.

Fatimah

, Rosadi

, Hakim

R.B.F.

and Alcantud

J.C.R.

, Probabilistic soft sets and dual probabilistic soft sets in decision-making, Neural Computing and Applications (2017). https://doi.org/10.1007/s00521-017-3011-y.

24.

Feng

, Soft rough sets applied to multicriteria group decision making, Annals of Fuzzy Mathematics and Informatics 2(1) (2011), 69–80.

25.

Feng

, Li

, Davvaz

and Ali

M.I.

, Soft sets combined with fuzzy sets and rough sets: A tentative approach, Soft Computing 14(9) (2010), 899–911.

26.

Feng

, Liu

, Leoreanu-Fotea

and Jun

Y.B.

, Soft sets and soft rough sets, Information Sciences 181(6) (2011), 1125–1137.

27.

Feng

, Jun

Y.B.

, Liu

and Li

, An adjustable approach to fuzzy soft set based decision-making, Journal of Computation and Applied Mathematics 234(1) (2010), 10–20.

28.

Herawan

and Deris

M.M.

, On multi-soft sets construction in information systems, Springer, Berlin, Heidelberg, 2009, pp. 101–110.

29.

Kong

, Gao

and Wang

, Comment on: A fuzzy soft set theoretic approach to decision-making problem, Journal of Computational and Applied Mathematics 223(2) (2009), 540–542.

30.

C.C.

, Dong

and Herrera

, A consensus model for largescale linguistic group decision making with a feedback recommendation based on clustered personalized individual semantics and opposing consensus groups, IEEE Transactions on Fuzzy Systems 27(2) (2019), 221–233.

31.

Liu

, Dong

, Liang

, Chiclana

and Herrera-Viedma

, Multiple attribute strategic weight manipulation with minimum cost in a group decision making context with interval attribute weights information, IEEE Transactions on Systems, Man, and Cybernetics: Systems (2018). doi: 10.1109/TSMC.2018.2874942.

32.

Liu

, Qin

and Pei

, A method for fuzzy soft sets in decisionmaking based on an ideal solution, Symmetry 9(10) (2017), 246. doi: 10.3390/sym9100246.

33.

, Zhan

, Ali

M.I.

and Mehmood

, A survey of decision making methods based on two classes of hybrid soft set models, Artificial Intelligence Review (2016), forthcoming, doi: 10.1007/s10462-016-9534-2.

34.

, Liu

and Zhan

, A survey of decision making methods based on certain hybrid soft set models, Artificial Intelligence Review 47 (2017), 507–530.

35.

Maji

P.K.

, Biswas

and Roy

A.R.

, An application of soft sets in a decision-making problem, Computers and Mathematics with Applications 44(8) (2002), 1077–1083.

36.

Maji

P.K.

, Biswas

and Roy

A.R.

, Soft set theory, Computers and Mathematics with Applications 45(4-5) (2003), 555–562.

37.

Maji

P.K.

, Biswas

and Roy

A.R.

, Fuzzy soft sets, Journal of Fuzzy Mathematics 9(3) (2001), 589–602.

38.

Molodtsov

D.A.

, Soft set theory- First results, Computers and Mathematics with Applications 37(4-5) (1999), 19–31.

39.

Pawlak

, Rough sets, International Journal of Computer and Information Sciences 11(5) (1982), 145–172.

40.

Pawlak

, Rough sets, rough relations and rough functions, Fundamenta Informaticae 27(2) (1996), 103–108.

41.

Roy

A.R.

and Maji

P.K.

, A fuzzy soft set theoretic approach to decision making problems, Journal of Computational and Applied Mathematics 203(2) (2007), 412–418.

42.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

43.

Zhan

and Alcantud

J.C.R.

, A survey of parameter reduction of soft sets and corresponding algorithms, Artificial Intelligence Review, forthcoming.

44.

Zhang

W.R.

, Bipolar fuzzy sets and relations: A computational framework for cognitive modeling and multi-agent decision analysis, Proc of IEEE Conf (1994), 305–309.

45.

Zhang

, Dong

and Chen

, The 2-rank consensus reaching model in the multigranular linguistic multiple-attribute group decision-making, IEEE Transactions on Systems, Man, and Cybernetics: Systems 48(12) (2018), 2080–2094.

46.

Zhang

, Dong

, Chiclana

and Yu

, Consensus efficiency in group decision making: A comprehensive comparative study and its optimal design, European Journal of Operational Research 275(2) (2019), 580–598.

47.

Zhang

, Dong

, Palomares-Carrascosa

and Zhou

, Failure mode and effect analysis in a linguistic context: A consensus-based multiattribute group decision-making approach, IEEE Transactions on Reliability (2018), in press. doi: 10.1109/TR.2018.2869787.

48.

Zou

and Xiao

, Data analysis approaches of soft sets under incomplete information, Knowledge-Based Systems 21(8) (2008), 941–945.

Hybrid multi-attribute decision-making model based on ( m,N )-soft rough sets

Abstract

Keywords

1 Introduction

2 Construction of (m, N)-soft sets

4 Applications

4.1 Selection of a restaurant

Table 13 Information obtained from the related data V c 1 c 2 c 3 v 1 ★★★ ★★ ★★ v 2 ★★★ ★★★ ★★ ★★ v 3 ★★★ ★★ ★★★ ★★★ ★ v 4 ★★★ ★ ★★★ ★★★

Table 29 Information obtained from the related data V c 1 c 2 c 3 q 1 ★★★ ★ ★★★ ★ q 2 ★★★ ★ ★ ★★ q 3 ★★★ ★★ ★★★ ★ ★★★ q 4 ★★★ ★ ★★ ★★

Table 42 Information obtained from the related data V z 1 z 2 y 1 ★★★ ★★ ★★★ y 2 ★★★ ★ ★★★ ★ y 3 ★★★ ★ ★★★ ★★ y 4 ★★★ ★ ★★★ ★★

Table 43 Information extracted from the related data V z 1 z 2 z 3 r 1 ★ ★★★ ★★ r 2 ★★★ ★ ★★★ ★★ r 3 ★★★ ★★ ★★★ ★ ★★ r 4 ★★★ ★★★ ★★★

6 Conclusion

Conflict of interest

Footnotes

Acknowledgments

References

Table 13
Information obtained from the related data

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Table 29
Information obtained from the related data

V c ₁ c ₂ c ₃

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q ₃ ★★★ ★★ ★★★ ★ ★★★

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Table 42
Information obtained from the related data

V z ₁ z ₂

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y ₂ ★★★ ★ ★★★ ★

y ₃ ★★★ ★ ★★★ ★★

y ₄ ★★★ ★ ★★★ ★★

Table 43
Information extracted from the related data

V z ₁ z ₂ z ₃

r ₁ ★ ★★★ ★★

r ₂ ★★★ ★ ★★★ ★★

r ₃ ★★★ ★★ ★★★ ★ ★★

r ₄ ★★★ ★★★ ★★★