Decision-making methods based on hybrid m F models

1 Introduction

The nature of uncertainties arising in complicated problems in different fields, including economics, engineering, social sciences, medical sciences, environmental sciences may be very different. Classical mathematical techniques cannot solve these problems. To solve these problems a lot of mathematical theories have been reported, including, fuzzy set theory [27]. Chen et al. [8] introduced the notion of mF sets as a generalization of bipolar fuzzy sets and proved that 2-polar fuzzy sets and bipolar fuzzy sets are cryptomorphic mathematical notions which can be computed concisely from one another. The idea behind this is that “multipolar information" (not just bipolar information which correspond to two-valued logic) exists because data for real world problems are sometimes from n agents (n ≥ 2). For example, the exact degree of telecommunication safety of mankind is a point in [0, 1] ⁿ (n ≈ 7 ×10⁹) because different person has been monitored different times. There are many examples such as truth degrees of a logic formula which are based on n logic implication operators (n ≥ 2), similarity degrees of two logic formula which are based on n logic implication operators (n ≥ 2), ordering results of a magazine, ordering results of a university and inclusion degrees (accuracy measures, rough measures, approximation qualities, fuzziness measures, and decision preformation evaluations) of a rough set. Akram et al. [1–3] promoted the work on mF graphs and introduced new concepts.

As a theory of data analysis and processing, the rough set theory proposed by Pawlak [19–21] in 1982, is a new mathematical tool to deal with incomplete, imprecise and uncertain information after probability, evidence theory and fuzzy set theory. For the fuzzy set theory introduced by Zadeh [27] in 1965, membership function plays the vital role. However, the selection of membership function is uncertain. Therefore, the fuzzy set theory is an uncertain tool to solve the uncertain problems, but in rough set theory two precise boundary lines are established to describe the imprecise concepts. Therefore, the rough set theory is a certain mathematical tool to solve uncertain problems. Rough set theory is based on an assumption that every object in the universe is associated with some information. Objects characterized by the same information are indiscernible. The indiscernibilty relation induced in this way forms the mathematical basis of the rough set theory. In general, the indiscernibility relation is also referred to as an equivalence relation. Then any subset of a universe can be characterized by two definable or observable subsets called upper and lower approximations. However, the equivalence relations in Pawlak rough set are too restrictive for many applications. In recent years, from both theoretical and practical needs, many authors have generalized the notion of Pawlak rough set by using non-equivalence binary relations. This has led to various generalized rough set models [7, 12]. The study of hybrid models combining rough set with other mathematical models is emerging as an active research topic of rough set theory. Dubois and Prade [9] first introduced the concepts of rough fuzzy sets and fuzzy rough sets by combining fuzzy sets and rough sets. A rough fuzzy set is the approximation of a fuzzy set in a crisp approximation space. Therefore, rough sets not only have applications in mathematical theories but also in real-world problems [13, 28].

Molodtsov [18] introduced a novel idea of soft set theory. Soft set theory plays a very substantial role in various fields including, data analysis [29], forecasting [23] and decision-making [10]. Some basic algebraic operations on soft set theory were presented by Maji et al. [15]. Ali et al. [5] introduced some new operations for soft sets. Maji et al. [17] generalized the idea of soft sets and presented a hybrid model fuzzy soft sets. Furthermore, Maji et al. [16] presented the concept of intuitionistic fuzzy soft sets. By combining the Pawlak rough sets and soft sets, Feng et al. [11] gave the novel idea of rough soft sets. Actually the decision making based on fuzzy sets, soft sets and fuzzy soft sets has been considered by many researchers, see [6, 10]. All mathematical models, including fuzzy sets, soft sets, fuzzy soft sets and rough sets have their pros and cons. But one of the major drawback of all these models is they have lack of sufficient number of parameters to deal with uncertain information. Akram et al. [4] introduced the notions of soft rough mF sets and mF soft rough sets. They proposed a multi-attribute decision-making method based on mF soft rough sets in [4]. In order to give a new approach to multi-attribute decision-making problems, we introduce new hybrid models, rough mF sets and mF soft sets by combining mF sets with rough sets and soft sets, respectively. We present some fundamental properties of these models and describe their applications to multiple criteria decision-making problems. We also develop algorithm in each multiple criteria decision-making problem.

The outline of the paper is as follows. Section 2 defines rough mF sets and also discuss basic operations on them in detail. Rough degrees of mF sets are computed. Section 3 describes the mF soft sets and illustrate some basic operations. The decision-making mechanisms that deals with these models are explained in section 4. We also apply these models to real examples in order to prove their implementability. We conclude in Section 5.

2 Rough mF sets

Definition 2.1. [8] An mF set on a universe S is a function X = (p₁ ∘ X (z), p₂ ∘ X (z), …, p _m ∘ X (z)) : S → [0, 1] ^m , where the i-th projection mapping is defined as p _i ∘ X : [0, 1] ^m → [0, 1]. The family of all mF sets of S is denoted by mF ^S . 0 = (0, 0, ⋯, 0) is the smallest element in [0, 1] ^m and 1 = (1, 1, ⋯, 1) is the largest element in [0, 1] ^m . For simplicity, X can be written as X = { ( p 1 ∘ X ( z ) , p 2 ∘ X ( z ) , … , p m ∘ X ( z ) ) z | z ∈ S } .

Definition 2.2. [19] Let S be a universe of discourse and let Q be an equivalence relation on S. A pair (S, Q) is called a Pawlak approximation space. For any X ⊆ S, the lower approximation Q_* (X) and upper approximation Q* (X) of X are defined by Q * ( X ) = { x ∈ S | [ x ] Q ⊆ X } , Q * ( X ) = { x ∈ S | [ x ] Q ∩ X ≠ ∅ } . If Q_* (X) = Q* (X), X is said to be definable in (S, Q), otherwise X is referred to as a rough set.

We now define rough mF sets.

Definition 2.3. Let S be a universe and M a crisp equivalence relation on S. A pair (S, M) is said to be a crisp approximation space(CAS, for short). For each X ∈ mF ^S , the lower and upper approximations of X, denoted by M_* (X) (u) and M* (X) (u), respectively, which are mF sets in S defined by M * ( X ) ( u ) = ⋀ v ∈ [ u ] M ( p i ∘ X ( v ) ) , M * ( X ) ( u ) = ⋁ v ∈ [ u ] M ( p i ∘ X ( v ) ) , for all u ∈ S, where [u] _M = {v | (u, v) ∈ M} is an equivalence class of u ∈ S and i = 1, 2, …, m. The operators M_* (X) and M* (X) are called the lower and upper approximation operators of X, respectively. If M_* (X) = M* (X), then X is said to be definable, otherwise X is called a rough mF set.

Example 2.4. Let S = {1, 2, 3, 4, 5, 6} be a universe, M = {(1, 3), (3, 1), (2, 6), (6, 2), (4, 5), (5, 4)} an equivalence relation on S. Then [1] _M = {1, 3} = [3] _M , [2] _M = {2, 6} = [6] _M , [4] _M = {4, 5} = [5] _M . Consider a 3-polar fuzzy set X as follows: X = { ( 0.8 , 0.8 , 0.8 ) 1 , ( 0.3 , 0.3 , 0.3 ) 2 , ( 0.5 , 0.5 , 0.5 ) 3 , ( 0.1 , 0.1 , 0.1 ) 4 , ( 0.4 , 0.4 , 0.4 ) 5 , ( 0.3 , 0.3 , 0.3 ) 6 } . By Definition 2.3, we have lower and upper approximations: M * ( X ) ( 1 ) = ( 0.5 , 0.5 , 0.5 ) = M * ( X ) ( 3 ) , M * ( X ) ( 1 ) = ( 0.8 , 0.8 , 0.8 ) = M * ( X ) ( 3 ) , M * ( X ) ( 2 ) = ( 0.3 , 0.3 , 0.3 ) = M * ( X ) ( 6 ) , M * ( X ) ( 2 ) = ( 0.3 , 0.3 , 0.3 ) = M * ( X ) ( 6 ) , M * ( X ) ( 4 ) = ( 0.1 , 0.1 , 0.1 ) = M * ( X ) ( 5 ) , M * ( X ) ( 4 ) = ( 0.4 , 0.4 , 0.4 ) = M * ( X ) ( 5 ) . Thus, M * ( X ) = { ( 0.5 , 0.5 , 0.5 ) 1 , ( 0.3 , 0.3 , 0.3 ) 2 , ( 0.5 , 0.5 , 0.5 ) 3 , ( 0.1 , 0.1 , 0.1 ) 4 , ( 0.1 , 0.1 , 0.1 ) 5 , ( 0.3 , 0.3 , 0.3 ) 6 } , M * ( X ) = { ( 0.8 , 0.8 , 0.8 ) 1 , ( 0.3 , 0.3 , 0.3 ) 2 , ( 0.8 , 0.8 , 0.8 ) 3 , ( 0.4 , 0.4 , 0.4 ) 4 , ( 0.4 , 0.4 , 0.4 ) 5 , ( 0.3 , 0.3 , 0.3 ) 6 } . Clearly, M_* (X) ≠ M* (X). Hence X is a rough 3-polar fuzzy set.

We now present operations on rough mF sets.

Theorem 2.5. Let (S, M) be a CAS and X, Y ∈ mF ^S . Then the lower and upper approximations of X and Y have the following properties:

M* (X ∪ Y) = M* (X) ∪ M* (Y),

proof.

From Definition 2.3, ∀ u ∈ S, we have M * ( X ∪ Y ) ( u ) = ⋁ v ∈ [ u ] M p i ∘ ( X ∪ Y ) ( v ) , = ⋁ v ∈ [ u ] M ( p i ∘ X ( v ) ∨ p i ∘ Y ( v ) ) , = ⋁ v ∈ [ u ] M p i ∘ X ( v ) ∨ ⋁ v ∈ [ u ] M p i ∘ Y ( v ) , = ( M * ( X ) ∪ M * ( Y ) ) ( u ) . Thus, M* (X ∪ Y) = M* (X) ∪ M* (Y).

Remark. For any X ∈ mF ^S ,

M_* (X) ⊆ X ⊆ M* (X),

Definition 2.6. Let H = {σ = (a₁, a₂, …, a _m ) | a _i ∈ [0, 1], i = 1, 2, …, m}. Then, σ is called an mF number. For all σ = (a₁, a₂, …, a _m ) ∈ H, we define the mediation value of σ as follows: σ A = a 1 + a 2 + … + a m m .

Definition 2.7. Let (S, M) be a CAS. For X ∈ mF ^S , σ ∈ H and 0 < σ < 1, the σ-upper level boundary of X is defined as ML * σ ( X ) = { u ∈ S | M * ( X ) ( u ) < σ } .

Definition 2.8. Let (S, M) be a CAS. For X ∈ mF ^S , σ ∈ H and 0 < σ < 1, the σ-lower level boundary of X is defined as ML σ * ( X ) = { u ∈ S | M * ( X ) ( u ) > σ } .

Definition 2.9. Let (S, M) be a CAS. For X, Y ∈ mF ^S , σ ∈ H, 0 < σ < 1, we define

W ( X , Y ) σ = { u ∈ S | M * ( X ∪ Y ) ( u ) = σ , u ∈ ML * σ ( X ) ∩ ML * σ ( Y ) } ,

Theorem 2.10. Let (S, M) be a CAS, then lower and upper approximations of mF sets X and Y have the following properties:

M* (X ∩ Y) = M* (X) ∩ M* (Y) ∩ M* (W) (X, Y),

Proof.

From Theorem 2.5 (5), we have M * ( X ∩ Y ) ⊆ M * ( X ) ∩ M * ( Y ) ∀ u ∈ S, set M* (X ∩ Y) (u) = σ. If there exists u ∈ W (X, Y) _σ , then M * ( W ) ( X , Y ) = σ . Otherwise M * ( W ) ( X , Y ) ( u ) = 1 . Thus, ∀ u ∈ S, M * ( X ∩ Y ) ( u ) ≤ M * ( W ) ( X , Y ) ( u ) . That is, M * ( X ∩ Y ) ⊆ M * ( W ) ( X , Y ) . If M* (W) (X, Y) (u) = 1, there exists σ such that M * ( X ∩ Y ) ( u ) = σ and u ∉ ML σ * ( X ) or u ∉ ML σ * ( Y ) , this implies that M * ( X ) ( u ) = σ or M * ( Y ) ( u ) = σ . That is, M * ( X ∩ Y ) ( u ) = M * ( X ) ( u ) ∩ M * ( Y ) ( u ) ∩ M * ( W ) ( X , Y ) ( u ) . If M * ( W ) ( X , Y ) ( u ) = σ ≠ 1 , there exists σ satisfying M* (X ∩ Y) (u) = σ and u ∈ ML σ * ( X ) , u ∈ ML σ * ( Y ) , followed by M * ( X ) ( u ) > σ and M * ( Y ) ( u ) > σ . Therefore, we prove that M * ( X ∩ Y ) ( u ) = M * ( X ) ( u ) ∩ M * ( Y ) ( u ) ∩ M * ( W ) ( X , Y ) ( u ) , = M * ( W ) ( X , Y ) ( u ) = σ . Hence, M* (X ∩ Y) = M* (X) ∩ M* (Y) ∩ M* (W) (X, Y).

Proposition 2.11. Let (S, M) be a CAS. Then the lower and upper approximations of mF sets X and Y satisfies the following laws:

∼ (M_* (X) ∪ M_* (Y)) = M* (∼ X) ∩ M* (∼ Y),

Proof. Proof is obvious.

Definition 2.12. Let (S, M) be a CAS. For X ∈ mF ^S , σ ∈ H, we define M * ( X ) σ = { u ∈ S | M * ( X ) ( u ) ≥ σ } , M * ( X ) σ = { u ∈ S | M * ( X ) ( u ) ≥ σ } .

Theorem 2.13. Let (S, M) be a CAS and X, Y ∈ mF ^S . Then,

M_* (X ∩ Y) _σ = M_* (X) _σ ∩ M_* (Y) _σ ,

Proof. Its proof follows immediate from the Theorems 2.5 and 2.10.

Definition 2.14. Let (S, M) be a CAS. For 0 < τ ≤ σ ≤ 1, we define the (σ, τ)-related accuracy degree of the mF set X as γ σ , τ = | M * ( X ) σ | | M * ( X ) τ | , and the corresponding (σ, τ)-related rough degree of the mF set X is given by ρ σ , τ = 1 - γ σ , τ = 1 - | M * ( X ) σ | | M * ( X ) τ | .

Definition 2.15. Let X ∈ mF ^S , if there exists v ∈ S such that X (v) = 1, then X is said to be a normal mF set. Suppose that the range of the mediations of X is Range ( X A ) = { a 1 , a 2 , … , a m } , where a _i > a_i+1 > 0, for i = 1, 2, …, m - 1.

Definition 2.16. Let X ∈ mF ^S , the mediation mass assignment(MMA, in short) of X is denoted by m and is defined as m ( ∅ ) = 1 m - a 1 , m ( X i ) = a i - a i + 1 , i = 1 , 2 , … , m , where X _i is a Pawlak set, and X i = { u ∈ S | X A ( u ) ≥ a i } , i = 1 , 2 , … , m . { X i } m = 1 n are the focal elements of m .

Next, the notion of MMA of mF sets is used to propose the parameter-free rough degree of the normal mF sets.

Definition 2.17. Let (S, M) be a CAS. For a normal mF set X, the parameter-free rough degree of X about (S, M) is defined as ρ ( X ) = ∑ i = 1 m m ( X i ) ( 1 - | M * ( X i ) | | M * ( X i ) | ) , = ∑ i = 1 m m ( X i ) ρ ( X i ) .

Remark.

X is called rough mF set ⇔ 0 ≤ ρ (X) ≤1,

Proposition 2.18. Let (S, M) be a CAS and X ∈ mF ^S , then ∀ a _i ∈ Range (X ^A ), we have

M* (X _i ) ⊆ M* (X) _i ,

Proof. Proof is obvious.

Definition 2.19. Let (S, M) be a CAS. For X ∈ mF ^S , the parameter-free rough degree of X about (S, M) is defined as ρ ( X ) = ∑ i = 1 m m ( X i ) ρ ( X i ) + m ( ∅ ) ρ ( ∅ ) .

If A is normal and ρ (∅) =0, then m ( ∅ ) ρ ( ∅ ) = 0 , but γ (∅) =1, by convention. So, the parameter-free rough accuracy degree of an mF set in a CAS is defined as follows.

Definition 2.20. Let (S, M) be a CAS. For X ∈ mF ^S , the parameter-free rough accuracy degree of X about (S, M) is defined as γ ( X ) = ∑ i = 1 m m ( X i ) γ ( X i ) , where X is a normal mF set. If X is not a normal mF set, then γ ( X ) = ∑ i = 1 m m ( X i ) γ ( X i ) + m ( ∅ ) γ ( ∅ ) .

Remark. For any normal mF set X ∈ mF ^S in (S, M), we have γ ( X ) = 1 - ρ ( X ) .

The parameter related rough degree and parameter-free rough degree are different. The difference between them is investigated through an example.

Now, It is clear from Example 2.21 that (σ, τ)-related rough degree in Definition 2.14 depends on σ and τ, and the rough degree in Definition 2.17 does not depends on parameters.

Definition 2.22. An information system is a pair I = (S, C), where S is a nonempty finite set of objects and C is a nonempty set of attributes. Every attribute r ∈ C is a mapping r : C ⟶ Y _r , where Y _r is the set of values of attribute r.

Definition 2.23. Let (S, C, {C _i }, D, {D _j }) be an mF objective decision information system, where (S, C, {C _i }) is an information system, and D _j is the set of m-polar decision values for attributes d _j , where d _j ∈ D, j = 1, 2,…, n. If the conditional attribute values {C _i } are mF numbers, then (S, C, {C _i }, D, {D _j }) is called an mF objective decision information system (mFODIS, in short).

Definition 2.24. Let (S, C, {C _i }, D, {D _j }) be an mFODIS and B ⊂ C, then B generates a CAS (S, M _B ). Decision attributes set {d₁, …, d _n } on S can be seen as an mF partition PB = {D₁, …, D _n } of S. Approximation of PB about M _B is denoted by η (PB) and is defined as η ( PB ) = 1 | S | ∑ j = 1 n ∑ K = 1 l m ( D j k ) | ( M * ) B ( D j k ) | ,(1) where m ( D j k ) and D j k represents the MMA of D _j and its focal elements, respectively.

Definition 2.25. Let (S, C, {C _i }, D, {D _j }) be an mFODIS, if S/M _C ≤ S/M _D , i.e., for every [v] _C , there exists [v] _D , such that [v] _C ⊆ [v] _D , then the mFODIS is called consistent. If S/M _C ≤ S/M _D , i.e., for every [v] _C , there exists [v] _D , such that [v] _C ⊈ [v] _D , then the mFODIS is called inconsistent.

Definition 3.1. Let S be a universe, T a set of parameters and N ⊆ T. Define ζ : N → mF ^S . Then (ζ, N) is called an mF soft set over a universe S, which is defined by, ( ζ , N ) = { ( x , p i ∘ N ϵ ( x ) ) : x ∈ S and ϵ ∈ N } .

Example 3.2. Let S = {b₁, b₂, b₃, b₄, b₅} be the set of five bungalows under consideration, and let T = {t₁, t₂, t₃, t₄} be the set of parameters, where the parameter, ‘t₁’ stands for the Location of Bungalow, ‘t₂’ stands for the Price of Bungalow, ‘t₃’ stands for the Architecture of Bungalow, ‘t₄’ stands for the Beauty of Bungalow. We give further characteristics of these parameters.

The parameter “Location of Bungalow" depends on centrality, neighborhood and commercial development.

Suppose that a family wants to buy a bungalow of S. They consider three parameters t₂, t₃, t₄ for the selection of a bungalow. Let N = {t₂, t₃, t₄} be subset of T. Then all available information on these bungalows under consideration can be formulated as a 3-polar fuzzy soft set (ζ, N).

( ζ , N ) = { ζ ( t 2 ) = { ( 0.7 , 0.5 , 0.3 ) b 1 , ( 0.6 , 0.8 , 0.5 ) b 2 , ( 0.6 , 0.4 , 0.7 ) b 3 , ( 0.5 , 0.6 , 0.3 ) b 4 , ( 0.7 , 0.6 , 0.4 ) b 5 } , ζ ( t 3 ) = { ( 0.9 , 0.6 , 0.7 ) b 1 , ( 0.6 , 0.5 , 0.6 ) b 2 , ( 0.5 , 0.6 , 0.7 ) b 3 , ( 0.7 , 0.8 , 0.6 ) b 4 , ( 0.4 , 0.3 , 0.6 ) b 5 } , ζ ( t 4 ) = { ( 0.8 , 0.6 , 0.4 ) b 1 , ( 0.6 , 0.5 , 0.5 ) b 2 , ( 0.4 , 0.3 , 0.5 ) b 3 , ( 0.6 , 0.6 , 0.5 ) b 4 , ( 0.7 , 0.6 , 0.7 ) b 5 } . } Thus (ζ, N) is a 3-polar fuzzy soft set in which we have chosen the Price, Architecture and Beauty of the Bungalow as desired parameters for the selection of Bungalow. For example, if we consider the parameter “Beauty of Bungalow", ( 0.6 , 0.5 , 0.5 ) b 2 shows that according to the family bungalow b₂ has 60% beautiful landscape, 50% stylish furniture and 50% good wall decorations.

Definition 3.3. An mF soft class is the collection of all mF soft sets on S with attributes from T. An mF soft set (ζ, N) over S is said to be a null mF soft set, denoted by ∅, if for all t ∈ N, ζ (t) is the null m-polar fuzzy set 0 of S, where 0 (u) = (0, 0, …, 0), for all u ∈ S. An mF soft set (ζ, N) over S is said to be an absolute mF soft set, denoted by ω, if for all t ∈ N, ζ (t) is the absolute m-polar fuzzy set 1 of S, where 1 (u) = (1, 1, …, 1), for all u ∈ S.

Definition 3.4. Let (ζ, N) and (ϱ, O) be two mF soft sets over the same universe S. Then (ζ, N) is said to be an mF soft subset of (ϱ, O), denoted by (ζ, N) Subset (ϱ, O), if

N ⊆ O,

We now present operations on mF soft sets.

Definition 3.5. The union of two mF soft sets (ζ, N) and (ϱ, O) over the universe S is an mF soft set (υ, P) = (ζ, N) ⋓ (ϱ, O), where P = N ∪ O and for all t ∈ P, υ ( t ) = { ζ ( t ) if t ∈ N \ O , ϱ ( t ) if t ∈ O \ N , ζ ( t ) ∪ ϱ ( t ) if t ∈ N ∩ O .

Definition 3.6. The intersection of two mF soft sets (ζ, N) and (ϱ, O) over the universe S is an mF soft set (υ, P) = (ζ, N) ⋒ (ϱ, O), where P = N∩ O ≠ ∅ and υ : P → mF ^S is defined by υ (t) = ζ (t) ∩ ϱ (t), for all t ∈ P.

Proposition 3.7. Let (ζ, N) be an mF soft set over a universe S. Then

(ζ, N) ⋓ (ζ, N) = (ζ, N).

Proof. Proof is obvious.

Lemma 3.8. Absorption property of mF soft sets (ζ, N) and (ϱ, O).

(ζ, N) ⋓ ((ζ, N) ⋒ (ϱ, O)) = (ζ, N),

Proof.

An intersection of two mF soft sets (ζ, N) and (ϱ, O) is (υ, P), where P = N ∩ O. ( υ , P ) = ( ζ , N ) ⋒ ( ϱ , O ) , where P = N ∩ O .(2)υ (t) = ζ (t) ∩ ϱ (t), if t ∈ P = N ∩ O.

Let mF soft set (ψ, V) is the union of two mF soft sets (ζ, N) and (υ, P) which is ( ψ , V ) = ( ζ , N ) ⋓ ( υ , P ) , where V = N ∪ P .(3) We define ψ ( t ) = { ζ ( t ) if t ∈ N \ P , υ ( t ) if t ∈ P \ N , ζ ( t ) ∪ υ ( t ) if t ∈ N ∩ P . Thus, there are the following three cases: Case 1: If t ∈ N \ P, then ψ ( t ) = ζ ( t ) if t ∈ N \ P , = ζ ( t ) if t ∈ N , ψ ( t ) = ζ ( t ) , ( ψ , V ) = ( ζ , N ) from ( ref eq 3 ) . Case 2: If t∈ P \ N = N ∩ O - N = ∅, then ψ ( t ) = υ ( t ) if t ∈ P \ N = ∅ , = ∅ if t ∈ ∅ , ψ ( t ) = ∅ if t ∈ ∅ , ( ψ , V ) = ∅ from ( ref eq 3 ) . Case 3: If t ∈ N ∩ P, then

ψ ( t ) = ζ ( t ) ∪ υ ( t ) if t ∈ N ∩ P and P = N ∩ O , = ζ ( t ) ∪ ( ζ ( t ) ∩ ϱ ( t ) from ( ref eqa ) , = ζ ( t ) since ( ζ ( t ) ∩ ϱ ( t ) ) ⊂ ζ ( t ) , ψ ( t ) = ζ ( t ) , ( ψ , V ) = ( ζ , N ) from ( ref eq 3 ) . All cases are satisfied. Hence, (ζ, N) ⋓ ((ζ, N) ⋒ (ϱ, O)) = (ζ, N).

Theorem 3.9 Commutative property of mF soft sets (ζ, N) and (ϱ, O).

(ζ, N) ⋒ (ϱ, O) = (ϱ, O) ⋒ (ζ, N)

Proof. Its proof is similar to the proof of Lemma 3.8.

Theorem 3.10. Associate law of mF soft sets (ζ, N), (ϱ, O) and (υ, P).

(ζ, N) ⋒ ((ϱ, O) ⋒ (υ, P)) = ((ζ, N) ⋒ (ϱ, O)) ⋒ (υ, P),

Proof. Proof is obvious.

Theorem 3.11. Distributive law over mF soft sets (ζ, N), (ϱ, O) and (υ, P).

(ζ, N) ⋒ ((ϱ, O) ⋓ (υ, P)) = ((ζ, N) ⋒ (ϱ, O)) ⋓ ((ζ, N) ⋒ (υ, P)),

Proof. Its proof is similar to Lemma 3.8.

Lemma 3.12. Let (ζ, N) and (ϱ, O) be two mF soft sets.

(ζ, N) ⊂ (ϱ, O) ⇒ (ζ, N) ⋒ (ϱ, O) = (ζ, N),

Proof. Proof is obvious.

Definition 3.13. The AND operation of two mF soft sets (ζ, N) and (ϱ, O) over a common universe S is denoted by (ζ, N) barwedge (ϱ, O) and is defined by (ζ, N) barwedge (ϱ, O) = (υ, N × O), where υ (t _j , t _k ) = ζ (t _j ) ∩ ϱ (t _k ) for all (t _j , t _k ) ∈ P = N × O, where ∩ is the intersection operation of mF sets.

Definition 3.14. The OR operation of two mF soft sets (ζ, N) and (ϱ, O) over a common universe S is denoted by (ζ, N) ⊻ (ϱ, O) and is defined by (ζ, N) ⊻ (ϱ, O) = (υ, N × O), where υ (t _j , t _k ) = ζ (t _j ) ∪ ϱ (t _k ) for all (t _j , t _k ) ∈ P = N × O, where ∪ is the union operation of mF sets.

Proposition 3.15. Idempotent property of two mF soft sets (ζ, N) and (ϱ, O).

(ζ, N) ⊻ (ζ, N) = (ζ, N),

Proof. Proof is obvious.

Definition 4.1. [22] A comparison table is a square table in which number of rows and number of columns are equal, and both are labeled by the object name of the universe such as u₁, u₂, u₃, …, u _n and the entries e _jk where e _jk = the number of parameters for which the value of e _j exceeds or equal to the value of e _k .

Multi-criteria decision-making is a very important tool that we can apply to many complex real life problems and decisions. It has all the characteristics of a useful decision support tool. It helps us focus on what is easy to use, important, consistent and logical. Rough mF sets and mF soft set have wide range of applications to deal with uncertainties in our real life problems. In this section, we describe applications of these models to decision-making. 1. Selection of flats: Nowadays, the selection of suitable flats for investment and personal use is a multi-criteria decision-making problem and is of great importance for the buyers. It is a very complicated decision due to high cost of reconfiguration and relocation. There are a number of factors to take into consideration for buying a flat such as location of the flat and size of the flat. These factors among many others influence house buyers before they even get to start thinking about buying a new flat because the location and size are two things about a property which cannot really be altered. Suppose a buyer(Mr. Ali) wants to find suitable flats to buy, one for live in and the others for the investment, simultaneously, in the city Z. Mr. Ali has 10 alternatives in the city Z. The alternatives are f₁, f₂, …, f₁₀. Let S = {f₁, f₂, f₃, f₄, f₅, f₆, f₇, f₈, f₉, f₁₀} be the set of flats, and let C = {c₁, c₂} be the set of attributes related to the flats in S, where,

c₁ stands for the Location of the flat,

We characterize these attributes into further more characteristics, which are represented by 1, 2 and 3.

The “Location” of the flat include, close to workplace, close to the city center and near to main road.

2. Selection of employees for promotion and bonus: The selection of employees for promotion and bonus is a multi-criteria decision-making problem. Suppose that a state department holds a meeting with higher officials of the concerned ministry for the selection of employees to promote and bonus, technically called departmental promotion committee (DPC). As per rules of promotion, 5E, i.e,. There are two selectors, one is representative of the director general of the department and the other represents the ministry under which department in functioning. The selection is made on the basis of evaluation reports submitted by the immediate bosses of the employees under consideration. Suppose that the set of employees under consideration is S = {e₁, e₂, e₃, e₄, e₅, e₆} and the set of attributes related to employees characteristics is Q = {r₁, r₂}, where

r₁ stands for the Personal characteristics,

3: Selection of an employee in an organization: The selection of an employee in an organization is the job of human resource department(shortly, HR). Depending on the nature of job, suitable candidates are selected out of many aspirants. It is very important to make sure that the candidate will fit in with your organization’s culture and the team. Suppose a multinational company XYZ wants to hire a manager. The HR department wants to select the most capable, effective, suited, experienced and qualified, person for the post of manager. Suppose S = {c₁, c₂, c₃, c₄, c₅} be the set of candidates who appear for the interview and T = {t₁, t₂, t₃, t₄} be the set of parameters related to the candidates of S, where, squot₁’ stands for the Personal Characteristics, squot₂’ stands for the Business Characteristics, squot₃’ stands for the Communication Qualities, squot₄’ stands for the Relationship Qualities. We give further characteristics of these parameters.