Modal-style operators on fuzzy soft sets and their application to decision making

Abstract

The hybrid soft sets such as, fuzzy soft sets and soft fuzzy rough sets, have been regarded as mathematical tools for handling uncertainties. The aim of this paper is to develop a novel decision making approach for fuzzy soft sets. The modal-style operators of formal context are introduced to fuzzy soft sets and some basic properties of these operators are discussed in detail. A novel fuzzy soft set based decision making approach is presented by using modal-style operators. Further, some shortcomings of an existing decision making method have been highlighted and overcome by the proposed decision making approach. Some numerical examples are employed to show the effectiveness of the approach presented in this study.

Keywords

Soft fuzzy rough set fuzzy soft set modal-style operator decision making

1 Introduction

Classical mathematical tools are not always successful in dealing with complex problems involved in uncertainty, imprecision and vagueness. In 1999, Molodtsov [4] initiated the theory of soft sets as a mathematical tool for dealing with uncertainties, which is not affected by the difficulties of existing methods. The soft set theory is different from traditional tools for dealing with uncertainties, such as fuzzy set theory [22] and rough set theory [33], is that it is free from the inadequacy of the parametrization tools of these theories. Afterwards, the generalized models of soft sets (hybrid soft sets) come forth rapidly and there has been an increasing interest in the practical applications of hybrid soft set theories [7 , 20], especially with regard to their applications in decision making [7 , 19].

Decision making is considered a cognitive-based human activity for selecting the best alternative. The decision making under soft environment often requires decision makers to provide evaluation information about the criteria and the alternatives with a hybrid soft set. The combinations of soft sets with generalized fuzzy sets are typical models of hybrid soft sets. In this direction, Maji et al. [23, 24] extended the theory of soft sets to the fuzzy soft sets and intuitionistic fuzzy soft sets respectively. Jiang et al. [30] presented the idea of interval valued fuzzy soft sets by combining soft sets with the interval-valued fuzzy sets. In terms of fuzzy soft set based decision making methods, Roy and Maji [1] provided a comparison score based method for decision making under fuzzy soft environment. In this approach, we compare the membership values of two objects with respect to a common attribute to determine which one relatively possesses that attribute. This idea is implemented by introducing the notions of comparison table and comparison score of an object. By means of level soft sets, Feng et al. [8] presented an adjustable approach to fuzzy soft set based decision making. In [34], a computational tool called D-score table is introduced to improve the fuzzy soft set based decision process of a classical approach and its convenience has been proved when attributes change across the decision process. In addition, a novel adjustable approach based on decision rules is introduced. Recently, decision making under fuzzy soft environment by using some novel aggregation operators were extensively studied. The aggregation operators are used to aggregate all the preferences of the decision maker into a collective value and hence to find a desirable alternative(s) according to the score values. Garg et al. [12] proposed the notion of fuzzy number intuitionistic fuzzy soft sets and discussed some basic operations on them. In [13], dual hesitant fuzzy soft weighted averaging and geometric operators on dual hesitant fuzzy soft sets are proposed and a multi-criteria decision making approach is presented based on these operators. Meanwhile, Garg and Arora [15] proposed generalized intuitionistic fuzzy soft power aggregation operator based on t-norm and discussed their application in multi-criteria decision making. Robust aggregation operators for multi-criteria decision making with intuitionistic fuzzy soft set environment was studied [14]. Peng and Garg [29] proposed some distance measure, similarity measure and entropy for interval-valued fuzzy soft sets. Three decision making algorithms for interval valued fuzzy soft sets are introduced by using these information measures. Arora and Garg [25] presented some novel correlation coefficients for measuring the relationship between two dual hesitant fuzzy soft set and proposed a multi-criteria decision making method based on the proposed correlation coefficients.

The hybrid soft set model involving soft sets and rough sets was initiated by Feng et al. [7, 9] and have been extensively studied [6 , 21]. In [9], Feng introduced the notions of soft rough sets and soft rough fuzzy sets. The key point is that a soft set instead of an equivalence relation is used to granulate the universe of discourse. Afterwards, Meng et al. [6] proposed the notion of soft fuzzy rough set, in which model a fuzzy soft set is employed to granulate the universe of discourse. Shabir et al. [21] introduced a kind of modified soft rough set, which has already been extended to fuzzy soft sets and Z-soft rough fuzzy sets was proposed [16]. Liu et al. [32] made a review and a comparative study of some existing soft fuzzy rough set models. Accordingly, some decision making approaches for fuzzy soft sets by using soft rough approximation operators and soft rough fuzzy approximation operators are presented [10 , 16–18]. Ma [28] made a review and comparative study of these decision making methods in detail. Sun [2] proposed a novel model of soft fuzzy rough set on the basis of double universe fuzzy rough sets. A novel approach to decision making problem is presented.

The theory of soft set and formal concept analysis (FCA) [26] have the similar initial data description. From a mathematical point of view, the notions of soft set and formal context are equivalent. The modal-style operators [11] (sufficiency operator, necessity operator and possibility operator) on formal context have a precise description of attributes possessed by objects, and thus can be used to evaluate the priority order of objects. We noticed that the existing decision making approaches for fuzzy soft sets can be roughly divided into two categories: one is based on some specific aggregation operators which are used to aggregate all the preferences to obtain the score values of alternatives. The other is based on some particular soft rough approximations. To the best of our knowledge, little work has been done on fuzzy soft sets based decision making approach by using modal-style operators. Additionally, the relationships among soft rough approximation operators and derivation operators used in FCA are also interesting issues to be addressed. In this paper, we conduct this issue and present the approaches for fuzzy soft sets based decision making by modal-style operators. Furthermore, we make a theoretic analysis of an existing decision making approach and point out that it suffers from some limitations. Some numerical examples are employed to show the effectiveness of the approach presented in this study.

The rest of the paper is classified as follows: For convenience of discussion, we first recall some notation and fundamental definitions related to fuzzy sets, rough sets and soft sets in Section 2. In Section 3, we make a theoretic analysis of soft rough fuzzy sets on basis of decision making in detail and use a numerical example to illustrate the limitations of the approach. Then, in Section 4, we present a novel concept of modal-style operators on fuzzy soft sets and discuss their application to decision making problem. In Section 5, some numerical examples are shown for comparative analysis in practical applications to proof the effectiveness of the approach presented in this study.

2 Preliminaries

This section proposes a brief review of a few fundamental concepts of fuzzy sets [22], soft sets [4], fuzzy soft sets [23] and rough sets [33].

Fuzzy set theory [22] offers a suitable framework for representing and processing fuzzy concepts according to permit partial memberships. Assumed that U is a nonempty set and is said to be universe. A fuzzy set θ on U is termed to be a membership function θ : U → [0, 1]. For h ∈ U, the membership value θ (h) actually specifies the degree where h pertains to the fuzzy set θ. In the following, P (U) and F (U) define respectively the family of all subsets of U and the family of all fuzzy sets of U. For any θ, η ∈ F (U), θ is termed to be a fuzzy subset of η, expressed by θ ⊆ η, if θ (h) ≤ η (h) for all h ∈ U. Clearly, θ = η if both θ ⊆ η and η ⊆ θ, i.e. θ (h) = η (h) for every h ∈ U.

Molodtsov [4] initiated the definition of soft sets in 1999. Assume that U is the universe set and E is the set of parameters associated with U like, attributes, properties, or characteristics of objects in U. (U, E) will be known as a soft space. According to [4], the concept of soft sets is presented as follows:

Definition 1. [4] A pair (f, S) is termed to be a soft set on U if S ⊆ E and f : S → P (U).

Videlicet, soft set in Definition 1 is a parameterized family of subsets of universe set. For s ∈ S, f (s) is treated as the set of s-approximate elements of the soft set (f, S).

Molodtsov [4] noted that Zadeh’s fuzzy set is treated as a special case of the soft set. Assume that θ is a fuzzy set on U. For β ∈ [0, 1], $f (β) = {h \in U; θ (h) \geq β}$ (1) is referred to as the β-level set of θ. If we realize the family {f (β) ; β ∈ [0, 1]}, we can compute θ (h) through the formula $θ (h) = \sup_{h \in f (β)} β$ , that is to say, θ = ∪ _β∈[0,1]β · f (β). This observation is usually generalized according to a representation theorem in fuzzy set theory, and the theorem establishes a one-to-one correspondence between a fuzzy set and a family of crisp sets fulfilling some conditions. Therefore, fuzzy set θ is treated as the soft set (f, [0, 1]).

Maji et al. [23] researched hybrid structures with respect to both fuzzy sets and soft sets. The definition of fuzzy soft sets was presented as a fuzzy generalization of soft sets.

Definition 2. [23] Assume that (U, E) is a soft space. A pair (g, S) is termed to be a fuzzy soft set on U if S ⊆ E and g : S → F (U).

In Definition 2, fuzzy sets on the universe U replace the crisp subsets of U. Thus, each soft set is regarded as a fuzzy soft set.

The rough set theory was shown by Pawlak [33]. The basic thought of the rough set theory contains two parts. In the first part, it shows concepts and rules according to the classification of relational database. In the second part, it discoveries knowledge by classifying the equivalence relation and the approximation of the target. Assume that R is equivalence relation on the universe of discourse U. The pair (U, R) is termed to be a Pawlak approximation space [33]. A partition U/R = {[h] _R ; h ∈ U} on U will be generated by R, where [h] _R is the equivalence class in regard to R containing h. Blocks (concepts) of rough approximations are built by these equivalence classes. For every H ⊆ U, the lower approximation $\underline{R} (H)$ and upper approximation $\bar{R} (H)$ [33] of H in regard to (U, R) are defined as

$\underline{R} (H) = {h \in U; [h]_{R} \subseteq H}$ (2) $\bar{R} (H) = {h \in U; [h]_{R} \cap H \neq \emptyset}$ (3)

H is termed to be definable in (U, R) if $\underline{R} (H) = \bar{R} (H)$ ; otherwise H is said to be a rough set. Therefore, a rough concept is characterized by several accurate concepts, scilicet, its lower approximation and upper approximation [33] in rough set theory.

According to take the place of the equivalence relation through a fuzzy arbitrary relation or an arbitrary relation, various generalized models of Pawlak rough set are build, see Ref [3 , 28]. Dubois [3] proposed the lower and upper approximations of fuzzy sets in Pawlak approximation space and fuzzy approximation space, and gained new concepts of rough fuzzy sets and fuzzy rough sets.

Definition 3. [3] Assume that (U, R) is a Pawlak approximation space. The lower and upper rough approximations of θ ∈ F (U) in (U, R) are marked as $\underline{R} (θ)$ and $\bar{R} (θ)$ , respectively, which are fuzzy subsets on U remarked as: $\underline{R} (θ) (h) = \land {θ (o); o \in [h]_{R}}$ (4) $\bar{R} (θ) (h) = \lor {θ (o); o \in [h]_{R}}$ (5) for every h ∈ U. $\underline{R} (θ)$ and $\bar{R} (θ)$ are known as rough fuzzy approximation operators.

Wu [27] extended fuzzy approximation space to generalized fuzzy approximation space on double universe. The generalized fuzzy rough approximation operators [27] on basis of double universe is defined in the following way:

Definition 4. [27] Supposed that U and V are two universes of discourse and R is a fuzzy relation from U to V. (U, V, R) is termed to be a generalized fuzzy approximation space. For each θ ∈ F (V), the lower and upper rough approximations of θ in (U, V, R) are marked as $\underline{R} (θ)$ and $\bar{R} (θ)$ , which are fuzzy subsets in U defined by: $\underline{R} (θ) (h) = \land_{o \in V} ((1 - R (h, o)) \lor θ (y))$ (6) $\bar{R} (θ) (h) = \lor_{o \in V} (R (h, o) \land θ (o))$ (7) for every h ∈ U. $\underline{R} (θ)$ and $\bar{R} (θ)$ are termed to be generalized fuzzy rough approximation operators on basis of double universe. The pair $(\underline{R} (θ), \bar{R} (θ))$ is said to be a generalized fuzzy rough set on basis of double universe.

Recently, Sun et al. [2] established the concept of soft fuzzy approximation operators and introduced the concept of pseudo soft set over the universe.

Definition 5. [2] Assume that (U, S) is a soft space. A pair (g^-1, S) is said to be a pseudo fuzzy soft set on U, if g^-1 : U → F (S) is a mapping from U into the set of all fuzzy subsets of S.

Intuitively, a pseudo fuzzy soft set on U is a family of fuzzy subsets of the parameters set S. For any o ∈ U, g^-1 (o) is treated as the set of u-approximate parameters. g^-1 (o) (s) is the degree with which u posses the attribute s. From a mathematical point of view, the definitions of pseudo fuzzy soft set and fuzzy soft set are equivalent. In fact, if (g, S) is a fuzzy soft set on U, (g^-1, S) is a pseudo fuzzy soft set on U where g^-1 : U → F (S) is given by g^-1 (o) (s) = g (s) (o) for each o ∈ U and s ∈ S. Conversely, assume that (g^-1, S) is a pseudo fuzzy soft set [12] on U. (g^-1, S) determines a fuzzy soft set (g, S) on U by g (s) (o) = g^-1 (o) (s) for each o ∈ U and s ∈ S.

Definition 6. [2] Assume that (U, S) is a soft space and (g^-1, S) is a pseudo fuzzy soft set on U. (U, S, g^-1) is termed to be the soft fuzzy approximation space. For every θ ∈ F (S), the lower approximation $\underline{g} (θ)$ and upper approximation $\bar{g} (θ)$ of θ concerning the soft fuzzy approximation space (U, S, g^-1) are fuzzy sets of U whose membership functions are defined, respectively, by

$\underline{g} (θ) (o) = \land_{s \in S} ((1 - g^{- 1} (o) (s)) \lor θ (s))$ (8) $\bar{g} (θ) (o) = \lor_{s \in S} (g^{- 1} (o) (s) \land θ (s))$ (9)

The pair $(\underline{g} (θ), \bar{g} (θ))$ is termed to be a soft fuzzy rough set and $\underline{g}$ and $\bar{g} : F (S) \to F (U)$ are termed to be a lower and upper soft fuzzy rough approximation operators, respectively.

In this definition, a fuzzy subset on E is approximated by two fuzzy subsets on the universe U.

Formal concept analysis (FCA) [26] provides a method of knowledge description and summarization. Formal concept analysis is calculated on basis of the concept of a formal context specifying which objects owns what properties or attributes. It is noted that soft set theory is closely related to formal concept analysis [26]. In fact, soft set theory and the concept lattices theory have the similar basis data description. From a mathematical point of view, the concepts of soft set and formal context are equivalent.

Definition 7. [26] A formal context (G, M, R) composed of G, M and a relation R between G and M. The elements of G are termed to be the objects and the elements of M are termed to be the attributes of the context. (k, e) ∈ R show that the object k has the attribute e, or the attribute e is possessed by the object k.

Assume that (G, M, R) is a formal context. For K ⊆ G, E ⊆ M, Wille [26] proposed the following derivation operators:

$K^{↑} = {e \in M; \forall k \in K ((k, e) \in R)}$ (10) $E^{↓} = {k \in G; \forall e \in E ((k, e) \in R)}$ (11)

Duntsch [11] introduced a pair of modal-style approximate operators ◊, □ on formal context (G, M, R) as follows: for O ⊆ G, E ⊆ M, $K^{◊} = {e \in M; \exists k \in K ((k, e) \in R)}$ (12) $K^{□} = {e \in M; \forall k \in G ((k, e) \in R \to k \in K)}$ (13) $E^{◊} = {k \in G; \exists e \in E ((k, e) \in R)}$ (14) $E^{□} = {k \in G; \forall e \in M ((k, e) \in R \to e \in E)}$ (15)

3 An argument on soft rough fuzzy sets based decision making

Since Molodtsov [5] initiated the theory of soft set, all kinds of generalized soft set models have been applied to handing decision making problems. Feng [9] researched hybrid structures with respect to both soft sets and rough sets. The notions of soft rough set and soft rough fuzzy set were proposed and their basic properties were discussed. Recently, soft rough set and soft rough fuzzy set on basis of decision making methods are growing very rapidly. Ma [28] made a review and comparative study of these methods in detail. Sun [2] proposed a novel model of soft fuzzy rough set on basis of double universe fuzzy rough sets. Accordingly, a novel approach to decision making problem is presented.

Algorithm 1. [2]

Give the fuzzy soft set (f, A) (or the pseudo fuzzy soft set (f^-1, A)).

Calculate the optimum normal decision object θ ∈ F (A) given by $θ (e) = \max_{o \in U} f (e) (o)$ for each e ∈ A.

Calculate the soft fuzzy rough lower approximation $\underline{f} (θ)$ and soft fuzzy rough upper approximation $\bar{f} (θ)$ .

Calculate the choice value σ (o) for every o ∈ U, where $σ (o) = \underline{f} (θ) (o) + \bar{f} (θ) (o)$ .

5.The optimal decision is to choose l ∈ U if $σ (h) = \max_{o \in U} σ (o)$ .

If l has multiple values, then any one may be selected.

The key points of this algorithm are Step 2, Step 3 and Step 4. Note that for the evaluation problem of a ceratin objects, it is desirable to find a decision making plan with the largest possible evaluation value in the universe. Thus, in Step 2, an optimum (ideal) decision θ is constructed. It is a fuzzy set on E and the membership degree θ (e) is the maximum value of f (e) (o) for each o ∈ U. In Step 3, the soft fuzzy rough approximations of θ are computed and accordingly the decision values of each object are computed in Step 4. For illustrate the basic ideal of Algorithm 1 [2], we provide an illustrate example as follows:

Example 1 [2] Assume that U = {v₁, v₂, v₃, v₄, v₅, v₆} is a set of six houses, and E = {s₁, s₂, s₃, s₄, s₅, s₆, s₇} is the set of parameters that Mr.X(say) is interested in buying a house, where s₁, s₂, s₃, s₄, s₅, s₆ and s₇ stand for "expensive", "beautiful", "wooden", "cheap", "in the surroundings", "modern" and "in good repair" respectively. The tabular representation of the fuzzy soft set was given in Table 1. Namely, in the available houses of U, Mr.X wants to buy houses that extremely conform to the attributes in S.

Table 1
Table for fuzzy soft set (f, A)

s ₁ s ₂ s ₃ s ₄ s ₅ s ₆ s ₇ $\underline{f} (θ)$ $\bar{f} (θ)$ σ (v)

v ₁ 0.3 0.1 0.4 0.4 0.1 0.1 0.5 0.6 0.5 1.1

v ₂ 0.3 0.3 0.5 0.1 0.3 0.1 0.5 0.5 0.5 1.0

v ₃ 0.4 0.3 0.5 0.1 0.3 0.1 0.6 0.5 0.6 1.1

v ₄ 0.7 0.4 0.2 0.1 0.2 0.1 0.3 0.6 0.7 1.3

v ₅ 0.2 0.5 0.2 0.3 0.5 0.5 0.4 0.5 0.5 1.0

v ₆ 0.3 0.5 0.2 0.2 0.2 0.3 0.3 0.5 0.5 1.0

	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇	$\underline{f} (θ)$	$\bar{f} (θ)$	σ (v)
v ₁	0.3	0.1	0.4	0.4	0.1	0.1	0.5	0.6	0.5	1.1
v ₂	0.3	0.3	0.5	0.1	0.3	0.1	0.5	0.5	0.5	1.0
v ₃	0.4	0.3	0.5	0.1	0.3	0.1	0.6	0.5	0.6	1.1
v ₄	0.7	0.4	0.2	0.1	0.2	0.1	0.3	0.6	0.7	1.3
v ₅	0.2	0.5	0.2	0.3	0.5	0.5	0.4	0.5	0.5	1.0
v ₆	0.3	0.5	0.2	0.2	0.2	0.3	0.3	0.5	0.5	1.0

By direct computation, the ideal normal decision object θ ∈ F (A) is given by θ = 0.7/_{s
₁} + 0.5/_{s
₂} + 0.5/_{s
₃} + 0.4/_{s
₄} + 0.5/_{s
₅} + 0.5/_{s
₆} + 0.6/_{s
₇}.

The lower approximation $\underline{f} (θ)$ , upper approximation $\bar{f} (θ)$ and the choice values σ (o) are given in Table 1. From this table, it is obviously that maximum value is 1.3, scored by v₄ and the optimal decision is to choose v₄.

However, Algorithm 1 may not successfully solve a few decision making problems. Now, we give the following example to illustrate and analyze.

Example 2. Take into account the following fuzzy soft set (g, A) given in Table 2. Compared with the fuzzy soft set (f, A) in Example 1, we notice that g (s_j) = f (s_j) for 2 ≤ j ≤ 7 and g (s₁) (v_i) = f (s₁) (v_i) for i ≠ 5. Furthermore, g (s₁) (v₅) =0.7 and f (s₁) (v₅) =0.2. By direct computation, $\underline{g} (θ)$ , $\bar{g} (θ)$ and the choice values are given in Table 2.

Table 2

Table for fuzzy soft set (g, A)

	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇	$\underline{g} (θ)$	$\bar{g} (θ)$	σ (v)
v ₁	0.3	0.1	0.4	0.4	0.1	0.1	0.5	0.6	0.5	1.1
v ₂	0.3	0.3	0.5	0.1	0.3	0.1	0.5	0.5	0.5	1.0
v ₃	0.4	0.3	0.5	0.1	0.3	0.1	0.6	0.5	0.6	1.1
v ₄	0.7	0.4	0.2	0.1	0.2	0.1	0.3	0.6	0.7	1.3
v ₅	0.7	0.5	0.2	0.3	0.5	0.5	0.4	0.5	0.7	1.2
v ₆	0.3	0.5	0.2	0.2	0.2	0.3	0.3	0.5	0.5	1.0

By the decision making Algorithm 1, v₄ is the optimal decision. On the other hand, we notice that g (s) (v₅) ≥ g (s) (v₄) for each s ∈ A. That is, v₅ is superior to v₄ with respect to each attribute s ∈ A, but the choice value of v₅ is smaller that it of v₄. Thus, it seems that the decision method [2] is not reasonable in this case. It’s worth noting that the optimal decision calculated through Maji’s the decision method is also v₅.

As was mentioned above, it is claimed that the rough lower approximation $\underline{f} (θ)$ and rough upper approximation [2] $\bar{f} (θ)$ are two most close to the approximated sets of the universe. So, $\underline{f} (θ) (v)$ and $\bar{f} (θ) (v)$ are two most close values to the decision alternative v. Accordingly, $σ (v) = \underline{f} (θ) (v) + \bar{f} (θ) (v)$ is taken as the choice value for object v. However, it seems that this is not the case.

By the definition, the optimum decision objective is a fuzzy set θ ∈ F (A) given by $θ (s) = \max_{v \in U} f (v) (s)$ for each s ∈ A. It follows that θ (s) ≥ f (s) (v) for each v ∈ U and consequently,

$\bar{f} (θ) (v) = \lor_{s \in E} (f^{- 1} (v) (s) \land θ (e)) = \lor_{s \in E} (f (s)$ (v) ∧ θ (e)) = ∨ _s∈Ef (s) (v) .

That is, $\bar{f} (θ) (v)$ is the maximum value of the row with respect to v.

For the lower approximation $\underline{f} (θ)$ , if v, y ∈ U such that v is superior than y for each attribute s ∈ A, i.e., f (s) (v) ≥ f (s) (y) for all s ∈ A, then

$\underline{f} (θ) (v) = \land_{s \in E} ((1 - f^{- 1} (v) (s)) \lor θ (s)) = \land_{s \in E} ((1 - f (s) (v)) \lor θ (s)) \leq \land_{s \in E} ((1 - f (s) (y)) \lor θ (s)) = \underline{f} (θ) (y) .$

Therefore, if we make decision partially based on $\underline{f} (θ)$ , we will get contradictory results, i.e., if v is superior than y associated with each attribute s ∈ A, then $\underline{f} (θ) (v) \leq \underline{f} (θ) (y)$ . Thus, take $\underline{f} (θ)$ as a most close set to the approximated set is not reasonable.

If v, y ∈ U such that v is superior than y with respect to each attribute s ∈ A and ∨_s∈Ef (s) (v) = ∨ _s∈Ef (s) (y), then, by (1) and (2), the choice value

$σ (v) = \underline{f} (θ) (v) + \bar{f} (θ) (v) \leq \underline{f} (θ) (y) + \bar{f} (θ) (y) = σ (y) .$

Thus, in this case, the decision method presented by Sun [2] is not reasonable. Especially, it will leads to contradictory results by using the lower approximation operator.

4 Using modal-style operators on fuzzy soft sets based decision making

As with most soft set decision problems, the existing results depend on the evaluation of all the decision alternations. A number of these problems are substantially humanistic and therefore subjective in nature. To avoid the influence of subjective factors, we should use data information as much as possible to replace subjective information. The theory of soft set and formal concept analysis (FCA) [26] have the similar initial data description. From a mathematical point of view, the notions of soft set and formal context are equivalent. The modal-style operators [11] (sufficiency operator, necessity operator and possibility operator) on formal context have a precise description of attributes possessed by objects, and thus can be used to evaluate the priority order of objects. Now, we apply the formal concept analysis [26] to the decision making analysis of fuzzy soft sets.

4.1 Modal-style operators on fuzzy soft sets

Different types of data relationships and structures can be defined, represented, and analyzed by using modal-style operators [11]. So, these modal-style data operators offer an unified approach to check, describe, and construct different types of knowledge. In this paper, we just consider modal-style operators based attributes.

Assume that S = (F, A) is a soft set on U. Defining the binary relation R_S on U and A as (o, a) ∈ R_S if and only if o ∈ f (a). Then (U, A, R_S) is a formal context. Conversely, each formal context determines a soft set in a similar way. For each L ⊆ A, by L^↓ = {g ∈ G ; ∀ m ∈ L ((g, m) ∈ R)} and L^◊ = {g ∈ G ; ∃ m ∈ L ((g, m) ∈ R)}, we know that ↓ and ◊ represent the two extremely cases in describing a set of attributes based on the related objects, namely, an object owns all properties in H and at least one property in H respectively:

o ∈ L^↓ ⇒ ∀ y ∈ M (y ∈ L → (o, y) ∈ R)

o ∈ L^◊ ⇒ ∃ y ∈ M (y ∈ L ∧ (o, y) ∈ R)

The pair (L^↓, L^◊) with L^↓ ⊆ L^◊ therefore offers a characterization of L in terms of objects. Alternatively, we have the following interpretation:

If o ∈ L^↓, y ∈ L, then o certainly has the property y;

If o ∈ L^◊, y ∈ L, then o probably has the property y.

Based on the above discussion, we consider the modal-style operators on fuzzy formal context for dealing with fuzzy soft sets. Assume that U is the universe set, E is the set of all possible parameters under consideration associated with U and (f, S) is a fuzzy soft set. (f, S) induces a fuzzy formal context (U, S, R) where R : U × S → [0, 1] is a fuzzy relation between U and S given by R (o, a) = f (a) (o) for each o ∈ U and a ∈ S. For any θ ∈ F (S), θ^↓ ∈ F (U) and θ^◊ ∈ F (U) are defined by: for each x ∈ U,

$θ^{↓} (o) = \land_{a \in S} (θ (a) \to R (o, a))$ (16) $θ^{◊} (o) = \lor_{a \in S} (θ (a) \otimes R (o, a))$ (17) where → is a fuzzy implication operator and ⊗ is a t-norm on [0, 1].

Since dependencies between the attributes can be described by implications, modal-style operators with respect to fuzzy soft set were constructed by t-norm and fuzzy implication operator. In fuzzy formal context, R (x, a) represents the degree to which x owning the attribute a, where the value R (x, a) is the number in [0,1]. For illustration, we will give an example.

Example 3 Take into account the following fuzzy soft set (k, A) given in Table 3. For simplicity, we use →_G(Godel) as fuzzy implication operator and $θ (s) = \max_{v \in U} k (s) (v)$ for each s ∈ A. The calculation results are given in Table 3, where p ⊗ _Gq = p ⋀ q and $p \to_{G} q = {\begin{matrix} 1, \\ q . \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} p \leq q . \\ p > q . \end{matrix}$

Table 3

Table for fuzzy soft set (k, A)

	s ₁	s ₂	s ₃	s ₄	θ ^↓	θ ^◊
v ₁	0.7	0.4	0.2	0.6	0.2	0.6
v ₂	0.3	0.5	0.5	0.3	0.3	0.5
v ₃	0.4	0.5	0.2	0.4	0.4	0.5

By direct computation, θ ∈ k (A) is given by θ = 0.7/s₁ + 0.5/s₂ + 0.5/s₃ + 0.6/s₄. According to the calculation formula (16) and (17), we have

θ^↓ (v₁) = ⋀ _{s_i∈A} (θ (s_i) → R (v₁, s_i)) = ⋀ _{s_i∈A} (θ (s_i) →k (v₁, s_i))

= (0.7 → 0.7) ∧ (0.5 → 0.4) ∧ (0.5 → 0.2) ∧ (0.6 → 0.6)

= 1∧0.4 ∧ 0.2 ∧ 1 =0.2 ;

θ^◊ (v₁) = ⋁ _{s_i∈A} (θ (s_i) ⊗ R (v₁, s_i)) = ⋁ _{s_i∈A} (θ (s_i) ⊗ k (v₁, s_i))

= (0.7 ⊗ 0.3) ⋁ (0.5 ⊗ 0.4) ⋁ (0.5 ⊗ 0.2) ⋁ (0.6 ⊗ 0.6)

= 0.3 ⋁ 0.4 ⋁ 0.2 ⋁ 0.6 = 0.6 .

Similarity, we also have θ^↓ (v₂) =0.3, θ^↓ (v₃) =0.4 and θ^◊ (v₂) =0.5, θ^◊ (v₃) =0.5 . For further understanding of θ^↓ and θ^◊, some basic properties of θ^↓ and θ^◊ are listed as follows:

If θ₁, θ₂ ∈ F (A) and θ₁ ⊆ θ₂, then $θ_{2}^{↓} \subseteq θ_{1}^{↓}$ and $θ_{1}^{◊} \subseteq θ_{2}^{◊}$ .

If there exists a ∈ A such that θ (a) =1, then θ^↓ ⊆ θ^◊.

For any θ₁, θ₂ ∈ F (A), $(θ_{1} \cup θ_{2})^{↓} = θ_{1}^{↓} \cap θ_{2}^{↓}$ and $(θ_{1} \cup θ_{2})^{◊} = θ_{1}^{◊} \cup θ_{2}^{◊}$ .

4.2 An approach using modal-style operators

In this subsection, we propose the following algorithm for handing with fuzzy soft set on basis of decision making problem by using modal-style operators.

Algorithm 2. A modal-style operator based decision making algorithm:

Give the fuzzy soft set (f, A)

Calculate θ^↓ and θ^◊.

Calculate the choice value σ (x) for each x ∈ U, where σ (x) = θ^↓ (x) + θ^◊ (x).

The decision is y ∈ U if $σ (y) = \max_{x \in U} σ (x)$ .

If x has multiple values, then any one may be selected.

Noted that in the step 2 of the above algorithm, the optimum decision represents maximum membership degree that the object set has an attribute. Hence, it will be made by

θ (s_i) = max {f^-1 (v_j) (s_i) |v_j ∈ U, s_i ∈ A},

where f^-1 (v_j) (s_i) is pseudo fuzzy soft set.

Now, we apply the above algorithm to the fuzzy soft set (g, A) described by Table 2. Let →_G(Godel) be fuzzy implication operator. By the step 2, we have optimum decision θ = 0.7/s₁ + 0.5/s₂ + 0.5/s₃ + 0.4/s₄ + 0.5/s₅ + 0.5/s₆ + 0.6/s₇. By simple calculation, we get

θ^↓ = 0.1/v₁+ 0.1/v₂ + 0.1/v₃ + 0.1/v₄ + 0.2/v₅ + 0.2/v₆ ;

θ^◊ = 0.5/v₁ + 0.5/v₂ + 0.6/v₃ + 0.7/v₄ + 0.7/v₅ + 0.5/v₆ .

Afterwards, we compute the related choice values σ = 0.6/v₁ + 0.6/v₂ + 0.7/v₃ + 0.8/v₄ + 0.9/v₅ + 0.7/v₆. It is obviously that the maximum choice value is 0.9, and v₅ is chosen as the optimal alternative.

The basic thought behind Algorithm 2 is to handle fuzzy soft set on basis of decision making problems by modal-style operators. By the formulas (16) and (17), we can compute θ^↓ and θ^◊ associated with fuzzy soft set (f, A). Similar to the case of L^↓ and L^◊ on formal context, θ^↓ and θ^◊ are equivalent to two extreme value on fuzzy formal context. Thus, we use θ^↓ and θ^◊ to compute the optimal choice value such that the decision-making results are more reasonable. Last, we choose the object y ∈ U with the maximum choice value as the optimum decision for handing with decision making problem. As was mentioned above, if there are some objects with the same maximum selection value, one of them is randomly selected as the optimal decision.

5 An illustrative example with comparative analysis

With the rapid development of the theory of soft set, the application of fuzzy soft sets in handing decision making problems has attracted many researchers’attention. In the previous section, we show a decision making method for fuzzy soft set by modal-style operators. The related decision making algorithm 2 is presented. To verify the feasibility and superiority of the decision making method proposed in this paper, we make a comparative study of Roy et al.’s algorithm, Feng’s algorithm, Zhan’s algorithm and Algorithm 2.

Roy et al. [1] initiated the following algorithm to handle fuzzy decision making problems on basis of fuzzy soft sets.

Algorithm 3.

Give the fuzzy soft set (f, S), (g, T) and (d, J).

Give the attribute set P as observed by the observer.

Calculate the corresponding resultant fuzzy soft set (x, P) from the fuzzy soft sets (f, S), (g, T) and (d, J) and lay it in tabular form.

Construct the comparison table of the fuzzy soft set (x, P) and calculate r_i and t_i for v_i, ∀i.

Calculate the score q_i = r_i - t_i of v_i, ∀i.

The decision object is v_k if $q_{k} = \max_{i} q_{i}$

If k has multiple values, then any one of v_k may be chosen.

Roy et al. [1] thought the object recognition problem is treated as a multi-observer decision making problem in which the final object identification stems from the input set of different observes who offer all object characterization through different parameter sets. The above-mentioned Algorithm 3 uses fuzzy soft set method to handle the recognition problem. However, Kong [34] argued that the above method was not correct since the decision result obtained by using “the score based method” is not always the object with the maximum choice value. To illustrate this point, a concrete example was given in the following:

Exmple 4. [34] Take into account the following fuzzy soft set (l, D) given in Table 4.

Table 4
Table for fuzzy soft set (l, D)

t ₁ t ₂ t ₃ t ₄ t ₅

v ₁ 0.1 0.5 0.3 0.4 0.3

v ₂ 0.3 0.5 0.2 0.3 0.6

v ₃ 0.1 0.7 0.4 0.5 0.1

v ₄ 0.7 0.2 0.2 0.2 0.3

v ₅ 0.2 0.6 0.3 0.2 0.3

v ₆ 0.9 0.2 0.1 0.1 0.8

	t ₁	t ₂	t ₃	t ₄	t ₅
v ₁	0.1	0.5	0.3	0.4	0.3
v ₂	0.3	0.5	0.2	0.3	0.6
v ₃	0.1	0.7	0.4	0.5	0.1
v ₄	0.7	0.2	0.2	0.2	0.3
v ₅	0.2	0.6	0.3	0.2	0.3
v ₆	0.9	0.2	0.1	0.1	0.8

According to Roy et al.’s algorithm, corresponding choice values and the score for each object are as in Table 5. It is obviously that the maximum score is 6 and so the optimal decision is to choose v₃. But here the idea choice value maxc_i = c₆ in Table 5, then v₆ is the optimal choice object which is contradictory to the result included the algorithm for fuzzy soft set.

Table 5

Table for choice values

	v ₁	v ₂	v ₃	v ₄	v ₅	v ₆
Choice value (c_i)	1.6	1.9	1.8	1.6	1.6	2.1
Score (q_i)	0	3	6	-6	1	-4

Now, we apply Algorithm 2 to fuzzy soft set (l, D). Supposed that →_G(Godel) is fuzzy implication operator and $θ (t) = \max_{v \in U} l (t) (v)$ for each t ∈ D. The optimum decision object θ ∈ l (D) is given by θ = 0.9/_{t
₁} + 0.7/_{t
₂} + 0.4/_{t
₃} + 0.5/_{t
₄} + 0.8/_{t
₅}. By simple calculation with respect to the Algorithm 2, we can get

θ^↓ = 0.1/v₁+ 0.2/v₂ + 0.1/v₃ + 0.2/v₄ + 0.2/v₅ + 0.1/v₆ ;

θ^◊ = 0.5/v₁ + 0.6/v₂ + 0.7/v₃ + 0.7/v₄ + 0.6/v₅ + 0.9/v₆ . It is obviously that σ₆ > σ₄ > σ₂ ≈ σ₃ ≈ σ₅ > σ₁ and we can see that the maximum choice value is 1.0, and v₆ is chosen as the optimal alternative. Similarity, the optimal decision of this algorithm is also v₆. Thus, the algorithm 3 is not suitable in some cases.

Feng [10] presented an adjustable approach to weighted fuzzy soft set on basis of decision making. Now, we simple review the algorithm [10].

Algorithm 4.[10]

Give the weighted fuzzy soft set Γ = (l, D, w).

Give a threshold fuzzy set ρ : D → [0, 1](or input a threshold value ɛ ∈ [0, 1] .; or select the mid-level decision rule; or select the top-level decision rule; or select the weight function decision rule) for decision making.

Calculate the level soft set τ ((l, D) ; ρ) of Γ associated with the threshold fuzzy set ρ (or the t-level soft set τ ((l, D) ; ɛ); or the mid-level soft set τ ((l, D) ; mid); or the top-level soft set τ ((l, D) ; max); or the level soft set τ ((l, D) ; w)).

Propose the level soft set τ ((l, D) ; ρ) (or τ ((l, D) ; ɛ); or τ ((l, D) ; mid); or τ ((l, D) ; max); or τ ((l, D) ; w)) in tabular form and calculate the weighted choice value ${\bar{c}}_{i}$ of v_i, ∀i.

The optimal decision is to choose v_k if ${\bar{c}}_{k} = \max_{i} {\bar{c}}_{i}$

If k has multiple values, then any one of v_k may be chosen.

Feng et al.’s method actually select the objects considering the attributes cooperate with them both in quality and in quantity by introducing levels for membership values. Though Feng’s algorithm has overcome defect of Roy et al.’s algorithm, but it requires the decision maker to select the thresholds in advance according to decision makers. Then the results will rely on the threshold values to some extent. The choice of threshold values depends mostly on the subjective factors of decision makers. Now, we consider the fuzzy soft set (l, D) in Example 4. Assume that a weight function w : D → [0, 1] is given by w₁ = w (t₁) =0.9, w₂ = w (t₂) =0.5, w₃ = w (t₃) =0.6, w₄ = w (t₄) =0.6, w₅ = w (t₅) =0.8. The related weighted fuzzy soft set (l, D, w) is given in Table 6.

Table 6

Table for weighted fuzzy soft set (l, D, w)

	t₁, w₁ = 0.9	t₂, w₂ = 0.7	t₃, w₃ = 0.4	t₄, w₄ = 0.5	t₅, w₅ = 0.8
v ₁	0.1	0.5	0.3	0.4	0.3
v ₂	0.3	0.5	0.2	0.3	0.6
v ₃	0.1	0.7	0.4	0.5	0.1
v ₄	0.7	0.2	0.2	0.2	0.3
v ₅	0.2	0.6	0.3	0.2	0.3
v ₆	0.9	0.2	0.1	0.1	0.8

The weight function w is used as the threshold the threshold to get the level soft set τ ((l, D) ; w), and its tabular representation is shown in Table 7. From Table 7, we obviously have ${\tilde{c}}_{6} > {\tilde{c}}_{3} > {\tilde{c}}_{1} \approx {\tilde{c}}_{2} \approx {\tilde{c}}_{4} \approx {\tilde{c}}_{5}$ . It is obviously that the maximum weighted choice value is ${\tilde{c}}_{6}$ and so the optimal decision is to choose v₆. Although we can get optimal decision by Algorithm 4, we find that ${\tilde{v}}_{1}, {\tilde{v}}_{2}, {\tilde{v}}_{4}$ and ${\tilde{v}}_{5}$ cannot be compared.

Table 7

Tabular representation of the level soft set τ ((l, D) ; w) with weighted choice values.

	t₁, w₁ = 0.9	t₂, w₂ = 0.7	t₃, w₃ = 0.4	t₄, w₄ = 0.5	t₅, w₅ = 0.8	choice value ${\tilde{c}}_{i}$
v ₁	0	0	0	0	0	0
v ₂	0	0	0	0	0	0
v ₃	0	1	1	1	0	1.6
v ₄	0	0	0	0	0	0
v ₅	0	0	0	0	0	0
v ₆	1	0	0	0	0	1.7

Similarity, we apply Algorithm 2 to fuzzy soft set (l, D). Supposed that →_G(Godel) is fuzzy implication operator and $θ (t) = \max_{v \in U} l (t) (v)$ for each t ∈ D. The optimum decision object θ ∈ l (D) is given by θ = 0.9/_{t
₁} + 0.7/_{t
₂} + 0.4/_{t
₃} + 0.5/_{t
₄} + 0.8/_{t
₅}. By simple calculation with respect to the Algorithm 2, the related choice values are listed in Table 8. It is obviously that σ₆ > σ₄ > σ₂ ≈ σ₃ ≈ σ₅ > σ₁ and we can see that the maximum choice value is 1.0, and v₆ is chosen as the optimal alternative. The ideal decision of algorithm 2 is the same as Feng’s algorithm, except that algorithm 2 is more precise. Thus, in some cases, modal-style operator based decision making method uses data information effectively to avoid errors caused by subjective factors.

Table 8

Table for modal-style operators with fuzzy soft set (l, D)

	t ₁	t ₂	t ₃	t ₄	t ₅	θ ^↓	θ ^◊	σ
v ₁	0.1	0.5	0.3	0.4	0.3	0.1	0.5	0.6
v ₂	0.3	0.5	0.2	0.3	0.6	0.2	0.6	0.8
v ₃	0.1	0.7	0.4	0.5	0.1	0.1	0.7	0.8
v ₄	0.7	0.2	0.2	0.2	0.3	0.2	0.7	0.9
v ₅	0.2	0.6	0.3	0.2	0.3	0.2	0.6	0.8
v ₆	0.9	0.2	0.1	0.1	0.8	0.1	0.9	1.0

Recently, Zhan [16] presented a way for decision making problem by using Z-soft fuzzy rough sets. The following is a brief explanation of the basic definitions. See references [12] for details.

Definition 8. Let ℏ = (f^-1, S) be a pseudo fuzzy soft set over U and (U, S, f^-1) be an soft fuzzy approximation space. For each θ ∈ F (U), the lower approximation of θ is remarked as $θ_{\underline{f^{- 1}}}$ and is given by

$θ_{\underline{f^{- 1}}} (o) = \land {θ (v) | v \in U, f^{- 1} (v) (s) = f^{- 1} (o) (s),$ ∀s ∈ S} for any o ∈ U.

Whereas upper approximation of θ is remarked as $θ_{\bar{f^{- 1}}}$ , and is shown as

$θ_{\bar{f^{- 1}}} (o) = \lor {θ (v) | v \in U, f^{- 1} (v) (s) = f^{- 1} (o) (s),$ ∀s ∈ S} for any o ∈ U.

If $θ_{\underline{f^{- 1}}} = θ_{\bar{f^{- 1}}}$ , then θ is said to be Z-soft fuzzy definable; else θ will be called a Z-soft fuzzy rough set. It is obviously that Z-soft fuzzy rough set associated with pseudo fuzzy set keeps the basic property of Pawlak rough sets, namely $θ_{\underline{f^{- 1}}} \subseteq θ \subseteq μ_{\bar{f^{- 1}}}$ by Definition 13. Now, we review the algorithm on basis of the Z-soft fuzzy rough set to handle decision making problem, which is presented by Zhan [17].

Algorithm 5. [16]

Give the pseudo fuzzy soft set (f^-1, S), the (U, S, f^-1) soft fuzzy approximation space and the presented fuzzy set θ.

Calculate $θ_{\underline{f^{- 1}}} (o_{i})$ and $θ_{\bar{f^{- 1}}} (o_{i})$ .

Find the choice value C_i = $\frac{θ_{\underline{f^{- 1}}} (o_{i}) \cdot θ_{\bar{f^{- 1}}} (o_{i})}{1 + (1 + θ_{\underline{f^{- 1}}} (o_{i})) \cdot (1 + θ_{\bar{f^{- 1}}} (o_{i}))}$ .

The result is o_k ∈ U if $C_{k} = \max_{i} C_{i} (i = 1, 2, . . ., | U |) .$

If k has multiple values, then we can choose any value o_k.

The above Algorithm is a feasible method to deal with decision making problem, while some defects are existed in practical application. It is remarkable that the so-called fuzzy set proposed in Step 1 is actually constructed by the decision maker, where its value implies an optimal result of the decision. The decision method is based on lower and upper approximation operators and thus a feasible method in general. However, the following example will verify that Algorithm 5 has some obstacles in the process of practical application in some cases. We give the fuzzy soft set (l, D) in Example 12. It becomes a pseudo fuzzy soft set (l^-1, D), and its tabular representation is shown in Table 9.

Table 9

Table for pseudo fuzzy soft set (l^-1, D)

	v ₁	v ₂	v ₃	v ₄	v ₅	v ₆
t ₁	0.1	0.3	0.1	0.7	0.2	0.9
t ₂	0.5	0.5	0.7	0.2	0.6	0.2
t ₃	0.3	0.2	0.4	0.2	0.3	0.1
t ₄	0.4	0.3	0.5	0.2	0.2	0.1
t ₅	0.3	0.6	0.1	0.3	0.3	0.8

Thus, we have

$l^{- 1} (v_{1}) = \frac{0.1}{t_{1}} + \frac{0.5}{t_{2}} + \frac{0.3}{t_{3}} + \frac{0.4}{t_{4}} + \frac{0.3}{t_{5}};$

$l^{- 1} (v_{2}) = \frac{0.3}{t_{1}} + \frac{0.5}{t_{2}} + \frac{0.2}{t_{3}} + \frac{0.3}{t_{4}} + \frac{0.6}{t_{5}};$

$l^{- 1} (v_{3}) = \frac{0.1}{t_{1}} + \frac{0.7}{t_{2}} + \frac{0.4}{t_{3}} + \frac{0.5}{t_{4}} + \frac{0.1}{t_{5}};$

$l^{- 1} (v_{4}) = \frac{0.7}{t_{1}} + \frac{0.2}{t_{2}} + \frac{0.2}{t_{3}} + \frac{0.2}{t_{4}} + \frac{0.3}{t_{5}};$

$l^{- 1} (v_{5}) = \frac{0.2}{t_{1}} + \frac{0.6}{t_{2}} + \frac{0.3}{t_{3}} + \frac{0.2}{t_{4}} + \frac{0.3}{t_{5}};$

$l^{- 1} (v_{6}) = \frac{0.9}{t_{1}} + \frac{0.2}{t_{2}} + \frac{0.1}{t_{3}} + \frac{0.1}{t_{4}} + \frac{0.8}{t_{5}} .$

We note that for any v_i ≠ v_j, there exists s such that l^-1 (v_i) (t) ≠ l^-1 (v_j) (t). Further, for any θ ∈ l^-1 (U), $θ_{\underline{l^{- 1}}} = θ = θ_{\bar{l^{- 1}}}$ . Thus, the superior order of objects determined by the choice values presented in Algorithm 5 is the same as that determined directly by θ. For example, let $θ = \frac{0.9}{v_{1}} + \frac{0.5}{v_{2}} + \frac{0.6}{v_{3}} + \frac{0.6}{v_{4}} + \frac{0.8}{v_{5}} + \frac{0.7}{v_{6}}$ . By direct computation, we have

$θ_{\underline{l^{- 1}}} = \frac{0.9}{v_{1}} + \frac{0.5}{v_{2}} + \frac{0.6}{v_{3}} + \frac{0.6}{v_{4}} + \frac{0.8}{v_{5}} + \frac{0.7}{v_{6}} = θ;$

$θ_{\bar{l^{- 1}}} = \frac{0.9}{v_{1}} + \frac{0.5}{v_{2}} + \frac{0.6}{v_{3}} + \frac{0.6}{v_{4}} + \frac{0.8}{v_{5}} + \frac{0.7}{v_{6}} = θ .$

By using step 3 of Algorithm 5, we have the choice values C₁ ≈ 0.176, C₂ ≈ 0.077, C₃ = C₄ ≈ 0.101, C₅ ≈ 0.151 and C₆ ≈ 0.126. Thus, the superior order of object is v₁ > v₅ > v₆ > v₃ ≈ v₄ > v₂. On the other hand, by directly using θ, we also have v₁ > v₅ > v₆ > v₃ ≈ v₄ > v₂. Consequently, in this case, there is no need to calculate the lower and upper approximations. We can make decision directly by using the fuzzy set θ given by decision makers. But by Algorithm 2, we can get σ₆ > σ₄ > σ₂ ≈ σ₃ ≈ σ₅ > σ₁ and v₆ is chosen as the optimal alternative. Thus, in some cases, modal-style operator based decision making method uses data information effectively to avoid errors caused by subjective factors.

Through the above comparative analysis, the Algorithm 2 using modal-style operators on fuzzy soft sets on basis of decision making can better handle the practical application in decision making, not only to avert the influence of subjective factors, but also make the calculation result more precise.

6 Conclusions

In this study, we analyzed an existing approach to handling decision making problem and pointed out that it suffers from some limitations. Furthermore, we presented a novel approach for fuzzy soft set based decision making problem by using the modal-operators and proposed the related decision making algorithm. We conduct a comparative study of the existing decision making approaches proposed by Roy et al. [1], Feng [10], Zhan [16] and the approach proposed in this paper through examples. The comparative results indicate that the new approach presented in this study performs best with respect to the accuracy. Thus it is easier to be applied in real life applications. In further research, one can consider to apply modal-style operator on fuzzy soft set to multi-criteria group decision making problem.

Footnotes

Acknowledgments

This work has been supported by the National Natural Science Foundation of China (Grant No. 61473239, 61372187).

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Modal-style operators on fuzzy soft sets and their application to decision making

Abstract

Keywords

1 Introduction

2 Preliminaries

4.1 Modal-style operators on fuzzy soft sets

5 An illustrative example with comparative analysis

Table 4 Table for fuzzy soft set (l, D) t 1 t 2 t 3 t 4 t 5 v 1 0.1 0.5 0.3 0.4 0.3 v 2 0.3 0.5 0.2 0.3 0.6 v 3 0.1 0.7 0.4 0.5 0.1 v 4 0.7 0.2 0.2 0.2 0.3 v 5 0.2 0.6 0.3 0.2 0.3 v 6 0.9 0.2 0.1 0.1 0.8

Footnotes

Acknowledgments

References

Table 4
Table for fuzzy soft set (l, D)

t ₁ t ₂ t ₃ t ₄ t ₅

v ₁ 0.1 0.5 0.3 0.4 0.3

v ₂ 0.3 0.5 0.2 0.3 0.6

v ₃ 0.1 0.7 0.4 0.5 0.1

v ₄ 0.7 0.2 0.2 0.2 0.3

v ₅ 0.2 0.6 0.3 0.2 0.3

v ₆ 0.9 0.2 0.1 0.1 0.8