A novel parameter reduction method for fuzzy soft sets

Abstract

Fuzzy soft set as a tool to deal with uncertainty can effectively handle decision making problems. However, there are many redundant parameters in the decision making process. In order to remove redundant parameters to improve the efficiency of decision making, different parameter reduction algorithms for fuzzy soft sets based on different decision criteria have been proposed. This paper focuses on the problem of parameter reduction of fuzzy soft sets based on choice value criteria. The restrictions of the strict conditions about parameter reduction lead to a very low applicability of some previous algorithms based on choice value criteria. To address this limitation, we introduce a flexible definition of parameter reduction for fuzzy soft sets. Further a difference-based parameter reduction algorithm for fuzzy soft sets is proposed. Compared with some previous algorithms based on choice value criteria, the proposed algorithm not only has wider applicability, but also can reduce more redundant parameters making the found parameter reduction with a lower cardinality, and it is easier to find the parameter reduction of fuzzy soft sets.

Keywords

Soft set fuzzy soft set parameter reduction difference

1 Introduction

With the development of the times, we often encounter many uncertainty problems in real life. Soft set theory [1] is an effective mathematical tool for dealing with uncertainty proposed by Molodtsov, which avoids the shortcomings of other theories dealing with uncertainty such as probability theory, rough set [2], and fuzzy set [3] lacking parametric tools. Fuzzy soft set [4], as the effective extension of soft set, not only inherits the characteristics of parameterization of soft set, but also enhances the flexibility of dealing with uncertainty, and has been widely used in many fields such as decision making, data mining and medical diagnosis and so on.

Decision making problems can be found everywhere in daily life, and the decision making process exists in all aspects of daily life. It is evident that decision making plays a crucial role in life. After Maji et al. [5] first applied soft sets to decision problems, many researchers have studied the application of soft sets and their extended models in decision making, especially the fuzzy soft sets based decision making has been widely concerned and studied and has been widely applied to many fields [6 –8]. Roy and Maji [9] proposed a decision making algorithm for fuzzy soft sets by constructing the comparison table of the fuzzy soft sets. Subsequently, the algorithm was improved in [10] and Feng et al. [11] proposed an adjustable approach to fuzzy soft sets based decision making using level soft sets. After that, many researchers have proposed different methods [12 –18] to solve the decision making problems based on fuzzy soft sets. Recently, Wang et al. [16] presented a novel fuzzy soft sets based decision making approach by using modal-style operators. The article of [17] proposed Union-Intersection decision making method based on fuzzy soft relationship and level fuzzy soft relationship. Alkhazaleh [18] introduced the concept of effective fuzzy soft sets and gave an application of this concept in decision making problem.

In recent years, more and more researchers are interested in parameter reduction in soft set theory. Parameter reduction plays a very important role in dealing with decision making problems in soft set theory. There are often many redundant and unimportant parameters in the decision making process. In order to improve the decision making efficiency, the redundant parameters can be deleted by parameter reduction so that the decision making ability does not change before and after the reduction. At present, many researchers have proposed different parameter reduction methods for soft sets. The concept of parameter reduction of soft sets was first introduced by Maji et al. [19], after which Chen et al. [20] improved the method proposed in [19]. Kong et al. proposed the normal parameter reduction algorithm in [21]. After that, many researchers have improved the normal parameter reduction algorithm for soft sets [22 –25]. However, the applicability of the algorithms in [21 –25] is low due to the overly stringent conditions about normal parameter reduction. To address this problem, Qin et al. [26] proposed a parameter reduction method based on chi square distribution for soft sets.

Different parameter reduction methods for fuzzy soft sets based on different decision criteria are proposed in [21 , 27–33]. Kong et al. proposed the normal parameter reduction algorithm for fuzzy soft sets in [21]. In order to remove more redundant parameters and improve the reduction effect, the proximate normal parameter reduction algorithm for fuzzy soft sets was proposed in [27]. After that, Ma and Qin [28] proposed the distance-based parameter reduction algorithm for fuzzy soft sets. Khan and Zhu [29] illustrated through specific examples that the algorithm proposed by Ma and Qin could not maintain the ranking order of decision alternatives, and proposed the rank-based parameter reduction algorithm of fuzzy soft sets and compared it with the above three algorithms [21 , 28] from different aspects. All of these four parameter reduction methods for fuzzy soft sets above are based on the choice value criteria. Kong et al. [30] proposed a new normal parameter reduction method for fuzzy soft sets based on the score decision criterion. Ma et al. [31] found that the parameter reduction method proposed by Kong et al. was complex and had high computational complexity, and in order to overcome this drawback, Ma et al. proposed a new improved normal parameter reduction method. Zhang [32] proposed a parameter reduction method for fuzzy soft sets based on soft fuzzy rough approximation operators. A new approach to parameter reduction of fuzzy soft sets based on the three-way decision methodology was proposed by Khameneh et al. [33]. Parameter reduction problem for fuzzy soft sets has been studied in [32, 33] from a different perspective than the decision criteria mentioned above. The parameter reduction problem has been extended not only to fuzzy soft sets, but also to other extended models of soft sets, such as N-soft sets [34].

In this paper, we will focus on a kind of parameter reduction problem, which is the parameter reduction of fuzzy soft sets based on the choice value criteria. The overly strict conditions about parameter reduction lead to a very low applicability of previous parameter reduction methods [21 , 29] for fuzzy soft sets based on choice value criteria. Moreover, these methods do not give specific steps to find the parameter subsets that satisfy the conditions. Therefore, when finding the parameter reduction, it is necessary to keep trying different parameter subsets to check whether they satisfy the conditions or not, which leads to the high computational complexity of these methods. And these methods may not be able to delete all redundant parameters. In order to overcome the shortcomings of these methods, we propose a difference-based parameter reduction method for fuzzy soft sets. The main contributions of this work can be summarized as follow.

To introduce the definition of the difference-based parameter reduction of fuzzy soft sets.

To propose a difference-based parameter reduction algorithm for fuzzy soft sets.

Compared with some previous parameter reduction algorithms for fuzzy soft sets based on choice value criteria, the proposed algorithm has a wider range of applicability, and is able to remove more redundant parameters while keeping the ranking order of all the alternatives invariant, and is more likely to find the parameter reduction for fuzzy soft sets.

The rest of this paper is organized as follows. Section 2 reviews some basic concepts related to soft sets and fuzzy soft sets. Section 3 briefly discusses the algorithms proposed in [21 , 29]. Section 4 presents a new method for parameter reduction of fuzzy soft sets based on the difference. Section 5 compares the proposed algorithm with the algorithms from references [21, 27] and [29] on certain aspects. Finally, Section 6 concludes the paper and outlook for potential future work.

2 Preliminaries

In this section, suppose U is an initial universe set and E is a set of parameters. We review some basic concepts related to soft sets and fuzzy soft sets.

Definition 2.1 A pair (F, E) is called a soft set over U, if F is a mapping of E into the set of all subsets of U.

Maji et al. [4] proposed fuzzy soft set by combining soft set and fuzzy set.

Definition 2.2 [4] Let U be the universe and A be the parameter set. P (U) denotes the set of all fuzzy subsets of U, a pair (F, A) is called a fuzzy soft set over U, where F : A → P (U) is a mapping from A into P (U).

Definition 2.3 [10] The fuzzy choice value of object u_i ∈ U under fuzzy soft set (F, E) is formulated by $f_{E} (u_{i}) = \sum_{j = 1}^{| E |} u_{ij}$ where | . | denotes the cardinality of a set and u_ij is the membership degree of u_i about parameter e_j, u_ij ∈ [0, 1].

To better understand the definition of fuzzy soft sets, we introduce the following example.

Example 2.1 Suppose there are four patients with COVID-19 and a doctor examines these patients and finds that they present with symptoms of fever, sore throat, cough, runny nose and fatigue. This doctor uses these symptoms to identify the patient most affected by SARS-CoV-2. We use fuzzy soft set to describe the evaluation system. Let U = {u₁, u₂, u₃, u₄} be the set of four patients and E = {e₁, e₂, e₃, e₄, e₅} be the set of symptoms presented by the four patients, where e_i stands for “fever”, “sore throat”,“cough”, “runny nose" and “fatigue”, respectively. The tabular form of such fuzzy soft set is represented in Table 1.

Table 1
Tabular representation of the fuzzy soft set (F, E)

(F, E) e ₁ e ₂ e ₃ e ₄ e ₅

u ₁ 0.1 0.6 0.3 0.2 0.4

u ₂ 0.5 0.7 0.1 0.5 0.6

u ₃ 0.4 0.8 0.2 0.1 0.7

u ₄ 0.2 0.5 0.4 0.3 0.4

(F, E)	e ₁	e ₂	e ₃	e ₄	e ₅
u ₁	0.1	0.6	0.3	0.2	0.4
u ₂	0.5	0.7	0.1	0.5	0.6
u ₃	0.4	0.8	0.2	0.1	0.7
u ₄	0.2	0.5	0.4	0.3	0.4

Based on Definitions 2.3, we can calculate the choice values of u_i (i = 1, 2, 3, 4) with respect to E and E - {e_l} (l = 1, 2, 3, 4, 5) as given in Table 2.

Table 2

Tabular representation of f_E (u_i) and f_{E-{e_l}} (u_i)

f_E (u_i)
f_{E-{e_l}} (u_i)	E	E - {e₁}	E - {e₂}	E - {e₃}	E - {e₄}	E - {e₅}
u ₁	1.6	1.5	1	1.3	1.4	1.2
u ₂	2.4	1.9	1.7	2.3	1.9	1.8
u ₃	2.2	1.8	1.4	2	2.1	1.5
u ₄	1.8	1.6	1.3	1.4	1.5	1.4

According to the choice values of u_i for all i with respect to E in Table 2, all patients can be ranked from largest to smallest as u₂ > u₃ > u₄ > u₁. Obviously, the choice value of u₂ is the largest and the choice value of u₁ is the smallest among all patients. Based on the choice value criteria, the object u₂ with the largest choice value is the patient most affected by SARS-CoV-2 and the object u₁ with the smallest choice value is the patient least affected by SARS-CoV-2.

3 Related work

In this section the algorithms proposed in [21 , 29] are briefly discussed.

3.1 The algorithm proposed in [21]

Definition 3.1 For fuzzy soft set (F, E), E = {e₁, e₂, ⋯ , e_m}. If there exists a subset $A = {e_{1}^{'}, e_{2}^{'}, \dots, e_{p}^{'}} \subset E$ satisfying ∑_{e_k∈A}u_1k = ∑_{e_k∈A}u_2k = ⋯ = ∑_{e_k∈A}u_nk, A is dispensable, otherwise, A is indispensable. B ⊂ E is a normal parameter reduction of E if B is indispensable and ∑_{e_k∈E-B}u_1k = ∑_{e_k∈E-B}u_2k = ⋯ = ∑_{e_k∈E-B}u_nk.

Based on the above definition, Kong et al. [21] proposed the normal parameter reduction algorithm of fuzzy soft sets given by Algorithm 1.

Algorithm 1 The algorithm of [21]
Step 1: Input the fuzzy soft set (F, E), the parameter set E;
Step 2: Find maximal subset A ⊆ E satisfying
∑_{e_k∈A}u_1k = ∑_{e_k∈A}u_2k = ⋯ = ∑_{e_k∈A}u_nk;
Step 3:Compute E - A as the optimal normal parameter reduction.

Example 3.1 Let (F, E) be the fuzzy soft set given in Table 3. The algorithm proposed in [21] is used to try to find the parameter reduction of fuzzy soft sets. Based on the steps given in the algorithm, although we have tried all the parameter subsets to check whether they satisfy the condition of Definition 3.1, we have not found a subset that satisfies the condition. It is clear that parameter reduction of fuzzy soft sets cannot be found using the algorithm proposed in [21].

Table 3
Tabular representation of the fuzzy soft set (F, E)

(F, E) e ₁ e ₂ e ₃ e ₄ e ₅

u ₁ 0.1 0.2 0.4 0.8 0.6

u ₂ 0.6 0.5 0.2 0.6 0.7

u ₃ 0.2 0.3 0.1 0.2 0.5

u ₄ 0.3 0.4 0.2 0.5 0.1

u ₅ 0.5 0.2 0.5 0.6 0.7

(F, E)	e ₁	e ₂	e ₃	e ₄	e ₅
u ₁	0.1	0.2	0.4	0.8	0.6
u ₂	0.6	0.5	0.2	0.6	0.7
u ₃	0.2	0.3	0.1	0.2	0.5
u ₄	0.3	0.4	0.2	0.5	0.1
u ₅	0.5	0.2	0.5	0.6	0.7

3.2 The algorithm proposed in [27]

Definition 3.2 For fuzzy soft set (F, E), U = {u₁, u₂, ⋯ , u_n}, E = {e₁, e₂, ⋯ , e_m}. The maximal deviation ɛ is defined by $ɛ = \min {| f_{E} (u_{i}) - f_{E} (u_{j}) | / 2, i \neq j, i, j = 1, 2, \dots, n}$

Definition 3.3[27] For fuzzy soft set (F, E), U = {u₁, u₂, ⋯ , u_n}, E = {e₁, e₂, ⋯ , e_m}, $A = {e_{1}^{'}, e_{2}^{'}, \dots, e_{p}^{'}} \subset E$ , the expected choice value of parameter set A is defined by ${\bar{C}}_{A} = \frac{1}{n} \sum_{i = 1}^{n} f_{A} (u_{i})$

Definition 3.4[27] For fuzzy soft set (F, E), U = {u₁, u₂, ⋯ , u_n}, E = {e₁, e₂, ⋯ , e_m}, if there exists a subset $A = {e_{1}^{'}, e_{2}^{'}, \dots, e_{p}^{'}} \subset E$ satisfying $| f_{A} (u_{i}) - {\bar{C}}_{A} | \leq ɛ$ , then E - A is the proximate normal parameter reduction of E, where ɛ is the maximal deviation, ${\bar{C}}_{A}$ is the expected choice value of parameter set A.

Based on the above definitions, the algorithm for the proximate normal parameter reduction of fuzzy soft sets is proposed in [27], as shown in Algorithm 2.

Algorithm 2 The algorithm of [27]
Step 1: Input the fuzzy soft set (F, E), the parameter set E;
Step 2: Compute the maximal deviation ɛ of parameter set E;
Step 3: Select combination A ⊂ E in E, compute ${\bar{C}}_{A}$
$({\bar{C}}_{A} = \frac{1}{n} \sum_{i = 1}^{n} f_{A} (u_{i}))$ ;
Step 4: If $\| f_{A} (u_{i}) - {\bar{C}}_{A} \| \leq ɛ$ , then A is the proximate normal
parameterization reduction and put A into a feasible
combination set, otherwise A is deleted from the
combination set;
Step 5: Find the maximum cardinality of A in feasible parameter
reduction set;
Step 6: Compute E - A as the optimal proximate normal
parameter reduction.

Example 3.2 Consider the fuzzy soft set (F, E) given in Table 3. We use the algorithm proposed in [27] to find the parameter reduction of fuzzy soft set (F, E). First, compute the maximal deviation ɛ = min {0.25, 0.4, 0.3, 0.2, 0.65, 0.55, 0.05, 0.1, 0.6, 0.5} =0.05. Then check one by one if there exists a parameter subset that satisfies the condition of Definition 3.4. Although all the parameter subsets are checked but no subset is found to satisfy the condition. Obviously we cannot find any parameter reduction of fuzzy soft set (F, E) using the algorithm proposed in [27].

3.3 The algorithm proposed in [29]

Definition 3.5 Let (F, E) be a fuzzy soft set over U where U = {u₁, u₂, ⋯, u_n}, and $u_{1}^{'} ⪰ u_{2}^{'} ⪰ \dots ⪰ u_{n}^{'}$ be the ranking order of objects in U such that $f_{E} (u_{i}^{'}) \geq f_{E} (u_{i + 1}^{'})$ for i = 1, 2, ⋯ , n - 1. The rank-based distance between two consecutive objects u_i and u_i+1 in the ranking order is defined by $d_{r} (u_{i}^{'}, u_{i + 1}^{'}) = f_{E} (u_{i}^{'}) - f_{E} (u_{i + 1}^{'}), i = 1, 2, \dots, n - 1$

Definition 3.6[29] Let (F, E) be a fuzzy soft set over U where U = {u₁, u₂, ⋯, u_n}, and $u_{1}^{'} ⪰ u_{2}^{'} ⪰ \dots ⪰ u_{n}^{'}$ be the ranking order of objects in U such that $f_{E} (u_{i}^{'}) \geq f_{E} (u_{i + 1}^{'})$ for i = 1, 2, ⋯ , n - 1. If there exists A ⊂ E satisfying $\begin{matrix} {\begin{matrix} | f_{A} (u_{i}^{'}) - f_{A} (u_{i + 1}^{'}) | < d_{r} (u_{i}^{'}, u_{i + 1}^{'}), if d_{r} (u_{i}^{'}, u_{i + 1}^{'}) \neq 0 \\ | f_{A} (u_{i}^{'}) - f_{A} (u_{i + 1}^{'}) | = d_{r} (u_{i}^{'}, u_{i + 1}^{'}), if d_{r} (u_{i}^{'}, u_{i + 1}^{'}) = 0 \end{matrix} \end{matrix}$ where i = 1, 2, ⋯ , n - 1, then E - A is called the rank-based parameter reduction of E.

According to Definition 3.5 and Definition 3.6, the rank-based parameter reduction algorithm of fuzzy soft sets is proposed in [29] as shown in Algorithm 3.

Algorithm 3 The algorithm of [29]
Step 1: Input the fuzzy soft set (F, E), the parameter set E;
Step 2: Compute the choice values of all u_i ∈ U and obtain
the ranking order of objects;
Step 3: Arrange the rows of fuzzy soft set table according to
the ranking order of objects and compute the
rank-based distances between all consecutive objects;
Step 4: Find maximal subset A ⊂ E in the ranked table
satisfying the condition of Definition 3.6;
Step 5: Compute E - A as as the rank-based parameter reduction.

Example 3.3 Consider the fuzzy soft set (F, E) given in Table 3. We use the algorithm proposed in [29] to find the parameter reduction of fuzzy soft set (F, E). Firstly compute the choice values of all the objects as f_E (u₁) =2.1, f_E (u₂) =2.6, f_E (u₃) =1.3, f_E (u₄) =1.5, f_E (u₅) =2.5, respectively. And get the ranking order of all the objects as u₂ > u₅ > u₁ > u₄ > u₃. Then calculate the rank-based distances between all consecutive objects as d_r (u₂, u₅) =0.1, d_r (u₅, u₁) =0.4, d_r (u₁, u₄) =0.6, d_r (u₄, u₃) =0.2, respectively. Further check the parameter subsets one by one whether there exists a parameter subset which satisfies the condition of Definition 3.6. However all the parameter subsets are checked but no parameter subset is found to satisfy the condition. Thus using the algorithm proposed in [29] the parameter reduction of fuzzy soft set (F, E) cannot be found.

Obviously, according to Examples 3.1-3.3, we can find that the algorithms proposed in [21 , 29] have low applicability due to overly stringent reduction conditions.

4 A novel difference-based parameter reduction method for fuzzy soft sets

In this section, the difference-based parameter reduction algorithm of fuzzy soft sets is presented. The algorithm avoids the disadvantage of the algorithms proposed in [21 , 29] which have very low applicability due to the overly stringent reduction conditions. And the algorithm proposed in this paper is applicable to all fuzzy soft sets with high applicability.

Assuming that (F, E) is a fuzzy soft set over U = {u₁, u₂, ⋯ , u_n}. Based on choice value criteria, the larger the choice value of an alternative with respect to E, the better the alternative is. Thus if the choice value of u_i with respect to E is larger than the choice value of u_j with respect to E, then u_i > u_j. Obviously if the difference between the choice values of u_i and u_j with respect to E is greater than 0, then u_i > u_j. If the difference between the choice values of u_i and u_j with respect to E is less than 0, then u_i < u_j. If the difference between the choice values of u_i and u_j with respect to E is equal to 0, then u_i = u_j. Therefore the preference orders of the alternatives can be discerned by the magnitude of the difference between the choice values of the alternatives with respect to E. Therefore, the preference orders of the alternatives can be discerned by the magnitude of the difference between the choice values of the alternatives with respect to E. If the magnitude of the difference between the choice values of all alternatives does not change in relation to 0 after deleting some of the parameters, then it means that the ranking of all alternatives before and after deleting the parameters has not changed.

Through the above analysis, we give the definition of difference between the choice values and present a new method for parameter reduction of fuzzy soft sets based on the difference.

Definition 4.1 Let (F, E) be a fuzzy soft set over U where U = {u₁, u₂, ⋯, u_n}, A ⊆ E. The difference between the choice values of u_i and u_j with respect to the parameter set A is defined as: $D_{A} (u_{i}, u_{j}) = f_{A} (u_{i}) - f_{A} (u_{j})$ where f_A (u_i) is the choice value of object u_i in parameter set A and i < j.

To better understand the concept of difference in Definition 4.1, the following example is given.

Example 4.1 Consider the fuzzy soft set (F, E) represented in Table 1. According to Definition 4.1 and Table 2, D_E (u_i, u_j) and D_{E-{e_l}} (u_i, u_j) (i < j, i, j = 1, 2, 3, 4, l = 1, 2, 3, 4, 5) are calculated as given in Table 4.

Table 4
Tabular representation of D_E (u_i, u_j) and D_{E-{e_l}} (u_i, u_j)

D_E (u_i, u_j)

D_{E-{e_l}} (u_i, u_j) E E - {e₁} E - {e₂} E - {e₃} E - {e₄} E - {e₅}

(u₁, u₂) –0.8 –0.4 –0.7 –1 –0.5 –0.6

(u₁, u₃) –0.6 –0.3 –0.4 –0.7 –0.7 –0.3

(u₁, u₄) –0.2 –0.1 –0.3 –0.1 –0.1 –0.2

(u₂, u₃) 0.2 0.1 0.3 0.3 –0.2 0.3

(u₂, u₄) 0.6 0.3 0.4 0.9 0.4 0.4

(u₃, u₄) 0.4 0.2 0.1 0.6 0.6 0.1

D_E (u_i, u_j)
(u₁, u₂)	–0.8	–0.4	–0.7	–1	–0.5	–0.6
(u₁, u₃)	–0.6	–0.3	–0.4	–0.7	–0.7	–0.3
(u₁, u₄)	–0.2	–0.1	–0.3	–0.1	–0.1	–0.2
(u₂, u₃)	0.2	0.1	0.3	0.3	–0.2	0.3
(u₂, u₄)	0.6	0.3	0.4	0.9	0.4	0.4
(u₃, u₄)	0.4	0.2	0.1	0.6	0.6	0.1

Definition 4.2 Let (F, E) be a fuzzy soft set over U where U = {u₁, u₂, ⋯, u_n}, ∀u_i, u_j ∈ U (i < j), if there exists A ⊆ E satisfying $\begin{matrix} {\begin{matrix} D_{E} (u_{i}, u_{j}) D_{E - A} (u_{i}, u_{j}) > 0, if D_{E} (u_{i}, u_{j}) \neq 0 \\ D_{E - A} (u_{i}, u_{j}) = 0, if D_{E} (u_{i}, u_{j}) = 0 \end{matrix} \end{matrix}$ then E - A is called the difference-based parameter reduction of E.

Clearly, if a parameter subset A satisfies the condition of Definition 4.2, then E - A must be the set that keeps the ranking order of all the alternatives unchanged.

Definition 4.2 provides a parameter reduction method for fuzzy soft sets according to the difference between the choice values with respect to E and E - A. However, it’s not easy to find a parameter subset A satisfying the condition of Definition 4.2 directly. To address this issue, we give the following theorem and corollary.

Theorem 4.1 Let (F, E) be a fuzzy soft set over U, where U = {u₁, u₂, ⋯, u_n} and E = {e₁, e₂, ⋯ , e_m}. ∀u_i, u_j ∈ U (i < j), if there exists a parameter subset ${e_{1}^{'}, e_{2}^{'}, \dots, e_{s}^{'}} \subseteq E (1 \leq s \leq m - 1)$ satisfying the following properties:

(1) If D_E (u_i, u_j) ≠0, then $\frac{\sum_{l = 1}^{s} D_{E - {e_{l}^{'}}} (u_{i}, u_{j})}{D_{E} (u_{i}, u_{j})} > s - 1$ (2) If D_E (u_i, u_j) =0, then $D_{E - {e_{1}^{'}, e_{2}^{'}, \dots, e_{s}^{'}}} (u_{i}, u_{j}) = 0$ Then $E - {e_{1}^{'}, e_{2}^{'}, \dots, e_{s}^{'}}$ is the difference-based parameter reduction of E.

Proof: Based on Definition 4.1, the following equation is obtained: $D_{E} (u_{i}, u_{j}) - \sum_{l = 1}^{s} D_{E - {e_{l}^{'}}} (u_{i}, u_{j}) = - (s - 2)$ $D_{E} (u_{i}, u_{j}) - D_{E - {e_{1}^{'}, e_{2}^{'}, \dots, e_{s}^{'}}} (u_{i}, u_{j})$ Then the following equation is hold: $D_{E - {e_{1}^{'}, e_{2}^{'}, \dots, e_{s}^{'}}} (u_{i}, u_{j}) = - (s - 1)$ $D_{E} (u_{i}, u_{j}) + \sum_{l = 1}^{s} D_{E - {e_{l}^{'}}} (u_{i}, u_{j})$ If D_E (u_i, u_j) ≠0, then $\frac{D_{E - {e_{1}^{'}, e_{2}^{'}, \dots, e_{s}^{'}}} (u_{i}, u_{j})}{D_{E} (u_{i}, u_{j})} = - (s - 1) +$ $\frac{\sum_{l = 1}^{s} D_{E - {e_{l}^{'}}} (u_{i}, u_{j})}{D_{E} (u_{i}, u_{j})}$ Since $\frac{\sum_{l = 1}^{s} D_{E - {e_{l}^{'}}} (u_{i}, u_{j})}{D_{E} (u_{i}, u_{j})} > s - 1$ , the following inequality is obtained: $\frac{D_{E - {e_{1}^{'}, e_{2}^{'}, \dots, e_{s}^{'}}} (u_{i}, u_{j})}{D_{E} (u_{i}, u_{j})} > 0$ Then the following inequality is hold: $D_{E} (u_{i}, u_{j}) D_{E - {e_{1}^{'}, e_{2}^{'}, \dots, e_{s}^{'}}} (u_{i}, u_{j}) > 0$ It is easy to see that ${e_{1}^{'}, e_{2}^{'}, \dots, e_{s}^{'}}$ satisfy the condition of Definition 4.2. Thus, $E - {e_{1}^{'}, e_{2}^{'}, \dots, e_{s}^{'}}$ is the difference-based parameter reduction of E.

According to Theorem 4.1, the following Corollary 4.1 can be induced.

Corollary 4.1 Let (F, E) be a fuzzy soft set over U, where U = {u₁, u₂, ⋯, u_n} and E = {e₁, e₂, ⋯ , e_m}. ∀u_i, u_j ∈ U (i < j), if there exists a parameter subset ${e_{1}^{″}, e_{2}^{″}, \dots, e_{q}^{″}} \subseteq E (1 \leq q \leq m - 1)$ satisfying the following properties:

(1) If D_E (u_i, u_j) ≠0, then $\frac{\sum_{l = 1}^{q} D_{E - {e_{l}^{″}}} (u_{i}, u_{j})}{D_{E} (u_{i}, u_{j})} < q$ (2) If D_E (u_i, u_j) =0, then $D_{{e_{1}^{″}, e_{2}^{″}, \dots e_{q}^{″}}} (u_{i}, u_{j}) = 0$ Then ${e_{1}^{″}, e_{2}^{″}, \dots, e_{q}^{″}}$ is the difference-based parameter reduction of E.

Proof: Suppose $E - {e_{1}^{″}, e_{2}^{″}, \dots, e_{q}^{″}} = {e_{q + 1}^{″}, e_{q + 2}^{″}, \dots, e_{m}^{″}}$ .

If D_E (u_i, u_j) ≠0, then $\frac{\sum_{l = q + 1}^{m} D_{E - {e_{l}^{″}}} (u_{i}, u_{j})}{D_{E} (u_{i}, u_{j})} =$ $\frac{\sum_{l = 1}^{m} D_{E - {e_{l}}} (u_{i}, u_{j}) - \sum_{l = 1}^{q} D_{E - {e_{l}^{″}}} (u_{i}, u_{j})}{D_{E} (u_{i}, u_{j})}$ $= \frac{\sum_{l = 1}^{m} D_{E - {e_{l}}} (u_{i}, u_{j})}{D_{E} (u_{i}, u_{j})} - \frac{\sum_{l = 1}^{q} D_{E - {e_{l}^{″}}} (u_{i}, u_{j})}{D_{E} (u_{i}, u_{j})}$

Based on Definition 4.1, we have $\frac{\sum_{l = 1}^{m} D_{E - {e_{l}}} (u_{i}, u_{j})}{D_{E} (u_{i}, u_{j})} = m - 1$

Since $\frac{\sum_{l = 1}^{q} D_{E - {e_{l}^{″}}} (u_{i}, u_{j})}{D_{E} (u_{i}, u_{j})} < q$ , we have $\frac{\sum_{l = q + 1}^{m} D_{E - {e_{l}^{″}}} (u_{i}, u_{j})}{D_{E} (u_{i}, u_{j})} > m - 1 - q = m - q - 1$

If D_E (u_i, u_j) =0, then $D_{E - {e_{q + 1}^{″}, e_{q + 2}^{″}, \dots, e_{m}^{″}}} (u_{i}, u_{j}) = 0$

Therefore, according to Theorem 4.1, we can get $E - {e_{q + 1}^{″}, e_{q + 2}^{″}, \dots, e_{m}^{″}} = {e_{1}^{″}, e_{2}^{″}, \dots, e_{q}^{″}}$ is the difference-based parameter reduction of E.

Example 4.2 Consider the fuzzy soft set (F, E) given in Table 1. Let q = 1, check if there exists a parameter subset A ⊆ E that satisfies the conditions of Corollary 4.1 for any u_i, u_j (i < j). By Table 4, we find a parameter subset {e₁} satisfying Corollary 4.1. Thus, {e₁} is the difference-based parameter reduction of E.

Corollary 4.1 provides a method to find a parameter subset A such that A is the difference-based parameter reduction of E. When q = a, we need to check if $C_{| E |}^{a}$ parameter subsets whose cardinality are a satisfy the conditions of Corollary 4.1. It’s easy to see that the larger |E| is, the greater the computational effort to find the parameter subset A. Therefore, when the number of parameters is big, finding the parameter reduction of E through Corollary 4.1 requires a large amount of computation.

In order to reduce the computational complexity of finding the parameter reduction, we give the following theorems and corollaries.

Theorem 4.2 Let (F, E) be a fuzzy soft set over U where U = {u₁, u₂, ⋯, u_n}, E = {e₁, e₂, ⋯ , e_m}, $e_{t}^{″} \in E (1 \leq t \leq m)$ , and $D_{E - {e_{1}^{″}}} (u_{i}, u_{j}) \leq D_{E - {e_{2}^{″}}} (u_{i}, u_{j}) \leq \dots \leq D_{E - {e_{m}^{″}}} (u_{i}, u_{j})$ . A is the difference-based parameter reduction of E, |A| = q (1 ≤ q ≤ m - 1). If there exist u_i, u_j ∈ U such that

(1) when D_E (u_i, u_j) >0, $\sum_{l = 2}^{q + 1} D_{E - {e_{l}^{″}}} (u_{i}, u_{j}) \geq {qD}_{E} (u_{i}, u_{j})$ , then $e_{1}^{″} \in A$ .

(2) when D_E (u_i, u_j) <0, $\sum_{l = m - q}^{m - 1} D_{E - {e_{l}^{″}}} (u_{i}, u_{j})$ ≤qD_E (u_i, u_j), then $e_{m}^{″} \in A$ .

Proof: (1) Suppose $e_{1}^{″} \notin A$ , then for u_i, u_j ∈ U, there does not exist any parameter subset of E whose cardinality is q satisfying the conditions of Corollary 4.1. Therefore there does not exist the difference-based parameter reduction of E whose cardinality is q, which is a contradiction. Thus $e_{1}^{″} \in A$ .

(2) It is derived by the similar process used in (1).

Theorem 4.3 Let (F, E) be a fuzzy soft set over U where U = {u₁, u₂, ⋯, u_n}, E = {e₁, e₂, ⋯ , e_m}, $e_{t}^{″} \in E (1 \leq t \leq m)$ , and $D_{E - {e_{1}^{″}}} (u_{i}, u_{j}) \leq D_{E - {e_{2}^{″}}} (u_{i}, u_{j}) \leq \dots \leq D_{E - {e_{m}^{″}}} (u_{i}, u_{j})$ . A is the difference-based parameter reduction of E, |A| = q (1 ≤ q ≤ m - 1). If there exist u_i, u_j ∈ U such that

(1) when D_E (u_i, u_j) >0, $\sum_{l = q}^{m - 1} D_{E - {e_{l}^{″}}} (u_{i}, u_{j}) \leq (m - q - 1) D_{E} (u_{i}, u_{j})$ , then $e_{m}^{″} \notin A$ .

(2) when D_E (u_i, u_j) <0, $\sum_{l = 2}^{m - q + 1} D_{E - {e_{l}^{″}}} (u_{i}, u_{j})$ ≥ (m - q - 1) D_E (u_i, u_j), then $e_{1}^{″} \notin A$ .

Proof: (1) Assume that $e_{m}^{″} \in A$ , i.e., $e_{m}^{″} \notin E - A,$ then for u_i, u_j ∈ U, there does not exist any parameter subset of E whose cardinality is m - q satisfying the conditions of Theorem 4.1. Therefore there does not exist a parameter subset B whose cardinality is m - q such that E - B is the difference-based parameter reduction of E. That is, there does not exist the difference-based parameter reduction of E whose cardinality is q, which is a contradiction. So $e_{m}^{″} \notin A$ .

(2) It is derived by the similar process used in (1).

Corollary 4.2 Let (F, E) be a fuzzy soft set over U, where U = {u₁, u₂, ⋯, u_n} and E = {e₁, e₂, ⋯ , e_m}, $e_{t}^{″} \in E (1 \leq t \leq m)$ . A is the difference-based parameter reduction of E, |A| = q (1 ≤ q ≤ m - 1), and $e_{1}^{″}, e_{2}^{″}, \dots, e_{k}^{″} \in A$ , $D_{E - {e_{k + 1}^{″}}} (u_{i}, u_{j}) \leq D_{E - {e_{k + 2}^{″}}} (u_{i}, u_{j}) \leq \dots \leq D_{E - {e_{m}^{″}}} (u_{i}, u_{j})$ .If there exist u_i, u_j ∈ U such that

(1) when D_E (u_i, u_j) >0, $\sum_{l = k + 2}^{q + 1} D_{E - {e_{l}^{″}}} (u_{i}, u_{j}) \geq {qD}_{E} (u_{i}, u_{j}) - \sum_{l = 1}^{k} D_{E - {e_{l}^{″}}}$ (u_i, u_j), then $e_{k + 1}^{″} \in A$ .

(2) when D_E (u_i, u_j) <0, $\sum_{l = m - q + k}^{m - 1} D_{E - {e_{l}^{″}}} (u_{i}, u_{j})$ $\leq {qD}_{E} (u_{i}, u_{j}) - \sum_{l = 1}^{k} D_{E - {e_{l}^{″}}}$ (u_i, u_j), then $e_{m}^{″} \in A$ .

Proof: (1) Suppose $e_{k + 1}^{″} \notin A$ , then for u_i, u_j ∈ U, there does not exist any parameter subset B ⊆ E whose cardinality is q - k such that $B ⋃ {e_{1}^{″}, e_{2}^{″}, \dots, e_{k}^{″}}$ satisfies the conditions of Corollary 4.1, that is, there does not exist any parameter subset whose cardinality is q satisfying the conditions of Corollary 4.1,and hence there does not exist the difference-based parameter reduction of E whose cardinality is q, which is a contradiction. Thus $e_{k + 1}^{″} \in A$ .

(2) It is derived by the similar process used in (1).

Corollary 4.3 Let (F, E) be a fuzzy soft set over U, where U = {u₁, u₂, ⋯, u_n} and E = {e₁, e₂, ⋯ , e_m}, $e_{t}^{″} \in E (1 \leq t \leq m)$ . A is the difference-based parameter reduction of E, |A| = q (1 ≤ q ≤ m - 1), and $e_{1}^{″}, e_{2}^{″}, \dots, e_{k}^{″} \in E - A$ , $D_{E - {e_{k + 1}^{″}}} (u_{i}, u_{j}) \leq D_{E - {e_{k + 2}^{″}}} (u_{i}, u_{j}) \leq \dots \leq D_{E - {e_{m}^{″}}} (u_{i}, u_{j})$ .If there exist u_i, u_j ∈ U such that

(1) when D_E (u_i, u_j) >0, $\sum_{l = q + k}^{m - 1} D_{E - {e_{l}^{″}}} (u_{i}, u_{j}) \leq (m - q - 1) D_{E} (u_{i}, u_{j}) - \sum_{l = 1}^{k} D_{E - {e_{l}^{″}}} (u_{i}, u_{j})$ , then $e_{m}^{″} \notin A$ .

(2) when D_E (u_i, u_j) <0, $\sum_{l = k + 2}^{m - q + 1} D_{E - {e_{l}^{″}}} (u_{i}, u_{j})$ $\geq (m - q - 1) D_{E} (u_{i}, u_{j}) - \sum_{l = 1}^{k} D_{E - {e_{l}^{″}}} (u_{i}, u_{j})$ , then $e_{k + 1}^{″} \notin A$ .

Proof: (1) Assume that $e_{m}^{″} \in A$ , i.e., $e_{m}^{″} \notin E - A,$ then for u_i, u_j ∈ U, there does not exist any parameter subset B ⊆ E whose cardinality is m - q - k such that $B ⋃ {e_{1}^{″}, e_{2}^{″}, \dots, e_{k}^{″}}$ satisfies the conditions of Theorem 4.1,that is, there does not exist any parameter subset whose cardinality is m - q satisfying the conditions of Theorem 4.1.Therefore there does not exist the difference-based parameter reduction of E whose cardinality is q, which is a contradiction. So $e_{m}^{″} \notin A$ .

(2) It is derived by the similar process used in (1).

Definition 4.3 Let (F, E) be a fuzzy soft sets over U where U = {u₁, u₂, ⋯, u_n}, E = {e₁, e₂, ⋯ , e_m}, $e_{t}^{″} \in E (1 \leq t \leq m)$ , and $D_{E - {e_{1}^{″}}} (u_{i}, u_{j}) \leq D_{E - {e_{2}^{″}}} (u_{i}, u_{j}) \leq \dots \leq D_{E - {e_{m}^{″}}} (u_{i}, u_{j})$ , define m_q (u_i, u_j) and M_q (u_i, u_j) as follows: $m_{q} (u_{i}, u_{j}) = \sum_{t = 1}^{q} D_{E - {e_{t}^{″}}} (u_{i}, u_{j})$ $M_{q} (u_{i}, u_{j}) = \sum_{t = m - q + 1}^{m} D_{E - {e_{t}^{″}}} (u_{i}, u_{j})$ where i, j = 1, 2, ⋯ , n, i < j, 1 ≤ q ≤ m - 1.

Theorem 4.4 Let (F, E) be a fuzzy soft sets over U where U = {u₁, u₂, ⋯, u_n}, E = {e₁, e₂, ⋯ , e_m}, $e_{t}^{″} \in E (1 \leq t \leq m)$ , and $D_{E - {e_{1}^{″}}} (u_{i}, u_{j}) \leq D_{E - {e_{2}^{″}}} (u_{i}, u_{j}) \leq \dots \leq D_{E - {e_{m}^{″}}} (u_{i}, u_{j})$ .If there exist u_i, u_j ∈ U such that

(1) when D_E (u_i, u_j) >0, M_q (u_i, u_j) ≥ qD_E (u_i, u_j) , then ${e_{m - q + 1}^{″}, e_{m - q + 2}^{″}$ , $\dots, e_{m}^{″}}$ is not the difference-based parameter reduction of E.

(2) when D_E (u_i, u_j) <0, m_q (u_i, u_j) ≤ qD_E (u_i, u_j) , then ${e_{1}^{″}, e_{2}^{″}, \dots, e_{m}^{″}}$ is not the difference-based parameter reduction of E.

Proof: This proof is easily obtained from Corollary 4.1.

4.1 Proposed difference-based parameter reduction algorithm

According to our proposed definitions and theorems, the difference-based parameter reduction algorithm of fuzzy soft sets is presented, as shown in Algorithm 4.

A sketch map of the proposed algorithm for parameter reduction of fuzzy soft sets is shown in Fig. 1.

Algorithm 4 The difference-based parameter reduction algorithm
Step 1: Input the fuzzy soft set (F, E), the parameter set E and q = 1;
Step 2: Compute the choice values f_E (u_i) and f_{E-{e_l}} (u_i), derive the differences D_E (u_i, u_j) and D_{E-{e_l}} (u_i, u_j), where i < j, i, j = 1, 2, ⋯ , n, l = 1, 2, ⋯ , m (Definition 2.3 and 4.1);
Step 3: Search for redundant parameters and put these parameters in the set N_q(Theorem 4.3 and Corollary 4.3);
Step 4: Verify if \|N_q\| < m - q, if yes, proceed to step 3 until all searches are complete and then proceed to step 5,otherwise verify whether E - N_q satisfies Corollary 4.1. If yes, let A = E - N_q, and go to Step 9,otherwise, let q = q + 1 and go to Step 3;
Step 5: Search for non-redundant parameters in E - N_q and put these parameters in the set S_q(Theorem 4.2 and Corollary 4.2);
Step 6: Verify if \|S_q\| < q, if yes, proceed to step 5 until all searches are complete and then proceed to step 7, otherwise verify whether S_q satisfies Corollary 4.1.If yes, then A = S_q and go to Step 9, otherwise, let q = q + 1 and go to Step 3;
Step 7: Choose any q - \|S_q\| parameters in E - N_q - S_q to put into S_q and put every S_q into L_q. Remove sets in L_q that are not the difference-based parameter reduction of E(Theorem 4.4);
Step 8: Check if there exists a parameter subset A ∈ L_q that satisfies Corollary 4.1. If it exists, go to Step 9.Otherwise, let q = q + 1 and go to Step 3;
Step 9: A is the difference-based parameter reduction of E.

Fig. 1

A sketch map of the proposed algorithm for parameter reduction.

By the proposed algorithm, the minimal subset of E that keeps the entire decision order constant can surely be found. The proposed algorithm not only gives the condition that a parameter subset is the parameter reduction that needs to be satisfied, but also gives the method to find the minimal parameter subset that satisfies that condition, such that the found parameter reduction is the subset with the smallest cardinality in the parameter subsets that keeps the decision order unchanged.

In order to show the applicability, feasibility and effectiveness of our proposed algorithm, we present the following examples.

Example 4.3 [29] Let (F, E) be the fuzzy soft set given in Table 5. To get parameter reduction of E by the proposed algorithm, the following steps are involved:

Table 5

Tabular representation of the fuzzy soft set (F, E)

(F, E)	e ₁	e ₂	e ₃	e ₄	e ₅
u ₁	0.4	0.5	0.9	0.35	0.4
u ₂	0.6	0.5	0.3	0.6	0.25
u ₃	0.8	0.3	0.1	0.1	0.2
u ₄	0.0	0.6	0.7	0.5	0.65

Step 1: Compute the choice value of u_i (i = 1, 2, 3, 4) with respect to E and E - {e_l} (l = 1, 2, 3, 4, 5) as given in Table 6.

Table 6

Tabular representation of f_E (u_i) and f_{E-{e_l}} (u_i)

f_E (u_i)
f_{E-{e_l}} (u_i)	E	E - {e₁}	E - {e₂}	E - {e₃}	E - {e₄}	E - {e₅}
u ₁	2.55	2.15	2.05	1.65	2.2	2.15
u ₂	2.25	1.65	1.75	1.95	1.65	2
u ₃	1.5	0.7	1.2	1.4	1.4	1.3
u ₄	2.45	2.45	1.85	1.75	1.95	1.8

Step 2: By Definition 4.1 and Table 6, D_E (u_i, u_j) and D_{E-{e_l}} (u_i, u_j) (i < j, i, j = 1, 2, 3, 4, l = 1, 2, 3, 4, 5) are calculated as given in Table 7.

Table 7

Tabular representation of D_E (u_i, u_j) and D_{E-{e_l}} (u_i, u_j)

D_E (u_i, u_j)
D_{E-{e_l}} (u_i, u_j)	E	E - {e₁}	E - {e₂}	E - {e₃}	E - {e₄}	E - {e₅}
(u₁, u₂)	0.3	0.5	0.3	–0.3	0.55	0.15
(u₁, u₃)	1.05	1.45	0.85	0.25	0.8	0.85
(u₁, u₄)	0.1	–0.3	0.2	–0.1	0.25	0.35
(u₂, u₃)	0.75	0.95	0.55	0.55	0.25	0.7
(u₂, u₄)	–0.2	–0.8	–0.1	0.2	–0.3	0.2
(u₃, u₄)	–0.95	–1.75	–0.65	–0.35	–0.55	–0.5

Step 3: Suppose A is the difference-based parameter reduction of E and |A| = q.Let q = 1.By Theorem 4.3, we can find parameters that must not belong to A.

For u₁, u₂ :

Since D_E-{e₁} (u₁, u₂) + D_E-{e₂} (u₁, u₂) + D_E-{e₃} (u₁, u₂) + D_E-{e₅} (u₁, u₂) = 0.65 < (5 - 1 -1) D_E (u₁, u₂) =0.9, then e₄ ∉ A.

For u₁, u₃ :

Since D_E-{e₂} (u₁, u₃) + D_E-{e₃} (u₁, u₃) + D_E-{e₄} (u₁, u₃) + D_E-{e₅} (u₁, u₃) = 2.75 < (5 - 1 -1) D_E (u₁, u₃) =3.15, then e₁ ∉ A.

For u₁, u₄ :

Since D_E-{e₁} (u₁, u₄) + D_E-{e₂} (u₁, u₄) + D_E-{e₃} (u₁, u₄) + D_E-{e₄} (u₁, u₄) = 0.05 < (5 - 1 -1) D_E (u₁, u₄) =0.3, then e₅ ∉ A.

Therefore, e₁, e₄, e₅ ∉ A. That is, e₁, e₄, e₅ are redundant and unimportant parameters, and their deletion does not change the decision making ability. The following determines the parameters that must belong to A from the remaining parameters according to Theorem 4.2.

For u₁, u₂ :

Since D_E-{e₂} (u₁, u₂) =0.3 = D_E (u₁, u₂) =0.3, then e₃ ∈ A.

If there exists a parameter reduction whose cardinality is 1, then A = {e₃}. Verify whether {e₃} is the difference-based parameter reduction of E by Corollary 4.1. Clearly {e₃} satisfies the conditions of Corollary 4.1, so {e₃} is the difference-based parameter reduction of E.

It is obvious to see that the ranking order of all the alternatives does not change after removing the redundant parameters, and remains u₁ > u₄ > u₂ > u₃. And {e₃} is the minimal subset of E that keeps the decision capability unchanged.

Example 4.4 Consider the fuzzy soft set (F, E) given in Table 3. To get parameter reduction of E by the proposed algorithm, the following steps are involved:

Step 1: Compute the choice values of all objects with respect to E and E - {e_l} (l = 1, 2, 3, 4, 5) as shown in Table 8.

Table 8

Tabular representation of f_E (u_i) and f_{E-{e_l}} (u_i)

f_E (u_i)
f_{E-{e_l}} (u_i)	E	E - {e₁}	E - {e₂}	E - {e₃}	E - {e₄}	E - {e₅}
u ₁	2.1	2	1.9	1.7	1.3	1.5
u ₂	2.6	2	2.1	2.4	2	1.9
u ₃	1.3	1.1	1	1.2	1.1	0.8
u ₄	1.5	1.2	1.1	1.3	1	1.4
u ₅	2.5	2	2.3	2	1.9	1.8

Step 2: Compute the differences D_E (u_i, u_j) and D_{E-{e_l}} (u_i, u_j) shown in Table 9.

Table 9

Tabular representation of D_E (u_i, u_j) and D_{E-{e_l}} (u_i, u_j)

D_E (u_i, u_j)
D_{E-{e_l}} (u_i, u_j)	E	E - {e₁}	E - {e₂}	E - {e₃}	E - {e₄}	E - {e₅}
(u₁, u₂)	–0.5	0	–0.2	–0.7	–0.7	–0.4
(u₁, u₃)	0.8	0.9	0.9	0.5	0.2	0.7
(u₁, u₄)	0.6	0.8	0.8	0.4	0.3	0.1
(u₁, u₅)	–0.4	0	–0.4	–0.3	–0.6	–0.3
(u₂, u₃)	1.3	0.9	1.1	1.2	0.9	1.1
(u₂, u₄)	1.1	0.8	1	1.1	1	0.5
(u₂, u₅)	0.1	0	–0.2	0.4	0.1	0.1
(u₃, u₄)	–0.2	–0.1	–0.1	–0.1	0.1	–0.6
(u₃, u₅)	–1.2	–0.9	–1.3	–0.8	–0.8	–1
(u₄, u₅)	–1	–0.8	–1.2	–0.7	–0.9	–0.4

Step 3: Assume that A is the difference-based parameter reduction of E and |A| = q. Let q = 1. According to Table 9 and Theorem 4.3, we can find parameters that must not belong to A.

For u₁, u₂ :

Since D_E-{e₁} (u₁, u₂) + D_E-{e₂} (u₁, u₂) + D_E-{e₃} (u₁, u₂) + D_E-{e₅} (u₁, u₂)

= -1.3 > (5 - 1 -1) D_E (u₁, u₂) = -1.5,

D_E-{e₁} (u₁, u₂) + D_E-{e₂} (u₁, u₂) + D_E-{e₄} (u₁, u₂) + D_E-{e₅} (u₁, u₂)

= -1.3 > (5 - 1 -1) D_E (u₁, u₂) = -1.5, then e₃, e₄ ∉ A.

For u₁, u₃ :

Since D_E-{e₁} (u₁, u₃) + D_E-{e₃} (u₁, u₃) + D_E-{e₄} (u₁, u₃) + D_E-{e₅} (u₁, u₃)

= 2.3 < (5 - 1 -1) D_E (u₁, u₃) =2.4,

D_E-{e₂} (u₁, u₃) + D_E-{e₃} (u₁, u₃) + D_E-{e₄} (u₁, u₃) + D_E-{e₅} (u₁, u₃)

= 2.3 < (5 - 1 -1) D_E (u₁, u₃) =2.4, then e₁, e₂ ∉ A.

If there exists a parameter reduction whose cardinality is 1, then A = {e₅}. Verify whether {e₅} is the difference-based parameter reduction of E by Corollary 4.1. Obviously, {e₅} does not satisfy the conditions of Corollary 4.1, so {e₅} is not the difference-based parameter reduction of E, that is, there is no parameter reduction whose cardinality is 1, i.e., |A|≠1.

Step 4: Let q = 2.First, by Theorem 4.3, we have:

For u₂, u₅:

Since D_E-{e₁} (u₂, u₅) + D_E-{e₄} (u₂, u₅) + D_E-{e₅} (u₂, u₅) =0 + 0.1 + 0.1 = 0.2 = (5 - 2 -1) D_E (u₂, u₅) =0.2, then e₃ ∉ A.

For u₃, u₄ :

Since D_E-{e₁} (u₃, u₄) + D_E-{e₂} (u₃, u₄) + D_E-{e₃} (u₃, u₄) = -0.1 - 0.1 - 0.1 = -0.3 > (5 - 2 -1) D_E (u₃, u₄) = -0.4, then e₅ ∉ A.

And then, by using Corollary 4.3, we can also find the parameters that must not belong to A.

For u₄, u₅ :

Since D_E-{e₄} (u₄, u₅) = -0.9 = (5 - 2 -1) D_E (u₄, u₅) - (D_E-{e₃} (u₄, u₅) + D_E-{e₅} (u₄, u₅)), then e₂ ∉ A.

Therefore, we have e₂, e₃, e₅ ∉ A. If there exists a parameter reduction whose cardinality is 2, then A = {e₁, e₄}. Finally, determine whether A is the difference-based parameter reduction of E according to Corollary 4.1.

Obviously, the ranking order of all the alternatives has not changed before and after parameter reduction, which is u₂ > u₅ > u₁ > u₄ > u₃. And {e₁, e₄} is the minimal subset of E that keeps the entire decision order invariant.

Example 4.5 Let (F, E) be the fuzzy soft set given in Table 10. To get parameter reduction of E by the proposed algorithm, the following steps are involved:

Table 10

Tabular representation of the fuzzy soft set (F, E)

(F, E)	e ₁	e ₂	e ₃	e ₄	e ₅	e ₆	e ₇	e ₈
u ₁	0.81	0.10	0.36	0.14	0.27	0.60	0.15	0.92
u ₂	0.61	0.48	0.87	0.03	0.80	0.20	0.01	0.39
u ₃	0.13	0.55	0.30	0.99	0.10	0.85	0.10	0.12
u ₄	0.40	0.96	0.99	0.01	0.90	0.10	0.65	0.35
u ₅	0.63	0.86	0.40	0.10	0.38	0.45	0.30	0.03
u ₆	0.34	0.35	0.5	0.85	0.15	0.2	0.32	0.45

Step 1: Compute the choice values of all objects with respect to E and E - {e_l} (l = 1, 2, ⋯ , 8) as shown in Table 11.

Table 11

Tabular representation of f_E (u_i) and f_{E-{e_l}} (u_i)

f_E (u_i)
f_{E-{e_l}} (u_i)	E	E - {e₁}	E - {e₂}	E - {e₃}	E - {e₄}	E - {e₅}	E - {e₆}	E - {e₇}	E - {e₈}
u ₁	3.35	2.54	3.25	2.99	3.21	3.08	2.75	3.2	2.43
u ₂	3.39	2.78	2.91	2.52	3.36	2.59	3.19	3.38	3
u ₃	3.14	3.01	2.59	2.84	2.15	3.04	2.29	3.04	3.02
u ₄	4.36	3.96	3.4	3.37	4.35	3.46	4.26	3.71	4.01
u ₅	3.15	2.52	2.29	2.75	3.05	2.77	2.7	2.85	3.12
u ₆	3.16	2.82	2.81	2.66	2.31	3.01	2.96	2.84	2.71

Step 2: Using Table 11, calculate D_E (u_i, u_j) and D_{E-{e_l}} (u_i, u_j) (i < j, i, j = 1, 2, ⋯ , 6) as shown in Table 12.

Table 12

Tabular representation of D_E (u_i, u_j) and D_{E-{e_l}} (u_i, u_j)

D_E (u_i, u_j)
D_{E-{e_l}} (u_i, u_j)	E	E - {e₁}	E - {e₂}	E - {e₃}	E - {e₄}	E - {e₅}	E - {e₆}	E - {e₇}	E - {e₈}
(u₁, u₂)	–0.04	–0.24	0.34	0.47	–0.15	0.49	–0.44	–0.18	–0.57
(u₁, u₃)	0.21	–0.47	0.66	0.15	1.06	0.04	0.46	0.16	–0.59
(u₁, u₄)	–1.01	–1.42	–0.15	–0.38	–1.14	–0.38	–1.51	–0.51	–1.58
(u₁, u₅)	0.2	0.02	0.96	0.24	0.16	0.31	0.05	0.35	–0.69
(u₁, u₆)	0.19	–0.28	0.44	0.33	0.9	0.07	–0.21	0.36	–0.28
(u₂, u₃)	0.25	–0.23	0.32	–0.32	1.21	–0.45	0.9	0.34	–0.02
(u₂, u₄)	–0.97	–1.18	–0.49	–0.85	–0.99	–0.87	–1.07	–0.33	–1.01
(u₂, u₅)	0.24	0.26	0.62	–0.23	0.31	–0.18	0.49	0.53	–0.12
(u₂, u₆)	0.23	–0.04	0.1	–0.14	1.05	–0.42	0.23	0.54	0.29
(u₃, u₄)	–1.22	–0.95	–0.81	–0.53	–2.2	–0.42	–1.97	–0.67	–0.99
(u₃, u₅)	–0.01	0.49	0.3	0.09	–0.9	0.27	–0.41	0.19	–0.1
(u₃, u₆)	–0.02	0.19	–0.22	0.18	–0.16	0.03	–0.67	0.2	0.31
(u₄, u₅)	1.21	1.44	1.11	0.62	1.3	0.69	1.56	0.86	0.89
(u₄, u₆)	1.2	1.14	0.59	0.71	2.04	0.45	1.3	0.87	1.3
(u₅, u₆)	–0.01	–0.3	–0.52	0.09	0.74	–0.24	–0.26	0.01	0.41

Step 3: Suppose there exists a difference-based parameter reduction A of E and |A| = q. Let q = 1.

First, we get that e₁, e₂, e₄, e₆, e₈ ∉ A by Theorem 4.3, and then we get that e₅, e₇ ∉ A by Corollary 4.3. If there exists a parameter reduction whose cardinality is 1, then A = {e₃}. Finally, it is determined by Corollary 4.1 that {e₃} is not the difference-based parameter reduction of E, i.e., |A|≠1.

Step 4: Let q = 2.

First by Theorem 4.3 we have e₄, e₆, e₈ ∉ A. Then by Corollary 4.3 we have e₂ ∉ A. The following determines the parameters in {e₁, e₃, e₅, e₇} that must belong to A. According to Theorem 4.2 we get that e₁, e₃ must belong to A. If there is the difference-based parameter reduction of E whose cardinality is 2, then A = {e₁, e₃}. Finally, use Corollary 4.1 to determine whether {e₁, e₃} is the difference-based parameter reduction. Clearly {e₁, e₃} is not the difference-based parameter reduction, so |A|≠2.

Step 5: Let q = 3. First, we get e₄, e₆ ∉ A according to Theorem 4.3. Then we find the parameter e₈ in {e₁, e₂, e₃, e₅, e₇, e₈} that must belong to A according to Theorem 4.2.

Let L₃ = {{e₁, e₂, e₈} , {e₁, e₃, e₈} , {e₁, e₅, e₈} , {e₁, e₇, e₈} , {e₂, e₃, e₈} , {e₂, e₅, e₈} , {e₂, e₇, e₈} , {e₃, e₅, e₈} , {e₃, e₇, e₈} , {e₅, e₇, e₈}}}.

When D_E (u_i, u_j) >0, calculate M₃ (u_i, u_j), and when D_E (u_i, u_j) <0, calculate m₃ (u_i, u_j). According to Theorem 4.4, we get that {e₁, e₂, e₈}, {e₁, e₅, e₈} , {e₁, e₇, e₈} , {e₂, e₇, e₈} are not the difference-based parameter reduction of E.

Let Z₃ = {{e₁, e₂, e₈} , {e₁, e₅, e₈} , {e₁, e₇, e₈} , {e₂, e₇, e₈}} , H₃ = L₃ - Z₃ = {{e₁, e₃, e₈}, {e₂, e₃, e₈} , {e₂, e₅, e₈} , {e₃, e₅, e₈} , {e₃, e₇, e₈} , {e₅, e₇, e₈}}.

The following check if there exists a parameter subset in H₃ that satisfies Corollary 4.1. By calculation we find that {e₂, e₃, e₈} and {e₃, e₅, e₈} satisfy the conditions of Corollary 4.1, so both {e₂, e₃, e₈} and {e₃, e₅, e₈} are the difference-based parameter reduction of E.

The above three examples illustrate that our proposed algorithm is efficient and feasible.

5 Comparative analysis

In this section, we will compare our proposed algorithm with the algorithms from references [21, 27] and [29] in terms of scope of application, computational complexity and cardinality of parameter reduction, respectively. Suppose (F, E) is a fuzzy soft set over U = {u₁, u₂, ⋯ , u_n}, E = {e₁, e₂, ⋯ , e_m}. In the following, we compare the above four algorithms for parameter reduction of this fuzzy soft set (F, E) from the above three aspects.

5.1 Scope of application

(i) The algorithms in [21 , 29]

The algorithms in [21 , 29] do not work for all fuzzy soft sets. For the algorithm proposed in [21], we can find parameter reduction of fuzzy soft set based on this algorithm only if there exists a parameter subset A satisfying f_A (u₁) = f_A (u₂) = ⋯ = f_A (u_n). For the algorithm proposed in [27], the parameter reduction of the fuzzy soft set can be found by this algorithm only if there exists a parameter subset A satisfying $| f_{A} (u_{i}) - {\bar{C}}_{A} | \leq ɛ$ for i = 1, 2, ⋯ , n, where ɛ = min {|f_E (u_i) - f_E (u_j) |/2, i ≠ j, i, j = 1, 2, ⋯ , n} $, {\bar{C}}_{A} = \frac{1}{n} \sum_{i = 1}^{n} f_{A} (u_{i})$ . For the algorithm proposed in [29], only if there exists a parameter subset satisfying the condition of Definition 3.6 we can use the algorithm to find parameter reduction of fuzzy soft sets. However, not all fuzzy soft sets with existential parameter reduction have a parameter subset that satisfy the condition of the algorithm proposed in [21] or [27] or [29]. Therefore, we cannot use the algorithms proposed in [21, 27] and [29] to find the parameter reduction of any fuzzy soft set for which parameter reduction exists.

(ii) Our proposed algorithm

Our proposed algorithm is applicable to any fuzzy soft set. The proposed algorithm shows that if there exists a parameter subset satisfying the conditions of Corollary 4.1, we can find parameter reduction of a fuzzy soft set. And as long as a fuzzy soft set exists parameter reduction, then we must be able to find a parameter subset satisfying the conditions of Corollary 4.1, which is parameter reduction of this fuzzy soft set. Therefore, as long as there exists parameter reduction for a fuzzy soft set, the parameter reduction for that fuzzy soft set can definitely be found using our proposed algorithm.

Through Examples 3.1–3.3 and Example 4.4 we can find that using the algorithms proposed in [21, 27] and [29] cannot find any parameter reduction of the fuzzy soft set shown in Table 3. Whereas using the algorithm proposed in this paper, we find the parameter reduction of this fuzzy soft set as {e₁, e₄}.

Next, we use the algorithms proposed in [21, 27] and [29] to find the parameter reduction of the fuzzy soft set in Example 4.5, respectively. In Example 4.5, we cannot find any parameter subset of E satisfying f_A (u₁) = f_A (u₂) = f_A (u₃) = f_A (u₄) = f_A (u₅) = f_A (u₆) or satisfying $| f_{A} (u_{i}) - {\bar{C}}_{A} | \leq ɛ$ for i = 1, 2, ⋯ , 6, or satisfying the condition of Definition 3.6, so we cannot find any parameter reduction of the fuzzy soft set using the algorithms proposed in [21, 27] and [29]. However, by our proposed algorithm, the parameter reduction of the fuzzy soft set is found, which is {e₂, e₃, e₈} or {e₃, e₅, e₈}.

5.2 Computational complexity

5.2.1 Reduction path

(i) The algorithms in [21 , 29]

It is known from the algorithms proposed in [21, 27] and [29] that if there exists a parameter subset A satisfying the conditions proposed in [21] or [27] or [29], then E - A is a parameter reduction of the fuzzy soft set. However, none of these algorithms gives a specific way to find A. When we use these algorithms to find the parameter reduction of a fuzzy soft set, we can only find the parameter reduction of a fuzzy soft set by continuously trying different parameter subsets to verify whether they satisfy the corresponding conditions until we find a parameter subset that meets the conditions. And the larger the cardinality of E is, the less easy it is to find the parameter subset A that satisfies the conditions, then the more difficult it is to find the parameter reduction of the fuzzy soft set using the algorithms proposed in [21, 27] and [29]. Obviously, these algorithms only give conditions to discriminate whether a parameter subset is a parameter reduction or not, and do not give a specific reduction path to facilitate us to find the parameter reduction of a fuzzy soft set.

(ii) Our proposed algorithm

Our proposed algorithm not only gives the conditions that need to be satisfied to discriminate whether a parameter subset is parameter reduction of a fuzzy soft set or not, but also gives a specific reduction path to find the parameter reduction. And the proposed algorithm gives a method to discriminate whether a parameter belongs to parameter reduction or not, which in turn reduces the difficulty of finding parameter reduction of fuzzy soft sets.

5.2.2 The number of parameter subsets checked

The algorithms proposed in [21 , 29] and the algorithm proposed in this paper all aim at finding the parameter reduction with the smallest cardinality. Therefore, it is assumed that the algorithms of [21 , 29] all search for the parameter subsets that satisfy the condition in the order of cardinality from the largest to the smallest, respectively. The search ends when a subset satisfying the condition is found. Since none of these algorithms gives a specific method to find the parameter subsets that satisfy the condition, it is necessary to keep trying different subsets to check whether they satisfy the condition, and thus the number of subsets to be checked is large, so the computational complexity of these algorithms is high. In contrast, our proposed algorithm is able to determine some redundant and non-redundant parameters, as well as the sets that are not the parameter reduction, and thus is able to determine the parameter reduction of the fuzzy soft set by checking fewer parameter subsets. Obviously the computational complexity of the proposed algorithm is less compared to the above three algorithms.

Next, in Examples 4.3–4.5, calculate how many parameter subsets need to be checked to find the parameter reduction of the fuzzy soft set using the algorithms proposed in [21 , 29] and the algorithm proposed in this paper, respectively.

In Example 4.3 the algorithm proposed in [21] checks all the parameter subsets but does not find any subset A that satisfies f_A (u₁) = f_A (u₂) = f_A (u₃) = f_A (u₄). Clearly the number of parameter subsets checked by this algorithm is $C_{5}^{4} + C_{5}^{3} + C_{5}^{2} + C_{5}^{1} = 30$ .

The algorithm proposed in [27] successively checks whether the parameter subsets with cardinality 4 and cardinality 3 satisfy $| f_{A} (u_{i}) - {\bar{C}}_{A} | \leq ɛ$ for i = 1, 2, 3, 4 respectively. Eventually, it is found that the parameter subset {e₁, e₂, e₅} with cardinality 3 satisfies the aforementioned condition. Thus the number of parameter subsets checked by this algorithm is $C_{5}^{4} + C_{5}^{3} = 15$ . Similarly, the number of parameter subsets checked by the algorithm proposed in [29] is also $C_{5}^{4} + C_{5}^{3} = 15$ .

In contrast, the algorithm proposed in this paper finds the parameter reduction of the fuzzy soft set by only checking whether {e₃} satisfies the conditions of Corollary 4.1. Thus the number of parameter subsets checked by the algorithm is 1.

In Example 4.4, based on Examples 3.1–3.3 we find that the algorithms proposed in [21 , 29] check all the parameter subsets without finding the parameter reduction of the fuzzy soft set. Obviously these three algorithms must check $C_{5}^{4} + C_{5}^{3} + C_{5}^{2} + C_{5}^{1} = 30$ parameter subsets respectively. Whereas the algorithm proposed in this paper finds the parameter reduction by checking only {e₅} and {e₁, e₄}, so the algorithm needs to check only 2 parameter subsets.

Similarly, in Example 4.5, the algorithms proposed in [21 , 29] all need to check $C_{8}^{7} + C_{8}^{6} + C_{8}^{5} + C_{8}^{4} + C_{8}^{3} + C_{8}^{2} + C_{8}^{1} = 254$ parameter subsets respectively. In contrast, our proposed algorithm can find the parameter reduction of fuzzy soft set by checking only the 8 parameter subsets {e₃}, {e₁, e₃}, {e₁, e₃, e₈}, {e₂, e₃, e₈}, {e₂, e₅, e₈}, {e₃, e₅, e₈}, {e₃, e₇, e₈}, {e₅, e₇, e₈}.

The above comparison on the number of parameter subsets checked is represented in Fig. 2.

Fig. 2

Comparison of the number of parameter subsets checked.

Table 14

Comparison table

Comparison	Proposed algorithm	Algorithm [21]	Algorithm [27]	Algorithm [29]
Parameter reduction	{e₃}	No result	{e₃, e₄}	{e₃, e₄}
for Example 4.3
Parameter reduction	{e₁, e₄}	No result	No result	No result
for Example 4.4
Parameter reduction	{e₂, e₃, e₈},	No result	No result	No result
for Example 4.5	{e₃, e₅, e₈}

5.3 Cardinality of parameter reduction

Compared with the algorithms proposed in [21, 27] and [29], our proposed algorithm is able to reduce more parameters and the final obtained parameter reduction has a smaller cardinality. Moreover, the parameter reduction of fuzzy soft sets derived by our proposed algorithm must be the subset with the smallest cardinality in the parameter subsets that keep the ranking order of all the alternatives unchanged.

It is known from [21] that the parameter reduction of the fuzzy soft set in Example 4.3 cannot be found using the algorithm proposed in [21]. The parameter reduction found using the algorithms proposed in [27] and [29] are all {e₃, e₄}. However the parameter reduction of the fuzzy soft set found using our proposed algorithm is {e₃}. Obviously, based on this example, it can be seen that our proposed algorithm is not only able to find the parameter reduction of this fuzzy soft set, but also to reduce more parameters making the cardinality of the parameter reduction smaller compared to the algorithms proposed in [21 , 29].

The results of the above comparison are summarized by Tables 13 and 14. It is obvious from Tables 13 and 14 that compared to the algorithms proposed in [21 , 29], our proposed algorithm has wider applicability, is able to delete more redundant parameters, and is easier to find parameter reduction of fuzzy soft sets.

Table 13
Comparison table

Comparison Scope of application Reduction path Cardinality of parameter reduction

Proposed algorithm All fuzzy soft sets A specific method is given to find parameter reduction without constantly trying different parameter subsets The obtained parameter reduction must be the one with the smallest cardinality

Algorithms from [21],[27] and [29] Not all fuzzy soft sets It is necessary to keep trying different parameter subsets until the required conditions are met in order to find parameter reduction The cardinality of the obtained parameter reduction is not necessarily the smallest

Comparison	Scope of application	Reduction path	Cardinality of parameter reduction
Proposed algorithm	All fuzzy soft sets	A specific method is given to find parameter reduction without constantly trying different parameter subsets	The obtained parameter reduction must be the one with the smallest cardinality
Algorithms from [21],[27] and [29]	Not all fuzzy soft sets	It is necessary to keep trying different parameter subsets until the required conditions are met in order to find parameter reduction	The cardinality of the obtained parameter reduction is not necessarily the smallest

6 Conclusion and future work

The purpose of this paper is to propose a new parameter reduction method for fuzzy soft sets based on the choice value criteria, aiming to overcome the drawbacks of existing parameter reduction methods for fuzzy soft sets with the same decision criteria.

6.1 The main contributions of this paper

(i) The definition of the difference-based parameter reduction of fuzzy soft sets has been introduced, which is easy to satisfy and relatively flexible.

(ii) A method is provided to identify partially redundant parameters, non-redundant parameters, and partially parameter subsets that are not parameter reduction, which reduces the computational complexity of our algorithm.

(iii) An algorithm with high applicability is proposed, which overcomes the shortcomings of the algorithms proposed in [21 , 29] that have low applicability due to overly stringent reduction conditions.

(iv) The parameter reduction of fuzzy soft sets with the smallest cardinality can definitely be found using the proposed algorithm.

6.2 Limitations of the proposed method

(i) We have not yet proposed a method to determine the range of the cardinality of parameter reduction in fuzzy soft sets. Therefore, we can only search for the parameter reduction in ascending order of the cardinality. When |E| increases, the proposed algorithm inevitably faces a greater workload.

(ii) Through the relevant theorems and corollaries proposed in this paper, only some redundant and non-redundant parameters in fuzzy soft sets can be determined, and how to find more redundant and non-redundant parameters needs further research.

6.3 Future work

(i) Finding a way to determine the range of the cardinality of parameter reduction in fuzzy soft sets as well as finding a method to identity remove more redundant and non-redundant parameters can reduce the computational complexity of the proposed algorithm, which is very interesting work in the future.

(ii) We will apply the proposed difference-based parameter reduction algorithm to soft sets and other extensions of soft set theory as future work. For example, based on the definition of parameter reduction of fuzzy soft sets introduced in this paper, we will propose the definition of the difference-based parameter reduction in soft sets, and further investigate the difference-based parameter reduction algorithm for soft sets. And we also will apply the proposed theory and methods to other extensions of soft set theory like (a, b)- fuzzy soft sets, recently defined in [35].

Footnotes

Acknowledgments

This work was partially supported by the 2022 Annual Project of Educational Science of Shanxi Province “the 14th Five-Year Plan” (GH-220769), the Natural Science Foundation of Shanxi Province (202103021224254) and the Research Project of Shanxi Technology and Business College (202258).

References

Molodtsov

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

Pawlak

, Rough sets, International Journal of Computer and Information Science 11 (1982), 341–356.

Zadeh

L.A.

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

Maji

P.K.

, Biswas

and Roy

A.R.

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

Maji

P.K.

, Roy

A.R.

and Biswas

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

Shraddha Harode , Manoj Jha , Namita Srivastava and Sujoy Das , Fuzzy soft set based decision approach for financial trading, Advances in Modelling and Analysis C 73(3) (2018), 102–111.

Fehmi Aziz , Faheem Tahir , Sadia Midhat , Shafaq Naz and Naveeda Akhtar Qureshi , Fuzzy soft set theory: Application of classification rules in decision making during medical diagnosis*, Journal of Intelligent & Fuzzy Systems 41(1) (2021), 2377–2385.

Amit Kumar Mishra , Ramakant Bhardwaj , Nisheeth Joshi and Iti Mathur , A fuzzy soft set based novel method to destabilize the terrorist network, Journal of Intelligent & Fuzzy Systems 43(1) (2022), 35–48.

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.

10.

Zhi Kong , Liqun Gao and Lifu Wang , Comment on “A fuzzy soft set theoretic approach to decision making problems”, Journal of Computational and Applied Mathematics 223(2) (2009), 540–542.

11.

Feng Feng , Young Bae Jun , Xiaoyan Liu and Lifeng Li , An adjustable approach to fuzzy soft set based decision making, Journal of Computational and Applied Mathematics 234(1) (2010), 10–20.

12.

Zhi Kong , Lifu Wang and Zhaoxia Wu , Application of fuzzy soft set in decision making problems based on grey theory, Journal of Computational and Applied Mathematics 236(6) (2011), 1521–1530.

13.

José Carlos Alcantud

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

14.

Hongxiang Tang , A novel fuzzy soft set approach in decision making based on grey relational analysis and Dempster-Shafer theory of evidence, Applied Soft Computing 31 (2015), 317–325.

15.

Qinrong Feng and Fenfen Wang , A discernibility matrix approach to fuzzy soft sets based decision making problems, Intelligent Data Analysis 22(3) (2018), 659–674.

16.

Lu Wang and Keyun Qin , Modal-style operators on fuzzy soft sets and their application to decision making, Journal of Intelligent & Fuzzy Systems 36(6) (2019), 6381–6392.

17.

Al blowi

S.A.

, El Sayed

and El Safty

M.A.

, Decision Making Based on Fuzzy Soft Sets and Its Application in COVID-19, Intelligent Automation & Soft Computing 30(3) (2021), 961–972.

18.

Shawkat Alkhazaleh , Effective Fuzzy Soft Set Theory and Its Applications, Applied Computational Intelligence and Soft Computing 2022 (2022), 1–12.

19.

Maji

P.K.

, Biswas

and Roy

A.R.

, An application of soft sets in a decision making problem, Computers and Mathematics with Applications 44(8-9) (2002), 1077–1083.

20.

Degang Chen , Tsang

E.C.C.

, Daniel Yeung

and Xizhao Wang , The parameterization reduction of soft sets and its applications, Computers and Mathematics with Applications 49(5-6) (2005), 757–763.

21.

Zhi Kong , Liqun Gao , Lifu Wang and Steven Li , The normal parameter reduction of soft sets and its algorithm, Computers and Mathematics with Applications 56(12) (2008), 3029–3037.

22.

Xiuqin Ma , Norrozila Sulaiman , Hongwu Qin , Tutut Herawan and Jasni Mohamad Zain , A new efficient normal parameter reduction algorithm of soft sets, Computers and Mathematics with Applications 62(2) (2011), 588–598.

23.

Zhi Kong , Wenhua Jia , Guodong Zhang and Lifu Wang , Normal parameter reduction in soft set based on particle swarm optimization algorithm, Applied Mathematical Modelling 39(16) (2015), 4808–4820.

24.

Banghe Han , Yongming Li and Shengling Geng , 0-1 linear programming methods for optimal normal and pseudo parameter reductions of soft sets, Applied Soft Computing 54 (2017), 467–484.

25.

Abid Khan and Yuanguo Zhu , An improved algorithm for normal parameter reduction of soft set, Journal of Intelligent & Fuzzy Systems 37(2) (2019), 2953–2968.

26.

Hongwu Qin , Qinghua Fei , Xiuqin Ma and Wanghu Chen , A new parameter reduction algorithm for soft sets based on chi-square test, Applied Intelligence 51(11) (2021), 7960–7972.

27.

Zhi Kong , Lifu Wang , Zhaoxia Wu and Dexuan Zou , A new parameter reduction in fuzzy soft sets, 2012 IEEE International Conference on Granular Computing (2012), 730–732.

28.

Xiuqin Ma and Hongwu Qin , A distance-based parameter reduction algorithm of fuzzy soft sets, IEEE Access 6 (2018), 10530–10539.

29.

Abid Khan and Yuanguo Zhu , A Novel Approach to Parameter Reduction of Fuzzy Soft Set, IEEE Access 7 (2019), 128956–128967.

30.

Zhi Kong , Jianwei Ai , Lifu Wang , Piyu Li , Lianjie Ma and Fuqiang Lu , New normal parameter reduction method in fuzzy soft set theory, IEEE Access 7 (2019), 2986–2998.

31.

Xiuqin Ma , Qinghua Fei , Hongwu Qin , Xiaoyan Zhou and Huifang Li , New Improved Normal Parameter Reduction Method for Fuzzy Soft Set, IEEE Access 7 (2019), 154912–154921.

32.

Zhiming Zhang , The parameter reduction of fuzzy soft sets based on soft fuzzy rough sets, Advances in Fuzzy Systems 2013 (2013), 1–12.

33.

Azadeh Zahedi Khameneh and Adem Kılıçman , Parameter reduction of fuzzy soft sets: An adjustable approach based on the three-way decision, International Journal of Fuzzy Systems 20(3) (2018), 928–942.

34.

Muhammad Akram , Ghous Ali , José Alcantud

C.R.

and Fatia Fatimah , Parameter reductions in N-soft sets and their applications in decision-making, Expert Systems 38(1) (2021), 1–15.

35.

Tareq Al-shami

, José Carlos Alcantud

and Abdelwaheb Mhemdi , New generalization of fuzzy soft sets: (a, b)-Fuzzy soft sets, AIMS Mathematics 8(2) (2023), 2995–3025.

A novel parameter reduction method for fuzzy soft sets

Abstract

Keywords

1 Introduction

2 Preliminaries

Table 1 Tabular representation of the fuzzy soft set (F, E) (F, E) e 1 e 2 e 3 e 4 e 5 u 1 0.1 0.6 0.3 0.2 0.4 u 2 0.5 0.7 0.1 0.5 0.6 u 3 0.4 0.8 0.2 0.1 0.7 u 4 0.2 0.5 0.4 0.3 0.4

3.1 The algorithm proposed in [21]

Table 3 Tabular representation of the fuzzy soft set (F, E) (F, E) e 1 e 2 e 3 e 4 e 5 u 1 0.1 0.2 0.4 0.8 0.6 u 2 0.6 0.5 0.2 0.6 0.7 u 3 0.2 0.3 0.1 0.2 0.5 u 4 0.3 0.4 0.2 0.5 0.1 u 5 0.5 0.2 0.5 0.6 0.7

3.3 The algorithm proposed in [29]

4 A novel difference-based parameter reduction method for fuzzy soft sets

5.1 Scope of application

5.2 Computational complexity

5.2.1 Reduction path

5.2.2 The number of parameter subsets checked

6.1 The main contributions of this paper

6.2 Limitations of the proposed method

6.3 Future work

Footnotes

Acknowledgments

References

Table 1
Tabular representation of the fuzzy soft set (F, E)

(F, E) e ₁ e ₂ e ₃ e ₄ e ₅

u ₁ 0.1 0.6 0.3 0.2 0.4

u ₂ 0.5 0.7 0.1 0.5 0.6

u ₃ 0.4 0.8 0.2 0.1 0.7

u ₄ 0.2 0.5 0.4 0.3 0.4

Table 3
Tabular representation of the fuzzy soft set (F, E)

(F, E) e ₁ e ₂ e ₃ e ₄ e ₅

u ₁ 0.1 0.2 0.4 0.8 0.6

u ₂ 0.6 0.5 0.2 0.6 0.7

u ₃ 0.2 0.3 0.1 0.2 0.5

u ₄ 0.3 0.4 0.2 0.5 0.1

u ₅ 0.5 0.2 0.5 0.6 0.7