Some novel decision making algorithms for intuitionistic fuzzy soft set

Abstract

This paper is designed to show deeper standpoints for dealing the decision making issues based on intuitionistic fuzzy soft set (IFSS). Firstly, we propose a novel definition and formula of similarity measure on intuitionistic fuzzy information, which can reserve more original judgment information. Afterwards, we present a combination weight that take the objective weight (obtained by grey system theory) and the subjective weight into consideration. Later, three intuitionistic fuzzy soft methods based on similarity measure, MABAC and EDAS are presented for dealing the decision making issues. Finally, the validity of methods are stated by some practical examples. The key characteristics of the developed methods have no strict requirements for decision data and possess a stronger ability in differentiating the best alternative.

Keywords

1. Introduction

Soft set (SS), explored by Molodtsov [1], is an effect parametric tool for managing uncertain issues [2 –5]. Zhang et al. [6] applied soft sets into pseudo-BCI algebras and discussed their relations. Li et al. [7] established the evaluation index set for investigating a novel screening alternative issue based on soft set. The study of compounded models uniting SS with existing mathematical models are an important academic topic. Maji et al. [8] originally developed fuzzy soft set (FSS), a more broader concept uniting fuzzy set and SS. Zhan and Zhu [9] presented the notion of Z-soft rough fuzzy sets of hemirings. Yang et al. [10] developed the notion of interval-valued fuzzy soft set (IVFSS) and employed them in decision making issue [11]. Uniting intuitionistic fuzzy set (IFS) [12] with SS, Maji et al. [13] defined intuitionsitic fuzzy soft set (IFSS). In addition, diverse application scenarios of IFSS are in rapid development such as decision making [14 –23], aggregation operators [24 –26], medical diagnosis [27], information measure [28, 29], image process [33].

Since the shortcomings [14 , 19] (discrimination issues and strict requirements for decision data) of the existing decision making approaches, it will lead to the difficulty in choosing the best alternative for decision makers or experts. Hence, the fundamental purpose of this paper is to solve the two drawbacks above by proposing three decision making methods, which not only possess the a stronger ability in differentiating the best alternative, but also has no strict requirements for decision data. The decision making approaches are presented and discussed as follows:

For the sake of computing the similarity measure of two IFSs, the novel definition of similarity measure is presented. The presented similarity measure can reserve more primitive decision information compared with the existing reference [34 –36]. Afterwards, we employ the developed similarity measure into decision making issues.

EDAS (Evaluation based on Distance from Average Solution), initially developed by Ghorabaee et al. [37], is a novel decision making algorithm. It is quite effective in the case of possessing some incompatible parameters. The optimal alternative possesses lesser distance from ideal solution with larger distance from nadir solution employing the EDAS. Ghorabaee et al. [38] employed the EDAS approach in supplier selection. EDAS approach has been successfully employed in many domains such as investment decision [39], software projects selection [40].

MABAC (Multi-Attributive Border Approximation area Comparison) approach is a fire-new approach developed by Pamucar and Cirovic [41], which has an oversimplified and systematic computation procedure, and a sufficient logic that shows certain basics of decision making. MABAC approach has been successfully employed in many domains such as R&D project selection [44], investment decision [43], outsourcing provider selection [44], computer language selection [45], selection of university web pages [46].

Considering that various parameter weights would affect the ordering results of the given alternatives, we explore a new approach to ascertain the parameter weights by uniting subjective elements with objective elements. Such model is differ from other than the existing approaches and can be split into two strategies: one is subjective weight assessment approach, the other is the objective weight determination approach, which could be calculated by grey system theory (GST) [47]. The subjective weighting approach [18–21 , 27] concentrates on the predilection information of decision makers (DMs), but ignores the objective weight information. The objective weighting approach [17] does not consider the preference of decision makers. In other words, such methods do not consider the risk attitude of DMs. Consequently, we propose a combination weight (combined objective weights with subjective weights) for achieving the each parameters’ weight.

To achieve the aim, the key contributions are listed in the following.

(i) A novel combination weight model is developed for alleviating the effect of subjective element and objective element.

(ii) The novel axiomatic definition and formula of similarity measure are presented for reserving more original judgment information.

(iii) Some explored approaches (EDAS, MABAC and similarity measure) are initiated for solving two challenges carried by the existing decision making approaches [14 , 19].

The rest of this paper is listed as follows: In Section 2, some essential notions of IFS, SS and IFSS. In Section 3, the novel axiomatic definition and formula of similarity measure based on IFS are designed. In Section 4, some intuitionistic fuzzy soft decision methods (similarity measure, EDAS and MABAC) are presented. In Section 5, a hotel group company selection example is shown for stating the validity of the developed methods. In Section 6, a comparison with the existing decision making approaches is constructed for stating the effectiveness of proposed methods. The paper arrivals at a conclusion in Section 7.

2. Preliminaries

In this section, we firstly retrospect some essential notions of IFS, SS, IFSS and their properties.

2.1. Intuitionistic fuzzy set

Definition 1. [12] Let X be a domain of discourse, the intuitionistic fuzzy set (IFS) A in X is summarized as follows: $A = {< x, μ (x), ν (x) > ∣ x \in X},$ (1) where μ (x) and ν (x) signify the membership degree and the non-membership degree, respectively, with the condition 0 ≤ μ (x) + ν (x) ≤1. Besides, π (x) =1 - μ (x) - ν (x) is called the indeterminacy degree of x to A. For simplicity, Xu and Yager [48] named a = (μ_a, ν_a) as intuitionistic fuzzy number (IFN).

Definition 2. [49] Let x = (μ_x, ν_x) and y = (μ_y, ν_y) be two IFNs, then the Euclidean distance can be signified as follows: $d_{e} (x, y) = \sqrt{\frac{1}{2} [(μ_{x} - μ_{y})^{2} + (ν_{x} - ν_{y})^{2} + (π_{x} - π_{y})^{2}]} .$ (2)

Definition 3. [12, 50] Let A and B be two IFSs on X, then

(1) A ⊆ B, iff μ_A (x) ≤ μ_B (x) and ν_A (x) ≥ ν_B (x) , ∀ x ∈ X;

(2) A^c = {< x, ν_A (x) , μ_A (x) > ∣ x ∈ X};

(3) A ⊕ B = {< x, μ_A (x) + μ_B (x) - μ_A (x) μ_B (x) , ν_A (x) ν_B (x) > ∣ x ∈ X};

(4) A ⊗ B = {< x, μ_A (x) μ_B (x) , ν_A (x) + ν_B (x) - ν_A (x) ν_B (x)} > ∣ x ∈ X};

(5) λA = {< x, 1 - (1 - μ_A (x)) ^λ, (ν_A (x)) ^λ} > ∣ x ∈ X} , λ > 0;

(6) A^λ = {< x, (μ_A (x)) ^λ, 1 - (1 - ν_A (x)) ^λ} > ∣ x ∈ X} , λ > 0.

Definition 4. [50] Let a_j = (μ_j, ν_j) (j = 1, 2, ⋯ , n) be a range of IFNs, then intuitionistic fuzzy weighted averaging (IFWA) operators of dimension n are mapping IFWA : Θⁿ → Θ, and signified as follows: $\begin{matrix} IFWA (a_{1}, a_{2}, \dots, a_{n}) \\ = \oplus_{j = 1}^{n} w_{j} a_{j} \\ = (1 - \prod_{j = 1}^{n} {(1 - μ_{j})}^{w_{j}}, \prod_{j = 1}^{n} {(ν_{j})}^{w_{j}}), \end{matrix}$ (3) where w = (w₁, w₂, ⋯ , w_n) be the weight vector with w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ .

Definition 5. [51] For an IFN a = (μ_a, ν_a), its score function can be defined as follows: $s (a) = \frac{e^{μ_{a} - ν_{a}}}{π_{a} + 1} .$ (4)

For two IFNs a₁ = (μ₁, ν₁) and a₂ = (μ₂, ν₂), if s (a₁) > s (a₂), then a₁ > a₂; if s (a₁) = s (a₂), then a₁ = a₂.

2.2. Intuitionistic fuzzy soft set

Definition 6. [13] A pair (F, A) is named an intuitionistic fuzzy soft set (IFSS) over U, where F is a mapping offered by F : A → H (U).

An IFSS is a mapping from some parameters to H (U), and it is not a set, but a parameterized family of IF subset of U. For e ∈ A, F (e) can be deemed as the e-approximate elements of the IFSS. Suppose F (e) (x) signified the membership value that object x holds parameter e, then F (e) can be written as an IFS that F (e) = {x/F (e) (x) ∣ x ∈ U} = {x/(μ_F (e) (x) , ν_F (e) (x)) ∣ x ∈ U}.

Example 1. Suppose that U = {x₁, x₂, x₃, x₄} and A = {e₁, e₂, e₃, e₄}. Let (F, A) be an IFSS over U, signified as follows:

F (e₁) = {x₁/(0.2, 0.7) , x₂/(0.2, 0.6) , x₃/(0.4, 0.6) , x₄/(0.2, 0.6)}, F (e₂) = {x₁/(0.1, 0.4) , x₂/(0.6, 0.3) , x₃/(0.3, 0.7) , x₄/(0.2, 0.7)}, F (e₃) = {x₁/(0.2, 0.4) , x₂/(0.2, 0.7) , x₃/(0.3, 0.6) , x₄/(0.4, 0.6)}, F (e₄) = {x₁/(0.2, 0.4) , x₂/(0.2, 0.6) , x₃/(0.4, 0.6) , x₄/(0.5, 0.4)}.

Then, the tabular form of (F, A) is shown in Table 1

Table 1
The intuitionistic fuzzy soft set (F, A)

e ₁ e ₂ e ₃ e ₄

x ₁ (0.2,0.7) (0.1,0.4) (0.2,0.4) (0.2,0.4)

x ₂ (0.2,0.6) (0.6,0.3) (0.2,0.7) (0.2,0.6)

x ₃ (0.4,0.6) (0.3,0.7) (0.3,0.6) (0.4,0.6)

x ₄ (0.2,0.6) (0.2,0.7) (0.4,0.6) (0.5,0.4)

	e ₁	e ₂	e ₃	e ₄
x ₁	(0.2,0.7)	(0.1,0.4)	(0.2,0.4)	(0.2,0.4)
x ₂	(0.2,0.6)	(0.6,0.3)	(0.2,0.7)	(0.2,0.6)
x ₃	(0.4,0.6)	(0.3,0.7)	(0.3,0.6)	(0.4,0.6)
x ₄	(0.2,0.6)	(0.2,0.7)	(0.4,0.6)	(0.5,0.4)

Definition 7. Let (F, A) and (G, B) be two IFSSs over the conjunct universe U. (F, A) is named as intuitionistic fuzzy soft subset of (G, B) if A ⊆ B, μ_F (e) (x) ≤ μ_G (e) (x) , ν_F (e) (x) ≥ ν_G (e) (x) , ∀ e ∈ A, x ∈ U. We signify it by (F, A) ⊆ (G, B).

3. A novel intuitionistic fuzzy similarity measure

In the sake of computing similarity measure (SM) of two IFSs, we develop novel axiomatic definition and formula of SM, which adopts IFN. Comparing with the existing SM [34 –36], the proposed SM can reserve more primitive judgment preference information.

Definition 8. Let A₁, A₂ and A₃ be three IFSs on X. A similarity measure S^Δ (A₁, A₂) is a mapping S^Δ : IFS (X) × IFS (X) → IFN, having the following characters:

(1) S^Δ (A₁, A₂) is an IFN;

(2) S^Δ (A₁, A₂) = (1, 0), iff A₁ = A₂;

(3) S^Δ (A₁, A₂) = S^Δ (A₂, A₁);

(4) If A₁ ⊆ A₂ ⊆ A₃, then S^Δ (A₁, A₂) ⊇ S^Δ (A₁, A₃) and S^Δ (A₂, A₃) ⊇ S^Δ (A₁, A₃).

Theorem 1. Let A_i and A_k be two IFSs, then S^Δ (A_i, A_k) is a similarity measure. $\begin{matrix} S^{Δ} (A_{i}, A_{k}) = (\min {L (A_{i}, A_{k}), M (A_{i}, A_{k})}, \\ 1 - \max {L (A_{i}, A_{k}), M (A_{i}, A_{k})}), \end{matrix}$ (5) where $L (A_{i}, A_{k}) = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{μ_{ij}}, e^{μ_{kj}}}}{\sum_{j = 1}^{n} w_{j} \max {e^{μ_{ij}}, e^{μ_{kj}}}}, M (A_{i}, A_{k}) = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{1 - ν_{ij}}, e^{1 - ν_{kj}}}}{\sum_{j = 1}^{n} w_{j} \max {e^{1 - ν_{ij}}, e^{1 - ν_{kj}}}}$ , and w_j is the weight of jth IFN.

Proof. In order to prove the S^Δ (A_i, A_k), it should comply with axiomatic requirements.

(1) Since $0 \leq L (A_{i}, A_{k}) = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{μ_{ij}}, e^{μ_{kj}}}}{\sum_{j = 1}^{n} w_{j} \max {e^{μ_{ij}}, e^{μ_{kj}}}} \leq 1$ ,

$0 \leq M (A_{i}, A_{k}) = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{1 - ν_{ij}}, e^{1 - ν_{kj}}}}{\sum_{j = 1}^{n} w_{j} \max {e^{1 - ν_{ij}}, e^{1 - ν_{kj}}}} \leq 1$ , so, 0 ≤ min {L (A_i, A_k) , M (A_i, A_k)} ≤1, 0 ≤ 1 - max {L (A_i, A_k) , M (A_i, A_k)} ≤1.

Furthermore,

0 ≤ min {L (A_i, A_k) , M (A_i, A_k)} +1 - max {L (A_i, A_k) , M (A_i, A_k)} ≤ 1.

Consequently, S^Δ (A_i, A_k) is an IFN.

(2)

① Necessity:

Since S^Δ (A_i, A_k) = (1, 0), we have

min {L (A_i, A_k) , M (A_i, A_k)} =1, max {L (A_i, A_k) , M (A_i, A_k)}=1.

It means that L (A_i, A_k) = M (A_i, A_k) =1.

Furthermore,

$L (A_{i}, A_{k}) = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{μ_{ij}}, e^{μ_{kj}}}}{\sum_{j = 1}^{n} w_{j} \max {e^{μ_{ij}}, e^{μ_{kj}}}} = 1$ ,

$M (A_{i}, A_{k}) = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{1 - ν_{ij}}, e^{1 - ν_{kj}}}}{\sum_{j = 1}^{n} w_{j} \max {e^{1 - ν_{ij}}, e^{1 - ν_{kj}}}} = 1$ .

Based on the randomicity of w_j, we can have μ_ij = μ_kj, ν_ij = ν_kj, i.e., A_i = A_k.

② Sufficiency:

Since A_i = A_k, we have μ_ij = μ_kj, ν_ij = ν_kj.

Furthermore,

$\sum_{j = 1}^{n} w_{j} \min {e^{μ_{ij}}, e^{μ_{kj}}} = \sum_{j = 1}^{n} w_{j} \max {e^{μ_{ij}}, e^{μ_{kj}}},$

$\sum_{j = 1}^{n} w_{j} \min {e^{1 - ν_{ij}}, e^{1 - ν_{kj}}} = \sum_{j = 1}^{n} w_{j} \max {e^{1 - ν_{ij}}, e^{1 - ν_{kj}}}$ .

Consequently, S^Δ (A_i, A_k) = (1, 0).

(3) It is obvious.

(4) If A₁ ⊆ A₂ ⊆ A₃, then ∀j, μ_1j ≤ μ_2j ≤ μ_3j and ν_1j ≥ ν_2j ≥ ν_3j.

Hence,

$L (A_{1}, A_{3}) = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{μ_{1 j}}, e^{μ_{3 j}}}}{\sum_{j = 1}^{n} w_{j} \max {e^{μ_{1 j}}, e^{μ_{3 j}}}} = \frac{\sum_{j = 1}^{n} w_{j} e^{μ_{1 j}}}{\sum_{j = 1}^{n} w_{j} e^{μ_{3 j}}} \leq \frac{\sum_{j = 1}^{n} w_{j} e^{μ_{1 j}}}{\sum_{j = 1}^{n} w_{j} e^{μ_{2 j}}} = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{μ_{1 j}}, e^{μ_{2 j}}}}{\sum_{j = 1}^{n} w_{j} \max {e^{μ_{1 j}}, e^{μ_{2 j}}}} = L (A_{1}, A_{2})$ ,

$M (A_{1}, A_{3}) = \frac{\sum_{j = 1}^{n} w_{j} \min {1 - e^{ν_{1 j}}, 1 - e^{ν_{3 j}}}}{\sum_{j = 1}^{n} w_{j} \max {1 - e^{ν_{1 j}}, 1 - e^{ν_{3 j}}}} = \frac{\sum_{j = 1}^{n} w_{j} (1 - e^{ν_{1 j}})}{\sum_{j = 1}^{n} w_{j} (1 - e^{ν_{3 j}})} \leq \frac{\sum_{j = 1}^{n} w_{j} (1 - e^{ν_{1 j}})}{\sum_{j = 1}^{n} w_{j} (1 - e^{ν_{2 j}})} = \frac{\sum_{j = 1}^{n} w_{j} \min {1 - e^{ν_{1 j}}, 1 - e^{ν_{2 j}}}}{\sum_{j = 1}^{n} w_{j} \max {1 - e^{ν_{1 j}}, 1 - e^{ν_{2 j}}}} = M (A_{1}, A_{2})$ .

Furthermore, min {L (A₁, A₂) , M (A₁, A₂)} ≥ min {L (A₁, A₃) , M (A₁, A₃)} , 1 - max {L (A₁, A₂) , M (A₁, A₂)} ≤1 - max {L (A₁, A₂) , M (A₁, A₂)}.

Consequently, S^Δ (A₁, A₂) ⊇ S^Δ (A₁, A₃).

Similarly, S^Δ (A₂, A₃) ⊇ S^Δ (A₁, A₃).

Theorem 2. Let A_i and A_k be two IFSs, then

(1) S^Δ (A_i, A_k) = S^Δ (A_i ∩ A_k, A_i ∪ A_k);

(2) S^Δ (A_i, A_i ∩ A_k) = S^Δ (A_k, A_i ∪ A_k);

(3) S^Δ (A_i, A_i ∪ A_k) = S^Δ (A_k, A_i ∩ A_k).

Proof. We only prove axiom (1), the axioms (2) and (3) can be proved in the same way.

(1) Since

$L (A_{i} \cap A_{k}, A_{i} \cup A_{k}) = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{\min {μ_{ij}, μ_{kj}}}, e^{\max {μ_{ij}, μ_{kj}}}}}{\sum_{j = 1}^{n} w_{j} \max {e^{\min {μ_{ij}, μ_{kj}}}, e^{\max {μ_{ij}, μ_{kj}}}}} = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{μ_{ij}}, e^{μ_{kj}}}}{\sum_{j = 1}^{n} w_{j} \max {e^{μ_{ij}}, e^{μ_{kj}}}} = L (A_{i}, A_{k}),$

$M (A_{i} \cap A_{k}, A_{i} \cup A_{k}) = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{1 - \min {ν_{ij}, ν_{kj}}}, e^{1 - \max {ν_{ij}, ν_{kj}}}}}{\sum_{j = 1}^{n} w_{j} \max {e^{1 - \min {ν_{ij}, ν_{kj}}}, e^{1 - \max {ν_{ij}, ν_{kj}}}}} = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{1 - ν_{ij}}, e^{1 - ν_{kj}}}}{\sum_{j = 1}^{n} w_{j} \max {e^{1 - ν_{ij}}, e^{1 - ν_{kj}}}} = M (A_{i}, A_{k})$ ,

so we can have S^Δ (A_i, A_k) = S^Δ (A_i ∩ A_k, A_i ∪ A_k).

4. Some algorithms for intuitionistic fuzzy soft decision making

4.1. Problem description

Let U = {x₁, x₂, ⋯ , x_m} be a fixed set of m alternatives, E = {e₁, e₂, ⋯ , e_n} be a series of n parameters, and the weight of parameter e_i is w_i,

w_{i} \in [0, 1], \sum_{i = 1}^{n} w_{i} = 1

. IFSS (F, E) can be presented in Table 2. F (e_j) (x_i) = (μ_F (e_j) (x_i) , ν_F (e_j) (x_i)) signify the given IFN of the ith alternative x_i satisfying the jth parameter e_j which are offered by the experts or decision makers.

Table 2
Tabular form of (F, E)

	e₁	e₂	⋯	e_n
x ₁	F (e₁) (x₁)	F (e₂) (x₁)	⋯	F (e_n) (x₁)
x ₂	F (e₁) (x₂)	F (e₂) (x₂)	⋯	F (e_n) (x₂)
⋮	⋮	⋮	⋱	⋮
x _m	F (e₁) (x_m)	F (e₂) (x_m)	⋯	F (e_n) (x_m)

4.2. The method of computing the combination weights

4.2.1. Calculating objective weights: grey relational analysis

Grey system method (GSM) [47] is a nice tool for dealing with small-sample and uncertain information. In theory, if the parameter information about other parameters is more matched in the mean information of parameters, then the parameter includes more decision making information, and the greater the weight is. Hence, a grey relational analysis method to determine the weight of the parameter is given in the following.

Definition 9. Assume R = (F (e_j) (x_i)) _n×m (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n) be an intuitionistic fuzzy soft matrix (IFSM), and S = (s_ij) _m×n (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n) be the score function s (defined in Eq.(4)) of R. Let ${\bar{s}}_{i} = \frac{1}{n} \sum_{j = 1}^{n} s_{ij}$ , then the weight ω_j is signified in the following. $ω_{j} = \frac{1 - \frac{1}{m} (\sum_{i = 1}^{m} (o_{ij}^{q}))^{\frac{1}{q}}}{n - \frac{1}{m} \sum_{j = 1}^{n} (\sum_{i = 1}^{m} (o_{ij}^{q}))^{\frac{1}{q}}},$ (6) where $o_{ij} = \frac{\min_{i} | s_{ij} - {\bar{s}}_{i} | + ξ \max_{i} | s_{ij} - {\bar{s}}_{i} |}{| s_{ij} - {\bar{s}}_{i} | + ξ \max_{i} | s_{ij} - {\bar{s}}_{i} |}$ is the degree of grey mean relational (ξ = 0.5).

In the sake of improving the efficient differentiation, the Euclidean distance (q = 2) is employed.

4.2.2. Calculating the combination weights: the non-linear weighted synthesis

The subjective weight, offered by the experts or DMs directly, is w = {w₁, w₂, ⋯, w_n}, where $\sum_{j = 1}^{n} w_{j} = 1, 0 \leq w_{j} \leq 1$ . The objective weight, calculated by Eq.(6), is ω = {ω₁, ω₂, ⋯ , ω_n}, where $\sum_{j = 1}^{n} ω_{j} = 1, 0 \leq ω_{j} \leq 1$ .

Hence, the combination weight ϖ = {ϖ₁, ϖ₂, ⋯ , ϖ_n} can be signified in the following. $ϖ_{j} = \frac{w_{j} * ω_{j}}{\sum_{j = 1}^{n} w_{j} * ω_{j}},$ (7) where $\sum_{j = 1}^{n} ϖ_{j} = 1, 0 \leq ϖ_{j} \leq 1$ .

The objective weight and subjective weight are summarized by nonlinear weighted synthesis method. Based on the multiplier effect, the greater the objective and subjective weights, the greater the combination weights, and vice versa. We can conclude that the advantage of Eq. (7) overcomes the limitation of Eq.(7), which only considers subjective or objective factors.

Next, the EDAS, MABAC and similarity measure methods with combined weights are employed in solving some decision making issues of IFSSs.

4.3. The EDAS method

In this subsection, a revised EDAS method is developed for handling the decision making issues in intuitionistic fuzzy soft domain. Hence, the arithmetic operations and notions of IFNs are employed for generalizing EDAS approach which is shown in Algorithm 1.

Algorithm 1: EDAS

Choose the significative parameters and potential alternatives, and achieve the IFSS (F, E) which is given in Table 2.

Standard the IFSS (F, E) into (F′, E) by Eq. (8), $\begin{array}{l} F^{'} (e_{j}) (x_{i}) = ({μ^{'}}_{F} (e_{j}) (x_{i}), {v^{'}}_{F} (e_{j}) (x_{i})) \\ = {\begin{cases} (μ_{F} (e_{j}) (x_{i}), v_{F} (e_{j}) (x_{i})), e_{j} \in B, \\ (v_{F} (e_{j}) (x_{i}), μ_{F} (e_{j}) (x_{i})), e_{j} \in C, \end{cases} \end{array}$ (8) where B is set of benefit parameters and C is set of cost parameters.

Calculate the combination weights ϖ by Eq.(7).

Calculate the mean solution for proprietary parameters, signified in the following. $AV = ({AV}_{j})_{1 \times n},$ (9) where $\begin{array}{l} A V_{j} = \frac{1}{m} \oplus_{i = 1}^{m} F (e_{j}) (x_{i}) \\ = \frac{1}{m} (1 - \prod_{i = 1}^{m} (1 - μ_{F} (e_{j}) (x_{i})), \prod_{i = 1}^{m} v_{F} (e_{j}) (x_{i})) \\ = (1 - {(\prod_{i = 1}^{m} (1 - μ_{F} (e_{j}) (x_{i})))}^{\frac{1}{m}}, {(\prod_{i = 1}^{m} v_{F} (e_{j}) (x_{j}))}^{\frac{1}{m}}) . \end{array}$ (10)

Determine the PDA (positive distance from average) with PDA = (P_ij) _m×n and the NDA (negative distance from average) with NDA = (N_ij) _m×n matrixes by means of diverse parameters, signified in the following. $P_{ij} = {\begin{matrix} \frac{\max {0, s (F (e_{j}) (x_{i})) - s ({AV}_{j})}}{s ({AV}_{j})}, e_{j} \in B, \\ \frac{\max {0, s ({AV}_{j}) - s (F (e_{j}) (x_{i}))}}{s ({AV}_{j})}, e_{j} \in C, \end{matrix}$ (11) $N_{ij} = {\begin{matrix} \frac{\max {0, s ({AV}_{j}) - s (F (e_{j}) (x_{i}))}}{s ({AV}_{j})}, e_{j} \in B, \\ \frac{\max {0, s (F (e_{j}) (x_{i})) - s ({AV}_{j})}}{s ({AV}_{j})}, e_{j} \in C, \end{matrix}$ (12) where s (AV_j) and s (F (e_j) (x_i)) are score function of AV_j and F (e_j) (x_i), respectively.

Calculate the weighted sum of NDA and PDA for proprietary alternatives, signified in the following. ${SP}_{i} = \sum_{j = 1}^{n} ϖ_{j} P_{ij},$ (13) ${SN}_{i} = \sum_{j = 1}^{n} ϖ_{j} N_{ij} .$ (14)

Standardize the values of SP_i and SN_i for proprietary alternatives, signified in the following. ${NSP}_{i} = \frac{{SP}_{i}}{\max_{i} {{SP}_{i}}}$ (15) ${NSN}_{i} = 1 - \frac{{SN}_{i}}{\max_{i} {{SN}_{i}}}$ (16)

Determine the appraisal score AS_i (i = 1, 2, ⋯ , m) for all alternatives, signified in the following. ${AS}_{i} = \frac{1}{2} ({NSP}_{i} + {NSN}_{i}),$ (17) where 0 ≤ AS_i ≤ 1.

Choose the optimal alternative(s) by maximization of their appraisal score AS_i.

4.4. The method of MABAC

The MABAC method [41] is a novel credible tool which has a oversimplified computation procedure and stable alternatives’ ranking for decision making. Next, a revised MABAC method under intuitionistic fuzzy soft environment is shown in Algorithm 2.

Algorithm 2: MABAC

Just like what did with the Step 1 to Step 3 presented in Algorithm 1.

Calculate the weighted matrix T = ( t_ij) _m×n by Eq.(18), $\begin{array}{l} t_{i j} = (μ_{t_{i j}}, v_{t_{i j}}) = ϖ_{j} ({μ^{'}}_{F} (e_{j}) (x_{i}), {v^{'}}_{F} (e_{j}) (x_{i})) \\ = (1 - {(1 - {μ^{'}}_{F} (e_{j}) (x_{j}))}^{ϖ j}, {({v^{'}}_{F} (e_{j}) (x_{i}))}^{ϖ j}) . \end{array}$ (18) where ϖ_j is combination weight.

Calculate the border approximation area matrix G = (g_j) _1×n. The border approximation area for each parameter is achieved by Eq.(19).

$\begin{matrix} g_{j} = \prod_{i = 1}^{m} (t_{ij})^{1 / m} \\ = & (\prod_{i = 1}^{m} (μ_{t_{ij}})^{1 / m}, 1 - \prod_{i = 1}^{m} (1 - ν_{t_{ij}})^{1 / m}) . \end{matrix}$ (19)

Compute the matrix D = (d_ij) _m×n by Eq.(20), $d_{ij} = {\begin{matrix} d_{e} (t_{ij}, g_{j}), if s (t_{ij}) > s (g_{j}), \\ 0, if s (t_{ij}) = s (g_{j}), \\ - d_{e} (t_{ij}, g_{j}), if s (t_{ij}) < s (g_{j}), \end{matrix}$ (20) where distance measure d_e is given in Eq.(2), and s (t_ij) , s (g_j) are score function of t_ij and g_j, respectively.

Choose the best alternative(s) by maximization of the Q_i. $Q_{i} = \sum_{j = 1}^{n} d_{ij}, i = 1, 2, \dots, m; j = 1, 2, \dots, n .$ (21)

4.5. The method of similarity measure

The notion of ideal point has been successfully employed in determining the optimal alternative in decision making process. While an ideal alternative does not arise in real issues, it does provide an effective theoretical framework for evaluating alternatives. Consequently, we denote the ideal alternative x^∗ as the IFN $x_{j}^{*} = (μ^{*}, ν^{*}) = (1, 0)$ for ∀j. By employing Eq.(5), the developed similarity measure S^Δ between an alternative x_i and the ideal alternative x^∗ presented by IFSs is shown as follows:

$\begin{matrix} S^{Δ} (x_{i}, x^{*}) & = & (\min {L (x_{i}, x^{*}), M (x_{i}, x^{*})}, \\ 1 - \max {L (x_{i}, x^{*}), M (x_{i}, x^{*})}) \end{matrix}$ (22) where $L (x_{i}, x^{*}) = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{μ_{F} (e_{j}) (x_{i})}, e^{1}}}{\sum_{j = 1}^{n} w_{j} \max {e^{μ_{F} (e_{j}) (x_{i})}, e^{1}}} = \sum_{j = 1}^{n} w_{j} e^{μ_{F} (e_{j}) (x_{i}) - 1}, M (x_{i}, x^{*}) = \frac{\sum_{j = 1}^{n} w_{j} \min {e^{1 - ν_{F} (e_{j}) (x_{i})}, e^{1}}}{\sum_{j = 1}^{n} w_{j} \max {e^{1 - ν_{F} (e_{j}) (x_{i})}, e^{1}}} = \sum_{j = 1}^{n} w_{j} e^{- ν_{F} (e_{j}) (x_{i})}$ .

Algorithm 3: Similarity measure

Just like what did with the Step 1 to Step 3 presented in Algorithm 1.

Compute the similarity measure S (x_i, x^∗) (i = 1, 2, ⋯ , m) by Eq.(22).

Calculate the score function s (S (x_i, x^∗)) of each alternative x_i by Eq.(4).

Choose the best alternative(s) by maximization of the s (S (x_i, x^∗)).

5. Application of the proposed algorithms for IFSSs in a practical problem

Example 2. A hotel group company wants to select a hotel to invest. Assume that there are four potential hotels. The company selects five parameters to evaluate the four hotels. Let U = {x₁, x₂, x₃, x₄} be a set of software projects, E = {e₁, e₂, e₃, e₄, e₅} be a set of parameters, e₁ represents housing area, e₂ represents room numbers, e₃ represents average letting rate, e₄ represents average house price, e₅ represents total capital cost. e₁, e₂, e₃, e₄ are benefit parameters and e₅ is cost parameter. The parameters’ weight is w = (0.3, 0.15, 0.1, 0.35, 0.1) ^T. The decision tabular given by expert is presented in Table 3.

Table 3
Tabular representation of (F, E)

	e ₁	e ₂	e ₃	e ₄	e ₅
x ₁	(0.6, 0.4)	(0.6, 0.3)	(0.7, 0.1)	(0.6, 0.3)	(0.4, 0.5)
x ₂	(0.7, 0.3)	(0.6, 0.2)	(0.6, 0.2)	(0.6, 0.2)	(0.4, 0.4)
x ₃	(0.6, 0.3)	(0.7, 0.2)	(0.6, 0.1)	(0.7, 0.2)	(0.3, 0.5)
x ₄	(0.6, 0.3)	(0.7, 0.2)	(0.7, 0.2)	(0.6, 0.1)	(0.2, 0.6)

Next, we employ the proposed approaches in selecting desirable hotel under intuitionistic fuzzy soft environment.

Algorithm 1: EDAS

Step 1. Choose the significative parameters and potential alternatives, and achieve the IFSS (F, E) which is given in Table 3.

Step 2. Standard the IFSS (F, E) into (F′, E) by Eq.(8), presented in Table 4.

Table 4

Normalized IFSS (F′, E)

	e₁	e₂	e₃	e₄	e₅
x ₁	(0.6, 0.4)	(0.6, 0.3)	(0.7, 0.1)	(0.6, 0.3)	(0.5, 0.4)
x ₂	(0.7, 0.3)	(0.6, 0.2)	(0.6, 0.2)	(0.6, 0.2)	(0.4, 0.4)
x ₃	(0.6, 0.3)	(0.7, 0.2)	(0.6, 0.1)	(0.7, 0.2)	(0.5, 0.3)
x ₄	(0.6, 0.3)	(0.7, 0.2)	(0.7, 0.2)	(0.6, 0.1)	(0.6, 0.2)

Step 3. Calculate the combination weights by Eq.(7) as follows:

ϖ₁ = 0.3049, ϖ₂ = 0.1523, ϖ₃ = 0.0958, ϖ₄ = 0.3463, ϖ₅ = 0.1006.

Step 4. Calculate the mean solution for proprietary parameters by Eq.(10), signified in the following.

AV₁ = (0.6278, 0.3224) , AV₂ = (0.6536, 0.2213) , AV₃ = (0.6536, 0.1414) , AV₄ = (0.6278, 0.1861) , AV₅ = (0.3299, 0.4949) .

Step 5. Determine the PDA = (P_ij) _4×5 and the NDA = (N_ij) _4×5 matrixes by Eqs.(11) and (12), signified in the following.

$PDA = (P_{ij})_{4 \times 5} = (\begin{matrix} 0.0000 & 0.0000 & 0.0963 & 0.0000 & 0.1401 \\ 0.1541 & 0.0000 & 0.0000 & 0.0000 & 0.1550 \\ 0.0000 & 0.0945 & 0.0000 & 0.1431 & 0.0000 \\ 0.0000 & 0.0945 & 0.0822 & 0.0000 & 0.0000 \end{matrix}),$

$NDA = (N_{ij})_{4 \times 5} = (\begin{matrix} 0.0551 & 0.1039 & 0.0000 & 0.0641 & 0.0000 \\ 0.0000 & 0.0922 & 0.1024 & 0.0519 & 0.0000 \\ 0.0507 & 0.0000 & 0.0843 & 0.0000 & 0.0544 \\ 0.0507 & 0.0000 & 0.0000 & 0.0328 & 0.2258 \end{matrix}) .$

Step 6. Calculate the weighted sum of PDA and NDA for proprietary alternatives by Eqs.(13) and (14), signified in the following.

SP₁ = 0.0233, SP₂ = 0.0626, SP₃ = 0.0639, SP₄ = 0.0223,

NP₁ = 0.0548, NP₂ = 0.0418, NP₃ = 0.0290, NP₄ = 0.0495.

Step 7. Standardize the values of SP_i and SN_i for proprietary alternatives by Eqs. (15) and (16), signified in the following.

NSP₁ = 0.3647, NSP₂ = 0.9784, NSP₃ = 1.0000, NSP₄ = 0.3482,

NSN₁ = -0.8907, NSN₂ = -0.4416, NSN₃ = 0.0000, NSN₄ = -0.7073.

Step 8. Determine the appraisal score AS_i (i = 1, 2, 3, 4) for proprietary alternatives by Eq. (17), signified in the following.

AS₁ = -0.2630, AS₂ = 0.2684, AS₃ = 0.5000, AS₄ = -0.1795 .

Step 9. Choose the hotel x_i by the decreasing values of AS_i in the following.

x₃ ≻ x₂ ≻ x₄ ≻ x₁.

Clearly, the x₃ is the best hotel among four nominated hotels.

Algorithm 2: MABAC

Step 1. Just like what did with the Step 1 to Step 3 presented in Algorithm 1.

Step 2. Calculate the weighted matrix T = (t_ij) _4×5 by Eq.(18), presented in Table 5.

Table 5

The weighed intuitionistic fuzzy soft matrix T = ( t_ij) _4×5

	e₁	e₂	e₃	e₄	e₅
x ₁	(0.2437, 0.7563)	(0.1303, 0.8325)	(0.1089, 0.8020)	(0.2719, 0.6591)	(0.0674, 0.9119)
x ₂	(0.3073, 0.6927)	(0.1303, 0.7826)	(0.0840, 0.8571)	(0.2719, 0.5727)	(0.0501, 0.9119)
x ₃	(0.2437, 0.6927)	(0.1675, 0.7826)	(0.0840, 0.8020)	(0.3409, 0.5727)	(0.0674, 0.8859)
x ₄	(0.2437, 0.6927)	(0.1675, 0.7826)	(0.1089, 0.8571)	(0.2719, 0.4505)	(0.0881, 0.8505)

Step 3. The BAA G = (g_j) _1×5 is computed by Eq.(19), we can have

g₁ = (0.2583, 0.7100) , g₂ = (0.1477, 0.7963) , g₃ = (0.0957, 0.8318) ,

g₄ = (0.2877, 0.5700) , g₅ = (0.0669, 0.8928).

Step 4. Calculate the distance matrix D = (d_ij) _4×5 by Eq.(20), presented in Table 6.

Table 6

The distance matrix D = (d_ij) _4×5

	e₁	e₂	e₃	e₄	e₅
x ₁	-0.040945	-0.031309	0.025835	-0.082343	0.019415
x ₂	0.043028	-0.027065	-0.021929	-0.014637	-0.018101
x ₃	-0.027580	0.017575	-0.036995	0.054654	0.006616
x ₄	-0.027580	0.017575	0.033924	-0.128098	0.036589

Step 5. Order the alternatives by Q_i (i = 1, 2, 3, 4) as follows:

Q₁ = -0.109347, Q₂ = -0.038704, Q₃ = 0.014270, Q₄ = -0.067591 .

Hence, x₃ > x₂ > x₄ > x₁, i.e., the optimal hotel is x₃.

Algorithm 3: similarity measure

Step 1. Just like what did with the Step 1 to Step 3 presented in Algorithm 1.

Step 2. Compute the similarity measure S (x_i, x^∗) (i = 1, 2, 3, 4) by Eq.(22), presented in the following.

S (x₁, x^∗) = (0.6706, 0.2721) , S (x₂, x^∗) = (0.6795, 0.2200) ,

S (x₃, x^∗) = (0.6990, 0.2047) , S (x₄, x^∗) = (0.6877, 0.1753) .

Step 3. Calculate score function s (S (A_i, A^∗)) of each hotel by Eq.(4), presented in the following.

s (S (x₁, x^∗)) =1.408830, s (S (x₂, x^∗)) =1.438755,

s (S (x₃, x^∗)) =1.495309, s (S (x₄, x^∗)) =1.468278 .

Step 4. Order the hotels by s (S (x_i, x^∗)) (i = 1, 2, 3, 4) as follows:

x₃ ≻ x₄ ≻ x₂ ≻ x₁ .

Clearly, x₃ is the best hotel among them.

It is easily obtained that the optimal hotel are consistent based on proposed algorithms (MABAC, EDAS and similarity measure). Consequently, the developed algorithms are resultful and feasible.

6. A comparison with some existing intuitionistic fuzzy soft decision making methods

As discussed in [14], the given approach is a regolabile approach which seizes a key function for imprecise decision making environment. Most of these issues are substantially humanistic, and therefore subjective in nature. Actually, there does not exist a unified standard for assessing alternatives. Clearly, the approaches in [14 , 19] can not be employed in decision making based IFSS in certain situations.

6.1. Jiang et al.’s method [14] and its limitation

Jiang et al. [14] applied a regolabile method in decision making based IFSS. They presented a definition of (s, t)-level SS as follows:

Definition 10. Let ϒ = (F, A) be an IFSS over U, where A ⊆ E and E is a series of parameters. For s, t ∈ [0, 1] , the (s, t)-level SS of ϒ is a crisp SS L (ϒ ; s, t) = (F_(s,t), A) denoted as follows:

$\begin{matrix} F_{(s, t)} (e_{j}) & = & L (F (e_{j}); s, t) = {x_{i} \in U ∣ μ_{F} (e_{j}) (x_{i}) \geq s, \\ ν_{F} (e_{j}) (x_{i}) \leq t}, \forall e_{j} \in A, \end{matrix}$ (23) where s ∈ [0, 1] can be viewed as a prescribed least threshold values on membership and t ∈ [0, 1] can be viewed as a prescribed greatest threshold values on non-membership.

Next, we present an example to state the drawback of the (s, t)-level SSs.

Example 3. [14] Let us think an IFSS (F, A) which depicts the “attractiveness of houses” that Mr. X is considered for purchase. Assume that there have six houses in U = {x₁, x₂, x₃, x₄, x₅, x₆} under consideration, and A = {e₁, e₂, e₃, e₄, e₅} is a series of decision parameters. The e_j (j = 1, 2, 3, 4, 5) stand for the parameters “green surroundings”, “expensive”, “large”, “convenient traffic” and “beautiful”, respectively. Table 7 gives the tabular form of the IFSS ϒ = (F, A).

Table 7

Tabular representation of intuitionistic fuzzy soft set ϒ = (F, A)

	e₁	e₂	e₃	e₄	e₅
x ₁	(0.9,0.1)	(0.8,0.1)	(0.6,0.2)	(0.4,0.5)	(0.9,0.1)
x ₂	(0.7,0.2)	(0.7,0.1)	(0.5,0.2)	(0.9,0.1)	(0.4,0.5)
x ₃	(0.8,0.2)	(0.6,0.3)	(0.4,0.5)	(0.7,0.1)	(0.8,0.1)
x ₄	(0.5,0.4)	(0.4,0.6)	(0.7,0.3)	(0.7,0.2)	(0.7,0.1)
x ₅	(0.6,0.3)	(0.8,0.2)	(0.8,0.2)	(0.8,0.1)	(0.4,0.5)
x ₆	(0.7,0.3)	(0.5,0.3)	(0.6,0.2)	(0.8,0.1)	(0.7,0.2)

Jiang et al. [14] selected s = 0.7 and t = 0.3, based on Eq.(23), we know that the bold decision value in Table 7 is not available. That is to say, the (s, t)-level soft set has a great limitation in massive data. Meanwhile, the following decision making is not in progress.

If we revise the unreasonable data and add some new data in Table 7, we can obtain the new IFSS ϒ′ = (F′, A) in Table 8.

Table 8

Tabular representation of intuitionistic fuzzy soft set ϒ′ = (F′, A)

	e₁	e₂	e₃	e₄	e₅
x ₁	(0.9,0.1)	(0.8,0.1)	(0.6,0.3)	(0.4,0.5)	(0.9,0.1)
x ₂	(0.7,0.2)	(0.7,0.1)	(0.8,0.2)	(0.6,0.3)	(0.4,0.5)
x ₃	(0.8,0.2)	(0.6,0.3)	(0.4,0.5)	(0.7,0.2)	(0.8,0.1)
x ₄	(0.5,0.4)	(0.7,0.1)	(0.6,0.3)	(0.7,0.2)	(0.7,0.1)
x ₅	(0.6,0.3)	(0.8,0.2)	(0.8,0.2)	(0.8,0.1)	(0.4,0.5)
x ₆	(0.7,0.3)	(0.6,0.3)	(0.6,0.3)	(0.8,0.1)	(0.7,0.2)

According to the Algorithm 1 proposed by Jiang et al. [14], we can obtain the results shown in Table 9. From Table 9, we can know that the conclusive results are the equal, i.e., it is unreasonable.

Table 9

Tabular representation of the (0.7, 0.3)-level soft set L (ϒ′ ; 0.7, 0.3) with choice values

	e₁	e₂	e₃	e₄	e₅	Choice value c_i
x ₁	1	1	0	0	1	c₁ = 3
x ₂	1	1	1	0	0	c₂ = 3
x ₃	1	0	0	1	1	c₃ = 3
x ₄	0	1	0	1	1	c₄ = 3
x ₅	0	1	1	1	0	c₅ = 3
x ₆	1	0	0	1	1	c₆ = 3

For comparing the effectiveness of our approaches, firstly, we can compute the objective weights by Eq.(7) instead of Step 3 in Algorithms 1–3, the later are the same. So we can obtain the comparison of results which is presented in Table 10.

Table 10

A comparison study with some existing methods in revised Example 3

Methods	The final ranking	The optimal house
Jiang et al . [14]	x₁ = x₂ = x₃ = x₄ = x₅ = x₆	∗
Li et al . [21]	x₁ ≻ x₄ ≻ x₃ ≻ x₅ ≻ x₆ ≻ x₂	x ₁
Algorithm1	x₁ ≻ x₆ ≻ x₅ ≻ x₃ ≻ x₄ ≻ x₂	x ₁
Algorithm2	x₁ ≻ x₄ ≻ x₅ ≻ x₃ ≻ x₆ ≻ x₂	x ₁
Algorithm3	x₁ ≻ x₅ ≻ x₆ ≻ x₃ ≻ x₂ ≻ x₄	x ₁

“*” denotes that there is no unied house to selected.

From Table 10, we know that the best house is x₁. It can make a distinction in 6 houses and choose the optimal house by our proposed approaches.

6.2. Mao et al.’s method [17] and its limitation

Mao et al. [17] also applied a regolabile approach in decision making based IFSS. They offered a threshold based on median could be denoted as follows:

Definition 11. Let ϒ = (F, A) be an IFSS and λ_med (A) be a median threshold vector over parameter set A, A ⊆ E and E is a series of parameters, expressed as follows:

$\begin{matrix} λ_{med} (A) & = & {λ_{med} (e_{j}), 1 \leq j \leq n} \\ = & {(μ_{med} (e_{j}), ν_{med} (e_{j})), 1 \leq j \leq n}, \end{matrix}$ (24) where for ∀e_j ∈ A, μ_med (e_j) is the median by ordering membership degree of proprietary alternatives by ranking from large to small, namely

$μ_{med} (e_{j}) = {\begin{matrix} μ (e_{j}) (x_{\frac{m + 1}{2}}), if m is an odd number, \\ (μ (e_{j}) (x_{\frac{m}{2}}) + μ (e_{j}) (x_{\frac{m}{2} + 1})) / 2, if m is an even number, \end{matrix}$ (25) and ν_med (e_j) is the median by ranking membership degree of all alternatives by ordering from large to small, namely

$ν_{med} (e_{j}) = {\begin{matrix} ν (e_{j}) (x_{\frac{m + 1}{2}}), if m is an odd number, \\ (ν (e_{j}) (x_{\frac{m}{2}}) + ν (e_{j}) (x_{\frac{m}{2} + 1})) / 2, if m is an even number, \end{matrix}$ (26)

According to median threshold vector λ_med (A), we can obtain λ_med-level soft set L (ϒ, λ_med (A)) = (F_{λ
_med}, A), namely for e_j ∈ A, we have

$\begin{matrix} F_{λ_{med}} (e_{j}) & = & {x_{i} \in U ∣ μ_{F} (e_{j}) \geq μ_{λ_{med}} (e_{j}), ν_{F} (e_{j}) \\ \leq & ν_{λ_{med}} (e_{j})} . \end{matrix}$ (27)

In the following, we give a revised example from [17] to state their drawback.

Example 4. Consider that the computer publisher hope to come out for a best-seller. The publishing editor give five latent computer language books to choose: Java, C, PHP, Python, C++. The computer publisher determines two parameters to assess the five computer language books. Let U = {x₁, x₂, x₃, x₄, x₅} be the set of books under consideration, E = {e₁, e₂} is the series of parameters, e₁ stands for popularity, e₂ stands for readability. In the two parameters, e₁ and e₂ are benefit parameters. The evaluation information is shown in Table 11.

Table 11

Tabular representation of (F, A) in Example 4

	e₁	e₂
x ₁	(0.36, 0.64)	(0.80, 0.10)
x ₂	(0.20, 0.78)	(0.40, 0.58)
x ₃	(0.24, 0.70)	(0.32, 0.66)
x ₄	(0.30, 0.63)	(0.30, 0.68)
x ₅	(0.32, 0.62)	(0.22, 0.76)
λ_med (A)	(0.30, 0.64)	(0.32, 0.66)

Based on Eq.(27), we know that the bold decision value in Table 11 is not available. That is to say, the λ_med-level soft set has a great limitation in massive data. Meanwhile, the following decision making is not in progress.

If we revise the unreasonable data which is shown in Table 11, we can obtain the new IFSS (F′, A) in Table 12.

Table 12

Tabular representation of (F′, A)

	e₁	e₂
x ₁	(0.32, 0.68)	(0.11, 0.73)
x ₂	(0.30, 0.69)	(0.22, 0.76)
x ₃	(0.24, 0.70)	(0.30, 0.68)
x ₄	(0.13, 0.63)	(0.32, 0.66)
x ₅	(0.11, 0.62)	(0.40, 0.58)
x ₆	(0.09, 0.55)	(0.80, 0.10)
λ_med (A)	(0.185, 0.665)	(0.31, 0.67)

From Table 13, we can know that the conclusive results are equal, i.e., it is unreasonable.

Table 13

λ_med-level soft set L (ϒ′ ; λ_med) with choice values

	e₁	e₂	Choice value c_i
x ₁	1	0	c₁ = 1
x ₂	1	0	c₂ = 1
x ₃	1	0	c₃ = 1
x ₄	0	1	c₄ = 1
x ₅	0	1	c₅ = 1

For comparing the effectiveness of our approaches, firstly, we can compute the objective weights by Eq.(6) instead of Step 3 in Algorithms 1-3, the later are the same. So we can obtain the comparison of results which is shown in Table 14.

Table 14

A comparison study with some existing methods in revised Example 4

Methods	The final ranking	The optimal house
Mao et al . [17]	x₁ = x₂ = x₃ = x₄ = x₅	∗
Li et al . [21]	x₃ ≻ x₅ ≻ x₂ ≻ x₄ ≻ x₁	x ₃
Algorithm1	x₃ ≻ x₅ ≻ x₂ ≻ x₄ ≻ x₁	x ₃
Algorithm2	x₃ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₄	x ₃
Algorithm3	x₃ ≻ x₅ ≻ x₂ ≻ x₄ ≻ x₁	x ₃

"∗" denotes that there is no unified alternative to selected.

From Table 14, we know that the optimal computer language book is x₃ (PHP). It can make a distinction in five computer language books and choose the optimal computer language book by our developed approaches.

6.3. Zhao et al.’s method [19] and its limitation

Zhao et al. [19] also applied a regolabile approach in decision making based IFSS. They provided four principles to choose best alternative, which could be expressed as follows:

Definition 12. An IFSS ϒ = (F, A) = {(x_i, μ_F (e_j) (x_i) , ν_F (e_j) (x_i)) ∣ x_i ∈ U, e_j ∈ A} (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n). π_A (x_i) is called the synthetical hesitation function of the alternative x_i ∈ U in whole parameters A of (F, A), which is defined by the following formula:

$\begin{matrix} π_{A} (x_{i}) & = & \sum_{j = 1}^{n} π_{F} (e_{j}) (x_{i}) \\ = & \sum_{j = 1}^{n} (1 - μ_{F} (e_{j}) (x_{i}) - ν_{F} (e_{j}) (x_{i})) . \end{matrix}$ (28)Definition 13. [19] An IFSS ϒ = (F, A) = {(x_i, μ_F (e_j) (x_i) , ν_F (e_j) (x_i)) ∣ x_i ∈ U, e_j ∈ A} (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n). $s_{A}^{'} (x_{i})$ is called the synthetical score function of the alternative x_i ∈ U in all parameters A of (F, A), which is denoted as follows:

$\begin{matrix} s_{A}^{'} (x_{i}) & = & \sum_{j = 1}^{n} s_{F}^{'} (e_{j}) (x_{i}) \\ = & \sum_{j = 1}^{n} (μ_{F} (e_{j}) (x_{i}) - ν_{F} (e_{j}) (x_{i})) . \end{matrix}$ (29)

Definition 14. [19] An IFSS ϒ = (F, A) = {(x_i, μ_F (e_j) (x_i) , ν_F (e_j) (x_i)) ∣ x_i ∈ U, e_j ∈ A} (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n). t_A (x_i) is called the synthetical accuracy function of the alternative x_i ∈ U in all parameters A of (F, A), which is defined by the following formula:

$\begin{matrix} t_{A} (x_{i}) & = & \sum_{j = 1}^{n} t_{F} (e_{j}) (x_{i}) \\ = & \sum_{j = 1}^{n} (μ_{F} (e_{j}) (x_{i}) + ν_{F} (e_{j}) (x_{i})) . \end{matrix}$ (30)

Zhao et al. [19] developed four principles for selecting optimal alternative, named as minimum hesitation principle, maximal score principle, maximal accuracy principle and maximal choice value principle.

In the following, we give an example to state the drawback of the approach [19].

Example 5. The DMs chooses two parameters to assess the four potential psycholinguistic schools: (1) e₁ is the teaching environment; (2) e₂ is the teaching management. The three possible psycholinguistic schools x_i (i = 1, 2, 3) are to be assessed employing the IFNs by the DMs under two parameters, and e₁, e₂ are benefit parameters. ϒ = (F, A) be an IFSS over a domain U = {x₁, x₂, x₃}, and a fix parameter set A = {e₁, e₂}. The (F, A) is presented in Table 15.

Table 15

Tabular representation of the (F, A) in Example 5

	e ₁	e ₂
x ₁	(0.2,0.4)	(0.3,0.5)
x ₂	(0.15,0.35)	(0.35,0.55)
x ₃	(0.1,0.3)	(0.4,0.6)

Based on above four principles, we can have the following results, which are shown in Tables 16–19.

Table 16

Hesitation of (F, A) in Example 5

	π _{e ₁}	π _{e ₂}	π _A
x ₁	0.4	0.2	0.6
x ₂	0.6	0.0	0.6
x ₃	0.5	0.1	0.6

Table 17

Score of (F, A) in Example 5

	$s_{e_{1}}^{'}$	$s_{e_{2}}^{'}$	$s_{A}^{'}$
x ₁	-0.2	-0.2	-0.4
x ₂	-0.2	-0.2	-0.4
x ₃	-0.2	-0.2	-0.4

Table 18

Accuracy of (F, A) in Example 5

	t _{e ₁}	t _{e ₂}	t _A
x ₁	0.6	0.8	1.4
x ₂	0.5	0.9	1.4
x ₃	0.4	1.0	1.4

Table 19

(0.7, 0.3)-level soft set L (ϒ ; 0.7, 0.3) with choice value in Example 5

	e ₁	e ₂	Choice values
x ₁	0	0	c₁ = 0
x ₂	0	0	c₂ = 0
x ₃	0	0	c₃ = 0

Based on above four principles with Tables 16–19 by Example 4, the decision maker will have to choose any one out of three. Hence, it is unreasonable.

For comparing the effectiveness of our approaches, firstly, we can compute the objective weights by Eq.(6) instead of Step 3 in Algorithms 1–3, the later are the same. So we can obtain the comparison of results which is shown in Table 20.

Table 20

A comparison study with some existing methods in Example 5

Methods	The final ranking	The optimal house
Zhao et al .	x₁ = x₂ = x₃	∗
Li et al . [24]	x₃ ≻ x₂ ≻ x₁	x ₃
Algorithm1	x₃ ≻ x₂ ≻ x₁	x ₃
Algorithm2	x₃ ≻ x₂ ≻ x₁	x ₃
Algorithm3	x₃ ≻ x₂ ≻ x₁	x ₃

"∗" denotes that there is no unified alternative to selected.

From Table 20, we know that the optimal alternative is x₃. It can make a distinction in 3 alternatives and choose the optimal alternative by our proposed approaches.

Example 6. A venture capital company hopes to invest a software development project (SDP). There are four potential SDP: (1) x₁ is take-out; (2) x₂ is game; (3) x₃ is browser; (4) x₄ is a film review. The venture capital company chooses four parameters to assess the four potential SDP: (1) e₁ is economic practicability; (2) e₂ is technological practicability; (3) e₃ is social practicability; (4) e₄ is staff practicability, and they are all benefit parameters. Suppose that the DMs gave the weight information as w = (0.3, 0.3, 0.3, 0.1). In order to avoid influence each other, the DMs are required to assess the potential four SDPs x_i (i = 1, 2, 3, 4) under the above four parameters in anonymity and the IFSS (F, A) which is presented in Table 21.

Table 21

Tabular representation of the (F, A) in Example 6

	e ₁	e ₂	e ₃	e ₄
x ₁	(0.8,0.1)	(0.9,0.1)	(0.7,0.1)	(0.3,0.1)
x ₂	(0.8,0.2)	(0.8,0.1)	(0.8,0.1)	(0.3,0.1)
x ₃	(0.9,0.1)	(0.7,0.1)	(0.7,0.1)	(0.3,0.2)
x ₄	(0.8,0.1)	(0.7,0.1)	(0.6,0.1)	(0.3,0.2)

From Table 22, we know that the optimal alternative is x₁. It can make a distinction in 4 alternatives and choose the optimal alternative by our proposed approaches. For Jiang et al. [14] and Mao et al. [17], it can’t give a decision values due to its limitation in massive data. For Zhao et al. [19], different principles (hesitation, Score, Accuracy) has diverse decision results. That is to say, it will not obtain credible results.

Table 22

A comparison study with some existing methods in Example 6

Methods	The decision values	Ranking	The optimal project
Jiang et al . [14]	∇	∇	∇
Mao et al . [17]	∇	∇	∇
Zhao et al . [19] (Hesitation)	π₁ = 0.9, π₂ = 0.8, π₃ = 0.9, π₄ = 1.1	x₂ ≻ x₁ ∼ x₃ ≻ x₄	x ₂
Zhao et al . [19] (Score)	$s_{1}^{'} = 2.3, s_{2}^{'} = 2.2, s_{3}^{'} = 2.1, s_{4}^{'} = 1.9$	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Zhao et al . [19] (Accuracy)	a₁ = 3.1, a₂ = 3.2, a₃ = 3.1, a₄ = 2.9	x₂ ≻ x₁ ∼ x₃ ≻ x₄	x ₂
Algorithm1	AS₁ = 0.8154, AS₂ = 0.7590, AS₃ = 0.2759, AS₄ = -0.5672	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm2	Q₁ = 0.1178, Q₂ = 0.0559, Q₃ = -0.0102, Q₄ = -0.2000	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm3	s₁ = 1.8081, s₂ = 1.7909, s₃ = 1.7138, s₄ = 1.5728	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁

∇ denotes that it can’t meet the requirement.

6.4. The advantages of the developed approaches

According to the above comparative discussions, the four advantages have been achieved in the following.

(1) The results calculated by the similarity measures in [34 –36] may lead to the information distortion and hence result in an unfair decision. But the proposed similarity measure can keep more original decision information.

(2) The proposed approaches (EDAS, MABAC and similarity measure) have more discrimination compared with the existing algorithms [17, 19]. That is to say, the optimal alternative cannot be selected by existing algorithms [17, 19] which are shown in Examples 2-4.

(3) For some special data, (s, t)-level soft set proposed by Jiang et al. [14] has a great limitation in massive data. That is to say, there is strict requirements for decision making data. But our algorithms have no requirements.

(4) The diverse parameters’ objective weights are calculated by GST, meanwhile, the presented combination weights can reveal both the subjective factor and the objective information which can reduce the random of weights by decision makers [14 , 19].

7. Conclusion

The goal of this paper is to give three algorithms for handling decision making issues under IFSS. For this, a new axiomatic definition of intuitionistic fuzzy distance measure and similarity measure has been proposed whose values are summarized by intuitionistic fuzzy numbers. Comparing with the existing similarity measure [34 –36], our similarity measure can keep more primitive decision information. Also, some properties of them are stated in detail. The proposed three intuitionistic fuzzy soft decision making approaches (EDAS, MABAC, similarity measure) have been illustrated with some numerical examples (Examples 1-5). From the study, it has been come to a conclusion that our algorithms can reduce the effect of unfair arguments on the ultimate results provided by DMs.

In the future, we will employ the similarity measure of IFSSs to other fields such as self-paced learning [52], deep learning and machine learning. Also we will utilize revised algorithms in other fuzzy environment [53 –67].

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Some novel decision making algorithms for intuitionistic fuzzy soft set

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1. Intuitionistic fuzzy set

Table 1 The intuitionistic fuzzy soft set (F, A) e 1 e 2 e 3 e 4 x 1 (0.2,0.7) (0.1,0.4) (0.2,0.4) (0.2,0.4) x 2 (0.2,0.6) (0.6,0.3) (0.2,0.7) (0.2,0.6) x 3 (0.4,0.6) (0.3,0.7) (0.3,0.6) (0.4,0.6) x 4 (0.2,0.6) (0.2,0.7) (0.4,0.6) (0.5,0.4)

4.1. Problem description

Table 2 Tabular form of (F, E)

4.2.1. Calculating objective weights: grey relational analysis

Table 3 Tabular representation of (F, E)

6.1. Jiang et al.’s method [14] and its limitation

7. Conclusion

References

Table 1
The intuitionistic fuzzy soft set (F, A)

e ₁ e ₂ e ₃ e ₄

x ₁ (0.2,0.7) (0.1,0.4) (0.2,0.4) (0.2,0.4)

x ₂ (0.2,0.6) (0.6,0.3) (0.2,0.7) (0.2,0.6)

x ₃ (0.4,0.6) (0.3,0.7) (0.3,0.6) (0.4,0.6)

x ₄ (0.2,0.6) (0.2,0.7) (0.4,0.6) (0.5,0.4)

Table 2
Tabular form of (F, E)

Table 3
Tabular representation of (F, E)