Hesitant fuzzy soft decision making methods based on WASPAS,MABAC and COPRAS with combined weights

Abstract

This paper presents three novel hesitant fuzzy soft set (HFSS) methods. First, the objective weights of various parameters are determined via Shannon entropy theory. Then, we develop the combined weights, which can show both the subjective information and the objective information. Later, we propose three algorithms to solve hesitant fuzzy soft decision making problem by Multi-Attributive Border Approximation area Comparison (MABAC), Weighted Aggregated Sum Product Assessment (WASPAS) and Complex Proportional Assessment (COPRAS). Finally, the effectiveness and feasibility of approaches are demonstrated by some numerical examples.

Keywords

Hesitant fuzzy soft set combined weighed MABAC WASPAS COPRAS

1 Introduction

Many complex and practical issues in economics, engineering, medical science, and environmental science involve uncertainty and indeterminacy. While a wide variety of existing theories such as probability theory, fuzzy set theory [1], and rough set theory [2] have been developed to model indeterminacy. However, each of these theories has its inherent difficulties as pointed out in [3]. The soft set theory, initiated by Molodtsov [3], is free from the inadequacy of the parameterization tools of those theories [1]. It has potential applications in many different fields such as rule mining [4], game theory [5], feature selection [6], decision making [7], parameter reduction [8], groups [9].

By combining soft sets with other mathematical models, many generalizations of soft set have been advanced. Maji et al. [10] firstly explored fuzzy soft sets, a more generalized notion combining fuzzy sets and soft sets. Alcantud [11] proposed a novel algorithm for fuzzy soft set based decision making from multiobserver input parameter data set. Meanwhile, some formal relationships among soft sets, fuzzy sets, and their extensions are discussed in detail [12]. Yang and Peng [13] developed the concept of interval fuzzy soft sets. Meanwhile, Peng applied interval-valued fuzzy soft sets in decision making [14, 15]. Peng et al. [16] presented Pythagorean fuzzy soft sets, and discussed their operations. Yang et al. [17] proposed bipolar multi-fuzzy soft sets which can describe the parameter more precisely. Peng and Liu [18] introduced neutrosophic soft decision making methods based on EDAS, new similarity measure and level soft set. Feng et al. [19] established a colorful connection between rough sets and soft sets. Jun [20] studied the application of soft sets in BCK/BCI-algebras. Wang et al. [21] presented the hesitant fuzzy soft sets by aggregating hesitant fuzzy set [22] with soft set, and developed an algorithm to solve decision making problems. Peng and Yang [23] further proposed interval-valued hesitant fuzzy soft set, presented an algorithm, and discussed its calculation complexity with others algorithms. Moreover, there have been some applications of hesitant fuzzy soft sets such as BCK/BCI-algebras [24], decision making [25 –30].

The Multi-Attributive Border Approximation area Comparison (MABAC) method is a new method initiated in [31]. It has a straightforward computation procedure, systematic process and a sound logic that shows the foundation of decision making. Combining with the advantages of Pythagorean fuzzy sets [32, 33], Peng and Yang [34] applied the MABAC to R&D project selection procedure to rank and obtain the desired project. Xue et al. [35] proposed a MABAC approach for material selection under interval-valued intuitionistic fuzzy environment. As far as we know, however, the study of the decision making problem based on MABAC method have not been reported in the existing academic literature. Therefore, it is a glamorous research topic to apply MABAC method in decision making to rank and obtain the best alternative under hesitant fuzzy soft environment.

The Weighted Aggregated Sum Product Assessment (WASPAS) method is a decision making method which was proposed and optimized by Zavadskas et al. [36]. This method has been applied to and extended in many decision making problems and environments. Chakraborty and Zavadsk [37] explored the WASPAS method as an effective MCDM tool while solving eight manufacturing decision making problems. Baleɘentis and Streimikiene [38] studied the effectiveness of multi-criteria ranking of energy generation scenarios with Monte Carlo simulation. Ghorabaee et al. [39] proposed multi-criteria evaluation of green suppliers using an extended WASPAS method with interval type-2 fuzzy environment.

The Complex Proportional Assessment (COPRAS) method, suggested by Zavadskas et al. [40], is a new method of decision making. The COPRAS method determines a solution and the ratio to the ideal solution and the ratio to the worst-ideal solution, and therefore can be regarded as a compromising method. The COPRAS method is applied for solving numerous problems by its exhibitors and their colleagues. Ghorabaee et al. [41] presented a multiple criteria group decision-making for supplier selection based on COPRAS method with interval type-2 fuzzy sets. Liou et al. [42] developed COPRAS decision making model for improving and selecting suppliers in green supply chain management.

As far as we know, however, the study of the decision making problem based on MABAC, COPRAS and WASPAS methods have not been reported in the existing academic literature. Meanwhile, each of proposed methods has its application prospect in corresponding case which can boost and replenish the research. Hence, it is an interesting research topic to apply them in decision making to rank and determine the best alternative under hesitant fuzzy soft environment. Through a comparison analysis of the given methods, their objective evaluation is carried out, and the methods which maintain consistency of their results are chosen.

Considering that different parameter weights will influence the ranking results of alternatives, we develop a new method to determine the parameter weights by combining the subjective elements with the objective ones. This model is different from the existing hesitant fuzzy soft methods, which can be divided into two tactics: one is the subjective weighting evaluation methods and the other is the objective weighting determine methods, which can be computed by Shannon entropy theory [43]. The subjective weighting methods focus on the preference information of the decision maker [25 , 27–29], while they ignore the objective information. The objective weighting determine methods do not take the preference of the decision maker into account, that is to say, these methods fail to take the risk attitude of the decision maker into account. The feature of our weighting model can show both the subjective information and the objective information. Hence, combining objective weights with subjective weights, a combined model to obtain parameter weights is proposed.

The remainder of this paper is organized as follows: In Section 2, we review some fundamental conceptions of hesitant fuzzy sets and hesitant fuzzy soft sets. In Section 3, three hesitant fuzzy soft decision making approaches based on MABAC, COPRAS and WASPAS are shown. In Section 4, a numerical example is given to illustrate the proposed methods. In Section 5, we compare the novel proposed approaches with the existing hesitant fuzzy soft set decision making approaches, and show the effectiveness of the proposed approaches. The paper is concluded in Section 6.

2 Preliminaries

This section recalls some preliminaries that are used throughout this work. Here we briefly describe some basic concepts and operational laws related to hesitant fuzzy set and hesitant fuzzy soft set.

2.1 Hesitant fuzzy set

Definition 1. [22] A hesitant fuzzy set on U is defined in terms of a function that returns a subset of [0, 1] when applied to U, i.e., $A = {< x, h (x) > ∣ x \in U}$ where h (x) is a set of some different values in [0,1], representing the possible membership degrees of the element x ∈ U to A. For convenience, Xia and Xu [44] called h (x) a hesitant fuzzy element (HFE) and H the set of all hesitant fuzzy elements (HFEs). If all the elements in HFE are 1, then we call Γ; if all the elements in HFE are 0, then we call Φ.

Definition 2. [44] Let h = ⋃ _γ∈h {γ} be a HFE, then the score function of h is defined as follows: $S (h) = \frac{1}{# h} \sum_{γ \in h} γ$ (1) where # h is the number of the elements in h. For two HFEs h₁, h₂, if S (h₁) > S (h₂), then h₁ > h₂; if S (h₁) = S (h₂), then h₁ = h₂.

Definition 3. [22, 44] Given three HFEs represented by h, h₁, and h₂, then the operational laws of HFEs are defined as follows:

h^c = ⋃ _r∈h {1 - r};

h₁ ∪ h₂ = ⋃ _{r₁∈h₁,r₂∈h₂}max {r₁, r₂};

h₁ ∩ h₂ = ⋃ _{r₁∈h₁,r₂∈h₂}min {r₁, r₂};

h₁ ⊕ h₂ = ⋃ _{r₁∈h₁,r₂∈h₂} {r₁ + r₂ - r₁r₂};

h₁ ⊗ h₂ = ⋃ _{r₁∈h₁,r₂∈h₂} {r₁r₂};

λh = ⋃ _r∈h {1 - (1 - r) ^λ} , λ > 0;

h^λ = ⋃ _r∈h {r^λ} , λ > 0.

Definition 4. [44] Suppose that a HFE h, and stipulate that h⁺ and h^- are the maximum and the minimum hesitant fuzzy values in the HFE h, then we call h^N = ηh⁺ + (1 - η) h^- an extension value, where η (0 ≤ η ≤ 1) is the parameter determined by the decision maker (DM) according to his/her principle preference.

Consequently, if l_α ≠ l_β, and α, β are corresponding HFEs, then we need add different values to the HFE which has the less elements using the parameter η via the DM’s principle preference until both of them have the same length, i.e.

when the DM’s principle preference is optimistic, we can add the extension valueh^N = h⁺;

when the DM’s principle preference is neutral, we can add the extension value $h^{N} = \frac{1}{2} (h^{+} + h^{-})$ ;

when the DM’s principle preference is pessimistic, we can add the extension valueh^N = h^-.

Obviously, the parameter η provided by the DM reflects his/her principle preference which can affect the final results. In what follows, we take the optimistic principle.

Based on the well-known Hamming distance, as well as the above operational laws, analogous to the distance measure for HFEs in [45]:

$d (h, g) = \frac{1}{2 l} \sum_{i = 1}^{l} (∣ h_{σ (i)} - g_{σ (i)} ∣)$ (2) where l is the number of max {# h, # g}.

Definition 5. [44] Let h_j (j = 1, 2, ⋯ , n) be a series of HFEs. A hesitant fuzzy weighted averaging (HFWA) operator is a mapping Hⁿ → H such that

$\begin{matrix} HFWA (h_{1}, h_{2}, \dots, h_{n}) = \oplus_{j = 1}^{n} (w_{j} h_{j}) \\ = ⋃_{r_{1} \in h_{1}, r_{2} \in h_{2}, \dots, r_{n} \in h_{n}} {1 - \prod_{j = 1}^{n} (1 - r_{j})^{w_{j}}} \end{matrix}$ (3)

However, we can see that the HFWA operator has drawbacks in some cases, described as follows.

Let h_j (j = 1, 2, ⋯ , n) be a collection of HFEs. If there is i such that h_i = Γ, then based on Equation (3), we can have HFWA (h₁, h₂, ⋯ , h_n) = Γ. This result may cause counter-intuitive phenomena in decision making. In other words, it only determines by h_i to make decision and the decision information of others can be neglected.

Hence, it is unreasonable and unsuitable to apply Equation (3) to aggregate the information in decision making when meet the special cases mentioned above.

Definition 6. [44] Let h_j (j = 1, 2, ⋯ , n) be a series of HFEs. A hesitant fuzzy weighted geometric (HFWG) operator is a mapping Hⁿ → H such that

$\begin{matrix} HFWG (h_{1}, h_{2}, \dots, h_{n}) = ⨂_{j = 1}^{n} (h_{j})^{w_{j}} \\ = ⋃_{r_{1} \in h_{1}, r_{2} \in h_{2}, \dots, r_{n} \in h_{n}} {\prod_{j = 1}^{n} (r_{j})^{w_{j}}} \end{matrix}$ (4)

However, we can see that the HFWG operator has drawbacks in some cases, described as follows.

Let h_j (j = 1, 2, ⋯ , n) be a collection of HFEs. If there is i such that h_i = Φ, then based on Equation (4), we can have HFWG (h₁, h₂, ⋯ , h_n) = Φ. This result may cause counter-intuitive phenomena in decision making. In other words, it only determines by h_i to make decision and the decision information of others can be neglected.

Hence, it is unreasonable and unsuitable to apply Equation (4) to aggregate the information in decision making when meet the special cases mentioned above.

2.2 Hesitant fuzzy soft set

Definition 7. [21] A pair (F, A) is called a hesitant fuzzy soft set (HFSS) over U, where F is a mapping given by F : A → H (U).

A HFSS is a mapping from parameters to H (U), and it is not a set, but a parameterized family of hesitant fuzzy subset of U. For e ∈ A, F (e) may be considered as the e-approximate elements of the hesitant fuzzy soft set (F, A).

Definition 8. [21] Let A, B ⊆ E. (F, A) and (G, B) be two HFSSs over U. (G, B) is called to be a hesitant fuzzy soft subset of (F, A) if

A ⊇ B;

∀e ∈ B, x ∈ U, S (h_F(e) (x)) ≥ S (h_G(e) (x)).

In this case, we write $(F, A) \tilde{\supseteq} (G, B) .$

3 Three algorithms for hesitant fuzzy soft decision making

3.1 Problem description

Let U = {x₁, x₂, ⋯ , x_m} be a finite set of m alternatives, E = {e₁, e₂, ⋯ , e_n} be a set of n parameters, and the weight of parameter e_i is w_i, $w_{i} \in [0, 1], \sum_{i = 1}^{n} w_{i} = 1$ . (F, E) is hesitant fuzzy soft set which can be expressed in Table 1. F (e_j) (x_i) represents the possible HFE of the ith alternative x_i satisfying the jth parameter e_j which is given by the decision maker.

Table 1
Tabular representation of (F, E)

e ₁ e ₂ ⋯ 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)

	e ₁	e ₂	⋯	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)

In the following, we will apply the MABAC, WASPAS and COPRAS methods to HFSS.

3.2 The method of computing the combined weights

3.2.1 Determining the objective weights: The Shannon entropy method

Shannon entropy [43] evaluates the expected information content of a certain message. The degree of uncertainty in information can be measured using the entropy concept. Information entropy idea can regulate decision making process because it is able to measure existent contrasts between sets of data and thus clarify the intrinsic information for decision maker.

The following procedure should be employed to determine integrated weights through Shannon entropy under hesitant fuzzy soft environment.

Step 1. Normalize decision matrix R = (F (e_j) (x_i)) _n×m into $\tilde{R} = (\tilde{F} (e_{j}) (x_{i}))_{n \times m}$ by Equation (5).

$\begin{matrix} \tilde{F} (e_{j}) (x_{i}) \\ = {\begin{matrix} ⋃_{r_{ij} \in F (e_{j}) (x_{i})} {r_{ij}}, C_{j} is benefit parameter, \\ ⋃_{r_{ij} \in F (e_{j}) (x_{i})} {1 - r_{ij}}, C_{j} is cost parameter . \end{matrix} \end{matrix}$ (5)

Step 2. Compute the score function $S (\tilde{F} (e_{j}) (x_{i}))$ of $\tilde{F} (e_{j}) (x_{i}) (i = 1, 2, \dots, m; j = 1, 2, \dots, n)$ by Equation (1).

Step 3. Normalize the score function $S (\tilde{F} (e_{j}) (x_{i}))$ by Equation (6), $P_{ij} = \frac{e^{S (\tilde{F} (e_{j}) (x_{i}))}}{\sum_{i = 1}^{m} e^{S (\tilde{F} (e_{j}) (x_{i}))}}$ (6)

Step 4. Calculate the entropy measure of the score function of normalized decision matrix asfollows: $E_{j} = - \frac{1}{ln m} \sum_{i = 1}^{m} P_{ij} ln P_{ij}$ (7)

Step 5. Obtain objective weights ω_j as follows: $ω_{j} = \frac{1 - E_{j}}{\sum_{j = 1}^{n} (1 - E_{j})} .$ (8)

3.2.2 Determining the combined weights: The linear weighted comprehensive method

Suppose that the vector of the subjective weight, given by the decision makers directly, is w = {w₁, w₂, ⋯, w_n}, where $\sum_{j = 1}^{n} w_{j} = 1, 0 \leq w_{j} \leq 1$ . The vector of the objective weight, computed by Equation (8) directly, is ω = {ω₁, ω₂, ⋯ , ω_n}, where $\sum_{j = 1}^{n} ω_{j} = 1, 0 \leq ω_{j} \leq 1$ .

Therefore, the vector of the combined weight ϖ = {ϖ₁, ϖ₂, ⋯ , ϖ_n} can be defined as follows: $ϖ_{j} = λ w_{j} + (1 - λ) ω_{j},$ (9) where λ is the key degree (based on the real decision cases, in this paper, we set λ = 0.5), $\sum_{j = 1}^{n} ϖ_{j} = 1, 0 \leq ϖ_{j} \leq 1$ .

The objective weight and subjective weight are aggregated by linear weighted comprehensive method. According to the addition effect, the larger the value of the subjective weight and objective weight are, the larger the combined weight is, or vice versa. At the same time, we can obtain that the Equation (9) overcomes the limitation of only considering either subjective or objective factor influence. The advantage of Equation (9) is that the attribute weights and rankings of alternatives can show both the subjective information and the objective information.

3.2.3 The hesitant fuzzy soft decision making approach based MABAC

The MABAC is a new MADM method presented in [31]. Due to its straightforward computation procedure and the steady (consistency) of solution, the MABAC method is a particularly practical and credible tool for decision making. In this subsection, a modified MABAC method within the hesitant fuzzy soft environment is introduced to help decision makers.

Algorithm 1: MABAC

Steps 1. Identify the alternatives and parameters, and obtain the hesitant fuzzy soft matrix R = (F (e_j) (x_i)) _m×n (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n) which is shown in Table 2.

Table 2
Tabular representation of (F, E)

e ₁ e ₂ e ₃

x ₁ {0.1, 0.2} {0.8, 0.9, 1.0} {0.1}

x ₂ {0.1, 0.2} {0.8, 0.9} {0.1}

x ₃ {0.0, 0.1, 0.2} {0.8, 0.9} {0.1}

x ₄ {0.1, 0.2, 0.3} {0.7, 0.8} {0.2}

	e ₁	e ₂	e ₃
x ₁	{0.1, 0.2}	{0.8, 0.9, 1.0}	{0.1}
x ₂	{0.1, 0.2}	{0.8, 0.9}	{0.1}
x ₃	{0.0, 0.1, 0.2}	{0.8, 0.9}	{0.1}
x ₄	{0.1, 0.2, 0.3}	{0.7, 0.8}	{0.2}

Steps 2. Normalize decision matrix R = (F (e_j) (x_i)) _n×m into $\tilde{R} = (\tilde{F} (e_{j}) (x_{i}))_{n \times m}$ by Equation (5).

Steps 3. Compute relative weight ϖ_j of parameter C_j by Equation (9).

Step 4. Compute the weighted matrix T = ( t_ij) _m×n by Equation (10). $t_{ij} = ϖ_{j} \tilde{F} (e_{j}) (x_{i}) = ⋃_{r_{ij} \in \tilde{F} (e_{j}) (x_{i})} {1 - (1 - r_{ij})^{ϖ_{j}}} .$ (10)

Step 5. Compute the border approximation area (BAA) matrix G = (g_j) _1×n. The BAA for each parameter is obtained by Equation (11).

$\begin{matrix} g_{j} & = & ⨂_{i = 1}^{m} (t_{ij})^{1 / m} \\ = & ⋃_{r_{1}^{j^{'}} \in t_{1 j}, r_{2}^{j^{'}} \in t_{2 j}, \dots, r_{mj}^{'} \in t_{mj}} {\prod_{i = 1}^{m} (r_{ij}^{'})^{1 / m}} \\ = & ⋃_{r_{1 j} \in \tilde{F} (e_{j}) (x_{1}), r_{2 j} \in \tilde{F} (e_{j}) (x_{2}), \dots, r_{mj} \in \tilde{F} (e_{j}) (x_{m})} \\ {\prod_{i = 1}^{m} (1 - (1 - r_{ij})^{ϖ_{j}})^{1 / m}} \end{matrix}$ (11)

Step 6. Reckon the distance matrix D = (d_ij) _m×n by Equation (12). $d_{ij} = {\begin{matrix} d (t_{ij}, g_{j}), if t_{ij} > g_{j}, \\ 0, if t_{ij} = g_{j}, \\ - d (t_{ij}, g_{j}), if t_{ij} < g_{j}, \end{matrix}$ (12) where distance measure d is defined in Equation (2).

Especially, alternative x_i will pertain to the BAA (G) if d_ij = 0, upper approximation area (G⁺) if d_ij > 0, and lower approximation area (G^-) if d_ij < 0. The upper approximation area (G⁺) is the area which includes the ideal alternative (A⁺) while the lower approximation area (G^-) is the area which includes the anti-ideal alternative (A^-) (see Fig. 1. [31]). For choosing alternative x_i as the best from the set, there is need as many parameters as possible pertaining to the upper approximatearea (G⁺).

Fig.1

Exhibition of the upper (G⁺), lower (G^-), and border (G) approximation areas.

Step 7. Rank the alternatives by V_i (i = 1, 2, ⋯ , m). The most desired alternative is the one with the biggest value of V_i. $V_{i} = \sum_{j = 1}^{n} d_{ij}, i = 1, 2, \dots, m; j = 1, 2, \dots, n .$ (13)

3.2.4 The hesitant fuzzy soft decision making approach based WASPAS

The application of WASPAS method, which is a unique combination of two well known decision making approaches, i.e. weighted sum model (WSM) and weighted product model (WPM) at first requires linear normalization of the decision matrix elements using the following two equations:

For benefit parameters, $\tilde{F} (e_{j}) (x_{i}) = ⋃_{r \in F (e_{j}) (x_{i})} {r}$ ;

For cost parameters, $\tilde{F} (e_{j}) (x_{i}) = ⋃_{r \in F (e_{j}) (x_{i})} {1 - r}$ .

In WASPAS method, a joint criterion of optimality is sought based on two criteria of optimality. The first criterion of optimality, i.e. criterion of a mean weighted success is similar to WSM method. It is a popular and well accepted decision making approach applied for evaluating a number of alternatives in terms of a number of decision parameters. Based on WSM method [46], the total relative importance of ith alternative is calculated as follows:

$\begin{matrix} Q_{i}^{(1)} & = & \oplus_{j = 1}^{n} w_{j} \tilde{F} (e_{j}) (x_{i}) \\ = & ⋃_{r_{i 1} \in \tilde{F} (e_{1}) (x_{i}), r_{i 2} \in \tilde{F} (e_{2}) (x_{i}), \dots, r_{in} \in \tilde{F} (e_{n}) (x_{i})} \\ {1 - \prod_{j = 1}^{n} (1 - r_{ij})^{w_{j}}} . \end{matrix}$ (14)

On the other hand, according to WPM method [46], the total relative importance of ith alternative is computed using the following expression:

$\begin{matrix} Q_{i}^{(2)} & = & ⨂_{j = 1}^{n} (\tilde{F} (e_{j}) (x_{i}))^{w_{j}} \\ = & ⋃_{r_{i 1} \in \tilde{F} (e_{1}) (x_{i}), r_{i 2} \in \tilde{F} (e_{2}) (x_{i}), \dots, r_{in} \in \tilde{F} (e_{n}) (x_{i})} \\ {\prod_{j = 1}^{n} (r_{ij})^{w_{j}}} . \end{matrix}$ (15)

There is an attempt to apply a joint criterion for determining a total importance of alternative, giving equal contribution of WSM and WPM for a total evaluation [47] $Q_{i} = 0.5 * Q_{i}^{(1)} \oplus 0.5 * Q_{i}^{(2)} .$ (16)

Based on previous research [9] and supposing the increase of ranking accuracy and, respectively, the effectiveness of decision making, the WASPAS method for ranking of alternatives is proposed in the current research $Q_{i} = λ * Q_{i}^{(1)} \oplus (1 - λ) * Q_{i}^{(2)} .$ (17)

It was proposed to measure the accuracy of WASPAS based on initial criteria accuracy and when λ ∈ [0, 1](when λ = 0, WASPAS is transformed to WPM; and when λ = 1, WASPAS is transformed to WSM). It was proved that accuracy when aggregating methods is larger comparing to accuracy of single ones.

The concrete steps are shown in the following.

Algorithm 2: WASPAS

Steps 1–3. It is similar to Steps 1–3 in Algorithm 1.

Step 4. Compute the WSM $Q_{i}^{(1)}$ by Equation (14).

Step 5. Compute the WPM $Q_{i}^{(2)}$ by Equation (15).

Step 6. Compute the contribution of WSM and WPM for a total evaluation of Q_i by Equation (17).

Step 7. Rank the alternatives by S (Q_i) (i = 1, 2, ⋯ , m) defined in Equation (1). The most desired alternative is the one with the biggest value of S (Q_i).

3.2.5 The hesitant fuzzy soft decision making approach based COPRAS

In this section, the COPRAS method is extended under the condition that information on the decision making problem has appeared in the form of the HFSS.

The concrete steps are shown in the following.

Algorithm 3: COPRAS

Steps 1–3. It is similar to Steps 1–3 in Algorithm 1.

Steps 4. Calculate the weighted matrix. According to (6) in Definition 3, the weighted matrix is calculated as $\bar{R} = (\bar{F} (e_{j}) (x_{i}))_{n \times m} (i = 1, 2, \dots, m; j = 1, 2, \dots, n)$ .

$\begin{matrix} \bar{F} (e_{j}) (x_{i}) = ϖ_{j} F (e_{j}) (x_{i}) \\ = ⋃_{r_{ij} \in F (e_{j}) (x_{i})} {1 - (1 - r_{ij})^{ϖ_{j}}} . \end{matrix}$ (18)

Steps 5. Sum the values of criteria for benefit. Let J be a set of benefit parameters. The higher values of which are better. Then, calculate the following index for each alternative: $P_{i} = \oplus_{j \in J} \bar{F} (e_{j}) (x_{i})$ (19)

Steps 6. Sum the values of criteria for cost. Let K be a set of cost parameters. The lower values of which are better. Then, calculate the following index for each alternative: $R_{i} = \oplus_{j \in K} \bar{F} (e_{j}) (x_{i}),$ (20)J ⋃ K is the all parameters.

Steps 7. Calculate the relative weight of each alternative Q_i: $Q_{i} = S (P_{i}) + \frac{\sum_{i = 1}^{m} e^{S (R_{i})}}{e^{S (R_{i})} \sum_{i = 1}^{m} \frac{1}{e^{S (R_{i})}}} .$ (21) where $S (R_{\min}) = \min_{i} S (R_{i})$ .

Steps 8. Calculate the quantitative utility U_i of each alternative: $U_{i} = \frac{Q_{i}}{\max_{i} {Q_{i}}} \times 100 %,$ (22)

The larger the value of U_i, the more preference of the alternative x_i.

4 A numerical example

Example 1. An internet company wants to select a software development project to invest. Assume that there are four software projects: e-commerce development project, game development project, browser development project and web development project. The company selects three parameters to evaluate the four software development projects. Let U = {x₁, x₂, x₃, x₄} be the set of software projects, E = {e₁, e₂, e₃} be the set of parameters, e₁ stands for economic feasibility, e₂ stands for technological feasibility, e₃ stands for staff feasibility. e₁ and e₂ are benefit parameters, e₃ is cost parameter. The weight vector of the parameters is given as w = (0.5, 0.4, 0.1) ^T. The decision tabular given by expert is presented in Table 2.

In what follows, we utilize the algorithms proposed above to select software development projects under hesitant fuzzy soft information.

According to Algorithms 1, 2 and 3 shown in Table 3, we can conclude that the final decision results are the same, i.e., x₁ is the most desirable investment software development project. Hence, the three approaches proposed above are effective and available.

Table 3
Final results and ranking

Algorithm Results Ranking Optimal alternative

Algorithm 1 Q₁ = 0.2903, Q₂ = 0.0607, Q₃ = 0.0604, Q₄ = -0.0542 x₁ ≻ x₂ ≻ x₃ ≻ x₄ x ₁

Algorithm 2 V₁ = 0.8024, V₂ = 0.7068, V₃ = 0.6508, V₄ = 0.6215 x₁ ≻ x₂ ≻ x₃ ≻ x₄ x ₁

Algorithm 3 U₁ = 1.0000, U₂ = 0.9101, U₃ = 0.9040, U₄ = 0.8283 x₁ ≻ x₂ ≻ x₃ ≻ x₄ x ₁

Algorithm	Results	Ranking	Optimal alternative
Algorithm 1	Q₁ = 0.2903, Q₂ = 0.0607, Q₃ = 0.0604, Q₄ = -0.0542	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 2	V₁ = 0.8024, V₂ = 0.7068, V₃ = 0.6508, V₄ = 0.6215	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 3	U₁ = 1.0000, U₂ = 0.9101, U₃ = 0.9040, U₄ = 0.8283	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁

5 Comparison of the newly proposed approaches with the other approaches to hesitant fuzzy soft set based decision making

5.1 Comparison of the newly proposed three approaches with their own

In the following, some comparisons of Algorithm 1, Algorithm 2 and Algorithm 3 are shown.

Comparison of computational complexity

We know that Algorithm 1 and Algorithm 2 will consume more computational complexity than Algorithm 3, especially in Step 5 (Algorithm 1), Steps 4 and 5 (Algorithm 2). So if we take the computational complexity into consideration, the Algorithm 3 is given priority to make decision.

Comparison of discrimination

Comparing the results in Algorithm 1, Algorithm 2 with Algorithm 3, we can find that the results of Algorithm 2 and Algorithm 3 are quite close and vary from 0.8283 to 1.0000 and 0.6215 to 0.8024. These result of decision values cannot clearly distinguish, in other words, the results obtained from Algorithm 2 and Algorithm 3 are not very convincing (or at least not applicable). That is to say, the Algorithm 1 has a clearly distinguish. So if we take the discrimination into consideration, the Algorithm 1 is given priority to make decision.

Comparison of applied situation

When the decision maker takes number of parameters into consideration in the decision process, the Algorithm 1 is given priority to make decision.

When the decision maker takes the effects of optimal λ values into consideration in the decision process, the Algorithm 2 is given priority to make decision.

When the decision maker takes ideal solution and the ratio to the worst-ideal solution into consideration in the decision process, the Algorithm 3 is given priority to make decision.

5.2 Comparison of the newly proposed three methods with other approaches

5.2.1 Babitha and John’s method [25] and its limitation

The main problem for Babitha and John’s [25] method comes from the use of average value and fuzzy choice values. It will lose some information after a simple average. It can only be seen as a synthesized measure to estimate each alternative by a fusion of all parameters.

Example 2. Assume that (F, A) is a hesitant fuzzy soft set and its weight is w = (0.3, 0.4, 0.3), which is shown in Table 4.

Table 4
Tabular representation of (F, A)

e ₁ e ₂ e ₃

x ₁ {0.3, 0.4, 0.5} {0.5} {0.1}

x ₂ {0.4} {0.5} {0.1}

x ₃ {0.2, 0.4, 0.6} {0.5} {0.1}

x ₄ {0.35, 0.4, 0.45} {0.5} {0.1}

	e ₁	e ₂	e ₃
x ₁	{0.3, 0.4, 0.5}	{0.5}	{0.1}
x ₂	{0.4}	{0.5}	{0.1}
x ₃	{0.2, 0.4, 0.6}	{0.5}	{0.1}
x ₄	{0.35, 0.4, 0.45}	{0.5}	{0.1}

From Table 5, we can see that the final result of all alternatives are the same, hence, by algorithm in [25], each of them could be selected as the best choice. That is to say, it is unreasonable. But in fact, we can choose an optimal alternative by our proposed algorithms. Suppose that we use Equation (9) to determine the combined weight. Meanwhile, suppose that e₃ is cost parameter. Then, we can have the decision results by our algorithms, which is shown in Table 5.

Table 5

A comparison study with some existing method in Example 2

Method	Results	The final ranking	The optimal alternative
Babitha and John [25]	a₁ = 0.1167, a₂ = 0.1167, a₃ = 0.1167, a₄ = 0.1167	x₁ ∼ x₂ ∼ x₃ ∼ x₄	∗
Algorithm 1	Q₁ = 0.0242, Q₂ = 0.0093, Q₃ = 0.0512, Q₄ = 0.0136	x₃ ≻ x₁ ≻ x₄ ≻ x₂	x ₃
Algorithm 2	V₁ = 0.3224, V₂ = 0.3221, V₃ = 0.3233, V₄ = 0.3222	x₃ ≻ x₁ ≻ x₄ ≻ x₂	x ₃
Algorithm 3	U₁ = 0.9970, U₂ = 0.9960, U₃ = 1.0000, U₄ = 0.9962	x₃ ≻ x₁ ≻ x₄ ≻ x₂	x ₃

“∗” denotes that there is no unified alternative to selected.

5.2.2 Wang et al.’s method [21] and its limitation

As mentioned in [21], the proposed approach is in fact an adjustable method which captures an important feature for decision making in an imprecise environment: some of these problems are essentially humanistic and thus subjective in nature; there actually does not exist a unique or uniform criterion for evaluating the alternatives. Obviously, the methods in [21] can not be applied to hesitant fuzzy soft set based decision making in some cases.

Example 3. Assume that (F, A) is a hesitant fuzzy soft set and its weight is w = (0.5, 0.1, 0.4) which is shown in Table 6.

Table 6
Tabular representation of (F, A)

e ₁ e ₂ e ₃

x ₁ {0.1, 0.5, 0.9} {0.6} {0.2}

x ₂ {0.5} {0.6} {0.2}

x ₃ {0.2, 0.5, 0.8} {0.6} {0.2}

x ₄ {0.4, 0.5, 0.6} {0.6} {0.2}

	e ₁	e ₂	e ₃
x ₁	{0.1, 0.5, 0.9}	{0.6}	{0.2}
x ₂	{0.5}	{0.6}	{0.2}
x ₃	{0.2, 0.5, 0.8}	{0.6}	{0.2}
x ₄	{0.4, 0.5, 0.6}	{0.6}	{0.2}

From Table 8, we can see that the final result of all alternatives are the same, hence, by Algorithm 5 in [21], each of them could be selected as the best choice. That is to say, it is unreasonable. But in fact, we can choose an optimal alternative by our proposed algorithms. Suppose that we use Equation (9) to determine the combined weight. Meanwhile, suppose that e₃ is cost parameter. Then, we can have the decision results by our algorithms, which is shown in Table 7.

Table 7

A comparison study with some existing method in Example 3

Method	Results	The final ranking	The optimal alternative
Wang et al . [21]	c₁ = 4, c₂ = 4, c₃ = 4, c₄ = 4	x₁ ∼ x₂ ∼ x₃ ∼ x₄	∗
Algorithm 1	Q₁ = 0.1846, Q₂ = 0.0325, Q₃ = 0.1217, Q₄ = 0.0450	x₁ ≻ x₃ ≻ x₄ ≻ x₂	x ₁
Algorithm 2	V₁ = 0.6488, V₂ = 0.6394, V₃ = 0.6436, V₄ = 0.6398	x₁ ≻ x₃ ≻ x₄ ≻ x₂	x ₁
Algorithm 3	U₁ = 1.0000, U₂ = 0.9714, U₃ = 0.9851, U₄ = 0.9727	x₁ ≻ x₃ ≻ x₄ ≻ x₂	x ₁

“∗” denotes that there is no unified alternative to selected.

Table 8

Tabular representation of (F, A)

	e ₁	e ₂	e ₃
x ₁	{0.0}	{1.0}	{0.0}
x ₂	{0.7, 0.8, 0.9}	{0.8, 0.9}	{0.1}
x ₃	{0.3, 0.4, 0.5}	{0.4, 0.5, 0.6}	{0.2}
x ₄	{0.1, 0.2, 0.3}	{0.2, 0.3}	{0.2}

5.2.3 Das and Kar’s method [28] and Wang et al. [29] and their limitations

Example 4. Assume that (F, A) is a hesitant fuzzy soft set and its weight is w = (0.5, 0.1, 0.4) which is shown in Table 8.

Suppose that e₃ is cost parameter. From the above results shown in Table 9, we can know that the final ranking of the four alternatives and optimal alternative are in agreement with the results of based HFWA. For algorithm based HFWG, the final ranking and the optimal alternative cannot be obtained due to its drawback that we have discussed in Definition 5.

Table 9
A comparison study with some existing method in Example 4

Method Results The final ranking The optimal alternative

Das and Kar, Wang et al. [28, 29] S (F (x₁)) =1.0000, S (F (x₂)) =0.8755, S (F (x₃)) =0.6919, S (F (x₄)) =0.6489 x₁ ≻ x₂ ≻ x₃ ≻ x₄ x ₁

Das and Kar, Wang et al. [28, 29] S (F (x₁)) =0.0000, S (F (x₂)) =0.8633, S (F (x₃)) =0.6111, S (F (x₄)) =0.4547 x₂ ≻ x₃ ≻ x₄ ≻ x₁ x ₂

Algorithm 1 Q₁ = 0.5662, Q₂ = 0.2954, Q₃ = 0.0124, Q₄ = -0.0586 x₁ ≻ x₂ ≻ x₃ ≻ x₄ x ₁

Algorithm 2 V₁ = 1.0000, V₂ = 0.8705, V₃ = 0.6545, V₄ = 0.5632 x₁ ≻ x₂ ≻ x₃ ≻ x₄ x ₁

Algorithm 3 U₁ = 1.0000, U₂ = 0.7467, U₃ = 0.5721, U₄ = 0.5210 x₁ ≻ x₂ ≻ x₃ ≻ x₄ x ₁

Method	Results	The final ranking	The optimal alternative
Das and Kar, Wang et al. [28, 29]	S (F (x₁)) =1.0000, S (F (x₂)) =0.8755, S (F (x₃)) =0.6919, S (F (x₄)) =0.6489	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Das and Kar, Wang et al. [28, 29]	S (F (x₁)) =0.0000, S (F (x₂)) =0.8633, S (F (x₃)) =0.6111, S (F (x₄)) =0.4547	x₂ ≻ x₃ ≻ x₄ ≻ x₁	x ₂
Algorithm 1	Q₁ = 0.5662, Q₂ = 0.2954, Q₃ = 0.0124, Q₄ = -0.0586	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 2	V₁ = 1.0000, V₂ = 0.8705, V₃ = 0.6545, V₄ = 0.5632	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 3	U₁ = 1.0000, U₂ = 0.7467, U₃ = 0.5721, U₄ = 0.5210	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁

“Bold” denotes unreasonable results.

Example 5. Assume that (F, A) is a hesitant fuzzy soft set and its weight is w = (0.5, 0.1, 0.4) which is shown in Table 10.

Table 10

Tabular representation of (F, A)

	e ₁	e ₂	e ₃
x ₁	{0.0}	{1.0}	{0.0}
x ₂	{0.7, 0.8, 0.9}	{0.8, 0.9}	{0.1}
x ₃	{0.3, 0.4, 0.5}	{0.4, 0.5, 0.6}	{0.2}
x ₄	{0.1, 0.2, 0.3}	{0.2, 0.3}	{0.3}

Suppose that e₃ is cost parameter. From the above results shown in Table 11, we can know that the final ranking of the four alternatives and optimal alternative are in agreement with the results of algorithm based on HFWG [28, 29]. For algorithm based on HFWA in [28, 29], the final ranking and the optimal alternative cannot be obtained due to its drawback that we have discussed in Definition 6.

Table 11

A comparison study with some existing method in Example 5

Method	Results	The final ranking	The optimal alternative
Das and Kar, Wang et al. [28, 29]	S (F (x₁)) =1.0000, S (F (x₂)) =0.8724, S (F (x₃)) =0.6770, S (F (x₄)) =0.6265	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Das and Kar, Wang et al. [28, 29]	S (F (x₁)) =0.3476, S (F (x₂)) =0.8595, S (F (x₃)) =0.5937, S (F (x₄)) =0.4281	x₂ ≻ x₃ ≻ x₄ ≻ x₁	x ₂
Algorithm 1	Q₁ = 0.0910, Q₂ = 0.3114, Q₃ = 0.0129, Q₄ = -0.0720	x₂ ≻ x₁ ≻ x₃ ≻ x₄	x ₂
Algorithm 2	V₁ = 1.0000, V₂ = 0.8673, V₃ = 0.6384, V₄ = 0.5386	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 3	U₁ = 0.7284, U₂ = 1.0000, U₃ = 0.7632, U₄ = 0.6899	x₂ ≻ x₃ ≻ x₁ ≻ x₄	x ₂

“Bold” denotes unreasonable results.

Example 6. Assume that (F, A) is a hesitant fuzzy soft set which is shown in Table 12.

Table 12

Tabular representation of (F, A)

	e ₁	e ₂	e ₃
x ₁	{0.6, 0.7}	{0.9}	{0.2}
x ₂	{0.7, 0.8}	{0.8}	{0.2}
x ₃	{0.3, 0.4, 0.5}	{0.7}	{0.2}
x ₄	{0.2, 0.3, 0.4}	{0.6}	{0.2}

Suppose that e₃ is cost parameter. From the above results shown in Table 13, we can know that the final ranking of the four alternatives and optimal alternative are influenced by the weight given by decision maker. Hence, to give a reasonable weight information is an important topic.

Table 13

A comparison study with some existing method in Example 6

Method	Results	The final ranking	The optimal alternative
w = {0.1, 0.7, 0.2}
Babitha and John [25]	a₁ = 0.8283, a₂ = 0.7942, a₃ = 0.6781, a₄ = 0.6147	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Wang et al. [21]	c₁ = 3, c₂ = 3, c₃ = 1, c₄ = 1	x₁ ∼ x₂ ≻ x₃ ∼ x₄	x₁, x₂
Das and Kar, Wang et al. [28, 29]	S (F (x₁)) =0.8473, S (F (x₂)) =0.7951, S (F (x₃)) =0.7013, S (F (x₄)) =0.6466	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Das and Kar, Wang et al. [28, 29]	S (F (x₁)) =0.8231, S (F (x₂)) =0.7938, S (F (x₃)) =0.6591, S (F (x₄)) =0.5905	x₁ ≻ x₂ ≻ x₃ ≻ x₁	x ₁
Algorithm 1	Q₁ = 0.1253, Q₂ = 0.0875, Q₃ = -0.0300, Q₄ = -0.0864	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 2	V₁ = 0.8358, V₂ = 0.7945, V₃ = 0.6811, V₄ = 0.6197	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 3	U₁ = 1.0000, U₂ = 0.9529, U₃ = 0.8683, U₄ = 0.8189	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
w = {0.7, 0.1, 0.2}
Babitha and John [25]	a₁ = 0.7583, a₂ = 0.7842, a₃ = 0.5881, a₄ = 0.5347	x₂ ≻ x₁ ≻ x₃ ≻ x₄	x ₂
Wang et al. [21]	c₁ = 3, c₂ = 3, c₃ = 1, c₄ = 1	x₁ ∼ x₂ ≻ x₃ ∼ x₄	x₁, x₂
Das and Kar, Wang et al. [28, 29]	S (F (x₁)) =0.7811, S (F (x₂)) =0.7858, S (F (x₃)) =0.6327, S (F (x₄)) =0.5882	x₂ ≻ x₁ ≻ x₃ ≻ x₄	x ₂
Das and Kar, Wang et al. [28, 29]	S (F (x₁)) =0.7521, S (F (x₂)) =0.7836, S (F (x₃)) =0.5560, S (F (x₄)) =0.4945	x₂ ≻ x₁ ≻ x₃ ≻ x₄	x ₂
Algorithm 1	Q₁ = 0.1350, Q₂ = 0.1351, Q₃ = -0.0279, Q₄ = -0.0818	x₂ ≻ x₁ ≻ x₃ ≻ x₄	x ₂
Algorithm 2	V₁ = 0.7675, V₂ = 0.7852, V₃ = 0.5971, V₄ = 0.5441	x₂ ≻ x₁ ≻ x₃ ≻ x₄	x ₂
Algorithm 3	U₁ = 0.9955, U₂ = 1.0000, U₃ = 0.8537, U₄ = 0.8112	x₂ ≻ x₁ ≻ x₃ ≻ x₄	x ₂

6 Conclusion and remarks

The major contributions in this paper can be summarized as follows:

A novel hesitant fuzzy soft decision making approach based on MABAC is explored, which has not been reported in the existing literature. The approach has a straightforward computation procedure, systematic process and a sound logic that shows the foundation of decision making.

A novel hesitant fuzzy soft decision making approach based on WASPAS is proposed, which unique combination of two well known decision making approaches, i.e. weighted sum model (WSM) and weighted product model (WPM) at first.

A novel hesitant fuzzy soft decision making approach based on COPRAS is proposed. It determines a solution and the ratio to the ideal solution and the ratio to the worst-ideal solution, and therefore can be regarded as a compromising method.

The subjective weighting methods pay much attention to the preference information of the decision maker [25 , 27–29], while they neglect the objective information. The objective weighting methods do not take into account the preference of the decision maker, in particular, these methods fail to take into account the risk attitude of the decision maker. The characteristic of our weighting model can reflect both the subjective considerations of the decision maker and the objective information.

In the future, we shall apply immediate probabilities methods [48, 49], Jeffrey’s Rule [50], decision making methods [51 –58], aggregation operators [59 –66], information measure [67, 68], algebraic structure [69, 70] into hesitant fuzzy soft set and solve more decision making problems.

Footnotes

Acknowledgments

The authors are very appreciative to the reviewers for their precious comments which enormously ameliorated the quality of this paper. Our work is sponsored by the National Natural Science Foundation of China (No. 61101134), the General Project of Shaoguan University (No. SY2016KJ11).

References

Zadeh

L.A.

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

Pawlak

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

Molodtsov

D.A.

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

Feng

, Cho

, Pedrycz

, Fujita

and Herawan

, Soft set based association rule mining, Knowledge-Based Systems 11 (2016), 268–282.

Deli

and Çağman

, Probabilistic equilibrium solution of soft games, Journal of Intelligent & Fuzzy Systems 30 (2016), 2237–2244.

Yang

D.L.

, Xiao

, Xu

, Wang

X.N.

and Pan

Y.C.

, A Novel Soft Set Approach for Feature Selection, International Journal of Database Theory and Application 9 (2016), 77–90.

Aktaş

and Çağman

, Soft decision making methods based on fuzzy sets and soft sets, Journalof Intelligent & Fuzzy Systems 30 (2016), 2797–2803.

Xie

N.X.

, An algorithm on the parameter reduction of soft sets, Fuzzy Information and Engineering 8(2016), 127–145.

Atagün

A.O.

and Aygün

, Groups of soft sets, Journal of Intelligent & Fuzzy Systems 30 (2016), 729–733.

10.

Maji

P.K.

, Biswas

and Roy

A.R.

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

11.

Alcantud

J.C.R.

, A novel algorithm for fuzzy soft set based decision making from multiobserver input parameterdata set, Information Fusion 29 (2016), 142'C148.

12.

Alcantud

J.C.R.

, Some formal relationships among soft sets, fuzzy sets, and their extensions, InternationalJournal of Approximate Reasoning 68 (2016), 45–53.

13.

Yang

and Peng

X.D.

, A revised TOPSIS method based on interval fuzzy soft set models with incomplete weightinformation, Fundamenta Informaticae 152 (2017), 297–321.

14.

Peng

X.D.

, Dai

J.G.

and Yuan

H.Y.

, Interval-valued fuzzy soft decision making methods based on MABAC, similaritymeasure and EDAS, Fundamenta Informaticae (2017). DOI: 10.3233/FI-2017-1505.

15.

Peng

X.D.

and Yang

, Algorithms for interval-valued fuzzy soft sets in stochastic multi-criteria decision making based on regret theory and prospect theory with combined weight, Applied Soft Computing 54 (2017), 415–430.

16.

Peng

X.D.

, Yang

, Song

J.P.

and Jiang

, Pythagoren fuzzy soft set and its application, Computer Engineering 41 (2015), 224–229.

17.

Yang

, Peng

X.D.

, Chen

and Zeng

, A decision making approach based on bipolar multi-fuzzy soft set theory, Journal of Intelligent & Fuzzy Systems 27 (2014), 1861–1872.

18.

Peng

X.D.

and Liu

, Algorithms for neutrosophic soft decision making based on EDAS, new similarity measure andlevel soft set, Journal of Intelligent & Fuzzy Systems 32 (2017), 955–968.

19.

Feng

, Liu

X.Y.

, Fotea

V.L.

and Jun

Y.B.

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

20.

Jun

Y.B.

, Soft BCK/BCI-algebras, Computers & Mathematics with Applications 56 (2008), 1408–1413.

21.

Wang

F.Q.

, Li

X.H.

and Chen

X.H.

, Hesitant fuzzy soft set and its applications in multicriteria decision making, Journal of Applied Mathematics 2014 (2014), 1–10.

22.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems 25 (2010), 529–539.

23.

Peng

X.D.

and Yang

, Interval-valued hesitant fuzzy soft sets and their application in decision making, Fundamenta Informaticae 141 (2015), 71–93.

24.

Jun

Y.B.

, Ahn

S.S.

and Muhiuddin

, Hesitant fuzzy soft subalgebras and ideals in BCK/BCI-algebras, The Scientific World Journal 2014 (2014), 1–7.

25.

Babitha

K.V.

and John

S.J.

, Hesitant fuzzy soft sets, Journal of New Results in Science 3 (2013), 98–107.

26.

Farhadinia

, Multicriteria decision making method based on the higher order hesitant fuzzy soft set, International Scholarly Research Notices 2014 (2014), 1–11.

27.

Zhou

X.Q.

, Xiao

Q.M.

and Li

, A group decision making approach based on hesitant fuzzy soft set theory, INFOR 53 (2015), 113–124.

28.

Das

and Kar

, The hesitant fuzzy soft set and its application in decision-making, Springer Proceedingsin Mathematics & Statistics (2015), 235–247.

29.

Wang

J.Q.

, Li

X.E.

and Chen

X.H.

, Hesitant fuzzy soft sets with application in multicriteria group decisionmaking problems, The Scientific World Journal 2015 (2015), 1–14.

30.

Beg

and Rashid

, Ideal solutions for hesitant fuzzy soft sets, Journal of Intelligent & Fuzzy Systems 30 (2016), 143–150.

31.

Pamucar

and Cirovic

, The selection of transport and handling resources in logistics centers using Multi-Attributive Border Approximation area Comparison (MABAC), Expert Systems with Applications 42(2015), 3016–3028.

32.

Yager

R.R.

and Abbasov

A.M.

, Pythagorean membership grades, complex numbers, and decision making, International Journal of Intelligent Systems 28 (2013), 436–452.

33.

Peng

X.D.

and Yang

, Some results for Pythagorean fuzzy sets, International Journal of Intelligent Systems 30 (2015), 1133–1160.

34.

Peng

X.D.

and Yang

, Pythagorean fuzzy choquet integral based MABAC method for multiple attribute groupdecision making, International Journal of Intelligent Systems 31 (2016), 989–1020.

35.

Xue

Y.X.

, You

J.X.

, Lai

X.D.

and Liu

H.C.

, An interval-valued intuitionistic fuzzy MABAC approach for materialselection with incomplete weight information, Applied Soft Computing 38 (2016), 703–713.

36.

Zavadskas

E.K.

, Turskis

, Antucheviciene

and Zakarevicius

, Optimization of weighted aggregated sumproduct assessment, Electronics and Electrical Engineering 6 (2012), 3–6.

37.

Chakraborty

and Zavadskas

E.K.

, Applications of WASPAS method in manufacturing decision making, Informatica 25 (2014), 1–20.

38.

Baležentis

and Streimikiene

, Multi-criteria ranking of energy generation scenarios with Monte Carlo simulation, Applied Energy 185 (2017), 862–871.

39.

Ghorabaee

M.K.

, Zavadskas

E.K.

, Amiri

and Esmaeili

, Multi-criteria evaluation of green suppliers using anextended WASPAS method with interval type-2 fuzzy sets, Journal of Cleaner Production 137(2016), 213–229.

40.

Zavadskas

E.K.

, Kaklauskas

and Sarka

, The new method of multicriteria complex proportional assessment ofprojects, Technological and Economic Development of Economy 1 (1994), 131–139.

41.

Ghorabaee

M.K.

, Amiri

, Sadaghiani

J.S.

and Goodarzi

G.H.

, Multiple criteria group decision-making for supplierselection based on COPRAS method with interval type-2 fuzzy sets, Int J Adv Manuf Technol 75(2014), 1115–1130.

42.

Liou

J.J.H.

, Tamošaitienė

, Zavadskas

E.K.

and Tzeng

, New hybrid COPRAS-G MADM Model for improvingand selecting suppliers in green supply chain management, International Journal of Production Research 54 (2016), 114–134.

43.

Shannon

C.E.

, A mathematical theory of communication, Bell System Technical Journal 27 (1948), 379–423.

44.

Xia

M.M.

and Xu

Z.S.

, Hesitant fuzzy information aggregation in decision making, International Journal ofApproximate Reasoning 52 (2011), 395–407.

45.

Z.S.

and Xia

M.M.

, Distance and similarity measures for hesitant fuzzy sets, Information Sciences 181 (2011), 2128–2138.

46.

Triantaphyllou

and Mann

S.H.

, An examination of the effectiveness of multi-dimensional decision-makingmethods: A decision-making paradox, Decision Support Systems 5 (1989), 303–312.

47.

Saparauskas

, Zavadskas

E.K.

and Turskis

, Selection of facade’s alternatives of commercial and publicbuildings based on multiple criteria, International Journal of Strategic Property Management 15(2011), 189–203.

48.

Yager

R.R.

, Engemann

K.J.

and Filev

D.P.

, On the concept of immediate probabilities, International Journalof Intelligent Systems 10 (1995), 373–397.

49.

Yager

R.R.

and Alajlan

, Sugeno integral with possibilistic inputs with application to multi-criteria decisionmaking, International Journal of Intelligent Systems 31 (2016), 813–826.

50.

Yager

R.R.

and Alajlan

, On a role for copula’s in jeffrey’s rule with an application to decision making, International Journal of Intelligent Systems 30 (2015), 1117–1132.

51.

S.M.

, Wang

and Wang

J.Q.

, An interval type-2 fuzzy likelihood-based MABAC approach and its application inselecting hotels on the tourism website, International Journal of Fuzzy Systems 19 (2017), 47–61.

52.

Peng

J.J.

, Wang

J.Q.

and Wu

X.H.

, Novel multi-criteria decision-making approaches based on hesitant fuzzy setsand prospect theory, International Journal of Information Technology and Decision Making 15 (2016), 621–643.

53.

Zhang

and Wei

G.W.

, Extension of VIKOR method for decision making problem based on hesitant fuzzy set, Applied Mathematical Modelling 37 (2013), 4938–4947.

54.

Wei

G.W.

and Zhang

, A multiple criteria hesitant fuzzy decision making with Shapley value-based VIKOR method, Journal of Intelligent and Fuzzy Systems 26 (2014), 1065–1075.

55.

Wei

G.W.

, Wang

H.J.

, Zhao

X.F.

and Lin

, Approaches to hesitant fuzzy multiple attribute decision making withincomplete weight information, Journal of Intelligent and Fuzzy Systems 26 (2014), 259–266.

56.

Wang

, Wang

J.Q.

and Zhang

H.Y.

, A likelihood-based TODIM approach based on multi-hesitant fuzzy linguisticinformation for evaluation in logistics outsourcing, Computers and Industrial Engineering 99(2016), 287–299.

57.

S.M.

, Wang

and Wang

J.Q.

, An extended TODIM approach with intuitionistic linguistic numbers, International Transactions in Operational Research (2016). DOI: 10.1111/itor. 12363

58.

Peng

J.J.

, Wang

J.Q.

and Yang

W.E.

, A multi-valued neutrosophic qualitative flexible approach based on likelihoodfor multi-criteria decision-making problems, International Journal of Systems Science 48 (2017), 425–435.

59.

Peng

J.J.

, Wang

J.Q.

, Wu

X.H.

and Tian

, Hesitant Intuitionistic Fuzzy Aggregation Operators based on theArchimedean t-norms and t-conorms, International Journal of Fuzzy Systems (2017). DOI: 10.1007/s40815-017-0303-4

60.

Z.S.

, Xia

M.M.

and Chen

, Some hesitant fuzzy aggregation operators with their application in group decision making, Group Decision and Negotiation 22 (2013), 259–279.

61.

Zhang

Z.M.

, Hesitant fuzzy power aggregation operators and their application to multiple attribute group decisionmaking, Information Sciences 234 (2013), 150–181.

62.

Zhou

L.Y.

, Zhao

X.F.

and Wei

G.W.

, Hesitant fuzzy hamacher aggregation operators and their application tomultiple attribute decision making, Journal of Intelligent and Fuzzy Systems 26 (2014), 2689–2699.

63.

Wei

G.W.

, Hesitant Fuzzy prioritized operators and their application to multiple attribute group decision making, Knowledge-Based Systems 31 (2012), 176–182.

64.

Wei

G.W.

and Zhao

X.F.

, Induced hesitant interval-valued fuzzy einstein aggregation operators and theirapplication to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 24(2013), 789–803.

65.

Wei

G.W.

, Xu

X.R.

and Deng

D.X.

, Interval-valued dual hesitant fuzzy linguistic geometric aggregation operatorsin multiple attribute decision making, International Journal of Knowledgebased and Intelligent Engineering Systems 20 (2016), 189–196.

66.

Nie

R.X.

, Wang

J.Q.

and Li

, 2-tuple linguistic intuitionistic preference relation and its application insustainable location planning voting system, Journal of Intelligent and Fuzzy Systems (2017), DOI: 10.3233/JIFS-162139

67.

Z.S.

and Xia

, On distance and correlation measures of hesitant fuzzy information, InternationalJournal of Intelligence Systems 26 (2011), 410–425.

68.

Zhou

, Wang

J.Q.

and Zhang

H.Y.

, Multi-criteria decisionmaking approaches based on distance measures forlinguistic hesitant fuzzy sets, Journal of the Operational Research Society (2016). DOI: 10.1057/jors.2016.41

69.

Zhan

J.M.

, Liu

and Davvaz

, A new rough set theory: Rough soft hemirings, Journal of Intelligent and Fuzzy Systems 28 (2015), 1687–1697.

70.

Zhan

J.M.

, Yu

and Fotea

, Characterizations of two kinds of hemirings based on probability spaces, Soft Computing 20 (2016), 637–648.

Hesitant fuzzy soft decision making methods based on WASPAS,MABAC and COPRAS with combined weights

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Hesitant fuzzy set

3 Three algorithms for hesitant fuzzy soft decision making

3.1 Problem description

Table 1 Tabular representation of (F, E) e 1 e 2 ⋯ 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 )

3.2.1 Determining the objective weights: The Shannon entropy method

Table 2 Tabular representation of (F, E) e 1 e 2 e 3 x 1 {0.1, 0.2} {0.8, 0.9, 1.0} {0.1} x 2 {0.1, 0.2} {0.8, 0.9} {0.1} x 3 {0.0, 0.1, 0.2} {0.8, 0.9} {0.1} x 4 {0.1, 0.2, 0.3} {0.7, 0.8} {0.2}

5.1 Comparison of the newly proposed three approaches with their own

5.2 Comparison of the newly proposed three methods with other approaches

5.2.1 Babitha and John’s method [25] and its limitation

Table 4 Tabular representation of (F, A) e 1 e 2 e 3 x 1 {0.3, 0.4, 0.5} {0.5} {0.1} x 2 {0.4} {0.5} {0.1} x 3 {0.2, 0.4, 0.6} {0.5} {0.1} x 4 {0.35, 0.4, 0.45} {0.5} {0.1}

Table 6 Tabular representation of (F, A) e 1 e 2 e 3 x 1 {0.1, 0.5, 0.9} {0.6} {0.2} x 2 {0.5} {0.6} {0.2} x 3 {0.2, 0.5, 0.8} {0.6} {0.2} x 4 {0.4, 0.5, 0.6} {0.6} {0.2}

Footnotes

Acknowledgments

References

Table 1
Tabular representation of (F, E)

e ₁ e ₂ ⋯ 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)

Table 2
Tabular representation of (F, E)

e ₁ e ₂ e ₃

x ₁ {0.1, 0.2} {0.8, 0.9, 1.0} {0.1}

x ₂ {0.1, 0.2} {0.8, 0.9} {0.1}

x ₃ {0.0, 0.1, 0.2} {0.8, 0.9} {0.1}

x ₄ {0.1, 0.2, 0.3} {0.7, 0.8} {0.2}

Table 4
Tabular representation of (F, A)

e ₁ e ₂ e ₃

x ₁ {0.3, 0.4, 0.5} {0.5} {0.1}

x ₂ {0.4} {0.5} {0.1}

x ₃ {0.2, 0.4, 0.6} {0.5} {0.1}

x ₄ {0.35, 0.4, 0.45} {0.5} {0.1}

Table 6
Tabular representation of (F, A)

e ₁ e ₂ e ₃

x ₁ {0.1, 0.5, 0.9} {0.6} {0.2}

x ₂ {0.5} {0.6} {0.2}

x ₃ {0.2, 0.5, 0.8} {0.6} {0.2}

x ₄ {0.4, 0.5, 0.6} {0.6} {0.2}