Algorithms for hesitant fuzzy soft decision making based on revised aggregation operators,WDBA and CODAS

Abstract

Realistically, it isn’t invariably the case that decision makers (DMs) or experts are capable of evaluating alternatives by means of exact values. With the uncertain information achieved, the DMs tend to offer their evaluations by giving various values in corresponding parameters. Hesitant fuzzy soft sets (HFSSs) permit each element to possess diverse number of parameter values of those parameters are denoted by multiple conceivable membership values. This paper develops three novel decision making methods in hesitant fuzzy soft text. First, the revised hesitant fuzzy aggregation operators are proposed for avoiding the counterintuitive phenomena. Later, the objective weights of diverse parameters are counted by deviation-based method. Then, we introduce the combination weights, which can reveal both the subjective decision information and the objective decision information. Afterwards, we present three methods for dealing hesitant fuzzy soft decision making issue via revised aggregation operators, WDBA (Weighted Distance Based Approximation) and CODAS (COmbinative Distance-based ASsessment). Finally, the feasibility and effectiveness of algorithms are stated by some numerical examples. The notable traits of the developed algorithms, compared to the existing hesitant fuzzy soft decision making algorithms, are (1) they can achieve the best alternative out of counterintuitive phenomena; (2) they have a stronger ability in differentiating the best alternative; (3) they can forbear the parameter selection issues.

Keywords

Hesitant fuzzy soft set decision-making approach revised aggregation operators WDBA CODAS

1. Introduction

Many complex and practical problems in real-life involve indeterminacy and uncertainty. Meanwhile, plentiful existing theories or methods such as fuzzy sets (FSs) [1], and rough sets (RSs) [2] have been presented to simulate indeterminacy. Nevertheless, each of these theories or methods has its intrinsic deficiencies as discussed in [3]. The soft sets (SSs), developed by Molodtsov [3], is averted from the shortcomings of the parameterization means of those methods ortheories [1]. It has latent use cases in various domains such as feature selection [4], rule mining [5], decision making [6], parameter reduction [7], game theory [8]. Uniting soft sets with the existing uncertain theories, various generalizations of SSs have been explored such as fuzzy soft sets (FSSs) [9 –11], interval-valued fuzzy soft sets (IVFSSs) [12 –15], Pythagorean fuzzy soft sets (PFSSs) [16], intuitionistic fuzzy soft sets (IFSSs) [17], neutrosophic soft sets (NSSs) [18], rough soft sets (RSSs) [19 –21].

The above models are conceived to dispose of uncertainties by taking virtues of soft matrix [22 –25]. In above extensions of SSs, the membership value is either a monodromy. As a matter of fact, the value of membership may be multiple conceivable values in SSs, so uniting the hesitant fuzzy sets (HFSs) [26] with SSs, Wang et al. [27] initiated the notion of hesitant fuzzy soft sets (HFSSs) which can denote diverse preferences from different decision-makers and avoid overlooking any subjective intentions of decision-makers, and introduced a decision making method for handling some decision making issues. Later, the hesitant fuzzy soft aggregation operators [28 –30] were explored for dealing with the multi-criteria decision making (MCDM) issues and multi-criteria group decision making (MCGDM) issues, respectively. Beg and Rashid [31] presented a revised TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) algorithm for MCGDM in hesitant fuzzy soft text. Das et al. [32] pioneered an algorithm by applying correlation coefficient in MCGDM issue. Peng and Dai [33] examined three approaches to deal hesitant fuzzy soft decision making issue by MABAC (Multi-Attributive Border Approximation area Comparison) [34], COPRAS (Complex Proportional Assessment) and WASPAS (Weighted Aggregated Sum Product Assessment). Babitha and John [35] conceived the choice value methods in dealing MCDM issues in hesitant fuzzy soft text. In addition, some extensions of hesitant fuzzy soft sets [36, 37] have been widely applied in various domains.

Due to the shortcomings (counterintuitive phenomena [28–30 , 33], discrimination problem [28 –30], parameter selection problems [33]) of some existing MCDM or MCGDM algorithms for HFSS, it may be hard for experts to choose best or optimal alternative. Hence, the goal of this paper is to solve the two dilemmas discussed above by introducing three MCDM algorithms (revised aggregation operators, WDBA [38] and CODAS [43]) to deal with decision information for HFSSs, which not only own a stronger ability in differentiating the best alternative, but also can achieve the best alternative without parameter selection issues and counterintuitive cases.

Considering that diverse weights of parameters will affect the final ordering results of the given alternatives, inspired by Peng and Dai [33], a novel algorithm to count the parameters’ weights by uniting the objective ones with the subjective ones is proposed. This weighting method is different from the existing methods in hesitant fuzzy soft text, which can be classified into two sides: one is the subjective weighting counting approaches and the other is the objective weighting counting algorithms, which can be calculated by deviation-based method. The subjective weighting counting methods concentrate on the preference information setting in advance of the experts [28–30 , 35], while they defiance the objective decision information. The objective weighting counting methods are out of the consideration of the preference information of experts. The characteristic of our weighting counting model can reveal both the subjective preference information and the objective preference information. Therefore, uniting the objective weights with subjective weights, a combination weight model is presented. In addition, our weight counting algorithm has achieved the consistency with the reference [33] when employing the same level MCDM method for obtaining the conclusive ranking results.

To reach above targets, the key contributions are listed in the following.

(1) Some revised hesitant fuzzy aggregation operators (R-HFWA and R-HFWG) are presented which can avoid the counterintuitive phenomena [28–30 , 33].

(2) A novel weight counting method is presented for averting the effect of subjective way and objective way.

(3) Three developed methods (revised aggregation operators, WDBA and CODAS) with some existing MCDM methods [28–30 , 33] are contrasted by certain examples.

The rest paper is listed as follows: In Section 2, some essential notions of HFSs and HFSSs are briefly reviewed. In Section 3, three hesitant fuzzy soft MCDM methods based on revised aggregation operators, WDBA and CODAS are developed. In Section 4, an emergency decision making example is presented to state the developed algorithms. In Section 5, a comparison with existing MCDM methods is given for showing the validity of developed methods. The paper gives some conclusions in Section 6.

2. Preliminaries

2.1. Hesitant fuzzy set

Definition 1. [26, 47] A hesitant fuzzy set on U is denoted in terms of a function that returns a subset of [0, 1] when applied to U, i.e., $A = {< x, h (x) > ∣ x \in U},$ (1)

where h (x) is a series of some diverse values in [0,1], representing the conceivable membership degrees of the element x ∈ U to A. For the sake of simplicity, Xia and Xu [47] called h (x) a hesitant fuzzy element (HFE). If all the elements in HFE are 1, then Peng and Dai [33] called Γ; if all the elements in HFE are 0, then Peng and Dai [33] called Φ.

Definition 2. [47] Let h = ⋃ _r∈h {γ} be an HFE, then the score function of h is defined as follows: $S (h) = \frac{1}{# h} \sum_{r \in h} γ,$ (2) where # h is the number of the elements in h. For two HFEs h₁, h₂, if S (h₁) > S (h₂), then h₁ > h₂.

Definition 3. [26, 47] Let h, h₁, and h₂ be three HFEs, then

(1) h^c = ⋃ _r∈h {1 - r};

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

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

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

(5) h₁ ⊗ h₂ = ⋃ _{r₁∈h₁,r₂∈h₂} {r₁r₂};

(6) λh = ⋃ _r∈h {1 - (1 - r) ^λ} , λ > 0;

(7) h^λ = ⋃ _r∈h {r^λ} , λ > 0.

Definition 4. [48] Let h and g be two HFEs, then the distance measure for HFEs is denoted as follows: $d (h, g) = \frac{1}{2 l} \sum_{i = 1}^{l} (∣ h_{σ (i)} - g_{σ (i)} ∣),$ (3) where l is the number of max {# h, # g}.

Definition 5. [47] Let h_j (j = 1, 2, ⋯ , n) be a collection of HFEs. The hesitant fuzzy weighted averaging (HFWA) operator is a mapping Hⁿ → H such that

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

However, Peng and Dai [33] pointed that HFWA operator has shortcomings in certain cases, stated as follows:

Let h_j (j = 1, 2, ⋯ , n) be a set of HFEs. If there is i such that h_i = Γ, then based on Eq. (3), we can obtain HFWA (h₁, h₂, ⋯ , h_n) = Γ. This result may lead to counter-intuitive phenomena in some MCDM issues. That is to say, it only determines by h_i to make ultimate decision and the decision information of others can be overlooked.

Consequently, it is irrational and unsuitable to apply Eq. (3) to integrate the information in decision making when meet the special cases discussed above.

Definition 6. [47] Let h_j (j = 1, 2, ⋯ , n) be a set of HFEs. The hesitant fuzzy weighted geometric (HFWG) operator is a mapping Hⁿ → H such that

$\begin{matrix} HFWG (h_{1}, h_{2}, \dots, h_{n}) = ⨂_{j = 1}^{n} (h_{j})^{w_{j}} \\ = ⋃_{r_{1} \in h_{1}, r_{2} \in h_{2}, \dots, r_{n} \in h_{n}} {\prod_{j = 1}^{n} (r_{j})^{w_{j}}} \end{matrix}$ (5)

Nevertheless, Peng and Dai [33] pointed that the HFWG operator has shortcomings in certain cases, stated as follows.

Let h_j (j = 1, 2, ⋯ , n) be a set of HFEs. If there is i such that h_i = Φ, then based on Eq.(4), we can have HFWG (h₁, h₂, ⋯ , h_n) = Φ. This result may lead to counter-intuitive phenomena in MCDM issues. That is to say, it only determines by h_i to make decision and the decision information of others can be overlooked.

Hence, it is irrational and unsuitable to apply Eq.(4) to integrate the information in decision making when meet the special cases discussed above.

For solving the above drawbacks, we propose two revised methods in the following.

Definition 7. Let h = ⋃ _r∈h {r} be an HFE, then

(1) The Δ-revised of weighted averaging: $h^{Δ} = {\begin{matrix} ⋃_{r \in h} {r - Δ}, r = 1 \\ ⋃_{r \in h} {r}, r \neq 1 \end{matrix}$ (6)

(2) The Δ-revised of weighted geometric: $h^{Δ} = {\begin{matrix} ⋃_{r \in h} {r + Δ}, r = 0 \\ ⋃_{r \in h} {r}, r \neq 0 \end{matrix}$ (7)

where Δ is a positive fuzzy number and far less than any nonzero r.

Definition 8. Let h_j (j = 1, 2, ⋯ , n) be a collection of HFEs. A Δ-revised hesitant fuzzy weighted averaging (R-HFWA) operator is a mapping Hⁿ → H such that

$\begin{matrix} R - HFWA (h_{1}, h_{2}, \dots, h_{n}) = \oplus_{j = 1}^{n} (w_{j} h_{j}^{Δ}) \\ = ⋃_{r_{1}^{Δ} \in h_{1}, r_{2}^{Δ} \in h_{2}, \dots, r_{n}^{Δ} \in h_{n}} {1 - \prod_{j = 1}^{n} (1 - r_{j}^{Δ})^{w_{j}}} \end{matrix}$ (8)

Definition 9. [47] Let h_j (j = 1, 2, ⋯ , n) be a series of HFEs. A Δ-revised hesitant fuzzy weighted geometric (HFWG) operator is a mapping Hⁿ → H such that

$\begin{matrix} R - HFWG (h_{1}, h_{2}, \dots, h_{n}) = ⨂_{j = 1}^{n} (h_{j}^{Δ})^{w_{j}} \\ = ⋃_{r_{1}^{Δ} \in h_{1}, r_{2}^{Δ} \in h_{2}, \dots, r_{n}^{Δ} \in h_{n}} {\prod_{j = 1}^{n} (r_{j}^{Δ})^{w_{j}}} \end{matrix}$ (9)

Theorem 1. Let h_j (j = 1, 2, ⋯ , n) be a collection of HFEs. The aggregation result by R-HFWA(Eq. (7)) is monotonically decreasing when Δ is monotonically increasing and the aggregation result by R-HFWG(Eq. (8)) is monotonically decreasing when Δ is monotonically increasing.

Proof. Divide the possible value into two cases (case 1 and case 2) when r_j = 1. Case 1 has t₁ HFEs which have revised by Eq.(5) with j ∈ m_{t
₁}. Case 2 has t₂ HFEs which keep original values with j ∈ m_{t
₂}. It is obvious that t₁ + t₂ = n.

Based on Eq.(7), we can have

$R - HFWA (h_{1}, h_{2}, \dots, h_{n}) = \oplus_{j = 1}^{n} (w_{j} h_{j}^{Δ}) = ⋃_{r_{1}^{Δ} \in h_{1}, r_{2}^{Δ} \in h_{2}, \dots, r_{n}^{Δ} \in h_{n}} {1 - \prod_{j = 1}^{n} (1 - r_{j}^{Δ})^{w_{j}}} = ⋃_{r_{1}^{Δ} \in h_{1}, r_{2}^{Δ} \in h_{2}, \dots, r_{n}^{Δ} \in h_{n}} {1 - \prod_{j \in m_{t_{1}}} (1 - r_{j} + Δ)^{w_{j}} \prod_{j \in m_{t_{2}}} (1 - r_{j})^{w_{j}}}$ .

Suppose that Δ₁ < Δ₂, then

$1 - \prod_{j \in m_{t_{1}}} (1 - r_{j} + Δ_{1})^{w_{j}} \prod_{j \in m_{t_{2}}} (1 - r_{j})^{w_{j}} - (1 - \prod_{j \in m_{t_{1}}} (1 - r_{j} + Δ_{2})^{w_{j}} \prod_{j \in m_{t_{2}}} (1 - r_{j})^{w_{j}}) = \prod_{j \in m_{t_{2}}} (1 - r_{j})^{w_{j}} (\prod_{j \in m_{t_{1}}} (1 - r_{j} + Δ_{2})^{w_{j}} - \prod_{j \in m_{t_{1}}} (1 - r_{j} + Δ_{1})^{w_{j}}) > 0$ .

Consequently, we have the aggregation result by R-HFWA(Eq. (7)) is monotonically decreasing when Δ is monotonically increasing.

Similarly, we can have the aggregation result by R-HFWG(Eq. (8)) is monotonically decreasing when Δ is monotonically increasing.

2.2. Hesitant fuzzy soft set

Definition 10. [27] 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).

Definition 11. [27] Let A, B ⊆ E. (F, A) and (G, B) be two HFSSs over U. (G, B) is called a hesitant fuzzy soft subset of (F, A) if

(1) A ⊇ B;

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

Example 1. Suppose that U = {x₁, x₂, x₃} is a set of houses and E = {e₁ =cheap, e₂ =beautiful, e₃=size, e₄ = location} is a set of parameters. In this case, HFSS can describe the characteristics of the house under the hesitant fuzzy information and it is shown in Table 1.

Table 1
Tabular representation of (F, E) in Example 1

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

F (x) = F (y) iff F (e) (x) = F (e) (y) for all e ∈ E . HFSS theory is a generalization of HFS theory. If set of parameters is singleton set then every HFSS on U is also HFS on U. The advantage of using the HFSS theory regarding the HFS theory is their enough parameters which can overcome shortcomings of the parameterization.

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 shown in Table 2. 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 DMs or experts.

Table 2
Tabular representation 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)

In the following, we will apply the revised aggregation operators, WDBA and CODAS methods to HFSS.

3.1.1 The hesitant fuzzy soft decision making approach based WDBA

WDBA (Weighted Distance Based Approximation), developed by Rao and Singh [38], gauges the distance from the optimum point (maximum value of alternatives) and non-optimum point (minimum value of alternatives). Last, the alternatives are ordered by the suitability index (SI). The alternative owning the minimum value of SI is ordered at final place and with the maximum value at first place. WDBA method has been successfully applied in diverse domains such as COTS component selection [39], selection of E-learning websites [40, 41], selection of software effort estimation [42].

To address the MCDM issue with hesitant fuzzy soft text, we attempt to develop a hesitant fuzzy soft method based on WDBA.

First of all, we take normalize information into account due to certain cost parameters and benefit parameters in MCDM matrix. These two diverse parameters are opposite. That is to say, the larger value means the better behavior of a benefit parameter but reports the worse behavior of a cost parameter. Thus, for the sake of the compatibility of all parameters, we proceed to transform the value of cost parameter into the value of benefit parameter via the following equation:

$\begin{matrix} \tilde{F} (e_{j}) (x_{i}) = \\ {\begin{matrix} ⋃_{r_{ij} \in F (e_{j}) (x_{i})} {r_{ij}}, e_{j} is benefit parameter, \\ ⋃_{r_{ij} \in F (e_{j}) (x_{i})} {1 - r_{ij}}, e_{j} is cost parameter . \end{matrix} \end{matrix}$ (10) Based on this equation, we can obtain the normalized hesitant fuzzy soft matrix $R^{'} = (\tilde{F} (e_{j}) (x_{i}))_{n \times m}$ .

Next, we compute the score function t_ij (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n) of $\tilde{F} (e_{j}) (x_{i})$ by Eq. (11), $t_{ij} = \frac{1}{# \tilde{F} (e_{j}) (x_{i})} \sum_{γ_{ij} \in \tilde{F} (e_{j}) (x_{i})} γ_{ij} .$ (11)

In order to show the standardized matrix SM_ij visually, we define it in an average value matrix ${\bar{V}}_{j} (j = 1, 2, \dots, n)$ and revised deviation matrix SD_j (j = 1, 2, ⋯ , n). The detailed process is shown as follows: ${SM}_{ij} = \frac{t_{ij} - {\bar{V}}_{j}}{{SD}_{j}},$ (12) ${\bar{V}}_{j} = \frac{1}{m} \sum_{i = 1}^{m} t_{ij},$ (13) ${SD}_{j} = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} (t_{ij} - {\bar{V}}_{j})^{2}} .$ (14)

The explored WDBA algorithm is based on the notion that the chosen alternative (greatest) own the shortest distance measure from the ideal solution (best alternative) and be farthest from the anti-ideal solution (worst alternative). The ideal points are a series of values ideally longing for. The anti-ideal points are a series of values ideally not longing for. The ideal points, defined by SM⁺ and anti-ideal points, defined by SM^- are found from decision matrix, which can be denoted in the following. ${SM}^{+} = {\max_{i} {{SM}_{ij}} ∣ j = 1, 2, \dots, n},$ (15) ${SM}^{-} = {\min_{i} {{SM}_{ij}} ∣ j = 1, 2, \dots, n},$ (16) where i = 1, 2, ⋯ , m.

Fig.1

Euclidean distance of an alternative to ideal and anti-ideal points in 2D space (case of two parameters).

Figure 1 presents Euclidean distance of the alternative to ideal and anti-ideal solutions in 2D space in case of two parameters e₁ and e₂. The real domain is presented inside the rectangular box. The Euclidean distance between points P and Q in n dimensional space is the length of the line segment, PQ.

In order to obtain the objective weight information from the decision data, the deviation-based method is proposed as follows: $ζ_{j} = \frac{{SD}_{j}}{\sum_{j = 1}^{n} {SD}_{j}} .$ (17)

Combining the subjective weight w_j (j = 1, 2, ⋯ , n) (AHP determination method) [49] given by decision maker with above objective weight ζ_j, the combined weight ϖ_j of criteria is shown as follows: $ϖ_{j} = β w_{j} + (1 - β) ζ_{j},$ (18) where β is the aggregating coefficient which could be changed in the range of 0 to 1. In this paper, we take β = 0.5 .

Weighted Euclidean distance (WED) between an alternative A_i and ideal point SM⁺ is denoted as ${WED}_{i}^{+}$ and between an alternative A_i and anti-ideal point is shown as ${WED}_{i}^{-}$ . ${WED}_{i}^{+} = \sqrt{\sum_{j = 1}^{n} ϖ_{j} ({SM}_{ij} - {SM}^{+})^{2}},$ (19) ${WED}_{i}^{-} = \sqrt{\sum_{j = 1}^{n} ϖ_{j} ({SM}_{ij} - {SM}^{-})^{2}},$ (20) where i = 1, 2, ⋯ , m.

The suitability index (SI) indicates the relative closeness of a particular alternative to the ideal point. The bigger the index scores for a particular alternative, the nearer the alternative to the ideal point. The greater the SI, the higher the rank of that alternative. The SI is calculated in the following. ${SI}_{i} = \frac{{WED}_{i}^{-}}{{WED}_{i}^{+} + {WED}_{i}^{-}} .$ (21)

It is easy to see that the WDBA is similar to the wide used TOPSIS method. The difference between the WDBA and the classical TOPSIS is that WDBA focuses on its standard process shown in Eq. (12). That is to say, our WDBA has a better data presentation.

In general, hesitant fuzzy soft WDBA approach involves the following steps:

Algorithm 1 :WDBA

1: Obtain the hesitant fuzzy soft decision matrix R = (F (e_j) (x_i)) _n×m (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n).

2: Transform the matrix R = (F (e_j) (x_i)) _m×n into a normalized hesitant fuzzy soft matrix $R^{'} = (\tilde{F} (e_{j}) (x_{i}))_{n \times m}$ by Eq. (10).

3: Calculate the score matrix T = (t_ij) _m×n of $R^{'} = (\tilde{F} (e_{j}) (x_{i}))_{n \times m}$ by Eq. (11).

4: Form the standardized matrix SM = (SM_ij) _m×n by Eq. (12).

5: Compute the ideal points SM⁺ and anti-ideal points SM^- by Eqs. (15) and (16).

6: Calculate the combined weight by Eq.(18).

7: Compute the ${WED}_{i}^{+}$ and ${WED}_{i}^{-}$ by Eqs. (19) and (20).

8: Obtain the suitability index value SI_i of each alternative A_i by Eq. (21).

9: Determine the ordering of the alternatives by the SI.

Remark 1. It is obbligato to state the hesitant fuzzy soft WDBA decision making method, which the information in decision matrix is denoted as HFEs to show the experts’ opinions. Voiced by the degree of membership with a set of conceivable values, HFEs are very efficient for seizing the vagueness and uncertainty of the experts in the MCDM issue. Moreover, the hesitant fuzzy soft WDBA method is an inestimable tool to deal the MCDM or MCGDM issues with HFEs having a great power in differentiating the best alternative and achieving the best alternative without parameter selection problems and counterintuitive phenomena (It has no the process of aggregation operators discussed in Definitions 5 and 6). But other methods [28–30 , 35] do not own this desirable character.

3.1.2 The hesitant fuzzy soft decision making approach based CODAS

CODAS (Combinative Distance-based Assessment), initiated by Ghorabaee et al. [43], gauges the whole performance of one alternative by Hamming distance and Euclidean distance from the negative-ideal point. The CODAS employs the Euclidean distance as the first measure of evaluation. If the Euclidean distance of two alternatives is quite near to each other, Hamming distance is used to compute and compare them. The closeness degree of Euclidean distance is setting by a threshold parameter. The Hamming distance and Euclidean distance are gauged for l¹-norm and l²-norm indifference spaces, respectively [44]. Consequently, for CODAS algorithm, we firstly evaluate the alternatives in an l²-norm indifference space. If given alternatives are out of comparison in current space, we continue to go to an l¹-norm indifference space. To run this process, we should compare each pair of alternatives. CODAS method has been successfully applied in emergency decision making [45] and market segmentevaluation [46].

For solving the MCDM issue with hesitant fuzzy soft text, we attempt to develop a hesitant fuzzy soft CODAS method. It is a novel and resultful MCDM algorithm. Nevertheless, the Euclidean distance and Hamming distance are denoted in a crisp number and we cannot employ them in dealing with hesitant fuzzy soft set issues. The goal of this paper is to develop a hesitant fuzzy soft CODAS algorithm. For reaching this goal, we utilize the hesitant fuzzy weighted Hamming distance and hesitant fuzzy weighted Euclidean distance which instead of the traditional distances.

First of all, we take normalize information into consider by Eq. (10) and compute the score function t_ij of F (e_j) (x_i) (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n) by Eq. (11).

In order to calculate the weighted normalized decision matrix R = (r_ij) _m×n, we denote it in weighted normalized performance values, shown as follows: $r_{ij} = ϖ_{j} t_{ij},$ (22) where w_j (0 < ϖ_j < 1) denotes the combined weight of jth parameter, and $\sum_{j = 1}^{n} ϖ_{j} = 1$ .

The developed CODAS method is based on the negative-ideal solution (NIS). We denote a negative-ideal solution in the following. $NIS = ({nis}_{j})_{1 \times n},$ (23) ${nis}_{j} = \min_{i} {r_{ij}},$ (24) where i = 1, 2, ⋯ , m.

Later, we calculate the Euclidean distance measure E = (E_i) _1×m and Hamming distance T = (T_i) _1×m of alternatives from negative-ideal solution in the following. $E_{i} = \sqrt{\sum_{j = 1}^{n} (r_{ij} - {nis}_{j})^{2}},$ (25) $T_{i} = \sum_{j = 1}^{n} ∣ r_{ij} - {nis}_{j} ∣ .$ (26)

According to the above two distances, we can obtain the relative assessment (RA) matrix, shown in the following. $RA = ({ra}_{ik})_{m \times m},$ (27) ${ra}_{ik} = (E_{i} - E_{k}) + (ψ (E_{i} - E_{k}) \times (T_{i} - T_{k})),$ (28) where k ∈ 1, 2, ⋯ , m and ψ express a threshold function to recognize the equality of the Euclidean distance of two alternatives, and is shown in the following. $ψ (x) = {\begin{matrix} 1, if ∣ x ∣ \geq Θ, \\ 0, if ∣ x ∣ < Θ . \end{matrix}$ (29)

In above function, Θ is the threshold parameter that can be set by experts or DMs. It is advised to set this parameter at a value between 0.01 and 0.05 which a value close to 0. If the discrepancy between Euclidean distance of two alternatives is greater than Θ, these two alternatives are also computed by the Hamming distance. It has been validated that when Θ=0.02, the results are kept in with the primitive data [43].

Next, we can aggregate the evaluation score of each alternative. The assessment score value RA_i of an alternative is the greatest is the optimal alternative which can be chosen for the considered MCDM issue. ${RA}_{i} = \sum_{k = 1}^{m} {ra}_{ik} .$ (30)

To depict the developed method, Ghorabaee et al. [43] used a simple situation with seven alternatives {x₁, x₂, x₃, x₄, x₅, x₆, x₇} and two parameters {e₁, e₂}. Suppose that weighted normalized performance values r_ij have been computed. These values are ranged between 0 and 1. Figure 2 [43] shows the position of all alternatives by means of these values.

Fig.2

A simple graphical example with two parameters.

In general, hesitant fuzzy soft CODAS algorithm involves the following steps:

Algorithm 2 :CODAS

1: It is the same as the Step 1 to Step 3 in Algorithm 1.

2: Calculate the combined weight by Eq.(18).

3: Compute the weighted normalized decision matrix r_ij by Eq. (22).

4: Determine the negative-ideal solution NIS by Eq. (23).

5: Compute the Euclidean distance E and Hamming distance T from the NIS by Eqs. (25) and (26).

6: Compute the relative assessment matrix RA by Eq. (27).

7: Compute the assessment score of each alternative RA_i by Eq. (30).

8: Rank the alternatives by the decreasing values of assessment RA.

Remark 2. It is obbligato to state the hesitant fuzzy soft CODAS decision making method, which the information in decision matrix is denoted as HFEs to show the experts’ opinions. Moreover, the hesitant fuzzy soft CODAS method is an inestimable tool to deal the MCDM or MCGDM issues with HFEs having a great power in differentiating the best alternative and achieving the best alternative without parameter selection problems and counterintuitive phenomena (It has no the process of aggregation operators discussed in Definitions 5 and 6). But other methods [28–30 , 35] do not own this desirable character.

3.1.3 The hesitant fuzzy soft decision making approach based revised aggregation operators

In general, hesitant fuzzy soft revised aggregation operators algorithm involves the following steps:

Algorithm 3 :Revised aggregation operators

1: It is the same as the Step 1 to Step 3 in Algorithm 1.

2: Compute the weighted normalized decision matrix r_ij by Eq. (22).

3: Utilize the R-HFWA operator

$\begin{matrix} R (A_{i}) = R - HFWA (F (e_{1}) (x_{i}), \\ F (e_{2}) (x_{i}), \dots, F (e_{n}) (x_{i})) \end{matrix}$ (31) or

Utilize the R-HFWG operator

$\begin{matrix} R (A_{i}) = R - HFWG (F (e_{1}) (x_{i}), \\ F (e_{2}) (x_{i}), \dots, F (e_{n}) (x_{i})) \end{matrix}$ (32) to get the integrated preference value.

4: Determine the score function s (R (A_i)) of the whole values R (A_i) (i = 1, 2, ⋯ , m).

5: Rank the alternatives by the decreasing values of assessment RA.

4. A numerical example

We will think over the emergency decision making issues of mine accidents utilizing the initiated decision making methods based on CODAS and WDBA. Mine explosions are one of the most dangerous hazards in mine accidents. The mine explosion overwhelmingly threatens the safety of life and endangers the safety production of mines. In virtue of the explosion accidents always occur suddenly and unexpectedly, it is hard to divine the accident to a crumb and own adequate preparation and emergency actions in advance. Consequently, the simulations of the accidents and the emergency response plans are an essential method in disaster preparedness and advisable responses. The high feasibility and quality of the emergency plans will immediately impact the subsequent emergency actions, and influence the evolution of disasters. Therefore, the assessment and decision of the emergency plans with simulations is deemed essentially for disaster management in mine accidents [50].

Example 2. Suppose that there are four emergency plans U = {x₁, x₂, x₃, x₄} to be taken into account for an explosion accident in the coal mine. The specialist chooses the optimal parameters set E = {e₁, e₂, e₃} to be e₁ (denoted as gas), e₂ (denoted as casualty) and e₃ (denoted as facility). According to the general evolving principle and the feature of the mine accidents, we can judge that e₁ and e₂ are benefit parameters, and e₃ is cost parameter. Assume that the specialist has the prior weight: w = (w₁, w₂, w₃) = (0.5, 0.4, 0.1). The assessments for emergency plans obtaining from questionnaire investigation to the specialist and constructing a hesitant fuzzy soft set (F, E) shown in Table 3.

Table 3
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}

Next, we use the methods developed above (revised aggregation operators, WDBA and CODAS) to choose emergency plans under hesitant fuzzy soft environment.

By means of Algorithms 1 and 2 shown in Table 4, we can see that the final decision results are the alike, that is to say, x₁ is the most desirable emergency plans. In other words, the three approaches presented above are available and effective because of the same results of mutual authentication.

Table 4

Final results and ranking.

Algorithms	Results	Ranking	Optimal alternative
Algorithm 1 : WDBA	SI₁ = 0.8001, SI₂ = 0.7392, SI₃ = 0.4828, SI₄ = 0.4638	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 2 : CODAS	RA₁ = 0.0054, RA₂ = 0.0006, RA₃ = -0.0021, RA₄ = -0.004	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 3 : R - HFWA	s (RA₁) =0.8238, s (RA₂) =0.7476, s (RA₃) =0.74, s (RA₄) =0.6372	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 3 : R - HFWG	s (RA₁) =0.4605, s (RA₂) =0.454, s (RA₃) =0.2897, s (RA₄) =0.4479	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁

5. A comparison with the existing decision making algorithms based on hesitant fuzzy soft set

5.1. Comparison of the newly proposed two algorithms with their own

Next, some comparisons of Algorithm 1 and Algorithm 2 are shown.

(1) Comparison of computational complexity

We can find that R-HFWA and R-HFWG (Algorithm 3) will outlay more computational complexity than Algorithm 1, especially in Step 3 (Algorithm 3). The WDBA (Algorithms 1) may be given priority if the computational complexity are taken into consideration.

(2) Comparison of discrimination

Comparing the decision results shown in WDBA (Algorithm 1) with CODAS (Algorithm 2) and R-HFWA and R-HFWG (Algorithm 3), we can see that the decision results of CODAS (Algorithm 2) are very close and go from -0.0040 to 0.0054. These result of decision values can’t efficaciously differentiate (See Fig. 3), that is to say, the results obtained from CODAS (Algorithm 2) are not very credible. Meanwhile, the R-HFWA and R-HFWG (Algorithm 3) also has the small variations of all alternatives. That is to say, the WDBA (Algorithm 1) has a good distinction degree. So if we take the discrimination into consideration, the WDBA (Algorithm 1) should be given priority for making decision.

(3) Comparison of applied situation

When the decision maker only takes negative-ideal solution into consideration during the decision process, the Algorithm 2 is given priority.

Fig.3

The comparison of developed algorithms.

5.2. Comparison of the newly proposed two methods with other approaches

5.2.1 The aggregation operators methods [28 –30] and their limitations

Remark 4. However, we can find that the above algorithm will encounter the counterintuitive phenomena discussed in Definitions 5 and 6 (See Examples 3 and 4).

Algorithm 4 :HFWA and HFWG

1: It is similar to Algorithm 1 in Steps 1 and 2.

2: Aggregate all hesitant fuzzy values $\tilde{F} (e_{j}) (x_{i}) (i = 1, 2, \dots, m; j = 1, 2, \dots, n)$ into F (x_i) by HFWA or HFWG for the alternatives U = {x₁, x₂, ⋯ , x_m}.

3: Compute the score function S (F (x_i)) of F (x_i) by Eq. (2).

4: Rank all the alternatives and select the best one(s) in accordance with the ranking of S (F (x_i)).

5.2.2 Peng and Dai’s methods [33] and their limitations

Remark 5. However, for the parameter λ in Eq. (3), we can find that it may achieve various decision results when λ assigns diverse values [51]. In other words, the decision results will be suspect (See Figs. 4 and 5). Meanwhile, it also meet the similar counterintuitive phenomena condition discussed in Definitions 5 and 6(See Examples 3 and 4).

Remark 6. However, for Eq. (5) or Eq. (6), we can find that it has no mechanism to deal the decision data when the parameters are all benefit or cost. That is to say, its scope of application is limited (See Example 5). Meanwhile, it also meet the similar counterintuitive phenomena condition discussed in Definitions 5 and 6 (See Examples 3-5).

Algorithm 5 :WASPAS

1: It is similar to Algorithm 1 in Steps 1 and 2.

2: Compute the WSM $Q_{i}^{(1)}$ by Eq.(33).

$\begin{matrix} Q_{i}^{(1)} = \oplus_{j = 1}^{n} w_{j} \tilde{F} (e_{j}) (x_{i}) \\ = ⋃_{r_{i 1} \in \tilde{F} (e_{1}) (x_{i}), r_{i 2} \in \tilde{F} (e_{2}) (x_{i}), \dots, r_{in} \in \tilde{F} (e_{n}) (x_{i})} \\ {1 - \prod_{j = 1}^{n} (1 - r_{ij})^{w_{j}}} . \end{matrix}$ (33)

3: Compute the WPM $Q_{i}^{(2)}$ by Eq.(34).

$\begin{matrix} Q_{i}^{(2)} = ⨂_{j = 1}^{n} (\tilde{F} (e_{j}) (x_{i}))^{w_{j}} \\ = ⋃_{r_{i 1} \in \tilde{F} (e_{1}) (x_{i}), r_{i 2} \in \tilde{F} (e_{2}) (x_{i}), \dots, r_{in} \in \tilde{F} (e_{n}) (x_{i})} \\ {\prod_{j = 1}^{n} (r_{ij})^{w_{j}}} . \end{matrix}$ (34)

4: Compute the contribution of WSM and WPM for a total evaluation of Q_i by Eq.(35). $Q_{i} = λ * Q_{i}^{(1)} \oplus (1 - λ) * Q_{i}^{(2)} .$ (35)

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

Remark 7. However, the MABAC algorithm also meet the similar counterintuitive phenomena condition discussed in Definition 5 (See Example 3).

Fig.4

The comparison with existing algorithms in Example 2.

Fig.5

The comparison with existing algorithms in Example 4.

5.2.3 The comparison of the computational complexity

We only consider the times of division, multiplication, root, absolute value and power, and the computational times of addition and subtraction are neglected on account of their fast calculation in computer. To sum up, we can know that the computation complexity of the existing methods and proposed methods are shown in Table 5.

From Table 5, we can find that the proposed methods (WDBA and CODAS) have lower computation complexity than the existing methods [28–30 , 33].

Algorithm 6 :COPRAS

1: Input the hesitant fuzzy soft decision matrix R = (F (e_j) (x_i)) _n×m (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n).

2: Calculate the weighted matrix $\bar{R} = (\bar{F} (e_{j})$ (x_i)) _n×m by Eq. (4)

$\begin{matrix} \bar{F} (e_{j}) (x_{i}) = ϖ_{j} F (e_{j}) (x_{i}) \\ = ⋃_{r_{ij} \in F (e_{j}) (x_{i})} {1 - (1 - r_{ij})^{ϖ_{j}}} . \end{matrix}$ (36)

3: Sum the values of benefit parameters by Eq. (5). (Let J be a set of benefit parameters.) $P_{i} = \oplus_{j \in J} \bar{F} (e_{j}) (x_{i})$ (37)

4: Sum the values of cost parameters by Eq. (6). (Let K be a set of cost parameters.) $R_{i} = \oplus_{j \in K} \bar{F} (e_{j}) (x_{i}),$ (38) where J ⋃ K is the all parameters.

5: Calculate the relative weight of each alternative Q_i by Eq. (7). $Q_{i} = S (P_{i}) + \frac{\sum_{i = 1}^{m} e^{S (R_{i})}}{e^{S (R_{i})} \sum_{i = 1}^{m} \frac{1}{e^{S (R_{i})}}} .$ (39)

6: Calculate the quantitative utility U_i of each alternative: $U_{i} = \frac{Q_{i}}{\max_{i} {Q_{i}}} \times 100 %,$ (40)

7: The larger the value of U_i, the more preference of the alternative x_i

5.3. Some illustrative examples

Example 3. An investment company wants to invest a company for increasing income; there are four potential and competitive companies as alternatives U = {x₁, x₂, x₃, x₄}. There are three parameters that are used to assess these alternatives, including e₁: the risk analysis; e₂: the growth condition; e₃: the social-political impact. The parameters’ weight is w = (0.5, 0.1, 0.4). The evaluations for alternatives obtaining from the CEO are denoted by hesitant fuzzy soft set (F, A), shown in Table 6.

Table 5
A comparison for computational complexity of some existing methods

Methods	Counterintuitive	Number of executions	Computation complexity
WDBA	No	mn (2 log ₂m + 7) + m log ₂m + 6n + 3m	o (mn log ₂m)
CODAS	No	mn (log ₂m + 4) + m² + m log ₂m + 3n + m	o (mn log ₂m)
R - HFWA	No	m (2n - 1) (# h) ⁿ + m (1 + log ₂m) +3n	o (mn (# h) ⁿ)
R - HFWG	No	m (2n - 1) (# h) ⁿ + m (1 + log ₂m) +3n	o (mn (# h) ⁿ)
HFWA [28 –30]	Yes	m (2n - 1) (# h) ⁿ + m (1 + log ₂m) +3n	o (mn (# h) ⁿ)
HFWG [28 –30]	Yes	m (2n - 1) (# h) ⁿ + m (1 + log ₂m) +3n	o (mn (# h) ⁿ)
WASPAS [33]	Yes	m (# h) ⁿ [(# h) ⁿ + 4n] + m log ₂m + 3n	o (m (# h) ²ⁿ)
COPRAS [33]	Yes	m [(J - 1) (# h) ^J + (K - 1) (# h) ^K + n # h	o (mJ (# h) ^J + mK (# h) ^K)
		+ (m + 1) log ₂m + 2m + 5] +3n
MABAC [33]	Yes	(2m + 1) n (# h) ^m + mn (# h + 2) + m log ₂m + 3n	o (mn (# h) ^m)

# h is the number of the elements in hesitant fuzzy set h. J is the the number of benefit parameters. K is the the number of cost parameters.

Assume that e₃ is cost parameter. From the decision results shown in Table 7, we can find that the ultimate ordering of the four alternatives and best alternative are same as the results of HFWG [28 –30]. For HFWA [28 –30], MABAC [33], COPRAS [33] and WASPAS (λ = 0.5) [33], the ultimate ordering and the best alternative cannot be got on account of its drawback that we have stated in Definition 5.

Table 6

Tabular representation of (F, A) in Example 3

	e ₁	e ₂	e ₃
x ₁	{0.3}	{1.0}	{0.1}
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}

Table 7

A comparison study with some existing methods in Example 3

Method	Results	The final ranking	The optimal alternative
Algorithm 1 : WDBA	SI₁ = 0.5406, SI₂ = 0.9017, SI₃ = 0.2898, SI₄ = 0.0000	x₂ ≻ x₁ ≻ x₃ ≻ x₄	x ₂
Algorithm 2 : CODAS	RA₁ = 0.2799, RA₂ = 1.2346, RA₃ = -0.5694, RA₄ = -0.9450	x₂ ≻ x₁ ≻ x₃ ≻ x₄	x ₂
R - HFWA	S (F (x₁)) =0.8520, S (F (x₂)) =0.8558, S (F (x₃)) =0.5682, S (F (x₄)) =0.4457	x₂ ≻ x₁ ≻ x₃ ≻ x₄	x ₂
Algorithm3 : HFWA [28 –30]	S (F (x₁)) =1.0000, S (F (x₂)) =0.8560, S (F (x₃)) =0.5682, S (F (x₄)) =0.4455	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm3 : HFWG [28 –30]	S (F (x₁)) =0.5668, S (F (x₂)) =0.8424, S (F (x₃)) =0.5030, S (F (x₄)) =0.2983	x₂ ≻ x₁ ≻ x₃ ≻ x₄	x ₂
Algorithm 4 : WASPAS [33]	S (Q₁) =1.0000, S (Q₂) =0.8516, S (Q₃) =0.5378, S (Q₄) =0.3770	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 5 : COPRAS [33]	U₁ = 1.0000, U₂ = 0.8770, U₃ = 0.6748, U₄ = 0.5863	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 6 : MABAC [33]	V₁ = 0.3564, V₂ = 0.3549, V₃ = -0.0160, V₄ = -0.1456	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁

“Bold” denotes unreasonable results. Δ = 0.02.

In order to discuss the impact of the parameter λ in the optimal alternative selection, we give the four alternatives with different λ values in the range between 0 and 1 shown in Fig. 4. From Fig. 4, we can know that the best alternative is x₂ when λ = 0. When λ ∈ [0.1, 1], the best alternative is x₁. Meanwhile, the decision values of three alternatives (x₂, x₃, x₄) are increasing when the increasing of λ is increasing. For alternative x₁, when λ ∈ [0.1, 1], it keeps a steady statue with no change of decision values.

Example 4. Continue to Example 3. Suppose that the evaluations for alternatives arising from another CEO are presented by hesitant fuzzy soft set (F, A), shown in Table 8.

Table 8

Tabular representation of (F, A) in Example 4

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

Suppose that e₃ is cost parameter. From the decision results shown in Table 9, we can find that the ultimate ordering of the four alternatives and best alternative are same as the results of HFWA [28 –30], WASPAS (λ = 0.5) [33], MABAC [33]. Foralgorithms based on HFWG in [28 –30] and COPRAS [33], the ultimate ordering and the best alternative cannot be achieved on account of its shortcoming that we have stated in Definition 6.

Table 9

A comparison study with some existing methods in Example 4

Method	Results	The final ranking	The optimal alternative
Algorithm 1 : WDBA	SI₁ = 0.6637, SI₂ = 0.3979, SI₃ = 0.3643, SI₄ = 0.3363	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 2 : CODAS	RA₁ = 2.1198, RA₂ = -0.6367, RA₃ = -0.7076, RA₄ = -0.7756	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
R - HFWG	S (F (x₁)) =0.8322, S (F (x₂)) =0.2325, S (F (x₃)) =0.1784, S (F (x₄)) =0.1196	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm3 : HFWA [28 –30]	S (F (x₁)) =0.8290, S (F (x₂)) =0.2315, S (F (x₃)) =0.1777, S (F (x₄)) =0.1194	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm3 : HFWG [28 –30]	S (F (x₁)) =0.0000, S (F (x₂)) =0.2106, S (F (x₃)) =0.1702, S (F (x₄)) =0.1142	x₂ ≻ x₃ ≻ x₄ ≻ x₁	x ₂
Algorithm 4 : WASPAS [33]	S (Q₁) =0.5865, S (Q₂) =0.2213, S (Q₃) =0.1740, S (Q₄) =0.1168	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 5 : COPRAS [33]	U₁ = 0.8725, U₂ = 1.0000, U₃ = 0.9755, U₄ = 0.9490	x₂ ≻ x₃ ≻ x₄ ≻ x₁	x ₂
Algorithm 6 : MABAC [33]	V₁ = 0.4487, V₂ = 0.0124, V₃ = -0.0318, V₄ = -0.0585	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁

“Bold” denotes unreasonable results. Δ = 0.02.

Algorithm 7 :MABAC

1: It is similar to Steps 1-2 in Algorithm 1.

2: Compute the weighted matrix T = ( t_ij) _m×n by Eq.(41). $t_{ij} = ϖ_{j} \tilde{F} (e_{j}) (x_{i}) = ⋃_{r_{ij} \in \tilde{F} (e_{j}) (x_{i})} {1 - (1 - r_{ij})^{ϖ_{j}}} .$ (41)

3:Compute the border approximation area (BAA) matrix G = (g_j) _1×n. The BAA for each parameter is obtained by Eq.(42). $\begin{matrix} g_{j} = ⨂_{i = 1}^{m} (t_{ij})^{1 / m} \\ = ⋃_{r_{1 j}^{'} \in t_{1 j}, r_{2 j}^{'} \in t_{2 j}, \dots, r_{mj}^{'} \in t_{mj}} {\prod_{i = 1}^{m} (r_{ij}^{'})^{1 / m}} \end{matrix}$ (42) $\begin{matrix} = ⋃_{r_{1 j} \in \tilde{F} (e_{j}) (x_{1}), r_{2 j} \in \tilde{F} (e_{j}) (x_{2}), \dots, r_{mj} \in \tilde{F} (e_{j}) (x_{m})} \\ {\prod_{i = 1}^{m} (1 - (1 - r_{ij})^{ϖ_{j}})^{1 / m}} \end{matrix}$ (43)

4: Reckon the distance matrix D = (d_ij) _m×n by Eq.(43). $d_{ij} = {\begin{matrix} d (t_{ij}, g_{j}), if t_{ij} > g_{j}, \\ 0, if t_{ij} = g_{j}, \\ - d (t_{ij}, g_{j}), if t_{ij} < g_{j}, \end{matrix}$ (44) where distance measure d is defined in Eq.(3).

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

In order to discuss the impact of the parameter λ in the optimal alternative selection, we give the four alternatives with different λ values in the range between 0 and 1 shown in Fig. 5. From Fig. 5, we can know that the best alternative is x₂ when λ = 0. When λ ∈ [0.1, 1], the best alternative is x₁. Meanwhile, the decision values of four alternatives are increasing when the increasing of λ is increasing.

Example 5. An investment company wants to seek an invest in software development project in the optimal option. There is a panel with four potential and competitive alternatives: (1) x₁ is mail project; (2) x₂ is game project; (3) x₃ is browser project; (4) x₄ is music player project. The investment company selects four parameters to assess the four potential and competitive software projects: (1) e₁ is technological viability; (2) e₂ is social viability; (3) e₃ is staff viability. Assume that the expert owns the following weight: w = (0.4, 0.3, 0.3). For the sake of avoiding influencing each other, the DMs or experts are required to assess the four potential software development projects x_i (i = 1, 2, 3, 4) under the above three parameters in anonymity and the hesitant fuzzy soft set (F, A) which is shown in Table 10.

Assume that all parameters are benefit parameters. From the decision results shown in Table 11, we can see that the of the four alternatives and best alternative are same as the decision results of HFWA [28 –30], HFWG [28 –30], WASPAS (λ = 0.5) [33], MABAC [33]. For COPRAS [33], we cannot compute due to "the mechanism problem" discussed in Remark 6, hence no best alternative can be selected. For better understanding the trend of the alternatives, we can see Fig. 6. In Fig. 6, the proposed method WDBA has a great power in differentiating the best alternative.

Table 10

Tabular representation of (F, A) in Example 5

	e ₁	e ₂	e ₃
x ₁	{0.88}	{0.75}	{0.85}
x ₂	{0.87, 0.88}	{0.74}	{0.86}
x ₃	{0.86}	{0.73, 0.74}	{0.87}
x ₄	{0.85, 0.86}	{0.72}	{0.87}

Table 11

A comparison study with some existing methods in Example 5

Method	Results	The final ranking	The optimal alternative
Algorithm 1 : WDBA	SI₁ = 1.0000, SI₂ = 0.6554, SI₃ = 0.3846, SI₄ = 0.0000	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 2 : CODAS	RA₁ = 0.0005, RA₂ = 0.0001, RA₃ = -0.0002, RA₄ = -0.0004	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm3 : HFWA [28 –30]	S (F (x₁)) =0.7350, S (F (x₂)) =0.7277, S (F (x₃)) =0.7118, S (F (x₄)) =0.7033	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm3 : HFWG [28 –30]	S (F (x₁)) =0.5113, S (F (x₂)) =0.4987, S (F (x₃)) =0.4935, S (F (x₄)) =0.4798	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 4 : WASPAS [33]	S (Q₁) =0.6401, S (Q₂) =0.6306, S (Q₃) =0.6179, S (Q₄) =0.6071	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁
Algorithm 5 : COPRAS [33]	No results	No ranking	*
Algorithm 6 : MABAC [33]	V₁ = 0.0154, V₂ = 0.0096, V₃ = -0.0038, V₄ = -0.0135	x₁ ≻ x₂ ≻ x₃ ≻ x₄	x ₁

“No ranking” denotes it cannot compute due to “the mechanism problem”, and “*” presents no optimal alternative to be selected.

The subjective weighting counting methods focus on the preference decision information of the experts or DMs [28 , 35], while they ignore the objective decision information. The objective weighting counting methods fail to take the attitude of the experts or DMs. The feature of our weighting counting method not only can own the subjective information given by DMs but also can carry objective information which the data transmit. Meanwhile, we give the comparison of the combined weight with [33] shown in Table 12. From Table 12, we can find that our weight model can obtain same decision results compared with Peng and Dai [33]. In other words, our combination weight counting model is effective and feasible.

Fig.6

The comparison with existing algorithms in Example 5.

Table 12

The comparison with weigh information.

WDBA/CODAS	Subjective weights	Shannon entropy method : ϖ_s [33]	The final ranking by ϖ_s	deviation-based method ϖ_d	The final ranking by ϖ_d
Example2	w = (0.5, 0.4, 0.1)	ϖ_s = (0.3528, 0.3736, 0.2736)	x₁ ≻ x₂ ≻ x₃ ≻ x₄	ϖ_d = (0.3901, 0.3929, 0.2170)	x₁ ≻ x₂ ≻ x₃ ≻ x₄
Example3	w = (0.5, 0.1, 0.4)	ϖ_s = (0.3915, 0.3020, 0.3065)	x₂ ≻ x₁ ≻ x₃ ≻ x₄	ϖ_d = (0.4502, 0.3058, 0.2440)	x₂ ≻ x₁ ≻ x₃ ≻ x₄
Example4	w = (0.5, 0.1, 0.4)	ϖ_s = (0.3333, 0.2496, 0.4171)	x₁ ≻ x₂ ≻ x₃ ≻ x₄	ϖ_d = (0.4885, 0.2785, 0.2329)	x₁ ≻ x₂ ≻ x₃ ≻ x₄
Example5	w = (0.4, 0.3, 0.3)	ϖ_s = (0.3637, 0.3520, 0.2843)	x₁ ≻ x₂ ≻ x₃ ≻ x₄	ϖ_d = (0.3804, 0.3439, 0.2758)	x₁ ≻ x₂ ≻ x₃ ≻ x₄

6. Conclusion

The key contributions can be summarized in the following.

(1) Some revised hesitant fuzzy aggregation operators (R-HFWA and R-HFWG) are discussed in detail (Tables 7 and 9) which can avoid the counterintuitive phenomena [28–30 , 33].

(2) Three novel hesitant fuzzy soft MCDM methods based on revised aggregation operators, WDBA and CODAS are developed, which have not been discussed in the existing papers. Also they have a great power in differentiating the best alternative (Fig. 6), obtain the best alternative without counterintuitive phenomena (Tables 7 and 9) and can avert the parameter selection issues (Table 11).

(3) A combination weight counting model is studied. In addition, our weight counting model can obtain the consistent results as reference [33] when employing the same level MCDM method for obtaining the ultimate decision results (Table 12).

The current research suffers some limitations as well. These result in the following future work:

(1) The proposed decision making algorithms are not efficient enough to deal some real applications such as gene selection [52].

(2) The proposed methods (revised aggregation operators, WDBA and CODAS) just deal the hesitant fuzzy soft environment. It would be also interesting if we can deal other uncertain environment [53 –86].

Footnotes

Acknowledgements

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. 61462019), MOE (Ministry of Education in China) Project of Humanities and Social Sciences (No.18YJCZH054), Natural Science Foundation of Guangdong Province (No. 2018A030307033), Social Science Foundation of Guangdong Province (No. GD18CFX06).

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Algorithms for hesitant fuzzy soft decision making based on revised aggregation operators,WDBA and CODAS

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1. Hesitant fuzzy set

Table 1 Tabular representation of (F, E) in Example 1

3.1. Problem description

Table 2 Tabular representation of (F, E)

Table 3 Tabular representation of (F, E)

5.1. Comparison of the newly proposed two algorithms with their own

5.2.1 The aggregation operators methods [28–30] and their limitations

5.2.2 Peng and Dai’s methods [33] and their limitations

Table 5 A comparison for computational complexity of some existing methods

Footnotes

Acknowledgements

References

Table 1
Tabular representation of (F, E) in Example 1

Table 2
Tabular representation of (F, E)

Table 3
Tabular representation of (F, E)

5.2.1 The aggregation operators methods [28 –30] and their limitations

Table 5
A comparison for computational complexity of some existing methods