A multi-attribute group decision making method based on novel distance measures and regret theory under probabilistic dual hesitant fuzzy sets

2.1 Probabilistic dual hesitant fuzzy sets

In this subsection, we review some basic conceptions, score functions, deviation degrees, comparison methods and aggregation operators for PDHFS.

Definition 1 [12]. Let X be a non-empty set, a PDHES on X is recorded as:

A = { < x i , h ( x ) | p i ( x ) , g i ( x ) | q i ( x ) > | x i ∈ X }(1)

The components h i ( x ) | p i ( x ) = { γ i λ | p i λ } ( λ = 1 , 2 , … , # h i , ∑ λ = 1 # h i p i λ = 1 ) and g i ( x ) | q i ( x ) = { η i λ | q i λ } ( λ = 1 , 2 , … , # g i , ∑ λ = 1 # g i q i λ = 1 ) are two sets of some possible elements where γ i λ and η i λ represent the hesitant fuzzy membership and non-membership degrees to the set X of x _i , respectively. p i λ and q i λ are the corresponding probabilistic information for these two types of degrees. There is 0 ⩽ γ i λ , η i λ ⩽ 1 , 0 ⩽ γ i λ , η i λ ⩽ 1 , 0 ⩽ max λ γ i λ + max λ η i λ ⩽ 1 .

If ∑ λ = 1 # h i p i λ ⩽ 1 , ∑ λ = 1 # g i q i λ ⩽ 1 , , then A is called a generalized PDHFS. The element of PDHES α = < h (x) |p (x) , g (x) |q (x)  > is called PDHFE, and the elements in h (x) |p (x) and g (x) |q (x) are arranged in ascending order of MD γ ^λ and NMD η ^λ, respectively. The complementary set of α is α ^c  =〈 g (x) |q (x) , h (x) |p (x) 〉.

Definition 2 [12]. Let α =〈 h (x) |p (x) , g (x) |q (x) 〉 be a PDHFE, then we define the score function of the PDHFE as:

s ( α ) = ∑ λ = 1 # h γ λ p λ - ∑ λ = 1 # g η λ q λ(2)

Definition 3 [12]. Let α =〈 h (x) |p (x) , g (x) |q (x) 〉 be a PDHFE, then we define the deviation degree of the PDHFE as:

σ ( α ) = ( ∑ λ = 1 # h ( γ λ - s ( α ) ) 2 p λ + ∑ λ = 1 # g ( η λ - s ( α ) ) 2 q λ ) 1 2(3)

Definition 4 [12]. Let α _i  (i = 1, 2) be two PDHFEs, then:

If s (α₁) > s (α₂), then the PDHFE α₁ is superior to α₂, denoted by α₁ > α₂. On the contrary, there is α₁ < α₂.

If s (α₁) = s (α₂), then, { σ ( α 1 ) > σ ( α 2 ) → α 1 > α 2 σ ( α 1 ) = σ ( α 2 ) → α 1 = α 2 σ ( α 1 ) < σ ( α 2 ) → α 1 < α 2 .

Definition 5 [12]. Let be n PDHFEs and ω _i  (i = 1, 2, …, n) be the weight with ω _i  ∈ [0, 1] and ∑ i = 1 n ω i = 1 . Then PDHFWA operator is defined as:

PDHFWA ( α 1 , α 2 , … , α n ) = ⊕ j = 1 n ω k α i = ∪ γ 1 ∈ h 1 , γ 2 ∈ h 2 , … , γ n ∈ h n , η 1 ∈ g 1 , η 2 ∈ g 2 , . . , η n ∈ g n { { ( 1 - ∏ i = 1 n ( 1 - γ i ) ω i ) | ∏ i = 1 n p γ i } , { ∏ i = 1 n η i ω i | ∏ i = 1 n q η i } }(4)

Regret theory, also known as regret theory, was proposed in literature [42, 43], respectively, and its essential idea is that in the decision-making process, the DMs will not only focus on the results obtained from the chosen alternative, but also compare the chosen alternative with the unchosen alternative. If the DMs find that the unchosen alternative brings better value than the alternative already chosen, the DMs regret it. If the DMs find that the unchosen alternative brings a worse value than the alternative already chosen, the DMs will feel elated. The DMs will anticipate possible regret and elation during the decision-making process and will obviously avoid the alternative that will cause them to feel regret. The perceived utility of the DMs after choosing the alternative is:

u ab = v a + R ( Δ v ) = v a + R ( v a - v b )(5)

Where v _a refers to the utility value brought about by the DM choosing alternative a, Δv refers to the gain/loss value of DMs choosing alternative a instead of alternative b, and R(Δv) refers to the regret-elation value of the DMs choosing alternative a instead of alternative b. When R(Δv)>0, a elation value is generated, and when R(Δv)<0, it means that the regret value is generated. The regret- rejoice function R(*) is a monotonically increasing concave function satisfying R(0) = 0, R'(*)>0, and R” (*)<0.

In this study, the regret- rejoice function presented by Caspar [44] is chosen:

R ( Δ v ) = 1 - exp ( - ɛ ( Δ v ) )(6)

Where ɛ(ɛ>0) refers to the regret avoidance parameter, and a larger ɛ indicates a higher degree of regret avoidance by the DMs.

Xia [45] proposed the group satisfaction index as a utility function in regret theory in MADM problems with evaluated values of hesitant fuzzy numbers. In this study, it is extended to the MADM problem with PDHFE.

Definition 6 [45]. Let α =〈 h (x) |p (x) , g (x) |q (x) 〉 be a PDHFE, then we define the group satisfaction index of the PDHFE as:

φ ( α ) = s ( α ) 1 + σ ( α )(7)

Where s(α) is the score function of PDHFE and σ(α) is the deviation function of PDHFE, reflecting the degree of disagreement of the decision group.

3.1 Classical distance measures of PDHFEs

The definition and formulae for the distance measure proposed by Ning and Wei et al. [28] are given in Definition 7 - Definition 8.

Definition 7 [28]. Let α₁ =〈 h₁ (x) |p₁ (x) , g₁ (x) |q₁ (x) 〉 and α₂ =〈 h₂ (x) |p₂ (x) , g₂ (x) |q₂ (x) 〉 be 2 PDHFEs, then d (α₁, α₂) is named as distance measure between two PDHFEs, if d (α₁, α₂) satisfies:

0 ⩽ d (α₁, α₂) ⩽ 1;

Definition 8 [28]. Let α₁ =〈 h₁ (x) |p₁ (x) , g₁ (x) |q₁ (x) 〉 and α₂ =〈 h₂ (x) |p₂ (x) , g₂ (x) |q₂ (x) 〉 be 2 PDHFEs, then the Euclidean distance between the two is:

d ( α 1 , α 2 ) = { 1 2 [ 1 2 ( 1 Г ∑ λ = 1 Г | γ 1 λ p 1 λ − γ 2 λ p 2 λ | μ + 1 Φ ∑ λ = 1 Φ | η 1 λ q 1 λ − η 2 λ q 2 λ | μ ) + max λ ( | γ 1 λ p 1 λ − γ 2 λ p 2 λ | μ , | η 1 λ q 1 λ − η 2 λ q 2 λ | μ ) ] } 1 μ(8)

Where Γ  =   max(# h₁,   # h₂) , Φ  =   max(# g₁, # g₂). If μ = 1, d (α₁, α₂) is the Hemming distance d _HD  (α₁, α₂) for both, and if μ = 2, d (α₁, α₂) is the Euclidean distance d _ED  (α₁, α₂) for both.

The definition and formulae for the distance measure presented by Ning and Lei et al. [29] are given in Definition 9 - Definition 10.

Definition 9 [29]. Let α₁ =< h₁ (x) |p₁ (x) , g₁ (x) |q₁ (x) > and α₂ =< h₂ (x) |p₂ (x) , g₂ (x) |q₂ (x) > and α₃ =< h₃ (x) |p₃ (x) , g₃ (x) |q₃ (x) > be 3 PDHFEs, then d (α₁, α₂) is named as distance measure between two PDHFEs, if they satisfy:

0 ⩽ d (α₁, α₂) ⩽ 1;

Definition 10 [29]. Let and α₂ =〈 h₂ (x) |p₂ (x) , g₂ (x) |q₂ (x) 〉 be 2 PDHFEs, then the Euclidean distance between the two is:

d ( α 1 , α 2 ) = { 1 2 [ 1 2 ( ( ∑ λ = 1 # h 1 γ 1 λ p 1 λ − ∑ λ = 1 # h 2 γ 2 λ p 2 λ ) μ + ( ∑ λ = 1 # g 1 η 1 λ q 1 λ − ∑ λ = 1 # g 2 η 2 λ q 2 λ ) μ ) + 1 2 ( ( ∑ λ = 1 # h 1 ( γ 1 λ p 1 λ − ∑ λ = 1 # h 1 γ 1 λ p 1 λ ) μ − ∑ λ = 1 # h 2 ( γ 2 λ p 2 λ − ∑ λ = 1 # h 2 γ 2 λ p 2 λ ) μ ) μ + ( ∑ λ = 1 # g 1 ( η 1 λ q 1 λ − ∑ λ = 1 # g 1 η 1 λ q 1 λ ) μ − ∑ λ = 1 # g 2 ( η 2 λ q 2 λ − ∑ λ = 1 # g 2 η 2 λ q 2 λ ) μ ) μ ) ] } 1 μ(9)

If μ = 1, d (α₁, α₂) is the Hemming distance d _HD  (α₁, α₂) for both, and if μ = 2, d (α₁, α₂) is the Euclidean distance d _ED  (α₁, α₂) for both.

Example 1. There are 2 PDHFEs α₁ =< { 0.2|0.3, 0.6|0.7 } , { 0.1|0.2, 0.4|0.3 } > and α₂ =< { 0.3|0.2, 0.7|0.6 } , { 0.2|0.1, 0.3|0.4 } >, and the distance d (α₁, α₂) = 0 between α₁ and α₂ is obtained according to the distance formulas in literature [28] and [29]. But α₁, α₂ are two different PDHFEs, so it is necessary to improve the distance measure of PDHFEs given in literature [28] and [29].

The distance formulas given in the existing literature only consider the distances between the products of the MDs and the corresponding probabilities and between the products of the NMDs and the corresponding probabilities, ignoring the specific distribution of the MD and NMD in PDHFEs, and the MD and NMD in PDHFEs does not play a role in the existing distance formulas. Therefore, MD and NMD in PDHFEs should be regarded as important features of the distance measure.

Definition 11. Let α₁ =< h₁ (x) |p₁ (x) , g₁ (x) |q₁ (x) > and α₂ =< h₂ (x) |p₂ (x) , g₂ (x) |q₂ (x) > be 2 PDHFEs, then d (α₁, α₂) is named as distance measure between two PDHFEs, if d (α₁, α₂) satisfies:

0 ⩽ d (α₁, α₂) ⩽ 1;

Definition 12. Let α₁ =< h₁ (x) |p₁ (x) , g₁ (x) |q₁ (x) > and α₂ =< h₂ (x) |p₂ (x) , g₂ (x) |q₂ (x) > be 2 PDHFEs, then the distance between the two is:

d ( α 1 , α 2 ) = { 1 4 [ τ μ ( p ) + τ μ ( q ) 2 + ζ μ ( p ) + ζ μ ( q ) 2 + υ μ ( p ) + υ μ ( q ) 2 + ξ μ ( p ) + ξ μ ( q ) 2 ] } 1 μ(10) τ ( p ) = ∑ λ = 1 # h 1 γ 1 λ p 1 λ - ∑ λ = 1 # h 2 γ 2 λ p 2 λ , τ ( q ) = ∑ λ = 1 # g 1 η 1 λ q 1 λ - ∑ λ = 1 # g 2 η 2 λ q 2 λ . ζ ( p ) = ∑ λ = 1 # h 1 ( γ 1 λ p 1 λ - ∑ λ = 1 # h 1 γ 1 λ p 1 λ ) 2 - ∑ λ = 1 # h 2 ( γ 2 λ p 2 λ - ∑ λ = 1 # h 2 γ 2 λ p 2 λ ) 2 , ζ ( q ) = ∑ λ = 1 # g 1 ( η 1 λ q 1 λ - ∑ λ = 1 # g 1 η 1 λ q 1 λ ) 2 - ∑ λ = 1 # g 2 ( η 2 λ q 2 λ - ∑ λ = 1 # g 2 η 2 λ q 2 λ ) 2 . υ ( p ) = ∑ λ = 1 # h 1 γ 1 λ p 1 λ p 2 λ - ∑ λ = 1 # h 2 γ 2 λ p 1 λ p 2 λ , υ ( q ) = ∑ λ = 1 # g 1 η 1 λ q 1 λ q 2 λ - ∑ λ = 1 # g 2 η 2 λ q 1 λ q 2 λ . ξ ( p ) = ∑ λ = 1 # h 1 ( γ 1 λ - s ( α 1 ) ) 2 p 1 λ - ∑ λ = 1 # h 2 ( γ 2 λ - s ( α 2 ) ) 2 p 2 λ , ξ ( q ) = ∑ λ = 1 # g 1 ( η 1 λ - s ( α 1 ) ) 2 q 1 λ - ∑ λ = 1 # g 2 ( η 2 λ - s ( α 2 ) ) 2 q 2 λ .

Where s ( α i ) = ∑ λ = 1 # h i γ i λ p i λ - ∑ λ = 1 # g i η i λ q i λ . If μ = 1, d (α₁, α₂) is the Hemming distance d _HD  (α₁, α₂) for both, and if μ = 2, d (α₁, α₂) is the Euclidean distance d _ED  (α₁, α₂) for both.

Theorem 1. In Definition 12, the novel distance measure determined by Equation (10) satisfies the three conditions of the distance measures of PDHFEs given in Definition 11.

Proof 1. Obviously, the novel distance formula proposed in this paper satisfies the nonnegativity and exchangeability in Definition 11; therefore, we next prove condition (3): transitivity.

1) If d (α₁, α₂) = 0, we have { τ ( p ) = τ ( q ) = 0 ζ ( p ) = ζ ( q ) = 0 υ ( p ) = υ ( q ) = 0 ξ ( p ) = ξ ( q ) = 0 ,

then { ∑ λ = 1 # h 1 γ 1 λ p 1 λ = ∑ λ = 1 # h 2 γ 2 λ p 2 λ ∑ λ = 1 # g 1 η 1 λ q 1 λ = ∑ λ = 1 # g 2 η 2 λ q 2 λ ∑ λ = 1 # h 1 γ 1 λ = ∑ λ = 1 # h 2 γ 2 λ ∑ λ = 1 # g 1 η 1 λ = ∑ λ = 1 # g 2 η 2 λ ∑ λ = 1 # h 1 ( γ 1 λ - s ( α 1 ) ) 2 p 1 λ = ∑ λ = 1 # h 2 ( γ 2 λ - s ( α 2 ) ) 2 p 2 λ ∑ λ = 1 # g 1 ( η 1 λ - s ( α 1 ) ) 2 q 1 λ = ∑ λ = 1 # g 2 ( η 2 λ - s ( α 2 ) ) 2 q 2 λ ,

then { ∑ λ = 1 # h 1 γ 1 λ p 1 λ - ∑ λ = 1 # g 1 η 1 λ q 1 λ = ∑ λ = 1 # h 2 γ 2 λ p 2 λ - ∑ λ = 1 # g 2 η 2 λ q 2 λ ∑ λ = 1 # h 1 ( γ 1 λ - s ( α 1 ) ) 2 p 1 λ - ∑ λ = 1 # g 1 ( η 1 λ - s ( α 1 ) ) 2 q 1 λ = ∑ λ = 1 # h 2 ( γ 2 λ - s ( α 2 ) ) 2 p 2 λ - ∑ λ = 1 # g 2 ( η 2 λ - s ( α 2 ) ) 2 q 2 λ ,

and from the Definitions 2, we get α₁ =α₂.

2) On the other hand, if α₁ =α₂, the above arguments can be reversed to obtain d (α₁, α₂) = 0.

The new distance measure is an improvement on the existing distance measures, which retains the characteristics of the existing distance measure and overcomes its shortcomings. To illustrate the superiority of the proposed novel distance measure, the PDHFEs in Example 1 were selected for calculation, and d (α₁, α₂) = 0.0759 was obtained, which can effectively clarify the distance between the two.

Definition 13. Let M and N be any two PDHFSs on A = {α₁, α₂, …, α _n }, then d (M, N) is named as distance measure between two PDHFEs, if d (M, N) satisfies:

0 ⩽ d (M, N) ⩽ 1;

Definition 14. Let M and N be any two PDHFSs on A = {α₁, α₂, …, α _n }, then the distance between M and N is:

d ( M , N ) = ∑ j = 1 n w j d ( α Mj , α Nj )(11)

Where w _j refers to the importance of each criterion and 0 < w _j  ⩽ 1, ∑ j n w j = 1 .

Theorem 2. In Definition 14, the novel distance measure determined by Equation (11) satisfies the three conditions of the distance measures of PDHFEs given in Definition 13.

4.1 Similarity-trust analysis method to determine expert weights

In MAGDM, the evaluation values of the group are obtained by aggregating the evaluation values of each expert. Similarity reflects the consistency of experts’ decisions, and trust reflects the trust of experts in themselves and among experts. In this subsection, we use the similarity of experts and other experts to determine the similarity weight of experts and use trust to determine the trust weight of experts. Finally the comprehensive expert weights are obtained.

The similarity between expert e _k and expert e _t under the j-th attribute is:

ρ j ( e k , e t ) = 1 - 1 m ∑ i = 1 m d ED ( α kij , α tij )(12)

The similarity between expert e _k and the group under the j-th attribute is:

ρ j ( e k , e D ) = ∑ t = 1 l ( 1 - 1 m ∑ i = 1 m d ED ( α kij , α tij ) ) l - 1(13)

The similarity weight of expert e _k under the j-th attribute is:

ω kj 1 = ρ j ( e k , e D ) ∑ k = 1 l ρ j ( e k , e D )(14)

Experts express trust in each other with the same ambiguity and hesitation, and experts use PDHFEs to express trust that is more in line with the actual expression preferences of experts. T _ktj refers to the expression of trust of the k-th expert to the t-th expert under the j-th attribute, and the trust relationship matrix is obtained from the trust values given by the experts, the total trust of expert e _k under the j-th attribute is:

T j ( e k ) = ∑ t = 1 l s ( T kij ) l(15)

s (T _kij ) is the score function value of the assessed value T _ktj .

The trust weight of expert e _k under the j-th attribute is:

ω kj 2 = T j ( e k ) ∑ k = 1 l T j ( e k )(16)

The comprehensive weight of the expert e _k under the j-th attribute is obtained as:

ω kj = b ω kj 1 + ( 1 - b ) ω kj 2(17)

Where b is the preference parameter of expert weights, 0≤b≤1.

The attribute weights of PDHFEs are determined based on the idea of the maximum deviation method, and a nonlinear programming model of the attribute weights is developed.

maxD ( w ) = ∑ j = 1 n ∑ i 1 = 1 m ∑ i 2 = 1 , i 2 ≠ i 1 m w j d ED ( α i 1 j , α i 2 j )(18)

s . t . { w j ⩾ 0 , j = 1 , 2 , … , n ∑ j = 1 n w j 2 = 1(19)

The optimal weights of the attributes are obtained by solving the formula using Lagrange’s theorem and normalizing it.

w j = ∑ i 1 = 1 m ∑ i 2 = 1 , i 2 ≠ i 1 m d ED ( α i 1 j , α i 2 j ) ∑ j = 1 n ∑ i 1 = 1 m ∑ i 2 = 1 , i 2 ≠ i 1 m d ED ( α i 1 j , α i 2 j )(20)

In this subsection, we consider the regret-avoidance psychology of experts in the actual decision-making process, calculate the regret-rejoice values and perceived utility values of PDHFSs, and then construct a multi-attribute decision model by combining the EDAS method.

Step 1: Unify the types of attribute assessment values. Cost type was transformed into benefit type according to the method proposed by Zhao et al. [46], α kij → α kij c .

Step 2: Calculate the comprehensive weights of the experts under the j-th attribute in Section 4.2, and the group decision matrix [α _ij ] _m×n is obtained by the aggregation of PDHFEs by Equation (4).

Step 3: Calculate attribute weights W = (w₁,w₂, . . .  ,w _j , . . .  ,w _n )^T in Section 4.3.

Step 4: Calculate the regret-rejoice value R _ij and perceived utility value u _ij of the group decision by Equations (2-3, 21-22), and the perceived utility matrix [u _ij ] _m×n can be obtained.

R ij = ∑ k = 1 . k ≠ i m ( 1 - exp ( - ɛ ( φ ij - φ kj ) ) )(21)

u ij = φ ij + R ij = s ( α ij ) 1 + σ ( α ij ) + R ij(22)

Step 5: Extend EDAS to MAGDM of PDHFSs.

(1) Calculate the average alternative evaluation value [ u ¯ j ] 1 × n of the group perceived utility value under the j-th attribute.

(2) Calculate the positive distance from average (PDA) and negative distance from average (NDA) between the perceived utility value of each alternative and the perceived utility value of the average alternative by Equation (23).

{ PDA ij = max ( 0 , u ij - u ¯ j ) u ¯ j NDA ij = max ( 0 , u ¯ j - u ij ) u ¯ j(23)

(3) Aggregate attribute weights to obtain the weighted sum of PDA (SP) and weighted sum of NDA (NP) by Equation (24).

{ SP i = ∑ j = 1 n w j PDA ij SN i = ∑ j = 1 n w j NDA ij(24)

(4) Calculate the normalized SP (NSP) and normalized SN (NSN), and obtain the final appraisal score (AS) of the perceived utility for all alternatives by Equations (25-26).

{ NSP i = SP i max i ( SP i ) NSN i = 1 - SN i max i ( SN i )(25)

AS i = 1 2 ( NSP i + NSN i )(26)

Step 6: Rank the alternatives according to the final evaluation score to obtain the optimal solution.

A flowchart of the proposed MAGDM method based on the novel distance measure and regret theory is shown in Fig. 1.

OPEN IN VIEWER

Fig. 1

The flowchart of the proposed MAGDM method based on the novel distance measure and regret theory.

5.1 Decision-making for the protection of radiation from nuclear power plants

To verify the effectiveness of the proposed method, we took the protection of radiation under the PWR5 accident source of nuclear power plant D as an example, the choice of radiation protection strategy involves social, public, and environmental issues, and belongs to the category of MAGDM problems; therefore, three experts were invited to make decisions on radiation protection strategies. Experts evaluated the radiation protection effects of nuclear accidents under four attributes: negative psychosocial impact c₁, economic cost c₂, maximum preventable individual dose c₃, and preventable collective dose c₄, in which the attribute weights were completely unknown. The PDHFE evaluation values of the alternatives provided by experts under different attributes are listed in Table 1–3.

The trust matrix given by the experts is shown in Table 4, where experts in the columns are evaluators, and experts in the rows are evaluated.

Step 1: The attributes of the evaluation values given by the three experts were unified, among which the negative psychosocial impact c₁ and economic cost c₂ belong to cost type, and them were transformed into benefit type: α ki 1 → α ki 1 c and α ki 2 → α ki 2 c .

Step 2: The comprehensive weight matrix of the experts under the j-th attribute was calculated taking b = 0.5:ω _kj  = [[0.3316,0.3960,0.2699], [0.3475,0.3851,0.2676], [0.2764,0.3288,0.4011],[0.3564,0.4106,0.4804]]^T, and the group decision matrix of PDHFEs was obtained by Equation (4), as listed in Table 5.

Step 3: The attribute weights were calculated as described in Section 4.3: W = [0.2449,0.20023,0.2684,0.2864]^T.

Step 4: The perceived utility values of group decision were calculated taking ɛ= 0.3 by Equations (2-3, 21-22), and the perceived utility matrix was obtained and shown in Table 6.

Step 5: The EDAS method was used for calculation based on the obtained perceived utility matrix.

(1) The average evaluation value matrix of the perceived utility value of group decisions was calculated under the j-th attribute: [ u ¯ j ] 1 × n = [[0.0603,0.0371,0.0964,0.0267],[–0.0883,0.0602,–0.0292,0.0888],[–0.0774,0.0545,–0.0802,0.1346],[–0.1040,0.0157,–0.1361,0.1443]]^T.

(2) The PDAs and NDAs were calculated and listed in Table 7.

(3) The SPs and SNs were calculated: SP =  [3.2999,–2.2638,–1.2287,1.4600]^T,SN = [–3.6471,1.0022,0.3692,–1.5400]^T.

(4) The NSPs and NSNs are calculated and the AS of the perceived utility of the alternatives were obtained: AS = [0.5000,0.0196,0.2632,0.9323]^T.

Step 6: The alternatives were ranked according to the final evaluation score: x₄ > x₁ > x₃ > x₂; therefore, the optimal solution for radiation protection was alternative 4.

To better express the psychological behavior of DMs in probabilistic dual hesitant fuzzy multi-attribute decision-making, a sensitivity analysis was conducted on the regret avoidance parameter in regret theory to test the influence of the psychological behavior of DMs on the decision outcome under different regret avoidance parameters.

As the regret avoidance parameter increases, the ranking of alternatives changes from x₄ > x₂ > x₁ > x₃ to x₄ > x₁ > x₃ > x₂, then to x₁ > x₄ > x₃ > x₂, and finally to x₄ > x₃ > x₂ > x₁, indicating that the regret avoidance behavior of DMs affects the choice of alternatives and the degree of regret avoidance of DMs has an impact on the choice of alternatives. Therefore, considering the regret avoidance behavior of DMs has important practical significance in the actual decision-making process.

5.3 Comparative analysis

To further illustrate the effectiveness and feasibility of the multi-attribute decision-making method proposed in this paper, it was compared with the methods in the literature [27, 28], and [29]. The psychological behaviors of DMs are not considered in the literature [27], where the alternatives are ranked using a score function after aggregating the alternative evaluation values given by experts. The expert weights and attribute weights of the decision-making method proposed by Ning and Wei et al. [28] were given directly, and the regret avoidance psychology of the DMs was not considered. Two ranking methods have been proposed by Ning and Wei et al. [28]: (1) the same attribute weights for MD and NMD in PDHFE; (2) the experts assigned different weights to MD and NMD in PDHFE. For the first case, we used the expert and attribute weights calculated in the decision-making method proposed in this study to calculate the similarity of each alternative to the ideal solution using the distance formula of PDHFEs proposed by Ning and Wei et al. [28]. For the second case, we used the expert weights calculated in the decision-making method proposed in this study and attribute weights [0.25,0.1,0.3,0.35]^T under NMD to calculate the similarity of each alternative to the ideal solution using the distance formula of PDHFEs proposed by Ning and Wei et al. [28]. The alternatives were ranked for comparison using the distance formula proposed in literature [29]. Table 9 presents the results of the comparative analysis.

Compared with the method in literature [27], we found that the optimum in the ranking of alternatives obtained in literature [27] was the same as the optimum in the ranking of alternatives in the method proposed in this paper with regret avoidance coefficients ɛ= 0.7, 0.9, but the other alternatives were ranked differently. The distance measure proposed by Garg et al. [27] was smaller than the actual distance between PDHFSs, which may lead to a different ranking of alternatives.

Compared with the method in literature [28], we found that the ranking of alternatives obtained in literature [28] was the same as the ranking of alternatives when the regret avoidance parameters ɛ= 0.7, 0.9, and ɛ= 0.3, 0.5, in the decision-making method proposed in this study, indicating that the decision-making method proposed in this study has some rationality. This study considered the influence of DMs’ regret avoidance psychology on the decision result, which is more comprehensive than that considered by Ning and Wei et al. [28], and presented the calculation methods of the expert weights and attribute weights, which are not in the method proposed by Ning and Wei et al. [28]. This study ranked the alternatives based on the EDAS method, which reduced the influence of extreme values to a certain extent compared to the ranking method proposed by Ning and Wei et al. [28], which only considered the distances between alternatives and the ideal alternative.

Compared with the method in the literature [29], we found that the ranking of alternatives obtained in literature [29] was the same as the ranking of alternatives when the regret aversion coefficients ɛ= 0.3, 0.5 in the method proposed in this paper, which also showed that the decision-making method proposed in this paper has certain rationality. This study improved the distance measure given in literature [29] by considering the distances between MDs and the distances between NMDs, which could compare the distance between different PDHFSs more effectively. The method proposed in this paper considers experts’ preference for risk, which is more relevant to actual decision-making.

The decision-making method proposed in this paper not only improved distance measures, which can compare the distances between different PDHFSs more effectively, but also adopted a similarity-trust analysis method based on the novel distance measure to calculate the expert weights and adopted the deviation-maximization method based on the novel distance measure to calculate the attribute weights, which is more objective and accurate than the directly given expert weights and attribute weights, making the final group opinion more reasonable. The method proposed in this study also considers the DMs’ risk preference, which is more in line with the fact that DMs are irrational individuals in the actual decision-making process.