Interactive multi-criteria group decision-making with probabilistic linguistic information for emergency assistance of COVID-19.

This paper develops a new method for interactive multi-criteria group decision-making (MCGDM) with probabilistic linguistic information and applies to the emergency assistance area selection of COVID-19 for Wuhan. First, a new possibility degree for PLTSs is defined and a new possibility degree algorithm is devised to rank a series of probabilistic linguistic term sets (PLTSs). Second, some new operational laws of PLTSs based on the Archimedean copulas and co-copulas are defined. A generalized probabilistic linguistic Choquet (GPLC) operator and a generalized probabilistic linguistic hybrid Choquet (GPLHC) operator are developed and their desirable properties are discussed in details. Third, a tri-objective nonlinear programming model is constructed to determine the weights of DMs. This model is transformed into a linear programming model to solve. The fuzzy measures of criterion subsets are derived objectively by establishing a goal programming model. Fourth, using the probabilistic linguistic Gumbel weighted average (PLGWA) operator, the collective normalized decision matrix is obtained by aggregating all individual normalized decision matrices. The overall evaluation values of alternatives are derived by the probabilistic linguistic Gumbel hybrid Choquet (PLGHC) operator. The ranking order of alternatives is generated. Finally, an emergency assistance example is illustrated to validate the proposed method of this paper.

Introduction

In 2020, a new coronavirus COVID-19 broke out all over the world. Wuhan in China was also suffering COVID-19. As an important transportation hub with large population flow, Wuhan has great difficulty to prevent and control the epidemic. To win the battle of the epidemic earlier and better, a lot of national medical support teams provide emergency assistance to the hospitals of Wuhan. How to select an appropriate area to assist is an urgent issue. Recently, lots of emergency events become more serious and urgent. Some methods [1], [2], [3] have been proposed to solve emergency decision-making, which can be ascribed as a type of the multi-criteria group decision-making (MCGDM) problems. Since the emergency assistance area selection of COVID-19 can be evaluated from different aspects, the emergency event of assistance area selection can also be regarded as a MCGDM problem.

MCGDM is an important component of decision science. Because of the uncertainty and ambiguity of human thinking, it is difficult for decision makers (DMs) to deliver accurate information for complex decision-making problems. Zadeh [4] introduced fuzzy sets (FSs) to express the uncertain information. FSs have been extensively applied to MCGDM. However, it is hard for DMs to quantify the information characters of MCGDM problems directly in some specified situations. DMs tended to express their preferences in vague qualitative linguistic terms such as “excellent”, “good”, “bad” rather than precise quantitative numerical values. Currently, some extension models of linguistic terms have been proposed, such as 2-tuple fuzzy linguistic representation model [5], uncertain linguistic model [6], intuitionistic fuzzy linguistic model [7] and hesitant fuzzy linguistic model [8]. Xu [9] proposed a subscript-symmetric additive linguistic term set (LTS). However, due to the discreteness of information features, it is not convenient to calculate and analyze many information features. Then, Xu [10] further extended the discrete LTS to a continuous LTS. Rodriguez et al. [11] proposed a conception of hesitant fuzzy linguistic term set (HFLTS). Different with traditional LTS, HFLTS uses several linguistic terms provided by DMs to describe the evaluation information taking the hesitant degree of DMs into account. For a HFLTS, the weights of linguistic terms provided by DMs are equal. However, it is impractical to require DMs to provide linguistic terms with equal weight. HFLTSs may not be suitable to express linguistic terms with different weights. Consequently, Pang et al. [12] proposed the concept of probabilistic linguistic term set (PLTS) in 2016. PLTS can not only contain more than one possible linguistic term, but also reflect the corresponding probability information, which can avoid the information loss. To reflect the DM’s cognitive certainty and uncertainty in the group decision making (GDM) process, the probabilistic linguistic preference relations [13], [14] have received great attention.

Up to now, the investigations of PLTSs have made fruitful achievements. The existing progresses on the PLTSs mainly contain the following four aspects:

(1) Ranking of PLTSs. Pang et al. [12] compared the PLTSs by the score function and deviation degree of PLTSs. However, the ranking order of two PLTSs obtained by Pang et al. [12] may turn to be the opposite when giving a small perturbation of the probability information. In order to gain robustness, Bai et al. [15] ranked the PLTSs by using lower and upper bounds of the linguistic terms and corresponding probability information. Wu & Liao [16] ranked the PLTSs by gained and lost dominance score functions, which considered the individual regret values and the group utility values of alternatives. Feng et al. [17] ranked PLTSs by using the new probability formula obtained from the main structure of QUALIFLEX (Qualitative flexible) multiple-criteria method. Zhang et al. [18] built a probabilistic linguistic-based deviation model to identify the decision results in multi-expert multi-criteria decision making (MEMCDM).

(2) Information measures for PLTSs. Lin et al. [19] proposed the distance measure of PLTSs and used the probability degree to rank the alternatives for MCGDM. To enrich the calculation of PLTSs, Lin et al. [20] developed a novel distance measure for PLTSs and proposed an entropy measure to measure the uncertainty degree of PLTSs. Wu et al. [21] advanced three kinds of probabilistic linguistic distance measures reflecting on the difference of linguistic terms and probabilities, which do not need to add linguistic terms to the smaller one with the improved Borda rule. To improve the accuracy and visualization of comparison, Xian et al. [22] compared multiple PLTSs by calculating the novel similarity measure based on the RRD (relative repetition degree) and DD (diversity degree). Tang et al. [23] proposed inclusion measure for PLTSs and presented the relationships among the distance, similarity, entropy and inclusion measures.

(3) Operational laws and aggregation operators for PLTSs. Mao et al. [24] defined some new operational laws for PLTSs by using Archimedean triangular norm (t-norm) and triangular conorm (t-conorm or s-norm) and then defined a generalized probabilistic linguistic Hamacher ordered weighted averaging (GPLHOWA) operator. Liu & Teng [25] combined Muirhead mean aggregation operators with PLTSs based on the Archimedean t-conorm and t-norm and linguistic scale functions. Liu & Li [26] proposed a probabilistic linguistic-dependent weighted average (PLDWA) operator based on the prospect theory. Liu & Li [27] extended the generalized Maclaurin symmetric mean (GMSM) operator into probabilistic linguistic information and proposed four new GMSM operators for PLTSs for multi-criteria decision making (MCDM).

(4) Methods for probabilistic linguistic decision-making method. Pang et al. [12] extended TOPSIS method to the probabilistic linguistic group decision environment. Liu & Teng [28] extended PL-TODIM method to evaluate alternative products through consumer opinions regarding product performance. Ahmad et al. [29] proposed a MCGDM method based on VIKOR. Liao et al. [30] introduced the distillation process and Borda rule into the algorithm of the probabilistic linguistic ELECTRE III (PL-ELECTRE III) method. Wu & Liao [31] extended the quality function deployment (QFD) into the probabilistic linguistic context to get the fuzzy weights of design requirements. Furthermore, Wu & Liao [31] developed a probabilistic linguistic ORESTE (organísation, rangement et Synthèse de données relarionnelles, in French) method to obtain the preference, indifference and incomparability relations between the alternatives.

Although the above methods are valid for solving MCGDM with PLTSs, none of them considered the interactions among criteria. However, there are many interactions among criteria in some real decision situations. Consider the emergency assistance area selection based on supply medical support capacity, medical supply delivery speed and other criteria. A stronger supply medical support capacity often needs faster medical supply delivery speed. It is easily seen that supply medical support capacity and medical supply delivery speed are complementary interaction. Additionally, the supply medical support capacity is a qualitative criterion. The evaluation of the supply medical support capacity can be represented by a PLTS , which means that the possible linguistic term of the evaluation of supply medical support capacity may be or or . Meanwhile, 0.2, 0.3 and 0.2 are the corresponding probabilities of the linguistic terms , and . Therefore, the emergency assistance area selection may be ascribed to a kind of the interactive MCGDM with PLTSs. The Choquet integral is suitable to describe the interactions among criteria. It is necessary to develop some Choquet integral operators of PLTSs and investigate some new methods for solving such problems. To achieve this goal, this paper first proposes a new possibility degree algorithm for ranking a series of PLTSs. Then, some new operational laws of PLTSs based on the Archimedean copulas and co-copulas are defined. Considering the interactions among criteria, we develop a probabilistic linguistic Gumbel weighted average (PLGWA) operator, a generalized probabilistic linguistic Choquet (GPLC) operator and a probabilistic linguistic Gumbel hybrid Choquet (PLGHC) operator. Finally, a new method for the interactive MCGDM with PLTSs is put forward and applied to the emergency assistance area selection of COVID-19 for Wuhan. The main contributions of this paper are clarified as follows:

(1) A new possibility degree of PLTSs is defined and then a new possibility degree algorithm is proposed to rank a series of PLTSs. A new similarity degree of PLTSs is defined considering the linguistic scale function of linguistic terms.

(2) Some new operational laws of PLTSs based on the Archimedean copulas and co-copulas are defined. Considering the interactions among criteria, the generalized probabilistic linguistic Choquet (GPLC) operator and generalized probabilistic linguistic hybrid Choquet (GPLHC) operator are developed. Some desirable properties for these operators are discussed in details.

(3) Motivated by TOPSIS, a tri-objective nonlinear programming model is constructed to determine the weights of DMs. This model is transformed into a linear programming model to solve. To derive the fuzzy measures of criteria subsets, an optimization model is built and transformed into a goal programming model for resolution.

(4) Using the PLGWA operator, the collective normalized decision matrix is obtained by aggregating all individual normalized decision matrices. The overall evaluation values of alternatives are derived by the PLGHC operator. The ranking order of alternatives is then generated by the proposed possibility degree algorithm of PLTSs. Thereby, a new method for the interactive MCGDM with PLTSs is put forward.

Section 2 briefly reviews some concepts of PLTSs. Section 3 presents a new possibility degree algorithm to rank PLTSs and defines a new similarity degree of PLTSs. Section 4 defines some new operational laws of PLTSs based on the Archimedean copulas. Section 5 develops some generalized Choquet integral operators of PLTSs. Section 6 proposes a new method for interactive MCGDM with probabilistic linguistic information. Section 7 provides an emergency assistance area selection of COVID-19 to illustrate the validity of the method proposed in this paper. Section 8 draws some conclusions and ends this paper.

Preliminaries

In this section, some preliminaries about LTS, PLTS, Archimedean copulas and co-copulas are briefly reviewed to facilitate the discussions.

Linguistic term set

[9]

Let be a finite and totally ordered discrete LTS, where represents a possible value for a linguistic term, and is a positive integer. Especially, and denote the lower and the upper limits of linguistic terms, respectively. Furthermore, any two linguistic terms satisfy that if and only if .

To preserve all given linguistic information, Xu [10] extended the discrete LTS to a continuous LTS , where is a sufficiently large positive integer. If , then is called an original linguistic term. If , then is called a virtual linguistic term.

Probabilistic linguistic term set

[12]

Let be a LTS, a PLTS is defined as: where represents the linguistic term associated with the probability , and is the number of all different linguistic terms in .

Given a PLTS with , the normalized PLTS is defined as: where for all .

Let and be two PLTSs, where and are the numbers of linguistic terms in and respectively. If , then add linguistic terms to . Moreover, the added linguistic terms are the smallest linguistic term in and the probabilities of added linguistic terms are zero.

[32]

Given a PLTS , where is the subscript of . An ascending ordered normalized PLTS can be obtained in the following:

(1) If all elements in a PLTS are with different values of , then all the elements are arranged according to the values of in ascending order.

(2) If there are two or more elements with equal value of , then

(i) When the subscripts are unequal, such the elements are arranged according to the values of in ascending order;

(ii) When the subscripts are equal, such the elements are arranged according to the values of in ascending order.

According to above, after all elements of is ordered by Definition 5, a PLTS is transformed into an ascending ordered normalized PLTS .

[33]

Let be a LTS, if is a numeric value, then the linguistic scale function is mapped from to , which is represented as follows: where reflects the preference of the DMs and .

Wang et al. [33] presented three different forms of linguistic scale functions.

(Form 1) The evaluation scale of the linguistic information is divided on average:

(Form 2) The absolute deviation between adjacent linguistic subscripts increases from the middle of the linguistic term set to both ends (The value of can be determined using a subjective method):

The linguistic scale varies with the value of . The value of can be obtained experimentally or subjectively [34]. On the one hand, through large number of experimental research data [35], it can be concluded that is most likely to be obtained within the interval [1.36,1.4]. On the other hand, can also be determined by using a subjective method. Assuming that indicator A is far more important than indicator B, the importance ratio is , then ( represents the scale level) and . At present, vast majority of researchers believe that is the upper limit of the importance ratio [35]. In general, if the scale level is 7, then  [34].

(Form 3) The absolute deviation between adjacent linguistic subscripts decreases from the middle of the linguistic term set to both ends: where . If , then Form 3 is reduced to Form 1.

Let and be two normalized PLTSs. By using Definition 4, Definition 5, we can turn and into two ascending ordered normalized PLTSs and with equal number of linguistic terms. For simplicity, we still denote the two ascending ordered normalized PLTSs by and .

The distance between two ascending ordered normalized PLTSs and is defined as where , and , is the number of linguistic terms in the LTS .

Based on Definition 7, we define Minkowski distance between and as follows: where is a parameter. If , Eq. (8) is reduced to Hamming distance; if , Eq. (8) is reduced to Euclidean distance; if , Eq. (8) is reduced to Chebyshev distance.

Archimedean copulas and co-copulas

[36]

A copula is named as an Archimedean copula, which is denoted by , , if the generated functions and satisfy the following conditions:

(1) The generated function is strictly decreasing and continuous function from to with ;

(2) The function from to is defined as follows:

Considering the special situation where is a strictly increasing function on , and on , Genest & Mackay [37] proposed a special Archimedean copula as follows:

[38]

Let be a copula. The co-copula is defined as:

Possibility degree and similarity degree of PLTSs

This section develops a new possibility degree algorithm to rank PLTSs and defines a new similarity degree of PLTSs.

Existing ranking methods of PLTSs

[24]

Let and be two normalized PLTSs. A binary relation between and is defined as follows: The possibility degree is defined as follows [24]:

Let be the subscript of in a normalized PLTS , and are the lower and upper bounds of the subscripts of , and are the corresponding probabilities, respectively. The range value is defined as [24]:

Using Eqs. (12), (13), a preorder of normalized PLTSs and is presented as follows [24]:

(i) If , then is bigger than , denoted by ;

(ii) If , then

(a) if , then ;

(b) if , then is indifferent to , denoted by .

Let be a LTS, , , , be four normalized PLTSs.

By Eq. (13), the probability degrees of and are calculated as:

It is easy to see that the probability degree of Eq. (13) cannot distinguish the difference between and .

A new possibility degree algorithm for ranking PLTSs

As in Example 1, there are still some different PLTSs that cannot be distinguished. Similar to Eq. (12), a new binary relation between and is defined as follows:

To get the precise ranking result, a new possibility degree algorithm is designed in the following.

Let and be two normalized PLTSs. A new possibility degree ( is not inferior to ) is defined as follows:

Let and be two normalized PLTSs, where . The desirable properties of the new possibility degree in Definition 11 are presented as follows:

(i) (Normalization) .

(ii) (Visualization) If , then ; if , then ( and are the lower and upper bounds of the linguistic terms ( ).

(iii) (Complementarity) .

(iv) (Transitivity) If and , then .

It is easy to prove the properties (i), (iii), and (iv) in Property 1 by Eq. (15), (16). For the property (ii) of visualization, consider two PLTSs and . Similar to the comparison between interval numbers, if , then , namely ; if , then , namely .

If , it is hard to distinguish and . According to the statistical method, when the mean values of two sets of numbers are equal, the variance values can be used to further compare two sets of numbers. Then, a new range value of PLTSs is defined in the sequel.

For a normalized PLTS , the new range value is defined as: where is the linguistic scale function value of linguistic term . Let and be the lower and upper bounds of , where and are the corresponding probabilities, respectively.

A preorder of two normalized PLTSs and is defined as follows:

(i) If , then is bigger than , namely ;

(ii) If , then

(a) If , then is bigger than , namely ;

(b) If , then is indifferent to , namely .

The possibility degree algorithm for ranking a series of PLTSs is designed as:

Step 1. Calculate the possibility degree by Eq. (16) and the range value by Eq. (17) respectively.

Step 2. Aggregate into the ranking value as follows:

Step 3. Rank PLTSs by the ranking values in descending order. If the ranking of some PLTSs is equal, reorder these PLTSs by the range value in ascending order.

Continue to consider the four normalized PLTSs in Example 1, namely, , , , .

By Eq. (13), the possibility degree in Definition 10 is

By Eq. (16), the new probability degrees are calculated as follows:

It has by the new probability degrees, which shows that is different from .

Hence, Eq. (16) can identify the difference between and rationally and validly. Additionally, the intensiveness of is superior to . Hence, the distinguishing power of the new possibility degree in Definition 11 is stronger than that of the possibility degree in Definition 10.

To compare with the Mao et al.’s possibility degree method in [24], the new possibility degree method proposed in this paper is used to solve Example 3, Example 4, Example 5 in [24].

Consider three PLTSs , and of Example 3 in [24].

By Eq. (16), one has and , thus and , which are the same as the ranking results of Example 3 in [24]. Thus, the possibility degree defined in this paper still ensures that the order between and remains unchanged with respect to small disturbances of , which verifies the robustness of the proposed possibility degree of this paper.

Consider two PLTSs and of Example 4 in [24].

By Eq. (16), it yields that . Thus, the ranking order is , which is same as the result of Example 4 in [24].

Consider four PLTSs , , and based on of Example 5 in [24].

Step 1. Calculate the possibility degree by Eq. (16) as follows:

Calculate the range value by Eq. (17) based on Form 1 (i.e., Eq. (4)) as:

Calculate the range value by Eq. (17) based on Form 2 (i.e., Eq. (5)) (Let ) as:

Calculate the range value by Eq. (17) based on Form 3 (i.e., Eq. (6)) (Let )as:

Step 2. Aggregate into the ranking value as follows:

Step 3. Rank PLTSs by in descending order.

The ranking order based on Form 1 (i.e., Eq. (4)) is generated as .

The ranking order based on Form 2 (i.e., Eq. (5)) is generated as .

The ranking order based on Form 3 (i.e., Eq. (6)) is generated as .

By Eqs. (13), (14), one has , , , and . The ranking order obtained by method [24] is , which is the same as that obtained by the proposed new possibility degree algorithm of this paper. Although the ranking result between and obtained by this proposed method is same as that obtained by method [24], the range values and are different from and obtained by [24].

From the above examples, it is easily seen that the proposed possibility degree algorithm can take DM’s different preferences of linguistic scale functions into account, which is more robust and more in accordance with real situations.

Similarity degree of PLTSs

Xian et al. [22] defined the RRD (relative repetition degree) of linguistic terms between PLTSs and as where is the linguistic term vector, satisfying .

The DD (diversity degree) of probability between PLTSs and is defined as [22]: where is the linguistic term vector, satisfying .

The similarity degree between PLTSs and is defined as [22]:

Consider three PLTSs , and based on LTS . By Eq. (19), the similarity degree . However, the similarity degree between and is remarkably different from that between and intuitively. Thus, the similarity degree may be a little unreasonable. To overcome this drawback, a new similarity degree is given below.

Let be a LTS, and be two PLTSs, where and be the linguistic scale function value of linguistic term . A similarity degree between and is defined as follows:

The similarity degree between and satisfies:

(i) ; (ii) ;

(iii) If , then .

and can be regarded as two vectors and , respectively. For property (i) in Property 2, the similarity degree between and is equivalent to the cosine between and , namely . Properties (i) and (ii) denote the boundedness and symmetric respectively. For property (iii), if , then , namely the angle between and is 0. Therefore, , which completes the proof of property (iii).

Consider three PLTSs , and in Example 6. By Eq. (20), the similarity degrees are obtained as and , which are significantly different from that obtained by Eq. (19). Thus, Eq. (20) can overcome this deficiency of Eq. (19) and get a reasonable result.

New operational laws of PLTSs based on the Archimedean copulas and co-copulas

This section defines some new operational laws of PLTSs based on the Archimedean copulas and co-copulas. Some desirable properties of the new operational laws are discussed. Then, some specific cases are presented with respect to four different generated functions.

New operational laws of PLTSs based on the Archimedean copulas and co-copulas

This subsection defines some new operational laws of PLTSs based on the Archimedean copulas and co-copulas. Some desirable properties of the new operational laws are discussed.

Let be a LTS, , and be three PLTSs. The inverse function of is denoted by . . The new operational laws of PLTSs based on Archimedean copulas and co-copulas are defined as follows:

(1) Additive operation: where .

(2) Multiplication:

(3) Scalar-multiplication: ;

(4) Power operation: .

The linguistic terms of PLTSs based on Archimedean copulas and co-copulas still belong to the LTS and thus the operation results of PLTSs with these new operational laws in Definition 15 are still PLTSs.

For any , it holds that and when and . Hence, the linguistic terms still belong to the LTS.

By Definition 15, it is easily seen that the operation results of PLTSs with these new operational laws are still PLTSs.

Let , and be any three PLTSs, be the positive real numbers. Some desirable properties of the new operational laws in Definition 15 are satisfied as follows:

The proof of Theorem 2 is presented in Appendix A.

Some different types of operational laws for PLTSs based on common Archimedean copulas

Wang et al. [39] summarized four different types of common Archimedean copulas including Gumbel copula, Clayton copula, Frank copula and Joe copula, which are shown in Table 1.

Table 1

Four different types of common Archimedean copulas.

Types	Function	Copulas	Parameter
Gumbel	Ge(x)=(−lnx)^ε	Cp(x_1,x_2)=exp{−[(−lnx_1)^ε]+[(−lnx_2)^ε]^1∕ε}	ε≥1
Clayton	Ge(x)=x^−ε−1	Cp(x_1,x_2)=(x1−ε+x2−ε−1)^1∕ε	ε≥−1,ε≠0
Frank	Ge(x)=−lne^−εx−1e^−ε−1	Cp(x_1,x_2)=−1εln(1+(e^−εx_1−1)(e^−εx_2−1)e^−ε−1)	ε≠0
Joe	Ge(x)=−ln[1−(1−x)^ε]	Cp(x_1,x_2)=1−[(1−x_1)^ε+(1−x_2)^ε−(1−x_1)^ε(1−x_2)^ε]^1∕ε	ε≥1

Note: In Table 1, and are defined in Definition 8, and is the parameter of the function .

In the following, some different types of operational laws for PLTSs based on common Archimedean copulas can be obtained by Definition 15 and are listed in Table 2.

Table 2

Some different types of operational laws for PLTSs based on common Archimedean copulas.

Type	Function	New operational law
Gumbel	Ge(x)=(−lnx)^ε(ε≥1)	L_1(p)⊕_CpL_2(p)=⋃_k_1=1,2,…,#L_1(p)k_2=1,2,…,#L_2(p)g^−1[1−exp(−((−ln(1−g(L1(k_1))))^ε+(−ln(1−g(L2(k_2))))^ε)^1∕ε)]((p1(k_1)+p2(k_2)−p1(k_1)p2(k_2))L)L_1(p)⊗_CpL_2(p)=⋃_k_1=1,2,…,#L_1(p)k_2=1,2,…,#L_2(p)g^−1[exp(−((−ln(g(L1(k_1))))^ε+(−ln(g(L2(k_2))))^ε)^1∕ε)](p1(k_1)p2(k_2))λ⊙_CpL(p)=⋃_k=1,2,…,#L(p)g^−1[1−exp(−(λ(−ln(1−g(L^(k_1))))^ε)^1∕ε)](p^(k))(L(p))^λ=⋃_k=1,2,…,#L(p)g^−1[exp(−(λ(−ln(g(L^(k))))^ε)^1∕ε)](p^(k))
Clayton	Ge(x)=x^−ε−1(ε≥−1,ε≠0)	L_1(p)⊕_CpL_2(p)=⋃_k_1=1,2,…,#L_1(p)k_2=1,2,…,#L_2(p)g^−1[1−((1−g(L1(k_1)))^−ε+(1−g(L2(k_2)))^−ε−1)^−1∕ε]((p1(k_1)+p2(k_2)−p1(k_1)p2(k_2))L)L_1(p)⊗_CpL_2(p)=⋃_k_1=1,2,…,#L_1(p)k_2=1,2,…,#L_2(p)g^−1[(g(L1(k_1))^−ε+g(L2(k_2))^−ε−1)^−1∕ε](p1(k_1)p2(k_2))
		λ⊙_CpL(p)=⋃_k=1,2,…,#L(p)g^−1[1−(λ((1−g(L^(k)))^−ε−1)+1)^−1∕ε](p^(k))(L(p))^λ=⋃_k=1,2,…,#L(p)g^−1[(λ((g(L^(k)))^−ε−1)+1)^−1∕ε](p^(k))
Frank	Ge(x)=−lne^−εx−1e^−ε−1(ε≠0)	L_1(p)⊕_CpL_2(p)=⋃_k_1=1,2,…,#L_1(p)k_2=1,2,…,#L_2(p)g^−1[1+1εln(1+(exp(−ε(1−g(L1(k_1))))−1)(exp(−ε(1−g(L2(k_2))))−1)exp(−ε)−1)]((p1(k_1)+p2(k_2)−p1(k_1)p2(k_2))L)L_1(p)⊗_CpL_2(p)=⋃_k_1=1,2,…,#L_1(p)k_2=1,2,…,#L_2(p)g^−1[−1εln(1+(exp(−εg(L1(k_1)))−1)(exp(−εg(L2(k_2)))−1)exp(−ε)−1)](p1(k_1)p2(k_2))
		λ⊙_CpL(p)=⋃_k=1,2,…,#L(p)g^−1[1+1εln((exp(−ε(1−g(L^(k))))−1exp(−ε)−1)^λ+1)](p^(k))(L(p))^λ=⋃_k=1,2,…,#L(p)g^−1[−1εln((exp(−ε(1−g(L^(k))))−1exp(−ε)−1)^λ+1)](p^(k))
Joe	Ge(x)=−ln[1−(1−x)^ε](ε≥1)	L_1(p)⊕_CpL_2(p)=⋃_k_1=1,2,…,#L_1(p)k_2=1,2,…,#L_2(p)g^−1[(g(L1(k_1))^ε+g(L2(k_2))^ε−g(L1(k_1))^εg(L2(k_2))^ε)^1∕ε]((p1(k_1)+p2(k_2)−p1(k_1)p2(k_2))L)L_1(p)⊗_CpL_2(p)=⋃_k_1=1,2,…,#L_1(p)k_2=1,2,…,#L_2(p)g^−1[1−((1−g(L1(k_1)))^ε+(1−g(L2(k_2)))^ε−(1−g(L1(k_1)))^ε(1−g(L2(k_2)))^ε)^1∕ε](p1(k_1)p2(k_2))λ⊙_CpL(p)=⋃_k=1,2,…,#L(p)g^−1[(1−(1−g(L^(k_1))^ε)^λ)^1∕ε](p^(k))(L(p))^λ=⋃_k=1,2,…,#L(p)g^−1[1−(1−(1−(1−g(L^(k_1)))^ε)^λ)^1∕ε](p^(k))

Note: In Table 2, , , and , , are defined in Theorem 2.

Comparison with Mao et al. ’s operational laws

Let be a LTS, Consider two PLTSs and . To simplify the calculation process, let () (i.e., Gumbel type) and .

The computation results based on Mao et al.’s operational laws and the proposed operational laws of this paper can be obtained and presented in Table 3.

Table 3

Computation results.

Operational laws	Results
Mao et al. [24]	L_1(p)⊕_AL_2(p)={s_1.75(0.19),s_2.5(0.24),s_3.25(0.27),s_3.5(0.30)}L_1(p)⊗_AL_2(p)={s_0.25(0.12),s_0.5(0.18),s_0.75(0.28),s_1.5(0.42)}2⊙_AL_1(p)={s_1.75(0.4),s_3(0.6)}(L_1(p))^2={s_0.25(0.4),s_1(0.6)}
This paper with Form 1 of linguistic scale function	L_1(p)⊕_CpL_2(p)={s_1.75(0.19),s_2.5(0.24),s_3.25(0.27),s_3.5(0.30)}L_1(p)⊗_CpL_2(p)={s_0.25(0.12),s_0.5(0.18),s_0.75(0.28),s_1.5(0.42)}2⊙_CpL_1(p)={s_1.75(0.4),s_3(0.6)}(L_1(p))^2={s_0.25(0.4),s_1(0.6)}
This paper with Form 2 of linguistic scale function (a=94≈1.73)	L_1(p)⊕_CpL_2(p)={s_2.23(0.19),s_2.89(0.24),s_3.38(0.27),s_3.57(0.30)}L_1(p)⊗_CpL_2(p)={s_0.26(0.12),s_0.43(0.42),s_0.62(0.28),s_1.11(0.18)}2⊙_CpL_1(p)={s_2.23(0.4),s_3.26(0.6)}(L_1(p))^2={s_0.26(0.4),s_0.74(0.6)}
This paper with Form 3 of linguistic scale function (α=β=0.8)	L_1(p)⊕_CpL_2(p)={s_1.67(0.19),s_2.29(0.24),s_3.20(0.27),s_3.48(0.30)}L_1(p)⊗_CpL_2(p)={s_0.22(0.12),,s_0.52(0.18)s_0.80(0.28),s_1.71(0.42)}2⊙_CpL_1(p)={s_1.67(0.4),s_2.84(0.6)}(L_1(p))^2={s_0.22(0.4),s_1.16(0.6)}

As shown in Table 3, the computation results obtained by Mao et al.’s operational laws [24] are just as same as those obtained by the Gumbel operational laws (when ) and the linguistic scale function Form 1 (i.e., Eq. (4)). Therefore, the proposed operational laws based on Archimedean copulas and co-copulas greatly generalize Mao et al.’s operational laws based on Archimedean t-corm and t-conorm. The proposed operational laws use three different forms of linguistic scale functions to obtain the calculation results, whereas Mao et al. [24] only used Form 1 (i.e., Eq. (4)) to get the calculation results. In addition, DMs can select different linguistic scale functions and different types of Archimedean copulas according to their preference, which greatly enhances the flexibility of decision.

New aggregation operators of PLTSs based on the Archimedean copulas

This section develops a generalized probabilistic linguistic Choquet (GPLC) operator and a generalized probabilistic linguistic hybrid Choquet (GPLHC) operator. Some attractive properties of the proposed operators are discussed in details.

Probabilistic linguistic weighted averaging operator

Let be a set of PLTSs. Then, the probabilistic linguistic weighted average (PLWA) operator is defined as: where is the weight vector of , satisfying , . Especially, if , then the PLWA operator reduces to the probabilistic linguistic average (PLA) operator.

According to the new operational laws of Definition 15, the PLWA operator can be converted to different operators. Take Gumbel copula function as an example, when , the PLWA operator is converted to a probabilistic linguistic Gumbel weighted average (PLGWA) operator as follows: where , is the probability associated with the linguistic term , .

Generalized probabilistic linguistic Choquet operator

[40]

Let be a set of criteria. The set of function is fuzzy measure on if the following conditions are satisfied:

(1) (Boundary conditions) and ;

(2) (Monotonicity) If and , then , where is the power set of .

In the MCGDM problem, the properties of interactions among criteria can be represented by . Consider as the standard of subjective importance of criteria set . For any pair of criteria subsets with , three types of the properties are described as follows:

(1) (Simple additive measure) If , and are independent.

(2) (Super additive measure) If , and are positive interaction.

(3) (Sub additive measure) If , and are negative interaction.

[41]

Let be a positive real function on and be a fuzzy measure on . Then, the discrete Choquet integral of on is defined as follows: where , and .

Let be the set of fuzzy numbers, which is denoted as . The discrete Choquet integral of on can be obtained as follows:

Inspired by the probabilistic linguistic Choquet integral operator in [42], a GPLC operator is proposed below.

Let be a set of PLTSs. The GPLC operator is defined as: where , is a permutation of such that according to the proposed possibility ranking algorithm in Section 3.2, and .

Let be a set of PLTSs. The result by using the GPLC operator is obtained as: where , , .

The proof of Theorem 3 is shown in Appendix B.

Idempotency

Let be a set of PLTSs. If , then .

Monotonicity

Let and be two sets of PLTSs. If , then

Boundedness

Let be a set of PLTSs, where . and are two special PLTSs, where and are the minimal and maximal linguistic terms of in , respectively.

Then, one has , where and .

The proofs of Properties 3–5 are shown in Appendix C .

Let be a set of PLTSs.

(1) When, it easily follows from Eq. (25) that which is degenerated to a probabilistic linguistic geometric ordered weighted Choquet (PLGOWC) operator;

(2) When, it easily follows from Eq. (25) that which is degenerated to a probabilistic linguistic Choquet integral (PLC) operator;

(3) When, it easily follows from Eq. (25) that which is degenerated to a max operator of PLTSs .

Generalized probabilistic linguistic hybrid Choquet operator

Let be a set of PLTSs. The GPLHC operator is defined as: where , is obtained by weighting the PLTS , i.e., , is the weight of satisfying , is the -th largest PLTS of according to the proposed possibility ranking algorithm in Section 3.2, and .

Let be a set of PLTSs. The result by using the GPLHC operator is obtained as where , , .

Similar to Theorem 3, it is easy to prove Theorem 4.

The GPLHC operator has similar properties to the GPLC operator as follows:

(1) When , it easily follows from Eq. (27) that which is degenerated to a geometric probabilistic linguistic hybrid Choquet integral (GPLHCI) operator;

(2) When , it easily follows from Eq. (27) that which is degenerated to a probabilistic linguistic hybrid Choquet integral (PLHC) operator;

(3) When , it easily follows from Eq. (27) that which is degenerated to a max operator of PLTSs .

In terms of four different types of common Archimedean copulas functions, the GPLHC operator can be converted into different forms.

Case 1. (Gumbel type) When , the GPLHC operator is called a generalized probabilistic linguistic Gumbel hybrid Choquet (GPLGHC) operator as follows: where , .

Especially, when , GPLGHC operator is degenerated to a PLGHC operator as where , .

Case 2. (Clayton type) When , the GPLHC operator is called a generalized probabilistic linguistic Clayton hybrid Choquet (GPLCHC) operator as follows: where , .

Case 3. (Frank type) When , the GPLHC operator is called a generalized probabilistic linguistic Frank hybrid Choquet (GPLHC) operator as follows: where , .

Case 4. (Joe type) When , the GPLHC operator is called a generalized probabilistic linguistic Joe hybrid Choquet (GPLJHC) operator as follows: where , .

A new method for interactive MCGDM with probabilistic linguistic information

In this section, a new method for interactive MCGDM with PLTSs is proposed.

Problem description

The notation clarifications of Probabilistic linguistic MCGDM problem are shown below:

is a set of alternatives, where denotes the -th alternative.

is a set of criteria, where denotes the -th criterion. There exist interactions among criteria.

is a set of DMs, where denotes the -th DM.

is the weight vector of DMs, where denotes the weight of DM , satisfying and .

is the fuzzy measure of criteria, where denotes the fuzzy measure of criterion subset and .

is an individual decision matrix given by DM , where is a PLTS and denotes the evaluation of alternative on criterion provided by DM .

is an individual ascending ordered normalized decision matrix given by DM , where is an ascending ordered normalized PLTS of .

is a collective normalized decision matrix, where is a collective evaluation of alternative on criterion .

is a PLTS and denotes the collective comprehensive value of alternative .

Incomplete information structure

This subsection depicts the incomplete information structure of DMs’ weights and criteria fuzzy measures.

Incomplete information structure of DMs’ weights

Due to the complexity of realistic decision-making problems and the incomprehensive experience and knowledge of DMs, it is hard to determine the weight vector of DMs. Therefore, the information of the DMs’ weights is incomplete. Let be the incomplete information structure of DMs’ weights, which may consist of several basic forms [43] (please see [43] for more details):

(Form 1) A weak ranking: ;

(Form 2) A strict ranking: ;

(Form 3) A ranking of differences: ;

(Form 4) A ranking with multiples: ;

(Form 5) A interval ranking: .

Incomplete information structure of criteria fuzzy measures

In some real decision situations, DMs tend to specify their preferences on criteria fuzzy measures according to their knowledge and judgment. Therefore, the information of the criteria fuzzy measures is incomplete [44]. Let be the incomplete information of criteria fuzzy measures, which may consist of several forms [44] (please see [44] for more details).

Determination of the weights of DMs

Inspired by Yue [45], this subsection introduces the TOPSIS method to obtain the weights of DMs.

(1) Determine the individual ascending ordered normalized probabilistic linguistic decision matrix .

Normalize probabilistic linguistic decision matrix to corresponding ascending ordered normalized probabilistic linguistic decision matrix , where is an ascending ordered normalized PLTS defined in Definition 5.

(2) Determine the ascending ordered normalized positive ideal decision matrix , where

The ascending ordered normalized negative ideal decision matrix is divided into left ascending ordered normalized negative ideal decision matrix and right ascending ordered normalized negative ideal decision matrix .

(3) Determine the left ascending ordered normalized negative ideal decision matrix , where is the corresponding probability value of .

(4) Determine the right ascending ordered normalized negative ideal decision matrix , where is the corresponding probability value of .

By Eq. (34), the similarity degree between and is defined as follows: where .

By Eq. (35), the similarity degree between and is defined as follows: where .

By Eq. (36), the similarity degree between and is defined as follows: where .

Then, the individual relative closeness degree of alternative on criterion for DM is defined as follows:

The global relative closeness degree of alternative on criterion is defined as follows: where denotes the weight of DM .

By Eq. (40), the deviation between the individual relative closeness degree and other individual relative closeness degrees on criterion for DM can be calculated as follows:

By Eqs. (40), (41), the deviation between the individual relative closeness degree and global relative closeness degree on criterion for DM can be calculated as follows:

By Eq. (40), the deviation between the individual relative closeness degrees on the criterion for DM and other DMs can be calculated as follows:

To determine DMs’ weight , it is reasonable to maximize the deviation between the individual relative closeness degree and other individual relative closeness degrees on criterion for DM , while minimize the deviation between the individual relative closeness degree and global relative closeness degree on criterion for DM , and minimize the deviation between the individual relative closeness degrees on the criterion for DM and other DMs. Hence, a tri-objective nonlinear programming model can be constructed as follows: Then, (Mod 1) is transformed into a single objective nonlinear programming model as follows:

(Mod 2)

To solve (Mod 2), let

Thus, (Mod 2) is transformed into a linear programming model as follows:

(Mod 3)

The weight vector of DMs can be derived by solving (Mod 3).

A new method for interactive MCGDM with probabilistic linguistic information

Based on the above analyses, a new method for interactive MCGDM with probabilistic linguistic information is summarized as follows:

Step 1. Elicit the individual probabilistic linguistic matrix .

Step 2. Acquire the individual ascending ordered normalized probabilistic linguistic decision matrix by Definition 5.

Step 3. Determine the weight of DMs by solving (Mod 3).

Step 4. Compute the collective normalized decision matrix .

Based on the weight vector of DMs , aggregate all individual ascending ordered normalized probabilistic linguistic decision matrices into a collective normalized decision matrix by PLGWA operator (i.e., Eq. (22)). where , .

Step 5. Calculate the collective comprehensive value of alternative .

Based on the fuzzy measures of criteria subsets, aggregate the th line elements of collective normalized probabilistic linguistic decision matrix by PLGHC operator (i.e., Eq. (30)) to obtain as: where , , , is the weight vector of criteria, is the th largest PLTS of according to the proposed possibility ranking algorithm in Section 3.2.

Step 6. Define the positive ideal solution (PIS).

The positive ideal solution (PIS) can be defined as . The Hamming distance between alternative and PIS can be calculated by Eq. (8) with as follows: where .

Step 7. Determine the fuzzy measures of criteria subsets.

To determine the fuzzy measures of criteria subsets, it is reasonable to minimize the distance between each alternative and PIS . Then, an optimization model is constructed as follows:

(Mod 4) To solve (Mod 4), let

Thus, (Mod 4) is converted into a goal programming model (Mod 5).

(Mod 5)

The fuzzy measures of criteria subsets can be obtained by solving (Mod 5).

Step 8. Rank the alternatives by ranking PLTSs using the possibility ranking algorithm proposed in Section 3.2.

The decision-making flowchart of the new proposed method can be depicted in Fig. 1.

Fig. 1

Decision-making flowchart of the new proposed method.

Once we have proposed a new method for solving a MCGDM problem, the complexity effort of the new method should be analyzed by the number of bits in the input and the dimension of the problem (in terms of the O-notation), since the requirement of time in emergency decision making is very important. In this paper, (Mod 3) and (Mod 5) determine the complexity degree of the proposed method. The numbers of decision variables in (Mod 3) and (Mod 5) are and respectively, where is the number of DMs and is the number of criteria. Using Karmarkar’s algorithm [46], the time complexity of (Mod 3) is calculated as , where denotes the number of bits in the input. Similarly, the time complexity of (Mod 5) is . Hence, the time complexity of the proposed method is . Despite the time complexity is a little high, (Mod 3) and (Mod 5) only consume very little computational time by using some mature software packages (e.g., LINGO and MATLAB).

Emergency assistance case study for COVID-19

In this section, an emergency assistance case is presented to demonstrate the rationality and robustness of the proposed method. Sensitivity analysis and comparative analysis are conducted to measure and compare the evaluation results of different type generated functions.

Emergency assistance area selection of COVID-19 for Wuhan

In 2020, a new coronavirus COVID-19 broke out all over the world. Wuhan in China was also suffering COVID-19. How to select an appropriate area to assist is an urgent issue. Taking five Hardest-hit Wuhan areas into account, whose confirmed and suspected cases of pneumonia were in the top five. There exist four criteria that affect the best and optimal assistance, such as supply medical support capacity , medical supply delivery speed , living material support capacity , medical personnel transport capacity . Preset LTS ,. The first aid comes from four national medical support teams, including medical support team , , and . The weights of criteria are determined as after discussion and negotiation. Based on the four criteria, four probabilistic linguistic evaluation matrices are constructed by these four medical support teams. The incomplete information structure of DMs’ weights is provided by all medical support teams as follows:

The incomplete information of fuzzy measures of criteria subsets is given as

Next, the proposed method of this paper is employed to solve this example.

Step 1. The decision matrices , , and are constructed in Table 4.

Table 4

Decision matrices , , and given by four medical support teams.

		C_1	C_2	C_3	C_4
E_1	A_1	{s_3(0.1),s_4(0.3),s_5(0.4)}	{s_2(0.4),s_3(0.3)}	{s_4(0.1),s_5(0.2),s_6(0.4)}	{s_3(0.2),s_4(0.4),s_5(0.2)}
	A_2	{s_3(1)}	{s_2(0.3),s_3(0.3),s_4(0.4)}	{s_4(1)}	{s_2(0.1),s_3(0.2),s_4(0.2)}
	A_3	{s_2(0.4),s_3(0.6)}	{s_2(0.7),s_3(0.3)}	{s_3(0.1),s_4(0.4),s_5(0.2)}	{s_3(0.4),s_6(0.2)}
	A_4	{s_3(0.4),s_4(0.5)}	{s_2(0.3),s_3(0.3)}	{s_4(0.3),s_5(0.4)}	{s_2(0.1),s_3(0.4)}
	A_5	{s_1(0.1),s_2(0.3),s_3(0.3)}	{s_2(0.4),s_3(0.2)}	{s_4(0.3),s_5(0.4)}	{s_2(1)}
E_2	A_1	{s_5(0.5),s_6(0.1)}	{s_3(0.1),s_4(0.2),s_5(0.3)}	{s_5(0.3),s_6(0.5)}	{s_3(0.2),s_4(0.3),s_5(0.2)}
	A_2	{s_4(0.1),s_5(0.3),s_6(0.4)}	{s_3(0.1),s_4(0.4),s_5(0.2)}	{s_5(0.4),s_6(0.1),s_7(0.1)}	{s_3(0.3),s_4(0.6)}
	A_3	{s_4(0.2),s_5(0.1),s_6(0.4)}	{s_3(0.8),s_4(0.1)}	{s_5(0.4),s_6(0.2)}	{s_1(0.2),s_2(0.4),s_3(0.4)}
	A_4	{s_5(0.3),s_6(0.2)}	{s_2(0.4),s_3(0.5)}	{s_4(0.1),s_5(0.1),s_6(0.4)}	{s_2(0.4),s_3(0.5)}
	A_5	{s_5(0.4),s_6(0.4)}	{s_2(0.2),s_3(0.3),s_4(0.2)}	{s_5(0.7),s_6(0.1)}	{s_2(0.2),s_3(0.5),s_4(0.2)}
E_3	A_1	{s_4(0.2),s_5(0.4),s_6(0.3)}	{s_3(1)}	{s_4(0.2),s_5(0.3),s_6(0.5)}	{s_4(0.1),s_5(0.4)}
	A_2	{s_4(0.7),s_5(0.1)}	{s_5(0.4),s_6(0.1)}	{s_5(0.2),s_6(0.1),s_7(0.5)}	{s_4(0.1),s_5(0.3),s_6(0.1)}
	A_3	{s_3(0.1),s_4(0.4),s_5(0.3)}	{s_3(0.5),s_4(0.4)}	{s_5(0.1),s_6(0.5)}	{s_3(1)}
	A_4	{s_3(0.3),s_4(0.3)}	{s_4(0.3),s_5(0.2),s_6(0.3)}	{s_4(1)}	{s_2(0.1),s_3(0.3),s_4(0.1)}
	A_5	{s_4(0.4),s_5(0.3)}	{s_6(0.1),s_7(0.5)}	{s_4(0.1),s_5(0.4),s_6(0.2)}	{s_1(0.3),s_2(0.2),s_3(0.3)}
E_4	A_1	{s_5(0.1),s_6(0.6),s_7(0.1)}	{s_3(0.1),s_4(0.3),s_5(0.1)}	{s_3(0.1),s_4(0.3),s_5(0.3)}	{s_5(0.4),s_6(0.1)}
	A_2	{s_3(0.3),s_4(0.3),s_5(0.3)}	{s_3(0.1),s_4(0.4),s_5(0.1)}	{s_4(0.1),s_5(0.1),s_6(0.4)}	{s_4(1)}
	A_3	{s_4(0.4),s_5(0.2)}	{s_2(0.5),s_3(0.4)}	{s_4(0.3),s_5(0.5),s_6(0.1)}	{s_3(0.4),s_4(0.4),s_5(0.1)}
	A_4	{s_3(0.3),s_4(0.4)}	{s_3(0.2),s_4(0.4)}	{s_4(0.1),s_5(0.6)}	{s_3(0.3),s_4(0.3)}
	A_5	{s_3(0.3),s_4(0.6)}	{s_2(0.1),s_3(0.5),s_4(0.3)}	{s_4(0.4),s_5(0.4)}	{s_3(0.1),s_4(0.3),s_5(0.1)}

Step 2. The corresponding ascending ordered normalized matrices , , and are obtained in Table 5.

Table 5

Ascending ordered normalized decision matrices , , and .

		C_1	C_2	C_3	C_4
E_1	A_1	{s_3(1∕8),s_4(3∕8),s_5(1∕2)}	{s_2(0),s_2(4∕7),s_3(3∕7)}	{s_4(1∕7),s_5(2∕7),s_6(4∕7)}	{s_3(1∕4),s_4(1∕2),s_5(1∕4)}
	A_2	{s_3(0),s_3(0),s_3(1)}	{s_2(3∕10),s_3(3∕10),s_4(2∕5)}	{s_4(0),s_4(0),s_4(1)}	{s_2(1∕5),s_3(2∕5),s_4(2∕5)}
	A_3	{s_2(0),s_2(2∕5),s_3(3∕5)}	{s_2(0),s_2(7∕10),s_3(3∕10)}	{s_3(1∕7),s_4(4∕7),s_5(2∕7)}	{s_3(0),s_3(2∕3),s_6(1∕3)}
	A_4	{s_3(0),s_3(4∕9),s_4(5∕9)}	{s_2(0),s_2(1∕2),s_3(1∕2)}	{s_4(0),s_4(3∕7),s_5(4∕7)}	{s_2(0),s_2(1∕5),s_3(4∕5)}
	A_5	{s_1(1∕7),s_2(3∕7),s_3(3∕7)}	{s_2(0),s_2(2∕3),s_3(1∕3)}	{s_4(0),s_4(3∕7),s_5(4∕7)}	{s_2(0),s_2(0),s_2(1)}
E_2	A_1	{s_5(0),s_5(5∕6),s_6(1∕6)}	{s_3(1∕6),s_4(1∕3),s_5(1∕2)}	{s_5(0),s_5(3∕8),s_6(5∕8)}	{s_3(2∕7),s_4(3∕7),s_5(2∕7)}
	A_2	{s_4(1∕8),s_5(3∕8),s_6(1∕2)}	{s_3(1∕7),s_4(4∕7),s_5(2∕7)}	{s_5(2∕3),s_6(1∕6),s_7(1∕6)}	{s_3(0),s_3(1∕3),s_4(2∕3)}
	A_3	{s_4(2∕7),s_5(1∕7),s_6(4∕7)}	{s_3(0),s_3(8∕9),s_4(1∕9)}	.{s_5(0),s_5(2∕3),s_6(1∕3)}.	{s_1(1∕5),s_2(2∕5),s_3(2∕5)}
	A_4	{s_5(0),s_5(3∕5),s_6(2∕5)}	{s_2(0),s_2(4∕9),s_3(5∕9)}	{s_4(1∕6),s_5(1∕6),s_6(2∕3)}	{s_2(0),s_2(4∕9),s_3(5∕9)}
	A_5	{s_5(0),s_5(1∕2),s_6(1∕2)}	{s_2(2∕7),s_3(3∕7),s_4(2∕7)}	{s_5(0),s_5(7∕8),s_6(1∕8)}	{s_2(2∕9),s_3(5∕9),s_4(2∕9)}
E_3	A_1	{s_4(2∕9),s_5(4∕9),s_6(1∕3)}	{s_3(0),s_3(0),s_3(1)}	{s_4(1∕5),s_5(3∕10),s_6(1∕2)}	{s_4(0),s_4(1∕5),s_5(4∕5)}
	A_2	{s_4(0),s_4(7∕8),s_5(1∕8)}	{s_5(0),s_5(4∕5),s_6(1∕5)}	{s_5(1∕4),s_6(1∕8),s_7(5∕8)}	{s_4(1∕5),s_5(3∕5),s_6(1∕5)}
	A_3	{s_3(1∕8),s_4(1∕2),s_5(3∕8)}	{s_3(0),s_3(5∕9),s_4(4∕9)}	{s_5(0),s_5(1∕6),s_6(5∕6)}	{s_3(0),s_3(0),s_3(1)}
	A_4	{s_3(0),s_3(1∕2),s_4(1∕2)}	{s_4(3∕8),s_5(1∕4),s_6(3∕8)}	{s_4(0),s_4(0),s_4(1)}	{s_2(1∕5),s_3(3∕5),s_4(1∕5)}
	A_5	{s_4(0),s_4(4∕7),s_5(3∕7)}	{s_6(0),s_6(1∕6),s_7(5∕6)}	{s_4(1∕7),s_5(4∕7),s_6(2∕7)}	{s_1(3∕8),s_2(1∕4),s_3(3∕8)}
E_4	A_1	{s_5(1∕8),s_6(3∕4),s_7(1∕8)}	{s_3(1∕5),s_4(3∕5),s_5(1∕5)}	{s_3(1∕7),s_4(3∕7),s_5(3∕7)}	{s_5(0),s_5(4∕5),s_6(1∕5)}
	A_2	{s_3(1∕3),s_4(1∕3),s_5(1∕3)}	{s_3(1∕6),s_4(2∕3),s_5(1∕6)}	{s_4(1∕6),s_5(1∕6),s_6(2∕3)}	{s_4(0),s_4(0),s_4(1)}
	A_3	{s_4(0),s_4(2∕3),s_5(1∕3)}	{s_2(0),s_2(5∕9),s_3(4∕9)}	{s_4(1∕3),s_5(5∕9),s_6(1∕9)}	{s_3(4∕9),s_4(4∕9),s_5(1∕9)}
	A_4	{s_3(0),s_3(3∕7),s_4(4∕7)}	{s_3(0),s_3(1∕3),s_4(2∕3)}	{s_4(0),s_4(1∕7),s_5(6∕7)}	{s_3(0),s_3(1∕2),s_4(1∕2)}
	A_5	{s_3(0),s_3(1∕3),s_4(2∕3)}	{s_2(1∕9),s_3(5∕9),s_4(1∕3)}	{s_4(0),s_4(1∕2),s_5(1∕2)}	{s_3(1∕5),s_4(3∕5),s_5(1∕5)}

Step 3. Determine the weights of DMs.

According to (Mod 3), a linear programming model of the DMs’ weights is built as:

Then, consider the Form 1 (i.e., Eq. (4)), the DMs’ weights are obtained by solving Eq. (52) as follows:

Step 4. Compute the collective normalized decision matrix . To simplify the calculation process, set . By Eq. (48), the collective normalized decision matrix is obtained and listed in Table 6.

Table 6

A collective normalized decision matrix .

	C_1	C_2	C_3	C_4
A_1	{s_4.20(0.19),s_5.12(0.46),s_6.19(0.35)}	{s_2.67(0.15),s_3.18(0.40),s_4.03(0.45)}	{s_3.80(0.19),s_4.68(0.37),s_5.70(0.44)}	{s_4.00(0.20),s_4.38(0.41),s_5.40(0.39)}
A_2	{s_3.32(0.18),s_3.80(0.40),s_4.56(0.42)}	{s_3.19(0.23),s_3.92(0.45),s_4.94(0.32)}	{s_4.33(0.36),s_5.06(0.18),s_5.93(0.46)}	{s_3.29(0.16),s_3.82(0.38),s_4.52(0.46)}
A_3	{s_3.18(0.17),s_3.52(0.41),s_4.56(0.42)}	{s_2.32(0),s_2.32(0.55),s_3.32(0.45)}	{s_4.03(0.19),s_4.68(0.41),s_5.70(0.40)}	{s_2.83(0.23),s_3.29(0.36),s_4.13(0.41)}
A_4	{s_3.25(0),s_3.25(0.50),s_4.27(0.50)}	{s_2.81(0.17),s_3.10(0.39),s_4.15(0.44)}	{s_4(0.09),s_4.11(0.34),s_4.95(0.57)}	{s_2.37(0.10),s_2.57(0.44),s_3.57(0.46)}
A_5	{s_3.89(0.07),s_3.16(0.46),s_4.19(0.47)}	{s_3.18(0.16),s_3.56(0.42),s_4.72(0.42)}	{s_4.11(0.07),s_4.33(0.49),s_5.34(0.44)}	{s_2.19(0.25),s_2.89(0.35),s_3.64(0.40)}

Step 5. Combine the weights of criteria to calculate the collective comprehensive value of alternative by Eq. (49) as follows:

Step 6. Define the PIS .

Step 7. Determine the fuzzy measures of criteria subsets.

For simplicity, denote the fuzzy measure of criterion subset by . For example, , . Set . According to (Mod 5), a goal programming model of the fuzzy measures is built as follows:

(Mod 6)

By solving (Mod 6), the fuzzy measures of criteria subsets are obtained and presented in Table 7.

Table 7

Fuzzy measures of criteria subsets.

Subsets	Γ	Subsets	Γ	Subsets	Γ	Subsets	Γ	Subsets	Γ
C_1	1.0000	C_4	1.0000	C_1,C_4	1.0000	C_3,C_4	1.0000	C_1,C_3,C_4	0.0000
C_2	1.0000	C_1,C_2	0.7273	C_2,C_3	1.0000	C_1,C_2,C_3	0.1818	C_2,C_3,C_4	1.0000
C_3	0.0000	C_1,C_3	0.7273	C_2,C_4	1.0000	C_1,C_2,C_4	1.0000	C_1,C_2,C_3,C_4	1.0000

Then, the collective comprehensive value of alternative is obtained, as shown in Table 8.

Table 8

Collective comprehensive value of each alternative.

A_1	A_2	A_3	A_4	A_5
{s_3.20(0.24),s_3.51(0.38),s_4.32(0.38)}	{s_2.63(0.28),s_3.06(0.35),s_3.61(0.37)}	{s_2.26(0.21),s_2.63(0.40),s_3.31(0.39)}	{s_1.90(0.15),s_2.06(0.41),s_2.86(0.44)}	{s_1.76(0.20),s_2.31(0.40),s_2.91(0.40)}

Step 8. To rank PLTSs , calculate the possibility degree by Eq. (16). The results are presented in Table 9.

Table 9

Possibility degree.

P12′	P13′	P14′	P15′	P21′	P23′	P24′	P25′	P31′	P32′	P34′	P35′	P41′	P42′	P43′	P45′	P51′	P52′	P53′	P54′
0.77	0.91	1	1	0.23	0.64	0.88	0.89	0.09	0.36	0.73	0.67	0	0.12	0.27	0.38	0	0.11	0.33	0.62

The ranking values are obtained by Eq. (16) as .

Therefore, the ranking order of alternatives is and the best alternative is .

To illustrate the influence of the linguistic scale function in this example, rank the alternatives based on the Form 2 (i.e., Eq. (5)) and Form 3 (i.e., Eq. (6)). The ranking values are shown below.

For the Form 2 (i.e., Eq. (5)) (Let ), it has

The ranking order of alternatives is .

For the Form 3 (i.e., Eq. (6)) (Let ), it has

The ranking order of alternatives is .

Sensitivity analyses

To manifest that the ranking result is universal, this subsection analyzes the influences of four different types of generator based on common Archimedean copulas and the parameter .

The ranking results with four different types of common Archimedean copulas based on Form 1 (i.e., Eq. (4)) are presented in Table 10.

Table 10

Ranking results for four different types of common Archimedean copulas of PLTSs.

Type	Function	P1′	P2′	P3′	P4′	P5′	Ranking
Gumbel	Ge(x)=(−lnx)^ε(ε=1)	4.18	3.14	2.36	1.26	1.56	A_1≻A_2≻A_3≻A_5≻A_4
Clayton	Ge(x)=x^−ε−1(ε=1)	4.18	3.25	2.25	1.26	1.56	A_1≻A_2≻A_3≻A_5≻A_4
Frank	Ge(x)=−lne^−εx−1e^−ε−1(ε=1)	4.18	3.25	2.25	1.26	1.56	A_1≻A_2≻A_3≻A_5≻A_4
Joe	Ge(x)=−ln[1−(1−x)^ε](ε=1)	4.18	3.14	2.36	1.26	1.56	A_1≻A_2≻A_3≻A_5≻A_4

As can be seen from Table 10, the rank results are the same for four different types of common Archimedean copulas. No matter which generator is, the ranking order of alternatives keeps unchanged. The ranking result of Gumbel copula function is the same as that of Joe copula function. The main reason is that the PLGHC operator is degraded to PLJHC operator when . From the above analysis, the ranking results are highly similar for these four different types of common Archimedean copulas.

Then, taking Gumbel generator into consideration, the ranking results for difference values of parameter are listed in Table 11 (based on Form 1 (i.e., Eq. (4))). No matter the value of is, the ranking order remains unchanged. It is easy to see that any tiny intervention on evaluations would not affect the ranking results.

Table 11

Ranking results for different values of parameter .

ε	P1′	P2′	P3′	P4′	P5′	Ranking
1	4.18	3.14	2.36	1.26	1.56	A_1≻A_2≻A_3≻A_5≻A_4
3	4.18	3.14	2.36	1.26	1.56	A_1≻A_2≻A_3≻A_5≻A_4
5	4.18	3.14	2.36	1.26	1.56	A_1≻A_2≻A_3≻A_5≻A_4
15	4.18	3.14	2.36	1.26	1.56	A_1≻A_2≻A_3≻A_5≻A_4
25	4.18	3.14	2.36	1.26	1.56	A_1≻A_2≻A_3≻A_5≻A_4
45	4.18	3.14	2.36	1.26	1.56	A_1≻A_2≻A_3≻A_5≻A_4
95	4.18	3.14	2.36	1.26	1.56	A_1≻A_2≻A_3≻A_5≻A_4

The aforesaid sensitivity analyses reveal that the method proposed in this paper is robust.

Comparative analysis

In this subsection, comparative analyses with Pang et al.’s method [12] and Liu et al.’s method [26] are conducted to illustrate the advantages of the proposed method.

Comparison with Pang et al. ’s method

Pang et al.’s method [12] is used to solve the example in Section 7.1. The steps are described below.

Step 1. The individual decision matrices , , and are aggregated into the collective decision matrix , as shown in Table 12.

Table 12

Collective decision matrix .

	C_1	C_2	C_3	C_4
A_1	{s_3(0.03),s_4(0.15),s_5(0.48),s_6(0.31),s_7(0.03)}	{s_2(0.14),s_3(0.45),s_4(0.23),s_5(0.18)}	{s_3(0.04),s_4(0.19),s_5(0.35),s_6(0.42)}	{s_3(0.13),s_4(0.28),s_5(0.54),s_6(0.05)}
A_2	{s_3(0.33),s_4(0.33),s_5(0.21),s_6(0.13)}	{s_2(0.08),s_3(0.15),s_4(0.41),s_5(0.31),s_6(0.05)}	{s_4(0.29),s_5(0.27),s_6(0.24),s_7(0.20)}	{s_2(0.05),s_3(0.18),s_4(0.57),s_5(0.15),s_6(0.05)}
A_3	{s_2(0.10),s_3(0.18),s_4(0.37),s_5(0.21),s_6(0.14)}	{s_2(0.31),s_3(0.55),s_4(0.14)}	{s_3(0.04),s_4(0.22),s_5(0.42),s_6(0.32)}	{s_1(0.05),s_2(0.10),s_3(0.38),s_4(0.44),s_5(0.03)}
A_4	{s_3(0.34),s_4(0.41),s_5(0.15),s_6(0.10)}	{s_2(0.24),s_3(0.35),s_4(0.26),s_5(0.06),s_6(0.09)}	{s_4(0.18),s_5(0.65),s_6(0.17)}	{s_2(0.21),s_3(0.61),s_4(0.18)}
A_5	{s_1(0.04),s_2(0.11),s_3(0.19),s_4(0.31),s_5(0.23),s_6(0.12)}	{s_2(0.27),s_3(0.33),s_4(0.15),s_6(0.04),s_7(0.21)}	{s_4(0.27),s_5(0.63),s_6(0.10)}	{s_1(0.09),s_2(0.37),s_3(0.28),s_4(0.21),s_5(0.05)}

Step 2. Calculate the weights of criteria by Eq. (26) in [12] as follows:

Step 3. Determine the PIS and the NIS by Definition 17, Definition 18 in [12] as follows:

Step 4. Calculate the deviation degrees between each alternative and the PIS (NIS) by Eqs. (28), (29) in [12], respectively. Then, and can be determined as follows:

Step 5. Derive the closeness coefficient of each alternative by Eq. (32) in [12].

Step 6. Rank the alternatives according to as follows:

Thus, ranking order obtain by method in [12] is slightly different from the ranking obtained by the proposed method based on Form 1 (i.e., Eq. (4)), but completely the same as the ranking obtained by the proposed method based on Form 2 (i.e., Eq. (5)). The best alternative obtained by method in [12] is the same as that obtained by the proposed method of this paper based on Forms 1 and 2. Compares with Pang et al.’s method [12], the proposed method of this paper has some merits:

(1) This paper considers the different importance among the weights of DMs and the interactions among criteria. The weights of DMs are determined objectively by constructing a tri-objective nonlinear programming model and the fuzzy measure of criteria subsets are obtained by constructing a multi-objective optimization model, which makes the decision results more reasonable. However, Pang et al. [12] considers the DMs’ weights and the criteria weights by a simple calculation.

(2) This paper uses linguistic scale function to develop the PLGHC operator, which considers the interactions among criteria. However, Pang et al. [12] used linguistic variables labels rather than linguistic scale function to aggregate probabilistic linguistic terms, which ignored the interactions among criteria and may lose the linguistic evaluation information.

Comparison with Liu et al. ’s method

Liu et al.’s method [26] is used to solve the example in Section 7.1. The steps are listed as follows:

Step 1. Calculate the dependent weights for the criterion with respect to the alternative of DMs by PLDWA operator (i.e., Eq. (11)) in [26], where , and .

The decision matrices , , and are transformed into the collective decision matrix , as shown in Table 13.

Table 13

Collective decision matrix .

	C_1	C_2	C_3	C_4
A_1	{s_4(0.12),s_5(0.60),s_6(0.28)}	{s_3(0.09),s_4(0.34),s_5(0.57)}	{s_4(0.12),s_5(0.35),s_6(0.53)}	{s_3(0.13),s_4(0.46),s_5(0.41)}
A_2	{s_3(0.11),s_4(0.43),s_5(0.46)}	{s_3(0.14),s_4(0.60),s_5(0.26)}	{s_4(0.28),s_5(0.12),s_6(0.60)}	{s_2(0.10),s_3(0.35),s_4(0.54)}
A_3	{s_3(0.11),s_4(0.43),s_5(0.46)}	{s_3(0),s_3(0.67),s_4(0.33)}	{s_4(0.11),s_5(0.47),s_6(0.42)}	{s_2(0.15),s_3(0.35),s_4(0.50)}
A_4	{s_3(0),s_3(0.50),s_4(0.50)}	{s_2(0.12),s_3(0.37),s_4(0.51)}	{s_4(0),s_4(0.21),s_5(0.79)}	{s_2(0.06),s_3(0.45),s_4(0.49)}
A_5	{s_3(0.03),s_4(0.47),s_5(0.50)}	{s_2(0.10),s_3(0.43),s_4(0.47)}	{s_4(0.04),s_5(0.60),s_6(0.36)}	{s_2(0.21),s_3(0.35),s_4(0.44)}

Step 2. Determine the ranking results using the PL-PT-MULTIMOORA method [26]. The evaluation values of the alternatives and the ranking results obtained by the PL-PT-ratio system method, PL-PT-reference point method, and PL-PT-full multiplicative method are presented in Table 14. (Set , , , , and ).

Table 14

Ranking results obtained using the PL-PT-MULTIMOORA method.

	PL-PT-ratio system method		PL-PT-reference point method		PL-PT-full multiplicative method
	TSi′	Rank	TSi′′	Rank	TSi′′′	Rank
A_1	0.5823	1	0.1251	3	3.5691	2
A_2	0.3861	3	0.1732	1	3.7655	1
A_3	0.4358	2	0.1388	2	3.1437	5
A_4	0.3859	4	0.0934	4	3.3597	3
A_5	0.2813	5	0.0932	5	3.2430	4

Step 3. Calculate the final ranking of the alternatives.

(1) The normalized results by using the PL-PT-MULTIMOORA method are obtained in the matrix:

The weights of the three methods are obtained by Eq. (24) and are obtained by Eq. (25) in [26].

(2) The ranking results by using the PL-PT-MULTIMOORA method are presented in matrix

By Eq. (26) in [26], the synthesized ranking value of alternative is obtained as

(3) By Eq. (27) in [26], the final ranking value of alternative is obtained as

Step 4. Rank the alternatives according to as follows:

The ranking result obtained by method [26] is slightly different from the ranking obtained by the proposed method based on Form 1 (i.e., Eq. (4)), but completely the same as the ranking obtained by the proposed method based on Form 2 (i.e., Eq. (5)). The best alternative obtained by method in [26] is , which is the same as that obtained by the proposed method based on Forms 1 and 2. Compared with Liu et al.’s method in [26], the proposed method of this paper has some merits:

(1) The PLDWA operator in [26] provides the lower weight to the too small or too large evaluations, but the effect of the evaluations is not obvious. This paper considers the different importance among the weights of DMs. The DMs’ weights are obtained objectively by constructing a tri-objective nonlinear programming model, which makes the decision results more reasonable.

(2) For the PLDWA operator in [26] and the PLGHC operator in the proposed method, two operators both consider the interactions among criteria. The ranking result of Archimedean copula function of this paper is similar to the ranking result of method in [26]. Besides the above, this paper also considers the interrelation among input arguments, which reflects the robustness and rationality of the proposed method of this paper.

The ranking results obtained by Pang et al.’s method [12], Liu et al.’s method [26], the proposed method with Gumbel type (PLGHC), the proposed method with Clayton type (PLCHC), the proposed method with Frank type (PLFHC) and the proposed method with Joe type (PLJHC) based on Form 1 (i.e., Eq. (4)) are visually plotted in Fig. 2.

Fig. 2

Ranking results obtained by six different methods.

It can be seen clearly from Fig. 2, is the best alternative, is the second best alternative and is the third best alternative for the above six methods. The ranking results based on common Archimedean copulas are the same, which is . Thus, the ranking results verify the rationality and robustness of the proposed method.

Conclusion

To minimize the loss of people’s lives and property and to maintain social stability in the disaster areas, it is of great importance to ensure the efficient and orderly emergency assistance after the occurrence of new coronavirus COVID-19. This paper develops a new method for interactive MCGDM with PLTSs and applies to the emergency assistance area selection of COVID-19 for Wuhan. The primary work and features of this paper are outlined as follows:

(1) In this paper, a new possibility degree of PLTSs is defined and then a new possibility degree algorithm is proposed to rank a series of PLTSs. The proposed possibility degree algorithm can take DM’s different preferences of linguistic scale functions into account, which is more robust and more in accordance with real situations.

(2) Some new operational laws of PLTSs based on the Archimedean copulas and co-copulas are defined. Archimedean copulas functions are suitable to characterize probability distributions. They have been extended to PLTSs. The proposed operational laws greatly generalize that defined by Mao et al. [24].

(3) Archimedean copulas are monotone non-decreasing, and they can be viewed as aggregation functions on one certain set. Considering the interactions among criteria, the PLGC operator and PLGHC operator are developed. Besides, the properties of these operators are studied, including idempotency, monotonicity and boundedness.

(4) To determine the weights of DMs, a tri-objective nonlinear programming model is constructed and transformed into a linear programming model for resolution. To derive the fuzzy measures of criteria subsets, an optimization model is built and transformed into a goal programming model for resolution. Th DMs’ weights and the fuzzy measures of criteria subsets are obtained objectively, which can make the decision results more reasonable and objective.

(5) Use the PLGWA operator to determine the collective normalized decision matrix. Use the PLGHC operator to derive the overall evaluation values of alternatives. The ranking order of alternatives is generated by the proposed possibility degree algorithm of PLTSs. Thereby, a new method for the interactive MCGDM with PLTSs is put forward. The proposed method considers the different importance among the weights of DMs and the interactions among criteria, which is more in accordance with the real decision making situations.

Although the emergence assistance example is provided to illustrate the validity of the proposed method, it can be employed to solve many practical decision-making problems, such as supply chain management, college evaluation, plant siting selection, etc. For future research, based on the defined new operational laws of PLTSs, some new generalized Choquet geometric operators of PLTSs will be developed for MCGDM.

CRediT authorship contribution statement

Shu-Ping Wan: Supervision, Data curation, Writing - original draft, Writing review & editing, Validation, Funding acquisition. Wen-Bo Huang Cheng: Conceptualization, Software, Writing - original draft, Writing - review & editing. Jiu-Ying Dong: Resources, Investigation, Methodology, Formal analysis, Project administration.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.