A regret theory-based GEDM method with heterogeneous probabilistic hesitant information and its application to emergency plans selection of COVID-19

Abstract

Emergency events are happening with increasing frequency, inflicting serious damage on the economic development and human life. A reliable and effective emergency decision making method is great for reducing various potential losses. Hence, group emergency decision making (GEDM) has drawn great attention in past few years because of its advantages dealing with the emergencies. Due to the timeliness and complexity of GEDM, vagueness and regret aversion are common among decision makers (DMs), and decision information usually needs to be expressed by various mathematical forms. To this end, this paper proposes a novel GEDM method based on heterogeneous probabilistic hesitant information sets (PHISs) and regret theory (RT). Firstly, the PHISs with real numbers, interval numbers and linguistic terms are developed to depict the situation that decision group sways precariously between several projects and best retain the original assessment. In addition, the score functions, the divergence functions and some operations of the three types of PHISs are defined. Secondly, the normalization model of PHISs is presented to remove the influence of different dimensions on information aggregation. Thirdly, group satisfaction degree (GSD) based on the score functions and the divergence functions is combined with RT for completely portraying the regret perception of decision group. Then, we introduce Dempster-Shafer (DS) theory to determine the probabilities of future possible states for emergency events. Finally, an example of coronavirus disease 2019 (COVID-19) situation is given as an application for the proposed GEDM method, whose superiority, stability and validity are demonstrated by employing the comparative analysis and sensitivity analysis.

Keywords

Group emergency decision making heterogeneous decision information probabilistic hesitant information sets regret theory group satisfaction degree

1 Introduction

The frequent occurrences of emergency events around the world, such as severe acute respiratory syndrome in 2003, Wenchuan earthquake in 2008, Japan’s nuclear leakage event caused by earthquake in 2011 and coronavirus disease 2019 (COVID-19), causing serious damages on economic development and human health, have drawn increasing attention in recent years. Therefore, governments and scholars put emergency response on the agenda. The emergency decision making (EDM) has a crucial position in emergency response, and an appropriate EDM method plays an important role in reducing or avoiding various losses and potentially harmful risks [1]. Given the complicacy of emergency event, it is difficult for a single expert to get a view of the whole emergency event on account of the limitation of individual in some aspects, such as knowledge and experience. Hence, group emergency decision making (GEDM) involving multiple decision makers (DMs) is good for obtaining better solution.

Owing to the inherent fuzziness and uncertainty of emergency events, both qualitative criteria and quantitative criteria usually need to be considered in most practical GEDM problems. However, it is often hard to express the performance of multiple conflicting criteria in a uniform mathematical form. More specifically, numeric cases, such as crisp numbers, interval numbers and triangular fuzzy numbers, are generally used to portray the quantitative criteria, while linguistic cases are better forms for representing qualitative criteria values that are too complex to be determined by utilizing numeric cases. To sum up, the criteria values may be represented by various forms (e.g., real numbers, interval numbers, and linguistic terms) in the process of GEDM, which is also in line with the features of heterogeneous MCGDM (HMCGDM) problems. The HMCGDM methods have been successfully applied to the fields of sustainable project selection [2], supplier selection [3], group decision making [4], etc. Therefore, the study on applying HMCGDM methods to GEDM problems is of great importance.

It is universally known that the uncertainty and complexity of things are inevitable in the real world, the estimation errors and inaccuracies exist in the GEDM environment due to the ambiguous thoughts of DMs. Specifically, decision group or individuals may sway among several possible criteria values, and show different degrees of preference for these possible values when evaluating emergency alternatives. Probabilistic hesitant fuzzy sets (PHFSs) [5], probabilistic linguistic term sets (PLTSs) [6], and their extensions [7, 8] that include not only several possible membership degrees or linguistic terms but also the corresponding probabilistic information are developed to depict the irresolution and probabilistic preference of human. Though these theories have been well introduced to GEDM problems, some scholars found it difficult to determine the membership function of a fuzzy set, which is hard to describe the real performance of criteria [9]. Additionally, better decision results can be obtained by considering heterogeneous evaluation information. Consequently, we present the concept of probabilistic hesitant information sets (PHISs) by borrowing PHFSs and PLTSs, in which the membership degrees are replaced by the original criteria values. The complete feature of PHISs is modeled as {s₁ (0.2) , s₂ (0.5) , s₃ (0.3)}, where s₁, s₂, and s₃ represent the original evaluation information in heterogeneous forms, such as crisp numbers, interval numbers, linguistic terms and so on. The figure “0.2”, “0.5” and “0.3” are the probabilities of s₁, s₂, and s₃ respectively.

Example 1. When an evaluation criterion of an alternative is “cost", real numbers are usually used to represent the performance of this criterion, an expert may express that he/she is 30% sure that the criterion value is 1000, and 70% sure that the criterion value is 1500. Then the corresponding evaluation value can be represented by PHFS as {1000(0.3), 1500(0.7)}.

In addition, PHFSs can also express the hesitation and preference of decision group.

Example 2. A hundred experts are invited to evaluate the service quality of a third-party reverse logistics provider. Suppose that thirty experts consider that it is bad, fifty experts deem it good, and twenty experts think that it is excellent. In this case, the evaluation information can be denoted as {bad(0.3), good(0.5), excellent(0.2)}.

PHISs describe decision information more intuitively and accurately, and avoid the subjectivity when the experts determine the membership function in the decision process. Moreover, for a criterion of an alternative in group decision making, the preference of decision group is clearer when evaluation values given by experts are aggregated into a PHIS rather than that is displayed separately, and the aggregation of evaluation information is simpler due to the preliminary integration by PHISs, especially when many members participate in the assessment. In addition, the process of GEDM have multiple-format criteria values (such as real numbers, interval numbers, linguistic terms, etc.), so PHISs are very suitable to solve GEDM problems considering ambiguous and real human judgments.

Given the dynamic development of emergency events, it is difficult for DMs to make completely accurate decision in GEDM, and over time, DMs may find that the abandoned alternative is better than the being implemented one, which is a catalyst for regret emotion of DMs. In short, DMs hate regret emotion. Thus, DMs will select the alternative that minimizes their regret when they conduct GEDM. Emergency events are characterized by unpredictability and strong destructive power, which asks DMs to find an appropriate emergency solution in a short time. However, research shows that regret perceptions of DMs prolong the time of decision making and have an adverse effect on the decision-making quality [10]. In contrast, DMs will rejoice when the chosen alternative has better performance than the abandoned one. Such possible feelings of regret or rejoice actually affect DMs’ judgement. Hence, DMs usually want to avoid this regret emotion in the current decision-making process by choosing the alternative with the least regret. Regret theory (RT) [11] is an effective tool to portray such emotions. At present, some MCGDM methods based on RT have been proposed. Zhang et al. [12] developed a novel MCGDM approach considering regret aversion of DMs and incomplete weight information. Liu and Li [13] proposed an improved failure mode and effect analysis method based on group decision making and RT to conduct the risk assessment of cold chain green logistics. Zheng et al. [14] proposed a dynamic approach using case retrieval and MCGDM to deal with the problem of emergency alternative generation. Liu et al. [15] constructed a new RT-based group decision-making framework to obtain appropriate cloud service provider. Bai and Sarkis [16] established a hybrid group decision model to evaluate blockchain technology integrating RT and hesitant fuzzy set. The abovementioned literature provides important enlightenment for the application of RT in GEDM.

During the process of GEDM, the future possible states of emergency event should be predicted to support the evaluation of alternatives because the anticipated performance of alternatives may be different under different situations of emergency event. Dempster-Shafer (DS) theory [17] is an appropriate tool to portray the uncertainty of a project and a good method to determine the future possible states and corresponding probabilities.

To sum up, we propose PHISs to describe the irresolution and preference of decision group or individual considering the heterogeneous criteria values expressed by real numbers, interval numbers, and linguistic terms. RT is integrated into PHISs for measuring the regret and rejoice emotion of DMs. DS theory is used to acquire the probability distribution of future possible states of the emergency event. On this foundation, a new RT-based HMCGDM method is developed to evaluate and select emergency solutions. The rest of this article is arranged as follows. Section 2 introduces the basic knowledge about PHISs and RT. In Section 3, a new GEDM method is given. Section 4 illustrates the effectiveness of the proposed method through an example of COVID-19, comparative analysis and sensitivity analysis are used to prove the superiority of this method. Finally, Section 5 provides the conclusions of the proposed approach.

2 Preliminaries

In this section, we resolve the essential knowledge about PHISs and RT.

2.1 Probabilistic hesitant information sets

Definition 1. Let φ ={ φ₁, φ₂, ⋯ , φ_v } be a given set, a PHIS A on φ is defined as

$A = {A^{(q)} (p^{(q)}) | A^{(q)} \in φ, 0 ⩽ p^{(q)} ⩽ 1, q = 1, 2, \dots, Z_{A}, \sum_{q = 1}^{Z_{A}} p^{(q)} = 1},$ (1) where A^(q) is the basic element of the PHIS A, abbreviated as probabilistic hesitant element (PHE). A^(q) (p^(q)) denotes the qth PHE A^(q) associated with the probability p^(q), and Z_A is the number of all different PHEs in A.

Due to the complexity and uncertainty of the GEDM environment, it is difficult for DMs to accurately represent the decision information by a single format of criteria value. In general, multiple-format values are simultaneously used to describe the performance of all criteria, such as real numbers, interval numbers, and linguistic terms. This paper expresses criteria values in different mathematical forms: real numbers, interval numbers and linguistic terms. The three types of PHISs are denoted as follows:

Definition 2. If PHE is a real number, let $A = \dot{A}$ , $A^{(q)} = {\dot{A}}^{(q)}$ , and $p^{(q)} = {\dot{p}}^{(q)}$ . φ is a given set, $\dot{A} = {{\dot{A}}^{(q)} ({\dot{p}}^{(q)}) | {\dot{A}}^{(q)} \in φ}$ , where ${\dot{A}}^{(q)}$ is a real number, $0 ⩽ {\dot{p}}^{(q)} ⩽ 1$ , $q = 1, 2, \dots, Z_{\dot{A}}$ , and $\sum_{q = 1}^{Z_{\dot{A}}} {\dot{p}}^{(q)} = 1$ . We can generally assume that ${\dot{A}}^{(q)} ⩾ 0$ .

For a PHIS $\dot{A}$ , the score function is defined as $S (\dot{A}) = \sum_{q = 1}^{Z_{\dot{A}}} {\dot{A}}^{(q)} {\dot{p}}^{(q)},$ (2) and the divergence function is $E (\dot{A}) = \sqrt{\sum_{q = 1}^{Z_{\dot{A}}} {\dot{p}}^{(q)} ({\dot{A}}^{(q)} - S (\dot{A}))^{2}} .$ (3)

Definition 3. If PHE is an interval number, let A = $\overset{⌢}{A}$ , A^(q) = $\overset{⌢}{A}$ ^(q), and p^(q) = $\overset{⌢}{p}$ ^(q). φ is a given set, $\overset{⌢}{A}$ ={ $\overset{⌢}{A}$ ^(q) ( $\overset{⌢}{p}$ ^(q)) | $\overset{⌢}{A}$ ^(q) ∈ φ }, where $\overset{⌢}{A}$ ^(q) = [A^L(q), A^U(q)], 0 ⩽ $\overset{⌢}{p}$ ^(q) ⩽ 1, q = 1, 2, ⋯ , Z_{$\overset{⌢}{A}$}, and $\sum_{q = 1}^{Z_{\overset{⌢}{A}}} {\overset{⌢}{p}}^{(q)} = 1$ . We can generally suppose that A^U(q) ⩾ A^L(q) ⩾ 0.

The interval number comes from the calculation of random sampling, so it can be considered that [A^L(q), A^U(q)] is relatively immobile, but the actual evaluation value x is randomly selected in the interval number [A^L(q), A^U(q)] and obeys a certain distribution.

Suppose that the probability density function of the interval value [A^L(q), A^U(q)] in the PHIS $\overset{⌢}{A}$ is f (x), then the score function S ( $\overset{⌢}{A}$ ) is expressed by: $S (\overset{⌢}{A}) = \sum_{q = 1}^{Z_{\overset{⌢}{A}}} {\overset{⌢}{p}}^{(q)} \int_{Lq}^{U (q)} xf (x) dx,$ (4)

and the divergence function E ( $\overset{⌢}{A}$ ) is $E (\overset{⌢}{A}) = \sqrt{\sum_{q = 1}^{Z_{\overset{⌢}{A}}} {\overset{⌢}{p}}^{(q)} \int_{L (q)}^{U (q)} (x - S (\overset{⌢}{A}))^{2} f (x) dx} .$ (5)

In real life, uniform distribution and normal distribution are the most common distribution types, so we only consider these two distributions [18]. When x obeys uniform distribution, f (x) is defined as $f (x) = {\begin{matrix} \frac{1}{A^{U (q)} - A^{L (q)}} A^{L (q)} ⩽ x ⩽ A^{U (q)} \\ 0, otherwise \end{matrix}$ (6)

If x conforms to normal distribution φ (μ ( $\overset{⌢}{A}$ ^(q)) , σ ( $\overset{⌢}{A}$ ^(q)) ²), x belongs to [A^L(q), A^U(q)] with a 99.73% probability according to the principle of 3σ in probability statistics, where μ ( $\overset{⌢}{A}$ ^(q)) = (A^L(q) + A^U(q))/ - 2 and σ ( $\overset{⌢}{A}$ ^(q)) = (A^U(q) - A^L(q))/ - 6. f (x) is defined as $f (x) = \frac{1}{\sqrt{2 π σ ({\overset{⌢}{A}}^{(q)})}} exp [\frac{- (x - μ ({\overset{⌢}{A}}^{(q)}))^{2}}{2 (σ ({\overset{⌢}{A}}^{(q)}))^{2}}] .$ (7)

Definition 4 [6, 19]. Let G ={ g_r|r = 0, 1, ⋯ , T } be a linguistic term set (LTS), where g_r denotes the n + 1th linguistic term of G, and T is an even number.

g_r should satisfy some operational laws as follows:

(1) Negation operator: If τ = T - r, neg (g_τ) = g_r.

(2) The set is ordered: If r > τ, then g_r is superior to g_τ, i.e., g_r ≻ g_τ.

(3) Max operator and min operator: If g_r ≻ g_τ, then max { g_τ, g_r } = g_r and min { g_τ, g_r } = g_τ.

If PHE is a linguistic term, then $A = \tilde{A}$ , $A^{(q)} = {\tilde{A}}^{(q)}$ , and $p^{(q)} = {\tilde{p}}^{(q)}$ . φ is a given set, $\tilde{A} = {{\tilde{A}}^{(q)} ({\tilde{p}}^{(q)}) | {\tilde{A}}^{(q)} \in φ}$ , where ${\tilde{A}}^{(q)} = g_{r}^{(q)}$ and $g_{r}^{(q)} \in G$ . ${\tilde{p}}^{(q)}$ satisfies $0 ⩽ {\tilde{p}}^{(q)} ⩽ 1$ and $\sum_{q = 1}^{Z_{\tilde{A}}} {\tilde{p}}^{(q)} = 1$ , for all $q = 1, 2, \dots, Z_{\tilde{A}}$ .

For a PHIS $\tilde{A}$ , the score function is defined as $S (\tilde{A}) = \sum_{q = 1}^{Z_{\tilde{A}}} r^{(q)} {\tilde{p}}^{(q)},$ (8) and the divergence function is defined as $E (\tilde{A}) = \sqrt{\sum_{q = 1}^{Z_{\tilde{A}}} {\tilde{p}}^{(q)} (r^{(q)} - S (\tilde{A}))^{2}},$ (9) where r^(q) is the subscript of linguistic term $g_{r}^{(q)}$ .

Definition 5. Given two PHISs A₁ and A₂, then a method can be proposed to compare A₁ and A₂ as

(1) If S (A₁) > S (A₂), then A₁ > A₂;

(2) If S (A₁) < S (A₂), then A₁ < A₂;

(3) If S (A₁) = S (A₂), then

a) If E (A₁) < E (A₂), then A₁ > A₂;

b) If E (A₁) > E (A₂), then A₁ < A₂;

c) If E (A₁) = E (A₂), then A₁ ∼ A₂.

2.2 Some operations and aggregation operators of PHISs

According to the elaborations above, PHISs are more convenient and intuitive. As we all know, the aggregation operators are strong tools for the integration of decision information. Some operations and aggregation operators about three-format PHISs are given as follows.

2.2.1 Real-number PHISs

Definition 6. Let $\dot{A}$ , ${\dot{A}}_{1}$ and ${\dot{A}}_{2}$ be three PHISs with real-number PHEs, ω is a real number, then

(1) $ω \dot{A} = \underset{{\dot{A}}^{(q)} \in \dot{A}}{\cup} {ω {\dot{A}}^{(q)} ({\dot{p}}^{(q)})}$ ;

(2) ${\dot{A}}^{ω} = \underset{{\dot{A}}^{(q)} \in \dot{A}}{\cup} {({\dot{A}}^{(q)})^{ω} ({\dot{p}}^{(q)})}$ ;

(3) ${\dot{A}}_{1} \oplus {\dot{A}}_{2} = \underset{{\dot{A}}_{1}^{(q)} \in {\dot{A}}_{1}, {\dot{A}}_{2}^{(e)} \in {\dot{A}}_{2}}{\cup} {({\dot{A}}_{1}^{(q)} + {\dot{A}}_{2}^{(e)}) ({\dot{p}}_{1}^{(q)} {\dot{p}}_{2}^{(e)})}$ ;

(4) ${\dot{A}}_{1} \otimes {\dot{A}}_{2} = \underset{{\dot{A}}_{1}^{(q)} \in {\dot{A}}_{1}, {\dot{A}}_{2}^{(e)} \in {\dot{A}}_{2}}{\cup} {({\dot{A}}_{1}^{(q)} {\dot{A}}_{2}^{(e)}) ({\dot{p}}_{1}^{(q)} {\dot{p}}_{2}^{(e)})}$ .

Theorem 1. Given three real-number PHISs ${\dot{A}}_{1}$ , ${\dot{A}}_{2}$ and ${\dot{A}}_{3}$ , ω is a real number, then

(1) ${\dot{A}}_{1} \oplus {\dot{A}}_{2} = {\dot{A}}_{2} \oplus {\dot{A}}_{1}$ ;

(2) $({\dot{A}}_{1} \oplus {\dot{A}}_{2}) \oplus {\dot{A}}_{3} = {\dot{A}}_{1} \oplus ({\dot{A}}_{2} \oplus {\dot{A}}_{3})$ ;

(3) ${\dot{A}}_{1} \otimes {\dot{A}}_{2} = {\dot{A}}_{2} \otimes {\dot{A}}_{1}$ ;

(4) $({\dot{A}}_{1} \otimes {\dot{A}}_{2}) \otimes {\dot{A}}_{3} = {\dot{A}}_{1} \otimes ({\dot{A}}_{2} \otimes {\dot{A}}_{3})$ .

Proof. $\begin{matrix} (1) {\dot{A}}_{1} \oplus {\dot{A}}_{2} \\ = \underset{{\dot{A}}_{1}^{(q)} \in {\dot{A}}_{1}, {\dot{A}}_{2}^{(e)} \in {\dot{A}}_{2}}{\cup} {({\dot{A}}_{1}^{(q)} + {\dot{A}}_{2}^{(e)}) ({\dot{p}}_{1}^{(q)} {\dot{p}}_{2}^{(e)})} \\ = \underset{{\dot{A}}_{1}^{(q)} \in {\dot{A}}_{1}, {\dot{A}}_{2}^{(e)} \in {\dot{A}}_{2}}{\cup} {({\dot{A}}_{2}^{(e)} + {\dot{A}}_{1}^{(q)}) ({\dot{p}}_{2}^{(e)} {\dot{p}}_{1}^{(q)})} \\ = {\dot{A}}_{2} \oplus {\dot{A}}_{1} . \end{matrix}$ $\begin{matrix} (2) ({\dot{A}}_{1} \oplus {\dot{A}}_{2}) \oplus {\dot{A}}_{3} \\ = \underset{{\dot{A}}_{1}^{(q)} \in {\dot{A}}_{1}, {\dot{A}}_{2}^{(e)} \in {\dot{A}}_{2}}{\cup} {({\dot{A}}_{1}^{(q)} + {\dot{A}}_{2}^{(e)}) ({\dot{p}}_{1}^{(q)} {\dot{p}}_{2}^{(e)})} \oplus {\dot{A}}_{3} \\ = \underset{{\dot{A}}_{1}^{(q)} \in {\dot{A}}_{1}, {\dot{A}}_{2}^{(e)} \in {\dot{A}}_{2}, {\dot{A}}_{3}^{(c)} \in {\dot{A}}_{3}}{\cup} {({\dot{A}}_{1}^{(q)} + {\dot{A}}_{2}^{(e)} + {\dot{A}}_{3}^{(c)}) ({\dot{p}}_{1}^{(q)} {\dot{p}}_{2}^{(e)} {\dot{p}}_{3}^{(c)})} \\ = {\dot{A}}_{1} \oplus \underset{{\dot{A}}_{2}^{(e)} \in {\dot{A}}_{2}, {\dot{A}}_{3}^{(c)} \in {\dot{A}}_{3}}{\cup} {({\dot{A}}_{2}^{(e)} + {\dot{A}}_{3}^{(c)}) ({\dot{p}}_{2}^{(e)} {\dot{p}}_{3}^{(c)})} \\ = {\dot{A}}_{1} \oplus ({\dot{A}}_{2} \oplus {\dot{A}}_{3}) . \end{matrix}$ $\begin{matrix} (3) {\dot{A}}_{1} \otimes {\dot{A}}_{2} \\ = \underset{{\dot{A}}_{1}^{(q)} \in {\dot{A}}_{1}, {\dot{A}}_{2}^{(e)} \in {\dot{A}}_{2}}{\cup} {({\dot{A}}_{1}^{(q)} {\dot{A}}_{2}^{(e)}) ({\dot{p}}_{1}^{(q)} {\dot{p}}_{2}^{(e)})} \\ = \underset{{\dot{A}}_{1}^{(q)} \in {\dot{A}}_{1}, {\dot{A}}_{2}^{(e)} \in {\dot{A}}_{2}}{\cup} {({\dot{A}}_{2}^{(e)} {\dot{A}}_{1}^{(q)}) ({\dot{p}}_{2}^{(e)} {\dot{p}}_{1}^{(q)})} \\ = {\dot{A}}_{2} \otimes {\dot{A}}_{1} . \end{matrix}$ $\begin{matrix} (4) ({\dot{A}}_{1} \otimes {\dot{A}}_{2}) \otimes {\dot{A}}_{3} \\ = \underset{{\dot{A}}_{1}^{(q)} \in {\dot{A}}_{1}, {\dot{A}}_{2}^{(e)} \in {\dot{A}}_{2}}{\cup} {({\dot{A}}_{1}^{(q)} {\dot{A}}_{2}^{(e)}) ({\dot{p}}_{1}^{(q)} {\dot{p}}_{2}^{(e)})} \otimes {\dot{A}}_{3} \\ = {\dot{A}}_{1} \otimes \underset{{\dot{A}}_{2}^{(e)} \in {\dot{A}}_{2}, {\dot{A}}_{3}^{(c)} \in {\dot{A}}_{3}}{\cup} {({\dot{A}}_{2}^{(e)} {\dot{A}}_{3}^{(c)}) ({\dot{p}}_{2}^{(e)} {\dot{p}}_{3}^{(c)})} \\ = {\dot{A}}_{1} \otimes ({\dot{A}}_{2} \otimes {\dot{A}}_{3}) . \end{matrix}$

Based on the above operations, the real-number probabilistic hesitant information weighted averaging (RNPHIWA) operator can be given, which is good for the decision making with probabilistic hesitant information.

Definition 7. Let ${\dot{A}}_{i} = {{\dot{A}}_{i}^{(q)} ({\dot{p}}_{i}^{(q)}) | q = 1, 2, \dots,$ Z_A } (i = 1, 2, ⋯ , m) be m real-number PHISs, and w = (w₁, w₂, ⋯ , w_m) be the weight vector, w_i ∈ [0, 1] , i = 1, 2, ⋯ , m, and $\sum_{i = 1}^{m} w_{i} = 1$ . Then, RNPHIWA can be expressed by

$\begin{matrix} RNPHIWA ({\dot{A}}_{1}, {\dot{A}}_{2}, \dots, {\dot{A}}_{m}) = \oplus_{i = 1}^{m} w_{i} {\dot{A}}_{i} = \\ \underset{{\dot{A}}_{1}^{(q)} \in {\dot{A}}_{1}, {\dot{A}}_{2}^{(q)} \in {\dot{A}}_{2}, \dots, {\dot{A}}_{m}^{(q)} \in {\dot{A}}_{m}}{\cup} {(\sum_{i = 1}^{m} w_{i} {\dot{A}}_{i}^{(q)}) (\prod_{i = 1}^{m} {\dot{p}}_{i}^{(q)})} . \end{matrix}$ (10)

Particularly, if w = (1/ - m, 1/ - m, ⋯ , 1/ - m), then the RNPHIWA operator reduces to the real-number probabilistic hesitant information averaging (RNPHIA) operator.

2.2.2 Interval -number PHISs

Definition 8. Let $\overset{⌢}{A}$ , $\overset{⌢}{A}$ ₁ and $\overset{⌢}{A}$ ₂ be three PHISs with interval-number PHEs, ω is a real number, then $(1) ω \overset{⌢}{A} = {\begin{matrix} \underset{[A^{L (q)}, A^{U (q)}] \in \overset{⌢}{A}}{\cup} {[ω A^{L (q)}, ω A^{U (q)}] ({\overset{⌢}{p}}^{(q)})}, ω > 0 \\ \underset{[A^{L (q)}, A^{U (q)}] \in \overset{⌢}{A}}{\cup} {[ω A^{U (q)}, ω A^{L (q)}] ({\overset{⌢}{p}}^{(q)})}, ω < 0 \end{matrix};$ $(2) {\overset{⌢}{A}}^{ω} = \underset{[A^{L (q)}, A^{U (q)}] \in \overset{⌢}{A}}{\cup} {[(A^{L (q)})^{ω}, (A^{U (q)})^{ω}] ({\overset{⌢}{p}}^{(q)})};$ $(3) {\overset{⌢}{A}}_{1} \oplus {\overset{⌢}{A}}_{2} = \underset{[A_{1}^{L (q)}, A_{1}^{U (q)}] \in {\overset{⌢}{A}}_{1}, [A_{2}^{L (e)}, A_{2}^{U (e)}] \in {\overset{⌢}{A}}_{2}}{\cup} {[A_{1}^{L (q)} + A_{2}^{L (e)}, A_{1}^{U (q)} + A_{2}^{U (e)}] ({\overset{⌢}{p}}_{1}^{(q)} {\overset{⌢}{p}}_{2}^{(e)})};$ $\begin{matrix} (4) {\overset{⌢}{A}}_{1} \otimes {\overset{⌢}{A}}_{2} = \underset{[A_{1}^{L (q)}, A_{1}^{U (q)}] \in {\overset{⌢}{A}}_{1}, [A_{2}^{L (e)}, A_{2}^{U (e)}] \in {\overset{⌢}{A}}_{2}}{\cup} {[A_{1}^{L (q)} A_{2}^{L (e)} \land A_{1}^{L (q)} A_{2}^{U (e)} \land A_{1}^{U (q)} A_{2}^{L (e)} \land A_{1}^{U (q)} A_{2}^{U (e)}, \\ A_{1}^{L (q)} A_{2}^{L (e)} \lor A_{1}^{L (q)} A_{2}^{U (e)} \lor A_{1}^{U (q)} A_{2}^{L (e)} \lor A_{1}^{U (q)} A_{2}^{U (e)}] ({\overset{⌢}{p}}_{1}^{(q)} {\overset{⌢}{p}}_{2}^{(e)})} . \end{matrix}$

Theorem 2. Let $\overset{⌢}{A}$ ₁, $\overset{⌢}{A}$ ₂, and $\overset{⌢}{A}$ ₃ be three PHISs, ω is a real number, then $\begin{matrix} (1) {\overset{⌢}{A}}_{1} \oplus {\overset{⌢}{A}}_{2} = {\overset{⌢}{A}}_{2} \oplus {\overset{⌢}{A}}_{1}; \\ (2) ({\overset{⌢}{A}}_{1} \oplus {\overset{⌢}{A}}_{2}) \oplus {\overset{⌢}{A}}_{3} = {\overset{⌢}{A}}_{1} \oplus ({\overset{⌢}{A}}_{2} \oplus {\overset{⌢}{A}}_{3}); \\ (3) {\overset{⌢}{A}}_{1} \otimes {\overset{⌢}{A}}_{2} = {\overset{⌢}{A}}_{2} \otimes {\overset{⌢}{A}}_{1}; \\ (4) ({\overset{⌢}{A}}_{1} \otimes {\overset{⌢}{A}}_{2}) \otimes {\overset{⌢}{A}}_{3} = {\overset{⌢}{A}}_{1} \otimes ({\overset{⌢}{A}}_{2} \otimes {\overset{⌢}{A}}_{3}) . \end{matrix}$

Proof. $\begin{matrix} (1) {\overset{⌢}{A}}_{1} \oplus {\overset{⌢}{A}}_{2} \\ = \underset{[A_{1}^{L (q)}, A_{1}^{U (q)}] \in {\overset{⌢}{A}}_{1}, [A_{2}^{L (e)}, A_{2}^{U (e)}] \in {\overset{⌢}{A}}_{2}}{\cup} {[A_{1}^{L (q)} + A_{2}^{L (e)}, A_{1}^{U (q)} + A_{2}^{U (e)}] ({\overset{⌢}{p}}_{1}^{(q)} {\overset{⌢}{p}}_{2}^{(e)})} \\ = \underset{[A_{1}^{L (q)}, A_{1}^{U (q)}] \in {\overset{⌢}{A}}_{1}, [A_{2}^{L (e)}, A_{2}^{U (e)}] \in {\overset{⌢}{A}}_{2}}{\cup} {[A_{2}^{L (e)} + A_{1}^{L (q)}, A_{2}^{U (e)} + A_{1}^{U (q)}] ({\overset{⌢}{p}}_{2}^{(e)} {\overset{⌢}{p}}_{1}^{(q)})} \\ = {\overset{⌢}{A}}_{2} \oplus {\overset{⌢}{A}}_{1} . \end{matrix}$ $\begin{matrix} (2) ({\overset{⌢}{A}}_{1} \oplus {\overset{⌢}{A}}_{2}) \oplus {\overset{⌢}{A}}_{3} \\ = \underset{[A_{1}^{L (q)}, A_{1}^{U (q)}] \in {\overset{⌢}{A}}_{1}, [A_{2}^{L (e)}, A_{2}^{U (e)}] \in {\overset{⌢}{A}}_{2}}{\cup} {[A_{1}^{L (q)} + A_{2}^{L (e)}, A_{1}^{U (q)} + A_{2}^{U (e)}] ({\overset{⌢}{p}}_{1}^{(q)} {\overset{⌢}{p}}_{2}^{(e)})} \oplus {\overset{⌢}{A}}_{3} \\ = \underset{[A_{1}^{L (q)}, A_{1}^{U (q)}] \in {\overset{⌢}{A}}_{1}, [A_{2}^{L (e)}, A_{2}^{U (e)}] \in {\overset{⌢}{A}}_{2}, [A_{2}^{L (c)}, A_{2}^{U (c)}] \in {\overset{⌢}{A}}_{3}}{\cup} {[A_{1}^{L (q)} + A_{2}^{L (e)} + A_{3}^{L (c)}, A_{1}^{U (q)} + A_{2}^{U (e)} + A_{3}^{U (c)}] ({\overset{⌢}{p}}_{1}^{(q)} {\overset{⌢}{p}}_{2}^{(e)} {\overset{⌢}{p}}_{3}^{(c)})} \\ = {\overset{⌢}{A}}_{1} \oplus \underset{[A_{2}^{L (e)}, A_{2}^{U (e)}] \in {\overset{⌢}{A}}_{2}, [A_{3}^{L (c)}, A_{3}^{U (c)}] \in {\overset{⌢}{A}}_{3}}{\cup} {[A_{2}^{L (e)} + A_{3}^{L (c)}, A_{2}^{U (e)} + A_{3}^{U (c)}] ({\overset{⌢}{p}}_{2}^{(e)} {\overset{⌢}{p}}_{3}^{(c)})} \\ = {\overset{⌢}{A}}_{1} \oplus ({\overset{⌢}{A}}_{2} \oplus {\overset{⌢}{A}}_{3}) . \end{matrix}$ $\begin{matrix} (3) {\overset{⌢}{A}}_{1} \otimes {\overset{⌢}{A}}_{2} \\ = \underset{[A_{1}^{L (q)}, A_{1}^{U (q)}] \in {\overset{⌢}{A}}_{1}, [A_{2}^{L (e)}, A_{2}^{U (e)}] \in {\overset{⌢}{A}}_{2}}{\cup} {[A_{1}^{L (q)} A_{2}^{L (e)} \land A_{1}^{L (q)} A_{2}^{U (e)} \land A_{1}^{U (q)} A_{2}^{L (e)} \land A_{1}^{U (q)} A_{2}^{U (e)}, \\ A_{1}^{L (q)} A_{2}^{L (e)} \lor A_{1}^{L (q)} A_{2}^{U (e)} \lor A_{1}^{U (q)} A_{2}^{L (e)} \lor A_{1}^{U (q)} A_{2}^{U (e)}] ({\overset{⌢}{p}}_{1}^{(q)} {\overset{⌢}{p}}_{2}^{(e)})} \\ = \underset{[A_{1}^{L (q)}, A_{1}^{U (q)}] \in {\overset{⌢}{A}}_{1}, [A_{2}^{L (e)}, A_{2}^{U (e)}] \in {\overset{⌢}{A}}_{2}}{\cup} {[A_{2}^{L (e)} A_{1}^{L (q)} \land A_{2}^{L (e)} A_{1}^{U (q)} \land A_{2}^{U (e)} A_{1}^{L (q)} \land A_{2}^{U (e)} A_{1}^{U (q)}, \\ A_{2}^{L (e)} A_{1}^{L (q)} \lor A_{2}^{L (e)} A_{1}^{U (q)} \lor A_{2}^{U (e)} A_{1}^{L (q)} \lor A_{2}^{U (e)} A_{1}^{U (q)}] ({\overset{⌢}{p}}_{2}^{(e)} {\overset{⌢}{p}}_{1}^{(q)})} \\ = {\overset{⌢}{A}}_{2} \otimes {\overset{⌢}{A}}_{1} . \end{matrix}$ $\begin{matrix} (4) ({\overset{⌢}{A}}_{1} \otimes {\overset{⌢}{A}}_{2}) \otimes {\overset{⌢}{A}}_{3} \\ = \underset{[A_{1}^{L (q)}, A_{1}^{U (q)}] \in {\overset{⌢}{A}}_{1}, [A_{2}^{L (e)}, A_{2}^{U (e)}] \in {\overset{⌢}{A}}_{2}}{\cup} {[A_{1}^{L (q)} A_{2}^{L (e)} \land A_{1}^{L (q)} A_{2}^{U (e)} \land A_{1}^{U (q)} A_{2}^{L (e)} \land A_{1}^{U (q)} A_{2}^{U (e)}, \\ A_{1}^{L (q)} A_{2}^{L (e)} \lor A_{1}^{L (q)} A_{2}^{U (e)} \lor A_{1}^{U (q)} A_{2}^{L (e)} \lor A_{1}^{U (q)} A_{2}^{U (e)}] ({\overset{⌢}{p}}_{1}^{(q)} {\overset{⌢}{p}}_{2}^{(e)})} \otimes {\overset{⌢}{A}}_{3} \\ = \underset{[A_{1}^{L (q)}, A_{1}^{U (q)}] \in {\overset{⌢}{A}}_{1}, [A_{2}^{L (e)}, A_{2}^{U (e)}] \in {\overset{⌢}{A}}_{2}, [A_{3}^{L (c)}, A_{3}^{U (c)}] \in {\overset{⌢}{A}}_{3}}{\cup} {[A_{1}^{L (q)} A_{2}^{L (e)} A_{3}^{L (c)} \land A_{1}^{L (q)} A_{2}^{U (e)} A_{3}^{U (c)} \land A_{1}^{U (q)} A_{2}^{L (e)} A_{3}^{L (c)} \land \\ A_{1}^{U (q)} A_{2}^{U (e)} A_{3}^{U (c)}, A_{1}^{L (q)} A_{2}^{L (e)} A_{3}^{L (c)} \lor A_{1}^{L (q)} A_{2}^{U (e)} A_{3}^{U (c)} \lor \\ A_{1}^{U (q)} A_{2}^{L (e)} A_{3}^{L (c)} \lor A_{1}^{U (q)} A_{2}^{U (e)} A_{3}^{U (c)}] ({\overset{⌢}{p}}_{1}^{(q)} {\overset{⌢}{p}}_{2}^{(e)} {\overset{⌢}{p}}_{3}^{(c)})} \\ = {\overset{⌢}{A}}_{1} \otimes \underset{[A_{2}^{L (e)}, A_{2}^{U (e)}] \in {\overset{⌢}{A}}_{2}, [A_{3}^{L (c)}, A_{3}^{U (c)}] \in {\overset{⌢}{A}}_{3}}{\cup} {[A_{2}^{L (e)} A_{3}^{L (c)} \land A_{2}^{L (e)} A_{3}^{U (c)} \land A_{2}^{U (e)} A_{3}^{L (c)} \land A_{2}^{U (e)} A_{3}^{U (c)}, \\ A_{2}^{L (e)} A_{3}^{L (c)} \lor A_{2}^{L (e)} A_{3}^{U (c)} \lor A_{2}^{U (e)} A_{3}^{L (c)} \lor A_{2}^{U (e)} A_{3}^{U (c)}] ({\overset{⌢}{p}}_{2}^{(e)} {\overset{⌢}{p}}_{3}^{(c)})} \\ = {\overset{⌢}{A}}_{1} \otimes ({\overset{⌢}{A}}_{2} \otimes {\overset{⌢}{A}}_{3}) . \end{matrix}$

Definition 9. Let ${\overset{⌢}{A}}_{i} = {{\overset{⌢}{A}}_{i}^{(q)} ({\overset{⌢}{p}}_{i}^{(q)}) | q = 1, 2, \dots,$ Z_A } (i = 1, 2, ⋯ , m) be m interval-number PHISs, then the interval-number probabilistic hesitant information weighted averaging (INPHIWA) operator can be obtained as: $\begin{matrix} INPHIWA ({\overset{⌢}{A}}_{1}, {\overset{⌢}{A}}_{2}, \dots, {\overset{⌢}{A}}_{m}) = \oplus_{i = 1}^{m} w_{i} {\overset{⌢}{A}}_{i} \\ = \underset{{\overset{⌢}{A}}_{1}^{(q)} \in {\overset{⌢}{A}}_{1}}{\cup} {([w_{1} A_{1}^{L (q)}, w_{1} A_{1}^{U (q)}) ({\overset{⌢}{p}}_{1}^{(q)})} \\ \oplus \underset{{\overset{⌢}{A}}_{2}^{(q)} \in {\overset{⌢}{A}}_{2}}{\cup} {([w_{2} A_{2}^{L (q)}, w_{2} A_{2}^{U (q)}) ({\overset{⌢}{p}}_{2}^{(q)})} \\ \oplus \dots \oplus \underset{{\overset{⌢}{A}}_{i}^{(q)} \in {\overset{⌢}{A}}_{i}}{\cup} {([w_{i} A_{i}^{L (q)}, w_{i} A_{i}^{U (q)}) ({\overset{⌢}{p}}_{i}^{(q)})} . \end{matrix}$ (11)

Especially, if w = (1/ - m, 1/ - m, ⋯ , 1/ - m), then the INPHIWA operator reduces to the INPHIA operator.

2.2.3 Linguistic PHISs

When PHEs in PHISs are the language terms, the PHISs are the same as the probability linguistic term sets (PLTSs). Therefore, the operations and aggregation operators of linguistic PHISs and PLTSs are identical, more details are shown in paper [6].

2.2 Regret theory

Regret theory was independently proposed by Bell [11]. They believe that a DM will compare the results of his/her own choice with others’ in the decision-making process. Then if it is inferior to the results obtained by choosing other options, the DM will feel regret, otherwise, he/her will feel like rejoicing. Therefore, DMs will try to avoid choosing the alternatives that may lead to regret during the comparison process.

The perceived utility function of the alternatives is composed of a utility function and a regret-rejoice function according to the classical RT. Assume that the results of choosing alternative A and alternative B are a and b, respectively, then the perceived utility of DM for alternative A is: $I (a, b) = K (a) + D (K (a) - K (b)),$ (12) where K (a) and K (b) denote the utility value of alternative A and B, respectively, and D (K (a) - K (b)) represents the regret-rejoice value obtained by DM, while choosing alternative A to give up alternative B. D (·) denotes regret-rejoice function with D′ (·) >0, D^′′ (·) <0 and D (0) =0 [20].

Quiggin [21] extended the classical RT to the selection problem of multiple alternatives. After that, many scholars applied RT to the field of multi-criteria decision making. Then, we construct the perceived utility function based on the model proposed by Quiggin (1994): $I (A_{i}) = K (A_{i}) + D (K (A_{i}) - K (A^{*})),$ (13) where K (A_i) denotes the utility value of the project A_i, K (A^*) is the utility value of the best project, and D (K (A_i) - K (A^*) represents the regret-rejoice value of the project A_i minus the best project A^*.

3 A GEDM method considering heterogeneous probabilistic hesitant information and regret aversion

This paper proposes a new HMCGDM method based on PHISs and RT to solve the emergency alternative selection problem. In this section, the statement of GEDM problem is given. Then, we normalize the three kinds of PHISs to eliminate the influence of different dimensions on the calculation. Furthermore, the probabilities of the future possible states are calculated by using DS theory, and the criteria weights are determined based on analytic hierarchy process (AHP) and principal component analysis (PCA). Finally, the GEDM procedure is summarized, and ranking orders of all alternatives are obtained through the overall perceived utility values.

3.1 Problem description

For the GEDM problems considering heterogeneous information and possible states, let M ={ 1, 2, ⋯ , m }, N ={ 1, 2, ⋯ , n } and O ={ 1, 2 ⋯ , o }. Assume thatB ={ B₁, B₂, ⋯ , B_m } is the set of alternatives, where B_i (i ∈ M) denotes the ith alternative for assessment according to n criteria. Suppose that C ={ C₁, C₂, ⋯ , C_n } is the set of criteria, where C_j (j ∈ N) denotes the jth criterion. Let the subscript sets of the benefit-type and cost-type criteria be denoted as N₁ and N₂ respectively, which satisfy N₁ ∪ N₂ = N and N₁∩ N₂ = Ø. Assume that w = (w₁, w₂, ⋯ , w_n) is the criteria weight vector, where w_j denotes the weight of criterion C_j such that w_j ⩾ 0 and $\sum_{j = 1}^{n} w_{j} = 1$ . Suppose that φ ={ φ₁, φ₂, ⋯ , φ_o } is the set of the possible states in the future, where φ_k (k ∈ O) is the kth states. ɛ = (ɛ₁, ɛ₂, ⋯ , ɛ_o) is the probability vector of φ ={ φ₁, φ₂, ⋯ , φ_o }, and ɛ_k is the probability of the kth state such that ɛ_k ⩾ 0 and $\sum_{k = 1}^{o} ɛ_{k} = 1$ .

In summary, the evaluation information can be represented by the group decision matrix A_k = [A_ijk] _m×n as Equation (12). A_ijk denotes the evaluation value of alternative B_i concerning criterion C_j under state φ_k. $A_{k} = {(A_{ijk})}_{m \times n} = (\begin{matrix} A_{11 k} & \dots & A_{1 nk} \\ ⋮ & ⋱ & ⋮ \\ A_{m 1 k} & \dots & A_{mnk} \end{matrix}) .$ (14)

3.2 Normalization approach

The GEDM problems have multiple criteria with different dimensions. In order to eliminate the influence of different dimensions on the integrated calculation of PHIS A_ijk, we normalize A_ijk to Y_ijk. Different types of PHISs have different normalization methods, the criteria set C is divided into three independent subsets: C¹ ={ C₁, C₂, ⋯ , C_{j
₁} }, C² ={ C_j₁+1, C_j₁+2, ⋯ , C_{j
₂} } and C³ ={ C_j₂+1, C_j₂+2, ⋯ , C_n }, which satisfy 1 ⩽ j₁ ⩽ j₂ ⩽ n. The criteria of alternatives in C¹, C², and C³ are denoted as real numbers, interval numbers, and linguistic terms, respectively. H₁, H₂, and H₃ represent the subscript sets of C¹, C², and C³, respectively, such that H₁ ={ 1, 2, ⋯ , j₁ }, H₂ ={ j₁ + 1, j₁ + 2, ⋯ , j₂ }, H₃ ={ j₂ + 1, j₂ + 2, ⋯ , n }, and H₁ ∪ H₂ ∪ H₃ = N. The normalization processes of evaluation value A_ijk are as follows:

(1) Real-number PHIS. When the jth evaluation criterion C_j (j ∈ H₁) is unambiguously represented by real numbers, the probabilistic hesitant element ${\dot{A}}_{ijk}^{(q)} (j \in N^{1})$ can be normalized by the following: $Y_{ijk}^{(q)} = {\begin{matrix} ({\dot{A}}_{ijk}^{(q)} - χ_{jk}^{\min}) / - (χ_{jk}^{\max} - χ_{jk}^{\min}) j \in H_{1} \cap N_{1} \\ (χ_{jk}^{\max} - {\dot{A}}_{ijk}^{(q)}) / - (χ_{jk}^{\max} - χ_{jk}^{\min}) j \in H_{1} \cap N_{2} \end{matrix},$ (15) where ${\begin{matrix} χ_{jk}^{\max} = max (max ({\dot{A}}_{ijk}^{(q)})) j \in H_{1} \\ χ_{jk}^{\min} = min (min ({\dot{A}}_{ijk}^{(q)}) j \in H_{1} \end{matrix}$ (16)

(2) Interval-valued PHIS. If the evaluation criterion C_j (j ∈ H₂) is hardly denoted but will most probably fall under a certain numerical interval, then it can be described as an interval number, and the corresponding PHIS is A_ijk = $\overset{⌢}{A}$ _ijk, such that . The normalized ${\overset{⌢}{A}}_{ijk}^{(q)}$ can be obtained as follows:

$Y_{i j k}^{(q)} = [Y_{^{i j k}}^{L (q)}, Y_{^{i j k}}^{U (q)}] = {\begin{matrix} [(A_{^{i j k}}^{L (q)} - χ_{j k}^{u p}) / (χ_{j k}^{u p} - χ_{j k}^{l o w}), (A_{^{i j k}}^{U (q)} - χ_{j k}^{u p}) / (χ_{j k}^{u p} - χ_{j k}^{l o w})], j \in N^{2} \cap N_{1} \\ [(χ_{j k}^{u p} - A_{^{i j k}}^{L (q)}) / (χ_{j k}^{u p} - χ_{j k}^{l o w}), (χ_{j k}^{u p} - A_{^{i j k}}^{U (q)}) / (χ_{j k}^{u p} - χ_{j k}^{l o w})], j \in N^{2} \cap N_{2} \end{matrix},$ (17) where ${\begin{matrix} χ_{jk}^{up} = max (max (A_{ijk}^{U (q)})) j \in H_{2} \\ χ_{jk}^{low} = min (min (A_{ijk}^{L (q)})) j \in H_{2} \end{matrix}$ (18)

(3) Linguistic PHIS. When the criterion C_j (j ∈ H₃) needs to be qualitatively evaluated by using linguistic terms, the probabilistic hesitant element ${\dot{A}}_{ijk}^{(q)} (j \in H_{3})$ can be normalized as follows: $Y_{ijk}^{(q)} = {\begin{matrix} ({\tilde{A}}_{ijk}^{(q)} - χ_{jk}^{le}) / - (χ_{jk}^{re} - χ_{jk}^{le}) j \in H_{3} \cap N_{1} \\ (χ_{jk}^{re} - {\tilde{A}}_{ijk}^{(q)}) / - (χ_{jk}^{re} - χ_{jk}^{le}) j \in H_{3} \cap N_{2} \end{matrix},$ (19) where ${\begin{matrix} χ_{jk}^{re} = max (max ({\tilde{A}}_{ijk}^{(q)})) j \in H_{3} \\ χ_{jk}^{le} = min (min ({\tilde{A}}_{ijk}^{(q)})) j \in H_{3} \end{matrix}$ (20)

3.3 A new perceived utility function based on group satisfaction degree

Different DMs have discrepant preference information due to the differences in the knowledge structure and preferences. Liao and Xu [22] proposed the satisfaction degree to express the comprehensive performance of criteria under hesitant fuzzy environment using the score function. On this foundation, Liu et al. [23] presented a new group satisfaction degree (GSD) function by introducing the divergence of DM into satisfaction degree function, which makes full use of decision information and effectively avoids the subjective randomness of selecting reference points. After that, some scholars used GSD to portray the overall characteristic of information in hesitant decision environment [24 –26]. In order to more comprehensively describe the opinions of the decision group, we apply GSD to the group decision making with PHISs. According to the abovementioned literature, the GSD of PHIS Y_ijk is denoted as follows: $η (Y_{ijk}) = \frac{S (Y_{ijk})}{1 + E (Y_{ijk})},$ (21) where S (Y_ijk) denotes the score of PHIS Y_ijk, such that the GSD of Y_ijk is higher when S (Y_ijk) is larger, E (Y_ijk) expresses the divergence of Y_ijk, then the GSD of Y_ijk is larger when E (Y_ijk) is smaller.

The GSD utilizes decision information fully, and is the comprehensive evaluation of decision group for a project. The combination of GSD and RT is beneficial to make the better decisions, in which decision group has minimum regret emotion. Then, the perceived utility function based on GSD can be given by the following steps.

Step 1. We need to construct the group utility function K (·), DMs are usually sensitive to losses and show the characteristic of risk aversion in the real risk MCGDM problems, so K (·) is a monotonically increasing concave function [11] such that K′ (·) >0, K^′′ (·) <0, and K (0) =0. The power function is employed as the group utility function of PHIS Y_ijk, then the GSD-based group utility function is as follows: $K (A) = η (Y_{ijk})^{α},$ (22) where α is the coefficient of risk aversion with 0 < α < 1. The smaller the α, the greater the risk aversion of decision group.

Step 2. The regret-rejoice function is established. DMs are risk-averse for both regret and rejoice, so D (ΔK) is a monotonically increasing concave function with $D^{'} (K (Y_{ijk}) - K (Y_{ijk}^{*})) > 0$ and $D^{''} (K (Y_{ijk}) - K (Y_{ijk}^{*})) < 0$ . If $K (Y_{ijk}) = K (Y_{ijk}^{*})$ , then $D (K (Y_{ijk}) - K (Y_{ijk}^{*})) = 0$ . According to the existing studies [11, 16], D (ΔK_i) can be defined as $D (Δ K_{i}) = 1 - e^{- θ (K (Y_{ijk}) - K (Y_{ijk}^{*}))},$ (23) where θ represents the coefficient of regret aversion with θ > 0, the regret aversion of decision group is greater when θ is larger.

Step 3. The perceived utility function is determined by $I (Y_{ijk}) = η (Y_{ijk})^{α} + 1 - e^{- θ (η (Y_{ijk})^{α} - η (Y_{ijk}^{*})^{α})} .$ (24)

3.4 Estimating the probability of possible states with Dempster-Shafer theory

The evaluation value of a criterion may be different under different states, decision group can usually determine the possible states of an emergency event in the future, but the probabilities of states are difficult to obtain directly. Motivated by [27], the Pignistic transformation from the D-S theory is an effective tool to acquire the probabilities of possible states. For the event F, which is the subset of ψ, if there is a set function h: 2^ψ → [0, 1], then it satisfies the condition below. ${\begin{matrix} \sum_{F \subseteq ψ} h (F) = 1 \\ h (ι) = 0 \end{matrix},$ (25) where h is the basic probability assignment function, ι represents the empty set. h (F) is the exact trust degree from the decision group for F. Pignistic probabilistic transformation approach is used to average such trust degree base on Equation. $ɛ_{k} = \sum_{φ_{k} \in F} \frac{h (φ_{k})}{| F |},$ (26) where ɛ_k is the probability of the state φ_k, and |F| is the number of focal elements in F.

Example 3. Assume that the emergency event has three possible states in the future such that φ = {φ₁, φ₂, φ₃}, which denotes the “bad”, “general”, and “good”, respectively. The decision group considers that the basic probability assignment functions are described as: h (F₁) =0.25, h (F₂) =0.15, h (F₃) =0.20, h (F₄) =0.40, where F₁ = {φ₁, φ₂}, F₂ = {φ₁, φ₃}, F₃ = {φ₂, φ₃}, and F₄ = {φ₁, φ₂, φ₃}. Based on this, the probability of state φ₁ can be calculated as $ɛ_{1} = \frac{h (F_{1})}{| F_{1} |} + \frac{h (F_{2})}{| F_{2} |} + \frac{h (F_{4})}{| F_{4} |} = \frac{0.25}{2} + \frac{0.15}{2} + \frac{0.40}{3} = 0.33$ . Similarly, ɛ₂ = 0.36 and ɛ₃ = 0.31.

3.5 Calculating the criteria weights using AHP and PCA

Scientific weight determination methods are important to the rationality of decision results. It is not only necessary to concern the preferences and perceptions of experts, but also significant to mine the data structure. In other words, both subjective weights and objective weights of criteria should be considered in HMCGDM process. AHP is perhaps the most popular methodology that is utilized to obtain the objective criteria weights in many MCDM problems. The factor loadings in PCA can be used to calculate the objective weights through the contribution rates of criteria values to decision results. In this paper, the criteria weights are computed through AHP and PCA.

3.5.1 Subjective weights calculation by AHP

AHP, developed by Saaty [28], divides the evaluation objects into goal hierarchy, criterion hierarchy and index hierarchy. In this methodology, the criteria weights are determined according to the designer’s knowledge. Compared to the methods for objective weights calculation, the experts can adjust the criteria weights assignment through the pairwise comparison matrix during the implementation of AHP, which can largely avoid the situation where the obtained weights are far from the practical conditions. The steps of this approach are shown as follows.

Step 1. Obtain the evaluation matrix V with the result of the pairwise comparison on n criteria based on the standardized comparison scale of nine levels (Table 1). The element v_tj represents the quotient of weights of the criteria.

Table 1
Nine-point intensity of importance scale and its description

Intensity of importance Definition

1 Equally important

3 Moderately more important

5 Strongly more important

7 Very strongly more important

9 Extremely more important

2, 4, 6, 8 Intermediate values

Intensity of importance	Definition
1	Equally important
3	Moderately more important
5	Strongly more important
7	Very strongly more important
9	Extremely more important
2, 4, 6, 8	Intermediate values

$\begin{matrix} V = {(v_{tj})}_{n \times n} = (\begin{matrix} v_{11} & v_{12} & \dots & v_{1 n} \\ v_{21} & v_{22} & \dots & v_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ v_{n 1} & v_{n 2} & \dots & v_{nn} \end{matrix}), \\ v_{tt} = 1, v_{tj} = 1 / - v_{jt}, v_{tj} \neq 0 \end{matrix}$ (27)

Step 2. Find the relative criteria weight z_tj using $z_{tj} = v_{tj} / - \sqrt{\sum_{t = 1}^{n} v_{tj}^{2}}$ .

Step 3. Calculate the subjective criteria weights by ${\tilde{w}}_{j} = \frac{\sum_{t = 1}^{n} z_{tj}}{\sum_{j = 1}^{n} \sum_{t = 1}^{n} z_{tj}} .$ (28)

Step 4. In AHP, consistency index (CI) is provided to analyze and test the consistency of the pairwise matrix based on the eigenvalue λ_max. $CI = \frac{λ_{\max} - n}{n - 1} .$ (29) where $λ_{\max} = \sum_{j = 1}^{n} v_{tj} w_{j} / - w_{t}$ , and n is the number of criteria.

Step 5. The consistency ratio (CR), which is determined as the ratio of the CI and the random index (RI), is used to conclude whether the evaluations are sufficiently consistent. The formula can be expressed as CR = CI/ - RI. The number 0.1 is the acceptance limit of CR. If the value of CR is greater than 0.1, it is suggested to implement the pairwise comparison again to make the evaluations more consistent.

3.5.2 Objective weights calculation by PCA

PCA can decompose the original multiple criteria into independent single criterion and implement diversified statistics. Then, the criteria weights after dimensionality reduction are obtained according to variance contributions of principal components (PCs). During the implementation of PCA, PCs are the linear combination of the original criteria variables and extract useful decision information. The coefficients of the variables, namely the factor loadings, represent the contribution of original criteria to the PCs or decision results. Hence, many researchers use the factor loadings to determine the objective weights in decision making [29]. The procedure is summarized as follows.

Step 1. Based on score S (Y_ijk), the matrix of standardized variables is constructed by ${\tilde{X}}_{k} = (\begin{matrix} {\tilde{x}}_{1 k}^{T} \\ {\tilde{x}}_{2 k}^{T} \\ ⋮ \\ {\tilde{x}}_{nk}^{T} \end{matrix}) = (\begin{matrix} {\tilde{x}}_{11 k} & {\tilde{x}}_{12 k} & \dots & {\tilde{x}}_{1 nk} \\ {\tilde{x}}_{21 k} & {\tilde{x}}_{22 k} & \dots & {\tilde{x}}_{2 nk} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ {\tilde{x}}_{n 1 k} & {\tilde{x}}_{n 2 k} & \dots & {\tilde{x}}_{nnk} \end{matrix}) .$ (30) where ${\tilde{x}}_{jk} (i = 1, 2, \dots, n; j = 1, 2, \dots, n)$ is the standardized variable. ${\tilde{x}}_{ijk} = (S (Y_{ijk}) - \bar{S} (Y_{jk})) / - s_{jk}$ , $\bar{S} (Y_{ijk})$ and s_jk are the mean value and the standard deviation of S (Y_jk), respectively.

Step 2. Establish the correlation matrix R_k = (ς_ijk) _n×n through the matrix ${\tilde{X}}_{k}$ . $R_{k} = (ς_{ijk})_{n \times n} = \frac{{\tilde{X}}_{k}^{T} {\tilde{X}}_{k}}{n - 1} .$ (31)

Step 3. Calculate the eigenvalue λ_jk based on |R_k - λ_jkE| = 0.

Step 4. Obtain the single contribution rate (SCR) and cumulative contribution rate (CCR). The required number of PCs is the smallest value of l for which the chosen percentage of total variation or the cumulative contribution rate (CCR), say 85 % , is exceeded, ${SCR}_{jk} = λ_{jk} / - \sum_{j = 1}^{n} λ_{jk}$ , and ${CCR}_{lk} = \sum_{j = 1}^{l} λ_{jk} / - \sum_{j = 1}^{n} λ_{jk}$ .

Step 5. Determine the score coefficient u_ijk of PC y_jk. Given $u_{ijk} = {\tilde{x}}_{ik}^{T} d_{jk}^{0}, j = 1, 2, \dots, n$ , where $d_{jk}^{0} = d_{jk} / - ∥ d_{jk} ∥$ and d_jk is acquired through R_kd = λ_jkd. In addition, the score coefficient u_ijk can also be calculated by using the factor loadings ξ_ijk given by SPSS. $u_{ijk} = \frac{ξ_{ijk}}{\sqrt{λ_{jk}}} .$ (32)

Step 6. Compute the comprehensive score coefficient κ_jk. $κ_{jk} = \frac{\sum_{j = 1}^{l} | u_{ijk} | {CR}_{jk}}{{CCR}_{lk}} .$ (33)

Step 7. Calculate the original objective criteria weights ${\bar{w}}_{j}$ as follows. ${\bar{w}}_{j} = \sum_{k = 1}^{o} \frac{κ_{jk}}{\sum_{j = 1}^{n} κ_{jk}} \cdot \frac{1}{o}$ (34)

According to the subjective weight ${\tilde{w}}_{j}$ and the objective weight ${\bar{w}}_{j}$ , the overall criteria weight w_j can be given by using $w_{j} = ϖ {\tilde{w}}_{j} + (1 - ϖ) {\bar{w}}_{j}$ , where ϖ is the importance coefficient of ${\tilde{w}}_{j}$ for DMs.

3.6 Group emergency decision making procedure

A brief algorithm for the proposed GEDM method with heterogeneous probabilistic hesitant information based on the regret theory and DS theory is presented as follows:

Step 1. Form the decision group and identify the basic information such as the feasible alternatives, criteria, possible natural states of the emergency event, etc.

Step 2. Obtain the evaluation information of all emergency alternatives, and construct the decision matrix A_k = [A_ijk] _m×n with PHISs.

Step 3. Normalize decision matrix A_k = [A_ijk] _m×n into Y = [Y_ijk] _m×n through Equations (15)–(20).

Step 4. Determine the score S (Y_ijk) and the divergence degree E (Y_ijk) of Y_ijk according to Equations (2)–(9), and compute the perceived utility value I (Y_ijk) by using Equations (19)–(22).

Step 5. Obtain the probabilities of future possible states of the emergency event based on Equations (25) and (26).

Step 6. Calculate the overall perceived utility values of all the alternatives based on the criteria weights determined by Equations (27)–(34). $ζ_{i} = \sum_{k = 1}^{o} \sum_{j = 1}^{N} ɛ_{k} I (Y_{ijk}) w_{j} .$ (35)

The optimal alternative can be selected according to the overall perceived utility values of alternatives: The priority of the alternative B_i is greater when the overall perceived utility value ζ_i is higher.

Figure 1 is provided to concisely describe the algorithm of the proposed method.

Fig. 1

The flowchart of the proposed RT-based GEDM method.

4 Case study

To illustrate the applicability and effectiveness of the proposed method, a real case study about COVID-19 is provided as an application.

On January 2^nd, 2021, another wave of COVID-19 epidemic broke out in Shijiazhuang, Hebei, China, and then the number of infections increased rapidly in a few days. The infections have attended many large weddings without any prevention and control measures. Hence, the risk of COVID-19 spreading further is very high in Shijiazhuang. Moreover, most of the infections come from densely populated urban villages, which makes the prevention and control of COVID-19 more difficult. There is no doubt that the economic development and human health will be greatly threatened if COVID-19 epidemic spins out of control. In such situation, it is essential to provide an appropriate emergency plan for avoiding additional losses and to prevent more deaths.

The performance of most COVID-19 emergency plans is evaluated by following three aspects.

Implementation cost (C₁): The effective implementation of COVID-19 emergency plan requires sufficient supplies, isolation rooms and so on, which cost very much. The costs of different emergency plans are unequal, and the financial situation of different local governments is usually not the same. Therefore, governments may give diverse importance about implementation cost when choosing emergency plans.

Cost support and economic relief (C₂): When a person with poor financial status is infected with COVID-19, he/she may not go to see a doctor due to the high cost of COVID-19 treatment [30]. In addition, emergency plans often inhibit the normal production and life of human beings, resulting in the reduction of their income, people may violate the emergency plan under implementation to deal with the decreasing income [31]. Both situations will greatly increase the transmission risk of COVID-19. Therefore, the economic relief and the cost support of testing and treatment are very important for the effective implementation of emergency plans.

Effectiveness (C₃): Different emergency plans usually have different abilities of decreasing COVID-19 to be transmitted from person to person, for instance, some emergency plans ask people to be isolated at home, others allow people to go out freely, but they must wear masks. Hence, the effectiveness of preventing COVID-19 transmission is an important criterion in the process of COVID-19 emergency plans selection.

4.1 The decision steps

Step 1. After the outbreak of the COVID-19 in Shijiazhuang, relevant department sets up a decision group composed of twenty DMs with the same weight to carry out the epidemic prevention and control work. Five epidemic prevention and control alternatives {B₁, B₂, B₃, B₄, B₅} are screened out for selection. The decision group judges the advantages and disadvantages of the epidemic prevention and control alternatives according to three attributes {C₁, C₂, C₃}, where evaluation values of C₁ are represented by real-number PHISs (Unit: $1000; Cost type), the performance of C₂ is expressed by interval-number PHISs (Unit: $10,000; Benefit type), the interval number obeys uniform distribution, the criteria values of C₃ are represented by linguistic PHISs (Cost type). The experts give the evaluation matrix V as follows, and the subjective weight vector is determined using Equations (27)–(29) as $\tilde{w} = (0.0545, 0.5712, 0.3743)$ , CR = 0.0634. ϖ is usually taken as 0.5, then the objective weight vector can be computed based on Equations (30)–(34) as $\bar{w} = (0.3449, 0.3271, 0.3280)$ . $V = (\begin{matrix} 1 & 1 / - 8 & 1 / - 9 \\ 8 & 1 & 2 \\ 9 & 1 / - 2 & 1 \end{matrix}) .$

The vector of criteria weights is determined by DMs using PCA and AHP as w = (0.20, 0.45, 0.35). We set T = 8 for convenience of evaluation and calculation. G is then defined as follows: $\begin{matrix} G = {g_{0} = Extremly Low (EL), \\ g_{1} = Very Low (VL), g_{2} = Low (L), \\ g_{3} = Medium Low (ML), g_{4} = Medium (M), \\ g_{5} = Medium High (MH), \\ g_{6} = High (H), g_{7} = Very High (VH), \\ g_{8} = Extremly High (EH)} \end{matrix}$

After the outbreak of the COVID-19 in this area, the relevant experts determine the possible evolution states {φ₁, φ₂, φ₃} of the epidemic, which respectively indicated under control, uncertainty, and out of control. The detailed information is exhibited in Table 2 by using Equation (25) and Equation (26).

Table 2
The basic probabilistic information of possible states

The basic probability assignment Event

h (F₁) =0.2 F₁ = {φ₁, φ₂}

h (F₂) =0.36 F₂ = {φ₁, φ₂, φ₃}

h (F₃) =0.16 F₃ = {φ₂, φ₃}

h (F₄) =0.28 F₄ = {φ₁, φ₃}

The basic probability assignment	Event
h (F₁) =0.2	F₁ = {φ₁, φ₂}
h (F₂) =0.36	F₂ = {φ₁, φ₂, φ₃}
h (F₃) =0.16	F₃ = {φ₂, φ₃}
h (F₄) =0.28	F₄ = {φ₁, φ₃}

Step 2. Based on the information in step 1 of section 4.1, the experts determine decision matrix as shown in Table 3.

Table 3

The evaluation information of alternatives provided by decision group

		C ₁	C ₂	C ₃
φ ₁	B ₁	{300(0.50), 350(0.50)}	{(0.80), [600,700](0.20)}	{VL(0.60), L(0.20), M(0.20)}
	B ₂	{500(0.20), 550(0.20), 570(0.20), 580(0.40)}	{[1000,1200](0.20), [1100,1250](0.80)}	{L(1.00)}
	B ₃	{400(0.25), 420(0.25), 560(0.50)}	{[900,1100](0.50), [1000,1150](0.50)}	{L(0.80), ML(0.20)}
	B ₄	{100(0.40), 120(0.40), 130(0.20)}	{[500,550](0.20), [500,600](0.60), [550,600](0.20)}	{H(0.75), VH(0.25)}
	B ₅	{700(0.60), 740(0.20), 750(0.20)}	{[1100,1200](0.60), [1200,1350](0.40)}	{EL(0.80), VL(0.20)}
φ ₂	B ₁	{400(0.20), 450(0.60), 500(0.20)}	{[650,800](0.40), [700,800](0.40), [800,850](0.20)}	{L(0.60), ML(0.40)}
	B ₂	{500(0.40), 600(0.60)}	{[1100,1250](0.40), [1150,1250](0.40), [1200,1300](0.20)}	{ML(0.6), M(0.4)}
	B ₃	{450(0.40), 500(0.20), 600(0.20), 700(0.20)}	{[1200,1300](0.60), [1300,1350](0.20), [1300,1400](0.20)}	{EL(0.20), L(0.40), ML(0.40)}
	B ₄	{150(0.20), 160(0.60), 270(0.20)}	{[700,800](0.60), [750,850](0.40)}	{VH(0.80), EH(0.20)}
	B ₅	{800(0.40), 850(0.40), 860(0.20)}	{[1400,1500](0.20), [1450,1500](0.60), [1450,1600](0.20)}	{VL(0.60), L(0.40)}
φ ₃	B ₁	{500(0.60), 600(0.40)}	{[900,1000](0.20), [950,1000](0.40), [1000,1100](0.20), [1000,1200](0.20)}	{ML(0.25), M(0.50), H(0.25)}
	B ₂	{500(0.20), 600(0.40), 650(0.40)}	{[1000,1100](0.80), [1100,1200](0.20)}	{M(0.20), H(0.60), MH(0.20)}
	B ₃	{700(0.75), 950(0.25)}	{[1500,1700](0.80), [1600,1700](0.20)}	{EL(0.20), VL(0.80)}
	B ₄	{200(0.80), 370(0.20)}	{[1000,1100](0.50), [1050,1200](0.50)}	{EH(1.00)}
	B ₅	{1000(0.20), 1050(0.60), 1100(0.20)}	{[1400,1700](0.80), [1600,1700](0.20)}	{VL(0.50), L(0.50)}

Step 3. The normalized decision matrix is calculated by utilizing Equations (15)–(20), and is listed in Table 4.

Table 4

The normalized evaluation values

		C ₁	C ₂	C ₃
φ ₁	B ₁	{0.69(0.50), 0.62(0.50)}	{(0.80), [0.12,0.24](0.20)}	{0.86(0.60), 0.71(0.20), 0.43(0.20)}
	B ₂	{0.38(0.20), 0.31(0.20), 0.28(0.20), 0.26(0.40)}	{[0.59,0.82](0.60), [0.71,0.88](0.40)}	{0.71(1.00)}
	B ₃	{0.54(0.25), 0.51(0.25), 0.29(0.50)}	{[0.47,0.71](0.50), [0.59,0.76](0.250)}	{0.71(0.80), 0.57(0.20)}
	B ₄	{1.00(0.40), 0.97(0.40), 0.95(0.20)}	{[0.00,0.06](0.20), [0.00,0.12](0.60), [0.06,0.12](0.20)}	{0.14(0.75), 0.00(0.25)}
	B ₅	{0.08(0.60), 0.02(0.20), 0.00(0.20)}	{[0.71,0.82](0.60), [0.82,1.00](0.40)}	{1.00(0.80), 0.86(0.20)}
φ ₂	B ₁	{0.65(0.20), 0.58(0.60), 0.51(0.20)}	{[0.00,0.16](0.40), [0.05,0.16](0.40), [0.16,0.21](0.20)}	{0.75(0.60), 0.63(0.40)}
	B ₂	{0.51(0.40), 0.37(0.60)}	{[0.47,0.63](0.40), [0.53,0.63](0.40), [0.58,0.68](0.20)}	{0.63(0.60), 0.50(0.40)}
	B ₃	{0.58(0.40), 0.51(0.20), 0.37(0.20), 0.23(0.20)}	{[0.58,0.68](0.60), [0.68,0.74](0.20), [0.68,0.79](0.20)}	{1.00(0.20), 0.75(0.40), 0.63(0.40)}
	B ₄	{1.00(0.20), 0.99(0.60), 0.83(0.20)}	{[0.05,0.16](0.60), [0.11,0.21](0.40)}	{0.13(0.80), 0.00(0.20)}
	B ₅	{0.08(0.40), 0.01(0.40), 0.00(0.20)}	{[0.79,0.89](0.20), [0.84,0.89](0.60), [0.84,1.00](0.20)}	{0.88(0.60), 0.75(0.40)}
φ ₃	B ₁	{0.67(0.60), 0.56(0.40)}	{[0.00,0.13](0.20), [0.06,0.13](0.40), [0.13,0.25](0.20), [0.13,0.38](0.20)}	{0.63(0.25), 0.50(0.50), 0.25(0.25)}
	B ₂	{0.67(0.20), 0.56(0.40), 0.50(0.40)}	{[0.13,0.25](0.80), [0.25,0.38](0.20)}	{0.50(0.20), 0.25(0.60), 0.13(0.20)}
	B ₃	{0.47(0.75), 0.17(0.25)}	{[0.75,1.00](0.80), [0.88,1.00](0.20)}	{1.00(0.50), 0.88(0.50)}
	B ₄	{1.00(0.80), 0.81(0.20)}	{[0.13,0.25](0.50), [0.19,0.38](0.50)}	{0.00(1.00)}
	B ₅	{0.11(0.20), 0.06(0.60), 0.00(0.20)}	{[0.63,1.00](0.80), [0.88,1.00](0.20)}	{0.88(0.50), 0.75(0.50)}

Step 4. The score and divergence degree of Y_ijk are obtained based on Equations (2)–(9) as shown in Table 5. Here, we set α = 0.88 and θ = 0.3 as in Tversky and Kahneman [32] and Zhang et al. [12] (which used experimental verification, the parameters are generally consistent with the psychologies of human under uncertainty). According to Equations (21)–(24), the perceived utility value I (Y_ijk) can be computed in Table 5.

Table 5

The scores, divergence degrees, and the perceived utility values of the normalized evaluation values

		The scores			The divergence degrees			The perceived utility values
		C ₁	C ₂	C ₃	C ₁	C ₂	C ₃	C ₁	C ₂	C ₃
φ ₁	B ₁	0.654	0.082	0.743	0.038	0.065	0.167	0.666	0.105	0.672
	B ₂	0.298	0.741	0.714	0.046	0.075	0.000	0.332	0.721	0.744
	B ₃	0.408	0.632	0.686	0.116	0.074	0.057	0.412	0.628	0.683
	B ₄	0.978	0.059	0.107	0.018	0.034	0.062	0.965	0.080	0.133
	B ₅	0.049	0.824	0.971	0.034	0.083	0.057	0.069	0.786	0.928
φ ₂	B ₁	0.577	0.111	0.700	0.045	0.052	0.061	0.594	0.138	0.693
	B ₂	0.423	0.579	0.575	0.069	0.047	0.061	0.442	0.593	0.583
	B ₃	0.451	0.668	0.750	0.137	0.054	0.137	0.443	0.670	0.693
	B ₄	0.958	0.126	0.100	0.064	0.040	0.050	0.912	0.156	0.126
	B ₅	0.039	0.874	0.825	0.037	0.037	0.061	0.056	0.860	0.801
φ ₃	B ₁	0.622	0.138	0.469	0.054	0.081	0.136	0.629	0.163	0.459
	B ₂	0.556	0.213	0.275	0.061	0.071	0.122	0.566	0.241	0.290
	B ₃	0.375	0.888	0.938	0.120	0.071	0.063	0.382	0.848	0.896
	B ₄	0.962	0.234	0.000	0.076	0.066	0.000	0.907	0.264	0.000
	B ₅	0.056	0.838	0.813	0.035	0.110	0.063	0.076	0.780	0.790

Step 5. Based on Equation (25) and Equation (26), the probabilities of possible states are calculated as ɛ₁ = 0.36, ɛ₂ = 0.3, and ɛ₃ = 0.34.

Step 6. The overall perceived utility values of alternatives are determined by using Equation (35) as ζ₁ = 0.246, ζ₂ = 0.398, ζ₃ = 0.607, ζ₄ = 0.097, ζ₅ = 0.605. Furthermore, the rankings of emergency alternatives are obtained as B₃ ≻ B₅ ≻ B₂ ≻ B₁ ≻ B₄, where “≻” represents “be better than”. Hence, alternative B₃ that is considered as the best emergency solution should be implemented for fighting COVID-19. Alternative B₅ and B₂ should be considered as the alternatives in turn in case of abnormal situation, and alternatives B₁ and B₄ with low overall perceived utility value should be abandoned.

4.2 Comparison with some existing methods

In order to further illustrate the effectiveness of the proposed method, several comparisons are executed as follows.

(1) The psychological behaviors of DMs, such as regrets and rejoicings, have an indelible effect on the decision results. To demonstrate this influence more persuasively, the proposed method is compared with the TOPSIS-based HMCGDM method [33], the VIKOR-based HMCGDM method [34], and the proposed method with θ = 0, respectively. The first two methods are developed base on the strict assumption that DMs are absolutely rational. θ = 0 means that the decision group has no regret emotions in the decision-making process.

(2) To make the proposed method more convincing, it is important to compare this method with other approaches that DMs are boundedly rational. Hence, the case study in Section 4.1 is implemented by the HMCGDM model [35], which ranks alternatives based on TODIM (an acronym in Portuguese of interactive and multiple attribute decision making) method.

(3) The proposed method takes the GSD of evaluation values into consideration, which makes full use of decision information and gives enough consideration to the opinions of decision group. The proposed method is compared with the proposed method that replaces GSD with the score in Table 5.

(4) The existing HMCGDM methodologies integrate evaluation information of an alternative according to different ways. The proposed method is compared with the approaches [36 –38] for showing the advantages in information fusion.

In order to compare the different methods, the above example in Section 4.1 is solved by using other five approaches. The used methods and corresponding decision results are shown in Table 6. In the meanwhile, the overall perceived utility values according to different methods are graphically depicted in Fig. 2. Some important conclusions and managerial implications are summarized as follows.

Table 6
Ranking results using different methods

The proposed The proposed method TOPSIS-based VIKOR-based TODIM-based The proposed

method with θ = 0 HMCGDM HMCGDM HMCGDM method without

method method method considering GSD

Overall perceived Overall perceived Closeness Compromise Comprehensive Overall perceived

utility value utility value (Rank order) degree dominance utility value

(Rank order) (Rank order) (Rank order) degree (Rank order) (Rank order)

B ₁ 0.246 (4) 0.399 (4) 0.372 (4) 0.853 (4) 0.281 (4) 0.272 (4)

B ₂ 0.398 (3) 0.512 (3) 0.513 (3) 0.581 (3) 0.386 (3) 0.416 (3)

B ₃ 0.607 (1) 0.669 (2) 0.725 (2) 0.027 (1) 1.000 (1) 0.654 (1)

B ₄ 0.097 (5) 0.290 (5) 0.247 (5) 1.000 (5) 0.000 (5) 0.098 (5)

B ₅ 0.605 (2) 0.671 (1) 0.742 (1) 0.183 (1) 0.770 (2) 0.644 (2)

	The proposed	The proposed method	TOPSIS-based	VIKOR-based	TODIM-based	The proposed
B ₁	0.246 (4)	0.399 (4)	0.372 (4)	0.853 (4)	0.281 (4)	0.272 (4)
B ₂	0.398 (3)	0.512 (3)	0.513 (3)	0.581 (3)	0.386 (3)	0.416 (3)
B ₃	0.607 (1)	0.669 (2)	0.725 (2)	0.027 (1)	1.000 (1)	0.654 (1)
B ₄	0.097 (5)	0.290 (5)	0.247 (5)	1.000 (5)	0.000 (5)	0.098 (5)
B ₅	0.605 (2)	0.671 (1)	0.742 (1)	0.183 (1)	0.770 (2)	0.644 (2)

Fig. 2

The overall perceived utility values according to different methods.

(1) It can be seen from Table 6 and Fig. 2 that the rankings of the five alternatives determined by the proposed method with θ = 0 are the same as that obtained by the TOPSIS-based HMCGDM method [33] and the VIKOR-based HMCGDM method [34], which further shows the rationality and effectiveness of the proposed method. However, the proposed method (θ = 0.3) considers B₃ is the best alternative, while the other three methods think that B₅ is the best alternative. We analyze the principle and calculation process of the four methods: The proposed method with θ = 0, the TOPSIS-based HMCGDM method, and the VIKOR-based HMCGDM method assume that the DMs are completely rational, and the advantages and disadvantages of each COVID-19 prevention and control alternative are not overestimated or underestimated. The proposed method considers that the DMs are bounded rational with the psychological characteristics of regret aversion. The DMs in decision group will create regret perception when the criteria of alternatives are worse than the positive ideal solution, which further leads to the overestimation of the “loss” of each COVID-19 prevention and control alternative. In conclusion, the proposed method considers that DMs are bounded rational in the GEDM process, which is more in line with the practical risk decision-making environment. This paper provides a useful reference for other researchers that explore the GEDM problems under the assumption of bounded rationality.

(2) TODIM and RT have very similar fundamental principles, they all quantify the impact of DMs’ bounded rationality on decision results. From Table 6, it can be observed that the ranking results obtained by the proposed method and the TODIM-based HMCGDM method [35] are same, which proves the validity of the proposed method.

(3) Table 6 and Fig. 2 show that the alternatives’ rankings given by the proposed method and the proposed method without considering GSD are same. However, from the overall perceived utility values in Table 6, compared with the evaluation value determined by the proposed method except GSD, the evaluation value of alternative B₄ decreases by 0.01 considering GSD, and the value of alternative B₃ decreases by 0.47. This indicates that the decision group has high consistency on the performance of B₄, but has great divergence on the performance of B₃. Thus it can be seen that contemplating GSD in GEDM process is necessary for acquiring the better decision results.

(4) We analyze the logical function of HMCGDM frameworks [36 –38], they list the decision matrix of each decision expert, and aggregate these matrices at the end of the decision-making process. The proposed method preprocesses the heterogeneous information given by decision experts through PHISs, in which the evaluation values of an alternative with respect to a criterion gather in a PHIS. This significantly reduces the complexity of processing information. Hence, the decision group can determine the optimal alternative in a shorter time, which is in accord with the urgency of GEDM.

4.3 Sensitivity analysis with respect to parameter θ

To observe the influence of regret avoidance coefficient θ on the rankings of the alternatives, an investigation are made by considering distinct values of θ. The regret avoidance coefficient θ is taken continuously from the range [0,2], the calculation results of overall perceived utility values changing with the regret avoidance coefficient value are drawn as shown in Fig. 3.

Fig. 3

The change of the overall perceived utility values for the five alternatives with θ ∈ [0, 2].

From Fig. 3, it is indicated that the θ values have a great impact on the overall perceived utility values of the five alternatives. In particular, the proposed method provides the same ranking orders as the completely rational HMCGDM methods [33, 34] when θ is equal to 0 or very close to 0, which stands for the assumption of rational DMs. With the increase of the θ values, the overall perceived utility values of alternatives always decrease. Moreover, the best alternative is B₅ when θ ⩽ 0.228, and the optimal alternative is B₃ when θ > 0.228. In more detail, for the losses with respect to the positive ideal alternative, the performance of B₅ is worse than B₃. Hence, the regret perception of B₅ rises faster than B₃. Thus it can be seen that the decision considering the psychology of regret aversion of decision group is more convincing. Furthermore, the ranking orders of alternatives are invariant when θ > 0.228, which interpreted the robustness of the proposed method.

5 Conclusion

For the GEDM problems, the performance of multiple evaluation criteria is usually represented by heterogeneous values, the regret emotions of DMs have indelible effects on the decision results. PHISs have significant advantages in solving the GEDM problems with complexity and uncertain information. Regret theory is a powerful and effective tool to depict the psychological behaviors of DMs. A method combining PHISs and RT is very valuable for dealing with the general uncertainty of the GEDM problems. Therefore, we proposed a novel GEDM method based on PHISs and RT. The key contribution of our work can be summarized as follows.

Multi-format PHISs are developed to represent the heterogeneous information and to preliminarily integrate the evaluation values given by different experts, which simplifies the group decision-making process and is good for selecting the optimal alternative in a short time. Moreover, the maximum retention of accurate decision information in PHISs contributes to the authenticity and accuracy of the final decision result. Additionally, the preference and hesitation of decision group can be well portrayed by using PHISs.

GSD is used to describe the performance of alternatives, and fully reflects the overall level and group divergence. By combination of GSD and RT, the perceived utility function is proposed to portray the overall perception of decision group concerning alternatives.

The proposed method is demonstrated by an empirical case study of COVID-19, and the given criteria are for emergency experts’ reference. The comparison with some existing methods (e.g., TOPSIS-based HMCGDM, VIKOR-based HMCGDM and TODIM-based HMCGDM) is implemented to prove the effectiveness of the proposed method. Sensitivity analysis emphasizes the importance of considering regret aversion psychology in GEDM process.

In the further, we shall study new applications of the proposed method in more realistic GEDM problems. In addition, we have introduced three-format PHISs in this paper, other types of PHISs will be discussed. Furthermore, we will try to propose other HMCGDM methods with the better information transformation.

Funding statement

This The work was supported National Natural Science Foundation of China (No. 71401027), the S&T Program of Hebei (No. 215576116D), the Science and technology research and development plan of Qinhuangdao City (202005A068).

Conflicts of interest

The authors declare that they have no conflicts of interest to report regarding the present study.

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A regret theory-based GEDM method with heterogeneous probabilistic hesitant information and its application to emergency plans selection of COVID-19

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Probabilistic hesitant information sets

2.2.1 Real-number PHISs

2.2 Regret theory

3.1 Problem description

3.5.1 Subjective weights calculation by AHP

Table 1 Nine-point intensity of importance scale and its description Intensity of importance Definition 1 Equally important 3 Moderately more important 5 Strongly more important 7 Very strongly more important 9 Extremely more important 2, 4, 6, 8 Intermediate values

4.1 The decision steps

Table 2 The basic probabilistic information of possible states The basic probability assignment Event h (F1) =0.2 F1 = {φ1, φ2} h (F2) =0.36 F2 = {φ1, φ2, φ3} h (F3) =0.16 F3 = {φ2, φ3} h (F4) =0.28 F4 = {φ1, φ3}

Funding statement

Conflicts of interest

References

Table 1
Nine-point intensity of importance scale and its description

Intensity of importance Definition

1 Equally important

3 Moderately more important

5 Strongly more important

7 Very strongly more important

9 Extremely more important

2, 4, 6, 8 Intermediate values

Table 2
The basic probabilistic information of possible states

The basic probability assignment Event

h (F₁) =0.2 F₁ = {φ₁, φ₂}

h (F₂) =0.36 F₂ = {φ₁, φ₂, φ₃}

h (F₃) =0.16 F₃ = {φ₂, φ₃}

h (F₄) =0.28 F₄ = {φ₁, φ₃}