Large group decision-making considering multiple classifications for participators: a method based on preference information on multiple elements of alternatives

Abstract

In large group decision-making, participators with different knowledge structures, backgrounds, and other characteristics are unlikely to accurately evaluate alternatives. For this, it is necessary to decompose alternatives into several elements, and consider the participators’ preferences for elements of alternatives and the multiple classifications for participators according to their characteristics. However, related studies are still scarce. The objective of this paper is to propose a multi-elemental large group decision-making method, in which the desirable alternative(s) are selected from a set of feasible alternatives according to the preference information on multiple elements of alternatives provided by participators from multiple subgroups, and multiple classifications for participators are considered. In the method, according to the strict preference ordering of elements provided by participators, the percentage distributions on preferences of each subgroup concerning each element are firstly presented under each classification for participators. Secondly, the decision weight of each subgroup is determined by three factors, i.e., the consensus of preferences provided by each subgroup, the organizer’s preference for each subgroup, and the number of participators in each subgroup. Then, the comprehensive preference concerning each element is determined by combing the preference information from multiple subgroups and the decision weights of multiple subgroups, the overall preference vector can be obtained under each classification, and the virtual alternatives are determined by normalizing the overall preference vector. Further, considering multiple classifications for participators, the overall dominant degrees of alternatives can be obtained by calculating the similarity degrees between each virtual alternative and each alternative, thus the ranking order of alternatives can be obtained based on the overall dominant degrees of alternatives. Finally, an example is given to confirm the feasibility of the proposed method. The results of the sensitivity and comparative analyses show that the proposed method is applicable and effective. The proposed method can further enrich and improve the theory and approach of large group decision-making with multiple elements considering multiple classifications for participators.

Introduction

Large group decision-making (LGDM) is a practical group decision-making problem based on the opinions of a large number of participators with different characteristics [32, 33, 36]. Typically, participators can be divided into multiple subgroups according to characteristics of participators, and the LGDM analysis is based on the opinions of multiple subgroups [32, 38], as shown in Fig. 1. However, participators usually have different characteristics, and subgroups and the opinions of subgroups are different when participators are classified according to different characteristics. Therefore, in the LGDM analysis, multiple classifications for participators based on characteristics of participators should be considered. For example, a company plans to design some incentives to motivate employees. In order to select a desirable alternative, all the employees of the company take part in the decision process. Employees can be classified according to different characteristics, such as levels of positions and age, and then multiple subgroups can be obtained and the opinions of subgroups are different. In this LGDM analysis, levels of positions can be regarded as the first classification for employees, and age can be regarded as the second classification. And the opinions of employees with the two classifications should be integrated and processed to select a desirable alternative. Therefore, how to provide a novel method considering different classifications for participators is a valuable research topic.

Fig. 1

The LGDM problem

In recent years, many scholars conducted studies on LGDM from consensus building [34, 59], (fuzzy) multi-criteria decision analysis [1,2,3], group clustering ([11, 70]), and other perspectives. Wang et al. [47] proposed a linguistic LGDM method based on the cloud model and applied it in the context of the Belt and Road Initiative of China. Zhou et al. [69] proposed a LGDM method for the social recommendations for cyber-enabled online services. Xu et al. [56] proposed a method for large-group emergency decision-making in major emergencies. Liu et al. [26] proposed a method for LGDM problems in social network contexts by giving a trust relationship-based conflict detection and elimination decision-making model. Song and Li [41] developed a LGDM model to handle the problem that stakeholders may only provide partial preference information. He et al. [18] proposed a LGDM method based on a shadowed set-based TODIM, and the feasibility of the proposed method was illustrated by a case study about assembly factory site selection. Du and Shan [14] proposed a dynamic intelligent integration recommendation method for the LGDM problem. Yu et al. [59] developed a method for probabilistic linguistic LGDM problems by providing a hierarchical punishment-driven consensus model. In addition, there are studies on the influence of disturbances, modelling errors, and various uncertainties in the real systems, which show the vibrancy of this research field. Stojanovic and Nedic [40] proposed a robust algorithm for identifying output error model with constrained output in the presence of non-Gaussian noise. Xu et al. [55] studied the exponential stability of nonlinear impulsive systems with double state-dependent delays. Zhang et al. [61] investigated the fuzzy security filtering for a class of nonlinear partial differential equation systems with dual cyber-attacks.

It is necessary to point out that, although prior studies have made significant contributions to LGDM analysis, multiple classifications for participators have not been considered. In most of the existing studies, the participators are regarded as independent individuals or only one classification for participators is considered. Nevertheless, in some practical LGDM, decision outcomes are often related to the interests of multiple participators with different characteristics, and participators with different characteristics usually have different opinions. Thus, in the decision analysis, multiple classifications for participators according to characteristics of participators should be considered, so as to take into account the preferences and needs of participators from different subgroups under different classifications. In addition, the LGDM methods provided in prior studies are based on the opinions of participators on alternatives, rather than on multiple elements of alternatives. However, in many practical LGDM problems, it is usually quite difficult for participators to provide accurate opinions on alternatives because of the professionalism and complexity of alternatives. Therefore, in order to obtain more accurate opinions of participators, the alternative can be decomposed into several elements, and participators can provide their opinions on elements of the alternative. For example, an urban park is planned to be built in one city, and several alternatives are provided and a large number of citizens are allowed to take part in the decision process. Because the alternatives are usually highly professional and complex, citizens can’t give accurate opinions on alternatives. If the alternative is decomposed into several elements, such as park greening, children’s play facilities, body-building facilities, leisure squares and so on, then citizens can provide more accurate opinions on these elements. However, the LGDM method based on the opinions of participators on multiple elements of alternatives considering multiple classifications for participators has not been found. Therefore, it is necessary to develop a novel method for solving the multi-elemental large group decision-making (MELGDM) problem. This is the research motivation of this paper.

The objective of this paper is to propose a novel method for the MELGDM problem. In the method, the desirable alternative(s) can be selected according to the preference information on multiple elements of alternatives, the preference information is provided by a large number of participators from multiple subgroups, and multiple classifications for participators are considered. The main contributions of this paper are listed as follows. (1) This paper focuses on a new LGDM problem, i.e., the MELGDM problem. In the problem, the decision analysis is based on the preference information on multiple elements of alternatives, and multiple classifications for participators are considered. To solve the MELGDM problem, a resolution procedure is presented. It is different from the traditional LGDM analysis and lays a foundation for further research on LGDM methods. (2) A new method is proposed to solve the MELGDM problem. Compared with these existing methods [32, 33, 45], the proposed method not only considers the multiple classifications for participators, but also decomposes alternatives into several elements, considering participators’ opinions on elements of alternatives, and improving the accuracy of participators’ evaluation of alternatives. Moreover, a detailed example and comparative analyses are presented to demonstrate the validity of the method. (3) A method for determining decision weights for subgroups that considers multiple dependable factors is proposed. Many existing LGDM methods provide weights in advance or consider that each subgroup has the same weight [12]. However, in practical decision problems, the decision weights of subgroups depend on multiple factors, so this weight determination method is more reasonable.

The rest of this paper is arranged as follows. Sect. "Literature review" presents a review of the related literature on LGDM. Sect. "The MELGDM problem and resolution process" describes the MELGDM problem and presents the resolution process for the problem. Sect. "The proposed method" provides a new method according to the resolution process. Sect. "Illustrative example" gives an example to illustrate the use of the proposed method. In Sect. "Conclusions", the major contributions of this paper are summarized.

Literature review

In recent years, some studies on LGDM can be found. The research on LGDM mainly focuses on three aspects [15, 32, 33, 41, 54], i.e., cluster methods for classifying participators, consensus-based methods in LGDM and LGDM methods. In the following, a brief literature review will be given with respect to the three aspects.

Cluster methods for classifying participators

In order to effectively alleviate the complexity of LGDM problems, clustering methods are used to divide a large number of participators into several subgroups [23, 50, 62]. There have been many studies on clustering methods in LGDM, and the studies mainly include two aspects: one is the classical clustering algorithm and its extension algorithm based on the similarity or correlation of preference information,and the other is the community detection method that considers social relations and preference mixed information. Liu et al. [25] constructed a partial binary tree DEA-DA cyclic classification model to achieve the multiple groups’ classification of decision makers. Zhu et al. [70] developed a clustering procedure combining three-dimensional gray relational analysis and the concept of hierarchical clustering to solve LGDM problems with double information. Wu et al. [49] developed linguistic principal component analysis and fuzzy equivalence clustering method to classify the attributes and decision makers, respectively. Ding et al. [11] proposed a sparse representation-based intuitionistic fuzzy clustering method, which was more robust and efficient for LGDM problems. Ma et al. [35] proposed a clustering method for LGDM with multistage hesitant fuzzy linguistic terms based on expert similarity. Du et al. [13] developed a clustering method that considers opinion similarity and trust relationship, so as to implement clustering operations in LGDM events in a social network. Zhong and Xu [63] proposed a clustering method for integrating the correlation and consensus of hesitant fuzzy linguistic information. Zhong et al. [64] proposed a clustering method based on the K-means clustering algorithm by combining the cardinal distance and ordinal distance of experts to reduce the complexity of the decision process. Liu et al. [30] used probabilistic k-means clustering algorithm to divide decision makers with similar features into different subgroups to support LGDM. Trillo et al. [44] created a classifier to detect the aggressiveness and positivity of experts, so as to cluster and create a LGDM system. Zhong et al. [66] proposed a double clustering method combining the similarity degrees of experts’ fuzzy preference relations and criteria weights to classify experts.

Consensus-based methods in LGDM

To improve the effectiveness of collective opinions, consensus reaching process (CPR) is used to remove conflicting preferences among participators and achieve a collective solution as close to a unanimous agreement as possible [5, 17, 22, 28]. To date, different CRPs have been explored in LGDM, such as non-cooperative behavior analysis, the minimum cost consensus model, CRP-based consistency management, and CRP in social networks. Xu et al. [53] proposed an improved consensus model for multi-criteria large-group emergency decision-making to manage minority opinions and non-cooperative behaviors. Chao et al. [7] proposed a consensus model for multi-criteria LGDM considering non-cooperative behaviors and minority opinions, which was applied to rank Go players available at Go4Go.net. Wu and Xu [51] proposed a consensus model for large-scale group decision-making with hesitant fuzzy information and changeable clusters. Zhang et al. [60] developed a consensus reaching model for the heterogeneous large-scale GDM with individual and collective satisfactions. Xu et al. [54] presented a confidence consensus-based model for LGDM, and provided a novel approach to address non-cooperative behaviors. Liu et al. [31] presented a consensus model which considers both the fuzzy preference relations and self-confidence. Zhong et al. [65] proposed a non-threshold consensus model that combines the minimum cost and maximum consensus increasing for multi-attribute LGDM. Lu et al. [34] developed a consensus management process, which was based on social network clustering and distrust behaviors with incomplete hesitant fuzzy preference relations. Rodríguez et al. [37] proposed a comprehensive minimum cost consensus model for LGDM for circular economy measurement. Liu et al. [29] proposed a clustering and maximum consensus-based resolution framework with linguistic distribution for social network LGDM problems. Yuan et al. [58] proposed an optimization model that considers both cardinal consistency and ordinal consistency to estimate unknown preferences in incomplete fuzzy preference relations. Jin et al. [19] proposed a three levels consensus, which can gradually improve the value of group consensus by constantly adjusting the most inconsistent evaluation information.

LGDM methods

The purpose of the LGDM method is to determine the ranking of alternatives or select the best alternative by aggregating the opinions or evaluations provided by participators from multiple subgroups [32, 33, 41, 48]. In recent years, LGDM methods have gained increased attention from a number of scholars. For example, Srdjevic [38] aggregated AHP and social choice methods to support group decision-making in water management. Alonso et al. [4] presented a fuzzy group decision-making model for large groups of individuals, and the model took some dynamic aspects of the actual decision process into account. Liu et al. [24] proposed a method for the complex multi-attribute LGDM problems based on partial least squares path modelling, which effectively addressed the relativity problem in the decision attributes. Liu et al. [32] developed a method for solving the LGDM problems with participators from multiple groups. Liu et al. [33] proposed a method for LGDM considering both the collective evaluation and the fairness of the alternative. Zhong et al. [67] proposed a statistical method incorporating decision risk and risk attitude into LGDM. Teng and Liu [43] proposed a new probabilistic linguistic LGDM method to analyze the interactions among interest subgroups and the interrelationships among criteria. Tang and Liao [42] developed an efficient multi-attribute LGDM model, the model can effectively address evaluation problems of circular economy activities involving a large group of experts. Chen et al. [8] developed an integrated multi-criteria LGDM framework to address the problem of sustainable building material selection under uncertain contexts. Cao et al. [6] proposed a method based on topic sentiment analysis to address the problem of risk control of the intuitionistic fuzzy preference in large group emergency decision-making. Zhou et al. [68] proposed an approach for handling Pythagorean fuzzy information under the risk attitude of decision-makers, so as to address the loss of decision information. Wan et al. [46] proposed a new personalized individual semantic for LGDM and implemented it into the selection of COVID-19 surveillance plans. Li et al. [21] presented a method considering the additive consistency and consistency to solve the situation where the multi-criteria decision-making has problems with hesitant fuzzy preference relations.

Table 1 presents a summary of the main studies on LGDM.

Table 1 Summary of the main studies on LGDM

The existing studies have made significant contributions to MELGDM. However, there are still some limitations. On the one hand, participators are regarded as independent individuals in most existing studies, but the decision results in LGDM usually relate to the interests of participators from multiple subgroups. Therefore, decision analysis should be conducted based on the preferences of participators from different subgroups. Although several LGDM methods based on opinions or preferences provided by participators from different subgroups can be found [32, 33, 38, 43], multiple classifications for participators are overlooked and the impact of different characteristics of participators on decision results is not considered. In reality, the decision result is determined by the opinions or preferences provided by participators from different subgroups. Since the opinions or preferences of the participators are affected by their social background, knowledge structure and other characteristics, participators from subgroups with different characteristics have different opinions or preferences. Thus, in the LGDM analysis, multiple classifications for participators based on different characteristics of participators should be considered. On the other hand, prior studies are conducted based on opinions or preferences of participators on alternatives [32, 33, 45]. However, in many practical LGDM problems, it is usually quite difficult for participators to provide accurate opinions on alternatives because of the professionalism and complexity of the alternatives. Therefore, in order to improve the accuracy of participators’ evaluation, the alternative can be decomposed into several elements, and participators provide their opinions on the elements of the alternative. Nevertheless, the method considering multiple classifications for participators and preferences on elements of alternatives has not been found. In view of this, it is necessary to propose a new method to solve the MELGDM problem considering multiple classifications for participators.

The MELGDM problem and resolution process

Description of the MELGDM problem

The MELGDM problem concerned in this paper is how to select the desirable alternative(s) from a set of feasible alternatives according to the preference information on multiple elements of alternatives. In the MELGDM problem, the preference information is provided by a large number of participators from multiple subgroups, and multiple classifications for participators are considered. The MELGDM problem is vividly shown in Fig. 2.

Fig. 2

The MELGDM problem concerned in this paper

Let $A = \{ A_{1} ,A_{2} ,...,A_{n} \}$ be a finite alternative set, where $A_{i}$ denotes the $i$ th alternative, $i = 1,2,...,n$; $E = \{ E_{1} ,E_{2} ,...,E_{k} \}$ be a finite element set, where $E_{r}$ denotes the $r$ th element, $r = 1,2,...,k$. Let ${\varvec{b}}_{i} = (b_{i1} ,b_{i2} ,...,b_{ik} )$ be an element value vector of $A_{i}$, where $b_{ir}$ is the element value for alternative $A_{i}$ with respect to element $E_{r}$, $i = 1,2,...,n$, $r = 1,2,...,k$. Let $L_{h}$ be the $h$ th classification for participators, $h = 1,2,...,s$; $G_{h} = \{ G_{h}^{1} ,G_{h}^{2} ,...,G_{h}^{{m_{h} }} \}$ be a subgroup set, where $G_{h}^{{j_{h} }}$ is the $j_{h}$ th subgroup according to $L_{h}$, and $C_{h} = \{ c_{h}^{1} ,c_{h}^{2} ,...,c_{h}^{{m_{h} }} \}$ be the number set of subgroups according to $L_{h}$, where $c_{h}^{{j_{h} }}$ is the number of participators in subgroup $G_{h}^{{j_{h} }}$, $j_{h} = 1,2,...,m_{h}$, $h = 1,2,...,s$. $P_{h}^{{j_{h} }} = \{ P_{h}^{{j_{h} (1)}} ,P_{h}^{{j_{h} (2)}} ,...,P_{h}^{{j_{h} (c_{h}^{{j_{h} }} )}} \}$ is the participator set in subgroup $G_{h}^{{j_{h} }}$, where $P_{h}^{{j_{h} (l_{h} )}}$ is the $l_{h}$ th participator in subgroup $G_{h}^{{j_{h} }}$,$j_{h} = 1,2,...,m_{h}$, $h = 1,2,...,s$, $l_{h} = 1,2,...,c_{h}^{{j_{h} }}$. Let $P_{l}$ be the $l$ th participator, $P_{l} \in P_{h}^{1} \cup P_{h}^{2} \cup \cdots \cup P_{h}^{{m_{h} }}$, $P_{h}^{1} \cap P_{h}^{2} \cap \cdots \cap P_{h}^{{m_{h} }} = \emptyset$, and $R_{l}$ is the strict preference ordering of elements provided by $P_{l}$, $l = 1,2,...,\sum\nolimits_{{j_{h} = 1}}^{{m_{h} }} {c_{h}^{{j_{h} }} }$. Each participator should provide the strict preference ordering on the elements of alternatives in this paper. Usually, each participator ranks the elements in order of importance, and the most important element is ranked at Number One.

Resolution process for the MELGDM problem

To solve the MELGDM problem mentioned above, a method is proposed and the resolution procedures are shown in Figs. 3 and 4. It can be seen from Figs. 3 and 4 that the resolution procedures can be divided into four parts. A brief description of each part is given below.

• (1) Determine the percentage distributions concerning each element of alternatives under each classification for participators. According to the strict preference ordering of elements provided by each participator, the preference scores of elements can be calculated. Then the preference matrix of each subgroup concerning each element can be obtained under each classification for participators. Thus, the percentage distributions on preferences of each subgroup concerning each element can be presented.

• (2) Determine the comprehensive preference concerning each element under each classification for participators. Under each classification for participators, the collective preference concerning each element of alternatives is firstly determined by combing the preference information from multiple subgroups concerning each element and the decision weights of multiple subgroups, in which the decision weight of each subgroup is determined by three factors, i.e., the consensus of preferences provided by each subgroup, the organizer’s preference for each subgroup, and the number of participators in each subgroup. Then the comprehensive preference concerning each element under each classification can be obtained.

• (3) Determine the virtual alternative under each classification for participators. In view of the comprehensive preference concerning each element under each classification, the overall preference vector which denotes the importance degrees of elements considering the opinions of all participators can be obtained under each classification, and then the virtual alternatives can be obtained by normalizing the overall preference vector.

• (4) Ranking of the alternatives. According to the virtual alternatives under each classification and the element values of alternatives, the similarity degrees between each virtual alternative and each alternative can be calculated, then the overall dominant degrees of alternatives can be obtained by constructing the dominant matrix of alternatives, thus the ranking order of alternatives can be obtained according to the overall dominant degrees, and the most desirable alternative(s) can be selected.

Fig. 3

The resolution procedure for the determining of the virtual alternative under each classification

Fig. 4

The resolution procedure for the ranking of alternatives based on the virtual alternative under each classification

The proposed method

According to the resolution process proposed in Sect. "Resolution process for the MELGDM problem", a detailed description of the proposed method is given below.

Determine the percentage distributions concerning each element of alternatives under each classification for participators

Usually, the strict preference ordering $R_{l}$ for elements of alternatives provided by $P_{l}$ is transformed into preference scores, i.e., $R_{l} \to E_{(1)} \succ \;E_{(2)} \succ ... \succ E_{(k)}$. $E_{(1)}$ is the element ranked at Number One, $E_{(2)}$ is the element ranked at Number Two, and so on. $E_{(k)}$ is the element ranked last (at Number $k$). Then, $E_{(1)}$ is assigned a preference score $k$, $E_{(k)}$ is assigned a preference score $1$, $E_{(2)} ,\;E_{(3)} ,\; \cdots \;$ and $E_{(k - 1)}$ are respectively assigned preference scores $k - 1$, $k - 2$,$\cdots$, and 2. $S_{h}^{{j_{h} (l_{h} )r}}$ is the preference score of participator $P_{h}^{{j_{h} (l_{h} )}}$ in subgroup $G_{h}^{{j_{h} }}$ with respect to element $E_{r}$,$j_{h} = 1,2,...,m_{h}$, $h = 1,2,...,s$, $l_{h} = 1,2,...,c_{h}^{{j_{h} }}$, $r = 1,2,...,k$. Let $S = \{ S^{(1)} ,S^{(2)} ,...,S^{(k)} \}$ be the set of preference scores, where $S^{(t)}$ is the preference score of the element that ranked at Number $t$ according to $R_{l}$ and

$$ S^{(t)} = k - (t - 1),\quad t = 1,2,...,k $$

(1)

To facilitate the understanding, the preference score $S^{(t)}$ is shown in Fig. 5. Obviously, the lower the ranking is, the lower the preference score will be, $S^{(t)} \in [1,k]$.

Fig. 5

The preference score for elements of alternatives

Then, the preference matrix ${\varvec{I}}_{h}^{{j_{h} r}}$ of subgroup $G_{h}^{{j_{h} }}$ with respect to element $E_{r}$ can be obtained, i.e.,

(2)

where

$$ I_{h}^{{j_{h} (l_{h} )rS^{(t)} }} = \left\{ \begin{gathered} 1,\;\;\quad S_{h}^{{j_{h} (l_{h} )r}} = S^{(t)} ,\quad \quad j_{h} = 1,2,...,m_{h} ,\;\;l_{h} = 1,2,...,c_{h}^{{j_{h} }} \hfill \\ 0,\;\;\quad otherwise,\quad \quad \quad r,\;t = 1,2,...,k,\;\;h = 1,2,...,s \hfill \\ \end{gathered} \right. $$

(3)

Next, let $\sigma_{h}^{{j_{h} rS^{(t)} }}$ be the number of participators from subgroup $G_{h}^{{j_{h} }}$ with preference score $S^{(t)}$ concerning element $E_{r}$. According to the preference matrix ${\varvec{I}}_{h}^{{j_{h} r}} = [I_{h}^{{j_{h} (l_{h} )rS^{(t)} }} ]_{{c_{h}^{{j_{h} }} \times k}}$, $\sigma_{h}^{{j_{h} rS^{(t)} }}$ can be calculated by

$$ \sigma_{h}^{{j_{h} rS^{(t)} }} = \sum\limits_{{l_{h} = 1}}^{{{c_{h}{j_{h} }} }} {I_{h}^{{j_{h} (l_{h} )rS^{(t)} }} } ,\quad j_{h} = 1,2,...,m_{h} ,\;\;r,\;t = 1,2,...,k,\;\;h = 1,2,...,s $$

(4)

Further, let $p_{h}^{{j_{h} rS^{(t)} }}$ be the percentage of participators from subgroup $G_{h}^{{j_{h} }}$ with preference score $S^{(t)}$ concerning element $E_{r}$. According to $\sigma_{h}^{{j_{h} rS^{(t)} }}$, $p_{h}^{{j_{h} rS^{(t)} }}$ can be calculated by

$$ \begin{aligned} p_{h}^{{j_{h} rS^{(t)} }} = \frac{{\sigma_{h}^{{j_{h} rS^{(t)} }} }}{{c_{h}^{{j_{h} }} }},\quad j_{h} = 1,2,...,m_{h} , \hfill \\ r,\;t = 1,2,...,k,\;\;h = 1,2,...,s \end{aligned}$$

(5)

Obviously, $0 \le p_{h}^{{j_{h} rS^{(t)} }} \le 1$ and $\sum\nolimits_{{S^{(t)} = 1}}^{k} {p_{h}^{{j_{h} rS^{(t)} }} } = 1$, $j_{h} = 1,2,...,m_{h} ,\;\;r,\;t = 1,2,...,k,\;\;h = 1,2,...,s$.

Thus, according to Eq. (5), the percentage distributions on preferences of subgroup $G_{h}^{{j_{h} }}$ concerning element $E_{r}$ can be obtained, as presented in Table 2.

Table 2 The percentage distributions on preferences of $G_{h}^{{j_{h} }}$ concerning $E_{r}$

Determine the comprehensive preference concerning each element under each classification for participators

For the MELGDM problem we studied, the collective preference concerning each element under each classification for participators is firstly determined, and then the comprehensive preference concerning each element under each classification can be obtained.

Under each classification for participators, the collective preference concerning each element of alternatives can be obtained by combining the preference information from multiple subgroups concerning each element with the decision weights of multiple subgroups. In this paper, three factors are considered in the determining of the decision weight of each subgroup, i.e., the consensus of preferences provided by each subgroup concerning each element of alternatives, the organizer’s preference for each subgroup concerning each element of alternatives, and the number of participators in each subgroup. Firstly, since the number of participators from each subgroup is large and the preference information provided by each subgroup concerning each element of alternatives is different, the consensus of preferences provided by each subgroup concerning each element of alternatives should be considered in the determining of the decision weights of subgroups. The higher consensus implies that the participators from the same subgroup have the same benefit demand [20, 27]. Thus, if the consensus of preferences provided by a subgroup concerning each element of alternatives is higher, the decision weight of the subgroup will be greater [32]. The consensus of preferences $CI_{h}^{{j_{h} r}}$ from subgroup $G_{h}^{{j_{h} }}$ concerning element $E_{r}$ can be obtained by

$$ CI_{h}^{{j_{h} r}} = 1 - \frac{{1 - \tilde{E}_{h}^{{j_{h} r}} }}{{\sqrt {\sum\nolimits_{{j_{h} = 1}}^{{m_{h} }} {(1 - \tilde{E}_{h}^{{j_{h} r}} )^{2} } } }},\quad r = 1,2,...,k,\quad h = 1,2,...,s. $$

(6)

where $CI_{h}^{{j_{h} r}}$ has a positive correlation with entropy value $\tilde{E}_{h}^{{j_{h} r}}$ [9]. $\tilde{E}_{h}^{{j_{h} r}}$ can be obtained as follows

$$ \tilde{E}_{h}^{{j_{h} r}} = - \frac{{\sum\nolimits_{t = 1}^{k} {p_{h}^{{j_{h} rS^{(t)} }} \ln (p_{h}^{{j_{h} rS^{(t)} }} )} }}{\ln k},\quad r = 1,2,...,k,\quad h = 1,2,...,s. $$

(7)

It is noted that the differences of preferences among participators from a subgroup should be smaller if $CI_{h}^{{j_{h} r}}$ is higher.

The consensus of preferences $CI_{h}^{{j_{h} r}}$ can be normalized as $WCI_{h}^{{j_{h} r}}$, in which $WCI_{h}^{{j_{h} r}}$ is calculated by

$$ WCI_{h}^{{j_{h} r}} = \frac{{CI_{h}^{{j_{h} r}} }}{{\sum\nolimits_{{j_{h} = 1}}^{{m_{h} }} {CI_{h}^{{j_{h} r}} } }},,\quad r = 1,2,...,k,\quad h = 1,2,...,s. $$

(8)

Secondly, since participators from different subgroups often have different knowledge structures, past experiences, backgrounds, and benefit demands, the organizer may pay different attention to the preferences provided by participators from different subgroups concerning each element of alternatives. For example, a city provides several urban park construction schemes, and a large number of citizens are allowed to participate in the decision process for selecting the desirable alternative. Due to the intensification of population aging, the elderly have become an important group that needs attention in urban construction. At the same time, as an important place to carry the leisure activities of the elderly, the demands of the elderly should be primarily considered in green space planning, fitness equipment and other elements in the construction of urban parks. Generally, in order to better describe the organizer’s preference for subgroups, experts are invited by the organizer to estimate the participators according to actual needs in MELGDM and assign different weights to the subgroups. Let $WP_{h}^{{j_{h} r}}$ denote the weight of subgroup $G_{h}^{{j_{h} }}$ provided by experts concerning element $E_{r}$.

In addition, in order to embody the fairness among participators from different subgroups, the influence of the number of participators in each subgroup should be considered in the determining of the decision weight of each subgroup. For instance, there are two subgroups $G_{h}^{{j_{h} }}$ and $G_{h}^{{j_{h}^{\prime } }}$, if $c_{h}^{{j_{h} }} > c_{h}^{{j_{h}^{\prime } }}$, then the consensus of preferences of $G_{h}^{{j_{h} }}$ concerning element $E_{r}$ may be lower than that of $G_{h}^{{j_{h}^{\prime } }}$ concerning element $E_{r}$. However, the subgroups with more participators provided more preference information concerning each element of alternatives, and subgroup $G_{h}^{{j_{h} }}$ with more participators should be given priority consideration. Thus, if the number of participators in a subgroup is greater, the decision weight of the subgroup will be greater. Let $WQ_{h}^{{j_{h} }}$ denote the weight of subgroup $G_{h}^{{j_{h} }}$ concerning the number of participators, and $WQ_{h}^{{j_{h} }}$ can be calculated by

$$ WQ_{h}^{{j_{h} }} = \frac{{c_{h}^{{j_{h} }} }}{{\sum\nolimits_{{j_{h} = 1}}^{{m_{h} }} {c_{h}^{{j_{h} }} } }},\quad h = 1,2,...,s. $$

(9)

Let $w_{h}^{{j_{h} r}}$ denote the decision weight of subgroup $G_{h}^{{j_{h} }}$ concerning element $E_{r}$. Considering the above factors, $w_{h}^{{j_{h} r}}$ can be obtained by

$$\begin{aligned} & w_{h}^{{j_{h} r}} = \alpha \cdot WCI_{h}^{{j_{h} r}} + \beta \cdot WP_{h}^{{j_{h} r}} + \gamma \cdot WQ_{h}^{{j_{h} }},\\ & r = 1,2,...,k,\quad h = 1,2,...,s.\end{aligned} $$

(10)

where $0 \le w_{h}^{{j_{h} r}} \le 1$ and $\sum\nolimits_{{j_{h} = 1}}^{{m_{h} }} {w_{h}^{{j_{h} r}} } = 1$, $h = 1,2,...,s$, $r = 1,2,...,k$. $\alpha$, $\beta$ and $\gamma$ are coefficients, which reflect the importance of the preference consensus, the organizer’s preference, and the number of participators in MELGDM, respectively, $0 \le \alpha ,\beta ,\gamma \le 1$, $\alpha + \beta + \gamma = 1$.

According to Eqs. (6)–(11), the collective preference $P_{hr} = (p_{{hrS^{(t)} }} )_{1 \times k} = (p_{hrk} ,p_{hr(k - 1)} ,\;...,$ $p_{hr2} ,p_{hr1} )$ concerning element $E_{r}$ under classification $L_{h}$ can be obtained by aggregating the percentage distributions and the decision weight $w_{h}^{{j_{h} r}}$ of subgroup $G_{h}^{{j_{h} }}$. $p_{{hrS^{(t)} }}$ is the collective distribution concerning element $E_{r}$ under classification $L_{h}$ when the preference score is $S^{(t)} \;(t = 1,2,...,k)$, $p_{{hrS^{(t)} }}$ can be obtained by

$$ \begin{aligned} & p_{{hrS^{(t)} }} = \sum\limits_{{j_{h} = 1}}^{{m_{h} }} {w_{h}^{{j_{h} r}} } p_{h}^{{j_{h} rS^{(t)} }} = w_{h}^{1r} p_{h}^{{1rS^{(t)} }} + w_{h}^{2r} p_{h}^{{2rS^{(t)} }} + \cdots + w_{h}^{{m_{h} r}} p_{h}^{{m_{h} rS^{(t)} }} ,\\ & \quad h = 1,2,...,s,\quad r = 1,2,...,k,\;\;t = 1,2,...,k.\end{aligned}$$

(11)

Then, the comprehensive preference $D_{hr}$ concerning element $E_{r}$ under classification $L_{h}$ can be obtained by

$$ D_{hr} = \sum\limits_{t = 1}^{k} {S^{(t)} p_{{hrS^{(t)} }} } ,\quad h = 1,2,...,s,\quad r = 1,2,...,k. $$

(12)

Determine the virtual alternative under each classification for participators

In view of the comprehensive preference $D_{hr}$ concerning element $E_{r}$ under classification $L_{h}$, the overall preference vector ${\mathbf{D}}_{h} = (D_{h1} ,D_{h2} ,...,D_{hk} )$ under classification $L_{h}$ can be obtained. Obviously, for the MELGDM problem we studied, ${\mathbf{D}}_{h} = (D_{h1} ,D_{h2} ,...,D_{hk} )$ denotes the importance degree of elements $E_{1} ,\;E_{2} ,\;...$ and $E_{k}$ considering the opinions of all participators under classification $L_{h}$. Thus, if the element values in an alternative are given according to the proportion in ${\mathbf{D}}_{h} = (D_{h1} ,D_{h2} ,...,D_{hk} )$, then this alternative will make all participators most satisfied under classification $L_{h}$.

Then, the overall preference vector under classification $L_{h}$ can be normalized as ${\tilde{\mathbf{D}}}_{h} = (\tilde{D}_{h1} ,\tilde{D}_{h2} ,...,\tilde{D}_{hk} )$, in which $\tilde{D}_{hr}$ is calculated by

$$ \tilde{D}_{hr} = \frac{{D_{hr} }}{{\sum\nolimits_{r = 1}^{k} {D_{hr} } }},\quad h = 1,2,...,s. $$

(13)

In this study, let ${\tilde{\mathbf{D}}}_{h} = (\tilde{D}_{h1} ,\tilde{D}_{h2} ,...,\tilde{D}_{hk} )$ denote the virtual alternative under classification $L_{h}$, $h = 1,2,...,s$.

Ranking of the alternatives

For the convenience of calculation, the element value vector ${\varvec{b}}_{i} = (b_{i1} ,b_{i2} ,...,b_{ik} )$ of alternative $A_{i}$ is firstly normalized as $\tilde{\user2{b}}_{i} = (\tilde{b}_{i1} ,\tilde{b}_{i2} ,...,\tilde{b}_{ik} )$, in which $\tilde{b}_{ir}$ is calculated by

$$ \tilde{b}_{ir} = \frac{{b_{ir} }}{{\sum\nolimits_{r = 1}^{k} {b_{ir} } }},\quad i = 1,2,...,n. $$

(14)

Then, the similarity degree ${\varvec{S}}({\tilde{\mathbf{D}}}_{h} ,\tilde{\user2{b}}_{i} )$ between each virtual alternative and each alternative can be calculated [39, 52, 57], i.e.,

$$ \begin{aligned} & {\varvec{S}}({\tilde{\mathbf{D}}}_{h} ,\tilde{\user2{b}}_{i} ) = \frac{{{\tilde{\mathbf{D}}}_{h} {\tilde{\mathbf{b}}}_{i} }}{{||{\tilde{\mathbf{D}}}_{h} || \cdot ||{\tilde{\mathbf{b}}}_{i} ||}} \\ &\quad = \frac{{\sum\nolimits_{r = 1}^{k} {\tilde{D}_{hr} \tilde{b}_{ir} } }}{{\sqrt {\sum\nolimits_{r = 1}^{k} {(\tilde{D}_{hr} )^{2} } } \cdot \sqrt {\sum\nolimits_{r = 1}^{k} {(\tilde{b}_{ir} )^{2} } } }},\\ & \quad h = 1,2,...,s,\quad i = 1,2,...,n. \end{aligned} $$

(15)

Obviously, the greater ${\varvec{S}}({\tilde{\mathbf{D}}}_{h} ,\tilde{\user2{b}}_{i} )$ is, the better alternative $A_{i}$ will be under classification $L_{h}$. For alternatives $A_{i}$ and $A_{j}$, $A_{i} \succ A_{j}$ if ${\varvec{S}}({\tilde{\mathbf{D}}}_{h} ,\tilde{\user2{b}}_{i} ) > {\varvec{S}}({\tilde{\mathbf{D}}}_{h} ,\tilde{\user2{b}}_{j} )$ under classification $L_{h}$, and the number is one. $A_{i} \prec A_{j}$ if ${\varvec{S}}({\tilde{\mathbf{D}}}_{h} ,\tilde{\user2{b}}_{i} ) < {\varvec{S}}({\tilde{\mathbf{D}}}_{h} ,\tilde{\user2{b}}_{j} )$ under classification $L_{h}$, and the number is negative one. The number is zero if ${\varvec{S}}({\tilde{\mathbf{D}}}_{h} ,\tilde{\user2{b}}_{i} ) = {\varvec{S}}({\tilde{\mathbf{D}}}_{h} ,\tilde{\user2{b}}_{j} )$ under classification $L_{h}$. Let $DM_{ij}$ be the sum of the number according to the relationship of size between ${\varvec{S}}({\tilde{\mathbf{D}}}_{h} ,\tilde{\user2{b}}_{i} )$ and ${\varvec{S}}({\tilde{\mathbf{D}}}_{h} ,\tilde{\user2{b}}_{j} )$ under all classifications, $h = 1,2,...,s$, $i,\;j = 1,2,...,n$. Therefore, $DM_{ij}$ is the dominant degree of $A_{i}$ over $A_{j}$, and the dominant matrix ${\varvec{DM}}$ of alternatives can be obtained, i.e.,

(16)

Therefore, the overall dominant degree vector $\user2{\tilde{D}\tilde{M}} = [\tilde{D}\tilde{M}_{i} ]_{1 \times n}$ can be obtained, where $\tilde{D}\tilde{M}_{i}$ is the overall dominant degree of alternative $A_{i}$. It can be determined as follows:

$$ \tilde{D}\tilde{M}_{i} = \sum\limits_{j = 1}^{n} {DM_{ij} } ,\quad i = 1,2,...,n. $$

(17)

Obviously, the greater $\tilde{D}\tilde{M}_{i}$ is, the better alternative $A_{i}$ will be. Thus, according to the overall dominant degrees $\tilde{D}\tilde{M}_{1}$, $\tilde{D}\tilde{M}_{2}$, $\cdots$, $\tilde{D}\tilde{M}_{n}$, the ranking order of $n$ alternatives can be obtained, and the most desirable alternative(s) can be selected.

Procedure for the proposed method

In this subsection, the procedure for solving the MELGDM problem in this study is given below. The flowchart of the proposed method is presented, as shown in Fig. 6.

Fig. 6

The flow chart of the proposed method

Step 1. Under each classification for participators, determine the percentage distribution $p_{h}^{{j_{h} rS^{(t)} }}$ on preferences from each subgroup concerning each element of alternatives using Eqs. (1)–(5), $j_{h} = 1,2,...,m_{h} ,\;\;r = 1,2,...,k,\;\;h = 1,2,...,s$.

Step 2. Under each classification for participators, determine the collective preference $P_{hr}$ and the comprehensive preference $D_{hr}$ concerning each element using Table 2 and Eqs. (6)–(12), $r = 1,2,...,k,\;\;h = 1,2,...,s$.

Step 3. Determine the virtual alternative ${\tilde{\mathbf{D}}}_{h} = (\tilde{D}_{h1} ,\tilde{D}_{h2} ,...,\tilde{D}_{hk} )$ under classification $L_{h}$ for participators using Eq. (13), $h = 1,2,...,s$.

Step 4. Calculate the dominant matrix ${\varvec{DM}} = [DM_{ij} ]_{n \times n}$ of alternatives using Eqs. (14)–(16).

Step 5. Calculate the overall dominant degree $\tilde{D}\tilde{M}_{i}$ of alternative $A_{i}$ using Eq. (17), $i = 1,2,...,n$.

Step 6. Determine the ranking order of alternatives according to the obtained overall dominant degrees.

Obviously, there are differences between the proposed method and the LGDM method. In the proposed method, participators are classified based on different characteristics of participators, and the subgroups under different classifications are different. However, multiple classifications of participators are usually not considered in LGDM methods. Additionally, in the proposed method, participators provide preference information on elements of alternatives rather than on alternatives, this is also different from previous studies.

Illustrative example

In this section, an example is given to illustrate the use of the proposed method. Company FC plans to set aside 10 percent of profits (i.e., 6 million CNY) to motivate employees by designing some incentives, which include the improvement of working conditions, the provision of more ways of relaxing (to set up coffee corners, to purchase more sports facilities and so on), the organizing of tours, the distributing of festival gifts and the provision of the opportunity for training to improve professional ability. That is, five elements ($E_{1} ,\;E_{2} ,\;E_{3} ,\;E_{4} ,\;E_{5}$) of alternatives to be considered, i.e.,

$E_{1}$: the improvement of working conditions;

$E_{2}$: the provision of more ways of relaxing;

$E_{3}$: the organizing of tours;

$E_{4}$: the distributing of festival gifts;

$E_{5}$: the provision of the opportunity for training to improve professional ability.

By the analyses of relevant departments in this company, three feasible alternatives can be considered, as presented in Table 3. The numbers in Table 3 are the amounts of alternatives for each element (Unit: ten thousand CNY), i.e., ${\varvec{b}}_{1} = (150,\;120,\;90,\;140,\;100)$, ${\varvec{b}}_{2} = (90,\;100,\;180,\;180,\;50)$, ${\varvec{b}}_{3} = (180,\;100,\;80,\;80,\;160)$.

Table 3 The feasible alternatives

Application of the proposed method

To select a desirable alternative, all the employees of Company FC take part in the decision process through the WeChat platform. Employees provide personal preferences concerning each element, i.e., each employee provides strict ordering of the five elements. 662 employees take part in the decision process and provide their preferences on the elements. This company implements the system of position appointment, and there are nine levels of positions. Positions from the first level to the third level are senior positions, positions from the fourth level to the sixth level are intermediate positions, and positions from the seventh level to the ninth level are junior positions. Senior managers and senior technical professionals are usually in senior positions, and they usually have higher incomes. The middle managers and intermediate technical personnel are usually in intermediate positions. The employees in junior positions usually have lower incomes. In addition to the levels of positions, the ages of employees are also different. Therefore, age is regarded as the first classification for employees (i.e., L₁ = “age”) and levels of positions are regarded as the second classification for employees (i.e., L₂ = “levels of positions”). Then, three subgroups can be obtained according to $L_{1}$, i.e.,

$G_{1}^{1}$: employees under 30 years old;

$G_{1}^{2}$: employees between 30 and 45 years old (including 30 and 45 years old);

$G_{1}^{3}$: employees above the age of 45.

The number set of employees according to $L_{1}$ is $C_{1} = \{ 262,280,120\}$. According to $L_{2}$, three subgroups can be obtained, i.e.,

$G_{2}^{1}$: employees at the junior positions;

$G_{2}^{2}$: employees at the intermediate positions;

$G_{2}^{3}$: employees at the senior positions.

The number set of employees according to $L_{2}$ is $C_{2} = \{ 162,300,200\}$. The number of subgroups under each classification is shown in Fig. 7.

Fig. 7

The number of subgroups under each classification

Each employee provides strict ordering of the five elements, according to Eqs. (1)–(5), the percentage distributions on preferences of subgroup $G_{h}^{{j_{h} }}$ concerning element $E_{r}$ under classification $L_{1}$ and classification $L_{2}$ can be presented, which are shown in Tables 4 and 5.

Table 4 The percentage distributions on preferences of $G_{1}^{{j_{h} }} \;(j_{h} = 1,2,3)$

Table 5 The percentage distributions on preferences of $G_{2}^{{j_{h} }} \;(j_{h} = 1,2,3)$

According to Eqs. (6)–(8), the normalized preference consensus $WCI_{h}^{{j_{h} r}}$ of the subgroup $G_{h}^{{j_{h} }}$ concerning element $E_{r}$ can be obtained, as shown in Table 6. Experts are invited by this company to estimate employees according to actual needs and assign different weights to subgroups concerning each element of alternatives, and $WP_{h}^{{j_{h} r}}$ ($h = 1,2;\;j_{h} = 1,2,3;\;r =$$1,2,3,4,5$) is shown in Table 7. And according to Eq. (9), $WQ_{h}^{{j_{h} }}$($h = 1,2;\;j_{h} = 1,2,3$) can be obtained, as shown in Table 8. According to Eq. (10), the decision weight $w_{h}^{{j_{h} r}}$ of the subgroup $G_{h}^{{j_{h} }}$ concerning element $E_{r}$ can be obtained (here $\alpha = 0.6$, $\beta = 0.2$, $\gamma = 0.2$), as shown in Table 9. And according to Eqs. (11)–(12), the comprehensive preference $D_{hr}$ concerning element $E_{r}$ classification $L_{1}$ and classification $L_{2}$ can be presented, as shown in Table 10.

Table 6 The normalized preference consensus $WCI_{h}^{{j_{h} r}}$ of the subgroup $G_{h}^{{j_{h} }}$ concerning element $E_{r}$

Table 7 The weight $WP_{h}^{{j_{h} r}}$ assigned to the subgroup $G_{h}^{{j_{h} }}$ concerning element $E_{r}$

Table 8 The proportion of the number of participators in the subgroup $G_{h}^{{j_{h} }}$

Table 9 The decision weight $w_{h}^{{j_{h} r}}$ of the subgroup $G_{h}^{{j_{h} }}$ concerning element $E_{r}$

Table 10 The comprehensive preference $D_{hr}$ concerning element $E_{r}$

Then, according to Eq. (13), the virtual alternatives ${\tilde{\mathbf{D}}}_{1} = (\tilde{D}_{11} ,\tilde{D}_{12},\tilde{D}_{13},\tilde{D}_{14},\tilde{D}_{15} ) =(0.213,0.216,0.178,0.189,\;0.204)$ and ${\tilde{\mathbf{D}}}_{2} = (\tilde{D}_{21} ,\tilde{D}_{22} ,\tilde{D}_{23} ,\tilde{D}_{24} ,\tilde{D}_{25} ) = (0.238, 0.245,\;{\kern 1pt} 0.165,{\kern 1pt} \;0.143,{\kern 1pt} \;0.209)$ can be obtained. And the dominant matrix of alternatives can be obtained using Eqs. (14)–(16), i.e.,

$$ {\varvec{DM}} = [DM_{ij} ]_{3 \times 3} = \;\left( {\begin{array}{*{20}c} 0 &\quad {DM_{12} } &\quad {DM_{13} } \\ {DM_{21} } &\quad 0 &\quad {DM_{23} } \\ {DM_{31} } &\quad {DM_{32} } &\quad 0 \\ \end{array} } \right)\; = \left( {\begin{array}{*{20}c} 0 &\quad 2 &\quad 2 \\ { - 2} &\quad 0 &\quad { - 2} \\ { - 2} &\quad 2 &\quad 0 \\ \end{array} } \right) $$

Finally, the overall dominant degrees of alternatives can be determined using Eq. (17), i.e., $\tilde{D}\tilde{M}_{1} = 4$, $\tilde{D}\tilde{M}_{2} = - 4$, $\tilde{D}\tilde{M}_{3} = 0$. Obviously, the ranking order of alternatives is $A_{1} \succ A_{3} \succ A_{2}$, and the most desirable alternative is $A_{1}$. The ranking results of alternatives are shown in Fig. 8.

Fig. 8

The ranking results of alternatives

Sensitivity analysis

A sensitivity analysis is conducted to observe the influence of the coefficients $\alpha$, $\beta$ and $\gamma$ on the ranking order of the alternatives. In order to analyze the influence of the three coefficients clearly, the coefficient $\gamma$ is fixed and the other two coefficients are changed relatively. The results are shown in Figs. 9, 10, 11, 12, 13.

Fig. 9

Effect of $\alpha$ and $\beta$ on the ranking of alternatives ($\gamma = 0$)

Fig. 10

Effect of $\alpha$ and $\beta$ on the ranking of alternatives ($\gamma = 0.2$)

Fig. 11

Effect of $\alpha$ and $\beta$ on the ranking of alternatives ($\gamma = 0.4$)

Fig. 12

Effect of $\alpha$ and $\beta$ on the ranking of alternatives ($\gamma = 0.6$)

Fig. 13

Effect of $\alpha$ and $\beta$ on the ranking of alternatives ($\gamma = 0.8$)

From Figs. 9, 10, 11, 12, 13, we can observe that the optimal alternative(s) are different when the coefficients $\alpha$, $\beta$ and $\gamma$ change. Specifically, if $\alpha \ge 0.5$ or $\beta \le 0.3$ or $\gamma \ge 0.8$, the most desirable alternative is $A_{1}$. Moreover, it can be found that $A_{1}$ and $A_{3}$ are the optimal alternatives if $\alpha \le 0.4$ and $\beta \ge 0.5$ or $\beta ,\gamma \ge 0.4$. Therefore, the results of sensitivity analysis show that different values of $\alpha$, $\beta$ and $\gamma$ have an impact on the ranking of alternatives, and different coefficient values lead to different decision results.

To distinguish the influence of the three coefficients, we can change one coefficient and fix the other two coefficients relatively so that the other two coefficients are equal. The results are shown in Figs. 14, 15, 16. Specifically, if $\alpha \ge 0.3$ or $\beta \le 0.4$, the most desirable alternative is $A_{1}$. If $\alpha \le 0.2$ or $\beta \ge 0.5$, the desirable alternatives are $A_{1}$ and $A_{3}$. The ranking of the alternatives is stable when $\gamma$ changes and the other two parameters are equal, and the optimal alternative is $A_{1}$. It can be found that the best alternatives are different when the coefficients $\alpha$ and $\beta$ change, and the change of $\gamma$ has less impact on the ranking of alternatives.

Fig. 14

Effect of $\alpha$ on the ranking of alternatives

Fig. 15

Effect of $\beta$ on the ranking of alternatives

Fig. 16

Effect of $\gamma$ on the ranking of alternatives

Comparative analyses

To present the rationality and advantages of the proposed method, comparative analyses are conducted in this subsection. The LGDM method based on the expectations of decision-makers provided by Guo et al. [16] is introduced for comparison. For simplicity, the method proposed by Guo et al. [16] is denoted as ${\text{CM - I}}$. Based on the example data in this paper, the collective prospect values of subgroups obtained by applying ${\text{CM - I}}$ are presented in Table 11, and the comprehensive prospect values (denoted as $CP_{i}$) for the alternatives are calculated as $CP_{1} = - 0.0368$, $CP_{2} = - 0.0909$, $CP_{3} = - 0.0480$. According to the comprehensive prospect values, the ranking results of alternatives can be obtained, i.e., $A_{1} \succ A_{3} \succ A_{2}$. It can be seen that the decision-making results of ${\text{CM - I}}$ and the proposed method are the same, the alternative $A_{1}$ most closely meets the preferences or expectations of participators. Although ${\text{CM - I}}$ considers the expectations of participators concerning different attributes in the decision process, it is entirely based on the similarity between participators when dividing participators, only considering a single factor and ignoring the impact of other factors on classification. The method proposed in this paper considers multiple characteristics of participators in the classification process, thereby taking into account the preferences and needs of subgroups with different characteristics in the decision process and improving the rationality of the decision results. In terms of determining subgroup weights, ${\text{CM - I}}$ only considers the number of participators, while this paper considers multiple factors, which is more in line with the actual situation.

Table 11 The collective prospect values of subgroups concerning elements of alternatives

Considering that the LGDM method proposed by Li et al. [32] (denoted as ${\text{CM - II}}$) has some similarities with our application process, the two methods are compared to verify the effectiveness and improvements of the proposed method. According to the resolution process of ${\text{CM - II}}$, the collective percentage distribution can be obtained, as shown in Table 12, and the relative dominance degree (denoted as $\Phi (A_{i} )$) of each alternative can be calculated, i.e., $\Phi (A_{1} ) = 0.0216$, $\Phi (A_{2} ) = - 0.0131$, $\Phi (A_{3} ) = - 0.0085$. According to the obtained relative dominance degrees, the ranking results of alternatives are as follows: $A_{1} \succ A_{3} \succ A_{2}$. It can be seen that the ranking results of alternatives ($A_{1} \succ A_{3} \succ A_{2}$) obtained by our method are completely consistent with those obtained by ${\text{CM - II}}$. However, ${\text{CM - II}}$ is based on the evaluation information of alternatives in the decision process, and has strong casualness for the evaluation of complex alternatives. In some real decision scenarios, it is obviously difficult for individuals to directly evaluate alternatives and give precise scores. For this, this paper decomposes the alternatives into multiple elements, and then the participators evaluate each element of alternatives, so that the accuracy of the evaluation results obtained can be improved and the rationality of the decision results can be increased. In addition, ${\text{CM - II}}$ considers multiple groups, but groups according to a single feature, while the proposed method performed multiple classifications based on different features of the participators.

Table 12 The percentage distributions on preferences of $L_{h}$

Compared with the above two methods, the advantages of the proposed method in this paper are as follows. The proposed method considers multiple classifications for participators, the preferences of participators concerning elements of alternatives, and the impact of multiple factors on subgroup weights, making the obtained evaluation information more accurate and decision-making results more reasonable.

Meanwhile, Wilcoxon test [10] is conducted to verify whether there are significant differences between the proposed method of this paper and ${\text{CM - II}}$, as well as between the proposed method and ${\text{CM - I}}$. The P value obtained from the experiment is greater than 0.05, indicating that there are no significant differences between the results of the proposed method and those of ${\text{CM - II}}$, as well as between the results of the proposed method and those of ${\text{CM - I}}$. Therefore, the results obtained by the proposed method are accurate and reliable, and the proposed method is feasible. It should be pointed out that ${\text{CM - I}}$ and ${\text{CM - II}}$ are difficult to solve the MELGDM problem and do not consider multiple classifications for participators, while the proposed method in this paper considers multiple classifications of participators and can effectively solve the MELGDM problem.

Conclusions

This paper has presented a novel method for solving MELGDM problems, in which the desirable alternative(s) are selected from a set of feasible alternatives according to the preference information on multiple elements of alternatives, the preference information is provided by a large number of participators from multiple subgroups, and multiple classifications for participators are considered. The percentage distributions on preferences of each subgroup concerning each element are firstly presented under each classification for participators, then the weights of multiple subgroups can be determined by the consensus of preferences provided by each subgroup, the organizer’s preference for each subgroup, and the number of participators in each subgroup. By normalizing the overall preference vector, the virtual alternatives are determined. Considering multiple classifications for participators, the overall dominant degrees of alternatives can be obtained by calculating the similarity degrees between each virtual alternative and each alternative, thus the ranking order of alternatives can be obtained according to the overall dominant degrees of alternatives. The main theoretical and practical contributions of this paper are as follows.

In the theoretical aspect, firstly, this paper studies the LGDM problem with many practical backgrounds from a new perspective, i.e., the MELGDM problem. In the problem, participators provide preference information on the elements of alternatives rather than on the alternatives, this is also different from prior studies. Participators providing preference information on elements of alternatives can improve the accuracy of the assessment results obtained. Therefore, the proposed method can provide more accurate decision results according to the preference information on multiple elements of alternatives. Meanwhile, the preference information provided by participators from each subgroup under each classification is presented in the form of percentage distributions, which has the advantages of easy description and convenient information processing. Secondly, a novel method for solving MELGDM problems is proposed. The method is different from the traditional LGDM method. Existing methods often lack the consideration of multiple classifications for participators, while multiple classifications for participators are considered in this paper. The subgroups are different when participators are classified according to different characteristics of participators, and the preference information of subgroups is also different. This method provides new insights into research on the LGDM. Thirdly, a technology for determining the decision weight of each subgroup is proposed. In the LGDM, how to determine the decision weights of the subgroups is an issue to be considered, and the decision weights of the subgroups usually depend on multiple factors and play an important role in the decision process. The decision weight of each subgroup is determined by considering three factors in this paper, which is more reasonable. Moreover, the proposed method has some advantages compared to other methods in the current literature. ${\text{CM - I}}$ and ${\text{CM - II}}$ classify the large group of participators entirely based on a single feature, while the proposed method uses multiple characteristics as the classification basis to make the classification more comprehensive and detailed. ${\text{CM - II}}$ only considers the situation where participators in the subgroup give rating values for alternatives, which has strong casualness for the evaluation of complex alternatives. The proposed method decomposes the alternatives into multiple elements, which can improve the accuracy of evaluation results based on participators’ preferences for elements.

In the practical aspect, the proposed method has a wide range of practical application background, which provides a new method and idea for solving many LGDM problems in reality, such as the optimization or adjustment of livelihood engineering projects, the reform of university teachers’ appointment system and the selection of enterprise employees’ incentive scheme. Then, the proposed method selects the optimal alternative by considering participators’ opinions on multiple elements of alternatives and multiple classifications for participators. This not only helps to improve the accuracy of information on the evaluation of alternatives, but also makes the decision results more consistent with the preferences of groups with different characteristics. Besides, the proposed method has a clear concept, a simple calculation, easy implementation and good availability.

This study also has limitations to be further studied in the future. First of all, the preference information on elements of alternatives provided by the participators in this paper is complete. However, due to the cognitive limitations of participators, they may not be able to provide complete preference information. Therefore, the proposed method can be extended to solve MELGDM problems based on incomplete preference information. Secondly, to make the use of the proposed method more convenient and efficient, the support system based on the proposed method needs to be developed, which is a valuable direction for future research.