A two-layer weight determination method for complex multi-attribute large-group decision-making experts in a linguistic environment

Abstract

We propose a two-layer weight determination model in a linguistic environment, when all the clustering results of the experts are known, to objectively obtain expert weights in complex multi-attribute large-group decision-making (CMALGDM) problems. The linguistic information considered in this paper involves both linguistic terms and linguistic intervals. We assume that, for CMALGDM problems, the final expert weights should be determined based on the expert weight in the cluster and on the cluster weights. This is mainly because experts in the same cluster will certainly make varying contributions to the cluster’s overall consensus, and different clusters will also obtain the distinctive “cluster information quality”. Hence, a Minimized Variance Model and an Entropy Weight Model are proposed to determine the expert weights in the cluster and the cluster weights, respectively. We then synthesize these two types of weights into the final objective weights of the CMALGDM experts. The feasibility of the two-layer weight determination model method for the CMALGDM problems is illustrated using a case study of salary reform for professors at a university.

Introduction

Because of the continuing democratization of countries as well as the increasing complexity of major societal and economic decision-making problems, citizens have become important stakeholders in decision-making problems such as the preliminary decision-making on a major investment project and the adjustment of tax policy in the country. In addition, because of innovations in Internet-based systems by which decisions that involve a tremendous number of decision makers can be made, public participation is becoming an indispensable and significant component of the decision-making process. Clearly, academic circles have already taken interest in this new type of decision-making problem. Efremov et al. [11] proposed a framework to support participatory decision-making through the Internet that does not require significant sophistication from the participants. Focusing on participatory public decision-making processes, Kim [17] proposed a framework for group support systems and discussed a number of related research issues. Palomares et al. [25] presented a consensus-reaching model to help manage large groups of decision makers or experts. Furthermore, Chen [3] termed the new type of decision-making problem the complex multi-attribute large-group decision-making (CMALGDM) problem and summarized its four features as follows: (a) the group size is relatively large, often containing more than 20 experts that have been involved in the decision-making process, and there exist differences among the group members, including certain conflicts of interest; (b) the experts in the group can make decisions at relatively different times and in relatively different places with the aid of Internet-based systems; (c) the decision attributes tend to be complex because of their large size and the connections among them; and (d) because of the complexity of the decision-making problem as well as of the experts themselves, the preference information provided by the experts tend to be fuzzy or uncertain.Furthermore, Liu et al. [21], [22] proposed the partial binary tree DEA-DA classification model and the IVIF-PCA model to address the complexity of the experts and attributes in CMALGDM problems, respectively. In summary, it seems that the CMALGDM problem has a profound and wide application perspective compared to traditional group decision-making (GDM). Hence, we focus on solving the issues found in CMALGDM problems in this paper.

As mentioned above, the preference information of the experts is normally fuzzy in CMALGDM problems. Hence, it is critical to first choose the suitable information representation for the evaluation values of the attributes. Currently, research into GDM problems in a fuzzy environment mainly adopts three modes to describe attribute evaluation values, i.e., fuzzy numbers [39], linguistic terms [40] and intuitionistic fuzzy numbers [1], [2]. Compared with the other two fuzzy information representation approaches, linguistic information is more suitable to describe the features of human decision-making, and it has been broadly applied in real-world decision-making problems [5], [14], [41], [42]. Because methods of handling linguistic information such as those based on the extension principle [4] and those based on symbols [6] either result in the loss of and distortion of information or fail to produce evaluation results that are consistent with the original linguistic term set, Herrera and Martínez [12] proposed the 2-tuple linguistic (2TL) representation model consisting of a linguistic term and a real number to avoid these drawbacks during the processing of linguistic information. As an effective approach to representing linguistic information, this model has been widely applied to the decision-making area ever since it had been proposed [9], [10], [13], [15], [23], [24], [28], [29]. However, completely considering the complexity of human’s though process and reasoning as well as that of the decision-making problem, experts may find that it is too difficult to express their opinion on the attribute simply using a linguistic term in some cases. They may prefer a linguistic interval instead. For instance, when evaluating the comfort degree of a given type of automobile, the expert may deem that the comfort degree is approximately located between “Medium” and “High”. Indeed, in early 2004, Xu [30] had already noticed this situation, and he defined the “uncertain linguistic variable” to address this issue. However, as the boundaries (i.e., the lower and upper limits) of the “uncertain linguistic variable” of the aggregation result may also not consistent with any linguistic term in the predefined linguistic term set, the symbolic values are difficult to explain [27], [33].To overcome this shortcoming, both Lin et al. [19] and Zhang [43] proposed the interval-valued 2-tuple linguistic (IV2TL) representation model as the generalization of the 2TL representation model. It is worth noting that Zhang’s model, which can handle the linguistic information coming from different multi-granularity linguistic term sets without transforming the multi-granularity linguistic term sets into a basic linguistic term set [43], was developed partially based on Lin’s model. Because, in this paper, we assume that all the experts share the same linguistic term set, we use Lin’s IV2TL representation model to denote the linguistic interval information. To summarize, we would like to set the linguistic environment as our CMALGDM setting, and more specifically, by taking into account the expression preferences among the experts, both the 2TL and the IV2TL representation models would be considered in this paper.

Expert weight determination is always an important issue to be addressed in GDM problems. In reality, the weights of decision experts may vary because of such factors as their knowledge structures and levels, social environment and personal experience; therefore, in the process of GDM, the accuracy and validity of the decision-making result is largely determined by how reasonable the determination of the expert weights is. As an extension of the conventional GDM, the CMALGDM faces the same issue. A number of expert weight determination methods in GDM have been devised that can be roughly divided into three categories: subjective, objective, and combined weighting methods. Subjective weighting methods, such as the AHP [26] and the Delphi [20] methods, require the experts to have a relatively high level of familiarity with the problem to be solved. Objective weighting methods, including the entropy-based method [35], TOPSIS-based methods [36], [37], and the projection method [38], determine expert weights based on the evaluations and judgment information the experts provide. The combined weighting methods [16], [18] consider both subjective weights and objective weights and synthesize them into comprehensive weights. However, these methods cannot be directly applied in CMALGDM problems, which is mainly because these methods failed to fully reflect the clustering information of the decision experts. In fact, one significant difference between CMALGDM and GDM is the clustering of the decision experts. The CMALGDM method usually first divides the numerous decision experts into several clusters based on expert preferences or evaluation values to reflect the similarity degree of these experts [22], [31], [32], [34], thereby reducing the complexity of the computation process and making the information aggregation procedure more accurate as well. Hence, when determining the expert weights in the CMALGDM problem, we feel that, in general, two stages should be involved: (a) expert clustering and (b) expert weight determination. Because Liu et al. [22] provided an efficient approach to clustering the experts (although their model is based on the interval-valued intuitionistic fuzzy setting, their main idea could be directly applied to the linguistic case), this paper will mainly focus on the second stage of the process for determining the expert weights in the CMALGDM problem. More specifically, this paper will discuss how we would obtain the experts’ weights given their clustering results.

We find that not every expert in the same cluster will have a high consistency degree with the cluster’s overall preference; therefore, we deem that experts in the cluster should also be allocated the appropriate weights so that the cluster can obtain the maximum consensus. In addition, we believe different clusters should be assigned distinct weights not just because their group sizes vary but also because the useful information each group provides is different. Based on these two considerations, this paper proposes a two-layer weight determination model for CMALGDM experts in a linguistic environment. The basic concept of this model can be summarized as follows: We first propose a Minimized Variance Model to determine each expert’s weight in the cluster for the first layer (i.e., the expert layer). Next, an Entropy Weights Model is utilized to obtain the cluster weights for the second layer (i.e., the cluster layer). Finally, by combining the weights of these two layers, we objectively obtain the final weights for the experts.

To fully demonstrate the two-layer weight determination method, the remainder of this paper is organized as follows. In Section 2, we first show a number of basic concepts related to the 2TL representation model and the IV2TL representation model. Section 3 first introduces CMALGDM problems in the linguistic environment. Next, the derivation of the two-layer weight determination model is demonstrated in detail. Finally, the model is constructed in the 2TL and IV2TL environment, respectively. Using the proposed method, Section 4 provides a practical example to illustrate the expert weight determination procedures. In Section 5, we conclude our paper.