Local Interpretable Model-agnostic Explanations (LIME) is an interpretable method used to explain the predictions of machine learning models. It generates perturbed samples around an instance and fits a simple surrogate model, such as linear regression, to approximate the local behavior of the black-box model. This paper introduces Additive Local Interpretable Model (ALIME), a LIME variant based on generalized additive models (GAMs), where feature effects are modeled by penalized splines (P-splines), providing flexibility for capturing nonlinear relationships. Experimental results show that ALIME outperforms the original LIME method in terms of local fidelity. In addition, the shape functions generated by ALIME can clearly capture local feature contributions, providing insight into the relationship between features and model outputs, and enhancing overall interpretability.

multiple attribute group decision making (MAGDM) aims to assist a group in evaluating multiattribute alternatives for seeking the most satisfactory one(s). To improve the decision quality of MAGDM, various consensus models were suggested to deal with opinion differences among experts and achieve consensual decision outcomes. This study proposes a novel ordinal consensus framework for MAGDM with incomplete social network trust preference relations (TPRs). In this framework, incomplete linguistic preference relation are first utilized to represent experts' social network TPRs. After that, a consistency-driven two-stage optimization approach is designed to deal with incomplete TPRs for obtaining individual trust levels and expert weight information. Then, a distance-based ordinal consensus measure is designed with the integration of the obtained expert weight information and the basic idea that the higher-ranked alternatives should have greater importance than the lower-ranked ones. When the consensus degree among experts is unacceptable, an opinion dynamic-based feedback adjustment mechanism is devised by integrating the obtained individual trust levels to provide reasonable opinion modification suggestions for accelerating the consensus reaching in MAGDM. Otherwise, the selection process is used to make a selection. A simulation experiment is designed to investigate the effect of key parameters on consensus efficiency. Next, examples of project investment and software supplier selection demonstrate the usability of the proposed consensus framework. Meanwhile, a comparison analysis and in-depth discussions are presented to justify our proposal. The main contributions of this study are twofold. First, a new perspective on managing incomplete social network TPRs is suggested for MAGDM. Second, a novel ordinal consensus process is designed to enhance the effectiveness of the consensual decision outcome. These results can offer new insights into the consensus building for practice social network MAGDM problems.

The use of social network to model the evolution of credit scores of networked enterprises is still a challenging task. This article develops an opinion dynamics model of the evolution of credit scores of enterprises in a social network. Firstly, based on the number of potential cooperated enterprises and the initial credit scores, the leader and follower enterprises are identified. Then, taking into consideration the cooperated benefit and discrimination cost, the cooperated utility between any two enterprises is calculated, which is used to compute the weights that one enterprise assigns to other enterprises. An opinion dynamics model on the evolution of credit scores of enterprises, inspired on the classical Friedkin–Johnsen’s social network model, is developed. Some desirable properties of the proposed opinion dynamics model are theoretically stated and proved. Finally, a numerical example is provided to illustrate the feasibility of the proposed opinion dynamics model, while a simulation analysis to investigate the joint influences of the connection probabilities and the network structure on the evolution of credit scores of enterprises is reported.

Consistency is an important issue in linguistic decision making with various consistency measures and consistency improving methods available in the literature. However, existing linguistic consistency studies omit the fact that words mean different things for different people, that is, decision makers' personalized individual semantics (PISs) over their expressed linguistic preferences are ignored. Therefore, the aim of this article is to propose a novel consistency improving approach based on PISs in linguistic group decision making. The proposed approach combines the characteristics of personalized representation and integrates the PIS-based model in measuring and improving the consistency of linguistic preference relations. A detailed numerical and comparative analysis to support the feasibility of the proposed approach is provided.

Consensus reaching process (CRP) is important and present in a wide range of application areas. In practical CRP, the managers (e.g., enterprise) often hire some informed individuals (e.g., persuaders) to promote the efficiency of consensus reaching. This article proposes a CRP with minimum cost of informed individuals and time constraint in large-scale group decision-making (LSGDM) with bounded confidence effects. The consensus model with bounded confidence effects (CBC model) is formulated. Then, desirable properties of the CBC model are discussed to facilitate its resolution. Next, an extended particle swarm optimization algorithm is designed to solve the CBC model. Finally, a numerical analysis, a comparison analysis, and a simulation analysis are provided to illustrate the feasibility and effectiveness of the proposed approach.

Abstract Multiple attribute decision making (MADM) is used to rank the alternatives according to evaluation information based on multiple attributes, and many MADM methods have been studied to deal with the MADM problems. In existing MADM methods, when setting different attribute weights, the ranking of alternatives are different. And ranking range can be used to measure a lower bound and an upper bound of rankings of alternatives with the change of the attribute weights. Also, in some real MADM problems, the information on attribute weights may be unknown or partially known, which is called incomplete attribute weight information. Then, this study investigates the ranking range models (RRMs) under incomplete attribute weight information in the selected six MADM methods: Weighted geometric averaging (WGA), Ordered weighted geometric averaging (OWGA), TOPSIS, VIKOR, PROMETHEE and ELECTRE. Particularly, we can construct several 0‐1 mathematical programming models to compute the ranking range of alternatives under incomplete attribute weight information for the selected six MADM methods. Then, two case studies on project investment and Academic Ranking of World Universities (ARWU) are used to justify the validity of the RRMs under incomplete attribute weight information in the selected six MADM methods.

When people express their opinions, they often cannot provide exact opinions but express uncertain opinions, such as numerical interval opinions. Moreover, due to the differences in the cultural background and character of agents, people who encounter numerical interval opinions often show different uncertainty tolerances. By taking different numerical interval opinions and different uncertainty tolerances into account, in this article, we propose a numerical interval opinion dynamics model to investigate the process of forming collective opinions in a group of interaction agents under an uncertain and social network context. We propose the theoretical analysis and algorithms to identify the stable agents whose opinions will be becoming stable and the oscillation agents whose opinions will always be fluctuating in the opinion evolution process. Furthermore, we study the conditions under which a consensus opinion can be built among the stable agents and estimate the opinion ranges of the oscillation agents. Finally, numerical examples and analysis are used to show the feasibility and effectiveness of the proposed theories and algorithms.

Opinion dynamics are investigated extensively to describe the process of opinion formation in groups of individuals. Most of the existing opinion dynamics models assume that the individuals express numerical opinions. However, this assumption does not consider the fact that people often express their opinions in a linguistic way. Particularly, for linguistic opinions, an important point to be highlighted in computing with words (CW) is that words mean different things to different people. In this article, following the idea of personalized individual semantics (PIS) model, we propose the PIS-based linguistic opinions dynamics model (PIS-LOD model) in the framework of bounded confidence effects. Then, some desired properties in the PIS-LOD model are discussed in detail. Furthermore, we design the detailed simulation experiments to show that the individuals' familiarity (which refers to the knowledge on others' semantics) and the PISs' differences have a great influence on the stabilized time, distributions of the extreme and moderate opinions, linguistic opinions distribution, semantics distribution, number of clusters, consensus reaching and the extremely small clusters. The results in this article are very helpful for us to understand the process of group opinion formation in a linguistic context.

Nowadays, online social networks, such as “Facebook” and “WeChat,” facilitate the expression, diffusion, and interactions of individuals' opinions regarding various issues. In this environment, individuals' opinions are liable to be influenced by others and then evolve over the time. In this paper, we propose an approach based on minimum adjustments to manage the consensus in the group decision making (GDM) with opinions evolution. First, inspired by the idea proposed in DeGroot model, we establish a novel GDM model with opinions evolution, and then discuss its consensus conditions. Based on this, we propose an algorithm to achieve the network partition, and then provide a consensus model with minimum adjustments to obtain the optimal adjusted initial opinions and collective consensus opinion. Finally, we provide a numerical example to demonstrate the feasibility and effectiveness of the proposed theoretical results, and design comparative simulations to explore the effects of the opinions evolution on the final consensus solution.

In the bounded confidence model, it is widely known that individuals rely on the opinions of their close friends or people with similar interests. Meanwhile, the decision maker always hopes that the opinions of individuals can reach a consensus in a required time. Therefore, with this idea in mind, this article develops a consensus reaching model with time constraints and minimum adjustments in a group with bounded confidence effects. In the proposed consensus approach, the minimum adjustments rule is used to modify the initial opinions of individuals with bounded confidence, which can further influence the opinion evolutions of individuals to reach a consensus in a required time. The properties of the model are studied, and detailed numerical examples and comparative simulation analysis are provided to justify its feasibility.

Managing comparative linguistic expressions (CLEs) information is a key issue in group decision-making (GDM). A transformation approach has been previously defined to convert CLEs into hesitant fuzzy linguistic terms sets (HFLTSs). However, it is noted that the occurring possibilities of the linguistic terms in the HFLTSs are assumed equal. This assumption might sometimes not capture the real opinions of the decision makers. Linguistic distribution assessments (LDAs) are an effective way to deal with this issue. This paper develops a linguistic distribution-based optimization approach for converting CLEs into LDAs, in which we assume that decision makers provide their opinions using preference relations with CLEs. Particularly, the proposed optimization approach is based on the use of a consistency-driven methodology, which seeks to minimize the inconsistency level of LDA preference relations obtained by transforming the original CLE preference relations elicited from decision makers. The linguistic distribution-based optimization approach is further developed to transform CLEs into interval LDAs to increase their flexibility. Moreover, society and technology trends make it possible to involve and manage large groups of decision makers in GDM environment. Therefore, a large-scale GDM framework with CLE information is designed based on the linguistic distribution-based optimization approach. To justify the effectiveness and applicability of the proposed methodology, it is applied to solve a real large-scale GDM problem, pertaining the selection of the best sustainable disinfection technique for wastewater reuse projects. A comparison against a baseline method is likewise provided to highlight the advantages and innovations of our proposal.

Pursuing a consensus-based solution in group decision making has been widely studied. However, most of the existing consensus models have overlooked one important issue, that is, the decision makers' willingness to accept these suggestions in a consensus reaching process. This issue is explained by the bounded confidence model in the field of opinion dynamics. In this line, this study develops an optimization-based consensus model that considers individual bounded confidences and provides decision makers with more acceptable suggestions based on their bounded confidences. Finally, a numerical example is presented to show the use of the model.

Different feedback mechanisms have been developed in large-scale group decision-making (GDM) to provide the decision-makers with advices for preference adjustment with the aim of improving the group consensus level. However, the willingness of the decision-makers to accept these advices is rarely considered in the extant feedback mechanisms. In the field of opinion dynamics, this issue is studied by the bounded confidence model, which shows that the decision-makers only consider the preferences that differ from their own preferences not more than a certain confidence level. Following this idea, this article proposes a large-scale consensus model with a bounded confidence-based feedback mechanism to promote the consensus level among decision-makers with bounded confidences. Specifically, this feedback mechanism classifies the decision-makers into different clusters and provides the corresponding clusters with more acceptable advices based on a bounded confidence-based optimization approach. Finally, through the numerical example and the simulation analysis, the use of the model is introduced, and the effectiveness of the model is justified.

When people express their opinions about a certain issue, they often give uncertain opinions rather than exact opinions. Particularly, these uncertain opinions will evolve in social networks. Therefore, in this paper, we focus on investigating uncertain opinion formation with social networks under bounded confidence. Specifically, we define the uncertain opinions by numerical interval opinions, whose ranges are between zero and one, and the larger width of numerical interval opinions means the more uncertainty of the opinions. Meanwhile, we describe social network structures by ER random graphs with different agents' scales and network connected probabilities. Then, we present the detailed simulation experiments to reveal the strong impact of social network structures on uncertain opinion formation. Simulation results show that: 1) larger agents' scales will yield the smaller ratios of agents expressing the uncertain opinions and larger average widths of uncertain opinions; 2) the average stable time starts increasing and then decreases with the increase in the network connected probabilities; and 3) larger network connected probabilities will yield less opinion clusters and the smaller ratios of the extremely small clusters in all clusters. The obtained results are helpful for the government and public opinion management departments to understand and manage uncertain public opinion evolution effectively.

In multiple attribute decision making (MADM), strategic weight manipulation is understood as a deliberate manipulation of attribute weight setting to achieve a desired ranking of alternatives. In this paper, we study the strategic weight manipulation in a group decision making (GDM) context with interval attribute weight information. In GDM, the revision of the decision makers' original attribute weight information implies a cost. Driven by a desire to minimize the cost, we propose the minimum cost strategic weight manipulation model, which is achieved via optimization approach, with the mixed 0-1 linear programming model being proved appropriate in this context. Meanwhile, some desired properties to manipulate a strategic attribute weight based on the ranking range under interval attribute weight information are proposed. Finally, numerical analysis and simulation experiments are provided with a twofold aim: 1) to verify the validity of the proposed models and 2) to show the effects of interval attribute weights information and the unit cost, respectively, on the cost to manipulate strategic weights in the MADM in a group decision context.