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.

The inverse analysis of the graph model for conflict resolution (GMCR) determines the preferences required to ensure that a desired resolution is an equilibrium. However, existing studies have yet to examine the transition mechanism from the current state to the desired resolution. To address this issue, this study introduces an integrated minimum cost conflict mediation path (I-MCCMP) model within the GMCR framework. This model not only identifies the necessary preferences to establish the desired resolution as an equilibrium but also determines the optimal conflict mediation path to achieve it. To demonstrate its applicability, the proposed model is applied to the real-world Lake Gisborne Conflict.

Nowadays the emergence of digital technology and artificial intelligence brings a lot of extra information and convenience for group decision activities, such as rich historical decision data, mobile decision platform, artificial interactive device and others. However, it also results in high complexity and uncertainty of the decision environments, modeling and analysis, and the preference of decision maker is crucial to the design of group decision and consensus mechanism. Meanwhile, the information transparency results in more attention being paid to fairness concerns in group decision making. Based on the historical group decision data, this study proposes a novel minimum cost consensus model with preference learning and fairness concerns (MCCM-PLFC), which characterizes the modification motivation of decision makers in a utility-driven way. A preference learning framework is first proposed to determine the parameters of utility functions of decision makers, where fairness concerns are modeled based on the return bias among the decision makers. Then, in a Stackelberg game consensus framework, the MCCM-PLFC is constructed to minimize the consensus cost of the moderator with consideration of the utility-driven behavior of decision makers. In addition, a best-response update algorithm is designed to solve the equilibrium solution among the decision makers and an adaptive differential evolution algorithm is used to solve the MCCM-PLFC. Finally, a throughout of experimental study is performed to quantify the validity of the proposed consensus model.

The opinions of experts often exhibit initial variance in group decision-making process due to the differences in background, knowledge, stance, and other influential factors. Thus, an incentive mechanism is critical to motivate experts to adjust individuals’ opinions and achieve a consensus solution. The incentive mechanism is usually costly and included in the consensus reaching process (CRP), and its effect relies on the behavior interaction between the moderator and experts. Within the Stackelberg game framework, we present the maximum-experts and minimum-cost consensus models with multiple incentives and cost budget. First, from a two-side perspective of the moderator and experts, a consensus model with maximum-return modification and maximum-experts feedback (MRMECM) is built to pursue the maximum number of consensus experts under an established cost budget, where the incentive mechanism is realized with the allocation of unit return to each return-driven expert. Then, a double incentive mechanism (DIM) is designed with the modification and shared return incentives. Subsequently, the MRMECM with DIM is constructed to improve the utilization efficiency of cost budget. Finally, the DIM module is integrated into the maximum-return modification and minimum-cost feedback consensus model (MRMCCM), which aims to minimize the consensus cost in the feedback mechanism. All the proposed consensus models are built as bi-level programming models in a unified Stackelberg game framework. We propose a hybrid approach that combines the best-response update strategy with the differential evolution (DE) algorithm to solve the equilibrium solutions. Consequently, several experimental studies are performed to validate the effectiveness of the proposed models.

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.

In the large-scale group decision making (LSGDM) problems, some experts may adopt strategic manipulation behaviors which can be reflected in trust and preference values. These behaviors can bias or hinder the large-scale consensus reaching process (LCRP). This paper proposes a novel consensus framework to deal with these strategic manipulation behaviors from the perspective of historical data of trust and preference values in LSGDM. In the proposed consensus framework, the experts are classified into several clusters using the historical data of trust and preference values. Next, the strategic manipulation behaviors of trust and preference values of clusters are identified respectively. Then, we take the penalty strategy against experts in clusters with two kinds of strategic manipulation behaviors by updating experts' weights. Simulation and comparison studies are employed to show the validity of the proposed consensus framework against traditional frameworks for managing strategic manipulation behaviors in LSGDM.

Failure modes and effect analysis (FMEA) is a promising reliability management approach widely used to prioritize the failure modes. Different backgrounds and levels of knowledge of FMEA participants may lead to substantial variation between their risk ratings, making the implementation of a consensus mechanism within FMEA to assist FMEA participants in reaching acceptable collective risk ratings worthwhile. Since social trust has an influence on individual's willingness to change risk ratings, this study discusses the role of social-trust-driven rating modifications willingness in prioritizing failure modes during the consensus reaching process. FMEA participants’ rejection of rating modifications suggestions significantly different from their trusted FMEA participants’ risk ratings is formulated as the basis assumption. Based on this assumption, a two-stage social-trust-driven consensus model is proposed to assist FMEA participants with linguistic distribution assessment in reaching consensus willingly. The proposed framework implements a social-trust-driven consensus model with a minimum number of rating modifications in its first stage and a social-trust-driven consensus model with a minimum distance of rating modifications in its second stage. A case study, related to the reliability management of automatic transmission of new energy vehicles, and a simulation analysis are presented and analyzed to validate the proposed FMEA approach.

Consistency is usually associated with transitive properties, among which the additive transitivity is one of the most popular methods. Although various linguistic additive consistency measurements have been proposed in the literature, the existing additive consistency measurements are usually directly proposed without a general definition with axiomatic design, and thus there is a lack of axiomatic comparisons among them. Moreover, the existing additive consistency measurements seldom consider the general case of personalized individual semantics (PIS) where a word has different numerical meanings to different people. In this study, we aim to develop a comparative study for the linguistic additive consistency measurements in the PIS context from an axiomatic perspective. Specifically, we first revisit the most widely used additive consistency measurements of simple linguistic preference relations (SLPRs) and hesitant fuzzy linguistic preference relations (HFLPRs) in the PIS context. Next, we present several rational axioms that the additive consistency measurements with PIS in SLPRs and HFLPRs should satisfy. Finally, we compare these additive consistency measurements by verifying the satisfaction of the proposed axioms. The results in this article show that the existing linguistic additive consistency measurements behave differently from the axiomatic perspective: they perform well in SLPRs but cannot satisfy all axioms in HFLPRs.

The graph model is devoted to game conflicts arising from incongruent pursued objectives among conflicting parties. Considering that each conflicting party is composed of multiple individuals, preference conflicts stemming from differing cognitive levels and knowledge backgrounds exist among internal individuals. This scenario simultaneously involving game conflicts and preference conflicts is termed dual conflict decision-making problem. Tailored to effectively address this problem, this study proposes an enhanced graph model that incorporates internal consensus and external stability. The best-worst method, incorporating comparative linguistic expressions, is devised to effectively elicit individual preferences over game states. To mitigate preference conflicts inherent to internal individuals within conflicting party concerning game states, a consensus reaching model minimizing preference information loss is introduced. By this way, collective preferences are obtained. Based on these, the concept of “game consensus” is proposed to manage the game conflicts and the diverse behaviors exhibited by conflicting party. Finally, a case study regarding price conflict within a dual-channel supply chain, accompanied by a comparative analysis, is presented to validate the effectiveness of the proposal. Compared to existing graph model, the proposal effectively grapples with consensus issues and heterogeneous behaviors within conflicting parties, making it more valuable in practice.

In group decision making (GDM), opinion dynamics is a useful tool to investigate consensus formation. Notably, consensus convergence speed is of key importance to manage the consensus formation in GDM with opinion dynamics. Recently, social network DeGroot (SNDG) model has been widely used in opinion dynamics. Based on this, this article dedicates to study how agents’ high self-confidence levels affect the consensus convergence speed in SNDG model. Interestingly, using theoretical analysis, we prove that: 1) the speed of consensus reaching is subject to the largest self-confidence level of opinion followers and 2) the speed of consensus reaching is also subject to the top two self-confidence levels of opinion leaders. Furthermore, through extensive simulation’, we find that the theoretical results are robust to the topological structure and the size of social networks.

A key issue in social network group decision making (SNGDM) is to determine the weights (i.e., social influences) of individuals. Notably, in some SNGDM scenarios, the social influences of individuals may evolve over time. Meanwhile, consensus reaching is another important issue in SNGDM. In this article, we are dedicated to disclosing the natural evolution process of social influence, and further to discussing the consensus reaching issue in SNGDM. First, we establish the social influence evolution model, where the individual's social influence is obtained by combining his/her intrinsic influence and network influence. Afterward, we design the consensus reaching process based on social influence evolution (CRP-SIE) to assist the individuals to reach a consensus. Furthermore, we use a hypothetical application to show the applicability of the proposed CRP-SIE. Finally, simulation analysis is adopted to investigate the effects of social influence evolution on consensus reaching in SNGDM, and comparative analysis is conducted to demonstrate the advantages of our proposal.

This study proposes a minimum cost consensus-based failure mode and effect analysis (MCC-FMEA) framework considering experts' limited compromise and tolerance behaviors, where the first behavior indicates that a failure mode and effect analysis (FMEA) expert might not tolerate modifying his/her risk assessment without limitations, and the second behavior indicates that an FMEA expert will accept risk assessment suggestions without being paid for any cost if the suggested risk assessments fall within his/her tolerance threshold. First, an MCC-FMEA with limited compromise behaviors is presented. Second, experts' tolerance behaviors are added to the MCC-FMEA with limited compromise behaviors. Theoretical results indicate that in some cases, this MCC-FMEA with limited compromise and tolerance behaviors has no solution. Thus, a minimum compromise adjustment consensus model and a maximum consensus model with limited compromise behaviors are developed and analyzed, and an interactive MCC-FMEA framework, resulting in an FMEA problem consensual collective solution, is designed. A case study, regarding the assessment of COVID-19-related risk in radiation oncology, and a detailed sensitivity and comparative analysis with the existing FMEA approaches are provided to verify the effectiveness of the proposed approach to FMEA consensus-reaching.

When making decisions, individuals often express their preferences linguistically. The computing with words methodology is a key basis for supporting linguistic decision making, and the words in that methodology may mean different things to different individuals. Thus, in this article, we propose a continual personalized individual semantics learning model to support a consensus-reaching process in large-scale linguistic group decision making. Specifically, we first derive personalized numerical scales from the data of linguistic preference relations. We then perform a clustering ensemble method to divide large-scale group and conduct consensus management. Finally, we present a case study of intelligent route optimization in shared mobility to illustrate the usability of our proposed model. We also demonstrate its effectiveness and feasibility through a comparative analysis.

Most existing opinion formation processes apply one opinion dynamics model. However, this article combines opinion formation and complex networks to innovatively develop two new opinion dynamics models to more realistically describe the opinion evolution process: 1) an opinion similarity mixed (OSM) model and 2) a structural similarity mixed (SSM) model, both of which include characteristics from the DeGroot model and the Hegselmann–Krause bounded confidence model. In addition, the strong and weak relations between individuals are considered. The network dynamically changes by two developed network updating algorithms based on opinion similarity and structural similarity. Simulations are then conducted using artificial and real-world networks, which are Erdös-Rényi random networks, random regular networks, scale-free networks, and the Twitter network. It is found that compared with static networks, the opinion evolution in dynamic networks produces fewer opinion clusters and smaller opinion variances. The dynamic network mechanism reduces the weak relations between agents and improves the global clustering coefficient in the ER random networks but not in the Twitter network, which means that the network topology has an impact on results. Therefore, it is concluded that agents’ subjective behaviors significantly influence the outcome of opinion evolution and networks, which is consistent with real life.

In group decision making (GDM), considering the individuals’ satisfaction (i.e., utilities) about the consensus result is important. However, this issue is ignored by the extant minimum cost consensus models (MCCMs). Altruism describes a way of behaving that the individuals unselfishly concern for the utilities of others. Inspired by this, in this article we discuss the minimum cost consensus problem in social network GDM (SNGDM), where the utility constraints based on altruism are first incorporated. In SNGDM, altruism is reflected as the individuals will take both their personal utility and the utilities of others who connected with them into account to form their total utility. Based on this, we first define the individuals’ altruistic utility functions in the consensus process of SNGDM, and we propose several interesting properties of the proposed utility function. Afterward, we present a novel MCCM in SNGDM with opinion dynamics and altruistic utility (MCCM-OD-AU). Notably, in the proposed model, the deviation of opinion adjustment in the consensus process is measured as a psychological distance perceived by the individuals. Furthermore, some simulation studies and comparative analysis are conducted to investigate the effects of the altruistic behavior and the psychological distance of opinion adjustment on the consensus reaching results. Finally, two numerical studies, including an illustrative example and an application in energy-saving target formulation with real-world social network data, are provided to justify the performance of our proposal.

Successful new product development advocates for collaboration among different institutions in which technology transfer dispute widely exists. Although several studies have discussed conflict modeling and resolution in technology transfer disputes, scant research attempted to model third-party (or mediator) mediation, let alone develop effective approaches to minimize cost in the conflict resolution process. This study uses a graph model and minimum cost to investigate the conflict resolution and mediation in the technology transfer dispute of new product collaborative development. On the one hand, the conflict in technology transfer of new product collaborative development is modeled using the graph model theory, in which the stakeholders (or decision makers), their options, the feasible states, and the preferences of decision makers are analyzed. On the other hand, an inverse graph model with minimum cost is designed to tackle the problem of specifying which decision-makers’ preferences lead to a desired solution, thereby making it easier for a mediator or other third party to influence the course of the conflict. In the inverse graph model with minimum cost, two 0–1 mixed linear approaches are constructed to judge the Nash and general merataionality stabilities within the graph model, and several optimization-based models that minimize mediation cost are designed for the mediator to guide the technology transfer conflict resolution process to achieve the desired solution. Finally, the proposed methodology is applied to a technology transfer dispute case study.

In group decision making (GDM), consensus level is regarded as a critical criterion to measure the effectiveness and availability of the final group decision solution. Consensus model is aimed at conducting the decision group to reach agreement through the process of group negotiation, advice feedback, and opinion modification, which is time-consuming and rests with the willingness and behavior of individual decision makers. Thus, to guide the shift in the opinions of decision makers within a limited time, it is essential to design an effective, interpretable, and fair consensus mechanism in GDM, which is particularly vital when a mass of decision makers (e.g., more than 30) are involved in the decision process, viz., we encounter a large-scale GDM (LSGDM). With the involvement of information granulation, this study presents a rule-based consensus model in LSGDM by optimally allocating the level of information granularity to each decision maker. The opinions of decision makers in LSGDM are divided into different clusters by engaging the fuzzy$C$-means method. Inspired by a generic fuzzy rule-based model, the radius of the individual preference granule (PG) is calculated by a weighted linear combination of the granularity levels allocated to the clusters. Then, a consensus model with the optimal allocation of information granularity (CMOIG) is built to determine the granularity level for each cluster by minimizing the sum of radii of individual PG. An interactive consensus reaching process is proposed with the proposed CMOIG and fuzzy modification rules. The CMOIG and fuzzy modification rules simultaneously guarantees high efficiency and interpretability, and the generation method of PGs leads to high fairness due to low discrepancy among the decision group. Finally, numerical and comparative experiments are conducted in detail to verify the validity and superiority of the presented models in terms of the efficiency, interpretability, and fairness.

In the analytic hierarchy process (AHP), the reciprocal matrix is generated based on the pairwise comparisons completed among all the alternatives or attributes under consideration. To ensure reliability and validity of the decision solution, a certain modification of entries of the matrix is usually needed to improve the consistency of the reciprocal matrix. This study aims to present a consistency improvement method by admitting some level of information granularity in the evaluation process. This gives rise to a granular rather than numeric matrix of pairwise comparisons. First, with a given average level of information granularity, we present an optimal granularity model that is characterized by maximal consistency. One can maximize the consistency degree by invoking a process of allocation of information granularity across the corresponding modifications of the reciprocal matrix. Based on the optimal granularity model, an interactive consistency improvement process is presented with the involvement of the decision maker. Then, an adaptive differential evolution algorithm is applied to optimize entries of the modified reciprocal matrix. Detailed experiments along with a thorough comparative analysis are completed to demonstrate the effectiveness of the proposed method.

In linguistic decision-making problems, there may be cases when decision makers will not be able to provide complete linguistic preference relations. However, when estimating unknown linguistic preference values in incomplete preference relations, the existing research approaches ignore the fact that words mean different things for different people, that is, decision makers have personalized individual semantics (PISs) regarding words. To manage incomplete linguistic preference relations with PISs, in this article, we propose a consistency-driven methodology both to estimate the incomplete linguistic preference values and to obtain the personalized numerical meanings of linguistic values of the different decision makers. The proposed incomplete linguistic preference estimation method combines the characteristic of the personalized representation of decision makers and guarantees the optimum consistency of incomplete linguistic preference relations in the implementation process. Numerical examples and a comparative analysis are included to justify the feasibility of the PISs-based incomplete linguistic preference estimation method.

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.

Inspired by the continuous opinion and discrete action (CODA) model, bounded confidence and social networks, the bounded confidence evolution of opinions and actions in social networks is investigated and a social network opinions and actions evolutions (SNOAEs) model is proposed. In the SNOAE model, it is assumed that each agent has a CODA for a certain issue. Agents' opinions are private and invisible, that is, an individual agent only knows its own opinion and cannot obtain other agents' opinions unless there is a social network connection edge that allows their communication; agents' actions are public and visible to all agents and impact other agents' actions. Opinions and actions evolve in a directed social network. In the limitation of the bounded confidence, other agents' actions or agents' opinions noticed or obtained by network communication, respectively, are used by agents to update their opinions. Based on the SNOAE model, the evolution of the opinions and actions with bounded confidence is investigated in social networks both theoretically and experimentally with a detailed simulation analysis. Theoretical research results show that discrete actions can attract agents who trust the discrete action, and make agents to express extreme opinions. Simulation experiments results show that social network connection probability, bounded confidence, and the opinion threshold of action choice parameters have strong impacts on the evolution of opinions and actions. However, the number of agents in the social network has no obvious influence on the evolution of opinions and actions.

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.

Existing consensus models focus on improving the group consensus level, but ignore whether a higher group consensus level means higher mutual acceptance of decision makers. In the field of opinion dynamics, the bounded confidence model asserts that the decision makers will accept the preferences of others within a neighborhood of theirs with width a certain confidence level. Inspired by this research methodology, this article develops a consensus model to address the acceptance issue based on individual bounded confidences. Specifically, a bounded confidence-based consensus measure is designed to measure the level of group mutual acceptance, and a multi-stage optimization feedback mechanism based on individual bounded confidences is proposed to maximize the group mutual acceptance and minimize the amount of preference adjustment. A numerical example and a simulation analysis are included to illustrate the use of the model and to justify its effectiveness, respectively.

Social network group decision making (SNGDM) has emerged as a new decision tool to effectively model the social trust relationships among decision makers. The impact of the social trust relationships on assessments-modifications in the consensus reaching in the SNGDM is seldom considered. This study aims at addressing this issue. The main starting point is the assumption that a decision maker will not be willing to accept the assessments-modifications suggestions that significantly differ from his/her trusted decision makers’ assessments in a social trust network. Thus, this study proposes a social trust driven minimum adjustments consensus model (STDMACM) for SNGDM. Simultaneously, a social trust driven consensus maximum optimization model (STDCMOM) is proposed for maximizing the consensus level among decision makers under the above assumption. Based on both STDCMOM and STDMACM, an interactive consensus reaching process is presented, in which the assessments-modifications suggestions generated from the STDMACM are used, when the maximum consensus level obtained from STDCMOM is acceptable, as the references for guiding the consensus reaching; otherwise, assessments-modifications suggestions are generated from the designed STDCMOM. The validity of the social trust driven consensus reaching process with respect to its consensus convergence rate and consensus success ratio is verified with a simulation and comparison analysis.

Linguistic multiple attribute decision making (LMADM) has been widely used in different decision contexts to achieve a final solution, e.g., obtaining the ranking of alternatives. Evaluation information using linguistic scales in LMADM is usually expressed in simple discrete linguistic terms. In this article, we consider that each simple linguistic term in LMADM actually has a continuous representation (denoted as a linguistic two tuples) whose rounding operation matches the linguistic term. Under such conditions, the ranking of alternatives based on the simple linguistic terms may not be consistent with that based on the linguistic two tuples. Thus, this article studies the linguistic scale ranking consistency issue in LMADM under seven classical and commonly used decision rules: weighted averaging, ordered weighted averaging, weighted geometric averaging, ordered weighted geometric averaging, preference ranking organization method for enrichment evaluation, technique for order preference by similarity to an ideal solution, and elimination et choice translating reality. We first define the concept of the linguistic scale ranking consistency in LMADM. Afterward, several consistency conditions are presented analytically for the selected decision rules to guarantee the linguistic scale ranking consistency. Finally, we present the detailed theoretical and simulation-based comparisons. The theoretical comparisons show the roles of decision rules and granularity in the consistency conditions, and the simulation-based comparisons demonstrate the performance of the selected decision rules in defending against the ranking inconsistency in LMADM.

In computing with words, it has been stressed that words mean different things for different people, which entails that decision makers (DMs) have personalized individual semantics (PISs) attached to linguistic expressions in linguistic group decision making (GDM). In particular, the PISs of DMs are not fixed, and they will be changing during the consensus building process, which indicates the necessary of continual PIS learning. Therefore, in this article, we propose a continual PIS-learning-based consensus approach in linguistic GDM. Specifically, a continual PIS learning model with the consistency-driven methodology is proposed to update the PISs taking into account all the linguistic preference data given by DMs during the consensus process. Then, the consensus measurement and feedback recommendation based on PIS are developed to detect the consensus process. Finally, numerical examples and simulation analysis are presented to illustrate and justify the use of the continual PIS-learning-based consensus 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.

Offline map matching identifies corresponding roads to a GPS trajectory represented by a series of recorded geographic coordinates (GPS points) to the road network. This paper defines matching error as cost on the corresponding road-link to matched GPS points and formulates the offline map matching problem as a shortest path problem with resource constraints. By regarding matched points on one link as a type of resource consumed, the resource constraint indicates that the number of matched GPS points equals the total number of points in the given trajectory. We propose an offline map matching algorithm based on shortest paths by calculating the matching error on each link and extending the classic label-setting shortest path algorithm to find the path with the minimum total matching error for all GPS points. We use real-world taxi trajectories to compare our algorithm with three state-of-the-art map matching algorithms. Our algorithm outperforms all benchmark algorithms in terms of both matching accuracy and computational efficiency. Our algorithm achieves greater matched length (5.36 to 12.27% larger) and lower mis-matched length (3.72 to 75.30% smaller) at a very high matching speed (60.59 points per second on average over thirteen sampling intervals).

With the development of the information and Internet technology, the public opinions with big data will rapidly emerge in an online-offline social network, and an inefficient management of public opinions often will lead to the security crisis for either firms or governments. To unveil the interaction mechanism among a large number of agents between the online and offline social networks, in this paper we propose the public opinion dynamics model in an online-offline social network context. Next, in the theory aspect we investigate the analytical conditions to form a consensus in the public opinion dynamics model. Furthermore, we conduct the extensive simulations to investigate how the online agents impact the dynamics of public opinion formation, and unfold that the online agents shorten the steady-state time, decrease the number of opinion clusters, and smoothen the opinion changes in the opinion dynamics. The increase in the size of the online agents often enhances these effects. The results in this paper can provide a basis for the management of the public opinions in the Internet age.

In this article, we propose the concept of flexible linguistic preference relations (FLPRs), in which the flexible linguistic expressions, a more flexible way to form linguistic expressions, are employed. Further, we present a method to rank alternatives based on preference information in FLPRs by exploring the linguistic distribution (LD) and priority-based approximation (PA) of FLPRs. In the LD-based approximation, we first present a two-stage optimization process to approximate FLPRs to distribution linguistic preference relations (DLPRs) following the principles of minimum preference loss and maximum consistency. Then in the PA process, we derive priority vectors from the DLPRs with some desired properties. Finally, a comparative analysis of the priority vectors derived from different kinds of linguistic preference relations is presented to illustrate our proposal.

In group decision-making, the consensus reaching process (CRP) is usually designed as an interactive discussion and negotiation process to guide decision-makers in modifying their opinions. In CRP, consensus measure may mean agreement among most but not all the decision-makers, and the actual or virtual moderator may provide modification suggestions with his/her own interests besides the consensus-oriented suggestions. With consideration of fuzzy consensus and private interest of the moderator, this article presents two optimal feedback mechanisms with private interest (PIFM), which aim to minimize the consensus cost with an established consensus threshold and maximize the fuzzy consensus measure with a given cost budget. First, the fuzzy consensus measure is defined by means of specificity and coverage of information granule. Then, two PIFMs based on the minimum consensus cost and maximum fuzzy consensus are proposed to obtain the optimal modified individual opinions, and these two PIFMs are equivalently transformed into the mixed 0-1 programming models. In what follows, we develop an interactive CRP based on the proposed PIFMs, where the optimal modified individual opinions are considered as the modification references in the feedback mechanism. Finally, detailed experiment studies are conducted to demonstrate the effectiveness and validity of the proposed 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.

In this article, a novel framework to prevent manipulation behavior in consensus reaching process under social network group decision making is proposed, which is based on a theoretically sound optimal feedback model. The manipulation behavior classification is twofold: first, “individual manipulation” where each expert manipulates his/her own behavior to achieve higher importance degree (weight); and second, “group manipulation” where a group of experts force inconsistent experts to adopt specific recommendation advices obtained via the use of a fixed feedback parameter. To counteract “individual manipulation,” a behavioral weights assignment method modeling sequential attitude ranging from “dictatorship” to “democracy” is developed, and then a reasonable policy for group minimum adjustment cost is established to assign appropriate weights to experts. To prevent “group manipulation,” an optimal feedback model is investigated where objective function is the individual adjustments cost and constraints related to the group threshold of consensus. This approach allows the inconsistent experts to balance group consensus and adjustment cost, which enhances their willingness to adopt the recommendation advices and consequently the group reaching consensus on the decision-making problem at hand. A numerical example is presented to illustrate and verify the proposed optimal feedback model.

The analytic hierarchy process (AHP) is widely employed to guide the decision-maker to rank or evaluate the alternatives in decision activities. Its fuzzy set-based version, i.e., the fuzzy AHP, has also been widely studied and applied since its inception. The essential distinction between the AHP and fuzzy AHP comes from the diverse transformation methods between the linguistic and numeric judgments. In this article, we conduct a thorough comparative study between the AHP and fuzzy AHP methods in the framework of two linguistic models, i.e., the linguistic model based on the membership functions and two-tuple linguistic model. First, four AHP and three fuzzy AHP methods are revisited with the involvement of two linguistic models. Then, the comparison criteria are involved by calculating the cardinal or ordinal deviation between the original information and decision solutions, and the effects of the transitivity of the reciprocal matrix are also discussed in the comparative study. Finally, the detailed experiments along with a thorough comparative analysis are conducted based on the random and publicly available data to show the difference between the AHP and fuzzy AHP methods.

Comparative linguistic expression preference relations (CLEPRs) are an effective tool to represent uncertain opinions of decision makers in group decision making (GDM). Nevertheless, multiple self-confidence levels are not considered by existing research on CLEPRs. Thus, this article proposes CLEPRs with self-confidence by considering multiple self-confidence levels and presents a way to measure their consistency level. Meanwhile, personalized individual semantics (PIS), indicating that words mean different things for different people, have been highlighted and investigated in the GDM with linguistic assessment information. Considering PIS in comparative linguistic expressions, this article proposes an optimization model based on the consistency-driven methodology to assess individual semantics in CLEPRs with self-confidence. Particularly, the PIS are described and addressed by setting different numerical scales of linguistic terms for different decision makers. Finally, an optimization-based consensus model is proposed to obtain a consensual collective solution, which seeks to minimize the information loss between the decision makers' preference relations with self-confidence and corresponding individual preference vectors.

Quite often, complex systems or phenomena are observed from various points of view yielding the particular subsets of data usually being composed of locally available attributes. Such datasets give rise to individual models. As is reflective of the local behavior of the system (global data), each model can produce different, albeit similar results. A critical issue is to aggregate the results coming from the individual models. In virtue of the diversity of the produced results, the aggregation process has to be reflective of this variety. Equally important is a way of quantifying the diversity of the individual results. In this article, we provide an efficient and original way of aggregation of the results by engaging a principle of justifiable granularity and in this manner leading to interval-valued results summarizing the results produced by a collection of models. We develop an overall design process and discuss the associated optimization mechanism leading to a granular fuzzy model of a global nature. The detailed scheme of the principle of justifiable granularity is discussed along with the related performance indexes; in particular, two modes of design of information granules are investigated. The quality of the granular model is quantified with the aid of the criteria of coverage and specificity.

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.

To date, a large number of consensus reaching processes (CRPs) have been reported in group decision making (GDM). Trust relationships should be an essential element in interactions among a group of individuals, leading to the evolution of individuals’ preferences. Therefore, in this article, we present a trust relationships CRP with a feedback mechanism which consists of two approaches of facilitating consensus reaching: 1) the leader-based preference adjustment and 2) the trust relationships improvement. In the trust relationships CRP, we build a bridge between opinion dynamics and GDM to highlight the role of the leaders and trust relationships improvements in the GDM problems. Furthermore, we present a new strategic manipulation issue, called trust relationship manipulation, and discuss some clique-based strategies to manipulate trust relationships to obtain the desired ranking of the alternatives in the GDM problems. Finally, the detailed simulation experiments are proposed to justify our proposal.

This article provides a brief tour through the main fuzzy and linguistic decision-making trends, studies, methodologies, and models developed in the last 50 years. Fuzzy and linguistic decision-making approaches allow to address complex real-world decision problems where humans exhibit vagueness, imprecision, and/or use natural language to assess decision alternatives, criteria, etc. The aim of this article is threefold. First, the main fuzzy set theory and computing with words-based representation paradigms of decision information, with their different levels of expressive richness and complexity, are reviewed. Second, three core decision-making frameworks are examined: 1) multicriteria decision making; 2) group consensus-driven decision making; and 3) multiperson multicriteria decision making. Third, the article discusses new complex decision-making frameworks that have emerged in recent years, where decisions are guided by the “wisdom of the crowd”: their associated challenges are highlighted and considerations on much needed key guidelines for future research in the field are provided.

Different feedback mechanisms have been reported in consensus reaching models to provide advices for preference adjustment to assist decision makers to improve their consensus levels. However, most feedback mechanisms do not consider the willingness of decision makers to accept these advices. In the opinion dynamics discipline, the bounded confidence model justifies well that in the process of interaction a decision maker only considers the preferences that do not exceed a certain confidence level compared to his own preference. Inspired by this idea, this article proposes a new consensus reaching model with personalized feedback mechanism to help decision makers with bounded confidences in achieving consensus. Specifically, the personalized feedback mechanism produces more acceptable advices in the two cases where bounded confidences are known or unknown, and the unknown ones are estimated by a learning algorithm. Finally, numerical example and simulation analysis are presented to explore the effectiveness of the proposed model in reaching consensus.

Various linguistic expressions have been presented to model the flexibility of linguistic preference expressions and to support the consensus reaching in linguistic group decision-making (GDM). In this paper, we propose the concept of flexible linguistic expressions (FLEs) as a general linguistic preference expression format to improve the flexibility of the construction of complex linguistic expressions and the elicitation of linguistic preferences and, then, we develop a new linguistic GDM model with FLEs, referred to as FLE-based GDM (FLEGDM). In the FLEGDM, an FLE aggregation process with accurate constraints is developed to improve the quality (i.e., accuracy) of the collective result as well as guarantee the principle of minimum preference-loss through a mixed 0-1 linear programming model. Meanwhile, the consensus rules with minimum preference-loss are designed to support the consensus reaching process (CRS) in the FLEGDM. Finally, we present the detailed comparative analysis involving different linguistic GDM models to show the advantages of the FLEGDM.

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.

Preference relations have been widely used in group decision-making (GDM) problems. Recently, a new kind of preference relations called fuzzy preference relations with self-confidence (FPRs-SC) has been introduced, which allow experts to express multiple self-confidence levels when providing their preferences. This paper focuses on the analysis of additive consistency for FPRs-SC and its application in GDM problems. To do that, some operational laws for FPRs-SC are proposed. Subsequently, an additive consistency index that considers both the fuzzy preference values and self-confidence is presented to measure the consistency level of an FPR-SC. Moreover, an iterative algorithm that adjusts both the fuzzy preference values and self-confidence levels is proposed to repair the inconsistency of FPRs-SC. When an acceptable additive consistency level for FPRs-SC is achieved, the collective FPR-SC can be computed. We aggregate the individual FPRs-SC using a self-confidence indices-based induced ordered weighted averaging operator. The inherent rule for aggregation is to give more importance to the most self-confident experts. In addition, a self-confidence score function for FPRs-SC is designed to obtain the best alternative in GDM with FPRs-SC. Finally, the feasibility and validity of the research are demonstrated with an illustrative example and some comparative analyses.

On-line platforms foster the communication capabilities of the Internet to develop large-scale influence networks in which the quality of the interactions can be evaluated based on trust and reputation. So far, this technology is well known for building trust and harnessing cooperation in on-line marketplaces, such as Amazon (www.amazon.com) and eBay (www.ebay.es). However, these mechanisms are poised to have a broader impact on a wide range of scenarios, from large scale decision making procedures, such as the ones implied in e-democracy, to trust based recommendations on e-health context or influence and performance assessment in e-marketing and e-learning systems. This contribution surveys the progress in understanding the new possibilities and challenges that trust and reputation systems pose. To do so, it discusses trust, reputation and influence which are important measures in networked based communication mechanisms to support the worthiness of information, products, services opinions and recommendations. The existent mechanisms to estimate and propagate trust and reputation, in distributed networked scenarios, and how these measures can be integrated in decision making to reach consensus among the agents are analysed. Furthermore, it also provides an overview of the relevant work in opinion dynamics and influence assessment, as part of social networks. Finally, it identifies challenges and research opportunities on how the so called trust based network can be leveraged as an influence measure to foster decision making processes and recommendation mechanisms in complex social networks scenarios with uncertain knowledge, like the mentioned in e-health and e-marketing frameworks.

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 linguistic large-scale group decision making (LSGDM), it is often necessary to achieve a consensus. Particularly, when computing with words and linguistic decision, we must keep in mind that words mean different things to different people. Therefore, to represent the specific semantics of each individual, we need to consider the personalized individual semantics (PIS) model in linguistic LSGDM. In this paper, we propose a consensus model based on PIS for LSGDM. Specifically, a PIS process to obtain the individual semantics of linguistic terms with linguistic preference relations is introduced. A consensus process based on PIS, including the consensus measure and feedback recommendation phases, is proposed to improve the willingness of decision makers who follow the suggestions to revise their preferences in order to achieve a consensus in linguistic LSGDM problems. The consensus measure defines two opposing consensus groups with respective acceptable and unacceptable consensus. In the feedback recommendation phase, a PIS-based clustering method to get decision makers with similar individual semantics is proposed. Recommendation rules design a feedback for decision makers with unacceptable consensus, finding suitable moderators from the decision makers with acceptable consensus based on cluster proximity.

The individual consistency and the consensus degree are two basic measures to conduct group decision making with reciprocal preference relations. The existing frameworks to manage individual consistency and consensus degree have been investigated intensively and follow a common resolution scheme composed by the two phases: the consistency improving process, and the consensus reaching process. But in these frameworks, the individual consistency will often be destroyed in the consensus reaching process, leading to repeat the consistency improving process, which is time consuming. In order to avoid repeating the consistency improving process, a consensus reaching process with individual consistency control is proposed in this paper. This novel consensus approach is based on the design of an optimization-based consensus rule, which can be used to determine the adjustment range of each preference value guaranteeing the individual consistency across the process. Finally, theoretical and numerical analysis are both used to justify the validity of our proposal.

Failure mode and effect analysis (FMEA) is an effective risk-management tool, which has been extensively utilized to manage failure modes (FMs) of products, processes, systems, and services. Almost all FMEA models are concerned with how to get a complete risk order of FMs from highest to lowest risk. However, in many situations, it may be sufficient to classify the FMs into several ordinal risk classes. Meanwhile, generating a consensual decision is crucial for the FMEA problem because 1) reaching consensus will enhance the connections among FMEA participants, and 2) a highly accepted group solution to the FMEA problem can be generated. Thus, this study proposes a consensus-based group decision-making framework for FMEA with the aim of classifying FMs into several ordinal risk classes in which we assumed that FMEA participants provide their preferences in a linguistic way using possibilistic hesitant fuzzy linguistic information. In the FMEA framework, a consensus-driven methodology is presented to generate the weights of risk factors. Following this, an optimization-based consensus rule guided by a minimum adjustment distance policy is devised, and an interactive model for reaching consensus is developed to generate consensual FM risk classes. In order to justify its validity of the proposal, our framework is applied for the risk evaluation of proton beam radiotherapy.

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.

In this study, we propose a social network uncertain opinion formation model in the framework of bounded confidence to investigate the process of forming collective opinions in a group of interaction agents under uncertain and social networks context. By taking different uncertain opinions and different uncertainty tolerances into account, the simulations analysis conducted on data describing users' relationships in social media platforms in opinion formation from two aspects: different social networks connection probability and the numbers of agents. Based on simulation analysis, we provide the explanations of the observations obtained.

Large-scale group decision making (LGDM) is becoming more and more common, and how to assure the security and quality of the decision process has become a hot topic. Supply chain risk mitigation is a LGDM problem which involves many stakeholders. In the decision making process, a group of experts aims at reaching a consensus among alternatives in which non-cooperative behaviors often appear. Some experts might designedly form a small alliance and change their preferences in a direction against consensus with the aim to foster the alliance's own interests. In this study, we present a novel large-scale consensus reaching framework based on a self-management mechanism to manage non-cooperative behaviors. In the proposed framework, experts are classified into different subgroups using a clustering method, and they provide their evaluation information, i.e., the multi-criteria mutual evaluation matrices (MCMEMs), regarding the obtained subgroups based on their performance. The subgroups' weights are generated dynamically from the MCMEMs, which are in turn used to update experts' weights. This self-management mechanism allows penalizing the weights of the experts with non-cooperative behaviors. Detailed comparison analysis is presented to verify the validity of the proposed consensus framework.

Computing with Words (CW) is a methodology which takes an essential role in decision making problems, thus the first aim of this paper is to highlight the importance of CW in decision making. There are many proposals of CW models that can be classified in two main fields: approaches based on membership functions and approaches based on qualitative scales. This paper focuses on the qualitative scales since it provides a closer environment to human knowledge and more accurate results avoiding a loss of information. An analysis and discussion of the qualitative scales of linguistic values in CW for decision making is provided. Finally, we highlight the open challenges for the qualitative scales in linguistic decision making.

In social network group decision making (SNGDM), the consensus reaching process (CRP) is used to help decision makers with social relationships reach consensus. Many CRP studies have been conducted in SNGDM until now. This paper provides a review of CRPs in SNGDM, and as a result it classifies them into two paradigms: (i) the CRP paradigm based on trust relationships, and (ii) the CRP paradigm based on opinion evolution. Furthermore, identified research challenges are put forward to advance this area of research.

In large-scale group decision making (GDM), noncooperative behavior in the consensus reaching process (CRP) is not unusual. For example, some individuals might form a small alliance with the aim to refuse attempts to modify their preferences or even to move them against consensus to foster the alliance's own interests. In this paper, we propose a novel framework based on a self-management mechanism for noncooperative behaviors in large-scale CRPs (LCRPs). In the proposed consensus reaching framework, experts are classified into different subgroups using a clustering method, and experts provide their evaluation information, i.e., the multicriteria mutual evaluation matrices (MCMEMs), regarding the subgroups based on subgroups' performance (e.g., professional skills, cooperation, and fairness). The subgroups' weights are dynamically generated from the MCMEMs, which are in turn employed to update the individual experts' weights. This self-management mechanism in the LCRP allows penalizing the weights of the experts with noncooperative behaviors. Detailed simulation experiments and comparison analysis are presented to verify the validity of the proposed framework for managing noncooperative behaviors in the LCRP.

Nowadays, societal and technological trends demand the management of large scale of decision makers in group decision-making (GDM) contexts. In a large-scale GDM, decision makers often have individual concerns and satisfactions, and also they will use heterogeneous preference representation structures to express their preferences. Meanwhile, it is difficult to set the numerical consensus threshold to judge whether a consensus degree can be acceptable or not in the consensus reaching process in a large-scale GDM. This study proposes a novel consensus reaching model for the heterogeneous large-scale GDM with the individual concerns and satisfactions. In this consensus reaching model, a selection process is proposed to obtain the individual preference vectors, to divide decision makers into different clusters, and to yield the preference vector of the large group. Following this, a consensus measure method that considers the individual concerns on alternatives is defined for measuring the consensus degree, and a linguistic approach is developed to measure the individual and collective satisfactions regarding the consensus degree. Finally, a feedback adjustment process is proposed and utilized to help decision makers adjust their preferences. A practical example and a simulation analysis are presented to demonstrate the validity of the proposed consensus reaching model.

With the accessibility to information, users often face the problem of selecting one item (a product or a service) from a huge search space. This problem is known as information overload. Recommender systems (RSs) personalize content to a user's interests to help them select the right item in information overload scenarios. Group RSs (GRSs) recommend items to a group of users. In GRSs, a recommendation is usually computed by a simple aggregation method for individual information. However, the aggregations are rigid and overlook certain group features, such as the relationships between the group members' preferences. In this paper, it is proposed a GRS based on opinion dynamics that considers these relationships using a smart weights matrix to drive the process. In some groups, opinions do not agree, hence the weights matrix is modified to reach a consensus value. The impact of ensuring agreed recommendations is evaluated through a set of experiments. Additionally, a sensitivity analysis studies its behavior. Compared to existing group recommendation models and frameworks, the proposal based on opinion dynamics would have the following advantages: 1) flexible aggregation method; 2) member relationships; and 3) agreed recommendations.

In the multiple-attribute group decision-making (MAGDM), the decision objective is to obtain a complete ranking of the alternatives from best to worst. In the real world, however, obtaining such a complete ranking of alternatives is very time-consuming, and it is sometimes not necessary. There are many MAGDM problems that we need to assign two rank levels only, so as to create a ranking of one subset of alternatives above another subset. This type of MAGDM problem is called a 2-rank MAGDM problem. This paper investigates the 2-rank MAGDM problem under the multigranular linguistic context, and proposes a 2-rank consensus reaching framework with the minimum adjustments. In the consensus reaching framework, a 2-rank selection process is proposed to find the individual and collective 2-rank preference vectors, and a 2-rank consensus reaching process with the minimum adjustments is put forwarded to help decision makers achieve a consensus. To demonstrate the consensus efficiency of the 2-rank consensus reaching model, a comparison analysis between our proposal and the classical MAGDM models is presented. The main improvement of this paper is to provide a flexible framework to constitute a better approximate decision model to real-world MAGDM problems.

In decision making dealing with computing with words, the importance of the statement that words mean different things for different people has been highlighted. In this paper, we focus on personalizing numerical scales of linguistic terms in decision making with hesitant fuzzy linguistic preference relations (HFLPRs). First, an average consistency measure for HFLPRs is provided, and then an optimization-based model to personalize individual semantics via numerical scales is presented, aiming at maximizing the average consistency of HFLPRs. Numerical examples are used to illustrate the proposal.

In some real-world multiple attribute decision making (MADM) problems, a decision maker can strategically set attribute weights to obtain her/his desired ranking of alternatives, which is called the strategic weight manipulation of the MADM. Sometimes, the attribute weights are given with imprecise or partial information, which is called incomplete information of attribute weights. In this study, we propose the strategic weight manipulation under incomplete information on attributes weights. Then, a series of mixed 0-1 linear programming models (MLPMs) are proposed to derive a strategic weight vector for a desired ranking of an alternative. Finally, a numerical example is used to demonstrate the validity of our models.

The linguistic distribution is becoming a popular tool to model linguistic expressions in group decision making. Due to the knowledge limitation, it is difficult for decision makers to provide complete linguistic distribution information and partial ignorance exists in practical group decision making problems. Meanwhile, in group decision making it is hoped to find a group opinion whose distribution information is complete and the preference loss between this group opinion and individual opinions is the minimum. To tackle these issues, this paper introduces the concept of incomplete linguistic distributions and proposes a new model called the minimum preference loss model (MPLM), aiming at minimizing the preference loss between the group opinion and individual opinions in the group decision making with incomplete linguistic distributions. Finally, a numerical example is provided to demonstrate our model.

The study of consistency is a very important problem in decision making using preference relations. This paper focuses on measuring the consistency of hesitant fuzzy linguistic preference relations (HFLPRs). In this paper we propose the optimization-based approach to estimate the range of consistency degree in a HFLPR. The underlying idea of the proposed approach consists in measuring the pessimistic consistency index (PCI) of HFLPRs, and also the optimistic consistency index (OCI) of HFLPRs. The PCI of HFLPRs is determined by its linguistic preference relation with the worst consistency degree, and the OCI of HFLPRs is determined by its linguistic preference relation with the best consistency degree. Furthermore, numerical examples are provided to show the use of the proposed consistency measure.

Preference relations are very useful to express decision makers' preferences over alternatives in the process of decision-making. However, multiple self-confidence levels are not considered in existing preference relations. In this study, we propose a new type of preference relations: fuzzy preference relations with self-confidence. A linear programming model is proposed for estimating priority vectors of this new type of preference relations. Finally, two numerical examples are provided to demonstrate the linear programming model, and a comparative analysis is used to show the influence of self-confidence levels on the decision-making results.

In the consensus reaching process (CRP), different decision makers will concern with different alternatives. As a result, decision makers will naturally use individual consensus methods to measure individual consensus degrees. Meanwhile, it is difficult to set the consensus threshold to judge whether a consensus degree can be acceptable. In order to develop the individual consensus methods and avoid setting the consensus threshold, this study proposes a novel consensus reaching model with individual satisfactions in group decision making (GDM). In this consensus reaching model, the novel individual consensus measure methods are defined by taking into account the individual concerns on the alternatives. Then, based on numerical individual consensus degrees, a linguistic index is developed to evaluate the satisfaction degree. Finally, the feedback adjustment process is used to help decision makers to modify their opinions. A numerical example is provided to show the application of the proposed consensus reaching model.

The 2-tuple linguistic modeling is a popular tool for computing with words in decision making. In order to deal with the linguistic term sets that are not uniformly and symmetrically distributed, the numerical scale model has been developed to generalize the 2-tuple linguistic modeling. In the numerical scale model, the key task of the 2-tuple based models is the definition of a numerical scale function that establishes a one to one mapping between the linguistic information and numerical values. In this paper, we propose a consistency-driven automatic methodology to set interval numerical scales of 2-tuple linguistic term sets in the decision making problems with linguistic preference relations. This consistency-driven methodology is based on a natural premise regarding the consistency of preference relations. If linguistic preference relations provided by experts are of acceptable consistency, the corresponding transformed numerical preference relations by the established interval numerical scale are also consistent. Compared with the existing approach based on canonical characteristic values, the consistency-driven methodology provides a new way to set the interval numerical scale without the need of the semantics defined by interval type-2 fuzzy sets. Meanwhile, interval multiplicative preference relations are used in the pairwise comparisons method and the presented theory can be utilized in the pairwise comparisons method as it provides a novel approach to automatic construct interval multiplicative preference relations. Finally, we present the framework for the use of the consistency-driven automatic methodology in linguistic group decision making problems and two numerical examples are given to illustrate the feasibility and validity of this proposal.

The methodology for selecting the individual numerical scale and prioritization method has recently been presented and justified in the analytic hierarchy process (AHP). In this study, we further propose a novel AHP-group decision making (GDM) model in a local context (a unique criterion), based on the individual selection of the numerical scale and prioritization method. The resolution framework of the AHP-GDM with the individual numerical scale and prioritization method is first proposed. Then, based on linguistic Euclidean distance (LED) and linguistic minimum violations (LMV), the novel consensus measure is defined so that the consensus degree among decision makers who use different numerical scales and prioritization methods can be analyzed. Next, a consensus reaching model is proposed to help decision makers improve the consensus degree. In this consensus reaching model, the LED-based and LMV-based consensus rules are proposed and used. Finally, a new individual consistency index and its properties are proposed for the use of the individual numerical scale and prioritization method in the AHP-GDM. Simulation experiments and numerical examples are presented to demonstrate the validity of the proposed model.

Herrera and Martinez initiated a 2-tuple fuzzy linguistic representation model for computing with words (CWW). In addition to the Herrera and Martinez model, two different models based on linguistic 2-tuples (i.e., the model of Herrera et al. and the numerical scale model) have been developed to deal with term sets that are not uniformly and symmetrically distributed, i.e., unbalanced linguistic term sets (ULTSs). Both the model of Herrera et al. and the numerical scale model can deal with ULTSs, so a challenge is naturally proposed to analysts: how to compare these two different models. In this study, we provide a connection between the model of Herrera et al. and the numerical scale model. The results show that the model of Herrera et al. provides a new approach to set a numerical scale. Furthermore, we prove the equivalence of the linguistic computational models between the model of Herrera et al. and the numerical scale model, if the numerical scale is set based on the model of Herrera et al.

In this study, we present a novel selection process to solve the multiperson decision making (MPDM) problems with different preference representation structures. This selection process is based on the prospect theory, which is one of the most influential psychological behavior theories, and seeks to maximize the satisfactory of all decision makers. Specifically, the individual selection methods associated with different preference structures are used to obtain individual preference orderings. Then, the preference-approval structures are used to determine the reference points of the prospect theory, according to the obtained individual preference orderings. Next, the gains and losses are calculated based on the prospect theory and the established reference points. Finally, the prospect values of the alternatives are obtained to rank the alternatives.

Herrera and Martínez initiated a 2-tuple fuzzy linguistic representation model for computing with words. Moreover, Wang and Hao further developed a new 2-tuple fuzzy linguistic representation model to deal with the linguistic term sets that are not uniformly and symmetrically distributed. This study proposes another linguistic computational model based on 2-tuples and intervals, which we call an interval version of the 2-tuple fuzzy linguistic representation model. The proposed model possesses three steps: 1) interval numerical scale; 2) computation based on interval numbers; and 3) a generalized inverse operation of the interval numerical scale. The first step transforms linguistic terms into interval numbers, based on which the second step is executed with output as an interval number. Finally, this number is then mapped into the interval of the linguistic 2-tuples by the generalized inverse operation. This study also generalizes the numerical scale approach, presented in the Wang and Hao model, to set the interval numerical scale, by considering the context where semantics of linguistic terms are defined by interval type-2 fuzzy sets (IT2 FSs). In order to compare the proposed model with the existing linguistic computational model based on IT2 FSs, we have conducted extensive simulations. The simulations demonstrate that the results obtained by our proposal are consistent with the results of the linguistic computational model based on IT2 FSs (in some sense) in a vast majority of cases.

The validity of the priority vector used in the analytic hierarchy process (AHP) relies on two factors: the selection of a numerical scale and the selection of a prioritization method. The traditional AHP selects only one numerical scale (e.g., the Saaty scale) and one prioritization method (e.g., the eigenvector method) for each particular problem. For this traditional selection approach, there is disagreement on which numerical scale and prioritization method is better in deriving a priority vector. In fact, the best numerical scale and the best prioritization method both rely on the content of the pairwise comparison data provided by the AHP decision makers. By defining a set of concepts regarding the scale function and the linguistic pairwise comparison matrices (LPCMs) of the priority vector and by using LPCMs to unify the format of the input and output of AHP, this paper extends the AHP prioritization process under the 2-tuple fuzzy linguistic model. Based on the extended AHP prioritization process, we present two performance measure criteria to evaluate the effect of the numerical scales and prioritization methods. We also use the performance measure criteria to develop a 2-tuple fuzzy linguistic multicriteria approach to select the best numerical scales and the best prioritization methods for different LPCMs. In this paper, we call this type of selection the individual selection of the numerical scale and prioritization method. We also compare this individual selection with traditional selection by using both random and real data and show better results with individual selection.

In group decision making, consensus models are decision aid tools and help experts modify their individual opinions to reach a closer agreement. Based on the concept of minimum-cost consensus, this paper proposes a novel framework to achieve minimum-cost consensus under aggregation operators. Analytical results indicate that the proposed framework reduces to the consensus model of Ben-Ariehwhen the selected aggregation operator is the ordered weighted averaging (OWA) operator with weight vector$(1/2, \ldots, 0, \ldots, 1/2)^{T}$. Furthermore, this paper closely examines the minimum-cost consensus models with a linear cost function under the common aggregation operators (e.g., the weighted averaging operator and the OWA operator). Linear-programming-based approaches are also developed to solve these models. The results of this paper significantly contribute to efforts to develop the consensus model of Ben-Arieh

When using linguistic approaches to solve decision problems, we need the techniques for computing with words (CW). Together with the 2-tuple fuzzy linguistic representation models (i.e., the Herrera and MartÍnez model and the Wang and Hao model), some computational techniques for CW are also developed. In this paper, we define the concept of numerical scale and extend the 2-tuple fuzzy linguistic representation models under the numerical scale. We find that the key of computational techniques based on linguistic 2-tuples is to set suitable numerical scale with the purpose of making transformations between linguistic 2-tuples and numerical values. By defining the concept of the transitive calibration matrix and its consistent index, this paper develops an optimization model to compute the numerical scale of the linguistic term set. The desired properties of the optimization model are also presented. Furthermore, we discuss how to construct the transitive calibration matrix for decision problems using linguistic preference relations and analyze the linkage between the consistent index of the transitive calibration matrix and one of the linguistic preference relations. The results in this paper are pretty helpful to complete the fuzzy 2-tuple representation models for CW.

In group decision making, because the decision-makers usually represent different interest backgrounds, it is worth to study how to make the different decision makers coordinate and cooperate for aggregating group opinions. In this paper, based on the analytic hierarchy process, we propose a nonlinear program model to obtain consensus priority vector, and point that the model can make decision-makers reach consensus by improving compatibility of judgement matrices. Moreover, we use the genetic-simulated annealing algorithm to obtain its optimal solution. Finally, a numerical example is presented to illustrate the application of this method.

Similar to the concept of on-line auctions presented by Ron Lavi and Noam Nisan [3], this paper discusses pricing algorithms for on-line reverse auction which bidders arrive one by one and on-line buyer must be required to make a decision immediately about each bid as it is received. For online buyer in a reverse auction, we propose on-line mean pricing algorithm and on-line randomized pricing algorithm, and then prove that the two algorithms are competitive and incentive compatible. Moreover, as the bid prices concentrated in a small domain, by competitive analysis for the two algorithms, we find their merits which can avoid the results of purchasing failure or more cost caused by reservation price algorithm. Finally, an example is obtained to illustrate their application.