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.

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.

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.

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 the light bulb problem, one is given as input vectors $x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{n} \in\{-1,1\}^{d}$ which are all uniformly random. They are all chosen independently except for a planted pair $\left(x_{i^{*}}, y_{j^{*}}\right)$ which is chosen to have correlation $\rho$ for some constant $\rho\gt 0$. The goal is to find the planted pair. The light bulb problem was introduced over 30 years ago by L. Valiant, and is known to have many applications in data analysis, statistics, and learning theory. The naive algorithm runs in $\Omega\left(n^{2}\right)$ time, and algorithms based on Locality-Sensitive Hashing approach quadratic time as $\rho \rightarrow 0$. In 2012, G. Valiant gave a breakthrough algorithm running in time $O\left(n^{(5-\omega)} /(4-\omega)\right)\lt O\left(n^{1.615}\right)$, no matter how small $\rho\gt 0$ is, by making use of fast matrix multiplication. This was subsequently refined by Karppa, Kaski, and Kohonen in 2016 to running time $O\left(n^{2 \omega / 3}\right)\lt $ $O\left(n^{1.582}\right)$, but is essentially the only known approach for this important problem. In this paper, we propose a new approach based on replacing fast matrix multiplication with other variants and generalizations of matrix multiplication, which can be computed faster than matrix multiplication, but which may omit some terms one is supposed to compute, and include additional error terms. Our new approach can make use of a wide class of tensors which previously had no known algorithmic applications, including tensors which arise naturally as intermediate steps in border rank methods and in the Laser method. We further show that our approach can be combined with locality-sensitive hashing to design an algorithm whose running time improves as $\rho$ gets larger. To our knowledge, this is the first algorithm which combines fast matrix multiplication with hashing for the light bulb problem or any closest pair problem, and it leads to faster algorithms for small $\rho\gt 0$. We then focus on tensors for “multiplying“ $2 \times 2$ matrices; using such small tensors is typically required for practical algorithms. In this setting, the best prior algorithm, using Strassen’s algorithm for matrix multiplication, yields a running time of only $O\left(n^{1.872}\right)$. We introduce a new such low-rank tensor we call $T_{2112}$, which has omissions and errors compared to matrix multiplication, and using it, we design a new algorithm for the light bulb problem which runs in time $O\left(n^{1.797}\right)$. We also explain why we are optimistic that this approach could yield asymptotically faster algorithms for the light bulb problem.

We study graph spanners for point-set in the high-dimensional Euclidean space. On the one hand, we prove that spanners with stretch $\lt \sqrt{2}$ and subquadratic size are not possible, even if we add Steiner points. On the other hand, if we add extra nodes to the graph (non-metric Steiner points), then we can obtain $(1+\epsilon)$-approximate spanners of subquadratic size. We show how to construct a spanner of size $n^{2-\Omega\left(\epsilon^{3}\right)}$, as well as a directed version of the spanner of size $n^{2-\Omega\left(\epsilon^{2}\right)}$. We use our directed spanner to obtain an algorithm for computing $(1+\epsilon)$-approximation to Earth-Mover Distance (optimal transport) between two sets of size n in time $n^{2-\Omega\left(\epsilon^{2}\right)}$.

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.

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.

Abstract. This paper considers Bayesian revenue maximization in the [Formula: see text]-unit setting, where a monopolist seller has [Formula: see text] copies of an indivisible item and faces [Formula: see text] unit-demand buyers (whose value distributions can be nonidentical). Four basic mechanisms among others have been widely employed in practice and widely studied in the literature: Myerson auction, sequential posted-pricing, [Formula: see text]-th price auction with anonymous reserve, and anonymous pricing. Regarding a pair of mechanisms, we investigate the largest possible ratio between the two revenues (also known as the revenue gap), over all possible value distributions of the buyers. Divide these four mechanisms into two groups: (i) the discriminating mechanism group, Myerson auction and sequential posted-pricing, and (ii) the anonymous mechanism group, anonymous reserve and anonymous pricing. Within one group, the involved two mechanisms have an asymptotically tight revenue gap of [Formula: see text]. In contrast, any two mechanisms from the different groups have an asymptotically tight revenue gap of [Formula: see text].

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.

This paper considers Bayesian revenue maximization in the k-unit setting, where a monopolist seller has k copies of an indivisible item and faces n unit-demand buyers (whose value distributions can be non-identical). Four basic mechanisms among others have been widely employed in practice and widely studied in the literature: Myerson Auction, Sequential Posted-Pricing, (k + 1)-th Price Auction with Anonymous Reserve, and Anonymous Pricing. Regarding a pair of mechanisms, we investigate the largest possible ratio between the two revenues (a.k.a. the revenue gap), over all possible value distributions of the buyers. Divide these four mechanisms into two groups: (i) the discriminating mechanism group, Myerson Auction and Sequential Posted-Pricing, and (ii) the anonymous mechanism group, Anonymous Reserve and Anonymous Pricing. Within one group, the involved two mechanisms have an asymptotically tight revenue gap of 1 + Θ(1 / √k). In contrast, any two mechanisms from the different groups have an asymptotically tight revenue gap of Θ(łog k).

The fastest known LP solver for general (dense) linear programs is due to [Cohen, Lee and Song’19] and runs in O*(nω +n2.5−α/2 + n2+1/6) time. A number of follow-up works [Lee, Song and Zhang’19, Brand’20, Song and Yu’20] obtain the same complexity through different techniques, but none of them can go below n2+1/6, even if ω=2. This leaves a polynomial gap between the cost of solving linear systems (nω) and the cost of solving linear programs, and as such, improving the n2+1/6 term is crucial toward establishing an equivalence between these two fundamental problems. In this paper, we reduce the running time to O*(nω +n2.5−α/2 + n2+1/18) where ω and α are the fast matrix multiplication exponent and its dual. Hence, under the common belief that ω ≈ 2 and α ≈ 1, our LP solver runs in O*(n2.055) time instead of O*(n2.16).

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.

We present improved distributed algorithms for variants of the triangle finding problem in the model. We show that triangle detection, counting, and enumeration can be solved in rounds using expander decompositions . This matches the triangle enumeration lower bound of by Izumi and Le Gall [PODC’17] and Pandurangan, Robinson, and Scquizzato [SPAA’18], which holds even in the model. The previous upper bounds for triangle detection and enumeration in were and , respectively, due to Izumi and Le Gall [PODC’17]. An -expander decomposition of a graph is a clustering of the vertices such that (i) each cluster induces a subgraph with conductance at least and (ii) the number of inter-cluster edges is at most . We show that an -expander decomposition with can be constructed in rounds for any and positive integer . For example, a -expander decomposition only requires rounds to compute, which is optimal up to subpolynomial factors, and a -expander decomposition can be computed in rounds, for any arbitrarily small constant . Our triangle finding algorithms are based on the following generic framework using expander decompositions, which is of independent interest. We first construct an expander decomposition. For each cluster, we simulate algorithms with small overhead by applying the expander routing algorithm due to Ghaffari, Kuhn, and Su [PODC’17] Finally, we deal with inter-cluster edges using recursive calls.

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.

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.

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.

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.

We present improved distributed algorithms for triangle detection and its variants in the CONGEST model. We show that Triangle Detection, Counting, and Enumeration can be solved in Õ(n1/2) rounds. In contrast, the previous state-of-the-art bounds for Triangle Detection and Enumeration were Õ(n2/3) and Õ(n3/4), respectively, due to Izumi and LeGall (PODC 2017). The main technical novelty in this work is a distributed graph partitioning algorithm. We show that in Õ(n1–δ) rounds we can partition the edge set of the network G = (V, E) into three parts E = Em ∪ Es ∪ Er such that Each connected component induced by Em has minimum degree Ω(nδ) and conductance Ω(1/polylog(n)). As a consequence the mixing time of a random walk within the component is O(polylog(n)). The subgraph induced by Es has arboricity at most nδ. |Er| ≤ |E|/6. All of our algorithms are based on the following generic framework, which we believe is of interest beyond this work. Roughly, we deal with the set Es by an algorithm that is efficient for low-arboricity graphs, and deal with the set Er using recursive calls. For each connected component induced by Em, we are able to simulate CONGESTED-CLIQUE algorithms with small overhead by applying a routing algorithm due to Ghaffari, Kuhn, and Su (PODC 2017) for high conductance graphs.

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.

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.

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.

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 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.

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.