Minimum Cost Consensus Model Research Articles

Data-driven robust cost consensus model with individual adjustment willingness in group decision-making

In the consensus reaching process, the moderator, as the representative of collective interests, needs to consume certain resources to help decision-makers (DMs) reach a consensus. However, in real life, when the adjustment opinions provided by the moderator are too different from the initial opinions of the DMs, they may not accept the adjustment. On the other hand, it is difficult for the moderator to determine the unit consensus cost of every DM. However, existing consensus methods ignore the above two problems. Therefore, a data-driven robust minimum cost consensus model with individual adjustment willingness (DDR-MCCM-IAW) is proposed. First, an IAW function based on the approximation degree between the adjustment opinion and the original opinion is provided to construct the MCCM-IAW. Subsequently, principal component analysis and kernel density estimation are combined to obtain a data-driven uncertainty set of unit consensus cost, to reduce the conservatism of classical robust cost consensus models. Meanwhile, the tractable DDR-MCCM-IAW is obtained by using linear dual theory. Finally, to test the effectiveness and rationality of the proposed method, some experiments and comparative analysis are presented.

A Minimum-Cost Consensus Model in Social Networks Derived From Uncertain Preferences

The influence of mutual interaction behaviors on the opinions and consensus process has gradually emerged due to the information sharing in social networks. Currently, the cost of making individual decisions and the similarity between experts’ decision behaviors are relatively less addressed in the social network decision process. Therefore, in this study, a consensus approach with the trust relationships and adjustment cost is proposed to fill in such a gap. The method is divided into three stages: 1) trust propagation; 2) weight allocation; and 3) consensus reaching. In the trust propagation phase, uninorm is extended to the uncertain theory and employed in the transmission and integration problems of trust relationships. In the weight allocation stage, the comprehensive weight is assigned based on network structure and strength of relationship. In the consensus-reaching process, two levels of consensus are considered: 1) consensus among individuals and 2) consensus between individuals and the collective group, and chance-constrained programming models are constructed to obtain collective decision opinions. Moreover, a comparative analysis is performed to clarify the effectiveness and advancement of the proposed consensus method.

A Fuzzy-set based formulation for minimum cost consensus models

Minimum Cost Consensus (MCC) models are a popular approach to obtaining consensus in Group Decision-Making (GDM) problems, but previous extensions of these models have not been thoroughly analyzed in terms of their relationships and generalizability, which limits the practical application of these models. This paper presents a reformulation of MCC models for GDM problems using the Fuzzy Set Theory. The proposed fuzzy-based MCC (FZZ-MCC) framework offers a clearer understanding of MCC models and their extensions, as well as a rigorous and flexible methodology for addressing various types of GDM problems. The applicability of the FZZ-MCC framework is demonstrated through three practical examples related to e-democracy, personnel selection, and green supplier selection.

Consensus Reaching in Group Decision Making With Linear Uncertain Preferences and Asymmetric Costs

Consensus reaching process (CRP), as an essential part of group decision making (GDM), can facilitate more effective consensus by taking human behaviors into account. Extending the established research on uncertain minimum cost consensus models (MCCMs), this article continues to adopt linear uncertainty distributions (LUDs) to represent decision-maker’s (DM’s) preference, but considers asymmetric costs into a new framework of CRP, where DM’s preference and weight are both adjusted according to democratic consensus. Moreover, in light of the uncertain distance measure, two novel optimization-based consensus models are built in this article: one is to obtain a minimum cost consensus by simultaneously considering asymmetric costs, aggregation function, and consensus measure; while the other provides a more flexible way to address GDM problems without presetting a specific consensus level (CL) threshold. Some 0–1 binary variables are further introduced to reduce the calculation complexity resulted from piecewise functions in the new multicoefficient goal programming models. Finally, an illustrative example and further discussion reveal the feasibility and superiority of our new method.

A Linguistic Metric for Consensus Reaching Processes Based on ELICIT Comprehensive Minimum Cost Consensus Models

Linguistic Group Decision-Making (LiGDM) aims at solving decision situations involving human decision-makers (DMs) whose opinions are modeled by using linguistic information. To achieve agreed solutions that increase DMs' satisfaction towards the collective solution, Linguistic Consensus Reaching Processes (LiCRPs) have been developed. These LiCRPs aim at suggesting DMs to change their original opinions to increase the group consensus degree, computed by a certain consensus measure. In recent years, these LiCRPs have been a prolific research line, and consequently numerous proposals have been introduced in the specialized literature. However, it has been pointed out the non-existence of objective metrics to compare these models and decide which one presents the best performance for each LiGDM problem. Therefore, this paper aims at introducing a metric to evaluate the performance of LiCRPs that takes into account the resulting consensus degree and the cost of modifying DMs' initial opinions. Such a metric is based on a linguistic Comprehensive Minimum Cost Consensus (CMCC) model based on ELICIT (Extended Comparative Linguistic Expressions with Symbolic Translation) information that models DMs' hesitancy and provides accurate Computing with Words processes. Additionally, the linguistic CMCC optimization model is linearized to speed up the computational model and improve its accuracy.

Minimum cost consensus model with altruistic preference

The minimum cost consensus model (MCCM) aims at reaching group consensus for either conflict or polarized opinions evaluated in group decision making. Decision makers (DMs) consider both their own and other interests in real-world minimum cost consensus problems, exhibiting the altruistic preference behavior. To quantify the behavior, we define a satisfaction degree function to reflect the interaction among DMs. On this basis, the MCCM with altruistic preference (MCCM-AP) is built, in which a novel consensus measurement for concentrated or scattered opinions is introduced. Furthermore, the MCCM-AP based satisfaction improvement model is established. Finally, an illustrative example and a practical case study are carried out to illustrate the performance of the proposed models, together with sensitivity and comparative analyses are conducted to explore the impact of altruistic preference and discuss the merits of our proposal. The findings indicate that total adjustment cost is nonincreasing with the increase of altruistic preference degree, which provides the decision support for managers to handle the altruistic preference behavior and reduce the consensus cost.

A Minimum Adjustment Cost Consensus Framework Considering Harmony Degrees and Trust Propagation for Social Network Group Decision Making

In consensus-based social network group decision-making (SN-GDM) problems, the low degree of harmony caused by opinion discrepancies and weak trust relationships may lead to high consensus costs and conflict among experts. To address these problems, we design a minimum adjustment cost consensus framework considering harmony degrees in SN-GDM with incomplete 2-tuple linguistic trust. First, considering the decay of trust through its propagation process, a discount-factor-based 2-tuple linguistic trust propagation operator is proposed to estimate the unknown trust relationships among experts. Then, the harmony degrees are measured based on opinion discrepancies and 2-tuple linguistic trust values, and experts’ weights could be obtained. To improve the consensus levels of the inconsistent experts, a minimum adjustment cost consensus model considering harmony degrees is explored to generate interval recommendations. The existence of optimal solutions and the convergence of the proposed consensus model are also proved. Finally, the validity of the proposed consensus framework is verified by the case of base station selection. With the higher overall harmony degree and individual consensus level, we extend the classical minimum cost consensus model and incorporate the harmony degrees among experts into the consensus reaching process.

Consensus Decision Modeling with Distributionally Robust Chance Constraint in Uncertain Environments

Group decision-making (GDM) in an ambiguous environment has consistently become a research focus in the decision science field during the past decade. Existing minimum cost consensus models either control the total budget in a deterministic context or focus on improving the utility of decision makers. In this study, a novel consensus model with a distributionally robust chance constraint (DRO-MCCM) is explored. First, two distributionally robust chance constraints consensus models are developed based on the varied utility preferences of decision-makers and taking into consideration the uncertainty of the unit adjustment cost. Next, construct conditional value-at-risk (CVaR) to approximate the cost chance constraint, simulate the viewpoint of decision makers with ambiguous preferences such as utility function and Gaussian distribution, and convert the model into a feasible semidefinite programming problem using dual theory and the moment method. Finally, the supply chain management scenario involving new product prices employs these models. Comparison and sensitivity analyses demonstrates the model’s superiority and effectiveness.

Unit consensus cost-based approach for group decision-making with incomplete probabilistic linguistic preference relations

In group decision-making, the incomplete probabilistic linguistic preference relation (InPLPR) has been widely studied because it can flexibly reflect the decision psychology of decision makers (DMs). In addition, to help DMs revise their opinions and reach a consensus, a large number of scholars have conducted research on the consensus reaching process. Therefore, this study proposes a group consensus decision approach based on InPLPR, considering consistency, the social trust network (STN), and unit cost consensus adjustment. First, an InPLPR consistency measure formula is proposed, based on which a multi-stage missing value estimation method based on optimal consistency and STN is offered for the missing information problem in InPLPR. Thereafter, the hesitation index and unit consensus cost measurement formulas are presented for incomplete probabilistic linguistic term set. Subsequently, consensus reaching and ranking selection are performed based on the proposed unit minimum cost consensus model. Finally, a numerical example of the future development choice problem for new energy vehicle enterprises is provided to demonstrate the applicability of the decision process, and the advantages of the method are derived through comparative analysis.

Social network utility consensus model with empathic and fuzzy interactions

In social network group decision-making, interactions between decision makers are the basis for the formation of a sufficient consensus. Based on the traditional minimum cost consensus models, this study adopts Choquet integral and empathy theory to consider interaction constraints that exist in the integration of a consensus opinion and the evolution of opinion adjustments. We employ social network structures, interaction characteristics, and influence indices as weight constraints to study the convergence law of social network cost consensus, and apply piecewise-linear utility functions to fit the heterogeneity and stage characteristics of decision makers’ attitudes. Three cost consensus models of social utility maximization are constructed with consideration of fuzzy interaction, empathic interaction, and a mixture of fuzzy and empathic interactions, respectively. And this paper extracts the correlation between traditional cost consensus models and cost consensus models with interaction. A case analysis of emergency material reserve decision-making explains the good applicabilities the constructed models in this paper.

Distributionally Robust Optimization Model for a Minimum Cost Consensus with Asymmetric Adjustment Costs Based on the Wasserstein Metric

When solving the problem of the minimum cost consensus with asymmetric adjustment costs, decision makers need to face various uncertain situations (such as individual opinions and unit adjustment costs for opinion modifications in the up and down directions). However, in the existing methods for dealing with this problem, robust optimization will lead to overly conservative results, and stochastic programming needs to know the exact probability distribution. In order to overcome these shortcomings, it is essential to develop a novelty consensus model. Thus, we propose three new minimum-cost consensus models with a distributionally robust method. Uncertain parameters (individual opinions, unit adjustment costs for opinion modifications in the up and down directions, the degree of tolerance, and the range of thresholds) were investigated by modeling the three new models, respectively. In the distributionally robust method, the construction of an ambiguous set is very important. Based on the historical data information, we chose the Wasserstein ambiguous set with the Wasserstein distance in this study. Then, three new models were transformed into a second-order cone programming problem to simplify the calculations. Further, a case from the EU Trade and Animal Welfare (TAW) program policy consultation was used to verify the practicability of the proposed models. Through comparison and sensitivity analysis, the numerical results showed that the three new models fit the complex decision environment better.

A multistage decision-making method with quantum-guided expert state transition based on normal cloud models

Multistage group decision problems are characterized by a changeable decision-making environment and a complex group consensus process, which make it difficult to obtain decision-making results. However, few methods have considered the phenomenon of expert state transition between manageable subgroups in multistage problems. In this paper, we propose a multistage decision-making framework considering quantum-guided expert transition based on normal cloud models. First, a minimum cost consensus model considering individual compromise intervals and the cost function is established based on the normal cloud models to obtain consensual subgroup opinions. Second, an attribute weight optimization model is established based on adjustment satisfaction and fairness satisfaction to aggregate the subgroups. Third, we define the satisfaction determination method of experts in the transition process and then apply the quantum framework to update their state in the next stage. Finally, the feasibility and applicability of the proposed method are illustrated via a power grid emergency evaluation example by conducting analyses and comparisons of the decision results.

Robust minimum cost consensus models with various individual preference scenarios under unit adjustment cost uncertainty

In group decision making (GDM), individual opinions and unit adjustment costs are essential factors in reaching a collective consensus. However, in previous studies, most scholars only considered the uncertainty of individual opinions while ignored the uncertainty of unit adjustment costs or had an inaccurate description of unit adjustment cost uncertainty, which increases the risk for decision makers (DMs). This study constructs four types of uncertainty sets to describe the uncertainty of unit adjustment costs more accurately to solve this problem. Moreover, based on multi-role individual preference scenarios, we adopt a robust optimisation (RO) method to reduce the risk of the model, and we propose robust minimum cost consensus models with various individual preference scenarios under the uncertainty of unit adjustment costs. Furthermore, the proposed robust models were applied to numerical experiments on marine ranching in Weihai, China. The results showed that the proposed robust models were more potent than the original model. Finally, a sensitivity analysis is presented and the characteristics of the proposed models are revealed.

Minimum cost consensus modeling under dynamic feedback regulation mechanism considering consensus principle and tolerance level

Since the hard consensus is difficult and costly to achieve in group decision making (GDM), it is reasonable and necessary for decision makers (DMs) to reach a soft consensus within a certain level of tolerance and consensus. To achieve this purpose, the concepts and definitions of consensus principle, tolerance level and consensus level are proposed in this paper. The consensus principle keeps the consensus reaching process (CRP) always moving in a better direction, and the tolerance level characterizes the individual’s psychological receptivity, while the consensus level guarantees the quality of consensus results, then the minimum cost consensus models (MCCMs) based on these three points are proposed. Thereafter, we add a feedback regulation mechanism based on tolerance threshold rewards to the proposed MCCMs, the mechanism connects individual weight with tolerance threshold, avoiding the subjective disadvantage caused by the two values need to be given in advance. Meanwhile, the framework and specific algorithm of CRP under the tolerance threshold reward feedback regulation mechanism are given, providing a precise method for solving the realistic consensus problems. Finally, the consensus reaching case of the carbon emission benchmark shows the reasonable effectiveness of the two different models proposed in this paper, and the comparative analysis shows the differences and applicability of the two models.

Enhanced Minimum-Cost Consensus: Focusing on Overadjustment and Flexible Consensus Cost

Minimum-cost consensus (MCC) using the optimization-based consensus rule (OR) has been developed and widely applied to various group decision-making (GDM) contexts because resources for the consensus-reaching process are generally limited. In view of the shortcomings of traditional MCC, including the overadjustment of individual opinions and reliance on single type of consensus constraints, this study proposes several enhanced MCC (EMCC) models. First, we provide a visual analytical solution for opinion adjustment in MCC using explicit adjustment paths, through which the adjustment direction and feedback coefficient of each individual opinion are embodied and quantified. According to the obtained feedback coefficients, the criterion for determining overadjustment is defined. By incorporating explicit adjustment paths into traditional MCC, several EMCC models with different types of consensus constraints are developed, all of which can effectively avoid overadjustment. Then, the connection between the identification-direction consensus rule and OR is established based on EMCC. We propose the concept of coordination elasticity and discuss the performance of EMCC from the perspective of consensus cost and overadjustment in order to analyze the using conditions of EMCC. Flexible EMCC provides a negotiated solution for dealing with overadjustment by following the principle of individual consensus cost priority. Finally, the characteristics and advantages of the proposed EMCC models are revealed through a case study and comparison with traditional MCC.

Relationship between the distance consensus and the consensus degree in comprehensive minimum cost consensus models: A polytope-based analysis

Agreement in Group Decision-Making problems has recently been tackled through the use of Minimum Cost Consensus (MCC) models, which are associated with solving convex optimization problems. Such models minimize the cost of changing experts’ preferences towards reaching a mutual consensus, and establish that the distance between the modified individual preferences and the collective opinion must be bounded by the threshold ε>0. A recent MCC-based model, called the Comprehensive Minimum Cost Consensus (CMCC) model, adds another constraint related to a parameter γ∈[0,1] to the above constraint related to the parameter ε to enforce modified expert preferences in order to achieve a minimum level of agreement dictated by the consensus threshold 1−γ∈[0,1]. This paper attempts to analyze the relationship between the aforementioned constraints in the CMCC models from two different perspectives. The first is based on inequalities and allows simple bounds to be determined to relate the parameters ε and γ. The second one is based on Convex Polytope Theory and provides algorithms that compute more precise bounds to relate these parameters, and could also be applied to other similar optimization problems. Finally, several examples are provided to illustrate the proposal.

Minimum cost consensus model with loss aversion based large-scale group decision making

In minimum cost consensus problems, to accomplish the group consensus, decision makers accommodate their initial discrepant opinions based on the unit adjustment costs. During this process, decision makers may exhibit loss aversion, which results in extra expenses when participants feel loss in contrast to their reference opinion adjustments. Existing minimum cost consensus models pay little attention to the loss-averse preference. Hence, to fill up this gap, a minimum cost consensus model with loss aversion (MCCM-LA) is established and the desired properties are analyzed. To manage large-scale group decision making problems, we first propose an integrated opinion similarity, connectivity similarity and behavior similarity clustering algorithm to divide decision makers into multiple subgroups. Balancing the individual adjustment willingness and consensus reaching efficiency, a two-stage consensus reaching mechanism is further designed based on MCCM-LA to realize the accordant opinion. Finally, the effectiveness and feasibility of the proposed method are demonstrated by sensitivity and comparative analyses with an illustrative example.

A Robust Minimum-Cost Consensus Model With Uncertain Aggregation Weights Based on Data-Driven Method

Due to the existence of cognitive bias, it is difficult to accurately obtain the decision-makers’ weights in the aggregating process, which leads to great fluctuations in the optimal decision-making. Therefore, the purpose of this article is to establish a data-driven robust cost consensus model induced by uncertain weights to deal with the adverse effects of uncertainty in the aggregating process. First, the probability density function of the uncertain weight in the aggregation operator is obtained by the kernel density estimation method, and the confidence level is used to control the perturbation range of the uncertainty. Second, a new uncertainty set based on the data-driven robust optimization method is established, which is called the flexible uncertainty set, and two classes of data-driven robust consensus models under uncertain environments are proposed. In the application of power emergency management, by using the data-driven method to traverse the historical data of weights, the new model can effectively deal with the uncertainty caused by the deviation and improve the quality of the aggregation operator, which proves the applicability of the proposed model.

Trust Cop-Kmeans Clustering Analysis and Minimum-Cost Consensus Model Considering Voluntary Trust Loss in Social Network Large-Scale Decision-Making

Social network large-scale decision-making (SN-LSDM) has attracted much attention in the field of decision science. Clustering and consensus building are two important processes for solving SN-LSDM problems. Traditional clustering methods are typically based on the opinion similarity. When extending decision-making to social network contexts, we argue that trust relationships among decision-makers (DMs) serve as a reliable resource for clustering. This study proposes a trust Cop-Kmeans clustering algorithm, which is a semisupervised clustering technique using prior knowledge of trust constraints generated by trust relationships. The consensus-reaching process (CRP) is an effective tool for reducing differences of opinion. In general, the costs and resources associated with the CRP are limited. Therefore, minimum-cost consensus (MCC) models have been developed and used widely in various group decision-making contexts. As an important resource influencing decision-making, trust provides a common-sense perception that the opinion of a DM with a high trust degree is considered to be widely recognized by others. We hold that the DM with high trust and low consensus can reduce the consensus cost by voluntarily losing some trust. Consequently, an improved MCC model considering voluntary trust loss is developed. Finally, a numerical example is presented to illustrate the feasibility of the proposed clustering algorithm and consensus model. A comparative analysis is implemented to explore the influence of the trust constraint on clustering and the influence of trust loss on the CRP.

Consensus models with aggregation operators for minimum quadratic cost in group decision making.

In group decision making (GDM), to facilitate an acceptable consensus among the experts from different fields, time and resources are paid for persuading experts to modify their opinions. Thus, consensus costs are important for the GDM process. Notwithstanding, the unit costs in the common linear cost functions are always fixed, yet experts will generally express more resistance if they have to make more compromises. In this study, we use the quadratic cost functions, the marginal costs of which increase with the opinion changes. Aggregation operators are also considered to expand the applications of the consensus methods. Moreover, this paper further analyzes the minimum cost consensus models under the weighted average (WA) operator and the ordered weighted average (OWA) operators, respectively. Corresponding approaches are developed based on strictly convex quadratic programming and some desirable properties are also provided. Finally, some examples and comparative analyses are furnished to illustrate the validity of the proposed models.