Additive Preference Relations Research Articles

Modeling consistency and consensus in social network group decision making: The role of limited dual tolerance and compromise behaviors

In interactive social network group decision making (SNGDM), decision-makers frequently consider both their own preferences and the preferences of trusted members when adjusting preferences. This dual consideration results in the formation of limited dual tolerance and compromise behaviors, as reflected in the cost and willingness of decision-makers to adjust their preferences. This paper investigates the role of limited dual tolerance and compromise behaviors in the consistency and consensus management in SNGDM with additive preference relations. First, this paper introduces the concepts of limited dual tolerance and compromise behaviors. Then, this paper presents an optimization-based model, tailored to manage consistency and consensus in the interactive SNGDM. This model takes into account the two behaviors mentioned, with the goal of assisting decision-makers in adjusting their preferences to achieve consistency and consensus while minimizing associated costs. Given that limited dual compromise behavior may impede decision-makers from achieving consistency and consensus, this paper designs a model to help decision-makers adjust their dual compromise values. Further, this paper develops an interactive consistency and consensus management process with limited dual tolerance and compromise behaviors. Finally, an illustrative example involving sustainable supplier selection is applied, along with the simulation analysis and a discussion, to validate the effectiveness of the proposed framework.

A two-stage stochastic preference analysis on group decision making with interval additive preference relations

Interval additive preference relation (IAPR) is an efficient tool for describing the uncertain pairwise comparisons between alternatives in group decision-making (GDM). Existing GDM methods with IAPRs view the preference value as a whole and establish concrete conflict/consensus mechanisms for fusing individual opinions while neglecting the stochastic preference and aggregation modes to meet the different demands of decision. Thus, to reflect the stochastic characteristic in both preference representation and group preference aggregation, this paper presents a novel two-stage stochastic preference analysis on GDM with IAPRs. In first stage, the method involves extracting stochastic additive preference relations (SAPRs) from the IAPRs. A preference relations matrix is then established using a logarithmic least squares model applied to the SAPRs. In second stage, the weighted preference matrix is synthesized by combining the preference relations matrix with the decision makers’ weight space. This leads to the computation of positive and negative ideal preference solutions, as well as positive and negative preference distances. Subsequently, the relative preference closeness is calculated based on these distances. Relative ranking results are determined by an algorithm evaluating pairwise comparisons among alternatives, resulting in transitional preference rankings. The second stage involve deriving ranking acceptance indices and ranking confidence degrees through a continuous extraction process, facilitating the establishment of the ultimate best preference rankings. The proposed method’s effectiveness and validity are demonstrated by numerical examples and Monte Carlo simulations.

Double stochastic preference analysis on group decision making with interval additive preference relations

Interval additive preference relations (IAPRs) provide a valuable framework for representing uncertain pairwise comparisons among alternatives in group decision-making (GDM). Conventional GDM methods that incorporate IAPRs often treat preference values as uniform entities and establish dedicated conflict or consensus mechanisms for amalgamating individual opinions while overlooking the stochastic preferences and lacking the flexibility to address diverse decision requirements. Therefore, to account for the stochastic characteristics in both preference representation and group preference aggregation, this paper introduces an innovative double stochastic preference analysis for GDM with IAPRs. In the initial phase, stochastic additive preference relations are extracted from IAPRs using a specific density function. A best preference matrix is then constructed through a logarithmic consistency minimum deviation model. In the second phase, preference comparison intervals are synthesized to encapsulate the collective viewpoints of decision-makers. Subsequently, ranking acceptability indices and preference confidence degrees are computed through processing of the preference ranking judgment space, leading to the derivation of the final preference rankings. The experimental results, obtained through Monte Carlo simulations for a site selection case in Anhui Province, China, demonstrate the effectiveness and validity of the proposed method. The ranking results achieved a confidence degree exceeding 89%, offering insights into decision-makers’ weight distribution for various ranking scenarios.

An efficiency-based interactive dynamic technique with interval-valued hesitant fuzzy constraint cone for rescue route planning

This paper develops an efficiency-based interactive dynamic technique to design the post-disaster rescue route by combining the data envelopment analysis (DEA), traveling salesman problem (TSP) and interval-valued hesitant fuzzy constraint cone (IVHFCC). The traditional DEA potentially presumes that the weights of the input (output) criteria are unrestricted, resulting in its inapplicability to some complex decision problems. To restrict the weights of input (output) criteria legitimately, this paper constructs an IVHFCC based on expert’s subjective preferences in the form of interval-valued hesitant fuzzy sets. This paper defines the additive consistent normal interval-valued hesitant fuzzy preference relation (NIVHFPR) and analyses its desirable properties. Based on these properties, three linear programming models are proposed to repair the incomplete NIVHFPR, judge the additive consistency of the repaired NIVHFPR and generate the additive consistent NIVHFPR. By incorporating the TSP, DEA and the constructed IVHFCC, an efficiency-based interactive dynamic model is originally proposed to generate the optimal rescue route for the rescue team. Some desirable properties of this efficiency-based interactive dynamic model are analyzed. The validity of this model is demonstrated with a real post-disaster rescue route planning problem of Wenchuan earthquake. Some comparative analyses are provided to show the superiorities of the proposed technique.

Improving consistency based on regret theory: A multi-attribute group decision making method with linguistic distribution assessments

Multi-attribute group decision making (MAGDM) problems widely exist in our daily life. In such problems, multiple decision makers (DMs) are involved and elicit their evaluations to assess a set of possible alternatives. Their complexity usually implies to deal with uncertain contexts and the use of linguistic preference modelling for the elicited assessments has been common. Linguistic distribution assessments (LDAs) applied to MAGDM, involving several linguistic terms associated with their symbolic proportions, have been effective to describe such assessments. However, to ensure a reasonable decision making result, the consistency of LDAs should be investigated and a corresponding MAGDM method proposed. This paper proposes a MAGDM method based on regret theory, not only to take the psychological behaviors of DMs into account, but also to reflect the consistency of assessments. Hence, the additive regret preference relation is firstly defined and then four algorithms are devised to improve the consistency of assessments. On this basis, two weighting determination models are built to calculate the weights of attributes and DMs respectively. Furthermore, alternatives can be ranked according to their overall synthetic values. The proposed MAGDM method is applied to an emergency assistance area selection problem and its result verifies the feasibility and effectiveness of the proposal.

Consensus Reaching Process With Multiobjective Optimization for Large-Scale Group Decision Making With Cooperative Game

Large-scale group decision making (LSGDM) is a common decision-making activity in which experts elicit their preferences through preference relations with consistencies based on additive properties. LSGDM has become a popular topic in decision making because of its necessity and applicability in multiple fields. In LSGDM, the consensus reaching process (CRP) ensures that decision makers (DMs) agree on the final decision-making results. Therefore, it is significant to investigate the CRP to improve the LSGDM process. In this article, a new method for LSGDM that includes a clustering algorithm, a weight determination method, and a CRP is developed. First, based on the consensus and the consistency of the additive preference relation, the DMs are classified into different subgroups by using the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -means clustering algorithm. Then, because different subgroups have distinct decision-making interests, a weight determination method based on a cooperative game is proposed. Furthermore, to measure the consensus of each subgroup more comprehensively, the intra- and interconsensus levels are defined. These consensus levels are divided into four scenarios (acceptable–acceptable, acceptable–unacceptable, unacceptable–acceptable, and unacceptable–unacceptable) according to predetermined thresholds. Furthermore, different feedback mechanisms based on multiobjective optimization models corresponding to different CRP scenarios are presented. Finally, an illustrative example and comparative analyses are provided to verify the feasibility and validity of the proposed method.

A personalized individual semantics model for computing with linguistic intuitionistic fuzzy information and application in MCDM

This paper develops a personalized individual semantics (PISs) model for computing with linguistic intuitionistic fuzzy information and applies to evaluate different brands of mobile phones. First, considering that a linguistic term means different things to different decision-makers, a consistency-driven optimization model for checking the additive consistent linguistic intuitionistic fuzzy preference relations (LIFPRs) is constructed by considering the PISs model. Besides, several optimization models are built to determine the PISs of linguistic terms with LIFPRs and obtain the acceptable additive consistent LIFPRs. Second, a new definition of Hamming distance for measuring linguistic intuitionistic fuzzy numbers (LIFNs) is developed by considering the PISs model, and several desirable properties are discussed. Then, the method of deriving the weight vectors of criteria is calculated based on the proposed distance measure. Subsequently, a framework of group decision-making (GDM) process with LIFPRs is offered, and the application of the proposed method is illustrated by using a multi-criteria decision-making (MCDM) problem about evaluating different brands of mobile phones. Finally, the comparative analysis is conducted to show the feasibility of proposed method.

From numerical to heterogeneous linguistic best–worst method: Impacts of personalized individual semantics on consistency and consensus

This paper investigates the impacts of personalized individual semantics (PIS) on consistency and consensus in Best–Worst method (BWM) with heterogeneous linguistic preference information. Firstly, a PIS driven consistency measurement and information transformation method is presented to analyze the consistency level of linguistic preference information (i.e., best-to-others and others-to-worst vectors) of each decision maker, and this is conducted by maximizing the consistency level of additive preference relation that converted from linguistic preference information via personalized numerical scales. Secondly, a PIS driven maximum consensus optimization model within the BWM framework is designed to yield the maximum consensus level among the additive preference relations generated from heterogeneous linguistic preference information by means of personalized numerical scales. Thirdly, a PIS-based heterogeneous linguistic consensus reaching process is put forward to promote the consensus establishment among decision makers. Finally, the validity of the proposed BWM framework is verified by a case study, a sensitive analysis, and a comparison analysis.

Additive consistent triangular fuzzy preference relation and likelihood comparison algorithm based group decision making

Additive consistent triangular fuzzy preference relation and likelihood comparison algorithm based group decision making

New additive-consistency-driven methods for deriving two types of normalized utility vectors from additive reciprocal preference relations

In multi-criteria decision making, importance weights of criteria are usually derived from pairwise comparisons and requested for adding up to one. The framework of additive reciprocal preference relations (ARPRs) is a commonly used tool representing pairwise comparisons under the unit interval scale. This paper illustrates that existing additive-consistency-driven utility derivation methods fail to acquire a satisfactory additively normalized importance weight vector of criteria. A normalization framework is introduced under an Abelian linearly ordered group and new methods are proposed to generate additively consistent ARPRs respectively from vectors satisfying new normalization conditions and additively normalized vectors. Two formulas are devised to obtain the two types of normalized utility vectors associated with an additively consistent ARPR. Three optimization models are built and their analytical solutions are found to derive the two types of normalized optimal utility vectors from ARPRs. A geometrically additive consistency index is presented and its approximated thresholds are suggested to check acceptably additive consistency of ARPRs. A stepwise procedure is developed to deal with multi-criteria decision making problems. Two numerical illustrations with comparative analysis and an international exchange postgraduate student selection problem are offered to demonstrate the performance and practicality of the presented models.

Additive consistency analysis and normalized optimal utility vector derivation for triangular fuzzy additive reciprocal preference relations

This paper focuses on extending Tanino’s additive consistency to triangular fuzzy additive reciprocal preference relations (TFARPRs) and deriving normalized optimal fuzzy utility vectors from TFARPRs. Spread based indices are introduced to measure vagueness of [0, 1]-valued triangular fuzzy judgments and TFARPRs. A transitivity equation with parameters is devised to define additive consistency of TFARPRs. Values of parameters are then determined to introduce an equivalent definition of additively consistent TFARPRs. Pivotal properties of additively consistent TFARPRs are put forward and a computational formula is designed to obtain additive consistency indices of TFARPRs. A novel method is proposed to generate additively consistent TFARPRs from [0, 1]-valued triangular fuzzy utility vectors. By examining the equivalency of [0, 1]-valued triangular fuzzy utility vectors, a framework is presented to normalize triangular fuzzy utility vectors. A goal programming model is subsequently built to derive normalized triangular fuzzy utility vectors from TFARPRs. This optimization model is transformed into a linear program and its analytical solution is achieved for additively consistent TFARPRs. The performance and validity of the proposed models are demonstrated by three illustrations with comparative analysis.

Theorem Verification of the Quantifier-Guided Dominance Degree with the Mean Operator for Additive Preference Relations

Deciding which film is the best or which portfolio is the best for investment are examples of decisions made by people every day. Decision-making systems aim to help people make such choices. In general, a decision-making system processes and analyses the available information to arrive at the best alternative solution of the problem of interest. In the preference modelling framework, decision-making systems select the best alternative(s) by maximising a score or choice function defined by the decision makers’ expressed preferences on the set of feasible alternatives. Nevertheless, decision-making systems may have logical errors that cannot be appreciated by developers. The main contribution of this paper is the provision of a verification theorem of the score function based on the quantifier-guided dominance degree (QGDD) with the mean operator in the context of additive preference relations. The provided theorem has several benefits because it can be applied to verify that the result obtained is correct and that there are no problems in the programming of the corresponding decision-making systems, thus improving their reliability. Moreover, this theorem acts on different parts of such systems, since not only does the theorem verify that the order of alternatives is correct, but it also verifies that the creation of the global preference relation is correct.

Managing Consistency and Consensus Issues in Group Decision-Making with Self-Confident Additive Preference Relations and Without Feedback: A Nonlinear Optimization Method

Managing Consistency and Consensus Issues in Group Decision-Making with Self-Confident Additive Preference Relations and Without Feedback: A Nonlinear Optimization Method

The breaking of additively reciprocal property of fuzzy preference relations and its implication to decision making under uncertainty

The existing works usually assume that fuzzy preference relations (FPRs) exhibit additively reciprocal property. In this study, we investigate the situation that FPRs have no additively reciprocal property and address its implication to decision making under uncertainty. First, the considered situation is captured by proposing the novel concept of additively reciprocal property breaking (ARPB) for FPRs. It is proved that FPRs with ARPB, interval additive reciprocal preference relations (IARPRs) and intuitionistic fuzzy preference relations (IFPRs) are transformed each other. Second, the equivalence between FPRs with ARPR and additive reciprocal matrices (ARMs) is defined by keeping the inconsistency level unchanged. An optimization model is proposed to adjust an FPR without weak transitivity to an ARM with weak transitivity. Third, a novel decision making model is developed by considering weak transitivity of FPRs as the minimum requirement of rational choices. A new algorithm is elaborated on where decision information could be expressed as FPRs with ARPR, IARPRs and IFPRs, respectively. Finally, numerical results are reported to show the advantages of the developed model by comparing with the existing ones. The observations reveal that the concept of ARPB offers a novel understanding for uncertainty in decision information.

Group decision making based on the modified probability calculation method and DEA cross-efficiency with probabilistic hesitant fuzzy preference relations

In this paper, we propose a novel group decision making (GDM) method with probabilistic hesitant fuzzy preference relations (PHFPRs) based on the modified probability calculation method and data envelopment analysis (DEA) cross-efficiency. The primary advantage of the method is that it can reduce information loss and ensure the reliability of the decision-making result. A modified probability calculation method for hesitant fuzzy preference relations (HFPRs) is first proposed to obtain complete and normalized PHFPRs. Then, for the consistent additive preference relation (APR), we construct an output-oriented DEA model to derive the priority vector, in which each alternative is regarded as a decision-making unit (DMU). A DEA cross-efficiency model is further proposed to yield the priority vector, irrespective of whether the desired consistency level is achieved. Furthermore, a stochastic analysis method is developed to obtain the expected priority vector for any PHFPR, and a mathematical programming model is established to obtain the weight vector of decision makers based on group consensus. Finally, the procedure of the GDM method based on the modified probability calculation and the DEA cross-efficiency is designed. We also show the applicability and effectiveness of the proposed GDM method using illustrative examples.

Optimal improvement of the best and worst consistency levels for interval additive preference relations

The best consistency index (BCI) and the worst consistency index (WCI) were introduced into interval additive preference relations (IAPRs) to guarantee that any APR within an IAPR would have a satisfactory best or worst consistency. However, the methods for controlling the BCI and WCI consistencies were not clear, and consensus reaching problems based on the BCI and WCI, which are a key issue in group decision making using IAPRs, have not yet been addressed. Therefore, to address this particular omission, this paper suggests various optimization approaches for the modeling of both consistency and consensus. First, an APR that corresponds to the worst consistency is determined using the IAPR boundary elements, after which an optimization model is introduced to improve the BCI and WCI that minimizes the amount of change. It was found that if all individual IAPRs had acceptable WCIs, the group IAPR also had an acceptable WCI for the group decision making. An optimal consensus model is also proposed to deal with the consistency and consensus. These models were found to be able to effectively manage the worst consistency in the decision process. Some numerical examples and comparison analyses are given to validate the efficiency of the developed decision models.

A two-stage minimum adjustment consensus model for large scale decision making based on reliability modeled by two-dimension 2-tuple linguistic information

Consensus reaching processes (CRPs) have been required to assure the consensus in large scale group decision making (LSGDM). Opinion reliability detection has been demanded to ensure the trustworthiness of the original information and different information modeling approaches have facilitated it in which two dimensional linguistic (TDL) information has an outstanding place. The reliability degree of original opinions elicited by TDL expressions is often given in advance as subjective evaluation, and after adjustment during CRP, the reliability of the adjusted opinions is often neglected especially for automatic CRP, which may lead to unreliable decisions. In real decision making, considering the interest of decision makers (DMs) themselves, the self-assessment of the DMs on the reliability of the given opinions could be easily manipulated by DMs. To reduce the subjectivity of the decision making, we propose a method to obtain objectively the reliability of the adjusted opinions through a two-stage minimum cost consensus model based on 2-tuple linguistic additive preference relations. Firstly, a support degree (SD)-based clustering method will be developed for classifying DMs into several subgroups to make more manageable the large number of DMs. Subsequently, a two-stage minimum adjustment consensus model will be presented to improve the consensus level (CL) gradually. Eventually, the adjusted opinions will be presented as two-dimension 2-tuple linguistic (TD2L) information. A comparative performance analysis of this CRP based LSGDM approach will be provided to show its effectiveness.

Managing classification-based consensus in social network group decision making: An optimization-based approach with minimum information loss

This study proposes a classification-based consensus framework in social network group decision making, which aims to classify alternatives into several ordinal classes from best to worst. In the classification-based consensus framework, a maximum consensus-based optimization model is devised to determine the weight of decision makers by linearly combining three reliable sources: in-degree centrality, consistency and similarity indexes. This is done by maximizing the consensus level among decision makers regarding the collective classification of alternatives. Following this, a minimum information loss-based optimization model is constructed to generate the consensual collective classification of alternatives. It seeks to minimize the information loss between the additive preference relations provided by decision makers and their preference vectors. Particularly, the proposed optimization models are converted into 0–1 mixed linear programming models to easily find their optimal solutions. Finally, a numerical example and a detailed comparison analysis are provided to show the effectiveness of the proposed approach.

Personalized individual semantics-based approach for linguistic failure modes and effects analysis with incomplete preference information

Failure Modes and Effects Analysis (FMEA) is a very useful reliability-management instrument for detecting and mitigating risks in various fields. The linguistic assessment approach has recently been widely used in FMEA. Words mean different things to different people, so FMEA members may present Personalized Individual Semantics (PIS) in their linguistic assessment information. This article presents the design of a PIS-based FMEA approach, in which members express their opinions over failure modes and risk factors using Linguistic Distribution Assessment Matrices (LDAMs) and also provide their opinions over failure modes using incomplete Additive Preference Relations (APRs). A preference information preprocessing method with a two-stage optimization model is presented to generate complete APRs with acceptable consistency levels from incomplete APRs. Then, a deviation minimum-based optimization model is designed to personalize individual semantics by minimizing the deviation between APR and the numerical assessment matrix derived from the corresponding LDAM. This is followed by the development of a ranking process to generate the risk ordering of failure modes. A case study and a detailed comparison analysis are presented to show the effectiveness of the PIS-based linguistic FMEA approach.

Group Decision Making Based on Flexibility Degree of Fuzzy Numbers Under a Confidence Level

There is always vagueness experienced by experts in practical decision making processes. Fuzzy sets are introduced for their ability to model objects and phenomena with a flexible degree. Then, it is more advisable to utilize fuzzy numbers to express the opinions of decision makers (DMs) for the purpose of reflecting the flexibility of DMs in a decision making process. Of much importance is how to quantify the flexibility degree of fuzzy numbers. In this article, a novel method for computing the flexibility degree of fuzzy numbers is generally presented by using the concept of α-cut sets. In particular, some novel formulas are proposed to quantify the flexibility degrees of triangular and trapezoidal fuzzy numbers by equipping a bounded universe, respectively. The flexibility degree of a preference relation with triangular fuzzy numbers is computed by considering the effects of the applied scale and the reciprocal property. Furthermore, a new group decision making (GDM) model is formed when triangular fuzzy additive reciprocal preference relations (TFARPRs) are used to evaluate the judgments of DMs. A flexibility degree induced ordered weighted averaging operator is constructed to aggregate individual TFARPRs by offering more importance to that with less flexibility. Finally, some numerical results are reported to illustrate the new definitions and the proposed model. The sensitivity of confidence levels to the final decision reached by a group of experts is analyzed. The observations reveal that the flexibility degrees of DMs under various confidence levels are worth to be considered in the GDM problem with a dominant position, and the existing shortcomings are overcome.