Individual Preference Relations Research Articles

Three-way group consensus method based on probabilistic linguistic preference relations with acceptable inconsistency

In the realm of addressing consensus challenges within group decision-making (GDM) that encompass individual preferences, this article presents an innovative approach. This article introduces a three-way group consensus methodology grounded in probabilistic linguistic preference relations (PLPRs) that ensures an acceptable level of inconsistency by applying the three-way decision (TWD) principle. This methodology is referred to as the TWD-GC-AIC approach. The TWD-GC-AIC approach consists of two core components, each dedicated to distinct aspects: ensuring the internal consistency of individual expert preference relations and achieving consensus within the expert group. The first component is dedicated to enhancing the consistency of individual expert preferences. This process commences with the determination of expert weights and the development of a novel aggregation formula. Subsequently, the consistency metric is refined through the incorporation of a newly devised distance formula. Once the predefined consistency threshold is reached, this study seamlessly integrates a planning model with an iterative algorithm. This integration serves a dual purpose: minimizing adjustment costs while retaining as much of the original information provided by the experts as possible. In the consensus-reaching phase, known as the consensus reaching process (CRP) of the expert group, this study employs iterative algorithms that incorporate TWD within the feedback mechanism. This strategic approach aims to comprehensively consider the emotional attitudes of the experts when adjusting the evaluation information, ultimately leading to a more informed and rational final decision outcome. Finally, this paper empirically validates the effectiveness and efficiency of the TWD-GC-AIC method, particularly in scenarios involving probabilistic linguistic term sets (PLTSs). The method is applied to a practical example and rigorously benchmarked against various other consensus methodologies to assess its performance.

A Novel Approach on Decision Support System Based on the Aczel-Alsina Aggregation Operators and Their Applications to Supplier Selection Problems

Induced aggregation operators are more suitable for aggregating the individual preference relations into a collective fuzzy preference relation. Therefore, in this paper, we introduce the notion of some new types of induced aggregation operators, based on the Aczel-Alsina operations. We construct some induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina ordered weighted averaging/geometric (I-IVITrFAAOWA/G) operator, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid averaging/geometric (I-IVITrFAAHA/G) operator. Moreover, some dominant properties of these developed operators are studied in detail. Based on these proposed approaches, a model is a build up for multicriteria decision making (MCDM), and their stepwise algorithm is being presented. Finally, in utilizing the developed approach, an illustrative example is solved with the help of proposed operators. In the end, we utilize an applicable example for supplier selection to prove the proposed methods and compare the result with existing methods, which shows the superiority, competence, and ability of the developed model.

Local minimum adjustment for the consensus model with distribution linguistic preference relations considering preference reliability

Preference reliability analysis for preference relations reflected by the ordinal consistency and the cardinal consistency was intensively studied. The ordinal consistency is a minimum condition to guarantee that the decision maker’s pairwise comparison preferences between any triple alternatives are transitive. These issues for group decision-making with distribution linguistic preference relations have not been well addressed. This paper aims to provide a consensus model with distribution linguistic preference relations that controls both the cardinal consistency and the ordinal consistency. To do so, a new ordinal consistency for the distribution linguistic preference relations is defined, which is based on the transformation from the distribution linguistic preference relations to its expectation-based matrixes and then to its numerical scale matrixes. Two mixed integer linear programming models are respectively designed to eliminate the ordinal inconsistencies, and to simultaneously control the ordinal and cardinal consistencies so as to ensure the rationality of individual preference relations. Then, a consensus optimization model is developed in which only the individuals with consensus levels lower than the given threshold are allowed to modify their preferences. Preference reliability is also analyzed in the proposed consensus model. Finally, some classical examples are given to illustrate the proposed models. Comparative analysis validates the feasibility and effectiveness of the proposed approaches.

Determination of Break-Even Point through Consumer Preference Relation

Social choice theory beliefs about how the consumers function to chose their interested goods and services. Preference relation with affine indifference curves that has a concave representation has a linear utility representation. This study asks how individual preference relations might be combined to give a single ordering which captures the overall wishes of the group of individuals. There are certain properties that one would like such a utility rule, utility have thus become a more abstract concept that is not necessarily solely based on the satisfaction or pleasure received. Concept of cardinal utility is studied in three different situations Debreu (1958) gave quite different approach. This study maintains link between mathematical theory and financial concept to determine break-even point through the consumers’ preference relation.

Transitivity and approximate consistency threshold determination for reciprocal preference relations in group decision making

In a group decision-making process, contradictory pairwise comparisons may exist in individual preferences or the group preferences even though the consensus level is reached. To avoid such a contradictory phenomenon, this study presents an approximate transitivity-based consistency threshold for reciprocal preference relations to ensure the reliability of the ranking of alternatives with reciprocal preference relations in group decision making. The natural inconsistency or intransitivity of reciprocal preference relations is analyzed and verified by numerical experiments. Then, using the results of numerical experiments, approximate transitivity-based consistency thresholds are introduced based on the objectives of minimising the Type I error, Type II error and total error in statistics. Moreover, a transitivity checking process regarding individual reciprocal preference relations and group reciprocal preference relations is incorporated in the group decision-making process. A transitivity-checking integrated group decision-making model is given for application. An example about the station selection for high-speed railway line is provided to show the necessity of the approximate transitivity-based consistency threshold in checking the transitivity of reciprocal preference relations for group decision making.

Sustainable competitiveness evaluation of container liners based on granular computing and social network group decision making

In the shipping industry, the sustainability of the shipping companies has received more attention besides the economic benefits. A suitable evaluation model can provide a credible sustainable improvement test and guide the operation and management of shipping companies. This study is concerned with a granular evaluation model for the sustainable competitiveness evaluation of container liners, where the interval linguistic version of the analytic hierarchy process is designed for comparing alternatives comprehensively. To obtain ideal consistent preference relations, we propose a modification method by constructing information granules and using particle swarm optimization. Moreover, based on the combination of individual preference relations and experts’ weights obtained from the social network, the aggregated overall opinion of alternatives is achieved. Furthermore, a case study evaluating the sustainable competitiveness of container liners is conducted to support the proposed evaluation framework’s feasibility and practicality.

Some induced generalized geometric aggregation operators based on interval-valued Pythagorean fuzzy numbers

Induced aggregation operators are more suitable for aggregating the individual preference relations into a collective fuzzy preference relation. Therefore, in this paper, we introduce the notion of some new types induced aggregation operators, namely induced interval-valued Pythagorean fuzzy ordered weighted geometric aggregation operator, induced interval-valued Pythagorean fuzzy hybrid geometric aggregation operator, induced generalized interval-valued Pythagorean fuzzy ordered weighted geometric aggregation operator and induced generalized interval-valued Pythagorean fuzzy hybrid geometric aggregation operator. Furthermore, these operators are applied to decision-making problems in which experts provide their preferences in the Pythagorean fuzzy environment to show the validity, practicality and effectiveness of the new approach. We also study the applicability in a decision-making problem concerning strategic selection of the best information system and give an illustrative example to show the effectiveness of the developed methods and operators.

Approaches to Some Induced Einstein Geometric Aggregation Operators Based on Interval-Valued Pythagorean Fuzzy Numbers and Their Application

Interval-valued Pythagorean fuzzy set is one of the successful extensions of the interval-valued intuitionistic fuzzy set for handling the uncertainties in the data. Under this environment, in this paper we introduce the notion of induced interval-valued Pythagorean fuzzy Einstein ordered weighted geometric (I-IVPFEOWG) operator and induced interval-valued Pythagorean fuzzy Einstein hybrid geometric (I-IVPFEHG) operator along with their some basic properties such as, idempotency, boundedness, commutatively, monotonicity. The main advantage of the induced aggregation operators is that, these operators are more suitable to aggregate the individual preference relations into a collective preference relation. Therefore these methods play a vigorous role in daily life problems. Furthermore, a method for multi-attribute group decision-making (MAGDM) problems based on these operators was developed, and the operational procedures were explained in detail.

Consistency- and Consensus-Based Group Decision-Making Method With Incomplete Probabilistic Linguistic Preference Relations

The use of incomplete probabilistic linguistic term sets (InPLTSs) can enrich the flexibility of qualitative decision-making information expression, especially in decision-making situations with high time pressure and insufficient knowledge. In this article, we develop a method for group decision-making (GDM) with incomplete probabilistic linguistic preference relations (InPLPRs), considering consistency and consensus simultaneously. First, to fully explore the ability of InPLTSs to express uncertain information, InPLTSs are specifically classified. Then, an expected multiplicative consistency of InPLPRs is introduced, which is conducive to estimating the missing information more accurately and effectively. Subsequently, considering the consensus of GDM problems, a consensus index, which considers the principle of majority and minority, is developed to measure the agreement degree among multiple individuals. Because individual InPLPRs may not all meet acceptable consistency after reaching consensus, a consistency- and consensus-improving mathematical programming model considering information distortion is presented. Then, to aggregate all individual preference relations into a collective one, a reliability-induced ordered weighted geometric operator is introduced, whose induced variable reliability is determined by the confidence degree and consistency index of individual preference relations. Furthermore, a multiphase algorithm with InPLPRs is developed to solve GDM problems. Finally, a numerical example of fire emergency decisions is presented to illustrate the applicability of the proposed method, and a detailed validity test and comparative analysis are conducted to highlight the advantages of the proposed method.

A New Group Decision Making Method With Distributed Indeterminacy Form Under Neutrosophic Environment: An Introduction to Neutrosophic Social Choice Theory

We present a novel social choice theory based multi-criteria decision making method under neutrosophic environment and a new form of truth representation of neutrosophic theory called Distributed Indeterminacy Form (DIF). Our hybrid method consists of classical methods and an aggregation operator used in social choice theory. In addition to this, we also use DIF function to provide a more sensitive indeterminacy approach towards accuracy functions. We also consider reciprocal property for all individuals. This provides, as in intuitionistic fuzzy decision making theory, a consistent decision making for each individual. The solution approach presented in this paper in group decision making is treated under neutrosophic individual preference relations. These new approaches seem to be more consistent with natural human behaviour, hence should be more plausible and feasible. Moreover, the use of a similar approach to develop some deeper soft degrees of consensus is outlined. Finally, we give a Python implementation of our work in the Appendix section.

A consensus model for group decision making under additive reciprocal matrices with flexibility

A consensus model is proposed for group decision making (GDM) with additive reciprocal preference relations based on the technique for order preference by similarity to an ideal solution (TOPSIS). The initial positions of decision makers are expressed as additive reciprocal matrices. A method of obtaining a collective ideal additive reciprocal matrix is proposed in terms of individual ones. A novel fitness function is constructed through a linear combination of two objective functions involving the consensus degree within the group of experts and the consistency measure of individual preference relations. A new algorithm based on particle swarm optimization (PSO) is proposed to model the dynamic and iterative consensus process of GDM with flexibility. Numerical results are reported to illustrate the proposed model through some comparisons with the existing ones. The weight variance is proposed to characterize the dispersion degree of the weights of alternatives, and analyzed by considering the effects of the flexibility degree and the constraint conditions. The observations reveal that with an increasing of the flexibility degree, the dominant alterative may be chosen, or any solution may not be reached by a group of experts.

An Optimization-Based Approach to Social Network Group Decision Making with an Application to Earthquake Shelter-Site Selection.

The social network has emerged as an essential component in group decision making (GDM) problems. Thus, this paper investigates the social network GDM (SNGDM) problem and assumes that decision makers offer their preferences utilizing additive preference relations (also called fuzzy preference relations). An optimization-based approach is devised to generate the weights of decision makers by combining two reliable resources: in-degree centrality indexes and consistency indexes. Based on the obtained weights of decision makers, the individual additive preference relations are aggregated into a collective additive preference relation. Further, the alternatives are ranked from best to worst according to the obtained collective additive preference relation. Moreover, earthquakes have occurred frequently around the world in recent years, causing great loss of life and property. Earthquake shelters offer safety, security, climate protection, and resistance to disease and ill health and are thus vital for disaster-affected people. Selection of a suitable site for locating shelters from potential alternatives is of critical importance, which can be seen as a GDM problem. When selecting a suitable earthquake shelter-site, the social trust relationships among disaster management experts should not be ignored. To this end, the proposed SNGDM model is applied to evaluate and select earthquake shelter-sites to show its effectiveness. In summary, this paper constructs a novel GDM framework by taking the social trust relationship into account, which can provide a scientific basis for public emergency management in the major disasters field.

Additive consistency‐based priority‐generating method of q ‐rung orthopair fuzzy preference relation

The q-rung orthopair fuzzy set, whose membership function and nonmembership function belong to the interval [0,1], is more powerful than both intuitionistic fuzzy set and Pythagorean fuzzy set in expressing imprecise information of decision-makers. The aim of this paper is to investigate a method to determine the priority weights from individual or group q-rung orthopair fuzzy preference relations (q-ROFPRs). To do so, firstly, a new definition of additively consistent q-ROFPR is presented based on the preference relation of alternatives given by decision-makers. Afterward, according to individual and group q-ROFPRs, two kinds of goal programming models are proposed, respectively, to generate the q-rung orthopair fuzzy priority weight vector of the given q-ROFPR(s). Finally, two numerical examples are given to illustrate the effectiveness and superiority of the method proposed in this paper.

A series of generalized induced Einstein aggregation operators and their application to group decision-making process based on Pythagorean fuzzy numbers

Induced aggregation operators are more suitable for aggregating the individual preference relations into a collective fuzzy preference relation. Therefore, in this paper, we introduce the notion of some new types induced Einstein aggregation operators namely, induced Pythagorean fuzzy Einstein ordered weighted averaging (I-PFEOWA) operator, induced Pythagorean fuzzy Einstein hybrid averaging (I-PFEHA) operator, induced generalized Pythagorean fuzzy Einstein ordered weighted averaging (I-GPFEOWA) operator, and induced generalized Pythagorean fuzzy Einstein hybrid averaging (I-GPFEHA) operator. Some desirable properties, such as idempotency, boundedness, and monotonicity corresponding to the proposed operators have been investigated. The main advantage of the proposed operators is that these operators are able to reflect the complex attitudinal character of the decision-maker using order inducing variables and provide much more complete information for decision-making. Furthermore, these operators are applied to decision-making problems in which experts provide their preferences in the Pythagorean fuzzy environment to show the validity, practicality, and effectiveness of the new approach.

A Method Adjusting Consistency and Consensus for Group Decision-Making Problems with Hesitant Fuzzy Linguistic Preference Relations Based on Discrete Fuzzy Numbers

In each hesitant fuzzy linguistic preference relation, experts may express their opinions through comparison linguistic information combined with a discrete fuzzy number. In this paper, a hesitant fuzzy linguistic computational model based on discrete fuzzy numbers whose support is a subset of consecutive natural numbers is proposed, which enriches the flexibility of group decision-making. First, some main concepts related to discrete fuzzy numbers and an aggregation function of individual subjective linguistic preference relations are introduced. Then, a consistency measure is presented to check and improve the consistency in a given matrix. Further, in order to achieve the predefined degree of consensus and to arrive at the final result, a consensus-reaching process based on the interactive feedback mechanism is defined. Meanwhile, a revised formula is introduced to calculate the consistency and the degree of consensus in a preference relation matrix. Besides, an illustrative example and comparative analysis are conducted through the proposed calculation process and the optimization algorithm. Finally, the analysis on the threshold values is made to help the decision-maker determine critical consensus level. The proposed method can address both consistency and consensus, and the results confirmed the effectiveness of the proposed method and its potential use in the qualitative decision-making problems.

On the road toward innovative solutions: Supporting agents via some fuzzy group decision models

We deal with an important, yet a very challenging problem of whether, and how the process of attaining some innovative solution in a particular setting can be modeled in a relatively intuitively appealing, simple and constructive way, and then used for the support of such processes. We assume a multiagent group decision making setting, and a possibility of imprecise information and testimonies provided by the agents. For simplicity and generality, fuzzy preferences and fuzzy majority are assumed. Basically, we assume a (finite, relatively small) set of options and a set of agents, human beings or software entities, who present their testimonies as fuzzy preference relations which are quite general and can encompass many different forms. We seek and option or a set options that is somehow best acceptable to the whole group of agents, with respect to their preferences. This setting, with a relatively small number of options and agents is characteristic for many real world problems exemplified by making decisions by humans in committees, groups of experts, etc. We also assume that a fuzzy majority is employed. It is meant in Kacprzyk’s sense as equated with a fuzzy linguistic quantifier exemplified by “most”, “almost all”, “much more than a half”, etc. dealt with using, for instance, Zadeh’s calculus of linguistically quantified propositions, Yager’s OWA operators, etc. We consider some group decision solution concepts in such a setting, concentrating of Kacprzyk’s fuzzy Q-cores which is a fuzzy set of options which are not (strongly) defeated in pairwise comparisons (in the sense that other options are preferred over them) by the required fuzzy majority Q, e.g. “most”, of (important) agents. In this setting, we take into account all preference relations of the particular agents, just adding some relevance of options and importance of agents which are assumed to be given, for instance by domain experts. This can be viewed as a reflection of a so called status quo bias which, well known from psychology, which is an emotional bias with a preference for the current state of affairs, the principle of minimal change, etc. We develop some additional setting that could explicitly or implicitly make it possible to allow innovative group decisions, with the innovation considered to be an opposite to the status quo. We develop a simple yet effective and efficient approach in which – in addition to the relevance of options and importance of agents – there is another aspect taken into account. Namely, we derive first a social fuzzy preference relation, derived by some aggregation of the individual fuzzy preference relations, that represents the group testimony, which can be viewed as representing some “consensus”. Then, we calculate for each agent’s individual fuzzy preference relations its distance to that group (consensory) preference relation. Then, we assign additional weights to the particular individual preference relations, in addition to the weights corresponding to the relevance of options and importance of individuals. One can then argue that, from the innovation point of view, a solution that is best acceptable by, for instance, most agents whose preferences are close to the consensory group fuzzy preference relation, would rather reflect a conservative attitude, we reformulate the concept of the fuzzy Q-core that would more take into account the agents’ individual fuzzy preference relations that are farer from the consensory social fuzzy preference relation. Our analysis is exemplified by an application for real estate decision making.

Compatibility measurement-based group decision making with interval fuzzy preference relations

In this paper, we put forward a ratio-based compatibility degree between any two ]0,1[-valued interval numbers to measure how proximate they approach to each other. A compatibility measurement is presented to evaluate the compatibility degree between a pair of ]0,1[-valued interval fuzzy preference relations (IFPRs). By employing the geometric mean, a measurement formula is proposed to calculate how close one interval fuzzy preference relation is to all the other interval fuzzy preference relations in a group. We devise an induced interval fuzzy ordered weighted geometric (IIFOWG) operator to aggregate ]0,1[-valued interval numbers, and apply the induced interval fuzzy ordered weighted geometric operator to fuse interval fuzzy preference relations into a collective one. Based on the compatibility measurement between two interval fuzzy preference relations, a notion of acceptable consensus of interval fuzzy preference relations is introduced to check the consensus level between an individual interval fuzzy preference relation and a collective interval fuzzy preference relation, and a novel procedure is developed to handle group decision-making problems with interval fuzzy preference relations. A numerical example with respect to the evaluation of e-commerce websites is provided to illustrate the proposed procedure.

A preference degree for intuitionistic fuzzy values and application to multi-attribute group decision making

This paper proposes a novel preference degree algorithm to rank intuitionistic fuzzy values (IFVs) and applies it to multi-attribute group decision making (MAGDM) with IFVs. First, the concept of preference degree of IFVs is defined according to the geometrical representation of IFVs. Some useful properties of the defined preference degree are discussed. Combining the preference degree with the hesitation margin of IFVs, a novel preference degree algorithm is designed to rank a series of IFVs. In MAGDM problems with IFVs, the attribute weights would be usually diverse for different decision makers (DMs). To derive the attribute weights for each DM objectively, a multi-objective programming model is constructed and transformed into a linear program to resolve. Then, the consensus indices are calculated from three levels based on the individual fuzzy reciprocal preference relations and the collective one. To determine DMs’ weights, a linear goal programming model is established by maximizing the consensus index of group of DMs. Using the intuitionistic fuzzy weighted average operator, the individual comprehensive values of alternatives are obtained and further integrated into the collective comprehensive values of alternatives. The ranking order of alternatives is generated by the designed preference degree algorithm. At length, the validity of the proposed method is illustrated with a construction organization selection example.

A Framework of Group Decision Making with Hesitant Fuzzy Preference Relations Based on Multiplicative Consistency

The purpose of this paper is to develop a general framework of group decision making with hesitant fuzzy preference relations based on the multiplicative consistency. First, we define a consistency index to measure whether or not a hesitant fuzzy preference relation is of acceptably multiplicative consistency. A consistency improving process is developed to improve the consistency level of a hesitant fuzzy preference relation with unacceptably multiplicative consistency until it is acceptably multiplicative. Then, we present a group consensus index to measure the deviation measure between the individual hesitant fuzzy preference relations and their collective hesitant fuzzy preference relation. A consensus reaching process is developed to help the decision makers improve the consensus level among hesitant fuzzy preference relations until they meet a predefined consensus level. Subsequently, we establish a complete framework of group decision making with hesitant fuzzy preference relations. In this framework, individual multiplicative consistency and group consensus are simultaneously highlighted and the acceptably multiplicative consistency of each hesitant fuzzy preference relation is still maintained in the achievement of the predefined consensus level. Finally, an illustrative numerical example is given to verify the effectiveness and practicality of the developed method.

Generalized Edgeworth–Pareto principle

The general multicriteria choice problem with m individual preference relations and an asymmetric collective preference relation is considered. The concept of a k-effective alternative is introduced, which coincides with an effective alternative for k = 1 and represents a weakly effective alternative for k = m. For the other integer values of k, it lies somewhere in between. In terms of the general multicriteria choice problem, the Pareto axiom and the exclusion axiom for dominated alternatives are stated. Assuming that these axioms hold, a generalized Edgeworth–Pareto principle is established, which was earlier introduced by the author in the special case k = 1. The results are extended to a fuzzy collective preference relation and to a fuzzy set of initial alternatives.