Reciprocal Preference Relations Research Articles

A Local Adjustment Method to Improve Multiplicative Consistency of Fuzzy Reciprocal Preference Relations

Preferences that verify the transitivity property are usually referred to as rational or consistent preferences. Existent methods to improve the consistency of inconsistent fuzzy reciprocal preference relations (FPRs) fail to retain the original preference values because they always derive a new FPR. This article presents a new inconsistency identification and modification (IIM) method to detect and rectify only the most inconsistent elements of an inconsistent FPR. As such, the proposed IIM can be considered a local adjustment method to improve multiplicative consistency (MC) of FPRs. The case of inconsistent FPRs with missing values, i.e., incomplete FPRs, is addressed with the estimation of the missing preferences with a constrained nonlinear optimization model by the application of the IIM method. The implementation process of the proposed algorithms is illustrated with numerical examples. Simulation experiments and comparisons with existent methods are also included to show that the new method requires fewer iterations than existent methods to improve the MC of FPRs and achieves better MC level, while preserving the original preference information as much as possible than the existent methods. Thus, the results presented in this article demonstrate the correctness, effectiveness, and robustness of the proposed method.

Average consistency index based consensus model for a group decision making problem dealing with ELICIT expressions

A group decision making (GDM) problem is often modeled by using a reciprocal preference relation (RPR) structure. Linguistic information is widely used in RPRs to depict its components because it facilitates its understanding and management of uncertainty. Among the existing linguistic representation models, the extended comparative linguistic expressions with symbolic translation (ELICIT) model improves the accuracy of computational processes and maintains the comprehensibility and interpretability of the results. Therefore, we introduce a new linguistic RPR (L-RPR) structure using ELICIT information, denoted as ELICIT-RPR, whose elements are uniformed as 2-tuple linguistic intervals to simplify and facilitate computations between ELICIT expressions. In addition, the average consistency index (ACI) of ELICIT-RPR is proposed together an algorithm for improving the ACI values of ELICIT-RPRs based on their best and worst consistency indices. To avoid conflicts and obtain agreed solutions, consensus reaching processes (CRPs) have been widely used in GDM. Thus, a novel CRP based on ACI values is developed using a new distance between linguistic intervals. Finally, a case study and a comparison analysis are presented to demonstrate the validity and applicability of the proposal.

Deriving priorities from the fuzzy best-worst method matrix and its applications: A perspective of incomplete reciprocal preference relation

Preference relation is an effective tool in multi-criteria decision making (MCDM). The fuzzy best-worst method (FBWM), which is an extension of the BWM, is proposed to determine weights for criteria. In the FBWM, the Fuzzy Best-to-others Vector (FBV) and Fuzzy others-to-Worst Vector (FWV) are given. The FBV and FWV can intrinsically formulate one incomplete reciprocal preference relation (IRPR), which we call the FBWM matrix. As the FBWM is designed mainly to determine the weights, and the existing FBWM only uses the min–max problem to derive the weights. Therefore, it is important to develop other effective methods to derive the weights from the FBWM matrix. Deriving the weights from FBWM matrix is converted into deriving the priorities from the corresponding IRPR. In this view, two groups of methods are proposed. One group is for a single FBWM matrix, and the other group is for a group of FBWM matrices. To show the effectiveness and performance of the developed methods, Monte Carlo simulations were implemented. Finally, two examples were used to show how these methods are applied in real decision-making problems. Comparative analyses were performed to show the differences and usefulness of the proposed methods.

An improved multiplicative acceptability consistency-driven group decision making with triangular fuzzy reciprocal preference relations

Triangular fuzzy reciprocal preference relation (TFRPR) is one of the most effective tools to express the vague and uncertain preferences of decision makers in group decision making (GDM). Consistency of TFRPRs is a crucial premise for reasonable and reliable decision-making. To do so, this study develops an improved multiplicative acceptability consistency-driven GDM with TFRPRs. By analyzing the composition of TFRPRs, a multiplicative consistency index and the corresponding threshold are defined to measure whether a TFRPR is acceptably consistent. The consistency measurement reflects the essential traits of triangular fuzzy numbers, which fully considers the multiplicative consistency of modal values and the multiplicative consistency of geometric mean based on central tendency. Subsequently, three inconsistent TFRPR cases are analyzed, and the algorithm based on mathematical derivation and a linear programming model is presented to repair the inconsistency of TFRPRs. As per the multiplicative acceptability consistency degree (MACD) of TFRPRs, an MACD-induced ordered weighted averaging (MACD-IOWA) operator is proposed to aggregate individual TFRPRs into group TFRPR. The essence principle is that the larger the MACD value, the higher the decision maker’s weight. Finally, a case study, comparative analysis and discussions are given to verify the feasibility and validity of the proposed methods.

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.

Adaptive multi-criteria group decision-making model based on consistency and consensus with intuitionistic reciprocal preference relations: a case study in energy storage technology selection

Aim: This study aims to present a multi-criteria group decision-making model with intuitionistic reciprocal preference relations that incorporates a new feedback mechanims adapting its operation to the decision-makers' consistency. Methods: A new adaptive feedback mechanism is developed within a multi-criteria group decision-making model. Assuming different kinds of decision-makers classified according to their consistency, this feedback mechanism customizes the recommendations given by adopting three strategies to identify the problematic assessments that the decision-makers should change if they want to collaborate to increase the consensus. Results: The performance of the proposed model is shown in a case study for the selection of an energy storage technology. It shows that the decision-makers providing assessments with low consistency must modify more assessments, allowing the most consistent decision-makers to be the leaders of the negotiation and persuade the others to adjust their assessments to improve the agreement achieved. Conclusion: In this study, a novel multi-criteria group decision-making model is developed. It includes an adaptive feedback mechanism adjusting the advice provided by considering the decision-makers' consistency. It allows that the assessments of the most consistent decision-makers never be strongly modified during the decision process.

Bounded Rational Reciprocal Preference Relation for Decision Making

Fuzzy relations have been widely applied in decision making process. However, the application process requires people to have a high level of ability to compute and infer information. As people usually have limited ability of computing and inferring, the fuzzy relation needs to be adapted to fit the abilities of people. The bounded rationality theory holding the view that people have limited rationality in terms of computing and inferring meets such a requirement, so we try to combine the fuzzy relation with the bounded rationality theory in this study. To do this, first of all, we investigate four properties of fuzzy relations (i.e. reflexivity, symmetry, transitivity and reciprocity) within the bounded rationality context and find that these properties are not compatible with the bounded rationality theory. Afterwards, we study a new property called the bounded rational reciprocity of fuzzy relations, to make it possible to combine a fuzzy relation with the bounded rationality theory. Based on the bounded rational reciprocity, the bounded rational reciprocal preference relation is then introduced. A rationality visualization technique is proposed to intuitively display the rationality of experts. Finally, a bounded rationality net-flow-based ranking method is presented to solve real decision-making problems with bounded rational reciprocal preference relations, and a numerical example with comparative analysis is given to demonstrate the advantages of the proposed methods.

An adaptive algorithm for consensus improving in group decision making based on reinforcement learning

ABSTRACT In group decision-making problems with reciprocal preference relations, the process of improving individual consistency and consensus degree among decision-makers is dynamic and iter ative. Traditional automatic consensus-reaching process has some problems, such as adopting a fixed strategy in a deterministic environment without considering the dynamics of the decision-making environment, the destruction of individual consistency, etc. To solve these problems, an adaptive consensus-reaching model in a dynamic environment is proposed in this paper. Firstly, a Q-learning algorithm is used to build an environment model for different decision-making states of matrix modification. On the premise of modifying the preference matrix with a small matrix deviation, the optimal modification strategy is learned to improve the consensus degree among decision-makers. Second, we propose a method to control individual consistency in the process of consensus reaching by using a reward function. Finally, several numerical examples are used to illustrate the effectiveness and feasibility of the proposed algorithm. The experimental results show that the proposed algorithm significantly improves the consensus degree of the decision-makers in a small matrix deviation and ensures that the decision-maker’s individual consistency is not destroyed.

New additive consistency framework and utility derivation for interval fuzzy reciprocal preference relations

The interval fuzzy reciprocal preference relation (IFRPR) is one of commonly used frameworks characterizing decision-makers’ indeterminate preferences in multi-criteria decision making. Existing additive consistency frameworks and methods of generating additively consistent IFRPRs from interval fuzzy utility vectors as well as interval utility derivation methods often fail to obtain a satisfactory solution. To settle these challenging issues, this paper devises parametric transitivity equations to define additive consistency of IFRPRs. The parameter values are then determined and an equivalent definition is proposed for additively consistent IFRPRs. Important properties of consistent IFRPRs are developed and an additive consistency index is designed. A notion of equivalent interval fuzzy utility vectors is introduced and a novel framework is presented to normalize interval fuzzy utility vectors. Computational formulas are established for the generation between additively consistent IFRPRs and normalized interval fuzzy utility vectors. A goal programming model is built and converted into a linear program for deriving normalized interval fuzzy utility vectors from IFRPRs. Three illustrations including comparative analysis are provided to expose how the proposed models are utilized to identify whether an IFRPR is additively consistent and determine unknown preferences in an incomplete IFRPR as well as obtain normalized interval fuzzy utility vectors from IFRPRs.

General consistency conditions for fuzzy interval-valued preference relations

We define four types of general consistency conditions for fuzzy interval-valued additively reciprocal preference relations. These are far-reaching generalizations, on one hand, of the Krejčí's conditions, Krejčí (2017) [5] and, on the other hand, of the FG-transitivities defined by Świtalski (2003) [11]. We use the concept of the so-called consistency set, which gives possibility of considering more general conditions than the traditional additive consistency of Tanino. We prove theorems characterizing all the conditions by easily verifiable inequalities. Our characterization results for weak consistency and consistency generalize the results of Krejčí. We define consistency indexes related to considered consistency sets and show that in some situations these indexes can give more subtle characterization of consistency status of a given relation than the conditions presented by Krejčí.

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.

An indirect weak transitivity standard for inconsistent multiplicative reciprocal preference relations

An indirect weak transitivity standard for inconsistent multiplicative reciprocal preference relations

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.

New decision-making methods with interval reciprocal preference relations: A new admissible order relation of intervals

Interval reciprocal preference relations (IRPRs) have been extensively applied to real-life decision-making problems. This paper aims at introducing two new decision-making methods with IRPRs. To overcome drawbacks of several extant order relations of intervals, a new admissible order relation of intervals is proposed. An algorithm is then introduced to rank a series of intervals. We define a new consistency index and the satisfactory consistency of IRPRs. For improving the consistency level of IRPRs, a linear programming model is built. Subsequently, a new individual decision-making (IDM) method with an IRPR is introduced. For group decision making, a group consensus index is proposed. To improve the group consensus degree, an interactive convergent iterative algorithm is designed. Decision makers’ weights are determined by combining the logarithmic Manhattan distance between two IRPRs with the consistency indices of IRPRs. Accordingly, a novel consistent and consensus-based group decision-making (GDM) method with IRPRs is presented. Eventually, a decision support system is developed based on the proposed IDM method and GDM method. Illustrative examples and simulation experiments are provided to illustrate the superiority of the proposed IDM method and GDM method.

A characterization of reciprocal fuzzy preference structures and its compatibility with standard fuzzy preference structures

Fuzzy relations R:A2→[0,1] are widely used in decision-making in order to represent degrees of preference between alternatives in A. Examining fuzzy relations, this paper first reflects on an inherent incompatibility inside the domain of reciprocal fuzzy preference relations and preference structures. Aiming at clarifying the semantics of reciprocal fuzzy relations by means of that of weak fuzzy relations, a general frame is then proposed for studying the transformation between them, based on the notion of reciprocal functions. Next, we show that reciprocal relations can be naturally endowed with a preference structure of their own, containing two relations respectively showing a semantics of strict preference and lack of preference. Finally, we study the compatibility between these reciprocal preference structures and the standard preference structures of weak fuzzy preference relations by means of an axiomatic approach, leading to a system of functional equations that is shown to admit different solutions, allowing to explicitly assign one of the available semantics for reciprocal preference relations.

Analysis and classification of companies on tehran stock exchange with incomplete information

Due to uncertainty and large number of companies in financial market, it has become difficult to choose the right stock to investments. Identifying and classifying stocks using fundamental criteria help investors to better understand the risks involved in selecting companies and better manage their own capital, thereby rapidly and accurately choose their preferred stock and make more secure profit. The main concern that capital market investors are facing difficulty to choosing the right stock despite the uncertainties in the market. Uncertainties in the market that lead to incomplete information are presented in this article to complete the reciprocal preference relation method. The purpose of this paper is to present a method for completing information to reduce the uncertainties in the market and finally classify companies in each industry based on fundamental criteria. The classification method used is acceptability/reject ability which is based on distance fuzzy analysis yields more accurate results. Finally, a case study on one of the most critical industries in Tehran Stock Exchange is presented to show the effectiveness of the proposed approach.

A decision making model based on the leading principal submatrices of a reciprocal preference relation

A completed comparison matrix is the basic tool in the analytic hierarchy process (AHP). In practice, the process of forming a comparison matrix is complex and it is worth being investigated carefully. In this study, by decomposing pairwise comparison process of alternatives, the leading principal submatrices (LPSMs) of a completed comparison matrix are used as the basis for decision analysis. Based on the particle swarm optimization (PSO), a novel method for improving consistency of inconsistent comparison matrices is proposed. The fitness function is constructed by considering the acceptable consistency of pairwise comparison matrices and the similarity degree between the initial and the adjusted decision-making information. A new algorithm is elaborated on for solving a decision making problem with multiplicative reciprocal matrices. Numerical results are reported to show the advantages of the proposed model by comparing with the other methods. The observations reveal that the proposed method and algorithm are effective when dealing with inconsistent comparison matrices and the corresponding decision making problems.

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.

Matrix representation of stability definitions for the graph model for conflict resolution with reciprocal preference relations

Within the framework of the graph model, a matrix formulation is developed to model and analyze conflicts in which decision makers (DMs) may have reciprocal preferences. Specifically, matrix expressions are employed to represent DMs' reciprocal preference relations, unilateral movements (UMs) and fuzzy unilateral improvements (FUIs), as well as joint UMs and joint FUIs for a coalition of two or more DMs. Furthermore, a matrix methodology is provided to calculate whether a state in a conflict model, or scenario, is stable for a particular DM under various solution concepts, or stability definitions, that reflect the diversity of possible behavioral patterns for a DM in a conflict when preferences can be reciprocal. Five solution concepts associated with reciprocal preferences, Fuzzy Nash Stability, Fuzzy Symmetric Metarationality, Fuzzy General Metarationality, Fuzzy Sequential Stability, and Fuzzy Symmetric Sequential Stability, are redefined for matrix representations of both two-DM and multiple-DM conflict models. To illustrate how the matrix representation can be conveniently employed in practice, it is applied to two real-world conflicts.