Fuzzy 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.

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.

Comments on “A prioritization approach of non-reciprocal fuzzy preference relations and its extension”

A recently published paper by Liu, Yang and Hu (2022) introduced a concept of non-reciprocal fuzzy preference relations (NrFPRs) and brought forward double-variance-based measuring formulas to compute additive consistency indices of NrFPRs and interval reciprocal fuzzy preference relations as well as built a double-variance-based minimization model with three parameters to acquire priority weight vectors from NrFPRs. This study employs five numerical illustrations to expose that some properties proposed by Liu, Yang and Hu (2022) are not true and the measuring formulas fail to obtain correct outcomes as well as the double-variance-based minimization model is based on an incorrect assumption. NrFPRs are redefined as particular cases of fuzzy preference relations. An additive transitivity equation is established to define additive consistency of NrFPRs and a square deviation based formula is developed to calculate additive consistency indices of NrFPRs. A quadratic programming model is set up to acquire priority weight vectors with interval fuzzy elements from NrFPRs. A closed-form solution based approach is put forward to acquire optimal priority weight vectors from additively consistent NrFPRs.

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.

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.

New consensus measure for group decision-making based on Spearman’s correlation coefficient for reciprocal fuzzy preference relations

ABSTRACT In group decision-making, reaching to a level of agreement about the decision between the group members is important. Indeed, it is considered as the main challenge to be solved in any group decision-making problem. Typically, similarity and distance measures are used to measure the degree of similarities of preference intensities in group decision-making. In this paper, we propose a new consensus measure based on Spearman’s correlation coefficient for group decision-makers with reciprocal preference relations. We rely on ranked preference intensities of a decision-maker and find its correlation with respect to other decision-makers in the group. The novelty of this work relies on considering the coherence of decision-maker preference degree ranks as a whole rather than in a pairwise manner. This work does not rely directly on similarity/distance functions, as do most of the consensus models in the literature, but rather adopts the idea of measuring the monotonic degree among them. In addition, based on this model, we propose a feedback mechanism to act as a mediator to guide the group into the consensus solution. We illustrate the new model by presenting a numerical example. Moreover, the model results show robustness when validated on several problems.

Two new methods for decision-making with incomplete reciprocal fuzzy preference relations based on additive consistency

ABSTRACT In this paper, we propose two new methods to solve the problem of incomplete reciprocal fuzzy preference relations. Sometimes decision-maker(s) might not be able to provide complete information about their preferences on the alternatives. In general, an incomplete preference relation can be completed if at least a set of nonleading diagonal preference values are known and each one of the alternatives is compared directly or indirectly at least once. However, this is not always the case. Sometimes, the decision-maker(s) might not be able to provide any information with regards to at least one of the alternatives, which is called the ignorance situation. We propose two new methods based on additive consistency for solving these two problems in a multi-attributes/group decision-making environment. The first method is based on a system of equations, which is suitable for estimating missing information in the general case. The second method, which is an extension of the system of equations, utilizes a goal programming model to address the ignorance situations. Our validation of these two methods shows that the proposed methods generate a high consistency level irrespective of the nature of the problem under study. The proposed methods outperform other comparable methods without a need to modify or change the original decision-maker(s) preferences.

A Bayesian approach to incomplete fuzzy reciprocal preference relations in consensus reaching process and its application in project performance evaluations

Due to the fact that individual knowledge is often inadequate and deficient in group decision-making problems, it is commonly seen that the preference information provided by the decision makers are incomplete. To encounter this problem lots of rele

The graph model for conflict resolution with incomplete fuzzy reciprocal preference relations

Within the paradigm of the Graph Model for Conflict Resolution (GMCR), fuzzy reciprocal preference relations (FPRs) can be usually employed to handle decision makers' (DMs') certain and uncertain preferences. Due to lack of time, limited expertise, or dynamic factors, the DMs' fuzzy preferences over the alternatives or states may be incomplete. Thus, a new preference relation, the incomplete fuzzy reciprocal preference relation (IFPR), is proposed to deal with both complete and incomplete fuzzy preferences within GMCR. In this study, one key contribution is that, according to DMs' attitude to risk associated with incomplete fuzzy preferences, four extended sets of stability definitions are put forward for providing a nuanced description of human behavior for both two-DM and n-DM conflict models. Furthermore, the interrelationships of solution concepts within and across the four extensions of stability definitions are investigated. Finally, an illustrative example, the potential large-scale export of water from Lake Gisborne, Canada, is used to show the usefulness of GMCR with incomplete fuzzy reciprocal preference relations.

Visualizing and rectifying different inconsistencies for fuzzy reciprocal preference relations

Consistency is related with reasonableness. This paper proposes a graphical method to visualize and rectify different inconsistencies for fuzzy reciprocal preference relations (FPRs), and the consistency includes the ordinal and cardinal consistencies. Ordinal inconsistency implies cardinal inconsistency, but not vice versa, and cardinal consistency of FPRs includes additive consistency and multiplicative consistency. Therefore, there are four types of inconsistencies, includes: (1) the additive inconsistency; (2) the multiplicative inconsistency; (3) the ordinal and additive inconsistencies, and (4) the ordinal and multiplicative inconsistencies. First, Gower plots are proposed to visualize the ordinal and cardinal inconsistencies on a two-dimensional plane. Then, some optimization models are formulated to rectify the four different types of inconsistences, respectively. Some examples are illustrated to show how the models work, and comparisons with the existing methods are also offered to show the advantages of the proposed methods.

A two-stage consensus reaching model for group decision making with reciprocal fuzzy preference relations

Reciprocal fuzzy preference relation (FPR) is a hot topic in the research field of group decision making (GDM). Within the framework of GDM, a two-stage consensus reaching model is proposed to modify the decision makers’ (DMs’) FPRs. In the first stage which is called consistency improving stage, a consistency index and an iterative algorithm are employed to measure and improve individual consistency, respectively. In each iteration of the algorithm, only one pair of preference values, which deviates the most from the corresponding elements in the consistent FPR matrix, are modified to keep the DM’s original information as much as possible. In the second stage which is called consensus reaching stage, individual consensus index and group consensus index are applied to measure consensus. A convex quadratic programming model is developed to help DMs reach consensus. Some desired properties of the model are also investigated and testified. What’s more, the final individual consistency is also limited to an acceptable threshold. Finally, a practical example is used to illustrate the validity of the proposed consensus model.

Geo-uninorm consistency control module for preference similarity network hierarchical clustering based consensus model

In order to avoid misleading decision solutions in group decision making (GDM) processes, in addition to consensus, which is obviously desirable to guarantee that the group of experts accept the final decision solution, consistency of information should also be sought after. For experts’ preferences represented by reciprocal fuzzy preference relations, consistency is linked to the transitivity property. In this study, we put forward a new consensus approach to solve GDM with reciprocal preference relations that implements rationality criteria of consistency based on the transitivity property with the following twofold aim prior to finding the final decision solution: (A) to develop a consistency control module to provide personalized consistency feedback to inconsistent experts in the GDM problem to guarantee the consistency of preferences; and (B) to design a consistent preference network clustering based consensus measure based on an undirected weighted consistent preference similarity network structure with undirected complete links, which using the concept of structural equivalence will allow one to (i) cluster the experts; and (ii) measure their consensus status. Based on the uninorm characterization of consistency of reciprocal preferences relations and the geometric average, we propose the implementation of the geo-uninorm operator to derive a consistent based preference relation from a given reciprocal preference relation. This is subsequently used to measure the consistency level of a given preference relation as the cosine similarity between the respective relations’ essential vectors of preference intensity. The proposed geo-uninorm consistency measure will allow the building of a consistency control module based on a personalized feedback mechanism to be implemented when the consistency level is insufficient. This consistency control module has two advantages: (1) it guarantees consistency by advising inconsistent expert(s) to modify their preferences with minimum changes; and (2) it provides fair recommendations individually, depending on the experts’ personal level of inconsistency. Once consistency of preferences is guaranteed, a structural equivalence preference similarity network is constructed. For the purpose of representing structurally equivalent experts and measuring consensus within the group of experts, we develop an agglomerative hierarchical clustering based consensus algorithm, which can be used as a visualization tool in monitoring current state of experts’ group agreement and in controlling the decision making process. The proposed model is validated with a comparative analysis with an existing literature study, from which conclusions are drawn and explained.

An approach to decision-making with triangular fuzzy reciprocal preference relations and its application

ABSTRACTTriangular fuzzy reciprocal preference relations (TFRPRs) are powerful tools to denoting decision-makers’ fuzzy judgments, which permit the decision-makers to apply triangular fuzzy ratio rather than real numbers to express their judgements. Consistency analysis is one of the most crucial issues in preference relations that can guarantee the reasonable ranking order. However, all previous consistency concepts cannot well address this type of preference relations. Based on the operational laws on triangular fuzzy numbers, this paper introduces an additive consistency concept for TFRPRs by using quasi TFRPRs, which can be seen as a natural extension of the crisp case. Using this consistency concept, models to judging the additive consistency of TFRPRs and to estimating missing values in complete TFRPRs are constructed. Then, an algorithm to decision-making with TFRPRs is developed. Finally, two numerical examples are offered to illustrate the application of the proposed procedure, and comparison analysis is performed.

Investigating rank reversal in reciprocal fuzzy preference relation based on additive consistency: Causes and solutions

Rank reversal is a common phenomenon in decision making. Rank reversal occurs when a new alternative is added to (or removed from) a set of alternatives, which causes change in the ranking order of the alternatives. This paper studies the possible causes of rank reversal in reciprocal preference relation based on additive consistency. Our investigation reveals that inconsistency of information is the main cause of this phenomena in preference relations followed by ranking score aggregation. We propose score aggregation methods to address the phenomenon of rank reversal. The proposed methods are illustrated using numerical examples. The results are better than other tested methods.

A Group Decision Making Approach Based on Newly Defined Additively Consistent Interval-Valued Intuitionistic Preference Relations

Because of the complexity of interval-valued intuitionistic fuzzy set, there are few literatures investigating the additive consistency of the interval-valued intuitionistic preference relation. This paper proposes a group decision making approach with interval-valued intuitionistic preference relations. Firstly, two additive reciprocal fuzzy preference relations and two fuzzy non-preferred relations are obtained by dividing an interval-valued intuitionistic preference relation in order to avoid the operations of interval-valued intuitionistic fuzzy numbers. Here, the fuzzy non-preferred relation is interpreted as the non-preferred intensity of one alternative over another one. Moreover, the rational concept of additive consistency of interval-valued intuitionistic preference relation is defined. Secondly, optimizing models to derive priority weight vector from interval-valued intuitionistic preference relation are constructed. In particular, the priority weight vector is in terms of interval-valued intuitionistic fuzzy numbers, which is rational under the interval-valued intuitionistic fuzzy environment. Thirdly, an extended approach to address group decision making (GDM) problems is proposed. Each individual expert’s consistency index is applied to measure his/her degree of importance. Finally, an illustrative example of supplier selection is provided to illustrate the proposed approach addressing GDM problems. The feasibility to derive priority weight vector in terms of interval-valued intuitionistic fuzzy numbers using the developed approach is verified. And the proposed approach is compared with the other conventional existing ones to show its advantages.

Deriving the priority weights from trapezoidal fuzzy reciprocal preference relations based on uncertainty ratio and geometric mean

Deriving the priority weights from trapezoidal fuzzy reciprocal preference relations based on uncertainty ratio and geometric mean

On multiplicative consistency of interval and fuzzy reciprocal preference relations

Extension of Saaty’s definition of consistency to interval and fuzzy reciprocal preference relations is studied in the paper. The extensions of the definition to interval and triangular reciprocal preference relations proposed by Wang (2005), Liu (2009), Liu et al. (2014) and Wang (2015a, 2015b) are reviewed and some shortcomings in the definitions are pointed out. Particularly, as was already shown by Wang (2015a, 2015b), the definitions of consistency proposed by Liu (2009) and Liu et al. (2014) are not invariant under permutation of compared objects. Wang’s (2015a, 2015b) definitions rectify this drawbacks. However, as is pointed out in this paper, Wang’s definitions of consistent interval and triangular reciprocal preference relations do not keep the reciprocity of pairwise comparisons, which is the substance of reciprocal preference relations. In this paper, definitions of consistent interval, triangular and trapezoidal reciprocal preference relations invariant under permutation of compared objects and preserving the reciprocity of pairwise comparisons are introduced. Useful tools for verifying the consistency are proposed and some properties of consistent interval and fuzzy reciprocal preference relations are derived. Furthermore, the new definition of consistency for interval reciprocal preference relations is compared with the definition of consistency proposed by Wang et al. (2005), and numerical examples are provided to illustrate the difference between the consistency definitions.

The additive consistency measure of fuzzy reciprocal preference relations

Fuzzy reciprocal preference relations (FPR) are one of the most common preference relations which decision makers (DMs) express their comparison information in decision making, and the consistency of preference relations is an important step for reasonable and reliable decision making. Based on the concept of deviation between two matrices, we develop some consistency measures for FPRs to ensure that the DMs are being neither random nor illogical. A consistency index $$(CI)$$ and the threshold $$(\overline{CI})$$ of FPRs are defined to measure whether a FPR is of acceptable consistency. For FPRs with unacceptable consistency, an optimization method and two iterative algorithms are presented to improve its consistency and the process terminates until the $$CI$$ is controlled within the threshold $$\overline{CI}.$$ Furthermore, one of algorithms is extended to handle group decision making (GDM) of FPRs. Finally, two examples and comparative analysis are furnished to demonstrate the effectiveness of the developed methods.

An axiomatic approach to approximation-consistency of triangular fuzzy reciprocal preference relations

Under the assumption of rational economics, the consistency of judgments is one of the important issues in multiple criteria decision making (MCDM) methods. We propose three axiomatic properties of the consistent judgments in relative measurements being considered from the perspective of strict logic and rationality: (A1) the reciprocal property of pairwise comparisons, (A2) the invariance of consistency with respect to permutations of alternatives, and (A3) the robustness of the ranking of alternatives. These properties are further applied to analyze the consistency of the judgments expressed by positive real numbers and triangular fuzzy numbers, respectively. The inconsistency of comparison matrices encountered in the Analytic Hierarchy Process (AHP) can be considered to be the case of weakening the axiomatic property (A3). Fuzzifying the judgments is due to the weakening of axiomatic property (A1). A method of weakening the axiomatic properties for triangular fuzzy reciprocal matrices is proposed and a new concept of approximation-consistency is put forward to distinguish from the typical consistency. By considering the permutations of alternatives, an illustrative example is presented to show the use of the proposed procedures for solving the decision making problems with triangular fuzzy reciprocal preference relations.

Modeling and applying credible interval intuitionistic fuzzy reciprocal preference relations in group decision making

Intuitionistic fuzzy preference relations are powerful techniques used to express uncertain preference information. However, simultaneously providing the exact priority and non-priority intensities could be difficult in real applications. A credible interval intuitionistic fuzzy number (CIIFN) is introduced and a credible interval intuitionistic fuzzy reciprocal preference relation (CIIFRPR) is developed to solve this issue. Unlike intuitionistic fuzzy preference relations, the new preference relations use the CIIFNs to express the preference information such that the decision makers simply provide the priority intensity with interval-valued numbers and calculate the non-preference intensity with the transformed method, which avoids a complex evaluation of non-priority information. Furthermore, some basic operations and comparison laws are investigated, based on which three credible interval intuitionistic fuzzy aggregation operators are proposed. Two models are presented to manage the group decision-making. Finally, a practical case is used to demonstrate the feasibility and reasonability of the proposed preference relations and aggregation operators.