Interval-valued Intuitionistic Fuzzy Preference Relations Research Articles

Interactively iterative group decision-making method with interval-valued intuitionistic fuzzy preference relations based on a new additively consistent concept

Interval-valued intuitionistic fuzzy preference relation (IVIFPR) is a powerful instrument for describing uncertain judgement of expert in preference group decision making (PGDM). Nevertheless, the extant additive consistency definitions for IVIFPRs violate the invariance regarding arrangement of objectives. In addition, the modification for comparative judgments in the consensus reaching process (CRP) of existing PGDMs with IVIFPRs do not involve the interaction or communication with experts. Therefore, we propose an PGDM to avoid these situations. First, a new additive consistency definition for IVIFPRs is given satisfying the invariance regarding arrangement of objectives. Then, an interactively iterative algorithm considering minimum information loss is developed to enhance the consistency of IVIFPRs. Next, as to group decision making (GDM) with IVIFPRs, a bi-objective optimization model is constructed to determine the experts’ weights, and an interactively iterative algorithm considering minimum information loss is developed to enhance the consensus level among experts. Furthermore, a fuzzy optimization model is constructed to derive the interval priority weights of objectives from an IVIFPR. Finally, the feasibility and superiority of the proposed GDM method are verified by application examples and analyses. The proposed GDM method has obtained an average of 0.0432 for information deviations and 22/72 for modified judgments on two datasets, respectively, which is one of the best results compared to state-of-art.

The consistency and consensus analysis for group decision-making with incomplete linguistic interval-valued intuitionistic fuzzy preference relations

The consistency and consensus analysis for group decision-making with incomplete linguistic interval-valued intuitionistic fuzzy preference relations

Interactively Iterative Group Decision-Making Method with Interval-Valued Intuitionistic Fuzzy Preference Relations Based on a New Additive Consistent Concept

Interactively Iterative Group Decision-Making Method with Interval-Valued Intuitionistic Fuzzy Preference Relations Based on a New Additive Consistent Concept

Some Methods Considering Multiplicative Consistency and Consensus in Group Decision Making with Interval-Valued Intuitionistic Fuzzy Preference Relations

In decision making with interval-valued intuitionistic fuzzy preference relations (IVIFPRs), the consistency and its improvement are key issues. The aim of this paper is to investigate a new multiplicative consistency of IVIFPRs based on interval arithmetic, and propose goal programming approach to improve consistency and consensus. Based on analysis of the drawbacks of the existing consistency definitions of IVIFPR, new definition for multiplicative consistent IVIFPR is introduced. Based on the proposed multiplicative consistency and analysis of the relationship between interval-valued intuitionistic fuzzy weights and multiplicative consistent IVIFPR, goal programming-based model is developed for deriving interval-valued intuitionistic fuzzy weights from IVIFPR. To measure the multiplicative consistency degree of IVIFPRs, a consistency index is introduced and then an acceptable consistency is defined. For an IVIFPR with unacceptable consistency, a mathematical programming approach is proposed to improve its consistency. By considering the experts’ subjective and consensus weights, the experts’ comprehensive weights are determined. Some numerical examples illustrate the proposed models.

Optimization-based group decision making using interval-valued intuitionistic fuzzy preference relations

In this paper, we propose an optimization-based group decision making (GDM) method using interval-valued intuitionistic fuzzy preference relations (IVIFPRs). First, the concept of consistency of intuitionistic fuzzy preference relations (IFPRs) is provided. Moreover, the consistency index for IFPRs is presented. Subsequently, by splitting an IVIFPR into two IFPRs, an additive consistency is proposed for IVIFPRs. Afterward, a consensus index is presented for GDM. When the consistency and the consensus do not achieve the requirement, we propose several models to reach the requirement. Furthermore, individual IVIFPRs are integrated into a collective IVIFPR. After that, a procedure is offered to obtain the interval-valued intuitionistic fuzzy (IVIF) priority weights of the alternatives. Moreover, a new GDM method with IVIFPRs is offered. Finally, some application examples are offered. The proposed GDM method can conquer the shortcomings of the existing GDM methods. It offers us a useful way for GDM in the IVIF context.

Analysis of Acceptably Multiplicative Consistency and Consensus for Incomplete Interval-Valued Intuitionistic Fuzzy Preference Relations

This article investigates group decision-making (GDM) problems, where the decision makers’ (DMs) preference information is represented by incomplete interval-valued intuitionistic fuzzy preference relations (IVIFPRs). First, a multiplicative consistency property and an acceptably multiplicative consistency property for IVIFPRs are offered. Then, an optimization model to estimate the missing values in an incomplete IVIFPR is constructed. Subsequently, two optimization models are, respectively, established to derive a perfectly consistent IVIFPR and an acceptably consistent IVIFPR from a given inconsistent IVIFPR. Furthermore, a model is offered to gain the DMs’ weights. Afterward, the consensus index is defined. When the consensus for IVIFPRs is unacceptable, a model is presented to reach the consensus requirement. Moreover, a novel GDM method for incomplete IVIFPRs is presented. Finally, the presented method is applied to an illustrative example that shows the feasibility of the offered method.

A comprehensive group decision-making method with interval-valued intuitionistic fuzzy preference relations

This paper investigates the group decision making (GDM) with interval-valued intuitionistic fuzzy (IVIF) preference relations (IVIFPRs). Considering decision maker’s (DM’s) risk attitude, the new score and accuracy functions for an IVIF value (IVIFV) are defined and a new order relation is proposed to rank IVIFVs. By transforming an IVIFPR into the direct and indirect IFPRs, a new additive consistency of IVIFPR is defined considering the uncertainty and ambiguity. Combining the direct IFPRs and the indirect IFPRs extracted from the IVIFPR, an algorithm is designed to determine the comprehensive IVIF priority weights of IVIFPR. For GDM with IVIFPRs, a multi-objective programming model is constructed to derive DMs’ weights by combining TOPSIS and cross entropy, which is solved by Lagrange function method. Based on the determination of DMs’ weights, the determination of priority weights and the new order relation of IVIFVs, a comprehensive method is proposed for GDM with IVIFPRs. Finally, a ventilation system selection example is analyzed to verify the effectiveness of the proposed method.

A Group Decision Making Method with Interval-Valued Intuitionistic Fuzzy Preference Relations and Its Application in the Selection of Cloud Computing Vendors for SMEs

To solve the problem of choosing the appropriate cloud computing vendors in small and medium-sized enterprises (SMEs), this paper boils it down to a group decision making (GDM) problem. To facilitate the judgment, this paper uses preference relation as the decision making technology. Considering the situation where uncertain positive and negative judgments exist simultaneously, interval-valued intuitionistic fuzzy preference relations (IVIFPRs) are employed to express the decision makers’ judgments. In view of the multiplicative consistency and consensus analysis, a new GDM algorithm with IVIFPRs is offered. To accomplish this goal, a new multiplicative consistency is first defined, which can avoid the limitations of the previous ones. Then, a programming model is built to check the consistency of IVIFPRs. To deal with incomplete IVIFPRs, two programming models are constructed to determine the missing values with the goal of maximizing the level of multiplicative consistency and minimizing the total uncertainty. To achieve the minimum adjustment of original preference information, a programming model is established to repair inconsistent IVIFPRs. In addition, programming models for getting the decision makers (DMs)’ weights and improving the consensus degree are offered. Finally, a practical decision making example is given to illustrate the effectiveness of the proposed method and to compare it with previous methods.

Call center performance measurement using intuitionistic fuzzy sets

PurposeThe companies are struggling to collect invoices due to the decrease in the economic growth. This global trend does not only affect undeveloped countries, but it also has a strong impact on the developed countries. Improving the debt collection process become a significant element to maintain financial stability. The institutions that are specialized on collecting payments, debt collection agencies and their call centers, with their expertise in the field can improve the payment process. Yet, managing evaluating the performance of debt collection agencies is a very hard process that involves uncertainty and imprecision. Performance measurement (PM) is a combination of numerically expressed characteristics which give insight about the success or degree of accomplishment of an activity. PM can be handled in various levels such as individual, team, department or company. The aim of this study is to present a systematic and objective PM method for call centers.Design/methodology/approachIn this study, first an exploratory approach is used to understand the call center measurement problem. Several meetings are done with the representatives of both call center firms and the parent firms that outsource debt collection process. Simultaneously, a broad literature review is conducted. An iterative approach is selected to reach deeper knowledge on the process. New meetings are planned and scope of the literature review has changed based on this iterative approach. After these steps, the problem has been considered as the multi-criteria decision-making problem since more than one criteria should be considered for evaluating the performances of call centers. The result of the literature review and the meetings with experts show that defining the weights for the criteria is very crucial for evaluating the performances accurately. Collecting human judgment for defining the weights of call center criteria necessitates dealing with vagueness and uncertainty. The intuitionistic fuzzy sets excellent tools for representing uncertainty. Interval valued intuitionistic fuzzy sets can easily represent the human judgments. Thus, in this study, an intuitionistic fuzzy multi-criteria decision making approach is used to design the proposed methodology. Incomplete interval-valued intuitionistic preference relations are used to determine the weights of the indicators aggregating linguistic evaluations of the decision makers.FindingsThe proposed approach provides an objective calculation of performance measurement. In order to provide objectivity, indicator performance functions are proposed for the first time in this study. Nine different functions and related parameters are defined to objectively measure indicator performances.Originality/valueThe paper proposes an objective and easy-to-modify approach for call-center PM, which can be used by call center managers. It presents a new fuzzy multi-criteria decision-making (MCDM) method for call center performance evaluation, which can consider the multi-experts' judgments under vagueness and impreciseness, which may be conflicting and incomplete interval-valued intuitionistic fuzzy preference relations. Also nine new functions are defined for indicator performance.

Analysis of the consistency and consensus for group decision-making with interval-valued intuitionistic fuzzy preference relations

This paper aims to offer a new group decision-making (GDM) method based on interval-valued intuitionistic fuzzy preference relations (IVIFPRs). To furnish this goal, a new additive consistency definition of IVIFPRs is first proposed. Then, a programming model is built to check the additive consistency of IVIFPRs. For incomplete IVIFPRs, two programming models are constructed, which aim at maximizing the consistency and minimizing the uncertainty of missing information. To achieve the minimum total adjustment, a goal programming model is established to repair inconsistent IVIFPRs. Considering the consensus, a programming model for improving the consensus degree is established, which permits different IVIFVs to have different adjustments and makes individual IVIFPRs have the smallest total adjustment to remain more original information. Based on these results, a consistency- and consensus-based GDM method is proposed. At length, a practical example for screening new majors of a private college in China is offered to illustrate the feasibility and efficiency of proposed method.

A procedure for group decision making with interval-valued intuitionistic linguistic fuzzy preference relations

To express the uncertain preferred and non-preferred qualitative judgments of decision makers, interval-valued intuitionistic linguistic fuzzy sets (IVILFSs) are proposed in a similar way as Atanassov and Gargov’s interval-valued intuitionistic fuzzy sets. Considering the application of IVILFSs, the concept of interval-valued intuitionistic linguistic fuzzy variables (IVILFVs) is defined and a ranking order is offered. Then, we introduce interval-valued intuitionistic linguistic fuzzy preference relations (IVILFPRs) whose elements are IVILFVs. Furthermore, an additive consistency concept is presented and a model for judging the consistency of IVILFPRs is built. Meanwhile, optimization models for deriving additively consistent IVILFPRs and for determining missing linguistic variables in incomplete IVILFPRs are constructed, respectively. For group decision making, optimization models for determining the weights of the decision makers and for improving the consensus level are established, respectively. A procedure for group decision making with IVILFPRs is developed that can cope with inconsistent and incomplete IVILFPRs. Finally, a practical group decision-making problem about evaluating express companies is selected to show the application of the new results.

An Algorithm for Interval-Valued Intuitionistic Fuzzy Preference Relations in Group Decision Making Based on Acceptability Measurement and Priority Weight Determination

Group decision making with intuitionistic fuzzy preference information contains two key issues: acceptability measurement and priority weight determination. In this paper, we investigate the above two issues with respect to multiplicative interval-valued intuitionistic fuzzy preference relation (IVIFPR). Firstly, a consistency index is defined to measure the multiplicative consistency degree of IVIFPR and an optimization model is established to improve the consistency degree of IVIFPR to an acceptable one. Next, in terms of priority weight determination, an error-analysis-based extension method is proposed to obtain priority weight vector from the acceptable IVIFPR. For GDM problems, decision makers’ weights are derived by the proposed multiplicative consistency index. Subsequently, the collective IVIFPR is obtained by using an interval-valued intuitionistic fuzzy (IVIF) weighted averaging operator. Finally, a step-by step algorithm for GDM with IVIFPRs is given, and an example of enterprise innovation partner selection is analyzed, and comparative analyses with existing approaches are performed to demonstrate that the proposed algorithm is both effective and practical in dealing with GDM problems.

Additively Consistent Interval-Valued Intuitionistic Fuzzy Preference Relations and Their Application to Group Decision Making

This paper aims to propose an innovative approach to group decision making (GDM) with interval-valued intuitionistic fuzzy (IVIF) preference relations (IVIFPRs). First, an IVIFPR is proposed based on the additive consistency of an interval-valued fuzzy preference relation (IVFPR). Then, two mathematical or adjusted programming models are established to extract two special consistent IVFPRs. In order to derive the priority weight of an IVIFPR, after taking the two special IVFPRs into consideration, a linear optimization model is constructed by minimizing the deviations between individual judgments and between the width degrees of the interval priority weights. For GDM with IVIFPRs, the decision makers’ weights are generated by combining the adjusted subjective weights with the objective weights. Subsequently, using an IVIF-weighted averaging operator, the collective IVIFPR is obtained and utilized to derive the IVIF priority weights. Finally, a practical example of a supplier selection is analyzed to demonstrate the application of the proposed method.

Decision making with interval-valued intuitionistic fuzzy preference relations based on additive consistency analysis

This paper discusses the theory and application of interval-valued intuitionistic fuzzy preference relations (IVIFPRs). To do this, we first introduce an additive consistency concept for IVIFPRs that can address the issues in previous ones. Then, several additive consistency-based programming models for judging the consistency of IVIFPRs and for determining missing values are constructed. Furthermore, an approach for calculating the interval-valued intuitionistic fuzzy priority weight vector is offered. With respect to group decision making with IVIFPRs, a consensus index is offered, and an interactive algorithm for improving the consensus level is proposed that can ensure the additive consistency. After that, a consensus analysis-based programming model for determining the weights of decision makers is built. Furthermore, a consistency-and-consensus-based group decision-making method that can address inconsistent and incomplete IVIFPRs is developed. Finally, a practical decision-making problem about evaluating talents in universities is offered to show the feasibility and efficiency of the new method.

A group decision-making method considering both the group consensus and multiplicative consistency of interval-valued intuitionistic fuzzy preference relations

This paper investigates a group decision-making (GDM) method that considers group consensus and multiplicative consistency of interval-valued intuitionistic fuzzy (IVIF) preference relations (IVIFPRs). First, the mean and variance of IVIF values (IVIFVs) are defined and a ranking method for IVIFVs is proposed considering the risk attitude of the expert. Then, the group consensus is presented by the individual similarity between experts. An iteration algorithm is designed to improve the group consensus. A statistical comparative analysis validates this algorithm. Subsequently, a new multiplicative consistency of IVIFPR is defined based on the multiplicative consistency of interval fuzzy preference relation. Two single-objective programming models are established to extract the most optimistic and pessimistic interval priority weight vectors from an IVIFPR, respectively. In particular, if the feasible domains of these two models are empty, two adjusted programs are constructed to replace the originals. Combining the most optimistic and pessimistic interval priority weights, the IVIF priority weights are generated. Further, expert weights are derived from Markov model and used to derive the collective IVIFPR for generating the IVIF priority weights. Therefore, a new method is proposed to solve the GDM with IVIFPRs. Finally, two cases are analyzed to verify the effectiveness of the proposed method.

A Three-Phase Method for Group Decision Making With Interval-Valued Intuitionistic Fuzzy Preference Relations

This paper develops a new method for solving group decision making (GDM) problems with interval-valued intuitionistic fuzzy preference relations (IVIFPRs). First, an additive consistency of an IVIFPR is defined by the additive consistency of intuitionistic fuzzy preference relation (IFPR). Based on the additive consistency definition of IVIFPR, two linear programming models are established to extract the most optimistic and pessimistic consistent IFPRs from an IVIFPR, respectively. Especially, if the feasible regions of these two models are empty, two adjusted programming models are constructed. Afterwards, a risk attitudinal-based consistent IFPR is determined considering decision maker's (DM's) risk attitude. To derive the intuitionistic fuzzy priority weights from the risk attitudinal-based consistent IFPR, a multiobjective programming model is established and transformed into a linear goal program to resolve. Subsequently, combining DMs’ subjective and objective importance degrees, the comprehensive importance degrees of DMs are generated. Using comprehensive importance degrees as order inducing variables, a new comprehensive importance interval-valued intuitionistic fuzzy induced ordered weighted averaging (CI-IVIF-IOWA) operator is defined to aggregate the individual IVIFPRs into a collective one. Thereby, a three-phase method is proposed for GDM with IVIFPRs. An example of network system selection is examined to illustrate the practicability and effectiveness of the proposed method.

A programming-based algorithm for interval-valued intuitionistic fuzzy group decision making

Interval-valued intuitionistic fuzzy preference relations (IVIFPRs) are powerful to express the uncertain preferred and non-preferred judgments of decision makers simultaneously. After reviewing previous researches about IVIFPRs, we find that several limitations exist. Especially, previous methods are insufficient to address inconsistent and incomplete cases. Considering these issues, this paper first analyzes the limitations of the previous consistency concepts for IVIFPRs and then introduces a new one that avoids the disadvantages of previous ones. 0–1 mixed programming models are built for judging the multiplicative consistency of IVIFPRs. Subsequently, several multiplicative consistency-based 0–1 mixed programming models are constructed for determining missing values that can address the situation where ignored objects exist. For group decision making, a distance measure on IVIFPRs is defined, by which the weights of the decision makers are derived. Meanwhile, a consensus index is offered. A multiplicative consistency and consensus based algorithm to group decision making with IVIFPRs is proposed. Finally, two practical decision-making problems are offered to show the application of the new algorithm, and theoretical and numerical analysis of several related methods is made.

A Novel Approach to Group Decision-Making with Interval-Valued Intuitionistic Fuzzy Preference Relations via Shapley Value

This paper proposes a novel weighting approach to group decision-making (GDM) with interval-valued intuitionistic fuzzy preference relations (IVIFPRs) based on a fuzzy cooperative game method and the continuous interval-valued intuitionistic fuzzy ordered weighted averaging (CIVIFOWA) operator. First of all, a continuous IVIFPR (CIVIFPR) is defined based on the CIVIFOWA operator, and then considering the contribution of each decision-maker (DM), an iterative algorithm is designed to redistribute weights of DMs by using cooperative method. Moreover, a logarithm least optimal model is developed to deriving interval priority weights of IFPR and a two-stage resolution process is proposed for the GDM with IVIFPRs. Finally, a practical example with cooperation and competition is provided to verify the feasibility and efficiency of the proposed method. The characteristics of the proposed method are as follows: (1) the iterative algorithm is devoted to deriving DMs’ weights in GDM by using fuzzy cooperative game based on the CIVIFOWA operator in which the contribution of each DM’s opinion to the group indicates the rationality and importance in GDM of their opinions; (2) the weighting algorithm can be adjusted by modifying the attitude parameter based on the CIVIFOWA operator, which makes the proposed method more flexible.

Likelihood-based assignment methods for multiple criteria decision analysis based on interval-valued intuitionistic fuzzy sets

The aim of this paper is to develop useful likelihood-based assignment methods for addressing multiple criteria decision-making problems within the environment of interval-valued intuitionistic fuzzy sets. Based on the likelihoods of interval-valued intuitionistic fuzzy preference relations, this paper determines the mean likelihoods of outranking relations and presents a mean likelihood determination method for generating a set of criterion-wise rankings of alternatives. By employing the concepts of rank frequency matrices and (ordinary) rank contribution matrices, this paper establishes a likelihood-based linear assignment model for multiple criteria decision analysis in the interval-valued intuitionistic fuzzy context. Additionally, this paper propounds two likelihood-based assignment models for handling incomplete and conflicting certain information of importance weights. These models can transform the criterion-wise ranks into the overall ranks for determining the optimal priority ranking of the alternatives. The feasibility and applicability of the proposed methods are illustrated with a practical problem of selecting a bridge construction method which involves various preference types. Finally, this paper conducts a comparative analysis with previous assignment-based methods in an interval-valued intuitionistic fuzzy setting to validate the effectiveness and advantages of the proposed methods.

Multiple criteria decision analysis using a likelihood-based outranking method based on interval-valued intuitionistic fuzzy sets

The purpose of this paper is to develop a likelihood-based outranking method for handling multiple criteria decision analysis problems based on interval-valued intuitionistic fuzzy sets. Using the concept of likelihood of the interval-valued intuitionistic fuzzy preference relations, this paper determines certain generalized criteria and the corresponding likelihood-based preference functions. Based on some useful concepts of comprehensive preference indices, concordance indices, counter-likelihood-based preference functions, and discordance indices, this paper conducts a concordance–discordance analysis with concordance and discordance outranking relationships to determine a global Boolean matrix and acquire partial ranking orders of the alternatives. Alternatively, this paper determines complete ranking orders of the alternatives using the concepts of net concordance indices, net discordance indices, and mean outranking values. The feasibility and applicability of the proposed method are illustrated with a practical multiple criteria decision-making application concerning the selection of a suitable bridge construction method. Finally, comparative discussions with different decision-making methods are conducted to verify the effectiveness and advantages of the proposed method in aiding decision making.