Geometric Averaging Operator Research Articles

Trapezoidal neutrosophic aggregation operators and its application in multiple attribute decision making process

The aim of this paper is to introduce interval trapezoidal neutrosophic set which is a combination of trapezoidal fuzzy numbers and interval neutrosophic set. The paper presents some operational rules, the score and accuracy functions of interval trapezoidal neutrosophic numbers. Then, some aggregating operators under interval trapezoidal neutrosophic information which are called interval trapezoidal neutrosophic number weighted arithmetic averaging (ITNNWAA) operator and interval trapezoidal neutrosophic number weighted geometric averaging (ITNNWGA) operator, are proposed, and their properties are investigated in detail. Furthermore, based on these operators a multi-attribute decision making method is developed. Finally, a numerical example is presented to illustrate the application and effectiveness of the proposed method.

Innovative method to evaluate quality management system audit results’ using single value neutrosophic number

Using Quality Management System (QMS) enables the organization to fulfil its purpose and mission to meet customers and regulatory requirements and improve its effectiveness and efficiency on a continuous basis, Quality Audit is very critical tool to ensure that the standards, policies and/or procedures are applied in the organization as planned in QMS, in real world applications, the results of the Quality Audit process depend on the auditors’ human judgments that characterized with doubts sometimes, estimating values another times, and using general impressions in his judgment, which motivate to introduce for the first time the Neutrosophic Quality Audit (NQA) model which can be extended to manage and audit in the future. NQA will be used to enhance the way of evaluating the quality audit results while many uncertainty aspects and indeterminacy do exist, the proposed model using Single Value Neutrosophic Numbers (SVNN) which is an instance of neutrosophic set, to overcome the weakness of using a crisp value to present the result of the audit findings, in addition, using Simplified Neutrosophic Weighted Geometric Average Operator (SNWGAO), and cosine similarity degree, an illustrative example is provided to validate the proposed model.

New form of single valued neutrosophic uncertain linguistic variables aggregation operators for decision-making

A neutrosophic uncertain linguistic variable is composed of an uncertain linguistic variable and a neutrosophic number, which has more advantages than both of uncertain linguistic variable and neutrosophic number in expression. Therefore, the decision-making theory and its related methods based on neutrosophic uncertain linguistic variables have received wide attention of scholars. This paper mainly studies the new expression and operations of single value neutrosophic uncertain linguistic variables and its application in multiple attribute group decision-making (MAGDM). First, a new form of single value neutrosophic uncertain linguistic set (NFSVNULS) and its operational rules are defined. Then, a new form of single value neutrosophic uncertain linguistic variable weighted arithmetic average (NFSVNULVWAA) operator and a new form of single value neutrosophic uncertain linguistic variable weighted geometric average (NFSVNULVWGA) operator are proposed. Furthermore, a MAGDM method based on the proposed aggregation operators is put forward. Finally, an example of investment is used to demonstrate the feasibility and effectiveness of the proposed method.

Multiple Attribute Decision-Making Method Using Linguistic Cubic Hesitant Variables

Linguistic decision making (DM) is an important research topic in DM theory and methods since using linguistic terms for the assessment of the objective world is very fitting for human thinking and expressing habits. However, there is both uncertainty and hesitancy in linguistic arguments in human thinking and judgments of an evaluated object. Nonetheless, the hybrid information regarding both uncertain linguistic arguments and hesitant linguistic arguments cannot be expressed through the various existing linguistic concepts. To reasonably express it, this study presents a linguistic cubic hesitant variable (LCHV) based on the concepts of a linguistic cubic variable and a hesitant fuzzy set, its operational relations, and its linguistic score function for ranking LCHVs. Then, the objective extension method based on the least common multiple number/cardinality for LCHVs and the weighted aggregation operators of LCHVs are proposed to reasonably aggregate LCHV information because existing aggregation operators cannot aggregate LCHVs in which the number of their hesitant components may imply difference. Next, a multi-attribute decision-making (MADM) approach is proposed based on the weighted arithmetic averaging (WAA) and weighted geometric averaging (WGA) operators of LCHVs. Lastly, an illustrative example is provided to indicate the applicability of the proposed approaches.

Operations and Aggregation Methods of Single-Valued Linguistic Neutrosophic Interval Linguistic Numbers and Their Decision Making Method

To comprehensively describe uncertain/interval linguistic arguments and confident linguistic arguments in the decision making process by a linguistic form, this study first presents the concept of a single-valued linguistic neutrosophic interval linguistic number (SVLN-ILN), which is comprehensively composed of its uncertain/interval linguistic number (determinate linguistic argument part) and its single-valued linguistic neutrosophic number (confident linguistic argument part), and its basic operations. Then, the score function of SVLN-ILN based on the attitude index and confident degree/level is presented for ranking SVLN-ILNs. After that, SVLN-ILN weighted arithmetic averaging (SVLN-ILNWAA) and SVLN-ILN weighted geometric averaging (SVLN-ILNWGA) operators are proposed to aggregate SVLN-ILN information and their properties are investigated. Further, a multi-attribute decision-making (MADM) method based on the proposed SVLN-ILNWAA or SVLN-ILNWGA operator and the score function is established under consideration of decision makers’ preference attitudes (pessimist, moderate, and optimist). Lastly, an actual example is given to show the applicability of the established MADM approach with decision makers’ attitudes.

Multi-criteria decision making method based on interval-valued intuitionistic fuzzy sets

Multi-criteria group decision making is a widely used efficient decision methodology to improve quality of the decision. In this paper, the interval-valued intuitionistic fuzzy weighted arithmetic average operator, the interval-valued intuitionistic fuzzy weighted geometric average operator, and an accuracy function of interval-valued intuitionistic fuzzy value are introduced. The proposed aggregation operators with a accuracy function is more efficient to take decision. Finally, an example is provided to illustrate the application of the developed approach. The results show that the proposed new approach is more comprehensive and flexible by comparing with the other existing aggregation operators and accuracy functions.

Dombi Aggregation Operators of Linguistic Cubic Variables for Multiple Attribute Decision Making

A linguistic cubic variable (LCV) is comprised of interval linguistic variable and single-valued linguistic variable. An LCV contains decision-makers’ uncertain and certain linguistic judgments simultaneously. The advantage of the Dombi operators contains flexibility due to its changeable operational parameter. Although the Dombi operations have been extended to many studies to solve decision-making problems; the Dombi operations are not used for linguistic cubic variables (LCVs) so far. Hence, the Dombi operations of LCVs are firstly presented in this paper. A linguistic cubic variable Dombi weighted arithmetic average (LCVDWAA) operator and a linguistic cubic variable Dombi weighted geometric average (LCVDWGA) operator are proposed to aggregate LCVs. Then a multiple attribute decision making (MADM) method is developed in LCV setting on the basis of LCVDWAA and LCVDWGA operators. Finally, two illustrative examples about the optimal choice problems demonstrate the validity and the application of this method.

Operations and aggregation method of neutrosophic cubic numbers for multiple attribute decision-making

A neutrosophic cubic set (NCS) consists of both an interval neutrosophic set and a single-valued neutrosophic set. Hence, NCSs are very suitable for expressing the hybrid information of both the interval neutrosophic set (related to the uncertain part of the information) and the single-valued neutrosophic set (related to the certain part) simultaneously. However, there is little research on operations and applications of NCNs in existing literature. First, therefore, this paper proposes the operations of neutrosophic cubic numbers (NCNs), which are basic elements in NCSs. Second, we define the score, accuracy, and certainty functions, along with a ranking method for NCNs. Third, we develop a neutrosophic cubic number weighted arithmetic averaging (NCNWAA) operator and a neutrosophic cubic number weighted geometric averaging (NCNWGA) operator in order to aggregate NCN information and indicate their properties. Then, a multiple attribute decision-making method based on the NCNWAA or NCNWGA operator and the ranking method of NCNs is established under a NCN environment. Finally, an illustrative example about the selection problem of investment alternatives is provided in order to demonstrate the application and feasibility of the developed approach.

Dombi Aggregation Operators of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making

The neutrosophic cubic set can describe complex decision-making problems with its single-valued neutrosophic numbers and interval neutrosophic numbers simultaneously. The Dombi operations have the advantage of good flexibility with the operational parameter. In order to solve decision-making problems with flexible operational parameter under neutrosophic cubic environments, the paper extends the Dombi operations to neutrosophic cubic sets and proposes a neutrosophic cubic Dombi weighted arithmetic average (NCDWAA) operator and a neutrosophic cubic Dombi weighted geometric average (NCDWGA) operator. Then, we propose a multiple attribute decision-making (MADM) method based on the NCDWAA and NCDWGA operators. Finally, we provide two illustrative examples of MADM to demonstrate the application and effectiveness of the established method.

Comments on “A group decision-making model with interval multiplicative reciprocal matrices based on the geometric consistency index”

Abstract A very recent paper published by Liu et al. [Liu, F., Zhang, W. G., & Shang, Y. F. (2016). A group decision-making model with interval multiplicative reciprocal matrices based on the geometric consistency index. Computers & Industrial Engineering, 101, 184–193.] defined a geometric consistency index (GCI) to measure the inconsistency level of an interval multiplicative reciprocal matrix (IMRM) and to devise a GCI induced ordered weighted geometric averaging (GCI-IOWGA) operator, and proposed a method for determining the associated weight vector of the GCI-IOWGA operator. This note reveals that such a GCI and the GCI-IOWGA operator as well as the associated weight determination method are questionable. A new consistency index called a weak GCI and an associated weight determination approach are put forward to rectify defects in the existing results. A notion of acceptably weak consistency of IMRMs is introduced to develop a weak GCI induced IMRM ordered weighted geometric averaging operator for aggregating individual IMRMs with acceptably weak consistency. Some important properties are explored for the proposed weak GCI and the weak GCI induced aggregation operator.

A Novel Multi-Attribute Decision Making Approach: Addressing the Complexity of Time Dependent and Interdependent Data

An innovative multi-attribute decision making (MADM) approach is put forward in this paper, which is demonstrated to be more precise than other methods currently available, especially, when the evaluations given by the decision maker(s) involve the complexity of time dependent and interdependent data. As for dynamic MADM interval-valued intuitionistic fuzzy environments with the time and attribute weights completely unknown, the new approach achieves the precision improvement by overcoming the drawbacks, the traditional geometric average operator existing: these fail to consider the relationships of the integrated data as well as the influences of attribute weights and time preference factor. The new method is systematically defined in a seven step approach; steps within the approach consist of the definition of multi-period power-weighted geometric average operators and relative optimization models to compute attribute weights and time weights. A case analysis on the option of a superior company about venture capital funding is provided to demonstrate the method proposed; validation studies present a quantitative cross-comparison study and a sensitivity analysis of the new approach.

New Operators for Aggregating Intuitionistic Fuzzy Information With Their Application in Decision Making

Aggregation of intuitionistic fuzzy information is a hot topic in Atanassov’s intuitionistic fuzzy set theory, which has attracted much interest from researchers in recent years. In this paper, a series of new aggregation operators and weighted averaging operators are proposed for aggregating intuitionistic fuzzy information. First, some basic laws for operations on intuitionistic fuzzy values are presented together with their properties. Then, we propose intuitionistic fuzzy weighted arithmetic averaging operator and intuitionistic fuzzy weighted geometric averaging operator to aggregate intuitionistic fuzzy information. Inspired by the idea of ordered weighted averaging and hybrid weighted averaging, we further develop intuitionistic fuzzy ordered weighted arithmetic averaging operator, intuitionistic fuzzy ordered weighted geometric averaging operator, intuitionistic fuzzy hybrid weighted arithmetic averaging (IFHWAA) operator, and intuitionistic fuzzy hybrid weighted geometric averaging (IFHWGA) operator. It is proved that all proposed weighted averaging operators have the properties of idempotency, boundary, monotonicity, and commutativity. Finally, we propose new methods based on IFHWAA and IFHWGA operators, respectively, to solve multi-attribute group decision making under intuitionistic fuzzy environment. Some examples are applied to illustrate the performance of the proposed methods. The experimental results show the effectiveness and advantages of the developed method by comparing with the other methods.

Study on the ranking problems in multiple attribute decision making based on interval-valued intuitionistic fuzzy numbers

This paper puts forward a new ranking method for multiple attribute decision-making problems based on interval-valued intuitionistic fuzzy set (IIFS) theory. First, the composed ordered weighted arithmetic averaging operator and composed ordered weighted geometric averaging operator are extended to the IIFSs in which they are, respectively, named interval-valued intuitionistic fuzzy composed ordered weighted arithmetic averaging (IIFCOWA) operator and interval-valued intuitionistic composed ordered weighted geometric averaging (IIFCOWG) operator. Afterwards, to compare interval-valued intuitionistic fuzzy numbers, we define the concepts of the maximum, the minimum, and ranking function. Some properties associated with the concepts are investigated. Using the IIFCOWA or IIFCOWG operator, we establish the detailed steps of ranking alternatives (or attributes) in multiple attribute decision making. Finally, an illustrative example is provided to show that the proposed ranking method is feasible in multiple attribute decision making.

Linguistic Neutrosophic Cubic Numbers and Their Multiple Attribute Decision-Making Method

To describe both certain linguistic neutrosophic information and uncertain linguistic neutrosophic information simultaneously in the real world, this paper originally proposes the concept of a linguistic neutrosophic cubic number (LNCN), including an internal LNCN and external LNCN. In LNCN, its uncertain linguistic neutrosophic number consists of the truth, indeterminacy, and falsity uncertain linguistic variables, and its linguistic neutrosophic number consists of the truth, indeterminacy, and falsity linguistic variables to express their hybrid information. Then, we present the operational laws of LNCNs and the score, accuracy, and certain functions of LNCN for comparing/ranking LNCNs. Next, we propose a LNCN weighted arithmetic averaging (LNCNWAA) operator and a LNCN weighted geometric averaging (LNCNWGA) operator to aggregate linguistic neutrosophic cubic information and discuss their properties. Further, a multiple attribute decision-making method based on the LNCNWAA or LNCNWGA operator is developed under a linguistic neutrosophic cubic environment. Finally, an illustrative example is provided to indicate the application of the developed method.

Multi-attribute decision-making based on the SPIFGIA operators

The work considers the prioritized relationships between different criteria with priority labels, and develops the scaled prioritized intuitionistic fuzzy geometric interaction averaging (SPIFGIA) operator, the scaled prioritized intuitionistic fuzzy weighted geometric interaction averaging (SPIFWGIA) operator, the scaled prioritized intuitionistic fuzzy hybrid geometric interaction averaging (SPIFHGIA) operator, and the scaled prioritized intuitionistic fuzzy hybrid weighted geometric interaction averaging (SPIFHWGIA) operator. We investigate the properties of the new SPIFGIA and SPIFWGIA operators, and the prioritized relationships between criteria in different situations for decision-making problems, and we also discuss the constructions of intuitionistic fuzzy sets with granularity. Then, we develop new approaches to intuitionistic fuzzy decision-making problems based on the presented SPIFGIA, SPIFWGIA, SPIFHGIA, and SPIFHWGIA operators. Finally, the paper shows the effectiveness and feasibility of the new methods by an example, and gives some comparisons.

A Generalized Triangular Intuitionistic Fuzzy Geometric Averaging Operator for Decision-Making in Engineering and Management

Triangular intuitionistic fuzzy number (TIFN) is a more generalized platform for expressing imprecise, incomplete, and inconsistent information when solving multi-criteria decision-making problems, as well as for expressing and reflecting the evaluation information in several dimensions. In this paper, the TIFN has been applied for solving multi-criteria decision-making (MCDM) problems, first, by defining some existing triangular intuitionistic fuzzy geometric aggregation operators, and then developing a new triangular intuitionistic fuzzy geometric aggregation operator, which is the generalized triangular intuitionistic fuzzy ordered weighted geometric averaging (GTIFOWGA) operator. Based on these operators, a new approach for solving multicriteria decision-making problems when the weight information is fixed is proposed. Finally, a numerical example is provided to show the applicability and rationality of the presented method, followed by a comparative analysis using similar existing computational approaches.

A consensus model for group decision making under trapezoidal fuzzy numbers environment

Natural linguistic terms can preferably express the opinions of decision makers in complicated decision environment. Group decision making with multiplicative trapezoidal fuzzy preference relations transforming from natural linguistic terms attracts the attention of researchers for its important research significant. The developed approach is based on consensus improving process by using a new similarity measure and a trapezoidal fuzzy power ordered weighted geometric averaging (TFPOWGA) operator. In order to introduce this approach, firstly, trapezoidal fuzzy power geometric averaging operator and TFPOWGA operator are presented to aggregate trapezoidal fuzzy numbers (TFNs). Secondly, a new similarity measure for TFNs is introduced by combining centroids and areas of TFNs. We further propose a new consensus improving algorithm that consists of consensus measure and a dynamic feedback mechanism containing a multi-objective optimization model and some indirect rules. And then a selection stage is described to rank the alternatives. At last, an example is implemented to demonstrate effectiveness of the approach.

Multiple Attribute Group Decision-Making Method Based on Linguistic Neutrosophic Numbers

Existing intuitionistic linguistic variables can describe the linguistic information of both the truth/membership and falsity/non-membership degrees, but it cannot represent the indeterminate and inconsistent linguistic information. To deal with the issue, this paper originally proposes the concept of a linguistic neutrosophic number (LNN), which is characterized independently by the truth, indeterminacy, and falsity linguistic variables. Then, we define the basic operational laws of LNNs and the score and accuracy functions of LNN for comparing LNNs. Next, we develop an LNN-weighted arithmetic averaging (LNNWAA) operator and an LNN-weighted geometric averaging (LNNWGA) operator to aggregate LNN information and investigate their properties. Further, a multiple attribute group decision-making method based on the proposed LNNWAA or LNNWGA operator is established under LNN environment. Finally, an illustrative example about selecting problems of investment alternatives is presented to demonstrate the application and effectiveness of the developed approach.

A fuzzy group decision making model with trapezoidal fuzzy preference relations based on compatibility measure and COWGA operator

This paper proposes a fuzzy group decision-making model based on a logarithm compatibility measure with multiplicative trapezoidal fuzzy preference relations (MTFPRs) based on a continuous ordered weighted geometric averaging (COWGA) operator. New concepts are presented to measure deviation between MTFPR and its expected fuzzy preference relation. Then, an iterative algorithm is developed to help individual MTFPR reach acceptable compatibility. To determine the weights of decision makers, an optimal model is constructed using group logarithm compatibility index COWGA operator. Finally, we illustrate an example to show how it works and compare it with the existing methods. The main advantages of the proposed approach are the following: (1) The COWGA operator makes decision making more flexible; (2) an iterative and convergent algorithm is proposed to improve the compatibility of MTFPR; (3) decision makers’ weights in group decision making are determined by an optimal model based on a logarithm compatibility measure.

Some Single-Valued Neutrosophic Dombi Weighted Aggregation Operators for Multiple Attribute Decision-Making

The Dombi operations of T-norm and T-conorm introduced by Dombi can have the advantage of good flexibility with the operational parameter. In existing studies, however, the Dombi operations have so far not yet been used for neutrosophic sets. To propose new aggregation operators for neutrosophic sets by the extension of the Dombi operations, this paper firstly presents the Dombi operations of single-valued neutrosophic numbers (SVNNs) based on the operations of the Dombi T-norm and T-conorm, and then proposes the single-valued neutrosophic Dombi weighted arithmetic average (SVNDWAA) operator and the single-valued neutrosophic Dombi weighted geometric average (SVNDWGA) operator to deal with the aggregation of SVNNs and investigates their properties. Because the SVNDWAA and SVNDWGA operators have the advantage of good flexibility with the operational parameter, we develop a multiple attribute decision-making (MADM) method based on the SVNWAA or SVNWGA operator under a SVNN environment. Finally, an illustrative example about the selection problem of investment alternatives is given to demonstrate the application and feasibility of the developed approach.