Uncertain Linguistic Numbers Research Articles

A VIKOR Method Based on the Relative Closeness on Interval Linguistic Neutrosophic Uncertain Linguistic Numbers

In actual multi-attribute group decision-making problems, due to the complexity and uncertainty of objective things and the ambiguity of human thinking, decision-makers find it hard to give accurate evaluation information by crisp numbers. Even the weights of attribute values and decision-makers are ambiguous. At this time, we are more inclined to adopt the intuitive form of linguistic variables such as "excellent", "good" or "bad" to describe attribute values and weights. So in this paper, based on a neutrosophic set (NS), we further propose interval linguistic neutrosophic uncertain linguistic number (ILNULN). ILNULN combines interval linguistic neutrosophic and uncertain linguistic numbers, and it has the advantages of both. At the same time, due to the weighted arithmetic Bonferroni mean operator considering the interrelationship between aggregation parameters, therefore we combine the ILNULN and weighted arithmetic Bonferroni mean operator to propose the interval linguistic neutrosophic uncertain linguistic weighted arithmetic Bonferroni mean (ILNULWABM) operator. Finally, under the environment of interval linguistic neutrosophic and uncertain linguistic numbers, this article uses the linguistic weights and ILNULWABM operator to make VIKOR decision based on the relative closeness, and gives a practical example.

Bonferroni Mean Operator of Interval Linguistic Neutrosophic Uncertain Linguistic Number and its Application in Multi-Attribute Group Decision-Making

This paper proposes the concept of interval linguistic neutrosophic uncertain linguistic number (IL-NULN). As a new and effective way of NS expression, ILNULN combines interval linguistic neutrosophic numbers and uncertain linguistic numbers, which can better handle uncertain and inconsistent information. In the ILNULN, the first part “uncertain linguistic number” reflects the attitude of the decision maker (DM) towards the evaluation object, and the second part “interval linguistic neutrosophic number” reflects the subjective linguistic judgment of the DM on the given uncertain linguistic number. In addition, considering the weighted arithmetic Bonferroni mean (WABM) operator integrates the correlation of aggregation parameters, this paper combines the ILNULN and WABM operator to propose the interval linguistic neutrosophic uncertain linguistic weighted arithmetic Bonferroni mean (ILNUL-WABM) operator. Finally, under the environment of interval linguistic neutrosophic uncertain linguistic number, this paper uses the ILNULWABM operator to make VIKOR decision based on the relative closeness, and gives a practical example to solve multi-attribute group decision making (MAGDM) problems.

Intuitionistic uncertain linguistic prioritized aggregation operators and their application to multi-criteria decision making

The intuitionistic uncertain linguistic information aggregation problems considering different priority levels of criteria are investigated. Firstly, we extended the prioritized averaging (PA) operator to intuitionistic uncertain linguistic environment, defined two new prioritized aggregation operators called the intuitionistic uncertain linguistic prioritized weighted average (IULPWA) operator and the intuitionistic uncertain linguistic prioritized weighted geometric (IULPWG) operator, and established various desirable properties of the proposed operators. Secondly, we developed a multi-criteria decision making (MCDM) approach based on the IULPWA operator (or the IULPWG operator) to deal with the MCDM problems in which the criterion values take the form of intuitionistic uncertain linguistic numbers (IULNs) and the criteria are in different priority levels. Finally, an example is given to illustrate the feasibility and effectiveness of the proposed method, and a comparison analysis is conducted to make clear the differences among the IULPWA operator, the IULPWG operator, the intuitionistic uncertain linguistic number weighted averaging (IULNWA) operator and the intuitionistic uncertain linguistic weighted geometric average (IULWGA) operator.

T-spherical uncertain linguistic MARCOS method based on generalized distance and Heronian mean for multi-attribute group decision-making with unknown weight information

The T-spherical uncertain linguistic (TSUL) sets (TSULSs) integrated by T-spherical fuzzy sets and uncertain linguistic variables are introduced in this article. This new concept is not only a generalized form but also can integrate decision-makers’ quantitative evaluation ideas and qualitative evaluation information. The TSULSs serve as a reliable and comprehensive tool for describing complex and uncertain decision information. This paper focuses on an extended MARCOS (Measurement of Alternatives and Ranking according to the Compromise Solution) method to handle the TSUL multi-attribute group decision-making problems where the weight information is completely unknown. First, we define, respectively, the operation rules and generalized distance measure of T-spherical uncertain linguistic numbers (TSULNs). Then, we develop two kinds of aggregation operators of TSULNs, one kind of operator with independent attributes is T-spherical uncertain linguistic weighted averaging and geometric (TSULWA and TSULWG) operators, and the other is T-spherical uncertain linguistic Heronian mean aggregation operators (TSULHM and TSULWHM) considering attributes interrelationship. Their related properties are discussed and a series of reduced forms are presented. Subsequently, a new TSUL-MARCOS-based multi-attribute group decision-making model combining the proposed aggregation operators and generalized distance is constructed. Finally, a real case of investment decision for a community group-buying platform is presented for illustration. We further test the rationality and superiorities of the proposed method through sensitivity analysis and comparative study.

Some Single-Valued Neutrosophic Uncertain Linguistic Maclaurin Symmetric Mean Operators and Their Application to Multiple-Attribute Decision Making

The Maclaurin symmetric mean (MSM) operator has a good aggregation effect. It can capture the relationships between multiple input parameters, and the neutrosophic uncertain linguistic numbers can well represent some indeterminate and incomplete information. In this paper, we combine the MSM operator with the singled-valued neutrosophic uncertain linguistic set and propose some MSM operators based on single-valued neutrosophic uncertain linguistic environment, such as single-valued neutrosophic uncertain linguistic Maclaurin symmetric mean(SVNULMSM) operator and single-valued neutrosophic uncertain linguistic generalized Maclaurin symmetric mean(SVNULGMSM) operator. First of all, according to the neutrosophic set and uncertain linguistic numbers, we propose the single-valued neutrosophic uncertain linguistic numbers and give some operating rules. Furthermore, considering the influence of attribute weight on the results, we introduce the weighted SVNULMSM operator and weighted SVNULGMSM operator. Then, we propose a method to deal with MSDM problems and give the specific steps to solve the problem. Finally, an investment example is used to verify the effectiveness of our method, and the superiority of the method is proved by comparing with other methods.

Hamacher heronian mean operators for multi-critria decision-making under multi-valued picture fuzzy uncertain lingsuitic environment

Picture fuzzy set (PFS) and linguistic term set (LTS) are two significant notions in multi-criteria decision-making (MCDM). In practice, decision-makers sometimes need utilize the multiple probable membership degrees for an uncertain linguistic term to express evaluation information. Motivated by these, to better convey the vagueness and uncertainty of cognitive information, multi-valued picture fuzzy uncertain linguistic set combining picture hesitant fuzzy set with uncertain linguistic term set is proposed. We firstly define the concepts of multi-valued picture fuzzy uncertain linguistic set and multi-valued picture fuzzy uncertain linguistic number. Hamacher operations are more general and flexible in information fusion, thus, Hamacher operations and comparison method are developed at the same time. Improved generalized Heronian Mean operator can simultaneously reflect correlations between values and prevent the redundant calculation. Then, two operators of improved generalized weighted Heronian mean and improved generalized geometric weighted Heronian mean in view of Hamacher operations are proposed. Meanwhile, some distinguished properties and instances of two operators are explored as well. Moreover, a novel MCDM approach applying the developed operators is constructed. Ultimately, an illustrative example on vendor selection is performed, and sensitivity analysis and comparison analysis are provided to verify the powerfulness of the proposed method.

Spherical uncertain linguistic Hamacher aggregation operators and their application on achieving consistent opinion fusion in group decision making

PurposeThe aim of this research is to establish a new type of aggregation operator based on Hamacher operational law of spherical uncertain linguistic numbers (SULNs).Design/methodology/approachFirst, the authors define spherical uncertain linguistic sets and develop some operational laws of SULNs. Furthermore, the authors extended these operational laws to the aggregation operator and developed spherical uncertain linguistic Hamacher averaging and geometric aggregation operators.FindingsThe authors were limited in achieving a consistent opinion on the fusion in group decision-making problem with the SULN information.Originality/valueIn order to give an application of the introduced operators, the authors first constrict a system of multi-attribute decision-making algorithm.

An Approach of Interval-Valued Picture Fuzzy Uncertain Linguistic Aggregation Operator and Their Application on Supplier Selection Decision-Making in Logistics Service Value Concretion

With respect to multiple criteria group decision-making (MCGDM) problems in which both the criteria weights and the expert weights take the form of crisp numbers and attribute values take the form of interval-valued picture fuzzy uncertain linguistic numbers, some new group decision-making analysis methods are developed. Firstly, some operational laws, expected value, and accuracy function of interval-valued picture fuzzy uncertain linguistic numbers are introduced. Then, an interval-valued picture fuzzy uncertain linguistic averaging and geometric aggregation operators are developed. Furthermore, some desirable properties of the developed operators, such as commutativity, idempotency, and monotonicity, have been studied. Based on these operators, an approach to multiple criteria group decision-making with interval-valued picture fuzzy uncertain linguistic information has been proposed. Finally, a practical example of 3PL supplier selection in logistics service value concretion is taken to test the defined method and to expose the effectiveness of the defined model.

CODAS methods for multiple attribute group decision making with interval-valued bipolar uncertain linguistic information and their application to risk assessment of Chinese enterprises’ overseas mergers and acquisitions

Bipolar fuzzy set theory has been successfully applied in some areas, but there are situations in real life which can’t be represented by bipolar fuzzy sets. However, all the existing approaches are unsuitable to describe the positive and negative membership degree an element to an uncertain linguistic label to have an interval value, which can reflect the decision maker’s confidence level when they are making an evaluation. In order to overcome this limit, we propose the definition of interval-valued bipolar uncertain linguistic sets (IVBULSs) to solve this problem based on the bipolar fuzzy sets and uncertain linguistic information processing models. In this paper, we extend the traditional information aggregating operators to interval-valued bipolar uncertain linguistic sets (IVBULSs) and propose some IVBUL aggregating operators. Then, we extend the CODAS method to solve multiple attribute group decision making (MAGDM) issues with interval-valued bipolar uncertain linguistic numbers (IVBULNs) based on these operators. An example for risk assessment of Chinese enterprises’ overseas mergers and acquisitions (M&As) is given to illustrate the proposed methodology.

Interval-Valued Intuitionistic Uncertain Linguistic Cloud Petri Net and Its Application to Risk Assessment for Subway Fire Accident

This article proposes a risk assessment method based on interval intuitionistic integrated cloud Petri net (IIICPN). The cloud model is widely used in data mining and knowledge discovery, especially in risk assessment problems with linguistic variables. However, the cloud models proposed in the literature do not express interval-valued intuitionistic linguistic satisfactorily, and the reasoning methods based on the cloud models cannot perform risk assessment well. The work in this article includes the definition of IIIC and IIICPN, the method of converting the interval-valued intuitionistic uncertain linguistic numbers into IIIC, and the reasoning method of IIICPN. As proofs, a subway fire accident model is adopted to confirm the feasibility of the proposed method, and comparison experiments between the IIICPN with general fuzzy Petri net and the trapezium cloud model are conducted to verify the superiority of the proposed model. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This work deals with the subway fire risk assessment problem. It proposes a cloud model based on interval-valued intuitionistic uncertain linguistic and builds a cloud-based Petri net model. The methods of fire risk assessment use the existing fault trees or aggregation operators to combine all the factors into consideration, but they do not take the interaction of factors. The goal of this work is to assess the risk of subway fire accident of subway, using fuzzy linguistic decision variables. The simulation results indicate that the proposed method is highly effective. The obtained results can help assessors better determine which factors may cause the disaster.

Multiple-Valued Neutrosophic Uncertain Linguistic Sets With Dombi Normalized Weighted Bonferroni Mean Operator and Their Applications in Multiple Attribute Decision Making Problem

In order to take into account quantitative and qualitative information in real complex decision making issue, a multiple-valued neutrosophic uncertain linguistic set (MVNULS) is initially proposed, which includes the uncertain linguistic part and the multiple-valued neutrosophic set (MVNS). Consequently, it has the advantages of them in expressing evaluation information. In the article, we initially define some concepts containing a multiple-valued neutrosophic uncertain linguistic set (MVNULS) and a multiple-valued neutrosophic uncertain linguistic element (MVNULE). Next, we present some basic operational rules regarding Dombi t-conorm and t-norm, which is more flexibility with a general parameter, and introduce a comparison method for multiple-valued neutrosophic uncertain linguistic numbers (MVNULNs). Since the aggregating operator of normalized weighted Bonferroni mean (NWBM) cannot only capture the interrelationship among multiple arguments, but also consider weight vector on each criterion and has the reducibility property, the NWBM operator has gained many attentions of scholars. Hence, in order to deal with the aggregation of MVNULEs, we put forward the multiple-valued neutrosophic uncertain linguistic Dombi normalized weighted Bonferroni mean (MVNULDNWBM) operator on account of Dombi t-conorm and t-norm, and discuss the desirable properties of the MVNULDNWBM operator. Furthermore, we investigate a multiple attribute decision making (MADM) approach in view of the aggregating operator to handle the MADM problem with multiple-valued neutrosophic uncertain linguistic information. Eventually, we conduct a concrete instance to validate the effectiveness and application of the developed approach.

Cubic uncertain linguistic powered Einstein aggregation operators and their application to multi-attribute group decision making

The cubic uncertain fuzzy linguistic variable can easily express the fuzzy information, and the power average (PA) operator is a useful tool which provides more versatility in the information aggregation procedure. In this paper, we will combine the PA operator and Einstein operations to cubic uncertain linguistic environment and propose some new PA operators. Firstly, the definition and some basic operations of cubic uncertain linguistic number, power aggregation (PA) operator and Einstein operations are introduced. Then, we propose cubic uncertain linguistic fuzzy powered Einstein averaging operator, cubic uncertain linguistic fuzzy powered Einstein weighted (CULFPEWA) operator, cubic uncertain linguistic fuzzy Einstein geometric operator and cubic uncertain linguistic fuzzy Einstein weighted geometric (CULFPEWG) operator and discuss some properties of these in detail. Furthermore, we develop the decision-making methods for multi-attribute group decision-making problems with cubic uncertain linguistic information and give the detail decision steps. At last, an illustrate example is given to show the process of decision making and the effectiveness of the proposed method.

Induced Choquet Integral Aggregation Operators with Single-Valued Neutrosophic Uncertain Linguistic Numbers and Their Application in Multiple Attribute Group Decision-Making

Induced Choquet Integral Aggregation Operators with Single-Valued Neutrosophic Uncertain Linguistic Numbers and Their Application in Multiple Attribute Group Decision-Making

Induced Choquet Integral Aggregation Operators with Single-Valued Neutrosophic Uncertain Linguistic Numbers and Their Application in Multiple Attribute Group Decision-Making

For real decision-making problems, aggregating the attributes which have interactive or correlated characteristics by traditional aggregation operators is unsuitable. Thus, applying Choquet integral operator to approximate and simulate human subjective decision-making process, in which independence among the input arguments is not necessarily assumed, would be suitable. Moreover, using single-valued neutrosophic uncertain linguistic sets (SVNULSs) can express the indeterminate, inconsistent, and incomplete information better than FSs and IFSs. In this paper, we studied the MAGDM problems with SVNULSs and proposed two single-valued neutrosophic uncertain linguistic Choquet integrate aggregation operators where the interactions phenomena among the attributes or the experts are considered. First, the definition, operational rules, and comparison method of single-valued neutrosophic uncertain linguistic numbers (SVNULNs) are introduced briefly. Second, induced single-valued neutrosophic uncertain linguistic Choquet ordered averaging (I-SVNULCA) operator and induced single-valued neutrosophic uncertain linguistic Choquet geometric (I-SVNULCG) operator are presented. Moreover, a few of its properties are discussed. Further, the procedure and algorithm of MAGDM based on the above single-valued neutrosophic uncertain linguistic Choquet integral operator are proposed. Finally, in the illustrative example, the practicality and effectiveness of the proposed method would be demonstrated.

Some neutrosophic uncertain linguistic number Heronian mean operators and their application to multi-attribute group decision making

Heronian mean (HM) is a useful aggregation operator which can deal with the interrelations of the aggregated arguments, and the neutrosophic uncertain linguistic set is a good tool to express the incomplete, indeterminate, and inconsistent information. In this paper, we combine the HM and the neutrosophic uncertain linguistic set and propose some HM operators based on neutrosophic uncertain linguistic numbers. Firstly, we introduce some definition and properties of uncertain linguistic numbers, the single-valued neutrosophic set, and some HM operators including the generalized weighted Heronian mean operator, the improved generalized weighted Heronian mean operator, and the improved generalized geometric weighted Heronian mean operator. Then, we propose the single-valued neutrosophic uncertain linguistic set by combining the uncertain linguistic numbers and the single-valued neutrosophic set. Further, the neutrosophic uncertain linguistic number improved generalized weighted Heronian mean operator and the neutrosophic uncertain linguistic number improved generalized geometric weighted Heronian mean operator are developed, and the properties of them are analyzed. Furthermore, we develop the decision making methods for multi-attribute group decision making problems with neutrosophic uncertain linguistic information and give the detailed decision steps. At last, an illustrate example is given to show the process and the effectiveness of the decision-making method.

The symmetrical interval intuitionistic uncertain linguistic operators and their application to decision making

Interval intuitionistic uncertain linguistic sets are an important generalization of fuzzy sets, which well cope with the experts’ qualitative preferences as well as reflect the interval membership and non-membership degrees of the uncertain linguistic term. This paper first points out the issues of the operational laws on interval intuitionistic uncertain linguistic numbers in the literature, and then defines some alternative ones. To consider the relationship between interval intuitionistic uncertain linguistic sets, the expectation and accuracy functions are defined. To study the application of interval intuitionistic uncertain linguistic sets, two symmetrical interval intuitionistic uncertain linguistic hybrid aggregation operators are defined. Meanwhile, models for the optimal weight vectors are established, by which the optimal weighting vector can be obtained. As a series of development, an approach to multi-attribute decision making under interval intuitionistic uncertain linguistic environment is developed, and the associated example is provided to demonstrate the effectivity and practicality of the procedure.

Some new intuitionistic linguistic aggregation operators based on Maclaurin symmetric mean and their applications to multiple attribute group decision making

With respect to multiple attribute group decision making (MAGDM) problems in which the attributes are dependent and the attribute values take the forms of intuitionistic linguistic numbers and intuitionistic uncertain linguistic numbers, this paper investigates two novel MAGDM methods based on Maclaurin symmetric mean (MSM) aggregation operators. First, the Maclaurin symmetric mean is extended to intuitionistic linguistic environment and two new aggregation operators are developed for aggregating the intuitionistic linguistic information, such as the intuitionistic linguistic Maclaurin symmetric mean (ILMSM) operator and the weighted intuitionistic linguistic Maclaurin symmetric mean (WILMSM) operator. Then, some desirable properties and special cases of these operators are discussed in detail. Furthermore, this paper also develops two new Maclaurin symmetric mean operators for aggregating the intuitionistic uncertain linguistic information, including the intuitionistic uncertain linguistic Maclaurin symmetric mean (IULMSM) operator and the weighted intuitionistic uncertain linguistic Maclaurin symmetric mean (WIULMSM) operator. Based on the WILMSM and WIULMSM operators, two approaches to MAGDM are proposed under intuitionistic linguistic environment and intuitionistic uncertain linguistic environment, respectively. Finally, two practical examples of investment alternative evaluation are given to illustrate the applications of the proposed methods.

Some Neutrosophic Uncertain Linguistic Number Heronian Mean Operators and Their Application to Multi-Attribute Group Decision Making

Heronian mean (HM) is a useful aggregation operator which is marked by catching the interrelations of the aggregated arguments and the neutrosophic uncertain linguistic set can be better to express the incomplete, indeterminate and inconsistent information. In this paper, we combine the Heronian mean and the neutrosophic uncertain linguistic set and proposed some Heronian mean operators based on neutrosophic uncertain linguistic numbers. Firstly, we introduce some definition and properties of uncertain linguistic numbers, the single valued neutrosophic set, and some heronian mean (HM) operators including the generalized weighted Heronian mean (GWHM) operator, the improved generalized weighted Heronian mean (IGWHM) operator, the improved generalized geometric weighted Heronian mean (IGGWHM) operator. Then, we propose the single valued neutrosophic uncertain linguistic set by combining the uncertain linguistic numbers and the single valued neutrosophic set. Further, the neutrosophic uncertain linguistic number improved generalized weighted Heronian mean (NULNIGWHM) operator and the neutrosophic uncertain linguistic number improved generalized geometric weighted Heronian mean (NLUNIGGWHM) operator are developed and the properties of them are analyzed. Furthermore, we develop the decision making methods for multi-attribute group decision making (MAGDM) problems with neutrosophic uncertain linguistic information and give the detail decision steps. At last, an illustrate example is given to show the process of decision making and the effectiveness of the proposed method.

INTUITIONISTIC UNCERTAIN LINGUISTIC POWERED EINSTEIN AGGREGATION OPERATORS AND THEIR APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING

The intuitionistic uncertain fuzzy linguistic variable can easily express the fuzzy information, and the power average (PA) operator is a useful tool which provides more versatility in the information aggregation procedure. At the same time, Einstein operations are a kind of various t-norms and <i>t</i>-conorms families which can be used to perform the corresponding intersections and unions of intuitionistic fuzzy sets (IFSs). In this paper, we will combine the PA operator and Einstein operations to intuitionistic uncertain linguistic environment, and propose some new PA operators. Firstly, the definition and some basic operations of intuitionistic uncertain linguistic number (IULN), power aggregation (PA) operator and Einstein operations are introduced. Then, we propose intuitionistic uncertain linguistic fuzzy powered Einstein averaging (IULFPEA) operator, intuitionistic uncertain linguistic fuzzy powered Einstein weighted (IULFPEWA) operator, intuitionistic uncertain linguistic fuzzy Einstein geometric (IULFPEG) operator and intuitionistic uncertain linguistic fuzzy Einstein weighted geometric (IULFPEWG) operator, and discuss some properties of them in detail. Furthermore, we develop the decision making methods for multi-attribute group decision making (MAGDM) problems with intuitionistic uncertain linguistic information and give the detail decision steps. At last, an illustrate example is given to show the process of decision making and the effectiveness of the proposed method.

A Group Decision Making Approach Based on Interval-Valued Intuitionistic Uncertain Linguistic Aggregation Operators

This paper investigates group decision making problems in which the criterion values take the form of interval-valued intuitionistic uncertain linguistic numbers (IIULNs). First, some additive operational laws of IIULNs are defined. Subsequently, some new arithmetic aggregation operators, such as the interval-valued intuitionistic uncertain linguistic weighted averaging (IIULWA) operator, interval-valued intuitionistic uncertain linguistic ordered weighted averaging (IIULOWA) operator and interval-valued intuitionistic uncertain linguistic hybrid aggregation (IIULHA) operator, are proposed which are based on the operational laws. Furthermore, an approach to group decision making with interval-valued intuitionistic uncertain linguistic information is developed, which is based on the IIULWA and IIULHA operators. Finally, an illustrative example is provided to demonstrate the feasibility and effectiveness of the proposed method.