Interval-valued Intuitionistic Fuzzy Values Research Articles

Multiattribute decision making based on nonlinear programming methodology, the Euclidean distance between IVIFVs, and new score function of IVIFVs

In this paper, we propose a new multiattribute decision making (MADM) method based on the proposed nonlinear programming methodology, the Euclidean distance between interval-valued intuitionistic fuzzy values (IVIFVs), and the proposed score function of IVIFVs. Firstly, a new score function of IVIFVs is proposed to overcome the shortcomings of the existing score functions of IVIFVs. Then, we construct the score matrix based on the proposed score function of IVIFVs and the decision matrix given by the decision maker. Then, a nonlinear programming model is proposed based on the Euclidean distance between IVIFVs and the interval-valued intuitionistic fuzzy weights of the attributes provided by the decision maker. Then, we solve the proposed nonlinear programming model to obtain the optimal weights of the attributes. Then, the weighted score of each alternative is calculated based on the obtained score matrix and the obtained optimal weights of the attributes. Finally, the alternatives are ranked according to the obtained weighted scores of the alternatives. The MADM method proposed in this paper can conquer the shortcomings of the existing MADM methods in the environments of IVIFVs.

Multiattribute decision making based on nonlinear programming methodology, novel score function of interval-valued intuitionistic fuzzy values, and the standard deviations of the score values in the score matrix

In this paper, we propose a new multiattribute decision making (MADM) method using the proposed nonlinear programming (NLP) model, the proposed score function (SCF) of interval-valued intuitionistic fuzzy values (IVIFVs), and the standard deviation of the score values appeared in each column of the constructed score matrix (SX). The proposed SCF of IVIFVs can overcome the shortcomings of the existing SCFs of IVIFVs. Firstly, we construct the SX using the decision matrix given by the decision maker and the proposed SCF of IVIFVs. Then, we construct a NLP model using the obtained SX, the standard deviation of the score values appeared in each column of the SX, the interval-valued intuitionistic fuzzy weights of the attributes, the largest ranges of IVIFVs, and the concept of deviation variables. Then, we solve the NLP model to get the optimal weights of the attributes, respectively. Then, based on the constructed SX and the obtained optimal weights of the attributes, we calculate the weighted score of each alternative. Finally, we rank the alternatives according to the obtained weighted scores. The proposed MADM method can overcome the drawbacks of the existing MADM methods.

Multiple attribute decision making based on novel nonlinear programming model, the distance between score values, and novel score function of interval-valued intuitionistic fuzzy values

In this paper, we propose a novel multiple attribute decision making (MADM) method based on the proposed nonlinear programming (NLP) model, the distance between the score values appeared in the constructed score matrix (SCMX), and the proposed score function (SF) of interval-valued intuitionistic fuzzy values (IVIFVs), where the NLP model is used to get the optimal weights (OWs) of the attributes. Firstly, we propose a novel SF to conquer the shortcomings of the existing SFs of IVIFVs. Then, we use the proposed SF to construct the SCMX from the decision matrix (DM) given by the decision maker (DK). Then, we propose a NLP model to obtain the OWs of the attributes based on the distance between the score values appeared in the constructed SCMX, the interval-valued intuitionistic fuzzy weights (IVIFWs) of the attributes provided by the DK, the concept of deviation variables, and the largest range of the IVIFW of each attribute. Then, we calculate the weighted score (WS) of each alternative based on the obtained SCMX and the obtained OWs of the attributes. Finally, we rank the alternatives according to the WSs of the alternatives. The proposed MADM method can conquer the shortcomings of the existing MADM methods.

Multiattribute decision making method based on nonlinear programming model, cosine similarity measure, and novel score function of interval-valued intuitionistic fuzzy values

In this paper, we propose a new multiattribute decision making (MADM) method based on the proposed score function (SF) of interval-valued intuitionistic fuzzy values (IVIFVs), the cosine similarity measure of IVIFVs, and the proposed nonlinear programming (NLP) model. The proposed SF of IVIFVs overcomes some shortcomings of the existing SFs of IVIFVs. The proposed MADM method overcomes some shortcomings of the existing MADM methods based on IVIFVs. Firstly, we propose a novel SF of IVIFVs to overcome the shortcomings of some existing SFs of IVIFVs. Then, we construct a score matrix based on the proposed score function of IVIFVs and the decision matrix given by the decision maker. Then, we build a NLP model based on the cosine similarity measure of IVIFVs and the interval-valued intuitionistic fuzzy weights of the attributes. Then, we solve the NLP model to obtain the optimal weights (OWs) of the attributes, respectively. Based on the obtained OWs of the attributes and the constructed score matrix, we compute the weighted scores (WSs) of the alternatives, respectively. Finally, we rank the alternatives based on the obtained WSs. The proposed MADM method provides us a very useful approach for MADM in the interval-valued intuitionistic fuzzy context.

Multiattribute decision making based on nonlinear programming methodology, score function of interval-valued intuitionistic fuzzy values, and the dispersion degree of score values

In this paper, we propose a new multiattribute decision making (MADM) method based on the proposed nonlinear programming (NLP) model, the proposed score function (SF) of interval-valued intuitionistic fuzzy values (IVIFVs), and the dispersion degree of the score values appeared in each column of the score matrix (SMX). Firstly, we propose a new SF of IVIFVs to construct the SMX. The proposed SF of IVIFVs can overcome the drawbacks of the existing SFs of IVIFVs. Then, we calculate the dispersion degree of the score values appeared at each column of the SMX. Then, we construct a NLP model to get the optimal weight (OW) of each attribute based on the obtained SMX, the dispersion degree of the score values appeared at each column of the SMX, and the interval-valued intuitionistic fuzzy weights of the attributes given by the decision maker. Then, we calculate the weighted score (WS) of each alternative based on the obtained SMX and the obtained OWs of the attributes. Finally, we get the preference order (PFO) of the alternatives based on the obtained WSs of the alternatives. The proposed MADM method can overcome the drawbacks of some existing MADM methods in an interval-valued intuitionistic fuzzy setting.

Multiattribute decision making based on nonlinear programming model, the Gini coefficient, and novel score function of interval-valued intuitionistic fuzzy values

In this paper, we propose a new multiattribute decision making (MADM) method based on the proposed nonlinear programming (NLP) model, the Gini coefficient, and the proposed score function (SF) of interval-valued intuitionistic fuzzy values (IVIFVs). Firstly, we propose a novel SF of IVIFVs to overcome the shortcomings of the existing SFs of IVIFVs. Then, we construct a score matrix (SMX) based on the proposed SF of IVIFVs and the decision matrix given by the decision maker (DMK). Then, we construct a NLP model based on the constructed SMX, the Gini coefficient, and the interval-valued intuitionistic fuzzy (IVIF) weights of the attributes given by the DMK. After solving the constructed NLP model, we obtain the optimal weight (OW) of each attribute. Then, we compute the weighted score (WS) of each alternative based on the constructed SMX and the obtained OWs of the attributes. Finally, we rank the alternatives based on the obtained WSs of the alternatives. The proposed MADM method can overcome the shortcomings of the existing MADM methods in IVIF environments.

Multiple attribute decision making based on novel score function of interval-valued intuitionistic fuzzy values, score matrix, and nonlinear programming model

This paper proposes a new multiple attribute decision making (MADM) method based on the proposed score function (SF) of interval-valued intuitionistic fuzzy values (IVIFVs), the score matrix (SMT), and the proposed nonlinear programming model. Firstly, we use the proposed SF of IVIFVs to construct the SMT, where the proposed SF of IVIFVs can overcome the drawbacks of the existing SFs of IVIFVs. Then, we calculate the average value of the values appeared at each column of the SMT. Then, we construct the nonlinear programming model using the obtained SMT, the obtained average value of the values appeared at each column of the SMT, the concept of deviation variables, and the interval-valued intuitionistic fuzzy weight of each attribute offered by the decision maker. Then, we solve the nonlinear programming model to obtain the optimal weight (OW) of each attribute. Then, we calculate the weighted score (WTS) of each alternative using the obtained SMT and the OWs of the attributes. Finally, we rank the alternatives according to the obtained WTSs. The bigger the WTS of an alternative, the better the preference order (PO) of the alternative. Our proposed MADM method can overcome the shortcomings of the existing MADM methods.

Multiattribute group decision making based on novel score function considering both the angle and fuzzy information under interval-valued intuitionistic fuzzy environment

The interval-valued intuitionistic fuzzy values (IVIFVs) have been widely employed for multiattribute group decision making (MAGDM). However, the existing decision making method using IVIFVs has some drawbacks. The recognition degree and information utilization suffer from a gap in the evaluation of alternatives. Therefore, the need for a reliable and effective decision method is obvious. In this paper, a novel score function considering both the angle and fuzzy information of IVIFVs is proposed, which ensures the effective fusion of the fuzzy information measure and angle measure and overcomes the shortcomings of the existing methods. Then, the weights of decision makers (DMs) are set as IVIFVs and subjective values are determined by the optimization model to ensure the objectivity and authenticity of the weights of DMs. Furthermore, a novel MAGDM method based on the proposed score function is proposed. Finally, the proposed method is implemented on the regional innovation resource allocation efficiency evaluation. The results show that our method possesses a higher recognition degree than the other methods, and it ensures relatively high ranking accuracy. The ideology and results of this research will help the DMs to evaluate the alternatives effectively.

A multiattribute group decision-making method based on a new aggregation operator and the means and variances of interval-valued intuitionistic fuzzy values

The development of information measures associated with interval-valued intuitionistic fuzzy values (IVIFVs) has been an important research area over the past few decades. In the literature, the existing decision -making method using IVIFVs has some drawbacks, and the identification degree and information utilization suffer from a gap in the evaluation of alternatives. Therefore, the need for a reliable, useful, and comprehensive decision method is obvious. To obtain more accurate and reliable evaluation results, multiattribute group decision-making (MAGDM) problems, where the same attribute weights given by different decision-makers are different, are studied in this paper. First, the novel operational laws of IVIFVs and a new interval-valued intuitionistic fuzzy weighted arithmetic aggregation operator are defined to overcome the drawbacks of the IIFWA aggregation operator and avoid losing or distorting the original decision information in the process of aggregation. Second, the mean and variance of the possibility degrees of IVIFVs are defined based on the concept of a definite integral. Third, a novel MAGDM method based on the new aggregation operator and the mean and variance of the possibility degrees of IVIFVs is proposed to improve the identification of the evaluation results and ensure the effectiveness of the ranking order. Finally, the effectiveness and practicability of the proposed method are verified by an air combat training accuracy assessment example. This example can be used to assist decision-makers in evaluating air combat training hits in a timely and efficient manner, providing an objective, scientific basis for the realization and application of air combat training hit assessment and a new method and idea for MAGDM problems in an interval-valued intuitionistic fuzzy environment.

Group decision making based on advanced interval-valued intuitionist fuzzy weighted averaging aggregation operator and score function of interval-valued intuitionist fuzzy values

This paper proposes a new group decision making (GDM) approach in the environments of interval-valued intuitionistic fuzzy values (IVIFVs). Firstly, we propose a new score function of IVIFVs, where the proposed score function of IVIFVs can overcome the drawbacks of the existing score function of IVIFVs. The properties of the proposed score function of IVIFVs are also presented. Then, we propose the advanced interval-valued intuitionist fuzzy averaging (AIVIFA) aggregation operator of IVIFVs. We also provide the proofs of the properties of the proposed AIVIFA aggregation operator. Then, we propose the advanced interval-valued intuitionist fuzzy weighted averaging (AIVIFWA) aggregation operator of IVIFVs. Then, we propose a new GDM approach based on the proposed AIVIFWA aggregation operator of IVIFVs and the proposed score function of IVIFVs. Finally, we apply the proposed GDM approach to deal with a real-world application of cloud service selection. The proposed GDM approach can overcome the drawbacks of the existing GDM approach, which is unable to distinguish the ranking orders of alternatives in some situations. The proposed GDM approach gives us a very useful way to deal with GDM problems in the environments of IVIFVs.

A Comparative Analysis of the Ranking Functions for the IVIFVs and A New Score Function

The ranking of interval-valued intuitionistic fuzzy values (IVIFVs) has an important role in real-life decision-making problems. Even though there are many approaches related to the ranking methods of the IVIFVs, some of them have some shortcomings. In this study, the disadvantages of the existing ranking functions of the IVIFVs are discussed. It is revealed that both the most popular ranking functions and their recently improved versions may lead to unacceptable results. Furthermore, in this article, a new score function is offered to cope with the shortcomings of the ranking functions. The importance and performance of this new score function are proven with the help of many examples. Also, the decision-making algorithm adapted by integrating the new function is presented to illustrate the applicability of the new score function in the decision-making problems.

Power Muirhead mean operators of interval-valued intuitionistic fuzzy values in the framework of Dempster–Shafer theory for multiple criteria decision-making

Multiple criteria decision-making (MCDM) based on interval intuitionistic fuzzy value (IVIFV) is a process of aggregating decision criteria represented by multiple interval-valued intuitionistic fuzzy numbers to select the optimal alternative. Among them, an aggregation operator is an indispensable tool, and the properties of an aggregation operator directly affect the decision results. Existing aggregation operators based on IVIFV have satisfactory results in eliminating the correlation between criteria and removing the influence of outliers on the results. However, there are some unreasonable results due to some undesired properties of IVIFVs. In this paper, IVIFV operation under the Dempster-Shafer theory (DST) framework is applied to combine the power average and Muirhead mean operators and interval intuitionistic fuzzy power Muirhead mean operators under DST framework are presented. Then a method based on the presented operators for MCDM problems is proposed. Finally, a set of numerical experiments are conducted to demonstrate the proposed method. The experimental results suggest that the proposed method not only retains the robustness of the power average operator and the capability of the Muirhead mean operator, but also eliminates a shortcoming that existing interval intuitionistic fuzzy operators cannot handle the case where the weights are in IVIFVs.

Multiattribute decision making based on nonlinear programming methodology and novel score function of interval-valued intuitionistic fuzzy values

In this paper, we propose a novel multiattribute decision making (MADM) method using the nonlinear programming (NLP) methodology and the proposed score function of interval-valued intuitionistic fuzzy values (IVIFVs). Firstly, we propose a new score function of IVIFVs to conquer the drawbacks of the existing score functions of IVIFVs. Then, we construct the converted matrix based on the proposed score function of IVIFVs by calculating the score value of each IVIFV in the decision matrix (DM) offered by the decision maker (DMK). Then, we construct the NLP model via the obtained converted matrix and the interval-valued intuitionistic fuzzy (IVIF) weight of each attribute given by the DMK. Then, we solve the NLP model to obtain the optimal weight for each attribute. Then, based on the obtained converted matrix and the obtained optimal weight of each attribute, we calculate the weighted score of each alternative. Finally, the alternatives are ranked on the basis of the obtained weighted scores of the alternatives. The larger the weighted score of an alternative, the better the preference order of the alternative. The proposed MADM method can overcome the drawbacks of the existing MADM methods. It offers us a very useful approach for MADM in IVIF settings.

Multiattribute decision making based on novel score function and the power operator of interval-valued intuitionistic fuzzy values

In this paper, we propose a novel multiattribute decision making (MADM) method on the basis of the proposed novel score function (SF) of interval-valued intuitionistic fuzzy values (IVIFVs) and the power operator of IVIFVs. The proposed SF of IVIFVs can conquer the shortcomings of the existing SFs of IVIFVs. Based on the proposed SF of IVIFVs and the power operator of IVIFVs, we propose a new MADM method in the interval-valued intuitionistic fuzzy context. The proposed MADM method can conquer the shortcoming of the existing MADM methods. It provides us a very useful approach for MADM in interval-valued intuitionistic fuzzy environments.

An extension of rule base evidential reasoning in the interval-valued intuitionistic fuzzy setting applied to the type 2 diabetes diagnostic

This paper presents an interval-valued intuitionistic fuzzy extension of rule base evidential reasoning which generally is a synthesis of fuzzy logic, the Dempster–Shafer theory of evidence (DST) and Atanassov’s intuitionistic fuzzy sets (A-IFS) theory redefined in the framework of DST. A lot of attention is paid to situations when in the solution of decision making problem the competing fuzzy classes with intersecting membership functions in antecedent parts of fuzzy rules, e.g., such as Low and Moderate play an important role in the problem formulation. As a result, a new mathematical object “belief interval bounded belief interval” (BIBBI) is obtained. The properties of BIBBI make it possible to use it as a more reliable representation of interval-valued intuitionistic fuzzy value than that formulated in terms of A-IFS. Using the introduced DST based definition of interval-valued A-IFS, the corresponding operations with BIBBIs and the DST representation of rule base evidential reasoning in the intuitionistic fuzzy setting, a new approach to the interval-valued intuitionistic fuzzy extension of rule base evidential reasoning is developed. To prove the correctness and applicability of this approach to the solution of decision making problems, a case study of its use for the type 2 diabetes diagnostics is analyzed.

Multiattribute decision making based on new score function of interval-valued intuitionistic fuzzy values and normalized score matrices

In this paper, we propose a novel multiattribute decision making (MADM) method on the basis of the proposed score function of interval-valued intuitionistic fuzzy values (IVIFVs) and normalized score matrices. Firstly, it calculates the score value of each IVIFV in the decision matrix (DM) based on the proposed score function of IVIFVs to construct the score matrix. Then, it constructs the normalized score matrix based on the obtained score matrix. Then, it computes the optimal weight of the interval-valued intuitionistic fuzzy (IVIF) weight of each attribute. Then, based on the obtained normalized score matrix and the obtained optimal weight of the IVIF weight of each attribute, it constructs the weighted normalized DM. Then, based on the obtained weighted normalized DM, it calculates the weighted score value of each alternative. Finally, it ranks the alternatives based on the weighted score values of the alternatives. The larger the weighted score value of an alternative, the better the preference order (PO) of the alternative. The proposed MADM method overcomes the drawbacks of the existing MADM methods. The proposed MADM method provides a very useful way to us for dealing with MADM problems in IVIF environments.

Multiattribute decision making using novel score function of interval-valued intuitionistic fuzzy values and the means and the variances of score matrices

In this paper, we propose a new multiattribute decision making (MADM) method based on the proposed score function of interval-valued intuitionistic fuzzy values (IVIFVs) and the means and the variances of score matrices, where the proposed score function can conquer the shortcomings of the existing score functions. Firstly, it computes the score value of each IVIFV in the decision matrix based on the proposed score function to construct the score matrix. Then, it calculates the mean and the variance of the obtained score matrix. Then, it builds the standard score matrix based on the obtained score matrix. Then, it converts the interval-valued intuitionistic fuzzy (IVIF) weight of each attribute into a crisp weight based on the proposed score function. Finally, based on the obtained standard score matrix and the obtained converted IVIF weights, it calculates the weighted score of each alternative to rank the alternatives. The larger the weighted score of an alternative, the better the preference order (PO) of the alternative. The proposed MADM method can conquer the shortcomings of the existing MADM methods. It offers us a very useful way for MADM in IVIF environments.

Multiple attribute decision making using Beta distribution of intervals, expected values of intervals, and new score function of interval-valued intuitionistic fuzzy values

In this paper, we propose a new multiple attribute decision making (MADM) method using the Beta distribution of intervals, the expected values of intervals, and the proposed new score function of interval-valued intuitionistic fuzzy values (IVIFVs). Firstly, it transformed each IVIFV in the decision matrix (DM) into three intervals to get the transformed matrix (TM). Then, it calculates the expected values of the obtained three intervals for each triplet in the TM, respectively, and then uses the proposed new score function to calculate the score value of each IVIFV in the DM to obtain the score matrix. Then, it transforms the interval-valued intuitionistic fuzzy (IVIF) weight of each attribute into three intervals and calculates the expected values of the obtained three intervals, respectively. Then, it uses the proposed new score function to calculate the score value of the IVIF weight of each attribute to obtain the normalized optimal weight of each attribute. Finally, based on the obtained score matrix and the obtained optimal weight of each attribute, it computes the weighted score of each alternative to get the preference order of alternatives. The proposed MADM method can overcome the drawbacks of the existing MADM methods in IVIF environments.

Multiattribute decision making based on converted decision matrices, probability density functions, and interval-valued intuitionistic fuzzy values

In this paper, we propose a new multiattribute decision making method based on converted decision matrices, probability density functions and interval-valued intuitionist fuzzy values. First, it obtains the converted decision matrix from the interval-valued intuitionistic fuzzy decision matrix given by the decision maker. Then, it computes the standard deviations and the mean values of the intervals appear at each row of the converted decision matrix, respectively, by using probability density functions. Then, by using the mean values and the standard deviations of the alternatives and the converted decision matrix, it gets the z-score decision matrix. Afterwards, the optimal weights of the attributes are calculated from the interval-valued intuitionist fuzzy weights of the attributes. Finally, it computes the value of overall performance of each alternative by using the z-score decision matrix and the optimal weights of the attributes for ranking the alternatives. The proposed method can conquer the shortcomings of the existing methods for interval-valued intuitionistic fuzzy multiattribute decision making.

Multiattribute decision making based on interval-valued intuitionistic fuzzy values, score function of connection numbers, and the set pair analysis theory

This paper proposes a new multiattribute decision making (MADM) method based on the proposed score function of connection numbers (CNs) and the set pair analysis (SPA) theory in the interval-valued intuitionist fuzzy (IVIF) context. Firstly, we develop a score function for ranking CNs. The various notable characteristics of the proposed score function of CNs are also presented. Then, we propose a new MADM method based on interval-valued intuitionist fuzzy values (IVIFVs), the proposed score function of CNs and the SPA theory, where we convert IVIFVs into CNs and the optimal weights of attributes are calculated from the IVIF weights of attributes. Finally, the proposed MADM method is applied for MADM in the IVIF context, where the preference orders (POs) of the alternatives obtained by the proposed MADM method are compared with the ones obtained by the existing MADM methods. The proposed MADM method can overcome the drawbacks of the existing MADM methods.