Proposed Entropy Measure Research Articles

An Entropy Measure for n-Cylindrical Fuzzy Neutrosophic Sets

n-Cylindrical fuzzy neutrosophic set (n-CyFNS) is a new variant of fuzzy neutrosophic sets. In this paper, our aim is to introduce an entropy measure for n-CyFNS. Here, we explained its properties along with examples. Two real life applications, one on better way of shopping and the second on teacher evaluation, based on this proposed entropy measure are also illustrated.

New information measures for linear Diophantine fuzzy sets and their applications with LDF-ARAS on data storage system selection problem

In addition to membership and non-membership degrees, linear Diophantine fuzzy set (LDFS) incorporates two reference parameters in fuzzy set definition to represent human judgment more comprehensively. It also provides flexibility in updating the decision setting by changing the meaning of the reference parameters when required by the decision-maker. Information measures in fuzzy sets have many application areas in the literature. Especially, distance and entropy measures are utilized in multiple attribute decision-making (MADM) field. Distance measures are needed by distance-based methods like TOPSIS and EDAS because their alternative ranking procedure is based on measuring the distances between alternatives and some artificial ideal or average solutions. Entropy measure is a very important tool being used in objective attribute weighting procedures of MADM applications. Thus, new distance and entropy measure propositions for LDFS are the first contribution of the study. Also, Additive Ratio Assessment (ARAS) method is extended into LDFS (LDF-ARAS) for the first time in the literature as a second contribution. In this extension, we selected ARAS as one of the most cited tools because it is found flexible and easily adaptable by researchers who do not have any background in MADM. The proposed entropy measure is integrated with the objective attribute weighting procedure of the novel LDF-ARAS approach. The third contribution is the newly developed decision model proposed for a real-life data storage system (DSS) selection problem. Three DSS options were evaluated with respect to thirteen attributes by five skilled and well-educated experts and LDF-ARAS was performed to rank alternatives. The entropy-based LDF-ARAS approach was validated by comparing the results of six different applications. In the first three, different entropy formulas are used and in the last three, these entropy measures are integrated with LDF-TOPSIS. All six comparisons gave similar alternative rankings.

TOPSIS Method Based on Entropy Measure for Solving Multiple-Attribute Group Decision-Making Problems with Spherical Fuzzy Soft Information

A spherical fuzzy soft set (SFSS) is a generalized soft set model, which is more sensible, practical, and exact. Being a very natural generalization, introducing uncertainty measures of SFSSs seems to be very important. In this paper, the concept of entropy, similarity, and distance measures are defined for the SFSSs and also, a characterization of spherical fuzzy soft entropy is proposed. Further, the relationship between entropy and similarity measures as well as entropy and distance measures are discussed in detail. As an application, an algorithm is proposed based on the improved technique for order preference by similarity to an ideal solution (TOPSIS) and the proposed entropy measure of SFSSs, to solve the multiple attribute group decision-making problems. Finally, an illustrative example is used to prove the effectiveness of the recommended algorithm.

Distance-Based Knowledge Measure and Entropy for Interval-Valued Intuitionistic Fuzzy Sets

The knowledge measure or uncertainty measure for constructing interval-valued intuitionistic fuzzy sets has attracted much attention. However, many uncertainty measures are measured by the entropy of interval-valued intuitionistic fuzzy sets, which cannot adequately reflect the knowledge of interval-valued intuitionistic fuzzy sets. In this paper, we not only extend the axiomatic definition of the knowledge measure of the interval-valued intuitionistic fuzzy set to a more general level but also establish a new knowledge measure function complying with the distance function combined with the technique for order preference by similarity to ideal solution (TOPSIS). Further, we investigate the properties of the proposed knowledge measure based on mathematical analysis and numerical examples. In addition, we create the entropy function by calculating the distance from the interval-valued fuzzy set to the most fuzzy point and prove that it satisfies the axiomatic definition. Finally, the proposed entropy is applied to the multi-attribute group decision-making problem with interval-valued intuitionistic fuzzy information. Experimental results demonstrate the effectiveness and practicability of the proposed entropy measure.

Time-sequential hesitant fuzzy entropy, cross-entropy and correlation coefficient and their application to decision making

In decision making problems, the hesitance of the decision maker to the same decision object is reflected not only in the length of multiple possible memberships under, but also in the time dimension. The newly proposed time-sequential hesitant fuzzy set (TSHFS) can show the hesitance in decision making information from the view of time dimension, and in which the concept of hesitance of hesitant fuzzy sets is improved from a constructive perspective. However, it is difficult to further explore the application prospects of TSHFS in decision making as the current measurement methods, such as entropy, cross-entropy, and correlation coefficient, could not be leveraged directly in the TSHFS environment. For that entropy, cross-entropy, and correlation coefficient are important metric components of fuzzy sets and in related scenarios, we propose corresponding methods for TSHFS in this work. Meanwhile, to make full use of the importance of attribute information in the decision matrix and make the decision results have better discrimination ability, hesitant fuzzy diffusion decision model is proposed. The proposed entropy measures are constructed on trigonometric functions, the new cross-entropy measure and correlation coefficients are derived from classical counterparts, which enrich ways to deal with problems under the TSHFS environment, especially referring to decision making. In hesitant fuzzy diffusion decision model, the prior ordered value is defined as the final decision result to measure relationships among alternatives. The effectiveness and advantages of proposed entropy measures, cross-entropy measure, correlation coefficients, and the proposed decision model are demonstrated through the application of decision making.

Biparametric Q Rung Orthopair Fuzzy Entropy Measure for Multi Criteria Decision Making Problem

In this study we propose a measure of the entropy of the norm (R, S) for q-row orthopair fuzzy sets (qROFS). The proposed entropy measure is validated both theoretically and practically to ensure validity. We also propose a simple methodology for the purpose of solving a multi-criteria decision-analysis problems using the introduced entropy measure. This method takes into account different circumstances of criteria weights, such as unknown weights, as well as other cases when the weights are not fully known. Finally, a demonstration with numerical examples for the proposed entropy has been provided to show how to apply the novel methodologies.

A new belief entropy and its application in software risk analysis

The measurement of uncertainty has been an important topic of research. In Dempster’s framework, Deng entropy serves as a reliable tool for such measurements. However, it fails to consider more comprehensive information, resulting in the loss of critical data. An improved belief entropy is proposed in this paper, which preserves all the merits of Deng entropy. When there is only a single element, it can be degraded to Shannon entropy. When dealing with multiple elements, the partitioning method employed for mass functions makes it more responsive and efficient than alternative measures of uncertainty. Some numerical examples are given to further illustrate the effectiveness and applicability of the proposed entropy measure. Additionally, a case study is conducted on software risk analysis, demonstrating the practical value and relevance of the proposed method in real-world scenarios.

An Advanced Entropy Measure of IFSs via Similarity Measure

The Entropy measure of an intuitionistic fuzzy set (IFS) plays a significant role in decision making sciences, for instance, medical diagnosis, pattern recognition, criminal investigation, etc. The inadequate nature of an entropy measure may lead to some invalid results. Therefore, it is significant to use an efficient entropy measure for studying various decision-making problems under IFS environment. This paper first proposes a novel similarity measure for IFS. Based on the proposed similarity measure, an advanced entropy measure is defined with a different axiomatic approach. This axiomatic approach allows us to measure an IFS's entropy with the help of a similarity measure. To show the efficiency of the proposed similarity measure, a comparative study is performed with the existing similarity measures. Some structural linguistic variables are taken as examples to show the validity and consistency of the proposed entropy measure along with the existing entropy measures. Finally, based on the proposed entropy measure, a multi-criteria decision-making problem is performed.

Some novel q‐rung orthopair fuzzy similarity measures and entropy measures with their applications

AbstractIn this paper, we have introduced some new similarity measures for ‐rung orthopair fuzzy sets and discussed their various features. These similarity measures have been constructed with the help of t‐conorms. The suggested similarity measures are more effective than the available ones in computing the similarity between distinct ‐rung orthopair fuzzy sets. With the help of these suggested similarity measures, we have also developed some novel entropy measures for ‐rung orthopair fuzzy sets. The newly proposed entropy measures are more effective than the existing ones in handling linguistic hedges. The applicability of the suggested similarity measures has been demonstrated in pattern analysis and the findings have been found to be better than due to existing ones. The traditional decision‐making method namely, the technique for order preference by similarity to the ideal solution gives us a compromise solution that is either closest to the positive ideal solution and farthest from the negative ideal solution but not both. So, to overcome this major drawback, we suggest a novel multi‐attribute decision‐making method in the ‐rung orthopair fuzzy environment and apply it to a decision‐making problem.

A New Insight into Entropy Based on the Fuzzy Operators, Applied to Useful Information Extraction from Noisy Time-Frequency Distributions

In this paper, we study the connections between generalized mean operators and entropies, where the mean value operators are related to the strictly monotone logical operators of fuzzy theory. Here, we propose a new entropy measure based on the family of generalized Dombi operators. Namely, this measure is obtained by using the Dombi operator as a generator function in the general solution of the bisymmetric functional equation. We show how the proposed entropy can be used in a fuzzy system where the performance is consistent in choosing the best alternative in the Multiple Attribute Decision-Making Problem. This newly defined entropy was also applied to the problem of extracting useful information from time-frequency representations of noisy, nonstationary, and multicomponent signals. The denoising results were compared to Shannon and Rényi entropies. The proposed entropy measure is shown to significantly outperform the competing ones in terms of denoising classification accuracy and the F1-score due to its sensitivity to small changes in the probability distribution.

A Novel Entropy Measure with its Application to the COPRAS Method in Complex Spherical Fuzzy Environment

A complex spherical fuzzy set (CSFS) is a generalization of the spherical fuzzy set (SFS) to express the two-dimensional ambiguous information in which the range of positive, neutral and negative degrees occurs in the complex plane with the unit disk. Considering the vital importance of the concept of CSFSs which is gaining massive attention in the research area of two-dimensional uncertain information, we aim to establish a novel methodology for multi-criteria group decision-making (MCGDM). This methodology allows us to calculate both the weights of the decision-makers (DMs) and the weights of the criteria objectively. For this goal, we first introduce a new entropy measure function that measures the fuzziness degree associated with a CSFS to compute the unknown criteria weights in this methodology. Then, we present an innovative Complex Proportional Assessment (COPRAS) method based on the proposed entropy measure in the complex spherical fuzzy environment. Besides, we solve a strategic supplier selection problem which is very important to maximize the efficiency of the trading companies. Finally, we present some comparative analyses with some existing methods in different set theories, including the entropy measures, to show the feasibility and usefulness of the proposed method in the decision-making process.

Group decision making based on entropy measure of Pythagorean fuzzy sets and Pythagorean fuzzy weighted arithmetic mean aggregation operator of Pythagorean fuzzy numbers

In this paper, we propose a new entropy measure of Pythagorean fuzzy sets (PFSs). The proposed entropy measure of PFSs can conquer the shortcomings of the existing entropy measure of PFSs. We also propose the Pythagorean fuzzy weighted arithmetic mean (PFWAM) aggregation operator (AO) of Pythagorean fuzzy numbers (PFNs). The proposed PFWAM AO of PFNs can conquer the shortcomings of the existing sine trignometry Pythagorean fuzzy weighted averaging (ST-PFWA) AO and the existing sine trignometry Pythagorean fuzzy weighted geometric (ST-PFWG) AO of PFNs. Based on the proposed entropy measure of PFSs and the proposed PFWAM AO of PFNs, we propose a new group decision making (GDM) approach in the environment of PFNs. The proposed GDM approach can conquer the shortcomings of existing GDM approaches, where they cannot distinguish the ranking orders (ROs) of alternatives in some conditions. It offers us a very useful approach to deal with GDM problems in the environment of PFNs.

Reliability Analysis of Poll Data with Novel Entropy Information Measure in Multicriteria Decision-Making Based upon Picture Fuzzy Environment

The polling system has a considerable role in the democratic nation. The uncertainty of the people’s participation in polling generally affects the electoral-based system. Therefore, PFS (picture fuzzy set) is the furthermost efficient and useful extension of IFS (intuitionistic fuzzy set) in a fuzzy system capable of precisely handling human perception in the decision-making system. The PFS structure involves the different degrees, i.e., membership, nonmembership, neutral, and hesitancy which are comprehensively applied to such types of complex practical problems in the real-life scenario. This advantage of PFS motivates the author to propose PFSs centred novel entropy measure via this communication, which is comparatively more generalized, reliable, and simplified in place of the existing uncertain measures. The practicability of the proposed present research work is to deal with real-world problems pertaining to the relative importance of the attributes. Therefore, certainly, the proposed novel entropy developed a different approach to handle the uncertainty more precisely as a part of the existing approaches. The validation proof of the proposed entropy measure is proved in an organized manner and practically employed in the perspective of the polling data outcomes about the people’s opinions with the VIKOR-TOPSIS approach.

Pythagorean fuzzy entropy measure-based complex proportional assessment technique for solving multi-criteria healthcare waste treatment problem.

With the increasing risk to human health and environmental issues, the selection of appropriate management and treatment of healthcare waste has become a major problem, especially in developing countries. There are various alternatives to dispose of health care waste. The important is to assess the best alternative among them. The assessment of each alternative should be done based on public health, psychological, economic, environmental, technological, and operational aspect. The selection of the best health care waste treatment (HCWT) alternative is a complicated, multi-criteria decision-making (MCDM) problem involving numerous disparate qualitative and quantitative features. Hence, in this research article, the MCDM method is presented for estimating and choosing the best alternative of HCWT by COPRAS technique in a Pythagorean fuzzy set (PFS). Here, in this paper, first of all, a new entropy measure on PFSs is proposed and its validity is studied. Thereafter, the MCDM technique Complex Proportional Assessment (COPRAS) is discussed in which the criteria weights are assessed by the proposed entropy measure and score function to enhance an efficacy and efficiency of the proposed technique. Furthermore, the above-defined technique is employed to resolve the real-life problem to obtain the best treatment alternative to disposal of the health care waste. Finally, sensitivity analysis is presented to rationale the proposed viewpoint for prioritizing HCWT alternatives.

Estimation of Entropy for Log‐Logistic Distribution under Progressive Type II Censoring

Entropy is a useful indicator of information content that has been used in a number of applications. The Log‐Logistic (LL) distribution is a probability distribution that is often employed in survival analysis. This paper addresses the problem of estimating multiple entropy metrics for an LL distribution using progressive type II censoring. We derive formulas for six different types of entropy measurements. To obtain the estimators of the proposed entropy measures, the maximum likelihood approach is applied. Approximate confidence intervals are calculated for the entropy metrics under discussion. A numerical evaluation is performed using various censoring methods and sample sizes to characterize the behavior of estimator’s measures using relative biases, related mean squared errors, average interval lengths, and coverage probabilities. Numerical analysis revealed that the accuracy measures improve with sample size, and the suggested entropy estimates approach the genuine values as censoring levels decrease. Finally, an actual dataset was evaluated for demonstration purposes.

Fermatean fuzzy Heronian mean operators and MEREC‐based additive ratio assessment method: An application to food waste treatment technology selection

Uncertainty is often occurred in real-life decision-making problems due to the lack of complete information, imprecise data, and the vagueness of decision making experts in qualitative judgment, thus, the crisp values of criteria may be insufficient to handle such types of complex real situations. As the extension of fuzzy set, intuitionistic fuzzy set and Pythagorean fuzzy set, the Fermatean Fuzzy Set (FFS) has been demonstrated as a powerful tool to handle the uncertainty arisen in practical decision-making problems. Thus, this study aims to introduce an integrated Fermatean fuzzy information-based decision-making method by combining method based on the removal effects of criteria (MEREC) and additive ratio assessment (ARAS) methods with the application in a food waste treatment technology selection problem. By using Fermatean fuzzy numbers, the suggested approach successfully handle the qualitative data and uncertain information that often occur in practical situations. This study consists of four phases. First, entropy measure is developed for FFS and further utilized for determining the experts’ weights. Second, some Fermatean fuzzy Heronian mean operators and their properties are introduced to aggregate the Fermatean fuzzy information. These operators can provide us a valuable means to handle practical multicriteria decision-making problems on FFSs context. Third, an extended MEREC technique is originated to assess objective criteria weights within FFS context. Fourth, an integrated ARAS method is introduced with the combination of proposed entropy measure, generalized weighted Fermatean fuzzy Heronian mean operator and MEREC technique to evaluate and rank the alternatives. To confirm the reasonableness and practicality of the proposed methodology, an empirical case study of food waste treatment technology selection is discussed on FFSs settings. Further, a comparison with extant models and a sensitivity investigation are performed to confirm the validity and robustness of the obtained outcomes.

A New Entropy Measurement for the Analysis of Uncertain Data in MCDA Problems Using Intuitionistic Fuzzy Sets and COPRAS Method

In this paper, we propose a new intuitionistic entropy measurement for multi-criteria decision-making (MCDM) problems. The entropy of an intuitionistic fuzzy set (IFS) measures uncertainty related to the data modelling as IFS. The entropy of fuzzy sets is widely used in decision support methods, where dealing with uncertain data grows in importance. The Complex Proportional Assessment (COPRAS) method identifies the preferences and ranking of decisional variants. It also allows for a more comprehensive analysis of complex decision-making problems, where many opposite criteria are observed. This approach allows us to minimize cost and maximize profit in the finally chosen decision (alternative). This paper presents a new entropy measurement for fuzzy intuitionistic sets and an application example using the IFS COPRAS method. The new entropy method was used in the decision-making process to calculate the objective weights. In addition, other entropy methods determining objective weights were also compared with the proposed approach. The presented results allow us to conclude that the new entropy measure can be applied to decision problems in uncertain data environments since the proposed entropy measure is stable and unambiguous.

New Pythagorean Entropy Measure with Application in Multi-Criteria Decision Analysis.

The purpose of this paper is to propose a new Pythagorean fuzzy entropy for Pythagorean fuzzy sets, which is a continuation of the Pythagorean fuzzy entropy of intuitionistic sets. The Pythagorean fuzzy set continues the intuitionistic fuzzy set with the additional advantage that it is well equipped to overcome its imperfections. Its entropy determines the quantity of information in the Pythagorean fuzzy set. Thus, the proposed entropy provides a new flexible tool that is particularly useful in complex multi-criteria problems where uncertain data and inaccurate information are considered. The performance of the introduced method is illustrated in a real-life case study, including a multi-criteria company selection problem. In this example, we provide a numerical illustration to distinguish the entropy measure proposed from some existing entropies used for Pythagorean fuzzy sets and intuitionistic fuzzy sets. Statistical illustrations show that the proposed entropy measures are reliable for demonstrating the degree of fuzziness of both Pythagorean fuzzy set (PFS) and intuitionistic fuzzy sets (IFS). In addition, a multi-criteria decision-making method complex proportional assessment (COPRAS) was also proposed with weights calculated based on the proposed new entropy measure. Finally, to validate the reliability of the results obtained using the proposed entropy, a comparative analysis was performed with a set of carefully selected reference methods containing other generally used entropy measurement methods. The illustrated numerical example proves that the calculation results of the proposed new method are similar to those of several other up-to-date methods.

Entropy Measures for Interval Neutrosophic Vague Sets and Their Application in Decision Making

Entropy measure is an important tool in measuring uncertain information and plays a vital role in solving Multi Criteria Decision Making (MCDM). At present, various entropy measures in literature are developed to measure the degree of fuzziness. However, they could not be used to deal with interval neutrosophic vague set (INVS) environment. In this study, two kinds of entropymeasures are proposed as the extension of the entropy measure of single valued neutrosophic set (SVNS). First, we construct the axiomatic definition of INVS and propose a new formula for the entropy measure of INVS. Based on this measure, we develop two multi criteria decision making methods. Subsequently, an illustrative example of investment problems under INVS is given todemonstrate the proposed entropy measures. Finally, a comparative analysis is presented to illustrate the rationality and effectiveness of the proposed entropy measures.

NOVEL ENTROPY MEASURE OF A FUZZY SET AND ITS APPLICATION TO MULTICRITERIA DECISION MAKING WITH FUZZY TOPSIS

Fuzzy entropy is being used to measure the uncertainty with high precision and accuracy than classical crisp set theory. It plays a vital role in handling complex daily life problems involving uncertainty. In this manuscript, we first review several existing entropy measures and then propose novel entropy to measure the uncertainty of a fuzzy set. We also construct an axiomatic definition based on the proposed entropy measure. Numerical comparison analysis is carried out with existing entropies to show the reliability and practical applicability of our proposed entropy measure. Numerical results show that our suggested entropy is reasonable and appropriate in dealing with vague and uncertain information. Finally, we utilize our proposed entropy measure to construct fuzzy TOPSIS (Technique for Ordering Preference by Similarity to Ideal Solution) method to manage Multicriteria decision-making problems related to daily life settings. The final results demonstrate the practical effectiveness and applicability of our proposed entropy measure