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2010, Entropy
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Entropy weight method (EWM) is a commonly used weighting method that measures value dispersion in decision-making. e greater the degree of dispersion, the greater the degree of differentiation, and more information can be derived. Meanwhile, higher weight should be given to the index, and vice versa. is study shows that the rationality of the EWM in decision-making is questionable. One example is water source site selection, which is generated by Monte Carlo Simulation. First, too many zero values result in the standardization result of the EWM being prone to distortion. Subsequently, this outcome will lead to immense index weight with low actual differentiation degree. Second, in multi-index decision-making involving classification, the classification degree can accurately reflect the information amount of the index. However, the EWM only considers the numerical discrimination degree of the index and ignores rank discrimination. ese two shortcomings indicate that the EWM cannot correctly reflect the importance of the index weight, thus resulting in distorted decision-making results.
2017
At present, the choice of the best solutions out of many possible under conditions of uncertainty is the actual economic task, arising and to be solved in many economic situations. Famous classical approaches to its solution are based on various assessments of decision-making practical situations. However, they often give insufficiently accurate or incorrect results, and do not satisfy sustainability requirements, when the only invariant calculation result relative to calculation methodology is a reliable one and a corresponding to the reality result. This article describes an alternative approach to the justification of decisions under conditions of uncertainty without the construction and use of assumptions about the decision-making situation and in conformity with the approaches of the stability theory. The problem of multi-criteria decision-making in conditions of complete uncertainty, wherein structuring of alternatives is performed using the fuzzy entropy, has been formulated ...
International Journal of Fuzzy System Applications, 2016
Decision making involves various attributes along with several decision takers. Recently it has become more complex. This gives raise to uncertainty and associated with the information provided. So it may be appropriate to suggest that uncertainty demonstrates itself in numerous forms and of different types. Uncertainties may arise due to human behaviour, fluctuations of information, unknown facts. Fuzzy set theory is tool to deal with uncertainty in a better way. Both Fuzzy set theory and information theory are involved in dealing with various real-world problems such as segmentation of images, medical diagnosis, managerial decision making etc. Several methods and concepts dealing with imprecision and uncertainty have been proposed by many researchers. In the present communication, the authors have proposed a parametric generalization of entropy introduced by De Luca and Termini along with its basic properties. Further, a new measure of weighted coefficient of correlation is develo...
Procedia Computer Science, 2013
2015 Iran Workshop on Communication and Information Theory (IWCIT), 2015
In this paper, first we propose a new approach for mathematical multiple criteria decision making (MCDM) methods using information theoretic measures, entropy and divergence. Using the concept of entropy, we determine the impact of each criterion in decision making process. The Shannon's entropy has been previously employed for this purpose. In this paper we use Renyi's entropy and the concept of information potential of each criterion for weight assessment. Next, we introduce divergence as new separation measure for MCDM methods. The results indicate that the new measure outperforms the conventional Euclidean distance measure. These techniques employed in MCDM methods are new and may be of independent interest. Also, we introduce a new perspective for the decision making problems. We propose time as a new dimension for conventional mathematical MCDM methods. This dimension opens a new horizon for future MCDM methods which are based on given decision matrix. To the best of our knowledge, no prior work has studied the MCDM methods from this point of view. Using the criteria values for each candidate at different time instances, we estimate the probability distribution of each criterion in order to accommodation to the criteria value uncertainty. Finally, by utilizing the generalized correlation function, i.e. correntropy, we study the statistical dependencies of criteria in the same and different time instances.
Communications in Statistics - Theory and Methods, 2019
Cross entropy is an important index for determining the divergence between two sets or distributions. Most existing cross entropy are proposed in a fuzzy environment and undefined in some uncertain situations (e.g., Dempster-Shafer theory). This study proposes an extended cross entropy measure of belief values based on a belief degree using available evidence. Thus, a new aspect of belief functions represents in the name of a belief set. Then, a new cross entropy measure between two belief sets is defined. Furthermore, the application of the cross-entropy measure in multi-criteria decision making (MCDM) is provided with belief valued information.
The aim of this study is to propose an objective method based on a new measure of intuitionistic fuzzy information, called knowledge measure for determining weights of attributes (also called criteria) in a real-world multicriteria decision making problem under intuitionistic fuzzy environment. To address this issue, we first analyze the existing entropy measures and show that their use in objective weight determination process may lead us to produce unreliable weights of attributes by citing appropriate examples. Then, a new method to determine weight of criteria is developed on the basis of knowledge measure where information about criteria weight is completely unknown and partly known. Finally, a practical example is presented to illustrate the proposed weight determination method.
Journal of Statistical Theory and Applications
This paper presents a new structure as a simple method at two uncertainties (i.e., aleatory and epistemic) that result from variabilities inherent in nature and a lack of knowledge. Aleatory and epistemic uncertainties use the concept of the entropy and Dempster-Shafer (D-S) theory, respectively. Accordingly, we propose the generalized Shannon entropy in the D-S theory as a measure of uncertainty. This theory has been originated in the work of Dempster on the use of probabilities with upper and lower bounds. We describe the framework of our approach to assess upper and lower uncertainty bounds for each state of a system. In this process, the uncertainty bound is calculated with the generalized Shannon entropy in the D-S theory in different states of these systems. The probabilities of each state are interval values. In the current study, the effect of epistemic uncertainty is considered between events with respect to the non-probabilistic method (e.g., D-S theory) and the aleatory uncertainty is evaluated by using an entropy index over probability distributions through interval-valued bounds. Therefore, identification of total uncertainties shows the efficiency of uncertainty quantification.
Applied Mathematical Modelling, 2010
A multicriteria fuzzy decision-making method based on weighted correlation coefficients using entropy weights is proposed under interval-valued intuitionistic fuzzy environment for the some situations where the information about criteria weights for alternatives is completely unknown. To determine the entropy weights with respect to a decision matrix provided as interval-valued intuitionistic fuzzy sets (IVIFSs), we propose two entropy measures for IVIFSs and establish an entropy weight model, which can be used to determine the criteria weights on alternatives, and then propose an evaluation formula of weighted correlation coefficient between an alternative and the ideal alternative. The alternatives can be ranked and the most desirable one(s) can be selected according to the values of the weighted correlation coefficients. Finally, two applied examples demonstrate the applicability and benefit of the proposed method: it is capable for handling the multicriteria fuzzy decision-making problems with completely unknown weights for criteria.
We introduce the concepts of entropy and cross-entropy for hesitant fuzzy information, and discuss their desirable properties. Several measure formulas are further developed, and the relationships among the proposed entropy, cross-entropy, and similarity measures are analyzed, from which we can find that three measures are interchangeable under certain conditions. Then we develop two multiattribute decision-making methods in which the attribute values are given in the form of hesitant fuzzy sets reflecting humans' hesitant thinking comprehensively. In one method, the weight vector is determined by the hesitant fuzzy entropy measure, and the optimal alternative is obtained by comparing the hesitant fuzzy cross-entropies between the alternatives and the ideal solutions; in another method, the weight vector is derived from the maximizing deviation method and the optimal alternative is obtained by using the TOPSIS method. An actual example is provided to compare our methods with the existing ones. C 2012 Wiley Periodicals, Inc.
Entropy
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 relia...
Axioms, 2021
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 weight...
The main purpose of this paper is to investigate the multi-criteria decision making (MCDM) under the completely unknown attributes weights. As the information collected from the various resources related to different criteria for assessing the best alternatives is always imprecise in nature. Thus, to handle the impreciseness in the data, fuzzy set theory has been used during the analysis and representation each attribute in the form of triangular fuzzy numbers. Moreover, the attribute weight vectors, used for aggregating the decision maker’s preferences, are found by using an entropy function. Finally, a house selection example, has been taken for demonstrating the approach.
This paper presents a survey about different types of fuzzy information measures. A number of schemes have been proposed to combine the fuzzy set theory and its application to the entropy concept as a fuzzy information measurements. The entropy concept, as a relative degree of randomness, has been utilized to measure the fuzziness in a fuzzy set or system. However, a major difference exists between the classical Shannon entropy and the fuzzy entropy. In fact while the later deals with vagueness and ambiguous uncertainties, the former tackles probabilistic uncertainties (randomness).
International journal of fuzzy system applications, 2017
The objective of this manuscript is divided into two fold. Firstly, a more generalized intuitionistic fuzzy entropy measure of order and degree has been presented for measuring the degree of fuzziness of the set with a proof of its validity. A structured linguistic variable has been taken as an illustrative example to show its validity and superiority than the existing measures. Furthermore, based on this measure, an approach to deal with multi-criteria decision making (MCDM) problem is developed. Finally, a practical example is provided to illustrate the decision making process. A computed result is compared with the help of existing results. A sensitivity analysis on the different values of the parameters will make a decision maker more choice for accessing their results.
Advances in Fuzzy Systems
Fuzzy entropy means the measurement of fuzziness in a fuzzy set and therefore plays a vital role in solving the fuzzy multicriteria decision making (MCDM) and multicriteria group decision making (MCGDM) problems. In this study, the notion of the measure of distance based entropy for uncertain information in the context of interval-valued intuitionistic fuzzy set (IVIFS) is introduced. The arithmetic and geometric average operators are firstly used to aggregate the interval-valued intuitionistic fuzzy information provided by the decision makers (DMs) or experts corresponding to each alternative, and then the fuzzy entropy of each alternative is calculated based on proposed distance measure. Several numerical examples are solved to demonstrate the application to MCDM and MCGDM problems to show the effectiveness of the proposed approach.
International Journal of Fuzzy Computation and Modelling, 2017
The present paper introduces generalised entropy for intuitionistic fuzzy entropy of order α with the evidences of its validity along with some of its properties. The proposed measure is a generalisation of the entropy given by De Luca and Termini (1972). It has been used to determine weights of both experts and attributes in intuitionistic fuzzy environment using decision matrices in multi-attributes decision-making problem with unknown weights. Further, weighted correlation coefficient is defined using the proposed entropy measure and correlation coefficients between alternatives and ideal point are determined. The value of correlation coefficient is used to rank the alternative and the alternative with greatest weighted correlation coefficient is selected as an optimal solution. Finally, an illustrative example describes its application on multi-attributes decision-making problem with undefined weights.
2016
In this paper, we have considered an intuitionistic fuzzy entropy with both the uncertainty and hesitancy degree of IF sets. Based on this IF entropy, a new decision-making method of a multi-attribute decision making problem (MADM) has been introduced in which attribute values are expressed with IF values. In the case of attribute weight, a case with partially known attribute weights is discussed and a method is developed to determine the attribute weights. This method is an extension of ordinary entropy weight method. At last, an air-codition example is used to illustrate the application of the proposed model.
Journal of Intelligent & Fuzzy Systems, 2015
The aim of this paper is to develop some measures of entropy for linguistic information and uncertain linguistic information. Based on some given functions with required monotonicity and some kinds of distance measures, we propose two groups of entropies of linguistic fuzzy sets and uncertain linguistic fuzzy sets, respectively. Then, as an application of the proposed entropy measures, we put forward two algorithms for obtaining the objective weight vector of attributes in multiple attributes decision making by utilizing the entropy weight approach and the maximum entropy method. Finally, two cases of subjective evaluation of the building environment are illustrated to show the application of entropies for linguistic information in real world.
2021
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 entropy measures 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 to demonstrate the proposed entropy measures. Finally, a comparative analysis is presented to illustrate the rationality and effectiveness of the proposed entropy measures.
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