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Today, the editor brings you journal papers
4. "Behavioral Tripartite Decision Making Based on Fuzzy Mahalanobis Distance: Application to Supply Chain Management Problems" 4. FF Mahalanobis distance (1): derivation of related concepts and attribute weights.
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Today, the editor brings you the
“4. FF Marginal distance (1): derivation of related concepts and attribute weights of the journal paper
＇Behavioural three-way decision making with Fermatean fuzzy Mahalanobis distance:
Application to the supply chain management problems＇＂.
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一、内容摘要（Summary of content）
In this issue, we will introduce the relevant concepts and attribute weights of FF Mahalanobis distance in the journal paper "Behavioral Tripartite Decision Making Based on Fuzzy Mahalanobis Distance: Application in Supply Chain Management Problems" from three aspects: mind mapping, intensive reading content, and knowledge supplementation.
This tweet will introduce the relevant concepts of FF marginal distance and derivation of attribute weights of the journal paper " Behavioural three-way decision making with Fermatean fuzzy Mahalanobis distance: Application to the supply chain management problems " from three aspects: mind mapping, intensive reading content, and knowledge supplementation.
二、思维导图（Mind mapping）
三、精读内容（Intensive reading content）
In this section, the authors present some new concepts of Fermatean fuzzy bias, variance, covariance, and correlation coefficient, and then propose a new method of attribute weighting.
In this section, the author gives some new concepts of Fermatean fuzzy bias, variance, covariance and correlation coefficient, and then proposes a new attribute weighting method.
(一)Fermatean fuzzy偏差(Fermatean fuzzy bias)
This section gives examples of how these definitions can be applied to a specific scenario, the land selection problem, shows how to construct a decision matrix, calculate fuzzy averages, and determine fuzzy bias.
This section provides an example of how to apply these definitions to a specific scenario - a land selection problem - showing how to construct a decision matrix, calculate fuzzy averages, and determine fuzzy biases.
The process is designed to provide a structured approach to complex decision-making problems with vague or imprecise information, helping decision-makers better understand and weigh the pros and cons of different options.
The entire process is designed to provide a structured approach to complex decision-making problems with vague or imprecise information, helping decision makers better understand and weigh the pros and cons of different options.
（二）Fermatean fuzzy方差与协方差（Fermatean fuzzy variance and covariance）
Fuzzy variance is used to measure the degree of dispersion of the fuzzy deviation of an attribute in FFDM, which can be obtained by calculating the average of the square of the fuzzy deviation of each object for that attribute.
Fuzzy variance is used to measure the dispersion of the fuzzy deviation of an attribute in FFDM, which can be calculated by calculating the average value of each object for the square of the fuzzy deviation of the attribute.
Fuzzy covariance is a measure of the correlation between the fuzzy bias of two attributes in FFDM, which can be obtained by calculating the average of the products of the fuzzy bias of each object on the two attributes.
Fuzzy covariance is used to measure the correlation of fuzzy deviation between two attributes in FFDM, which can be obtained by calculating the average value of the fuzzy deviation product of each object on the two attributes.
The fuzzy covariance matrix shows how to calculate the fuzzy covariance matrix of FFDM based on the previous definition, where the diagonal elements represent the fuzzy variance of the attributes, while the non-diagonal elements represent the fuzzy covariance between the attributes.
The fuzzy covariance matrix shows how to calculate the fuzzy covariance matrix of FFDM according to the previous definition. The diagonal elements of the matrix represent the fuzzy variance of each attribute, while the non-diagonal elements represent the fuzzy covariance between attributes.
These definitions and calculation methods provide quantitative analysis tools for dealing with MADM problems with fuzzy uncertainty, which are helpful to more comprehensively understand the relationship between attributes in the decision matrix and the variability within each attribute, so as to assist decision-makers to make more rational and accurate decisions.
These definitions and computational methods provide quantitative analysis tools for dealing with MADM problems with fuzzy uncertainties, which help to understand the relationship between attributes in the decision matrix and the internal variability of each attribute more comprehensively, so as to assist decision makers in making more rational and accurate decisions.
（三）一种评估属性权重的新方法（A new method for evaluating attribute weight）
This section elaborates an innovative attribute weight evaluation method, which is extended based on the CRITIC method in the framework of fuzzy decision analysis. The core is to introduce the concept of fuzzy correlation coefficient to measure the correlation between attributes cv and cw in FFDM, and then evaluate the contrast strength and divergence of attributes through fuzzy variance and fuzzy covariance.
This section details an innovative approach to attribute weight assessment that is based on an extension of the CRITIC method within the fuzzy decision analysis framework. The core lies in the introduction of the concept of fuzzy correlation coefficient to measure the correlation between attributes cv and cw in FFDM, which in turn assesses the strength of contrast and degree of divergence of attributes through fuzzy variance and fuzzy covariance.
UNLIKE THE TRADITIONAL CRITIC METHOD, WHICH REQUIRES THE DATA TO BE NORMALIZED, THIS METHOD DIRECTLY USES THE RAW DATA TO SIMPLIFY THE CALCULATION PROCESS. At the same time, it uses fuzzy variance and correlation coefficient to quantify the amount of attribute information, instead of relying only on the complement of standard deviation and correlation coefficient, so that the determination of weights is more in line with the decision-making needs in fuzzy environments. An example is shown how to calculate the amount of information for each attribute and then determine its normalized weight, which effectively reflects the application value of this method in practical decision-making problems.
Different from the traditional CRITIC method, which requires normalization of data, this method directly uses the original data source to simplify the calculation process. At the same time, it uses fuzzy variance and correlation coefficients to quantify the amount of attribute information, rather than just relying on the compensation of standard deviation and correlation coefficients, so that the determination of weights is more in line with the decision-making needs in fuzzy environments. Through an example, it shows how to calculate the amount of information of each attribute, and then determine its normalized weight, which effectively reflects the application value of this method in practical decision-making problems.
四、知识补充——CRITIC方法（Knowledge supplement —CRITIC method）
The CRITIC method is a weighting technique in multi-criteria decision analysis, which was proposed by Diakoulaki et al. in 1995. This method is mainly used to determine the relative importance of the attributes in the decision matrix, i.e., the weights, and it takes into account the intrinsic connections and interactions between the attributes, rather than just based on expert judgment or a single statistical measure.
The CRITIC method is a weight determination technique in multi-criteria decision analysis, which was proposed by Diakoulaki et al in 1995. This method is mainly used to determine the relative importance of each attribute in the decision matrix, that is, the weight. It takes into account the intrinsic relationship and mutual influence between attributes, rather than just based on expert judgment or a single statistical measure.
The basic idea of the CRITIC method is to determine the objective weight of the indicator, which is based on two basic concepts. The first is the contrast intensity, which indicates the size of the value gap between the evaluation schemes of the same index, which is expressed in the form of standard deviation, that is, the size of the standardized difference indicates the size of the value gap between the schemes within the same index, and the larger the standard deviation, the greater the value gap of each scheme. The second is the conflict between the evaluation indicators, which is based on the correlation between the indicators, such as a strong positive correlation between the two indicators, indicating that the conflict between the two indicators is low.
The basic idea of the CRITIC methodology is that the objective weights of the indicators are determined on the basis of two basic concepts. First, the intensity of comparison, which indicates the size of the gap between the values of the various evaluation programs of the same indicator, expressed in the form of standard deviation, that is, the size of the standardized difference indicates the size of the gap between the values of the various programs within the same indicator, the larger the standard deviation, the larger the gap between the values of the various programs. The second is the conflict between the evaluation indicators, the conflict between the indicators is based on the correlation between the indicators, such as the two indicators have a strong positive correlation between the two indicators, indicating that the two indicators have a low conflict.
A significant advantage of the CRITIC method is its objectivity, as it determines weights based on the statistical properties of the data rather than subjective judgments. In addition, it takes into account the interaction between attributes, which in many cases makes more sense than determining weights based solely on the difference of individual attributes.
A significant advantage of the CRITIC method is its objectivity, as it determines weights based on the statistical properties of the data rather than subjective judgment. In addition, it takes into account interactions between attributes, which in many cases makes more sense than determining weights based solely on the degree of variance of individual attributes.
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If you have a unique idea about the article,
please leave us a message,
and let us meet tomorrow.
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参考资料：百度百科、Chat GPT
参考文献：Mondal A, Roy S K. Behavioural three-way decision making with Fermatean fuzzy Mahalanobis distance: Application to the supply chain management problems[J]. Applied Soft Computing, 2024, 1(151): 1-20.
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