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《基于模糊馬氏距離的行為三方決策:在供應鍊管理問題中的應用》的4.FF馬氏距離（1）：相關概念與屬性權值的推導。
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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）
本期推文将從思維導圖、精讀内容、知識補充三個方面介紹期刊論文《基于模糊馬氏距離的行為三方決策:在供應鍊管理問題中的應用》的FF馬氏距離的相關概念及屬性權值的推導。
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、思維導圖（Mind mapping）
三、精讀内容（Intensive reading content）
本節作者給出了Fermatean fuzzy偏差、方差、協方差和相關系數的一些新概念，接着提出了一種新的屬性權重方法。
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）
本部分舉例說明了如何應用這些定義到一個具體的情景——土地選擇問題，展示了如何建構決策矩陣，計算模糊平均值，以及如何确定模糊偏差。
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.
整個過程旨在提供一種結構化方法，用于處理帶有模糊或不精确資訊的複雜決策問題，幫助決策者更好地了解和權衡不同選項之間的優劣。
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方差與協方差（Fermatean fuzzy variance and covariance）
模糊方差用于衡量FFDM中某個屬性的模糊偏差的離散程度，可以通過計算各對象對于該屬性模糊偏差平方的平均值得出。
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.
模糊協方差是用于衡量FFDM中兩個屬性之間模糊偏差的相關性，可以通過計算各對象在兩屬性上的模糊偏差乘積的平均值得出。
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.
模糊協方差矩陣展示了如何根據前面的定義計算出FFDM的模糊協方差矩陣，該矩陣對角線元素代表各個屬性的模糊方差，而非對角線元素代表屬性間的模糊協方差。
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.
這些定義和計算方法為處理帶有模糊不确定性的MADM問題提供了定量分析工具，有助于更全面地了解決策矩陣中屬性間的關系和各屬性内部的變異性，進而輔助決策者做出更加理性和精準的決策。
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）一種評估屬性權重的新方法（A new method for evaluating attribute weight）
本節詳細闡述了一種創新的屬性權重評估方法，該方法基于模糊決策分析架構内的CRITIC方法進行了拓展。核心在于引入了模糊相關系數的概念，用以衡量FFDM中屬性cv和cw之間的相關性，進而通過模糊方差和模糊協方差評估屬性的對比強度和分歧程度。
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.
不同于傳統的CRITIC方法需要對資料進行歸一化處理，本方法直接利用原始資料，簡化了計算流程。同時，它采用了模糊方差和相關系數來量化屬性資訊量，而非僅依賴于标準差和相關系數的補，使得權重的确定更符合模糊環境下的決策需求。通過一個示例，展示了如何計算各屬性的資訊量，進而确定其歸一化權重，有效地展現了該方法在實際決策問題中的應用價值。
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、知識補充——CRITIC方法（Knowledge supplement —CRITIC method）
CRITIC方法是一種多準則決策分析中的權重确定技術，它由 Diakoulaki 等人在1995年提出。這種方法主要用于确定決策矩陣中各屬性的相對重要性，即權重，它考慮了屬性之間的内在聯系和互相影響，而不僅僅是基于專家判斷或單一的統計度量。
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.
CRITIC方法的基本思路是确定名額的客觀權數以兩個基本概念為基礎。一是對比強度，它表示同一名額各個評價方案取值差距的大小，以标準差的形式來表現，即标準化差的大小表明了在同一名額内各方案的取值差距的大小，标準差越大各方案的取值差距越大。二是評價名額之間的沖突性，名額之間的沖突性是以名額之間的相關性為基礎，如兩個名額之間具有較強的正相關，說明兩個名額沖突性較低。
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.
CRITIC 方法的一個顯著優點是它的客觀性，因為它基于資料的統計特性而不是主觀判斷來确定權重。此外，它還考慮了屬性間的互相作用，這在很多情況下比僅僅基于單個屬性的差異度來确定權重更為合理。
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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參考資料：百度百科、Chat GPT
參考文獻：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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