分享兴趣，传播快乐，增长见闻，留下美好。
亲爱的您，
这里是LearningYard学苑！
今天小编给大家带来期刊论文精读，
欢迎您的用心访问！
本期推文阅读时长大约6分钟，请您耐心阅读。
Share interest, spread happiness, increase knowledge, and leave beautiful.
Dear you,
this is the LearningYard Academy!
Today, the editor brings you intensive reading of journal papers,
welcome your visit!
This tweet usually takes about 6 minutes to read. Please be patient and read.
今天小编将从思维导图、精读内容、知识补充三个板块为大家带来论文《属性关联的双极容度多属性决策 VIKOR方法》步骤四，接下来我们开始今天的学习吧！
Today, the editor will bring you the paper "The Bipolar Tolerance Multiple Attribute Decision Making VIKOR Method of Attribute Association" from three sections: mind mapping, intensive reading content, and knowledge supplement. Step 4, let's begin today's study!
思维导图
本节内容思维导图如下所示：
A mind map of the contents of this section is shown below.
精读内容
在学习论文的步骤四过程中，我们从论文参考文献[18]中Kojadinovic和Marichal定义了双极容度熵的概念以度量双极容度的不确定性，并证明了双极容度Marichal熵具有Shannon熵的特性。同时我们还了解到双极容度Marichal熵具有一个基本性质，它可以根据属性集上的最大链来写，如下所示：
In the process of learning step four of the paper, we defined the concept of bipolar capacity entropy from Kojadinovic and Marichal in the paper reference [18] to measure the uncertainty of bipolar capacity, and proved that bipolar capacity Marichal entropy has the characteristics of Shannon entropy. At the same time, we also understand that the bipolar capacity Marichal entropy has a fundamental property that can be written based on the maximum chain on the attribute set, as follows:
并且，量Hm(u)可以简单地看作是通过香农熵计算的概率分布p的均匀性值在N上的平均值。换句话说，Hm(u)可以被解释为u在所有最大链上的单调性的平均正则性的度量。
Moreover, the quantity Hm (u) can be simply regarded as the average of the uniformity values of the probability distribution p calculated by Shannon entropy over N. In other words, Hm (u) can be interpreted as a measure of the average regularity of the monotonicity of u on all the largest chains.
其中，Marichal熵算法是一种常用的计算模型模糊测度的方法，它通过数据学习得到模糊测度，比较客观，其具体计算思路是采用Marichal熵构建目标函数，建立优化模型，通过最大化Marichal熵求解各优势函数的重要程度。
The Marichal entropy algorithm is a commonly used method for calculating model fuzzy measures. It obtains fuzzy measures through data learning, which is relatively objective. The specific calculation idea is to use Marichal entropy to construct an objective function, establish an optimization model, and solve the importance of each dominant function by maximizing Marichal entropy.
在Kojadinovic和Marichal的论文《Entropy of bi-capacities》中，我们可以看到Marichal熵的公式可以写为下面这种形式：
In Kojadinovic and Marichal's paper "Entropy of bi cities", we can see that the formula for Marichal's entropy can be written in the following form:
这与论文中的步骤四中的模型倒数第二个约束条件一致，接下来我们继续对其中的参数进行理解。
This is consistent with the second to last constraint condition of the model in step four of the paper, and we will continue to understand the parameters in it.
针对上图的红框中的公式有如下解释：
The formula in the red box above has the following explanation:
针对双极容度的choquet积分算子问题，我们可以将choquet积分算子写为上图的公式。
For the problem of the choquet integral operator with bipolar capacity, we can write the choquet integral operator as the formula shown in the figure above.
知识补充
接下来，和小编一起了解一下两点分布的熵模型吧！
Next, let's learn about the entropy model of two-point distribution with the editor!
首先，举个【两点分布】的例子。
Firstly, give an example of a two point distribution.
即当p=1/2时，对应的熵就是最大值啦。而当p=0 或者p=1时，对应的熵都为0，意味着这就是个必然事件了。
When p=1/2, the corresponding entropy is the maximum value. When p=0 or p=1, the corresponding entropy is both 0, which means that this is an inevitable event.
今天的分享就到这里了。
如果您对今天的文章有独特的想法，
欢迎给我们留言，
让我们相约明天，
祝您今天过得开心快乐！
That's it for today's sharing.
If you have a unique idea about today’s article,
welcome to leave us a message,
let us meet tomorrow,
I wish you a nice day today!
参考资料：DeepL翻译、百度百科
参考文献：
[1]林萍萍,李登峰,江彬倩,余高锋,韦安鹏.属性关联的双极容度多属性决策VIKOR方法[J].系统工程理论与实践,2021,41(08):2147-2156.
文案 |Yuan
排版 |Yuan
审核 |Wang