Because I didn't watch the news these days, a classmate in the background of WeChat asked me to find a book, and I couldn't reply after more than 48 hours. QQ said sorry to this friend. If you see it, you can add my friend directly ~ Other students who want to find books can also add me to WeChat directly ... After all, if it is not a free e-book that can be found online, the books of the National Library Association are basically three yuan, and you pay for it yourself ... Taobao generally charges five yuan for a book, which is the labor cost.
Well, before it's too late, let's talk about an evaluation model-fuzzy comprehensive evaluation model today.
For the record, it's my first time to get in touch with this knowledge, and I may not be able to explain the principle well. I will write down the application process.
(About the entropy weight method mentioned in the last article, there is also the grey correlation analysis mentioned earlier, and I will make up for it later. )
First of all, explain the following "fuzzy mathematics". Fuzzy mathematics is a mathematical theory and method to study and deal with fuzzy phenomena. In real life, many concepts are difficult to be described by deterministic sets. For example, the concept of "youth" is 15 30 young or 18 25 young? Everyone may have different views on this issue, and it is difficult to give a precise scope. We can understand it as a vague concept.
The concepts of big and small, long and short, beauty and ugliness that are often mentioned in life are all vague concepts. Actually, it's quite easy to identify. You can ask yourself, how old are you? How small is it? How long is it? This kind of question feels a bit like herdsman's, but it is precisely because there is no precise scope that we can only ask this kind of question. Corresponding to this vague concept, it is a definite concept, such as gender, which is generally not male or female, and basically has an accurate basis for division; Another example is height, which is 180 and 190 respectively. It is also very accurate, and there won't be too many ambiguities. Note that "height" is a definite concept, while "height" is a vague concept. Think for yourself, hhh.
Fuzzy mathematics is used to deal with problems involving fuzzy concepts. Try to quantify fuzzy concepts in some way, which is convenient for processing and calculation. Fuzzy comprehensive evaluation is naturally a great application of fuzzy mathematics in evaluation, that is, dealing with evaluation problems involving fuzzy concepts.
In fact, we can also find that one of the cores of the evaluation problem is to quantify various evaluation indicators, then add them by weight, and so on. Basically, the difference is not too big, and so is the fuzzy comprehensive evaluation model, which is not too difficult to understand and practice. (This only refers to the evaluation model I have contacted, and I don't know if it is too advanced. )
In order to better explain the latter model, it is necessary to introduce some related concepts in fuzzy mathematics.
First, review the classic series. When we were in high school, we came into contact with the concept of set: a set of things with the same attributes. This set of classics has some basic attributes, such as certainty. Given a set and an arbitrary element, this element either belongs to this set or does not belong to this set. There is no third case.
In the fuzzy comprehensive evaluation model, we don't use this classic set, because we have to deal with fuzzy concepts, so we need to use fuzzy sets. Fuzzy set is a set used to describe fuzzy concepts. One of the differences between fuzzy sets and classical sets is that fuzzy sets have no certainty. For example, 35 years old, we can think of it as "young" or "middle-aged", and there is no precise definition.
Therefore, unlike traditional collections, an element either belongs to a collection or does not. We use "membership degree" to express the relationship between elements and fuzzy sets, that is, the degree to which elements belong to fuzzy sets. When it comes to membership, it is necessary to mention membership function, which is a very important concept. Simply put, the membership function is the function of membership degree to each element, the domain is the element we study, and the function value is the membership degree. The range of membership degree is that the greater its value, the more it belongs to this set. (In fact, the membership function is not described by domain values, but it is convenient to understand qwq. )
Give a simple example. We have to measure the concept of "youth", so it is not good to draw a line directly between 0- 150 years old to distinguish between young and not young. Therefore, for each integer age between 0- 150, we give a corresponding value, namely membership degree, to judge its relationship with the "young" set. In order to give such a value more conveniently, we design a function and membership function with the elements to be studied-here is an integer between 0- 150. The membership function is defined as follows.
Where A stands for fuzzy set, which is the "young" set here, and X stands for the elements in the set, which is the age between 0- 150, we can draw a function image.
It can be found that when the age is less than 20, the corresponding membership degree is 1, that is, we think that less than 20 years old must belong to the young category; When the age is between 20 and 40, the degree of membership gradually decreases with the increase of age; When we are over 40 years old, we think that we are basically out of the category of youth, and our membership is all zero. If a person is 30 years old, we can't be sure whether he is young or not, but we use the membership degree of 0.5 to think that 50% of the 30-year-old person belongs to the young category and 50% does not. 0.5 Measure the degree to which 30 years old belongs to the youth collection, and express the relationship between 30 and "youth".
We can also understand the degree of membership from the perspective of probability. In real life, the determination of the degree of membership is often achieved through investigation. For example, 100 people are asked whether they are young at the age of 30. If 40 people answer yes, their membership can be determined as: the greater the total number of surveys, the closer this value is to the true membership. Is it like "frequency approximation probability"? As for the above membership function, it is just constructed at will for easy understanding, which is not the same as the real investigation result, but it still reflects the subjective thought of the constructor. In fact, the membership function is not unique. Different people and samples of different sizes may get different membership functions.
Well, the basic concepts, namely fuzzy set, membership function and membership degree, are generalized here. Because the contact time is not long, it may not be clear and accurate. Simply put, the degree of membership I understand is the degree to which an element belongs to a fuzzy set, and the membership function is the function used to determine the degree of membership, that's all. There is no need to be too entangled, and it does not affect the specific application behind it.
Generally speaking, there are three main types of fuzzy sets, small, medium and large. In fact, it is similar to the maximum, minimum, intermediate and interval indicators in TOPSIS method, and there is nothing special. For example, "young" is a small fuzzy set, because the younger the age, the greater the degree of membership, the younger; "Old age" is a big fuzzy set. The older you get, the more members you have, and the older you get. And "middle age" is an intermediate set. Only when the age is in a certain intermediate range, the greater the degree of membership. To sum up, it is to consider the relationship between "element" and "membership degree", and so on, it is to consider the monotonicity of membership function. The following figure can represent the membership function images of three fuzzy sets: young, middle-aged and old. Take a look and you'll see what I mean.
Why do you want to know the classification of fuzzy sets? Because in the fuzzy comprehensive evaluation model, it is necessary to determine whether the corresponding fuzzy concept belongs to large scale, small scale or intermediate type, and then use the corresponding membership function to find the appropriate membership degree. Again, no matter what kind of fuzzy set, the greater the degree of membership, the greater the degree of belonging to this set, remember?
These are just three common types. In fact, if you think about it, there should be quite a lot of shapes, as long as one element corresponds to a membership degree, and the range is between. The above three kinds are only common ones, and they are also the types of fuzzy sets often involved in evaluation problems.
Of course, there may be some doubts. For example, for the collection of "young" and "old", we take age as the element of our research, and age can be quantified into numbers. Similarly, the vague concepts of fast and slow can be quantified by speed, deep and shallow can be quantified by depth, and so on. So, what should be used to quantify beauty and ugliness? I don't know ... I don't think there is a common variable that can be used to quantify beauty and ugliness. In the general evaluation model, this kind of pit problem should not be involved (no, no). If you are interested, please check it yourself. ...
Determining the membership function is actually giving a fuzzy set, and then by some methods, giving the membership degree of the elements we need to study relative to the fuzzy set. For example, for the "young" fuzzy set, we should try to determine the membership degree of each age between 0 and 150 and draw an image, which is the image of the membership function.
There are three ways to determine the membership function.
1. fuzzy statistical method
The principle of fuzzy statistics is to find multiple people to describe the same fuzzy concept, and define the membership degree with membership degree. Similar to finding probability, we can approximate probability with frequency. For example, as we mentioned above, if you want to know the degree of membership of 30 years old relative to "youth", then ask someone. If one of them thinks that 30 years old belongs to the category of "youth", it can be regarded as the degree of membership of 30 years old relative to "youth". The bigger the truth, the more accurate it is. If you ask other ages in this way, you can draw a function image.
Well, this method is more in line with the actual situation, but it is often investigated by issuing questionnaires or other means. Mathematical modeling competition, time may not be enough, so it is just an introduction, basically not needed. (But now Taobao fills in the questionnaire very quickly, so it's good to have money. )
2. With the help of the existing objective scale
For some fuzzy sets, we can use the existing indicators as the membership degree of elements. For example, for the fuzzy set of "well-off families", we need to determine the membership degree of 100 families, so we can use Engel coefficient to measure the corresponding membership degree. Engel coefficient = total food expenditure/total household expenditure. Obviously, the closer a family is to a well-off level, the lower its Engel coefficient should be, and the greater the "1- Engel coefficient", so we can regard "1- Engel coefficient" as the degree of membership of a family relative to a well-off family. However, this is just an analogy. After all, for wealthy families, Engel's coefficient is very small and the degree of membership is very large, but whether wealthy families are "well-off families" remains to be discussed.
Similarly, for the fuzzy set with "complete equipment", the membership degree can be measured by the equipment completeness rate, and for the fuzzy set with "stable quality", the membership degree can be measured by the genuine product rate. If you encounter problems, you can search Baidu first, and you may find a good indicator someday.
However, it should be noted that the membership degree is between, so when looking for indicators, we should also pay attention to between. If not, it can be normalized, as mentioned earlier.
This method can be used in modeling, depending on the specific topic.
3. Distribution method
This is a subjective method, that is, according to the subjective will, after determining the classification of fuzzy sets, assign them a membership function and get the membership degree of elements. It sounds subjective, but it is also one of the most commonly used methods in competitions. You can easily get the membership function by choosing.
I post the commonly used function forms below.
The picture may not be very clear, but it can basically be seen that for a small fuzzy set, the membership function as a whole is decreasing, that is, the greater a certain feature of an element, the smaller the membership degree; For large sets, the membership function is generally increasing, that is, the greater a certain feature of an element, the greater the membership degree; For the intermediate set, the membership function generally increases first and then decreases, and the middle part or a certain point takes the maximum value.
In the actual modeling competition, for the convenience of calculation, trapezoidal distribution membership function is the most commonly used (as I said in class). Of course, specific problems still need to be analyzed in detail. The membership function should be smoother and steeper, and the extreme value of the middle part or a certain point should be selected according to the specific situation, but in general, this is the way it is.
Let's take a look at the membership function image of trapezoidal distribution.
These are several methods to determine the membership function. There are other methods, such as Delphi method, binary comparison sorting method, comprehensive weighting method and so on. You can consult them yourself if you are interested.
After such a long time of preparation, we can finally solve the problem by this method.
First of all, we should introduce some concepts.
For example, if we want to evaluate a student's performance, according to the aforementioned analytic hierarchy process (AHP) or TOPSIS method, we will find the indicators, make a comprehensive score, and often compare the performance of several students to give a ranking. The above evaluation indicators actually correspond to the factors set here. We can use four indicators of factor clustering to evaluate a student's comprehensive performance.
The comment set is the evaluation result of the corresponding object, which is similar to the "scoring result" mentioned above. The difference is that the comment set is not a collection of scores, but a comment composed of fuzzy concepts. For example, to evaluate students' performance, we can set the annotation set to. These three comments in the comment set are vague concepts, but when dealing with specific problems, we can also put the scheme into the comment set to choose the best scheme.
The weight setting is the weight you want. Give each index a weight and use it for comprehensive evaluation, so I won't say much. Here, we can take the weight set as the weight of the four indicators in the factor set.
What problem does the fuzzy comprehensive evaluation model solve? Well, in fact, it is to give an object, evaluate it with the index of factor set, and find the most suitable comment from the comment set. If the focus of the comment is on the scheme, it is to choose the most suitable scheme. So what is the measure of "fit"? Obviously, it is membership, the membership of a fuzzy set.
Well, to sum up, for example, we now have a student, a factor set, a weight set and a comment set. Our aim is to give students an appropriate comment after some operations. Do you understand ~
The first-level fuzzy comprehensive evaluation model, that is, the factor concentration has only one layer of evaluation indicators, and there is no nesting, which is also the most basic situation.
There are several main steps to solve this problem.
Well, so far, we have learned the problem-solving steps of the first-level fuzzy comprehensive evaluation. This should also be recognized. The most important thing is to make clear the judgment matrix and weight vector. Multiply by two, and the comprehensive membership vector comes out. Choose the largest one. As the weight vector said before, how to find the membership degree of the judgment matrix, or the judgment matrix? The method of determining the membership function is also mentioned above. By using the membership function, the membership degree can be obtained. In practical modeling, we often use "assignment method" to specify a membership function that conforms to practical problems, or we can use other methods. As long as the judgment matrix and weight vector are known, the evaluation problem is basically solved.
In fact, the steps to solve the problem are quite simple, but there are too many preparations in front, so there are many things to write and it is not very complicated to do. I will find an example of a large open online course in China University to show the problem-solving process. Well, it's a waste of time to play all by hand, so I'll just take the map.
This is the topic, that is, the concentration of pollutants and the weight of each pollutant in the air quality grade are given. Let's determine the air quality level of this day.
The following figure shows the evaluation criteria.
The concentration of pollutants is the factor set of this question, and the four grades of air quality are the comment set, which is also a fuzzy concept. For example, when the concentration of TSP is 0.20, we can't simply determine whether the air quality grade is Grade I or Grade II from the perspective of TSP, but we can determine the degree of membership relative to each grade.
How to confirm membership? Here, the assignment method can be used to specify the membership functions of four fuzzy sets, and the most commonly used trapezoidal distribution membership function is more in line with the meaning of the question. It can be found that "first class" should be a very small fuzzy concept, that is, the lower the concentration of pollutants, the greater the degree of belonging to "first class"; "Secondary" and "tertiary" should be intermediate concepts. When the pollutant concentration is in a certain range in the middle, the corresponding membership degree is larger; "Grade 4" is a big concept. The greater the concentration of pollutants, the greater the degree of belonging to "Grade 4". After we determine the types of fuzzy concepts in the comment set, we can give the corresponding trapezoidal distribution membership function. As shown below.
This corresponds to the values when the concentration of each pollutant in the above table is just in the four grades of 1234. From the point of view of membership function, when the pollutant concentration is equal to the value in this table, the membership degree relative to the corresponding air quality grade is exactly 1. It shouldn't be hard to understand, just think about it.
The membership function is determined, and the concentration of each pollutant in this day is directly brought into the membership function, so as to obtain the membership degree and the judgment matrix.
Using the judgment matrix and weight vector, the comprehensive membership vector can be directly calculated.
Obviously, the air quality on this day belongs to Grade II to the greatest extent, so we think that the air quality on this day is Grade II.
Well, examples are also ready. You can go to the large-scale open network course of China University to search the mathematical modeling course of Huazhong Agricultural University, and give a more detailed explanation of fuzzy comprehensive evaluation. This example also comes from this course. Well, there are other modeling methods.
Multi-level fuzzy comprehensive evaluation is actually equivalent to adding several layers of factor sets. For example, we have to deal with 20 evaluation indicators at the same time, and it will be more troublesome to determine the weight. Then we can divide these 20 indicators into four categories, determine the weights of indicators once in each category, and then determine the weights of the four categories. This will be more convenient. If there are many indicators, it can be embedded in several layers, that is, multi-level fuzzy comprehensive evaluation.
The student evaluation model shown in the above figure is a two-level comprehensive evaluation model, and the numbers behind the indicators represent the weights of the corresponding levels. How do we determine the judgment matrix at this time? The judgment matrix of the first layer can't be determined as soon as it comes up, and it needs to be pushed up step by step from the last layer.
For example, when we examine the subordinate vector corresponding to academic performance, we need to examine its next-level indicators, namely professional course performance and non-professional course performance. For example, the score of Z's professional course is 90, so from this indicator, the membership vector of Z's student is that the comment set is still "excellent, good and poor". Then look at the scores of non-professional courses and get a membership vector. Using these two vectors, we can construct a matrix to represent the judgment matrix composed of two secondary indicators under the academic performance index. Then how is the membership vector of the first-level index of academic performance relative to the comment set obtained? It's simple. I thought there was a weight vector. If we use it, we can get a new vector, which is naturally the membership vector of Z students relative to the comment set from academic performance indicators. Well, in addition, it is the weighted sum of the membership degrees of the two secondary indicators, which should not be difficult to understand.
Similarly, the membership vectors of other first-level indicators are obtained to form the judgment matrix of first-level indicators, and then weighted to get the comprehensive membership vector.
Well, in fact, it is to get the judgment matrix of the first-level indicators, get the membership vector of the first-level indicators, and then form a judgment matrix with the membership vector of the first-level indicators to get the membership vector of the first-level indicators, which is the membership vector used for comprehensive evaluation.
Well, that's it ~
As for the limitations, I won't talk about it. I only know the conditions of use and don't know what to say. That's it. See you next time ~
By the way, one last reminder, if you want to find a pdf, it is similar to the business on Taobao. You can leave a message directly in the message applet in the official WeChat account Twitter, or you can add me to WeChat. If you reply backstage, if you don't see it in time, you can't reply after 48 hours. Well, there is no labor fee. You have to pay three yuan for the website yourself.
Above.