Understand the relevant knowledge of fuzzy mathematics. The fuzzy comprehensive evaluation algorithm is a method used for comprehensive evaluation and decision-making. The coefficient of variation method can be used to determine the weight vector. The relative deviation method and the relative dominance method can obtain the evaluation matrix R
related information
fuzzy mathematics
Many phenomena and relationships in the real world are uncertain. These uncertainties manifest themselves in various forms, such as randomness, grayness, fuzziness, roughness, etc.
Fuzzy mathematics is a branch of mathematics that uses fuzzy sets and their operations to study and deal with fuzzy and uncertain phenomena and relationships.
Many mathematical constructs Modular problems include fuzzy phenomena and relationships, and such problems can often be dealt with using fuzzy mathematical methods.
The following introduces fuzzy sets and fuzzy comprehensive evaluation.
fuzzy set
Many phenomena and relationships in reality are relatively vague. Phenomena such as tall and short, long and short, big and small, more and less, poor and rich
do not satisfy the "either-or" law of excluded middle, but have the ambiguity of "either this or that".
Fuzzy uncertainty is different from random uncertainty. Random uncertainty is the uncertainty caused by the damage of the law of causality, while fuzzy uncertainty is the uncertainty caused by the damage of the law of excluded middle
Let xi represent the i (i=1,2,...,30) line segment, then the universe of discourse U={x1, X2,…,X30}. If A is a set of "long line segments", then the membership of line segment xi as set A is the degree of membership of xi to A.
A membership function of A is established below. Because the longer the line segment, the greater the degree of belonging to A, so the length of the line segment can be used as the membership degree of A.
Fuzzy set operations
Since there is no absolute membership relationship between elements and sets in fuzzy sets, the operation of fuzzy sets is completed through membership functions.
Determination of membership function
From the concept of fuzzy sets, we can see that the basic idea of fuzzy mathematics is the degree of membership, so the key to establish a mathematical model using fuzzy mathematics is to establish a membership function that conforms to reality. However, how to determine the membership function of a fuzzy set is still an unsolved problem. The
common method to determine the degree of membership is the fuzzy distribution method.
The fuzzy distribution method regards the membership function as a fuzzy distribution. First, select the appropriate fuzzy distribution according to the nature of the problem, and then Then determine the parameters in the distribution based on relevant data
. The following is a brief introduction to the trapezoidal distribution commonly used in fuzzy distribution.
The small size is generally suitable for describing the fuzzy phenomenon of the small degree such as "small", "less", "shallow" and light; the large size is just the opposite of the small size; and the intermediate type is generally suitable for describing the fuzzy phenomenon in the middle state.
fuzzy comprehensive evaluation
The evaluation of a thing usually involves multiple factors or multiple indicators. Evaluation is a comprehensive evaluation based on the interaction of multiple factors.
There are many methods of comprehensive evaluation, such as gray evaluation method, analytic hierarchy process, fuzzy comprehensive evaluation method , data envelopment analysis method, artificial neural network evaluation method, ideal solution method and so on. Sometimes, the two evaluation methods can also be integrated into a combined evaluation method.
Various evaluation methods have different starting points, different ideas for solving problems, different applicable objects, and each has its own advantages and disadvantages
. Different evaluation methods will produce different evaluation conclusions, and sometimes even the conclusions are at odds , that is, the result of comprehensive evaluation is not the only
fuzzy comprehensive evaluation as a specific application of fuzzy mathematics . Comprehensive evaluation of the hierarchical status of the things being evaluated is based on multiple factors. The specific steps are: first determine the factor set and evaluation set of the evaluated object, then determine the weights of each factor and their membership degree vectors respectively, obtain the fuzzy evaluation matrix, and finally perform fuzzy operations on the fuzzy evaluation matrix and the weight vectors of factors and combine them. Normalized to obtain the comprehensive results of fuzzy evaluation. The fuzzy comprehensive evaluation method is simple and easy to master, and has good evaluation effect on complex problems with multiple factors and levels. It is difficult to be replaced by other evaluation methods.
Fuzzy comprehensive evaluation steps
The basic idea of the above synthesis is: perform some appropriate fuzzy operation on the evaluation matrix R and the weight vector A, and synthesize the two into a fuzzy direction B={b1,b2,…,bn}, that is, B=AR, and then B After comprehensive analysis according to certain rules, the final fuzzy comprehensive evaluation result can be obtained.
Commonly used fuzzy synthesis operators are:
In fact, M can also be taken as ordinary matrix multiplication, and the synthesis is a weighted average. Which operator to choose depends on the nature of the problem and the characteristics of the operator.
Generally, the results of using principal factor prominence and weighted average algorithms are similar. However, in practice, we should still pay attention to the characteristics of these two types of algorithms. The
main factor prominent type is suitable for situations where the data in the fuzzy matrix are very different , while the weighted average type is often used in situations where there are many factors to avoid information loss.
Example
The results show that the students recognized the teacher most in terms of "familiarity with the teaching material", followed by "clear and easy to understand", and
not in the neat and clear writing on the blackboard. Obviously, Example 3 is too simple. Not only the fuzzy comprehensive evaluation matrix R is given, but also the weight vector A is directly given.
In fact, practical problems often only provide a series of evaluation objects and several evaluation indicators for each object, and these indicators may vary greatly in value. The properties are also different.
At this time, not only the weight vector A of the index needs to be determined according to an appropriate method, but even the evaluation matrix R must be obtained after processing the evaluation index in a certain way. The common method for determining the weight vector A is the
aforementioned variation coefficient method , and the commonly used methods for processing evaluation indicators to obtain evaluation matrix R are relative deviation method and relative superiority method. These two methods are simple and practical, and can be considered in combination with gray relational analysis in modeling competitions.
These three methods are introduced below
coefficient of variation method
The design principle of the coefficient of variation method is: if the numerical difference of a certain indicator is large, it can clearly distinguish the objects being evaluated, indicating that the indicator has rich discriminating information, so a greater weight should be given to the indicator; conversely, if each If the numerical difference between the evaluated objects on a certain index is small, then the ability of this index to distinguish between various evaluation objects is weak, so this index should be given a smaller weight.
The weight of an indicator calculated using the coefficient of variation method and the importance of the indicator in the evaluation system are two concepts. The function
of the coefficient of variation method is only to improve the resolving power of the indicator and facilitate ranking. In fact, the premise for using the coefficient of variation method is that all indicators are equally important in the evaluation system. In other words, when the importance of indicators in the evaluation system varies greatly , it is not necessarily appropriate to use the coefficient of variation method to determine the weight.
Relative deviation fuzzy matrix evaluation method
The relative deviation fuzzy matrix evaluation method is somewhat similar to gray correlation analysis. First, virtualize an ideal plan u , then establish the deviation matrix R between each plan and u according to a certain method, then determine the weight A of each evaluation index, and finally use A to weight the average of R to obtain the comprehensive distance F between each plan and the weights A and u. , then the plans can be sorted according to F.
code
A=[1000 120 5000 1 50 1.5 1
700 60 4000 2 40 2 2
900 60 7000 1 70 1 4
800 70 8000 1.5 40 0.5 6
800 80 4000 2 30 2 5];
[m,n]=size(A);%找出多少行多少列
maxA=max(A);%找出每列最大值
minA=min(A);%找出每列最小值
G=maxA-min(A);%最大值减去最小值
A1=max(A(:,1));%A1为效益型
A2=min(A(:,2:n-1));%A2~A6为成本型
A3=max(A(:,7));%A7为效益型
u=[A1,A2,A3];
R=zeros(m,n);%将模糊综合矩阵初值设置为0
% 如下是得出模糊综合矩阵
for i=1:m
for j=1:n
R(i,j)=abs(A(i,j)-u(j))/G(j);
end
end
%利用变异系数计算权向量
x=mean(A);
s=std(A);%求每一列的方差
v=s./x;%权向量的初值
v2=sum(v);
c=zeros(1,7);
for i=1:7
c(i)=v(i)/v2;
end
FF=R*c'
operation result
Relative excellence fuzzy matrix evaluation method
The evaluation basis of the relative deviation method is the deviation of each plan from the ideal plan, and the basic idea of the relative superiority evaluation method is : first, use appropriate methods to convert all indicators (benefit type, cost type, fixed type) into benefit type ( cost type), obtain the superiority matrix R, and then determine the weight A of each evaluation index. Finally, use A to weight R and average it to get the comprehensive superiority F of each scheme. Then the schemes can be sorted according to F.
summary
Since gray correlation analysis, relative deviation method and relative superiority method are all comprehensive evaluation methods, and they all solve the same problem, a question naturally arises: Are the conclusions of these three methods to evaluate the same problem completely consistent? First,
respectively Use relative deviation method and relative superiority method to evaluate Example 5 and Example 4.
