1 Principles and steps of AHP
1.1 The principle of AHP
In the late 1970s, American operations researcher and professor at the University of Pittsburgh, TL Saaty, proposed the Analysis Hierarchy Process (AHP). It divides people's thinking process into target layer, criterion layer and plan layer, and analyzes it with the help of mathematical model. It is a practical decision analysis method that effectively combines qualitative judgment and quantitative calculation of decision makers. The method is systematic, flexible and simple to use, and is suitable for large-scale and complex organized systems. Especially when the system is large in scale, complex in structure, diverse in attributes and objectives, and many elements and indicators in the system have only qualitative relationships, it is very efficient to use AHP for evaluation and decision-making. The basic principle of Analytic Hierarchy Process is to divide complex problems into hierarchical structures according to the dominance relationship, and each level is composed of various elements that interact with each other. The relative importance of each element in the hierarchy is quantified by pair-by-pair comparison method, and finally the overall ranking of relative importance is carried out.
1.2 Application steps of AHP
When using the AHP method to make decisions, you need to go through the following 4 steps:
1.2.1 Establish a multi-level hierarchical structure model
According to the dominance relationship, the evaluation index system is established in three layers from top to bottom:
(1) The highest level: also known as the goal level or the target level, it is the goal or result that the system wants to achieve, and it is the primary criterion for system evaluation (such as evaluating the teaching quality of college teachers).
(2) Criterion layer: It is the criteria, sub-criteria, etc. established to achieve the target layer.
(3) The bottom layer: also known as the program layer. It is a variety of programs, measures, etc. taken to achieve the goal.
1.2.2 Constructing a pairwise comparison judgment matrix
For the elements belonging to the same level, the elements of the previous level are used as the criterion to compare them one by one to establish a judgment matrix.
Compare the influence of n elements B=(B1, B2, Bn) on the target layer element A: use a pairwise comparison, and use aij to represent the ratio of the influence degree of the element Bi and the element Bj on the target A.
To quantify the judgment, the relative importance of each element was determined on a 1-9 scale.
1.2.3 Weight calculation
(1) Calculate the approximate value of the eigenvectors of the judgment matrix by the method of finding roots.
(2) After normalizing the feature vector, the weight vector W=(W1,W2,…,Wn)T is obtained.
1.2.4 Consistency check
In order to ensure the correctness and rationality of the obtained weights, a consistency check is also required.
Calculate the consistency index CI
in,
Obviously, the larger the n, the larger the error of the CI. Therefore, the random consistency ratio CR is introduced in the test
When n=1,2,…,15, the value of RI is shown in the following table.
When the random consistency ratio
It is considered that the calculated hierarchical ranking weights are correct and reasonable, otherwise, the judgment matrix needs to be readjusted until the consistency check is passed.
1.2.5 Calculation of comprehensive importance
The scheme with the largest weight is the optimal choice to achieve the goal.
2 The evaluation index system of teaching quality of college teachers
The purpose of teaching quality evaluation of college teachers is to make valuable judgments on the object being evaluated, in order to promote teaching. The primary problem of doing a good job in teaching quality evaluation is to establish a scientific evaluation index system. The evaluation index system should be able to accurately and sensitively reflect the current teaching level of teachers.
According to the characteristics and operability of the evaluation index system of teachers' teaching quality in colleges and universities, four criterion layers and three program objects are set up.
For the criterion layer: teaching preparation, teaching thought, teaching execution, teaching effect and characteristics, we can construct such a 4*4 judgment matrix:
The diagonal line is the judgment of each indicator. For example, for [Teaching Preparation] and [Teaching Preparation], its importance is 1, because the comparison of the indicators themselves must be 1:1. For the second row and first column, that is, the comparison between [Teaching Thought] and [Teaching Preparation], maybe I think [Teaching Preparation] is obviously more important than [Teaching Thought], so the value can be marked as 0.7, and so on, until the construction A complete judgment matrix.
The normalized weight W is:
where A*W is:
AW ：
λmax=AW1/W1+AW2/W2+AW3/W3+···+AWn/Wn=x
Maximum eigenvalue λmax=x/matrix order=4.0489
After the maximum eigenvalue λmax is solved, the CI value is much better.
According to the CI value formula, λmax=4.0489, n=4, and the CI value can be obtained by substituting it in = 0.0163
Calculate the CR value according to the CI and RI values, and judge whether the consistency is passed.
The RI value can be known by looking up the table. This is the random consistency index RI value table obtained by Satty simulation 1000 times (as shown in the following table):
And our matrix is 4th order (number of criterion layer factors), when the matrix order is 4, the corresponding RI value is 0.90, which is substituted into the formula:
Therefore, when CR=0.018<0.1, it indicates that the consistency of the judgment matrix A is considered to be within the allowable range, and the eigenvectors of A can be used to carry out the weight vector calculation; logical error.
For the scheme layer: Zhang San, Li Si, Wang Wu, we can construct such a 3*3 judgment matrix:
Through the comparison of the teaching preparation of each scheme by the single-level ranking, we can get the weight of Zhang San, Li Si, and Wang Wu, then this weight can be used as the score of Zhang San, Li Si, and Wang Wu in teaching preparation.
By analogy, we construct the score matrix of Zhang San, Li Si, and Wang Wu in terms of teaching preparation, teaching thought, teaching execution, teaching effect and characteristics:
Calculate its score as:
The weight of Zhang San, Li Si, and Wang Wu on teaching preparation is [0.5954, 0.2764, 0.1283]
The weight of Zhang San, Li Si, and Wang Wu on teaching ideas is [0.0819, 0.2363, 0.6817]
The weight of Zhang San, Li Si, and Wang Wu's teaching execution is [0.6337, 0.1919, 0.1744]
The weights of Zhang San, Li Si, and Wang Wu on the teaching effect and characteristics are [0.1667, 0.1667, 0.6667]
PS: All the above judgment matrices need to be checked for consistency.
Then for program object B1 (Zhang San), its total score is:
Zhang San's score on teaching preparation * weight of teaching preparation + Zhang San's score on teaching thought * weight of teaching thought + Zhang San's teaching execution Score * weight of teaching execution + Zhang San's score on teaching effect and characteristics * weight of teaching effect and characteristics = 0.5954*0.2082+0.819*0.199+0.6337*0.4247+0.1667*0.1681=0.5841
and so on, it can be calculated Scheme object B2 (Li Si) is:
0.2764*0.2082+0.2363*0.199+0.1919*0.4247+0.1667*0.1681=0.2141
Scheme object B3 (Wang Wu):
0.1283*0.2082+0.6817*0.199+0.1744*0.42647+0.6667 0.3485
Therefore, Zhang San scored the highest, followed by Wang Wu, and finally Li Si.
3. Cases and tool implementation
3.1 The most important factors in evaluating teaching quality
The function here is actually to find the factor weights, so only the single-level sorting is calculated, and there is no scheme level, that is, there is no need for a total hierarchical sorting.
3.1.1 Use the tool
SPSSPRO (free online, all functions are free) —>【Analytical Hierarchy Process (AHP simplified version)】
3.1.2 Case operation
step1: Select [Analytic Hierarchy Process (AHP Simplified Version)];
step2: Select the judgment matrix hierarchy
step3: Set the judgment matrix (the judgment matrix is a symmetric matrix)
step4: Click [Start Analysis] to complete all operations.
3.1.3 Interpretation
of analysis
results
This is the judgment matrix filled in on the previous operation page
Output 2: AHP Analytic Hierarchy Process
The weight calculation results of AHP (square root method) show that the weight score of teaching preparation is 0.2082, the weight score of teaching thought is 0.199, the weight score of teaching execution is 0.4247, and the weight score of teaching effect and characteristics is 0.1681.
Output 3: Consistency check result
The calculation result of AHP shows that the maximum eigenroot is 4.0814, and the corresponding RI value is 0.882 according to the RI table, so CR=CI/RI=0.0308<0.1, a one-time test shows the rationality of the weight determination method , no need to modify the judgment matrix.
3.1.4 Summary
It can be seen from 3.1.3 that the most important factor in evaluating teaching quality is teaching execution, with a weighted score of 0.4247, followed by teaching preparation, teaching ideas, and finally teaching effects and characteristics.
3.2 Select the best teachers
The function here is actually to obtain the quantitative score of the scheme, so it is necessary to rank the calculation level of the criterion layer individually, and perform a total hierarchical order of the scheme layer.
3.2.1 Use the tool
SPSSPRO (free online, all functions are free) —>【Analytical Hierarchy Process (AHP Professional Edition)】
3.2.2 Case Operation
Step1: Choose Analytic Hierarchy Process (AHP Professional Edition);
Step2: Choose to build a decision model;
Step3: Enter the constructed evaluation index;
Step4: Enter the final plan;
Step5: Confirm to enter the next index score (only integers are accepted);
Step6: Enter the importance value of the pairwise comparison between the indicators;
Step7: Input the importance evaluation of the corresponding evaluation values of different schemes;
3.1.3 Interpretation of
analysis results
Based on the single ranking of the index level and the total ranking of the program level, the best program for teachers' teaching quality evaluation is Wang Wu, and its quantitative score is 0.9561.
Output 2: Hierarchical Decision Model
It can be seen from the figure that the two most important determinants are teaching execution and teaching preparation, while teaching ideas, teaching effects and characteristic situations are of low weight.
Output 3: Judgment matrix summary results
The maximum eigenvalue of the judgment matrix is 4.117, the CI value is 0.039, the RI value is 0.882, and the CR value is 0.0442, and the consistency verification is passed.
Output result 4: Summary results of the judgment matrix of the scheme layer
The above table shows the weight calculation results of the scheme layer of the AHP (that is, the total ranking of the hierarchy). A judgment matrix with the number of leaf node indicators is constructed to analyze the weight of each indicator, and the consistency test results are displayed by showing the results. It is used to judge whether there is a logical problem of constructing the judgment matrix in the weight matrix of the scheme layer.
Since the score of the upper-level node can be calculated according to the score * weight of its child nodes, when constructing the judgment matrix of the scheme layer, only the leaf nodes are constructed, that is, N judgment matrices are constructed for N leaf nodes, which are used to synthesize the two-to-two judgment matrix. Comparing the situation, get the score of the scheme layer for a certain leaf node;
The consistency test results require that the CR value (CR=CI/RI) is less than 0.1. It can be seen that the scores of the schemes all meet the consistency test.
3.2.4 Summary
It can be known from 3.2.3 that the teacher with the highest evaluation score of teaching quality is Wang Wu, whose quantitative score is 0.956
4 Conclusion
The use of AHP in teaching evaluation can greatly reduce the subjective factors. On the one hand, when constructing the judgment matrix in the AHP, it needs the relative ratio of the pairwise comparison of the influencing factors rather than the determined value, which greatly reduces the subjective arbitrariness of the evaluation experts; When the subjective judgment of experts deviates from the objective reality, the consistency of the test judgment matrix is used to analyze and evaluate, so as to make correct adjustments. However, AHP also has some problems, such as the inaccurate calculation of the weights of influencing factors. This is because not all judgment matrices can meet the consistency requirements in traditional AHP. Aiming at this problem, this paper proposes a new method of combining man-machine on the basis of the traditional AHP, which improves the accuracy and efficiency when judging the consistency of the matrix, ensures the accuracy of the weight calculation, and effectively improves the traditional method. Analytic Hierarchy Process.