Recently, many small partners asked in the group what is sensitivity analysis? Does each model require sensitivity analysis? How to do sensitivity analysis? So today we will take everyone to understand the sensitivity analysis.
By reading the O-principal papers of Meisai over the years, we can find that most of the articles will have sensitivity analysis at the end, more or less, and Article (5) of Meisai’s official review rules also has a clear requirement: whether the model has been tested for stability Therefore, sensitivity analysis is very important for the evaluation of competition questions.
We know that it is very important to make some necessary assumptions about the problem in mathematical modeling. Assumptions generally refer to factors that have a certain influence on the problem but cannot be measured or predicted in detail. Therefore, they are often set as constants or not. I think about it, but there is little guarantee that these assumptions are completely correct. Therefore, we need to consider how sensitive the results are to each hypothesis. This sensitivity analysis is an important aspect of mathematical modeling. The specific content is related to the modeling method used, but it is not necessary to conduct a sensitivity test for every construction, and the parameters tested are often parameters that cannot be accurately measured or calculated.
table of Contents
What's the use of sensitivity?
Under what circumstances do sensitivity analysis?
Case Study-First Prize of 2020 National Championship C
Sensitivity analysis-excellent paper
Classic Case
Let me talk about a classic case first, and I hope it will help you understand the sensitivity analysis.
Case description: A cow weighs 200 jin, gains 5 jin a day, and costs 45 yuan a day for feed. The market price of the cow is 65 yuan per catty, but the current market is not good, and it drops 1 yuan a day to find the best time to sell this cow.
This is a very simple univariate optimization problem. Seeking the best selling time for this cow is the time when the selling price is the highest. In order to find this time, we assume: selling time t, cow weight w, cow price p, feed cost C, and selling cow Income R, net income P then have, P=RC, where R=P*w, C=45t, the net income model is as follows:
P=(65-t)(200+5t)-45t...... (1)
It is easy to draw that when t=8, P takes the maximum value of 13,320 yuan.
In fact, in most cases, the problem has ended here. But in real life, if you are a little more careful, we will find that among the parameters of the above model, the weight of the cow, the current market price, and the daily feed price are all easily measured. That is, the certainty is greater, but the growth of the cow The two parameters, the rate and the rate of decline of market prices, are not so easy to determine. Although the example stipulates that the rate of decline of market prices r=1 yuan/day, in reality, the daily r is different.
So we have a question- is this model applicable to all market price decline rates?
In order to eliminate doubts, we randomly select several numbers around the original rate of decrease (1 yuan/day) r=0.8, 0.9, 1.0, 1.1, 1.2, and according to formula (1), we get t=15, 11, 8 respectively , 5, 3, get the following picture:
What's the use of sensitivity?
Under what circumstances do sensitivity analysis?
The successful application of sensitivity analysis usually requires good judgment. It is usually neither possible to calculate the sensitivity coefficient for every parameter in the model, nor is there such a special requirement. We need to select those parameters with greater uncertainty for sensitivity analysis . The interpretation of the sensitivity coefficient also depends on the uncertainty of the parameters . The degree of uncertainty in the data in the original question will affect our confidence in the answer. In this problem of selling pigs, we usually think that the growth rate of pigs g is more reliable than the rate of price decline r. If we observe the growth of pigs or other similar animals in the past, a 25% error in g would be very unusual, but it is not surprising that an estimate of r has a 25% error.
Therefore, sensitivity analysis is also to make the model you build more convincing. If the core parameters in the model you build are only good for solving the problem at a certain point, other values will not work. Then the universality of this model is not strong.
Case Study-First Prize of 2020 National Championship C
The following is the paper of the first prize of the first prize of the 2020 national competition C problem. In the fifth part, the sensitivity analysis is carried out. When calculating the loan strategy, the bank’s annual loan is limited in advance (because each bank’s loan The amount is often impossible to calculate accurately), but will the change in the total annual loan have an impact on the credit strategy? In the sensitivity analysis part, the author has selected different total credits to measure the effect of credit strategy (based on the bank's annual profit). The results show that the change in bank's annual profit has no obvious relationship with its annual credit total, so the sensitivity of the model can be explained. Lower, stronger stability.
During the modeling process, if the result is found to change with the change of the parameter, and the magnitude of the change is large, it means that the changed parameter has a greater impact on the model, and it is not easy to treat it as a constant. If the change is small, it means the parameter The impact on the results is small and can be ignored separately.
So in modeling, what models or problems will make us do sensitivity analysis? In fact, sensitivity analysis itself is not specific to any model. It is aimed at model assumptions. If there are certain parameters in your assumptions that are limited due to insufficient observation conditions, difficulty in data acquisition, etc., sensitivity analysis is often required , And use different values for the parameter to determine the degree of change in the result.
Sensitivity analysis is common in optimization or prediction problems . In optimization, the parameters in front of the decision variables are often directly valued, such as the price of freight, the cost of materials, etc., but in fact, factors such as price cost are subject to market fluctuations, so it is necessary Carry out stability tests; prediction problems often limit relevant indicators due to the uncertainty of the future situation, such as mortality, immigration rate, etc., which require sensitivity analysis.
Sensitivity analysis-excellent paper
In order to allow everyone to learn sensitivity analysis better, the Scientific Research Exchange Studio found 10 excellent papers for the first prize of the national competition and the O prize of the Meisai. There are content about the sensitivity analysis. You can take a look.
[Link: https://pan.baidu.com/s/1KhLy9j_HQKZhNQP9GT43bA extraction code: zjxs]
