1, clear the problem
The problems of mathematical modeling are usually practical problems in various fields, and these problems are often vague, so it is difficult to directly find the key points and clearly put forward what methods to use. Therefore, the primary task of establishing the model is to identify the problem, analyze the related conditions and problems, make the problem as simple as possible at first, and then improve it step by step according to the purpose and requirements.
2. Reasonable assumptions
Making reasonable assumptions is the key step of modeling. A practical problem can hardly be directly transformed into a mathematical problem without simplification and assumption, even though it may be too complicated to be solved. Therefore, the problem should be simplified reasonably according to the characteristics of the object and the purpose of modeling.
The function of reasonable assumption is not only to simplify the problem, but also to limit the application scope of the model.
The basis of making assumptions is usually the understanding of the inherent laws of the problem, or the analysis of data or phenomena, or the combination of the two. When putting forward a hypothesis, we should not only use the professional knowledge related to the problem, such as physics, chemistry, biology, economy, machinery, etc., but also give full play to our imagination, insight and judgment, identify the priorities of the problem and simplify the problem as much as possible.
In order to ensure the rationality of the hypothesis, in the case of data, we should test the hypothesis and the inference of the hypothesis, and pay attention to the implied hypothesis.
Step 3 build a model
Modeling is to establish the relationship between variables according to the basic principles or laws of practical problems.
To describe the change of one variable with another, the easiest way is to draw a picture, or draw a table, or use a mathematical expression. In modeling, one form is usually transformed into another. It is easy to convert mathematical expressions into graphs and tables, and vice versa.
Through the combination of some simple typical functions, various function forms can be formed. Using functions to solve specific practical problems, it is better to give the values of various parameters and seek practical explanations of these parameters, which can often grasp some essential characteristics of the problem.
4. Solve the model
The solution of the model often involves the professional knowledge of different disciplines. The development of modern computer science provides powerful auxiliary tools, and many software packages and simulation tools can be used for engineering numerical calculation and mathematical derivation. Mastering the simulation tools of mathematical modeling can greatly enhance the modeling ability.
Different mathematical models have different difficulties in solving. Generally speaking, many practical problems cannot be solved by analytical methods. Therefore, it is necessary to use computer to solve the problem numerically. Before coding, it is necessary to make clear the algorithm and calculation steps, and understand the influence of initial value, step size and other factors on the results.
5, analysis and inspection
After finding the solution of the model, it is necessary to analyze the model and "solution", what is the scope of application of the model solution, how stable and reliable the model is, whether it achieves the modeling purpose and solves the problem?
Compared with the objective reality, the mathematical model will inevitably bring some errors. On the one hand, it is necessary to determine the allowable range of errors according to the purpose of modeling, on the other hand, it is necessary to analyze the sources of errors and find ways to reduce them.
Common errors come from the following sources and need to be carefully analyzed and tested:
Model hypothesis error: Generally speaking, it is difficult for the model to fully reflect the objective reality, so different assumptions need to be made. When analyzing the model, it is necessary to carefully test these assumptions and analyze and compare the effects of different assumptions on the results.
Error of approximate solution: generally speaking, it is difficult to get the analytical solution of the model, and the numerical calculation method itself will have errors when solving it by numerical method. Many of these errors can be controlled.
Rounding error of calculation tools: When using a calculator or computer for numerical calculation, rounding error is inevitable due to the limited word length of the machine. If a large number of operations are carried out, the accumulation of these errors can not be ignored.
Measurement error of data: when obtaining data through sensors, questionnaires and other methods, we should pay attention to the error of the data itself.
6. Model interpretation
The final stage of mathematical modeling is to translate the model into real-world language, which is very important for users to deeply understand the results of the model. Whether the model reconciliation has practical significance and is consistent with the actual evidence. This step is a key step to make the mathematical model have practical value.
7. Model application
Flexible application of mathematical model in practical application.
Characteristics of analytic hierarchy process
(1) hierarchy weight decision analysis
② Less quantitative information
③ Multi-objective, multi-criteria or no structural features.
④ It is suitable for complex systems that are difficult to quantify completely.