First, master the basic mathematical knowledge necessary for modeling (such as first-class mathematics, advanced mathematics, etc.). ), and the methods commonly used in mathematical modeling but not yet learned, such as graph theory method, optimization, probability statistics, and several methods in operational research.
Second, according to the characteristics of modeling, combined with typical modeling problems, this paper focuses on the use and general development of some practical mathematical software (such as Mathematica, Matlab, Lindo, Lingo, SPSS), paying special attention to the problem that the same mathematical model can be solved by multiple softwares.
For example, the problem of buying a house by loan: someone borrows 80,000 yuan to buy a house, and the monthly repayment is 880.87 yuan, with a monthly interest rate of 1%.
(1) The loan has been repaid for six years. After six years of repayment, someone wants to know how much he still owes the bank. Please tell him.
(2) This person has forgotten how many years the loan term is. Please tell him.
We can use Mathematica, Matlab, Lindo, Lingo and other different software packages to solve this problem.
2. The process and method of modeling
Mathematical modeling is a very creative and challenging activity, and it is impossible to use some rules to stipulate how to build various models. But generally speaking, modeling mainly involves two aspects: one is to transform practical problems into theoretical models; Secondly, the theoretical model is calculated and analyzed. In short, it is the process of establishing mathematical models to solve various practical problems. This process can be represented by the following figure 1
3 design of common algorithms
Modeling and calculation are the two cores of mathematical model. When the model is established, calculation becomes the key factor to solve the problem, and the quality of the algorithm will directly affect the speed of operation. According to the characteristics of the competition questions and the experience of previous winners, it is suggested that you use mathematical software (Mathematica, MATLAB, Maple, Lindo, Lingo, SPSS, etc. ) to design the algorithm. Here are some commonly used mathematical modeling algorithms.
(1) Monte Carlo algorithm (this algorithm, also known as random simulation algorithm, is an algorithm to solve problems through computer simulation, and at the same time, it can test the correctness of its own model through simulation, which is an essential method in the competition, and is usually realized by Mathematica and Matlab software).
(2) Data processing algorithms such as data fitting, parameter estimation and interpolation (a large amount of data usually needs to be processed in the competition, and the key to data processing lies in these algorithms, and Matlab is usually used as a tool).
(3) Linear programming, integer programming, multivariate programming, quadratic programming and other planning problems (most of the problems in modeling competitions belong to optimization problems, and many times these problems can be described by mathematical programming algorithms, usually realized by Lindo and Lingo software).
(4) Graph theory algorithm (this kind of algorithm can be divided into many kinds, including shortest path algorithm, network flow algorithm, bipartite graph algorithm, etc. Problems related to graph theory can be solved by these methods, which require careful preparation, usually using Mathematica and Maple as tools).
(5) Computer algorithms such as dynamic programming, backtracking, divide-and-conquer algorithm and branch-and-bound algorithm (these algorithms are commonly used in algorithm design and can be used in competitions on many occasions, and are generally realized by Lingo software).
(6) Image processing algorithm (there is a kind of problem related to graphics in the competition, even if it has nothing to do with graphics, there should be many pictures in the paper. How to display and process these graphics is a problem that needs to be solved, generally using Matlab).
(7) Three non-classical algorithms of optimization theory: simulated annealing method, neural network and genetic algorithm (these problems are used to solve some difficult optimization problems, which are very helpful to some problems, but they are difficult to implement and need to be used carefully, generally using Lingo, Matlab and SPSS software).
4 paper structure, writing characteristics and requirements
The answer sheet (paper) is the written form of the crystallization of the results of the competition, and it is the only basis for evaluating the performance, level and award-winning level of the competition. Therefore, it is particularly important to write a good paper on mathematical modeling in competition activities, which is also something that students must master. In order to make students better master the essentials of writing competition papers, (1) requires students to seriously study and master the latest paper format requirements formulated by the Organizing Committee of the National Mathematical Modeling Competition for College Students, and read more scientific and technological documents. (2) By analyzing the excellent papers of previous modeling competitions (such as the paper of China People's Liberation Army School of Information Engineering and Zhao, who won the national first prize in 2004: MS network design scheme around Olympic venues as an example), the general structure and writing points of modeling papers are summarized for learning and discussion.
Problems needing attention in participating in the national mathematical modeling competition for college students
First, you should have a "bottom" in your heart.
First of all, it is really difficult to predict which actual territory the title comes from, but it is definitely not too "specialized". After all, it has been simplified. Most competition questions can only be understood by consciousness, and the actual background of a few competition questions may be unfamiliar. You only need to look up some information to understand the meaning of the question. Secondly, of course, all the competition questions need to use mathematical knowledge, but they must not be too advanced. Some theories and methods of operational research, probability statistics, calculation methods, discrete mathematics, differential equations, etc. Be widely used. These contents are all to be learned before the pre-competition training, and they are really used. I always know what to look for.