Mathematical Modeling Study Notes

Understanding Mathematical Modeling

What is Mathematical Modeling?

1. What is a Mathematical Model?

A mathematical model is a mathematical structure that represents the characteristics or quantitative dependency relationships of a system, using mathematical language to abstractly or approximately describe these characteristics. This structure captures the pure relational structure of a system through mathematical symbols. In a broad sense, a mathematical model includes various concepts, formulas, and theories in mathematics.

2. Mathematical Modeling

Mathematical modeling can be viewed as the process of transforming a problem definition into a mathematical model. Simply put, to solve various real-life problems using the mathematical knowledge we’ve learned, we need to build relevant models and use mathematics as a tool to address these practical issues. This is the core of modeling.

3. The Concept of Mathematical Modeling

Mathematical Modeling Competitions

1. National Undergraduate Mathematical Contest in Modeling (CUMCM)

Participants: Undergraduate students
Time: Every September
Introduction: The “Higher Education Press Cup” is currently the most influential and recognized mathematical modeling competition, commonly known as the “National Contest.” In 2020, a total of 45,680 teams (41,826 undergraduate teams and 3,854 junior college teams) from 1,470 institutions and campuses in China, the United States, the United Kingdom, and Malaysia, with more than 130,000 participants, registered for the competition. At some universities, the national competition is highly valued, and national awards are extremely prestigious.
Official Website: National Undergraduate Mathematical Contest in Modeling

2. Mathematical Contest in Modeling (MCM/ICM)

Participants: Undergraduate students
Time: Around February each year
Introduction: The Mathematical Contest in Modeling (MCM/ICM) is organized by the Consortium for Mathematics and Its Applications (COMAP). It is the only international mathematical modeling competition and is one of the most influential modeling competitions worldwide. The competition topics cover various fields, including economics, management, environment, resources, ecology, medicine, and security.
Official Website: Mathematical Contest in Modeling

3. Shenzhen Cup Mathematical Modeling Challenge

Participants: Undergraduate and graduate students
Time: Problems are released in mid-April each year, and the submission deadline is around September (there is no fixed time frame, which differs slightly from other competitions).
Introduction: The Shenzhen Cup is a prestigious competition, and due to the slightly higher difficulty of the problems, it requires stronger abilities from the participants.
Official Website: Shenzhen Cup Mathematical Modeling Challenge

How to Prepare for Mathematical Modeling

(1) Main Modeling Team Member

The main modeling team member should first have a general understanding of various commonly used models, including recognizing and understanding different types of models. You should have a certain level of familiarity with common models such as elementary models, optimization models, mathematical programming models, differential equation models, and difference equation models. Since modeling is a highly complex process, it’s impossible to master everything, but having a broad understanding of various models is crucial. When needed, the ability to self-learn by referring to materials is most beneficial.

(2) Main Programming Team Member

The programming team member is equally important. No matter how good the model is, if it cannot be solved, it becomes awkward.
Common tools used in mathematical modeling include MATLAB, Mathematica, SPSS, Lingo, and Lindo. These software programs focus on data processing, analysis, and solving various models. These tools are quite convenient and widely used in mathematical modeling.
Additionally, programming languages like C/C++ and Python can also be used for solving models. Regardless of which tool is used, having good programming skills is essential for solving models.

(3) Main Paper Writing Team Member

The main paper writing team member must have good writing skills, and also a certain level of expertise in formatting and beautifying the paper. Currently, various mathematical modeling competitions allow papers to be written using Office or WPS.
There are also other tools available for writing papers, such as LaTeX, which is similar to a programming-based paper writing tool. Those interested can explore it further. However, since learning LaTeX takes a significant amount of time, which could otherwise be spent on learning models and algorithms, it is not highly recommended to specialize in it.