First of all, pay full attention to the writing of abstract.
It plays an important role in the whole mathematical model paper and is the first impression of the judges on your paper. In the national mathematical modeling competition for college students, the organizing Committee put forward special requirements for the abstract of the paper, and repeatedly reminded the participants to pay attention to the writing of the abstract. In the evaluation of papers, the abstract is the decisive factor for your paper to get a good ranking, and the judges will decide whether to continue reading your paper through your abstract. In other words, even if your paper is well written in other aspects and the abstract is not well written, your paper will not be taken seriously or there will be no judges to see your paper at all.
Six aspects must be highlighted: problems, methods, models, algorithms, conclusions and characteristics. In short, the summary should reflect what methods you used, what problems you solved and what conclusions you reached. To avoid subjective comments, we must highlight the key points, so that people can know at a glance what the purpose of this paper is, what work has been done, what methods have been used, what achievements have been made, and what innovations and characteristics are there. Only such a summary can be successful.
The time for writing the abstract is generally arranged after the paper is basically completed, and a team member is responsible for it. After writing the first draft, other team members read it in turn and put forward suggestions for revision until everyone is satisfied.
A good summary contains two common characteristics: simplicity and clarity. The length is less than one page.
Example 1: optimization model of bus dispatching scheme
abstract
In this paper, an optimization model of bus dispatching scheme is established, which enables bus companies to give an ideal departure schedule and the least number of vehicles on the premise of meeting certain social benefits and obtain the maximum economic benefits. And provide better suggestions on collecting operational data.
In model ⅰ, a model for solving the 1 problem is established, such as the maximum passenger capacity, bus times, departure time interval, etc., and the maximum passenger capacity in each time period is given by using the decision-making method. Then, compared with the maximum passenger capacity of the vehicle, the minimum number of passengers in this time group is 462. Considering the convenience of operation and departure density, the timetable of the whole distribution vehicle and the minimum number of vehicles required are given. The fuzzy analysis model is established in the second model, and combined with the analytic hierarchy process, it is found that the daily satisfaction brought by the first model to the company and passengers is (0.94 1, 0.8 1 1). According to the range and degree of satisfaction of both parties, it is found that the optimal daily satisfaction of both parties (0.8807 and 0.8807) is reached at the same time, and the result at this time is 474 times for 50 cars. Starting from * * *, considering the least number of vehicles, the result is 484 times and 45 vehicles.
For question 2, the comprehensive benefit target model and linear programming method are explained.
For question 3, the collection method is to follow the law that the front door enters and the middle door exits. Two automatic recorders are used to record the number of people getting on and off and the automatic station announcer (adding time information) gives accurate data, which are stored in the company's general dispatching room according to the date after returning to the station.
Keywords: fuzzy optimization method of bus dispatching, analytic hierarchy process, satisfaction degree
Example 2: Optimal Decision of Lottery Issuance Scheme
abstract
At present, lottery has developed rapidly and healthily in China, which has made great contributions to the development of China's welfare undertakings. Aiming at various popular lottery issuance schemes at present, this paper comprehensively analyzes the influence of various factors such as the possibility of awards, the setting and amount of awards, and the attractiveness to lottery players on each scheme, and establishes three models.
Model 1: Based on the hypergeometric distribution principle, the first prize expectation model is established. According to this model, the traditional lottery scheme and the lottery scheme are obtained (that is, the design is reasonable; Generally speaking, the expectation of the first prize in the lottery scheme is the greatest and the scheme design is the most reasonable.
Model 2: A multi-objective decision-making model of winning probability, total winning probability, award setting and bonus distribution of high-grade awards is established by comprehensively considering various factors affecting the rationality of the scheme. The solutions are as follows: ① The weighted objective function value of scheme 19 is the largest, which is the most reasonable scheme among all schemes; ② In the "traditional" lottery scheme 1 ~ 4, scheme 4 is more reasonable; ③ The weighted objective function value of traditional lottery scheme (1 ~ 4) is generally smaller than that of lottery scheme (5 ~ 29). Generally speaking, the lottery scheme is relatively superior to the traditional scheme. (4) For the (selected) schemes, when they are the same, their rationality decreases in turn at 35, 30, 32, 33 and 34.
Model 3: Considering the relationship between supply and demand in the lottery market, combined with the satisfaction of lottery management departments and lottery players, a multi-objective optimal decision-making model is established. Through the supply and demand trend of the lottery market with sales, we can find the equilibrium point, and at the same time, through computer programming, we can find a better lottery distribution scheme.
This paper also accurately analyzes the sensitivity of the model from the change, and further discusses the formulation of lottery issuance plan from the aspects of changing single note to multiple notes and appropriately increasing the total prize amount.
Finally, according to the model, more active and practical suggestions are put forward for the lottery management department. And from the full understanding of lottery, the motivation and mentality of entering the market, strategies and skills, etc., it puts forward scientific reference opinions for lottery players to contact and vote for lottery tickets.
Keywords: probability expectation multi-objective decision-making hypergeometric distribution satisfaction
Example 3: Mathematical Model of Temporary MS Supermarket Store Design for Olympic Games
abstract
Based on the statistical analysis of the survey data, the percentage and distribution law of people flow in each business circle are found out, and then MS outlets are designed, three models are established and simulated.
Regarding the first question, it is found that people of different ages have great differences in travel, catering and consumption. Therefore, according to age and gender, eight laws of audience probability distribution are summarized from three aspects: tourism, catering and consumption.
On the second question, using the principle of BP neural network, the network is divided into three levels according to age-gender-business circle-import and export, and the chain analysis is carried out from two aspects: dining habits and entering and leaving venues, and the passenger flow model under the shortest path of each venue is established. Through programming, the distribution (%) of people flow in 20 business districts is obtained: from A 1 to A 10, and the business districts are1/0.887, 7.62 1, 8.540,10.30 respectively. B 1 to B6 business district are 1 1.686, 13.932, 18.760,1.686 and/kloc-respectively. C 1 to C4 business district are 18.75, 20.9843, 18.75, 410.5157 respectively. After calculating the distribution of people flow, the symmetry theorem is summarized, that is, the people flow is obliquely symmetrical with the connecting line of the entrance and exit of the venue as the axis, and the detailed proof is given.
In the third question, the related factors of audience's purchasing desire are analyzed in detail, the mathematical expressions of purchasing desire, age and consumption amount are established, the desire matrix is obtained, and the purchasing power is calculated by fuzzy method. Then, according to the two basic constraints of meeting the shopping demand and the basic balance of distribution during the Olympic Games, a mathematical expression is established, and a nonlinear multi-objective decision-making model with profit as the objective function is established:
With Lingo programming, a reference design scheme of MS outlets is obtained: the number of large MS in the business circle from A 1 to A 10 is 3, 1, 0, 1, 3, 1, 0 1, and small MS respectively. From B 1 to B6 business district, the number of big MS is 1, 2,3, 1, 2,3, and the number of small MS is 2, 1, 1, 2, 1. From C 1 to C4 business circle, the large MS numbers are 2, 4, 2 and 1 respectively, and the small MS numbers are 2, 0, 2 and 1 respectively.
Considering the arrangement of the Olympic Games schedule, the actual flow, consumption and profit will change with time. In order to further optimize the network design scheme, according to the principle of system dynamics, the traffic and profit model is systematically simulated by computer based on Venple5.3 technology, and the model is tested and evaluated by mode adjustment, thus verifying the rationality, scientificity and practicability of the model.
Finally, some suggestions are put forward for the economic income, tourism and hardware construction of Beijing 2008 Olympic Games.
Key words: probability, flow of people, symmetry, desire matrix, multi-objective decision-making system, dynamic system simulation.
Example 4: Comprehensive Evaluation and Predictive Control of Yangtze River Water Quality
abstract
Based on the statistical analysis of the survey data, the water quality of the Yangtze River in recent two years was comprehensively evaluated, the main areas of permanganate and ammonia nitrogen pollution sources were found out, the water pollution in the future 10 was predicted, the control scheme was put forward, and a series of scientific anti-pollution suggestions were given.
Firstly, the water quality of the main monitoring sections in the Yangtze River basin in recent two years 17 was sampled, and the statistics were made according to the order of temporal and spatial interaction, and a probabilistic statistical evaluation model was established. The results show that from 2003 to 2005, 85% of the Yangtze River sections met the requirements of Class I-III water quality, 12% belonged to Class IV and V water quality, and 3% was worse than Class V water quality. In recent two years, the local water quality of the Yangtze River has changed greatly, and the whole is relatively stable, but the high-quality water is declining and the water quality exceeding the standard is on the rise. In order to find the pollution source, we take seven sections of the main stream of the Yangtze River as the basic observation points, and establish the pollution source feedback index according to the water flow, water flow speed and degradation coefficient:
Through calculation, it is found that the permanganate pollution in Nanjing, Jiangsu Province and Yueyang, Hunan Province is the most serious, and Yueyang, Hunan Province is also the main area of ammonia nitrogen pollution, followed by Anqing, Anhui Province and Nanjing, Jiangsu Province, but there is a big difference compared with the same period last year.
Secondly, according to the principle of GM( 1, 1), the grey prediction model is established for the main statistical data in recent years. After normalization, the predicted value and trend function of water quality category are calculated by DPS mathematical statistics software. The analysis shows that the total water quality of the Yangtze River in categories I, II and III shows a downward trend, among which categories I and III show a downward trend. In order to effectively control the deterioration trend of pollution and prevent the water quality from rising beyond the standard, quadratic polynomial stepwise regression analysis is used to obtain the function of total wastewater discharge to various water quality percentages. After planning and operation, we put forward the Yangtze River sewage treatment scheme. The sewage treatment capacity in the coming 10 year is 0, 0, 2.66, 5. 14, 5.76, 8.2 1, 10.86, 13.5438+0,/.
Finally, based on the comprehensive evaluation of the water quality of the Yangtze River and the prediction of the future pollution trend, and according to the actual investigation of the "Protecting the Yangtze River" investigation team, we deeply realize that the water ecological environment in the Yangtze River basin is being destroyed day by day, and the prospect is not optimistic. In order to prevent the "canceration" of the Yangtze River, we put forward several concepts of water environmental protection: education first, efforts to arouse people's awareness of environmental protection; Insist on water control according to law and legislate for the protection of the Yangtze River; Implementing scientific planning and taking the road of sustainable development; Advocate humanistic environmental protection and build a harmonious ecosystem and human settlement environment.
Keyword monitoring part; Probability and statistical evaluation; Pollution source feedback; Grey prediction; Gradual regression; Humanism and environmental protection;
Second, the main body of the paper should be distinct and the structure should be complete.
According to the characteristics of the mathematical model paper, the main part of the paper includes the following contents:
(1) problem proposition-clear problem
I won't explain this part much. Generally, it is enough to copy the original competition questions directly, but I think it can be summarized appropriately when there is enough time. So you can write some background knowledge about this problem.
Defining the problem is the preparatory stage of modeling. To establish a mathematical model of real problems, we must first have a clear expression of the problems to be solved. Usually, a practical problem we encounter is vague in the initial stage and has a practical background. Therefore, before modeling, we must conduct a comprehensive and in-depth understanding and investigation of the problem, consult relevant literature, and start collecting relevant data. When collecting data, we should test the arrangement of data in advance, such as using tables or charts. In the meantime, the existing data and conditions should be carefully analyzed to further clarify the problem. What information did you get from the data? Is the data source reliable? What is the significance of the given conditions? What conditions are essential? Those conditions are variable, and so on. The analysis of data and conditions will further enhance our understanding of the problem, so as to better grasp the essence and characteristics of the problem and lay a good foundation for the next modeling.
(2) Model assumption-reasonable assumption
As the prototype of the topic, it is complex and concrete, and it is the unity of quality and quantity, phenomenon and essence, accident and necessity. If such a prototype is not abstracted and simplified, it is difficult for people to understand it and grasp its essential attributes, and the modeling hypothesis is to abstract and simplify the model according to the purpose of modeling. Abstract the forms, quantities and their relationships that reflect the essential attributes of the problem, simplify those non-essential factors, get rid of the concrete and complicated forms of the prototype, and form useful information resources and preconditions for modeling.
However, how to make reasonable assumptions about the problem is a difficult problem. This is because if it is too simple, the model will be far from reality and cannot be used to solve real problems. If it is too detailed, trying to take all factors into account, the model will be very complicated, even difficult to establish, and it will also complicate our calculation. The general model assumes the following principles:
(1) purpose principle, abstract the factors related to modeling purpose from the prototype, and simplify the irrelevant factors or irrelevant factors.
(2) Simplicity principle, the assumptions given should be simple and accurate, which is conducive to the construction of the model.
(3) the principle of authenticity, assuming that the terms should be reasonable and the errors caused by simplification should be within the allowable range of practical problems.
(4) The principle of comprehensiveness, while making assumptions about the prototype itself, also gives the environmental conditions in which the prototype is located.
The simplest method: Hypothetical conditions can generally be mined from the topic.
(1) Make assumptions according to the conditions in the topic.
(2) Make assumptions according to the requirements in the topic.
It should be pointed out that:
① The factors that have no influence (or little influence) on the problems we solve but can simplify the model should be reflected in the assumptions.
(2) not to simplify the problem and a large number of assumptions (make the solution of the problem itself inconsistent with the original intention), we should pay attention to the number and degree of assumptions.
(3) Symbolic interpretation-indispensable.
There are bound to be a large number of mathematical symbols in your paper, so you should make a brief explanation of these symbols in this part, which can be explained from the aspects of symbols, types (variables, constants), units, meanings, etc. (as shown in the following table):
sign
type
unit
meaning
It should be noted that the unit quantity outline is unified, and the meaning explanation should be accurate and clear.
(4) problem analysis-clear thinking, illustrated.
From topic to model is a thinking process from concrete to abstract, and this part is the embodiment of this process. This part should be a highlight of the main body of the paper. It is suggested that while explaining the text, list the thinking process with figures or charts to make your thoughts clear and clear. In addition, this part should make an overall analysis of the topic, make full use of the information and conditions in the topic, and determine what method to establish what model. Experience tells us that we can get some preliminary judgments of the problem from the topic: for example, we can get the maximum output in extreme cases and spend the shortest time. In this way, the final solution we get cannot exceed (or be lower than) the amount we analyze here. This part should reflect the embryonic form of our solution to the original problem. In short, the role of problem analysis in the whole paper lies in connecting the preceding with the following, and it can also reflect the comprehensive level of the contestants.
(5) Modeling-Mathematical Language
Mathematical models include: mathematical formulas, charts, schemes, etc.
The establishment of the model is to abstract the original problem into the expression of mathematical language, and its establishment method will vary with the understanding and emphasis of the problem. In recent years, there are two main directions in mathematical modeling competition: one is probability statistics; The first is the problem of operational optimization. Therefore, it is very important to master the above two aspects of knowledge for establishing the model.
In addition, I also think that we should pay attention to the clear explanation of each model formula, and the mathematical symbols in it must be consistent with the previous explanation.
The basic method is:
On the basis of modeling hypothesis, this paper further analyzes the conditions of modeling hypothesis. First, it distinguishes between constants, variables and known and unknown. Then, find out the position, function and relationship of various quantities, choose appropriate mathematical tools and methods to construct models to express them, and construct mathematical models to describe practical problems.
Here we should pay attention to two points: first, to build a model of a specific problem is to be as simple as possible, and then compare it with the actual problem, plus the next important factors, and gradually approach the reality to modify the model to make it perfect, so as to form a mathematical model that gradually pushes the reality. Second, we should be good at learning from existing mathematical models. Although there are different phenomena and backgrounds, many practical problems have the same model. For example, Newton's second law describing the relationship between force, mass and acceleration in mechanics, F= M a, and C= p q describing the relationship between unit price, sales volume and sales volume in economics. The mathematical models are all Y = K X, so we should learn to observe and analyze, see the essence of the problem, grasp the essential characteristics and correct our existing models.
(6). Model solving-software help
Different models need different mathematical tools to solve, such as solving equations, drawing, proving theorems, logical operations, numerical operations and other traditional and modern mathematical methods. With the development of modern modeling, most models are usually solved by software programming. You should be familiar with at least one of the three major softwares (Matlab, Maple, Mathematic) and learn some special softwares. For example, DPS, SAS and SPSS are used to solve probability and statistical problems; Argot, Lindo, etc. Who will solve the problem of operation optimization? Skillful use of these mathematical software will bring us quick and convenient answers. Secondly, try to solve it in different ways, which can not only reflect your open mind, but also indirectly verify the correctness of your solution. In addition, some brief steps of the main algorithm, methods to deal with or simplify problems, and appropriate application of tables or images.
Finally, I need to remind you that you can give mathematical proof when necessary, which will add a lot of color to your paper.
(7). Model (result analysis)-Test and correction
The purpose of establishing mathematical model is to solve practical problems. Therefore, the results of the model must return to the actual problem. If the result of the model is consistent with the actual problem, it means that the model is consistent with the actual problem after testing, otherwise it will not work and cannot be directly applied to the actual problem. At this time, if there is no problem in the establishment of the mathematical model, it is necessary to consider whether the assumptions made during the modeling are reasonable and check whether the factors that should not be ignored or should not be kept are ignored. Make necessary amendments to the assumptions and repeat the previous modeling process until the model can reflect the given practical problems.
The usual practice is:
Because some secondary factors affecting the problem are ignored in the model assumption, this simplifies the problem more or less, but it will inevitably produce some errors; In addition, there are many ways to solve problems, and only one or two of them may be used in the paper, so the ideas may be limited; And the pattern itself will have its advantages and disadvantages. Therefore, what we should do in this part mainly includes the following three points:
A. whether it can be solved in other ways or methods.
B. analysis of the advantages and disadvantages of the model.
C. error analysis or sensitivity analysis of the model.
Doing the above work well is not only a supplementary explanation to the original question, but also a rigorous thinking and logic, so that your paper can be completed in one go.
(8) Evaluation and popularization of the model
What kind of mathematical model is good? Generally speaking, a good model should have the following five points:
(1) gives a comprehensive consideration to the given problem. In an experimental problem, there are often many factors acting on the studied object at the same time, so these factors should be considered comprehensively in mathematical description. This work can be divided into three steps:
(1) List various factors;
② Select the main factors to be included in the model;
③ Considering the influence of other factors, modify the model.
(2) The existing model is creatively improved. Mathematical model is the product of abstraction and idealization of real objects. It is not unique to the domain to which the object belongs and can be transferred to another domain. Modeling in ecological, economic, social and other fields often borrows models from physical fields. Whether the existing model can be creatively transformed is an important sign to consider the advantages and disadvantages of a mathematical model.
(3) Be good at grasping the essence of the problem and simplifying the relationship between variables. Mathematical model should be an essential description of practical problems. If the model is too complex, it can't be solved or difficult to solve, otherwise it can't objectively reflect the objective reality.
(4) Pay attention to the result analysis and consider its rationality in practice. Mathematical model is a process from reality to mathematics, and then from mathematics to practical problems. Because the current model only depends on the data in the problem, if the results obtained from the model are consistent with the reality, then the model is successful, otherwise it is a failure, which requires us to further modify.
(5) It has good stability. Mathematical model is established by relying on existing data and other information, and its value lies in being able to predict unknown things from known information. Therefore, the results of a good mathematical model have a good dependence on the original data, that is, small changes in the original data and parameters will not cause great changes in the results, ensuring the adaptability and effectiveness of the model.
Due to the limitations of the paper itself, some problems can be discussed in depth here, which is another highlight of the article, and the stronger team can give full play to it. The role of this part in the whole paper is to make the finishing point. In addition, we have discussed and extended the problem in many aspects: we can relax the assumptions to consider the problem appropriately; You can improve your algorithm and so on, but I think qualitative analysis is enough here. Finally, it is mainly the horizontal and vertical divergence problem. Because the judges' evaluation work has basically ended.
9. References
Pay attention to the format here. Access conditions are clearly defined:
A the description of the book is: [No.] author, title, place of publication: publishing house, year of publication.
B. The expression of periodical papers in the references is: [No.] author, paper name, magazine name, issue number, page number and publication year.
C. The expressions of network resources in references are: [No.] author, resource title, website address and access time.
As for the appendix, it is enough to attach the relevant program and operation results and mathematical proof.
Finally, pay attention to the overall sense of the paper, especially whether the text expression is accurate and rigorous.
Third, write programs with general mathematical software.
When writing computer programs, the basic principle is to use common and familiar software, so that the results can be produced as soon as possible, and even mistakes can be found and corrected quickly. The general mathematical software is based on certain theoretical basis and algorithm, and its calculation results have certain credibility. Therefore, the programs written by matlab, mathematicas, lindo, lingo and other mathematical software can increase the credibility of the model results. In addition, some secondary development programs can also be used. Such as TSP, EXCEL, DPS and so on.
Fourth, be good at using charts rationally.
When writing a paper, you must pay attention to the use of charts as much as possible. Using charts is clearer and more direct than using words. Charts can usually replace a long boring text, and illustrations can also add color to paper. You know, most of the judges are old professors and experts. In order to teach experts' eyes and reduce their writing pain, it is definitely a good choice to use more charts. It should be noted that the reference of the chart should be standardized, and attention should be paid not to put it in the wrong place when cross-referencing. To this end, each chart should be numbered, and the number of the whole chart should be continuous. As far as possible, the charts appear alternately in the paper, and they should also be in the middle of the page when typesetting, so as to avoid appearing at the top, which can increase the visual beauty of the article.
Fifth, give full play to the role of the team.
In the competition, the cooperation between the players is very important. Everyone should have a clear and unified understanding of the strengths of their own group, what kind of problems they are good at and what kind of problems they are not good at. This will not delay too much time when choosing a topic.
The principle of division of labor:
Modeling: Deriving mathematical model, with strong mathematical ability.
Programming: Strong computer skills.
Thesis writing: Strong writing ability.
Secondly, there should be a core player in the team, and his role is equivalent to the CPU in the computer. If the core players play well, they can drive a team to work normally and effectively. Whether it is topic selection, discussion, writing, coordination or even emotion, the core players should give full play to their leading role, so that the whole team can complete the game confidently and efficiently, otherwise it may lead to team depression, lack of confidence and even give up all previous efforts.
Sixth, reasonably control the writing progress.
To do anything, a reasonable time schedule is very important, and so is modeling. A plan should be made in advance. The paper is divided into ten parts: abstract, problem raising, model hypothesis, problem analysis, model hypothesis, model establishment, model solution, result analysis, model evaluation and popularization, references and appendices. Generally, it is necessary to determine which sections of work our team members need to complete every day, so as to make the work get cold feet, ensure that the paper writing can be completed within the specified time, and avoid the passive situation that the paper can't be completed because the time runs out and the task is not completed.
The usual game schedule:
The first day: morning: determine the topic and consult the literature.
Afternoon: Start analysis and establish a preliminary model.
Evening: programming, get the preliminary calculation result of 65438+ rest at 02: 00 pm.
The next day: morning: get the reasonable result of the first model.
Afternoon: Start writing a paper and consider improving the first model.
Evening: Get the preliminary results of the second model. Rest at 0/2: 00 in the afternoon/kloc.
The third day: morning: get the reasonable result of the second model.
Afternoon: Consider further optimizing the first two models and get the third mathematical model.
Or verify the correctness of the first two models.
Evening: Get the final grade and finish the whole paper.