2016年美国大学生数学建模大赛A题获奖论文A Hot Bath
2016年美国大学生数学建模竞赛C题H奖论文
Contents1.Introduction (1)1.1 Background (1)1.2 Foundation & ROI (1)2 Task (1)3 Fundamental assumptions (2)4 Definitions and Notations (2)5 Models (3)5.1 Filter data (3)5.2 Object Selection Model (Grey Relational Analysis) (4)5.2.1 Model analysis (4)5.2.2 Model solution (4)5.3 ROI Model (Principal Component Analysis) (5)5.3.1 Model analysis (5)5.3.2 Model solution (6)5.4 Verify the possibility (9)5.4.1 Comparison (9)5.4.2 External factor (10)5.5 Investment Forecast Model (11)5.5.1 Linear Regression Forecasting Model (11)5.5.2 School potential Prediction (TOPSIS) (12)5.5.3 Final investment (TOPSIS) (13)6 Conclusions (16)7 Strengths and Weaknesses (18)7.1 Strengths (19)7.2 Weaknesses (20)8 Letter to Mr. Alpha Chiang (21)9 References (22)Team # 44952 Page 1 of 221 Introduction1.1 BackgroundThe Goodgrant Foundation is a charitable organization that wants to help improve educational performance of undergraduates attending colleges and universities in the United States. To do this, the foundation intends to donate a total of $100,000,000 (US100 million) to an appropriate group of schools per year, for five years, starting July 2016. In doing so, they do not want to duplicate the investments and focus of other large grant organizations such as the Gates Foundation and Lumina Foundation.Our team has been asked by the Goodgrant Foundation to develop a model to determine an optimal investment strategy that identifies the schools, the investment amount per school, the return on that investment, and the time duration that the organi zation’s money should be provided to have the highest likelihood of producing a strong positive effect on student performance. This strategy should contain a 1 to N optimized and prioritized candidate list of schools you are recommending for investment bas ed on each candidate school’s demonstrated potential for effective use of private funding, and an estimated return on investment (ROI) defined in a manner appropriate for a charitable organization such as the Goodgrant Foundation.1.2 Foundation & ROIFoundation (charitable foundation) refers to the nonprofit legal person who uses the property of the natural persons, legal persons or other organizations to engage in public welfare undertakings. In terms of its nature, foundation is a kind of folk non-profit organizations.ROI is a performance measure used to evaluate the efficiency of an investment or to compare the efficiency of a number of different investments. ROI measures the amount of return on an investment relative to the investment’s cost. To calculate ROI, the benefit (or return) of an investment is divided by the cost of the investment, and the result is expressed as a percentage or a ratio.2 Task●One-page summary for our MCM submission●Using our models to achieve the candidate list of schools●Calculate the time durati on that the organization’s money should be provided to have thehighest likelihood of producing a strong positive effect on student performance●Calculate the investment amount Goodgrant Foundation would pay for each school●Calculate the ROI of the Goodgrant Foundation●Forecast the development of this kind of investment mode●Write a letter to the CFO of the Goodgrant Foundation, Mr. Alpha Chiang, that describesthe optimal investment strategy。
2016年美国大学生数学建模E题英文版论文正稿
In this paper, a model is established to provide a measure of the ability of a region to provide clean water to meet the needs of its population, and find out the reason for the lack of water resources. Specific tasks are as follows:For Task 1: We establish a model. In the model, we think the supply of clean water depends on the amount of surface water, underground water and sewage purification. The water requirements are decided by the amount of life water, agricultural water and industrial water in the region. In water supply, surface water is affected by the annual average temperature, annual average precipitation and forest coverage rate. The groundwater is impacted by the annual average temperature, annual average precipitation. The agricultural water is affected by the population of the region and annual average precipitation. The GDP of the region influences the industrial water consumption. We use the principle of multivariate nonlinear regression to find out the regression coefficient. And then make sure its function. The ratio of water supply and water requirements is used as a measure of the ability of a region to provide clean water. We find that the ability of a region to provide clean water is good or not by comparing the ratio with 1.For Task 2: The model selects the Shandong Province of China as the testing region. We analyse the data of China's Shandong area between 2005 and 2014, and then crystallize the model through the thought of the function fitting and multivariate nonlinear regression. By the model, we think Shandong province's ability to provide clean water is weak. And then from two aspects which physical shortage and shortage of economic, this paper analyses the causes of water shortage in Shandong Province, and thus test the applicability of the model.For Task 3: We select several factors affecting water supply and water demand badly, which is annual precipitation, annual average temperature, the forest coverage rate and population forecast. We analyse the data of China's Shandong area between 2005 and 2014, according which to predict the changes of those factors in 15 years. After that this paper uses the model to analyse the situation of Shandong’s water in 15 years.For Task 4: According to the model in Task 1 and the analysis of the Task 2. We find the main factors influencing the ability to provide clean water in China's Shandong province. By these factors we make the intervention program. In view of the low average annual rainfall, increase the average annual rainfall by artificial rainfall. In view of the forest coverage rate, forest plantation and protect vegetation is came up with. For sewage purification capacity, putting forward to improve sewage treatment technology and improve the sewage conversion rate and increases daily sewage quantity. In view of the total population, we put forward the policy of family planning for water consumption per capita, putting forward to set the daily water consumption per person. And putting forward the industrial wastewater must reach the indexes of the rules, developing seawater desalination technology to increase the supply of clean water.Water has always been the hot spot in the world.The future is also not exceptional. Only finding out the problem, we can suit the remedy to the case.The model measure the ability of a region to provide clean water by analysing the cases which influence the supply and remand of water. Based on this, make a good intervention program. Offering helps to solve global water issues.1 Introduction (4)1.1 Problem Statement (4)1.2Problem Analysis (4)1.2.1Task 1 Analysis (4)1.2.2Task 2 Analysis (4)1.2.3Task 3 Analysis (5)1.2.4Task 4 Analysis (5)1.2.5Task 4 and 5 Analysis (5)2 Assumptions and Notations (6)2.1 Assumptions (6)2.2 Notations (6)3 Model Establishment and Solution (7)3.1 The effect of single factor on the water supply in a certain area (7)3.1.1Effects of annual average temperature, annual average precipitation andforest coverage on surface water resources in a certain area (7)3.1.2 Effects of annual average temperature and annual precipitation ongroundwater resources in a certain area (8)3.1.3 Influence of total population and per capita water consumption on dailywater consumption in a certain area (8)3.1.4 The influence of average annual rainfall and total population onagricultural water consumption in a certain area (9)3.1.5 Effect of average annual rainfall and population in an area of agriculturalwater use (9)3.2 Function Arrangement (9)3.2.1 Water supply function (9)3.2.2 Water demand function (10)3.2.3 The ability of a region to provide clean water (10)3.3 In order to test the accuracy and usability of the model, this model is selectedas a test area in Shandong Province, China. (10)3.3.1 Total surface water resources (11)3.3.2 Total groundwater resources (14)3.3.3 Total industrial water consumption function (16)3.3.4 Total agricultural water consumption function (17)3.3.5 Assessment of water supply capacity (18)3.4.2 Remediation Measures (19)3.5 Forecast for the next 15 years (20)3.5.1 Forecast of average annual rainfall (20)3.5.2 Prediction of annual temperature (21)3.5.3 Prediction of forest cover (22)3.5.4 Prediction of population (23)3.6 Intervention Program (24)3.6.1 Present ofthe Intervention Program (24)3.6.2 Implement ofthe Intervention Program (25)4 Advantages and Shortcoming of the model (26)4.1Advantages： (26)4.2 Shortcoming (26)5 Improvement of model (26)6 Reference (27)7 Appendices (28)7.1 Data used in task 2 (28)7.2 Matlab Source Code (30)1 Introduction1.1 Problem StatementOn the earth, the water that human beings can use the water directly or indirectly, is an important part of natural resources. At present, the total amount of the earth's water is about billion cubic meters, of which the ocean water is billion cubic meters, accounting for about 96.5% of the total global water. In the remaining water, the surface water accounts for 1.78%, 1.69% of the groundwater. The fresh water that human mainly use of is about billion cubic meters, accounting for only 2.53% in the global total water storage. Few of them is distributed in lakes, rivers, soil and underground water, and most of them are stored in the form of glaciers, permafrost and permafrost, The glacier water storage of about billion cubic meters, accounting for 69% of the world's total water, mostly stored in the Antarctic and Greenland, the available clean water in the dwindlingwith time going by.In order to assess the ability to provide clean water of an area, we set up an assessment model.1.2Problem Analysis1.2.1Task 1 AnalysisTask 1 requires establishing a model to measure the ability of a region to provide clean water. At the same time,we also need to provide a measure standard.This paper make the ratio of water supply and water requirements of a region as the measure standard, by which to measure the ability of a region to provide clean water.A region's main source of water is groundwater, surface water and sewagepurification.The model assumes the volume of groundwater in a region is mainly affected by average annual temperature, annual precipitation;Thevolume of surface water is mainly affected by the average annual temperature, annual precipitation, the forest coverage rate.These factors decide water supply of an area.The waterdemand of an area mainly includes living water, agriculture water and industrial water. We assume living water is affected by the population and per capita consumption decision;Agricultural waterdepends an annual precipitation and population decision;Industrial water is mainly decided by a gross regional product.The above factors decide the water demand in a certain area.1.2.2Task 2 AnalysisAccording to the information provided in the map, in Asia, China's Shandong Province is the region meeting the requirements.Through the data collection of Shandong Province, we can find the annual temperature, annual precipitation, the forest coverage rate, groundwater, surface water, sewage treatment capacity, water, agricultural water, industrial water, population,per capita consumption and GDP data.And then according to the model of Task 1,we analyze the ratio by using multivariate nonlinear regression to make sure that Shandong province is a water-deficient area.After proving that Shandon province is short of water through the model, we analyze the reasons for lack of water from two aspects: physical shortage and economic shortage.1.2.3Task 3 AnalysisBecause we already have the relevant data, we can function to fit the relationship between the variables and the year.Thus it is possible to predict in Shandong Provinc e’s data in the 15 years, then input the data into the model to achieve the purpose of prediction.In addition, it can be combined with the actual situation and the selected areas of the corresponding policy to analysis which factors will have a great change in the15 years.We still can analyze from two aspects of society and the environment.Socialaspects includes the promotion of water conservation, population growth;Environmental aspects includes policy changes to the environment, sewage purification capacity enhancement and so on.1.2.4Task 4 AnalysisFormulating plans for intervention mainly start from the perspective of the main model. According to the content of the model, we can still divide all of factors into two types: the social and environmental factors. The intervention programs can be developed based on two types of factors that affect the supply of water, reducing as much as possible the negative impact of the factors that control factors and intensifying the development of a positive impact. In addition, because Shandong Province is beside the sea, desalination and other measures can be developed to increase clean water supply sources.1.2.5Task 4 and 5 AnalysisTask 4 intervention programs indirectly impact the water supply and demand water through a direct impact on GDP model of forest cover, annual precipitation, annual temperature, water emissions, sewage treatment capacity, population growth and the region.2 Assumptions and Notations2.1 Assumptions●The water resources in a region are derived from the purification of surfacewater, groundwater and sewage, and the demand of water resources comes from domestic water, industrial water and agricultural water.●The surface water supply in a certain area is affected only by the average annualtemperature, annual precipitation and forest coverage. The groundwater supply is affected by the annual average temperature and annual precipitation.●The region's water consumption of a certain region depends on the populationand per capita water consumption; Agricultural water consumption is affected by the average annual precipitation and the numberof people. Industrial water is mainly determined by a regional GDP.● A certain region will not suddenly increase or decrease the population largely.●There will not be a serious natural disasters in a region in the next periodof time.2.2 Notations3 Model Establishment and SolutionThe model established here is a use of a region's water supply and water demand ratio to determine whether the water shortage in the region, the main variables involved.3.1 The effect of single factor on the water supply in a certain area3.1.1Effects of annual average temperature, annual average precipitation and forest coverage on surface water resources in a certain areaDue to the average annual temperature, annual precipitation, the forest coverage rate and surface water of linear or nonlinear relationship exists, so first in order to determine the average annual temperature, annual precipitation, the forest coverage rate and surface water, and then the nonlinear multiple regression analysis method to determine the functional relationship between the three factors and surface water.The surface water content is 1y , Average annual precipitation, annual average temperature and forest coverage rate are 1x ,2x ,3x , Using nonlinear regression statistical methods, the use of MATLAB fitting toolbox were identified 1x ,2x ,3x of the highest regression power(MATLAB fitting toolbox of the highest fitting function is the 9 power, greater than the 9 power function is too complex, not much research value), According to the decision coefficient R 2of the regression equation, the corresponding probability value of the statistic P, the regression coefficients β,0β,1n β,2n β, get the regression equation:35612412312345699999910123123111 (1)n n n n n n n n n n n n n n n y x x x x x x βββββ====++++∑∑∑∑∑∑3.1.2 Effects of annual average temperature and annual precipitation on groundwater resources in a certain areaThere is a linear or nonlinear relationship between the average annual temperature, average annual rainfall and the supply of groundwater, according to the idea of 5.1.1, the relationship between the average annual amount of groundwater supply, the average annual precipitation and the supply of groundwater is calculated. And the regression coefficient is determined, and the function relationship between the average annual temperature, average annual precipitation and the supply of groundwater is based on the regression coefficient:Design of underground water for 2y , with an average annual precipitation, annual average air temperature respectively 1x ,2x , using nonlinear regression statistical methods, according to the regression equation with coefficient of determination R2, F statistic corresponds to the probability value p, to determine the regression coefficients β,0β,1n β,2n β, got the regression equation:312412123499992012121111n n n n n n n n n n y x x x x ββββ=====+++∑∑∑∑ （2）3.1.3 Influence of total population and per capita water consumption on daily water consumption in a certain areaThe total population of a region and the amount of water consumption per capita and the daily use of the product of the relationship between the amount of water = total amount *Water usage per person consumption.Set daily water consumption is 5y , the total population, per capita water consumption were 5x , Q , 5y ,Q ,5x , the function of the relationship between the:55y Qx =（3）3.1.4 The influence of average annual rainfall and total population on agricultural water consumption in a certain areaDue to the annual precipitation and the total population and the area of agriculture of area of a water there is a linear or nonlinear relationship, according to the thought of multivariate nonlinear regression can be calculated average annual precipitation and the total population and the area of agriculture with the function relationship between water and to determine the regression coefficient and regression coefficient write GDP and industrial functional relationship between Gross domestic product GDP and industrial water consumption.Let industrial water consumption of 3y , gross production set 4x , using statistical nonlinear regression, regression equation based on the coefficient of determination 2R , F statistical probability value p corresponding to the amount determined regression coefficients 0β,1n β, the regression equation:11193011n n n y x ββ==+∑（4）3.1.5 Effect of average annual rainfall and population in an area of agricultural water useAgricultural water consumption of 4y , design with an average annual rainfall of 1x , 5x , using statistical nonlinear regression, regression equation based on the coefficient of determination 2R , F statistical probability value p corresponding to the amount determined regression coefficients 0β,1n β, the regression equation:3124121234999940151511n n n n n n n n n n y x x x x ββββ===+++∑∑∑∑（5） 3.2 Function Arrangement3.2.1 Water supply functionThe model takes into account a region's water supply from three aspects: surfacewater resources, groundwater resources and the amount of sewage treatment. The function relation between surface water resources, groundwater resources, sewage treatment and water supply is the function, that is, the amount of water supply = surface water resources + groundwater resources.The amount of water supply isX , and the sewage treatment capacity is *Q , by(2)(1): *12X y y Q =++（6）3.2.2 Water demand functionThe model takes into account the need for a region from three aspects: daily water consumption, industrial water consumption and agricultural water consumption. Daily water consumption, industrial water and agricultural water consumption and water demand is a function of the relationship between the function and the function, that is: water demand = daily water consumption + industrial water + agricultural water consumption.A demand for Y , by (3) (4) (5) to:345Y y y y =++（7）3.2.3 The ability of a region to provide clean waterA region to provide clean water and the area of water supply and water demand about, if water supply is greater than demand, the region provide clean water ability strong; on the contrary, the region provide clean water ability is weak. This model provides that a region to provide clean water capacity by the area of water supply and water demand ratio λ determined by (6) (7) available:(1) 1λ>: the region's ability to provide clean water;(2) 1λ=: the area provides a warning of the ability to provide clean water;(3) 1λ<: the ability of the region to provide clean water is weak;3.3 In order to test the accuracy and usability of the model, this model is selected as a test area in Shandong Province, China.Provide the capacity of water resources in China's Shandong Province, we collected in 2005 to 2014 this decade, Shandong Province, the total water supply, surface water resources amount, quantity of groundwater resources, sewage treatment capacity, agricultural water consumption, industrial water, living water, sewage emissions, forest coverage, total population, per capita water use, annual precipitation, GDP (see Appendix for the specific data).3.3.1 Total surface water resourcesSurface water resources amount 1y , groundwater resources quantity is 2y , industrial water use 3y , agricultural water use for 4y , with an average annual precipitation 1x , with an average annual temperature 2x , the forest coverage rate 3x , GDP for 4x , with a total population of 5x .The factors 1y that is affected by 1x , 2x , 3x , in order to determine the relationship between the 1y , and 1x , 2x , 3x ,first use the data in the appendix table to make 1y with 1x , 2x , 3x scatter plots, such as the figure:Figure 1 Surface water resources and average annual rainfallFigure 2 Surface water resources and annual temperatureFigure 3 Surface water resources and forest coverFigure 1 is obtained by MATLAB fitting curve, the fitting found that 1x and 1y is the 6 power function model (εfor random error).（9）Figure 2 is obtained by MATLAB fitting curve, the fitting found that 2x and 1y isthe 61011n n n y x ββε==++∑8 power function model（10）Figure 3 is obtained by MATLAB fitting curve, the fitting found that 3x and 1y is the 8 power function model（11）Combined with the above analysis, the model (9) (10) (11) established the following regression model（12）Directly using the MATLAB statistics toolbox in the command regress solution, the format is[b,bint,r,rint,stats]=regress(x,y,alpha)Output value of b for the estimation of the regression coefficient β, bint is the confidence intervals for b and r is the residual vector, rint is the confidence interval of r , stats is regression model test statistics, in the first number is a regression equation with coefficient of determination 2R ;The regression coefficients of the model (12) are estimated and their confidence intervals (confidence level α=0.05), test statistic 2R , F , ρ, and the results are shown in table.Table 1 Surface water regression coefficientCan get the regression coefficient from the figure, the estimated value of the regression coefficient into the model (12) forecast equation81021n n n y x ββε==++∑31031n n n y x ββε==++∑33121212312456683999103331231131111n n n n n n n n n n n n n n n y x x x x x x βββββε=======+++++∑∑∑∑∑∑（13）3.3.2 Total groundwater resourcesFactors that affect the 2y include 1x ,2x , in order to determine the relationship between 2y and 1x ,2x , first use the data in the appendix table to make the A3 and A4 and A5 of the scatter diagram, as shown in figure:Figure 4 the amount of groundwater resources Figure 5 the amount of groundwater resources and annual average temperatureand the average annual rainfallFigure 4 is obtained by MATLAB fitting curve, the fitting found that 1x and 2y is the 6 power function model (ε for random error),（14）Figure 5 is obtained by MATLAB fitting curve, the fitting found that 2x and 2y is the 8 power function model.（15）Combined with the above analysis, the model (9) (10) (11) established the following regression model.（16） Directly using the MATLAB statistics toolbox in the command regress solution, the format ^9665444111138273322223724 1.7610 1.0610 1.8910 1.18103.8410 2.610 3.2910 1.0110y x x x x x x x x x --------=+⨯-⨯+⨯+⨯-⨯+⨯-⨯+⨯62011n n n y x ββε==++∑82021n n n y x ββε==++∑121212123468992013121111n n n n n n n n n n y x x x x ββββε=====++++∑∑∑∑is[b,bint,r,rint,stats]=regress(x,y,alpha)Output value of b for the estimation of the regression coefficient β, bint is the confidence intervals for b ,and 2R is the residual vector, rint is the confidence interval of r , stats is regression model test statistics, in the first number is a regression equation with coefficient of determination 2R ;The regression coefficients of the model (12) are estimated and their confidence intervals (confidence level α=0.05), test statistic 2R , F , ρ, and the results are shown in table.Table 2 Regression coefficients of groundwater resourcesCan get the regression coefficient from the figure, the estimated value of the regression coefficient into the model (16) forecast equation（17）Its image is shown in Figure 6^9665445111123821000 1.8210 1.3410 3.37108.49101.3610y x x x x x -----=+⨯-⨯+⨯+⨯-⨯Figure 6 groundwater resources3.3.3 Total industrial water consumption functionFactors that affect 3y is 4x , in order to determine the relationship between 3y and 4x , the first use of the data in the appendix table to make the X and the scatter diagram, as shown in figure:Figure 7 industrial water consumption and GDPFigure 8 industrial water useFigure 7 is obtained by MATLAB fitting curve, the fitting found that 4x and 3y is a function model (εfor random errors),（18）The regression coefficient can be got from the following chartTable 3 Regression coefficient of industrial water consumptionAccording to the above analysis, combined with the model to establish the following regression model, regression coefficient estimation values are substituted into the model (18) to forecast equation.（19）Image as figure 8 3014y x ββε=++^443105.2888410y x -=+⨯3.3.4 Total agricultural water consumption functionFactors that affect the 4y are 1x , 5x , in order to determine the relationship between 4y and 1x , 5x , first using the data in the appendix table to make the 4y and 1x , 5x scatter diagram, as shown in figure:Figure 9 total agricultural water consumption Figure 10 the amount of agricultural water and the average annual rainfalland populationFigure 9 is obtained by MATLAB fitting curve, the fitting found that 4x and 4y is a function model (εfor random errors),（20）Figure 10 is obtained by MATLAB fitting curve, the fitting found that 5x and 4y is a function model (εfor random errors),（21）Combined with the above analysis, the model (20) (21) established the following regression model.（22）Directly using the MATLAB statistics toolbox in the command regress solution, the format is[b,bint,r,rint,stats]=regress(x,y,alpha)Output value of b for the estimation of the regression coefficient β, bint is the confidence intervals for b ,and 2Ris the residual vector, rint is the confidence 3014y x ββε=++74051n n n y x ββε==++∑121212123468994013121111n n n n n n n n n n y x x x x ββββε=====++++∑∑∑∑interval of r , stats is regression model test statistics, in the first number is a regression equation with coefficient of determination 2R ;The regression coefficients of the model (12) are estimated and their confidence intervals (confidence level α=0.05), test statistic 2R , F , ρ, and the results are shown in table.Table 4 regression coefficients of agricultural water useAccording to the above analysis, combined with the model to establish the following regression model, regression coefficient estimation values are substituted into the model (22) to forecast equation.（22）Its image is shown in Figure 11Figure 11 function of agricultural water3.3.5 Assessment of water supply capacityAccording to the data model obtained in 3.2.3, Shandong Province in China, therelevant ^737424155514 1.010 2.110510910 2.56810y x x x x ----=⨯-⨯+⨯-⨯+⨯data and the above function is brought into the model and calculated results:By the conclusion of the model, 1λ< shows that the ability to provide clean water in Shandong province is weak.3.4 Cause Analysis and Treatment Measures Water Shortage.3.4.1 the causes of water shortage in Shandong.（1） Water and soil erosion in hilly areas is serious, and water cannot be brought together into a river（2） Shandong is a temperate monsoon climate. Instability is one of the characters of the monsoon climate. Shandong is located in a part a Plain, and it is short of water. It is a big agricultural province. The water used in industry and agriculture is a lot.（3） Water shortage is the basic situation in the province of Shandong, the contradiction between water supply and demand have become increasingly prominent.（4） Total water resources shortage, average, low mu water resources, less water and more and more people, water resources and population, cultivated land resources serious imbalance, which is the main reason caused by a very prominent contradiction between water supply and demand in Shandong.（5） Have a great relationship with the natural geographical location. Shandong is located at the junction of the north and the south, which is a warm temperate monsoon climate. From the rainfall, the first is the uneven distribution of rainfall during the year.（6） As to rainfall distribution, in the southeast of Shandong Province annual rainfall average is up to 8.5 mm, and northwest region's annual average rainfall is only 550 millimeters, basically showing decreases from the southeast Shandong Province to the northwest of successive trend.（7） East Province is a coastal province, but the sea is not the water for drinking.A lot of rain in the coastal areas is typhoon. The available water in these areas is actually very little.（8） Groundwater levels continue to decline due to over exploitation of underground water in many places. The eastern provinces have formed a number of super mining areas.A series of environmental geological problems, such as groundwater pollution, are formed by the formation of the super mining area.（9）Water must not lack of water in the Yellow River in Shandong province. However, the amount of water in the Yellow River is declining year by year, and the available amount is decreasing.（10） Water conservancy project aging, degradation, water supply reduction3.4.2 Remediation Measures（1）With more rain and floods, water conservation, improvement of water cycle, reserve of groundwater resources, to achieve the use of abundant dry.（2）In strict accordance with the requirements of the state on the implementation of 216.030.741289.69X Y λ===<。
2016年东华大学数学建模竞赛A题(研究生)
2016年度东华大学数学建模竞赛（研究生组）A题基于核心指标的学科评估日前，教育部学位与研究生教育发展中心正式印发《全国第四轮学科评估邀请函》，邀请全国学位授予单位参加全国第四轮一级学科整体水平评估。
这也意味着全国第四轮学科评估工作正式启动了。
全国学科评估工作于2002年首次开展，在这之前已完成三轮评估。
第一轮评估于2002-2004年进行，第二轮评估于2006-2008年进行，第三轮评估于2012年举行。
本次启动的是全国第四轮学科评估。
本次评估是在贯彻落实国务院《统筹推进世界一流大学和一流学科建设总体方案》的背景下进行的，因此受到广泛关注。
根据教育部学位中心的文件（见附件1），本次评估按学科特点的不同分为9个门类。
各门类在师资队伍与资源, 人才培养质量, 科学研究水平，社会服务与学科声誉四个一级指标下设17~18个三级指标统计信息，其中部分信息从公共数据库获得，部分信息由学校填报，部分信息由学位中心组织问卷调查获得，在此基础上计算分析形成学科评估的最终结论。
由于学位中心的评估指标体系很复杂，且带有很多的主观因素，因此是否可以根据一些简单、客观的核心指标来快速进行学科评估结果的预测是一个有价值的课题。
比如，反映师资队伍与资源和人才培养质量两个一级指标的核心指标是该学科是否有博士学位授予权，以及该大学是否为985、211。
而反映社会服务与学科声誉的核心指标是大学的综合排名和专业排名。
对于理工类学科来说，体现科学研究水平的核心指标是该学科的ESI高被引论文数和国家自然科学基金获得情况。
而对于人文社科类学科来说，体现科学研究水平的核心指标是该学科的A类期刊论文数、国家社会科学基金和国家自然科学基金获得情况。
现要求你的团队完成以下工作：（1）根据第三轮评估的结果，建立用上文中的核心指标来对进行学科评估结果预测的数学模型，并分析评价其有效性。
（2）从SCI引文数据库、科学基金网络信息系统等搜集有关核心指标的数据，给出0202应用经济学、0701数学、0702物理学、0805材料科学与工程、0811控制科学与工程、0810信息与通信工程、1201管理科学与工程等7个一级学科的学科排名。
全国大学生数学建模大赛国家一等奖论文A题
=
− − ( − 1)′
, = 1, 2, · · ·, 210
当逐渐增大，锚链受到的竖直向下方向的合力与支持力之差先逐渐接近于0，
再等于0，直至小于0。当合力小于0时，锚链以海床接触，此时海床提供向上的支持
力，其大小与′ 相等。因此可将小于0 的值都作零处理，故锚链接触海床时，
对于问题二，首先考虑第一个子问题，将风速36/直接代入问题一的模型中，
得出此条件下的吃水深度为0.723，各钢管倾斜角度(度)依次为8.960、9.014、9.068
、9.123，钢桶倾斜角(度)为9.179，锚链链接处的切线方向与海床的夹角(度)为18.414，
游动区域半径为18.80。发现此条件下，水声通讯系统设备的工作效果较差,且锚被
计与应用对海上科学发展有重要意义。
1.2 问题的提出
已知某近浅海传输节点(如图1所示)，将浮标视作底面直径2为、高为2、质量
为1000的圆柱体，锚的质量为600，钢管共4节，每节长度为1，直径为50，
每节钢管的质量为10。水声通讯系统安装在一个长为1、外径为30的密封圆
柱形钢桶内，设备和钢桶总质量为100。
Step1: 遍历求解
令吃水深度ℎ的初始值为0.1，以0.0005为单位逐步增加至2。( 浮标高度为2，
完全浸没时吃水深度ℎ则为2 )，记录对应的数据，选取水下物体竖直方向高度和
与海域水深最接近的组别，进一步进行计算，结果如下表所示(具体程序见附录)：
表 1: 不同风速的相关结果表
以风速24/的情况为例，绘制游动区域图：
题意的变量临界值。以水深16、系统各部分递推关系式和钢桶与竖直方向夹角小
于5°为约束条件，将多目标优化转化为单目标优化。通过调节决策变量中锚链的型
2016高教社杯全国大学生数学建模竞赛题目A题解题思路
2016高教社杯全国大学生数学建模竞赛题目A题系泊系统的设计分析初稿，旨在交流，有各种做题思路，大家自由发挥！不保证正确,如有错误,欢迎指正!注意1：程序为最初稿，只是证明解的存在性，可以使用二分法、牛顿法等进行进一步求解！2：剩下的可以使用锚链线等更复杂的理论：请继续查阅文献，给文章加分3：此外可以化下面的流程图，解释求解程序，给文章加分4：剩下题目问题原则上是把问题做的更复杂，考虑更多的受力，请大家自行脑补。
5：第一天说了对系缆力的计算，目前主要有三种模型：悬链线模型（我们下面说的第三种静力学分析）、以多体动力学理论为基础的集中质量一弹簧模型（我们下面说的第二种，需要matlab做常微分方程数值解）以及细长杆模型（我们下面说的第一种，力学有限元分析））。
查阅参考文献《深海系泊系统动力特性研究进展》，请大家自行选择各类方法。
1. 某型传输节点选用II型电焊锚链22.05m，选用的重物球的质量为1200kg。
现将该型传输节点布放在水深18m、海床平坦、海水密度为1.025×103kg/m3的海域。
若海水静止，分别计算海面风速为12m/s和24m/s时钢桶和各节钢管的倾斜角度、锚链形状、浮标的吃水深度和游动区域。
1. 某型传输节点选用II型电焊锚链22.05m，选用的重物球的质量为1200kg。
现将该型传输节点布放在水深18m、海床平坦、海水密度为1.025×103kg/m3的海域。
若海水静止，分别计算海面风速为12m/s和24m/s时钢桶和各节钢管的倾斜角度、锚链形状、浮标的吃水深度和游动区域。
分析:为简化起见, 按平浮处理，风引起的水平力x F()()220.625,0.6252x F v S h r h h v θ'==⨯-浮力f F 为2f F g r h ρπ'=其中h '为正浮吃水深度。
则对浮标的方程有 1111011011sin ,cos sin ,cos x f x f F T F T G F T F G T θθθθ==+=-= (1)其中0G 为浮标自重，00G m g =，0m 为浮标的质量为1000kg 。
【2016年高教社杯全国大学生数学建模竞赛赛题】CUMCM2016-Problem-C-Chinese-version
全国大学生数学建模竞赛真题试卷复习材料2016年高教社杯全国大学生数学建模竞赛题目
（请先阅读“全国大学生数学建模竞赛论文格式规范”）
C题电池剩余放电时间预测
铅酸电池作为电源被广泛用于工业、军事、日常生活中。
在铅酸电池以恒定电流强度放电过程中，电压随放电时间单调下降，直到额定的最低保护电压（Um，本题中为9V）。
从充满电开始放电，电压随时间变化的关系称为放电曲线。
电池在当前负荷下还能供电多长时间（即以当前电流强度放电到Um的剩余放电时间）是使用中必须回答的问题。
电池通过较长时间使用或放置，充满电后的荷电状态会发生衰减。
问题1 附件1是同一生产批次电池出厂时以不同电流强度放电测试的完整放电曲线的采样数据。
请根据附件1用初等函数表示各放电曲线，并分别给出各放电曲线的平均相对误差（MRE，定义见附件1）。
如果在新电池使用中，分别以30A、40A、50A、60A和70A电流强度放电，测得电压都为9.8伏时，根据你获得的模型，电池的剩余放电时间分别是多少？
问题2 试建立以20A到100A之间任一恒定电流强度放电时的放电曲线的数学模型，并用MRE评估模型的精度。
用表格和图形给出电流强度为55A时的放电曲线。
问题3 附件2是同一电池在不同衰减状态下以同一电流强度从充满电开始放电的记录数据。
试预测附件2中电池衰减状态3的剩余放电时间。
2016年高教社杯全国大学生数学建模竞赛题目A.B
2016年高教社杯全国大学生数学建模竞赛题目（请先阅读“全国大学生数学建模竞赛论文格式规”）A题系泊系统的设计近浅海观测网的传输节点由浮标系统、系泊系统和水声通讯系统组成（如图1所示）。
某型传输节点的浮标系统可简化为底面直径2m、高2m的圆柱体，浮标的质量为1000kg。
系泊系统由钢管、钢桶、重物球、电焊锚链和特制的抗拖移锚组成。
锚的质量为600kg，锚链选用无档普通链环，近浅海观测网的常用型号及其参数在附表中列出。
钢管共4节，每节长度1m，直径为50mm，每节钢管的质量为10kg。
要求锚链末端与锚的处的切线方向与海床的夹角不超过16度，否则锚会被拖行，致使节点移位丢失。
水声通讯系统安装在一个长1m、外径30cm的密封圆柱形钢桶，设备和钢桶总质量为100kg。
钢桶上接第4节钢管，下接电焊锚链。
钢桶竖直时，水声通讯设备的工作效果最佳。
若钢桶倾斜，则影响设备的工作效果。
钢桶的倾斜角度（钢桶与竖直线的夹角）超过5度时，设备的工作效果较差。
为了控制钢桶的倾斜角度，钢桶与电焊锚链处可悬挂重物球。
图1 传输节点示意图（仅为结构模块示意图，未考虑尺寸比例）系泊系统的设计问题就是确定锚链的型号、长度和重物球的质量，使得浮标的吃水深度和游动区域及钢桶的倾斜角度尽可能小。
问题1某型传输节点选用II型电焊锚链22.05m，选用的重物球的质量为1200kg。
现将该型传输节点布放在水深18m、海床平坦、海水密度为1.025×103kg/m3的海域。
若海水静止，分别计算海面风速为12m/s和24m/s时钢桶和各节钢管的倾斜角度、锚链形状、浮标的吃水深度和游动区域。
问题2在问题1的假设下，计算海面风速为36m/s时钢桶和各节钢管的倾斜角度、锚链形状和浮标的游动区域。
请调节重物球的质量，使得钢桶的倾斜角度不超过5度，锚链在锚点与海床的夹角不超过16度。
问题3 由于潮汐等因素的影响，布放海域的实测水深介于16m~20m之间。
美赛一等奖论文-中文翻译版
目录问题回顾 (3)问题分析： (4)模型假设： (6)符号定义 (7)4.1---------- (8)4.2 有热水输入的温度变化模型 (17)4.2.1模型假设与定义 (17)4.2.2 模型的建立The establishment of the model (18)4.2.3 模型求解 (19)4.3 有人存在的温度变化模型Temperature model of human presence (21)4.3.1 模型影响因素的讨论Discussion influencing factors of the model (21)4.3.2模型的建立 (25)4.3.3 Solving model (29)5.1 优化目标的确定 (29)5.2 约束条件的确定 (31)5.3模型的求解 (32)5.4 泡泡剂的影响 (35)5.5 灵敏度的分析 (35)8 non-technical explanation of the bathtub (37)Sｕmｍary人们经常在充满热水的浴缸里得到清洁和放松。
本文针对只有一个简单的热水龙头的浴缸，建立一个多目标优化模型，通过调整水龙头流量大小和流入水的温度来使整个泡澡过程浴缸内水温维持基本恒定且不会浪费太多水。
首先分析浴缸中水温度变化的具体情况。
根据能量转移的特点将浴缸中的热量损失分为两类情况:沿浴缸四壁和底面向空气中丧失的热量根据傅里叶导热定律求出；沿水面丧失的热量根据水由液态变为气态的焓变求出。
因涉及的参数过多,将系数进行回归分析的得到一个一元二次函数。
结合两类热量建立了温度关于时间的微分方程。
加入阻滞因子考虑环境温湿度升高对水温的影响,最后得到水温度随时间的变化规律(见图＊*)。
优化模型考虑保持水龙头匀速流入热水的情况。
将过程分为浴缸未加满和浴缸加满而水从排水口溢出的两种情况，根据能量守恒定律优化上述微分方程,建立一个有热源的情况下水的温度随时间变化的分段模型，（见图*＊)接下来考虑人在浴缸中对水温的影响。
2016年数学建模国赛A题
2016年高教社杯全国大学生数学建模竞赛题目（请先阅读“全国大学生数学建模竞赛论文格式规范”）A题系泊系统的设计近浅海观测网的传输节点由浮标系统、系泊系统和水声通讯系统组成(如图1所示）。
某型传输节点的浮标系统可简化为底面直径2m、高2m的圆柱体，浮标的质量为1000kg。
系泊系统由钢管、钢桶、重物球、电焊锚链和特制的抗拖移锚组成。
锚的质量为600kg，锚链选用无档普通链环，近浅海观测网的常用型号及其参数在附表中列出。
钢管共4节,每节长度1m，直径为50mm，每节钢管的质量为10kg。
要求锚链末端与锚的链接处的切线方向与海床的夹角不超过16度，否则锚会被拖行，致使节点移位丢失。
水声通讯系统安装在一个长1m、外径30cm 的密封圆柱形钢桶内，设备和钢桶总质量为100kg。
钢桶上接第4节钢管，下接电焊锚链。
钢桶竖直时，水声通讯设备的工作效果最佳.若钢桶倾斜，则影响设备的工作效果。
钢桶的倾斜角度（钢桶与竖直线的夹角）超过5度时，设备的工作效果较差。
为了控制钢桶的倾斜角度,钢桶与电焊锚链链接处可悬挂重物球。
图1 传输节点示意图（仅为结构模块示意图，未考虑尺寸比例) 系泊系统的设计问题就是确定锚链的型号、长度和重物球的质量，使得浮标的吃水深度和游动区域及钢桶的倾斜角度尽可能小。
问题1某型传输节点选用II型电焊锚链22。
05m，选用的重物球的质量为1200kg。
现将该型传输节点布放在水深18m、海床平坦、海水密度为1。
025×103kg/m3的海域。
若海水静止，分别计算海面风速为12m/s和24m/s时钢桶和各节钢管的倾斜角度、锚链形状、浮标的吃水深度和游动区域。
问题2在问题1的假设下,计算海面风速为36m/s时钢桶和各节钢管的倾斜角度、锚链形状和浮标的游动区域。
请调节重物球的质量，使得钢桶的倾斜角度不超过5度，锚链在锚点与海床的夹角不超过16度.问题3 由于潮汐等因素的影响,布放海域的实测水深介于16m~20m之间。
2016数学建模A题 省二
六、模型求解
6.1 问题一求解： （1 ） （2） （3） （4） 对方程组 、 、 、 联立，使用 MATLAB 软件编程求解，最后求得： 12m/s 时: 钢桶的倾斜角度为 1.0090° 从下向上四节钢管的倾斜角度依次为 1.0032° 0.9975° 0.9919° 0.9863° 浮标吃水深度为 0.7529 m 浮标游动圆形区域半径为 21.3338 m 锚链末端与锚的链接处的切线方向与海床的夹角为 3.44°
系泊系统的设计
摘要
随着系泊技术在海洋石油开发中的广泛应用[1],系泊技术逐步趋入成熟， 对各种系泊 方式的设计及优化也开始深入研究。根据不同海域的情况，设计出安全又经济的系泊系 统，是现在海洋工程和船舶工程领域亟须解决的问题。 对于问题一，我们将系统分成锚链、钢桶、钢管、浮标四部分，然后对锚链和钢管 建立了微分方程模型，对钢桶和浮标进行受力平衡分析，建立了方程组。我们采用了逐 步近似法，使用 MATLAB 软件对所列的方程组求解，得到结果为：风速 12m/s 时:钢桶的 倾斜角度为 1.0090°，从下向上钢管的倾斜角度为 1.0032°、0.9975°、0.9919°、 0.9863°，浮标吃水深度为 0.7529 m，浮标游动圆形区域半径为 21.3338 m，锚链末端 与锚的链接处的切线方向与海床的夹角为 3.44°， 水下锚链形状见图 5 （见 P7） ； 24m/s 时:钢桶的倾斜角度为 3.7183°，从上而下钢管的倾斜角度为 3.6979°、3.6776°、 3.6576°、 3.6378°， 浮标吃水深度为 0.7678m， 浮标游动圆形区域的半径为 21.7316m， 锚链末端与锚的链接处的切线方向与海床的夹角为 3.44°，水下锚链形状见图 6（见 P8）。 对于问题二，我们根据问题一的模型，通过 MATLAB 编程不断改变重物球的质量反 复迭代即可得到结果： 当钢桶倾斜角小于 5°且锚链在锚点与海床的夹角不超过 16°时， 重物球临界质量为 1745kg，钢桶的倾斜角度为 4.9998°，从下而上钢管的倾斜角度为 4.9797°、4.9598°、4.9400°、 4.9204°，浮标吃水深度为 0.9320m，浮标游动圆形 区域半径为 21.8912m，锚链形状见图 7（见 P9）。当重物球小于这个质量时，钢桶倾角 会大于 5°。 对于问题三，我们在前两问模型的基础上，将其划分成不同情况分别求解。锚链型 号一共为五种，并将每种分为 16m 和 20m 两种情况，我们分别计算出它们在临界时的重 物球质量、锚链长度、浮标吃水情况以及浮标游动范围，对每个情况进行分析，得出最 优解。最后设计的系泊系统应选用 III 型号的锚链，长度为 22m，重物球质量最少为 3547kg，可以适当增加重物球质量，以确保倾角不会大于 5°。
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Summary Our experience of everyday life tells us that, bathing quality is relevant to the temperature of the water. Hence the optimality of the water-adding strategy in bathing process has become an issue. In this article, we establish two models. One is water temperature variation and distribution model and the other one is finding best strategy model. We put forward some acceptable hypothesis to simplify the model. What’s more, we clear the meaning of the word “noticeably”. The first part has five sections: air’s heat radiation, bathtub wall’s heat radiation, person in, hot water in, bubble existed. We discuss some factors that affect water temperature, such as the shape and the volume of the bathtub and person, especially the motions made by the person in the bathtub because the temperature in the bathtub has a great connection with person. Finally, we get the water temperature variation and distribution model. The second part aims to finding the best water strategy. Different strategies are considered for adding hot water and the amount of the hot water. The key of strategies is how to choose the moment of opening the faucet and the lasting time. By developing an optimization model of the strategy, we manage to determine a strategy for different users and extend the model with a finite volume method. Under the condition of the same bathing time, we study what’s the ideal time to add the hot water to make it to the initial temperature.
For office use only T1 ________________ T2 ________________ T3 ________________ T4 ________________ Team Control Number 46731 Problem Chosen A For office use only F1 ________________ F2 ________________ F3 ________________ F4 ________________
2016 MCM/ICM Summary Sheet (Your team's summary should be included as the first page of your electronic submission.) Type a summary of your results on this page. Do not include the name of your school, advisor, or team members on this page. Team # 46731 Page 2 of 19 We transform the weight p. With the restrict condition of the temperature in the bath and the amount of the hot water, the question can be transformed to a nonlinear optimization. We get the conclusion and optimal results through the PDE (Partial Differential Equations) toolbox of MATLAB. And the result shows a clear superiority for choosing a suitable strategy for different users.
Key words: bathtub; heat-conduction; temperature variation and distribution;
water strategy; finite volume method; Team # 46731 Page 3 of 19 Contents Summary ................................................................................................................ 1 I. Introduction ...................................................................................................... 4 II. Analysis ........................................................................................................ 4 III. Assumptions ................................................................................................. 6 IV. Notation ........................................................................................................ 7 V. Models .......................................................................................................... 8 Part I Water Temperature Variation and Distribution Model ......................... 8 Air’s Heat Radiation .......................................................................... 8 Bathtub Wall’s Heat Radiation ........................................................... 9 Person In ............................................................................................ 9 Hot Water In ..................................................................................... 10 Bubble Existed ................................................................................. 11 Water Temperature Variation and Distribution Model ..................... 11 Part II Finding the best strategy ................................................................... 12 VI. Results ........................................................................................................ 12 Part I ............................................................................................................. 12 Results of Air’s Heat Radiation Model ............................................ 12 Results of Bathtub Wall’s Heat Radiation ........................................ 13 Results of “Person In” ...................................................................... 13 Results of “Hot Water In” ................................................................ 13 Results of bubble existed model ...................................................... 15 Results of Water Temperature Variation and Distribution Model .... 15 Part II ........................................................................................................... 16 VII. Strengths and Weaknesses....................................................................... 17 VIII. Explanation for users .............................................................................. 18 IX. References ...................................................................................................... 19