2013美赛获奖作品.doc

Team#17696 Page 30 of 31For office use onlyT1_T2_T3_T4_Team Control Number17696Problem ChosenBFor office use onlyF1_F2_F3_F4_2013Mathematical Contest in Modeling (MCM/ICM) Summary SheetSummaryFresh water is the limiting constraint for development in much of the world. Take China into consideration, the nations fresh water supplies can no longer quench its thirst. We build Model One and Model Two to predict the fresh water needed and available in 2025, from which we can get a visualized conclusion of water shortage in future China. Based on the conclusion we set up Model Three and Model Four to put forward strategies to solve the problem of saving water in different fields and increase the sources of fresh water. In order to find the best strategies, Model Five was set up to evaluate the different strategies based on the indexes of cost and water saving. Finally we came up the best strategy according to our models.The first model predicts the fresh water available in 2025 based on the assumption that the main sources of fresh water are stable in the short period. Model Two assumes that sweeping reforms will not be implemented in terms of industry, agriculture and society and predicts the amount of fresh water needed based on the usage of different fields and areas in order to make the prediction more convincing. Based on the thought of “strategic sourcing initiative”, the third model is built to cut the water expenditure. Regression Model and Analytic Hierarchy Process are employed to solve the problem of saving fresh water in industry, agriculture and domestic. Model Four is employed to solve the problem of increasing the source of fresh water through desalinization and sewage recycling.The evaluation of different strategies takes cost and goal into consideration. Principal Component Analysis is employed to evaluate the strategies. We divided the strategies into “Business as usual, Governance, Strong and At all Costs”, and the strong scores highest in the comprehensive evaluation of all strategies. We proposed our water strategy based on the analysis and evaluation above.In a nutshell, we predicted the fresh water shortage roundly and put forward a strategy based on the comprehensive evaluation in distinctive thoughts.Key words: mathematic models, Chinese water strategy, water resourceContentWater, Water, EverywhereMathematic Models of China Water Strategy1.Introduction32.Assumptions43.Key Terminology and Symbols44.Predictions of fresh water54.1 How much fresh water is available in China in 2025?64.2 How much fresh water will be needed in China in 2025?74.3.1 Predictions based on different fields84.3.2 Predictions based on different areas94.3 Conclusion115.Solutions to the Problem115.1 Save fresh water 115.1.1 Save fresh water used in industry 125.1.2 Save fresh water used in agriculture135.1.3 Save fresh water in domestic use165.2 Increase the sources of fresh water 185.2.1 Sea water desalination185.2.2 Turn sewage to fresh water185.3 Adjust the distribution of water both in time and area205.3.1Water transfers205.3.2 Conservation216.Evaluation of Water Strategies216.1 Principal component analysis (PCA) model216.2 Evaluation of strategies based on PCA model226.3 Which strategy is the best? 287.Strengths and Weaknesses 287.1 Strengths287.2 Weaknesses298.Reference291. IntroductionFresh water is the limiting constraint for development in much of the world. Take China into consideration, the water resources of China are affected by both severe water quantity shortages and severe water quality pollution. A growing population and rapid economic development as well as lax environmental oversight have increased water demand and pollution1 /wiki/Water_resources_of_the_Peoples_Republic_of_China.RainfallSurface waterIndustryDomesticUnderground waterSea water desalinationSewage treatmentAgricultureFresh waterEcological water utilizationUnevenly distributed in area and timeSourceConsumeAcross China, the nations fresh water supplies can no longer quench its thirst. Chinese experts warned of future or current water shortages. Water resource usage was expected to peak in 2025 when the population peaks. Below is the figure of the analysis of the supply and cost of fresh water. Figure 1 Source and consume of the fresh waterWhatever the use of freshwater (industry, agriculture, domestic use), huge saving of water and improving of water management is possible. Legislative actions should be sought to get municipalities to use water in a rational, planned, orderly way. In addition to water and conservation storage/movement, desalinization technology that remove excess salt and other minerals from water holds promise to convert salt water into fresh water suitable for human consumption or irrigation. Due to the water problems, as well as for future exports, China is building up its desalination technological abilities and plans to create an indigenous industry. Some cities have introduced extensive water conservation and recycling programs and technologies2 MICHAEL WINES, China Takes a Loss to Get Ahead in the Business of Fresh Water, October 25, 2011,. One focus of desalinization is to develop cost-effective ways of providing fresh water for human use in regions where the availability of fresh water is limited. Large-scale desalinization typically uses extremely large amounts of energy as well as specialized, expensive infrastructure, making it very costly compared with the use of fresh water from rivers or groundwater.2. Assumptionsl There will be no war and natural disasters in China during this period.l The states we choose for modeling will not experience mass migrations of population. The water use at the state level is closely related to the state population. Our water use estimation model does take population change into consideration, while the change follows the current demographic trends. l Sweeping reforms will not be implemented in terms of industry, society, economy, policy, culture and environment. Our estimation model prefers that the future development in terms of industry, society, economy, policy, culture and ecosystem can keep a steady pace through 2013 to 2025. l The unit cost of desalination is the same in different areas.l The unit cost of water transfer is the same in different areas.l The residents water consumption is the same in same area.l The increase in fresh water storage from the thaw of glaciers caused by climate change and global warming is not considered. Climate change and global warming are major obstacles human beings are facing. Many a research has been done on this topic. We expect that effective measures can be taken and the thaw of glaciers will be ceased one day before 2025. And such increase is what we do not hope for. 3. Key Terminology and Symbols3.1 Gray Model (GM)Gray Model is such a model that it establishes a gray differential prediction model through a small amount of incomplete information, and makes a fuzzy long-term description of the development of the object.3.2 Exponential Smoothing Forecasting Method (ESFM)Exponential Smoothing Forecasting Method refers to a method that introduces a simplified weighted factor that uses some indicators of current actual number and current forecast number as the foundation, namely smoothing coefficient, to achieve an average index smooth forecast method. It is a change of weighted moving average prediction method.3.3 Regression Model (RM)Regression Model is such a model that through the establishment of the mathematical model between the numbers that have related relationships according to the statistical data in the past years, conducts a natural regression modeling to analyze the change law between the different quantities.3.4 Analytic Hierarchy Process (AHP)Analytic Hierarchy Process is a decision-making method that decomposes the elements that are always related into goals, standards and scheme level which are based on to analyze the qualitative and quantitative relations.3.5 SymbolstTimesVarianceCharacteristic valueThe first principal componentThe secondary principal componentThe third principal component4. Predictions of Fresh Water Chinas water resources include 2 711.5 cubic kilometers of mean annual run-off in its rivers and 828.8 cubic kilometers of groundwater recharge. As pumping water draws water from nearby rivers, the total available resource is less than the sum of surface and groundwater, and thus is only 2 821.4 cubic kilometers. 80% of these resources are in the South of China3 The Comprehensive Report of Chinas Sustainable Development of Water Resource Strategy Research.2008.In this part, we employed Gray Model to predict the amount of fresh water that is needed and available in 2025.4.1 How much fresh water is available in China in 2025?Figure 2 The amount of fresh water in ChinaWe drew a figure of the total amount of water resources in China from 2003 to 2011. By observing the figure above, we can find that the total amount of fresh water in China moved in steady cycles, and the average amount is around 2700 cubic kilometers.Analogously, the surface water and underground water in China can be regarded as unchanged until 2025.Figure 3 The average annual precipitation in different regions of ChinaMost of the rain China experiences during the year occurs during the summer months. We searched the data of rainfall in China, and rainfall differs from different regions.The picture above suggests that the annual rainfall in China is nearly stable during a short period, and it is different from different regions.Based on these conclusions and combining with other documents, we can calculate the fresh water that is available in 2025:Equation 1Relationship between the amount of total fresh water and available fresh waterWhere Va : the volume of available fresh water; Vt : the volume of total fresh water; : the percentage that the available fresh water take in the total fresh water.After the substitution of numerical values, we can figure out the volume of available fresh water in 2025:Va=27000.2763=746.01 cubic kilometers4.2 How much fresh water will be needed in China in 2025?Total water withdrawals were estimated at 554 cubic kilometers in 2005, or about 20% of renewable resources. Demand is from the following sectors: 65% agriculture 23% industry 12% domesticFigure 4 the amount of fresh water supplyIn 2006, 626 000 square kilometers were irrigated4 FAO Aquastat. China Profile, Version 2010. According to the data published by the National Bureau of Statistics of China, we drew a figure of the amount of the water supply in China from 2003 to 2011 (water was supplied for industry, agriculture, and domestic). By observing the figure above and analyzing the data from Chinese government, we can find that the supply of fresh water increased every year, namely the need of fresh water is increased over time. Chinese experts warned of future or current water shortages. Water resource usage was expected to peak in 2030 when the population peaks.Considering that the trend of increasing will not experience large fluctuations, we chose Gray Model to predict the need of fresh water in 2025, since GM can solve stable time series problems.4.2.1 Predictions based on different fieldsUse Gray Model to predict the fresh water demand in 2025l Class-compare verification We counted the total water resource years from 2003 to 2011 as follows：(4-1)u Calculate the class-compareTo ensure the feasibility of Gray model, we firstly performed data examination: (4-2)u Class-compare judgmentPut the calculated class-compare to substitute (4-3)Since all ，k=2,3,9, so we can useto make a content GM(1,1) model.l Build GM（1,1）modelu One-accumulate sequence of the original data (4-4)u Build data of matrix B and vector Y u Calculate the value of (4-5)u Build model (4-6)So that equation has response particular solution (4-7)Figure 5 the comparison between simulation and real dataTable 1 Model testl GM(1.1) Model error testSeriesYearThe original dataSimulation valueResidual errorRelated error1200353205320002200455475684-37.180.00673200556335657-24.910.0044420065794573261.370.010652007581858089.6820.0017620085909588523.990.004172009596559622.2850.00048201060216041-20.440.00349201161076121-14.210.0023The average of related error is 0.0042Table 2 Common Accuracy ClassrankingLevel 1Level 2Level 3Level 4The average of related error0.010.050.100.20By the comparison of error test value and the common accuracy class, we can see that the rank of the model is level 1, which means the model has high accuracy and can be used for prediction.ResultThe result is that the fresh water in demand in 2025 will be 745.2 cubic kilometers.4.2.2 Predictions based on different areasl The information of six strict in China u North: Beijing, Tianjin, Hebei, Shanxi, Inner Mongolia u Northeast: Liaoning, Jilin, Heilongjiang u East: Shanghai , Jiangsu, Zhejiang, Anhui, Fujian, Jiangxi u South: Henan, Hubei, Hunan, Guangdong, Hainan, Guangxi u Southwest: Chongqing，Sichuan，Guizhou, Yunnan，Tibet u Northwest: Shaanxi，Gansu ，Qinghai，Ningxia Xinjiang l We employed GM (1.1) to predict the water needed in different regions in 2025.Table 3 Fresh water needed in different regions in 2025DistrictWater demand in 2025 (cubic kilometer)North54.2Northeast101.7East 244.9South 186.5Southwest72.4Northwest93.3Total753.0From the table we can see the total amount of fresh water needed in 2025 is 753.0 cubic kilometers, comparing to the prediction based on different fields which is 745.2 cubic kilometers, we conclude that the result is correct.4.3 ConclusionAccording to the results of predicting the amount of the available fresh water and fresh water needed in 2025, we drew some conclusions as follow:l In China, the amount of fresh water needed in 2025 almost stretches to the limit of the water available, since the former is about 745 cubic kilometers and the latter is about 746 cubic kilometers. So take steps to save fresh water is necessary for the government of China, and adopt an appropriate water strategy is crucial for the development.l In China, fresh water is unevenly distributed in both area and time, making water transfer and conservation necessary and important.5. Solutions to the ProblemFrom the conclusion above we have known that take necessary steps to save fresh water is very important in China. According to the problems in the conclusion, we put forward solutions from three aspects, namely saving fresh water, increasing the source of fresh water and adjusting the distribution of water in time and space. 5.1 Save fresh water Fresh water demand is mainly from industry, agriculture and domestic, so that the measures to save fresh water will be adopted in three aspects.5.1.1 Save fresh water used in industry The industry can be divided into high water consumption and low water consumption industry, readjusting the industrial structure can remarkably save fresh water consumption.By decreasing the percentage that high water consumed enterprises take in the whole industry and at the same time increasing the percentage of low water consumed enterprises, the fresh water can be saved yet the cost of industrial restructuring will increase.We accessed relevant information and summed up a relationship among the reduction high water consumption industry, fresh water saving and the increasing of cost.We employed Exponential Smoothing Forecasting Method (ESFM) to predict the amount of cost and water saving.The mathematical expression of the relationship between the percentage of high water consumption industry reduction and the cost is: (5-1)In order to choose an appropriate value of , we calculated the standard deviation of different forecasting based on different .Table 4 Standard deviation S of different forecastingStandard deviation of different forecasting36.206524.294217.7608From the table we can find that the when =0.8, the S got minimum value 17.7608, so we choose =0.8.The mathematical expression is: (5-2)Using Regression Model, we summed up the mathematical expression of the relationship between the percentage of high water consumption industry reduction and the percentage of fresh water saving: (5-3)Where : regression coefficient, : random error.We can figure out the value of 0, iand i, so the ultimate mathematic expression is: (5-4)The coefficient of determination 0.8985, it shows that the regression effect is pretty good. F inspection was used to conduct significance analysis, and the value of P is 0.000, the level of significance is acceptable. By accessing relevant information and the building mathematic models above, we summed up a relationship among the reduction high water consumption industry, fresh water saving and the increasing o