2016美賽E題參考答案

1、Team # 52766 For office use onlyT1_T2_T3_T4_Team Control Number52766 Problem ChosenEFor office use onlyF1_F2_F3_F4_2021MCM/ICMSummary SheetIn order to predict the water scarcity and optimize the configuration reasonably, we analyze the situation of water scarcity by establishing a mathematical

2、model, and propose the feasible suggestions on optimization. All the work is based on the sufficient data we collect.Firstly, the local situation of water scarcity is estimated by introducing the water lacking rate index. Secondly, the local water consumption is predicted from personal living, indus

3、try, agriculture and ecology. Meanwhile, the local water consumption is predicted through establishing a compound model which based on an improved Logistic Model and the statistical regression analysis. Thirdly, the Gray Prediction Metabolism Model is used for predicting the amount of local water su

4、pply. At last, we comprehensively analyze the experimental results, and predict the ability of water supply in this local area.To verify the availability of the model, we choose the North China as the object of study. We conclude that this area is seriously scarce before 2021 on the basis of mass da

5、ta. The water scarcity will steadily remit and reach balance in 2025. This is due to China has finished the South-to-North Water Diversion and the North China gains large water resources from outside. The results call inside with the truth, so the model is reliable.Then we try to optimize the water

6、supplying and demanding structure in the North China, so that it can realize the internal self-sufficiency. We use analytic hierarchy process (AHP) to assess the four schemes of water storage, water transfer, wastewater treatment and desalination from four aspects of timeliness, sustainability, econ

7、omic, environmental benefits, so a more scientific water supplying system is developed. Finally, by means of adjusting the industrial structure, optimizing the mode of agricultural irrigation and improving the water conservation awareness of citizens, we propose a water resources allocation model to

8、 optimize the water supply system in the North China. In this way, the water scarcity in the North China can be solved five years ahead of the original schedule.Key wordsWater scarcity; water lacking rate index; improved Logistic Model; Gray Prediction Metabolism Model; statistical regression; analy

9、tic hierarchy process (AHP) Contents1 Introduction & Backgrounds12 Problem Analysis12.1 Problem Restatement12.2 Problem solving23 Assumptions24 Notations35 Basic model35.1 Model of the water consumption35.1.1 Compound population Model based on Logistic Model45.1.2 Model of Industrial water consu

10、mption55.1.3 Model of total water consumption of a region65.2 Model of the Gray Metabolism Model GM (1, 1)65.2.1 Principle of common Gray GM (1, 1) Model65.2.2 Principle of Gray Metabolism Model GM (1, 1)285.2.3 Accuracy testing85.3 Model of water supply capacity of a region95.4 Strengths & Weak

11、ness96 Choose a region to analyze106.1 Brief introduction106.2 Physical scarcity:116.3 Economical scarcity:127 Prediction model for the North China127.1 Model of water consumption in North China.127.2 Water supply model in the North China177.3 The comprehensive evaluation to the future water resourc

12、es in the North China187.4 Strengths and Weaknesses218 Optimization of water resources Allocation218.1 Optimization of water supply allocation218.1.1 Construct the model of hierarchical structure218.1.2 Construction of comparison matrix of target-criterion layer.228.1.3 Construction of comparison ma

13、trix criterion - target layer238.1.4 Total sorts of hierarchy and consistency check278.1.5 Interpretation of result288.2 Optimization of water consumption allocation298.2.1 Construction of water resources allocation model298.2.2 Main constraint equations308.2.3 Model Solution31Conclusions33Further d

14、iscussions34Reference34Appendix35Strategy of Conquering ThirstTeam #52766 Page 42 of 42 1 Introduction & BackgroundsAn effective plan of solving the water scarcity problem is crucial to human society. According to the United Nations, today more than one billion people lack access to safe, clean

15、drinking water, and just 10 countries share 60 percent of the worlds natural, renewable water resources; whats more, water use has been growing at twice the rate of population over the last century. A model of water scarcity of the world shows the serious situation. (See Figure 1) Figure 1: the map

16、of the worlds water scarcityAt the same time, our societal and economic growth is largely driven by the productive use of water. Actually, the world tripled its water use in the last 50 years alone. Our world population is increasing, yet we still share one water resource and its limited. If were go

17、ing to meet the agricultural, industrial and residential needs of this growing world, we must use our water in effective, efficient ways. 2 Problem Analysis2.1 Problem RestatementDevelop a model that provides a measure of the ability that a region cab provide clean water to meet the needs of its pop

18、ulation. Doing all this work with considering the dynamic nature of the factors that affect both supply and demand.Pick one country or region where water is either heavily or moderately overloaded. Explain why and how water is scarce in that region. Show what the water situation will be in 15 years,

19、 and predict how this situation impact the lives of citizens of this region. Design an intervention plan taking all the drivers of water scarcity into account according to the situation, to help with the water scarcity and optimizing the model.Estimate the optimized model and predict the results.2.2

20、 Problem solvingWe address the problem of optimizing water use of a region through analyzing the water situation, which provides a measure of the ability of a region to provide clean water to meet the needs of its population. The model consists of two big modules: water supply and water consumption.

21、 Water supply module is mainly composed of surface water and groundwater, the degree of environmental governance, storage capacity, precipitation, etc. Water module is mainly composed of personal living water, agricultural water, industrial water and ecological water use, etc. Finally we compare the

22、 model of water supply with the model of water consumption, then take a certain evaluation index to measure the region's water supply capacity.Next, since the region of North China (include Beijing, Tianjin, Hebei, Shanxi, Inner Mongolia) meets the requirement of being heavily or moderately over

23、loaded on using water, we choose the North China as the researching region. We explain the social and environmental reasons of the scarcity from physical and economical sides, to dissect what facts impact the water scarcity.Then we make quantitative and qualitative analysis of the water supply and c

24、onsumption in this region, to get what the characteristics of the situation in 15 years.At last, according to the fourth and fifth question, we optimize the model from improving water supply and reducing water consumption, to solve the problem of water shortage. We consider about improving the water

25、 supply by the construction of reservoirs, water diversion works, water desalination, wastewater treatment, and political, economic and social considerations. On the other side, we reduce the water consumption from raising awareness about saving water and improving agricultural irrigation methods, a

26、djusting the structure of industrial water, improving ecological aspects to consider autonomy ability. Then under the optimization of water supply and water consumption we set up a water allocation optimization model, which is used for exploitation and utilization planning of water resources, meanwh

27、ile predict the situation of water resources using in the future.3 AssumptionsIgnore the impacts of the extreme disasters.Ignore the impacts of the migration of population.Ignore the administration cost in late period of water diversion projectIgnore the effects of the other polluting factors in the

28、 wastewater when consider about the polluted degree.Ignore the transportation cost of the water supply when consider about the sea water desalination.we assume that the gross of the population is large enough.We assume that there is no big revolution in the configuration of water supply.4 NotationsT

29、able 1: Notations and DescriptionsNotations Descriptionst time unitN(t) Gross population of time tN(t0) Gross population of time t0M Allowed max population in a region Natural population growth rateP(t) Population net growth rate of time Coefficient of lifeS Relative standard deviationW Water consum

30、ption per capitaA(t) Gross of the personal water consumptionu Water lacking rateU() Upstream collection of objectsXZTL Sewage water back to the remaining amountXRSV Reservoir capacity of the surface waterPZBC1 Agricultural production function coefficientXZGO Lateral groundwater runoffPZGU Groundwate

31、r mining upper limitPRSF InflowPCSD Channel of water proportionPZWE River ecological water requirementPNSF At the end of the river channel control inflow XCSO The dodgers river water amount5 Basic model5.1 Model of the water consumptionThe water consumption of one region mainly consists of personal

32、consumption，agricultural consumption，industrial consumption，ecological consumption and etc. And the agricultural consumption，industrial consumption changes with the environment changing, population increasing, and economics developing. Therefore we need to develop mathematical model for the four fac

33、ts to research.Personal consumption refers to a single person use water to drink, bath and so on for daily life.5.1.1 Compound population Model based on Logistic Model Logistic Model fits for continuous population growth, but also can appear negative growth in some countries. We establish a compound

34、 population model based on Logistic Model, then build the model of population net growth rate in one-place linear，by researching the law of population growth.According to Logistic Model, the model of the population growth rate can express as the initial value problem of differential equation1： Postu

35、late , and denotes the population net growth rate, denotes the natural population growth rate, denotes the coefficient of life, denotes the gross population, denotes the gross population of initial time. Formula (1) represents an initial value problems of Bernoulli equation, and we can obtain the so

36、lution. However, due to the limited data information, we cannot confirm the parameters and . Thus even if we have obtained the solutions, it made no contribution to predict the gross of population. Therefore we assume that the population net growth rate has linear relationship with time; that is the

37、 model of the population net growth rate: are uncertain constant. The results of a multiply b is less than zero. Putting equation (2) into equation (1), we can get the model that reflects the gross of the population. By solving the initial value of the differential equation problem, we can get the c

38、omposite model of population growth: In this differential equation，guessing，because is greater than zero, becomes the only stable point of function , then we draw the conclusion that when , gets the maximum absolute value point.When constant a is less than zero, and constant b is greater than zero,

39、we get is a monotonic increasing function in, and a monotonic decreasing function in, so achieves the maximum value when ; When a is greater than zero, and b is less than zero, is a monotonic decreasing function in , and a monotonic increasing function in , so achieves the minimum value when.Since t

40、he gross of personal water consumption, then we put (4) into it, we get the equation of the gross of personal water consumption: 5.1.2 Model of Industrial water consumptionAccording to the national industrial water consumption statistics from various countries, the value presents a linear trend. We

41、determine the industrial water consumption through linear fitting.We use least square method for fitting. The method makes parameter in function reaches the minimum value. and are called the Least squares estimators. Due to the necessary condition of the extremum in the calculus, we obtain a and b:A

42、ccording to the industrial water consumption in each year, we can fit the industrial water consumption. Model of total water consumption of a regionSince it is the same for the agricultural consumption and ecological consumption to calculate as it does in industrial consumption, we can determine the

43、 model of total water consumption. Based on the model of personal consumption, industrial consumption, agricultural consumption, and ecological consumption, we can determine the model of total water consumption of a region as follows: 5.2 Model of the Gray Metabolism Model GM (1, 1)5.2.1 Principle o

44、f common Gray GM (1, 1) ModelGray System theory holds the view that all the random quantities are gray variables and process within certain range and time interval. The model is established after processing these data in certain ways and ranking into regular time series. Gray System Prediction Model

45、 GM (1, 1) is a first order differential equation with one variable，it is fit for prediction to the development of systematic behavior eigenvalue. Gray System Prediction Model GM (1, 1) produces random number and transforms them into ordered data, and then establishes differential equation, later, i

46、t seeks for the regulation of producing the data and then restore the operating results. The specific steps are following:We accumulate the variable to get hereinto，we get .So we can establish a differential equation in the form of an albino as fellow:The bleaching solution of differential equation

47、are as followsdisperse response:Parameter k denotes time series, can be year, season or month.Mark parameter sequences as , .We obtain from these equations: While represents data matrix， denotes data column. Because we get cumulative amount by GM Model is for once, and it is the predicted value when

48、, we must restore the obtained data (orÙ) to (or ) through repeatedly minus with determining (IAGO): Since , we get .5.2.2 Principle of Gray Metabolism Model GM (1, 1)2After making a gray prediction and getting the latest information，it adds this information into the original data series and wi

49、pes off the oldest information at the same time. Then, using the new one as original series, it repeats the above step 1.1 to set up GM1,1 Model，so on and so forth, until the fulfillment of all the prediction objectives, and that is the Gray Metabolism Model we wanted。5.2.3 Accuracy testingRelative

50、error and posterior difference ratio C are two most commonly used way to test the model, and its basic process is following: is original series， is the series simulated by GM Model and is residual sequence。Within it is , the relative error sequence is , and thus the total water resource amount in ye

51、ar could be obtained. Hereinto we have, and is the simulated relative error of the point, andis the average relative error. is defined as the prediction accuracy, which is displayed in percentage.,， Where is the mean variance of residual； is the mean variance of original series; is the posterior dif

52、ference ratio.Here is a reference table attached that illustrates the model accuracy classification in details:Table 2: The accuracy of the modelAccuracy of model Relative error/%Average relative accuracy (, %)Value of First class1Second class5Third class10disqualification20Thus, the method of predi

53、cting water resource amount in one place and checking the accuracy could be gained.5.3 Model of water supply capacity of a regionWe define water scarcity as:In order to estimate the situation of the region3, we lead in variable water lacking rate :For measuring the degree of water lacking in this re

54、gion, we set 4th level evaluation on the basis of water shortage ratethe standard see table 3:Table 3: The classificatory standard Scarce situationshortage rateMain problemsSufficient<5Achieves the balance between supply and demand of water resourcesLight510Normal years in balance, but in the ave

55、rage year, the water supply may occur.Middle1020Not only in design level year of water shortage occurs, also in normal timesHeavy>20Water supply gap is extremely serious.Wading activities are severely restricted5.4 Strengths & WeaknessStrengths It overcomes the traditional method on the forecasting water supply, facing the problems of the shortage of samples, difficult implementation and high requirement. In the prediction