1、Team # 52766 For office use onlyT1 _T2 _T3 _T4 _Team Control Number52766Problem ChosenEFor office use onlyF1 _F2 _F3 _F4 _2016MCM/ICMSummary SheetIn order to predict the water scarcity and optimize the configuration reasonably, we analyze the situation of water scarcity by establishing a mathematica
2、l 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, ind
3、ustry, 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
4、supply. 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 2010 on the basis of mass
5、data. 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 wate
6、r 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, ec
7、onomic, 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
8、to 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 wordsTeam # 52766 Water scarcity; water lacking rate index; improved Logistic Model; Gray Prediction Metabolism Model; statistical re
9、gression; analytic hierarchy process (AHP) Contents1 INTRODUCTION 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 of the worlds water scarcityAt the same
10、 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 going to meet the agricultural, industria
11、l 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 population. Doing all this work with consi
12、dering 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, and predict how this situation impact
13、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.Team #52766 Page 2 of 42 2.2 Problem solvi
14、ngWe 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. Water supply
15、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 model of wate
16、r supply with the model of water consumption, then take a certain evaluation index to measure the regions 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 overloaded on using wat
17、er, 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 consumption in this
18、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 supply by the cons
19、truction 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, adjusting the struct
20、ure 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, meanwhile predict the sit
21、uation 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 wastewater when co
22、nsider about the polluted degree.Ignore the transportation cost of the water supply when consider about the sea Team #52766 Page 3 of 42 water desalination.we assume that the gross of the population is large enough.()NtWe assume that there is no big revolution in the configuration of water supply.4
23、NotationsTable 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 t Coefficient of lifeS Relative standard deviationW
24、Water consumption 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 runoffPZG
25、U Groundwater 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 consumptionTeam #52766 Page 4 of 42 The water consumption of on
26、e region mainly consists of personal 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 develo
27、p mathematical model for the four facts 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 so
28、me countries. We establish a compound 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 pr
29、oblem of differential equation1：* MERGEFORMAT 0()()dNttN(1)Postulate , and denotes the population net growth rate, ()tPtdenotes the natural population growth rate, denotes the coefficient of life, ()Ntdenotes the gross population, denotes the gross population of initial time. 0()NtFormula (1) repres
30、ents an initial value problems of Bernoulli equation, and we can obtain the solution. 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
31、 the population net growth rate has linear relationship with time; that is the model of the population net growth rate:* PtabMERGEFORMAT (2)are uncertain constant. The results of a multiply b is less than zero. ,abPutting equation (2) into equation (1), we can get the model that reflects the gross o
32、f the population.Team #52766 Page 5 of 42 * 0()(dNtabtMERGEFORMAT (3)By solving the initial value of the differential equation problem, we can get the composite model of population growth:* 2200()expexpatatNtbbMERGEFORMAT (4)In this differential equation，guessing ，because is dNtatNtgreater than zero
33、, becomes the only stable point of function t, then we btadraw the conclusion that when , Nt gets the maximum absolute value point.tWhen constant a is less than zero, and constant b is greater than zero, we get is a monotonic increasing function in , and a monotonic decreasing Nt 0function in , so N
34、t achieves the maximum value when b bN; When a is greater than zero, and b is less than zero, tis a monotonic tdecreasing function in 0,b , and a monotonic increasing function in ,b, so Ntachieves the minimum value when .NbtaSince the gross of personal water consumption , then we put (4) ()*()AtWNti
35、nto it, we get the equation of the gross of personal water consumption:* 2200expexpatatWNbbAtMERGEFORMAT (5)Team #52766 Page 6 of 42 5.1.2 Model of Industrial water consumptionAccording to the national industrial water consumption statistics from various countries, the value presents a linear trend.
36、 We determine the industrial water consumption through linear fitting.()BtWe use least square method for fitting. The method makes parameter in Qfunction reaches the minimum value. and are called the 21tiiiQYabXabLeast squares estimators. Due to the necessary condition of the extremum in the calculu
37、s, we obtain a and b: 120tiiidQYabX1tiiiib1122t tii iii it ti ii iXYXYaYbAccording to the industrial water consumption in each year, we can fit the industrial water consumption . ()Bt5.1.3 Model of total water consumption of a regionSince it is the same for the agricultural consumption and ecologica
38、l ()Ctconsumption to calculate as it does in industrial consumption , we can ()Dt Bdetermine the 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 consu
39、mption of a region as follows:()Et* 2200*exp()*exp()()(atattWNbbBtCDtMERGEFORMAT (6)Team #52766 Page 7 of 42 5.2 Model of the Gray Metabolism Model GM (1, 1)5.2.1 Principle of common Gray GM (1, 1) ModelGray System theory holds the view that all the random quantities are gray variables and process w
40、ithin 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 GM (1, 1) is a first order differential equation with one variable，it is fit for prediction to the development of systematic
41、behavior eigenvalue. Gray System Prediction Model GM (1, 1) produces random number and transforms them into ordered data, and then establishes differential equation, later, it seeks for the regulation of producing the data and then restore the operating results. The specific steps are following:We a
42、ccumulate the variable to get (0)()(0)(0)1,2,xxN(1)()(1)(1),2,xxNhereinto，we get .(1)(0)(1),2,)tixxtnSo we can establish a differential equation in the form of an albino as fellow:(1)()daxutThe bleaching solution of differential equation are as follows（disperse response）: (1)(0)(1)atuxteParameter k denotes time series, can be year, season or month.Mark parameter sequences as , .UbWe obtain from these equations: 1()TaByWhile represents data matrix， denotes data column.By(0)(0)23xyN(1)(1)2(1)(1)232xNBecause we get cumulative amount by GM Model is for once, and it is the
