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2016美赛E题参考答案

2016美赛E题参考答案
2016美赛E题参考答案

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52766

Problem Chosen

E

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2016MCM/ICM

Summary Sheet

In order to predict the water scarcity and optimize the configuration reasonably, we analyze the situation of water scarcity by establishing a mathematical 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, industry, 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 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 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 water 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, economic, 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 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 words

Water scarcity; water lacking rate index;improved Logistic Model;Gray Prediction Metabolism Model; statistical regression; analytic hierarchy process (AHP)

Contents

1I NTRODUCTION &B ACKGROUNDS (1)

2P ROBLEM A NALYSIS (1)

2.1 Problem Restatement (1)

2.2 Problem solving (2)

3A SSUMPTIONS (2)

4N OTATIONS (3)

5B ASIC MODEL (3)

5.1 Model of the water consumption (3)

5.1.1 Compound population Model based on Logistic Model (4)

5.1.2 Model of Industrial water consumption (5)

5.1.3 Model of total water consumption of a region (6)

5.2 Model of the Gray Metabolism Model GM (1, 1) (6)

5.2.1 Principle of common Gray GM (1, 1) Model (6)

5.2.2 Principle of Gray Metabolism Model GM (1, 1)[2] (8)

5.2.3 Accuracy testing (8)

5.3 Model of water supply capacity of a region (9)

5.4 Strengths & Weakness (10)

6C HOOSE A REGION TO ANALYZE (10)

6.1 Brief introduction (10)

6.2 Physical scarcity: (11)

6.3 Economical scarcity: (12)

7P REDICTION MODEL FOR THE N ORTH C HINA (12)

7.1 Model of water consumption in North China. (12)

7.2 Water supply model in the North China (17)

7.3 The comprehensive evaluation to the future water resources in the North China (18)

7.4 Strengths and Weaknesses (21)

8O PTIMIZATION OF W ATER RESOURCES A LLOCATION (21)

8.1 Optimization of water supply allocation (21)

8.1.1 Construct the model of hierarchical structure (21)

8.1.2 Construction of comparison matrix of target-criterion layer. (22)

8.1.3 Construction of comparison matrix criterion - target layer (23)

8.1.4 Total sorts of hierarchy and consistency check (27)

8.1.5 Interpretation of result (28)

8.2Optimization of water consumption allocation (29)

8.2.1 Construction of water resources allocation model (30)

8.2.2 Main constraint equations (30)

8.2.3 Model Solution (32)

C ONCLUSIONS (34)

F URTHER DISCUSSIONS (34)

R EFERENCE (34)

A PPENDIX (35)

StrategyofConquering Thirst

1 Introduction & Backgrounds

An 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 drinking water, and just 10 countries share 60 percent of the world’s natural, renewable water resources; what’s 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 world’s water scarcity At 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 it’s limited. If we’re going to meet the agricultural, industrial and residential needs of this growing world, we must use our water in effective, efficient ways.

2 Problem Analysis

2.1 Problem Restatement

Develop 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 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, 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.

2.2Problem solving

We 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 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 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 overloaded 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 consumption 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 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, adjusting 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, meanwhile predict the situation of water resources using in the future.

3 Assumptions

☆Ignore the impacts of the extreme disasters.

☆Ignore the impacts of the migration of population.

☆Ignore the administration cost in late period of water diversion project

☆Ignore the effects of the other polluting factors in the wastewater when consider about the polluted degree.

☆Ignore the transportation cost of the water supply when consider about the sea water desalination.

N t is large enough.

☆we assume that the gross of the population ()

☆We assume that there is no big revolution in the configuration of water supply.

4 Notations

Table 1:Notations and Descriptions

Notations Descriptions

t time unit

N(t)Gross population of time t

N(t0)Gross population of time t0

M Allowed max population in a region

αNatural population growth rate

P(t)Population net growth rate of time t

βCoefficient of life

S Relative standard deviation

W Water consumption per capita

A(t) Gross of the personal water consumption

u Water lacking rate

U()Upstream collection of objects

XZTL Sewage water back to the remaining amount

XRSV Reservoir capacity of the surface water

PZBC1 Agricultural production function coefficient

XZGO Lateral groundwater runoff

PZGU Groundwater mining upper limit

PRSF Inflow

PCSD Channel of water proportion

PZWE River ecological water requirement

PNSF At the end of the river channel control inflow

XCSO The dodger’s river water amount

5Basic model

5.1 Model of the water consumption

The water consumption of one 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 develop 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.

5.1.1CompoundpopulationModel based on Logistic Model

Logistic Model fits for continuous population growth, but also can appear negative growth in some 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 problem of differential equation [1]:

[]00()

()()()

()dN t N t N t d t N t N αβ?=-?

??=?

(1)

Postulate ()()N t P t αβ-=, and ()P t denotes the population net growth rate,

αdenotes the natural population growth rate, β denotes the coefficient of life,

()N t denotes the gross population, 0()N t denotes the gross population of initial

time.

Formula (1) represents 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 αand β . 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 model of the population net growth rate:

()()P t at b =+(2)

,a b 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.

00()

()()()dN t at b N t dt

N t N ?=+?

??=?

(3) By solving the initial value of the differential equation problem, we can get the

composite model of population growth:

2

2000()exp exp 22at at N t N bt bt ??????

=*-+*+?? ? ????

???(4)

In this differential equation ,guessing

()()()

dN t at b N t dt

=+,because ()N t is greater than zero,b

t a

=-

becomes the only stable point of function ()N t , then we draw the conclusion that when b

t a

=-

,()N t gets the maximum absolute value point. When constant a is less than zero, and constant b is greater than zero, we get

()N t is a monotonic increasing function in 0,b a -??

???, and a monotonic decreasing

function in ,b a -??+∞????, so ()N t achieves the maximum value b N a -??

???

when

b

t a

=-

; When a is greater than zero, and b is less than zero, ()N t is a monotonic decreasing function in 0,b a -?? ??? , and a monotonic increasing function in ,b a -??

+∞????

,

so ()N t achieves the minimum value b N a -??

???

when b t a =-.

Since the gross of personal water consumption ()*()A t W N t =, then we put (4) into it, we get the equation of the gross of personal water consumption:

()2

2000exp exp 22at at W N bt bt A t ??????

=**-+*+?? ? ????

???(5)

5.1.2Model of Industrial water consumption

According to the national industrial water consumption statistics from various

countries, the value presents a linear trend. We determine the industrial water consumption ()B t through linear fitting.

We use least square method for fitting. The method makes parameter Q in function ()2

1t

i i i Q Y a bX ==-+????∑reaches the minimum value. a and b are called

the Least squares estimators. Due to the necessary condition of the extremum in the calculus, we obtain a and b :

()120t

i i i dQ

Y a bX da ==--+=????∑ ()120t

i i i i dQ

Y a bX X db ==--+*=????∑

()()()

()

(

)

1

1

12

2

1

1

t

t

i

i i

i

i i t t

i i i i X

X Y Y X

X Y b X X

X X

====---=

=

--∑∑∑∑

a Y bX =-

According to the industrial water consumption in each year, we can fit the industrial water consumption ()B t .

5.1.3Model of total water consumption of a region

Since it is the same for the agricultural consumption ()C t and ecological consumption ()D t to calculate as it does in industrial consumption ()B t , we can determine 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 consumption of a region

()E t as follows:

22

000()**exp[()]*exp()()()()22

at at

E t W N bt bt B t C t D t =-+++++(6)

5.2Model of the Gray Metabolism Model GM (1, 1)

5.2.1 Principle of common Gray GM (1, 1) Model

Gray 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 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, it seeks for the

regulation of producing the data and then restore the operating results. The specific steps are following:

We

accumulate

the

variable

(0)(0)(0)(0){(1),(2),,()}x x x x N = toget

(1)(1)(1)(1){(1),(2),,()}x x x x N =

hereinto ,we get (1)

(0)(1)()()(1),(1,2,...,)t

i x t x i x t t n ==-=∑.

So we can establish a differential equation in the form of an albinoas fellow:

The bleaching solution of differential equation are as follows (disperse

response ):

Parameter k denotes time series, can be year, season or month.

Mark parameter sequences as U , a U b ??

=????.

We obtain U from these equations:

1?()?T T a

U B B B y b -??==????

While B represents data matrix ,y denotes data column.

Because we get cumulative amount by GM Model is for once, and it is the

predicted value when {1,2,...}t n n ∈++, we must restore the obtained data (1)?(1)x t +(or (1)?()x t ) to (0)?(1)x t +(or (0)?()x t ) through repeatedly minus withdetermining (I —AGO):

(1)

(0)1??()()t

i x

t x i ==∑ 1

(0)(0)1

??()()t i x

i x t -==+∑ (1)

(1)dx ax u dt +=(1)(0)(1)()[(1)]a t u u

x t x e a a --=-+

(0)(0)(0)(2)(3)()x x y x N ?????

?=?????? (1)(1)

12(1)(1)1(1)(1)12[(2)(1)]1[(3)(2)]11[()(1)]1x x x x B x N x N ??-+??-+??=????-+-????

1

(1)

(1)

(0)1

???()()()t i x

t x k x i -==+∑ Since 1

(1)

(0)1

??(1)()t i x

t x i -=-=∑, we get (1)(1)(1)???(1)(1)(),(0,1,2,...)x t x t x t t +=+-=. 5.2.2 Principle of Gray Metabolism Model GM (1, 1)[2]

After making a gray prediction and getting the latest information ,it adds this

information into the original data series and wipes 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 GM (1,1) 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 testing

Relative error and posterior difference ratio C are two most commonly used way to test the model, and its basic process is following:

(0)x is original series ,(0)?x

is the series simulated by GM Model and εis residual sequence 。Within it is (0)(0)?()()()t x t x

t ε=-, the relative error sequence is 1p =-?, and thus the total water resource amount in 1t + year 12(,,...,)n ?=??? could be obtained. Hereinto we have 0()

|

|()

t t x k ε?=, and t ? is the simulated relative error of the

point, and 1

1n

k t n =?=?∑is the average relative error.1p =-?is defined as the

prediction accuracy, which is displayed in percentage.

1S =

,i

n

εε∑=

2S =,(0)(0)()

x k x n

∑=

1

2

S C S =

Where 1S is the mean variance of residual ;2S is the mean variance of original series;C is the posterior difference ratio.

Here is a reference table attached that illustrates the model accuracy

classification in details:

Table 2: The accuracy of the model

the accuracy could be gained.

5.3Model of water supply capacity of a region

We define water scarcity ()F t as:

In order to estimate the situation of the region [3], we lead in variable water lacking rate u :

For measuring the degree of water lacking in this region, we set 4th level

evaluation on the basis of water shortage rate u the standard see table 3:

Table 3: The classificatory standard

()()()()()2

2000exp exp 22at at F t Y t W N bt bt B t C t D t ??????

=-**-+*+---?? ? ????

???()

*100%()F t u E t =

5.4 Strengths & Weakness

Strengths

☆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 process, we can weed out the old data and add new data constantly, maintaining the higher prediction precision.

☆It makes up the situation of Logistic population grows negatively.

☆The model is fit for many areas.

☆the models of industry, agriculture, ecological water consumption are concluded from the change rules, are sufficient in scientific nature.

Weakness

☆Without considering the uncertainty of social development, the model not always reflects the future water resources condition.

☆The model can only be used over a period of time for prediction. It is limited for extreme conditions.

☆the models of the subsystem may not form a linear relationship in various countries, unless the situation is close to China.

6Choose a regionto analyze

In addition to the model, we also choose the region of North China to analyze.

6.1 Brief introduction

The North China is an important part of the Great Plains in the east of China,which located at north latitude 32°~40°and in the east longitude 114°~121°. North China's border is very large.The border of north is the south of Yan Mountains. The southern border is on the north side of the Dabie Mountains .The western boundary in Taihang to the Funiu Mountain Adjacent to the East is the Bohai Sea and the Yellow Sea, also with a total area of 300 thousand square kilometers. Plain has too much advantage such as flat terrain, many rivers, convenient transportation and advanced economy. From ancient times to the present, North China is the center of China in the economy, politics and culture area. And also, Beijing, the capital of China, is in its northern part.(see Figure 2)

Figure 2: the region in red curve is the chosen area.

North China is one of the most water-lacking area of the nation, with only 6% gross、11% per capita of theaverage level. Why it is scarce in water, and how? We conclude the reasons from the physical and economic aspects, including social and environmental objects.

6.2 Physical scarcity:

The amounts of the water resources is poor. The region is alluvial plain, and lacked of rivers and lakes. What’s more, the veget ation coverage is not enough, the rain fall flows away with the solid. Thus the water conservation is poor. For many years, Water resources per capita average volume is 335 cubic meters, reaches nearly 15% of the national per capita.

Water resources are uneven distributed of time and space. Annually, it mainly performs in the event of a wet year and dry year phenomenon; within one year, mainly in summer and autumn, the precipitation and water resources quantity is more, while less rainfall in winter and spring. The uneven distribution of water resources in space mainly reflect that the water resources quantity change with zonal and surface evaporation is uneven distribution.

This region belongs to monsoon climate of medium latitudes, the rainfall is comparatively scarce.

These above are environmental causes.

The huge population and giant consumptions. The plain population accounts for about 20% of the total population, with only 6% gross、15% per capita of theaverage level.

The urbanization and developing economics increase the demand of water in unit area. And this obviously stress the water supplying.

These above are social causes.

6.3Economical scarcity:

Serious pollution and waste, wastewater treatment rate is low. Industrial development and popularization of various kinds of agricultural chemicals leads to serious pollution. Set an example Deterioration of water quality is an important fact of water shortage in north China. Such as the Yong ding River upstream of the Guan Ting reservoir in Beijing, because of the serious pollution of water upstream reservoir has lost the function of water supply, adding to the water shortage situation in Beijing. What’s more, sewage treatment capacity in the region is not different, Beiji ng and Tianjin municipal wastewater treatment rate is higher, while other urban wastewater treatment rate is low.

Human activities. On the one hand, the development and utilization of water resources, and on the other hand, is characterized by the influence of land use. Due to the shortage of water resources and they have to excessive exploitation of groundwater. The region exploit 54.78% of the amount of the groundwater in China. Urbanization caused rainfall infiltration to groundwater recharge, and expanding agricultural irrigation area, increase the water evaporation consumption. The quantity of surface water and groundwater in the natural system greatly reduced—this forms a vicious circle in obtaining water.

7 Prediction model for the North China

According to the model of task 1 and the data statistics, which used to express water supply in North China, in task 2. Respectively. The model of water consumption and water supply were calculated.

7.1Model of water consumption in North China.

Table 4:the total population from 2008 to 2014 in the North

China[4](unit:thousand)

Figure 3: annual average growth rate scattered points

From Figure 3 one can seethat the average population growth rate is linear with time, it also verifies the average population growth rate is linear with time in the task 1. But in 2010 there was anoutlier point, which may be related to the policy, economic and social.We could exclude the point without affecting the overall trend of population.

By the task 1, we know the average annual growth rate model ()P t at b =+(1,2.14t =?),which means when t equals 1, it represents year 2008.According to the census of North China, we could be derived from the population average annual growth rate of () .P t

Using the software Matlab ,we get 0.000734,0.017604a b =-=. Therefore the average annual growth rate of population is:()0.0007340.017604P t t =-+The approximate coefficient is 0.9729 which close to 1. Model fits well.

The fitting results are shown in Figure 4.

Figure 4: Annual average growth rate of population

The fitting results could give us the fact that the annual average growth rate of population

Presented a negative correlation with time, on the other hand, the growth of the population in North China will be more slower, even there is zero growth at some time. This has a great relationship with the long-term family planning in China.

Put 0.000734,0.017604a b =-=into the complex model (4), we get the model of population growth in the North China.

2

20000.0007340.000734()exp 0.017604exp 0.01760422t t N t N t t ??????

--=*-+*+?? ? ?

??????

Choose the initial condition (17)147010N =, therefore

20.000734()147010exp(1.043)exp 0.0176042t N t t ??

-=**+ ?

??

Since 0.0007340,0.0176040a b =-<=>, we obtain the max value of ()

N t when 24b

t a

=-

=. This represents that the population of the North China will reach zero growth in 2032. Using the software Matlab to draw the curve of the population growth of the North China. See Figure 5:

Figure 5:The curve of the population growth in the North China

According to the Figure 5, the data will reach zero growth in 2032 and turn to run downward. At the same time, the current growth rate of population is decreasing. After analyzing the in-depth causes, the conclusion has been drawn in two aspects. Culture is the first cause. With the development of scientific and educational standard, people receive a lengthened education which results in the postponing of average marriage age. More attention will be paid to the development of all qualities one should have and the next generation as well. Lowering the growth rate and uplifting educational quality. Raising only one child rather than raising too many with the limited money. These are why the population growth did not see effective and profound change when China lifted the restriction on one-child policy in 2014. The disconcerted population construction of northern China is another cause such as the aging population. According to the statistics, the old over 60 years take up 9.5 percent of the population in northern China which leads to the increasing of the local death rate, another cause affecting the natural increasing of population.

According to the average volume-36.5 cubic meters [5]-of water using in northern China, we can get the water using model:

()A t unit: billion cubic meter

According to the analysis of the model in task one with the data of Table 5, we

20.000734() 5.3659exp(1.043)exp 0.0176042t A t t ??

-=**+ ?

??

can get industrial water using model through Matlab programming

=+

()0.1199 4.4068

B t t

The approximate coefficient is 0.9735 which close to 1, Model fits well.

The agricultural water using model

=+

C t t

()0.168119.8899

The approximate coefficient is 0.9522which close to 1, Model fits well.

The ecological water using model

()0.16420.585

=+

D t t

The approximate coefficient is 0.9648 which close to 1, Model fits well.

The statistics of the water usage in northern China from 2008 to 2014 are displayed in diagram 1

The industrial, agricultural and ecological water usages are displayed,see Figure 6:

Table 5:the water usage in the North China from 2008 to 2014

Unit: billion cubic meter

Figure 6:industrial, agricultural and ecological water usage

According to Figure 6, the agricultural water usage is on the rise much more than industrial and ecological water usage. Also, the growth rate is extremely high in accordance with the demand of agricultural production. The development of science and technology improves the output of agriculture and increase the demand of water as well. The low growth rate of industrial and ecological water usage comes from the intensive efforts put in environmental protection. The control on industrial sewage lowers the demand of water in this sector. The rise of the proportion of ecological water usage explains that the government in the North China put increasing emphasis on ecological water-supplying project. This is of great significance in improving local environment.

We can get the water consumption model in the North China from population, industrial, agricultural and ecological water using models.

7.2Water supplymodel in the North China

20.000734() 5.3659exp(1.043)exp 0.0176040.452224.8817

2t E t t t ??

-=**+++ ???

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