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2013年美赛B题国际一等奖论文

时间:2013-04-06


For office use only T1 ________________ T2 ________________ T3 ________________ T4 ________________

Team Control Number

18680
Problem Chosen

B

For office use only F1 ________________ F2 ________________ F3 ________________ F4 ________________

2013
Mathematical Contest in Modeling (MCM/ICM) Summary Sheet (Attach a copy of this page to your solution paper.)

Abstract
We build four models to devise national water strategy mainly considering five aspects: storage, movement, de-salinization, conservation and water price. Model One deals with prediction of freshwater resources. We collect total and demand amount of freshwater resources and analyze water deficit situation from global perspective . Then, in accordance with internationally recognized standards,we divide the whole nation into five subareas with water resources per capita. By calculating the amount of water deficit, we predict subarea Ⅲ and Ⅳ will face with extreme deficit of freshwater with GM(1,1). Model Two focuses on cost-efficient and feasible strategy considering storage movement,de-salinization and conservation.On the basis of conclusion drew from the first problem, we construct Minimumspanning Tree to obtain cost-efficient path and the minimum expenditure is 67.4928billion yuan. In Model Three, we discuss water price adjustment and based on comprehensive fuzzy evaluation.According to the Capital Asset Pricing Model of supply and demand, we calculate the water reduced due to price rise. To illustrate the water price adjustment process, we take Beijing as example. And we conclude that the water price for Beijing is 2.96yuan/ m3 in 2011, yet the highest water price affordable can reach 4.8075yuan, which suggests that certain price rise is acceptable. Besides, the amount of water consumption will decrease by 19.61% . The effect of saving water is obvious, so water price rise is an effective measure to case the deficit of freshwater. We build Model Four to analyze the economic, physical, and environmental implications of the strategy. Make online survey to acquire original data valuate the effect using modified Analytical Hierarchy Process with the aid of information entropy and the exponential transformation technology. We get the result that the water strategy pose greatest significance on economic impact, secondarily environment impact.

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CONTENTS

1. Introduction.....................................................................................................................3 2. Problem Analysis............................................................................................................ 3 3. Assumptions .................................................................................................................3 4. Model One: Prediction of Freshwater Resources......................................................... 4 4.1 GM(1,1) Model.....................................................................................................4 4.2 Solution and Comprehensive Evaluation............................................................6 4.3.1 National Analysis......................................................................................6 4.3.2 Subarea Analysis.......................................................................................7 4.3.3 Conclusion.................................................................................................9 5. Model Two: Minimum Spanning Tree Model.............................................................. 9 5.1 Water Transfer...................................................................................................... 9 5.2 Water Diversion Project..................................................................................... 11 6. Model Three: Water Price Adjustment Model............................................................ 13 6.1 Theory Introduction........................................................................................... 13 6.1.1Comprehensive Fuzzy Evaluation.......................................................... 13 6.1.2 Price Calculation Theory........................................................................13 6.1.3 Capital Asset Pricing Model.................................................................. 14 6.2 Comprehensive Evaluation................................................................................14 6.2.1 Decisive Indicators................................................................................. 14 ? Water Quality Index..................................................................................14 ? Population Density and GDP Evaluation................................................ 15 ? Degree of Satisfaction Evaluation........................................................... 15 6.2.2 Calculate the price vector....................................................................... 16 6.2.3 Water Cost............................................................................................... 16 6.3 Conclusion.......................................................................................................... 16 7. Model Four: Modified Analytical Hierarchy Process................................................ 17 7.1 Summary of Modified AHP...............................................................................17 7.2 Model Construction............................................................................................18 8. Strength and Weakness.................................................................................................19 8.1 Strength:..............................................................................................................19 8.2 Weakness:........................................................................................................... 19 9.Position Paper for the National People's Congress......................................................19 10. Appendix..................................................................................................................... 21 10.1 Tables................................................................................................................ 21 10.2 Programs........................................................................................................... 22

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Best Water Strategy Choice
1. Introduction
Fresh water is the constraint for development in countries. China is a country with serious water deficit.The amount of freshwater resources is 2.8 trillion cubic meters, only inferior to Brazil ,Russia and Canada, accounting for 6% of global water resources , ranking fourth in the world. But water resources per capita is only 2200 cubic meters, only 1/4 of the average globally, 1/5 of the United States, ranked 121 in the world, and is one of the most hydropenic countries in the world. Distribution of water resources in China is not balanced and can not match with population, land and economic layout. In recent years, contradiction of water resources uneven distribution regionally intensifies due to extreme climate occurring frequently in our country . The problem of water deficit in China becomes more and more prominent, and the deficit of water resources has produced adverse influence on normal living and industry. Faced with serious water deficit, it is essential that perfect water strategy be revised to store , transfer, save and control water pollution.

2. Problem Analysis
We tackle four main sub-problems: ? prediction of freshwater resources, ? design cost-efficient strategy considering storage movement,de-salinization and conservation, ? determine effective water price for saving water, and ? analyze the economic, physical, and environmental implications of the strategy To tackle the first problem, firstly, we collect total and demand amount of freshwater resources and analyze water deficit situation from global perspective . Then we divide the whole nation into five subareas in accordance with internationally recognized standards. Last, we predict which subarea will be faced with extreme deficit of freshwater with GM(1,1). For the second problem, on the basis of conclusion drew from the first problem, we construct the shortest path model to obtain cost-efficient path and calculate the expenditure. As for the third problem, we adjust the water price based on comprehensive fuzzy evaluation. According to the Capital Asset Pricing Model of supply and demand, we calculate the water reduced due to price rise. Eventually, we make online survey about the effect of water storage, transfer, desalination, price rise, high-tech and purification on environment, cultural, economic and purification impacts. With the aid of information entropy and the exponential transformation technology, we evaluate the effect with modified AHP.

3. Assumptions
? ? ? ? Suppose that there is not abnormal weather change such as drought in predicting the the water deficit. In desalination process, we suppose that the amount of desalination increases at certain rate over time. In the diverting process, all water is transferred through pipe. In the sewage process, the amount is a constant for some cities in North China

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4. Model One: Prediction of Freshwater Resources
4.1 GM(1,1) Model
Gray theory is a new method[1]for predicting uncertainty with poor information. It is applied widely for high accuracy and feasibility. And we introduce this method with three steps below: Step 1: Definition and Derivation ( 0) (0) (0) (0) Set the original data series: x = x (1) , x ( 2 ) ,? , x ( n ) .

(

)

The new sequence x is generated by accumulating the original data series.

(1)

x(1) = x (1) (1) , x (1) ( 2 ) ,? , x (1) ( n ) .
Where x(1) ( k ) = ∑ x(0) (i) ( k = 1, 2? n) .
i =1 k

(

)

And we define Gray derivative as dx

(1)

( k ) = x (0) ( k ) = x (1) ( k ) ? x (1) ( k ? 1) .

Let z(1) be the close series of mean x(1) ,namely

z ( ) ( k ) = 0.5 x( ) ( k ) + 0.5 x( ) ( k ?1) ( k = 1, 2, ?, n) .
1 1 1

(4.1)

() () () () Then z = z ( 2 ) , z ( 3) ,? , z ( n ) . So gray differential equation model of GM(1,1) is
1 1 1 1

(

)

dx ( ) ( k ) + az ( ) ( k ) = b .
1 1

(4.2)

Namely x

(0)

( k ) + az(1) ( k ) = b .
1

() Where x (0) ( k ) is gray differential, a is developing coefficient, z ( k ) is white background value, b is gray action value. (0) (1) Put the time value k = 2,3, ? , n into x ( k ) + az ( k ) = b ,we get

? x (0 ) ( 2 ) + az (1) ( 2 ) = b ? ? x (0 ) ( 3) + az (1) (3 ) = b ? ??? ? (0 ) (1) ? x ( n ) + az ( n ) = b Let

(4.3)

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? ? z (1) ( 2 ) ,1? ? ? T ? ? z (1) ( 3) ,1 ? ( 0) ( 0) ( 0) T Y = x ( 2 ) , x ( 3) ,? , x ( n ) , u = ( a, b ) , B = ? ?. ?? ? ? (1) ? ? ? z ( n ) ,1?

(

)

Where Y is data vector, B is data matrix, u is parameter vector. Then GM(1,1) model can be expressed as matrix equation : Y = Bu . With least square method we obtain ?= a ? ?, b u

( ) = ( B B)
T

T

?1

BT Y .

(4.4)

Step 2: Calculation Calculate gray differential equation and we obtain the predictive model u ? ak u ? (1) (0) ? x ( k + 1) = ( x (1) ? a )e + a ? (0) ?? ? (k ) ? (1) ? x ( k + 1) = exp( x ( k + 1) ? x ( k )), k = 1, 2,3,? , n Meanwhile, by serial-down calculation we get the original series. (0) 1 1 ? x ( i ) = x( ) ( i ) ? x( ) ( i ? 1)

(4.5)

(4.6)

Thus, we get basic formulas of gray prediction, namely, Equ. (4.5) and Equ.(4.6). Step 3: Test of Gray Prediction To test the accuracy of Grey prediction, we set four parameters and list them below: x0 ( k ) ? xi ( k ) × 100% ? Relative error ε ( k ) = x0 ( k ) 1 n ∑| ε (k ) | . n ?1 k =2 So the precision of GM(1,1) can be expressed as p = 1 ? ε . 1 n Correlation degree ri = ∑η k . n k =1 min min | x0 (t ) ? xs (t ) | + ρ min min | x 0 (t ) ? x s (t )| s t s t η k = ( ) Where, i . | x0 ( k ) ? xi ( k ) | + ρ min min | x0 (t ) ? x s (t ) | In addition, mean residual is ε =
s t

?

ρ ∈ [ 0,1] is distinguishing coefficient. Generally speaking, the closer this
? coefficient is to 1, the better is the correlation degree. Variance ratio c = S1 S0 .
n
2 Where, S0 = ∑ ( x ( k ) ? x) is variance between x ( k ) and x

.

k =1

?

Little probability of error p p = { ε (k ) ? ε < 0.6745S1 } .The greater is the frequency of falling into the

? interval ? ?ε ? 0.6745S1 , ε + 0.6745S1 ? , the better is the model. And the precision
is divided into four levels below.

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Level Level 1 (good) Level2 (qualified) Level 3 (not very qualified) Level4 (not qualified)

Table 1. Precision Level of Prediction References Table Mean Relative Correlation Variance Little Probability Error Degree Ratio of Error
0~0.01 0.01~0.05 0.05~0.1 0.1~0.2 0.8~0.9 0.7~0.8 0~0.35 0.35~0.5 0.5~0.65 0.65~0.8 0.95~1 0.8~0.95 0.7~0.8 0~0.6

0.6~0.7
0~0.6

4.2 Solution and Comprehensive Evaluation
4.3.1 National Analysis According to the data sources (derived from National Statistics Database), we tabulated the national total amount and demand of freshwater resources from the year 2002 to 2011. (detailed in Appendix) We put the data into Matlab, and we obtain the following results: ? Total Amount of Freshwater Resources: (1) ? x ( k + 1) = ?19662746.249× e?0.0013417×k + 19691007.5491 ? Demand of Freshwater Resources: (1) ? x ( k + 1) = 353407.7571× e?0.015285×k ? 347910.4771 And the test results are listed as follows: Table 2. Test Results of GM(1,1) Mean Relative Correlation Variance Ratio Error Degree
0.0672 0.0072 0.6215 0.6713 55.36% 15.53%

Standard
Total Amount Demand

Little Probability of Error
80% 100%

Hence,we can infer that the GM(1,1) model prediction is acceptable. According to the predicted formula above, we can predict the national total amount and demand of freshwater resources available from the year 2013 to 2025. And we show the results with a three-dimensional column diagram.

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Predicted Results of Total Amount and Demand

2025 2024 15000 10000 2022 2021 2020 2019 0 2017 1 2015 2 2013 2014 2016 2018 2023

Amount

5000

Year

Figure 1. Predicted Results(1:total amount;2:demand amount) After thorough analysis of the diagram We conclude that total amount is always larger than the demand of freshwater from the year 2012 to 2025. And the difference between them decrease over time, which suggests that supply and demand become serious on a whole. The predicted results can not reflect the fact that many provinces of China are faced with severe deficit of freshwater. It maybe caused by ignoring the population distribution and geography conditions. So we divide the whole nation into several subareas for detailed discussion. 4.3.2 Subarea Analysis In accordance with internationally recognized standards(as in Table 3), we can divide the whole nation into five subareas by analyzing water resources per capita of 31 provinces. Table 3. Level of deficit Water Resources per Below 500 500-1000 1000-2000 2000-3000 capita/ m3 Level of deficit mild moderate severe extreme And the following Chinese map shows the results:

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Figure 2. Provincial Water Usage per capita Conditions (The picture, available from the Internet, is reproduced by filling 31 provinces with five different colors.) For each subarea, we list the total amount and demand of provinces included from the year 2002 to 2011.(detailed in Appendix). Then we put the data into Matlab and obtain the following results: Table 4. Test Results of GM(1,1) Correlation Variance Ratio Degree
0.6373 0.6298 55.92% 49.38% 61.53% 42.79%

Subarea
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ

Mean Relative Error
0.0633 0.0639 0.0822 0.1473 0.1204

Little Probability of Error
80% 90% 80% 100%

0.6493
0.5554 0.6528

45.63%

90%

From the table, we can concluded that test of subarea Ⅳ and Ⅴ is not acceptable. Considering abnormal heavy storm of arid region in 2004 and less mean annual precipitation in 2006, we ignore the total amount of freshwater Ⅳ in 2004 and Ⅴ in 2006.And the following table shows that the prediction test of subarea ⅣandⅤ is acceptable. Added Prediction Test of References:

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Subarea
Ⅳ Ⅴ

Mean Relative Error 0.027 0.0043

Correlation Degree 0.6526 0.6189

Variance Ratio 36.49% 11.08%

Little Probability of Error 90% 100%

Referring to materials relevant[2], we know that

Va = Total amount × 40% Vl = Vn ? Va Where V a is available water, and Vl is the amount of deficit water. So we process data of total amount (detailed in Appendix) by multiplying 0.4and then we calculate amount of deficit in five subareas.(detailed in Appendix).
4.3.3 Conclusion It is conclude that the subarea Ⅲ Ⅳ will be faced with extreme deficit of freshwater, including Ningxia, Hebei, Shandong, Henan, Shanxi, Tianjin , Gansu, Liaoning, Jiangsu and Shanghai. And the predicted result is consistent with the fact that North China is lack in water resources severely.

5. Model Two: Minimum Spanning Tree Model
5.1 Water Transfer
Vl = Va + Vd + Vs + Vt Where, Vl is the amount of water shortage Va is the amount of available water Vd is the amount of desalination Vs is the amount of sewage treatment Vt is the amount of water movement
ed Water D eficit ? Predict redicted On the basis of conclusion drew from Model One, we analyze the water deficit situation in the Beijing, Tianjin, Hebei and Shandong Province.We predict water deficit of these regions with the same method and the results are shown below: Table 5. Predicted Water Deficit 2015 2017 2019 23.85 23.33 22.76 17.79 17.83 17.86 133.40 131.34 129.28 129.90 139.82 149.08 151.78 176.71 200.81

Province Beijing Tianjin Hebei Shandong Henan

2013 24.32 17.73 135.47 119.26 125.80

2021 22.13 17.88 127.24 157.74 224.26

2023 21.44 17.88 125.19 165.85 247.26

2025 20.70 17.87 123.16 173.46 269.96

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We can conclude that water deficit increases over time in Tianjin, Shandong, Hebei and Henan province. In Beijing water deficit almost remains unchanged, and Henan province is in the most serious situation. ? Available Water Resources Table 6. Available Water Resources 2015 2017 2019 2021 30.73 32.76 34.93 37.24 15.87 16.77 17.71 18.71 146.21 147.59 148.99 150.40 242.48 222.30 203.80 186.84 268.99 237.91 210.43 186.12

Province Beijing Tianjin Hebei Shandong Henan

2013 28.82 15.03 144.84 264.49 304.12

2023 39.70 19.77 151.82 171.29 164.62

2025 42.33 20.88 153.26 157.04 145.60

? Amount of Desalination After thorough research of materials[3]-[7];, we know that desalination increases at a certain rate p in Tianjin, Hebei and Shandong province. And the amount of desalination doubles per decade. (1 + p)10 = 2 , p = 0.0718 So we can get the amount of desalination of the next n -th year from 2012: i Vdn = Vdi0 × (1 + p ) n i = 1, 2,3 Where , Vdi0 is the amount of desalination in 2012
i Vdn is the amount of desalination the next n -th year from 2012 i = 1,2,3 means Tianjin, Hebei and Shandong province Then we predicted the amount of desalination in Tianjin, Hebei and Shandong province from 2013 to2025:
3

i Total amount: Vdn = ∑ Vdn = 1.72 ×108 m3 i

Area Tianjin Hebei Shandong

2013 0.7042 0.3912 0.6259

Table 6. Amount of Desalination 2015 2017 2019 2021 0.8089 0.9293 1.0675 1.2263 0.4494 0.5163 0.5931 0.6813 0.7190 0.8260 0.9489 1.0900

2023 1.4087 0.7826 1.2522

2025 1.6182 0.8990 1.4384

? The calculation of sewage treatment As for the amount of sewage treatment, the processing capacity among five provinces changed little every year, so we assume the processing capacity is a constant value[8]-[12], the data is as follow: Table 8. Amount of Sewage

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Area Amount of Sewage

Beijing 6.86

Tianjin 14.56

Hebei 8.51

Shandong 22.92

Henan 14.20

According to analysis above

Vl = Va + Vd + Vs + Vt Vt = Vl ? Va ? Vd ? Vs
Then we get the Transferred Water Table 9. Transferred Water 2017 2019 16.47 15.89 2.34 2.23 122.31 120.18 116.07 125.21 162.51 186.61

Area Beijing Tianjin Hebei Shandong Henan

2013 17.46 2.46 126.56 95.72 111.60

2015 16.99 2.41 124.44 106.26 137.58

2021 15.27 2.09 118.04 133.73 210.07

2023 14.58 1.91 115.90 141.68 233.06

2025 13.83 1.69 113.75 149.10 255.76

5.2 Water Diversion Project

Figure 3. Water DiversionDiagram We assume that the project use pipeline to move water, so the water evaporation can be ignored. And we estimate the price: r =P/L Where, r is the price of one cubic meter transferring 1 km P is the price of one cubic meter L is the length of pipeline r = 2.506 × 103 The weight of every segment: wkj = Vs j × Lkj × r

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Where, wkj is the weight move from the j -th city to the i -th city.

Vs j is the demand of the j -th city. Lkj is the distance from the j -th city to the k -th city.
We use the Kruskal algorithm in minimum spanning tree model to solve the problem, the algorithm is: 1. Choose e1 ∈ E ( G ) to make e1 brim with minimum weight 2. If e1 , e2 ,…, ei was chosen, then choose ei +1 from E (G) ? {e1 , e2 , …, ei } to make: (1) there is no circle in {e1 , e2 ,…, ei , ei +1} (2) ei +1 is the brim with minimum weight in E (G) ? {e1 , e2 , …, ei } 3. Do until choose ev ?1 Where ei stands for the brim of two point E (G ) stands for the brim collection

Figure 4. Weighted Graph (The blue lines bold reflect the shortest path.)
10 10

Smin = ∑ Si = ∑ vi × li × r
i =1 i =1

Where , S min is the minimum expenditure

Si is the expenditure of i -th path transferring water vi is the flux of i -th
After calculation, we get Smin = 67.4928×10 yuan.
8

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6. Model Three: Water Price Adjustment Model
6.1 Theory Introduction
omprehensive Fuzzy Evaluation 6.1.1C 6.1.1Comprehensive For each factor, the subordinated function can be confirmed by using the formula below [14]: ? 1 x ≤ xi1 ? ? x ?x ?i1 ( x ) = ? i 2 xi1 < x < xi 2 x ? x i 2 i 1 ? ? x ≥ xi 2 ? 0 ? x ? xi. j ?1 xi , j ?1 ≤ x ≤ xij ? ? xij ? xi , j ?1 ? ? xi , j +1 ? x ?ij ( x ) = ? xij < x < xi , j +1 . i = 1, 2,3, 4; j = 2,3, 4, j. x ? x i , j + 1 ij ? ? 0 x < xi , j ?1 , x ≥ xi , j +1 ? ? ? ? 1 ? ? x ? xi 4 ?i 5 ( x ) = ? ? xi5 ? xi4 ? 0 ?

x ≥ xi 5 xi 4 < x < xi 5 x ≤ xi 4

Where i is for evaluated element, j is the subordinated function of element.

V = W ? R ,where V is the evaluation result, W is the weight vector and R is the fuzzy evaluation matrix.
6.1.2 Price Calculation Theory ?

P = A?E C ? D
where, P is the upper limit of water price, A is the endurance index[15],

E is average income, C is water consumption and D is the cost of water supply.
?

S = ( P, P 1, P 2, P 3, 0 )

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Where, S is price vector and Pi (i = 1, 2,3) is obtained by dividing P with the interval P / 4 . ?

WRP = V ? S
Where WRP is the water cost. 6.1.3 Capital Asset Pricing Model

D.J. James and R.R. Lee proposed Capital Asset Pricing Model of supply and demand[17]:
?P? Q2 = Q1 ? ? 1 ? ? P2 ?
E

Where P 1 is the original price, P 2 is the adjusted price, Q 1 is the original amount of water consumption, Q2 is the adjusted amount and E is price elasticity index of water resources.

Thus, we can get %(reduced ) =

Q1 ? Q2 × 100% . Q1

6.2 Comprehensive Evaluation
In order to evaluate water price rise quantitively,we take the capital city Beijing in 2011 for example.(the data required below is available from the Beijing Statistical Bureau) 6.2.1 Decisive Indicators ? Water Quality Index In accordance with GB3838-2002 Environmental Quality Standards of Surface Water [13], we select main pollution parameters as evaluation factors. Build the factor set and determine the evaluation set with relevant level of water quality. Chaobai River is the key water source base[14]. So we select 9 parameters as evaluation factors and divide the water quality into five levels. Table 10. Water Quality Index of Different Detecting Segments in Chaobai River(mg.L-1) CODCr BOD5 NH3-N TP Cu Zn Cd 3.673 1.82 0.345 0 0.007 0.014 0.003

No. 1

DO 9.383

Pb 0.034

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2 3 4 5 6 7 8 9

10.072 8.081 10.847 0 10.804 0 10.825 3.673 10.33 2.204 8.883 8.815 8.965 6.612 7.243 44.077

1.558 0 0 2.87 0.99 2.761 1.05 0.963

0.387 0.333 0.274 0.37 0.336 0.223 0.279 0.733

0 0 0 0 0 0 0 0.036

0.004 0.003 0.004 0.004 0.002 0.007 0.004 0.001

0.002 0.012 0.024 0.01 0 0.009 0.013 0.014

0.003 0.005 0.006 0.003 0.003 0.007 0.003 0.004

0.019 0.037 0.028 0.008 0.013 0.055 0.055 0.007

With the method illustrated above(detailed in 6.1), we obtain the result:

R1 = {0.220, 0.283, 0.283, 0, 0}
? Population Density and GDP Evaluation Then, we process the data collected from the National Statistical Bureau with the same method above. Table 11. Population Density and GDP Evaluation Criterion value evaluation Very high High Moderate Low Population Density/km 9266 5000 3126 1500 Per Capita GDP/dollar 5000 3800 2600 1400 We obtain the result as follows: R2 = {0.144, 0.852, 0, 0, 0} R3 = {0, 0, 0.3899, 0.6101, 0} ? Degree of Satisfaction Evaluation Table 12. Degree of Satisfaction Evaluation Criterion value evaluation Very low Low Moderate High Very high Degree Not care Moderate Not high Higher Unbearable Standard Value 2.0 26.5 51.0 75.5 100 We obtain the result as follows: R4 = {0, 0, 0.917, 0.083, 0} To sum up, we get the evaluation matrix: 0 0? ? R1 ? ? 0.220 0.283 0.283 ? R ? ? 0.144 0.852 0 0 0? 2? ? ? ? R= = ? R3 ? ? 0 0 0.3899 0.6101 0? ? ? ? ? 0 0.917 0.083 0? ? R4 ? ? 0

Very low 557 200

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And we determine the weights by building the matrix reciprocal matrix by pair comparison.

W = ( 0.4 0.3 0.15 0.15 )
So the evaluation result is

V = W ? R = ( 0.1312 0.3688 0.3092 0.1040 0)
6.2.2 Calculate the price vector the upper limit of water price P = 0.03 × 32903 ? 2.8 = 9.6 79.5

S = ( P, P 1, P 2, P 3, 0 ) S = (9.6 5.82 3.88 1.94 0)
6.2.3 Water Cost

WRP = V ? S = V ? ( P, P1 , P2 , P3 , 0 )
= ( 0.1312 0.3688 0.3092 0.1040 0) ? ( 9.6 5.82 3.88 1.94 0 ) = 4.8075
T

? 2.96 ? 8 3 Q2 = 35.96 × 108 ? ? = 28.9093× 10 m ? 4.8075 ? 35.96 ? 28.9093 %(reduced ) = × 100% = 19.61% 35.96

0.45

6.3 Conclusion
On one hand ,the water price for Beijing is 2.96yuan/ m3 in 2011,yet from analysis above, the highest water price affordable can reach 4.8075,which suggests certain rise is acceptable. On the other hand, the amount of water consumption will decrease by 19.61% . The effect of saving water is obvious, so water price rise is an effective measure to case the deficit of freshwater. Besides, increasing block water tariff is also beneficial for saving water.

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7. Model Four: Modified Analytical Hierarchy Process
7.1 Summary of Modified AHP
According to different roles of each layer in the decision making process and model structure, the Modified AHP choose different statistical techniques and index to improve the AHP model. We make online survey to acquire statistical data and determine the evaluation of various weight indicators objectively. The following diagram shows the process: In the diagram, m is the number of criterion and n is number of evaluation indicators.
bij = xik ? x jk

X = ( xik )mn

pik =

xik
m

∑x
i =1

rij =

e ij
m

b

ik

∑e
i =1

bij

m

ek = ?∑ pik ln pik
i =1

m

w jk = ∑ r ij
j =1

wk =

uk

∑u ,
k k =1

n

uk =

1 ek

wik =

w ik
m

∑w
j =1

jk

n

∑w w
k k =1

ik

Optimal solution
Figure 5. Flow Chart of Modified AHP

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7.2 Model Construction
To acquire statistical data ,we make online survey about the effect of water storage, transfer, desalination, price rise, high-tech and purification on environment, cultural, economic and purification impacts. And we construct hierarchical structure as follows:

Effect Analysis

Storage

Transfer

Desalination Price rise

H ig h -T e c h Purification

Environment impact

Cultural impact

Economic impact

Physical impact

Figure 6. Hierarchical Structure

After thorough research of the data available from the survey, and we get the following results: Table 13. Score of Evaluation Indicator Environment Cultural Economic Physical Items impact impact impact impact Storage 9.3 9.6 9.8 9.9 Transfer 9 7.6 9.5 8.8 Desalination 8.5 7 9 8.5 Price rise 9.8 7.1 9.5 8.6 High-Tech 9.8 8 9.5 9.6 Purification 10 9.4 9.8 9.6 * full mark is 10. Firstly, calculate weight of criterion layer to the objective layer using the technology of information entropy. And we get the weight:

w1 = 0.166403 ; w2 = 0.166762 ; w3 = 0.166424 ; w4 = 0.166264 ; w5 = 0.166742 ; w6 = 0.166401 .
Then, do index transformation of score difference on each alternative layer. Last, construct judgment matrix to calculate weight of alternative layer to the criterion layer. And the result is below:

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Table 14. Relative Weight Items Environment impact Cultural impact Economic impact Physical impact Relative Weight 0.71 0.23 0.28 0.32
n

The comprehensive weight can be obtained with comprehensive weight of ith alternative to the object layer. Table 15. Grading Result Comprehensive Weight 0.31 0.14 0.33 0.24

∑w w
k k =1

ik

,which is the

Items Environment impact Cultural impact Economic impact Physical impact

Grading Result 2 4 1 3

8. Strength and Weakness
8.1 Strength:
? ? ? In Model Two, we simplify the problem by constructing wighted graph, which can be applied widely in our daily life. With the aid of information entropy and the exponential transformation technology, we determine weight of various indicators objectively. By means of investigation and statistical information entropy and the exponential transform, modified AHP avoid personal subjectivity in evaluation and selection .In addition, it provides a good solution to solve the difficulty in making intractable choices, and improve the accuracy of evaluation results.

8.2 Weakness:
? ? Due to limit of database, we just obtain the amount of freshwater from the year 2002 to 2011, so it may be not very accurate. Many factors may be not taken into consideration for the purpose of simplificatio

9.Position Paper for the National People's Congress
To authority relevant: Freshwater is the constraint for development in countries. China is a country with serious water deficit. The amount of freshwater resources is 2.8 trillion cubic meters, only inferior to Brazil, Russia and Canada, accounting for 6% of global water resources , ranking fourth in the world. But water resources per capita is only 2200

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cubic meters, only 1/4 of the average globally, 1/5 of the United States, ranked 121 in the world, and is one of the most hydropenic countries in the world. Distribution of water resources in China is not balanced and can not match with population, land and economic layout. So it is essential that perfect water strategy be revised to store , transfer, save and control water pollution. The water strategy includes the following parts: The South-to-North diversion project plays an important role in transferring water resources from water-sufficient regions to the North China. Moderate water price rise is acceptable. And water price rise is an effective measure to case the deficit of freshwater. Besides, increasing block water tariff is also beneficial for saving water. Other techniques, such as water purification and desalination, is also very important in addressing the problem of water deficit. Hope our strategy will be helpful to relieve the serious water deficit situation.

Yours sincerely, Team 18680

10. References
[1]Si,S.Q.,&Sun,Y.Q(Eds).(2011).MathematicalModeling.(pp.372-375).Beijing,China: Nati-onal Defense Industry Press. [2]Zuo,Q.T.(2011). Caculation of the utilization ratio of water and discussion of threshold value.Journal of Hydraulic Engineering,42(11),1342-1378. [3]…..Retrieved,February3,2013,from[url]http://www.dowater.com/info/2013-01-29/1 2 772 0.html[/url]. [4]…..Retrieved,February3,2013,from[url]http://hzs.ndrc.gov.cn/newhjyzyjb/t201207 02 _ 489224.h tm [/url]. [5]…..Retrieved,February3,2013,from[url]http://www.doc88.com/p-287604285665.ht m l[/url]. [6]…..Retrieved,February3,2013,from[url]http://www.doc88.com/p-99714745359.ht ml [/url] [7]…..Retrieved,February3,2013,from[url]http://finance.sina.com.cn/china/dfjj/ 201112 12/091410975294.shtm[/url]. [8]…..Retrieved,February3,2013,from[url]http://www.dowater.com/info/2013-01-29/1 27 720.html. [9]…..Retrieved,February3,2013,from[url]http://info.ep.hc360.com/2005/01/2617572 53 42.shtml[/url]. [10]…..Retrieved,February3,2013,from[url]http://news.iqilu.com/shandong/yaowen/2 010 / 11 14/360426.shtml[/url]. [11]…..Retrieved,February3,2013,from[url]http://news.h2o-china.com/html/2012/11/ 1 11 0 33_1.shtml[/url]. [12]…..Retrieved,February3,2013,from[url]http://www.zhb.gov.cn/zhxx/hjyw/200810 /t2 0081008_129694.htm[/url]. [13]Chai,H.Q, et al.,(2010).Reseach of the water price in Tianjin Based on the fuzzy co- mprehensive evaluation model. Acta Agriculturae Jiangxi,22(10),166-169.

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[14]…..Retrieved,February3,2013,from[url]http://wenku.baidu.com/view/354f5251ad 02d e 80d4d8 4003.html [15]Jiang,W.L.(1999).Value theory of water resource.Beijing,China:Science Press. [16]Jiang,W.L,et al.,(1994).Reseach the price of water resource. China Water and Wastewater.42(11),22-23.

10. Appendix
10.1 Tables
Table 1. Total Amount of Freshwater Resources Available Year
2002 2003 2004 2005 2006

Total Amount
28261.30 27460.19 24129.56 28053.10 25330.14

Year
2007 2008 2009 2010 2011

Total Amount
25255.16 27434.30 24180.20 30906.41 23258.53

* Total amount is in 108 m3 unit.
Table 2. Demand of Freshwater Resources Demand Year
28261.30 27460.19 24129.56 28053.10 25330.14 2007 2008 2009 2010 2011

Year
2002 2003 2004 2005 2006

Demand
25255.16 27434.30 24180.20 30906.41 23258.53

* Demand amount is in 108 m3 unit.
Table 3. Total amount Ⅱ Ⅲ 5750.36 7212.72 5304.53 7164.29 5066.44 5571.97 5428.25 7175.53 4450.70 6488.61 4780.90 6374.63 5230.60 6882.20 4642.70 6641.49 5438.40 8489.40 3992.40 6059.52 Table 4.

Subarea
2002 2003 2004 2005 2006 2007 2008 2009 2010 2011


14069.19 12373.38 11756.18 13075.74 12753.10 11958.00 13443.10 11187.90 14620.20 11148.60


566.60 1086.25 661.60 1113.76 850.40 986.10 831.50 780.30 1205.40 1029.40


656.05 1531.74 1073.31 1259.89 787.30 1155.50 1047.10 927.80 1152.80 1028.80

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Subarea 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

Ⅰ 1406.64 1381.1 1427.7 1471.2 1479.24 1523.8 1542.49 1533.66 1536.65 1550.37

Demand amount Ⅱ Ⅲ 605.46 1776.23 622.4 1742.2 628.3 1798.1 637.9 1825 642.84 1893.4 636.3 1895.4 633.14 1960.32 646.17 2012.67 656.89 2049.89 655.87 2096.97 Table 5. Test Results of GM(1,1) Correlation Variance Ratio Degree
0 0.596 0.5960 0.6636 0.6762 0.6526 0.8005 20.83% 21.85% 6.29% 36.49% 38.06%

Ⅳ 728.51 683.4 777.6 776 809.96 823.7 823.27 812.65 817.68 823.59

Ⅴ 980.43 891.5 916.2 923.3 969.55 939.4 950.72 959.99 960.89 980.4

Subarea
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ

Mean Relative Error
0.0112 0.0058 0.0039 0.027 0.0075

Little Probability of Error
100% 100% 100% 90% 90%

Area Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Area Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ

2013 3400.07 1119.81 658.386 -461.02 -642.63 2014 3382.696 1086.668 633.8215 -469.34 -664.636

2015 3365.04 1053.97 608.331 -477.79 -686.29

Table 6. Water deficit Conditions 2017 2019 3328.879 3291.551 989.9289 927.6099 554.4767 496.6193 -495.161 -513.135 -728.667 -769.86 2018 3310.363 958.5571 526.0615 -504.07 -749.403

2021 3253.028 866.9592 434.5454 -531.741 -809.991

2023 3213.278 807.9184 368.0313 -551 -849.168

2025 3172.26 750.430 296.842 -570.93 -887.49

2016 3347.104 1021.733 581.8915 -486.405 -707.637

2020 3272.441 897.0797 466.1231 -522.357 -790.051

2022 3233.309 837.2412 401.8578 -541.287 -829.693

2024 3192.931 778.9839 333.0362 -560.882 -868.43

10.2 Programs
GM(1,1) Prediction: function GM(X0) %format long ; [m,n]=size(X0); X1=cumsum(X0);

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X2=[]; for i=1:n-1 X2(i,:)=X1(i)+X1(i+1); end B=-0.5.*X2 ; t=ones(n-1,1); B=[B,t] ; YN=X0(2:end) ; P_t=YN./X1(1:(length(X0)-1)) A=inv(B.'*B)*B.'*YN.' ; a=A(1) u=A(2) c=u/a ; b=X0(1)-c ; X=[num2str(b),'exp','(',num2str(-a),'k',')',num2str(c)]; strcat('X(k+1)=',X) %syms k; for t=1:length(X0) k(1,t)=t-1; end k Y_k_1=b*exp(-a*k)+c; for j=1:length(k)-1 Y(1,j)=Y_k_1(j+1)-Y_k_1(j); end XY=[Y_k_1(1),Y] CA=abs(XY-X0) ; Theta=CA XD_Theta= CA ./ X0 h=mean(XD_Theta) AV=mean(CA); R_k=(min(Theta)+0.5*max(Theta))./(Theta+0.5*max(Theta)) ;% P=0.5 R=sum(R_k)/length(R_k) Temp0=(CA-AV).^2 ; Temp1=sum(Temp0)/length(CA); S2=sqrt(Temp1) ; %---------AV_0=mean(X0); Temp_0=(X0-AV_0).^2 ; Temp_1=sum(Temp_0)/length(CA); S1=sqrt(Temp_1) ; TempC=S2/S1*100; C=strcat(num2str(TempC),'%') %---------SS=0.675*S1 ; Delta=abs(CA-AV) ; TempN=find(Delta<=SS); N1=length(TempN); N2=length(CA);

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TempP=N1/N2*100; P=strcat(num2str(TempP),'%') The Shortest Path Model: clear,clc a=[ 0 263 193.7 inf inf 552.6 798.7 inf inf inf inf 263 0 62.6 340.9 376.4 inf inf inf inf inf inf 193.7 62.6 0 inf 374.5 357 inf inf inf inf inf inf 340.9 inf 0 267.4 inf inf inf 125.5 inf inf inf 376.4 374.5 267.4 0 51 inf 401.7 282 370 284 552.6 inf 357 inf 51 0 299.4 409 inf inf inf 798.7 inf inf inf inf 299.4 0 182.5 inf inf inf inf inf inf inf 401.7 409 182.5 0 inf inf inf inf inf inf 125.5 282 inf inf inf 0 143.1 inf inf inf inf inf 370 inf inf inf 143.1 0 109.6 inf inf inf inf 284 inf inf inf inf 109.6 0]; a(4,7)=42;a(5,6)=70; [i,j,b]=find(a); data=[i';j';b']; index=data(1:2,:); loop=length(a)-1; result=[]; while length(result)<loop temp=min(data(3,:)); flag=find(data(3,:)==temp); flag=flag(1); v1=index(1,flag); v2=index(2,flag); if v1~=v2 result=[result,data(:,flag)]; end index(find(index==v2))=v1; data(:,flag)=[]; index(:,flag)=[]; end result


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