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美赛训练题3

发布时间:2013-12-31 09:47:39  

Water, Water, Everywhere

Summary

In this paper, we determine a strategy for 2013 to meet the projected water needs of China in 2025. Firstly, we investigate the condition of water storage and usage in China and then we conduct a deep research on water transfer, seawater desalination and water resources optimal allocation.

For water storage, we build an Improved-Grey Prediction Model to predict water storage and usage in 2025. The prediction of water storage is 2418.4 billion ton (only approximately 40% available) and water usage is 719.3 billion ton. Although water storage is larger than usage, China is still in short of water because of water contamination and uneven distribution.

For water movement, we build a Transportation Model (based on Transportation Algorithm)to solve water transfer problem. In this model, we try to meet the needs of 2025 through national water transfer project. Through the analysis of data, there will be 14 provinces short of water with a total water quantity 114.196 billion ton and the 5 suitable water supply provinces canonlyprovide 111.823 billion ton in2025. The total cost of water transfer is 74.791086 billion Yuan.

For water desalination, we build an Optimal Location Model to choose the sites of seawater desalination plants. According to the result, we choose to construct the seawater desalination plants in Hebei, Tianjin, Shandong and Liaoning provinces.

For water conservation, we build a Water Resources Optimal Allocation Model based on Multi-Objective Genetic Algorithm. In this model, we assume that water resources are allocated to five water users: primary industry, secondary industry, tertiary industry, town and country. And we consider three object functions: economy, population and water circumstance function. Through this model, we obtain the optimal allocation water resources.

We draw out our Water Strategy for 2013 by combining the four models above. The strategy consists of two parts: Engineering Strategy and Management Strategy. Engineering Strategy includes Water Transfer, Seawater Desalination, Sewage Recovery and Rainwater Collection. Management Strategy includes Water Resources Optimal Allocation, Water Conservation and Reasonable Water Pricing System.

In the end, we analyze the economic, physical, and environmental implications of our strategy from the quantitative point of view by AHP Model. To better assess our model, we do the sensitivity test on Grey Prediction Model and Optimal Location Model. In a nutshell, our strategy is feasible, effective and of practical value.

Key words: Improved-Grey Prediction Model, Transportation Model, Water Resources Allocation Model, Water Strategy, sensitivity test

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Contents

1 Introduction ............................................................................................................................ 2

2 Problem Analyses ................................................................................................................... 2

3 Assumptions ........................................................................................................................... 4

4 Prediction of water consumption and water storage ............................................................... 4

4.1 Prediction of water consumption in 2025 .................................................................... 4

4.1.1 The Grey Prediction Model ............................................................................... 4

4.1.2 Solving the Model ............................................................................................. 6

4.1.3 Prediction of the Total Population ..................................................................... 9

4. 2 Prediction of water storage in 2025 ........................................................................... 11

5 Water transfer ........................................................................................................................ 11

5.1 Terminology and Definitions ...................................................................................... 11

5.2 Assumptions............................................................................................................... 12

5.3 Optimum Water Transfer ........................................................................................... 12

5.3.1 Transfer Cost Estimate .................................................................................... 12

5.3.2 Transportation Algorithm ................................................................................ 13

5.3.3 Solution and Evaluate...................................................................................... 14

6 Desalination of sea water...................................................................................................... 17

6.1 Sites ............................................................................................................................ 17

6.2 Desalination methods ................................................................................................. 17

6.3 Total cost ................................................................................................................... 18

6.4 Result analysis ........................................................................................................... 18

7 Optimal allocations of water resources ................................................................................ 18

7.1 Introduction ................................................................................................................ 18

7.2 The Multi-Objective Genetic Algorithm based on parataxis selection ...................... 18

7.3 The water resources optimal allocation based on Multi-Objective Genetic Algorithm

......................................................................................................................................... 19

7.3.1 Genetic coding of decision variable ................................................................ 19

7.3.2 Design of object function ................................................................................ 19

7.3.3 Restriction condition ....................................................................................... 20

7.4 Example analyses ....................................................................................................... 20

7.5 Conclusion ................................................................................................................. 21

7.6 Model evaluation ....................................................................................................... 22

8 Our water strategy and implications ..................................................................................... 22

8.1 Our water strategy for 2013 ....................................................................................... 22

8.2 Analysis of the implications of our strategy by AHP Model ..................................... 23

8.2.1 Establish the hierarchical structure model ....................................................... 23

8.2.2 Construct the judgment matrixes ..................................................................... 23

8.2.3 Calculate the characteristic value of maximum modulus and the characteristic

vector .........................................................................................................................24

8.2.4 Weight table ..................................................................................................... 25

9 Analysis and evaluation of the model .................................................................................. 25

9.1 Sensitivity analysis .................................................................................................... 25

9.2 Strengths and Weaknesses ......................................................................................... 25

Reference…………………………………………………………………………………......28

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1 Introduction

Fresh water is the limiting constraint for development in much of the world, not excepting China. The amount of water resources in China accounted for 6% of the world, but the support of 21% of the world's population. In recent years, the water resources of China face many crises, such as the serious water shortage in some areas and the severe problem of water pollution, although a lot of achievements have been made in the water saving and conservation of water and soil.

The current water resource strategy in our country includes: reasonable control of the total development, optimal allocation of water resources, high efficient utilization of water resources, and effective protection of water resources.

With comprehensive consideration of storage, movement, de-salinization, and conservation, we plan to accomplish goals as follows:

(1) To accurately predict the water consumption and water storage in 2025.

(2) To analyze the ways of solving the water problem we faced and Put forward effective and feasible measures to solve the concrete problems.

(3) To summarize the measures and the improved method we proposed, combined with our existing water resources strategy, and then list our water resources strategy which includes two aspects and 6 parts.

(4) To discuss the economic, physical, environmental implications of our strategy and provide a non-technical position paper to governmental leadership outlining our approach.

The approaches we use to realize such goals are:

(1) We use the Improved-Grey Prediction Model to predict the water consumption and water storage in 2025. Especially, we use two methods to predict the water consumption, one is directly prediction and the other is indirectly predicted by population. Finally we get the water shortage of all provinces in 2025.

(2) In this section, we build three models to solve the problem of water shortage in prediction. Firstly, we use the Transportation Model to determine the optimum

water diversion point and water transfer routine for water transfer problem. Secondly, we come out Optimal Location Model to predict the expense of water desalination, and then choose the optimum sites of water desalination plants. Lastly, we make use of Multi-Objective Genetic Algorithm to build Water Resources Optimal

Allocations Model. We obtain the method of water resources optimal Allocations.

(3) Aiming at water resources strategy we proposed, we analyze the influence and weight of 6 measures of our strategy and the three aspects, economic, physical and environmental to a reasonable water strategy through the Analytic Hierarchy Process.

2 Problem Analyses

The problem is to determine a strategy for 2013 to meet the projected water needs of China in 2025. We firstly investigate the condition of water storage and usage in

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China. And then make the prediction of water storage and usage in 2025. Secondly, we analyze the water shortage condition with the predicted data of 2025. Next we determine the strategy in the view of water source and usage. Consequently, there are two corresponding ways to solve the water shortage problem. One is to make the best use of existing water resources, including water resources optimal allocation, water conservation, water transfer and reasonable water price. The other is to ‘make water’ including seawater desalination, sewage recovery and rainwater collection. Afterwards we make a deep research on water transfer, seawater desalination and water resources optimal allocation in our model. After model analysis, we draw out our water strategy for 2013. The water strategy for 2013 consists of two parts: engineering strategy and management strategy. Engineering strategy includes water transfer, seawater desalination, sewage recovery, and rainwater collection. In the end, we discuss the economic, physical, and environmental implications of our strategy. Figure1 shows the details.

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Figure1. Flow chart

3 Assumptions

?All the data we obtain from the official sources are reliable.

?The nature disasters and wars are not taken into account in our model.

?The increase in fresh water storage from the thaw of glaciers caused by climate change and global warming is not considered.

4 Prediction of water consumption and water storage

Because of the special conditions of China, statistical data of national life index and social resources before the 21th Century is half-baked. We obtain the water consumption data of nationwide and each province from 2000 to 2010 by searching the official statistical yearbook [1]. We considered several prediction models, and then excluded the ARMA Model because of limited data samples and medium-term prediction required. Further, we found that the correlation of linear regression is not obvious through the test for several groups of data and excluded this method. Finally, we adopt the Grey Prediction Model, since its characteristics such as small amount of samples required, high accuracy of grey prediction, extensive application for long-term forecasts etc. We specially improved the GM(1,1) Model based on background value optimization in order to advance the precision.

4.1 Prediction of water consumption in 2025

4.1.1 The Grey Prediction Model

4.1.1.1 GM(1,1) Model

Predictions of variation and time for the development of things are called series forecasts. GM(1,1) is one of the frequently-used gray prediction models, which is a first-order linear dynamic model of single sequence.

The original given data sequence of time?X(i)?,(i?1,2,...,n)can’t be directly used for modeling, since the data is always random and irregular. By the accumulation of the original data sequence, we obtain new date sequence: (0)

?X(i)?,(i?1,2,...,n) (1) i(1)(0)X(i)??X(k) (2) (1)

k?1

Accumulation generates the curve that increases the regularity of original data sequence and weakens the randomness. Then, it is easy to be approximated by exponential curve. GM(1,1) model is developed based on the above principle. For the original data

By accumulation we get: X(0)??X(0)(1),X(0)(2),...,X(0)(n)? (3) X(1)??X(1)(1),X(1)(2),...,X(1)(n)? (4)

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Construct the mean generating sequence ofX(1)(k):

Z(1)(k)?0.5X(1)(k?1)?0.5X(1)(k),(k?2,3,...,n) (5)

And then the data matrix B:

?-Z(1)(2)1??-0.5[X(1)(1)?X(1)(2)]?(1)??-Z(3)1-0.5[X(1)(2)?X(1)(3)]???B==?……??……?(1)??(1)(1)-Z(n)1???-0.5[X(n?1)?X(n)]1??1? (6)1??1?

The date vectorYN:

YN??X(0)(2),X(0)(3),...,X(0)(n)? (7)T

And parameter vectoru?(a,b)T,

Then the GM(1,1) model can be expressed as the matrix equation:

YN?B?u (8)

-1Actually, ifexists, we have (BT?B) -1T?)T???(a?,bu(BT?B)BYN (9)

By the least square method, which can minimum the equation

?)?(YN?B?u?)T(YN?B?u?) J(u (10) Solving the above equations we obtain: ??ak?bb?(1)(0)?(k?1)?(X(1)?)e?,(k?1,2,...,n?1) X (11)??aa

4.1.1.2 Improved-Grey Model

The albinism differential equation correspond to the first-order module GM(1,1) generated by X(1)(k)is:

dX(1)(t)?aX(1)(t)?b (12) dt

Integrating the equation above in the interval [k, k+1], we get that:

X(1)(k?1)?X(1)(k)?a?kX(1)(t)dt?b,(k?1,2,...,n?1) (13)

From formula (11),we found that the precision of fitting and prediction of GM(1.1) Model is rested with constant a and b, moreover, constant a and blied on the background value of model. Thus, reasonable construction of the background value of model can raise the precision of prediction.

Let Z(1)(k?1)??kX(1)(t)dt be the background value in the interval [k, k+1], then

X(0)(k?1)?aZ(1)(k?1)?b (14) k?1k?1

We have the representation by solving the above equation:

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Z(k?1)??k

(1)

k?1

X(1)(k?1)?X(1)(k)

X(t)dt? (15) (1)(1)

lnX(k?1)?lnX(k)

(1)

After deriving the unknown parameters, we can get the discrete solution of albinism differential equation:

??ak?bb?(1)(0)?(k?1)?(X(1)?)e?,(k?1,2,...,n?1) X (16)

??aa

4.1.2 Solving the Model

Before predicting total water consumption in 2025, we have searched for the official data of water consumption in China [2] and we obtain the data from the official website of National Bureau of Statistics of China. Table1 shows that the total Water Consumption from 2000 to 2011 and four major component of it. Also, it shows the Per Capita Water Use. Figur2 shows the change with a bar graph.

lake, wetland and city environment.

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Figure2. Water Consumption in China from 2000 to 2011

◆The solution of the model

We bring the data from 2000 to 2010 into Improved-Grey Model to get the

prediction, and then test the model by 2011’s data. We can see from the table1 that water consumption has four major components: Agriculture water, Industry water, domestic water and Ecological protection water. So we have two methods to predict the total water consumption in 2025. One is using the total water consumption from 2000 to 2010 to predict water consumption in 2025 directly. Another is firstly predicting four major components in 2025 respectively, and then adds them up. We use MATLAB to calculate the data with our model and results are as follows:

The model of total water consumption is:

X(1)(k?1)?446880e0.011997k?441320,(k?1,2,...,n?1) (17)

WhereX(i)??X(0)(k)and K=1 here denotes year 2000. (1)

k?1i

◆ Accuracy test of Improved GM (1,1) model

We have two indexes to test the model accuracy:

Index1: testing on series of relative error [3] Table2 shows the series of relative error, so we can get Average Relative Error:

1n??|rel(k)|?0.011788 (18) nk?1

<0.05, the model precision is good and can be used for forecast. Table2. The series of relative error

-0.0105 0.0313 0.0071 -0.0383 -0.0078 -0.0045 0.0118 0.0039 0.0075 0.0048 0.0023 Index2: posterior-variance-test

Posterior Error Ratio of the model is:

C?0.41264

Little Probability of Error is:

p?0.81818

According to reference table of Accuracy test grade [5] in Table3, we can get that our model accuracy is qualified.

Table3. Reference table of Accuracy test grade

Model accuracy grade

Class 1(good)

Class 2(qualified)

Class 3(reluctantly) Posterior Error Ratio C C<=0.35 0.35<C<=0.5 0.5<C<=0.65 Little Probability of Error p 0.95<=p 0.80<=p<0.95 0.70<=p<0.80

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Class 4(unqualified) 0.65<C P<0.70

Combining two indexes, we can draw the conclusion that accuracy of our

model-Modified GM(1,1) Model is qualified and can be used for prediction. And prediction of 2013’s water consumption in this model is:

result2011=X(0)(12)?X(1)(12)?X(1)(11)?6080.9 (19)

Table1 shows the actual water consumption of 2011 is 6107.2, and the relative error

is:

6107.2?6080.9???100%?0.43% (20)

6107.2

We can see the relative error is quite small.

So, we can predict water consumption in 2025 using this model. According to the result we obtained in MATLAB, the prediction of water consumption in 2025 is:

result2025=X(0)(26)?X(1)(26)?X(1)(25)?7193 (21)

◆ Feasibility analysis of the other method

According to the result gotten in MATLAB, the model accuracy of four major

components is shown in Table4:

Table4. Model accuracy of four major components of water consumption

Average Relative Posterior Error

Error Ratio C

Agriculture water 0.023741 0.99553 Industry water 0.018434 0.24342 Domestic water 0.0036504 0.047851 Components

Little Probability

of Error P 0.54545 1 1 Model accuracy unqualified good good ecological water’s is reluctantly. What’s more, ecological water’s data from 2000 to 2002 is blank which is shown in Table1. So the prediction of agriculture water and ecological water cannot use this model. ◆Conclusion

We choose the former method to predict water consumption in our Improved GM(1,1) Model. By using this model, we can also predict every province’s water consumption in 2025 respectively. The prediction is shown in Table5.

Table5.Prediction of every province’s water resources and consumption in 2025

Per Capita

Water Use

Provinces Water Use

(100 million cu.m)

(cu.m/person)

Beijing 37.46 127.22 Tianjin 22.89 113.93 Hebei 179.52 229.84 Shanxi 79.21 203.47 Inner

203.20 768.27

Mongolia

Liaoning 175.42 372.97 Jilin 184.96 654.10

Per Capita

Water Resources

Water Resources

(100 million cu.m)

(100 million cu.m)

36.47 117.64 18.56 84.84 181.83 223.73 159.65 387.25 305.34 585.06 484.19

1135.30 1207.51 1703.36

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Heilong jiang Shanghai Jiangsu Zhejiang Anhui Fujian Jiangxi Shandong

Henan Hubei Hunan Guang dong Guangxi Hainan Chong qing Sichuan Guizhou Yunnan Tibet Shaanxi Gansu Qinghai Ningxia Xinjiang

525.03 144.73 625.99 189.35 664.30 252.95 361.95 240.07 301.21 415.65 320.48 482.05 304.47 42.09 151.14 275.43 116.56 156.25 45.87 101.40 119.78 30.89 63.77 626.25

1358.21 400.19 711.50 275.63 1186.96 622.23 749.39 232.04 326.22 766.22 505.33 319.29 642.56 427.11 556.64 375.95 386.08 313.01 1298.80 265.58 478.32 506.42 871.06 2297.89

968.46 50.47 832.52 1579.93 1295.61 1012.27 2059.58 255.38 283.16 1172.78 1024.54 2257.82 1438.69 3194.09 440.45 2079.26 675.97 1242.74 4105.29 1583.15 271.33 1005.23 7.36 962.39

2499.73 161.20 947.45 2304.01 2327.00 2514.73 4242.45 241.19 309.61 2167.86 1603.20 1645.98 3115.13 32568.93 1664.71 2890.90 2180.37 2486.59 111784.13 4132.38 1081.89 16204.99 99.58 3427.99

4.1.3 Prediction of the Total Population 4.1.3.1 Logistic Demographic Model

The most common population forecast models are the exponential growth model (Malthus Model) and the block growth model (Logistic Model). Malthus Model simply holds the view that the population growth rateris changeless, while Logistic Model assumes that growth rate of population is a function of the populationr(x), obviouslyr(x)reduces withx. A simple assumption is thatr(x)is a linear function ofx:

r(x)?r?sx,s?0 (22)

Consider the maximum number of population that natural resources and environmental conditions can hold, which is usually called the maximum population capacity. Under the linearization assumption, the growth rate is zero when x?xm and

x

r(x)?r(1?) (23)

xm

Here,r,xmare usually determined according to demographic or experience. Under this assumption, the exponential growth model can be modified as:

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x?dx??rx(1?)xm (24)?dt

?x(0)?x.0?

The equations above are named the block growth model (Logistic Model), its solution is: xmx(t)? (25)?rt1?(xm/x0?1)e

4.1.3.2 Solving the Model

Taking the special situation of China into account, like the three years of natural disasters and the family planning policy from 1983, we choose data of total population from 1980 to 2011 in order to achieve better prediction and analysis effect. We utilize the fitting function polyfit in Matlab and estimaterandxm:

r?0.0519 , xm?149.3793

Substituted in formula (25) and then get the fitting chart of the comparison between population and theoretical value from 1980 to 2011.

Figure 2.The comparison factual population and theoretical value in year

1980-2011

By the forecasting result we obtain the total amount of the population in 2025 is approximately142.3165(?107), i.e.1.4232billion.

4.1.3.3 The Prediction of Average Water Consumption

Similarly, we obtain the average annual water use by utilizing the mentioned Improved GM(1,1) Model and substituting the data of the last column of Table 1. The result is 493.48cu.m/person.

Combined with the forecast of the total population in 2025, we get the prediction of total water consumption:

result2025=142.32?107?493.48=7023?

108cu.m. (26)

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The relative error with the direct prediction is:

7193?7023

?=?100%?2.363% (27)

7193

We can see the result is in the error range permitted. 4. 2 Prediction of water storage in 2025

In addition, we also predict total amount of water resources in 2025 based on the data shown in Figure3. The prediction of total amount of water resources in 2025 is 24184 cu.M.

5 Water transfer

5.1 Terminology and Definitions

Transfer Cost (Ci j(k);1≤i ≤31, 1≤j≤31, 2000≤ k≤2025): The expense of water

transportation between two provinces in the year of k, with the label i and j that both references to the corresponding province(In table ?).

Water Shortage (Si(k); 1≤i ≤31, 2000≤ k≤2025): The total number of water that province i is in short in the year of k.

Water Abundance (Ai(k); 1≤i ≤31, 2000≤ k≤2025): The total number of water that province i can provide in the year of k.

Distance (Di j;1≤i ≤31, 1≤j≤31):The distance between two provinces of province i and province j;

2 3 4

Tianjin Hebei Shanxi

10 11 12

Jiangsu Zhejiang Anhui

18 19 20

Hunan Guangdong Guangxi

26 27 28

Tibet Shaanxi Gansu

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5 6 7 8

Inner Mongolia Liaoning Jilin Heilongjiang

13 14 15 16

Fujian Jiangxi Shandong Henan

21 22 23 24

Hainan Chongqing Sichuan Guizhou

29 30 31

Qinghai Ningxia Xinjiang

Table7. Water Transfer Model

5.2 Assumptions

? To simplify the model, we assume that the expense of water transportation is only determined by the distance, landscape, and immigration.

?The distance between two provinces [6] equals to the distance between the two corresponding capital.

?We use the mass Centre to present a set of provinces to get the length of the water transfer.

5.3 Optimum Water Transfer

As delivered above, we have some provinces in short of water as well as some others with abundant water. Then we come to an optimum water transfer plan in our model to make rational utilization of water resources. 5.3.1 Transfer Cost Estimate

According to South-to-North Water Diversion project, we obtain the data above.

Diversion project, we get an average allowance of 23 thousand Yuan per person for allowance and 15 thousand Yuan per citizen for infrastructure fee. We obtain a total average allowance money of 38 Yuan for per immigration.

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We simplify the transfer cost, with counting the length of diversion channel and

immigration alone. As for ERP (Eastern Route Project), we can get an average cost of 0.03412 billion per km. As to MRP (Middle Route Project), we obtain the total cost 12.54 billion for immigration and an average cost of 0.08201 billion per km.

5.3.2 Transportation Algorithm

The transportation algorithm follows the exact steps of the simplex method.

However, instead of using the regular simplex tableau, we take advantage of the special structure of the transportation model to organize the computations in a more convenient form [7].

The steps of the transportation algorithm are exact parallels of the simplex algorithm. Step1. Determine a starting basic feasible solution, and go to Step2.

Step2. Use the optimality condition of the simplex method to determine the entering variable from among all the non-basic variables. If the optimality condition is satisfied, stop. Otherwise, go to Step3.

Step3. Use the feasibility condition of the simplex method to determine the leaving variable from among all the current basic variables, and find the new basic solution. Return to Step2.

5.3.2.1Determination of the Starting Solution

A general transportation model with m sources and n destinations has m + n constraint equations, one for each source and each destination. However, because the

transportation model is always balanced (sum of the supply = sum of the demand), one of these equations is redundant. Thus, the model has m + n - 1 independent

constraint equations, which means that the starting basic solution consists of m + n - 1 basic variables.

The special structure of the transportation problem allows securing a non-artificial starting basic solution using one of three methods:

1. Northwest-corner method

2. Least-cost method

3. Vogel approximation method

In general, though not always, the Vogel method yields the best starting basic solution, and the northwest-corner method yields the worst. The tradeoff is the

northwest-corner method involves the least amount of computations.

◆Northwest-Corner Method. The method starts at the northwest-corner cell (route) of the tableau.

Step1. Allocate as much as possible to the selected cell, and adjust the associated amounts of supply and demand by subtracting the allocated amount.

Step2. Cross out the row or column with zero supply or demand to indicate that no further assignments can be made in that row or column. If both a row and a column net to zero simultaneously, cross out one only, and leave a zero supply (demand) in the uncrossed-out TOW (column).

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Step3. If exactly one row or column is left uncrossed out, stop. Otherwise, move to the cell to the right if a column has just been crossed out or below if a row has been crossed out. Go to Step 1.

◆Least-Cost Method. The least-cost method finds a better starting solution by

concentrating on the cheapest routes. The method assigns as much as possible to the cell with the smallest unit cost (ties are broken arbitrarily). Next, the satisfied row or column is crossed out and the amounts of supply and demand are adjusted

accordingly.

◆Vogel Approximation Method (VAM). VAM is an improved version of the least-cost method that generally, but not always, produces better starting solutions. Step1. For each row (column), determine a penalty measure by subtracting the smallest unit cost element in the row (column) from the next smallest unit cost element in the same row (column).

Step2. Identify the row or column with the largest penalty. Break ties arbitrarily.

Allocate as much as possible to the variable with the least unit cost in the selected row or column. Adjust the supply and demand, and cross out the satisfied row or column. If a row and a column are satisfied simultaneously, only one of the two is crossed out, and the remaining row (column) is assigned zero supply (demand).

Step3. a) If exactly one row or column with zero supply or demand remains

uncrossed out, stop. b) If one row (column) with positive supply (demand) remains uncrossed out, determine the basic variables in the row (column) by the least-cost method. Stop. c) If all the uncrossed out rows and columns have (remaining) zero

supply and demand, determine the zero basic variables by the least-cost method. Stop. d) Otherwise, go to step 1.

5.3.2.2 Iterative Computations of the Transportation Algorithm

After determining the starting solution, we use the following algorithm to determine the optimum solution:

Step1. Use the simplex optimality condition to determine the entering variable as the current non-basic variable that can improve the solution. If the optimality condition is satisfied, stop. Otherwise, go to Step 2.

Step2. Determine the leaving variable using the simplex feasibility condition. Change the basis, and return to Step 1.

The optimality and feasibility conditions do not involve the familiar row operations used in the simplex method. Instead, the special structure of the transportation model allows simpler computations.

5.3.3 Solution and Evaluate

As delivered above, we’ll have fourteen provinces (Beijing, Tianjin, Hebei, Shanxi, Inner Mongolia, Heilongjiang, Shanghai, Jiangsu, Anhui, Shandong, Henan, Gansu, Ningxia and Xinjiang) water in short in the year of 2025 from our model.

Consequently, we use water transfer project while taking Inner Mongolia, Gansu

Heilongjiang and Xinjiang out of consideration for geographic conditions. Further, we

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classify the left ten provinces into three kinds with Shanghai, Jiangsu and Anhui

belonging to Changjiang River System, Shanxi, Ningxia and Henan belonging to The Yellow River System 1 and the rest four provinces belonging to Yellow River System 2. Considering the river distribution and the geography conditions, five cities Jiangxi, Hubei, Hunan, Chongqing and Sichuan could take this burden.

Having the water prediction of supply and shortage as well as the water transfer cost, we use AMPL software to get the target transfer plan. From our prediction we find it that the total water supply quantization is a little smaller than the shortage, we calculate the total shortfall and leave all the water short provinces share and share alike. Thus, we obtain the following water transfer table.

Table9. The water transfer table

Table10. The starting solution

Table11. The optimum solution

Multiple the average channel cost, we get the final prediction payment of 74.791086 billion. The optimum solution table gives us the water transfer scheme, in which the three thirstiest provinces Shanghai, Anhui and Jiangsu could be satisfied only would all the five water abundance cities give a hand to leave their water pass through, and the main water source of Beijing, Tianjin, Hebei and Shandong is Jiangxi as well as the left three provinces Shanxi, Ningxia and Henan would get their necessary water

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from Sichuan.

According to the optimum solution table, we get three water supply weighed value points according to the water amount. Therefore, the longitude and latitude of water supply points are A1 (114.305, 30.593), A2 (109.864, 30.027), A3 (104.065, 30.659). Considering the geographic feature and the existing river way such as the Grand Canal from Beijing to Hangzhou and existing dams like Danjiangkou Reservoir, We then figure out the optimum water diversion point of Changjiang and Huaihe River as well as the optimum water transfer course. For the water quality of Yellow River, we choose three points in Changjiang and one point in Huaihe by the analysis of the output of water transfer optimum solution.

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Figure4. Water supply and short

Figure5. Water transfer plan

6 Desalination of sea water

6.1 Sites

We can see from Table1 that industry water accounts for nearly a quarter of total

water consumption. If we construct sea water desalination plants to provide industries with water, we can save so much water. And compared with water transfer, sea water desalination’ cost is lower and the cost will be lower and lower along with the development of science. Table6 shows the cost of the two ways for fresh water.

Table12. The cost of the two ways for fresh water

Ways for fresh water

Cost (RMB/cu.M)

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Water transfer

Water desalination Water Diversion Project from Luanhe River to Tianjin City South-to-North Water Transfer Sea water

Brackish water 2.3 (direct cost) 5-20 4-7 (Embedded Cost) 2-4(Embedded Cost)

From table12, we can see that cost of water desalination is lower than water transfer. And considering the distribution of provinces short of water which is shown in

Figure4 and that high transportation cost adds to the cost of desalinated water, coastal regions with high economic growth are the priority of our choice. So we choose Hebei, Tianjin, Shandong and Liaoning to construct sea water desalination plants as Figure5 shows.

6.2 Desalination methods

On the whole, desalination methods can be: Multiple-effect Distillation (MED),

Reverse Osmosis Membrane (RO), Vapor compression distillation (VC), Multistage Flash evaporation (MSF), Electro dialysis (ED). And different methods have different productivity which is shown in Figure6.

We can see that RO accounted for the largest proportion. To simplify the model, we select the RO as our method to put into service. Actually, it is reported seawater

desalination mostly adopt the RO technology and Original import of reverse osmosis membrane has recently been realize localization. And now the lowest cost of domestic seawater desalination is 3.5 Yuan per ton.

Figure6. The proportion of water yield with different methods [8]

6.3 Total cost

Considering the impact of science on cost reduction, we assume the cost of seawater desalination reduces by half in 2025. In other words, the cost is 1.75 Yuan per ton. Now we calculate the total cost of desalination in Hebei, Tianjin, Shandong and Liaoning.

Table4 shows the water deficit of every province and we can calculate that total water deficit of Hebei, Tianjin, Shandong and Liaoning is 21195 million ton.

So according to the prediction of desalination’s cost: 1.75 Yuan per ton, we can roughly estimate the total cost of seawater desalination in the four provinces:

W?1.75?21195?37091.25 million yuan?

37.09 billion yuan (28)

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Comparing with the water transfer cost, seawater desalination is apparently more economic.

6.4 Result analysis

In this model, we make the prediction of cost roughly and it just provides an intuitive understanding of the benefit of seawater desalination. Actually, there are already several desalination plants in use located in Qingdao and Tianjin. It coincides with our sites.

7 Optimal allocations of water resources

7.1 Introduction

Currently, the Sustainable Utilization of water resources relates to the harmonious development among mankind, society, and environment. Optimal allocation of water resources is an effective means which can promise Sustainable Utilization of water resources, so the research on water resources optimal allocation has great significance. The purpose of water resources optimal allocation is to allocate the water resources scientifically and rationally, making the limited water resources get reasonable, full exploitation, in order to acquire best possible comprehensive benefit, which includes economic benefit, ecosystem environment benefit and social benefit etc. [9] These benefits are usually mutual conflict and restriction, so the optimal allocation of water resources is a multi-objective optimization problem, MOPs. The definition of the MOPs is that in the feasible fields, a vector composed of decision variables is defined to make the objective functions, which have mutual conflict relationship, reach to the best at the same time. The essence of the multi-objective optimization is that in many instances, it is impossible to optimize all objective at the same time; we can only harmonize between all the objectives to get possible optimal solution [10].

7.2 The Multi-Objective Genetic Algorithm based on parataxis selection

Genetic Algorithms (GAs or GA) is inspired by Darwin's theory about evolution and it works very well on mixed (continuous and discrete), combinatorial problems. The parataxis selection algorithm is one method of the MOGA. The theory of it is that according to the number of the object function, the initial population is averaged, and each object function is distributed to each subgroup. In each subgroup the high fitness individuals are selected and a new population is composed of the subgroups which were selected. Carrying out crossover operation and mutation operation in new population and bringing to the next generation, thus carrying on "partition- parataxis selection- merger" operation continually. Finally, the Pareto optimization solutions are acquired [11].

7.3 The water resources optimal allocation based on Multi-Objective Genetic Algorithm

7.3.1 Genetic coding of decision variable

In this research, binary encoding is adopted. For decision variable (the water consumption of different sources distributing to different water users) xij(idenote water users, jdenote source), the length of the chromosome amounts to the number of decision variable. Every gene presents the corresponding distribution water

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following its gene-locus. Every chromosome corresponds to a certain allocation scheme of water resources.

7.3.2 Design of object function

According to the different needs, different object functions can be designed. In this study, water sources are allocated to five water users. The five water users are respectively primary industry, secondary industry, tertiary industry, town and country. The model as follows:

(1) Economy function: The water consumption of three industries divided by water consumption of output value which produced by ten thousand, the maximal result as the economy function.

?3104n?E?f1(xij)?max????Xij? (29) ?i?1Kij?1?

In the formula (29), the decision variable xij denotes the water consumption of different sources (j) distributing to different water users (i), (104m3), where i?15 denotes the five water users, j?indenotes the number of water sources and the symbolk1,k2,k3denote the water consumption of three industries respectively,(m3 / 104 Yuan); E denote total output value, (104 Yuan).

(2) Population function: The water consumption of town and country divided by water use quota, and the maximal number of population as the population function.

n?5103?P?f2(xij)?max????Xij? (30) ?i?4365?Kij?1?

In the formula (30) , the symbol of k4,k5denote the water use quota of town and country respectively, (L/(p﹒d)). The symbol P denotes population function,(104 p).

(3) Water circumstance function: The minimal contamination of COD discharge amount is used to represent water circumstance condition.

n?5?C?f3(xij)??max??0.01?di?pi??Xij? (31)

j?1?i?1?

In the formula (31), pidenotes sewage discharge rate, didenotes the amount of COD in per-unit wastewater) Cdenotes the total amount of COD, (t).

7.3.3 Restriction condition

(1) Restriction of water supply amount:

5?X

i?1

nij (32) ?Wjmax j?1~n wjmax denotes the maximal water supply amount. (2) Restriction of national economy: ?10

Eimin?j?14?Xij?Eimax i?1~3 (33)

EiminKidenotes the minimal production value of every industry, Eimaxdenotes the

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maximal production value of every industry. (3)Restriction of population:

103

Pimin???Xij?Pimax i?4~5 (34)

j?1365?Ki

Pdenotes the minimal number of population of town and country, Pmaximal number of population of town and country.

n

imin

imax

denotes the

(4) Restriction of circumstance:

??0.01?d?p?X

i

i

j?1i?1

n5

ij

?Cmax (35)

[12]

Cmaxdenotes the maximal amount of COD which can be accepted ,t.

7.4 Example analyses

Based on the above model and optimal method, water resource allocation of city is

optimized. Firstly, the total water supply should satisfy the demand of water users, and the economy object, population object and circumstance object are considered at the same time during allocating two water sources: surface water and groundwater .In this example, the quantity of two water sources are allocated to five water users, so the number of decision variables are ten, then the length of every chromosome is ten gene-locus. Other parameters are as follows:

The size of the populationPopsize?300, the maximal evolution generationmaxgen?50, the generation gap GGAP?0.9, the crossover probabilityPcross?0.7. Finally, fifteen optimal schemes are acquired. Here, we list six schemes among them for decision-making.

Table13. Results of Optimal Deployment of Water Resources (104m3/year)

Water resource allocation of surface water

Allocation schemes The first project The second project The third project The fourth project The fifth project The sixth project

Primary Secondary Tertiary

Town Country

industry industry industry 10120 10120 10080 10060 10190 10180

248760 249130 245300 245820 246940 247630

10030 10150 10110 10160 10130 10070

33600 33650 26720 28240 25070 25060

10190 10050 10010 10070 10020 10000

Existing Surface water 51000 51000 51000 51000 51000 51000

Difference values 550 1190 1210 530 540 540

Table14. Water resource allocation of groundwater

Allocation schemes

Water resource allocation of groundwater

Existing

Primary Secondary Tertiary

Town Country Ground

industry industry industry

water

Difference values

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The first project The second project The third project The fourth project The fifth project The sixth project

248400 248450 248450 248670 248680 248670

33680 33330 33610 33680 33600 33680

210 260 260 210 210 210

6200 6150 6150 6200 6200 6200

1250 1360 1360 1250 1250 1250

291400 291400 291400 291400 291400 291400

1660 1850 1570 1570 1460 1390

Table15. Results of Three Object Functions

Allocation schemes The first project The second project

The third project The forth project The fifth project The sixth project

Economy Function Production Value of Economy (108Yuan)

1853.00

1847.60 1687.90 1730.50 1658.10 1654.30

Population Function Total population

(104p) 822.64

836.20 937.13 934.80 787.43 772.25

Water circumstance

Function COD Discharge amount (t) 12217

12264 10311 10753 9856 9837

7.5 Conclusion

Water resources optimal allocation is an important content of water resources layout, because the system of water resource is multi-object, so it’s difficult by using

traditional method to solve this problem. In this model, in order to get the final plan, we take the economic, population and water circumstance benefits as the objective functions, restriction of water supply amount, national economy, population and circumstance as constraint conditions based on the Multi-objective genetic algorithm model.

In the optimal allocation of surface water resources, the weights of primary industry, secondary industry, tertiary industry, water for residents of town and country are 0.034, 0.817, 0.033, 0.083 and 0.033 respectively. Among the results, second industrial water consumption reduces more, water for residents of town reduces most, while primary industry and tertiary industry increase slightly. In the optimal allocation of ground water resources, the weights of primary industry, secondary industry, tertiary industry, water for residents of town and country are 0.858, 0.116, 0.001, 0.021 and 0.004 respectively. Among the results, the first industry increased slightly and the rest keep constant. See the final benefit optimization in Table15. 7.6 Model evaluation

The research on optimal allocation of water resources has already obtained certain achievement, but most research adopt single objective or traditional multi-objective optimal methods, such as, weighting method and restriction method etc., changing the complicated multi-objective optimal problem into single-object problem [13], but the

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disadvantage of these methods is that each time we can acquire only one result, after computing many times, we can acquire a group of Pareto optimization results. The essential difference between the single-objective optimization problem and MOPs is that the result of MOPs is not the one and only, but is a group of Pareto optimization results. In this model, Multi-Objective Genetic Algorithm is introduced to solve the problems of water resources optimal allocation.

8 Our water strategy and implications

8.1 Our water strategy for 2013

We have already predicted the water storage and consumption in 2025 and analyzed water transfer, seawater desalination and water resources optimal allocation with different models. We also take water conservation, reasonable water pricing system, and sewage recovery and rainwater collection into consideration. After combining the models and analyzing the results of different models comprehensively, we draw out our water strategy for 2013:

◆Engineering Strategy:

(1)Water Transfer

?The optimal water diversion points are three in Changjiang and one in Huaihe shown in Figure 5

(2)Seawater Desalination

?Construct seawater (or brackish water) desalination plants in Hebei, Tianjin, Shandong and Liaoning province.

?Vigorously develop reverse osmosis membrane on the present basis.

(3)Sewage Recovery and Rainwater Collection

?Formulate the standard of sewage utilization

?Improve the sewage utilization technology and establish a diversified market mechanism.

?Develop new rainwater collection technology and focus on market promotion. ◆Management Strategy:

(1)Water Resources Optimal Allocation

?Promote Water Resources Optimal Allocation through China.

?Use this model to solve competition of water between Agriculture, industry, households, and ecology

?Provide reasonable suggestions for water transfer.

(2)Water Conservation

?Formulate strict policy of sewage discharge.

?Vigorously develop and promote the water -saving irrigation technology.

(3)Reasonable Water Pricing System

?Formulate reasonable water price and put price adjusting leverage function into a full play.

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?Promote water pricing system’s transparency and openness. 8.2 Analysis of the implications of our strategy by AHP Model 8.2.1 Establish the hierarchical structure model

Figure7. The hierarchical structure model

8.2.2 Construct the judgment matrixes

Considered all the potential factors in the reference, we get the judgment matrix for water strategy and aspects of economy, order of nature, environment:

?11/51/4???A=?512? ?41/21???

Further, construct judgment matrix respectively as:

57371??1

??1/5121/231/4???1/71/211/321/4?B1=??

3131/2??1/32

?1/71/31/21/311/7????1?44271??

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97?1

?

1?1/91

?1/711B2=?

5?1/35

?1/711/3??1/91/31/5??1??5?2B3=?

?1?3??1/3?

1/51/2131/311/41/3

9??3?5?? 7?

1/512?

?

1/71/21??1431

1/3522

31/51/517135

3??7?4?? 3?

1/51/21/213?

?

1/71/41/31/31??

8.2.3 Calculate the characteristic value of maximum modulus and the characteristic vector

We use the power method in Matlab to solve the characteristic value of maximum

modulus and the corresponding characteristic vector of four matrixes. For matrix A, we have

?1?(0.0974,0.5695,0.3331)T , ?max?3.0246 , CI=0.0123 , CR=0.021<0.1 For matrixB1,

?2?(0.3667,0.0896,0.0587,0.1460,0.0397,0.2993)T , ?max?6.1112

CI=0.0222 , CR=0.018<0.1

For matrixB2,

?3?(0.5431,0.0603,0.0776,0.1810,0.0776,0.0603)T , ?max?6.3401

CI=0.0680 , CR=0.055<0.1

For matrixB3,

?4?(0.0811,0.4054,0.1622,0.0811,0.2432,0.0270)T , ?max?6.3615

CI=0.0723 , CR=0.058<0.1

8.2.4 Weight table

Table16. The weight table

The criterion layer Measure layer

P1 P2 P3 P4 P5 P6

C1 0.0974 0.3667 0.0896 0.0587 0.1460 0.0397 0.2993

C2 0.5695 0.5431 0.0603 0.0776 0.1810 0.0776 0.0603

C3 0.3331 0.0811 0.4054 0.1622 0.0811 0.2432 0.0270

The weight of measure layer 0.3720 0.1781 0.1039 0.1443 0.1291 0.0725

From this table, we can summarize that the proportion of economic, physical and environmental impact are 0.0974, 0.5695 and 0.3331. The proportions of six measures

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we proposed are 0.3720, 0.1781, 0.1039, 0.1443, 0.1291 and 0.0725.

9 Analysis and evaluation of the model

9.1 Sensitivity analysis

◆Discussion of available water resources proportion

In the Modified GM(1,1) Model, we use the available water resources proportion of the whole nation:39% to be the estimated data for every province in China. However, every province has different proportion actually. We have to look into the situation when the available water resources proportion is different. We change the proportion to 45%, and then we make the prediction of water resources and consumption of every province in 2025. Then we recalculate the expense of water transfer and seawater desalination.

The total water deficit of Hebei, Tianjin, Shandong and Liaoning changes to 149.53 ton. The total cost of seawater desalination is:W?1.75?149.53?26.17billion Yuan. Compared with 37.09 billion Yuan, it decreases by 29.4% when the available water resources proportion improves by 12.8%. We can obtain that the total cost of seawater desalination is a little sensitive to available water resources proportion, so the every province’s proportion should be estimate with reasonable methods to improve our model.

9.2 Strengths and Weaknesses

?We research the water prediction, water transfer, water desalination and water allocations by different models.

?The Grey Prediction Model can provide a better result for a small amount of samples. ?Two different models for the prediction of water consumption match quite well. ?The prediction of the provinces short of water accords with national situations.

?We optimize the water transfer plan as well as the optimal water diversion with considering both the distance factor and immigration factor.

?We can give predictions of water transfer cost and water desalination expense. ?The model does not give a precise course of water transfer.

The model does not have a good solution of getting the global optimum with all the factors considered

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References

[1]http://www.stats.gov.cn/tjsj/ndsj/2012/indexch.htm

[2]Han,Z.G.Methods and Applications of Mathematical Modeling.Peking:Higher education press.

[3]http://wenku.baidu.com/view/a7f4274cb307e87101f696da.html

[4] http://distancecalculator.globefeed.com/china_distance_calculator.asp.

[5]http://en.wikipedia.org/wiki/South-North_Water_Transfer_Project

[6]http://www.nsbd.gov.cn/zx/english/

[5]Hamdy A.Taha: Operations Research AnIntroduction (8th edition, 2007), p. 206-.212

[6][http://www.mem-china.com/hydt/zhxw/200912/18195.html]

[7][http://www.cecc-china.org/Item/12502.aspx]

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