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2013年美国大学生数学建模竞赛B题


Team 22942
Team Control Number For office use only T1 ________________ T2 ________________ T3 ________________ T4 ________________

22942
Problem Chosen

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

B
Water

Water

Everywhere

Summary
The year 2003 is ‘the United Nations International Year of Water Cooperation,and on March 22, 2013, the world water day will also come.Both of them are aimed at more efficient utilization of water resources, protection of water resources.Therefore, as a ubiquitous water resources, it is necessary for us through the mathematical modeling to develop a more reasonable water resources strategy. In this paper,we first use SPSS software to establish the mathematical model of water storage. As a basic model, it can be applied to integrate the data, carry on the regression analysis and forecast the data in 2025.Given the uneven distribution of water resources,we try on the water movement and transmission question discussion.With the help of AutoCAD,we develop a Chinese outline drawing with four different colors to illustrate the comparative water storage and water demand. Furthermore,we built a model of max spanning tree to get the shortest route of transfer. After a brief discussion of water desalination,it comes to the relationship analysis.Opening a new path,we successfully use excel for model processing.Then ,by refering to the related datas,it turns clearly that water resources do have a close relationship with the region’s economic, political, ecological environment and so on.Finally,With all the above models and discussions,we offer governmental leadership a non-technical position paper,in which we summarizes our water strategy and its unique characteristics. Our model is featured by its simplicity, feasibility and universality.However, due to the lack of statistical data and the limitation of both methods and computing tools,our models need to be further optimized,especially in the economic aspects.

Keywords regression analysis , spanning tree , water strategy 1

Team 22942

CONTENTS
1 Interoduction......................................................................................... 3 1.1 Background ................................................................................... 3 1.2 Restatement of the Problem........................................................... 3 2 Model Assumotion….……………....................................................... 3 3 MODEL Parameters Design………………………………………….3 4 Model Design…………………………………………………………..4 4.1 water storage……………………………………………………...4 4.2 water movement and transfer……………………………………..6 4.3 water desalinization and conservation……………………………9 5 Relationship Analysis…………………………………………………9 6 Position Paper for Leaders………………………………………….12 7 Reference….………………………………………………………….13

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Team 22942

1 Introduction
1.1 Background
Water, as a kind of natural precious resources which can be seen everywhere,is the human survival and all the creatures’ material foundation.To a certain extent,the quantity and quality of water resources determine a country's economic, political and cultural development.Thus,in order to identify the best water strategy,it ibecomes necessary to study to solve the water resources’ storage,movement,desalinization and conservation.

1.2 Restatement of the Problem
We based on China's national bureau of statistics of China for many years to regional water resources statistics data, set up effective and feasible strategic mathematical model of water resources. Things we know about the water: ? China has 31 provinces,and the 31 provinces of water consumption and water resources condition data is known; ? Water do have some relationship with the district’s every aspects; ? We get the national provinces water index in 1998. Requirements in developing schedules: ? Our mathematical model must address water storage and movement,de-salinization and conservation; ? we could use our model to discuss the economic, physical, and environmental implications of our strategy.

2 Model Assunption
? There is no wide range of flood and drought disasters caused by climate change. ? There is no mutation of population quantity or economic development conditions caused by political factors such as war. ? Seawater desalination technology won't have a major breakthrough in a short time. ? There is no substitutes for fresh water in a short time.

3 Model Parameters Design
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Team 22942
Table 1: Main Symbols and Meanings
Code y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 Province whole country Bei Jing Tian Jin He Bei Shan Xi Inner Mongolia Liao Ning Ji Lin Hei Long Jiang Shang Hai Jiang su Bei Jing Tian Jin Shi Jia Zhuang Tai Yuan Hohhot Shen Yang Chang Chun Harbin Shang Hai Nan Jing Capital Code y12 y13 y14 y15 y16 y17 y18 y19 y20 y21 y22 Province Zhe Jiang An Hui Fu Jian Jiang Xi Shan Dong He Nan Hu Bei Hu Nan Guang Dong Guang Xi Hai Nan Capital Hang Zhou He Fei Fu Zhou Nan Chang Ji Nan Zheng Zhou Wu Han Chang Sha Guang Zhou Nan Ning Hai Kou Code y23 y24 y25 y26 y27 y28 y29 y30 y31 y32 Province Capital Chong Qing Chong Qing Si Chuan Gui Zhou Yun Nan Tibet Shaan Xi Gan Su Qing Hai Ning Xia Xin Jiang Cheng Du Gui Yang Kun Ming Lhasa Xi An Lan Zhou Xi Ning Yin Chuan Urumchi

4 Model Design
4.1 water storage
For China's 31 provinces, through the access to China's national bureau of statistics website, we get every provinces' total water resources and water consumption datas from 2003 to 2010. First of all, we use SPSS software to data fitting, found that most of the provinces of the data are meet the linear correlation, for the few significant differences in the level of the undesirable, we think this is due to the less datas and the year interval nearer, therefore, we assume that all corresponding data are linearly dependent. Secondly, we use SPSS software for data to a linear regression analysis (linear regression), through analysis the variance and the regression coefficient of the significance test can we found, the established equation is significant. Table 2: The Original Data
time 2003 2004 2005 2006 2007 2008 2009 2010 Y1 27460.20 24129.60 28053.10 25330.10 25255.20 27434.30 24180.20 30906.40 Y2 18.40 21.30 23.20 22.10 23.80 34.20 21.80 23.10 Y3 10.60 14.30 10.60 10.10 11.30 18.30 15.20 9.20 Y4 153.10 154.20 134.60 107.30 119.80 161.00 141.20 138.90 Y5 134.90 92.50 84.10 88.50 103.40 87.40 85.80 91.50 y…… …… …… …… …… …… …… …… ……

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Team 22942

Finally, we use SPSS to predict the regression variance ,let us focus on the data of the whole country,We get five column datas based on the previous datas,The first column variables named pre_1 is the corresponding predicted datas.When we enter the year of 2025, with the corresponding regression equation that predicted water and total water resources for: 31536.90 and 7480.92. Table 3: The Processed Data

Using excel for Data integration,we can estimate the various provinces water consumption and water amount in 2025 as the table below.

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Team 22942
Table 4: Water Consumption and Water Amount in 2025
Provi -nce
y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11

Water resources situation
31536.90 38.93 16.63 122.75 27.19 124.23 702.66 778.16 892.80 91.78 633.86

Water consumpt ion
7480.92 36.72 26.62 181.70 70.63 211.20 182.32 146.18 489.22 153.59 641.17

Provi -nce
y12 y13 y14 y15 y16 y17 y18 y19 y20 y21 y22

Water resources situation
2349.91 768.70 2178.41 2768.50 161.91 410.66 1075.34 1429.70 3217.13 1770.97 1049.55

Water consumpti on
196.98 543.56 250.38 386.42 233.96 309.89 382.56 328.70 480.72 349.59 42.58

Provi -nce
y23 y24 y25 y26 y27 y28 y29 y30 y31 y32

Water resources situation
323.38 2113.72 1181.30 2150.30 3596.71 320.44 161.20 1063.18 7.29 974.06

Water consumpt ion
139.87 257.89 119.65 162.23 54.37 107.39 120.59 34.26 61.23 620.71

4.2 water movement and transfer
4.2.1 Conclusion of water resources Through the datas in the chart above,we know that different provinces face different situations. At the same time, through the comparison of the province's water usage and total water resources, we can clearly know whether the region faces water shortage or not. In order to make it obvious, we develop a Chinese outline drawing with four different colors to illustrate the comparative water storage and water demand. Let us have a look at the picture and the graphical representation below,we can draw the conclusion that north China generally lacks water, but the water in south is enough.

Figure 1: The Distribution of Water Resources

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Team 22942
What can appear in your mind, ‘South-to-north water transfer project?’.Yes,you are right.In recent years,’South-to-north water transfer project’as a national key project, alleviates the water at the imbalance problem dramatically in China.Thus,we also need to carry out the diversion of water resources at the moment. 4.2.2 Modeling-- Max Spanning Tree (MST) v??u, v ?? E ? Given that G = (V, E), (u, v) stands for the edge connecting point u and
v??u, v ?? E ? ,and

w(u, v) stands for the weight of the edge.

?T ? E
w(T ) ?

in a non-cyclic graph, and
?u ,v ??T

? w(u, v)

We choose from the above provinces major cities with high water supply or demand. We realize that these cities should maintain economic growth so as to distribute the transferred water to the rest area. As for water transfer within the provinces, the existing pipeline system within each province may serve the need. First,look at primary table of chinese provinces and their core cities,then review Figure 1, we can infer that north China faces serious water challenges, while the south China have surplus water storage.The wester province, though confronted with water shortage to some degree, are generally self-sufficient without large-scale water transfer,because there have less people.As the graph show, Blue letters are for water-surplus province and Red ones for water-deficient province. According to the MST model, we draft a cyclic graph and mark the weight of each edge on the graph. To simplify the problem, we connect the cities with straight lines. Don't consider the western inland, southeast mountain. Distance statistics are taken from Google Map. The ‘safe edge’ theory GENERIC-MST-FUNCTION (G,w) 1 T := Φ 2 while T is not yet a ‘tree’ 3 do look for a ‘safe edge’(u, v) for T 4 T := T U {(u, v)} 5 return T First ,we adopt Floyd algorithm to find the any two cities of the most short,using software matlab7.0.Then the Minimum Spanning Tree Model to

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Team 22942

Figure 2: The Process Diagram

find the shortest path,using lingo11.0 software.(Programs see appendix) A ‘safe edge’ is generated each time.

Figure 3: The Final Diagram

We obtain a max spanning tree through calculation. The whole pipeline system is approximately 5969.5 miles. Deviation might be between(0,150) miles.

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Team 22942 4.3 Water Desalination and conservation
Water desalination has a very broad prospect, but after decades of development, seawater desalination technology is still not very developed, and increasingly tend to nervous energy, seawater desalination this huge energy consumption process measures cost has been high. According to relevant data shows, seawater desalination cost generally includes several parts: chemical, electric power, membrane replacement cost, the workers salary welfare, fixed assets depreciation, equipment maintenance fees, management fees, loan interest and other parts. In 2013, China's domestic seawater desalination cost calculation are as follows:general chemical consumption in 0.3 ~ 0.5 yuan/ton water; Power consumption is about 2.2 ~ 2.5 yuan/ton water; Membrane replacement cost 0.3 ~ 0.5 yuan/ton water; The worker salary welfare about 0.2 yuan/ton water; Fixed assets depreciation cost 0.9 ~ 1.2 yuan/ton water; Equipment maintenance cost is 0.2 ~ 0.4 yuan/ton water; Management fee is less than 0.1 yuan/tons of water. In the not consider loan interest circumstances, the general method of membrane water desalination project system water cost is 4.2 ~ 5.4 yuan/tons of water, including electric power cost and depreciation is making water cost the largest of the two parts. Considering technology progress,seawater desalination price will have appropriate decline.But according only to to less than 4% of the rate decreased to 2025 annual cost is still greater than the water dispatching the small cost.In addition,the domestic water dispatching a large number of investment mainly in the earlier stage of development, the investment is small, and the late existing rivers, reservoirs, the use of long-term investment costs to lower.

5 Relationship Analysis
According to the above statement,we can easily find out that water resources do have a close relationship with the region’s economic, political, ecological environment and so on.Through the analysis, we can draw the following: ??Almost all of the provincial capital cities have relatively large rivers flowed through; ??Generally speaking, in the water content rich areas, its economic development level is higher; ??Water resources situation in China do have obvious differences, which is the direct causes of the difference of the ecological environment .

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Team 22942
Table 5:The Relationship between Water Resources and the Total Population
year 1998 1999 2000 2001 2002 2003 Water quantity per 2788.8 2280.7 2193.9 2112.5 2207.2 2131.3 total population 124761 125786 126743 127627 128453 129227 year 2004 2005 2006 2007 average Water quantity per 1856.3 2151.8 1932.1 1916.3 2157.1 total population 129988 130756 131448 132129 128691.8

Figure 4: The Relationship between Water resources and Total Population

Through the observation of above dates,we can be sure that they have linear relationship,so we set up linear regression analysis table below.
Table 6:Linear Regression Analysis
year 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 total average x 124761 125786 126743 127627 128453 129227 129988 130756 131448 132129 1286918 128691.8 y 2788.8 2280.7 2193.9 2112.5 2207.2 2131.3 1856.3 2151.8 1932.1 1916.3 21570.9 2157.1 y-yp 631.7 123.6 36.8 -44.6 50.1 -25.8 -300.8 -5.3 -225 -240.8 -0.1 -0.01 (y-yp)
2

x-xp -3930.8 -2905.8 -1948.8 -1064.8 -238.8 535.2 1296.2 2064.2 2756.2 3437.2 2.72848E-12 2.72848E-13

(x-xp)

2

399057.5 15279.43 1354.976 1988.268 2511.012 665.1241 90474.62 27.9841 50620.5 57979.82 619959.3 61995.92342

15451189 8443674 3797821 1133799 57025.44 286439 1680134 4260922 7596638 11814344 54521986 5452198.544

We establish a regression equation: y=-0.0869539x+13347.348.Through the regression analysis,we gat the decision coefficient: ,it

10

Team 22942
means that the model of data fitting is good.By carry the significance T test,we get t=2.3064,which reject to the null hypothesis,thus,we can get the conclution that water resources has a significant influence on the total amount of population . In order to define the relationship between them more clearly,we refer to the extra relevant materials. First of all, we compare the Chinese and part of high income countries per capita water use index.Through the data in the table, we can clearly see the water resources utilization rate differences.The economy developed country, its water resources utilization rate is higher,which means that the same number of water resources has created a different value .
Table 7 :China and some developed countries per capita water use index
Water per unit of GDP (m?/$10000)
185.7 210.5 266.1 215.8 517.9 109.6 255.7 575.1 757.7 651.5 375.0 5614 53 84.7 60.1 19---85 94*5

Added value of industrial water (m?/¥10000)
18.8 43.1 76.5 54.9 54.3 31.3

Countries

Dmestic water(m?)

Industrial water(m?)
243 405 459 316 266 158 20 203 273 221 256 90

Irrigation water(m?)
368 116 100 176 581 6 323 484 354 291 280 302

Total

(m?)
736 576 665 518 985 205 408 781 738 632 625 435

Japan Germany France Netherlands Italy UK Israel Spain Portugal South Korea Average China

125 58 106 26 138 41 65 94 111 120 88 43

Secondly,we found the national provinces water index in 1998.From the table,it becomes obviously that the higher the per capita GDP of provinces and cities, the lower its water consumption per capita and water consumption unit value of industrial output.It shows in per capita water consumption and the level of economic development, the per capita water resources, water resources development and utilization degree, water-saving level, etc has close relationship. In a word,through the above model and data analysis,we can clearly see the water to a region's economic, political, cultural, ecological environment is a significant impact.

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Team 22942

6 Position Paper for Leaders
Dear governmental leaderships :
Make an efficient, economic and sustainable water resources strategy is extremely urgent.Here are several offers:

? South-to-north water transfer project is very timely and useful; ? Have a clear cognition on water resources with the development of local economy, politics and its ecological environment; ? Take more measures to develop a water desalination program like encourage desalination plants to be located at favorable coastal area; ? Carry out diversified cooperation actively in all aspects; ? Try to use ladder-like water price to protect and save water; ? Depend on the high-tech to take full use of water resources.

The reason why should choose our model: ? Our model is based on the strict inference and calculation; ? The method we take is simple but accessible; ? SPSS,Matlab,Excel are all used in our modeling process; ? We comprehensively considering the influence of water resources to regional economy, politics and ecological environment; ? A large number of data were cited,which increased the authenticity and persuasiveness of the model. Believe us,and believe you!Take a look,take action!

Yours sincerely Team 22942 2013-2-4

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Team 22942

7 Reference
[1] Jing Guolin, Liu Yang,Han Chunjie,Zhu Qinghua.2009.Mathematical modeling for desalination by electrodialysis. [2] http://www.stats.gov.cn/ [3] Whipple W, Jr. 1998.Water Resources: A New Era for Coordination[M]. Reston, ASCE Press. [4] http://www.consulfrance-shanghai.org/ [5]Shiqi Peng.2011. Water resources strategy and agricultural

development in China.

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