美赛获奖作品之欧阳术创编

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Mathematical Contest in Modeling (MCM/ICM) Summary Sheet Summary
Fresh water is the limiting constraint for development in much of the world.Take China into consideration, the nation’s freshwater supplies can no longer quench its thirst.We build Model One and Model Two to predict the fresh water needed and available in 2025, from which we can get a visualized conclusion of water shortage in future China. Based on the conclusion we set up Model Three and Model Four to put forward strategies to solve the problem of saving water in different fields and increase the sources of fresh water. In order to find the best strategies, Model Five was set up to evaluate the different strategies based on the indexes of cost and water saving. Finally we came up the best strategy according to our models.
The first model predicts the fresh water available in 2025 based on the assumption that the main sources of fresh water are stable in the short period. Model Two assumes that sweeping reforms will not be implemented in terms of industry, agriculture and society
andpredicts the amount of fresh water needed based on the usage of different fields and areas in order to make the prediction more convincing.
Based on the thought of “strategic sourcing initiative”, the third model is built to cut the water expenditure. Regression Model and Analytic Hierarchy Processare employed to solve the problem ofsaving fresh water in industry, agriculture and domestic. Model Four is employed to solve the problem of increasing the source of fresh water through desalinization and sewage recycling.
The evaluation of different strategies takes cost and goal into consideration. Principal Component Analysisis employed to evaluate the strategies. We divided the strate gies into “Business as usual, Governance, Strong and At all Costs”, and the strong scores highest in the comprehensive evaluation of all strategies.We proposed our water strategy based on the analysis and evaluation above.
In a nutshell, we predicted the fresh water shortage roundly and put forward a strategy based on the comprehensive evaluation in distinctive thoughts.
Key words:mathematicmodels, Chinese water strategy, water resource
Content
Water, Water, Everywhere
Mathematic Models of China Water Strategy
1.Introduction3
2.Assumptions4
3.Key Terminologyand Symbols4
4.Predictions of fresh water5
4.1 How much fresh water is available in China in 2025?6
4.2 How much fresh water will be needed in China in 2025?7
4.3.1 Predictions based on different fields8
4.3.2 Predictions based on different areas9
4.3 Conclusion11
5.Solutions to the Problem11
5.1 Save fresh water 11
5.1.1 Save fresh water used in industry 12
5.1.2 Save fresh water used in agriculture13
5.1.3 Save fresh water in domestic use16
5.2 Increase the sources of fresh water 18
5.2.1 Sea water desalination18
5.2.2 Turn sewage to fresh water18
5.3 Adjust the distribution of water both in time and area20
5.3.1Water transfers20
5.3.2 Conservation21
6.Evaluation of Water Strategies21
6.1 Principal component analysis (PCA) model21
6.2 Evaluation of strategies based on PCA model22
6.3 Which strategy is the best? 28
7.Strengths and Weaknesses 28
7.1 Strengths28
7.2 Weaknesses29
8.Reference29
1.Introduction
Fresh water is the limiting constraint for development in much of the world. Take China into consideration, the water resources of China are affected by both severe water quantity shortages and severe water quality pollution. A growing population and rapid economic development as well as lax environmental oversight have increased water demand and pollution[1].
Across China, the nation’s freshwater supplies can no longer quench its thirst.Chinese experts warned of future or current water
shortages.Water resource usage was expected to peak in 2025 when the population peaks. Below is the figure of the analysis of the supply and cost of fresh water.
Whatever the use of freshwater (industry,agriculture, domestic use), huge saving of water and improving of water management is possible. Legislative actions should be sought to get municipalities to use water in a rational, planned, orderly way.
In addition to water and conservation storage/movement, desalinization technology that remove excess salt and other minerals from water holds promise to convert salt water into fresh water suitable for human consumption or irrigation. Due to the water problems, as well as for future exports, China is building up its desalination technological abilities and plans to create an indigenous industry. Some cities have introduced extensive water conservation and recycling programs and technologies[2].
One focus of desalinization is to develop cost-effective ways of providing fresh water for human use in regions where the availability of fresh water is limited. Large-scale desalinization typically uses extremely large amounts of energy as well as specialized, expensive infrastructure, making it very costly compared with the use of fresh water from rivers or groundwater.
2. Assumptions
●There will be no war and natural disasters in China during this
period.
●The states we choose for modeling will not experience mass
migrations of population. The water use at the state level is closely related to the state population. Our water use estimation
model does take population change into consideration, while the change follows the current demographic trends.
●Sweeping reforms will not be implemented in terms of industry,
society, economy, policy, culture and environment. Our estimation model prefers that the future development in terms of industry, society, economy, policy, culture and ecosystem can keep a steady pace through 2013 to 2025.
●The unit cost of desalination is the same in different areas.
●The unit cost of water transfer is the same in different areas.
●The resident’s water consumption is the same in same area.
●The increase in fresh water storage from the thaw of glaciers
caused by climate change and global warming is not considered.
Climate change and global warming are major obstacles human beings are facing. Many a research has been done on this topic.
We expect that effective measures can be taken and the thaw of glaciers will be ceased one day before 2025. And such increase is what we do not hope for.
3. Key Terminology and Symbols
3.1G ray Model (GM)
Gray Model is such a model that it establishes a gray differential prediction model through a small amount of incomplete information,and makesa fuzzy long-term description of the development of the object.
3.2E xponentialSmoothingForecastingMethod (ESFM)
Exponential Smoothing Forecasting Method refers to a method that introduces a simplified weighted factor that uses some indicators of current actual number and current forecast number as the foundation,namely smoothing coefficient, to achieve an average index smooth forecast method. It is a change of weighted moving average prediction method.
3.3R egression Model (RM)
Regression Model is such a model that through the establishment of the mathematical model between the numbers that have related relationships according to the statistical data in the past years, conducts a natural regression modeling to analyze the change law between the different quantities.
3.4A nalytic Hierarchy Process (AHP)
Analytic Hierarchy Process is a decision-making method that decomposes the elements that are always related into goals,
standards and scheme level which are based on to analyze the qualitative and quantitative relations.
3.5S ymbols
t Time
s Variance
Characteristic value
The first principal component
The secondary principal component
The third principal component
4. Predictions of Fresh Water
China's water resources include 2711.5 cubic kilometers of mean annual run-off in its rivers and 828.8 cubic kilometers of groundwater recharge. As pumping water draws water from nearby rivers, the total available resource is less than the sum of surface and groundwater, and thus is only 2821.4 cubic kilometers. 80% of these resources are in the South of China[3].
In this part, we employed Gray Model to predict the amount of fresh water that is needed and available in 2025.
4.1 How much fresh water is available in China in 2025?
We drew a figure of the total amount of water resources in China
from 2003 to 2011.
By observing the figure above, we can find that the total amount of fresh water in China moved in steady cycles, and the average amount is around 2700 cubic kilometers.
Analogously, the surface water and underground water in China can be regarded as unchanged until 2025.
Most of the rain China experiences during the year occurs during the summer months.We searched the data of rainfall in China, and
Figure 3 The average annual precipitation in different regions of
rainfall differs from different regions.
The picture above suggests that the annual rainfall in China is nearly stable during a short period, and it is different from different regions.
Based on these conclusions and combining with other documents, we can calculate the fresh water that is available in 2025:
Equation 1Relationship between the amount of total fresh water and
available fresh water
Where
: the volume of available fresh water;
: the volume of total fresh water;
: the percentage that the available fresh water take in the total fresh water.
After the substitution of numerical values, we can figure out the volume of available fresh water in 2025:
cubic kilometers
4.2How much fresh water will be needed in China in 2025?
Total water withdrawals were estimated at 554 cubic kilometers in 2005, or about 20% of renewable resources. Demand is from the following sectors:
•65% agriculture
•23% industry
•12% domestic
Figure 4 the amount of fresh water supply
In 2006, 626000 square kilometers were irrigated[4].
According to the data published by the National Bureau of Statistics of China, we drew a figure of the amount of the water supply in China from 2003 to 2011 (water was supplied for industry, agriculture, and domestic).
By observing the figure above and analyzing the data from Chinese government, we can find that the supply of fresh water increased every year, namely the need of fresh water is increased over time. Chinese experts warned of future or current water shortages. Water resource usage was expected to peak in 2030 when the population peaks.
Considering that the trend of increasing will not experience large fluctuations, we chose Gray Model to predict the need of fresh water in 2025, since GM can solve stable time series problems.
4.2.1 Predictions based on different fields
Use Gray Model to predict the fresh water demand in 2025
●Class-compareverification
We counted the total water resource years from 2003 to 2011 as follows：
(4-1)
◆Calculate the class-compare
To ensure the feasibility of Gray model, we firstly performed data examination:
(4-2)
◆Class-compare judgment
Put the calculated class-compare to substitute
(4-3)
Since all，k=2,3…,9, so we can use to make a content GM(1,1) model.
●Build GM（1,1）model
◆One-accumulate sequence of the original data
Table 1
(4-4)
◆Build data of matrix B and vector Y
◆Calculate the value of
(4-5)
◆Build model
(4-6)
So that equation has response particular solution
(4-7)
●GM(1.1) Model error test
The average of related error is 0.0042
Table 2 Common Accuracy Class
By the comparison of error test value and the common accuracy class, we can see that the rank of the model is level 1, which means the model has high accuracy and can be used for prediction.
Result
The result is that the fresh water in demand in 2025 will be 745.2 cubic kilometers.
4.2.2 Predictions based on different areas
●The information of six strict in China
◆North: Beijing, Tianjin, Hebei, Shanxi, InnerMongolia
◆Northeast: Liaoning, Jilin, Heilongjiang
◆East:Shanghai , Jiangsu, Zhejiang, Anhui, Fujian, Jiangxi
◆South: Henan, Hubei, Hunan, Guangdong, Hainan, Guangxi
◆Southwest: Chongqing，Sichuan，Guizhou, Yunnan，Tibet
◆Northwest: Shaanxi，Gansu ，Qinghai，Ningxia Xinjiang
●We employed GM (1.1) to predict the water needed in different
regions in 2025.
Table 3 Fresh water needed in different regions in 2025
From the table we can see the total amount of fresh water needed in 2025 is 753.0 cubic kilometers, comparing to the prediction based on different fields which is 745.2 cubic kilometers, we conclude thatthe result is correct.
4.3 Conclusion
According to the results of predicting the amount of the available fresh water and fresh water needed in 2025, we drew some conclusions as follow:
●In China, the amount of fresh water needed in 2025 almost
stretches to the limit of the water available, since the former is about 745 cubic kilometers and the latter is about 746 cubic kilometers. So take steps to save fresh water is necessary for the government of China, and adopt an appropriate water strategy is crucial for the development.
●In China, fresh water is unevenly distributed in both area and
time, making water transfer and conservation necessary and important.
5. Solutions to the Problem
From the conclusion above we have known that take necessary steps to save fresh water is very important in China. According to the problems in the conclusion, we put forward solutions from three aspects, namely saving fresh water, increasing the source of fresh water and adjusting the distribution of water in time and space.
5.1 Save fresh water
Fresh water demand is mainly from industry, agriculture and domestic, so that the measures to save fresh water will be adopted in three aspects.
5.1.1 Save fresh water used in industry
The industry can be divided into high water consumption and low water consumption industry, readjusting the industrial structure can remarkably save fresh water consumption.
By decreasing the percentage that high water consumed enterprises take in the whole industry and at the same time increasing the percentage of low water consumed enterprises, the fresh water can be saved yet the cost of industrial restructuring will increase.
We accessed relevant information and summed up a relationship among the reduction high water consumption industry, fresh water saving and the increasing of cost.
We employed Exponential Smoothing Forecasting Method (ESFM) to predict the amount of cost and water saving.
The mathematicalexpression of the relationship between the percentage of high water consumption industry reduction and the cost is:
(5-1) In order to choose an appropriate value of , we calculated the standarddeviation of different forecasting based on different .
Table 4 Standard deviation S of different forecasting
0.20.50.8
36.206524.294217.7608
From the table we can find that the when =0.8, the S got minimum value 17.7608, so we choose =0.8.The mathematical expression is:
(5-2)
Using Regression Model, we summed up the mathematical expression of the relationship between the percentage of high water consumption industry reduction and the percentage of fresh water
saving:
(5-3)
Where
: regression coefficient,
: random error.
We can figure out the value of , and , so the ultimate mathematic expression is:
(5-4)
The coefficientofdetermination0.8985, it shows that the regression effect is pretty good.F inspection was used to conduct significanceanalysis, and the value of P is 0.000, the level of significance is acceptable.
By accessing relevant information and the building mathematic
models above, we summed up a relationship among the reduction high water consumption industry, fresh water saving and the increasing of cost.
Table 5The relation among the water saving, industry adjustment and
the cost
The table suggests that the percentage of fresh water reduction and cost will both increase with the percentage of high water consumption industry reduction increasing.
5.1.2 Save fresh water used in agriculture
Agricultural water refers to water used for irrigation and rural cattle water, and water used for irrigation accounts for about 91.1%[5]. So improving water efficiency in irrigation is crucial to save agriculture water.
Once the quantitative and temporal characteristics of optimal water demand have been determined, a method that can make such
water available in the most effective way should be selected. There are three main irrigation methods, namely: Surface (or gravity) irrigation, Sprinkler irrigation, Drip irrigation.
Analysis of three irrigation methods
As we know that Surface irrigation involves the application of water by gravity flow to the surface of the field. Surface irrigation is the easiest and the least cost method yet consumes most of the water.
Sprinkler irrigation systems imitate natural rainfall. These systems are more efficient than surface irrigation, however, they are more costly to install and operate because of the need for pressurized water. This method can achieve efficiencies as high as 95% and water savings of up 70%.
Drip irrigation delivers water through the use of pressurized pipes and drippers that run close to the plants and that can be placed
on the soil surface or below ground. Drip irrigation is reported
to
Figure 6 Sprinkler irrigation
help achieve yield gains up to 100%, water savings up to 80%.
Figure 7 Drip irrigation
●Improving irrigation methods to save water
In order to save fresh water used in agriculture and increase the efficiency of irrigation, we suggest the government to popularize the Sprinkler and Drip irrigation methods.
The more the percentage of sprinkler and drip accounts for in irrigation methods increases, the more fresh water will be saved and in the same way the cost increases.
We build Regression models to analyze the quantitative relationship between the percentage of sprinkler/drip irrigation and water saving as well as the relationship between the percentage of better irrigation methods and cost.
●Regression model
According to the data of past, we drew a figure of the relationship between the percentage of Sprinkler/Drip irrigation method, and another figure is of the percentage of better irrigation method and the cost.
From the picture we can see that the independent variable has good linear relations with to dependent variables.
The regression model is:
(5-5)
Where
: regression coefficient,
: random error.
When processing data, we conducted deviation analysis and pick out outliers.
Figure 10deviation analysis 1
Figure 11deviation analysis 2
The pictures suggest that there is one abnormal point in picture 1 and two abnormal points in picture 2, after eliminating the points Figure 9 The relationship between the industry
adjustment and the cost
Figure 8 The relationship between the industry
adjustment and the water saving
we build a regression model again. And the ultimate relation between the increasing percentage of sprinkle/drip irrigation method and fresh water save is:
(5-6)
The coefficientofdetermination0.9234
Using the same model, only change the dependent variable, we can obtain the relation between the increasing percentage of sprinkle/drip irrigation method and the total cost:
(5-7)
The coefficientofdetermination0.8943
Assume the government will afford 10% of the total cost, the ultimate cost will be expressed as follow:
(5-8)
Result
By accessing relevant information and the building mathematic models above, we summed up a relationship among the increase of sprinkle /drip irrigation in agriculture, fresh water saving and the increasing of cost.
Table 6 The relation among agriculture improvement, water saving
and cost
122500
564800
10115200
15156510
20187800
25258900
30309500
353410962
5.1.3 Save fresh water in domestic use
Fresh water used in households can be affected with several factors such as water price, consciousness of saving water and the applying of water-saving device.
We employed Analytic Hierarchy Process (AHP) model to find the relations among the cost, fresh water saving and the factors that affect domestic water use and calculate the weights of each factor. 2.First, a judgment matrix was constructed
(5-9)
➢The matrix constructed is pretty subjective,soviability test is necessary.Solving matrix eigenvalues and choose the largest
one as one of the median of next step, the other one is
the size of the matrix.
➢Define a inconformity degree index ，to show the judgment matrix is realistic.
(5-10)
To this problem, the size of the matrix n is 3.
●Second, define a Random Consistency Index ，which is only
related to the size of the judgment matrix.
Table 7 Mean of random consistency index RI
Random consistency index RI
n2345678910
00.520.89 1.12 1.26 1.36 1.41 1.46 1.49
In this problem,0.52
●Third, calculate the proportion of the random consistency.
(5-11)
In this problem，CR= 0.07718
The way of judgment is: if CR<0.1, then the judgment matrix is satisfactory, otherwise the judgment matrix is unsatisfactory. Due to the result of CR, we can see the judgment matrix is acceptable.
●Fourth, calculate the eigenvector of the maximized eigenvalue.
Standardizing the eigenvector and make the sum of its components is 1.The ultimate standardized eigenvector is called weight vector:
(6-12)
By accessing relevant information and the building mathematic models above, we summed up a relationship among the factors of domestic water use, fresh water saving and the cost.
Table 8Relationships of the factors of domestic water use, fresh
water saving and cost.
Weight 0.6874Weight 0.1123Weight 0.2003
1110,50.7
5551,6 3.3
101010 2.7 5.4
151515 3.37.6
202020 3.510.3
2525254,212.4
303030 5.015.6
353535 5.617.3 The cost equation is:
Cost=0.1123(saving water conscious)+0.2003(Rising percent of applying water-saving device)-0.6874(water price rising);
The fresh water saving equation is:
Fresh water saving=0.1123(saving water
conscious)+0.2003(Rising percent of applying water-saving device)-0.6874(water price rising);
5.2 Increase the source of fresh water
5.2.1 Sea water desalination
Sea water desalination is an effective way to increase the quantity of water source, and it is also a strategic choice to solve the lack of fresh water resources.
The length of Chinese mainland coastline is more than 18 000 kilometers, there are more than 150 coastal cities and 6500 islands of which the areas is more than 500 square meters. This is the advantages of developing seawater desalination in China[6].
During China's " Eleventh five years plan" period, the growth rate of the annual capacity of sea water desalination is more than 60%, by the end of 2010, China built more than 70 sets of desalination devices, the designing capacity of desalination is 600,000 cubic meters / day[7].
According to the data above, we predicted that the capacity of desalination in China will increase by 900% in 2025, namely 1.971 cubic kilometers fresh water will be produced in 2025.
5.2.2 Turn sewage to fresh water
There are many ways to clean up polluted water, but at present most of the ways to deal with sewage aim at discharge instead of recycling.
According to the data of the amount of sewage process, we assumed the proportion of water can be used again, so the increasing volume of fresh water can be predicted in the future.
The paper analysis data of past and build Double Exponential Table 9The relation of sewage processing and cost and water
purification in past
Smoothing Model (DESM) to predict the amount of water can be used again from sewage.
Based on the data above, we used DESM to predict the water
amount.
，
,(5-13)
When t=9, we can calculate the value of a and b:a=13447, b=694.5906
The linear trend equation is:
(5-14) In order to calculate the amount of water purification of every year, we let T equal to 1.
(5-15)
Use this equation we figured out the amount of annual sewage purification from 2013 to 2025.
Considering the cost of the sewage will decrease by 3% each year, so the equation is:
（1+the percentage of water purification）(6-15)
Table 10 The prediction of the amount of annual sewage purification
and cost
201311 1.783610.3 1.1
201412 1.8612 4.40.3
201513 2.098611.20.7
201614 2.27468.3 1.0
201715 2.3356 6.6 1.3
201816 2.5567 5.2 1.4
201917 2.6745 5.8 1.5
202018 2.8902 6.0 1.6
202119 3.1005 6.6 1.8
202220 3.3678 6.0 1.9
202321 3.65787.0 2.0
202422 3.90007.4 2.1
202523 4.26788.6 2.3
By accessing relevant information and building mathematic models above, we know that 6% of the sewage treatment can be turned into fresh water. According to China’s 12th five-year plan and the data predicted by our model, the amount of water turned from sewage is 4.26780.25=1.0669 cubic kilometers.
5.3 Adjust the distribution of water both in time and area
5.3.1 Water transfers
Large-scale water transfers have long been advocated by Chinese planners as a solution to the country's water woes. The
South-North Water Transfer Project is being developed primarily to divert water from the Yangtze River to the Yellow River and Beijing.
The South–North Water Transfer Project is a multi-decade infrastructure project of the People's Republic of China to better utilize water resources available to China. This is because heavily industrialized Northern China has a much lower rainfall and its rivers are running dry. The project includes a Western, a Central and Eastern route. All three routes aim to divert water from the Yangtze River to the Yellow River and Hai River. The eastern route uses the course of the Grand Canal; the central route is from the upper reaches of the Han River (a tributary of Yangtze River) to Beijing and Tianjin; and the western route is in the western headwaters of the rivers where the Yangtze River and the Yellow River are closest to one another. This project will divert 44.8 billion cubic meters/year of water from South to North.
The idea for the South–North Water Transfer Project originated from Mao Zedong who said, "Southern water is plentiful, northern water scarce. If at all possible, borrowing some water would be good." The complete project is expected to cost $62bn – more than twice as much as the Three Gorges Dam[8].
Construction costs of the eastern and central routes were estimated to be $37.44bn in 2008. The government had budgeted only $7.9bn, less than a quarter of the total cost, at that time, including 26bn from the central government and special accounts, 8bn from local governments and almost 20bn in loans. As of 2008, around 30bn RMB had been spent for the construction of the eastern (5.66bn RMB) and central routes (24.82bn RMB). Costs of the projects have significantly increased[9]
5.3.2 Conservation
Water conservation encompasses the policies, strategies and activities to manage fresh water as a sustainable resource to protect the water environment and to meet current and future human demand. Population, household size and growth and affluence all affect how much water is used. Factors such as climate change will increase pressures on natural water resources especially in manufacturing and agricultural irrigation.
In implementing water conservation principles there are a number of key activities that may be beneficial.
1.Any beneficial reduction in water loss, use or waste
2.Avoiding any damage to water quality.。