1、Team #20300 Page 28 of 28For office use onlyT1_T2_T3_T4_ Team Control Number 20300Problem ChosenBFor office use onlyF1_F2_F3_F4_2013 Mathematical Contest in Modeling (MCM) Summary SheetSummary Save water, Save money, Save China ! China is a country with a serious water shortage. It is one of the poo

2、rest countries in per capita water.Whats worse, pollution and uneven distribution of water resources make the available water even less. In view of this, we set up an optimization model to obtain an optimal allocation strategy of water resources, hoping to contribute to Chinas endeavor for getting r

3、id of water shortage. Firstly, according to geography, climate and water resource conditions, we divide the whole country into 7 regions. Next, we predict the water demand of each region in China in 2025 by the method of GM(1,1). Then we establish Model 1-Protective Exploitation of Water Recourses,

4、finding out the maximum exploitation of water resources on condition that the environment can not be destroyed. Further, we work out the shortfall or surplus of water supply in each region. With the help of the data preparation above, we set up Model 2-Optimization of water transfer and supplement,

5、taking storage, movement and desalination into consideration. Besides, we set a constrain-Balance Degree which reflects the balance situation of water resources distribution in all regions. We solve the model, getting an optimal strategy to meet the water demand of each region with the lowest invest

6、ment: 68.7 billion yuan. We obtain two valuable consequences through detailed analysis and verification of our model. Firstly, balance degree has significant influence on the economic growth and environmental protection. The greater the balance degree is, the better the water strategy can promote to

7、 economy and environment. Secondly,we find that China should consider building desalination factories when the cost of it decreases to 1.5 yuan/m3. In this way, not only water demand are met, but also the investment cost gets smaller-dropping to 68.1 billion yuan. Finally,we discuss the feasibility

8、of our optimal strategy, which proves that it is effictive and practicable to help China to get rid of water shortage.Save water, Save money, Save China !content1. Introduction22. Model Assumptions23. Terminology And Analysis before Modeling33.1 Terminology3Explanation of nouns:3Explanation of Symbo

9、ls:33.2 The General Idea of Our Model44.Zoning And Data Preparation54.1 Zoning:54.2 Data Preparation:75. Models85.1 Medel 1: Protective exploitation of water recourses in each region85.2 Model 2: Optimization of water transfer and supplement105.2.1 Assumptions for Model 2105.2.2 The Establishment of

10、 Model 2115.2.3 Solution And Analysis of Model 2165.2.4 Water Resource Allocation in West Region186. Sensitivity Analysis of Our Model196.1 Sensitivity Analysis of Model 1196.2 Sensitivity Analysis of Model 2207. Verification of Model227.1 Economic Analysis22-Is it beneficial or not for Chinas econo

11、my to improve balance degree with huge investment?227.2 Realistic Analysis237.3 Protect Bohai Sea !248.Non-technical Position Paper269. Strengths And Weakness2710. Conclusion2711. Referrences281. Introduction China is a country with a serious water shortage. Although China has a total amount of fres

12、h water resources of 2.8 trillion cubic meters, ranking fourth in the world, the amount of water per capita is only 2200 Cubic meters - 1/4 of the world average. It is one of the poorest countries in per capita water.Whats worse, pollution and uneven distribution of water resources make the availabl

13、e water even less. China is now begin to construct some huge projects to improve this situation, such as the South-North Water Transfer Project, and the Three Gorges Dam. However, with the development of the country, water consumption also shows a rising trend. How to further improve the distributio

14、n of the countrys water resources, and solve water problems of water-deficient area for the next ten years? Its great challenge for the Chinese government at this stage. The rapid development of China, will inevitably result in the increase of water demand. We use Grey prediction and Model 1-Protect

15、ive exploitation of water resources, to forecast the water demand of each region of the country in 2025. In order to solve the problem of water shortfall in some regions， we set up Model 2-Optimization of water transfer and supplement to obtain the optimal water strategy, which contains how to trans

16、fer water among regions and how many reserviors and desalination devices should be built in each region. This strategy can meet the water demands for all the regions in China, while spending the least. Further more, we analyze the sensitivity of our model, and find that if we want the water supply m

17、ore fair on a national scale, which means to improve the balance degree, the cost will increase notably. However, in the long run, this balance of water distribution will bring a wonderful benefit for both the economy and the environment. In addition, we also find that the decrease of desalination c

18、osts can influence the optimal strategy, when this cost falls below a certain value, more desalination devices should be built. At last, we demonstrate the feasibility and the optimality of our water strategy, and give some suggestion for Chinas water policy to the government.2. Model Assumptions Ov

19、erall Assumptionl The geography and water conservancy information we collected is true and reliable;l In the future years, Chinas meteorological condition remains stable and severe flood or drought will not occur .Assumption for Model 1l Each region is supposed to exploit its own water resources in

20、the principle of sustainable development;l Weather condition changes little so that we can estimate the rainfall in 2025 with the data in 2012.Assumption for Model 2 l If one region transfers water to another, the transfer distance is centroid distance between two regions;l The altitude difference b

21、etween two regions is the difference of the average altitude of two regions;l Cost of water transfer between two regions consists of water conservancy construction and transporting costs. This cost can be represented with quantity of water transfer. Obviously, higher altitude and further distance ma

22、ke larger amount of cost ;l Storage in a reservoir mainly comes from rainwater and its cost of building can be roughly estimated by its capacity;l Desalination project is available for coastal regions only.3. Terminology And Analysis before Modeling3.1 TerminologyExplanation of nouns:Storage: Water

23、storing. In this paper, it refers to water storing of reservoirs. Movement: Transfer of water via canal or large pipe from one region to another.Explanation of Symbols: Exploitation quantity of a certain kind of water resources: Capacity of reservoir: Quantity of transfer water from region to region

24、 .: Distance between region and region .：Altitude difference between region and region .：Annual amount of precipitation of region .：Coastline length of region . ：Quantity of water after desalination of region . ：Surplus of water of region .：Shortfall of water of region .：Balance degree of water of t

25、he whole country.：Total cost of allocation strategy of water resources3.2 The General Idea of Our Model In order to make a good strategy, we find three ways of water exploitation, and they are storage of reservoirs, surface runoff and desalination of seawater. With reasonable use of these three ways

26、,we can solve water crisis of China. The following flow chart shows our general idea of modeling. Figure 1: Flow chart of modeling4.Zoning And Data Preparation4.1 Zoning: China is a big country with vast area, complex terrain and various climate. As a result, situation of each region is different .S

27、o we divide China into several regions. Within a region, geography and climate conditions is similar, so we can discuss several regions instead of 34 provinces. According to “Water resources distribution of China” and “Possession of water resources per capita of every province” as follows,Figure 2:

28、Water resources distribution of China1 Figure 3: Possession of water resources per capita of every province2We divide China into seven regions and they are: Inner Mongolia (IM): There is vast territory but less population, and one could hardly find a river in IM. The Northeast region (NE): This regi

29、on has dense population, moderate rainfall and flat terrain. There are Songhuajiang River and Heilongjiang River in this region. The Central region (CR): There is little rain in this high-terrain region. The Yellow River flows across it, but soil erosion often occurs at the same time. The North regi

30、on (NR): There is very dense population and developed economy,thus NR has a great demand of water, but it has less rain that concentrats in the summer . The South region (SR):SR has large population ,developed economy. Rainfall and rivers provide enough water for this region. The Southeast region (S

31、E): There is dense population, rich precipitation ,large number of rivers and lakes. The west region (WR): Though there is a large amount of river and rain, rugged and complex terrain and high altitude makes it hard to build water conservancy facilities.The following figure shows partition results:F

32、igure 4 : Partition results 4.2 Data Preparation: -Prediction of water demand of each region in 2025 According to China Statistical Yearbook, we can find water demand of each province from 2003 to 2011 3, combined with partition result we get, we obtain water demand of each region in these past year

33、s . Using Grey prediction GM（1,1）4,we predict their water demand in 2025. Grey prediction makes predictions by setting up model with little incomplete information. Here ,we choose Xinjiang as an example and build Grey prediction model: Where: ，. Employing Matlab, we get that: a =-0.0105，b =493.8847

34、and water demand after 2011 as the following figure shows:Figure 5: Water demand of Xinjiang after 2011 In the same way, we can obtain water demand of each region. In addition, we also get surface runoff, storage of reservoir and compensation amount of groundwater from China Statistical Yearbook 3 a

35、nd spread out these four items in the following table : unit:billion m3Table 1: Four items of water of different regionsRegionW.DSUR.RSTO.RCA.GNE88.45 98.63 41.44 29.60 NR85.7320.56 33.48 20.79 IM20.30 15.75 0.75 9.96 CR28.34 16.66 4.94 17.50 WR70.01 397.0940.60 167.24 SR173.32 695.09 76.71 232.65 S

36、E275.99 488.36 82.79 138.40 W.D: Water Demand; SUR.R: Surface runoff;STO.R: Storage of reservoir; CA.G: Compensation amount of groundwater5. Models 5.1 Medel 1: Protective exploitation of water recourses in each region In model 1,we are going to optimize the way of water supply of Regional internal.

37、 Supposed that water resources of every region can satisfy its own water demand in 2012 and water resources will not reduce by reasonable exploitation. As time goes by,water demand of each region increases as we find in the data preparation process. But considering the protection of water resources,

38、 taking the principles of sustainable development as constraints, we are supposed to limit water exploitation of three ways. Thus we can get an optimized exploitation plan and a maximum of water exploitation quantity. Assuming that in every region, quantity of water exploitation from storage of rese

39、rvoirs is . Similarly, from surface runoff and from underground water. Therefore a optimization model of protective exploitation can be set up: Objective : Maximum of water exploitation quantity in three ways For each region, we expect a maximum of water exploitation quantity: From the point of view

40、 of sustainable development, each kind of water resources can not be overexploited, for example,storage in reservoirs is not allowed to supply totally. We must leave some in case of drought and other unforeseen climate disasters. As for surface runoff, overexploitation will cause great damage to riv

41、er ecosystem and have effect on downstream regions as well. For underground water, its longer regeneration cycle than other kinds of water makes people have to exploit sustainably. Otherwise, damage like land subsidence will influence the development of cities.Therefore,we set three constraints:Cons

42、traint 1: Limit of exploitation of storage of reservoirs The quantity of water exploitation from reservoirs cannot exceed of the storage every yearConstraint 2: Limit of exploitation from surface runoff The quantity of water exploitation from surface runoff cannot exceed of the total runoff every ye

43、arWhere denotes the protection of rivers. Smaller value means better protection of rivers and Constraint 3: Limit of exploitation from underground water.Whereis compensation amount of groundwater. We can get it from China Statistical Yearbook 3 The value of ， can be different to show the different p

44、rotection of environment and smaller values make better protection. Here according to experience and international conventions5, we select 0.8，0.3 for calculation.In general, this model is: Thus,we can obtain the best water supply plan of each region and its maximum. With the help of work above,we c

45、an get water demand and water production of every region as the following table shows:unit:billion m3 Table 2: Result of model 1RegionW.D in 2025W.PW.P-W.DNE88.45 92.35 3.89 NR85.73 53.74 -31.99 IM20.30 15.28 -5.02 CR28.34 26.45 -1.89 WR70.01318.85 248.75 SR 173.32 502.55 329.22SE275.99 351.14 75.16

46、 W.D:Requirement water; P.W:Produced waterAnalysis of Results As we can see from the table above, in 2025, production of water of NR,CR and IM will not satisfy their water demand,while in NE,SE and SR it will be surplus. This result is of very important guiding significance for the overall strategy

47、of water resources in China5.2 Model 2: Optimization of water transfer and supplement It has been mentioned above that the Western Region has its special geographical conditions and water distribution. It is far from the sea, with small population and complex terrain. So the water there is enough fo

48、r itself, but can hardly be diverted to other regions, and desalination cant be adapt there. Therefore, in Model 2, we dont take the West Region into consideration, and we will study it alone in the end of this part. The research object of Model 2 is the other six regions: IM, NE,CR,NR,SR,SE. We wan

49、t a strategy to meet the water demands of all the regions. To achieve this goal, in this model, we discuss three methods. The first method we discuss here, is water transfer among regions. And there are two other implement methods for water supply-to build new reserviors and to build desalination de

50、vices. Note, desalination can only be adapted in the coastal regions: NE, NR, SE. Model 2 can find the best water strategy, which contain how to transfer water among regions, and how many reserviors and desalination devices should be built in each region. This strategy costs the least, but can meet

51、the water demands of all the regions.5.2.1 Assumptions for Model 2 l If one region transfer water to another, the conveying distance is the centroid distance between two regions;l The altitude difference between two regions is the difference of the average altitude;l Cost of water transfer between t

52、wo regions consists of construction, water conservancy and transporting costs. This cost can be represented with quantity of water transfer. Obviously,higher altitude and further distance make larger amount of cost;l Storages in the reservoirs mainly come from rain and cost of building reservoirs ca

53、n be roughly estimated by their capacity;l desalination project are available for coastal regions only.5.2.2 The Establishment of Model 2 Now，we are to express the costs and maximum capacity of the three methods above. In this way, we can find the decision variables and constraints of this optimizat

54、ion model.(1) Investment costs and maximum capacity of the three methodsWater transfer From the result of Model 1, we can see, there are three regions with water surplus, which are NE,SR, SE. We can transfer water from these regions to other regions with water shortfall. The transfer cost is determi

55、ned by three factors: transfer distances, lifting or drop heights and the quantity of transferring water.Figure 6: Water transfer route As the above figure shows, here, for regions with water surplus, we use i=1,2,3 to represent NE,SR, SE. For regions with water shortfall, we use i=1,2,3 to represen

56、t IM,CR,NR. denotes the quantity of transferring water between two regions. So we have this matrix of quantity of transferring water: (unit: trillion cubic meters) We are making a macroscopic planning for national water supply strategy, a huge project. For the sake of simplify, we assume that the distance of water transfer between two regions, is the straight-line distance between their centroids. With the help of AutoCAD (used for finding the centroid of regions) and Google eart