美国大学生数学建模大赛一等奖论文

1、 team control numberfor office use onlyt1 _t2 _t3 _t4 _ 20300problem chosenbsummary for office use only f1 _ f2 _ f3 _ f4 _ 2013 mathematical contest in modeling (mcm) summary sheetsave water, save money, save china !china is a country with a serious water shortage. it is one of the poorest countrie

2、s 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 rid of water s

3、hortage.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, finding out th

4、e 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, taking storage,

5、 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 investment: 68.7 bill

6、ion 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 economy and env

7、ironment. 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 of our optimal st

8、rategy, which proves that it is effictive and practicable to help china to get rid of water shortage. team #20300 page 1 of 28save water, save money, save china ! content 1. introduction . 22. model assumptions . 23. terminology and analysis before modeling . 33.1 terminology . 3explanation of nouns

9、: . 3explanation of symbols: . 33.2 the general idea of our model . 44.zoning and data preparation . 44.1 zoning: . 54.2 data preparation: . 75. models. 85.1 medel 1: protective exploitation of water recourses in each region . 85.2 model 2: optimization of water transfer and supplement . 105.2.1 ass

10、umptions for model 2 . 105.2.2 the establishment of model 2 . 115.2.3 solution and analysis of model 2 . 165.2.4 water resource allocation in west region . 186. sensitivity analysis of our model . 196.1 sensitivity analysis of model 1 . 196.2 sensitivity analysis of model 2 . 207. verification of mo

11、del . 227.1 economic analysis. 22 -is it beneficial or not for chinas economy to improve balance degree withhuge investment? . 227.2 realistic analysis . 237.3 protect bohai sea !. 248.non-technical position paper . 269. strengths and weakness . 2710. conclusion . 2711. referrences . 28 team #20300

12、page 2 of 281.introductionchina is a country with a serious water shortage. although china has a total amount of fresh 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 poor

13、est countries in per capita water.whats worse, pollution and uneven distribution of water resources make the available 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

14、 the development of the country, water consumption also shows a rising trend. how to further improve the distribution 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 devel

15、opment of china, will inevitably result in the increase of water demand. we use grey prediction and model 1-protective 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 mode

16、l 2-optimization of water transfer and supplement to obtain the optimal water strategy, which contains how to transfer 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 spe

17、nding the least.further more, we analyze the sensitivity of our model, and find that if we want the water supply more 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 wonderf

18、ul benefit for both the economy and the environment. in addition, we also find that the decrease of desalination costs 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

19、of our water strategy, and give some suggestion for chinas water policy to the government.2.model assumptionsoverall assumption l the geography and water conservancy information we collected is true andreliable;l in the future years, chinas meteorological condition remains stable and severeflood or

20、drought will not occur . assumption for model 1 team #20300 page 3 of 28 l each region is supposed to exploit its own water resources in the principle ofsustainable development;l weather condition changes little so that we can estimate the rainfall in 2025 withthe data in 2012. assumption for model

21、2 l if one region transfers water to another, the transfer distance is centroid distancebetween two regions;l the altitude difference between two regions is the difference of the averagealtitude of two regions;l cost of water transfer between two regions consists of water conservancyconstruction and

22、 transporting costs. this cost can be represented with quantity of water transfer. obviously, higher altitude and further distance make larger amount of cost ;l storage in a reservoir mainly comes from rainwater and its cost of building canbe roughly estimated by its capacity;l desalination project

23、is available for coastal regions only.3.terminology and analysis before modeling3.1 terminologyexplanation of nouns:storage: water 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:

24、x: exploitation quantity of a certain kind of water resources c: capacity of reservoirtij: quantity of transfer water from region i to region j .dij: distance between region i and region j .hij：altitude difference between region i and region j. team #20300 page 4 of 28 raink：annual amount of precipi

25、tation of region k.coastk：coastline length of region k.pk：quantity of water after desalination of region k.oi：surplus of water of region i.sj：shortfall of water of region j.b：balance degree of water of the whole country.w：total cost of allocation strategy of water resources3.2 the general idea of ou

26、r modelin 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,we can solve water crisis of china. the following flow chart shows our general idea of modeling.

27、 figure 1: flow chart of modeling team #20300 page 5 of 284.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 .so we divide china into several regions. within a region, geography and cl

28、imate 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: water resources distribution of china1 figure 3: possession of water res

29、ources per capita of every province2 team #20300 page 6 of 28 we 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 region has dense population, moderate rainfall and f

30、lat 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 region (nr):there is very dense population and develo

31、ped 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 (se):there is dense population, rich precipitation ,

32、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: figure 4 : partition results team #20300 page 7 of

33、284.2 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 years . using grey prediction g

34、m（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:b(a-)if=+bv ia(i-1)aegi=fi-fi-1where:f1=a1 ，i=1,2,3,22,23.employing matlab, we get that: a =-

35、0.0105，b =493.8847 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 stat

36、istical yearbook 3 and spread out these four items in the following table : team #20300 page 8 of 28 unit:billion m3table 1: four items of water of different regionsregion ne nr im cr wr sr sew.d 88.45 85.73 20.30 28.34 70.01 173.32 275.99sur.r 98.63 20.56 15.75 16.66 397.09 695.09 488.36sto.r 41.44

37、 33.48 0.75 4.94 40.60 76.71 82.79ca.g 29.60 20.79 9.96 17.50 167.24 232.65 138.40 w.d: water demand; sur.r: surface runoff;sto.r: storage of reservoir; ca.g: compensation amount of groundwater5. models5.1 medel 1: protective exploitation of water recourses in each regionin model 1,we are going to o

38、ptimize the way of water supply of regional internal. 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 proces

39、s. but considering the protection of water resources, 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,

40、 quantity of water exploitation from storage of reservoirs is x1. similarly, x2 from surface runoff and x3 from underground water.therefore a optimization model of protective exploitation can be set up:objective : maximum of water exploitation quantity in three waysfor each region, we expect a maxim

41、um of water exploitation quantity:maxx=x1+x2+x3from the point of view 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. a

42、s for surface runoff, overexploitation will cause great damage to river ecosystem and team #20300 page 9 of 28have 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 lan

43、d subsidence will influence the development of cities. therefore,we set three constraints: constraint 1: limit of exploitation of storage of reservoirsthe quantity of water exploitation from reservoirs cannot exceed d1of the storage every year x1d1reservoirconstraint 2: limit of exploitation from su

44、rface runoffthe quantity of water exploitation from surface runoff cannot exceed d2of the total runoff every yearx2d2riverwhere d2 denotes the protection of rivers. smaller value means better protection of rivers and d21constraint 3: limit of exploitation from underground water.x3groundwaterwheregro

45、undwateris compensation amount of groundwater. we can get it from china statistical yearbook 3the value of d1，d2 can be different to show the different protection ofenvironment and smaller values make better protection. here according to experience and international conventions5, we select d1=0.8，d2

46、=0.3 for calculation. in general, this model is:maxx=x1+x2+x3x1d1reservoirs.tx2d2river xgroundwater3 thus,we can obtain the best water supply plan of each region and its maximum. with the help of work above,we can get water demand and water production of every region as the following table shows: te

47、am #20300 page 10 of 28 unit:billion m3 table 2: result of model 1w.d in 2025 88.45 85.73 20.30 28.34 70.01 173.32 275.99 region ne nr im cr wr sr sew.p 92.35 53.74 15.28 26.45 318.85 502.55 351.14w.p-w.d 3.89 -31.99 -5.02 -1.89 248.75 329.22 75.16w.d:requirement water; p.w:produced water analysis o

48、f resultsas 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 of water resources in china5.2 model 2: optimization of water transfer and supplementit has been mentioned above that the western region has its special geographical conditions and water distribution. it is far