2015美国数学建模大赛一等奖作品.pdf

For office use only T1 T2 T3 T4 Team Control Number 36884 Problem Chosen D For office use only F1 F2 F3 F4 2015 Mathematical Contest in Modeling MCM ICM Summary Sheet Attach a copy of this page to your solution paper Type a summary of your results on this page Do not include the name of your school advisor or team members on this page Three dimension Sustainability Measuring Model Using Objective Weighting Method Based On Mean Square Deviation Summary In this paper we build a sustainability measuring model based on objective weighting methods Then we use the model to measure the condition of Afghanistan and design a 20 year plan We first establish an evaluation index system with three subsystems and twenty three indicators The three subsystems are sustainability systems of economy ecology and society We use indexes of three dimensions to measure sustainability Principle component analysis model is introduced to find the weight of each indicator in corresponding subsystem After applying our first model on the data from World Bank database we draw a preliminary conclusion We find that some indicators have been neglected and we cannot achieve the specific weight of each indicator Therefore we develop another model using objective weighting methods based on mean square deviation Additionally Coordination Degree is developed to measure how the three dimensions fit with each other We choose three countries America China and Afghanistan to implement our new model We can find America is more sustainable than the other two countries which proves our model reasonable After that we use world average level to define the division between sustainable and unsustainable and using a higher criterion to define the division between sustainable and highly sustainable In Task 2 we choose Afghanistan to conduct our measuring model We put forward two principles of development development with emphasis principle and balanced development principle By studying the current condition and predicting the future of Afghanistan we make a 20 year developing plan and propose some detailed policies programs and aids In Task 3 we give a table of quantitative effects additional factors have on our prospect of Afghanistan s condition By nonlinear regression we draw graphs to show the change of three sustainable indexes with and without our plan in the following 20 years After that we analyze the most effective policy in each subsystem The three policies are improving way of economic managing decrease natural resources depletion and increase life expectancy Finally we make a sensitivity analysis and discuss the strengths and weaknesses of our model It is meaningful to implement mathematics model on measuring sustainability We learn a lot from the process Team 368841 Contents 1Introduction2 2Basic Assumptions2 3Notations3 4Task 13 4 1set indicators 3 4 2standardize 4 4 3model 1 PCA 5 4 3 1PCA 5 4 3 2results of PCA model 5 4 3 3analysis of model 1 5 4 4Model 2 Weighting method based on the mean square deviation 6 4 4 1Weighting method based on the mean square deviation 6 4 4 2results and analysis 7 4 5Sustainability Division 8 5Task 210 5 1two principles 10 5 2analyze condition 10 5 3our plan 10 6Task 312 7Strengths and Weaknesses16 7 1Strengths 16 7 2Weaknesses 17 8Sensitivity Analysis17 Team 368842 1Introduction The Heads of State Government and high level representatives promised to re new our commitment to sustainable development in The Future We Want a solution adopted by the General Assembly But how we defi ne a country is sustainable or not The United Nations have developed some detailed indicators to measure the degree of sustainability Building a model taking advantage of these indicators to obtain a comprehensive measuring methodology is what we need to do Task 1Task1 requires us to build a model to evaluate a country s sustain ability and to defi ne sustainability and unsustainability clearly We separate sustainable development measurement into three subsystems economy ecology and society We choose 23 measuring indicators Firstly we build the evalua tion model of three dimension structure based on principal component analysis PCA However it is unreasonable that the larger the indicator value is the larg er the corresponding weight coeffi cient Therefore we use a weighting method based on the mean square deviation to optimize the model The new model can refl ect the importance of each indicator directly and objectively Task 2 It requires us to choose a country from the 48 LDC countries We need to evaluate its sustainability and design a sustainable development plan of 20 years for it We choose Afghanistan as our research object We use curve fi tting to predict the value of each indicators in the coming 20 years According to these indexes we calculate the three subsystems scores of next 20 years At last we design a sustainable development plan and relevant policy Task 3 Task 3 needs us to take the impact of some additional environmental factors into consideration In taht case we get predicted value fi tting the fact more and evaluate the real eff ect of our plan We can visually achieve the proportion of each index from the weighting method based on the mean square deviation The proportion of a index is larger its eff ect on the results is greater Then we can fi nd the most infl uential indexes to Afghanistan 2Basic Assumptions Assumption 1The measuring indicators under diff erent subsystems have no infl uence on each other Assumption 2The measuring indicators under the same subsystem have linear infl uence on its subsystem Assumption 3The larger the coordination degree is the better do the three dimension indexes coordinate with each other Assumption 4The larger a indicator s weighter is the more infl uence does it have on the model Team 368843 Assumption 5Extra environmental factors are unrelated to each other and their infl uences on each indicator can be added linearly 3Notations Table 1 Symbol Table Symbol Defi nition nnumber of countries pnumber of indicator types ythe sample matrix Rthe correlation matrix Zthe standardized matrix ejthe corresponding eigenvectors Athe programme set Gthe indicator set Dthe result set wweight of indicator 4Task 1 We believe that sustainable development measurement should be separated into three subsystems ecologic economy and society Traditional approach is sum the score of the three dimensions simply Actually the three factors are not independent of each other so the simple sum reach the sustainable stage but each index do not To overcome this weakness we measure the sustainable development on the three dimensional vector 4 1set indicators According to the index system for sustainable development put forward by the Scientifi c Committee on Problems on Ecologic SCOPE and the U N Ecologic Pro gramme UNEP we choose several indicators for each subsystem as shown in Table 2 The datum of each indicators for diff erent country in diff erent years are all get from The World Bank which ensure the accuracy of the datum The data of indicators for some countries are not available Thus we choose 20 coun tries based on two principles one is the country s datum of all indicators is available Team 368844 Table 2 Indicators to Measure Sustainability of Subsystems Num EconomyEcologicSociety 1CPIAmacroeconomic management rating Adjusted savings natu ral resources depletion Literacy rate adult to tal 2GDPpercapitaper year CO2 emissions metric tons per capita CPIA social protection rating 3Services etc value added of GDP Energy use kg of oil e quivalent per capita Unemployment total of total labor force 4Agriculture value added of GDP Arableland hectares per person Telephonelines per 100 people 5Net ODA received per capita current US Forest area of land area Access to electricity of population 6Foreigndirectinvest ment net infl ows BoP current US Renewableinternal freshwater resources per capita cubic meters Road density km of road per 100 sq km of land area 7Tax revenue of GDP Life expectancy at birth 8GDP growth annual Mortality rate infant 9GINI index the other is the country s indicators distribute randomly Then we standardize the indicators for these 20 countries We determine to use Principal Component Analy sis PCA to realize the dimension reduction of indicators that is to drop some less infl uential indicators 4 2standardize First we need to standardize the data of indicators If a indicator value is bigger the corresponding country tend to be more sustainable the indicator is defi ned as positive effi ciency indicators And indicators with contrary effi ciency are called nega tive effi ciency indicators We use n countries p types of indicators to create a sample matrix y of n rows and p columns For positive effi ciency indicators Zij yij ymin j ymax j ymin j i 1 2 n j 1 2 p 1 For negative effi ciency indicators Zij ymin j yij ymax j ymin j i 1 2 n j 1 2 p 2 Team 368845 Zijis the standardized indicator yijis the raw indicator ymin j and ymax j are the minimum and maximum value of the j column minimum and maximum value of a certain indicator in our model i and j give the position in the sample matrix 4 3model 1 PCA 4 3 1PCA Then we use PCA to reduce dimension of indicators Calculate the correlation matrix R for the standardized matrix Z fi nd the p eigenvalues fi nd the corresponding eigenvectors ej l1j l2j lpj represents the correlation value m j 1 j p j 1 j 0 85 3 According to 3 choose m principal components which refl ect 85 of the variance If we take traditional AHP sustainable development degree is measured by 4 Zj ejy l1jy1 l2jy2 lmjym 4 4 3 2results of PCA model We use PCA for each of the three subsystems Z1 0 2655y11 0 1566y12 0 1352y13 0 1172y14 0 1012y15 0 0972y16 Z2 0 3290y21 0 1785y22 0 1736y23 0 1625y24 0 1345y25 Z3 0 4053y31 0 1448y32 0 1084y33 0 1005y34 0 0945y35 Where Z1represents economic measuring index Z2represents ecological measuring index and Z3represents social measuring index yij i 1 2 3 j 1 2 6 repre sents jth principle component in ith subsystem The formulas of yij and each specifi c indicator are too complicated to give 4 3 3analysis of model 1 Strengths PCA is a kind of methodology serving to give objective weight It is better than methodologies calculating weight depending on man s subjective judgment such as Analytic Hierarchy Process AHP Team 368846 Weaknesses One of PCA s strengths is that it discards some unimportant components So we can simplify solution in some cases But in this task giving up unimportant component is malign to the integration of our analysis We cannot make out what each component in PCA represents Consequently we cannot know which is most or least signifi cant in a country s sustainable development Considering the analysis above we optimize our measure model using Weighting method based on the mean square deviation 4 4Model 2 Weighting method based on the mean square deviation 4 4 1Weighting method based on the mean square deviation In the multi index comprehensive evaluation model we assume that the programme set is A A1 A2 An the indicator set is G G1 G2 Gm the result set is D D1 D2 Dn In our model Ai i 1 2 n represents countries Gj j 1 2 m represents indicators Di i 1 2 n represents three dimension in dexes Weighting method based on mean square deviation is another kind of objective weighting method In this method if indicator Gjis the same to each program Gj has no eff ect on the ranking of programs This kind of indicators weight is 0 If Gj can make each program diff er a lot from each other it plays an important role in the ranking of the programs We give this kind of indicators bigger weights That is to say the weight of each indicator depends on the dispersion degree of programs under this indicator The bigger is the dispersion degree the bigger is the weight of the indicator If the programs property is the same under one indicator the weight of this indicator is 0 We assume that each indicator Gjis a random variable and dimensionless property value of each programme Aiis the value of corresponding random value for Gj We use mean square error to measure the dispersion degree of the random variables So we use weighting method based on mean square deviation to calculate the weights of indicators The average of the random variable is E Gi 1 n n i 1 zij 5 Team 368847 The mean square deviation is Gj v u u t n i 1 zij E G1 2 6 The weight of indicator wjis wj Gj m j 1 Gj 7 Then we get the fi nal score of the contry Di W m j 1 zijwi 8 4 4 2results and analysis We choose three country of great concern to complement our measure model The indicators and eventual indexes is shown in the following tables The indicators are data in 2004 from the World Bank database We replaced some data in vacant with world average Actually that conducts little impact on the ultimate index Table 3 Economic Sustainable Indicators Indicators12345678 CHN3 541490 3840 3813 391 323 2216 6910 09 USA3 5441921 7177 021 2973 231 199 603 79 AFG3 54220 0643 0230 2796 243 5316 691 06 Table 4 Ecological Sustainable Indicators Indicators123456 CHN4 965 151551 430 0821 292134 28 USA1 1719 357758 210 5433 069354 94 AFG1 930 082224 070 302 071789 43 Table 5 Social Sustainable Indicators Indicators123456789 CHN90 923 144 323 7999 719 4973 790 022139 55 USA78 093 145 660 1310066 5277 340 006940 57 AFG78 093 144 521 19411 4956 570 086239 55 Team 368848 Table 6 Three dimension Indexes of three countries Country IndexEconomyEcologySociety CHN0 35140 52780 7103 USA0 49120 52480 7694 AFG0 30520 49950 4349 World Average0 39850 55130 5981 As we can see in the Table 6 in terms of sustainable economy America shows great advantage above average level and far better than China and Afghanistan This phenomenon fi ts our common sense well As a rapidly developing country Chinas current condition is still not admirable America indicates a way of developing by adjusting the industrial structure Recently Afghanistan have just get rid of the war against terrorism Obviously its economy appears struggling throughout the past decade in war and demands concern In terms of sustainable ecology three countries are all below the average level with China ahead of America slightly which is much out of our prospect After observing each indicator the data is shown in Table 7 we found the answer Table 7 Ecological Sustainable Indicators after standardizing Indicators123456 USA0 98220 66840 60800 25350 34920 0009 CHN0 92030 91170 92200 03910 22490 0002 Although the fresh water resource the farmland resource the forest resource and the condition of damage of natural resources of America is better than China China does better in the aspects of CO2 emission and per capital use of energy which is attached much importance in our model Bigger emission of CO2 is inevitable in developed countries like America That alerts us that sometimes development means damage of ecology We may look for solution in low carbon economy In terms of sustainable society both China and America perform well while Afghanistan awaiting improvement More analysis can be found in the following context 4 5Sustainability Division We developed an approach to dividing sustainability indicating in the following table Team 368849 Table 8 Sustainability Division Economy 0 0 3985unsustainable 0 3985 0 48sustainable 0 48 1highly sustainable Ecology 0 0 5513unsustainable 0 5513 0 7sustainable 0 7 1highly sustainable Society 0 0 5981unsustainable 0 5981 0 75sustainable 0 75 1highly sustainable We defi ne world average level as the division ofsustainable and unsustainable Additionally we defi ne the division of sustainable and highly sustainable with a higher criterion Whats more we defi ne an index named Coordination Degree representing how the three dimensions of a country develop coordinately We can understand this index fi guratively with the following picture Figure 1 Sustainability Coordinate System X axis y axis and z axis are economic ecological and social sustainability separately We use the following formula to calculate coordination degree O is the worst condition with three indexes being 0 while T is the best condition with three indexes all being 1 S is the condition of the country we observe cos is the coordination degree The bigger is cos the better three indexes coordinate with each other cos OT2 OS2 TS2 9 OT2 3 10 OS2 x2 y2 z2 11 ST2 1 x 2 1 y 2 1 z 2 12 Team 3688410 In this way the condition of sustainability of three countries is like the table below Table 9 Sustainability of Three Countries EconomyEcologySocietyCoordinate Degree CHN unsustainableunsustainablesustainable0 825 USAhighly sustainableunsustainablehighly sustainable0 979 AFG unsustainableunsustainableunsustainable0 9814 5Task 2 5 1two principles We select Afghanistan as our study object from LDC country and create a 20 year sustainable development plan for it The three subsystems economy ecologic and society are interrelated and each has diff erent importance in diff erent stage of devel opment Therefore we adhere to Development with Emphasis Principle and Balanced Development Principle Development with Emphasis Principle When a country s economy eco logic and society indicators are all in the stage of unsustainable development we give priority to social development to satisfy people s vital needs and slow down the growth of economic and ecological development at the same time Balanced Development Principle Attaching importance to a certain aspect is not equal to ignoring other two aspects We emphasize the balanced develop ment of three aspects which is in accordance with the connotation of sustainable development 5 2analyze condition We obtain Afghanistan s indicator values between 2003 and 2013 from the World Bank database By using our sustainability measuring model we evaluate Afghanistan s economy ecologic and society sustainability score and its changing trend between 2003 and 2013 We fi nd that Afghanistan s sustainability score of the three a