这是我与同学组队在2017年2月参加美国数学建模大赛（Interdisciplinary Contest In Modeling ）的文章，比赛题目为：Sustainable Cities Needed
题目原文为：
Background:
Many communities are implementing smart growth initiatives in an effort to consider long range, sustainable planning goals. “Smart growth is about helping every town and city become a more economically prosperous, socially equitable, and environmentally sustainable place to live.”[2] Smart growth focuses on building cities that embrace the E’s of sustainability—Economically prosperous, socially Equitable, and Environmentally Sustainable. This task is more important than ever because the world is rapidly urbanizing. It is projected that by 2050, 66 percent of the world’s population will be urban—this will result in a projected 2.5 billion people being added to the urban population.[3] Consequently, urban planning has become increasingly important and necessary to ensure that people have access to equitable and sustainable homes, resources and jobs.
Smart growth is an urban planning theory that originated in 1990’s as a means to curb continued urban sprawl and reduce the loss of farmland surrounding urban centers. The ten principles for smart growth are
1 Mix land uses
2 Take advantage of compact building design
3 Create a range of housing opportunities and choices
4 Create walkable neighborhoods
5 Foster distinctive, attractive communities with a strong sense of place
6 Preserve open space, farmland, natural beauty, and critical environmental areas
7 Strengthen and direct development towards existing communities
8 Provide a variety of transportation choices
9 Make development decisions predictable, fair, and cost effective
10 Encourage community and stakeholder collaboration in development decisions
These broad principles must be tailored to a community’s unique needs to be effective. Thus, any measure of success must incorporate the demographics, growth needs, and geographical conditions of a city as well as the goal to adhere to the three E’s.
以下是我们小组的文章：
Evaluation and Solution of Urban Crawl
Abstract
In this paper, based on theproblems, we divide our thinking into 4 parts.
For task 1, we developa ’Successful Smart Growth’ index (SSG) to measure the governance of urbancrawl based on the impact of the ten principles on the three E’s and happiness.We used the rank correlation analysis and geometric compare scale to considerhow the factors were affected by each principle. Using this as a basis, wedetermine the influence level and furthermore the weights of the three E’s andhappiness in the SSG index. Finally we calculated the weighted sum of thenondimensionalized variance of three E’s and happiness and used that tocalculate our SSG index.
For task 2, we appliedour model to two cities: TAKASAKI and Mobile City. Based on our assumptions, weuse the financial condition of the annual statement and the variance ratio atthe national level to represent the changes in three E’s and happiness. Afterthat, we nondimensionalized the variance of three E’s and happiness by usingthe algebraic model to calculate the variance ratio. Finally, we gave score ofthe SSG index of TAKASAKI and Mobile City.
For task 3 & 4, weproposed six individual initiatives and developed a more robust model — FSGPindex based on the situation of TAKASAKI and Mobile City. According to theproblems in the cities, we proposed our smart growth plan based on tenprinciples. Using the scale comparison method and the weighting method, weobtained the SPG index. Then with the enlightenment of the theory of optimum population ( A. Sauvy ) , we calculated theproper population density for development by using the quadratic function andsymbolic-graphic combination. Finally, we used profit loss equation to defineprofit loss coefficient and optimized the SSG index by using the coefficient.
For task 5, using theconcept of task 4, we divided the result of population increase into twoconditions. And we analyzed the changes of our initiatives in two conditionsand prove the robustness of our model.
Synthesize tenprinciples, three E’s and happiness, we evaluated the success of smart growthof cities and their plans.
Key Words: SSG index SmartGrowth FSGP index
1 Introduction
1.1 Background
Urban development had come up to aseries of problems since 1970s. The boundary of the city was getting blurry. The original city planning could not adapt the change since more people werepurchasing houses in the suburban area. These phenomenon caused thedegeneration of natural resources, the loss in the open space, the aggravationof traffic, the large-scale expenditure of infrastructure. Scholars pointed outthat a large scale crash down would be inevitable if the city keep consumingresources in an unsustainable way.
Organizationslike the Environmental Protection Agency offer solutions in many aspects forthose problems, and finally concluded them into a concept with ten principlescalled ’smart growth’. After this strategy was published, according to a survey[1] carried by Smart Growth Network in Oct. 2000, 78% of the US citizenssupported that it is a necessity to curb the urban crawl.
1.2 Restatement
If we evaluate the progress of thesmart growth, the changes of the ten principles could be indistinct, but thegoal of the whole activity is clear, that is make the city a more economicallyprosperous, socially equitable and environmentally sustainable. When weinterpret the goal of smart growth, we believe the happiness(in the aspect ofcountry is GNH) would also be affected by the principles. According to theGeller(2003) [2], smart growth could curb the damage of the environment,improve the accessibility of infrastructure, promote the healthy life style andbuild a livable city so that the overall living quality can be improved. Thus,based on ten principles, we interpret:
• Economically prosperous
• Equitable society
• Environmentally sustainable
• Happiness of citizens as ourreference objects to evaluate the success of the smart growth.
2 Task 1: Model for Smart Growth Measurement
2.1 Outline of Our Approach
We came to an approach to measurethe success of smart growth by treating the reference objects as the factors inthe process of smart growth:
• Economy
• Equity
• Environment
• Happiness
By analyzing theobjects in order relation, we developed an index named: Successful Smart Growthindex (SSG)(Figure 1).
The ten principles are shown in Table 1 with corresponding codes from P1 to P10:
| Code | Principles |
|---|---|
| P1 | Mix land uses |
| P2 | Take advantage of compact building design |
| P3 | Create a range of housing opportunities and choices |
| P4 | Create walkable neighborhoods |
| P5 | Foster distinctive, attractive communities with a strong sense of place |
| P6 | Preserve open space, farmland, natural beauty, and critical environmental areas |
| P7 | Strengthen and direct development towards existing communities |
| P8 | Provide a variety of transportation choices |
| P9 | Make development decisions predictable, fair, and cost effective |
| P10 | Encourage community and stakeholder collaboration in development decisions |
Table 1: Codes of tenprinciples
Originatingfrom the problem of urban crawl, based on ten principles, different methods wasoffered to solve it. But if we want to evaluate the progress of the smartgrowth of a city, it is required to analyze the factors which could be affectedby the principles, that is our reference objects. Thus we interpret theprinciples as the influencing factors of the three E0s and happiness.
2.2 Modeling of SSG index
2.2.1 Model assumptions
There are a myraid of factors thatcould affect the process of smart growth, it is impossible to consider all ofthem, thus we list out the following assumptions as restrictions of the model:
• We would only consider theactions based on the ten principles, all the occasionality outside thedefinition of the principles would be ignored.
• As for a developed country, weassume the gap of GNH, EPI, social equity index between its cities and thecountry itself can be ignored. Thus we could use the data of the country torepresent the situation in a specific city.
• We consider the total effect ofeach principle to the three E0s and happiness is the same.
• We assume none of the varianceratio of three E0s and happiness would be zero.
• We ignore the interactionsamong ten principles as well as the three E0s and happiness.
2.2.2 Parameter setting
2.2.3 The solution to the model
Since different principles mayresult in different effects on the three E0s and happiness, we set weights tomeasure the progress of smart growth and create a fomula of SSG index.
(1)In this formula, the success of the smart growth of the city is evaluated by thevariance ratio of three E0s (∆E1 to∆E3) and happiness (in the fomula means ∆E4), along with different weights caused by the ten principles.Different principle plays a different role in the impact to the three E0s and happiness.[3] For example, as for mix land uses, it would mainly improve the availabilityof the land which may bring positive economic results. Thus, the economicfactor values most. Besides, the multi-used land could improve the livingconvenient level, which is the main factor of happiness. After that, thedecrese of living radius could improve the environment situation which givesthe environment the third place while this action may only present indirecteffect on social equity.
Now we have theranking to the effect on the three E0s and happiness. In the above example,the ranking order is economy, happiness, environment, equity. In order toemphasize the principal effect, we used a geometric progession (set thevariable r be the ranking of theimpact):
(2)The rest nine principles would beanalyzed with the same method as above. Uji is the element
in the j-th row and i-th column of Figure 2. Since each principle posesrelatively different effects on the three E0s and happiness, we have already gotthe rank of the priority in Figure 2.In the assumption we consider the the total effect of each principle on thethree E0s and happiness is the same. Thus we define the weight of the smartgrowth Wi is:
(3)After the calculation, we obtain theweight (Wi) of the three E0s and happiness in Figure 3:
Since the SSG index was created to measure the success of the smart growth of a city, thus, after outputing the result of the index, some analysis could be used:
•SSG > 0 The smart growth of the city is very likely to be a success.
•SSG < 0 The city doesnot matches the principles of smart growth.
•SSG = 0 The plan of the city attended to one thing and lost sight of another.
2.3 Evaluation of the Model
To assess whether a model cansolve a problem or not, we need to figure out the comprehensiveness. In orderto assess the success of the smart growth, we consulted 4 indexes:
• GDP (GrossDomestic Product)
• Gini Coefficient
• EPI (EnvironmentPerformance Index)
• GNH (Gross National Happiness)
Referring tothe contribution rate in Figure 2, weobtain the significance ordering of the indexes and give out a weighted sum. Asfor the success of smart growth, we interpret it as the combined effect of our4 reference objects: Economy, Equity, Environment, Happiness. Thus, the SSGindex includes all the aspects of smart growth.
2.3.1 Strengths
• Consider the aspects of tenprinciples can not be covered by three E0s, we chose happiness as the forthparameter to evaluate the success of smart growth.
• During the process of settingthe index, we determine the contribution rate of ten principles as thereference of deciding the weights of three E0s and happiness. The complexity of theproblem was simplified because of the decrease of dimension.
• When we decide the weight ofthree E0s and happiness by analyzing the contribution rate of tenprinciples, the use of order relations can represent different influence madeby different principle; the geometric compare scale shows our good handling onprimary effect and secondary effect.
2.3.2 Weeknesses
• The comparison scale of themodel to set the weight is relatively subjective.
• The number of variable factorsin the model is not enough and the model is rather simple.
Thus, this model may be affectedby other factors.
3 Task 2: Case Study
3.1 The smart growth plan of TAKASAKI City, Japan
From the annual report of TAKASAKICity[4] , we analyzed the actions that could pose positive effects based on thesmart growth principles. And three E0s and happiness are the referenceobjects of the principles. Thus, the Figure4 focuses on the actions related to the reference objects.
Firstly, as for the reforming of land distribution, it matches P1. This action would create a multi-function community and theutility of land. The government of TAKASAKI strengthen a lot of infrastructureand the quality of roads which would greatly improve the convenience of thecitizens and redistribute the social resources, which matches P7 , P4 and P8. In orderto make the environmental situation better, more budgets were invested tolandscape planning and afforestation. Both of them can be treated as the partof sustainable development and they match P6 and P5. The maintenance of parking lots and the restriction on the outdooradvertisements could give the citizens a more pleasant home. Thus we believethat these matches P5.
3.2 The smart growth plan of Mobile City, US
Similarly, we obtained the actionsof the Mobile City in the USA (Figure 5)from the annual report [5].
As for mobile city in the US,enhancing business sevice would make the companies in the city morecompetitive, thus, it matches P9. Trafficengineering includes the improvement of the road and public transportation, sothis area would benefit P4 and P7. The landfill ability is related to P6. Morebudget on the public safety area could improve the sense of safety among thecitizens while the increasing number of parks could build them a bettercommunity to live in, thus, they match P5 and P7.
3.3 Assessment of the plan in two cities by SSG index
In the condition when the societyhas spare resources and the salaries are relatively stable, from the theory ofmultiplier [6], the fiscal expenditure is positively correlated to economicgrowth. As for a developed country, we assume the gap of GNH, EPI, socialequity index between the its cities and the country itself can be ignored. Thuswe could use the data of the country to represent the situation in a specificcity.[7][8]
3.3.1 Calculation
Referring to our SSG index whichcould measure the success of the smart growth, we have already got the formulain equation (1).
we needto get a result based on the weights (which has been obtained in Figure 3 and the
variance ratio of the followingreference objects:
• Economically prosperous (Table 3): Based on the assumption,according to the theory of multiplier, we could approximately replace thevariance ratio of the economic growth with the variance ratio of fiscalexpenditure. From the financial statements of the city, we could get the dataand figure out the variance ratio of fiscal expenditure.
| FE2 | budget of fiscal expenditure of this year |
|---|---|
| FE1 | fiscal expenditure of last year |
(4)• Social justice (Table 4): Based on the definition of the Gini coefficient [9] andsocial justice [10], as well as the completeness of the Gini coefficient, weuse the variance ratio of Gini coefficient as the reference index of socialjustice.
| SJ2 | Gini index of this year |
|---|---|
| SJ1 | Gini index of last year |
(5)• Environment sustainable (Table 5): We evaluate the success of asustainable environment by using the variance ratio of EnvironmentalPerformance Index[8] as the parameter.
| EPI2 | EPI index of this year |
|---|---|
| EPI1 | EPI index of last year |
(6)• Happiness (Table 6): Based on the definition of the GNH (Gross NationalHappiness)[11], the measurement of happiness [12] could be judged by thevariance ratio of GNH.
| H2 | GNH index of this year |
|---|---|
| H1 | GNH index of last year |
(7)Using equation (1), we have Figure 6.
3.3.2 Result
In Figure 6, the SSG index of TAKASAKI city is -0.094097012, thus wedefine this city as The city does notmatches the principles of smart growth.
In the MobileCity the result is 0.869158697, thus we define it as The smart growth plan the city is very likely to be a success.
4 Task 3 & 4: Our Smart Growth Plan
Based on the ten principles of smartgrowth, we developed a plan with 6 initiatives: (consult table 1 about the codes of ten principles)
• build commercial center (P1)
• encourage the green commutingby introducing free bicycles (P8)
• timly restriction of communitytraffic to allow pedestrian-oriented activities (P4, P5)
• direct participation in policymaking in a small range (P10)
• improve medical benefits (P7)
• perfect the infrastructure inthe residential area (P7, P5)
4.1 TAKASAKI City
TAKASAKI City is becoming a centerof handicraft industry and trade in Kanto region [13]. And the city is in themiddle of the traffic network, which would attract the people to do bussinessactivities. Thus, the increase of number of the commercial centers would benecessary.
According to therelationship between natural landscape and human [14], we value the necessityof building a eco-friendly city. Plus, the topography of the city is gentle.Thus we encourage the green commuting by introducing free bicycles.
Moreover, tenprinciples emphasize the importance of an attractive community. So action 3, 5and 6 of our plan were made for it.
Lastly, inorder to have the citizens participate in the administration and discussion ofstate affairs, we decided to carry out the action of direct participation inpolicy making in a small range.
4.2 Mobile City
Mobile is a city rich in tourismresources [15]. Thus we believe the increase number of the commercial centerswould pose positive effect on tourism. Analyzing the map [16] of the Mobilecity, the topography of the city is convenient for bicycles introducing.
And there is acarnival in the city called Mardi Gras [17], thus we believe timly restrictionof community traffic to allow pedestrian-oriented activities would benecessary.
Following the ten principles,actions 4 and 5 would improve the happiness in the city.
Moreover, theoccurrence of hurricane in this area [18] is frequent, so the infrastructure inthe residential area aims to protect the citizens from the damage of thehurricane.
4.3 A more robust model to evaluatethe individual initiatives: FSGP index
4.3.1 Assumption
• The population factor wouldonly affect economy factor and happiness factor.
• Population factor canapproximately replace all the factors of smart growth in a city.
4.3.2 Parameter setting
4.3.3 The process of enhancing theSSG model
In order to evaluate whether a planadheres to the smart growth principles or not, we enhanced the original SSGindex.
For eachinitiative, we need to calculate the contribution to the three E0s and happiness.And thus, we developed an index named SGP (Smart Growth Plan) based on SSGindex:
(8)Assume the initiative involves principles Pj$_{1}$ … Pj$_{n}$ (n ≤ 10). Thus:
(9)and the value of Wicould be found in Figure 3.
Using theformula 8, we obtain the SGP indexfor each initiative and formed a table (shown in Figure 7:
Inspired by thetheory of optimum population, we considered the relationship of populationdensity and GNP, and tried to find out the best population density. Then, we comparedthe the best population density and the current population density so that theadjustment coefficient of the adjustment to SSG index. After that, wecalculated by using the adjustment algebraic model and obtained a new resultnamed Final Smart Growth Plan (FSGP).
Now we start to optimize the modelof FSGP.
Based on thetheory of optimum population, we decided to use quadratic function simulationto figure the relation between the population and GNP in the states of US andcounties in
Japan. And we have the figureshowing the results: (Figure 8, Figure 9)
We determinedthe proper population density for development as the population density of thestates or counties which is higher than the average GNP. Thus we used thestraight line: y = average GNP to cut out the figure of the correspondingcites.(Figure 10, Figure 11) (shown in next page)
We found outthe straight line and the fitting function intersects in 2 places (L and H).Andwe figured out the extreme point M. Thus we define the city with the populationdensity in
the interval of L to H (L, H) as thecondition which the economy development is effective to the smart growth of thecity. The population density of the city which does not locate in the intervalwould be treated as inapporopriate in development.
Futhermore,we introduced the limitation factor of population A. With a certain populationwhich is suitable for city development, a slightly increase in the populationdensity would bring relatively great GNP effect. When the population densityalmost approaches the optimized population density, a large scale of populationdensity increase would only bring slight effect to GNP.
This curve ofthe function of population density and GNP is convex, which proves thecorrectness of the fitting figure.
Since under theinfluence of population factor, the initiative of the plan on the economiclevel could not reach the ideal result. Thus, we defined the limitation factorA as the real number in the interval of [0, 1]. Thus, we concluded the theregulation of change and adjusted the economy