2009 美国数学建模竞赛B题特等奖论文(2)

Wireless Networks:An Easy Cell385

Wireless Networks:An Easy Cell

Jeff Bosco

Zachary Ulissi

Bob Liu

University of Delaware

Newark,DE

Advisor:Louis Rossi

Summary

The number of cellphones worldwide raises concerns about their en-ergy usage,even though individual usage is low(<10kWh/yr).We?rst model the change in population and population density until2050,with an emphasis on trends in the urbanization of America.We analyze the current cellular infrastructure and distribution of cell site locations in the U.S.By relating infrastructure back to population density,we identify the number and distribution of cell sites through2050.We then calculate the energy usage of individual cellphones calculated based on average usage patterns.

Phone-charging behavior greatly affects power consumption.The power usage of phones consumes a large part of the overall idle energy consump-tion of electronic devices in the U.S.

Finally,we calculate the power usage of the U.S.cellular network to the year2050.If poor phone usage continues,the system will require 400MW/yr,or5.6million bbl/yr of oil;if ideal charging behavior is adopted,this number will fall to200MW/yr,or2.8million bbl/yr of oil. Introduction

As energy becomes a growing issue,we are evaluating current infras-tructure to locate inef?ciencies in power consumption.The increase in cellphone usage in the past decade raises concern about greater energy consumption compared to landline phone networks.

By modeling subscriber growth and trends,we can get a clearer picture of the energy consequences of our mobile network.By correlating the The UMAP Journal30(3)(2009)385–402.c Copyright2009by COMAP,Inc.All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro?t or commercial advantage and that copies bear this notice.Abstracting with credit is permitted,but copyrights for components of this work owned by others than COMAP must be honored.To copy otherwise, to republish,to post on servers,or to redistribute to lists requires prior permission from COMAP.

386The UMAP Journal30.3(2009)

growth of mobile subscribers with changes in our mobile infrastructure, we can strategically develop our current communications network to meet energy-ef?cient guidelines.

Current Cellular Network Model Assumptions

?The FCC database contains all relevant and major cell sites in the U.S.?Cell sites serve areas of homogeneous population density,characterized by the population density at the exact location of the site.

?All cell sites can communicate to50km(approximately the limit of mod-ern technologies).

?The strength of a cell tower depends primarily on the number of antennas (we lack information about transmission power).

Communication Standards

CDMA and GSM,the two primary standards for mobile phones in the U.S.,require different antennas,so different cell sites exist for each standard. However,to simplify our models,we assume that all mobile phones use one generic standard.

Network Model and Component Power Usage

A simpli?ed cellular network model and corresponding energy usage requirements are shown in Figure1.Cellphones connect directly to cell sites,which may or may not be mounted on antenna towers.We consider each antenna mounted on a tower as a separate cell site.A tower can handle a range of calls at once(about200–500users,using600–1000W [Ericsson2007])and pass them along to Mobile Switching Stations(MSCs). Communication between MSCs and cell sites can be accomplished through ?ber-optic networks or microwave connections.Each MSC can handle approximately1.5million subscribers and consumes about200kW.MSCs connect directly into the communications backbone of the country.Since a ?ber-optic backbone is necessary in any scenario(or in any Pseudo U.S.), we do not consider it in energy estimates.

Cell Site Registration Databases

All cellular radio transmitters greater than200m in height are required to be registered in the FCC Universal Licensing System Database[Fed-

Wireless Networks:An Easy Cell 387

wireless

~0.9W

Cell Tower ~600-1000W

~0.9W

~0.9W

~0.9W

Cell Tower ~600-1000W

Mobile Switching Center 200kW/1.5M Users

wired Fiber Optic Backbone

(cell sites and MSCs)is assumed to be identical for all carriers and geographies.

eral Communications Commision 2009],ensuring that a majority (but not all)cell sites are included.The database contains approximately 20,000cell site locations comprising about 130,000individual cell sites.

Tower Location

We show cell-site location and population density in Figure 2.Interest-ingly,several cell towers seem to be in the Gulf of Mexico and in the Atlantic Ocean (either due to errors in registrations or to the use of ships and/or oil rigs).Also interesting is the single tower at the center of Dallas (northern Texas),which contains 25antennas and suggests a series of smaller sites spread throughout the city.

Antennas per Cell Site

Many cell sites in urban areas use more antennas and higher transmis-sion power.Although some Effective Radial Power (ERP)data is included in the FCC database,many sites have no published information and sev-eral have a negative ERP (impossible).Many sites have similar transmission power,likely due to FCC regulations.To quantify the power of a cell site,we use the number of antennas.While most sites have only a single antenna,many have several,and a few have as many as nine (Figure 3).

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(2009)

Figure3.Distribution of the number of antennas per tower.

Tower-Antenna-Population Density Relations

To calculate how many cell sites are used on average in regions of varying population density,we use the site locations to interpolate densities from the maps.Binning the data for population density,we get in Figure4the relationship between antenna density and population density.The initial portion of the graph approximately shows a steady increase in the number of towers,with one antenna per tower.However,above150people/km2, the number of towers levels off and the number of antennas per tower rises to compensate for the increased population.

Coverage Overlap

We investigated overlapping coverage by determining the number of nearby cell sites at a range of locations;the method is illustrated in Figure5.

Wireless Networks:An Easy Cell389

Figure4.Antenna density vs.population density.

Figure5.Illustration of algorithm to determine number of overlapping cell sites.The?gure does not represent the eccentricities of the grid due to changing longitudinal lengths.

For each cell in the population density grid,we construct a trial list of all towers within a reasonable range(towers within1?latitude,3?longitude, or approximately100–200km in each direction).For each candidate tower, we calculate the great-circle distance(in km)between the location(latitude δ1,longitudeλ1)and the tower(latitudeδ1,longitudeλ1)[Weisstein n.d.]:

d=6378cos?1[cosδ1cosδ2cos(λ1?λ2)+sinδ1sinδ2].

If the great circle distance is less than the maximum range of a tower (approximately50km),the region is considered to be in the tower’s plau-sible range.We thus calculate for each location the number of cell sites within range(Figure6).While some cities have a large degree of overlap, others accomplish full connectivity by using many smaller rooftop sites or higher-power antennas.Also noticeable are several regions in the Western U.S.with no current connectivity.

390The UMAP Journal30.3(2009)

Figure6.Results of overlap calculations for the known grid of cellsites as reported by the FCC. Most urban regions have higher overlap of cell towers to cope with an increased population load. Model for Cellphone Usage

Basic Assumptions

Our investigations uncover three main components of electricity con-sumption by cellphones:

?powering the phone during talking and standby,?powering the charger with a phone attached,and ?powering the charger without a phone attached.

Therefore,we model the cellphone usage of an average person as a function of three different characteristics:

?At what remaining battery level(0–100%)does the user recharge the cellphone?

?How long does the cellphone remain connected to the charger after the battery is completely charged?

?Does the user unplug the charger from the outlet upon completion of battery charging?

The possible power consumption states of a phone adapter are displayed in Table1.

Cellphone Information and Usage Behavior

Battery Capacity

Table2displays the average battery capacity,power consumption dur-ing talking,and standby power consumption for batteries of the nine largest

Wireless Networks:An Easy Cell391

Table1.

Cellphone charger states and energy consumption.

State Consumption(W)

Unplugged0

Plugged in,no phone0.5

Phone attached,not charging0.9

Phone attached,charging 4.0

mobile phone manufacturers in the U.S.We determined averages using manufacturer information about more than150popular cellphones,ap-proximately15phones per provider[IDC2008].Power consumption is calculated using battery capacity and estimates of talk time and standby time for individual phones,assuming each phone has a3.7V lithium-ion battery.The last line shows an overall average weighted by2008U.S.mar-ket share.

Table2.

Average capacity and energy consumption for popular U.S.cellphones.

Rank Manufacturer Market Battery Talk power,Standby power,

share(%)capacity(mAh)(W)(W) 1Samsung22.0980±2280.0138±0.00510.875±0.293 2Motorola21.6826±1220.0108±0.00230.655±0.292 3LG20.7890±1060.0116±0.00360.923±0.242 4RIM9.01216±2760.0145±0.0060 1.065±0.348 5Nokia8.51066±1920.0122±0.00320.735±0.334 6Sony Ericsson7.01015±2140.0085±0.00390.431±0.110 7Kyocera 5.0900±0000.0200±0.00300.970±0.080 8Sanyo 4.0810±890.0161±0.00370.908±0.152 9Palm 2.21500±3460.0167±0.0042 1.402±0.353 Weighted average960±1660.0127±0.00390.829±0.263

Cellphones Per Person

The average number of cellphones owned per person is determined using historical population and mobile phone data and extrapolated to the year2050[Federal Communications Commission2008;U.S.Census Bureau2008].Figure7a displays the total number of cellphone subscribers normalized by the population of the U.S.The historical data?t a sigmoidal curve,assuming that the ratio will eventually reach a value of1cellphone per person(complete saturation).Figure7b compares the yearly increase in U.S.population to that of cellphone users.By2015,the predicted number of cellphone owners reaches the total number of people and continues to grow with the population.

392The UMAP Journal 30.3(2009)

Year

% P o p u l a t i o n o w n i n g c e l l u l a r p h o n e

Figure 7a.Sigmoidal ?t for the average number of cellphones per person in the U.S.

8

Year

P o p u l a t i o n , N u m b e r o f P e o p l e

Figure 7b.Predicted growth and satura-tion of cellphone owners in the U.S.

Average Talk-Time per Person

The average talk time of an individual user between 1991and 2050is determined in a fashion similar to the average number of cellphones per owner.Figure 8displays the trends in landline and cellphone usage in terms of total minutes used per year between 1991and 2007[CTIA 2008;Federal Communications Commission 2008],together with our extrapo-lation.We assume that average usage will eventually saturate to some value,and a ?rst-order exponential growth function is employed to model this behavior.Figure 9displays the predicted growth of cellphone usage assuming saturation at 15,20,25,and 30minutes per person per day.

11

Year

T o t a l t a l k ?t i m e p e r y e a r , m i n

Figure 8.Historical behavior of landline and cellphone usage in the U.S.

Wireless Networks:An Easy Cell

393

Years

A v e r a g e M i n u t e s U s e d , m i n p e r s o n ?1 d a y ?1

Figure 9.Predicted saturation behavior of average daily mobile

cellphone usage.

Recharge Probability and Duration

We model the battery level at which a person is likely to charge their phone as a Gaussian distribution (Figure 10),based on cellphone behavior data [Banerjee et al.2007].Users tend to recharge their phone batteries at between 25%and 75%of full capacity .

020

4060

80100

Remaining Battery Charge, %

R e c h a r g e P r o b a b i l i t y

Figure 10.Fitted Gaussian distribution for recharging behavior of users.

The time to charge a lithium-ion battery is typically not proportional to the remaining charge to be added.Therefore,we assume that the battery charge increases exponentially as a function of charge time,as depicted in Figure 11.

394The UMAP Journal 30.3(2009)

Time, min

R e m a i n i n g B a t t e r y C h a r g e , %

Figure 11.Typical charge pro?le for lithium-ion battery .

Calculation of Average Energy Consumption

We calculate the energy consumed by the average cellphone user over the course of a year by employing the battery and usage behavior extrap-olations discussed earlier.We assume that the full range of remaining battery charge (0–100%)can occur before charging is initiated,depending on the type of user (“regular”or “ideal”).The total energy consumption is calculated from battery capacity and different power states of a charge-adapter.The duration that the adapter stays in a particular power state is determined by the frequency of charging (number of charge cycles per year),which is approximated by the power consumption during periods of cellphone talking and standby .Furthermore,the power consumption during talking/standby is weighted by the average number of minutes a person talks on the phone per day (see Figure 8).Finally ,the average energy consumption across the entire population of cellphone users is determined using a weighted sum of energy at each remaining battery level and the probability distribution that charging starts at that battery level.We assume that there are only two types of users:

?the “regular”user,who charges for 8hr at a time (at the probability given by the ?tted Gaussian distribution)and always leaves the charge-adapter plugged in;and

?the “ideal”user,who charges for only the time to reach 100%charge (at the probability distribution centered at 15–20%battery levels)and never leaves the charger plugged in when not charging.

Wireless Networks:An Easy Cell 395

Energy Usage of Cellphones

The yearly energy consumed by cellphone charging between 1991and 2050for the “regular”user and for the “ideal”user is displayed in Figure 12.The yearly consumption of the “ideal”user is less than one-?fth that of the “regular”user.This drastic difference is primarily a consequence of unplugging the charger after charge completion.As a result of the increased energy savings of the “ideal”behavior,we see an increased sensitivity to the cellular usage saturation at different values of minutes per person per day.These trends are more dif?cult to see with the regular behavior since the majority of energy consumption is wasted by the charger.

Year

E n e r g y U s a g e ? r e g u l a r p h o n e s u s e r s , T W h /y e a r

a.“Regular”

user.Year

E n e r g y U s a g e ? i d e a l p h o n e u s e r s , T W h /y e a r

b.“Ideal”user.

Figure 12.Yearly energy consumption of “regular”user and “ideal”user,assuming different user saturation times (15,20,25,and 30min/person/d).

Pseudo-U.S.Model

Assumptions

?A communication infrastructure is entirely nonexistent.?A power grid already exists.

?Each household must have television and Internet service.

?Each household has either one landline phone per person or one cell-phone per person.

Comparison of Fiber-optics to Wireless Networks

We compare the energy usage per person for an entirely wireless net-work to the cost of running a competitive ?ber-optic network.Since the

396The UMAP Journal30.3(2009)

choice of wireless vs.?ber optic affects the energy usage of TVs,computers, and phones in a household,we consider all three of these communication methods.The estimated power usage for each system is summarized in Table3.Based on current estimates for each electronic device[Rosen and Meier1999],a completely wireless approach could be energy competitive against a?ber-optic solution,due to the energy inef?cient link necessary in every household.

Table3.

Electricity usage for?ber-optic and wireless approaches,per household of2.5members with one

computer,one TV,and one phone per person.

Category Fiber-optic usage Wireless usage

General Fiber-optic link(16W)

TV DTV converter(5W)

Internet 2.5×WIMAX card(1W)

2.5×transmission(0.75W)

Phone 2.5×cellphone(0.75W)

2.5×transmission(0.75W)

Total16W13W

Energy to Oil Conversion

We determine the amount of electrical energy available per barrel of oil using historical data[Energy Information Administration2008;Taylor et al.2008].Figure13a shows the heat content per barrel of oil from1949to 2007with linear extrapolation to2050.Heat content is decreasing,possibly due to a decreasing proportion of energy-rich oil in the global market.The thermoelectric ef?ciency(i.e.,the ef?ciency of converting heat created by burning fuel into electricity)is displayed in Figure13b with extrapolation. Using the heat content and thermoelectric ef?ciency data,the total electric-ity produced per barrel of oil is obtained and displayed in Figure14.From the extrapolation,we?nd that one barrel of oil will produce approximately 628kWh of electricity in2050.

While a considerable amount oil is needed to create1TWh or more of electricity,it is very unlikely that oil would be used to create this electricity. In Figure15,we see that oil at its peak use(1977)accounted for only17% of the electricity produced in the U.S.Today,oil accounts for less than4% of electricity and this value appears to be decreasing slowly.

Wireless Networks:An Easy Cell

397

6

Year

O i l h e a t c o n t e n t , B T U /b a r r e l

a.Heat content.

?5

Year

T h e r m o e l e c t r i c e f f i c i e n c y , k W h /B T U

b.Thermoelectric ef?ciency .

Figure 13.Heat content and thermoelectric ef?ciency

data for oil,with extrapolations.

Year

O i l e l e c t r i c i t y c

o n v e r s i o n , k W h /B a r r e l

Figure 14.Electricity per barrel of oil,over time.

Year

% e l e c t r i c i t y p r o d u c e d u s i n g o i l

Figure 15.Trends in U.S.electricity production from oil.

398The UMAP Journal 30.3(2009)

Overall Charger Power Usage

To gauge the inef?ciency of cellphones compared to other electronics,we compare results of our analysis with Rosen and Meier [1999].With updating to re?ect 2008cellphone usage,the results are shown in Figure 16.Although the energy usage of cellphones chargers is signi?cant (2TWh/yr),it is only a small portion of the overall energy wasted by idle electronics (34TWh/yr),or 54million barrels per year using the conversions established above.

Television 12.7 TWh/yr

Audio 6.9 TWh/yr

Receivers 5.3 TWh/yr

Landline 2.9 TWh/yr

Mobile Phone 1.5 TWh/yr

Computers 3.2 TWh/yr

Figure https://www.wendangku.net/doc/ea10845212.html,age of various electronics according to Rosen and Meier [1999],with cellphone energy usage updated to 2008per our model.

Cellular Network Growth Through 2050

Assumptions

?No new (radically disruptive)technologies will be introduced past 3G (third generation of cellphones).Current technology will improve until a minimum necessary energy usage is achieved.

?Population density growth will follow similar trends to 2050.

?The number of towers necessary for a given population density will remain constant through 2050.

Technology Improvements

The power requirements of cellular networks has fallen drastically since the 1980s.Until 2050,similar reductions in power usage will be likely ,either through improvements in the electronics of cell sites (computers and such)or more-ef?cient communication strategies (antenna transmissions).To characterize this reduction in energy,we use information on energy usage

Wireless Networks:An Easy Cell399 of past technologies[Ericsson2007],as shown in Figure17.Technologies following the primary upgrade path(1G to2G and beyond)are leveling out in their minimum energy usage.Although the introduction of3G initially caused a large increase in power consumption,it seems to have a greater potential for reducing energy consumption.Since future technologies can-not be accurately quanti?ed,we assume that all future networks will be based on a variation of3G architecture.Calculated from Figure17,the relevant ef?ciencies for each decade are shown in Table4.

Figure17.Characterization of technological improvements in cellular infrastructures on en-ergy usage,for two different sets of technology, with corresponding exponential?ts of the form a exp(?bx)+c projecting to2050[Ericsson2007].

Table4.

Network technology ef?ciency. Year Relative power usage 20051

2010.85

2020.66

2030.63

2040.62

2050.62

Infrastructure Improvements

As the population grows and the use of cellphones increases,more cell sites and related infrastructure will be necessary.To model the increasing number of towers,we combine tower density/population density relations with population density predictions.The resulting increase in towers is seen in Figure18.These predictions assume that tower capacity will not grow directly but instead improve through energy ef?ciency.

Overall Energy Usage

We calculate total energy usage of the U.S.cellular network using the predicted increase in cell sites,observed trends in technology,predicted usage patterns,and recent energy statistics.Final predictions are shown for two usage scenarios in Figure19.If chargers are used inef?ciently power consumption will grow to approximately400MW,or5.6million

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Figure18.Predicted number of cellphone towers from2007to2050.

bbl/yr.However,if consumers utilize chargers ef?ciently,consumption by 2050will be approximately200MW/yr(2.8million bbl/yr of oil). Conclusion

We estimate power consumption of the U.S.cellular network,based on ?models of usage trends,

?current infrastructure,

?population projections,and

?technology improvements.

a.Inef?cient charger usage.

b.Ideal charger usage.

Figure19.Predictions for the energy usage of the U.S.cellphone network for two different charge scenarios.

Wireless Networks:An Easy Cell401 Technological developments will cause energy usage to decrease until 2015,after which increasing population will demand more power usage.

We assess the optimal communications network for a country similar to the U.S.A wireless network(to houses)comprising voice,data,and TV service would draw less electricity than a?ber-optic approach and hence be optimal,as long as wireless communication can provide suf?cient band-width(likely).

We compare energy consumption for“regular”users and“ideal”users in terms of charging practices.A“regular”user today wastes4.8kWh/yr through inef?cient charging.

We model energy wasted by various idle household electronics,includ-ing cellular network usage.A person today wastes125kWh/yr through idle electronics.

We model energy needs for phone service through2050and calculate the number of new cell towers and other infrastructure necessary.

If inef?cient charging strategies are used,cellular networks in2050will require400MW/yr of electricity(5.6million bbl/yr of oil).If more-ef?cient chargers are introduced or people change their habits,only200MW of power(2.8million bbl/yr of oil)will be required.

References

Banerjee,Nilanjan,Ahmad Rahmati,Mark D.Corner,Sami Rollins,and Lin https://www.wendangku.net/doc/ea10845212.html,ers and batteries:Interactions and adaptive power management in mobile systems.In UbiComp2007:Ubiquitous Computing, edited by J.Krumm et al.,217–234.Lecture Notes in Computer Sci-ence,vol.4717/2007.Berlin/Heidelberg,Germany:Springer.http: //https://www.wendangku.net/doc/ea10845212.html,/content/t2m30643713220k6/.

Center for International Earth Science Information Network(CIESIN),So-cioeconomic Data and Applications Center,Columbia University.2005. Gridded population of the world,version3(GPWv3):Population density grids.https://www.wendangku.net/doc/ea10845212.html,/gpw/global.jsp. CTIA:The Wireless Association.2008.2008CTIA semi-annual wireless industry survey.https://www.wendangku.net/doc/ea10845212.html,/pdf/CTIA_Survey_Mid_ Year_2008_Graphics.pdf.

Energy Information Administration.2008.Annual energy review(AER) 2007.Technical Report DOE/EIA-0384(2007).http://www.eia.doe. gov/aer/.

Ericsson.2007.Sustainable energy use in mobile communications. White paper,August2007.https://www.wendangku.net/doc/ea10845212.html,/campaign/ sustainable_mobile_communications/downloads/sustainable_ energy.pdf.

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Federal Communications Commi s sion,Industry Analysis and Technol-ogy Division,Wireline Competition Bureau.2008.Trends in tele-phone service.URL https://www.wendangku.net/doc/ea10845212.html,/edocs_public/ attachmatch/DOC-284932A1.pdf.

.Database downloads:Antenna structure registration:Cellular—47CFR Part22:Licenses.URL https://www.wendangku.net/doc/ea10845212.html,/antenna/ index.htm?job=uls_transaction&page=weekly.

Interactive Data Corp.2008.Worldwide quarterly mobile phone tracker. April2008.URL https://www.wendangku.net/doc/ea10845212.html,/getdoc.jsp?containerId= IDC_P8397.

Rosen,Karen,and Alan Meier.1999.Energy use of U.S.consumer electron-ics at the end of the20th century.Technical report,Lawrence Berkeley National Laboratory.https://www.wendangku.net/doc/ea10845212.html,/EA/Reports/46212/. Taylor,Peter,with Olivier Lavagne d’Ortigue,Nathalie Trudeau,and Michel Francoeur.2008.Energy ef?ciency indicators for public electricity production from fossil fuels.International Energy Agency.http://www. https://www.wendangku.net/doc/ea10845212.html,/textbase/papers/2008/En_Efficiency_Indicators.pdf. U.S.Census Bureau.2004.Tble HH-6.Average population per household and family:1940to https://www.wendangku.net/doc/ea10845212.html,/population/socdemo/ hh-fam/tabHH-6.pdf.

.2008.Population projections:U.S.interim projections by age, sex,race,and Hispanic origin:2000to2050.https://www.wendangku.net/doc/ea10845212.html,/ population/www/projections/usinterimproj/.

Weisstein,Eric W.n.d.Great circle.From MathWorld—A Wolfram Web Resource.https://www.wendangku.net/doc/ea10845212.html,/GreatCircle.html.

Advisor Louis Rossi with team members Bob Liu,Jeff Bosco,and Zachary Ulissi.

美国数学建模大赛比赛规则

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HiMCM不仅需要选手具备编程技巧，更强调数学，逻辑思维和论文写作能力。这项竞赛是借鉴了美国大学生数学建模竞赛的模式，结合中学生的特点进行设计的。 为什么要参加HiMCM 数学逻辑思维是众多学科的基础，在申请高中或大学专业的时候（如数学，经济学，计算机等），参加了优质的数学竞赛的经历都会大大提升申请者的学术背景。除了AMC这种书面数学竞赛，在某种程度上数学建模更能体现学生用数学知识解决各种问题的能力。

比赛形式 注意：HiMCM比赛可远程参加，无规定的比赛地点，无需提交纸质版论文。重要的是参赛者应注重解决方案的设计性，表述的清晰性。 1.参赛队伍在指定17天中，选择连续的36小时参加比赛。 2.比赛开始后，指导教师可登陆相应的网址查看赛题，从A题或B题中任选其一。 3.在选定的36小时之内，可以使用书本、计算机和网络，但不能和团队以外的任何人 员交流（包括本队指导老师） 比赛题目 1.比赛题目来自现实生活中的两个真实的问题，参赛队伍从两个选题中任选一个。比赛 题目为开放性的，没有唯一的解决方案。 2.赛事组委会的评审感兴趣的是参赛队伍解决问题的方法，所以不太完整的解决方案也 能提交。 3.参赛队伍必须将问题的解决方案整理成31页内的学术论文（包括一页摘要），学术 论文中可以用图表，数据等形式，支撑问题的解决方案 4.赛后，参赛队伍向COMPA递交学术论文，最终成果以英文报告的方式，通过电子 邮件上传。 表彰及奖励 参赛队伍的解决方案由COMPA组织专家评阅，最后评出： 特等奖（National Outstanding) 特等奖提名奖（National Finalist or Finalist） 一等奖（Meritorious)

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PROBLEM A: The Ultimate Brownie Pan When baking in a rectangular pan heat is concentrated in the 4 corners and the product gets overcooked at the corners (and to a lesser extent at the edges). In a round pan the heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges. However, since most ovens are rectangular in shape using round pans is not efficient with respect to using the space in an oven. Develop a model to show the distribution of heat across the outer edge of a pan for pans of different shapes - rectangular to circular and other shapes in between. Assume 1. A width to length ratio of W/L for the oven which is rectangular in shape. 2. Each pan must have an area of A. 3. Initially two racks in the oven, evenly spaced. Develop a model that can be used to select the best type of pan (shape) under the following conditions: 1. Maximize number of pans that can fit in the oven (N)

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