2017 HiMCM B题特等奖学生论文下载8121

2017 HiMCM B题特等奖学生论文下载8121

 

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论文摘要如下：

20th​ ​Annual​ ​High​ ​School​ ​Mathematical​ ​Contest​ ​in​ ​Modeling​ ​(HiMCM)

​Summary​ ​Sheet Team​ ​Control​ ​Number:​ ​8121

Problem​ ​Chosen:​ ​B:​ ​Ski​ ​Slope

Summary:

 

Wasatch​ ​Peaks​ ​Ranch​ ​is​ ​a​ ​prime​ ​location​ ​for​ ​a​ ​ski​ ​resort.​ ​​ ​Given​ ​the​ ​magnitude​ ​of​ ​the property​ ​and​ ​the​ ​randomness​ ​of​ ​nature,​ ​it​ ​is​ ​impossible​ ​to​ ​find​ ​a​ ​perfect​ ​ski​ ​resort​ ​layout.​ ​​ ​Our complete​ ​ski​ ​slope​ ​design​ ​is​ ​of​ ​Olympic​ ​caliber​ ​and​ ​will​ ​compete​ ​as​ ​one​ ​of​ ​the​ ​top​ ​ski​ ​resorts​ ​in all​ ​of​ ​North​ ​America.

 

Our​ ​model​ ​begins​ ​with​ ​extensive​ ​path​ ​generation​ ​which​ ​entails​ ​hundreds​ ​of​ ​trails​ ​being created,​ ​covering​ ​nearly​ ​60​ ​km.​ ​A​ ​subset​ ​of​ ​these​ ​trails​ ​were​ ​chosen​ ​based​ ​on​ ​their​ ​Trail​ ​Quality Index​ ​(TQI),​ ​a​ ​single​ ​value​ ​which​ ​determines​ ​the​ ​appropriateness​ ​of​ ​trails​ ​for​ ​skiers​ ​given​ ​their skill​ ​level,​ ​sharpness​ ​of​ ​curves,​ ​width​ ​of​ ​narrowest​ ​point,​ ​roughness,​ ​steepness,​ ​flatness, variability​ ​in​ ​directional​ ​derivatives,​ ​and​ ​points​ ​of​ ​kinetic​ ​energy​ ​loss.​ ​Using​ ​the​ ​data​ ​for​ ​the​ ​40 km​ ​subset​ ​of​ ​trails,​ ​we​ ​simulated​ ​120​ ​km​ ​of​ ​trails​ ​using​ ​normal​ ​distributions​ ​for​ ​each​ ​of​ ​the aforementioned​ ​values.​ ​Our​ ​final​ ​design​ ​boasts​ ​more​ ​than​ ​162​ ​km​ ​of​ ​slopes​ ​with​ ​38​ ​beginner trails​ ​(20.4%),​ ​74​ ​intermediate​ ​trails​ ​(39.8%),​ ​and​ ​74​ ​expert​ ​trails​ ​(39.8%).​ ​Our​ ​beginner​ ​and intermediate​ ​trails​ ​have​ ​an​ ​mean​ ​TQI​ ​of​ ​3.7​ ​compared​ ​to​ ​a​ ​TQI​ ​of​ ​2.4​ ​for​ ​Whistler​ ​Blackcomb’s premier​ ​ski​ ​trail.​ ​Any​ ​TQI​ ​values​ ​greater​ ​than​ ​0.5​ ​are​ ​considered​ ​acceptable.

 

The​ ​aggregate​ ​of​ ​all​ ​trails​ ​is​ ​our​ ​Wasatch​ ​Peaks​ ​Ranch​ ​ski​ ​resort​ ​design.​ ​To quantitatively​ ​compare​ ​our​ ​resort​ ​with​ ​others,​ ​we​ ​created​ ​a​ ​Resort​ ​Rating​ ​system​ ​which​ ​rates​ ​the resorts​ ​from​ ​0.1​ ​to​ ​10​ ​(10​ ​being​ ​the​ ​best),​ ​based​ ​on​ ​a​ ​variety​ ​of​ ​factors,​ ​including​ ​ski​ ​lifts,​ ​trail density,​ ​vertical​ ​drop,​ ​and​ ​direction​ ​that​ ​the​ ​slope​ ​is​ ​facing,​ ​which​ ​is​ ​critical​ ​for​ ​snow​ ​quality. Using​ ​this​ ​system,​ ​Wasatch​ ​Peaks​ ​Ranch​ ​achieved​ ​a​ ​Resort​ ​Rating​ ​score​ ​of​ ​7.44,​ ​ranking​ ​3rd among​ ​16​ ​top​ ​North​ ​American​ ​ski​ ​resorts.​ ​To​ ​put​ ​this​ ​into​ ​more​ ​context,​ ​Park​ ​City​ ​Mountain, known​ ​as​ ​one​ ​of​ ​the​ ​best​ ​ski​ ​resorts,​ ​and​ ​previous​ ​Winter​ ​Olympic​ ​site,​ ​received​ ​a​ ​score​ ​of​ ​8.67 using​ ​our​ ​rating​ ​and​ ​was​ ​ranked​ ​#1.​ ​Sugarloaf​ ​Mountain,​ ​ranked​ ​extremely​ ​poorly​ ​on ZRankings.com,​ ​received​ ​a​ ​4.74.​ ​The​ ​system​ ​is​ ​proven​ ​reliable​ ​through​ ​its​ ​strong​ ​correlation with​ ​existing​ ​ranking​ ​systems​ ​and​ ​its​ ​vulnerability​ ​to​ ​sensitivity​ ​analysis.

 

Our​ ​rating​ ​systems​ ​are​ ​based​ ​on​ ​weighted​ ​composite​ ​scores​ ​in​ ​the​ ​various​ ​factors​ ​we incorporated,​ ​so​ ​that​ ​it​ ​truly​ ​reflects​ ​what​ ​skiers​ ​are​ ​looking​ ​for.​ ​It​ ​also​ ​incorporates​ ​logistic models​ ​to​ ​make​ ​sure​ ​that​ ​ratings​ ​below​ ​the​ ​minimum​ ​and​ ​ratings​ ​above​ ​the​ ​maximum​ ​cannot happen.

 

To​ ​measure​ ​Olympic​ ​compatibility​ ​and​ ​readiness,​ ​we​ ​used​ ​a​ ​similar​ ​system,​ ​which resulted​ ​in​ ​our​ ​Olympic​ ​readiness​ ​score​ ​of​ ​7.8,​ ​well​ ​within​ ​the​ ​range​ ​of​ ​previous​ ​Olympic​ ​sites: Squaw​ ​Valley’s​ ​7.0​ ​to​ ​Park​ ​City​ ​Mountain’s​ ​8.3.

 

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