IV模糊層次分析模型1.模型的分析模糊層次分析的關(guān)鍵環(huán)節(jié)是建立判斷矩陣，判斷矩陣是否科學(xué)、合理直接影響到模糊層次分析的效果，而判斷矩陣的建立往往具有主觀性，并且判斷矩陣一致性的判斷標(biāo)準(zhǔn)：CR<0.1缺乏科學(xué)依據(jù)，而模糊層次分析法可以較好地規(guī)避這些問題。下面是先用模糊層次分析法得到課程性質(zhì)的權(quán)重向量，再根據(jù)公式：求出綜合成績。（在這里，將21門課的定義為權(quán)重向量W）2.模型的準(zhǔn)備建立模糊一致判斷矩陣，下表為模糊一致判斷矩陣的數(shù)量標(biāo)度：表5數(shù)量標(biāo)度標(biāo)度說明0.5兩元素相比，同等重要0.6兩元素相比，一元素稍微重要0.7兩元素相比，一元素明顯重要0.8兩元素相比，一元素重要得多0.9兩元素相比，一元素極端重要0.1,0.2,0.3,0.4若元素與元素相比較得到判斷，則元素與元素比較得到的判斷為運用表5中的數(shù)量標(biāo)度可得到如下模糊判斷矩陣,并根據(jù)模糊一致矩陣的充要條件進行調(diào)整，假設(shè)將第一行元素、視為有把握的，用R的第一行元素減去第二行對應(yīng)元素，若所得的一個差數(shù)為常數(shù)，不需調(diào)整第二行元素。否則，要對第二行元素進行調(diào)整，直到第一行元素減第二行的對應(yīng)元素之差為常數(shù)為止。用R的第一行元素減去第三行的對應(yīng)元素，若所得的n個差數(shù)為常數(shù)，則不需調(diào)整第三行的元素。否則，要對第三行的元素進行調(diào)整，直到第一行元素減去第三行對應(yīng)元素之差為常數(shù)為止。上面步驟如此繼續(xù)下去直到第一行元素減去第行對應(yīng)元素之差為常數(shù)為止。由以上步驟可以得到如下模糊一致性判斷矩：3.模型的建立設(shè)R是n階模糊矩陣,則R是模糊一致矩陣的充分條件是存在一n階非負(fù)歸一化的向量及一正數(shù)a,使得對于任意的i,j,有（1）若R是模糊一致矩陣,則其權(quán)重可由(2)式計算：，（i=1,2,n）（2）其中,(i=1,2,3,4,5)在本模型中，根據(jù)一致判斷矩陣，求得的權(quán)重向量（求解程序見附錄4），及五種課程性質(zhì)的比重再將學(xué)時比重和學(xué)分對權(quán)重向量的影響考慮進來得到權(quán)重向量（求解程序見附錄5）,利用MATLAB求得：W1=(0.0019,0.0013,0.0013,0.0013,0.0013,0.0007,0.0004,0.0004,0.0010,0.0008,0.0008,0.0008,0.0008,0.0008,0.0008,0.0006,0.0005,0.0005,0.0005,0.0002,0.0002);4.模型的求解在MATLAB中，建立一個41x21的成績矩陣A，用A與權(quán)重向量W相乘（求解程序見附錄6），并將綜合成績按降序排列。得到所有學(xué)生的綜合成績表如下：表6學(xué)生綜合成績綜合成績學(xué)生序號綜合成績學(xué)生序號1.4173701.3202101.4021301.3181631.3822861.3179541.374421.317691.3606601.3171621.3604751.3149181.3602201.3142531.3562641.3116961.3548841.3101441.3528121.3044921.3478721.299541.3465171.2994271.3435801.294111.3423991.2921291.3379511.2862811.3292331.2802691.3261731.2778911.3248131.250381.3245741.2313221.3233931.2149103從表6中，我們可以看到前十名的學(xué)生序號為：70,30,86,2,60,75,20,64,84,12。§5模型的誤差分析用四種方法得出的前10%的同學(xué)的學(xué)號如表7所示：表7四種方法得出的前10%學(xué)生學(xué)號方法一方法二方法三方法四51707070708686303030108686752027533646051227560513020272164808048464847212由表7可以看出不同方法得出的前10%學(xué)生學(xué)號不同，說明在用不同種方法進行處理的時候存在誤差。§6模型的優(yōu)缺點1.優(yōu)點(1）簡單加權(quán)平均值模型，簡潔易懂，有利于數(shù)據(jù)的篩選。(2）標(biāo)準(zhǔn)化模型所有的成績都轉(zhuǎn)化為0—1之間的數(shù)，使課程分?jǐn)?shù)域相同，這有效解決了各科老師給分習(xí)慣導(dǎo)致的評分標(biāo)準(zhǔn)不同的問題，使各科的成績可比性增強。(3）層次分析模型、模糊層次分析模型將研究對象看做一個系統(tǒng)，充分考慮了各種權(quán)重影響因素，解決了課程難度不均帶來的不公平的問題。(4）利用EXCEL軟件對數(shù)據(jù)進行處理并作出各種圖表，簡便，直觀，快捷；(5)運用多種數(shù)學(xué)軟件（如MATLAB等），取長補短，使計算結(jié)果更加準(zhǔn)確、明晰；(6)本文巧妙運用思路分解圖，將建模思路完整清晰的展現(xiàn)出來；2.缺點（1）簡單加權(quán)平均值模型可能會受到不同教師打分不同及標(biāo)準(zhǔn)差不同的問題、不同科目難度不同的問題的影響。（2）標(biāo)準(zhǔn)化模型的缺點是一些同學(xué)因為考取最低分而最終該科成績?yōu)?分，這種零分情況難以接受。（3）層次分析模型的判斷矩陣的建立有主觀性，不具有科學(xué)嚴(yán)謹(jǐn)性。（4）模糊層次分析模型直接采用分?jǐn)?shù)的比較，有可能會受到不同教師打分不同及標(biāo)準(zhǔn)差不同的問題、不同科目難度不同的問題。§7模型的推廣1.推行全面素質(zhì)教育，不局限于以學(xué)生考試成績作為評定的唯一標(biāo)準(zhǔn)，以競賽獲獎，宿舍衛(wèi)生情況等作為評定的輔助標(biāo)準(zhǔn)。2.根據(jù)聚類分析法依據(jù)學(xué)生每年的反映對課程學(xué)分進行動態(tài)調(diào)整，保證其先進性。3.此獎學(xué)金評定方案不僅適用于學(xué)校對學(xué)生的綜合評估，還可以推廣到教師、企業(yè)家等各類職員年度業(yè)績的評定。應(yīng)用本評定方案，也可對于年終個人評優(yōu)和獎金發(fā)放起到一定指導(dǎo)作用。§8參考文獻(xiàn)[1]陳恩水，王峰，數(shù)學(xué)建模與實驗[M]，北京：科學(xué)出版社，2008年6月：1-9，162-169。[2]趙靜，但琦，數(shù)學(xué)建模與數(shù)學(xué)實驗，北京：高等教育出版社，2000.11。[3]楊桂元，黃己立，數(shù)學(xué)建模[M]，合肥：中國科技技術(shù)大學(xué)出版社，2007.[4]揚啟帆,康旭升,等.數(shù)學(xué)建模[M].北京:高等教育出版社.2006年5月。[5]吳禮斌.經(jīng)濟數(shù)學(xué)實驗與建模[M].天津大學(xué)出版社.2009.8.[6]胡守信,李柏年,基于MATLAB的數(shù)學(xué)實驗,北京:科學(xué)出版社,2004.6第一版。[7]姜啟源等.數(shù)學(xué)模型（第三版）[M].高等教育出版社.2003.8.

附錄附錄1（求A的所有特征值）A=[12357;1/21236;1/31/2125;1/51/31/212;1/71/61/51/21];a=eig(A)附錄2（求A的最大特征值所對應(yīng)的特征向量）A=[12357;1/21236;1/31/2125;1/51/31/212;1/71/61/51/21];a=eig(A);[X,D]=eig(A);a1=X(:,1)附錄3（求所有學(xué)生的綜合成績）w=[0.1799,0.0386,0.0386,0.0386,0.0386,0.1028,0.0532,0.0532,0.1594,0.0277,0.0277,0.0277,0.0277,0.0277,0.0277,0.0463,0.0132,0.0132,0.0132,0.0225,0.0225];A=[75707480658780808475937075638296.7447.4482.0282.0296.7487.1488927893696860848575777979758461.538596.7496.74858579707977766469828373728977658496.7482.0282.0247.4487.1496.7473728476636562697364788684606096.7461.5385.5487.1496.7485.54969010073746671676873676679729168.5768.5768.5761.5347.4496.7498737182667485738089838380626696.7473.2761.5373.2761.5373.27100917774886060737475728671689547.4496.7477.9287.1461.5396.7477878888838496707762786076636396.7461.5379.1496.7479.1461.53697361781006266857765848284977296.7496.7496.7496.7496.7496.7485788586807286807574728674706277.9261.5347.4487.1496.7496.7484797686917069788170947181707096.7487.9487.9461.5396.7496.7460628779747381807165728473666880.3147.4480.3196.7480.3196.7477758078836585816961766880997747.4480.3196.7480.3180.3196.7473707478846590797873697984796481.881.881.887.1461.5396.7462878887906385738875917783877496.7496.7496.7496.7496.7496.7460796777878580778682978477807596.7461.5380.3187.1480.3196.7473798388917661846267866079798247.4480.396.7480.396.7480.361857573976495757565797486898596.7461.5396.7496.7489.796.7460699078716471838678738676766896.7496.7496.7496.7496.7496.7460728284877490776371817382887796.7485.5496.7487.1485.5461.5381827275936182827466738876767696.7496.7496.7496.7496.7496.7470867683886261887576666590768296.7496.7461.5387.1461.5380.7476857880637082798288726880697696.7461.5396.7487.9496.7487.9481688873868569797771677495717693.5496.7487.1496.7493.5493.5460758381746781838272737875867496.7461.5376.7487.1476.7461.5394748682837899757576757877838096.7496.7496.7496.7496.7496.7477818481838587816776887465696293.5496.7493.5487.1493.5496.7468798883756672867269817683696996.7496.7487.1494.3496.7494.3479888184736677798670727176807681.887.1481.861.5381.896.7462849287736078798771818267809887.1496.7491.9496.7487.1491.9468609190728969736863737981869196.7496.7496.7496.7496.7496.7484757977849286728061698374797764.4164.4161.5347.4487.1461.5386848481818077787280688077818861.5361.5396.7480.7496.7487.1477748686757992858174817369759096.7496.7496.7496.7496.7496.7460766581746978827380698265836896.7487.1496.7496.7494.3494.3471666685706089827981738773737496.7496.7494.3496.7494.3487.1477797481868677827365747970787096.7461.5387.9487.9496.7496.7472688786897277777573827674766296.7481.861.5381.887.1481.866839383857364867972857762928296.7477.1177.1187.1477.1147.4471749280907692787773696764617447.4447.4447.4447.4447.4447.44];C=A*w'附錄4(一致判斷矩陣R的權(quán)重向量的求解程序)R=[0.5,0.6,0.7,0.8,0.9;0.4,0.5,0.6,0.7,0.8;0.3,0.4,0.5,0.6,0.7;0.2,0.3,0.4,0.5,0.6;0.1,0.2,0.3,0.4,0.5];b=ones(5,1);w=1/5*(R*b+b-2*b)附錄5(權(quán)重向量W的求解程序)A=[0.50.50.50.50.50.60.40.40.40.30.30.30.30.30.30.20.20.20.20.10.1];B=[4/593/593/593/593/592/592/592/593/593/593/593/593/593/593/593/593/593/593/592/592/59];C=[3.5/613/613/613/613/612/612/612/613/613/613/613/613/613/613/613.5/613/613/613/613/613/61];D=A.*B.*C附錄6(綜合成績的求解程序)W1=[0.0019,0.0013,0.0013,0.0013,0.0013,0.0007,0.0004,0.0004,0.0010,0.0008,0.0008,0.0008,0.0008,0.0008,0.0008,0.0006,0.0005,0.0005,0.0005,0.0002,0.0002];A=[75707480658780808475937075638296.7447.4482.0282.0296.7487.1488927893696860848575777979758461.538596.7496.74858579707977766469828373728977658496.7482.0282.0247.4487.1496.7473728476636562697364788684606096.7461.5385.5487.1496.7485.54969010073746671676873676679729168.5768.5768.5761.5347.4496.7498737182667485738089838380626696.7473.2761.5373.2761.5373.27100917774886060737475728671689547.4496.7477.9287.1461.5396.7477878888838496707762786076636396.7461.5379.1496.7479.1461.53697361781006266857765848284977296.7496.7496.7496.7496.7496.7485788586807286807574728674706277.9261.5347.4487.1496.7496.7484797686917069788170947181707096.7487.9487.9461.5396.7496.7460628779747381807165728473666880.3147.4480.3196.7480.3196.7477758078836585816961766880997747.4480.3196.7480.3180.3196.7473707478846590797873697984796481.881.881.887.1461.5396.7462878887906385738875917783877496.7496.7496.7496.7496.7496.7460796777878580778682978477807596.7461.5380.3187.1480.3196.7473798388917661846267866079798247.4480.396.7480.396.7480.361857573976495757565797486898596.7461.5396.7496.7489.796.7460699078716471838678738676766896.7496.7496.7496.7496.7496.7460728284877490776371817382887796.7485.5496.7487.1485.5461.5381827275936182827466738876767696.7496.7496.7496.7496.7496.7470867683886261887576666590768296.7496.7461.5387.1461.5380.7476857880637082798288726880697696.7461.5396.7487.9496.7487.9481688873868569797771677495717693.5496.7487.1496.7493.5493.5460758381746781838272737875867496.7461.5376.7487.1476.7461.5394748682837899757576757877838096.7496.7496.7496.7496.7496.7477818481838587816776887465696293.5496.7493.5487.1493.5496.7468798883756672867269817683696996.7496.7487.1494.3496.7494.3479888184736677798670727176807