數(shù)學(xué)建模層次分析法旅游景點選址舉例

1、假期到了, 某學(xué)生打算做一次旅游, 有四個地點可供選擇, 假定他要考慮5個因素: 費用、景色、居住條件、飲食以及旅游條件. 由于該學(xué)生沒有固定收入, 他對費用最為看重, 其次是旅游點的景色, 至于旅游條件、飲食, 差不多就行, 住什么地方就更無所謂了. 這四個旅游點沒有一個具有明顯的優(yōu)勢, 而是各有優(yōu)劣. 該同學(xué)拿不定主意, 請用層次分析法幫助他找出最佳旅游點。正文：1、利用層次分析法構(gòu)造層次分析模型：準(zhǔn)則層目標(biāo)層方案層國外4省內(nèi)3省外2目的地本市周邊陪同人員旅途費用旅游交通旅途時間旅游范圍圖1-12、利用成對比較法對準(zhǔn)則層、方案層進(jìn)行列表主要影響因素對比目的地陪同人員旅途費用旅途時間旅游交通

2、目的地15393陪同人員1/511/221/2旅途費用1/32131旅途時間1/91/21/311/3旅游交通1/32131（表2-1）目的地對比本市周邊省內(nèi)省外國外本市周邊11/31/51/7省內(nèi)311/21/4省外5211/2國外1/7421（表2-2）景色對比本市周邊省內(nèi)省外國外本市周邊11/243省內(nèi)2155省外1/41/511國外1/31/511（表2-3）居住條件對比本市周邊省內(nèi)省外國外本市周邊1658省內(nèi)1/6112省外1/5117國外1/81/21/71（表2-4）飲食對比本市周邊省內(nèi)省外國外本市周邊111/31/3省內(nèi)111/21/5省外3211國外3511（表2-5）旅游條

3、件對比本市周邊省內(nèi)省外國外本市周邊1212省內(nèi)1/211/21省外1212國外1/211/21（表2-6）2.構(gòu)造成對比較判斷矩陣(1) 建立準(zhǔn)則層對目標(biāo)層的成對比較判斷矩陣(2) 建立方案層對準(zhǔn)則層的成對比較判斷矩陣 3.計算層次單排序權(quán)重向量并做一致性檢驗先利用mathematica計算矩陣a的最大特征值及特征值所對應(yīng)的特征向量.輸入a=1.0,5,3,9,3,1/5,1,1/2,2,1/2,1/3,2,1,3,1,1/9,1/2,1/3,1,1/3,1/3,2,1,3,1t=eigensystemj/chop輸出5.00974,-0.0048699+0.22084ä,-0.00

4、48699-0.22084ä,0,0,0.88126,0.167913,0.304926,0.0960557,0.304926,0.742882,-0.223286-0.278709ä,-0.165421+0.346134ä,0.151384-0.057689ä,-0.165421+0.346134ä,0.742882,-0.223286+0.278709ä,-0.165421-0.346134ä,0.151384+0.057689ä,-0.165421-0.346134ä,-0.993367,0,0.0

5、719207,0.0662245,0.0605282,0.884443,0,-0.380934,-0.0589629,0.263009得出a的最大特征值為 =5.00974,及其對應(yīng)的特征向量x=0.88126,0.167913,0.304926,0.0960557,0.304926輸入clearx;x=t2,1;w1=x/applyplus,x得到歸一化之后的的特征向量=0.502119,0.0956728,0.173739,0.0547301,0.173739計算一致性指標(biāo)， 故矩陣階數(shù)表矩陣階數(shù)1234567891011000.580.91.121.241.321.411.451.491

6、.51（表3-1）查表（見表3-1）得到相應(yīng)的隨機(jī)一致性指標(biāo)所以 通過了一致性檢驗,即認(rèn)為的一致性程度在容許的范圍之內(nèi), 可以用歸一化后的特征向量作為排序權(quán)重向量.下面再求矩陣的最大特征值及特征值所對應(yīng)的特征向量輸入b1=1.0,1/3,1/5,1/7,3,1,1/2,1/4,5,2,1,1/2,1/7,4,2,1b2=1,1/2,4,3,2,1,5,5,1/4,1/5,1,1,1/3,1/5,1,1b3=1,6,5,8,1/6,1,1,2,1/5,1,1,7,1/8,1/2,1/7,1b4=1,1,1/3,1/3,1,1,1/2,1/5,3,2,1,1,3,5,1,1b5=1,2,1,2,1

7、/2,1,1/2,1,1,2,1,2,1/2,1,1/2,1t1=eigensystemb1/chopt2=eigensystemb2/chopt3=eigensystemb3/chopt4=eigensystemb4/chopt5=eigensystemb5/chop輸出3.82325,0.0883772+0.544064ä,0.0883772-0.544064ä,0,0.111267,0.283002,0.536902,0.786934,-0.0248134-0.0681165ä,-0.141793+0.0729826ä,-0.154388+0.12

8、1345ä,0.964755,-0.0248134+0.0681165ä,-0.141793-0.0729826ä,-0.154388-0.121345 ä,0.964755,0,0.299667,-0.832409,0.4661494.02113,-0.0105652+0.291301ä,-0.0105652-0.291301ä,0,0.495852,0.84036,0.149575,0.159851,-0.234515+0.517899ä,0.805208,-0.109665-0.110941ä,0.04072

9、77 -0.0493071 ä,-0.234515-0.517899 ä,0.805208,-0.109665+0.110941 ä,0.0407277 +0.0493071 ä,0,-0.953463,-0.0953463,0.2860394.25551,-0.110262+1.03317ä,-0.110262-1.03317ä,-0.0349818,0.941183,0.179553,0.276018,0.0758271,0.898054,0.136097 +0.0728034 ä,-0.309669+0.2519 &#

10、228;,-0.0331642-0.0960598ä,0.898054,0.136097-0.0728034ä,-0.309669-0.2519ä,-0.0331642+0.0960598ä,0.958653,-0.256222,0.123505,-0.009047724.08009,-0.0400469+0.570251ä,-0.0400469-0.570251ä,0,0.214349,0.214031,0.59059,0.747963,0.00228339-0.0861419ä,-0.0895045+0.220107&#

11、228;,-0.388206-0.387638ä,0.796962,0.00228339+0.0861419ä,-0.0895045-0.220107ä,-0.388206+0.387638 ä,0.796962,-0.424264,0,0.565685,0.7071074.,0,0,0,0.632456,0.316228,0.632456,0.316228,0.116296,0.629208,-0.687356,-0.343678,-0.92582,0.154303,0.308607,0.154303,-0.92582,0.154303,0.30860

12、7,0.154303分別得出其最大特征值=3.82325,= 4.02113,= 4.25551,= 4.08009,= 4，以及其特征向量如下：b1=（0.111267,0.283002,0.536902,0.786934）b2=（0.495852,0.84036,0.149575,0.159851）b3=（0.941183,0.179553,0.276018,0.0758271）b4=（0.214349,0.214031,0.59059,0.747963）b5=（0.632456,0.316228,0.632456,0.316228）其中 為求出歸一化后的特征向量, 輸入clearb1,b

13、2,b3,b4,b5;b1=t12,1;w1=b1/applyplus,b1b2=t22,1;w2=b2/applyplus,b2b3=t32,1;w3=b3/applyplus,b3b4=t42,1;w4=b4/applyplus,b4b5=t52,1;w5=b5/applyplus,b5輸出4.,0,0,0,0.632456,0.316228,0.632456,0.316228,0.116296,0.629208,-0.687356,-0.343678,-0.92582,0.154303,0.308607,0.154303,-0.92582,0.154303,0.308607,0.1543

14、03w1= 0.0647614,0.164718,0.312497,0.458024w2=0.301313,0.510659,0.0908919,0.0971363w3= 0.639138,0.121931,0.187438,0.0514926w4= 0.121311,0.121132,0.334246,0.423311w5= 0.333333,0.166667,0.333333,0.166667計算一致性指標(biāo),其中,輸入lamda=t11,1,t21,1,t31,1,t41,1,t51,1ci=(lamda-4)/(4-1)/chop則可以得到-0.0589181, 0.00704344,0

15、.0851688,0.0266979,=0查表（見表3-1）得到相應(yīng)的隨機(jī)一致性指標(biāo)計算一致性比率,輸入cr=ci/0.90相應(yīng)可得到因通過了一致性檢驗. 即認(rèn)為的一致性程度在容許的范圍之內(nèi), 可以用歸一化后的特征向量作為其排序權(quán)重向量.4、計算層次總排序權(quán)重向量并做一致性檢驗購買個人電腦問題的第三層對第二層的排序權(quán)重計算結(jié)果列于表4-1k123450.06476140.3013130.6391380.1213110.3333330.1647180.5106590.1219310.1211320.1666670.3124970.09089190.1874380.3342460.3333330.4580240.09713630.05149260.4233110.1666673.823254.021134.255514.080094（表4-1）以矩陣表示第三層對第二層的排序權(quán)重計算結(jié)果為即是第三層對第二層的權(quán)重向量為列向量組成的矩陣. 最下層(第三層)對最上層(第一層)的總排序權(quán)向量為為了計算上式, 輸入w2=transposew1,w2,w