上海海事大學-2015優(yōu)化與統(tǒng)計建模試驗模擬考查題(答案

1、上海海事大學 2014 - 2015 學年第 二 學期研究生 優(yōu)化與統(tǒng)計建模試驗 課程考查模擬題專業(yè)： 學生姓名： 學號： 要求：1.本考查為無紙化形式,要求每位研究生獨立完成,禁止和他人交流。2.請將本文件更名(原文件名學號姓名)。3.必須在本文件內作答并緊隨相應試題后面，評閱老師只評閱本文件。4.所有問題僅限于采用本課程所學軟件: Lingo,Cplex,Spss,R進行求解. 其他方式求解將不被認可。5.作答主要內容：Lingo,Cplex,R要求源代碼及關鍵的輸出結果；Spss要求關鍵的輸出結果,一些重要的操作設置最好能加以說明。一. （共10分）線性規(guī)劃：已知線性規(guī)劃 Max z=x

2、12x2x3 x1+x2 +2x3 12 x1+x2 x3 1 x1,x2 ,x30. 1.分別用Lingo和Cplex求解該問題；最優(yōu)解x=(0,0,0)目標函數值 02.求對偶問題； min z=12*y1+y2 y1+y2=-1; y1+y2=-2; 2*y1-y2=-1; y1,y2=0; 3.解對偶問題，試驗影子價格； y=(0,0)4.對目標函數系數，約束右邊常量進行靈敏度分析。源代碼：Lingo代碼:model:sets:ii/1.3/:x,c;jj/1.2/:b;link(jj,ii):a;endsetsdata:c=-1,-2,-1;b=12 1;a=1 1 2 1 1 -1

3、;enddatamax=sum(ii(i):x(i)*c(i);for(jj(j):sum(ii(i):a(j,i)*x(i)=b(j);Cplex代碼: /* * OPL 5.5 Model * Author: zh * Creation Date: 2015/5/19 at 10:00 */range ii=1.3;range jj=1.2;float cii=-1,-2,-1;float bjj=12,1;float ajjii=1,1,2, 1,1,-1;dvar float+ xii;maximizesum(i in ii)ci*xi;subject to forall(j in j

4、j) sum(i in ii) aji*xi= - 1; X_2 y_2 + y_3 = - 2; X_3 2 * y_2 - y_3 = - 1; END主要輸出結果LingoX( 1) 0. 1. X( 2) 0. 2. X( 3) 0. 1.Cplex最終解決方案 目標 = 0:x = 0 0 0;對偶問題輸出結果 Variable Value Reduced Cost Y_2 0. 12.00000 Y_3 0. 1.靈敏度分析Ranges in which the basis is unchanged: Objective Coefficient Ranges: Current Al

5、lowable Allowable Variable Coefficient Increase Decrease X( 1) -1. 1. INFINITY X( 2) -2. 2. INFINITY X( 3) -1. 1. INFINITY Righthand Side Ranges: Current Allowable Allowable Row RHS Increase Decrease 2 12.00000 INFINITY 12.00000 3 1. INFINITY 1.二. （共15分）最短路： 已知線路網路如圖,兩點之間聯系上數字表示兩點間的距離。1. 求A1到A7所有最短路

6、。共兩條：1257 1457model:sets:node/1.7/;arcs(node,node):d,x,u;!d:distance,x:0,1,u:exist or not;endsetsdata:d=0 6 2 3 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 4 0 0 0 0 0 4 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1;u=0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

7、1;enddatamin=sum(arcs:d*x);!省略（i,j）!condition1: 進口路徑和出口路徑是唯一的;sum(node(i):x(1,i)=1;sum(node(i):x(i,7)=1;!condition2:進出口相同;for(node(j)|j#ne#1#and#j#ne#7:sum(node(i):x(j,i)=sum(node(i):x(i,j);!condition3:不可用路徑不顯示;for(arcs(i,j):x(i,j)=u(i,j);for(arcs(i,j):bin(x(i,j);end長度：92. 求A1到A6所有最短路。僅一條：146model:s

8、ets:node/1.7/;arcs(node,node):d,x,u;!d:distance,x:0,1,u:exist or not;endsetsdata:d=0 6 2 3 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 4 0 0 0 0 0 4 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1;u=0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1;enddatamin=sum(arcs

9、:d*x);!省略（i,j）!condition1: 進口路徑和出口路徑是唯一的;sum(node(i):x(1,i)=1;sum(node(i):x(i,6)=1;!condition2:進出口相同;for(node(j)|j#ne#1#and#j#ne#6:sum(node(i):x(j,i)=sum(node(i):x(i,j);!condition3:不可用路徑不顯示;for(arcs(i,j):x(i,j)=d(1);for(time(i)|i#Gt#1:x(i)+s(i-1)=d(i);s(1)=x(1)+10-d(1);for(time(i)|i#gt#1:s(i)=x(i)+s

10、(i-1)-d(i);end主要輸出結果：X( 1) 40.00000 0. X( 2) 50.00000 0. X( 3) 75.00000 0. X( 4) 25.00000 0.四. （共10分）統(tǒng)計描述性分析：對data1描述性分析: 求均值,方差,標準差,變異系數,偏度,峰度,常用分位數,極差,四分位差,直方圖,箱式圖,經驗分布圖,Q_Q圖源代碼： x=c( 0.28, 0.08,-0.97, 0.42, 1.22,-1.13, 0.37,-0.14, 0.2,-0.51,-0.29, 0.10,-0.14, -0.10,-0.09,-0.32, 0.38,-0.55, 0.39,

11、0.18,-1.00, 0.90, 0.47,-1.48, 1.13, 1.20, -1.08,-0.54, 1.63, 0.46,-1.53, 1.09, 1.26,-1.04,-0.17, 0.91, 0.16,-1.11, 0.25, 0.89,-0.46,-0.44, 0.77, 0.14,-0.87,-0.34, 0.50, 0.37,-1.19, 0.74, 0.17,-0.48, -0.16, 0.32,-0.65,-0.03,-0.20, 0.21,-0.35,-0.48, 0.30, 0.02,-0.88, 0.56,-0.21, 0.06, 0.54,-1.07, 0.36

12、, 0.90,-0.83, 0.12, 1.19,-0.42,-0.50, 0.08, 0.19,-0.89, 0.57, 0.31,-0.66, 0.39, 0.06,-0.90, 0.09, 0.39,-0.44,-0.12, 0.12,-0.56, 0.55, 0.15,-0.97, 0.88, 0.77,-1.89, 1.32, 0.95,-1.04, 0.44,-0.17, 0.01, 0.46,-0.48, -0.10,-0.21, 0.41,-0.73,-0.11, 0.43,-0.12,-1.00, 0.51, 0.79,-1.34, 0.55, 1.44, -1.17,-0.

13、17, 0.52, 0.23,-1.06, 0.35, 0.75,-0.64,-0.46, 0.69,-0.37, 0.08, 0.79, -0.82, 0.00, 0.09,-0.65, 0.12, 0.40,-1.17, 0.51, 0.57,-1.08, 0.33, 0.87,-0.59, -0.29, 1.22,-0.38,-0.51, 0.48, 0.21,-1.16, 0.85)mean(x)#均值#var(x)#方差#sd（x）#標準差#100*sd(x)/mean(x)#變異系數#all.moments(x,central=TRUE, order.max=4)all.momen

14、ts( x, order.max=4 )all.moments( x, absolute=TRUE, order.max=4 )skewness(x)#偏度#kurtosis(x)#峰度#max(x)-min(x)#極差#quantile(x, probs = c(0.1, 0.5, 1, 2.5, 5,0.75, 10, 50, NA)/100,type=1)y=ecdf(x)#經驗分布圖#plot(ecdf(x),verticals=TRUE,do.p=T) #do.p是邏輯變量=FALSE表示不畫點處的記號#x=seq(-2,2,0.01)#lines(x,pnorm(x,mean(x)

15、,sd(x),col=red)hist(x)#直方圖#boxplot(x)#箱式圖#boxplot(x,horizontal=T);qqnorm(x,pch=+,ylab=,main=)#q-q圖#qqline(x, col = 2)主要輸出結果： mean(x)1 -0. var(x)1 0. x=c(1,2,3) all.moments(x,central=TRUE, order.max=4)1 1. 0. 0. 0. 0. all.moments( x, order.max=4 )1 1. 2. 4. 12. 32. all.moments( x, central=TRUE, order

16、.max=4 )1 1. 0. 0. 0. 0. all.moments( x, absolute=TRUE, order.max=4 )1 1. 2. 4. 12. 32. skewness(x)1 0 kurtosis(x)1 1.5 quantile(x, probs = c(0.1, 0.5, 1, 2, 5, 10, 50, NA)/100,type=1)0.1% 0.5% 1% 2% 5% 10% 50% 1 1 1 1 1 1 2 NA y=ecdf(x) plot(ecdf(x),verticals=TRUE,do.p=T) #do.p是邏輯變量=FALSE表示不畫點處的記號

17、#x=seq(-2,2,0.01) #lines(x,pnorm(x,mean(x),sd(x),col=red) #boxplot(x) #boxplot(x,horizontal=T); qqnorm(x,pch=+,ylab=,main=) qqline(x, col = 2)五. （共10分）回歸分析：對data6中經漂吟霉素處理數據用指數增長模型非線性回歸1. 寫出回歸表達式，獲得回歸的檢驗結論；y1=192.095(1-exp(-11.385x)2. 比較Michaelis-Menten回歸，哪一個效果好，為什么？用R語言兩者求解系數顯著度差距不大，回歸效果差不多。見后面回歸檢驗結

18、果。(用SPSS比較R2值, Michaelis-Menten R2=0.96126, 而指數模型, R2=0.90144, 因而前者回歸效果好)3. 畫出回歸效果圖像。源代碼x=c(0.02 , 0.06, 0.11, 0.22, 0.56, 1.10);#底物濃度#y1=c(76, 47, 97, 107, 123, 139, 159, 152, 191, 201, 207, 200);#反應速度處理#xx=c(0.02 , 0.02 , 0.06, 0.06, 0.11, 0.11, 0.22, 0.22, 0.56, 0.56, 1.10, 1.10);x=xx;y=y1;plot(x

19、,y,pch=8); z=nls(ySSmicmen(x,Vm,K);#Michaelis-Menten模型summary(z);z=nls(ybeta1*(1-exp(-beta2*x),start=list(beta1 = 195, beta2=0.4);#混合反應模型summary(z);points(x,fitted(z),pch=e,col=red);#在前面最后一個plot，作圖基礎上添加擬合值主要輸出結果回歸檢驗輸出：Michaelis-Menten模型代碼：Formula: y SSmicmen(x, Vm, K)Parameters: Estimate Std. Error

20、t value Pr(|t|) Vm 2.127e+02 6.947e+00 30.615 3.24e-11 *K 6.412e-02 8.281e-03 7.743 1.57e-05 *-Signif. codes: 0 * 0.001 * 0.01 * 0.05 . 0.1 1Residual standard error: 10.93 on 10 degrees of freedomNumber of iterations to convergence: 0 Achieved convergence tolerance: 1.93e-06指數模型代碼：Formula: y beta1 *

21、 (1 - exp(-beta2 * x)Parameters: Estimate Std. Error t value Pr(|t|) beta1 192.095 8.176 23.495 4.42e-10 *beta2 11.385 1.628 6.992 3.75e-05 *-Signif. codes: 0 * 0.001 * 0.01 * 0.05 . 0.1 1Residual standard error: 17.44 on 10 degrees of freedomNumber of iterations to convergence: 19 Achieved converge

22、nce tolerance: 4.533e-06指數回歸效果圖六. （共15分）主成分分析：對data7 進行主成分分析，1. 要求獲得載荷矩陣，主成分得分矩陣，碎石圖；2. 各主成分命名。各企業(yè)總效應大小的綜合指標y1y1=0.32113x1+0.29516x2+0.38912x3+0.38472x4+0.37955x5+0.37087x6+0.31996x7+0.35546x8Spss主要輸出結果Communalities InitialExtractionVAR000011.000.812VAR000021.000.907VAR000031.000.984VAR000041.000.98

23、9VAR000051.000.988VAR000061.000.988VAR000071.000.709VAR000081.000.801Extraction Method: Principal Component Analysis.Total Variance ExplainedComponentInitial EigenvaluesExtraction Sums of Squared LoadingsTotal% of VarianceCumulative %Total% of VarianceCumulative %16.13776.70876.7086.13776.70876.7082

24、1.04213.02789.7341.04213.02789.7343.4365.44995.184 4.2202.75597.938 5.1521.89999.837 6.009.11099.948 7.003.03799.985 8.001.015100.000 Extraction Method: Principal Component Analysis.Component Matrix(a) Component12VAR00001.796.424VAR00002.731.610VAR00003.964-.235VAR00004.953-.285VAR00005.940-.323VAR0

25、0006.919-.379VAR00007.793.284VAR00008.881.160Extraction Method: Principal Component Analysis.a 2 components extracted.Component Score Coefficient Matrix Component12VAR00001.130.407VAR00002.119.585VAR00003.157-.225VAR00004.155-.273VAR00005.153-.310VAR00006.150-.364VAR00007.129.272VAR00008.143.154Extr

26、action Method: Principal Component Analysis.七. （共15分）時間序列分析：data2,3作時間序列分析(數據橫著讀)(要求用R語言)1. ARIMA模型的最佳參數data2 AR(3) data3 AR(2)2. 模型的檢驗3. 往后預測至少六期4. 畫圖源代碼: data2代碼：par(mfrow=c(1,1)#library(tseries)#無效命令x0=c(1000.7, 571.9, 573.6, 368.3, 146.6, 114.8, 122.3,389.1, 571.2, 647.6, 754.3, 1030.2, 733.8, 5

27、41.4,436.2, 250.9, 136.9, 453.9, 838.1, 1273.1, 1209.6,979, 797.9, 417.3, 367.4, 84.1, 237.8, 1110,1852.4, 1511.1, 1017.6, 817.1, 461.5, 273.6, 122,289.2, 994.4, 1584.3, 1570.9, 1417.3, 1078.7, 799,720.5, 562.8, 492, 255.2, 192.2, 76.7, 48.8,81.1, 173.7, 408, 540.4, 516.6, 569.6, 506.9,337.3, 120.6,

28、 97.7, 30.4, 0, 17, 59.4,146.3, 167.2, 424.8, 549.7, 492.7, 360.7, 287.3,188.1, 79.1, 48, 21.5, 102.5, 198.8, 435.3,596.5, 769.8, 804.3, 851.8, 573.7, 330.3, 102.3,158.9, 682.3, 1457.4, 1659.3, 1237.8, 1029.8, 758.3,441.6, 290.3, 128.1, 180, 480.7, 738, 1181.5,1491.8, 1150.4, 798.4, 774, 650.5, 468.

29、3, 246.8,80.5, 51.6, 273.3, 657.7, 1126, 1148.3, 926,709.3, 528.2, 563.4, 365.7, 195.5, 87.1, 447.5,886.8, 1669.3, 1334.4, 1220, 795.5, 535.8, 204.9,135.8, 147.3, 40.5, 71.5, 387.2, 651, 715.8,764.4, 761.4, 625.9, 304.5, 156.6, 81, 75.2,84.6, 427.5, 875.6, 1019.2, 936.1, 767.6, 501.4,314.9, 320.6, 1

30、45.3, 113.5, 32.9, 60.3, 292.6,503.4, 761.6, 646.3, 744.4, 582.5, 526.6, 223,68.4, 43.1, 17.3, 115.1, 568.4, 684.8, 1246.7,966.9, 763.3, 451.7, 313.6, 170.9, 69.3, 200.6,531.7, 766.7, 828.5, 933.5, 779.6, 428, 254.7,133.7, 67.9, 104.6, 432.7, 956.8, 1372.8, 1314.6,1065, 813.4, 569.7, 367.2, 195.9, 1

31、15.1, 397.1,1110.1, 1798.1, 1634.4, 1621.4, 1007.1, 837.1, 376.9,166.2, 52.9, 455.4, 1700.5, 2278.2, 2215.1, 1905,1347.3, 646.8, 451.2, 334.7, 122.4, 180.7)x0z=ts(x0,frequency=1,start=c(1742),end=c(1957)#起始時間和結束時間#acf(z)pacf(z)x - arima(z,order=c(3,0,0)tsdiag(x)#3個圖的出現#fore.mod-predict(object =x, n.

32、ahead = 12, se.fit = TRUE) fore.modforevalue=c( 275.1532, 324.2066 ,456.2553, 562.3887, 644.3462 ,663.5609, 639.0554, 581.7076 ,521.1731, 474.8483 ,455.7449, 462.8644)x00=c(x0,forevalue)zt=ts(x00,frequency=1,start=c(1742),end=c(1969)#產生新序列已設置后面plot格式pred.mod=z-x$residuals#計算擬和值par(mfrow=c(1,1)plot(z

33、t,col=green,lwd=2,xlab=,ylab=)#實值與預測值lines(pred.mod,col=green,lwd=2)#擬和值lines(z,col=red,lwd=2)#實值data3代碼：par(mfrow=c(1,1)#library(tseries)#無效命令x0=c(580.38, 581.86, 580.97, 580.8, 579.79, 580.39, 580.42, 580.82, 581.4, 581.32, 581.44, 581.68, 581.17, 580.53, 580.01, 579.91, 579.14, 579.16, 579.55, 57

34、9.67, 578.44, 578.24, 579.1, 579.09, 579.35, 578.82, 579.32, 579.01, 579, 579.8, 579.83, 579.72, 579.89, 580.01, 579.37, 578.69, 578.19, 578.67, 579.55, 578.92, 578.09, 579.37, 580.13, 580.14, 579.51, 579.24, 578.66, 578.86, 578.05, 577.79, 576.75, 576.75, 577.82, 578.64, 580.58, 579.48, 577.38, 576

35、.9, 576.94, 576.24, 576.84, 576.85, 576.9, 577.79, 578.18, 577.51, 577.23, 578.42, 579.61, 579.05, 579.26, 579.22, 579.38, 579.1, 577.95, 578.12, 579.75, 580.85, 580.41, 579.96, 579.61, 578.76, 578.18, 577.21, 577.13, 579.1, 578.25, 577.91, 576.89, 575.96, 576.8, 577.68, 578.38, 578.52, 579.74, 579.

36、31, 579.89, 579.96)x0z=ts(x0,frequency=1,start=c(1875),end=c(1972)acf(z)pacf(z)x - arima(z,order=c(2,0,0)tsdiag(x)fore.modB；乙-D；丙-E；丁-A；C任務無人做； 最小耗時數為：z=29+20+32+24=105 源代碼：model:sets:i0/1.9/;links(i0,i0):x,a,c;endsetsdata:a=0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0