上海海事大学优化与建模模拟考查题答案.docx
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上海海事大学优化与建模模拟考查题答案.docx
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上海海事大学优化与建模模拟考查题答案
一.(共10分)线性规划:
1.分别用Lingo和Cplex求解该问题。
答:
最优解x=(0,0,0)目标函数值:
0
2.求对偶问题。
答:
对偶问题为:
MIN=12*y1+y2+y3;
y1+y2+y3>=-1;
y1+y2>=-2;
2*y1-y2>=-1;
3.解对偶问题,试验影子价格;
答:
最优解y=(0,0,0)目标函数值:
0
4.对目标函数系数,约束右边常量进行灵敏度分析。
Rangesinwhichthebasisisunchanged:
ObjectiveCoefficientRanges:
CurrentAllowableAllowable
VariableCoefficientIncreaseDecrease
X
(1)-1.0000001.000000INFINITY
X
(2)-2.0000002.000000INFINITY
X(3)-1.0000001.000000INFINITY
RighthandSideRanges:
CurrentAllowableAllowable
RowRHSIncreaseDecrease
212.00000INFINITY12.00000
31.000000INFINITY1.000000
源代码:
Cplex模型代码:
rangeii=1..3;
rangejj=1..3;
floatc[ii]=[-1,-2,-1];
floatb[jj]=[12,1,1];
floata[jj][ii]=[[1,1,2],
[1,1,-1],
[1,0,0]];
dvarfloat+x[ii];
maximize
sum(iinii)
c[i]*x[i];
subjectto{
forall(jinjj)
sum(iinii)
a[j][i]*x[i]<=b[j];
}
Lingo代码:
model:
sets:
ii/1..3/:
x,c;
jj/1..3/:
b;
link(jj,ii):
a;
endsets
data:
c=-1,-2,-1;
b=1211;
a=112
11-1
100;
enddata
max=@sum(ii(i):
x(i)*c(i));
@for(jj(j):
@sum(ii(i):
a(j,i)*x(i))<=b(j));
主要输出结果:
VariableValueReducedCost
X
(1)0.0000001.000000
X
(2)0.0000002.000000
X(3)0.0000001.000000
RowSlackorSurplusDualPrice
10.0000001.000000
212.000000.000000
31.0000000.000000
41.0000000.000000
解对偶问题源代码:
model:
sets:
ii/1..3/:
x,c;
jj/1..3/:
b;
link(jj,ii):
a;
endsets
data:
c=1211;
b=-1,-2,-1;
a=111
110
2-10;
enddata
min=@sum(ii(i):
x(i)*c(i));
@for(jj(j):
@sum(ii(i):
a(j,i)*x(i))>=b(j));
主要输出结果:
VariableValueReducedCost
X
(1)0.00000012.00000
X
(2)0.0000001.000000
X(3)0.0000001.000000
RowSlackorSurplusDualPrice
10.000000-1.000000
21.0000000.000000
32.0000000.000000
41.0000000.000000
二.(共15分).最短路:
1.求A1到A7所有最短路。
答:
最短路,共两条:
1--2--5--7
1--4--5--7
长度:
9
2.求A1到A6所有最短路。
答:
最短路共一条:
1--4--6
长度:
4
源代码:
model:
sets:
p/1..7/:
f;
!
links(p,p):
x;
r(p,p)/1213142425364345465776/:
d,x;
Endsets
data:
d=62331454121;
Enddata
!
f(6)=0;
!
@for(p(j)|j#Lt#6:
f(j)=@min(r(j,i):
d(j,i)+f(i)));!
inverseorder;
f
(1)=0;
@for(p(j)|j#gt#1:
f(j)=@min(r(i,j):
d(i,j)+f(i)));
@for(r(i,j):
x(i,j)=@if(f(j)#eq#d(i,j)+f(i),1,0));
end
主要输出结果:
X(1,2)1.000000
X(1,3)1.000000
X(1,4)1.000000
X(2,4)0.000000
X(2,5)1.000000
X(3,6)0.000000
X(4,3)0.000000
X(4,5)1.000000
X(4,6)1.000000
X(5,7)1.000000
X(7,6)0.000000
三.(共10分)生产计划:
答:
第一季度,正常生产40条,库存10条;
第二季度,正常生产50条,库存0条;
第三季度,正常生产75条,库存0条;
第四季度,正常生产25条,库存0条;
总费用:
78450
源代码:
model:
sets:
time/1..4/:
x,s,d,c;
endsets
data:
d=40607525;
enddata
min=@sum(time(i):
c(i)+s(i)*20);
@for(time(i):
c(i)=@if(x(i)#le#40,x(i)*400,40*400+(x(i)-40)*450));
x
(1)+10>=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(i-1)-d(i));
end
主要输出结果
VariableValueReducedCost
X
(1)40.000000.000000
X
(2)50.000000.000000
X(3)75.000000.000000
X(4)25.000000.000000
S
(1)10.000000.000000
S
(2)0.0000000.000000
S(3)0.0000000.000000
S(4)0.000000420.0000
D
(1)40.000000.000000
D
(2)60.000000.000000
D(3)75.000000.000000
D(4)25.000000.000000
C
(1)16000.000.000000
C
(2)20500.000.000000
C(3)31750.000.000000
C(4)10000.000.000000
RowSlackorSurplusDualPrice
178450.00-1.000000
20.000000-1.000000
30.000000-1.000000
40.000000-1.000000
50.000000-1.000000
610.000000.000000
70.000000-20.00000
80.000000-70.00000
90.0000000.000000
100.000000430.0000
110.000000430.0000
120.000000380.0000
130.000000400.0000
四.(共10分)统计描述性分析:
以下数据描述性分析:
求均值,方差,标准差,变异系数,偏度,峰度,常用分位数,极差,四分位差,直方图,箱式图,经验分布图,Q_Q图
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,-0.55,0.39,0.18,-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,-1.07,0.36,0.90,-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,-0.17,0.52,0.23,-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
答:
均值,方差,标准差,变异系数,偏差,峰度:
-0.01218543,0.4908892,0.7006348,-0.01739198,-0.1545023,2.526147
常用分位数:
0.1%0.5%1%2%5%10%50%
-1.89-1.89-1.53-1.34-1.16-1.040.08NA
四分位差:
0.96
极差:
3.52
直方图:
箱式图:
经验分布图:
Q-Q图:
源代码:
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,-0.55,0.39,0.18,-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,-1.07,0.36,0.90,-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,-0.17,0.52,0.23,-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)
var(x)
all.moments(x,central=TRUE,order.max=4)
all.moments(x,order.max=4)
all.moments(x,central=TRUE,order.max=4)
all.moments(x,absolute=TRUE,order.max=4)
skewness(x)
kurtosis(x)
quantile(x,probs=c(0.1,0.5,1,2,5,10,50,NA)/100,type=1)
y=ecdf(x)
plot(ecdf(x),verticals=TRUE,do.p=T)#do.p是逻辑变量=FALSE表示不画点处的记号
#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)
主要输出结果:
>mean(x)
[1]-0.01218543
>var(x)
[1]0.4908892
>all.moments(x,central=TRUE,order.max=4)
[1]1.000000e+003.906000e-184.876383e-01-5.261161e-026.006953e-01
>all.moments(x,order.max=4)
[1]1.00000000-0.012185430.48778675-0.070439670.60369418
>all.moments(x,central=TRUE,order.max=4)
[1]1.000000e+003.906000e-184.876383e-01-5.261161e-026.006953e-01
>all.moments(x,absolute=TRUE,order.max=4)
[1]1.00000000.57059600.48778680.51008740.6036942
>
>skewness(x)
[1]-0.1545023
>kurtosis(x)
[1]2.526147
>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%
-1.89-1.89-1.53-1.34-1.16-1.040.08NA
>y=ecdf(x)
>plot(ecdf(x),verticals=TRUE,do.p=T)#do.p是逻辑变量=FALSE表示不画点处的记号
>#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)回归分析:
data6指数增长模型非线性回归
1.写出回归表达式,获得回归的检验结论。
答:
data6未处理情况下:
y1=128.842(1-exp(-20.469x))
处理情况下:
y1=192.095(1-exp(-11.385x))
2.比较两种回归,哪一个效果好,为什么?
答:
未处理情况下:
指数回归两估计参数都是一个高度显著(***),一个是显著度(*),残差标准误26.26而Michaelis-Menten两估计参数也是是高度显著(***),一个是显著度(*),残差标准误25.69相比较。
Michaelis-Menten效果较佳。
处理情况下(略)
3.画出回归效果图像。
答:
见后面图像:
未处理情况:
处理情况:
源代码:
x=c(0.02,0.06,0.11,0.22,0.56,1.10);
y1=c(76,47,97,107,123,139,159,152,191,201,207,200);
y2=c(67,51,84,86,98,115,131,124,144,158,160);
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,y,pch=8);
summary(z);
x=c(0.02,0.02,0.06,0.06,0.11,0.11,0.22,0.22,0.56,0.56,1.10);
y=y2;
points(x,y,pch=9);#xlab="cc",ylab="dd",可设置坐标名称,而且可以中文
#未处理情况非线性回归
x=xx;
y=y2;
z1=nls(y~beta1*x/(beta2+x),start=list(beta1=195,beta2=0.05));#Michaelis-Menten模型
summary(z1);
z2=nls(y~beta1*(1-exp(-beta2*x)),start=list(beta1=195,beta2=1));#指数模型
summary(z2);
plot(x,y,pch=8);
points(x,fitted(z1),pch=9,col=2);
points(x,fitted(z2),pch=10,col=3);
#处理情况非线性回归
x=xx;
y=y1;
z1=nls(y~beta1*x/(beta2+x),start=list(beta1=195,beta2=0.05));#Michaelis-Menten模型
summary(z1);
z2=nls(y~beta1*(1-exp(-beta2*x)),start=list(beta1=195,beta2=1));#指数模型
summary(z2);
plot(x,y,pch=8);
points(x,fitted(z1),pch=9,col=2);
points(x,fitted(z2),pch=10,col=3);
主要输出结果
未处理情况
>summary(z1);
Formula:
y~beta1*x/(beta2+x)
Parameters:
EstimateStd.ErrortvaluePr(>|t|)
beta1137.4874713.1489510.4561.05e-06***
beta20.029450.014382.0480.0677.
---
Signif.codes:
0‘***’0.001‘**’0.01‘*’0.05‘.’0.1‘’1
Residualstandarderror:
25.69on10degreesoffreedom
Numberofiterationstoconvergence:
3
Achievedconvergencetolerance:
1.666e-06
>z2=nls(y~beta1*(1-exp(-beta2*x)),start=list(beta1=195,beta2=1));#指数模型
Therewere50ormorewarnings(usewarnings()toseethefirst50)
>summary(z2);
Formula:
y~beta1*(1-exp(-beta2*x))
Parameters:
EstimateStd.ErrortvaluePr(>|t|)
beta1128.84210.75011.9852.96e-07***
beta220.4696.9942.9270.0151*
---
Signif.codes:
0‘***’0.001‘**’0.01‘*’0.05‘.’0.1‘’1
Residualstandarderror:
26.26on10degreesoffreedom
Numberofiterationstoconvergence:
12
Achievedconvergencetolerance:
4.28e-06
处理情况
>summary(z1);
Formula:
y~beta1*x/(beta2+x)
Parameters:
EstimateStd.ErrortvaluePr(>|t|)
beta12.127e+026.947e+0030.6153.24e-11***
beta26.412e-028.281e-037.7431.57e-05***
---
Signif.codes:
0‘***’0.001‘**’0.01‘*’0.05‘.’0.1‘’1
Residualstandarderror:
10.93on10degreesoffreedom
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