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1、数学建模作业-第五章 非线性最小二乘问题 (1)最小二乘方法无约束问题为:minz=编写lingo程序:sets:quantity/1.15/:x,y;endsetsdata: x=2,5,7,10,14,19,26,31,34,38,45,52,53,60,65; y=54,50,45,37,35,25,20,16,18,13,8,11,8,4,6;enddatamin=sum(quantity:(a+b*exp(x*c)-y)2);free(a);free(b);free(c);结果:Local optimal solution found. Objective value: 44.780
2、49 Extended solver steps: 5 Total solver iterations: 68 Variable Value Reduced Cost A 2.430177 0.000000B 57.33209 0.000000 C -0.4460383E-01 0.000000(2)最小一乘法无约束问题为:min(a,b) z=编写lingo程序写出相应的LINGO程序如下:sets:quantity/1.15/:x,y;endsetsdata: x=2,5,7,10,14,19,26,31,34,38,45,52,53,60,65; y=54,50,45,37,35,25,
3、20,16,18,13,8,11,8,4,6;enddatamin=sum(quantity:abs(a+b*exp(x*c)-y);free(a);free(b);free(c);运行结果:Local optimal solution found. Objective value: 20.80640 Extended solver steps: 2 Total solver iterations: 643 Variable Value Reduced CostA 3.398267 0.000000B 57.11461 0.000000 C -0.4752126E-01 0.000000(3)
4、最大偏差最小的方法编写程序:sets:quantity/1.15/: x,y;endsetsdata: x = 2,5,7,10,14,19,26,31,34,38,45,52,53,60,65; y = 54,50,45,37,35,25,20,16,18,13,8,11,8,4,6;enddatamin=max(quantity: abs(a+b*exp(c*x)-y);free(a); free(b); free(c);结果:Linearization components added: Constraints: 91 Variables: 76 Integers: 30 Local o
5、ptimal solution found. Objective value: 2.774408 Extended solver steps: 38 Total solver iterations: 7654 Variable Value Reduced CostA 2.885594 0.000000B 55.86246 0.000000C -0.4441314E-01 0.000000(4)画出散点图和曲线图编写matlab程序如下: x=2 5 7 10 14 19 26 31 34 38 45 52 53 60 65;y=54 50 45 37 35 25 20 16 18 13 8 1
6、1 8 4 6;scatter(x,y,k*);hold onx=0:0.1:100;y=2.43+exp(-0.0446*x).*57.33;plot(x,y, g);xlabel(x);ylabel(y);hold onx=0:0.1:100;y=3.40+exp(-0.048*x).*57.11;plot(x,y, r);xlabel(x);ylabel(y);hold onx=0:0.1:100;y=2.89+exp(-0.044*x).*55.86;plot(x,y, b);xlabel(x);ylabel(y);得到图形:分析:三条曲线的结果大致相同,但是略有差异。最小二乘法当回归
7、残差满足正态分布时,它有很好的统计性质,但是他的回归性质不稳定。最小一乘法和最小二乘法近似,更靠近主流数据。最大偏差最小回归法因为受到最大平偏差的影响,曲线始终偏移。 非线性优化问题:(1)设汽油由桶A类原油和桶B类原油化合而成,民用燃料油由桶A类原油和桶B类原油化合而成,汽油广告费为元,民用燃料的广告费为元。由题意知道,汽油产量为桶,销量为桶;民用燃料油产量为桶,销量为桶。约束条件:销售约束:,原料约束:,指标约束:,目标函数:利润:编写lingo程序如下:max=0.5*z1*250+z2*200-z1-z2;x1+x2=5000;y1+y2=8*(x1+y1);10*x2+5*y2=6*
8、(x2+y2);0.5*z1=x1+y1;z2=x2+y2;得到: Global optimal solution found. Objective value: 3230000. Total solver iterations: 2 Variable Value Reduced Cost Z1 10000.00 0.000000 Z2 10000.00 0.000000 X1 3000.000 0.000000 X2 2000.000 0.000000 Y1 2000.000 0.000000 Y2 8000.000 0.000000 Row Slack or Surplus Dual Pr
9、ice1 3230000. 1.000000 2 0.000000 297.0000 3 0.000000 174.5000 4 0.000000 -24.50000 5 0.000000 -24.50000 6 0.000000 248.0000 7 0.000000 199.0000结果分析:汽油由3000桶A类原油和2000桶B类原油化合而成;民用燃料油产品由2000桶A类原油和8000桶B类原油化合而成;广告费分别为1000元;石油公司获得最大利润3230000元。(2)根据问题的需要,设汽油中增加SQ量,民用燃料油增加SQ量.根据题意,新的约束条件为:销售约束:,SQ约束:,原料约束
10、:,指标约束:,新的目标函数:利润:编写lingo程序如下:在LINGO中输入:max=0.5*z1*250+z2*200-z1-z2-200*(k1+k2);x1+x2=5000;y1+y2=(8-k10.5)*(x1+y1);10*x2+5*y2=(6-0.6*k20.6)*(x2+y2);0.5*z1=x1+y1+k1;z2=x2+y2+k2;k1=(x1+y1)*0.05;k2=(x2+y2)*0.05;得到: Local optimal solution found.Objective value: 3756000. Total solver iterations: 26 Variable Value Reduced Cost