数学建模问题分析

问题分析：
我们的目标是通过决策好向不同的证卷分别投入多少的资金来实现税后收益最大化
限制条件有投资资金，必须购买的证券，平均信用等级，平均到期年限，以及其他证卷的50%的税率
虽然市政不收税并且相比税后的其他证券有很大的收益，但是有限制条件其他证券必须至少购进400万，那除了这400万是不是600万都买市政呢？但是市政的信用等级都高于1.4，所以必须还得在其他证券和市政上取得平衡，虽然其他证券中代办机构收入最高，但是到期年限也是最高的，所有在平均到期年限5年上也要做平衡
文字模型：
(1)
决策变量：
用x1,x2,x3,x4,x5分别表示A,B,C,D,E五中证券的购买量
决策目标：
税后收益为税前收益*税率
每种证券的购买量与到期税后收益的乘积的和就是总收益Z
max z = 0.043*x1 + 0.054*0.5*x2 + 0.050*0.5*x3 + 0.044*0.5*x4 + 0.045*x5;
约束条件：
金钱约束：总的投资资金只有1000万,得
x1+x2+x3+x4+x5<=1000;
信用约束：平均信用等级不超过1.4，将信用进行购买量加运算再取平均值
2*x1 + 2*x2 + 1*x3 + 1*x4 + 5*x5 < 1.4*(x1 + x2 + x3 + x4 + x5);
到期年限约束：平均到期年限不超过五年，同理加权求平均值
9*x1 + 15*x2 + 4*x3 + 3*x4 + 2*x5 < 5*(x1 + x2 + x3 + x4 + x5);
必须购买的证券限制：必须购买政府及代办机构至少购进400万
x2 + x3 + x4 > 400;
(2)
设借k万元资金
决策目标：
在原来的基础上减去k万的利率
max z = 0.043*x1 + 0.054*0.5*x2 + 0.050*0.5*x3 + 0.044*0.5*x4 + 0.045*x5 – 0.0275*k; 金钱约束：总的投资资金只有1000+k万,得
x1+x2+x3+x4+x5<=1000+k;
借钱约束：不超过100万
k<=100;
(3)
1. 只需要把决目标中相应的收益改为：
max z = 0.045*x1 + 0.054*0.5*x2 + 0.050*0.5*x3 + 0.044*0.5*x4 + 0.045*x5 ;
2. 只需要把决目标中相应的收益改为：
max z = 0.043*x1 + 0.054*0.5*x2 + 0.048*0.5*x3 + 0.044*0.5*x4 + 0.045*x5;
数学模型：
(1)
max = 0.043*x1 + 0.054*0.5*x2 + 0.050*0.5*x3 + 0.044*0.5*x4 + 0.045*x5;
x1 + x2 + x3 + x4 + x5 < 1000;
x2 + x3 + x4 > 400;
2*x1 + 2*x2 + 1*x3 + 1*x4 + 5*x5 < 1.4*(x1 + x2 + x3 + x4 + x5);
9*x1 + 15*x2 + 4*x3 + 3*x4 + 2*x5 < 5*(x1 + x2 + x3 + x4 + x5);
(2)
max = 0.043*x1 + 0.054*0.5*x2 + 0.050*0.5*x3 + 0.044*0.5*x4 + 0.045*x5 – 0.0275*k; x1 + x2 + x3 + x4 + x5 < 1000 + k;
k < 100;
x2 + x3 + x4 > 400;
2*x1 + 2*x2 + 1*x3 + 1*x4 + 5*x5 < 1.4*(x1 + x2 + x3 + x4 + x5);
9*x1 + 15*x2 + 4*x3 + 3*x4 + 2*x5 < 5*(x1 + x2 + x3 + x4 + x5);
(3)
1.
max = 0.045*x1 + 0.054*0.5*x2 + 0.050*0.5*x3 + 0.044*0.5*x4 + 0.045*x5;
x1 + x2 + x3 + x4 + x5 < 1000;
x2 + x3 + x4 > 400;
2*x1 + 2*x2 + 1*x3 + 1*x4 + 5*x5 < 1.4*(x1 + x2 + x3 + x4 + x5);
9*x1 + 15*x2 + 4*x3 + 3*x4 + 2*x5 < 5*(x1 + x2 + x3 + x4 + x5);
2.
max = 0.043*x1 + 0.054*0.5*x2 + 0.048*0.5*x3 + 0.044*0.5*x4 + 0.045*x5;
x1 + x2 + x3 + x4 + x5 < 1000;
x2 + x3 + x4 > 400;
2*x1 + 2*x2 + 1*x3 + 1*x4 + 5*x5 < 1.4*(x1 + x2 + x3 + x4 + x5);
9*x1 + 15*x2 + 4*x3 + 3*x4 + 2*x5 < 5*(x1 + x2 + x3 + x4 + x5);
求解结果：
(1)
Global optimal solution found.
Objective value: 29.83636
Infeasibilities: 0.000000
Total solver iterations: 3
Variable Value Reduced Cost
X1 218.1818 0.000000
X2 0.000000 0.3018182E-01
X3 736.3636 0.000000
X4 0.000000 0.6363636E-03
X5 45.45455 0.000000
Row Slack or Surplus Dual Price
1 29.83636 1.000000
2 0.000000 0.2983636E-01
3 336.3636 0.000000
4 0.000000 0.6181818E-02
5 0.000000 0.2363636E-02 (2)
Global optimal solution found.
Objective value: 30.07000
Infeasibilities: 0.000000
Total solver iterations: 3
Variable Value Reduced Cost
X1 240.0000 0.000000
X2 0.000000 0.3018182E-01 X3 810.0000 0.000000
X4 0.000000 0.6363636E-03 X5 50.00000 0.000000
K 100.0000 0.000000
Row Slack or Surplus Dual Price
1 30.07000 1.000000
2 0.000000 0.2983636E-01
3 0.000000 0.2336364E-02
4 410.0000 0.000000
5 0.000000 0.6181818E-02
6 0.000000 0.2363636E-02 (3)
1.
Global optimal solution found.
Objective value: 30.27273
Infeasibilities: 0.000000
Total solver iterations: 3
Variable Value Reduced Cost
X1 218.1818 0.000000
X2 0.000000 0.3436364E-01 X3 736.3636 0.000000
X4 0.000000 0.2727273E-03
X5 45.45455 0.000000
Row Slack or Surplus Dual Price
1 30.27273 1.000000
2 0.000000 0.3027273E-01
3 336.3636 0.000000
4 0.000000 0.6363636E-02
5 0.000000 0.2727273E-02
2.
Global optimal solution found.
Objective value: 29.42400
Infeasibilities: 0.000000
Total solver iterations: 3
Variable Value Reduced Cost
X1 336.0000 0.000000
X2 0.000000 0.3064000E-01
X3 0.000000 0.4400000E-03
X4 648.0000 0.000000
X5 16.00000 0.000000
Row Slack or Surplus Dual Price
1 29.42400 1.000000
2 0.000000 0.2942400E-01
3 248.0000 0.000000
4 0.000000 0.6360000E-02
5 0.000000 0.2440000E-02
结果分析：
x1~x5分别对应A~E的投资资金
row1为收入（万元）
(1)中A~E的投资分别为218.1818，0，736.3636，0，45.45455，可以收益29.83636 附件：。