数学模型实验三-线规划问题.doc

3 线性规划实验3.1 基本实验3.1.1生产计划安排（1）确定获利最大的生产方案；max=3*x1+x2+4*x3;6*x1+3*x2+5*x345;3*x1+4*x2+5*x330;gin(x1);gin(x2);gin(x3);求解得到输出：Global optimal solution found. Objective value: 27.00000 Objective bound: 27.00000 Infeasibilities: 0.000000 Extended solver steps: 0 Total solver iterations: 0 Variable Value Reduced Cost X1 5.000000 -3.000000 X2 0.000000 -1.000000 X3 3.000000 -4.000000 Row Slack or Surplus Dual Price 1 27.00000 1.000000 2 0.000000 0.000000 3 0.000000 0.000000结论：最佳的生产方案为A产品5件，不生产B产品，C产品3件，获利27元。（2）A、B、C产品的利润变化范围； Ranges in which the basis is unchanged: Objective Coefficient Ranges Current Allowable Allowable Variable Coefficient Increase Decrease X1 3.000000 1.800000 0.6000000 X2 1.000000 2.000000 INFINITY X3 4.000000 1.000000 1.500000 Righthand Side Ranges Row Current Allowable Allowable RHS Increase Decrease 2 45.00000 15.00000 15.00000 3 30.00000 15.00000 7.500000结论：A产品例润变化范围为（3-0.6,3+1.8）；B产品例润变化范围为（0,1+2）；C产品例润变化范围为（4-1.5,4+1）。（3）购买原料多少为宜；max=3*x1+x2+4*x3-0.4*z;6*x1+3*x2+5*x345;3*x1+4*x2+5*x3-z30;gin(x1);gin(x2);gin(x3);gin(z);求解得到输出：Global optimal solution found. Objective value: 30.00000 Objective bound: 30.00000 Infeasibilities: 0.000000 Extended solver steps: 0 Total solver iterations: 3 Variable Value Reduced Cost X1 0.000000 -3.000000 X2 0.000000 -1.000000 X3 9.000000 -4.000000 Z 15.00000 0.4000000 Row Slack or Surplus Dual Price 1 30.00000 1.000000 2 0.000000 0.000000 3 0.000000 0.000000结论：购买15份材料为宜。（4）D产品是否值得生产；max=3*x1+x2+4*x3+3*x4;6*x1+3*x2+5*x3+8*x445;3*x1+4*x2+5*x3+2*x430;gin(x1);gin(x2);gin(x3);gin(x4);求解得到输出： Global optimal solution found. Objective value: 27.00000 Objective bound: 27.00000 Infeasibilities: 0.000000 Extended solver steps: 0 Total solver iterations: 0 Variable Value Reduced Cost X1 5.000000 -3.000000 X2 0.000000 -1.000000 X3 3.000000 -4.000000 X4 0.000000 -3.000000 Row Slack or Surplus Dual Price 1 27.00000 1.000000 2 0.000000 0.000000 3 0.000000 0.000000结论：D产品不值得生产。3.1.3 工程进度安排max=z1+z2+z3+z4;a1+a2+a3=1;b1+b2+b3+b4=1;c1+c2+c3+c4+c5=1;d1+d2=1;5*a1+15*c13;5*a2+8*b1+15*c26;5*a3+8*b2+15*c3+1.2*d17;8*b3+15*c4+1.2*d27;8*b4+15*c57;z1=50*a1+50*(a1+a2)+50*2;z2=70*b1+70*(b1+b2)+70*(b1+b2+b3);z3=150*c1+150*(c1+c2)+150*(c1+c2+c3)+150*(c1+c2+c3+c4);z4=20*d1;求解得到输出：Global optimal solution found. Objective value: 503.7500 Infeasibilities: 0.000000 Total solver iterations: 9 Variable Value Reduced Cost Z1 180.0000 0.000000 Z2 15.75000 0.000000 Z3 288.0000 0.000000 Z4 20.00000 0.000000 A1 0.6000000 0.000000 A2 0.4000000 0.000000 A3 0.000000 0.000000 B1 0.000000 20.00000 B2 0.000000 10.00000 B3 0.2250000 0.000000 B4 0.7750000 0.000000 C1 0.000000 0.000000 C2 0.2666667 0.000000 C3 0.3866667 0.000000 C4 0.3466667 0.000000 C5 0.000000 18.75000 D1 1.000000 0.000000 D2 0.000000 8.000000 Row Slack or Surplus Dual Price 1 503.7500 1.000000 2 0.000000 -93.75000 3 0.000000 0.000000 4 0.000000 18.75000 5 0.000000 -2.500000 6 0.000000 38.75000 7 0.000000 28.75000 8 0.000000 18.75000 9 0.000000 8.750000 10 0.8000000 0.000000 11 0.000000 1.000000 12 0.000000 1.000000 13 0.000000 1.000000 14 0.000000 1.000000结论：总收入503.75（千万元）每年完成的比例：A1 0.6000000A2 0.4000000A3 0.000000B1 0.000000B2 0.000000B3 0.2250000B4 0.7750000C1 0.000000C2 0.2666667C3 0.3866667C4 0.3466667C5 0.000000D1 1.000000D2 0.0000003.1.3投资问题max =0.2*(X1A+ X2A + X3A)+0.5*X1B+0.6* X2C+0.4* X3D; X1A+X1B=30; X2A+X2C=1.2*X1A; X3A+X3B+X3D=1.2*X2A+1.5*X1B; bnd(0,X1B,20); bnd(0,X3B,20); bnd(0,X2C,15); bnd(0,X3D,10); 求解得到输出：Global optimal solution found. Objective value: 27.50000 Total solver iterations: 2 Variable Value Reduced Cost X1A 12.50000 0.000000 X2A 0.000000 0.6000000E-01 X3A 16.25000 0.000000 X1B 17.50000 0.000000 X2C 15.00000 -0.1000000 X3D 10.00000 -0.2000000 X3B 0.000000 0.2000000 Row Slack or Surplus Dual Price 1 27.50000 1.000000 2 0.000000 0.8000000 3 0.000000 0.5000000 4 0.000000 0.2000000 结论：第一年年初 投资A项目12.5万元，投资B项目17.5万元； 第二年年初 投资C项目15万元； 第三年年初 投资A项目16.25万元，投资D项目10万元； 第三年年年末可获最大收益27.5万元。3.1.4 生产计划与库存问题min=50*a1+60*b1+55*c1+75*a2+90*b2+80*c2+2*(a2-500)+1*(a1-a2)+2*(a2+b2-950)+1*(a1+b1-a2-b2);a1+b1+c1=1550;a2+b2+c2=1550;a2=a1;a2+b2=500;a2=500;a1+b1=950;a2+b2=950;0.6*a1=800;0.6*b1=700;0.6*c1=550;0.8*a2=1000;0.8*b2=850;0.8*c2=4; X8+ X9=4; X8+ X9+ X10=6; X9+ X10+ X11=6; X10+ X11+ X12=8;X11+ X12+ X13=8; X12+ X13+ X14=6; X13+ X14+ X15=6; X14+ X15+ X16=4; X15+ X16+ X17=4;X16+ X17+ X18=6; X17+ X18+ X19=6; X18+ X19+ X20=8; X19+ X20+ X21=8; 运行LINGO软件得到问题的最优解（只列出非零变量）： 最优目标函数值=32.00000X8=4.000000 X10=4.000000 X11=2.000000 X12=2.000000 X13=4.000000 X15=2.000000 X16=2.000000 X17=4.000000 X19=2.000000 X20=6.000000 根据运行结果，最优时间表确定如下，此时最少人数为32人时间8：009：0010：0011：0012：0013：0014：0015：0016：0017：0018：0019：0020：0021：00开始工作人数40422402240260需要人数44668866446688（2）没有志愿者愿意在12：00和18：00开始工作，即增加约束条件：X12=0； X18=0。LINGO Model如下：min= X8 +X9+ X10+ X11+ X12+ X13+ X14+ X15+ X16+ X17+ X18+ X19+ X20+ X21;X8=4; X8+ X9=4; X8+ X9+ X10=6; X9+ X10+ X11=6; X10+ X11+ X12=8;X11+ X12+ X13=8; X12+ X13+ X14=6; X13+ X14+ X15=6; X14+ X15+ X16=4; X15+ X16+ X17=4;X16+ X17+ X18=6; X17+ X18+ X19=6; X18+ X19+ X20=8; X19+ X20+ X21=8; X12=0; X18=0;运行LINGO软件得到问题的最优解（只列出非零变量）： 最优目标函数值=32.00000X8=4.000000 X10=6.000000 X11=2.000000 X13=6.000000 X16=4.000000 X17=2.000000 X19=4.000000 X20=4.000000 根据运行结果，最优时间表确定如下，此时最少人数为32人时间8：009：0010：0011：0012：0013：0014：0015：0016：0017：0018：0019：0020：0021：00开始工作人数40620600420440需要人数446688664466883.1.6 经济均衡问题（1）单一生产者，单一消费者min=15*x1+8*x2+5*x3+3*x4-2*A1-4*A2-6*A3-8*A4;A1+A2+A3+A4=x1+x2+x3+x4;A1=2;A2=2;A3=2;A4=2;x1=2;x2=2;x3=2;x4=0;A2=0;A3=0;A4=0;x1=0;x2=0;x3=0;x4=0;计算输出结果：Global optimal solution found. Objective value: -3.000000 Infeasibilities: 0.000000 Total solver iterations: 1 Variable Value Reduced Cost X1 0.000000 6.000000 X2 0.000000 1.500000 X3 2.000000 0.000000 X4 2.000000 0.000000 A1 0.000000 2.000000 A2 0.000000 1.000000 A3 2.000000 0.000000 A4 2.000000 0.000000 Row Slack or Surplus Dual Price 1 -3.000000 -1.000000 2 0.000000 3.000000 3 2.000000 0.000000 4 2.000000 0.000000 5 0.000000 0.000000 6 0.000000 1.000000 7 2.000000 0.000000 8 2.000000 0.000000 9 0.000000 0.000000 10 0.000000 0.5000000 11 0.000000 0.000000 12 0.000000 0.000000 13 2.000000 0.000000 14 2.000000 0.000000 15 0.000000 0.000000 16 0.000000 0.000000 17 2.000000 0.000000 18 2.000000 0.000000结论：A1=A2=x1=x2=0;A3=A4=x3=x4;清算价格为3万元。（2）两个生产者，两个消费者。min=15*x1+8*x2+5*x3+3*x4-2*A1-4*A2-6*A3-8*A4+9*y1+4.5*y2+3*y3+2.25*y4-1*B1-2*B2-3*B3-4*B4-0.6*BX-0.6*BY;AX+AY=A1+A2+A3+A4;BX+BY=B1+B2+B3+B4;AX+BX=x1+x2+x3+x4;AY+BY=y1+y2+y3+y4;A1=2;A2=2;A3=2;A4=2;B1=2;B2=2;B3=2;B4=2;x1=2;x2=2;x3=2;x4=2;y1=2;y2=2;y3=2;y4=0;A2=0;A3=0;A4=0;x1=0;x2=0;x3=0;x4=0;B1=0;B2=0;B3=0;B4=0;y1=0;y2=0;y3=0;y4=0;计算结果输出：Global optimal solution found. Objective value: -20.700000 Infeasibilities: 0.000000 Total solver iterations: 4 Variable Value Reduced Cost X1 0.000000 11.00000 X2 0.000000 4.000000 X3 0.000000 1.000000 X4 2.000000 0.000000 A1 0.000000 2.000000 A2 0.000000 0.000000 A3 2.000000 0.000000 A4 2.000000 0.000000 Y1 0.000000 5.000000 Y2 0.000000 0.5000000 Y3 2.000000 0.000000 Y4 2.000000 0.000000 B1 0.000000 2.400000 B2 0.000000 1.400000 B3 0.000000 0.4000000 B4 2.000000 0.000000