lingo软件求解线性规划及灵敏度分析注：以目标函数最大化为例进行讨论，对求最小的问题，有类似的分析方法！所有程序运行环境为lingo10。一、用lingo软件求解线性规划例1：max23..43103512,0zxystxyxyxy在模型窗口输入：model:max=2*x+3*y;4*x+3*y<=10;3*x+5*y<12;!theoptimalvalueis:7.454545;End如图所示：运行结果如下（点击工具栏上的‘solve’或点击菜单‘lingo’下的‘solve’即可）：Globaloptimalsolutionfound.Objectivevalue:7.454545（最优解函数值）Infeasibilities:0.000000Totalsolveriterations:2（迭代次数）Variable（最优解）ValueReducedCostX1.2727270.000000Y1.6363640.000000RowSlackorSurplusDualPrice17.4545451.00000020.0000000.9090909E-0130.0000000.5454545例2：12123124125max54..390280450zxxstxxxxxxxxxx在模型窗口输入：model:max=5*x1+4*x2;x1+3*x2+x3=90;2*x1+x2+x4=80;x1+x2+x5=45;end运行（solve）结果如下：Globaloptimalsolutionfound.Objectivevalue:215.0000Infeasibilities:0.000000Totalsolveriterations:3VariableValueReducedCostX135.000000.000000X210.000000.000000X325.000000.000000X40.0000001.000000X50.0000003.000000RowSlackorSurplusDualPrice1215.00001.00000020.0000000.00000030.0000001.00000040.0000003.000000例323123234235min2..223120zxxstxxxxxxxxxx在模型窗口输入：model:min=-x2+2*x3;x1-2*x2+x3=2;x2-3*x3+x4=1;x2-x3+x5=2;end运行结果如下：Globaloptimalsolutionfound.Objectivevalue:-1.500000Infeasibilities:0.000000Totalsolveriterations:2VariableValueReducedCostX22.5000000.000000X30.50000000.000000X16.5000000.000000X40.0000000.5000000X50.0000000.5000000RowSlackorSurplusDualPrice1-1.500000-1.00000020.0000000.00000030.0000000.500000040.0000000.5000000例4：（非线性）min..124xyzstxyxz在模型窗口输入：model:min=@abs(x)+@abs(y)+@abs(z);x+y<=1;2*x+z=4;@free(x);@free(y);@free(z);End求解器状态如下：（可看出是非线性模型！）运行结果为：Linearizationcomponentsadded:Constraints:12Variables:12Integers:3Globaloptimalsolutionfound.Objectivevalue:（最优解函数值）3.000000Objectivebound:3.000000Infeasibilities:0.000000Extendedsolversteps:0Totalsolveriterations:3Variable(最优解)ValueReducedCostX2.0000000.000000Y-1.0000000.000000Z0.0000000.000000RowSlackorSurplusDualPrice13.000000-1.00000020.0000001.00000030.000000-1.000000二、用lingo软件进行灵敏度分析实例例5：max6030208648421.52021.50.585,,0Sxyzxyzxyzxyzyxyz在模型窗口输入：Lingo模型：model:max=60*x+30*y+20*z;8*x+6*y+z<48;4*x+2*y+1.5*z<20;2*x+1.5*y+0.5*z<8;y<5;end（一）求解报告（solutionreport）通过菜单Lingo→Solve可以得到求解报告（solutionreport）如下：Globaloptimalsolutionfoundatiteration:0Infeasibilities:0.000000Objectivevalue:280.0000Totalsolveriterations:2VariableValueReducedCostX2.0000000.000000Y0.0000005.000000Z8.0000000.000000RowSlackorSurplusDualPrice1280.00001.000000224.000000.00000030.00000010.0000040.00000010.0000055.0000000.000000分析Value,ReducedCost，SlackorSurplus，DualPrice的意义如下：1、最优解和基变量的确定Value所在列给出了问题的最优解。由于基变量取值非零，因此Value所在列取值非零的决策变量x,z是基变量。2、差额成本ReducedCost（或opportunitycost）所在列的三个数值表示当决策变量取值增加一个单位时，目标函数值的减少量。例如：第2个数5表示当变量y增加一个单位时，最优目标函数值减少的量。例如：当y=1时，最优目标函数值为280-5=275。可通过如下模型可检验：model:max=60*x+30*y+20*z;8*x+6*y+z<48;4*x+2*y+1.5*z<20;2*x+1.5*y+0.5*z<8;y<5;y=1;end注：（1）换一个角度说，就是目标函数中变量y的系数至少增加5，那么生产y才会有利！（2）基变量的ReducedCost值为0，只有非基变量的ReducedCost值才可能...
