为什么我从 MILP 问题中得到非整数结果？

Question

我有一个用于饮食线性规划问题的 matlab 脚本。我正在尝试将 linearprog 或（intlinprog）的结果与工具箱 CPLEX 为 IBM 的 matlab 提供的 cplexlp（或 cplexmilp）函数的结果进行比较这是脚本

    %% Defining Variables 
clear;clc
Pnames = ["BEEF";
                    "CHK";
                    "FISH";
                    "HAM";
                    "MCH";
                    "MTL";
                    "SPG";
                    "TUR" ];
Packs = optimvar('Packs',Pnames,'Type','integer');
Packs.LowerBound = 2*ones(length(Pnames),1);
Packs.UpperBound = 10*ones(length(Pnames),1);
%% Setting the problem data
CostPerPack = [3.19;2.59;2.29;2.89;1.89;1.99;1.99;2.49];
VitA = [60;8;8;40;15;70;25;60];
VitC = [20;0;10;40;35;30;50;20];
VitB1 = [10;20;15;35;15;15;25;15];
VitB2 = [15;20;10;10;15;15;15;10];
NA = [938;2180;945;278;1182;896;1329;1397];
CAL = [295;770;440;430;315;400;370;450];
% Amount Per package table
TAMT = table(VitA,VitC,VitB1,VitB2,NA,CAL,...
    'VariableNames',{'A','C','B1','B2','NA','CAL'},...
    'RowNames',Pnames);
%% Objective 
TotCost = sum(CostPerPack .* Packs); % Objective 
obj = TotCost;
%% Constraints 
prob = optimproblem('Objective',obj,'ObjectiveSense','min');
prob.Constraints.c1 = sum(VitA.*Packs) >= 700;
prob.Constraints.c1a = sum(VitA.*Packs) <= 20000;
prob.Constraints.c2 = sum(VitC.*Packs) >= 700;
prob.Constraints.c2a = sum(VitC.*Packs) <= 20000;
prob.Constraints.c3 = sum(VitB1.*Packs) >= 700;
prob.Constraints.c3a = sum(VitB1.*Packs) <= 20000;
prob.Constraints.c4 = sum(VitB2.*Packs) >= 700;
prob.Constraints.c4a = sum(VitB2.*Packs) <= 20000;
prob.Constraints.c5 = sum(NA.*Packs) >= 0;
prob.Constraints.c5a = sum(NA.*Packs) <= 40000;
prob.Constraints.c5 = sum(CAL.*Packs) >= 16000;
prob.Constraints.c5a = sum(CAL.*Packs) <= 24000;
%% Putting the problem together and solving
problem = prob2struct(prob);
% LP
% [sol,fval,exitflag,output1] = linprog(problem);
% [sol2,fval2,exitflag2,output2] = cplexlp(problem);
% MILP problem
[sol,fval,exitflag,output1] = intlinprog(problem);
[sol2,fval2,exitflag2,output2] =cplexmilp(problem);
%% Display 
disp('Linear Prog function results')
if (~isempty(sol) )
T1 = table(sol,sol.*VitA,sol.*VitC,sol.*VitB1,sol.*VitB2,sol.*NA,sol.*CAL,sol.*CostPerPack,...
    'VariableNames',{'NuofPacks','PerVitA','PerVitC','PerVitB1','PerVitB2','NA','CAL','Cost'},...
    'RowNames',Pnames);
sumrow = array2table(sum(T1.Variables),...
    'VariableNames',...
    {'NuofPacks','PerVitA','PerVitC','PerVitB1','PerVitB2','NA','CAL','Cost'},...
    'RowNames',"Sum");
T1 = [T1;sumrow];
disp(T1)
else
    disp(['No feasible Solution with exit flag = ' ,num2str(exitflag)])
end
%% Display Cplex
disp('Cplex function results')
if (~isempty(sol2) )
T2 = table(sol2,sol2.*VitA,sol2.*VitC,sol2.*VitB1,sol2.*VitB2,sol2.*NA,sol2.*CAL,sol2.*CostPerPack,...
    'VariableNames',{'NuofPacks','PerVitA','PerVitC','PerVitB1','PerVitB2','NA','CAL','Cost'},...
    'RowNames',Pnames);
sumrow2 = array2table(sum(T2.Variables),...
    'VariableNames',...
    {'NuofPacks','PerVitA','PerVitC','PerVitB1','PerVitB2','NA','CAL','Cost'},...
    'RowNames',"Sum");
T2 = [T2;sumrow2];
disp(T2)
else
    disp(['No feasible Solution with exit flag = ' ,num2str(exitflag2)])
end

这里是 Cplex 函数结果

NuofPacks    PerVitA    PerVitC    PerVitB1    PerVitB2     NA       CAL       Cost 
        _________    _______    _______    ________    ________    _____    ______    ______

BEEF          2         120         40          20        30        1876       590      6.38
CHK          10          80          0         200       200       21800      7700      25.9
FISH          2          16         20          30        20        1890       880      4.58
HAM           2          80         80          70        20         556       860      5.78
MCH          10         150        350         150       150       11820      3150      18.9
MTL          10         700        300         150       150        8960      4000      19.9
SPG      7.3333      183.33     366.67      183.33       110        9746    2713.3    14.593
TUR           2         120         40          30        20        2794       900      4.98
Sum      45.333      1449.3     1196.7      833.33       700       59442     20793    101.01

来自 intlinprog 的这个 MILP 的结果似乎不错，但是从 cplexmilp 我得到了 SPG 包的非整数值。谁能帮我知道这里的问题？提前致谢

Answer 1

问题如下：当您调用 problem = prob2struct(prob); 时，来自 prob 的完整性信息被转移到 problem.intcon，但不幸的是它没有转移到 cplex。

这个问题以前出现过，但没有在 cplex 中直接解决，因为有一个简单的解决方法：替换调用

[sol2,fval2,exitflag2,output2] =cplexmilp(problem);

和

[sol2,fval2,exitflag2,output2] = ...
    cplexmilp(problem.f, problem.Aineq, problem.bineq, problem.Aeq, ...
              problem.beq, [], [], [], problem.lb, problem.ub, ...
              intcon_ctype(length(problem.lb),problem.intcon));

其中 intcon_ctype 函数是：

function ctype = intcon_ctype(numvars, intcon)
   ctype = char(ones(1, numvars) * 'C');
   for i = 1:length(intcon)
      ctype(intcon(i)) = 'I'
   end
end

请注意，这是一个非常简单的 intcon_ctype 转换函数。您可能需要检查 lb/ub 值并将类型设置为 'B' 如果变量是二进制，则可以指定半连续变量等

为什么我从 MILP 问题中得到非整数结果？

Why do I get non integer results from MILP problem?

matlab

cplex

mixed-integer-programming