How to get the optimum answer for this problem?

I solved the following problem using linprog and got optimum results,

     A= [-(1/0.07)     0         0        0        0     1; 
              0    -(1/0.45)     0        0        0     1; 
              0        0     -(1/0.37)    0        0     1;
              0        0         0    -(1/0.88)    0     1; 
              0        0         0        0    -(1/0.38) 1; 
              1        1         1        1        1     0];
    b=[0;0;0;0;0;30];
    lb=[0;0;0;0;0;0];
    ub=[30;30;30;30;30;30];
    f=-[0;0;0;0;0;1];
    [x_1]=linprog(f,A,b,[],[],lb,ub)

but for the want of integer results I’l have to solve it using genetic algorithm. I tried the following code,

   A= [-(1/0.07) 0 0 0 0 1; 0 -(1/0.45) 0 0 0 1; 0 0 -(1/0.37) 0 0 1;...
       0 0 0 -(1/0.88) 0 1; 0 0 0 0 -(1/0.38) 1; 1 1 1 1 1 0];
   b=[0;0;0;0;0;30];
   lb=[0;0;0;0;0;0];
   ub=[30;30;30;30;30;30];
   IntCon=[1 2 3 4 5];
   opts = gaoptimset('StallGenLimit',1000,'TolFun',1e-10,...
      'Generations',500,'PlotFcns',@gaplotbestf);
   [x] = ga(@data,6,A,b,[],[],lb,ub,[],IntCon,opts)

 (function declaration in a separate file)
 function scores = data(f)
 f=[0;0;0;0;0;1];
 scores = max(f);

but could not get an optimum result. Don’t know where I’ve gone wrong. Please help solving this one. Thank you

Answer 1

[Matt J]

Your fitness function data() always returns a constant value because you overwrite the input f with

f=[0;0;0;0;0;1];

Obviously, therefore, it cannot be meaningfully optimized.

In any case, you could solve the whole problem by exhaustive search. There are only 31465 combinations of integers between 0 and 30 that satisfy your constraint

 sum(x(1:5))<=30

[Matt J]

There are also other ways you could break down the problem. Note that once x(6) is specified, the remaining constraints reduce to simple bound constraints on x(1:5). The minimization problem then reduces to a Knapsack Problem

https://en.wikipedia.org/wiki/Knapsack_problem

which can be solved efficiently using dynamic programming. You could therefore loop over all possible values of x(6) and solve each sub-problem pretty quickly.

Though it wouldn't be a problem to loop over x(6) from 0 to 30, you can actually show that 0<=x(6)<=13. This is because your first five inequality constraints can be rewritten

 x(1)>=x(6)*0.07;
 x(2)>=x(6)*0.45;
 x(3)>=x(6)*0.37;
 x(4)>=x(6)*0.88;
 x(5)>=x(6)*0.38;

Summing these inequalities and combining with sum(x(1:5))<=30 leads to

 2.5 x(6) <= sum(x(1:5)) <=30

which cannot be satisfied for x(6)>13

[Nirmal]

Thank you for your suggestion, but the constraint should be,

 sum(x(1:5))=30

I have been directed to solve this using GA only, Can you please tell me what would be the fitness function to get the following results which I got using linprog, only for the first five rows to be integers while being run in GA.

 x_1 =

    0.9767
    6.2791
    5.1628
   12.2791
    5.3023
   13.9535

[Matt J]

Thank you for your suggestion, but the constraint should be, sum(x(1:5))=30

Then you should fix your posted code. It shows this as an inequality constraint. My remarks about exhaustive search remain unchanged, though, because I had equality in mind when I said there are 32465 combinations.

Can you please tell me what would be the fitness function to get the following results which I got using linprog,

 function scores = data(x)

 f=-[0;0;0;0;0;1];
 scores = dot(f,x);

[Nirmal]

Fantastic.... I have got the answer... You are a genius. Thank you very much