MATLAB: Intlinprog function – problem

intlinprog code matlab

I want to solve a math problem that give me a minimal value of a function. The function of the objective is:
min 64x1+40x2
The invariants are: 
3x1+5x2≤150
8X1+5X2≤300
0X1+1X2≤20
I wrote the code in Matlab:
f=-[64;40]; intcon = [1;2]; A=[3,5;8,5;0,1]; B=[150;300;20];
[x,fval] = intlinprog(f,intcon,A,B,[],[],lb)
When I run this code, I get the results:
 x =
 30.0000
 12.0000
 
 fval =
  -2.4000e+03
And this results are ok, but when I changes the value of the objective function from 64 to 64.01, I get the strongly changed values of X:
  x =
  35.0000
  4.0000
  
  fval =
  -2.4003e+03
I would like to know why the value of x varies so much.

Best Answer

The optimal solution of a linear program can be discontinuous as a function of the problem data, particular when you have integer constraints. Here is an example where it is easier to understand how integer constraints can induce this behavior.
>> LB=[0,0]; x=intlinprog([1,1],1:2,[],[],[],[],LB).'
x =
     0     0
One sees here that the optimum is achieved at the boundaries specified by LB. Clearly therefore, if I increase the lower bounds by even a small amount, the optimal x(i) must jump to the next integer:
>> x=intlinprog([1,1],1:2,[],[],[],[],LB+0.00002).'
x =
     1     1
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