I'm a little bit confused about a result that I got after a coefficients reduction on a constraint of a linear programming problem.
The problem is:
maximize z = x1 + x2 + x3 + x4 + x5 + x6
subject to: 6*x1 + 3*x2 - 5*x3 + 2*x4 + 7*x5 - 4*x6 <= 15
where:
1<=x1<=2 continuos
1<=x2<=2 continuos
1<=x3<=2 continuos
1<=x4<=2 continuos
1<=x5<=2 continuos
1<=x6<=2 continuos
After the coefficients reduction the contraints will be:
subject to: 3*x1 + 3*x2 - 3*x3 + 2*x4 + 3*x5 - 3*x6 <= 8
as stated in the Applied Integer Programming book (Der-San Chen - Robert G.Batson - Yu Dang) at page 96 (there is a little error at page 97. The x1 coefficient is 3 not 1).
After that I've tried to submit the problem to ampl with and without the coefficients reduction. But I got two different results:
[without coefficients reduction]
CPLEX 12.6.1.0: optimal integer solution; objective 11.57142857
display x;
x1 2
x2 2
x3 2
x4 2
x5 1.57
x6 2
[with coefficients reduction]
CPLEX 12.6.1.0: optimal integer solution; objective 11.33333333
display x;
x1 2
x2 2
x3 2
x4 2
x5 1.33
x6 2
why? can the solution be considered correct anyway even if the result for x5 is a little different? I've used three different solver (minos, gurobi, cplex) but they output the same results on the problem.
If you are referring to the technique in 4.4.3, then it's clear what's the problem here.
Suppose we are given a constraint of the form
a1*y1+ a2*y2 + ... + ai*yi < b
where yi = 0 or 1
You are not allowed to use this technique, as your coefficients are continuous ( in [1,2]) and not binary as needed here!