"Anna" wrote in message <email@example.com>... > I was just thinking that all these 3 constraints can be written as one > > x_ik+x_jk>=2*z_ijk, right? >
This condition does not exclude the case that z_ijk=0 although both x_ik and x_jk are equal to 1. So z_ijk is not equal to x_ik*x_jk. It depends on your objective function if z_ijk will automatically be chosen to be 1 if x_ik=1 and x_jk = 1.
> "Torsten" wrote in message <firstname.lastname@example.org>... > > Because you substitute x_ik*x_jk = min(x_ik,x_jk) , I think you will need z_ijk: > > z_ijk <= x_ik > > z_ijk <= x_jk > > z_ijk >= x_ik + x_jk - 1 > > Good point Torsten. Theoretically, I would need to index z over k but from other constraints I also know that there may be at most one k for which z_ijk=1 and I don't need to know the detailed information which k it is, but only the indication whether it happens at all. Would indexing z_ij be correct in that case? It's important to me to limit the size of the variables as well.
I don't think that this is possible. Imagine there are indices k1 and k2 such that, for fixed i and j, x_ik1*x_jk1 = 1 and x_ik2*x_jk2 = 0. Then, if you only introduce z_ij, your problem becomes infeasible. To save some variables, note that x_ik*x_jk = x_jk*x_ik. Thus z_ijk = z_jik.