Below I need solve for the binary variables $x_1,x_2,y_1,y_2,z_1,z_2$ that minimize the functions $f(x), f(y), f(z)$, subject to the 5 constraints that follow. By binary I mean they can only be 1 or 0. [Edit: $u_1,u_2,h$ are non-negative real valued. The functions to be minimized must also be non-negative. This is a much reduced version of a big workshift scheduling problem.]
I appreciate any advice as to what sort of strategy I might use. Thanks.
$x_1u_{1}+x_2u_{2}-h=f(x)$
$y_1u_{1}+y_2u_{2}-h=f(y)$
$z_1u_{1}+z_2u_{2}-h=f(z)$
$s.t.$
$x_1+x_2=1$
$y_1+y_2=1$
$z_1+z_2=1$
$x_1+y_1+z_1=1$
$x_2+y_2+z_2=1$
Assuming that $u_1,u_2,h \gt 0$
WLOG set $x_1=x_2=0$, $y_1=1$, $y_2=0$, $z_1=0$, and $z_2=1$.
This gives $f(x)=-h$, $f(y)=u_1-h$, $f(z)=u_2-h$
However, there are 5 other solutions that are equivalent to this one i.e. 3 choices for a variable to have both of its terms set to 0 and 2 choices of which remaining variable has its sub 1 equal to 1.
These 6 solutions appear to be the only ones that satisfy the constraints and for fixed $u_1,u_2,h$ they are all the same.