Math Is Fun Forum

Could anyone help me with the following problem? I am not familar with it, this problem is formulated   in my current project, so I am not sure if it is solvable or come up with any reasonable solution. Thank you in advance.

Contraints:
x1 + x2 + x3 = a
y1 + y2 + y3 = b
x1 <= m1, x2<= m2, x3 <= m3
y1 <= n1, y2<=n2, y3<=n3
a < m1+m2+m3
b < n1+n2+n3
x1, x2, x3, y1, y2, y3 positive

Objective function:
f(x1, x2, x3, y1, y2, y3) = n1*m2*m3 + n2*m1*m3 +
n3*m1*m2 - y1*x2*x3 - y2*x1*x3 - y3*x1*x2

Question: Is there any lower bound for the function f?
I.e., does there exist any E>0 such that f >= E for any
xi, yi satisfying the constraints. If yes, what is E?