  We consider ILPs, where each variable corresponds to an integral point within
a polytope $\mathcal{P}$, i. e., ILPs of the form $\min\{c^{\top}x\mid
\sum_{p\in\mathcal P\cap \mathbb Z^d} x_p p = b, x\in\mathbb Z^{|\mathcal P\cap
\mathbb Z^d|}_{\ge 0}\}$.
  The distance between an optimal fractional solution and an optimal integral
solution (called proximity) is an important measure. A classical result by Cook
et al.~(Math. Program., 1986) shows that it is at most $\Delta^{\Theta(d)}$
where $\Delta$ is the largest coefficient in the constraint matrix.
  Another important measure studies the change in an optimal solution if the
right-hand side $b$ is replaced by another right-hand side $b'$. The distance
between an optimal solution $x$ w.r.t.~$b$ and an optimal solution $x'$
w.r.t.~$b'$ (called sensitivity) is similarly bounded, i. e., $\lVert b-b'
\rVert_{1}\cdot \Delta^{\Theta(d)}$, also shown by Cook et al.
  Even after more than thirty years, these bounds are essentially the best
known bounds for these measures.
  While some lower bounds are known for these measures, they either only work
for very small values of $\Delta$, require negative entries in the constraint
matrix, or have fractional right-hand sides.
  Hence, these lower bounds often do not correspond to instances from
algorithmic problems.
  This work presents for each $\Delta > 0$ and each $d > 0$ ILPs of the above
type with non-negative constraint matrices such that their proximity and
sensitivity is at least $\Delta^{\Theta(d)}$.
  Furthermore, these instances are closely related to instances of the Bin
Packing problem as they form a subset of columns of the configuration ILP.
  We thereby show that the results of Cook et al. are indeed tight, even for
instances arising naturally from problems in combinatorial optimization.
