  The paper describes an algorithm that, given an initial design
$\mathcal{F}_n$ of size $n$ and a linear model with $p$ parameters, provides a
sequence $\mathcal{F}_n \supset \ldots \supset \mathcal{F}_{n-k} \supset \ldots
\supset \mathcal{F}_p$ of nested \emph{robust} designs. The sequence is
obtained by the removal, one by one, of the runs of $\mathcal{F}_n$ till a
$p$-run \emph{saturated} design $\mathcal{F}_p$ is obtained. The potential
impact of the algorithm on real applications is high. The initial fraction
$\mathcal{F}_n$ can be of any type and the output sequence can be used to
organize the experimental activity. The experiments can start with the runs
corresponding to $\mathcal{F}_p$ and continue adding one run after the other
(from $\mathcal{F}_{n-k}$ to $\mathcal{F}_{n-k+1}$) till the initial design
$\mathcal{F}_n$ is obtained. In this way, if for some unexpected reasons the
experimental activity must be stopped before the end when only $n-k$ runs are
completed, the corresponding $\mathcal{F}_{n-k}$ has a high value of robustness
for $k \in \{1, \ldots, n-p\}$. The algorithm uses the circuit basis, a special
representation of the kernel of a matrix with integer entries. The
effectiveness of the algorithm is demonstrated through the use of simulations.
