When designing structures (such as buildings, aerospace structures, etc.), it is important to estimate the displacements $x_j$ of different points of the structure that are caused by different external forces $f_i$ (these forces are also called loads). If the displacements are too high, the structure should be re-designed.
The displacements $x_j$ caused by the external forces $f_i$ can be determined from a system of linear equations $\sum_j K_{ij}x_j=f_i$; the coefficients $K_{ij}$ are called stiffnesses.
The exact values of loads are usually not known a priori; the probabilities of different loads are also usually not known. Normally, the only information that we have about each force $f_i$ is the interval of possible values of $f_i$. The stiffnesses $K_{ij}$ are also not known precisely. Due to the manufacturing and building tolerances, we only know the intervals of possible values of $K_{ij}$. Based on this interval information, we must find the intervals of possible values of $x_j$. In mathematical terms, we must solve a system of interval linear equations.
In principle, one can assume that the uncertainties $\Delta K_{ij}$ and $\Delta f_i$ in $K_{ij}$ and $f_i$ are small, neglect the terms that are quadratic in these uncertainties, and get a linearized formula for the dependency of $x_j$ on $\Delta K_{ij}$ and $\Delta f_i$. However, by neglecting quadratic terms, we add the risk that the computed interval will be narrower than the actual one and thus, if the actual interval is outside the bounds while the computed interval is still in, we will erroneously accept a faulty design.
For the vast majority of buildings, aerospace structures, etc., this risk in unacceptable, and therefore, we would prefer to have guaranteed estimates for $x_i$, even if they are overestimating enclosures. Because of this, the authors use the existing methods of solving interval linear systems to find the intervals of possible values of displacements.