In order to compute sensitivity, a small perturbation (e.g. 1 mm for the case study) is considered for each design variable. The perturbation can either increase the value of the design variable (PP) or decrease it (MP). Considering the Plackett-Burman design [1] with two levels and three factors, for each solution the cases in are considered:

Table 1 Plackett-Burman perturbations (2 levels, 3 factors)

The cases in lead to four runs for each considered solution, instead of six runs in the case of a full perturbation scheme. Correspondingly, in the objective function space the five dots (four from the Plackett-Burman perturbation scheme, and one being the non-perturbed solution) can originate different configurations, as shown in Figure 1, which refers to the optimal shape-design problem presented in the subsequent Section 3.

Figure 1 Two examples of solution perturbation: cross - perturbed solutions, circle - non-perturbed solution

Suppose to minimize a pair of objectives and simultaneously evaluate their sensitivity against small perturbations, *δ*x, around solution x. Two cases are possible: either the unperturbed solution (*i*.*e*. the nominal solution) *dominates* the perturbed ones (case A), or the unperturbed solution *is dominated* (case B, in which two solutions dominate the non-perturbed one).

In particular, referring to case B, it is interesting to note in Figure 1 that a perturbed solution might, by chance, dominate the nominal solution, and so be preferable to it. If this happened, the nominal solution, x, should be replaced by the dominant perturbed solution in the relevant optimization iteration.

In both cases the area, A, of the convex polygon associated with the nominal solution is considered as a metric criterion of sensitivity: the smaller the area, the less sensitive the solution.

In order to control the sensitivity, reducing the polygon area is not completely satisfactory, in fact, there could be “small” polygons exhibiting a substantial deformation in shape. In such a case, an anisotropic effect on sensitivity would arise, meaning that for the same amount of perturbation of the design variables, the corresponding variation in the objective space could be biased along a paramount direction. In contrast, an isotropic sensitivity would be preferable.

To this end, the deployment criterion, D, which rules the symmetry of the polygon in two-dimensional objective space, is introduced:

$$\begin{array}{}{\displaystyle D=\sqrt{{G}_{11}+{G}_{22}}}\end{array}$$(1)

where the square matrix G is computed as

$$\begin{array}{}{\displaystyle \mathit{G}\mathit{=}{\left({\mathit{H}}^{\mathit{T}}\mathit{H}\right)}^{\mathit{-}\mathit{1}}}\end{array}$$(2)

and, in turn, H is a (N,2) matrix calculated as follows

$$\begin{array}{}{\displaystyle H=\left[\begin{array}{cc}\frac{{f}_{1g}-{f}_{1i}}{{R}_{1i}}& \frac{{f}_{2g}-{f}_{2i}}{{R}_{2i}}\end{array}\right]\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}i=1,..,N}\end{array}$$(3)

where (f_{1g}, f_{2g}) are the coordinates of the center of gravity of the polygon, R_{1i} and R_{2i} are the scalarizing distances of the i-th vertex of the polygon from the center of gravity, and N is the number of vertices of the polygon.

The concept of deployment is inspired by the field of wireless indoor positioning systems [4] and [5].

The proposed approach gives rise to a cost-effective formulation [f_{k}(x), A(x,*δ*x), D(x,*δ*x)], k=1, n_{f}, incorporating only n_{f}+2 objectives. In fact, according to a standard formulation of design sensitivity, the k-th objective should be attributed the relevant variation *Δ* with respect to all variables [f_{k}(x), *Δ*f_{k}(x)], which gives rise to 2n_{f} objectives in total, against n_{f}+2 according to our approach.

A cost-effective formulation is particularly suitable for handling complex optimization problems e.g. multi-physics problems [6,7,8,9] or, in general, optimizations based on a forward problem the solution of which requires a FE simulation [10,11,12].

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