Iterative Model Adoption and Optimization Solution Strategy

Figure 3 presents an iterative model adoption framework to generate an optimization problem that can be solved efficiently. The strategy is to first employ the most accurate rigorous mechanistic model for property prediction. This is expected to provide a reliable solution. In case a rigorous model is not available, the relatively simple but less accurate short-cut model can be adopted. The surrogate model is used when there is no suitable mechanistic model. Through this strategy, the cosmetic formulation problem can be explicitly expressed as an MINLP optimization problem below.
\(\operatorname{}{q=f(V_{I_{A,1}},V_{I_{A,2}},\ldots,V_{I_{Z,z}},ms)}\)Sensorial rating (11)
s.t. \(PL^{k}\leq P^{k}\leq PU^{k}\), \(k\in K=MM\cup SM\) Design target
\(P^{m}=G^{m}(IM_{I_{A,1}}^{m},IM_{I_{A,2}}^{m}\ldots,IM_{I_{Z,z}}^{m})\),\(\text{IM}_{i}^{m}=IMG^{m}(V_{i},ms)\), \(m\in MM\) Mechanistic model
\(P^{s}=g^{s}(V_{I_{A,1}},V_{I_{A,2}},\ldots,V_{I_{Z,z}},ms)\),\(s\in SM\) Surrogate model
\(H\left(S_{I_{A,1}},S_{I_{A,2}},\ldots,S_{I_{Z,z}},V_{I_{A,1}},V_{I_{A,2}},\ldots,V_{I_{Z,z}}\right)\leq 0\)Heuristics in Table 3
\(msL\leq ms\leq msU\) Design variables
\(S_{i}\in\left\{0,1\right\}^{i}\), \(\sum_{i}{V_{i}=1}\),\(VL_{i}\bullet S_{i}\leq V_{i}\leq VU_{i}\bullet S_{i}\),\(i\in\left\{I_{A,1},I_{A,2},\ldots,I_{Z,z}\right\}\)
where \(P^{m}\) is the m -th property predicted using a (rigorous or short-cut) mechanistic model. MM is the set of properties predicted using mechanistic-based models. \(P^{s}\) is the s -th target property (\(P^{s}\)) predicted using a surrogate model.SM is the set of properties predicted using surrogate models.