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.