Emollients, nonionic surfactants, and thickeners (polymers or fatty
alcohols) are selected from an initial list with a total of 36
ingredients, as presented in Tables C-1 to 3 in Supplementary Material.
Additional restrictions were added to this problem formulation when
integrating the consumer assessment in each case. Other heuristics are
transformed into variables and correlated in equations employing
propositional logic, as described in a previous contribution from the
authors.31 Vectors of decision variables – binary
(\(y\)) and continuous (\(x\)) - both have a dimension of 36. The chosen
objective function was defined as the total cost of the formulation
(USD/kg), only considering the unit cost of each ingredient and
excluding fixed ingredients. Manufacture costs were not considered.
The restrictions involving RHLB and HLB are the only source of
non-linearity.
\(RHLB\ =\ \frac{\sum{x_{i}{RHLB}_{i}+\sum{x_{j}{RHLB}_{j}}+\sum{x_{k}{RHLB}_{k}}}}{\sum{y_{i} x_{i}}+\sum{y_{j} x_{j}}+\sum{y_{k} x_{k}}}\ ;\ HLB\ =\ \frac{\sum{x_{r}{HLB}_{r}}}{\sum{y_{r} x_{r}}}\) (4)
\(\left(\sum{x_{i}\text{RHLB}_{i}+\sum{x_{j}\text{RHLB}_{j}}+\sum{x_{k}\text{RHLB}_{k}}}\right)\ \left(\sum{y_{r} x_{r}}\right)-\left(\sum{y_{i} x_{i}}+\sum{y_{j} x_{j}}+\ \sum{y_{k} x_{k}}\right)\left(\sum{x_{r}\text{HLB}_{r}}\right)=0\ \ \ \ \ \) (5)
The resulting MINLP problem can be solved in less than one second using
the GAMS platform and the global solver BARON. The optimal solution is
selected in each case to be compared with the commercial sample.
Experimental tests and emulsions characterization : The complete
list of chemicals used in the prototyping of the alternative emulsions
is presented in the Table C-4 of the Supplementary Material. They were
used as obtained without further treatment.
Rheology and textural analysis: The rheological analysis was performed on a rotational viscometer ARES (TA Instruments, New Castle, USA). The flow behavior was studied by continuous shear investigations, which were performed to evaluate the shear response (Pa·s) as a function of shear rate (s-1) (from 0.01 to around 500 s−1) with a logarithmically increasing scale (5 points), using parallel plates (25 mm). The gap between the disc and the plate (0.5 mm) was carefully filled with a sample of the product, and any left material was removed using a metal spatula. All samples were tested one week after preparation, and the measurements were conducted at 20 ± 0.1°C, at least in duplicate.