Binary variables corresponding with the choice of ingredients (vector y) and continuous variables associated with their respective concentrations in the final product (i.e., vector x with mass fractions) are combined with product performance metrics (vector p), including well-defined physicochemical properties and parameters related to more subjective sensorial attributes. Product quality is evaluated in terms of the difference of these metrics with the target values (product performance specifications vector p*). The available property models, containing any relationship between metrics p and product composition, are represented by the set of equations h(x,y,p) = 0, also known as property function. The heuristic rules and other restrictions are incorporated as part of the set of constraints
represented by g1(x,y,p) ≤ 0 and g2(x,y,p) ≤ 0 .43 Finally, the global objective function to be minimized (f), which accounts for both product quality deviation and costs, allows to state the problem of optimal product formulation.
\(\min_{x,y}f\left(x,y,p,p*\right)\ \left[product\ performance\right]
\) (3)
\(s.t.\ \ \ \ h\left(x,y,p\right)=0\ \left[property\ function\right]\)
\(\ \ \ \ \ \ \ \ \ g_1\left(x,y,p\right)\le0\ \ \left[heuristic-related\ restrictions\right]\)
\(\ \ \ \ \ \ \ \ \ g_2\left(x,y,p\right)\le0\ \ \left[other\ restrictions\right]\)
As sketched in Phase II of Figure 2 above, one should arrive at an optimal formulation after successive cycles of optimization, in which updated models or new heuristic rules are reintegrated into the problem (3) as a result of the experimental tests. Through integer cuts, it is possible to obtain a ranking of several alternative formulations with increasing values of the objective function.31 In any case, these computer-generated alternatives meet all product specifications and also take into account the consumer assessment from Phase I. Most heuristic rules follow a linear formulation; however, some of them are likely to be non-linear, so problems in the form of (3) will be MINLP problems, as it is illustrated in the case study.
In summary, a general methodological approach is presented here for the design of formulated products. A fuzzy measure analysis for the consumer assessment, along with heuristics and property models incorporation for the product realization phase, are used to guide the selection of formulations that solve the original design problem, using the reformulations of cosmetic emulsions as case studies.
Cosmetic emulsions case study
The proposed methodology was tested in the design of cosmetic products, specifically skin moisturizers. These products are used to keep in good condition the skin by maintaining its balance of oil and water. 44 Skin moisturizers are generally oil in water (O/W) emulsions, and commercially they are usually divided into two main types: creams and lotions. Lotions are low-viscosity emulsions, while creams are much higher viscosity materials, generally presented as semi-solid emulsions.45 The O/W emulsions are characterized by a low internal phase ratio, typically containing 10 to 35% dispersed phase.46 For this reason, the addition of a suitable emulsifier agent along with the application of mechanical agitation is necessary to create a stable emulsion.47 In this work, apart from some essential ingredients like water or humectants (i.e., Glycerol) that are kept constant, three main types of components are considered for the modeling of O/W emulsions: Emollients, thickeners, and emulsifiers. The selection of these groups of ingredients is based on several studies that evaluated their impact on the sensorial, rheological and textural properties of cosmetic emulsions.20,47–51 When formulating cosmetic emulsions, the dispersed phase should be first selected,4 and emollients constitute the main component of this phase, particularly in O/W emulsions. Emollients are required in the dispersed phase because they help to prevent soaping, they improve spreadability, and they are responsible for the consumer-perceived benefits after evaporation of volatile materials.47,49 The performance of emollients is generally related to greasiness, and the emulsifying properties are dependent on other physical properties such as density, viscosity, melting point, and the required HLB (RHLB).52,53
Emulsifiers are essential ingredients to stabilize the emulsions.46 The type of surfactant and its physicochemical properties will influence the droplet size and stability of the emulsion.53 HLB is a used here to predict the emulsifying properties of surfactants,38 and to correlate with some sensorial properties of the emulsions, such as color, odor, and consistency.47,54
Thickeners are used to increase the viscosity of the continuous phase and to mitigate the phase separation.47,52 It has also been shown that thickeners could have a relevant impact on skin feeling, namely when removing cream from the container, and when spreading the product.55