[insert Table 1 here] and [insert Table 2 here]
Creating a qualified cosmetic formula with desirable attributes is
challenging because there exist a large number of cosmetic ingredients
leading to numerous possible recipes. Many attributes involve complex
physicochemical phenomena, some of which are not yet fully understood.
More importantly, it is hard to quantify or predict consumers’
sensations since they are elusive, subjective, and affected by consumer
status.7 In this case, the use of any single model or
tool cannot capture the cosmetic formulation problem in its
totality.8 The design of related personal care
products such as shampoo and toothpaste faces similar issues. Currently,
new cosmetics are usually developed by experimental trial-and-error.
This is expensive and time-consuming. The search space is limited and
there is no guarantee that an optimal formula is
found.5 For expediting new cosmetic formulation, it is
highly desirable to develop an effective model-based optimization
approach to complement the efforts of experienced cosmetic formulators.
Model-based computer-aided mixture/blend design
(CAMbD) methods have been applied extensively. The
ingredients are generated using the group contribution (GC) approach.
Linear and simple nonlinear mixing rules are applied to predict mixture
properties. The CAMbD methods are usually applied to
mixtures with less than six ingredients such as solvent
mixtures9–12 and blended
fuels.13–17 This is because much greater
computational effort is needed as the number of ingredients
increases.12,13 Since the number of ingredients in
cosmetic products is typically larger than 15 and can be up to
50,18 it is highly desirable to develop alternative
methods for cosmetic formulation.
From a product design perspective, several model-based methods have been
proposed and applied to cosmetics and the highly related personal care
products. Omidbakhsh et al.19 built statistical models
to design disinfectant. Disinfection effect is first correlated with
ingredient composition and then composition is optimized to design a new
disinfectant with maximal disinfection effect. Smith and
Ierapetritou20 optimized the formula of an under-eye
cream using regressed polynomial functions that correlate product
attributes with ingredient composition and operating conditions.
Bagajewicz et al.21 started with a base-case formula
of skin lotion and optimized its composition for maximum profit in a
competitive market. In these studies, cosmetic formulation is treated as
a nonlinear programming problem. Only the composition of pre-selected
ingredients is optimized without considering the selection of other
ingredient alternatives. Obviously, a more superior formula can be
easily missed without considering all the available ingredients. Conte
et al.3,22 combined computer-aided modeling with
experimental testing to formulate sunscreen spray. Ingredients are
selected using databases, knowledge-base, and GC methods. Kontogeorgis
et al.5 extended this integrated modeling-experimental
approach to formulate emulsified products. Zhang et
al.23 proposed an integrated framework for formulated
product design considering the optimal identification of ingredients,
composition, microstructure, etc. Arrieta-Escobar et
al.24 incorporated heuristics and mixed-integer
nonlinear programming (MINLP) to identify the optimal ingredients and
composition of hair conditioner. By integrating different methods and
tools, the above studies can properly select ingredients from a set of
candidates with optimized composition. However, only certain mixture
properties (e.g., color and greasiness) related to sensorial attributes
have been considered.5,24 Only recently has machine
learning been used to predict sensorial
perceptions.25–27 Even less is available on how
sensorial satisfaction can be explicitly quantified, modelled, and
incorporated into product formulation.25 By
integrating machine learning models, a grey-box optimization problem was
formulated and solved using genetic algorithm (GA) for food product
design.25 This method is applied with simplified
property models involving a limited number of equations, because GA is
inefficient in handling a large number of complex constraints that are
common in cosmetic formulation problem.23 In addition,
GA cannot guarantee \(\varepsilon\)-optimality.
To fill this gap, a novel optimization-based approach is developed for
the formulation of cosmetics. Figure 1 illustrates the overall
methodology. For given consumer needs and a set of potential chemical
ingredients, an MINLP problem is formed by integrating (rigorous and
short-cut) mechanistic models, data-driven surrogate models, and
mathematical equations derived from heuristics. The objective is to
maximize the sensorial perception. Then, a novel solution strategy that
involves an iterative adoption of a hierarchy of models and different
numerical techniques is applied to solve the optimization problem
efficiently. Then, the optimal formula can be verified by experiments.
The paper is organized as follows. A systematic procedure is first
introduced for problem formulation. Then, the iterative procedure for
model adoption and optimization solution strategy are described.
Finally, a perfume example is discussed to illustrate the applicability
of the proposed approach.