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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.