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\begin{document}
\title{Optimization-based Cosmetic Formulation: Integration of Mechanistic
Model, Surrogate Model, and Heuristics}
\author[1]{Xiang Zhang}%
\author[2]{Teng Zhou}%
\author[3]{Ka Ng}%
\affil[1]{Affiliation not available}%
\affil[2]{Max Planck Institute for Dynamics of Complex Technical Systems}%
\affil[3]{Hong Kong University of Science and Technology}%
\vspace{-1em}
\date{\today}
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\selectlanguage{english}
\begin{abstract}
Multiple functional and hard-to-quantify sensorial product attributes
that can be satisfied by a large number of cosmetic ingredients are
required in the design of cosmetics. To overcome this challenge, a new
optimization-based approach for expediting cosmetic formulation is
presented. It exploits the use of a hierarchy of models in an iterative
manner to refine the search for creating the highest-quality cosmetic
product. First, a systematic procedure is proposed for optimization
problem formulation, where the cosmetic formulation problem is defined,
design variables are specified, and a set of models for sensorial
perception and desired product properties are identified. Then, a
solution strategy that involves iterative model adoption and two
numerical techniques (i.e., generalized disjunctive programming
reformulation and model substitution) is applied to improve the
efficiency of solving the optimization problem and to find better
solutions. The applicability of the proposed procedure and solution
strategy is illustrated with a perfume formulation example.%
\end{abstract}%
\sloppy
\section*{Introduction}
{\label{introduction}}
The chemical industry designs and produces a vast number of chemical
products to serve the society. Chemical products are classified as
molecular products, formulated products, functional products, and
devices.\textsuperscript{1,2} Among them, formulated products such as
cosmetic and paint are formed by mixing selected ingredients in a
formula, which may possess certain microstructures of their own (e.g.,
powder and emulsion). The formula (i.e., ingredient selection and
composition) has a significant impact on formulated product quality.
Thus, the major aim of formulated product design is to find a formula
that exhibits consumer-desired properties.\textsuperscript{2,3}
As a major component of formulated products, cosmetics are applied to
the human body for cleansing, beautifying, promoting attractiveness, or
altering appearance. They are sold in many forms. Table 1 lists the
commonly used cosmetic product forms such as cream and gel. The global
cosmetic market valued at \$532 billion in 2017 is large, but
competitive and dynamic.\textsuperscript{4} Many cosmetic products exist
on the market but they tend to have short product life. To succeed in
this environment, rapid formulation of new and improved cosmetics is
crucial. The quality of cosmetics can be broadly represented by two
types of attributes. One is sensorial attributes (e.g., smell and sight)
perceived by five human senses during and after the application of
cosmetics. The other is functional attributes (e.g., stability and
safety), which ensures that cosmetics can be assuredly used with the
desired functions. Table 2 lists the relevant sensorial and functional
attributes of four cosmetic products with different product forms. For
instance, the senses of how lipstick is felt by the lips and how the
lips look after application are part of the lipstick quality. Meanwhile,
lipsticks should be stable, safe, and not broken in use. It is known
that the sensorial quality is the dominant consideration for consumers
to choose one cosmetic product over another.\textsuperscript{5,6} Thus,
it needs to be explicitly considered in cosmetic formulation.
\textbf{{[}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.\textsuperscript{7} In this case, the use of any single model or
tool cannot capture the cosmetic formulation problem in its
totality.\textsuperscript{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.\textsuperscript{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
(CAM\textsuperscript{b}D) 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 CAM\textsuperscript{b}D methods are usually applied to
mixtures with less than six ingredients such as solvent
mixtures\textsuperscript{9--12} and blended
fuels.\textsuperscript{13--17} This is because much greater
computational effort is needed as the number of ingredients
increases.\textsuperscript{12,13} Since the number of ingredients in
cosmetic products is typically larger than 15 and can be up to
50,\textsuperscript{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.\textsuperscript{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
Ierapetritou\textsuperscript{20} 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.\textsuperscript{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.\textsuperscript{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.\textsuperscript{5} extended this integrated modeling-experimental
approach to formulate emulsified products. Zhang et
al.\textsuperscript{23} proposed an integrated framework for formulated
product design considering the optimal identification of ingredients,
composition, microstructure, etc. Arrieta-Escobar et
al.\textsuperscript{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.\textsuperscript{5,24} Only recently has machine
learning been used to predict sensorial
perceptions.\textsuperscript{25--27} Even less is available on how
sensorial satisfaction can be explicitly quantified, modelled, and
incorporated into product formulation.\textsuperscript{25} By
integrating machine learning models, a grey-box optimization problem was
formulated and solved using genetic algorithm (GA) for food product
design.\textsuperscript{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.\textsuperscript{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.
\textbf{{[}insert Figure 1 here{]}}
\section*{Systematic Procedure for Optimization Problem
Formulation}
{\label{systematic-procedure-for-optimization-problem-formulation}}
Figure 2 shows the 3-step procedure. In Step 1, the cosmetic formulation
problem is defined where the desired product form, product attributes,
and product specifications and properties are determined based on market
study and product knowledge. In Step 2, potential ingredient candidates
and relevant microstructural descriptors (if applicable) are generated
to specify the design variables. Step 3 identifies a set of mechanistic
models and surrogate models and converts heuristics into mathematical
equations. For each step, the sources of the input are shown on the left
and the outputs are on the right. The details are described below.
\textbf{{[}insert Figure 2 here{]}}
\subsection*{Step 1: Problem Definition}
{\label{step-1-problem-definition}}
When a new cosmetic design project is launched, the product type and
form such as a facial powder or lipstick are first decided by the
marketing team based on the target market, potential consumers,
competing products, etc. The quality of the new cosmetic product depends
on its sensorial and functional attributes. Usually, the
consumer-desired attributes can be specified through interview and
survey with potential consumers. The sensorial perceptions given by a
cosmetic are the most essential for its satisfaction and repeated use by
consumers.\textsuperscript{6} In practice, sensorial perception is
assessed through sensorial evaluation. A number of panelists assess
various cosmetic samples using well-defined protocols and their
perceptions are quantified using sensorial ratings. Then, an overall
sensorial rating can be obtained to represent the degree of satisfaction
of the cosmetic.\textsuperscript{7} Note that in addition to perception,
other factors such as packaging and price affect consumer's purchase
decision. These factors are not considered in this work and the
objective function is to maximize the overall sensorial rating
(\(q\)).
\(\max\text{\ \ \ q}\) (1)
In addition, the functional attributes are also needed to be satisfied.
Each cosmetic has its unique functional attributes. For instance, a hair
spray should dry rapidly and perfume should be transparent (Table 2).
The product attributes can be translated into relevant physicochemical
properties (e.g., melting point for lipstick) and product specifications
(e.g., sun protection factor for sunscreen product) using engineering
know-how. For the four cosmetics in Table 2, the last column lists
various properties related to their sensorial and functional attributes.
How a lipstick is sensed by the lips is affected by its viscosity. The
pH of a skin cream affects its safety. Then, a set of design targets
(i.e., lower and upper bounds) on the properties can be identified based
on the engineering know-how and product in-house data. These bounds
serve as constraints in the optimization problem.
\(PL^{k}\leq P^{k}\leq PU^{k},\ \ \ k\in K\) (2)
where \(P^{k}\) is the \emph{k} -th desired property.
\(K\) is the set of properties. \(PL^{k}\) and
\(PU^{k}\) are the lower and upper bounds, respectively. Note
that the nomenclature is presented in Supporting Information.
\subsection*{Step 2: Ingredient Candidate
Generation}
{\label{step-2-ingredient-candidate-generation}}
To provide multiple desired attributes, many chemical ingredients are
needed. Cosmetic ingredients are classified into different types based
on their functionalities. Table S1 lists the ingredient types that are
widely used in various cosmetics and their
functions.\textsuperscript{7,28} For instance, an abrasive in a facial
cleanser is made up of solid particles used for physically cleaning hard
surface such as epidermis. Three types of moisturizers (i.e., emollient,
humectant, and occlusive) can be used to provide hydration effect.
Emollient can improve the skin's water-oil balance, humectant inhibits
water evaporation, and occlusive can form a water-repellent layer to
reduce water loss. For a cosmetic, the needed ingredient types can be
identified based on the fundamental formulation science and the desired
product attributes.
For each ingredient type, a set of ingredient candidates can be
generated using databases\textsuperscript{29,30} and computer-aided
tools.\textsuperscript{14,31} Regarding each of the ingredient types in
Table S1, the last column lists two commonly used ingredient candidates.
For instance, lactic acid and triethanolamine are often used as an
acidic and alkaline pH buffers, respectively. With the years of
development in the cosmetic industry, hundreds of ingredient candidates
exist for each ingredient type. To reduce the search space, the
candidates can be pre-screened using ingredient screening tools based on
cost, regulations, availability, etc. to generate a more organized pool
of ingredient candidates
\(\left.\ \par
\begin{matrix}I_{A}=\left\{I_{A,1},I_{A,2},\ldots,I_{A,a}\right\}\\
I_{B}=\left\{I_{B,1},I_{B,2},\ldots,I_{B,b}\right\}\\
\par
\begin{matrix}\cdots\\
I_{Z}=\left\{I_{Z,1},I_{Z,2},\ldots,I_{Z,z}\right\}\\
\end{matrix}\\
\end{matrix}\text{\ \ \ }\right\}\) (3)
where \(I_{A}\), \(I_{B}\),\ldots{},
\(I_{Z}\) are ingredient types.\(I_{A,1}\),
\(I_{A,2}\),\ldots{}, \(I_{A,a}\), etc. represent the
generated ingredient candidates. Here, the subscripts
\(a\), \(b\), and\(z\) denote
the number of candidates in each ingredient type. Each candidate has
different properties (e.g., density, solubility and pH) which can be
collected from the literature, database, and experiment. The selection
of ingredients is intuitively a discrete-continuous optimization
problem. Each ingredient candidate can be assigned a binary variable
\(S_{i}\) to control ingredient selection and a continuous
variable (e.g., volume fraction \(V_{i}\)) to denote its
composition. If the \emph{i} -th candidate is selected,
\(S_{i}\) is equal to 1 and\(V_{i}\) is constrained
by its lower (\(VL_{i}\)) and upper (\(VU_{i}\))
bounds. Otherwise, \(S_{i}\) and \(V_{i}\) are equal
to 0.
\(\sum_{i}{V_{i}=1}\) (4)
\(VL_{i}\bullet S_{i}\leq V_{i}\leq VU_{i}\bullet S_{i}\),\(i\in\left\{I_{A,1},I_{A,2},\ldots,I_{Z,z}\right\}\) (5)
In addition to ingredients, microstructure can affect the properties
when certain product forms are used. Typically, the major
microstructural features can be characterized by some geometric
descriptors that can be correlated with the mixture properties by
experiment and multi-scale modeling to account for the
microstructure-property relationship.\textsuperscript{32} The last
column of Table 1 lists the relevant microstructure descriptors for
various commonly used cosmetic product forms. For example, the oil
droplet size affects the viscosity and texture of a moisturizing lotion
in the form of an oil-in-water emulsion.\textsuperscript{33} The
emulsion type and particle shape can be decided using
heuristics.\textsuperscript{34} Geometric descriptors
ms such as particle size are continuous variables
\(msL\leq ms\leq msU\) (6)
where \(m\text{sL}\) and \(m\text{sU}\) are the lower and
upper bounds, respectively. The microstructure is decided by both the
formulation and manufacturing process design.\textsuperscript{35}
\subsection*{Step 3. Model
Identification}
{\label{step-3.-model-identification}}
\subsubsection*{Model for Sensorial
Perception}
{\label{model-for-sensorial-perception}}
Surrogate model that captures the input-output data is built to predict
the sensorial rating. After a surrogate model is trained, its analytical
form can be used for optimization.
\(q=f\left(V_{I_{A,1}},I_{I_{A,2}}\ldots,V_{I_{Z,z}},ms\right)\) (7)
The first task is to collect training data. The input data can be the
cosmetic recipes and the microstructures, namely (\(V_{I_{A,1}},V_{I_{A,2}},\ldots,V_{I_{Z,z}},ms\)).
The output data is the corresponding sensorial rating
(\(q\)). Here, the historical data of sensorial
evaluations can be utilized. When the historical data is scarce,
additional data sampling is required. By far, many efficient sampling
approaches have been used in the cosmetic industry such as
Latin-hypercube sampling, Plackett-Burman, full-fractional, etc.
Referring to the ``one in ten'' rule, the number of data samples is
preferably ten times more than the number of ingredient candidates. The
second task is to build an accurate surrogate model. Currently, multiple
types of surrogate models can be utilized such as linear regression,
kriging, artificial neural network (ANN), radial basis function, etc.
Among them, some surrogate models (e.g., random forest) cannot provide
available derivative information while the derivatives of many other
surrogate models are symbolically available such as linear regression,
ANN with tansig kernel function, etc.\textsuperscript{36} Here, a
surrogate model with available derivative information is preferred
because solving a discrete-continuous optimization problem with no
derivative information is very challenging. The hyperparameters of the
surrogate model structure should be carefully tuned. The heuristics and
experience reported in the literature can be
consulted.\textsuperscript{36,37} Afterward, model accuracy needs to be
validated. The widely used validation methods include K-fold cross
validation and holdout method. If the model is not sufficiently
accurate, the type of surrogate model and the hyperparameters should be
re-selected.
\subsubsection*{Models for Target
Properties}
{\label{models-for-target-properties}}
Three types of models can be applied for predicting the target
properties: rigorous mechanistic model, short-cut model, and surrogate
model. Typically, the formulation and application of cosmetics involve
various phenomena (e.g., kinetics, thermodynamics, and transport). For
any property, the associated phenomena should be first identified based
on the basic engineering sciences and domain knowledge, followed by the
identification of the relevant mechanistic models. Generally, rigorous
models are the most accurate but more complex and sometimes with unknown
parameters. The perfume diffusion model\textsuperscript{38} and
ingredient percutaneous absorption model\textsuperscript{39} are
examples. Instead of accounting fully the physical phenomena, simple
short-cut model captures the property's dependence on the most
influential factors. Usually, short-cut model is sufficiently accurate
within pre-specified conditions. Note that both rigorous and short-cut
models involve many intermediate variables for describing the relevant
phenomenon. The rigorous or short model for \emph{k} -th desired
property (\(P^{k}\)) can be represented as
\(P^{k}=G^{k}(IM_{I_{A,1}}^{k},IM_{I_{A,2}}^{k},\ldots,IM_{I_{Z,z}}^{k})\),\(k\in K\) (8)
\(\text{IM}_{i}^{k}=\text{IMG}^{k}\left(V_{i},ms\right)\),\(i\in\left\{I_{A,1},I_{A,2},\ldots,I_{Z,z}\right\}\) (9)
where \(\text{IM}_{i}^{m}\) denotes the intermediate variable related to
the \emph{i} -th ingredient candidate (e.g., vapor pressure and activity
coefficient). If there are no suitable mechanistic models but data are
available, surrogate models can be adopted,\textsuperscript{40} although
the model validity is often limited to the range of available data. The
input data should be the sampled cosmetic recipes and microstructure.
The output data are the target properties. For the \emph{k} -th property
(\(P^{k}\)), its surrogate model is
\(P^{k}=g^{k}(V_{I_{A,1}},I_{I_{A,2}}\ldots,V_{I_{Z,z}},ms)\),\(k\in K\) (10)
Accordingly, for any desired property, a set of models (rigorous,
short-cut, and surrogate) should be identified for use in the
optimization.
The use of heuristics is often inevitable in cosmetic
formulation.\textsuperscript{24,41} The reason is that some phenomena
have not been identified or are poorly understood. For instance, a
hydrocolloid thickener with a weak gel network structure is preferred
for use in emulsion-based product to generate thixotropic behavior,
although no formal justification has been given.\textsuperscript{34} In
addition, heuristics can effectively help reduce the search space. Many
heuristics, although not all, can be transformed into mathematical
design constraints for use in the optimization. Table 3 shows the widely
used forms of heuristics and the associated equations for formulated
product design. For instance, if the number of ingredients for certain
type of ingredient is suggested, an inequality
constraint\(TL\leq\sum_{i}S_{i}\leq TU,\ \ i\in I_{X}\) can be generated.
\textbf{{[}insert Table 3 here{]}}
\section*{Iterative Model Adoption and Optimization Solution
Strategy}
{\label{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 \emph{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 \emph{s} -th target property
(\(P^{s}\)) predicted using a surrogate
model.SM is the set of properties predicted using
surrogate models.
\textbf{{[}insert Figure 3 here{]}}
The computational difficulty of the MINLP problem depends on the number
of ingredient candidates and the complexity of the adopted models. A
large number of ingredient candidates often create a combinational
problem. Many rigorous models (e.g., thermodynamic and transport
phenomena models) involve nonlinear and nonconvex equations. Along with
complex surrogate models (e.g., neural network), the optimization
problem is prone to convergence failure if the problem is directly
solved using standard MINLP solvers. Some of these problems can be
handled with advanced algorithms. For instance, Schweidtmann and
Mitsos\textsuperscript{42,43} recently developed and applied an
efficient global solver for ANN embedded MINLP problems. Alternatively,
the problem can be resolved by reformulating the optimization
problem.\textsuperscript{44} Two techniques are proposed for enhancing
optimization convergence and finding better solutions (Figure 3). If the
problem can be directly solved, the solution is sent for experimental
validation. Otherwise, generalized disjunctive programming (GDP) can be
used because of the need to calculate the multiple intermediate
variables in the mechanistic models. If the GDP problem still cannot be
solved or better solutions are needed, the model(s) in use are replaced
with alternative model(s) and repeat the calculations.
\subsection*{GDP reformulation}
{\label{gdp-reformulation}}
As can be seen in Eq. 11, even if \emph{i} -th ingredient candidate is
not selected, its intermediate variables (\(IM_{i}^{m}\)) must be
calculated. Also, forcing \(V_{i}\) to 0 may lead to
singularity at\(V_{i}=0\) for some models (e.g., logarithmic
function). Thus, when mechanistic models are employed and multiple
intermediate variables are calculated through complex equations, the
redundant constraints and singularities can lead to convergence failure.
Similar to the tray selection problem in distillation column
design,\textsuperscript{44}cosmetic formulation problem can be
formulated using GDP. As an alternative way to program
discrete-continuous problem, GDP is a logic-based method containing
Boolean and continuous variables. Constraints are expressed as
disjunctions, algebraic equations, and logic
propositions.\textsuperscript{44} The following disjunction can be used
to express the bounds and intermediate variables for Eq. 11.
\(\par
\begin{bmatrix}Y_{i}\\
VL_{i}\leq V_{i}\leq VU_{i}\\
\par
\begin{matrix}IM_{i}^{m}=IMG^{m}(V_{i},ms)\\
\end{matrix}\\
\end{bmatrix}\bigvee\par
\begin{bmatrix}\neg Y_{i}\\
V_{i}=0\\
\par
\begin{matrix}IM_{i}^{m}=0\\
\end{matrix}\\
\end{bmatrix}\),\(i\in\left\{I_{A,1},I_{A,2},\ldots,I_{Z,z}\right\}\) (12)
where \(Y_{i}\) is the Boolean variable for ingredient
selection. If the\emph{i} -th ingredient is selected, the bounds on
\(V_{i}\) are fulfilled and its intermediate variables
\(IM_{i}^{m}\) are calculated. Otherwise, they are not calculated
and simply set as 0.
To solve a GDP problem, it is often transformed back into MINLP using
big-M or convex-hull relaxation to take advantage of standard MINLP
solvers. It is found that the big-M method is more appropriate in
solving mixture design problem since singularity issue can still occur
in the convex-hull relaxation.\textsuperscript{12} After transforming
the above disjunction using big-M approach, the cosmetic formulation
problem is reformulated below. \(S_{i}\) has a one-to-one
correspondence with \(Y_{i}\). bm is a
sufficiently large parameter.
\(\operatorname{}{q=f(V_{I_{A,1}},V_{I_{A,2}},\ldots,V_{I_{Z,z}},ms)}\)(13)
s.t. \(PL^{k}\leq P^{k}\leq PU^{k}\), \(k\in K=MM\cup SM\)
\(\left\{\par
\begin{matrix}VL_{i}-bm\bullet(1-S_{i})\leq V_{i}\leq VU_{i}+bm\bullet(1-S_{i})\\
-bm\bullet S_{i}\leq V_{i}\leq bm\bullet S_{i}\\
\par
\begin{matrix}\text{IM}G^{m}\left(V_{i},ms\right)-bm\left(1-S_{i}\right)\leq IM_{i}^{m}\leq IMG^{m}\left(V_{i},ms\right)+bm(1-S_{i})\\
-bm\bullet S_{i}\leq IM_{i}^{m}\leq bm\bullet S_{i}\\
\end{matrix}\\
\end{matrix}\right.\ \) Big-M constraint
\(P^{m}=G^{m}(IM_{I_{A,1}}^{m},IM_{I_{A,2}}^{m},\ldots,IM_{I_{Z,z}}^{m})\),\(m\in MM\)
\(P^{s}=g^{s}(V_{I_{A,1}},V_{I_{A,2}},\ldots,V_{I_{Z,z}},ms)\),\(s\in SM\)
\(H\left(S_{I_{A,1}},S_{I_{A,2}},\ldots,S_{I_{Z,z}},V_{I_{A,1}},V_{I_{A,2}},\ldots,V_{I_{Z,z}}\right)\leq 0\)
\(msL\leq ms\leq msU\),
\(S_{i}\in\left\{0,1\right\}^{i}\),\(\sum_{i}{V_{i}=1}\),\(i\in\left\{I_{A,1},I_{A,2},\ldots,I_{Z,z}\right\}\)
\subsection*{Model substitution}
{\label{model-substitution}}
Some rigorous mechanistic models are too complicated to be directly used
for optimization even if they are programmed using GDP. There is always
a trade-off between model accuracy and traceability. In this case, the
complicated but accurate rigorous models can be replaced by simple
short-cut model or surrogate model to reduce the computational effort
and to seek out even better solutions. A surrogate model is a good
choice when it is relatively easy to generate simulation data as
training data from the rigorous model. Although the model accuracy is
reduced, it is easier to solve and to obtain the global
solution.\textsuperscript{42,45--47} After model substitution, the newly
generated optimization problem should be solved and the optimal solution
obtained can be denoted as \(V_{i}^{*}\). This solution must be
validated using the original rigorous mechanistic models. If the
validation fails, the newly generated optimization problem should be
re-solved by adding the equation below to remove this solution
(\(V_{i}^{*}\)) that fails validation. Otherwise, the solution can
be sent for experimental verification.
\(\sum_{i}\left(V_{i}^{**}-V_{i}^{*}\right)^{2}\geq tol\) (14)
Here, \(V_{i}^{**}\) is the solution of a new round optimization.
The parameter tol is a small tolerance.
\section*{Case Study: Liquid Perfume}
{\label{case-study-liquid-perfume}}
As a popular cosmetic, perfume is a liquid mixture releasing pleasant
scents. The global perfume market is valued at \$31.4 billion in 2018.
Based on the volume fraction of fragrant compounds, perfume can be
classified into several types. Extrait contains 15-30\%, Eau de parfum
10-20\%, and Eau de cologne 3-5\%. For each type, thousands of products
exist on the market. Most perfumes are made from various synthetic
fragrances for easy quality control. The experienced perfumers create
new recipes by trial-and-error. Here, the proposed framework and
solution strategy are applied to formulate a new Eau de parfum.
\subsection*{Step 1: Problem definition}
{\label{step-1-problem-definition-1}}
Table 2 shows that the most critical sensorial attribute of perfume is
the smell. After applying the perfume, the fragrant compounds begin to
evaporate and are detected by an observer away from the location of
release. The scents change over time because each constituent is
released at a different rate. This process can last several hours. Based
on the order in which the odors appear, the released scents are
classified as: top note, middle note, and base note. Top note is
comprised of the scents perceived immediately after perfume application
and generally lasts 5-15 minutes. The scents in the middle note emerge
after the top note dissipates and remain for around an hour. The base
note appears close to the end of middle note and can last several hours.
During the sensorial evaluation of a perfume, each note is assessed and
rated. An average rating can then be obtained to represent consumer
preference.\textsuperscript{48} Thus, the objective function is to
maximize the overall sensorial rating on the smell of perfume
(\(q_{s}\)).
\(\operatorname{}q_{s}\) (15)
A perfume can be formulated to provide any specific scent with certain
intensity. In this study, it is assumed that the marketing team decides
that a lemon-like odor should dominate in the top note of the new Eau de
parfum. There are no specific odors required for the middle and base
notes as long as the overall sensorial rating is maximized. Thus, the
odor type with the highest intensity in the top note
(OTTN) is
\(OTTN=\mathrm{lemon-like}\) (16)
Moreover, since homogeneous liquid solution is transparent, all the
perfume ingredients must be completely miscible with each other. Perfume
safety is related to its toxicity and flammability. Toxicity can be
measured by the median lethal dose (\(LD_{50}\)). The larger
the\(LD_{50}\) is, the safer the perfume is. Since a solution
with\(LD_{50}\) larger than 5000 mg/kg can be regarded as
non-toxic, this is chosen as the design target as defined in Eq. 17.
Flammability depends on the flash point (\(T_{\text{fp}}\)), which is
the lowest temperature at which liquid vapor ignites given an ignition
source. A higher flash point indicates lower flammability. Here, the
flash point is required to exceed 15 \selectlanguage{ngerman}°C which is roughly the value of
existing perfume products.
\(LD_{50}\geq 5000\) mg/kg (17)
\(T_{\text{fp}}\geq 15\) (18)
Accordingly, four design targets are specified: constraints
on\(\text{\ L}D_{50}\) and flash point, a homogeneous solution, and a
dominant lemon-like odor in the top note.
\subsection*{Step 2: Ingredient candidate
generation}
{\label{step-2-ingredient-candidate-generation-1}}
Table 4 lists the four required ingredients types and their
functions.\textsuperscript{49} Various fragrances are used to provide
different scents. Based on the volatility, fragrance compounds can be
classified into three types in accordance with the top note, middle
note, and base note. For instance, the top note fragrances are most
volatile with a vapor pressure typically larger than 0.1 mmHg. The vapor
pressure of middle note and base note fragrances are 0.001--0.1 mmHg and
less than 0.001 mmHg, respectively.\textsuperscript{50}
Referring to the perfume manual,\textsuperscript{51} 48 common perfume
ingredients are generated in the four ingredient types (see Table 4). 17
candidates are top note fragrances, 16 candidates are middle note
fragrances, and 13 candidates are base note fragrances. Each candidate
has a different odor. For instance, as a top note fragrance, limonene
occurs naturally in the oil of citrus peels and offers a lemon-like
odor. Coumarin is the source of tonka bean's distinctive aroma and is
often added as a base note fragrance. An ethanol and water mixture is by
far the most common solvent in perfume.\textsuperscript{52} Ingredient
selection is controlled by the binary variable \(S_{i}\) and
ingredient composition is represented by volume fraction
\(V_{i}\). If the\emph{i} -th ingredient is not selected,
\(S_{i}\) and \(V_{i}\) are set to be 0. Otherwise,
\(S_{i}\) is equal to 1 and \(V_{i}\) is constrained
by its lower and upper bounds (\(VL_{i}\) and
\(VU_{i}\)).
\(\sum_{i}{V_{i}=1}\) (19)
\(VL_{i}\bullet S_{i}\leq V_{i}\leq VU_{i}\bullet S_{i}\) (20)
Their values as well as the properties of 48 candidates (e.g., density,
toxicity, etc.) are given in Table S2 in Supporting Information. These
are used as parameters in the optimization. Since perfume is a liquid
solution, no microstructural descriptors are considered.
\textbf{{[}insert Table 4 here{]}}
\subsection*{Step 3: Model
identification}
{\label{step-3-model-identification}}
The third step is to identify the models for the average sensorial
rating on perfume smell (\(q_{s}\)) and the four target
properties. The models are elaborated below.
\subsubsection*{ANN-based surrogate model for sensorial
rating}
{\label{ann-based-surrogate-model-for-sensorial-rating}}
A surrogate model is developed for predicting \(q_{s}\).
Perfume sensorial data are generated by matching the general consumers'
preferences reflected in various perfume review websites. Here, the data
is used to represent consumers' satisfaction. A total of 761 data
samples are uploaded in
https://github.com/zx2012flying/Perfume-Case-Study. These data samples
only involve the 48 ingredient candidates in Table 4. For each data
sample, the input data includes the selected ingredients and their
volume fractions. The output data is the overall sensorial rating. For
consistency, the ratings are scaled to {[}0, 100{]} with 100 denoting
the best smell. The minimum and maximum ratings for these samples are
50.2 and 89.7, respectively. Based on these data, several surrogate
models such as linear regression, artificial neuron network (ANN), and
support vector regression are built using the Surrogate Modeling
Toolbox, Pyrenn, and Scikit-learn packages in Python 3.6. The
hyperparameters are tuned manually and the model accuracy is evaluated
through 10-fold cross validation. A three-layer ANN model (i.e., one
input layer, one hidden layer, and one output layer) was found to offer
the highest accuracy. Figure S1 shows the schematic structure of the ANN
model. The tansig and purelin functions are applied in the hidden and
output layer, respectively. The number of neurons in the hidden layer is
tuned to be 8. Figure 4 presents the histogram of the absolute errors
between the true values and predicted values (\(q_{s}^{\text{true}}-q_{s}^{\text{pre}}\)). 90\%
of the deviations are less than 10. The mean average error (MAE) and
mean average percentage error (MAPE) are equal to 4.8 and 6.9\%,
respectively. This ANN model provides an accurate prediction of
\(q_{s}\), which is explicitly expressed as
\(q_{s}=\sum_{l=1}^{8}{wo_{l}\bullet\ \ f_{h}(ah_{l})}+bo\) (21)
\(f_{h}\left(ah_{l}\right)=1-\frac{2}{1+e^{2\times ah_{l}}},\ \ \ l=1,\ldots,8\)(22)
\(ah_{l}=\sum_{i=1}^{48}{\text{wh}_{l,\ i}\bullet V}_{i}+bh_{l},\ \ \ \ l=1,\ldots,8\)(23)
where \(wo_{l}\) and bo are the weights and bias
in the output layer, respectively. \(f_{h}\) is the tansig
function in the hidden layer. \(ah_{l}\) is the intermediate
variable in the hidden layer. \(\text{wh}_{l,\ i}\) and
\(bh_{l}\) are the weights and biases in the hidden layer,
respectively. These model parameters are provided in the Github platform
mentioned above.
\textbf{{[}insert Figure 4 here{]}}
\subsubsection*{\texorpdfstring{LD\textsubscript{50}}{LD50}}
{\label{ld50}}
\(LD_{50}\) of the perfume solution is calculated by Eq. 24. It
depends on the toxicity of ingredients (\(LD_{50,i}\)) and the mass
fraction (\(m_{i}\)) converted from the volume fraction
\(V_{i}\).
\(LD_{50}=\frac{1}{\sum_{i=1}^{48}\frac{m_{i}}{LD_{50,i}}}\) (24)
\(m_{i}=\frac{V_{i}\bullet\rho_{i}}{\sum_{j=1}^{48}{V_{j}\bullet\rho_{j}}}\)(25)
where \(LD_{50,i}\) and the density (\(\rho_{i}\)) for the 48
ingredient candidates are given in Table S2.
\subsubsection*{Flash point}
{\label{flash-point}}
The flash point (\(T_{\text{fp}}\)) of a flammable liquid mixture can
be theoretically determined based on the Le Chatelier's mixing
rule.\textsuperscript{53}
\(\sum_{i=1}^{48}\frac{\text{FP}P_{i}}{\text{FPLF}L_{i}}=1\) (26)
where \(\text{FP}P_{i}\) and \(\text{FPL}FL_{i}\) are the partial
pressure and lower flammability limit of the \emph{i} -th ingredient
candidate at the flash point, respectively. \(\text{FPL}FL_{i}\) is
calculated by
\(\text{FPL}FL_{i}=LFL_{i}^{*}-\frac{0.182\times(T_{\text{fp}}-298)}{Hc_{i}}\)(27)
where \(Hc_{i}\) and \(\text{LF}L_{i}^{*}\) are the heat of
combustion and lower flammability limit at 298 K (see Table S2),
respectively.\(\text{FP}P_{i}\) is calculated via the vapor-liquid
equilibrium in Eq. 28. The UNIFAC model is used to calculate the
activity coefficient\(\selectlanguage{greek}\text{FPγ}\selectlanguage{english}_{i}\) at the flash point. The mole
fraction \(x_{i}\) is converted from mass fraction
\(m_{i}\). \(\text{FPPsa}t_{i}\) is the saturated vapor pressure
at flash point, which is calculated using the Antoine equation in Eq.
31.
\(\text{FP}P_{i}\ =\selectlanguage{greek}\text{FPγ}\selectlanguage{english}_{i}\bullet x_{i}\bullet FPP\text{sat}_{i}\)(28)
\(\selectlanguage{greek}\text{FPγ}\selectlanguage{english}_{i}=f_{\text{unifac}}\left(x_{i},T_{\text{fp}}\right)\)(29)
\(x_{i}=\frac{m_{i}}{MW_{i}\bullet\sum_{j}\frac{m_{j}}{MW_{j}}}\)(30)
\(\operatorname{}{\text{FPPsa}t_{i}}=A_{i}-\frac{B_{i}}{C_{i}+T_{\text{fp}}}\)(31)
The molecular weight \(MW_{i}\), UNIFAC parameters, and Antoine
coefficients \(A_{i}\), \(B_{i}\), and
\(C_{i}\) for the 48 ingredient candidates are given in Table
S2.
\subsubsection*{Homogeneous solution}
{\label{homogeneous-solution}}
To ensure a homogeneous solution, the volume of selected organic
fragrances must be less than their volume solubility
(\(SV_{i,ew}\)) in the ethanol-water solvent system.
\(\frac{V_{i}}{(V_{47}+V_{48})}\leq SV_{i,ew},\ \ \ i=1,\ldots,46\)(32)
It is found that it is quite hard to calculate \(SV_{i,ew}\) using
rigorous thermodynamic models due to the many missing parameters. In the
literature, several short-cut models have been developed to
predict\(SV_{i,ew}\). The log-linear mixture rule below is widely
used.\textsuperscript{54}
\(\log{\text{SV}_{i,ew}=}\log{SV_{i,w}}+\beta\bullet\log\frac{SV_{i,e}}{SV_{i,w}},\ \ \ \ i=1,\ldots,46\)(33)
\(\beta=\frac{V_{47}}{V_{47}+V_{48}}\) (34)
\(\log\frac{SV_{i,e}}{SV_{i,w}}=M\bullet\log K_{ow,i}+N,\ \ \ \ i=1,\ldots,46\)(35)
where \(SV_{i,e}\) and \(SV_{i,w}\) are the volume
solubility in ethanol and water, respectively.\(\ K_{ow,i}\) is the
n-octanol/water partition coefficient of the \emph{i} -th candidate.
\(M\) and \(N\) are the cosolvent constants.
Based on experimental data, their values have been regressed as 0.81 and
0.85, respectively.
\subsubsection*{Odor type in top note}
{\label{odor-type-in-top-note}}
The fragrance molecules in a perfume solution first evaporate into the
air through the liquid-gas interface. Then, the molecules diffuse in the
air (assumed to be stagnant) and are detected at certain distance away.
The processes of evaporation, diffusion, and detection have been
modelled using chemical engineering principles and
psychophysics.\textsuperscript{38,52,55} Perfume evaporation is
simulated using Eq. 36 with an initial condition. The liquid molar
changes are equal to the moles of ingredients transported through the
interface (i.e., \(z=0\)).
\(\frac{dn_{i,t}}{\text{dt}}=C_{T}\bullet D_{i}\bullet A_{\lg}\ \bullet\left.\ \frac{\partial y_{i,t,z}}{\partial z}\right|_{z=0}\)(36)
Initial condition: \(n_{i,t=0}=n_{p}\bullet x_{i}\)
After discretization, Eq. 37 is obtained.
\(\frac{n_{i,t+t}-n_{i,t}}{t}=C_{T}\bullet D_{i}\bullet A_{\lg}\ \bullet\frac{{y_{i,t,z=z_{1}}-y}_{i,t,z=0}}{z_{1}}\)(37)
where \(n_{p}\) is the initial number of moles of perfume
solution.\(C_{T}=P/RT\) is a constant.\(\ D_{i}\) and
\(A_{\lg}\) are the diffusivity of \emph{i} -th candidate and
interfacial area, respectively.\(t\) and
\(z_{1}\)are the time interval and the first distance
interval, respectively. These parameters are given in Table
S2.\(n_{i,t}\) is the number of moles of the \emph{i} -th
candidate in the liquid at time \(t\). \(y_{i,t,z}\)
is the molar fraction of \emph{i} -th ingredient candidate in the air at
time \(t\) at distance \emph{z} . It is calculated via
vapor-liquid equilibrium.
\(y_{i,t,z=0}=\gamma_{i,t}\bullet x_{i,t}\bullet\frac{\text{Psa}t_{i}}{P}\)(38)
\(\gamma_{i,t}=f_{\text{unifac}}(x_{i,t},T_{r})\) (39)
\(x_{i,t}=\frac{n_{i,t}}{\sum_{i=1}^{48}n_{i,t}}\) (40)
where \(\gamma_{i,t}\) and \(x_{i,t}\) are the activity
coefficient and mole fraction of \emph{i} -th ingredient candidate at
time \emph{t} , respectively. \(\text{Psa}t_{i}\) is the saturated vapor
pressure at room temperature \(T_{r}=298\ K\).
After evaporation, fragrance diffusion is modelled based on Fick's 2nd
law of diffusion with one initial condition and two boundary conditions
(Eq. 41).
\(\frac{\partial y_{i,t,z}}{\partial t}=D_{i}\bullet\frac{\partial^{2}y_{i,t,z}}{\partial z^{2}}\)(41)
Initial condition: \(y_{i,t=0,z}=0\)
Boundary conditions: Eq. 38, \(y_{i,t,z=z_{\max}}=0\)
The initial condition assumes that no fragrances exist in the air before
diffusion begins (i.e., \(t=0\)). The boundary conditions
indicate that vapor-liquid equilibrium is maintained at the interface at
any time (i.e., Eq. 38) and no fragrances exist beyond the maximum
distance (\(z_{\max}=2m\)). This model is discretized using a
non-uniform distance grid (Table S2) for reducing the computational
difficulty. After discretization, we get
\(\frac{y_{i,t+t,z}-y_{i,t,z}}{t}=D_{i}\bullet\frac{\frac{y_{i,t,z+z_{j+1}}-y_{i,t,z}}{z_{j+1}}-\frac{y_{i,t,z}-y_{i,t,z-z_{j}}}{z_{j}}}{0.5\times(z_{j+1}+z_{j})}\),\(z\in[0,z_{\max}]\) (42)
where \(z_{j}\) and \(z_{j+1}\) are the distance
intervals, respectively.
Any fragrance with a different concentration leads to a different
intensity. Many theoretical models (e.g., Weber-Fenchner law, power law,
and linear law) have been proposed for quantifying odor intensity. The
power law is chosen here because it fits experimental data well. The
intensity of the \emph{i} -th odorant is defined as the ratio of its
concentration in the air (\(c_{i}\) in g/m\textsuperscript{3})
to its odor recognition threshold value (\(\text{OR}T_{i}\)), raised to
a power\(oe_{i}\).\textsuperscript{52} With this, the odor
intensity in the top note is determined based on the mole fraction of
fragrances in the air at 5 minutes (\(t_{\text{tn}}\)) after
application at a distance of 0.2 m (\(z_{\text{tn}}\)).
\(\psi_{i}=\left(\frac{c_{i}}{\text{OR}T_{i}}\right)^{oe_{i}}\) (43)
\(c_{i}=y_{i,t_{\text{tn}},z_{\text{tn}}}\bullet MW_{i}\bullet C_{T}\)(44)
Given multiple odorants, the one with the highest intensity is more
strongly sensed and can be regarded as the major odor type. Thus, the
dominant odor type in top note is expressed as
\(OTTN=i,\ \ if\ \psi_{i}=\psi_{\max}\operatorname{=}\left\{\psi_{i}\right\}\)(45)
\subsubsection*{Heuristics}
{\label{heuristics}}
Following Table 3, constraints for the Eau de parfum formulation are
derived from dozens of modern Eau de parfum
recipes.\textsuperscript{51}It is found that the suggested number of
ingredients for each fragrance note can be represented by Eq. 46-48. Eq.
49 shows that Eau de parfum usually contains 10-20\% organic fragrances.
The suggested volumetric proportions for top note and middle note are
15-25\% and 30-40\%, respectively (Eq. 50-51). The suggested volume
fraction of water is 9-13\%.\textsuperscript{49,52}
\(3\leq\sum_{i=1}^{17}S_{i}\leq 6\) (46)
\(3\leq\sum_{i=18}^{33}S_{i}\leq 6\) (47)
\(2\leq\sum_{i=34}^{46}S_{i}\leq 5\) (48)
\(0.1\leq\sum_{i=1}^{46}V_{i}\leq 0.2\) (49)
\(0.15\bullet\sum_{i=1}^{46}V_{i}\leq\sum_{i=1}^{17}V_{i}\leq 0.25\bullet\sum_{i=1}^{46}V_{i}\)(50)
\(0.3\bullet\sum_{i=1}^{46}V_{i}\leq\sum_{i=18}^{33}V_{i}\leq 0.4\bullet\sum_{i=1}^{46}V_{i}\)(51)
\(0.09\leq V_{48}\leq 0.13\) (52)
\subsection*{Iterative Model Adoption and Optimization Solution
Strategy}
{\label{iterative-model-adoption-and-optimization-solution-strategy-1}}
The identified rigorous mechanistic models for \(LD_{50}\),
flash point, and odor type, the short-cut model for transparency, the
surrogate model for sensorial rating as well as the heuristics in Eq.
46-52 are integrated to form the perfume formulation problem below.
\(\operatorname{}q_{s}\) (53)
s.t. Eq. 21-23 ANN-based surrogate model for \(q_{s}\)
Eq. 16-18 Design targets
Eq. 24-45 Mechanistic models
Eq. 46-52 Heuristics
Eq. 19-20 Design variables
This problem is implemented in GAMS 24.7 on a laptop with Intel 3.30 GHz
CPU. The global solver BARON is used first and then the local solver SBB
is employed if no optimal solutions are obtained from BARON.
\subsubsection*{GDP reformulation}
{\label{gdp-reformulation-1}}
Because of the complexity of the identified models and the number of
intermediate variables, the problem is directly programmed using GDP.
The disjunction is explicitly expressed as
\(\par
\begin{bmatrix}Y_{i}\\
\par
\begin{matrix}VL_{i}\leq V_{i}\leq VU_{i}\\
\par
\begin{matrix}Eq.25\\
\par
\begin{matrix}Eq.27-31\\
Eq.37-44\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{bmatrix}\bigvee\par
\begin{bmatrix}\neg Y_{i}\\
\par
\begin{matrix}V_{i}=0\\
\par
\begin{matrix}m_{i}=0\\
\par
\begin{matrix}\text{FPLF}L_{i},FPP_{i},FP\gamma_{i},x_{i},FPPsat_{i}=0\\
n_{i,t},y_{i,t,z},\gamma_{i,t},x_{i,t},c_{i},\psi_{i}=0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{bmatrix}\) (54)
The GDP problem is further reformulated using the big-M approach with
the solver JAMS and then solved by SBB. Different initial guesses are
utilized. The second column of Table 5 lists the computational
statistics. It contains 46 discrete variables, 9783 single variables,
and 18230 equations. It takes 3459 seconds to obtain a local optimal
solution.
\textbf{{[}insert Table 5 here{]}}
The perfume formula obtained is shown in the second column of Table 6.
The maximum sensorial rating is 92.4. The new perfume consists of 3
fragrances in top note, 4 fragrances in middle note, and 3 fragrances in
base note. Their volume fractions vary in the range of 0.3-1.9\% and the
total volume fraction of fragrances is 10.1\%. Furthermore, this recipe
fulfills the four design targets. The \(LD_{50}\) and flash point
are 6815 mg/kg and 15.1 \selectlanguage{ngerman}°C, respectively. These are higher than their
lowest acceptable design targets (5000 mg/kg and 15 °C in Eq. 17-18).
The volume fractions of the 10 fragrances are less than their volume
solubility shown in Table S3. Thus, a homogeneous and transparent
perfume solution can be obtained. Figure 5a shows the odor profile
during the first 350 seconds. The intensity of benzyl acetate
(jasmine-like) and octyl acetate (apple-like) are 2.6 and 0.5,
respectively. The limonene with a lemon-like odor has the maximum
intensity of 2.8 after 5 minutes. Note that since the odor intensities
of other fragrances are much less than those of top note fragrances,
they are not shown in the figure. Figure 6a shows the simulated
diffusion profile of top note fragrances within 2 meters at 5 minutes.
Obviously, the maximum intensities are located at \(z=0\).
The maximal intensity of limonene can reach 33.5. As the distance
increases, the intensity decreases and down to zero beyond 2 meters. The
simulated diffusion profiles of 4 middle note fragrances at 1 hour and 3
base note fragrances at 5 hours are illustrated in Figure S2a and S2b,
respectively.
\textbf{{[}insert Figure 5 here{]}, {[}insert Figure 6 here{]}, and
{[}insert Table 6 here{]}}
\subsubsection*{Model substitution}
{\label{model-substitution-1}}
The above result from GDP (92.4 in Table 6) is slightly larger than the
maximal sensorial rating (89.7) of the original 761 data samples.
Although a local optimal solution has already been obtained, the GDP
problem is still challenging to solve. In fact, the choice of the
initial values greatly affects whether feasible solutions can be
obtained and the quality of local solution. It is found that the major
computational difficulties come from the rigorous mechanistic models for
perfume evaporation (Eq. 36) and diffusion (Eq. 41), which requires the
handling of many highly nonlinear equations. For instance, the
vapor-liquid equilibrium and UNIFAC equations must be calculated at
every time point (i.e., Eq. 38-40). Thus, in order to solve the
formulation problem more efficiently and find better solutions, model
substitution is employed here.
Whether the top note of a perfume can be dominated by a lemon-like or
non-lemon-like scent is a binary decision. Thus, the prediction of the
odor type can be transformed into a classification problem. In other
words, the complex mechanistic models (Eq. 36-45) for predicting the
odor type in the top note is substituted by a classification-based
surrogate model. To do so, random sampling is applied to generate 15000
artificial perfume recipes that account for the heuristic rules in Eq.
46-52. Among them, 7500 recipes consist of 0.25-0.75\% limonene
(lemon-like), 5000 recipes contain 0.75-1.25\%, and 2500 recipes have
1.25-1.75\%. These recipes are used as the input data. For each recipe,
their odor intensities in the top note are calculated using Eq. 36-45.
If a lemon-like odor has the highest intensity, the output is set equal
to 1. Otherwise, it is equal to 0. Then, a support vector classification
(SVC) model with linear kernel function is trained. Through 10-fold
cross validation, the hyperparameter \emph{C} indicating the
regularization strength is tuned to be 10. Figure S3 presents the
classification error distribution. For the 7500 data samples containing
0.25-0.75\% limonene, the classification accuracy is 93.3\%. For the
other half samples, the accuracy is 98.9 \%. The overall accuracy is
96.1\%. These statistics indicate that this SVC model can serve as a
relatively simple surrogate for substituting the original complex
mechanistic models. The SVC model consists of 2126 support vectors and
is expressed as
\(OTTN=\sum_{c=1}^{2126}{\alpha_{c}\bullet K_{c}+bs}\) (55)
\(K_{c}=\sum_{i=1}^{48}{SV_{c,i}}\bullet\text{VN}_{i}\) (56)
\(VN_{i}=\frac{V_{i}-V_{i,min}}{V_{i,max}-V_{i,min}}\) (57)
where \(\alpha_{c}\) and bs are the weights for
support vector and a constant, respectively. \(SV_{c,i}\) is the
support vector.\(V_{i,max}\) and \(V_{i,min}\) are
normalization coefficients. These parameters are optimized automatically
during the training process and provided in the Github platform
mentioned above.
By substituting Eq. 36-45 with Eq. 55-57, the resulting perfume
formulation problem (MINLP-SVC) is solved using the global solver BARON.
Table 5 shows the computational statistics. It consists of 2860 single
variables, 2920 equations, and 2928 nonlinear matrix entries. Clearly,
the problem size and nonlinearity are much less than those of the GDP
problem. It takes 143 seconds to obtain the global solution given in the
last column of Table 6. The maximum sensorial rating is 98.3 which is
better than the GDP result. The new perfume formula consists of 13
different fragrances in different volume fractions. The total volume
fraction of fragrances is 20\%. Moreover, the design targets
on\(LD_{50}\) and flash point are fulfilled. As listed in Table
S4, all the ingredient's volume fractions are less than their volume
solubility in the ethanol-water solvent. In addition, the major odor
type in the top note is classified as 1 (i.e., lemon-like) by the SVC
model. As validated using the original mechanistic models (Eq. 36-45),
Figure 5b shows the odor intensity in the first 350 seconds. Again, only
the top note fragrances are plotted. It is clear that the lemon-like
fragrance limonene has the maximum odor intensity (around 3.5) which is
higher than those of other fragrances. This validates the SVC results as
well. In addition, Figure 6b shows the diffusion profile of 4 top note
fragrances at 5 minutes, which is simulated using the original
mechanistic models. Figure S4a and S4b present the simulated diffusion
of 5 middle note fragrances at 1 hour and 4 base note fragrances at 5
hours, respectively.
\section*{Conclusion}
{\label{conclusion}}
This paper presents a new optimization-based approach for cosmetic
formulation. A three-step procedure is proposed to formulate the
cosmetic formulation problem as an MINLP problem. For problem
definition, the objective function (i.e., sensorial perception) and
design targets are identified. Then, a pool of potential ingredient
candidates is generated for selection. Design variables include
ingredient selection, composition, and microstructure descriptors (not
include in the example). Next, models are identified for predicting the
sensorial rating and target properties. Meanwhile, common heuristics are
translated into mathematical equations which serve as constraints to
narrow down the search space. To improve the optimization convergence
and to find better solutions, a solution strategy that involves an
iterative model adoption and different numerical techniques is proposed.
The procedure and solution strategy are illustrated using a perfume case
study. Our approach is one of the first attempts to integrate multiple
(rigorous, short-cut, surrogate, and heuristic-based) models to account
for both sensorial and functional attributes for optimal cosmetic
formulation. It can be used for other cosmetics and personal care
products provided that the relevant models, data, heuristics, etc. are
available.
Product design involves a wide range of issues that include consumer
preference, ingredient selection, supply chain analysis, process design,
government regulations, economics, corporate social responsibility,
sustainability and so on.\textsuperscript{56} These issues interact in
an exceedingly complex manner as captured in the Grand Product Design
Model.\textsuperscript{57} While many detailed models exist to describe
the separate issues, it is a daunting task to solve the optimization
problem for product design when a number of disparate issues are
involved. It is interesting to study how the approach described in this
paper can be extended to the product design as a whole. Efforts in this
direction are underway.
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Figure 1. The General Methodology of Optimization-based Cosmetic
Formulation
\textbf{Hosted file}
\verb`image1.emf` available at \url{https://authorea.com/users/322113/articles/451197-optimization-based-cosmetic-formulation-integration-of-mechanistic-model-surrogate-model-and-heuristics}
Figure 2. Systematic Procedure for Optimization Problem Formulation
\textbf{Hosted file}
\verb`image2.emf` available at \url{https://authorea.com/users/322113/articles/451197-optimization-based-cosmetic-formulation-integration-of-mechanistic-model-surrogate-model-and-heuristics}
Figure 3. Iterative Model Adoption and Optimization Solution Strategy
\textbf{Hosted file}
\verb`image3.emf` available at \url{https://authorea.com/users/322113/articles/451197-optimization-based-cosmetic-formulation-integration-of-mechanistic-model-surrogate-model-and-heuristics}
Figure 4. Absolute Error Distribution of ANN Model for Predicting
Sensorial Rating
\textbf{Hosted file}
\verb`image4.emf` available at \url{https://authorea.com/users/322113/articles/451197-optimization-based-cosmetic-formulation-integration-of-mechanistic-model-surrogate-model-and-heuristics}
Figure 5. Odor Profile of Top Note Fragrance Calculated Using Rigorous
Mechanistic Models for Optimal Perfume Recipe from (a) GDP Formulation
(b) MINLP-SVC Formulation
\textbf{Hosted file}
\verb`image5.emf` available at \url{https://authorea.com/users/322113/articles/451197-optimization-based-cosmetic-formulation-integration-of-mechanistic-model-surrogate-model-and-heuristics}
(a)
\textbf{Hosted file}
\verb`image6.emf` available at \url{https://authorea.com/users/322113/articles/451197-optimization-based-cosmetic-formulation-integration-of-mechanistic-model-surrogate-model-and-heuristics}
(b)
Figure 6. Simulated Diffusion of Top Note Fragrances at 5 Minutes for
Optimal Perfume Recipe from (a) GDP Formulation (b) MINLP-SVC
Formulation
\textbf{Hosted file}
\verb`image7.emf` available at \url{https://authorea.com/users/322113/articles/451197-optimization-based-cosmetic-formulation-integration-of-mechanistic-model-surrogate-model-and-heuristics}
(a)
\textbf{Hosted file}
\verb`image8.emf` available at \url{https://authorea.com/users/322113/articles/451197-optimization-based-cosmetic-formulation-integration-of-mechanistic-model-surrogate-model-and-heuristics}
(b)
Table 1. Dosage Form of Typical Cosmetic Products and the
Microstructural Descriptors\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
\begin{minipage}[b]{0.22\columnwidth}\raggedright\strut
Dosage form\strut
\end{minipage} & \begin{minipage}[b]{0.22\columnwidth}\raggedright\strut
Dosage form\strut
\end{minipage} & \begin{minipage}[b]{0.22\columnwidth}\raggedright\strut
Typical cosmetic products\strut
\end{minipage} & \begin{minipage}[b]{0.22\columnwidth}\raggedright\strut
Relevant microstructural descriptors\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Solid\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Stick\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Lipstick, contour stick\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Droplet size\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Tablet\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Foundation tablet, eyeshadow\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Tablet size, porosity, pore size\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Powder/granule\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Facial powder, blush\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Porosity, pore size, particle size and shape\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Semi-solid\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Paste\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Facial-mask paste, skin paste\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Emulsion type, droplet size\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Gel\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Eye gel, aftershave gel\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
/\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Ointment\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Hair pomade, facial scrub\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Droplet size\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Cream\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Hair cream, hand cream\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Emulsion type, droplet size\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Liquid\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Lotion\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Body lotion, lip gloss\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
Droplet size\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Suspension\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Nail polish, mascara\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Particle size and shape\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Solution\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Perfume, makeup remover\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
/\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Gas\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Aerosol\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Hair spray, shaving foam\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Droplet size\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}
Table 2. Sensorial and Functional Attributes of Four Cosmetic Products\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
& Sensorial attributes & Functional attributes & Relevant
properties\tabularnewline
\midrule
\endhead
Lipstick & sight, touch & no surface defect, hard to break, stable, safe
& color, color intensity, viscosity, strength, homogeneity, melting
point, thixotropy, pH\tabularnewline
Skin cream & touch, smell, sight & moisturizing, skin protection, ease
of use, stable, safe & viscosity, oiliness, odor, color, moisture
content, anti-oxidation, adhesion, pH\tabularnewline
Perfume & smell & transparent, safe & odor intensity, odor type, flash
point, toxicity, homogeneity\tabularnewline
Hair spray & sight, smell & effective, rapid drying, easy to remove,
stable, safe & color, odor, adhesion, curl retention, drying time, flash
point, toxicity\tabularnewline
\bottomrule
\end{longtable}
Table 3. Typical Heuristics and the Translated Constraints for
Formulated Products\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
Heuristics & Constraints\tabularnewline
\midrule
\endhead
Suggested number of ingredients & \(L\leq\sum_{i}{S_{i}\leq}U,\ \ \ i\in TIC\)\tabularnewline
Suggested number of ingredients in certain type &
\(TL\leq\sum_{i}{S_{i}\leq}\text{TU},\ \ \ i\in I_{X}\)\tabularnewline
Ingredients with certain property is preferred &
\(\text{PL}_{i}\bullet S_{i}\leq P_{i}\bullet S_{i}\leq\text{PU}_{i}\bullet S_{i},\ i\in TIC\)\tabularnewline
Certain ingredients cannot be used simultaneously &
\(\sum_{j}{S_{j}=1}\)\tabularnewline
Certain ingredients should be used simultaneously &
\(S_{j}=S_{k}\)\tabularnewline
Suggested concentration for certain type &
\(VTL\leq\sum_{i}{V_{i}\leq VTU},\ \ i\in I_{X}\)\tabularnewline
Suggested concentration for certain candidate &
\(VL_{i}\leq V_{i}\leq VU_{i},\ i\in TIC\)\tabularnewline
Total ingredient candidate \(TIC=\left\{I_{A,1},I_{A,2},\ldots,I_{Z,z}\right\}\) Certain ingredient type
\(I_{X}=I_{A},\ I_{B},\ldots,\) or \(I_{Z}\) \(L\),
TL, \(\text{VT}L\), \(VL_{i}\), and
\(\text{PL}_{i}\) are lower bounds. \(U\),
TU, \(\text{VT}U\), \(VU_{i}\), and
\(\text{PU}_{i}\) are upper bounds. & Total ingredient candidate
\(TIC=\left\{I_{A,1},I_{A,2},\ldots,I_{Z,z}\right\}\) Certain ingredient type \(I_{X}=I_{A},\ I_{B},\ldots,\) or
\(I_{Z}\) \(L\), TL,
\(\text{VT}L\), \(VL_{i}\), and \(\text{PL}_{i}\) are
lower bounds. \(U\), TU,
\(\text{VT}U\), \(VU_{i}\), and \(\text{PU}_{i}\) are
upper bounds.\tabularnewline
\bottomrule
\end{longtable}
Table 4. Ingredient Types, Functions, and Ingredient Candidates for
Perfume Example\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
\begin{minipage}[b]{0.30\columnwidth}\raggedright\strut
Ingredient type\strut
\end{minipage} & \begin{minipage}[b]{0.30\columnwidth}\raggedright\strut
\textbf{Top note fragrance}\strut
\end{minipage} & \begin{minipage}[b]{0.30\columnwidth}\raggedright\strut
\textbf{Top note fragrance}\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
Function\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
very volatile, appear immediately to offer the first impression, and
last 5-15 minutes\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
very volatile, appear immediately to offer the first impression, and
last 5-15 minutes\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
Candidates\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
1. allyl amylglycolate (galbanum-like) 2. alpha-phellandrene
(pepper-like) 3. benzylidene acetal (green leaf-like) 4. grapefruit
acetal (grapefruit-like) 5. isoamyl propionate (apricot-like) 6. linayl
propionate (bergamot-like) 7. methyl 2-octynoate (violet-like) 8. methyl
benzoate (blackcurrant-like) 9. propyl octanoate (coconut-like)\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
10. amyl butyrate (pear-like) 11. benzyl acetate (jasmine-like) 12.
limonene (lemon-like) 13. estragole (anise-like) 14. nerol (neroli-like)
15. nonyl aldehyde (rose-like) 16. octanal (orange-like) 17. octyl
acetate (apple-like)\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
Ingredient type\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Middle note fragrance}\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Middle note fragrance}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
Function\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
The body of perfume, dominate after top notes fade, and last up to 1
hour\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
The body of perfume, dominate after top notes fade, and last up to 1
hour\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
Candidates\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
18. amylcinnamaldehyde (jasmine-like) 19. cinnamic alcohol
(cinnamon-like) 20. cyclohexylethanol (patchouli-like) 21. ethyl
4-phenylbutyrate (plum-like) 22. ethyl o-anisate (ylang ylang-like) 23.
gamma-decalacetone (peach-like) 24. methyl iso-eugenol (carnation-like)
25. phenethyl isobutyrate (rose-like)\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
26. 1-phenylethanol (gardenia-like) 27. 2-undecanone (orris root-like)
28. amyl phenylacetate (cacao-like) 29. cedryl acetate (woody-like) 30.
heliotropin (heliotrope-like) 31. lilyall (lily-like) 32. linalyl
salicylate (musk-like) 33. methyl anthranilate (neroli-like)\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
Ingredient type\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Base note fragrance}\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Base note fragrance}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
Function\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
Lowest volatility, appear close to the end of middle notes to offer the
lasting impression, and remain several hours\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
Lowest volatility, appear close to the end of middle notes to offer the
lasting impression, and remain several hours\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
Candidates\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
34. acetyl cedrene (woody-like) 35. alpha-ambrinol (amber-like) 36.
amyl-iso-eugenol (incense-like) 37. coumarin (tonke bean-like) 38.
patchouli alcohol (patchouli-like) 39. phenethyl phenylacetate
(musk-like) 40. sandal hexanol (sandalwood-like)\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
41. benzoin (benzoin-like) 42. cedrol (cedar-like) 43. ethyl vanillin
(vanilla-like) 44. maltol (caramel-like) 45. phenylacetic acid
(honey-like) 46. vetiverol (vetiver-like)\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
Ingredient type\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Solvent}\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Solvent}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
Function\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
Dilute organic fragrant to adjust odor release\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
Dilute organic fragrant to adjust odor release\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
Candidates\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
47. ethanol\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
48. water\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}
Table 5. Computational Results for the Perfume Case Study\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
& GDP formulation & MINLP-SVC formulation\tabularnewline
\midrule
\endhead
Number of discrete variables & 46 & 46\tabularnewline
Number of single variables & 9783 & 2860\tabularnewline
Number of equations & 18230 & 2920\tabularnewline
Number of nonlinear matrix entries & 42814 & 2928\tabularnewline
Solver & SBB & BARON\tabularnewline
CPU time (s) & 3459 & 143\tabularnewline
\bottomrule
\end{longtable}
Table 6. Optimal Perfume Formula Obtained from Two Optimization
Formulation\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
& GDP formulation & GDP formulation & MINLP-SVC formulation & MINLP-SVC
formulation\tabularnewline
\midrule
\endhead
& Recipe & \(V_{i}\)\textsuperscript{*} & Recipe &
\(V_{i}\)\textsuperscript{*}\tabularnewline
Top note fragrance & benzyl acetate & 1.6 & octanal & 2.0\tabularnewline
& limonene & 0.5 & benzyl acetate & 1.5\tabularnewline
& octyl acetate & 0.3 & limonene & 1.3\tabularnewline
& & & benzylidene acetal & 0.3\tabularnewline
Middle note fragrance & heliotropin & 1.9 & lilyall & 2.1\tabularnewline
& 2-undecanone & 1.6 & phenethyl isobutyrate & 1.7\tabularnewline
& 1-phenylethanol & 0.3 & heliotropin & 1.3\tabularnewline
& lilyall & 0.3 & amyl phenylacetate & 1.0\tabularnewline
& & & 1-phenylethanol & 0.7\tabularnewline
Base note fragrance & sandal hexanol & 1.9 & phenylacetic acid &
4.9\tabularnewline
& benzoin & 1.4 & maltol & 1.6\tabularnewline
& coumarin & 0.3 & benzoin & 1.3\tabularnewline
& & & phenylethyl phenylacetate & 0.3\tabularnewline
Solvent & ethanol & 77.6 & ethanol & 70.8\tabularnewline
& water & 12.3 & water & 9.2\tabularnewline
LD\textsubscript{50} & 6815 mg/kg & 6815 mg/kg & 7139 mg/kg & 7139
mg/kg\tabularnewline
Flash point & 15.1 \selectlanguage{ngerman}°C & 15.1 \selectlanguage{ngerman}°C & 15.1 \selectlanguage{ngerman}°C & 15.1 \selectlanguage{ngerman}°C\tabularnewline
Homogeneous & All fragrances dissolved \textsuperscript{**} & All
fragrances dissolved \textsuperscript{**} & All fragrances
dissolved\textsuperscript{***} & All fragrances
dissolved\textsuperscript{***}\tabularnewline
OTTN & Lemon-like & Lemon-like & Lemon-like &
Lemon-like\tabularnewline
Sensorial rating & 92.4 & 92.4 & 98.3 & 98.3\tabularnewline
\textsuperscript{*}Volume fraction in percentage (\%)
\textsuperscript{**}Volume solubility data in Table S3
\textsuperscript{***}Volume solubility data in Table S4 &
\textsuperscript{*}Volume fraction in percentage (\%)
\textsuperscript{**}Volume solubility data in Table S3
\textsuperscript{***}Volume solubility data in Table S4 &
\textsuperscript{*}Volume fraction in percentage (\%)
\textsuperscript{**}Volume solubility data in Table S3
\textsuperscript{***}Volume solubility data in Table S4 &
\textsuperscript{*}Volume fraction in percentage (\%)
\textsuperscript{**}Volume solubility data in Table S3
\textsuperscript{***}Volume solubility data in Table S4 &
\textsuperscript{*}Volume fraction in percentage (\%)
\textsuperscript{**}Volume solubility data in Table S3
\textsuperscript{***}Volume solubility data in Table S4\tabularnewline
\bottomrule
\end{longtable}
\selectlanguage{english}
\FloatBarrier
\end{document}