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.
Step 1: Problem
definition
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.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 °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.
Step 2: Ingredient candidate
generation
Table 4 lists the four required ingredients types and their
functions.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.50
Referring to the perfume manual,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.52 Ingredient
selection is controlled by the binary variable \(S_{i}\) and ingredient
composition is represented by volume fraction \(V_{i}\). If thei -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.