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.