[insert Figure 5 here], [insert Figure 6 here], and [insert Table 6 here]

Model substitution

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 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.