# Generation of Dissimilar Alternative Product Formulations Using Graphs

## Abstract

**:**

## 1. Introduction

## 2. Generation of Alternative Product Formulations

**Problem (P1)**:

## 3. Matrix and Graphical Representations of a Set of Product Formulations

## 4. Generation of Sets of Dissimilar Product Formulations

**Problem (P2)**). If $h$ and $g$ are linear functions, Problem (P2) is a MILP problem.

**Problem (P3)**—where the mean value of the design objective (or other appropriate measure) is minimized subject to $NEE\le $ ${NEE}^{*}$:

## 5. Example of a Cosmetic Emulsion

#### 5.1. Modelling of Heuristic Rules

**Heuristic 3.2.**Let $z\in \left\{\mathrm{0,1}\right\}$ and $z=1\u27fa{y}_{n}=0,\forall n$ (no thickening polymers are used). Then, the logical condition to be modelled is as follows:

**Heuristic 3.3.**In order to model Heuristic 3.3, the viscosity model has to be reformulated in such a way that if $z=1$ viscosity limits are obeyed. Let ${l\mu}_{1}=\mathrm{log}\left({\mu}_{1}\right)$ and ${l\mu}_{1c}={\sum}_{n}{(a}_{n}{x}_{n}+{b}_{n}{y}_{n}\left)\right]$. Then, the two following restrictions can describe both the viscosity model, ${l\mu}_{1}={(l\mu}_{1c}+c\varphi )$, and Heuristic 3.3:

#### 5.2. Sets of Dissimilar Alternative Formulations

^{®}Core™ i7-1065G7 processor.

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Matrix and graph representations of 7 product formulations (${f}_{1}$ to ${f}_{7}$), each one with components chosen from a pool of 18 possible components (${c}_{1}$ to ${c}_{18}$). A partition in three clusters (A, B, and C) is also shown. In the matrix, these clusters correspond to diagonal blocks; in the graph, the clusters are delineated by dashed lines.

**Figure 2.**Sets of formulations with maximum dissimilarity and minimum average cost. Set $F1$ was obtained with $NF=3$, set $F2$ was obtained with $NF=5$, and set $F3$ was obtained with $NF=7$. Each set is represented in matrix form, with each selected ingredient being depicted as a square; darker squares correspond to higher mass fraction values based on the grayscale shown at the bottom.

Design variables: Choice of ingredients (binary variables $y$) and respective mass fractions $x$; pair $\{y,x\}$ for each one of six subsets of ingredients: $i$, $j$, $k$, $m$, $n$, and $r$. |

Product performance specifications 1. Initial viscosity ${\mu}_{1}$ (perceived on the onset of flow of the product) between 1350 and 5000 Pa·s [27] 2. Final viscosity ${\mu}_{2}$ (at ~500 s, perceived during product application on hair) between 0.023 and 1.0 Pa·s [27] 3. Greasiness value ($\gamma $) between 2.0 and 2.4 [28] |

Property models 1. Initial viscosity: $\mathrm{log}\left({\mu}_{1}\right)={\sum}_{n}{(a}_{n}{x}_{n}+{b}_{n}{y}_{n})+c\varphi $; final viscosity: $\mathrm{log}\left({\mu}_{2}\right)={\sum}_{n}{(d}_{n}{x}_{n}+{e}_{n}{y}_{n})+f\varphi $, with log being decimal logarithm. Since, at most, one thickener is used, only one term of the sum ${\mathsf{\Sigma}}_{n}$ is non-zero. Parameters ${a}_{n},{b}_{n},{d}_{n}$, and ${e}_{n}$ are estimated from experimental data of aqueous solutions of $n$ [29] (see Table 2 below); $c$ and $f$ (effect of the dispersed phase) are estimated from the Yaron and Gal-Or theoretical model [30]: $c=1$; $f=2.$ 2. Greasiness value $\gamma $ given by a linear mixing rule using known $\gamma $ values for each emollient [31]. |

Heuristic rules ^{1}1. Oil phase mass fraction: $\mathsf{\varphi}={\sum}_{q}{x}_{q}\le 0.25,q=\{i,j,k,m,r\}$ 2. Cationic surfactants 2.1 Heuristic for “moist and soft” product [32]: no $r1$, $2{x}_{r3}\le {x}_{r2}\le 4{x}_{r3}$ 2.2 General limits: ${x}_{r1}\ge 0.01$, ${x}_{r2}\ge 0.01$, ${x}_{r1}+{x}_{r2}\le 0.03$ 2.3 Cationic surfactants at about 20% of the oil phase stabilizes the emulsion: $0.16\varphi \le {\sum}_{r}{x}_{r}\le 0.24\varphi $ 3. Fatty alcohols 3.1 Use at least one; 3 to 8% each; total lower than 8%: ${\sum}_{m}{y}_{m}\ge 1$; $0.03{y}_{m}\le {x}_{m}\le 0.08{y}_{m}$; ${\sum}_{m}{x}_{m}\le 0.08$ 3.2 When no thickening polymers are used, the concentration of fatty alcohols is at least twice that of cationic surfactants’ in a molar base [33] (mathematical formulation given in the main text). 3.3 If Heurist 3.2 holds, then ${\mu}_{1}$ and ${\mu}_{2}$ are expected to be within specifications (mathematical formulation in the main text). 4. Emollients 4.1 Use at least one emollient of each type [34]: ${\sum}_{i}{y}_{i}\ge 1$; ${\sum}_{j}{y}_{j}\ge 1$; ${\sum}_{k}{y}_{k}\ge 1$ 4.2 Use a minimum of 1% of each; total greater than 6%: $0.03{y}_{i}\le {x}_{i}\le {y}_{i}$, idem for $j$ and $k$; ${\sum}_{i}{x}_{i}+{\sum}_{j}{x}_{j}+{\sum}_{k}{x}_{k}\ge 0.06$ 5. Thickening polymers 5.1 Use only one: ${\sum}_{n}{y}_{n}\le 1$ 5.2 Limits: $0.0015{y}_{n1}\le {x}_{n1}\le 0.03{y}_{n1}$, $0.005{y}_{n1}\le {x}_{n1}\le 0.02{y}_{n1}$, $0.004{y}_{n1}\le {x}_{n1}\le 0.03{y}_{n1}$ |

^{1}Unreferenced heuristics result from the author and co-workers experience.

${a}_{n}$ | ${b}_{n}$ | ${d}_{n}$ | ${e}_{n}$ | |

$n1$ | 223.42 | 2.4839 | 77.181 | 0.7636 |

$n2$ | 271.13 | 2.0017 | 78.351 | 1.5196 |

$n3$ | 109.27 | 3.4731 | 174.74 | 0.2526 |

**Table 3.**Data regarding the sets of formulations of Figure 2. $NC$ is the number of selected components (out of a total of 32); $\overline{C}$ is the average cost in EUR/kg; CPU time is for Problem (P3) (Problem (P2) requires less than 0.3 s in all cases).

Set | $\mathit{N}\mathit{F}$ | $\mathit{N}\mathit{C}$ | $\mathit{N}\mathit{E}$ | $\mathit{N}\mathit{E}\mathit{E}$ | $\mathit{F}\mathit{E}\mathit{E}$ | $\overline{\mathit{C}}$ | CPU (s) |
---|---|---|---|---|---|---|---|

$F1$ | 3 | 14 | 19 | 5 | 0.263 | 1.05 | 0.3 |

$F2$ | 5 | 20 | 31 | 11 | 0.355 | 1.21 | 3.0 |

$F3$ | 7 | 25 | 43 | 18 | 0.419 | 1.36 | 395.8 |

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**MDPI and ACS Style**

Bernardo, F.P.
Generation of Dissimilar Alternative Product Formulations Using Graphs. *Processes* **2023**, *11*, 3152.
https://doi.org/10.3390/pr11113152

**AMA Style**

Bernardo FP.
Generation of Dissimilar Alternative Product Formulations Using Graphs. *Processes*. 2023; 11(11):3152.
https://doi.org/10.3390/pr11113152

**Chicago/Turabian Style**

Bernardo, Fernando P.
2023. "Generation of Dissimilar Alternative Product Formulations Using Graphs" *Processes* 11, no. 11: 3152.
https://doi.org/10.3390/pr11113152