Food product modelling

Among other information, the Open Food Facts database contains the ingredient list and nutritional composition of each product. Coupling this information to physical and regulatory constraints makes it possible to guess the recipe of the product. In order to do this, the following conceptual framework have been developed.

Let a product of mass \(F\) be composed of a set of ingredients \(I\). Let \(M\) be the total mass of ingredients used before processing. The ingredients processing may induce a water loss (cooking or drying for example), thus:

\[M \ge F\]

Let \(m_i\) be the mass of ingredient \(i \in I\) used and \(p_i\) its proportion of the total mass \(M\).

\[\begin{split}M = \sum_{i \in I}{m_i}\\ \forall i \in I, m_i = M \cdot p_i\\\end{split}\]

\(F\) is known but \(M\) is not and the proportions \(p_i\) are generally not. However, the impacts of the product depend on the masses of ingredients \(m_i\).

Note

This equation is implemented by _add_total_leaves_percentage_constraint()

Ingredients order

The ingredient list is given by decreasing proportion at the moment of incorporation in the product, if this proportion is superior to 2%.

\[\forall i \in I, p_{i+1} \le \max(p_i, 0.02)\]

Note

This equation is implemented by _add_mass_order_constraints() and _remove_decreasing_order_constraint_from_rank().

Compound ingredients

The ingredients composing a food product can themselves be composed by sub-ingredients. Let \(i\) be an ingredient of \(I\). Let \(J\) be the set of ingredients composing \(i\). Let \(m_j\) be the mass of ingredient \(j \in J\) used and \(p_j\) its proportion of the mass \(m_i\).

\[\begin{split}m_i = \sum_{j \in J}{m_j}\\ \forall j \in J, m_j = m_i \cdot p_j\\\end{split}\]

Note

This equation is implemented by _add_total_subingredients_percentages_constraint()

Nutriments balance

Let \(N\) be the set of nutrient types considered. Let \(F_n\) be the mass of the nutrient \(n \in N\) in the product according to its packaging information. This mass has an error margin depending on its value which can be relative (in percent) or absolute (in grams). Let \(M_n\) be the mass of nutriment \(n \in N\) actually contained in the product.

Nutriment information error margins (source: INCO EU regulation)

Nutrient	Content (g/100g)	Error margin
Carbohydrates, sugars, proteins, fibers	<10	±2g
	[10 – 40]	±20%
	>40	±8g
Fats	<10	±1.5g
	[10 – 40]	±20%
	>40	±8g
Saturated fats	<10	±1.5g
	[10 – 40]	±20%
	>40	±8g
Sodium	<0.5	±0.15g
	>0.5	±20%
Salt	<1.25	±0.375g
	>1.25	±20%

This error margin can be modeled as two separate margins \(\varepsilon_n\) and \(\delta_n\), respectively absolute and relative, with one and only one of them being non-zero. We note \(c_{n,i}\) (resp. \(m_{n,i}\)) the content (resp. mass) of nutrient \(n\) in the ingredient \(i\).

\[\begin{split}\forall n \in N, (1 - \delta_n) F_n - \varepsilon_n &\le M_n \le (1 + \delta_n) F_n + \varepsilon_n \\ (1 - \delta_n) F_n - \varepsilon_n &\le \sum_{i \in I}{m_{n,i}} \le (1 + \delta_n) F_n + \varepsilon_n \\ (1 - \delta_n) F_n - \varepsilon_n &\le M \sum_{i \in I}{p_i \cdot c_{n,i}} \le (1 + \delta_n) F_n + \varepsilon_n\end{split}\]

The nutrient content of ingredients is not necessarily known but it can be bounded by the values \(c_{min,n,i}\) and \(c_{max,n,i}\).

\[\forall n \in N, \forall i \in I, 0 \le c_{min,n,i} \le c_{n,i} \le c_{max,n,i} \le 1\]

Thus:

\[\begin{split}M\sum_{i \in I}{p_i\cdot c_{max,n,i}} \ge (1 - \delta) F_n - \varepsilon_n \\ M\sum_{i \in I}{p_i\cdot c_{min,n,i}} \le (1 + \delta) F_n + \varepsilon_n\end{split}\]

Note

These equations are implemented by _add_nutritional_constraints(). nutritional_error_margin() gives the relative and absolute margin for a nutriment and a value.

Water balance

Evaporation is modeled as a proportion \(E\) of the water in each ingredient that is lost during the product processing. Let \(c_{w,i}\) be the water content of ingredient \(i\).

\[\begin{split}F &= M - E \sum_{i \in I}{m_i \cdot c_{w,i}} \\ F &= M \left( 1- E \sum_{i \in I}{p_i \cdot c_{w,i}} \right)\end{split}\]

Considering that the water content \(c_{w,i}\) of ingredient \(i\) is between two bounds \(c_{min,w,i}\) and \(c_{max,w,i}\), we have:

\[\forall i \in I, 0 \le c_{min,w,i} \le c_{w,i} \le c_{max,w,i} \le 1\]

Thus:

\[M \left(1 - E \sum_{i \in I} p_i \cdot c_{max,w,i}\right) \le F \le M \left(1 - E \sum_{i \in I} p_i \cdot c_{min,w,i}\right)\]

Note

This equation is implemented by _add_evaporation_constraint()

Moreover, \(F\) can be used to bound the value of the total mass used \(M\). Indeed, in the case where the product is only made of water, we have:

\[F = M(1-E)\]

By extending it to the general case, we can deduce:

\[F \le M \le \frac{F}{1-E}\]

Note

This equation is implemented by _add_used_mass_constraint()

Mass balance

Considering that food products consist only of water and nutrients, we have:

\[\begin{split}F &= \sum_{i \in I}{m_i \cdot c_{w,i} \cdot (1-E) } + \sum_{n \in N}{F_n} \nonumber \\ F &= \sum_{i \in I}{m_i \cdot c_{w,i} \cdot (1-E) } + \sum_{i \in I}{\sum_{n \in N}{m_i \cdot c_{n,i}}} \nonumber \\ F &= \sum_{i \in I}{m_i \left( (1-E) c_{w,i} + \sum_{n \in N}{c_{n,i}}\right)} \nonumber \\ F &= M\sum_{i \in I}{p_i \left( (1-E) c_{w,i} + \sum_{n \in N}{c_{n,i}}\right)} \nonumber \\\end{split}\]

\[M\sum_{i \in I}{p_i \left( (1-E) c_{min,w,i} + \sum_{n \in N}{c_{min,n,i}}\right)} \le F \le M\sum_{i \in I}{p_i \left( (1-E) c_{max,w,i} + \sum_{n \in N}{c_{max,n,i}}\right)}\]

Note

This equation is implemented by _add_product_mass_constraint()

Product environmental impact

Let \(C\) be a set of environmental impact categories, we note \(\alpha_{c,i}\) the mass impact of ingredient \(i\) in category \(c\). Let us consider a product composed of a set of ingredients \(I\). The impact \(A_c\) of the product in category \(c\) is defined by the sum of the used mass of its ingredients weighted by their impact per mass unit.

\[\forall c \in C, A_c = \sum_{i\in I} m_i \cdot \alpha_{c,i}\]

Note

This equation is implemented by _compute_impact()