Scalability¶

Some Construction Products are available in varying sizes, each with a different MKI. An illustrative example is the variability in thickness among concrete tiles. The Environmental Declaration of such Construction Products contain one or more scalable Environmental Profiles. This scaling is based on a specific dimension, such as thickness.

The Score Matrix of a scaled Environmental Profile is defined as:

$\textbf{W}_\text{scaled profile,x} &= f(x) \cdot \textbf{W}_\text{reference}$

Scaled Environmental Profile $\textbf{W}_\text{scaled profile,x}$: Score matrix resulting from multiplication with the scale factor
Scaling formula $f(x)$: Expresses the scaling factor in terms of x
Dimension value $x$: Value of the dimension on which the scaling is based
Reference Environmental Profile $\textbf{W}_\text{reference}$: Score matrix to which the scale factor is applied

Note

$\textbf{W}_\text{reference}$ is produced using the inspected value, $x_0$ , as dimension. And thus, $f(x_0) = 1$ :

$\textbf{W}_{\text{scaled profile},x_0} &= f(x_0) \cdot \textbf{W}_\text{reference} \\ &= 1 \cdot \textbf{W}_\text{reference} \\ &= \textbf{W}_\text{reference}$

Scaling formula¶

Each Scalable Environmental Profile contains a formula-type and a set of parameters for this formula. There are 2 different sets of scaling formulas that are used: v3 and v4. Instrumenthouders are required to implement both to make full use of scaling.

V4 scaling formulas¶

As of January 2025, the v4 scaling formulas are used for new declarations. The v4 scaling formulas and their respective identifiers are:

Linear - v4_linear
Nonlinear - v4_nonlinear

The set of parameters is in the form $[a, b]$ for v4_linear or $[a, b, c, d]$ for v4_nonlinear.

Linear¶

$f(x) &= {ax + b \over {a x_\text{0} + b }}$

Nonlinear¶

$f(x) &= {ax^3 + bx^2 + cx + d \over {a x_\text{0}^3 + b x_\text{0}^2 + c x_\text{0} + d}}$

The inspected value $x_0$: The value of the scaling dimension used to determine the $\textbf{W}_\text{reference}$
Scaling parameter $a$: The first scaling parameter in the parameters list.
Scaling parameter $b$: The second scaling parameter in the parameters list.
Scaling parameter $c$: The third scaling parameter in the parameters list.
Scaling parameter $d$: The fourth scaling parameter in the parameters list.

V3 scaling formulas¶

Wherever v4 formulas are not used yet, v3 scaling formulas will be used instead. The v3 scaling formulas and their respective identifiers are:

Linear - v3_linear
Power - v3_power
Logarithmic - v3_logarithmic
Exponential - v3_exponential

The set of parameters is of the form $[p_1, p_2, p_3]$

Linear¶

$f(x) &= {ax + b \over {a x_\text{0} + b}}$

The inspected value $x_0$: The value of the scaling dimension used to determine the $\textbf{W}_\text{reference}$
Scaling parameter $p_1 = a$: The first scaling parameter $p_1$ in the parameters list. This parameter is referred to as ‘A’ in the legacy API.
Scaling parameter $p_2$: Unused for linear scaling. Should be ignored
Scaling parameter $p_3 = b$: The third scaling parameter $p_3$ in the parameters list. This parameter is referred to as ‘C’ in the legacy API

Power¶

$f(x) &= {ax^n + b \over {a x_\text{0}^n + b}}$

The inspected value $x_0$: The value of the scaling dimension used to determine the $\textbf{W}_\text{reference}$
Scaling parameter $p_1 = a$: The first scaling parameter $p_1$ in the parameters list. This parameter is referred to as ‘A’ in the legacy API.
Scaling parameter $p_2 = n$: The second scaling parameter $p_2$ in the parameters list, used to define the exponent in a power formula. This parameter is referred to as ‘B’ in the legacy API
Scaling parameter $p_3 = b$: The third scaling parameter $p_3$ in the parameters list. This parameter is referred to as ‘C’ in the legacy API

Logarithmic¶

$f(x) &= {a \cdot \ln(x) + b \over {a \cdot \ln(x_\text{0}}) + b}$

The inspected value $x_0$: The value of the scaling dimension used to determine the $\textbf{W}_\text{reference}$
Scaling parameter $p_1 = a$: The first scaling parameter $p_1$ in the parameters list. This parameter is referred to as ‘A’ in the legacy API.
Scaling parameter $p_2$: Unused for linear scaling. Should be ignored
Scaling parameter $p_3 = b$: The third scaling parameter $p_3$ in the parameters list. This parameter is referred to as ‘C’ in the legacy API

Exponential¶

$f(x) &= {a \cdot \exp(k x) + b \over {a \cdot \exp(k x_\text{0})} + b}$

The inspected value $x_0$: The value of the scaling dimension used to determine the $\textbf{W}_\text{reference}$
Scaling parameter $p_1 = a$: The first scaling parameter $p_1$ in the parameters list. This parameter is referred to as ‘A’ in the legacy API.
Scaling parameter $p_2 = k$: The second scaling parameter $p_2$ in the parameters list, used in the exponential formula. This parameter is referred to as ‘B’ in the legacy API.
Scaling parameter $p_3 = b$: The third scaling parameter $p_3$ in the parameters list. This parameter is referred to as ‘C’ in the legacy API

Two-dimensional scaling¶

Some Scalable Environmental Profiles define scaling on two dimensions (eg. length and width). In such cases, $x$ is defined as the product of the values in both scaling dimensions:

$x &= x_\text{dimension1} \cdot x_\text{dimension2}$