KPI Normalization

This section describes several normalization schemes used to convert KPI values into a score in the range \([0, 100]\). Throughout, \(x\) denotes the KPI value.

Linear Bounded Normalization

For KPIs with a bounded acceptable range \([a, b]\), with \(a < b\), the normalization function is defined as:

\[ \mathcal N(x, a, b) = 100 \, \left( \frac{x - a}{b - a} \right) \]

Here,

  • \(a\) corresponds to the worst score (0),

  • \(b\) corresponds to the best score (100).

Linear Half-Open Normalization

For KPIs that have a minimum acceptable value \(a\) but no finite upper limit, i.e. values in \([a, +\infty[\), we define:

\[ \mathcal N(x, a, m) = 100 \, \max \left(0, 1 - \frac{x-a}{m-a} \right) \]

where \(m > a\) is a parameter specifying the value of \(x\) at which the score reaches 0.

Here,

  • \(a\) corresponds to the best score (100),

  • \(m\) corresponds to the worst score (0).

Exponential Half-Open Normalization

For KPIs with a minimum acceptable value \(a\) and domain \([a, +\infty[\), we define an exponentially decaying score:

\[ \mathcal N(x, a, m) =100 \, \exp \left( - \frac{\ln 2 \, (x-a)}{m-a} \right). \]

This formulation ensures:

\[ \mathcal N(a,a,m) = 100, \qquad \mathcal N(m,a,m) = 50. \]

Thus, \(m\) is the KPI value at which the score is exactly 50. Here,

  • \(a\) corresponds to the best possible score (100),

  • \(m\) corresponds to the value at which the score has decreased to 50,

  • scores decay smoothly toward 0 as \(x \to +\infty\).

Symmetric Linear Open Normalization

For KPIs that have a symmetry around 0 but no finite limit, i.e. values in \([-\infty, +\infty[\), we define:

\[ \mathcal N(x, m) = 100 \, \max \left(0, 1 - \frac{|x|}{m} \right) \]

where \(m > a\) is a parameter specifying the value of \(x\) at which the score reaches 0.

Here,

  • \(a\) corresponds to the best score (100),

  • \(m\) corresponds to the worst score (0).

Symmetric Exponential Open Normalization

For KPIs that have a symmetry around 0 but no finite limit, i.e. values in \([-\infty, +\infty[\), we define:

\[ \mathcal N(x, m) =100 \, \exp \left( - \frac{\ln 2 \, |x|}{m} \right). \]

This formulation ensures:

\[ \mathcal N(0,m) = 100, \qquad \mathcal N(\pm m,m) = 50. \]

Thus, \(m\) is the KPI value at which the score is exactly 50. Here,

  • \(0\) corresponds to the best possible score (100),

  • \(m\) corresponds to the value at which the score has decreased to 50,

  • scores decay smoothly toward 0 as \(x \to \pm\infty\).