Standardizing Evaluation Indicators: Convert Small, Central, and Interval Metrics to Large Scores

1 Indicator Standardization

Indicator standardization means unifying the direction of evaluation metrics. For example, if a metric yields higher scores when the value is smaller, we transform it so that larger values correspond to higher scores.

In an evaluation metric system, there may be extremely large, extremely small, central, and interval-type indicators, each with distinct characteristics. When multiple types coexist, it is common to standardize them before comprehensive evaluation.

For instance, all indicators can be transformed into extremely large metrics; the usual approach is to convert non‑extremely large indicators into extremely large ones.

2 Transforming Extremely Small Indicators into Extremely Large Indicators

For an extremely small indicator, we can take its reciprocal to obtain an extremely large indicator.

Alternatively, a translation transformation can be applied:

where x_{max} is the maximum value of the j‑th indicator for the i‑th evaluation object.

Other monotonicity‑preserving transformations are also feasible.

3 Transforming Central Indicators into Extremely Large Indicators

For a central indicator, let a be the midpoint and apply a suitable scaling (e.g., shifting and stretching) so that the transformed value increases with performance, effectively converting the central indicator into an extremely large indicator.

4 Transforming Interval Indicators into Extremely Large Indicators

For an interval indicator, the optimal range is [L, U] . Values within this interval are best, and the farther the value from the interval, the worse the score. By applying an appropriate transformation (e.g., distance‑to‑interval mapping), the interval indicator can be turned into an extremely large indicator.

5 Summary

This article introduced three types of indicator standardization (extremely small, central, and interval) and presented methods to convert each into an extremely large indicator.

References

ThomsonRen https://github.com/ThomsonRen/mathmodels

司守奎，孙玺菁 Python数学实验与建模