How to Define the “Best” Object in Multi‑Criteria Evaluation Models

In daily life people often compare, rank, or score objects to judge how good they are, using the gap from the "best" as a basis for evaluation.

To build a mathematical evaluation model, we first define key terms:

Evaluation object : the items being assessed.

Evaluation indicator : a vector of attributes used to measure each object; each component reflects a specific aspect.

Weight coefficient : quantifies the relative importance of each indicator.

Indicator value : the observed value of each indicator for each object, assumed to be larger‑is‑better and already normalized.

1. Approach 1: “All Items Full Marks”

If we treat the maximum possible score for every indicator as the "best" reference, an object closer to these full marks is considered better.

2. Approach 2: “Combine All Advantages”

Instead of requiring full marks, we construct a virtual "ideal" object whose indicator values are the highest among all real objects (not necessarily the maximum possible). The distance of each real object from this ideal determines its ranking.

We can also use a grey relational coefficient to measure similarity, where a larger discrimination coefficient yields finer resolution.

3. Approach 3: TOPSIS – Near the Good, Far from the Bad

The TOPSIS method introduces both a positive ideal solution (the best values for each indicator) and a negative ideal solution (the worst values). By calculating the Euclidean distances of each object to these two ideals, a composite score is obtained; larger scores indicate better performance.

4. Approach 4: Data Envelopment Analysis (DEA)

DEA evaluates whether an object achieves the highest efficiency by weighting inputs to be no greater than those of the reference object while ensuring weighted outputs are at least as large. This method synthesizes all indicators into a single efficiency measure.

5. Summary

The article outlines several ways to define the "best" reference in evaluation models, presenting the corresponding modeling ideas and symbolic expressions.