Criteria Removal

Criteria Removal is not a ranking method on its own. You already hold a decision table, a set of weights and a baseline ranking built from the two; Criteria Removal is a robustness test laid over that ranking. DecisionMind uses classical TOPSIS as the base method for this test. The method removes each criterion from the table in turn, recomputes the ranking with the remaining criteria, and compares the new ranking with the original one. Its output is not a single "best alternative" but a stability score for every criterion: how much the ranking would move if that criterion were absent. It adapts the sensitivity-analysis tradition assembled systematically by Saltelli and colleagues (2004) to the decision table; within MCDM, Triantaphyllou and Sánchez (1997) examined a similar question, namely how changes to criteria and weights affect a ranking.

The idea resembles pulling the legs out from under a table one at a time. If the table stands on four legs, you remove one and see whether it still stands steady. Criteria Removal does the same: it pulls one criterion and checks whether the ranking still stands in the same shape. If removing a criterion leaves the ranking completely unchanged, that criterion either carries information already captured by the others or was never decisive to begin with. If removing a criterion reverses the ranking, the ranking in fact rests on that single criterion.

This perspective carries a consequence. Criteria Removal is neither compensatory nor eliminative; it is a diagnostic layer placed over an existing decision. It makes no claim of its own about the "best alternative", only an answer to the question "how solid is this claim". It should therefore be read in reports alongside the underlying baseline ranking, never on its own.

The method proceeds through four steps.

First, the baseline ranking. DecisionMind passes the decision table and the weights to classical TOPSIS and obtains a closeness score and a baseline rank for every alternative. This step follows TOPSIS's own five steps exactly (normalisation, weighting, ideal points, distance, closeness score).

Second, removing criteria one at a time. With n criteria, the method repeats n times. Each time, one criterion is dropped from the table, the remaining criteria's weights are scaled up proportionally so that they again sum to 1 (if two criteria remain with a combined weight of 0.60, each is divided by 0.60 to rescale), and TOPSIS is rerun on this smaller table.

Third, measuring stability. Every reduced ranking is compared with the baseline ranking through Spearman rank correlation. A correlation near 1 means the order barely moved once the criterion was dropped; a correlation near -1 means the order reversed completely.

Fourth, identifying the most influential criterion. The most influential criterion is the one whose removal produces the lowest correlation. DecisionMind reports this correlation value for every criterion as a stability profile.

The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

The stability score speaks not about the ranking itself but about how dependent that ranking is on a particular criterion. A criterion with a correlation of 1 is not "unimportant"; it merely failed to disturb the order when removed, because the information it carried may already be captured by other criteria. A criterion with a correlation near -1 is the true driver of the ranking, and shows that the decision has become overly dependent on that single measure; this does not mean the method has malfunctioned, it means the ranking is fragile. In small problems (two or three alternatives), Spearman correlation can only take a few discrete values (such as 1, 0, -1), so reading this score as a percentage "degree of importance" is mistaken.

Thus instead of writing:

"C1 is the most important criterion"

the report should read:

"Removing C1 completely reverses the ranking; this ranking is excessively dependent on C1"

Crisp data. In DecisionMind, this method currently rests only on classical TOPSIS and carries no extension of its own. You must already hold a complete decision table (alternatives, criteria, directions) and weights; at least two criteria are required, because removal becomes meaningless once only one is left. The practical range runs from three to twelve criteria; computational load grows in proportion to the number of criteria, because the ranking is rerun from scratch for each one.

Criteria Removal is a sound choice if you already hold a baseline MCDM ranking and need to answer, before presenting the report, "how many criteria does this order rest on". It strengthens the defensibility of a ranking especially in decisions likely to face objection, such as tender evaluation, public resource allocation or a thesis defence.

There are two situations where it should not be used. The first is when you have no baseline ranking yet; TOPSIS or another ranking method must be run first, and Criteria Removal applied only afterwards. The second is when criteria interact strongly with one another; the method removes criteria one at a time and does not see the effect of removing two together. This is exactly the core criticism Saltelli and colleagues (2004) direct at "one-at-a-time" methods: such methods can miss the joint effects between factors.

A baseline ranking is ready, the question is "which criterion does it depend on" → Criteria Removal

Criteria strongly influence one another, joint effects matter → variance-based sensitivity analysis (Sobol, Morris)

The question targets weights, not criteria → weight sensitivity analysis

No baseline ranking exists yet → run a ranking method such as TOPSIS first, then apply Criteria Removal

Criteria Removal's greatest strength is its transparency. Every step amounts to rerunning the same base method (TOPSIS) again and again; the decision-maker is shown, with concrete figures, what would happen if this piece of information were lost. It requires no extra assumption, performs no random sampling, carries a small computational load, and its result collapses into a single, interpretable correlation figure. This simplicity makes the method a strong tool particularly when presenting and defending a report.

Its limitations stem from the same structure. First, it removes only one criterion at a time; Saltelli and colleagues (2004) show that such methods can miss the joint effects between criteria and misorder their relative importance. Second, it depends on the base method; because TOPSIS is used here, TOPSIS's own limitations (the full-compensation assumption, the choice of normalisation) are carried across as well. Third, in small problems Spearman correlation can only take a few discrete values, so the stability profile gives a coarse signal and fails to distinguish fine differences. Fourth, the way weights are rescaled (proportional scaling up) is a single fixed rule; a different rescaling rule can produce a different stability profile (Triantaphyllou and Sánchez, 1997).

The most common mistake is reading the stability score as a measure of the method's own accuracy. A low correlation does not mean the method has malfunctioned; it means the ranking is sensitive to that criterion. A second mistake is automatically declaring the most influential criterion the "most important" one; a criterion whose effect looks large may in fact be poorly measured or have its direction marked wrongly. A third mistake is removing criteria only one at a time and assuming "every interaction has been tested"; the effect of removing two criteria together needs separate examination. A fourth mistake is applying this analysis's results directly when the base method is not TOPSIS (for instance, a ranking produced with VIKOR); this version of DecisionMind fixes the base at TOPSIS.

The governing principle is this:

Criteria Removal shows how robust a ranking is, not which ranking is correct; low stability is a signal that the report needs additional justification.