Chebyshev Distance (The Worst Single-Criterion Difference)

When a decision method wants to measure how far an alternative sits from the ideal, it must choose how to calculate that distance. Chebyshev distance is one such choice: it finds the difference between two points on every criterion in turn and takes only the largest of these as the distance, disregarding every other difference. Named after the 19th-century mathematician Pafnuty Chebyshev, this measure is also known in mathematics as the "L-infinity norm," and rests on the same formula as "chessboard distance," which measures how many moves a piece needs to travel from one square to another (Rosenfeld and Pfaltz, 1968). Among the studies that brought this measure into multi-criteria decision analysis is Yu's (1973) work on compromise solutions for group decision problems; the VIKOR method uses this distance directly in calculating its "individual regret" component (the R value) (Opricovic and Tzeng, 2004). Chebyshev distance is not, on its own, a decision method; it is a building block used within distance-based methods such as TOPSIS and VIKOR as an alternative to Euclidean distance.

The question behind Chebyshev distance is this: "Should the difference between two alternatives be determined by a single worst-case criterion, or by the combined contribution of all criteria?" Chebyshev distance chooses the former path. Even if an alternative sits very close to the ideal on nine criteria, if a large deviation exists on the tenth, Chebyshev distance sees only that tenth deviation; the closeness on the other nine criteria never changes the distance at all. This is a wholly non-compensatory (eliminatory) outlook, resting on the logic that "the weakest link determines the chain." Unlike Euclidean distance, poor performance on one criterion is never offset by good performance on another.

The procedure consists of a single step.

The single step, finding the largest difference. Between two alternatives (or an alternative and an ideal point), the difference on every criterion is calculated separately, the absolute value of each difference is taken, and the largest is chosen as the distance. Every other criterion difference plays no part in the calculation at all.

This step can be substituted for Euclidean distance in a ranking method's (such as TOPSIS or VIKOR) calculation of distance to the ideal and anti-ideal points. The formulas behind each step are given on the DecisionMind method page; this card carries no formulas.

Chebyshev distance gives the worst single-criterion difference between two alternatives; it is not a measure of total difference. A small Chebyshev distance does not mean "these two alternatives are close on every criterion," only that "even on the criterion where they diverge most, the difference is small." Equally, a large Chebyshev distance can also arise when the alternatives are similar on most criteria and diverge sharply on only one. This value cannot be directly compared with a value obtained from a different set of alternatives or a different distance measure (Euclidean, Manhattan).

Thus instead of writing:

"The Chebyshev distance is small, meaning these two alternatives are generally very similar"

the report should read:

"The difference on the single criterion where these two alternatives diverge most is small; the similarity or difference on the other criteria is not reflected in this figure"

It works with crisp data: a single number in every cell, no empty cells. It is generally calculated on a table that has first been normalised and weighted for use in a subsequent step; used directly on raw data, a single large-unit criterion (price, say) determines the whole distance on its own.

This is a building-block card; DecisionMind holds no extension (fuzzy, grey, intuitionistic version) of its own for it. Five other distance measures perform the same task by a different logic: Euclidean distance, Manhattan distance, Minkowski distance, Hamming distance and Mahalanobis distance; each is described on its own building-block card. At least two alternatives (or one alternative and one reference point) and at least one criterion are sufficient.

Chebyshev distance is appropriate where the worst case is decisive, that is, where a large deviation on one criterion must never be offset by the others. In methods such as VIKOR, which weigh both the total and the worst case together, it is used directly to calculate the worst-case component.

The situation in which it should not be used is one where the combined contribution of all criteria matters, and a slight deviation on one criterion should not overshadow the overall performance on the others; in such a situation Chebyshev distance causes a loss of information, because it disregards all data except the worst criterion.

The worst-criterion deviation is decisive, no compensation wanted → Chebyshev distance

The combined contribution of all criteria matters, compensation is accepted → Euclidean distance

The weighted sum of absolute differences across criteria is wanted → Manhattan distance

There is correlation between criteria and it must be accounted for → Mahalanobis distance

Chebyshev distance's most important strength is that its calculation is very simple and directly interpretable: the result is directly the answer to the question "what is the difference on the worst criterion." It prevents criteria with very strong performance from masking one with poor performance, a property that matters in safety- or threshold-focused decisions. Its computational load is very small and it runs just as fast on large tables.

Its limitations arise from this same property. First, all information other than the largest difference is disregarded; if two alternatives show a large difference on nine criteria and a small difference on one, Chebyshev distance sees only one of those nine large differences and ignores the other eight. Second, a measurement error or an outlier on a single criterion can determine the entire distance calculation on its own; this creates a particular fragility in the method. Third, two alternatives with the same Chebyshev distance can have very different profiles, which complicates interpretation. Fourth, there is no single founding paper specific to MCDM for the concept known as "Chebyshev distance"; the concept was borrowed from mathematics (Rosenfeld and Pfaltz, 1968) and carried into decision analysis afterwards (Yu, 1973; Opricovic and Tzeng, 2004).

The most common mistake is interpreting Chebyshev distance in the same way as Euclidean distance, that is, saying "a small distance means generally similar," when in fact Chebyshev distance sees only a single criterion. A second mistake is calculating Chebyshev distance directly on raw (non-normalised) data; a single large-unit criterion (price in millions, say) renders every other criterion meaningless and determines the distance on its own. A third mistake is giving only the number without reporting which criterion the Chebyshev distance came from; a decision-maker cannot interpret this number without knowing which criterion was decisive.

The governing principle is this:

Chebyshev distance measures only the worst single-criterion difference; if which criterion determines this difference is not reported clearly, the figure alone is misleading.