MINKOWSKI DISTANCE (Minkowski Distance)

Minkowski distance is a generalised distance measure that gauges the difference between two numerical vectors with a single parameter p. Its input is two equal-length vectors and a number p (p must be greater than or equal to 1), its output a single figure. Though it looks like a separate method, it is in fact a family: given p=1 it equals Manhattan distance, given p=2 it equals Euclidean distance, and as p approaches infinity it equals Chebyshev (largest-difference) distance. It is a building block that feeds the "how far is this alternative from the ideal" step of ranking methods such as TOPSIS and VIKOR.

Its origin lies in Hermann Minkowski's 1910 work Geometrie der Zahlen (Geometry of Numbers), where the general Lp norm was defined. In DecisionMind, Minkowski distance is the single general distance step in which the user chooses the p value; the other distance cards (Manhattan, Euclidean, Chebyshev) are fixed-p special cases of this family.

The question behind Minkowski distance is this: in assessing an alternative's deviation from the ideal, how much weight should be given to the way the deviations are distributed? At p=1 (Manhattan), every deviation is counted linearly, and none is penalised more than another. As p grows, large deviations are increasingly heavily penalised; once p becomes very large, only the single worst criterion determines the result, and small deviations on the other criteria lose their significance.

This makes it possible to view Minkowski distance as a slider: at low p the philosophy is "the total deviation matters, regardless of which criterion it comes from"; at high p the philosophy is "the weakest link matters most, and the others cannot compensate for it". A decision-maker's choice of p is, in effect, an implicit answer to the question "how much compensation between criteria is acceptable", and this answer must be stated explicitly in the report.

The calculation consists of a single step, but it requires a parameter choice.

One step: raising to the pth power, summing, then taking the pth root. The absolute difference between the two vectors is taken on each criterion, this difference is raised to the pth power, these values are summed across all criteria, and finally the pth root of the sum is taken. Given p=1, this reduces directly to Manhattan distance; given p=2, to Euclidean distance; both are special cases of Minkowski. In DecisionMind, the p value is set by the user; the default value is 2 (Euclidean).

The choice of p directly affects the result of the calculation, and this card's second and third cases demonstrate this with figures. The formula for this step and its implementation in DecisionMind are given on the method page; this card carries no formulas.

Minkowski distance is a single figure calculated for the chosen p value; two distances calculated with different p values cannot be compared with one another, because their scales differ. A value of zero means the two vectors are identical; a large value shows that the total deviation (weighted according to the chosen p) is large.

Stating only "the distance is 0.30" without reporting p is incomplete information, because the same two alternatives could come out equal at p=1 while diverging markedly at p=2. Minkowski distance must therefore always be reported together with the p value used; p is a decision that determines the result as much as the rest of the formula does.

Thus instead of writing:

"The distance is 0.30, so these two alternatives are equally far from the ideal"

it is correct to write:

"At p=1 the distances are equal (0.30); moving to p=2 breaks this equality because of how the deviations are distributed, since as p grows a single large deviation weighs more heavily than the others"

Minkowski distance works with crisp numerical data. Under this name, DM3 holds only the base Minkowski member; it has no separate extension, though the p=1 and p=2 special cases are also found in DM3 separately, under the names Manhattan and Euclidean distance. You need: two equal-length numerical vectors, preferably brought onto a common scale (normalised), and a p value greater than or equal to 1. The choice of p does not come from the data; it is a decision made by the decision-maker or analyst about how much compensation is allowed between criteria, and it must be justified in the report. A minimum of two criteria and two alternatives suffices; there is no upper limit. Minkowski produces no weights and requires none.

If you hold a clear view on how much compensation is acceptable between your criteria and want to express that view with a single number (p), Minkowski is a suitable choice; trying values between p=1 and p=2 (or a higher p) lets you see how sensitive the ranking is. If a standard distance (Manhattan or Euclidean) suffices, using that card directly, without the added trouble of choosing p, is simpler.

It should not be used where p is chosen by looking at the data or the desired result; this amounts to searching for p backwards from the result, and it is not honest. Where criteria are binary or categorical, Hamming distance should be used rather than Minkowski. Where criteria are strongly related (correlated) and this relationship should be taken into account, Mahalanobis distance is more suitable.

The meaning of p can be justified beforehand, sensitivity analysis wanted → Minkowski

A standard p=1 or p=2 suffices → Manhattan or Euclidean directly

Criteria binary/categorical → Hamming

Criteria are related to one another (correlated) → Mahalanobis

p is chosen by looking at the result → no distance measure at all; the justification for p must be settled first

Minkowski distance's greatest strength is its flexibility: a single parameter spans a wide family, from Manhattan to Euclidean to Chebyshev, letting the analyst tune the understanding of compensation between criteria with p. This flexibility can also be used to test how robust a ranking is: several different p values can be applied to the same data to see whether the order changes. It is simple to calculate and scales linearly as the number of criteria grows.

Its limitations are the price of its flexibility. First, the choice of p can be arbitrary; two analysts applying different p values to the same data can arrive at different rankings, and which is "correct" cannot be read off from the data, it is a decision. Second, as p grows, the result becomes increasingly sensitive to the single worst criterion alone; this is a desirable property in some situations and a loss of information in others. Third, different p values can rank the same table differently (Aruldoss, Travis and Venkatasamy, 2019); this sensitivity must be shown in the report. Fourth, it does not take the relationship (correlation) between criteria into account. Fifth, if p is not reported, the reader cannot tell which distance was used; this is a gap similar to not stating the form of normalisation in TOPSIS.

The most common mistake is saying only "Minkowski distance" without reporting the p value; the figure is meaningless without p. A second mistake is trying several results with different p values, choosing the p that gives the desired ranking, and not showing the others; this is searching for the parameter backwards from the result, and it is not honest. A third mistake is applying Minkowski distance directly to raw, unnormalised criteria; a large-scale criterion dominates the total whatever p is. A fourth mistake is assuming p=1 and p=2 results will always give the same order; where deviations are concentrated at a single point, the two p values can favour different alternatives.

The governing principle is this:

Minkowski distance offers not a single figure but a family; which member (which p) was used is information that must be reported as much as the calculation itself.