MANHATTAN DISTANCE (Manhattan Distance)

Manhattan distance is a distance measure that gives the total difference between two numerical vectors (an alternative and an ideal point, or two alternatives). Its input is two equal-length number sequences, its output a single figure: the sum of the absolute difference on each criterion. It is a building block that feeds the "how far is this alternative from the ideal" step of ranking methods such as TOPSIS, CODAS and MABAC; it is not a ranking method in its own right.

Its name comes from the distance covered on a city's grid-shaped streets by moving only horizontally and vertically, never diagonally; this geometry was introduced as "taxicab geometry" by Krause (1973). Mathematically, Manhattan distance is the p=1 special case of the Minkowski distance family; the roots of the general Lp norm lie in Minkowski's (1910) work. Some methods, CODAS among them, use Manhattan distance as a second measure alongside Euclidean distance.

The idea behind Manhattan distance is this: the overall difference between an alternative and the ideal is the plain sum of the deviation on each criterion, no more and no less. Euclidean distance squares each deviation before taking the root, inflating large deviations disproportionately; Manhattan does not do this, counting every deviation as it stands, linearly.

This choice has a consequence: Manhattan distance is more robust to outliers than Euclidean distance. An alternative that is very poor on one criterion and good on the others is heavily penalised under Euclidean distance because of that single poor criterion; under Manhattan distance, the same deviation enters the sum only in proportion to its own share. If the decision-maker's question is "what is the overall deviation, regardless of which criterion it comes from", Manhattan is the right tool; if it is "a large failure on a single criterion should overshadow everything else", Euclidean or a high-p Minkowski distance is more suitable.

The calculation consists of a single step.

One step: summing absolute differences. The two vectors are laid alongside one another in the same order; the difference between the two values on each criterion is taken and recorded regardless of sign (absolute value). These absolute differences are summed, and the result is the Manhattan distance. If the deviations on three criteria are 2, 2 and 30, the total is 34; if the deviations are 12, 12 and 12, the total is 36, and a more evenly spread total of deviations comes out slightly larger here than one concentrated at a single point. How the deviations are distributed changes the result far more sharply under Euclidean distance; the second case demonstrates this with figures.

The formula for this step and its implementation in DecisionMind are given on the method page; this card carries no formulas.

Manhattan distance is the total absolute deviation between two alternatives; it says nothing else. A value of zero means the two vectors are identical; a large value means the overall deviation is large. This figure is not a percentage and not a probability, and it is only meaningful between comparisons built with the same set of criteria and the same scale; if criteria are in different units (say, currency and days), they must first be brought onto a common scale (normalisation), otherwise a criterion with large figures dominates the sum on its own.

A further feature of Manhattan distance is that it conceals where the deviation comes from: a total deviation of 0.37 might come from a single criterion, or be shared equally across three. For this reason, Manhattan distance should not be reported alone by the host method; which criteria contribute most to the deviation should be shown separately.

Thus instead of writing:

"The Manhattan distance is 0.37, so this alternative is 63 per cent as good as the ideal"

it is correct to write:

"The total absolute deviation is 0.37; which criterion this deviation chiefly comes from must be stated separately, because the total alone does not show this"

Manhattan distance works with crisp numerical data. Under this name, DM3 holds only the base Manhattan member; it has no separate extension. You need: two equal-length numerical vectors, preferably already brought onto the same scale (normalised), because Manhattan distance itself performs no scaling. A minimum of two criteria and two alternatives suffices; there is no upper limit, and three to twelve criteria give a comfortably readable table. Manhattan produces no weights and requires none; weighting falls to the host method (TOPSIS's second step, for instance) and is generally applied before Manhattan distance is calculated.

If your criteria are numerical, have been brought onto a common scale, and you do not want a single outlying criterion to dominate the result on its own, Manhattan is a suitable choice. In some host methods, such as CODAS, Manhattan is automatically used as a second measure alongside Euclidean distance; in that case it need not be separately chosen, as it is part of the method's own definition.

It should not be used where a large failure on one criterion ought to weigh disproportionately on the decision; Euclidean or a high-p Minkowski distance is more suitable there, because these inflate large deviations by squaring (or a higher power). Where criteria are binary or categorical, Hamming distance should be used rather than Manhattan. Where criteria are strongly related (correlated) and this relationship should be taken into account, Mahalanobis distance is more suitable.

Numerical data, robustness to outliers wanted → Manhattan

A large deviation on a single criterion should be penalised → Euclidean or a high-p Minkowski

Criteria binary/categorical → Hamming

Criteria are related to one another (correlated) → Mahalanobis

Already used as a second measure alongside Euclidean, as in CODAS → no separate choice needed

Manhattan distance's greatest strength is its robustness to outliers: a single large deviation on one criterion does not inflate the total as sharply as under Euclidean distance, because deviations enter the sum directly rather than squared. It is simple to calculate and easily explained to a decision-maker as "the total absolute difference". Its computational burden grows linearly as the number of criteria and alternatives increases, and it runs quickly on large tables. Used alongside Euclidean distance, as in CODAS, the two measures' different sensitivities, one squared, one linear, complement one another.

Its limitations follow from its definition. First, it conceals where the deviation comes from; the same total can arise from very different deviation patterns, and this pattern remains invisible unless examined separately. Second, if criteria sit on different scales (one 0-1, one 0-1000), the total is dominated by the large-scale criterion; normalisation beforehand is essential. Third, different distance measures can rank the same table differently (Chen and Tsao, 2007); choosing Manhattan is itself a decision and must be justified in the report. Fourth, it does not take the relationship (correlation) between criteria into account; the deviation of two criteria that influence one another is summed separately as though they were independent. Fifth, its robustness to outliers is sometimes an unwelcome property: where a genuinely serious failure needs to be noticed, this robustness weakens the signal.

The most common mistake is applying Manhattan distance directly to raw, unnormalised data; when criteria on different scales are summed, the large-scale criterion determines the total on its own. A second mistake is assuming Manhattan and Euclidean distance will always give the same order; where deviations are concentrated in a single criterion, the two measures can favour different alternatives. A third mistake is interpreting the result, in a method that uses Euclidean and Manhattan together, such as CODAS, without knowing when the second measure comes into play (the threshold value); the switching rule between the two measures is part of the method's own definition and cannot be ignored. A fourth mistake is reading the total deviation as an "average deviation"; a total is not an average unless divided by the number of criteria, and totals from tables with different numbers of criteria cannot be compared.

The governing principle is this:

Manhattan distance counts the total deviation linearly and holds a single large deviation at equal weight with the others; if this is not exactly what you want, Euclidean or a Minkowski distance with a different p value is needed.