TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution)

TOPSIS is a ranking method for when you already hold a decision table filled with numbers ("which supplier," "which device," "which programme") and want the alternatives placed in a single order. Its output is a closeness score between 0 and 1 for every alternative, together with the rank that score produces. It does not sort alternatives into "acceptable / unacceptable" groups, and it does not generate criterion weights; weights are supplied from outside. Hwang and Yoon proposed it in 1981, and it remains one of the most widely applied multi-criteria decision methods; application surveys turn it up in almost every field, prominently in supply chains, energy, manufacturing and quality management.

TOPSIS's underlying idea is to turn "what is the best alternative" into "which alternative most resembles the ideal." Take the best value on each criterion in the table and gather them into one hypothetical alternative: the ideal, best in every respect, which does not actually exist. Do the same with the worst values to get the anti-ideal alternative. Real alternatives sit somewhere between these two poles, and the method looks for whichever is both close to the ideal and far from the anti-ideal. It does not look at closeness to the ideal alone, because between two alternatives equally close to the ideal, the safer choice is the one further from the anti-ideal.

This idea carries a philosophical consequence: TOPSIS is compensatory. A weakness on one criterion can be offset by strength on another, because distance is a single figure into which every criterion's contribution dissolves. If you can accept that "good at several things and excellent at one" may outrank "reasonably good at everything," TOPSIS is the right tool; if one criterion can never be traded away, its philosophy will clash with the decision's own.

The method proceeds through five steps, each of which can be followed on the table.

First, scale equalisation (normalisation). Criteria are expressed in different units: TL, days, points, kilograms. Left as they are, they cannot be added together; a criterion with large figures (price) would swamp one with small figures (a 1–9 scale). TOPSIS divides every column by its own magnitude: each value is divided by the square root of the sum of the squared values in its column. This is called vector normalisation. Every column becomes unit-free as a result, with values between 0 and 1, so that columns become comparable with one another. Other forms of normalisation exist (dividing by the maximum, scaling to a minimum–maximum range) and can give the same table a different order. DecisionMind fixes vector normalisation for classical TOPSIS and states this in the report.

Second, weighting. TOPSIS multiplies every equalised column by its criterion's weight, so that more important criteria carry more influence in the distance calculation. Weights come from outside TOPSIS itself and must sum to 1.

Third, two hypothetical alternatives. In the weighted table, the best value on each criterion (largest for a benefit, smallest for a cost) builds the ideal alternative, and the worst value builds the anti-ideal. Criterion direction matters here: for a "lower is better" criterion, the ideal is the smallest value.

Fourth, distance. TOPSIS measures every real alternative's distance to both the ideal and the anti-ideal. Classical TOPSIS uses straight-line, Euclidean, distance: it squares the difference on each criterion, sums the squares, and takes the square root. Other distance measures have been defined and can change the result; DecisionMind fixes Euclidean distance for the classical method.

Fifth, the closeness score. TOPSIS divides an alternative's distance to the anti-ideal by the sum of its two distances. For an alternative pinned to the ideal, this ratio tends towards 1; for one pinned to the anti-ideal, it tends towards 0. TOPSIS then ranks alternatives from the highest score to the lowest.

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 closeness score tells you how near an alternative sits to the ideal relative to the other alternatives in this particular analysis, and nothing more. A score of 0.64 does not mean "64 per cent good" or "64 per cent likely to be the best." It cannot be set against a 0.70 from a different analysis, because the ideal and anti-ideal points are built afresh, in every analysis, from that analysis's own alternatives; change the alternative set and the scores change with it. A score of 1 does not mean "perfect" but "the best in this set on every criterion"; a score of 0 means "the worst on every criterion."

The alternative ranked first is often not the outright best on any single criterion; it is simply the most balanced once weighted distance is taken into account. A narrow score gap between two alternatives, such as 0.64 against 0.61, does not signal a robust ranking but one that is sensitive to the weights. The report should therefore show which criteria the ranking depends on, and whether the ranking would change if the weights shifted only slightly.

Thus instead of writing:

"TOPSIS found the best alternative"

the report should read:

"With these weights and this alternative set, the alternative closest to the ideal is this one; the ranking is sensitive to the weight on these criteria"

Classical TOPSIS works with crisp data: one number per cell. If your data comes from expert judgement described as "approximate," is known only as a range, or is contradictory across experts, you change the data type rather than the method: TOPSIS has fuzzy, grey, intuitionistic, neutrosophic and other extensions, and DecisionMind holds twenty-nine TOPSIS family members alongside the base method. Which extension fits which data situation is explained on the relevant data-type cards; the tool asks about your data type and suggests the right member.

You need alternatives in rows, criteria in columns, one number per cell, and no empty cells; for every criterion, whether more is better or less is better (price, time and risk are "lower is better"); and criterion weights that sum to 1. TOPSIS does not produce weights, it asks for them. You can derive weights from expert opinion (AHP, BWM, SWARA) or from the data itself (Entropy, CRITIC); "equal to all" is also a decision and must be justified in the report. A minimum of two alternatives and two criteria is required; three to twelve criteria work comfortably, and going beyond that strains the weighting rather than the method itself. There is no upper limit on the number of alternatives.

TOPSIS is a sound choice if your criteria can be measured numerically, your table has no gaps, and you accept that a weakness on one criterion may be offset by strength on another. Its typical territory includes supplier and vendor evaluation, facility-location choice, technology and equipment selection, and programme or project prioritisation; its logic does not depend on the field.

There is a single case where it should not be used, and it follows from its own philosophy: if you will not compromise on one criterion, TOPSIS is the wrong tool. If you do not want a cheap device with a below-threshold safety rating to come out ahead simply because it is cheap, TOPSIS will not stop that on its own; you must first screen out anything below the threshold and rank only what remains with TOPSIS, or turn to methods built on elimination logic instead. Where criteria are strongly linked (one rising as another rises), that link needs handling first, because TOPSIS treats criteria as independent.

A numerical table, compensation accepted, the goal is ranking → TOPSIS

Same goal, but the data is fuzzy / grey / neutrosophic → the relevant TOPSIS extension

Not "the best" but "the compromise limiting the greatest regret" → VIKOR

No compromise allowed on one criterion, sub-threshold alternatives must be screened out → screening first, then ranking, or elimination-based methods

Not a ranking but weights are needed → AHP, BWM, SWARA (subjective); Entropy, CRITIC (objective)

TOPSIS's greatest strength is how easily it can be explained. The idea of "closeness to the ideal" is simple to convey to a decision-maker; every step can be followed on the table, and the result emerges not from a black-box formula but from two visible reference points. It brings criteria measured in different units into a single table, keeps the computational burden light, and works just as simply as the number of alternatives and criteria grows. Because it draws on both closeness to the ideal and distance from the anti-ideal, it is more balanced than a measure of "resemblance to the best" alone. These features explain why it remains one of the most widely used ranking methods in the applied literature.

Its limitations stem from the same structure. First, the ideal and anti-ideal points shift whenever the alternative set changes, so an alternative added or removed afterwards can change how the others rank relative to one another; this phenomenon, known in the literature as rank reversal, has been examined in detail for TOPSIS (García-Cascales and Lamata, 2012; Wang and Luo, 2009). Second, the assumption of full compensation means a serious weakness on one criterion can be papered over by others. Third, the choice of normalisation and distance measure affects the result; different normalisation forms can give the same table a different order (Roszkowska, 2011). Fourth, the method treats criteria as independent, so where criteria influence one another, weight is implicitly counted twice. Fifth, the quality of the weights lies outside the method itself; a flawless calculation built on poor weights still produces a poor ranking.

The most common mistake is marking criterion direction wrongly. Price, time and risk are "lower is better" criteria; mark them "higher is better" and the ideal point is built from the most expensive, slowest alternative, reversing the ranking. Every criterion's direction must be checked one by one.

A second mistake is assigning equal weights without justification. Equal weight does not mean "unimportant" but "equally important as everything else," which is itself a claim that needs defending in the report. A third is adding an alternative once the analysis is finished and being surprised the ranking shifts; the alternative set is fixed before the analysis begins, and adding one afterwards means starting again. A fourth is reading the closeness score as a percentage or a probability, or comparing scores taken from different analyses. A fifth is choosing a compensatory method for a situation where one criterion can never be traded away. A sixth is changing the normalisation form and expecting the same result.

The governing principle is this:

A TOPSIS result is a consistent summary of the directions, weights and alternative set you supplied; if any input is contested, the ranking is contested too, and the report must show this.