TOPSIS

TOPSIS was introduced by Ching-Lai Hwang and Kwangsun Yoon in their 1981 book "Multiple Attribute Decision Making: Methods and Applications." The method is grounded in a straightforward geometric intuition: the best alternative should have the shortest distance to the ideal solution (where every criterion is at its best possible value) and the longest distance from the anti-ideal solution (where every criterion is at its worst possible value).

The TOPSIS procedure follows six steps. First, construct a decision matrix with alternatives as rows and criteria as columns. Second, normalize the matrix so all criteria are on comparable scales, typically using vector normalization (dividing each value by the square root of the sum of squared values in its column). Third, apply criterion weights to the normalized matrix. Fourth, determine the ideal solution (best value for each criterion) and anti-ideal solution (worst value for each criterion). Fifth, calculate each alternative's Euclidean distance from both the ideal and anti-ideal solutions. Sixth, compute a relative closeness coefficient for each alternative — the ratio of the distance from the anti-ideal to the sum of distances from both ideal and anti-ideal. The alternative with the highest closeness coefficient ranks first.

TOPSIS has several practical advantages. It produces a clear numerical ranking, not just an ordinal ordering. The closeness coefficient ranges from 0 to 1, giving an intuitive measure of how close each alternative is to the theoretical best. The method handles both benefit criteria (higher is better) and cost criteria (lower is better) naturally. It scales well — adding more criteria or alternatives requires recalculation but no fundamental change in methodology.

The method is especially strong when performance data is quantitative and available for all alternatives across all criteria. This makes it popular in engineering, supply chain management, and manufacturing — domains where alternatives can be measured on objective scales. TOPSIS is less natural for purely qualitative criteria, though these can be converted to numerical scales with defined rating descriptors.

Common extensions include fuzzy TOPSIS (for handling imprecise or linguistic data), interval-valued TOPSIS (for ranges rather than point estimates), and group TOPSIS (for aggregating multiple decision-makers' evaluations). Researchers have also combined TOPSIS with AHP, using AHP to derive criterion weights and TOPSIS for the final ranking — a hybrid approach that leverages the strengths of both methods.