TOPSIS

TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) is a distance-based multi-criteria decision making method proposed by Ching-Lai Hwang and Kwangsun Yoon (1981, 'Multiple Attribute Decision Making: Methods and Applications', Springer). The intuition: the best alternative is the one geometrically closest to the positive-ideal solution (best value across every criterion) and farthest from the negative-ideal solution (worst value across every criterion). Steps: (1) column-normalize the decision matrix (commonly vector or min-max normalization), (2) multiply by criterion weights to form the weighted normalized matrix, (3) determine positive-ideal A+ and negative-ideal A- vectors (max on benefit criteria, min on cost criteria), (4) compute Euclidean distances S+ and S- of each alternative to A+ and A-, (5) score the relative closeness C* = S- / (S+ + S-); the highest C* is the best alternative. Strengths: computationally simple, scales well with the number of criteria, and unlike AHP requires no internal consistency check. Critiques: weights must be supplied exogenously (often via AHP or entropy weighting), and the choice of normalization can affect the ranking. Extensions: Fuzzy TOPSIS (Chen 2000) for vague judgments, Intuitionistic Fuzzy TOPSIS, and Interval TOPSIS. Complexity class: polynomial (O(m·n) for m alternatives and n criteria).