TOPSIS - Technique for Order of Preference by Similarity to Ideal Solution

Overview

C* ∈ [0,1]. Higher C* means the alternative is closer (in weighted Euclidean distance) to the positive-ideal A⁺ and farther from the anti-ideal A⁻. Rank alternatives in descending C* order.

Output

utility, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Supplier selection, vendor evaluation, facility location, technology selection

How it works

1. 1

   Vector normalisation: scale each column to unit Euclidean norm.

   $$r_{ij}=\frac{x_{ij}}{\sqrt{{\sum }_{i=1}^m{x}_{ij}^2}},i=1,\dots ,m;j=1,\dots ,n$$

   Hwang & Yoon 1981, p.130 Eq.(4.1); cross-ref Roszkowska 2011, p.206 Eq.(2.1)

2. 2

   Weighted normalised matrix: multiply each column by its weight.

   $$v_{ij}=w_j\cdot r_{ij}$$

   Hwang & Yoon 1981, p.130 Eq.(4.2); cross-ref Roszkowska 2011, p.207 Eq.(2.2)

3. 3

   Determine the positive-ideal A⁺ and negative-ideal A⁻ solutions per criterion direction.

   $$A^{+}=\{v_1^{+},\dots ,v_n^{+}\},v_j^{+}=\{\begin{array}{ll}{\max }_i{v}_{ij}& j\in J\\ {\min }_i{v}_{ij}& j\in J^{\prime }\end{array};A^{-}=\{v_1^{-},\dots ,v_n^{-}\},v_j^{-}=\{\begin{array}{ll}{\min }_i{v}_{ij}& j\in J\\ {\max }_i{v}_{ij}& j\in J^{\prime }\end{array}$$

   Hwang & Yoon 1981, p.130 Eqs.(4.3)-(4.4); cross-ref Roszkowska 2011, p.207 Eqs.(2.3)-(2.4)

4. 4

   Euclidean separation of each alternative from A⁺ and A⁻.

   $$d_i^{+}=\sqrt{\sum _{j=1}^n(v_{ij}-v_j^{+}{)}^2},d_i^{-}=\sqrt{\sum _{j=1}^n(v_{ij}-v_j^{-}{)}^2}$$

   Hwang & Yoon 1981, p.130 Eqs.(4.5)-(4.6); cross-ref Roszkowska 2011, p.208 Eqs.(2.5*)-(2.6*) (Euclidean p=2 special case of general p-norm Eqs.(2.5)-(2.6))

5. 5

   Relative closeness to the ideal solution; rank by C* descending.

   $$C_i^{\ast }=\frac{d_i^{-}}{d_i^{+}+d_i^{-}},0\le C_i^{\ast }\le 1$$

   Hwang & Yoon 1981, p.131 Eq.(4.7); cross-ref Roszkowska 2011, p.208 Eq.(2.7) (notated as R_i in Roszkowska)

Look elsewhere when

Assumptions to verify

• Criteria preferences are independent (no synergistic interactions)
• Compensation is acceptable: high score on one criterion can offset low on another
• Decision matrix is complete (no missing values)

Edge cases and pitfalls

• Forgetting to vector-normalise before weighting inflates magnitudes for criteria with large raw scales.

• Confusing cost vs. benefit criteria in Step 3 flips A⁺ and A⁻ for that column - always check criteria_types.

• Normalisation choice affects the ranking. Roszkowska 2011 (Eqs. 2.1, 2.1*, 2.1**) lists three options - vector (Σx²)^½, max x_ij, and min-max - and they are NOT interchangeable; this manifest fixes the canonical vector form Eq.(2.1).

• Distance metric choice. Roszkowska 2011 Eqs.(2.5)-(2.6) define a general p-norm separation; this manifest fixes p=2 (Euclidean, Eqs. 2.5*-2.6*). Other metrics (p=1 Manhattan, p=∞ Tchebychev) change both d⁺/d⁻ and the final ranking.

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