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

Overview

Distance-based (compromise). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Distance-based (compromise)

Limitations

• •Rank reversal known on alternative-set changes (ref: Belton-Gear-1983; Garcia-Cascales-Lamata-2012)
• •Assumes: Criteria preferences are independent (no synergistic interactions)
• •Assumes: Compensation is acceptable: high score on one criterion can offset low on another
• •Assumes: Decision matrix is complete (no missing values)

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)

When not to use

• •Criteria strongly correlated → consider DEMATEL/ANP for interdependence
• •Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE)

Edge cases

• •See F.steps and D.parameters for TOPSIS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criteria preferences are independent (no synergistic interactions)
• •Hatalı: 'TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Compensation is acceptable: high score on one criterion can offset low on another
• •Hatalı: 'TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix is complete (no missing values)
• •Hatalı: TOPSIS'yi 'Criteria strongly correlated → consider DEMATEL/ANP for interdependence' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: TOPSIS'yi 'Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Vector normalisation: scale each column to unit Euclidean norm. Formül: r_{ij} = \dfrac{x_{ij}}{\sqrt{\sum_{i=1}^{m} x_{ij}^{2}}}, \quad i=1,\ldots,m;\ j=1,\ldots,n Anchor: Hwang & Yoon 1981, p.130 Eq.(4.1); cross-ref Roszkowska 2011, p.206 Eq.(2.1)
2. 2.Adım 2 (F2): Step 2: Weighted normalised matrix: multiply each column by its weight. Formül: v_{ij} = w_{j} \cdot r_{ij} Anchor: Hwang & Yoon 1981, p.130 Eq.(4.2); cross-ref Roszkowska 2011, p.207 Eq.(2.2)
3. 3.Adım 3 (F3): Step 3: Determine the positive-ideal A⁺ and negative-ideal A⁻ solutions per criterion direction. Formül: A^{+} = \{v_{1}^{+}, \ldots, v_{n}^{+}\},\ v_{j}^{+} = \begin{cases} \max_{i} v_{ij} & j \in J \\ \min_{i} v_{ij} & j \in J' \end{cases};\quad A^{-} = \{v_{1}^{-}, \ldots, v_{n}^{-}\},\ v_{j}^{-} = \begin{cases} \min_{i} v_{ij} & j \in J \\ \max_{i} v_{ij} & j \in J' \end{cases} Anchor: Hwang & Yoon 1981, p.130 Eqs.(4.3)-(4.4); cross-ref Roszkowska 2011, p.207 Eqs.(2.3)-(2.4)
4. 4.Adım 4 (F4): Step 4: Euclidean separation of each alternative from A⁺ and A⁻. Formül: d_{i}^{+} = \sqrt{\sum_{j=1}^{n}(v_{ij}-v_{j}^{+})^{2}},\quad d_{i}^{-} = \sqrt{\sum_{j=1}^{n}(v_{ij}-v_{j}^{-})^{2}} Anchor: 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.Adım 5 (F5): Step 5: Relative closeness to the ideal solution; rank by C* descending. Formül: C_{i}^{*} = \dfrac{d_{i}^{-}}{d_{i}^{+}+d_{i}^{-}},\quad 0 \le C_{i}^{*} \le 1 Anchor: Hwang & Yoon 1981, p.131 Eq.(4.7); cross-ref Roszkowska 2011, p.208 Eq.(2.7) (notated as R_i in Roszkowska)

Commonly paired with

• •AHP + TOPSIS (high)
• •BWM + TOPSIS (high)
• •ENTROPY + TOPSIS (high)
• •CRITIC + TOPSIS (high)
• •SWARA + TOPSIS (high)

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