MARCOS: Measurement of Alternatives and Ranking according to Compromise Solution

Overview

Utility function (ideal + anti-ideal reference). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Utility function (ideal + anti-ideal reference)

Limitations

• •Rank reversal known on alternative-set changes (ref: general MCDM literature)
• •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 MARCOS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criteria preferences are independent (no synergistic interactions)
• •Hatalı: 'MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Compensation is acceptable: high score on one criterion can offset low on another
• •Hatalı: 'MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix is complete (no missing values)
• •Hatalı: MARCOS'yi 'Criteria strongly correlated → consider DEMATEL/ANP for interdependence' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MARCOS'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: Form extended decision matrix with AAI (anti-ideal) and AI (ideal) rows. Formül: X^{*} = [X^{T}\,;\,AAI\,;\,AI]^{T},\quad AAI_{j}=\min/\max,\ AI_{j}=\max/\min Anchor: Stević 2020, p.6 Eq.(1)
2. 2.Adım 2 (F2): Step 2: Normalise vs ideal AI per criterion direction. Formül: n_{ij} = \begin{cases} x_{ij}/x_{AI,j} & j\in J^{+} \\ x_{AI,j}/x_{ij} & j\in J^{-} \end{cases} Anchor: Stević 2020, p.6 Eq.(2)
3. 3.Adım 3 (F3): Step 3: Weighted normalised matrix v_ij = w_j · n_ij. Formül: v_{ij} = w_{j}\,n_{ij} Anchor: Stević 2020, p.6 Eq.(3)
4. 4.Adım 4 (F4): Step 4: Sum each alternative's weighted matrix row: S_i = Σ v_ij. Formül: S_{i} = \sum_{j=1}^{n} v_{ij} Anchor: Stević 2020, p.7 Eq.(4)
5. 5.Adım 5 (F5): Step 5: Utility ratios vs ideal K+ and anti-ideal K− rows. Formül: K^{+}_{i}=\dfrac{S_{i}}{S_{AI}},\quad K^{-}_{i}=\dfrac{S_{i}}{S_{AAI}} Anchor: Stević 2020, p.7 Eqs.(5)-(6)
6. 6.Adım 6 (F6): Step 6: Utility functions f(K+_i) and f(K−_i) of the utility ratios. Formül: f(K^{+}_{i})=\dfrac{K^{-}_{i}}{K^{+}_{i}+K^{-}_{i}},\quad f(K^{-}_{i})=\dfrac{K^{+}_{i}}{K^{+}_{i}+K^{-}_{i}} Anchor: Stević 2020, p.7 Eqs.(7)-(8)
7. 7.Adım 7 (F7): Step 7: Final utility f(K_i) and descending ranking. Formül: f(K_{i})=\dfrac{K^{+}_{i}+K^{-}_{i}}{1+\dfrac{1-f(K^{+}_{i})}{f(K^{+}_{i})}+\dfrac{1-f(K^{-}_{i})}{f(K^{-}_{i})}} Anchor: Stević 2020, p.7 Eq.(9)

Commonly paired with

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

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