MARA: Magnitude of the Area for the Ranking of Alternatives

Overview

Area-magnitude (integral) based. Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Area-magnitude (integral) based

Limitations

• •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 MARA-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'MARA bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criteria preferences are independent (no synergistic interactions)
• •Hatalı: 'MARA bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Compensation is acceptable: high score on one criterion can offset low on another
• •Hatalı: 'MARA bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix is complete (no missing values)
• •Hatalı: MARA'yi 'Criteria strongly correlated → consider DEMATEL/ANP for interdependence' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MARA'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: Decision matrix H = [h_ij]_{m×n}. Formül: H = \big[h_{ij}\big]_{m\times n} Anchor: Gligorić 2022, Eq.(1); Yürüyen-Ulutaş 2025 Eq.(1)
2. 2.Adım 2 (F2): Step 2: Linear normalisation. Benefit: h*_ij = h_ij/max(h_ij). Cost: h*_ij = min(h_ij)/h_ij. Formül: h^{*}_{ij} = \begin{cases} \dfrac{h_{ij}}{\max(h_{ij})} & j\in J^{+} \\ \dfrac{\min(h_{ij})}{h_{ij}} & j\in J^{-} \end{cases} Anchor: Gligorić 2022, Eqs.(12)-(13)
3. 3.Adım 3 (F3): Step 3: Weighted normalised matrix c_ij = w_jB · h*_ij. Formül: c_{ij} = w_{j}^{B}\cdot h^{*}_{ij} Anchor: Gligorić 2022, Eq.(14)
4. 4.Adım 4 (F4): Step 4: Optimal alternative components d_j = max_i c_ij per criterion. Formül: d_{j} = \max_{i}(c_{ij}),\quad D = \{d_{1},\ldots,d_{n}\} Anchor: Gligorić 2022, Eqs.(15)-(16)
5. 5.Adım 5 (F5): Step 5: Partition: D = D^max ⊎ D^min (k benefit + l cost = n). Formül: D = D^{\max}\sqcup D^{\min},\ |D^{\max}|=k,\ |D^{\min}|=l,\ k+l=n Anchor: Gligorić 2022, Eqs.(17)-(18)
6. 6.Adım 6 (F6): Step 6: Partition each alternative V_i = V_i^max ⊎ V_i^min. Formül: V_{i} = V_{i}^{\max}\sqcup V_{i}^{\min} Anchor: Gligorić 2022, Eqs.(19)-(20)
7. 7.Adım 7 (F7): Step 7: Density sums D_k, D_l, V_ik, V_il. Formül: D_{k} = \sum_{j\in J^{+}} d_{j},\ D_{l} = \sum_{j\in J^{-}} d_{j},\ V_{ik} = \sum_{j\in J^{+}} v_{ij},\ V_{il} = \sum_{j\in J^{-}} v_{ij} Anchor: Gligorić 2022, Eqs.(21)-(24)
8. 8.Adım 8 (F8): Step 8: Two linear functions on [0,1]: f^opt and f^i. Formül: f^{\text{opt}}(x) = (D_{l} - D_{k})x + D_{k};\quad f^{i}(x) = (V_{il} - V_{ik})x + V_{ik} Anchor: Gligorić 2022, Eqs.(25)-(26)
9. 9.Adım 9 (F9): Step 9: Area magnitude R_i = ∫f^opt − ∫f^i = (D_l−D_k)/2 + D_k − (V_il−V_ik)/2 − V_ik. Ascending ranking (smallest = best). Formül: R_{i} = \int_{0}^{1} f^{\text{opt}}(x)\,dx - \int_{0}^{1} f^{i}(x)\,dx = \Big[\dfrac{D_{l}-D_{k}}{2}+D_{k}\Big] - \Big[\dfrac{V_{il}-V_{ik}}{2}+V_{ik}\Big] Anchor: Gligorić 2022, Eqs.(27)-(29)

Commonly paired with

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

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