MARA - Magnitude of the Area for the Ranking of Alternatives

Overview

Smallest R = best alternative.

Output

utility, lower is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Alternative selection, Supplier evaluation

How it works

1. 1

   Decision matrix H = [h_ij]_{m×n}.

   $$H=[h_{ij}{]}_{m\times n}$$

   Gligorić 2022, Eq.(1); Yürüyen-Ulutaş 2025 Eq.(1)

2. 2

   Linear normalisation. Benefit: h*_ij = h_ij/max(h_ij). Cost: h*_ij = min(h_ij)/h_ij.

   $$h_{ij}^{\ast }=\{\begin{array}{ll}\frac{h_{ij}}{\max (h_{ij})}& j\in J^{+}\\ \frac{\min (h_{ij})}{h_{ij}}& j\in J^{-}\end{array}$$

   Gligorić 2022, Eqs.(12)-(13)

3. 3

   Weighted normalised matrix c_ij = w_jB · h*_ij.

   $$c_{ij}=w_j^B\cdot h_{ij}^{\ast }$$

   Gligorić 2022, Eq.(14)

4. 4

   Optimal alternative components d_j = max_i c_ij per criterion.

   $$d_j=\underset{i}{\max }(c_{ij}),D=\{d_1,\dots ,d_n\}$$

   Gligorić 2022, Eqs.(15)-(16)

5. 5

   Partition: D = D^max ⊎ D^min (k benefit + l cost = n).

   $$D=D^{\max }\bigsqcup D^{\min },|D^{\max }|=k,|D^{\min }|=l,k+l=n$$

   Gligorić 2022, Eqs.(17)-(18)

6. 6

   Partition each alternative V_i = V_i^max ⊎ V_i^min.

   $$V_i=V_i^{\max }\bigsqcup V_i^{\min }$$

   Gligorić 2022, Eqs.(19)-(20)

7. 7

   Density sums D_k, D_l, V_ik, V_il.

   $$D_k=\sum _{j\in J^{+}}{d}_j,D_l=\sum _{j\in J^{-}}{d}_j,V_{ik}=\sum _{j\in J^{+}}{v}_{ij},V_{il}=\sum _{j\in J^{-}}{v}_{ij}$$

   Gligorić 2022, Eqs.(21)-(24)

8. 8

   Two linear functions on [0,1]: f^opt and f^i.

   $$f^{\text{opt}}(x)=(D_l-D_k)x+D_k;f^i(x)=(V_{il}-V_{ik})x+V_{ik}$$

   Gligorić 2022, Eqs.(25)-(26)

9. 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).

   $$R_i={\int }_0^1{f}^{\text{opt}}(x)dx-{\int }_0^1{f}^i(x)dx=[\frac{D_l-D_k}{2}+D_k]-[\frac{V_{il}-V_{ik}}{2}+V_{ik}]$$

   Gligorić 2022, Eqs.(27)-(29)

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

• Applying MARA without verifying this assumption.

  Requirement: Criteria preferences are independent (no synergistic interactions)

• Applying MARA without verifying this assumption.

  Requirement: Compensation is acceptable: high score on one criterion can offset low on another

• Applying MARA without verifying this assumption.

  Requirement: Decision matrix is complete (no missing values)

• Using MARA when: Criteria strongly correlated → consider DEMATEL/ANP for interdependence.

  An alternative method is recommended in this situation.

• Using MARA when: Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE).

  An alternative method is recommended in this situation.

Works with

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