GRA: Grey Relational Analysis

Overview

Grey relational grade (reference sequence comparison). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Grey relational grade (reference sequence comparison)

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 GRA-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'GRA bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criteria preferences are independent (no synergistic interactions)
• •Hatalı: 'GRA bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Compensation is acceptable: high score on one criterion can offset low on another
• •Hatalı: 'GRA bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix is complete (no missing values)
• •Hatalı: GRA'yi 'Criteria strongly correlated → consider DEMATEL/ANP for interdependence' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: GRA'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: Normalise sequences per criterion type to [0,1]. Formül: x^{*}_{ij} = \begin{cases}\dfrac{x_{ij}-\min x_{ij}}{\max x_{ij}-\min x_{ij}} & j\in J^{+}\\\dfrac{\max x_{ij}-x_{ij}}{\max x_{ij}-\min x_{ij}} & j\in J^{-}\end{cases} Anchor: Deng 1989, p.4 Eq.(2)
2. 2.Adım 2 (F2): Step 2: Reference sequence x_0 = (1, 1, ..., 1) (post-normalisation ideal). Formül: x_{0,j} = 1,\quad j=1,\ldots,n Anchor: Deng 1989, p.5
3. 3.Adım 3 (F3): Step 3: Absolute differences Δ_ij = |x_0,j − x*_ij|. Formül: \Delta_{ij} = \lvert x_{0,j} - x^{*}_{ij}\rvert Anchor: Deng 1989, p.5 Eq.(3)
4. 4.Adım 4 (F4): Step 4: Grey relational coefficient ξ_ij with distinguishing coefficient ρ (typically 0.5). Formül: \xi_{ij} = \dfrac{\min_{i,j}\Delta_{ij} + \rho\,\max_{i,j}\Delta_{ij}}{\Delta_{ij} + \rho\,\max_{i,j}\Delta_{ij}} Anchor: Deng 1989, p.5 Eq.(4)
5. 5.Adım 5 (F5): Step 5: Grey Relational Grade Γ_i = Σ w_j · ξ_ij and descending ranking. Formül: \Gamma_{i} = \sum_{j=1}^{n} w_{j}\,\xi_{ij},\quad 0\le \Gamma_{i}\le 1 Anchor: Deng 1989, p.5 Eq.(5)

Commonly paired with

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

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