MACBETH: Measuring Attractiveness by a Categorical-Based Evaluation Technique

Overview

Weight_Subjective (qualitative pairwise judgment, linear programming, MAVT-based). Output typically weight (higher value = preferred).

Strengths

• •Method-specific: Weight_Subjective (qualitative pairwise judgment, linear programming, MAVT-based)
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •Assumes: Domain experts available
• •Assumes: Experts can express consistent comparisons

Assumptions to verify

• •Domain experts available
• •Experts can express consistent comparisons

When not to use

• •No experts available → use objective weighting
• •High inconsistency → discard and re-elicit

Edge cases

• •if ε_min = 0 ⟹ matrix is cardinally consistent and a basic MACBETH scale exists; otherwise solve LP2 (p.20) to identify the balanced cyclones to surface for revision.
• •If consistent, solve LP-MACBETH2011 (book §11.8.4 item 2): min (x_1 − x_n) s.t. (t_1) indifference: x_p − x_r = 0 ∀(a_p,a_r)∈I; (t_6) anchor: x_n = 0; (t_7) categorical lower bound: x_p − x_r ≥ i ∀(a_

Common pitfalls

• •Hatalı: 'MACBETH bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Domain experts available
• •Hatalı: 'MACBETH bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Experts can express consistent comparisons
• •Hatalı: MACBETH'yi 'No experts available → use objective weighting' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MACBETH'yi 'High inconsistency → discard and re-elicit' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Elicit qualitative pairwise difference judgements on the 7-level MACBETH scale (C_0=no preference, C_1=very weak, C_2=weak, C_3=moderate, C_4=strong, C_5=very strong, C_6=extreme). Fill the upper-triangular n×n matrix of judgements M where m_{p,r}=k iff (a_p, a_r) ∈ C_k. Formül: M = [m_{p,r}],\ m_{p,r}=k\ \Leftrightarrow\ (a_p, a_r) \in C_k,\ k \in \{0,1,2,3,4,5,6\} Anchor: Book Ch.11 §11.1 ('matrix of judgements'); 1997 Springer paper §1.1-§1.2 (pp.16-18, 6 non-zero categories + matrix M = [m_ab])
2. 2.Adım 2 (F2): Step 2: Test consistency by solving LP1 (1997 Springer paper p.20, Proposition 2): if ε_min = 0 ⟹ matrix is cardinally consistent and a basic MACBETH scale exists; otherwise solve LP2 (p.20) to identify the balanced cyclones to surface for revision. Formül: \text{LP1: } \min\ \varepsilon \ \text{s.t.} \ s_{k+1} \geq s_{k}+2,\ \mu(a)-\mu(b) \geq s_{k}+1-\varepsilon,\ \mu(a)-\mu(b) \leq s_{k+1}-1+\varepsilon \ \forall (a,b) \in C_{k} Anchor: 1997 Springer paper §2 pp.19-20 (LP1, LP2, Propositions 1-2)
3. 3.Adım 3 (F3): Step 3: If consistent, solve LP-MACBETH2011 (book §11.8.4 item 2): min (x_1 − x_n) s.t. (t_1) indifference: x_p − x_r = 0 ∀(a_p,a_r)∈I; (t_6) anchor: x_n = 0; (t_7) categorical lower bound: x_p − x_r ≥ i ∀(a_p,a_r)∈C_{i,j} with i ≤ j; (t_8 / t_9) cross-category dominance: x_p − x_r ≥ x_k − x_m + (i − j') for higher-category pair (a_p,a_r)∈C_{i,j} vs lower-category pair (a_k,a_m)∈C_{i',j'}; x_i ≥ 0. Produces the unique basic MACBETH scale {x_1, ..., x_n}. Formül: \min\ (x_1 - x_n)\ \text{s.t.}\ x_n=0,\ x_p - x_r \geq i\ \forall (a_p,a_r)\in C_{i,j},\ x_p - x_r \geq x_k - x_m + (i - j')\ \forall (a_p,a_r)\in C_{i,j},(a_k,a_m)\in C_{i',j'},\ i > j',\ x_i \geq 0 Anchor: Book Ch.11 §11.8.4 item 2 (LP-MACBETH2011), eqs. (t_1), (t_6), (t_7), (t_8), (t_9); hand-derivation Figs. 11.7-11.9
4. 4.Adım 4 (F4): Step 4: Anchor the basic MACBETH scale on two reference levels (typically x_best = 100, x_worst = 0) via positive affine transformation v(a_i) = 100·(x_i − x_worst) / (x_best − x_worst). For criterion-weighting workflows, the rescaled per-criterion values feed into additive aggregation U(a) = Σ_j w_j · v_j(a_ij). Formül: v(a_i) = 100\cdot\dfrac{x_i - x_{\text{worst}}}{x_{\text{best}} - x_{\text{worst}}};\ \ U(a)=\sum_{j=1}^{n}w_j\cdot v_j(a_{ij}) Anchor: Book Ch.11 §11.9 (anchoring discussion); §11.1 lines 29702 'additive value model'

Commonly paired with

• •MACBETH + TOPSIS (high)
• •MACBETH + VIKOR (high)
• •MACBETH + EDAS (high)
• •MACBETH + PROMETHEE (high)
• •MACBETH + ELECTRE-III (high)

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