This page is published in English.

Weight Subjective

MACBETH - Measuring Attractiveness by a Categorical-Based Evaluation Technique

Weight_Subjective (qualitative pairwise judgment, linear programming, MAVT-based)

Bana e Costa, C. A., Vansnick, J.-C.1994doi:10.1016/0969-6016(94)90010-8 ↗

Overview

For weights, compare concrete criterion swings from all-lower to one-criterion-upper profiles. Direct comparisons between criterion names are not MACBETH swing weighting.

Output: Weight, higher is better
Data: Crisp, expert input required
Weights: Derived internally, no weight source needed
Size: 2+ alternatives, 3-12 criteria works best
Used for: Qualitative judgment aggregation

How it works

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.
$M = [m_{p, r}], m_{p, r} = k \Leftrightarrow (a_{p}, a_{r}) \in C_{k}, k \in {0, 1, 2, 3, 4, 5, 6}$
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
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.
$LP1: min ε s.t. s_{k + 1} \geq s_{k} + 2, μ (a) - μ (b) \geq s_{k} + 1 - ε, μ (a) - μ (b) \leq s_{k + 1} - 1 + ε \forall (a, b) \in C_{k}$
1997 Springer paper §2 pp.19-20 (LP1, LP2, Propositions 1-2)
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}.
$min (x_{1} - x_{n}) s.t. x_{n} = 0, x_{p} - x_{r} \geq i \forall (a_{p}, a_{r}) \in C_{i, j}, x_{p} - x_{r} \geq x_{k} - x_{m} + (i - j^{'}) \forall (a_{p}, a_{r}) \in C_{i, j}, (a_{k}, a_{m}) \in C_{i^{'}, j^{'}}, i > j^{'}, x_{i} \geq 0$
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
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).
$v (a_{i}) = 100 \cdot \frac{x _{i} - x _{worst}}{x _{best} - x _{worst}}; U (a) = j = 1 \sum n w_{j} \cdot v_{j} (a_{ij})$
Book Ch.11 §11.9 (anchoring discussion); §11.1 lines 29702 'additive value model'

Fits when / Look elsewhere when

Fits when

•Native group-decision support (multi-DM aggregation built into the pipeline)

Look elsewhere when

•No experts available. Use objective weighting.
•High inconsistency. Discard and re-elicit.

Assumptions to verify

Domain experts available
Experts can express consistent comparisons

Edge cases and pitfalls

•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_

Category 0 means no difference; -1 means no judgment.
A local value scale of alternatives is a different estimand and must not be labelled criterion weights.

Works with

Commonly takes its weights from

MACBETH

Its derived weights can feed

AHPSORTAPLOCO ARAS AROMAN ARTASI COBRA COCOSO CODAS COMPROMISE-PROGRAMMING COPRAS CRADIS DEA DEA-CROSSEFF DNMA EDAS ELECTRE ELECTRE-I ELECTRE-II ELECTRE-III ELECTRE-IVELECTRE-TRIELECTRE-TRI-BEVAMIX EXPROM-I EXPROM-II FMEA FUCA GOAL-PROGRAMMING GRA HELLWIG IDOCRIW LINMAP LMAW MABAC MAIRCA MARA MARCOS MAUT MOORA MULTIMOORA NAIADE OCRA ORESTE PAM PAMSSEM-I PAMSSEM-II PROMETHEE PROMETHEE-I PROMETHEE-III PSI QUALIFLEX RAFSI RAPS RAT RAWEC REGIME RIM ROV SAW SECA SIR SMART SPOTIS TAXONOMY TODIM TOPSISTOPSIS-SORTUTAUTADISUTASTAR VIKOR WASPAS WEBIRA WEDBA WISP WPM

How to cite

Bana e Costa, C. A.; Vansnick, J.-C. (1994). MACBETH: An interactive path towards the construction of cardinal value functions. International Transactions in Operational Research. https://doi.org/10.1016/0969-6016(94)90010-8

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

MACBETH

Overview

For weights, compare concrete criterion swings from all-lower to one-criterion-upper profiles. Direct comparisons between criterion names are not MACBETH swing weighting.

Output

Weight, higher is better

Data

Crisp, expert input required

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-12 criteria works best

Used for

Qualitative judgment aggregation

How it works

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.

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}

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])

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.

LP1: min ε s.t. s_{k + 1} \geq s_{k} + 2, μ (a) - μ (b) \geq s_{k} + 1 - ε, μ (a) - μ (b) \leq s_{k + 1} - 1 + ε \forall (a, b) \in C_{k}

1997 Springer paper §2 pp.19-20 (LP1, LP2, Propositions 1-2)

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}.

min (x_{1} - x_{n}) 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

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

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).

v (a_{i}) = 100 \cdot \frac{x _{i} - x _{worst}}{x _{best} - x _{worst}}; U (a) = j = 1 \sum n w_{j} \cdot v_{j} (a_{ij})

Book Ch.11 §11.9 (anchoring discussion); §11.1 lines 29702 'additive value model'

Edge cases and pitfalls

•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_

Category 0 means no difference; -1 means no judgment.

A local value scale of alternatives is a different estimand and must not be labelled criterion weights.

How to cite

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

MACBETH