SMAA-2 - Stochastic extension of SMAA2

Overview

SMAA-2 (Lahdelma & Salminen 2001) extends base SMAA by computing the full rank acceptability index matrix b_i^r for all ranks r=1..m (not only rank 1) and aggregating into a holistic acceptability a_i^h = Σ_r α^r b_i^r (Eq.18) via monotone metaweights α^1 ≥ ... ≥ α^m ≥ 0, α^1=1. Monte Carlo samples (x^{(k)}, w^{(k)}) drawn from f_X and uniform W; per-sample additive utility (Eq.11) → rankings → b_i^r → a_i^h. Final ranking by a_i^h descending. No defuzzification - SMAA-2 is probabilistic, not fuzzy.

Output

rank acceptability index, higher is better

Data

Stochastic, decision matrix + weight distribution priors

Size

2+ alternatives, 3-10 criteria works best

Used for

Weight elicitation under uncertainty, robust ranking, public policy

How it works

1. 1

   Kriter değerleri için f_ij(ξ) dağılımları + ağırlık simpleksi W üzerinde f_W(w) (default: uniform) + meta-weights α^1 ≥ α^2 ≥ ... ≥ α^m ≥ 0 ile α^1=1 tanımla (paper §4.2). Default α: linear (m−r)/(m−1) ya da centroid Σ_{i=r}^m 1/i normalize.

2. 2

   K Monte Carlo örneği çek - her örnek için (ξ^(k), w^(k)) ~ f·f_W; K=10000 tipik.

3. 3

   Her örnek için additive utility u(ξ_i^(k), w^(k)) = Σ_j w_j^(k) · u_j(ξ_ij^(k)) (Eq.11). u_j alt-fayda fonksiyonu = paper §5 lineer min-max ölçekleme.

4. 4

   Her alternatif i ve her rank r için b_i^r = (örneklerde alternatif i'nin rank r aldığı sayı) / K (Eq.14). b_i^r ≥ 0, Σ_r b_i^r = 1, Σ_i b_i^r = 1.

5. 5

   Holistic acceptability a_i^h = Σ_{r=1}^m α^r · b_i^r (Eq.18). α=(1,0,...,0) elitistic → a_i^h = b_i^1 = base SMAA. Central weight w_i^c = E[w | u(ξ_i,w) maks] (Eq.7).

6. 6

   Confidence factor p_i^c = Pr(u(ξ_i, w_i^c) ≥ u(ξ_k, w_i^c) ∀k) (Eq.8). Final ranking: a_i^h azalan sırada.

Fits when / Look elsewhere when

Assumptions to verify

• Weight distribution (uniform Dirichlet, or DM-specified) is appropriate
• Sample size N sufficient for stable acceptability indices (N ≥ 10,000 typical)
• Underlying ranking method (TOPSIS, SAW, ...) is specified

Edge cases and pitfalls

• •Tüm DM'ler aynı ağırlığı kabul ederse: f_W dejenere → SMAA-2 ≡ deterministic weighted-sum (boş egzersiz)
• •α uniform (α^r = c) → Σ_r b_i^r = 1 her i için, dolayısıyla a_i^h = c (tüm alternatifler beraber). Pathological - paper §4.2 uyarır.
• •ZERO alternatif (paper §5): tüm kriterlerde nominal ortalama değer, deterministic anchor; p_i^c=100% bekleniyor (rakip alternatiflerden ayrılma yoksa)
• •K küçük (<1000) → b_i^r noisy, near-ties (örn. Helsinki IIB2/IIC2 a^h=14) sıra değiştirebilir

• Metaweight normalization confusion: paper §4.2 uses α^1 = 1 (best rank weight = 1), NOT Σ_r α^r = 1 (simplex). The latter would scale b_i^1 down for no reason.

• Conflating SMAA-2 with fuzzy methods: SMAA-2 is probabilistic (criteria distributions f_ij, weight density f_W). There is NO defuzzification step - final ranking is by holistic acceptability a_i^h directly.

• Monte Carlo sample size K too small: rank acceptabilities b_i^r have standard error ~√(b(1-b)/K). K=10,000 gives ±0.005 precision; K=1,000 only ±0.016. Use K≥10,000 for stable holistic ranking.

Works with

How to cite