SMAA: Stochastic Multiobjective Acceptability Analysis

Overview

Stochastic outranking/ranking: Stochastic element (distribution or scenario probabilities). Output typically rank_acceptability_index (higher value = preferred).

Strengths

• •Method-specific: Stochastic outranking/ranking: Stochastic element (distribution or scenario probabilities)
• •Preserves stochastic uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

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

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

When not to use

• •Exact weights known with certainty: use deterministic ranking
• •Computational budget very limited: use sensitivity perturbation instead

Edge cases

• •when no prior knowledge is available.

Common pitfalls

• •Hatalı: 'SMAA bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Weight distribution (uniform Dirichlet, or DM-specified) is appropriate
• •Hatalı: 'SMAA bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Sample size N sufficient for stable acceptability indices (N ≥ 10,000 typical)
• •Hatalı: 'SMAA bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying ranking method (TOPSIS, SAW, ...) is specified
• •Hatalı: SMAA'yi 'Exact weights known with certainty' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: SMAA'yi 'Computational budget very limited' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Define the decision problem with alternatives x_i (i=1..m) and criteria j=1..n; specify probability distributions f_X(x) for uncertain criteria measurements x_ij ~ f_{ij} and f_W(w) for the feasible weight space W={w∈ℝ^n_{≥0}: Σ w_j = 1}. Uniform (uninformative) priors are used when no prior knowledge is available. Formül: x_{ij} \sim f_{ij}(x),\quad w \sim f_{W}(w),\quad W = \{w \in \mathbb{R}^{n}_{\geq 0} : \sum_{j=1}^{n} w_{j} = 1\} Anchor: Lahdelma 1998 §2; Tervonen 2007 §3
2. 2.Adım 2 (F2): Step 2: Draw K Monte Carlo samples (x^{(k)}, w^{(k)}) from f_X and f_W (typically K∈[10^4,10^6]). Formül: (x^{(k)}, w^{(k)}) \sim f_{X}\times f_{W},\quad k=1,\ldots,K Anchor: Tervonen 2007 §4; Lahdelma 2004
3. 3.Adım 3 (F3): Step 3: For each sample compute the additive utility u(x_i^{(k)}, w^{(k)}) = Σ_j w_j^{(k)} · x_{ij}^{(k)} for every alternative. Formül: u(x_{i}^{(k)}, w^{(k)}) = \sum_{j=1}^{n} w_{j}^{(k)}\,x_{ij}^{(k)} Anchor: Lahdelma 1998 Eq.(2)
4. 4.Adım 4 (F4): Step 4: Rank alternatives within each sample (descending u). Estimate Rank Acceptability Indices b_i^r = (1/K)·Σ_k 𝟙[rank_k(i)=r], the probability that alternative i attains rank r over the weight/measurement uncertainty. Formül: b_{i}^{r} = \int_{w \in W_{i}^{r}} f_{W}(w)\,dw \;\approx\; \frac{1}{K}\sum_{k=1}^{K} \mathbf{1}\!\left[\mathrm{rank}_{k}(i)=r\right] Anchor: Lahdelma 1998 Eq.(3); Tervonen 2007 §3.2
5. 5.Adım 5 (F5): Step 5: Compute central weight vectors w_i^c = (∫_{w∈W_i^1} w·f_W(w)dw)/(∫_{w∈W_i^1} f_W(w)dw), the centre of gravity of the favourable weight space supporting rank-1 for alternative i. Formül: w_{i}^{c} = \dfrac{\displaystyle\int_{w \in W_{i}^{1}} w\,f_{W}(w)\,dw}{\displaystyle\int_{w \in W_{i}^{1}} f_{W}(w)\,dw} Anchor: Lahdelma 1998 Eq.(4); Tervonen 2007 §3.3
6. 6.Adım 6 (F6): Step 6: Confidence factor p_i^c = ∫_{x∈X_i^1(w_i^c)} f_X(x)dx: probability that alternative i is best given its central weights: and final analysis: alternatives with high b_i^1 and high p_i^c are recommended. Formül: p_{i}^{c} = \int_{x \in X_{i}^{1}(w_{i}^{c})} f_{X}(x)\,dx Anchor: Lahdelma 1998 Eq.(5); Tervonen 2007 §3.4

Commonly paired with

• •DM_uniform_prior + SMAA (high)

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