SMAA - Stochastic Multiobjective Acceptability Analysis

Overview

smaa extends SMAA to handle Stochastic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Stochastic element (distribution or scenario probabilities) algebra. The final scores are defuzzified via expected value E[x] = Σ p_k x_k before ranking.

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

   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.

   $$x_{ij}\sim f_{ij}(x),w\sim f_W(w),W=\{w\in {\mathbb{R}}_{\ge 0}^n:\sum _{j=1}^n{w}_j=1\}$$

   Lahdelma 1998 §2; Tervonen 2007 §3

2. 2

   Draw K Monte Carlo samples (x^{(k)}, w^{(k)}) from f_X and f_W (typically K∈[10^4,10^6]).

   $$(x^{(k)},w^{(k)})\sim f_X\times f_W,k=1,\dots ,K$$

   Tervonen 2007 §4; Lahdelma 2004

3. 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.

   $$u(x_i^{(k)},w^{(k)})=\sum _{j=1}^n{w}_j^{(k)}{x}_{ij}^{(k)}$$

   Lahdelma 1998 Eq.(2)

4. 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.

   $$b_i^r={\int }_{w\in W_i^r}{f}_W(w)dw\approx \frac{1}{K}\sum _{k=1}^{K}1\left[{\mathrm{rank}}_k(i)=r\right]$$

   Lahdelma 1998 Eq.(3); Tervonen 2007 §3.2

5. 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.

   $$w_i^c=\frac{{\int }_{w\in W_i^1}w{f}_W(w)dw}{{\int }_{w\in W_i^1}{f}_W(w)dw}$$

   Lahdelma 1998 Eq.(4); Tervonen 2007 §3.3

6. 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.

   $$p_i^c={\int }_{x\in X_i^1(w_i^c)}{f}_X(x)dx$$

   Lahdelma 1998 Eq.(5); Tervonen 2007 §3.4

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

• •when no prior knowledge is available.

• Value-space violation: ensure all entries satisfy Probabilities p_k ≥ 0, Σ p_k = 1 for each scenario before computation.

• Defuzzification method affects ranking: expected value E[x] = Σ p_k x_k is the canonical choice but alternatives exist.

Works with

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