Bayesian BWM: Probabilistic Group Best-Worst Method

Overview

Hierarchical Dirichlet posterior over weights via MCMC (JAGS): group decision. Output typically weight (higher value = preferred).

Strengths

• •Method-specific: Hierarchical Dirichlet posterior over weights via MCMC (JAGS): group decision
• •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

• •See F.steps and D.parameters for BWM-BAYESIAN-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'BWM-BAYESIAN bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Domain experts available
• •Hatalı: 'BWM-BAYESIAN bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Experts can express consistent comparisons
• •Hatalı: BWM-BAYESIAN'yi 'No experts available → use objective weighting' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: BWM-BAYESIAN'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: Collect from each DM k: best B_k, worst W_k, BO vector A_B^k, OW vector A_W^k. Formül: \forall k\in\{1,\ldots,K\}:\ B_{k}, W_{k}\in\{1,\ldots,n\},\ A_{B}^{k}=(a_{B1}^{k},\ldots,a_{Bn}^{k}),\ A_{W}^{k}=(a_{1W}^{k},\ldots,a_{nW}^{k}) Anchor: Mohammadi & Rezaei 2020, p.4 Sec.3.1
2. 2.Adım 2 (F2): Step 2: Hierarchical Dirichlet prior on aggregate weights w^agg and per-DM weights w^k. Formül: w^{agg} \sim \mathrm{Dir}(\boldsymbol{\alpha});\ w^{k} \mid w^{agg}, \gamma \sim \mathrm{Dir}(\gamma\, w^{agg})\ \text{for } k=1,\ldots,K;\ \gamma \sim \mathrm{Gamma}(a,b) Anchor: Mohammadi & Rezaei 2020, p.5 Eqs.(7)-(8)
3. 3.Adım 3 (F3): Step 3: Multinomial likelihood: OW counts proportional to w^k; reciprocals of BO entries proportional to w^k. Formül: A_{W}^{k} \mid w^{k} \sim \mathrm{Mult}\!\left(\textstyle\sum_{j} a_{jW}^{k},\ w^{k}\right);\ \big(1/A_{B}^{k}\big) \mid w^{k} \sim \mathrm{Mult}\!\left(\textstyle\sum_{j} 1/a_{Bj}^{k},\ w^{k}\right) Anchor: Mohammadi & Rezaei 2020, p.5 Eqs.(9)-(10)
4. 4.Adım 4 (F4): Step 4: Run MCMC (JAGS Gibbs sampler) for n_chains × n_iter iterations after n_burn burn-in. Collect posterior samples of (w^agg, w^1,...,w^K, γ). Formül: \{w^{agg,(s)}, w^{k,(s)}\}_{s=1}^{S} \sim p\!\left(w^{agg}, w^{1},\ldots,w^{K}, \gamma \,\middle|\, \{A_{B}^{k}, A_{W}^{k}\}_{k=1}^{K}\right) Anchor: Mohammadi & Rezaei 2020, p.6 Sec.3.3
5. 5.Adım 5 (F5): Step 5: Posterior mean weights and 95% credible intervals; aggregate w^agg is the group weight vector. Formül: \hat{w}_{j}^{agg} = \tfrac{1}{S}\sum_{s=1}^{S} w_{j}^{agg,(s)};\ CI_{95}(w_{j}^{agg}) = [Q_{0.025}, Q_{0.975}] Anchor: Mohammadi & Rezaei 2020, p.7 Eqs.(13)-(14)
6. 6.Adım 6 (F6): Step 6: Credal ranking matrix: P(c_i > c_j) = (1/S) Σ_s 1[w_i^{agg,(s)} > w_j^{agg,(s)}]. Apply credibility threshold to produce partial order. Formül: P(c_{i} \succ c_{j}) = \tfrac{1}{S}\sum_{s=1}^{S} \mathbf{1}\!\left[w_{i}^{agg,(s)} > w_{j}^{agg,(s)}\right];\ c_{i} \succ_{\tau} c_{j} \iff P(c_{i}\succ c_{j}) \geq \tau Anchor: Mohammadi & Rezaei 2020, p.8 Eqs.(15)-(16)

Commonly paired with

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

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