Ranking
BWM + Z-number + Zero-Sum Game: Best Worst Method weighting with Z-number payoff matrix and game-theoretic ranking
Adesina, K. A., Yazdi, M., Omidvar, M. · 2022
Overview
BWM-ZNUMBER-GAME üç katmanlı bir karar oyunu kurar: önce BWM ile kriter ağırlıkları (en iyi vs en kötü kritere göre) elde edilir; sonra her (alternatif, kriter) hücresine bir Z-sayısı (TFN değer + güven) atanır; en son alternatifler ile 'Doğa' (worst-case criterion player) arasında zero-sum bir oyun kurulup Nash dengesi LP ile çözülür. Sonuç: en kötü senaryoda bile en az kötü olan alternatif kazanır.
Strengths
- •Combines weight elicitation (BWM), confidence-aware payoff (Z-numbers), and worst-case optimization (Nash) in single pipeline
- •Nash equilibrium gives robust guarantee: alternative ranking holds under DM.2 (Nature) playing optimally adversarial criteria
- •Z-number confidence component B explicitly models 'how sure am I about this fuzzy estimate': distinguishes from pure fuzzy MCDM
- •BWM stage requires only 2n-3 comparisons (vs n(n-1)/2 for AHP): efficient weighting under expert fatigue
- •Iterative elimination handles degenerate G_i=0 payoff matrices systematically (paper p.145)
Limitations
- •Heavy assumption: criteria-DM = adversarial (Nature plays worst-case): may overstate risk in collaborative contexts
- •Z-number arithmetic (Step 2 Four facets) is non-standard; software support limited; symmetrization of asymmetric TFN is lossy
- •Paper §6.3 does NOT publish underlying payoff matrix P / BWM weights / CR: end-to-end reproduction impossible from paper alone
- •Nash LP assumes zero-sum (DM gain = Nature loss): restrictive for non-strictly-competitive scenarios
- •Single Best/Worst criterion shared across experts is a strong consensus requirement; multi-expert disagreement requires manual aggregation (not paper-canonical)
Method assistant
Grounded explanations: it explains the method, it does not compute.
Assumptions to verify
- •Zero-sum assumption holds (appropriate for adversarial/emergency scenarios)
- •BWM comparison vectors are sufficiently consistent (CR ≤ 0.2)
- •Experts can meaningfully provide both linguistic evaluations AND confidence levels (Z-number)
When not to use
- •Cooperative (non-adversarial) MCDM: zero-sum assumption inappropriate
- •No confidence/reliability information available for Z-number B component
- •Very large alternative sets (>15): game LP becomes computationally intensive with iterative elimination
Edge cases
- •G_i = 0 tüm alternatifler için → payoff matrix dejenere, Z-number ve BWM ağırlıkları kontrol et
- •Birden fazla G_i = 0 → paper p.145 iterative elimination (G_i≠0 satırlar ile residual matrix çöz)
- •BWM CR > 0.2 → uzmandan tutarlı karşılaştırma iste (warning, error değil)
- •Z-number B=0 (no confidence) → ε küçük pozitif ile değiştir (division-by-zero engelle)
- •Asymmetric TFN güven bileşeninde → Step 2 (iv) symmetrization önce uygula
- •Multi-expert C_B/C_W disagreement → paper varsayım single shared; uzlaşmazlık varsa majority vote (deviation log)
Common pitfalls
- •Hatalı: BWM-ZNUMBER-GAME = ZBWM (Aboutorab 2018) sanmak: ZBWM sadece BWM'i Z-number ile fuzz'lar, oyun katmanı YOK
- •Hatalı: 'Σ_j a_Bj = 1' diye normalize etmek: DOĞRUSU BWM 1-9 Saaty scalar, normalize sadece w_j ağırlıklarında (Σw=1)
- •Hatalı: Z-number B'yi olasılık sanmak: DOĞRUSU B linguistic confidence (subjective reliability), Bayesian probability değil
- •Hatalı: ν'yi 'game value' diye boş bırakıp Nash strategy değerlerini değişken sanmak: DOĞRUSU ν=1/Σc_i, Nash karma strateji c*_i=ν·c_i (LP solution'dan türetilir)
- •Hatalı: Tek uzmanla BWM uygulayıp 'expert aggregation done' demek: DOĞRUSU cross-expert mean of C_BO/C_OW vektörleri (paper §6.2 verbatim)
- •Hatalı: G_i=0 sonucunda 'method failed' diye raporlamak: DOĞRUSU paper iterative elimination prosedürü zaten bu durumu öngörüyor
Worked example
- 1.Adım 1 (F1, BWM): Her uzman k için Best kriter C_B ve Worst C_W seç. C_BO = (a_B1, ..., a_Bn) ve C_OW = (a_1W, ..., a_nW)^T 1-9 Saaty ölçeği. Cross-expert mean ile birleştir.
- 2.Adım 2 (F2, BWM Model 2 LP): min ξ s.t. |w_B/w_j − a_Bj| ≤ ξ, |w_j/w_W − a_jW| ≤ ξ, Σw=1, w≥0 (Eqs.6.1-6.4). Çözüm: w*_j optimal ağırlıklar + ξ* tutarlılık skoru. CR=ξ*/CI (Table 6.1: n=2→0.00, ..., n=9→5.23). CR≤0.2 kabul (Eq.6.5).
- 3.Adım 3 (F3, Z-number Step 2 'Four facets'): Her hücre (i,j) için: (i) linguistic A_ij + güven B_ij (örn. A='High' → TFN (0.5,0.7,0.9), B='Very Sure' → 0.9). (ii) B'yi crisp güvene çevir α_ij. (iii) weighted confidence: A'_ij = √α_ij · A_ij. (iv) asymmetric TFN'i sym hale getir. Sonuç: payoff matrix P[m×n] TFN-valued.
- 4.Adım 4 (F4, Zero-sum game setup): 5-tuple A = {DM.1 = decision maker, DM.2 = Nature, C_D = alternatif stratejileri, C_N = kriter stratejileri, P = payoff matrix}. DM.1 maksimizer, DM.2 minimizer.
- 5.Adım 5 (F5, Nash LP Model 3): min Σ_i c_i s.t. Σ_i p_ij·c_i ≥ 1 ∀j, c_i ≥ 0 (Eq.6.9). ν = 1/Σc_i oyun değeri. Nash karma strateji: c*_i = ν·c_i. Dual aynı şekilde C_N stratejisi c̃*_j üretir (Eq.6.10).
- 6.Adım 6 (F6, Expected G_i): G_i = c*_i · Σ_j (p_ij · c̃*_j) · w*_j. G_i azalan sırada ranking. Degenerate G_i=0 → iterative elimination (paper p.145): G_i≠0 satırları al, residual matrix üzerinde Model 1+2 yeniden çöz.
Commonly paired with
- •BWM + BWM-ZNUMBER-GAME (embedded)
How to cite
Adesina, K. A.; Yazdi, M.; Omidvar, M. (2022). Emergency Decision Making Fuzzy-Expert Aided Disaster Management System. in: Yazdi M. (ed.), Linguistic Methods Under Fuzzy Information in System Safety and Reliability Analysis, Studies in Fuzziness and Soft Computing, Vol. 414, Springer, Cham, pp. 139-150 (Chapter 6). https://doi.org/10.1007/978-3-030-93352-4_6