BWM + Z-number + Zero-Sum Game - Best Worst Method weighting with Z-number payoff matrix and game-theoretic ranking

Overview

G_i represents the expected payoff under Nash equilibrium. Higher G_i = better alternative. G_i = 0 alternatives are ranked iteratively by re-solving the game after removing G_i ≠ 0 rows. BWM weights w*_j modulate the relative importance of criteria in the payoff computation. Sensitivity analysis on confidence levels (B in Z-numbers) is recommended.

Output

expected payoff, higher is better

Data

Z-Number, z number pairs with confidence

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-7 criteria works best

Used for

Emergency decision making, Disaster management, Adversarial MCDM, Risk-based alternative selection under reliability uncertainty

How it works

1. 1

   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. 2

   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. 3

   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. 4

   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. 5

   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. 6

   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.

Fits when / Look elsewhere when

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)

Edge cases and pitfalls

• •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)

• Zero-sum game assumes strict antagonism between two players - appropriate for emergency/adversarial scenarios but not for cooperative MCDM.

• Z-number construction requires both a fuzzy evaluation AND a reliability/confidence assessment - partial information undermines Z-number reliability.

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