HF-TODIM: Hesitant Fuzzy TODIM via new measure function Z_δ (Zhang-Xu 2016 ITOR)

Overview

HF-TODIM, Gomes ve Lima (1991) tarafından önerilen TODIM (Tomada de Decisão Interativa e Multicriterio: Portekizce 'etkileşimli çok-kriterli karar') yönteminin hesitant fuzzy ortamda Zhang ve Xu (2016, ITOR) tarafından genişletilmiş halidir. TODIM'i diğer MCDM yöntemlerinden ayıran kritik özellik, Kahneman-Tversky (1979) prospect theory tabanlı ÇARPIK BİR DEĞER FONKSİYONU kullanmasıdır: alternatifler arasındaki kazanç (gain) ve kayıp (loss) ASİMETRİK ele alınır: kayıplar bir loss-aversion katsayısı θ ile büyütülür (θ>1 ise riskten-kaçınma, θ<1 ise risk-arayan davranış modellenir). HF uzantı şu adımları getirir: (i) Z_δ(h) hesitant ölçü fonksiyonu (Eq.2, δ ∈ (0,1] kullanıcı parametresi) ile HFE'leri sıralanabilir hâle getirir; (ii) referans kriter olarak max-ağırlıklı kriter seçilir (Adım 1, Eq.3); (iii) çift-yönlü Perceived Value Function ϕ_j(R_i, R_k) (Eq.4) kazançta sqrt(w_jR · d/Σw), kayıpta −1/θ · sqrt((Σw)·d/w_jR) formundadır; (iv) toplam PVF Ψ(R_i) tüm alternatiflerle pairwise karşılaştırmaların ağırlıklı toplamı, [0,1]'e normalize edilir (Eq.7). Tuojiang River Basin sürdürülebilir su yönetimi vakası, HF-TOPSIS'in eksenlerini ARTIRARAK uzman psikolojisini de yakalama kapasitesini gösterir.

Strengths

• •Method-specific: Prospect-theory pairwise dominance (Gomes-Lima 1992) extended to Hesitant Fuzzy Elements (HFE ⊂ [0,1]) via the parametric Z_δ measure function (Zhang-Xu 2016)
• •Preserves hesitant uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Rank reversal known on alternative-set changes (ref: Inherited from classical TODIM (Gomes-Lima 1992); HF extension additionally depends on the length-equalisation rule (risk-averse min-repeat vs risk-seeking max-repeat) and on the measure-function parameter δ.)
• •Assumes: Each (alternative, criterion) cell can be expressed as a non-empty finite set of [0,1] membership values
• •Assumes: Criterion weights are externally provided and sum to 1
• •Assumes: Expert preferences can be reasonably modelled by prospect-theoretic loss aversion (θ-attenuated loss branch)
• •Assumes: The set of compared alternatives is fixed during evaluation (Φ is set-dependent)

Assumptions to verify

• •Each (alternative, criterion) cell can be expressed as a non-empty finite set of [0,1] membership values
• •Criterion weights are externally provided and sum to 1
• •Expert preferences can be reasonably modelled by prospect-theoretic loss aversion (θ-attenuated loss branch)
• •The set of compared alternatives is fixed during evaluation (Φ is set-dependent)

When not to use

• •Crisp data suffices: use classical TODIM (Gomes-Lima 1991)
• •Intuitionistic / hesitancy structure (γ, ν) is needed: use IF-TODIM (Krohling-Pacheco-Siviero 2013)
• •Alternative set is dynamic (frequent additions/removals): min-max normalisation in Eq.(7) causes rank reversal across set changes (Zhang-Xu 2016 §3.3 ¶3 explicitly notes this)
• •Strict no-loss-aversion symmetric distance behaviour is required: use HF-TOPSIS (Xu-Zhang 2013) instead

Edge cases

• •θ=1 dejenere: kayıp-kazanç simetrik, prospect-theory etkisi kaybolur ve HF-TODIM ağırlıklı net-mesafe sıralayıcısına yakınsar. θ=2.25 (Kahneman-Tversky empirical) default; θ>>2 aşırı riskten-kaçınma, sıralama radikal değişebilir.
• •Tüm alternatifler bir kriterde aynı Z_δ değerine sahip: PVF o kriterde sıfır, kriter ayırt edici değil: diğer kriterler üzerinden ranking devam eder.
• •HFE uzunluk farkı: Xu-Xia 2011a extension rules: risk-averse min tekrarla, risk-seeking max tekrarla; engine bu seçimi parametre olarak alır (default risk-averse).
• •Referans kriter x_R seçimi: argmax_j w_j ile YALNIZ EN AĞIR kriter referans olur; eşitlik durumunda paper indeks-küçük olanı önerir (tie-break implicit).
• •Tuojiang case (paper §3): 14 region × 6 indicator; HF-TODIM HF-TOPSIS'ten farklı sıralama üretir (paper §4 karşılaştırma) çünkü psikolojik faktör eklenmiştir: bu epistemic katma değerin somut kanıtıdır.

Common pitfalls

• •Pitfall #1: θ loss-aversion parametresini 1 vermek. θ=1 'TODIM'i TODIM yapan' fonksiyonu düz lineer hâle getirir; sonuç prospect-theory etkisi olmadan ağırlıklı-mesafe sıralayıcısıdır, HF-TODIM iddiası boştur.
• •Pitfall #2: Z_δ(h) ölçü fonksiyonunu HFE score (1/l·Σγ) ile karıştırmak. Z_δ δ-power-mean'dir; δ=1 aritmetik ortalama, δ→0 geometrik ortalama. Score yalnızca Xia-Xu 2011'in özel hâlidir (δ=1). Genel formül Eq.(2)'dir.
• •Pitfall #3: Cost normalizasyonu HFE komplemantasyonu (γ → 1−γ) yerine TOPSIS'in min/max flip'i ile yapmak. TODIM'de cost handling Eq.(1)'le KOMPLEMENT yapılır (paper §2.2); flip kullanıldığında PVF'in 'profit' ve 'loss' yönleri ters döner.
• •Pitfall #4: Referans kriteri sabit (örn. kriter 1) seçmek. Paper Adım 1 EN AĞIR kriteri referans alır (argmax_j w_j); rasgele seçim relative weights w_jR'ı değiştirir, PVF büyüklüğünü saçmalaştırır.
• •Pitfall #5: HFE uzunluk eşitlemede risk-averse vs risk-seeking seçimini raporlamamak. Bu seçim sonucu doğrudan etkiler (paper §2 referansı Xu-Xia 2011a); thesis'te seçimin gerekçesi ve duyarlılık analizi olmalı.

Worked example

1. 1.Adım 1 (F1): Step 1 (Zhang-Xu 2016 p.752 Eq.(1)): Normalise the decision matrix to a uniform 'larger is better' orientation by taking the HFE complement on every cost criterion. For benefit criteria (j ∈ Ω_b) leave h_ij unchanged. For cost criteria (j ∈ Ω_c) replace h_ij by h_ij^c = {1 − γ | γ ∈ h_ij}. The result is a new HFE matrix H = (h_ij)_{n×m} with all columns interpreted as benefits. Formül: h_{ij}' = \begin{cases} h_{ij}, & j \in \Omega_b \ (\text{benefit}) \\ h_{ij}^{c} = \{ 1 - \gamma \mid \gamma \in h_{ij} \}, & j \in \Omega_c \ (\text{cost}) \end{cases} Anchor: Zhang-Xu 2016, p.752 Eq.(1)
2. 2.Adım 2 (F2): Step 2 (Zhang-Xu 2016 p.757 ¶ after Table 5): Length-equalisation within each column. Let l_j* = max_i l(h_ij'). For every cell with l(h_ij') < l_j*, extend the HFE to length l_j* under the risk-averse rule (default): repeat the smallest element γ_{ij}^σ(1) until the cell has l_j* elements. Alternative rule risk_seeking repeats the largest element. The resulting HFEs are written in non-decreasing order γ^σ(1) ≤ γ^σ(2) ≤ … ≤ γ^σ(l_j*) (sorted form required by Eqs.(2),(5)). Formül: l_{j}^{*} = \max_{i} l(h_{ij}'); \quad \tilde{h}_{ij} = h_{ij}' \cup \underbrace{\{ \gamma_{ij}^{\sigma(1)}, \ldots, \gamma_{ij}^{\sigma(1)} \}}_{l_{j}^{*} - l(h_{ij}') \ \text{copies}} \ (\text{risk-averse}); \quad \text{sort ascending} Anchor: Zhang-Xu 2016, p.757 ¶ following Table 5; Xu-Xia 2011b §3
3. 3.Adım 3 (F3): Step 3 (Zhang-Xu 2016 p.752 Eq.(2)): Compute the new measure function Z_δ for every cell. For δ ∈ (0,1] and HFE h̃_ij = {γ^σ(1),…,γ^σ(l_j*)} in ascending order, Z_δ(h̃_ij) = ((1/l_j*) Σ_q (γ^σ(q))^δ)^(1/δ). The measure function provides a scalar surrogate that orders HFEs and replaces the crisp score function used in earlier HF-TODIM variants (Qian-Wang-Feng 2013 used the arithmetic mean = δ=1 case). Formül: Z_{\delta}(\tilde{h}_{ij}) = \left( \dfrac{1}{l_{j}^{*}} \sum_{q=1}^{l_{j}^{*}} \bigl( \gamma_{ij}^{\sigma(q)} \bigr)^{\delta} \right)^{1/\delta}, \quad \delta \in (0, 1] Anchor: Zhang-Xu 2016, p.752 Eq.(2)
4. 4.Adım 4 (F4): Step 4 (Zhang-Xu 2016 p.753 Eq.(3)): Identify the reference criterion R = argmax_j w_j (tie-broken by smallest index) and compute relative weights w_jR = w_j / w_R for j = 1,…,m. Note w_RR = 1. Formül: w_{R} = \max_{j} w_{j}; \quad w_{jR} = \dfrac{w_{j}}{w_{R}}, \quad j = 1, \ldots, m Anchor: Zhang-Xu 2016, p.753 Eq.(3)
5. 5.Adım 5 (F5): Step 5 (Zhang-Xu 2016 p.761 Eq.(5); Xu-Xia 2011b): Compute the hesitant fuzzy Euclidean distance between every pair of cells in the same column. For ascending-sorted, length-equalised HFEs h̃_ij, h̃_kj of common length l_j*, d_E(h̃_ij, h̃_kj) = sqrt( (1/l_j*) Σ_q (γ_ij^σ(q) − γ_kj^σ(q))² ). Formül: d_{E}(\tilde{h}_{ij}, \tilde{h}_{kj}) = \sqrt{ \dfrac{1}{l_{j}^{*}} \sum_{q=1}^{l_{j}^{*}} \bigl( \gamma_{ij}^{\sigma(q)} - \gamma_{kj}^{\sigma(q)} \bigr)^{2} } Anchor: Zhang-Xu 2016, p.761 Eq.(5); precursor Xu-Xia 2011b
6. 6.Adım 6 (F6): Step 6 (Zhang-Xu 2016 p.753 Eq.(4)): Compute the per-indicator prospect-value function (PVF) φ_j(R_i, R_k) for every ordered pair (R_i, R_k) and every indicator x_j. Compare Z_δ(h̃_ij) and Z_δ(h̃_kj) and apply the three-case piecewise formula: GAIN (Z_δ(h̃_ij) > Z_δ(h̃_kj)): φ_j = +sqrt( w_jR · d_E² / Σ_j w_jR ); INDIFFERENCE (=): φ_j = 0; LOSS (<): φ_j = −(1/θ) · sqrt( (Σ_j w_jR) · d_E² / w_jR ). The asymmetry between gain and loss branches encodes prospect-theory loss aversion (Kahneman-Tversky 1979). Formül: \varphi_{j}(R_{i}, R_{k}) = \begin{cases} +\sqrt{\dfrac{w_{jR} \cdot d_{E}^{2}(\tilde{h}_{ij}, \tilde{h}_{kj})}{\sum_{j=1}^{m} w_{jR}}}, & Z_{\delta}(\tilde{h}_{ij}) > Z_{\delta}(\tilde{h}_{kj}) \ (\text{gain}) \\[8pt] 0, & Z_{\delta}(\tilde{h}_{ij}) = Z_{\delta}(\tilde{h}_{kj}) \\[6pt] -\dfrac{1}{\theta} \sqrt{\dfrac{\bigl( \sum_{j=1}^{m} w_{jR} \bigr) \cdot d_{E}^{2}(\tilde{h}_{ij}, \tilde{h}_{kj})}{w_{jR}}}, & Z_{\delta}(\tilde{h}_{ij}) < Z_{\delta}(\tilde{h}_{kj}) \ (\text{loss}) \end{cases} Anchor: Zhang-Xu 2016, p.753 Eq.(4)
7. 7.Adım 7 (F7): Step 7 (Zhang-Xu 2016 p.753 Eq.(6)): Integrate the per-indicator PVFs into the overall pairwise dominance ϑ(R_i, R_k) = Σ_{j=1}^m φ_j(R_i, R_k). The matrix Θ = (ϑ(R_i, R_k))_{n×n} has zero diagonal and is in general non-symmetric (paper Table 12). Formül: \vartheta(R_{i}, R_{k}) = \sum_{j=1}^{m} \varphi_{j}(R_{i}, R_{k}), \quad i, k = 1, \ldots, n Anchor: Zhang-Xu 2016, p.753 Eq.(6)
8. 8.Adım 8 (F8): Step 8 (Zhang-Xu 2016 p.753 Eq.(7) + p.759): Compute the overall prospect value Φ(R_i) by min-max normalising the row sums of Θ to [0,1]: let S_i = Σ_{k=1}^n ϑ(R_i, R_k); then Φ(R_i) = (S_i − min_k S_k) / (max_k S_k − min_k S_k). The alternative with max Φ is the best; rank descending. Formül: S_{i} = \sum_{k=1}^{n} \vartheta(R_{i}, R_{k}); \quad \Phi(R_{i}) = \dfrac{S_{i} - \min_{k} S_{k}}{\max_{k} S_{k} - \min_{k} S_{k}} \in [0, 1] Anchor: Zhang-Xu 2016, p.753 Eq.(7); paper p.759 verbal description of ranking R_2 ≻ R_1 ≻ R_5 ≻ R_4 ≻ R_3

Commonly paired with

• •HF-AHP + HF-TODIM (high)
• •ENTROPY + HF-TODIM (medium)
• •BWM + HF-TODIM (medium)

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