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

Overview

Φ ∈ [0,1] is the overall prospect value: 1 = best alternative under min-max normalisation, 0 = worst. Φ is NOT a probability or utility; it is a normalised aggregate of pairwise loss-averse dominance. Always report δ and θ alongside the ranking - different (δ, θ) pairs can produce different rankings on the same data (Zhang-Xu 2016 §5 'different methods of extension could produce different results'). For benchmark fidelity to the paper use δ=1, θ=1, risk_averse extension.

Output

prospect value, higher is better

Data

Hesitant, hfe per cell required

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Sustainable water management efficiency evaluation, supplier selection with expert disagreement, regional / project ranking under hesitant expert input, behavioural MCDM where loss aversion matters, Environmental impact assessment

How it works

1. 1

   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.

   $$h_{ij}^{\prime }=\{\begin{array}{ll}{h}_{ij},& j\in {\Omega }_b(\text{benefit})\\ h_{ij}^c=\{1-\gamma \mid \gamma \in h_{ij}\},& j\in {\Omega }_c(\text{cost})\end{array}$$

   Zhang-Xu 2016, p.752 Eq.(1)

2. 2

   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)).

   $$l_j^{\ast }=\underset{i}{\max }l(h_{ij}^{\prime });{\tilde{h}}_{ij}=h_{ij}^{\prime }\cup \underset{l_j^{\ast }-l(h_{ij}^{\prime })\text{copies}}{\underbrace{\{{\gamma }_{ij}^{\sigma (1)},\dots ,{\gamma }_{ij}^{\sigma (1)}\}}}(\text{risk-averse});\text{sort ascending}$$

   Zhang-Xu 2016, p.757 ¶ following Table 5; Xu-Xia 2011b §3

3. 3

   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).

   $$Z_{\delta }({\tilde{h}}_{ij})={\left(\frac{1}{l_j^{\ast }}\sum _{q=1}^{l_j^{\ast }}({\gamma }_{ij}^{\sigma (q)}{)}^{\delta }\right)}^{1/\delta },\delta \in (0,1]$$

   Zhang-Xu 2016, p.752 Eq.(2)

4. 4

   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.

   $$w_R=\underset{j}{\max }{w}_j;w_{jR}=\frac{w_j}{w_R},j=1,\dots ,m$$

   Zhang-Xu 2016, p.753 Eq.(3)

5. 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 ).

   $$d_E({\tilde{h}}_{ij},{\tilde{h}}_{kj})=\sqrt{\frac{1}{l_j^{\ast }}\sum _{q=1}^{l_j^{\ast }}({\gamma }_{ij}^{\sigma (q)}-{\gamma }_{kj}^{\sigma (q)}{)}^2}$$

   Zhang-Xu 2016, p.761 Eq.(5); precursor Xu-Xia 2011b

6. 6

   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).

   $${\phi }_j(R_i,R_k)=\{\begin{array}{ll}+\sqrt{\frac{w_{jR}\cdot d_E^2({\tilde{h}}_{ij},{\tilde{h}}_{kj})}{{\sum }_{j=1}^m{w}_{jR}}},& Z_{\delta }({\tilde{h}}_{ij})>Z_{\delta }({\tilde{h}}_{kj})(\text{gain})\\ 0,& Z_{\delta }({\tilde{h}}_{ij})=Z_{\delta }({\tilde{h}}_{kj})\\ -\frac{1}{\theta }\sqrt{\frac{\left({\sum }_{j=1}^m{w}_{jR}\right)\cdot d_E^2({\tilde{h}}_{ij},{\tilde{h}}_{kj})}{w_{jR}}},& Z_{\delta }({\tilde{h}}_{ij})<Z_{\delta }({\tilde{h}}_{kj})(\text{loss})\end{array}$$

   Zhang-Xu 2016, p.753 Eq.(4)

7. 7

   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).

   $$\vartheta (R_i,R_k)=\sum _{j=1}^m{\phi }_j(R_i,R_k),i,k=1,\dots ,n$$

   Zhang-Xu 2016, p.753 Eq.(6)

8. 8

   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.

   $$S_i=\sum _{k=1}^n\vartheta (R_i,R_k);\Phi (R_i)=\frac{S_i-{\min }_k{S}_k}{{\max }_k{S}_k-{\min }_k{S}_k}\in [0,1]$$

   Zhang-Xu 2016, p.753 Eq.(7); paper p.759 verbal description of ranking R_2 ≻ R_1 ≻ R_5 ≻ R_4 ≻ R_3

Fits when / Look elsewhere when

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)

Edge cases and pitfalls

• •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.

• Length-equalisation choice is consequential: risk_averse (repeat min) and risk_seeking (repeat max) generally produce different distances d_E and hence different φ_j and Φ. Document the choice in any reported result (paper §5 explicitly warns).

• Confusing Z_δ (measure function, Eq.(2)) with the arithmetic mean score function used in Qian-Wang-Feng 2013: they coincide only at δ=1 in the Z_δ family. Other HF-TODIM variants in the literature use the score-accuracy ordering or different aggregation operators and produce different rankings.

• Reference criterion R = argmax_j w_j enters as a denominator in w_jR - when w has tied maxima the tie-break (smallest index) is convention-dependent. For decisive results elicit distinct weights.

• Rank reversal: adding or removing alternatives can change min/max of row-sums and hence rescale Φ - paper §3.3 ¶3 explicitly notes 'when the number of sample regions changes, the ranking of Chengdu will change too'. Φ values are ONLY comparable within the same alternative set.

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