HF-TOPSIS: Hesitant Fuzzy TOPSIS with optional incomplete weight information (Xu-Zhang 2013 KBS)

Overview

HF-TOPSIS, klasik Hwang-Yoon (1981) TOPSIS'in mantığını: "en iyi alternatif aynı anda pozitif-ideale en yakın ve negatif-ideale en uzak olandır": hesitant fuzzy element'lere (HFE: aynı membership için bir küme [0,1] değer) Xu-Zhang (2013) yeniden inşa eder. Yöntemin gerçek yeniliği ranking adımında değil, ağırlık bilgisinin üç farklı eksiklik düzeyini tek bir iskelet içinde çözmesidir: ağırlıklar tamamen biliniyorsa kullanıcı w'sini korur; tamamen bilinmiyorsa Eq.(22)'nin kapalı-form maksimum-sapma çözümünü, kısmen biliniyorsa Model (M-2) doğrusal programını çağırır. Eşit-uzunluğa getirme (Def.2 η-kuralı) ve normalize hesitant Öklid uzaklığı d_1 (Eq.(2)) iki temel hazırlık ögesidir; bunlar olmadan HFE'ler karşılaştırılamaz.

Strengths

• •Method-specific: Distance-to-ideal ranking (Hwang-Yoon 1981) extended to Hesitant Fuzzy Elements (HFE ⊂ [0,1]) via the hesitant normalised Euclidean distance d_1 (Xu-Xia 2011b); supports three weight-information modes: fully specified, completely unknown (Eq.(22) maximizing deviation closed-form), and partly known (Model M-2 linear programme).
• •Preserves hesitant uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Rank reversal known on alternative-set changes (ref: Inherited from classical TOPSIS (Belton-Gear 1983, Wang-Luo 2009); HF extension additionally depends on the η-parametrized length-equalisation rule (Xu-Zhang 2013 Def.2) and on which weight-information mode is active. Switching between modes (fully_specified, completely_unknown via Eq.(22), partly_known via M-2 LP) can produce different rankings on the same matrix (paper §5 Case 1 vs Case 2 swap A1/A4).)
• •Assumes: Each (alternative, criterion) cell can be expressed as a non-empty finite set of [0,1] membership values
• •Assumes: Criterion weights are either externally provided OR the user accepts endogenous derivation via maximizing-deviation principle (Eq.(22) or Model M-2)
• •Assumes: Distance-to-ideal logic is appropriate (symmetric treatment of gains and losses); if loss aversion is needed, use HF-TODIM (Zhang-Xu 2016) instead
• •Assumes: The set of compared alternatives is fixed during evaluation (TOPSIS suffers from rank reversal across set changes)

Assumptions to verify

• •Each (alternative, criterion) cell can be expressed as a non-empty finite set of [0,1] membership values
• •Criterion weights are either externally provided OR the user accepts endogenous derivation via maximizing-deviation principle (Eq.(22) or Model M-2)
• •Distance-to-ideal logic is appropriate (symmetric treatment of gains and losses); if loss aversion is needed, use HF-TODIM (Zhang-Xu 2016) instead
• •The set of compared alternatives is fixed during evaluation (TOPSIS suffers from rank reversal across set changes)

When not to use

• •Crisp data suffices: use classical TOPSIS (Hwang-Yoon 1981)
• •Intuitionistic / hesitancy structure (γ, ν) is needed: use IF-TOPSIS
• •Linguistic ladder assessments instead of [0,1] memberships: use HFLTS-TOPSIS (Beg-Rashid 2013)
• •Interval-valued HFE [γ^L, γ^U] is needed: use IVHF-TOPSIS (Chen-Xu-Xia 2013)
• •Prospect-theory loss aversion is required: use HF-TODIM (Zhang-Xu 2016)
• •Alternative set is dynamic (frequent additions/removals): TOPSIS suffers rank reversal across set changes (Belton-Gear 1983, Wang-Luo 2009)

Edge cases

• •η seçimi sonucu etkiler: η=0 (default, riskten-kaçınma: eksik konumlara min tekrarlanır), η=½ (risk-nötr ortalama), η=1 (risk-arayan: max tekrarlanır). Paper §5.1 η=0 seçer; bunu değiştirmek Case 1'in A1↔A4 sıralamasını sallayabilir.
• •Tam-bilinmeyen ağırlık modu (Eq.(22)) ile kısmen-bilinen mod (Model M-2) AYNI matriste FARKLI sıralama verebilir: paper §5 Case 1 vs Case 2 bunu A1↔A4 swap'ı ile gösterir.
• •C_i payda d_i⁺+d_i⁻ = 0 dejenere durumu: alternatif TÜM kriterlerde hem PIS hem NIS ile aynı; bu yalnızca o satırın kolon-sabit matrise eşit olduğu durumda olur. Engine C_i=0 döndürür ve 'degenerate_closeness_warning' verir.
• •Cost kriteri için HFE kompleman ALINMAZ: yalnızca PIS/NIS extraction (Eq.(23)-(24)) max↔min çevirir. Bu, Zhang-Xu 2016 HF-TODIM'in Eq.(1) komplemantasyonundan ayrılır.

Common pitfalls

• •Pitfall #1: "HF-TOPSIS quadratic separation sqrt(Σ w·d²) kullanır" sanmak. Xu-Zhang 2013 KBS'in Eqs.(25)-(26)'sı LİNEER toplamdır; quadratic form Xia-Xu 2011 IJAR varyantıdır ve bu manifeste DAHİL DEĞİL.
• •Pitfall #2: η parametresini görmezden gelip eşitleme öncesi HFE'leri direkt karşılaştırmaya çalışmak. l(h_ij) farklıysa d_1 tanımsızdır; mutlaka Def.2 uygulanmalı.
• •Pitfall #3: Cost kriterinde HFE elemanlarını 1-γ ile komplemantasyon. Bu, HF-TODIM (Zhang-Xu 2016) konvansiyonudur; Xu-Zhang 2013 KBS'te cost handling SADECE F4 PIS/NIS max/min flip'i ile yapılır.
• •Pitfall #4: M-2 LP'nin kısıt setini boş bırakmak (Σ w_j=1 dışında hiçbir constraint yok) → çözüm tek bir kriteri 1, kalanları 0 yapan köşe noktası olur; sıralama saçmalaşır. LP anlamlı çalışsın diye en az birkaç sıralama veya aralık kısıtı verilmeli.
• •Pitfall #5: Alternatif kümesi değişirken (eleme/ekleme) sıralamayı yeniden hesaplamamak. Klasik TOPSIS rank-reversal (Belton-Gear 1983) burada da geçerlidir; sıralama dinamik alternatif setiyle stabil DEĞİLDİR.

Worked example

1. 1.Adım 1 (F1): Step 1 (Xu-Zhang 2013 §2 + §3 ¶1): Collect the Hesitant Fuzzy Decision Matrix H = (h_ij)_{m×n} with each cell an HFE in [0,1]. No HFE complement is applied here even on cost criteria (the cost/benefit asymmetry is handled at the PIS/NIS extraction step F4 by flipping max/min, NOT by complementing the cell: this is a key distinction from Zhang-Xu 2016 HF-TODIM which complements via Eq.(1)). Formül: \mathbf{H} = (h_{ij})_{m \times n}, \quad h_{ij} \subset [0,1] \ \text{non-empty finite} Anchor: Xu-Zhang 2013, §3 ¶1
2. 2.Adım 2 (F2): Step 2 (Xu-Zhang 2013 Def.2, p.55): η-parametrized 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 by repeatedly appending the value η·max(h_ij) + (1−η)·min(h_ij) until length l_j* is reached, then sort ascending. Default η=0 (risk-averse, repeats min: paper §5.1 choice). η=½ risk-neutral, η=1 risk-seeking. Formül: l_{j}^{*} = \max_{i} l(h_{ij}); \quad \tilde{h}_{ij} = h_{ij} \cup \underbrace{\{ \eta \cdot \max(h_{ij}) + (1-\eta) \cdot \min(h_{ij}), \ldots \}}_{l_{j}^{*} - l(h_{ij}) \ \text{copies}}; \quad \text{sort ascending} Anchor: Xu-Zhang 2013, Def.2 (p.55); §5.1 sets η=0
3. 3.Adım 3 (F3): Step 3 (Xu-Zhang 2013 §3: three-mode weight resolution): Resolve the criterion weight vector w according to weight_info_mode: (a) 'fully_specified': use the user-supplied w directly; (b) 'completely_unknown': compute Y_j = Σ_{i=1}^m Σ_{k=1}^m d_1(h̃_ij, h̃_kj) for each j and set w_j = Y_j / Σ_{j'=1}^n Y_{j'} (Eq.(22), closed-form maximizing deviation); (c) 'partly_known': solve the linear programme Model (M-2): max Σ_{j=1}^n Y_j w_j subject to user weight_constraints, Σ_{j=1}^n w_j = 1, w_j ≥ 0. The LP is feasible whenever weight_constraints define a non-empty subset of the simplex. After resolution, w is fixed and used identically downstream. Formül: (a) \ w = w_{\text{user}}; \quad (b) \ Y_{j} = \sum_{i=1}^{m} \sum_{k=1}^{m} d_{1}(\tilde{h}_{ij}, \tilde{h}_{kj}), \ w_{j} = \dfrac{Y_{j}}{\sum_{j'=1}^{n} Y_{j'}}; \quad (c) \ \max_{w} \sum_{j=1}^{n} Y_{j} w_{j} \ \text{s.t.} \ w \in \Delta, \sum_{j} w_{j} = 1, w_{j} \geq 0 Anchor: Xu-Zhang 2013, Eq.(22) (completely unknown); Model (M-2) (partly known); §3 ¶ on fully specified case
4. 4.Adım 4 (F4): Step 4 (Xu-Zhang 2013 Eqs.(23)-(24), p.56): Determine the Hesitant Fuzzy Positive-Ideal Solution h̃_j^+ and Negative-Ideal Solution h̃_j^− on the length-equalised, ascending-sorted matrix. For benefit criteria (j ∈ Ω_b): h̃_j^+ = (max_i γ_ij^σ(τ))_τ component-wise across τ=1,…,l_j*; h̃_j^− = (min_i γ_ij^σ(τ))_τ. For cost criteria (j ∈ Ω_c) swap max and min. NO multiplicative HFE weighting is applied before this step: the weights enter only in F6 as scalar coefficients of the per-criterion distances. Formül: \tilde{h}_{j}^{+} = \begin{cases} \bigl(\max_{i} \gamma_{ij}^{\sigma(\tau)}\bigr)_{\tau=1}^{l_{j}^{*}}, & j \in \Omega_{b} \\ \bigl(\min_{i} \gamma_{ij}^{\sigma(\tau)}\bigr)_{\tau=1}^{l_{j}^{*}}, & j \in \Omega_{c} \end{cases}; \quad \tilde{h}_{j}^{-} = \begin{cases} \bigl(\min_{i} \gamma_{ij}^{\sigma(\tau)}\bigr)_{\tau=1}^{l_{j}^{*}}, & j \in \Omega_{b} \\ \bigl(\max_{i} \gamma_{ij}^{\sigma(\tau)}\bigr)_{\tau=1}^{l_{j}^{*}}, & j \in \Omega_{c} \end{cases} Anchor: Xu-Zhang 2013, Eqs.(23)-(24)
5. 5.Adım 5 (F5): Step 5 (Xu-Zhang 2013 Eq.(2); Xu-Xia 2011b): Compute the hesitant normalised Euclidean distance d_1 between each cell h̃_ij and the per-criterion ideals h̃_j^+ and h̃_j^−. Both arguments share the column-common length l_j* and are ascending-sorted. d_1(h_a, h_b) = sqrt((1/l) Σ_τ (γ_a^σ(τ) − γ_b^σ(τ))²). Formül: d_{1}(\tilde{h}_{ij}, \tilde{h}_{j}^{\pm}) = \sqrt{ \dfrac{1}{l_{j}^{*}} \sum_{\tau=1}^{l_{j}^{*}} \bigl( \gamma_{ij}^{\sigma(\tau)} - \gamma_{j}^{\pm, \sigma(\tau)} \bigr)^{2} } Anchor: Xu-Zhang 2013, Eq.(2); Xu-Xia 2011b
6. 6.Adım 6 (F6): Step 6 (Xu-Zhang 2013 Eqs.(25)-(26), p.56): LINEAR-weighted separation: aggregate per-criterion distances into overall separations d_i^+ and d_i^− as the weighted SUM (not the square-rooted-sum-of-squares) of column distances. d_i^+ = Σ_{j=1}^n w_j · d_1(h̃_ij, h̃_j^+); d_i^− = Σ_{j=1}^n w_j · d_1(h̃_ij, h̃_j^−). NOTE: This linear-weighted form is a KEY ALGORITHMIC CHOICE of Xu-Zhang 2013 KBS; an alternative quadratic-weighted form d_i^± = sqrt(Σ_j w_j · d_1²) appears in Xia-Xu 2011 IJAR (the parallel HF-TOPSIS variant: see P.implementation_review.literature_disambiguation). DecisionMind implements ONLY the Xu-Zhang 2013 KBS linear form. Formül: d_{i}^{+} = \sum_{j=1}^{n} w_{j} \cdot d_{1}(\tilde{h}_{ij}, \tilde{h}_{j}^{+}), \quad d_{i}^{-} = \sum_{j=1}^{n} w_{j} \cdot d_{1}(\tilde{h}_{ij}, \tilde{h}_{j}^{-}), \quad i = 1, \ldots, m Anchor: Xu-Zhang 2013, Eqs.(25)-(26)
7. 7.Adım 7 (F7): Step 7 (Xu-Zhang 2013 Eq.(27), p.56): Compute the closeness coefficient C_i = d_i^− / (d_i^+ + d_i^−). C_i ∈ [0,1]; higher = closer to the positive-ideal solution. Degenerate case d_i^+ + d_i^− = 0 means alternative i is identical to BOTH ideals on every criterion (only possible if the matrix is column-constant); engine returns C_i = 0 and emits a 'degenerate_closeness_warning'. Formül: C_{i} = \dfrac{d_{i}^{-}}{d_{i}^{+} + d_{i}^{-}} \in [0, 1], \quad i = 1, \ldots, m Anchor: Xu-Zhang 2013, Eq.(27)
8. 8.Adım 8 (F8): Step 8 (Xu-Zhang 2013 §3 last ¶ + §5 case studies): Rank alternatives in descending order of C_i. The alternative with the largest C_i is the recommended choice. Formül: \text{rank}(A_{i}) = \text{argsort descending}_{i} \ C_{i} Anchor: Xu-Zhang 2013, §3 last paragraph; §5 Case 1 ranking A5≻A3≻A2≻A1≻A4, Case 2 ranking A5≻A3≻A2≻A4≻A1

Commonly paired with

• •HF-AHP + HF-TOPSIS (high)
• •ENTROPY + HF-TOPSIS (high)
• •BWM + HF-TOPSIS (medium)
• •endogenous_Eq22_maximizing_deviation + HF-TOPSIS (high)
• •endogenous_M2_LP_partly_known + HF-TOPSIS (medium)

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