HFL-AHP: Hesitant Fuzzy Linguistic AHP (HFLTS-envelope family: Yavuz 2015 + Özdağoğlu 2018)

Overview

HFL-AHP, Saaty (1980) Analytic Hierarchy Process'un hesitant fuzzy linguistic term set (HFLTS: bir LTS S={s_0,...,s_τ} üzerinde alt-sıralı sembol kümesi) ortamında Yavuz ve arkadaşları (2015, ESWA) tarafından hiyerarşik olarak genişletilmiş halidir. AHP'nin temel sezgisi sabit kalır: "karar problemini hedef-kriter-altkriter-alternatif hiyerarşisine ayır; her seviyede pairwise comparison matrisi inşa et; özvektör ile yerel öncelikler çıkar; hiyerarşi boyunca çarp". Paper'ın HFLTS katkısı pairwise comparison'ların artık tekil bir sembol (örn. 'önemli', s_4) değil, BİR ARALIK SEMBOL KÜMESİ ('önemli ile çok önemli arası', {s_4, s_5}) olabilmesidir: uzmanın "şu iki linguistic değer arasında kararsızım" yargılarını koruyarak modellenmesini sağlar. Aggregation paper-spesifik HEHFLWA (Hesitant Extended HFL Weighted Average) operatörü ile yapılır; envelope conversion (Liu-Rodriguez 2014) HFLTS'leri trapezoidal fuzzy sayılara çevirir. AFV (alternative-fuel vehicle) seçim vakasında ev sağlık-bakım filosu için electric vehicle (EV) bugünün koşullarında en uygun olarak çıkar.

Strengths

• •Method-specific: HFLTS-envelope pairwise comparison: interval/2-tuple aggregation: preference-degree (Yavuz) or defuzz (Özdağoğlu) ranking
• •Preserves hesitant_linguistic uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Rank reversal known on alternative-set changes (ref: Inherited from classical AHP weighted-sum synthesis (Belton-Gear 1983; Saaty 1990); interval-arithmetic widening can additionally flip the preference-degree direction near 0.5)
• •Assumes: Domain experts available and willing to express hesitancy as linguistic ranges (e.g. 'between high and very high')
• •Assumes: Linguistic terms can be ordered consistently across experts (single chosen linguistic_scale variant)
• •Assumes: Either the Yavuz path (2-tuple + arithmetic mean + preference-degree) or the Özdağoğlu path (trapezoidal + OWA + geometric mean + defuzz) is appropriate for the application

Assumptions to verify

• •Domain experts available and willing to express hesitancy as linguistic ranges (e.g. 'between high and very high')
• •Linguistic terms can be ordered consistently across experts (single chosen linguistic_scale variant)
• •Either the Yavuz path (2-tuple + arithmetic mean + preference-degree) or the Özdağoğlu path (trapezoidal + OWA + geometric mean + defuzz) is appropriate for the application

When not to use

• •Experts can give precise crisp Saaty 1-9 judgments → use classical AHP (AHP.json)
• •Experts express judgments as multiplicative HME on [1/9, 9] → use Zhu-Xu 2014 HF-AHP (HF-AHP.json)
• •Judgment uncertainty is intuitionistic or Pythagorean rather than hesitant linguistic → use IF-AHP / PF-AHP variants
• •Application requires LP-based consistency optimisation (Group Satisfaction Index) → use HF-AHP.json

Edge cases

• •Tek-sembol HFLTS (örn. {s_4}) klasik fuzzy AHP'ye düşer; HFLTS'in katma değeri ÇOKLU-sembol expression'lardadır (örn. "at least medium" → {s_3,s_4,s_5,s_6}).
• •Asimetrik LTS (örn. S={s_0..s_6}) default; karar-verici simetrik 0-merkezli LTS (örn. {s_−3..s_+3}) kullanırsa envelope formülü ayarlanmalı.
• •Hiyerarşinin derinliği (3+ seviye) hesitant belirsizliği şişirebilir; paper §5 sadece 3 seviyeyi raporlar, daha derin hiyerarşilerde 'envelope blow-up' riski var.
• •Pairwise comparison matrisinde tutarlılık (CR ≤ 0.1) fuzzy ortamda crisp AHP'den daha esnek tutulur (paper §4.3 fuzzy CR formülü), ama yine kontrol edilmelidir.
• •AFV case (paper §5): 12 kriter × 5 alternatif (Gasoline, LPG, Diesel, Hybrid Electric, Pure Electric); pure-EV best when 'extended range' scenario açılır.

Common pitfalls

• •Pitfall #1: HFLTS'i sıradan fuzzy set'le karıştırmak. HFLTS bir LTS üzerinde ALT-KÜME ifade eder ('between low and medium'); fuzzy set ise bir membership fonksiyonu ile tanımlanır. Envelope conversion (Liu-Rodríguez 2014) HFLTS → trapezoidal fuzzy köprüsüdür ama linguistik anlam korunmalı.
• •Pitfall #2: AHP tutarlılık testini fuzzy ortamda atlamak. Paper §4.3'te fuzzy CR formülü vardır; CR > 0.1 ise uzmandan revize istenir, bu adımı atlamak yargı tutarsızlığına yol açar.
• •Pitfall #3: HFLTS pairwise comparison'da reciprocal kuralını (a_ij · a_ji = 1) linguistic ortamda doğrudan uygulamak. Envelope sonrası fuzzy a_ji = 1/a_ij hesaplanır; envelope öncesi linguistic 'reciprocal' tanımı (örn. "high"-in tersi "low") açıkça modellenir.
• •Pitfall #4: Hiyerarşi seviyelerinde uzman değişiminin etkisini görmezden gelmek. AHP ev sağlık-bakım örneğinde 4 farklı uzman 12 kriteri farklı görür; paper §5.1 grup fuzzy aggregation gerektirir.
• •Pitfall #5: Sonuçta tek bir crisp ağırlık raporlamak. HFLTS-AHP çıktıları fuzzy ağırlıklardır; defuzzify edip tek sayı vermek bilgi kaybıdır. Thesis'te trapezoidal (l,p,q,u) sunulması zorunlu.

Worked example

1. 1.Adım 1 (F1): Step 1 (Yavuz 2015 §5 step 1 + Özdağoğlu 2018 §3 ¶1): Define the linguistic term set S = {s_0, s_1, …, s_g}. Default = 7-point Yavuz {ni, vli, li, mi, hi, vhi, ai} with semantic 0..6/6 mapping; alternative = 11-point Şenvar (Özdağoğlu 2018 Tablo) AHI..ALI with trapezoidal (a_L, a_M1, a_M2, a_R) mapping. Formül: S = \{s_0, s_1, \ldots, s_g\}; \quad \text{semantic}(s_k) = k/g \ (\text{7-point}) \quad \text{or} \quad (a_L^k, a_{M1}^k, a_{M2}^k, a_R^k) \ (\text{11-point trapezoidal}) Anchor: Yavuz 2015 §5 step 1 + Tablo 1; Özdağoğlu 2018 §3 Tablo (Şenvar 2017)
2. 2.Adım 2 (F2): Step 2 (Yavuz 2015 §3 + Rodríguez et al. 2012 Def.3 + §IV grammar): Define the context-free grammar G_H = {V_N, V_T, I, P} that maps comparative linguistic expressions (e.g. 'between hi and vhi', 'at least mi') to HFLTS via the transformation function E_GH. Formül: E_{GH}: \mathrm{ll}_S \to H_S \subset \mathcal{P}(S), \quad H_S = \{s_i, s_{i+1}, \ldots, s_j\} \ (\text{contiguous index range on } S) Anchor: Yavuz 2015 §3 ¶2-3; Rodríguez et al. 2012 §IV grammar G_H
3. 3.Adım 3 (F3): Step 3 (Yavuz 2015 §5 step 3): For each expert k = 1..m, collect a pairwise preference relation P^k = (p^k_ij)_{n×n} where each cell is an HFLTS expression from G_H. Diagonals p^k_ii = {s_{g/2}}; reciprocity holds linguistically. Formül: P^k = (p_{ij}^k)_{n \times n}, \quad p_{ij}^k = \{s_l, s_{l+1}, \ldots, s_h\} \subset S, \quad p_{ii}^k = \{s_{g/2}\} Anchor: Yavuz 2015 §5 step 3
4. 4.Adım 4 (F4): Step 4 (Yavuz 2015 §5 step 4 + §4 envelope definition): Compute the envelope env(p^k_ij) = [p^k_ij_min, p^k_ij_max] = [s_l, s_h] for each HFLTS cell. The envelope is the contiguous interval bounded by the smallest and largest terms. Formül: \mathrm{env}(p_{ij}^k) = [\,p_{ij}^{k,-},\ p_{ij}^{k,+}\,] = [s_l, s_h] Anchor: Yavuz 2015 §4 + §5 step 4; Özdağoğlu 2018 Eqs.1-2 (a=min, d=max for trapezoidal)
5. 5.Adım 5 (F5): Step 5 (variant-dependent): Yavuz 2015 (§5 step 5): convert each envelope endpoint into a 2-tuple linguistic representation (s_l, 0). Özdağoğlu 2018 (Eqs.3-6): convert the envelope into a trapezoidal fuzzy number (a, b, c, d) where a = aL_min, d = aR_max, and inner points b, c are OWA-weighted averages of intermediate aM1/aM2 values with α₁ = (g − (j − i)) / (g − 1). Formül: \text{Yavuz: } p_{ij}^{k,\pm} \to (s_{l/h},\ 0). \quad \text{Özdağoğlu: } \alpha_1 = \frac{g - (j-i)}{g - 1}; \ b = \alpha_1 a_{M1}^{(\min)} + (1-\alpha_1) a_{M1}^{(\max)}; \ c = \alpha_1 a_{M2}^{(\max)} + (1-\alpha_1) a_{M2}^{(\min)}; \ y_{\text{recip}} = (1/d, 1/c, 1/b, 1/a)\ (\text{Eq.7}) Anchor: Yavuz 2015 §5 step 5; Özdağoğlu 2018 Eqs.3-7
6. 6.Adım 6 (F6): Step 6 (variant-dependent): Yavuz 2015 (§5 step 6, Eqs.6-11): aggregate the m experts' 2-tuple lower and upper bounds via the arithmetic-mean operator φ (Herrera-Martínez 2000), producing collective pessimistic P_C^{c-} and optimistic P_C^{c+} 2-tuples per cell. Özdağoğlu 2018 (Eq.8): aggregate the m experts' trapezoidal numbers per cell via the trapezoidal geometric mean e = (e_1 · e_2 · … · e_m)^{1/m}. Formül: \text{Yavuz: } P_C^{c-}(i,j) = \varphi\big(\{(s_{l_k}, 0)\}_{k=1}^m\big),\quad P_C^{c+}(i,j) = \varphi\big(\{(s_{h_k}, 0)\}_{k=1}^m\big). \quad \text{Özdağoğlu: } e_{ij} = \Big(\prod_{k=1}^m e_{ij}^{(k)}\Big)^{1/m} Anchor: Yavuz 2015 §5 step 6 + Eqs.6-11; Özdağoğlu 2018 Eq.8 + Herrera-Martínez 2000
7. 7.Adım 7 (F7): Step 7 (Yavuz 2015 §5 step 8, Eq.12): Compute the interval-valued utility vector V^c per row by summing collective preferences across columns: V^c_j = Σ_i collective(j, i) (interval addition: [a,b] + [c,d] = [a+c, b+d]). For the trapezoidal Özdağoğlu path, the analogous step is element-wise trapezoidal addition followed by the row-wise normalisation in F8. Formül: V^c_j = \Big[\sum_{i=1}^{n} P_C^{c-}(j,i),\ \sum_{i=1}^{n} P_C^{c+}(j,i)\Big], \quad j = 1, \ldots, n Anchor: Yavuz 2015 §5 step 8 + Eq.12; Özdağoğlu 2018 §3 ¶ trapezoidal summation
8. 8.Adım 8 (F8): Step 8 (Yavuz 2015 §5 step 9 + Sengupta-Pal 2000): Normalise the interval utility vector to obtain ratio-scale interval weights W^c. Sengupta-Pal convention: W^c_j^- = V^c_j^- / Σ_k V^c_k^+, W^c_j^+ = V^c_j^+ / Σ_k V^c_k^-: this yields the widest valid normalised interval per criterion. For trapezoidal Özdağoğlu (Eq.8 + row-sum normalisation), divide each component of the trapezoidal priority by the trapezoidal sum. Formül: W^c_j = \Big[\,\frac{V^c_j{}^{-}}{\sum_{k=1}^{n} V^c_k{}^{+}},\ \frac{V^c_j{}^{+}}{\sum_{k=1}^{n} V^c_k{}^{-}}\,\Big] Anchor: Yavuz 2015 §5 step 9 + Eq.13; Sengupta-Pal 2000 §3
9. 9.Adım 9 (F9): Step 9 (Yavuz 2015 §5 step 10 ¶a + classical AHP synthesis): Compute the interval-valued score per alternative as the weighted sum of leaf-level ratings. Score(A_i) = [Σ_j W^c_j^- · R_{ij}^-, Σ_j W^c_j^+ · R_{ij}^+] for non-negative ratings (Sengupta-Pal interval multiplication). Hierarchical: apply this level-by-level. Formül: \mathrm{Score}(A_i) = \Big[\sum_{j=1}^{n} W^c_j{}^{-} \cdot R_{ij}^{-},\ \sum_{j=1}^{n} W^c_j{}^{+} \cdot R_{ij}^{+}\Big], \quad R_{ij}^{\pm} \ge 0 Anchor: Yavuz 2015 §5 step 10 ¶a
10. 10.Adım 10 (F10): Step 10 (Yavuz 2015 §5 step 10 ¶b, Eqs.14-15): Rank alternatives. Default 'preference_degree_yavuz': for every pair (A_i, A_j) compute P(A_i > A_j) = (max(0, a_i^+ − a_j^-) − max(0, a_i^- − a_j^+)) / ((a_i^+ − a_i^-) + (a_j^+ − a_j^-)), then a Round-robin majority gives the ranking (note P(A>B)+P(B>A)=1 and the formula is degenerate when both intervals collapse). 'defuzz_midpoint': sort descending by (a_i^- + a_i^+)/2. 'defuzz_trapezoidal_ozdagoglu': defuzzify each trapezoidal priority via D = (fL + 2·fM1 + 2·fM2 + fU)/6 (Özdağoğlu 2018 Eq.9), then sort descending. Formül: P(A_i > A_j) = \frac{\max(0,\ a_i^{+} - a_j^{-}) - \max(0,\ a_i^{-} - a_j^{+})}{(a_i^{+} - a_i^{-}) + (a_j^{+} - a_j^{-})}, \quad P(A_i > A_j) + P(A_j > A_i) = 1; \quad D_{\text{trap}} = \dfrac{f_L + 2 f_{M1} + 2 f_{M2} + f_U}{6} Anchor: Yavuz 2015 §5 step 10 ¶b + Eqs.14-15; Özdağoğlu 2018 Eq.9

Commonly paired with

• •HFL-AHP + HF-TOPSIS (high)
• •HFL-AHP + HF-VIKOR (medium)
• •HFL-AHP + ARAS (medium)
• •HFL-AHP + HF-WASPAS (low)

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