HF-GRA - Hesitant Fuzzy Grey Relational Analysis

Overview

HF-GRA ranks alternatives based on performance scores. Higher score = better rank.

Data

Hesitant

Weights

Needs a weight source

How it works

1. 1

   Construct HF decision matrix H = (h_ij)_{m×n}. Each h_ij is an HFE provided by DE(s). If group mode, perform Step F1b (HFSA Shapley aggregation per Li-Wei Eq.9 OR HFWA per Xia-Xu).

   $$H=(h_{i}j),h_{i}j\subseteq [0,1],|h_{i}j|\ge 1.$$

   Li-Wei 2014, Step 1; Guan 2018, Section 4 Step 1.

2. 2

   Step 1b (group mode only) - Aggregate per-DE matrices via HFSA Shapley-weighted operator (Li-Wei Eq.9) or HFWA (Xia-Xu) with DE weights φ_k.

   $$h_{i}j=HFSA(h_i^{1}j,...,h_i^{t}j)={\cup }_{{\gamma }^k}1-\prod _{k=1}^t(1-{\gamma }^k{)}^{{\phi }_k}where{\phi }_k=ShapleyvalueoverDEfuzzymeasure(Li-WeiEq.6);reducestoHFWAwhen{\phi }_k=w_{k}linearweights(Xia-Xu2011).$$

   Li-Wei 2014, Eq.(7) HFSA; Eq.(9) Step 2. Xia-Xu 2011 HFWA.

3. 3

   Step 2 - Extract HF Positive Ideal Solution (PIS) h+_j and HF Negative Ideal Solution (NIS) h-_j per criterion.

   $$Benefit{C}_j:h{+}_j={\cup }_{i}ma{x}_i{\gamma }_i{j}^{(}k);h{-}_j={\cup }_{i}mi{n}_i{\gamma }_i{j}^{(}k).Cost{C}_j:h{+}_j={\cup }_{i}mi{n}_i{\gamma }_i{j}^{(}k);h{-}_j={\cup }_{i}ma{x}_i{\gamma }_i{j}^{(}k).(Li-WeiStep3/Guan2018Eq.25)$$

   Li-Wei 2014, Step 3; Guan 2018, Eq.(25).

4. 4

   Step 3 - Compute hesitant normalised Euclidean distance d_hne(h_ij, h+_j) and d_hne(h_ij, h-_j) for every (i,j).

   $$d_{h}ne(h_A,h_B)=[(1/l){\Sigma }_{k=1}^l|{\gamma }_A^{\sigma }(k)-{\gamma }_B^{\sigma }(k){|}^2{]}^{1/2}wherel=max(|h_A|,|h_B|)and{\gamma }^{\sigma }(k)isthekthlargestvalue(descendingsort).(Guan2018Eq.6/Xu-Xia2011)$$

   Guan 2018, Eq.(6); Xu & Xia 2011 normalized Euclidean distance for HFEs.

5. 5

   Step 4 - Compute HFS grey relational coefficient ξ+_ij to PIS and ξ-_ij to NIS, using ρ ∈ (0,1] (default 0.5) and the global min/max over (i,j) of the corresponding distance matrix.

   $$\xi {+}_{i}j=[mi{n}_{i,j}{d}_{ij}^{+}+\rho \cdot ma{x}_{i,j}{d}_{ij}^{+}]/[d_{ij}^{+}+\rho \cdot ma{x}_{i,j}{d}_{ij}^{+}](Li-WeiEq.10/Guan2018Eq.5);\xi {-}_{i}jdefinedsymmetricallywith{d}_{ij}^{-}.\rho =0.5default.$$

   Li-Wei 2014, Eqs.(10)-(11); Guan 2018, Eq.(5).

6. 6

   Step 5 - Compute weighted HFS grey relational degrees ξ+_i (to PIS) and ξ-_i (to NIS) by linearly weighting per-criterion coefficients.

   $$\xi {+}_i={\Sigma }_{j=1}^n{w}_j\cdot \xi {+}_{i}j;\xi {-}_i={\Sigma }_{j=1}^n{w}_j\cdot \xi {-}_{i}j.(Li-WeiStep5/Guan2018Eq.8{\gamma }_{w}form)$$

   Li-Wei 2014, Step 5; Guan 2018, Eq.(8) γ_w(A,B).

7. 7

   Step 6 - Compute relative closeness η_i = ξ+_i / (ξ+_i + ξ-_i), a TOPSIS-style coefficient in [0,1].

   $${\eta }_i=\xi {+}_i/(\xi {+}_i+\xi {-}_i),{\eta }_i\in [0,1].(Li-WeiEq.14/Guan2018Eq.28)$$

   Li-Wei 2014, Eq.(14); Guan 2018, Eq.(28).

8. 8

   Step 7 - Rank alternatives in DESCENDING order of η_i (largest is best).

   $$A_i\succ A_{i^{\prime }}⟺{\eta }_i>{\eta }_{i^{\prime }};D\ast =argma{x}_i{\eta }_i.(Li-WeiStep7/Guan2018Step4)$$

   Li-Wei 2014, Step 7; Guan 2018, Section 4 Step 4.

Fits when

Edge cases and pitfalls

• •HF-GRA için kritik edge case'ler: (1) HFE uzunluk uyumsuzluğu - h_1 ve h_2 farklı kardinalitede ise paper §2 'pessimistic extension' (min değer ile uzat) tavsiye eder; engine bu default'u izler. (2) ρ=0.5 distinguishing kanonik; ρ→0 küçük farkları abartır, ρ→1 tüm grey coefficient'ları 1'e bastırır - duyarlılık analizi literatürün stabilite testi olarak önerilir. (3) Δ kümesi infeasible (M-2 çözümsüz) ise: kısıtlar çelişiyor, kullanıcıya iade - equal-weight default'a düşmek paper felsefesine aykırı. (4) PIS h^+_j = max_i max_{γ∈h_ij} ve NIS h^-_j = min_i min tanımı: tüm kriterler benefit varsayılır; cost kriterler için ön-normalize (complement: 1-γ) gerekir. (5) Shapley fuzzy measure 2^t-1 değer gerektirir; t>6 için uzman pratik olarak μ(K) tanımlayamaz - bu durumda λ-fuzzy measure (tek parametre) önerilir.

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