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

Overview

C_i ∈ [0,1] is the closeness coefficient: 1 = identical to PIS on every criterion (best), 0 = identical to NIS (worst). C_i is NOT a probability or utility; it is a normalised distance ratio. Always report the weight_info_mode AND the resolved weights w alongside the ranking - different modes can produce different rankings on the same data (Xu-Zhang 2013 §5 Case 1 vs Case 2 swap A1/A4). For benchmark fidelity to the paper use η=0 (risk-averse) length-equalisation. For η-sensitivity sweep see S.weight_perturbation and T.custom_extensions.

Output

closeness coefficient, higher is better

Data

Hesitant, hfe per cell required

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-10 criteria works best

Used for

Energy policy selection under expert disagreement, supplier selection with hesitant judgments, personnel / candidate ranking, project portfolio selection, any TOPSIS-fit problem where assessments come as sets of possible membership values rather than single numbers, decision problems where weights are only partially known (use partly_known LP mode)

How it works

1. 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)).

   $$\mathbf{H}=(h_{ij}{)}_{m\times n},h_{ij}\subset [0,1]\text{non-empty finite}$$

   Xu-Zhang 2013, §3 ¶1

2. 2

   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.

   $$l_j^{\ast }=\underset{i}{\max }l(h_{ij});{\tilde{h}}_{ij}=h_{ij}\cup \underset{l_j^{\ast }-l(h_{ij})\text{copies}}{\underbrace{\{\eta \cdot \max (h_{ij})+(1-\eta )\cdot \min (h_{ij}),\dots \}}};\text{sort ascending}$$

   Xu-Zhang 2013, Def.2 (p.55); §5.1 sets η=0

3. 3

   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.

   $$(a)w=w_{\text{user}};(b)Y_j=\sum _{i=1}^m\sum _{k=1}^m{d}_1({\tilde{h}}_{ij},{\tilde{h}}_{kj}),w_j=\frac{Y_j}{{\sum }_{j^{\prime }=1}^n{Y}_{j^{\prime }}};(c)\underset{w}{\max }\sum _{j=1}^n{Y}_j{w}_j\text{s.t.}w\in \Delta ,\sum _j{w}_j=1,w_j\ge 0$$

   Xu-Zhang 2013, Eq.(22) (completely unknown); Model (M-2) (partly known); §3 ¶ on fully specified case

4. 4

   (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.

   $${\tilde{h}}_j^{+}=\{\begin{array}{ll}({\max }_i{\gamma }_{ij}^{\sigma (\tau )}{)}_{\tau =1}^{l_j^{\ast }},& j\in {\Omega }_b\\ ({\min }_i{\gamma }_{ij}^{\sigma (\tau )}{)}_{\tau =1}^{l_j^{\ast }},& j\in {\Omega }_c\end{array};{\tilde{h}}_j^{-}=\{\begin{array}{ll}({\min }_i{\gamma }_{ij}^{\sigma (\tau )}{)}_{\tau =1}^{l_j^{\ast }},& j\in {\Omega }_b\\ ({\max }_i{\gamma }_{ij}^{\sigma (\tau )}{)}_{\tau =1}^{l_j^{\ast }},& j\in {\Omega }_c\end{array}$$

   Xu-Zhang 2013, Eqs.(23)-(24)

5. 5

   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).

   $$d_1({\tilde{h}}_{ij},{\tilde{h}}_j^{\pm })=\sqrt{\frac{1}{l_j^{\ast }}\sum _{\tau =1}^{l_j^{\ast }}({\gamma }_{ij}^{\sigma (\tau )}-{\gamma }_j^{\pm ,\sigma (\tau )}{)}^2}$$

   Xu-Zhang 2013, Eq.(2); Xu-Xia 2011b

6. 6

   (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.

   $$d_i^{+}=\sum _{j=1}^n{w}_j\cdot d_1({\tilde{h}}_{ij},{\tilde{h}}_j^{+}),d_i^{-}=\sum _{j=1}^n{w}_j\cdot d_1({\tilde{h}}_{ij},{\tilde{h}}_j^{-}),i=1,\dots ,m$$

   Xu-Zhang 2013, Eqs.(25)-(26)

7. 7

   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'.

   $$C_i=\frac{d_i^{-}}{d_i^{+}+d_i^{-}}\in [0,1],i=1,\dots ,m$$

   Xu-Zhang 2013, Eq.(27)

8. 8

   Rank alternatives in descending order of C_i. The alternative with the largest C_i is the recommended choice.

   $$\text{rank}(A_i)={\text{argsort descending}}_i{C}_i$$

   Xu-Zhang 2013, §3 last paragraph; §5 Case 1 ranking A5≻A3≻A2≻A1≻A4, Case 2 ranking A5≻A3≻A2≻A4≻A1

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 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)

Edge cases and pitfalls

• •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.

• Length-equalisation choice (η) is consequential: η=0 (risk-averse, paper §5.1) and η=1 (risk-seeking) generally produce different distances d_1 and hence different d^± and C. Document the choice in any reported result (paper §5 last ¶ explicitly warns).

• DO NOT confuse Xu-Zhang 2013 KBS (this manifest, LINEAR-weighted separation Σ_j w_j·d_1) with Xia-Xu 2011 IJAR (quadratic-weighted sqrt(Σ_j w_j·d_1²) + multiplicative HFE weighting 1−(1−γ)^w_j). They are DIFFERENT algorithmic families and produce different rankings in general. The legacy /tmp/HF_tex/HF-TOPSIS.tex description in this repo describes the Xia-Xu 2011 IJAR variant, NOT Xu-Zhang 2013 - see P.review_notes.

• Rank reversal across weight_info_mode: switching from 'completely_unknown' to 'partly_known' (or vice versa) on the same matrix can change the ranking - paper §5 Case 1 vs Case 2 explicitly swaps A1 and A4. Treat the choice of weight_info_mode as a substantive modelling decision, not a technical detail.

• Model (M-2) LP can be infeasible if weight_constraints are inconsistent (e.g. intervals that don't intersect the simplex). The engine raises an 'infeasible_weight_polytope_error'; check that Σ of lower bounds ≤ 1 ≤ Σ of upper bounds before LP submission.

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