Ranking

HF-MABAC: Hesitant Fuzzy Multi-Attributive Border Approximation area Comparison

Mishra, A.R., Saha, A., Rani, P., Pamucar, D., Dutta, D., Hezam, I.M. · 2022

Overview

Hesitant fuzzy border-approximation distance ranking. Output typically ranking.

Strengths

•Method-specific: Hesitant fuzzy border-approximation distance ranking
•Preserves hesitant uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

•Rank reversal known on alternative-set changes (ref: Inherited from crisp MABAC (Pamucar-Cirovic 2015); HF extension does not eliminate rank reversal under alternative addition/removal.)

Method assistant

Grounded explanations: it explains the method, it does not compute.

Edge cases

•If group mode, perform Step F1b (HFWA pre-aggregation).

Common pitfalls

•Bkz. HF-MABAC F.steps citation_anchor'lar ve P.verification_status.

Worked example

1.Adım 1 (F1): Step 1: Construct HF decision matrix D = (μ_ij^h)_{r×s}. Each μ_ij^h is an HFE provided by DE(s). If group mode, perform Step F1b (HFWA pre-aggregation). Formül: D = (μ_ij^h), μ_ij^h ⊆ [0,1], |μ_ij^h| ≥ 1. Anchor: Mishra 2022, Step 3 (p.8829); Torra 2010 Def.1; Xia & Xu 2011 HFE definition.
2.Adım 2 (F2): Step 1b (group mode only): Aggregate per-DE matrices into a single D via HFWA with DE weights ϖ_k. Formül: μ_ij^h = HFWA_ϖ(μ^{(1)}_ij,...,μ^{(ℓ)}_ij) = ∪_{α_k ∈ μ^{(k)}_ij} { 1 - ∏_{k=1}^ℓ (1-α_k)^{ϖ_k} } (Xia-Xu 2011 / Mishra 2022 Eq.(2)) Anchor: Mishra 2022, Eq.(2) HFWA; Xia & Xu 2011 HFWA operator.
3.Adım 3 (F3): Step 2: Normalize D by criterion type. Benefit: identity (μ̄_ij^h = μ_ij^h). Cost: pointwise complement (μ̄_ij^h = ∪{1-α : α ∈ μ_ij^h}). Formül: μ̄_ij^h = μ_ij^h if C_j ∈ Ω_b (benefit); μ̄_ij^h = ∪_{α ∈ μ_ij^h} {1-α} if C_j ∈ Ω_c (cost). (Mishra 2022 Eq.(12)) Anchor: Mishra 2022, Eq.(12); same operator family as HF-WASPAS / HF-COPRAS cost-complement.
4.Adım 4 (F4): Step 3: Compute weighted N-HF-DM cell-by-cell using single-criterion HFWA Eq.(13). Formül: ϑ̄_ij^h = ∪_{α ∈ μ̄_ij^h} { 1 - (1 - α)^{w_j} } (Mishra 2022 Eq.(13)) Anchor: Mishra 2022, Eq.(13) single-cell HFWA with weight w_j.
5.Adım 5 (F5): Step 4: Compute BAA matrix G = (ζ_j)_{1×s} via HFWG over alternatives for each criterion column. Formül: ζ_j = HFWG_r(ϑ̄_{1j}^h, ϑ̄_{2j}^h, ..., ϑ̄_{rj}^h) = ∪_{α_{1j} ∈ ϑ̄_{1j}, ..., α_{rj} ∈ ϑ̄_{rj}} { ∏_{i=1}^r (α_{ij})^{1/r} } (Mishra 2022 Eq.(14)) Anchor: Mishra 2022, Eq.(14); Xia & Xu 2011 HFWG operator.
6.Adım 6 (F6): Step 5: Compute HF distance matrix Q = (δ_ij) between WN-HF-DM and BAA via Cartesian-product absolute distance. Formül: δ_ij = ∪_{θ ∈ ϑ̄_{ij}^h} ∪_{η ∈ ζ_j} { |θ - η| } (Mishra 2022 Eq.(15)) Anchor: Mishra 2022, Eq.(15) HF Hamming-style absolute distance.
7.Adım 7 (F7): Step 6: Compute overall assessment value AV(A_i) as arithmetic mean of Cartesian-product per-criterion average distances. Formül: AV(A_i) = mean_{combo ∈ ∏_j δ_ij} [ (1/s) Σ_{j=1}^s β_j(combo) ] where β_j(combo) is the j-th element of the Cartesian-product combination. (Mishra 2022 Eq.(16)) Anchor: Mishra 2022, Eq.(16); Step 8 of HF-DEA-FOCUM-MABAC procedure.
8.Adım 8 (F8): Step 7: Rank alternatives in DESCENDING order of AV(A_i) (largest AV_i = best). Formül: A_i ≻ A_{i'} ⟺ AV(A_i) > AV(A_{i'}). (Mishra 2022 Step 9) Anchor: Mishra 2022, Step 9 (p.8829).

How to cite

Mishra, A.R.; Saha, A.; Rani, P.; Pamucar, D.; Dutta, D.; Hezam, I.M. (2022). Sustainable supplier selection using HF-DEA-FOCUM-MABAC technique: a case study in the Auto-making industry. Soft Computing. https://doi.org/10.1007/s00500-022-07192-8

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Worked example

1.Adım 1 (F1): Step 1: Construct HF decision matrix D = (μ_ij^h)_{r×s}. Each μ_ij^h is an HFE provided by DE(s). If group mode, perform Step F1b (HFWA pre-aggregation). Formül: D = (μ_ij^h), μ_ij^h ⊆ [0,1], |μ_ij^h| ≥ 1. Anchor: Mishra 2022, Step 3 (p.8829); Torra 2010 Def.1; Xia & Xu 2011 HFE definition.

2.Adım 2 (F2): Step 1b (group mode only): Aggregate per-DE matrices into a single D via HFWA with DE weights ϖ_k. Formül: μ_ij^h = HFWA_ϖ(μ^{(1)}_ij,...,μ^{(ℓ)}_ij) = ∪_{α_k ∈ μ^{(k)}_ij} { 1 - ∏_{k=1}^ℓ (1-α_k)^{ϖ_k} } (Xia-Xu 2011 / Mishra 2022 Eq.(2)) Anchor: Mishra 2022, Eq.(2) HFWA; Xia & Xu 2011 HFWA operator.

3.Adım 3 (F3): Step 2: Normalize D by criterion type. Benefit: identity (μ̄_ij^h = μ_ij^h). Cost: pointwise complement (μ̄_ij^h = ∪{1-α : α ∈ μ_ij^h}). Formül: μ̄_ij^h = μ_ij^h if C_j ∈ Ω_b (benefit); μ̄_ij^h = ∪_{α ∈ μ_ij^h} {1-α} if C_j ∈ Ω_c (cost). (Mishra 2022 Eq.(12)) Anchor: Mishra 2022, Eq.(12); same operator family as HF-WASPAS / HF-COPRAS cost-complement.

4.Adım 4 (F4): Step 3: Compute weighted N-HF-DM cell-by-cell using single-criterion HFWA Eq.(13). Formül: ϑ̄_ij^h = ∪_{α ∈ μ̄_ij^h} { 1 - (1 - α)^{w_j} } (Mishra 2022 Eq.(13)) Anchor: Mishra 2022, Eq.(13) single-cell HFWA with weight w_j.

5.Adım 5 (F5): Step 4: Compute BAA matrix G = (ζ_j)_{1×s} via HFWG over alternatives for each criterion column. Formül: ζ_j = HFWG_r(ϑ̄_{1j}^h, ϑ̄_{2j}^h, ..., ϑ̄_{rj}^h) = ∪_{α_{1j} ∈ ϑ̄_{1j}, ..., α_{rj} ∈ ϑ̄_{rj}} { ∏_{i=1}^r (α_{ij})^{1/r} } (Mishra 2022 Eq.(14)) Anchor: Mishra 2022, Eq.(14); Xia & Xu 2011 HFWG operator.

6.Adım 6 (F6): Step 5: Compute HF distance matrix Q = (δ_ij) between WN-HF-DM and BAA via Cartesian-product absolute distance. Formül: δ_ij = ∪_{θ ∈ ϑ̄_{ij}^h} ∪_{η ∈ ζ_j} { |θ - η| } (Mishra 2022 Eq.(15)) Anchor: Mishra 2022, Eq.(15) HF Hamming-style absolute distance.

7.Adım 7 (F7): Step 6: Compute overall assessment value AV(A_i) as arithmetic mean of Cartesian-product per-criterion average distances. Formül: AV(A_i) = mean_{combo ∈ ∏_j δ_ij} [ (1/s) Σ_{j=1}^s β_j(combo) ] where β_j(combo) is the j-th element of the Cartesian-product combination. (Mishra 2022 Eq.(16)) Anchor: Mishra 2022, Eq.(16); Step 8 of HF-DEA-FOCUM-MABAC procedure.

8.Adım 8 (F8): Step 7: Rank alternatives in DESCENDING order of AV(A_i) (largest AV_i = best). Formül: A_i ≻ A_{i'} ⟺ AV(A_i) > AV(A_{i'}). (Mishra 2022 Step 9) Anchor: Mishra 2022, Step 9 (p.8829).