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

Overview

HF-MABAC 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 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).

   $$D=({\mu }_i{j}^h),{\mu }_i{j}^h\subseteq [0,1],|{\mu }_i{j}^h|\ge 1.$$

   Mishra 2022, Step 3 (p.8829); Torra 2010 Def.1; Xia & Xu 2011 HFE definition.

2. 2

   Step 1b (group mode only) - Aggregate per-DE matrices into a single D via HFWA with DE weights ϖ_k.

   $${\mu }_i{j}^h=HFW{A}_{\varpi }({\mu }_i^{(1)}j,...,{\mu }_i^{(\ell )}j)={\cup }_{{\alpha }_k\in {\mu }_i^{(k)}j}1-\prod _{k=1}^{\ell }(1-{\alpha }_k{)}^{{\varpi }_k}(Xia-Xu2011/Mishra2022Eq.(2))$$

   Mishra 2022, Eq.(2) HFWA; Xia & Xu 2011 HFWA operator.

3. 3

   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}).

   $${\bar{\mu }}_i{j}^h={\mu }_i{j}^{h}if{C}_j\in {\Omega }_b(benefit);{\bar{\mu }}_i{j}^h={\cup }_{\alpha \in {\mu }_i{j}^h}1-\alpha if{C}_j\in {\Omega }_c(cost).(Mishra2022Eq.(12))$$

   Mishra 2022, Eq.(12); same operator family as HF-WASPAS / HF-COPRAS cost-complement.

4. 4

   Step 3 - Compute weighted N-HF-DM cell-by-cell using single-criterion HFWA Eq.(13).

   $${\bar{\vartheta }}_i{j}^h={\cup }_{\alpha \in {\bar{\mu }}_i{j}^h}1-(1-\alpha {)}^{w_j}(Mishra2022Eq.(13))$$

   Mishra 2022, Eq.(13) single-cell HFWA with weight w_j.

5. 5

   Step 4 - Compute BAA matrix G = (ζ_j)_{1×s} via HFWG over alternatives for each criterion column.

   $${\zeta }_j=HFW{G}_r({\bar{\vartheta }}_{1j}^h,{\bar{\vartheta }}_{2j}^h,...,{\bar{\vartheta }}_{rj}^h)={\cup }_{{\alpha }_{1j}\in {\bar{\vartheta }}_{1j},...,{\alpha }_{rj}\in {\bar{\vartheta }}_{rj}}\prod _{i=1}^r({\alpha }_{ij}{)}^{1/r}(Mishra2022Eq.(14))$$

   Mishra 2022, Eq.(14); Xia & Xu 2011 HFWG operator.

6. 6

   Step 5 - Compute HF distance matrix Q = (δ_ij) between WN-HF-DM and BAA via Cartesian-product absolute distance.

   $${\delta }_{i}j={\cup }_{\theta \in {\bar{\vartheta }}_{ij}^h}{\cup }_{\eta \in {\zeta }_j}|\theta -\eta |(Mishra2022Eq.(15))$$

   Mishra 2022, Eq.(15) HF Hamming-style absolute distance.

7. 7

   Step 6 - Compute overall assessment value AV(A_i) as arithmetic mean of Cartesian-product per-criterion average distances.

   $$AV(A_i)=mea{n}_{combo\in {\prod }_j{\delta }_{i}j}[(1/s){\Sigma }_{j=1}^s{\beta }_j(combo)]where{\beta }_j(combo)isthej-thelementoftheCartesian-productcombination.(Mishra2022Eq.(16))$$

   Mishra 2022, Eq.(16); Step 8 of HF-DEA-FOCUM-MABAC procedure.

8. 8

   Step 7 - Rank alternatives in DESCENDING order of AV(A_i) (largest AV_i = best).

   $$A_i\succ A_{i^{\prime }}⟺AV(A_i)>AV(A_{i^{\prime }}).(Mishra2022Step9)$$

   Mishra 2022, Step 9 (p.8829).

Fits when

Edge cases and pitfalls

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

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