MHF-TOPSIS - m-Polar Hesitant Fuzzy extension of TOPSIS

Overview

MHF-TOPSIS (Akram et al. 2019) extends classical TOPSIS to m-Polar Hesitant Fuzzy multi-criteria group decision-making (MCGDM). Each cell of the decision matrix is an m-tuple of HFEs, one per pole/sub-feature of the criterion; the m poles are independent (no DHF-style γ⁺+η⁺ constraint). The DM-chosen prolongation_mode (optimistic = repeat max, pessimistic = repeat min) equalises pole-HFE lengths to a common r. Weighting is literal scalar multiplication (H′ = w·H). Ideals mHP_IS and mHN_IS are pole-wise column extrema on the weighted matrix; distances average over q criteria × m poles × r positions with the 1/(rm) factor; rank by descending E_j′ = D⁻/(D⁺+D⁻).

Output

utility, higher is better

Data

Hesitant, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Multi-Hesitant Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Assemble the mHF decision matrix H = (ℏ_m^{jk})_{p×q} whose entries are mHFEs ℏ_m^{jk}(z) = ({ζ_h | ζ_h ∈ p_1 ∘ ℏ_m^{jk}(z)}, …, {ζ_h | ζ_h ∈ p_m ∘ ℏ_m^{jk}(z)}). j indexes alternatives a_j ∈ A (j=1,…,p), k indexes criteria c_k ∈ C (k=1,…,q).

   $$H=({\hslash }_m^{jk}{)}_{p\times q},{\hslash }_m^{jk}(z)=(\{{\zeta }_h\mid {\zeta }_h\in p_1\circ {\hslash }_m^{jk}(z)\},\dots ,\{{\zeta }_h\mid {\zeta }_h\in p_m\circ {\hslash }_m^{jk}(z)\})$$

   Akram et al. 2019, p.10 Step 1 (Table 1)

2. 2

   Step 1, continued (Akram et al. 2019, p.11 + §3.1 p.14 + §3.2 p.17) - Prolong each pole-HFE until all HFEs within H share the same cardinality r. If prolongation_mode='optimistic', repeat the maximum value of each HFE; if 'pessimistic', repeat the minimum. r is set to the largest pole-HFE cardinality observed across H (so that the maximal information length is preserved).

   $$r=\underset{j,k,i}{\max }|p_i\circ {\hslash }_m^{jk}(z)|;\text{optimistic: }{p}_i\circ {\hslash }_m^{jk}(z)\leftarrow p_i\circ {\hslash }_m^{jk}(z)\cup \{\max p_i\circ {\hslash }_m^{jk}(z){\}}^{r-|p_i\circ {\hslash }_m^{jk}(z)|};\text{pessimistic: }\text{repeat }\min \text{ instead}$$

   Akram et al. 2019, p.11 ¶3 + p.14 ¶1 (§3.1 optimistic) + p.17 ¶3 (§3.2 pessimistic)

3. 3

   Build the weighted mHF decision matrix H′ by scalar multiplication of each cell with its criterion weight w_k (no DHFE-style n·d power operator; literal element-wise scalar product of each ζ_h ∈ p_i ∘ ℏ_m^{jk}).

   $${\hslash }_m^{j{k}^{\prime }}=w_k{\hslash }_m^{jk}=(w_k\{{\zeta }_h\mid {\zeta }_h\in p_1\circ {\hslash }_m^{jk}(z)\},\dots ,w_k\{{\zeta }_h\mid {\zeta }_h\in p_m\circ {\hslash }_m^{jk}(z)\})=(\{{\zeta }_h^{{}^{\prime }}\mid {\zeta }_h^{{}^{\prime }}\in p_1\circ {\hslash }_m^{jk}(z)\},\dots ,\{{\zeta }_h^{{}^{\prime }}\mid {\zeta }_h^{{}^{\prime }}\in p_m\circ {\hslash }_m^{jk}(z)\})$$

   Akram et al. 2019, p.11 Table 2 + p.11 unnumbered equation

4. 4

   Steps 4-5 (Akram et al. 2019, pp.11-12 Eqs.(1)-(2)) - Compute the mHF positive-ideal solution mHP_IS by taking the column-wise maximum on each pole component of H′, and the mHF negative-ideal solution mHN_IS by the column-wise minimum on each pole component.

   $$mH{P}_{IS}=\{({\hslash }_m^{1^{\prime }}{)}^{+},({\hslash }_m^{2^{\prime }}{)}^{+},\dots ,({\hslash }_m^{q^{\prime }}{)}^{+}\},({\hslash }_m^{k^{\prime }}{)}^{+}=\underset{j}{\max }({\hslash }_m^{j{k}^{\prime }})=(\{({\zeta }_h^{{}^{\prime }}{)}^{+}\mid ({\zeta }_h^{{}^{\prime }}{)}^{+}\in p_1\circ {\hslash }_m^{jk}(z)\},\dots ,\{({\zeta }_h^{{}^{\prime }}{)}^{+}\mid ({\zeta }_h^{{}^{\prime }}{)}^{+}\in p_m\circ {\hslash }_m^{jk}(z)\});mH{N}_{IS}=\{({\hslash }_m^{1^{\prime }}{)}^{-},\dots ,({\hslash }_m^{q^{\prime }}{)}^{-}\},({\hslash }_m^{k^{\prime }}{)}^{-}=\underset{j}{\min }({\hslash }_m^{j{k}^{\prime }})$$

   Akram et al. 2019, p.11 Eq.(1) + p.12 Eq.(2)

5. 5

   Step 6 (Akram et al. 2019, p.12 Eqs.(3)-(4)) - Compute the mHF Euclidean distance of each alternative a_j from mHP_IS and mHN_IS. The distance averages over all q criteria, all m poles, and all r prolonged HFE positions; the 1/(rm) normalisation factor is verbatim from the paper.

   $$D_e^{{}^{\prime }}(a_j,mH{P}_{IS})=\sqrt{\frac{1}{rm}\sum _{k=1}^q\left[\sum _{i=1}^m\right\{({\zeta }_{h1}^{j{k}^{\prime }}-({\zeta }_{h1}^{k^{\prime }}{)}^{+}{)}^2+({\zeta }_{h2}^{j{k}^{\prime }}-({\zeta }_{h2}^{k^{\prime }}{)}^{+}{)}^2+\cdots +({\zeta }_{hr}^{j{k}^{\prime }}-({\zeta }_{hr}^{k^{\prime }}{)}^{+}{)}^2\left\}\right]};D_e^{{}^{\prime }}(a_j,mH{N}_{IS})=\sqrt{\frac{1}{rm}\sum _{k=1}^q\left[\sum _{i=1}^m\right\{({\zeta }_{h1}^{j{k}^{\prime }}-({\zeta }_{h1}^{k^{\prime }}{)}^{-}{)}^2+({\zeta }_{h2}^{j{k}^{\prime }}-({\zeta }_{h2}^{k^{\prime }}{)}^{-}{)}^2+\cdots +({\zeta }_{hr}^{j{k}^{\prime }}-({\zeta }_{hr}^{k^{\prime }}{)}^{-}{)}^2\left\}\right]}$$

   Akram et al. 2019, p.12 Eqs.(3)-(4)

6. 6

   Steps 7-8 (Akram et al. 2019, p.12 Eq.(5)) - Relative mHF closeness coefficient E_j′; rank alternatives in descending order of E_j′. The alternative with the highest E_j′ is the best.

   $$E_j^{{}^{\prime }}=\frac{D_e^{{}^{\prime }}(a_j,mH{N}_{IS})}{D_e^{{}^{\prime }}(a_j,mH{P}_{IS})+D_e^{{}^{\prime }}(a_j,mH{N}_{IS})},j=1,2,\dots ,p;0\le E_j^{{}^{\prime }}\le 1$$

   Akram et al. 2019, p.12 Eq.(5) + p.13 Algorithm 1 Steps 7-8

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Multi-Hesitant Fuzzy numbers/tuples
• Underlying crisp method's compensation assumption holds in uncertain space
• All decision-maker(s) and experts use the same linguistic/uncertainty scale

Edge cases and pitfalls

• •MHF-TOPSIS için kritik edge case'ler: (1) Pole uzunluğu eşitleme - DM farklı pole için farklı sayıda hesitancy değeri verirse, paper §3 'optimistic = add max' veya 'pessimistic = add min' politikası gerektirir; engine policy parametresi default = 'optimistic' (paper §3.1 brand-name kanonu). (2) m=1 özel durum - mHF-set indirgenir klasik HF-set; mHF-TOPSIS indirgenir HF-TOPSIS (Xu-Zhang 2013). (3) Pole sayısı m çok büyük (>5) - Euclidean Eq.3'te m faktörü payda 'rm' içine girer, normalize edilir; ama hesaplama maliyeti q·m·r mertebesindedir. (4) w_k normalizasyonu Σ w_k = 1, eksik veri durumunda paper 'equal weights' önerir - engine bu default'u izler. (5) PIS/NIS pole-bazlı max - Eq.1 (ℏ_m^k')+ = (max_j sup{ζ_h ∈ p_1 ℏ_m^jk}, ..., max_j sup{ζ_h ∈ p_m}); her pole bağımsız extremum, tek-skalar değil VEKTÖREL.

• Pole count m must be identical across all matrix cells; mixing m=3 and m=4 cells violates the mHF-set definition (Akram et al. 2019 Def. 1) and renders mHP_IS / mHN_IS extraction ill-defined.

• prolongation_mode is information-fusion-laden: 'optimistic' (repeat max) inflates pole-HFE scores upward, 'pessimistic' (repeat min) deflates them. The two modes can yield different rankings on the same raw matrix - fix prolongation_mode per the decision context (§3.1 uses optimistic for brand-name selection; §3.2 uses pessimistic for product-design selection).

• Weighting is literal scalar multiplication H′_{jk} = w_k · ℏ_m^{jk} (each ζ_h scaled), NOT the DHFE n·d power operator from Zhu et al. 2012. Applying n·d here over-deflates membership values and breaks Akram's numerical example (Table 5/Table 8 verification fails).

• The 1/(rm) factor in Eqs.(3)-(4) is averaging-style, not Σ-style; omitting it inflates distances by a factor of √(rm) but does not change the ranking (E_j′ is a ratio). Engine implementations should match the paper's formula verbatim for reproducibility.

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