MPF-HF-TOPSIS - m-Polar Hesitant Fuzzy TOPSIS (Akram, Adeel & Alcantud 2019, Symmetry 11(6):795) - multi-criteria group decision-making by extending TOPSIS to the m-polar hesitant fuzzy (mHF) set framework; pole-wise mHPIS/mHNIS extraction, mHF Euclidean distance and closeness coefficient ranking

Overview

Read the result as a complete descending ranking of alternatives a_j by the closeness coefficient E_j' ∈ [0,1] (Eq. 5). E_j' near 1 means the alternative is close to the m-polar hesitant fuzzy positive ideal (mHPIS) and far from the negative ideal (mHNIS); E_j' near 0 means the opposite. The §3.1 case study yields Bn_1 ≻ Bn_3 ≻ Bn_4 ≻ Bn_5 ≻ Bn_2 - Bn_1 is the perfect brand name. To interpret WHY one alternative dominates, inspect the pole-wise membership polygons (viz H4) and the mHPIS/mHNIS per criterion (viz H1).

Output

closeness coefficient, higher is better

Data

M-Polar Fuzzy, m polar hesitant tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Brand name selection (§3.1 case study, p. 13-16: p=5, q=3, m=4), Product design selection (§3.2 case study, p. 17-20: p=4, q=4, m=5), MCGDM under m-polar hesitant fuzzy information, Strategic-marketing decisions with multi-attribute hesitant evaluation by panels of experts

How it works

1. 1

   Ingest p alternatives, q criteria, m poles, r hesitancy length, and weight vector w=(w_1,…,w_q) with Σ w_k = 1 (Algorithm 1 Step 1). Compute / receive the mHF decision matrix H = (h̄_m^{jk})_{p×q} where each cell h̄_m^{jk} is an mHFE (Step 2 of Algorithm 1, Table 1). When hesitant-set lengths within a cell differ, prolong by the chosen padding_strategy (optimistic → max, pessimistic → min) until every hesitant set across the matrix has length r (Sec. 3 narrative on p. 11, executed in Table 4 of §3.1).

   $$H=({\bar{h}}_m^{jk}{)}_{p\times q},{\bar{h}}_m^{jk}(z)=(\{{\zeta }_h|{\zeta }_h\in p_1\circ {\bar{h}}_m^{jk}(z)\},\dots ,\{{\zeta }_h|{\zeta }_h\in p_m\circ {\bar{h}}_m^{jk}(z)\});w=(w_1,\dots ,w_q),w_k\in (0,1],\sum _{k=1}^q{w}_k=1.$$

   Algorithm 1 Steps 1-2 (p. 13); Sec. 3 narrative on padding (p. 11); §3.1 Tables 3-4 (p. 14)

2. 2

   Compute the weighted mHF decision matrix H' = (h̄_m^{jk'})_{p×q} by multiplying every membership value in every pole's hesitant set by the criterion weight w_k (Algorithm 1 Step 3, Table 2). The result is again an mHF matrix with the same shape p×q×m×r.

   $${\bar{h}}_m^{jk\prime }=w_k{\bar{h}}_m^{jk}=(w_k\{{\zeta }_h|{\zeta }_h\in p_1\circ {\bar{h}}_m^{jk}(z)\},\dots ,w_k\{{\zeta }_h|{\zeta }_h\in p_m\circ {\bar{h}}_m^{jk}(z)\})=(\{{\zeta }_h^{\prime }|{\zeta }_h^{\prime }\in p_1\circ {\bar{h}}_m^{jk}(z)\},\dots ,\{{\zeta }_h^{\prime }|{\zeta }_h^{\prime }\in p_m\circ {\bar{h}}_m^{jk}(z)\}).$$

   Algorithm 1 Step 3 (p. 13); Table 2 (p. 11); Table 5 (§3.1, p. 15)

3. 3

   Compute the m-polar hesitant fuzzy positive ideal solution (mHPIS) by pole-wise maximisation across alternatives per criterion (Algorithm 1 Step 4, Eq. 1). For each criterion k and pole r, mHPIS gathers the maximum weighted membership value across alternatives at hesitancy index l (l=1,…,r).

   $$\mathrm{mHPIS}=\{({\bar{h}}_m^{1\prime }{)}^{+},({\bar{h}}_m^{2\prime }{)}^{+},\dots ,({\bar{h}}_m^{q\prime }{)}^{+}\},({\bar{h}}_m^{k\prime }{)}^{+}=\underset{j}{\max }({\bar{h}}_m^{k\prime })=(\underset{j}{\max }\{{\zeta }_h^{\prime }|{\zeta }_h^{\prime }\in p_1\circ {\bar{h}}_m^{jk}(z)\},\dots ,\underset{j}{\max }\{{\zeta }_h^{\prime }|{\zeta }_h^{\prime }\in p_m\circ {\bar{h}}_m^{jk}(z)\}).$$

   Algorithm 1 Step 4 (p. 13); Eq. (1) (p. 11)

4. 4

   Compute the m-polar hesitant fuzzy negative ideal solution (mHNIS) by pole-wise minimisation across alternatives per criterion (Algorithm 1 Step 5, Eq. 2). Symmetric to F3 but with min instead of max at every hesitancy index.

   $$\mathrm{mHNIS}=\{({\bar{h}}_m^{1\prime }{)}^{-},({\bar{h}}_m^{2\prime }{)}^{-},\dots ,({\bar{h}}_m^{q\prime }{)}^{-}\},({\bar{h}}_m^{k\prime }{)}^{-}=\underset{j}{\min }({\bar{h}}_m^{k\prime })=(\underset{j}{\min }\{{\zeta }_h^{\prime }|{\zeta }_h^{\prime }\in p_1\circ {\bar{h}}_m^{jk}(z)\},\dots ,\underset{j}{\min }\{{\zeta }_h^{\prime }|{\zeta }_h^{\prime }\in p_m\circ {\bar{h}}_m^{jk}(z)\}).$$

   Algorithm 1 Step 5 (p. 13); Eq. (2) (p. 12)

5. 5

   Compute the mHF Euclidean distance of each alternative a_j from mHPIS (De+) and mHNIS (De-) via Eqs. 3-4 (Algorithm 1 Step 6). The distance double-sums over criteria k=1..q and over poles i=1..m (here written explicitly for hesitancy indices l=1..r as the sum across (ζ_{hl}^{jk'} − (ζ_{hl}^{k'})^{±})^2), normalised by 1/(r·m).

   $$D_e^{\prime }(a_j,\mathrm{mHPIS})=\sqrt{\frac{1}{rm}\sum _{k=1}^q\sum _{i=1}^m[({\zeta }_{h1}^{jk\prime }-({\zeta }_{h1}^{k\prime }{)}^{+}{)}^2+({\zeta }_{h2}^{jk\prime }-({\zeta }_{h2}^{k\prime }{)}^{+}{)}^2+\cdots +({\zeta }_{hr}^{jk\prime }-({\zeta }_{hr}^{k\prime }{)}^{+}{)}^2]};D_e^{\prime }(a_j,\mathrm{mHNIS})=\sqrt{\frac{1}{rm}\sum _{k=1}^q\sum _{i=1}^m[({\zeta }_{h1}^{jk\prime }-({\zeta }_{h1}^{k\prime }{)}^{-}{)}^2+({\zeta }_{h2}^{jk\prime }-({\zeta }_{h2}^{k\prime }{)}^{-}{)}^2+\cdots +({\zeta }_{hr}^{jk\prime }-({\zeta }_{hr}^{k\prime }{)}^{-}{)}^2]}.$$

   Algorithm 1 Step 6 (p. 13); Eqs. (3)-(4) (p. 12)

6. 6

   Compute the relative mHF closeness coefficient E_j' ∈ [0,1] for each alternative a_j via Eq. 5: E_j' = De'(a_j, mHNIS) / (De'(a_j, mHPIS) + De'(a_j, mHNIS)) (Algorithm 1 Step 7). Higher E_j' means closer to mHPIS and farther from mHNIS - therefore better.

   $$E_j^{\prime }=\frac{D_e^{\prime }(a_j,\mathrm{mHNIS})}{D_e^{\prime }(a_j,\mathrm{mHPIS})+D_e^{\prime }(a_j,\mathrm{mHNIS})},j=1,2,\dots ,p.$$

   Algorithm 1 Step 7 (p. 13); Eq. (5) (p. 12)

7. 7

   Rank the alternatives in descending order of E_j' (Algorithm 1 Step 8). The alternative with the highest mHF closeness coefficient is the best one. Ties at E_j' are broken (when needed) by the deviation degree ∆(h̄_m) of Definition 3 applied to the underlying aggregated mHFE; the seminal §3.1/§3.2 case studies exhibit no ties.

   $$a_{j_1}⪰a_{j_2}⪰\dots ⪰a_{j_p}⟺E_{j_1}^{\prime }\ge E_{j_2}^{\prime }\ge \dots \ge E_{j_p}^{\prime };\text{tie-break (if }{E}_j^{\prime }=E_{j^{\prime }}^{\prime }\text{): use }\Delta ({\bar{h}}_m)\text{ (Def. 3)}.$$

   Algorithm 1 Step 8 (p. 13); §3.1 final ranking Bn_1>Bn_3>Bn_4>Bn_5>Bn_2 (p. 16)

Fits when / Look elsewhere when

Assumptions to verify

• Every cell of H carries exactly m hesitant fuzzy elements (one per pole) and m is fixed across the matrix (§3.1 m=4, §3.2 m=5)
• After optimistic/pessimistic padding every HFE has equal length r across the matrix (Sec. 3 narrative on p. 11, Table 4 of §3.1)
• Criterion weights w_k ∈ (0,1] and Σ_k w_k = 1 (Step 2 of Algorithm 1, Eq. 6 of §3.1)
• All criteria interpreted as benefit-direction - the seminal paper does NOT handle cost-direction via complement
• Group-decision aggregation occurs BEFORE normalisation/weighting: r DMs jointly produce ONE mHF matrix upstream (mHF-TOPSIS does not define the aggregation operator itself)

Edge cases and pitfalls

• •When hesitant-set lengths within a cell differ, prolong by the chosen padding_strategy (optimistic → max, pessimistic → min) until every hesitant set across the matrix has length r (Sec. 3 narrative o
• •when needed) by the deviation degree ∆(h̄_m) of Definition 3 applied to the underlying aggregated mHFE; the seminal §3.1/§3.2 case studies exhibit no ties.

• Hesitancy-length parity: Eqs. 3-4 sum over fixed hesitancy indices l=1..r; if cells have unequal hesitant-set lengths the sum is undefined. The mandatory padding step (Sec. 3 p. 11, Table 4 of §3.1) MUST be applied before F3 and is part of the algorithm - skipping it produces incorrect distances.

• Optimistic vs pessimistic padding can reorder the ranking. The seminal §3.1 uses 'optimistic' (max-padding) explicitly because 'the company wants to base the perfect brand name on an optimistic spirit' (p. 14). Switching to 'pessimistic' (min-padding) re-pads with the minimum value of each HFE and may produce a different ranking - always document the strategy alongside the result.

• Pole-count uniformity: every cell of H must carry exactly m hesitant fuzzy elements; mixing m=3 in some cells and m=4 in others is undefined. The §3.1 case study uses m=4 throughout; §3.2 uses m=5 throughout.

• Weights must sum to 1 with w_k ∈ (0,1] (Step 2, Eq. 6 of §3.1). The seminal paper allows ANY method for weight elicitation (AHP, entropy, BWM, direct expert input, equal weights when no info) - but a non-normalised vector will silently bias the closeness coefficient.

• mHF distance treats every hesitancy index equally: in Eqs. 3-4 the l-th value of one cell is paired with the l-th value of the ideal (positional matching), not via a Hausdorff/aggregate over hesitant sets. Therefore the order in which the padding step prolongs the hesitant values matters mathematically - the seminal paper sorts ascendingly before padding (Tables 3-4 reflect this).

• Group-decision aggregation moment: the three DMs of §3.1 jointly produce ONE mHF matrix (no per-DM matrices are reported); aggregation happens BEFORE normalisation/weighting. If your workflow has per-DM matrices, you must average them into a single mHF matrix upstream - mHF-TOPSIS does NOT define the aggregation operator itself.

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