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

Distance-based ranking: m-polar hesitant fuzzy TOPSIS: pole-wise max/min ideals on a weighted mHF decision matrix (Eqs. 1-2), mHF Euclidean distance (Eqs. 3-4), closeness coefficient E_j' (Eq. 5). Output typically closeness_coefficient (higher value = preferred).

Strengths

• •Method-specific: Distance-based ranking: m-polar hesitant fuzzy TOPSIS: pole-wise max/min ideals on a weighted mHF decision matrix (Eqs. 1-2), mHF Euclidean distance (Eqs. 3-4), closeness coefficient E_j' (Eq. 5)
• •Preserves m_polar uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •Assumes: 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)
• •Assumes: After optimistic/pessimistic padding every HFE has equal length r across the matrix (Sec. 3 narrative on p. 11, Table 4 of §3.1)
• •Assumes: Criterion weights w_k ∈ (0,1] and Σ_k w_k = 1 (Step 2 of Algorithm 1, Eq. 6 of §3.1)
• •Assumes: All criteria interpreted as benefit-direction: the seminal paper does NOT handle cost-direction via complement

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)

When not to use

• •Crisp data sufficient: use base TOPSIS (Hwang-Yoon 1981) directly to avoid unnecessary m-polar+hesitancy layer
• •Single-pole (m=1): degenerates to standard HF-TOPSIS; use HF-TOPSIS directly
• •Criteria mix cost and benefit direction: seminal paper handles benefit-only, cost-complement is out of scope
• •Decision-makers cannot agree on a common padding strategy (optimistic vs pessimistic): the ranking can flip between the two strategies and the result becomes ambiguous

Edge cases

• •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.

Common pitfalls

• •Hatalı: 'MPF-HF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: 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)
• •Hatalı: 'MPF-HF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: After optimistic/pessimistic padding every HFE has equal length r across the matrix (Sec. 3 narrative on p. 11, Table 4 of §3.1)
• •Hatalı: 'MPF-HF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion weights w_k ∈ (0,1] and Σ_k w_k = 1 (Step 2 of Algorithm 1, Eq. 6 of §3.1)
• •Hatalı: 'MPF-HF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All criteria interpreted as benefit-direction: the seminal paper does NOT handle cost-direction via complement
• •Hatalı: 'MPF-HF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Group-decision aggregation occurs BEFORE normalisation/weighting: r DMs jointly produce ONE mHF matrix upstream (mHF-TOPSIS does not define the aggregation operator itself)
• •Hatalı: MPF-HF-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-HF-TOPSIS'yi 'Single-pole (m=1)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-HF-TOPSIS'yi 'Criteria mix cost and benefit direction' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): 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). Formül: H = (\bar{h}_m^{jk})_{p \times q}, \quad \bar{h}_m^{jk}(z) = \bigl(\{\zeta_h | \zeta_h \in p_1 \circ \bar{h}_m^{jk}(z)\}, \ldots, \{\zeta_h | \zeta_h \in p_m \circ \bar{h}_m^{jk}(z)\}\bigr); \quad w = (w_1, \ldots, w_q), \ w_k \in (0,1], \ \sum_{k=1}^{q} w_k = 1. Anchor: Algorithm 1 Steps 1-2 (p. 13); Sec. 3 narrative on padding (p. 11); §3.1 Tables 3-4 (p. 14)
2. 2.Adım 2 (F2): 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. Formül: \bar{h}_m^{jk\prime} = w_k\, \bar{h}_m^{jk} = \bigl(w_k \{\zeta_h | \zeta_h \in p_1 \circ \bar{h}_m^{jk}(z)\}, \ldots, w_k \{\zeta_h | \zeta_h \in p_m \circ \bar{h}_m^{jk}(z)\}\bigr) = \bigl(\{\zeta_h' | \zeta_h' \in p_1 \circ \bar{h}_m^{jk}(z)\}, \ldots, \{\zeta_h' | \zeta_h' \in p_m \circ \bar{h}_m^{jk}(z)\}\bigr). Anchor: Algorithm 1 Step 3 (p. 13); Table 2 (p. 11); Table 5 (§3.1, p. 15)
3. 3.Adım 3 (F3): 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). Formül: \mathrm{mHPIS} = \bigl\{(\bar{h}_m^{1\prime})^{+}, (\bar{h}_m^{2\prime})^{+}, \ldots, (\bar{h}_m^{q\prime})^{+}\bigr\}, \quad (\bar{h}_m^{k\prime})^{+} = \max_{j}(\bar{h}_m^{k\prime}) = \bigl(\max_{j}\{\zeta_h' | \zeta_h' \in p_1 \circ \bar{h}_m^{jk}(z)\}, \ldots, \max_{j}\{\zeta_h' | \zeta_h' \in p_m \circ \bar{h}_m^{jk}(z)\}\bigr). Anchor: Algorithm 1 Step 4 (p. 13); Eq. (1) (p. 11)
4. 4.Adım 4 (F4): 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. Formül: \mathrm{mHNIS} = \bigl\{(\bar{h}_m^{1\prime})^{-}, (\bar{h}_m^{2\prime})^{-}, \ldots, (\bar{h}_m^{q\prime})^{-}\bigr\}, \quad (\bar{h}_m^{k\prime})^{-} = \min_{j}(\bar{h}_m^{k\prime}) = \bigl(\min_{j}\{\zeta_h' | \zeta_h' \in p_1 \circ \bar{h}_m^{jk}(z)\}, \ldots, \min_{j}\{\zeta_h' | \zeta_h' \in p_m \circ \bar{h}_m^{jk}(z)\}\bigr). Anchor: Algorithm 1 Step 5 (p. 13); Eq. (2) (p. 12)
5. 5.Adım 5 (F5): 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). Formül: D_e'(a_j, \mathrm{mHPIS}) = \sqrt{\tfrac{1}{rm} \sum_{k=1}^{q} \sum_{i=1}^{m} \Bigl[(\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\Bigr]}; \quad D_e'(a_j, \mathrm{mHNIS}) = \sqrt{\tfrac{1}{rm} \sum_{k=1}^{q} \sum_{i=1}^{m} \Bigl[(\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\Bigr]}. Anchor: Algorithm 1 Step 6 (p. 13); Eqs. (3)-(4) (p. 12)
6. 6.Adım 6 (F6): 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. Formül: E_j' = \frac{D_e'(a_j, \mathrm{mHNIS})}{D_e'(a_j, \mathrm{mHPIS}) + D_e'(a_j, \mathrm{mHNIS})}, \quad j = 1, 2, \ldots, p. Anchor: Algorithm 1 Step 7 (p. 13); Eq. (5) (p. 12)
7. 7.Adım 7 (F7): 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. Formül: a_{j_1} \succeq a_{j_2} \succeq \ldots \succeq a_{j_p} \iff E_{j_1}' \ge E_{j_2}' \ge \ldots \ge E_{j_p}'; \quad \text{tie-break (if } E_{j}' = E_{j'}' \text{): use } \Delta(\bar{h}_m) \text{ (Def. 3)}. Anchor: Algorithm 1 Step 8 (p. 13); §3.1 final ranking Bn_1>Bn_3>Bn_4>Bn_5>Bn_2 (p. 16)

Commonly paired with

• •AHP + MPF-HF-TOPSIS (common)
• •ENTROPY + MPF-HF-TOPSIS (common)
• •BWM + MPF-HF-TOPSIS (occasional)

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