MPF-TOPSIS-LING: m-Polar Fuzzy Linguistic TOPSIS for MCGDM (Adeel, Akram & Koam 2019, Symmetry 11(6):735): multi-criteria group decision-making via m-polar fuzzy linguistic variables (mFLV), expert-aggregated m-PF linguistic decision matrix, aggregated linguistic-term-set weights, m-PF linguistic positive/negative ideal solutions (mPIS / mNIS), m-PF linguistic Euclidean distances, relative closeness coefficient E'_j descending ranking

Overview

Distance-based group-decision MCDM under m-polar fuzzy linguistic information: aggregated m-PF linguistic decision matrix d'^i_jk = (1/r) Σ_l d^l,i_jk, weighted matrix e^i_jk = w'_k d^i_jk, m-PF linguistic Euclidean distances (Eqs. 3-4) to mPIS / mNIS (Eqs. 1-2), closeness coefficient E'_j (Eq. 5) descending ranking. Output typically closeness_coefficient (higher value = preferred).

Strengths

• •Method-specific: Distance-based group-decision MCDM under m-polar fuzzy linguistic information: aggregated m-PF linguistic decision matrix d'^i_jk = (1/r) Σ_l d^l,i_jk, weighted matrix e^i_jk = w'_k d^i_jk, m-PF linguistic Euclidean distances (Eqs. 3-4) to mPIS / mNIS (Eqs. 1-2), closeness coefficient E'_j (Eq. 5) descending ranking
• •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

• •Rank reversal known on alternative-set changes (ref: Hwang-Yoon-1981-TOPSIS-rank-reversal-literature)
• •Assumes: Each decision-maker assigns m-PF linguistic ratings d^l,i_jk ∈ [0,1] (NOT raw 0-100 scores).
• •Assumes: All decision-makers use the same linguistic variable L_v, the same set of linguistic values V_k, and the same set of m pole-characteristics.
• •Assumes: Per-DM linguistic-term-set weights W^l = (w_1^l,…,w_q^l) are supplied as cardinal numbers in (0,1] with Σ_k w_k^l = 1 (label-to-number mapping done upstream).
• •Assumes: All criteria (linguistic values V_k) are benefit-direction: cost-direction handling is out of scope for this method as published.

Assumptions to verify

• •Each decision-maker assigns m-PF linguistic ratings d^l,i_jk ∈ [0,1] (NOT raw 0-100 scores).
• •All decision-makers use the same linguistic variable L_v, the same set of linguistic values V_k, and the same set of m pole-characteristics.
• •Per-DM linguistic-term-set weights W^l = (w_1^l,…,w_q^l) are supplied as cardinal numbers in (0,1] with Σ_k w_k^l = 1 (label-to-number mapping done upstream).
• •All criteria (linguistic values V_k) are benefit-direction: cost-direction handling is out of scope for this method as published.
• •Decision-makers receive equal weight 1/r (simple arithmetic mean aggregation, Sec. 3 top-of-p.7); non-uniform DM weights are not supported by the canonical algorithm.

When not to use

• •Single-pole crisp data is sufficient: use base TOPSIS (Hwang-Yoon 1981) directly to avoid unnecessary m-polar layer.
• •Decision cells are intuitionistic / Pythagorean / spherical fuzzy (use IF-TOPSIS / PF-TOPSIS / SF-TOPSIS instead: m-PF value space is [0,1]^m without complement-pair semantics).
• •Non-uniform decision-maker weights are required (use a different MCGDM aggregator such as IFWA-based TOPSIS-MCGDM).
• •User requires the multi-stage outranking semantics of ELECTRE: MPF-TOPSIS-LING produces a complete linear ranking, not a choice set or partial preorder.

Edge cases

• •Ties on E'_j are not addressed by the canonical paper: the §3.1 case study has no ties.

Common pitfalls

• •Hatalı: 'MPF-TOPSIS-LING bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Each decision-maker assigns m-PF linguistic ratings d^l,i_jk ∈ [0,1] (NOT raw 0-100 scores).
• •Hatalı: 'MPF-TOPSIS-LING bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-makers use the same linguistic variable L_v, the same set of linguistic values V_k, and the same set of m pole-characteristics.
• •Hatalı: 'MPF-TOPSIS-LING bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Per-DM linguistic-term-set weights W^l = (w_1^l,…,w_q^l) are supplied as cardinal numbers in (0,1] with Σ_k w_k^l = 1 (label-to-number mapping done upstream).
• •Hatalı: 'MPF-TOPSIS-LING bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All criteria (linguistic values V_k) are benefit-direction: cost-direction handling is out of scope for this method as published.
• •Hatalı: 'MPF-TOPSIS-LING bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision-makers receive equal weight 1/r (simple arithmetic mean aggregation, Sec. 3 top-of-p.7); non-uniform DM weights are not supported by the canonical algorithm.
• •Hatalı: MPF-TOPSIS-LING'yi 'Single-pole crisp data is sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-TOPSIS-LING'yi 'Decision cells are intuitionistic / Pythagorean / spherical fuzzy (use IF-TOPSIS / PF-TOPSIS / SF-TOPSIS instead' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-TOPSIS-LING'yi 'Non-uniform decision-maker weights are required (use a different MCGDM aggregator such as IFWA-based TOPSIS-MCGDM).' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Ingest the per-decision-maker m-PF linguistic rating cube d^l,i_jk = p_i◦d^l_jk(a_j, V_k) ∈ [0,1] for l=1,…,r decision-makers, j=1,…,p alternatives, k=1,…,q linguistic values of the mFLV L_v, and i=1,…,m pole-characteristics; ingest the per-DM linguistic-term-set weight vectors W^l = (w_1^l,…,w_q^l) ∈ (0,1]^q with Σ_k w_k^l = 1. Validate domain constraints E-1 through E-6. Formül: d^{l,i}_{jk} = p_i \circ d^l_{jk}(a_j, V_k) \in [0,1]; \quad W^l = (w_1^l, \ldots, w_q^l) \in (0,1]^q, \quad \sum_{k=1}^{q} w_k^l = 1, \quad l = 1, 2, \ldots, r. Anchor: Sec. 3 input preamble (p. 6) and Eq. (preceding Table 2)
2. 2.Adım 2 (F2): Aggregate the r per-DM m-PF linguistic rating matrices into a single aggregated m-PF linguistic decision matrix by simple arithmetic mean over decision-makers (Sec. 3, top-of-p. 7 formula and Table 3). For every alternative a_j, linguistic value V_k, and pole i, set d'^i_jk = (1/r) Σ_{l=1}^{r} d^l,i_jk. Simultaneously aggregate the per-DM linguistic-term-set weights by simple arithmetic mean: w'_k = (1/r) Σ_{l=1}^{r} w_k^l (Sec. 3, mid-p. 7); Σ_k w'_k = 1 is preserved. Formül: d'^{i}_{jk} = \frac{1}{r} \sum_{l=1}^{r} d^{l,i}_{jk}, \quad i = 1, \ldots, m;\ j = 1, \ldots, p;\ k = 1, \ldots, q. \qquad w'_k = \frac{1}{r} \sum_{l=1}^{r} w_k^l, \quad k = 1, \ldots, q; \quad \sum_{k=1}^{q} w'_k = 1. Anchor: Sec. 3, top-of-p. 7 aggregation formulas; Tables 3 and 6
3. 3.Adım 3 (F3): Form the weighted aggregated m-PF linguistic decision matrix E = [(e^1_jk, e^2_jk, …, e^m_jk)]_{p×q} by multiplying each pole component of the aggregated cell d'^i_jk by the aggregated weight w'_k of its linguistic value (Sec. 3, bottom-of-p. 7 formula and Table 7): e^i_jk = w'_k d^i_jk. The result is an m-PF number with every coordinate in [0,1] because w'_k ∈ (0,1] and d^i_jk ∈ [0,1]. Formül: E = \bigl[(e^1_{jk}, e^2_{jk}, \ldots, e^m_{jk})\bigr]_{p \times q}, \qquad e^i_{jk} = w'_k \, d^i_{jk}, \quad i = 1, \ldots, m;\ j = 1, \ldots, p;\ k = 1, \ldots, q. Anchor: Sec. 3, bottom-of-p. 7 weighted-matrix formula; Table 7
4. 4.Adım 4 (F4): Extract the m-PF linguistic positive ideal solution mPIS and the m-PF linguistic negative ideal solution mNIS column-wise per pole (Eqs. 1 and 2). For every linguistic value V_k (column) and every pole i, set (d^i_k)+ = max_j e^i_jk and (d^i_k)− = min_j e^i_jk. The mPIS / mNIS are q-tuples of m-PF numbers: mPIS = {((d^1_1)+,…,(d^m_1)+), …, ((d^1_q)+,…,(d^m_q)+)}; mNIS = {((d^1_1)−,…,(d^m_1)−), …, ((d^1_q)−,…,(d^m_q)−)}. All criteria are benefit-direction, so max → ideal positive and min → ideal negative everywhere. Formül: \mathrm{mPIS} = \Bigl\{\bigl((d^1_k)^+, (d^2_k)^+, \ldots, (d^m_k)^+\bigr)\Bigr\}_{k=1}^{q}, \quad (d^i_k)^+ = \max_{j=1,\ldots,p} e^i_{jk}; \qquad \mathrm{mNIS} = \Bigl\{\bigl((d^1_k)^-, (d^2_k)^-, \ldots, (d^m_k)^-\bigr)\Bigr\}_{k=1}^{q}, \quad (d^i_k)^- = \min_{j=1,\ldots,p} e^i_{jk}. Anchor: Eqs. (1) and (2), p. 8
5. 5.Adım 5 (F5): Compute the m-PF linguistic Euclidean distance of each alternative a_j from mPIS and from mNIS by the per-pole-equal-weighted Euclidean metric over q linguistic values and m poles (Eqs. 3 and 4): D_e(a_j, mPIS) = √((1/m) Σ_{k=1}^{q} [(e^1_jk − (d^1_k)+)^2 + (e^2_jk − (d^2_k)+)^2 + … + (e^m_jk − (d^m_k)+)^2]); D_e(a_j, mNIS) = √((1/m) Σ_{k=1}^{q} [(e^1_jk − (d^1_k)−)^2 + (e^2_jk − (d^2_k)−)^2 + … + (e^m_jk − (d^m_k)−)^2]). The (1/m) prefactor inside the square root applies per-pole-equal weighting; q linguistic values are summed without per-criterion reweighting because the criterion weights w'_k were already embedded into e^i_jk in F3. Formül: D_e(a_j, \mathrm{mPIS}) = \sqrt{\frac{1}{m} \sum_{k=1}^{q} \Bigl[(e^1_{jk} - (d^1_k)^+)^2 + (e^2_{jk} - (d^2_k)^+)^2 + \cdots + (e^m_{jk} - (d^m_k)^+)^2\Bigr]}; \qquad D_e(a_j, \mathrm{mNIS}) = \sqrt{\frac{1}{m} \sum_{k=1}^{q} \Bigl[(e^1_{jk} - (d^1_k)^-)^2 + (e^2_{jk} - (d^2_k)^-)^2 + \cdots + (e^m_{jk} - (d^m_k)^-)^2\Bigr]}. Anchor: Eqs. (3) and (4), p. 8
6. 6.Adım 6 (F6): Compute the relative m-PF linguistic closeness coefficient of each alternative a_j by Eq. (5): E'_j = D_e(a_j, mNIS) / (D_e(a_j, mPIS) + D_e(a_j, mNIS)) ∈ [0,1]. Rank the alternatives in descending order of E'_j; the alternative with the largest E'_j is the best. Ties on E'_j are not addressed by the canonical paper: the §3.1 case study has no ties. Formül: E'_j = \frac{D_e(a_j, \mathrm{mNIS})}{D_e(a_j, \mathrm{mPIS}) + D_e(a_j, \mathrm{mNIS})} \in [0,1], \quad j = 1, 2, \ldots, p; \qquad a_{j_1} \succeq a_{j_2} \succeq \cdots \succeq a_{j_p} \iff E'_{j_1} \ge E'_{j_2} \ge \cdots \ge E'_{j_p}. Anchor: Eq. (5), p. 8; Sec. 3.1 final ranking p. 11

Commonly paired with

• •AHP + MPF-TOPSIS-LING (common)
• •ENTROPY + MPF-TOPSIS-LING (occasional)

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