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

Read the result as a complete descending ranking of alternatives a_j by the relative m-PF linguistic closeness coefficient E'_j = D_e(a_j, mNIS) / (D_e(a_j, mPIS) + D_e(a_j, mNIS)) ∈ [0,1]. A value close to 1 means the alternative is simultaneously close to the m-PF linguistic positive ideal solution mPIS and far from the m-PF linguistic negative ideal solution mNIS, so higher E'_j is strictly better. The mPIS / mNIS are computed per-column per-pole over the weighted aggregated matrix E, so they always sit on the boundary of the alternative set and depend on which alternatives are present - removing or adding alternatives can shift the ideals (TOPSIS rank-reversal). The per-pole-equal-weighted Euclidean metric of Eqs. (3)-(4) treats all m poles symmetrically; if domain knowledge suggests one pole is more decisive (e.g. 'face features' in the §3.1 case), the method as published does not weight poles non-uniformly and a manual re-scoring is needed.

Output

closeness coefficient, higher is better

Data

M-Polar Fuzzy, m polar tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

M-Polar Fuzzy MCGDM with linguistic ratings (multi-expert evaluation), Personnel / candidate / model appearance evaluation (§3.1 case: models ranking by appearance with m=3 poles face/posture/figure), Engineering selection problems with multi-attribute criteria (§3.2 case: high-speed racing cars with m=4 poles engine/tires/condition/shape), Linguistic-variable MCDM where each linguistic value is independently characterised by m sub-attributes

How it works

1. 1

   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.

   $$d_{jk}^{l,i}=p_i\circ d_{jk}^l(a_j,V_k)\in [0,1];W^l=(w_1^l,\dots ,w_q^l)\in (0,1{]}^q,\sum _{k=1}^q{w}_k^l=1,l=1,2,\dots ,r.$$

   Sec. 3 input preamble (p. 6) and Eq. (preceding Table 2)

2. 2

   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.

   $$d_{jk}^{\prime i}=\frac{1}{r}\sum _{l=1}^r{d}_{jk}^{l,i},i=1,\dots ,m;j=1,\dots ,p;k=1,\dots ,q.w_k^{\prime }=\frac{1}{r}\sum _{l=1}^r{w}_k^l,k=1,\dots ,q;\sum _{k=1}^q{w}_k^{\prime }=1.$$

   Sec. 3, top-of-p. 7 aggregation formulas; Tables 3 and 6

3. 3

   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].

   $$E=[(e_{jk}^1,e_{jk}^2,\dots ,e_{jk}^m){]}_{p\times q},e_{jk}^i=w_k^{\prime }{d}_{jk}^i,i=1,\dots ,m;j=1,\dots ,p;k=1,\dots ,q.$$

   Sec. 3, bottom-of-p. 7 weighted-matrix formula; Table 7

4. 4

   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.

   $$\mathrm{mPIS}=\left\{\right((d_k^1{)}^{+},(d_k^2{)}^{+},\dots ,(d_k^m{)}^{+}){\}}_{k=1}^q,(d_k^i{)}^{+}=\underset{j=1,\dots ,p}{\max }{e}_{jk}^i;\mathrm{mNIS}=\{((d_k^1{)}^{-},(d_k^2{)}^{-},\dots ,(d_k^m{)}^{-}){\}}_{k=1}^q,(d_k^i{)}^{-}=\underset{j=1,\dots ,p}{\min }{e}_{jk}^i.$$

   Eqs. (1) and (2), p. 8

5. 5

   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.

   $$D_e(a_j,\mathrm{mPIS})=\sqrt{\frac{1}{m}\sum _{k=1}^q[(e_{jk}^1-(d_k^1{)}^{+}{)}^2+(e_{jk}^2-(d_k^2{)}^{+}{)}^2+\cdots +(e_{jk}^m-(d_k^m{)}^{+}{)}^2]};D_e(a_j,\mathrm{mNIS})=\sqrt{\frac{1}{m}\sum _{k=1}^q[(e_{jk}^1-(d_k^1{)}^{-}{)}^2+(e_{jk}^2-(d_k^2{)}^{-}{)}^2+\cdots +(e_{jk}^m-(d_k^m{)}^{-}{)}^2]}.$$

   Eqs. (3) and (4), p. 8

6. 6

   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.

   $$E_j^{\prime }=\frac{D_e(a_j,\mathrm{mNIS})}{D_e(a_j,\mathrm{mPIS})+D_e(a_j,\mathrm{mNIS})}\in [0,1],j=1,2,\dots ,p;a_{j_1}⪰a_{j_2}⪰\cdots ⪰a_{j_p}⟺E_{j_1}^{\prime }\ge E_{j_2}^{\prime }\ge \cdots \ge E_{j_p}^{\prime }.$$

   Eq. (5), p. 8; Sec. 3.1 final ranking p. 11

Fits when / Look elsewhere when

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.

Edge cases and pitfalls

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

• Pole-rating range: every coordinate d^l,i_jk must lie in [0,1] (Def. 1 of Chen-Li-Ma-Wang 2014). Raw scores on a 0-100 marking scale (as in Table 1 of the seminal paper) must be normalised to [0,1] before being fed to MPF-TOPSIS-LING; the method does not perform that normalisation.

• Equal pole count: every m-PF cell across the entire decision cube must share the same pole count m. The pole semantics (e.g. p_1=face features, p_2=postures, p_3=body figure in §3.1) must be identical across decision-makers, alternatives, and linguistic values - otherwise the column-wise max/min for mPIS/mNIS is incoherent.

• Linguistic-term-set weights need a cardinal mapping: §3.1 supplies the weights as numerical values (e.g. l_2=medium ≈ 0.23) that the DMs have already mapped from labels to numbers. If only labels {low, medium, high, extremely high} are available, the caller MUST first elicit a cardinal mapping such that Σ_k w_k^l = 1 - the method does not normalise label sequences.

• mPIS/mNIS sensitivity to outlier DMs: because mPIS/mNIS are computed AFTER expert-aggregation, a single contrarian DM whose ratings drag d'^i_jk towards a corner of [0,1] can shift the ideal solution for ALL alternatives. The aggregation moment is 'before_normalization' (A.group_aggregation_moment); to detect outlier-DM influence, run S.weight_perturbation with per-DM-removal in a downstream analysis.

• Symmetric per-pole Euclidean metric: Eqs. (3)-(4) place a uniform 1/m prefactor inside the square root and DO NOT carry per-criterion reweighting (the criterion weights w'_k were already embedded into e^i_jk in F3). A common mistake is to multiply the squared deviation by w'_k a second time - that double-weights the criterion and breaks the calibration against Table 8 of the seminal paper.

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