PL-TOPSIS - Probabilistic Linguistic extension of TOPSIS

Overview

pl-topsis extends TOPSIS to handle Probabilistic Linguistic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Probabilistic Linguistic Term Set (PLTS: {L_k|p_k}) algebra. The final scores are defuzzified via expected linguistic value E = Σ p_k · index(L_k) before ranking.

Output

utility, higher is better

Data

Probabilistic Linguistic, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Probabilistic Linguistic MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Define linguistic term set S={s_0,…,s_g}. Construct PL decision matrix where each cell L_ij(p_ij) = {L_ij^(k)(p_ij^(k))} is a PLTS; normalise so PLTSs in the same criterion have equal length and Σ_k p^(k) = 1.

   $$L_{ij}(p_{ij})=\{L_{ij}^{(k)}(p_{ij}^{(k)})\mid k=1,\dots ,\#L_{ij}\};\sum _k{p}_{ij}^{(k)}=1$$

   Report §4.1 Steps 1-3; Lu 2019 PL-TOPSIS

2. 2

   Weighted PLTS decision matrix via probabilistic-linguistic weighted aggregation: scale each PLTS by criterion weight w_j (subscript-scaled probabilities r(L^(k))·w_j) to produce v_ij(p).

   $$v_{ij}(p)=w_j\otimes L_{ij}(p_{ij})$$

   Report §4.1 Step 4 - weighted PL matrix

3. 3

   Determine the PL-positive ideal L^+(p^+) and PL-negative ideal L^-(p^-) per criterion direction, comparing PLTSs via expected value E(L(p))=Σ_k r(L^(k))·p^(k).

   $$E(L(p))=\sum _{k=1}^{\#L}r(L^{(k)})\cdot p^{(k)};L_j^{+}=\arg \max /\underset{i}{\min }E(v_{ij}),L_j^{-}=\arg \min /\underset{i}{\max }E(v_{ij})$$

   Report §4.1 Formulas 1, 4, 5 - expected value and PLPIS/PLNIS

4. 4

   PLTS distance d(L_1(p),L_2(p)) = (1/#L)·Σ_k |r(L_1^(k))·p_1^(k) − r(L_2^(k))·p_2^(k)|; aggregate separations D_i^+ = Σ_j d(L_ij,L_j^+) and D_i^- = Σ_j d(L_ij,L_j^-).

   $$d(L_1(p),L_2(p))=\frac{1}{\#L}\sum _{k=1}^{\#L}\left|r(L_1^{(k)})\cdot p_1^{(k)}-r(L_2^{(k)})\cdot p_2^{(k)}\right|;D_i^{\pm }=\sum _{j=1}^{n}d(L_{ij}(p_{ij}),L_j^{\pm }(p_j^{\pm }))$$

   Report §4.1 Formulas 3, 6 - PLTS distance and separation

5. 5

   Closeness coefficient CC_i = D_i^- / (D_i^+ + D_i^-); rank in descending order of CC_i.

   $$C{C}_i=\frac{D_i^{-}}{D_i^{+}+D_i^{-}};\text{rank}={\text{argsort}}_{\text{desc}}(C{C}_i)$$

   Report §4.1 Formula 7 - closeness coefficient

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Probabilistic Linguistic 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

• •ties r(L^(k))·w_j) to produce v_ij(p).

• Value-space violation: ensure all entries satisfy PLTS: {L_k|p_k} where L_k is linguistic label, Σ p_k ≤ 1 before computation.

• Defuzzification method affects ranking: expected linguistic value E = Σ p_k · index(L_k) is the canonical choice but alternatives exist.

Works with

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