PL-TOPSIS: Probabilistic Linguistic extension of TOPSIS

Overview

Probabilistic Linguistic outranking/ranking: Probabilistic Linguistic Term Set (PLTS: {L_k|p_k}). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Probabilistic Linguistic outranking/ranking: Probabilistic Linguistic Term Set (PLTS: {L_k|p_k})
• •Preserves linguistic_probabilistic 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: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •Assumes: Decision matrix entries are valid Probabilistic Linguistic numbers/tuples
• •Assumes: Underlying crisp method's compensation assumption holds in uncertain space
• •Assumes: All decision-maker(s) and experts use the same linguistic/uncertainty scale

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

When not to use

• •Crisp data sufficient: use base TOPSIS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

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

Common pitfalls

• •Hatalı: 'PL-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Probabilistic Linguistic numbers/tuples
• •Hatalı: 'PL-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PL-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PL-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PL-TOPSIS'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 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. Formül: L_{ij}(p_{ij}) = \{L_{ij}^{(k)}(p_{ij}^{(k)}) \mid k=1,\dots,\#L_{ij}\};\ \ \sum_{k} p_{ij}^{(k)} = 1 Anchor: Report §4.1 Steps 1-3; Lu 2019 PL-TOPSIS
2. 2.Adım 2 (F2): Step 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). Formül: v_{ij}(p) = w_{j} \otimes L_{ij}(p_{ij}) Anchor: Report §4.1 Step 4: weighted PL matrix
3. 3.Adım 3 (F3): Step 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). Formül: E(L(p)) = \sum_{k=1}^{\#L} r(L^{(k)})\cdot p^{(k)};\ \ L_{j}^{+} = \arg\max/\min_{i} E(v_{ij}),\ L_{j}^{-} = \arg\min/\max_{i} E(v_{ij}) Anchor: Report §4.1 Formulas 1, 4, 5: expected value and PLPIS/PLNIS
4. 4.Adım 4 (F4): Step 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^-). Formül: d(L_{1}(p), L_{2}(p)) = \dfrac{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})) Anchor: Report §4.1 Formulas 3, 6: PLTS distance and separation
5. 5.Adım 5 (F5): Step 5: Closeness coefficient CC_i = D_i^- / (D_i^+ + D_i^-); rank in descending order of CC_i. Formül: CC_{i} = \dfrac{D_{i}^{-}}{D_{i}^{+} + D_{i}^{-}};\ \ \text{rank} = \text{argsort}_{\text{desc}}(CC_{i}) Anchor: Report §4.1 Formula 7: closeness coefficient

Commonly paired with

• •n_a + PL-TOPSIS (common)

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