L2T-TOPSIS: Linguistic extension of L2T-TOPSIS

Overview

Linguistic outranking/ranking: 2-Tuple Linguistic Variable (2TL: (s_i, α)). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Linguistic outranking/ranking: 2-Tuple Linguistic Variable (2TL: (s_i, α))
• •Preserves linguistic_2tuple 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 Linguistic 2-Tuple 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 Linguistic 2-Tuple 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

• •See F.steps and D.parameters for L2T-TOPSIS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'L2T-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Linguistic 2-Tuple numbers/tuples
• •Hatalı: 'L2T-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'L2T-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: L2T-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: L2T-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_q} and construct the 2-tuple decision matrix where each x̃_ij = (s_i, α_ij), α ∈ [-0.5, 0.5). Formül: S = \{s_{0}, s_{1}, \ldots, s_{q}\};\ \tilde{D} = [(s_{ij}, \alpha_{ij})]_{m\times n},\ \Delta(\beta) = (s_{i}, \alpha),\ i = \text{round}(\beta),\ \alpha = \beta - i;\ \Delta^{-1}(s_{i}, \alpha) = i + \alpha Anchor: Report §3.1 Steps 1-2; Herrera-Martínez 2000 Δ/Δ⁻¹
2. 2.Adım 2 (F2): Step 2: Weighted 2-tuple aggregation per criterion via Δ⁻¹. Formül: \bar{x}_{j} = \Delta\!\left(\sum_{i=1}^{m} w_{i}\cdot \Delta^{-1}(s_{ij}, \alpha_{ij})\right) Anchor: Report §3.1 Formula 4: weighted 2-tuple aggregation
3. 3.Adım 3 (F3): Step 3: 2-Tuple Linguistic Positive Ideal Solution (TLPIS) A⁺ and Negative Ideal Solution (TLNIS) A⁻. Formül: A^{+} = \{(s_{1}^{+}, \alpha_{1}^{+}), \ldots, (s_{n}^{+}, \alpha_{n}^{+})\};\ \ A^{-} = \{(s_{1}^{-}, \alpha_{1}^{-}), \ldots, (s_{n}^{-}, \alpha_{n}^{-})\} Anchor: Report §3.1 Formulas 5-6: TLPIS / TLNIS
4. 4.Adım 4 (F4): Step 4: Separation measures using Δ⁻¹-based 2-tuple distance d((s_i,α_i),(s_j,α_j)) = |Δ⁻¹(s_i,α_i) − Δ⁻¹(s_j,α_j)|. Formül: d((s_{i},\alpha_{i}),(s_{j},\alpha_{j})) = |\Delta^{-1}(s_{i},\alpha_{i}) - \Delta^{-1}(s_{j},\alpha_{j})|;\ \ D_{i}^{+} = \sqrt{\sum_{j=1}^{n} d^{2}((r_{ij},\alpha_{ij}),(s_{j}^{+},\alpha_{j}^{+}))},\ \ D_{i}^{-} = \sqrt{\sum_{j=1}^{n} d^{2}((r_{ij},\alpha_{ij}),(s_{j}^{-},\alpha_{j}^{-}))} Anchor: Report §3.1 Formulas 3 & 7: distance & separation
5. 5.Adım 5 (F5): Step 5: Relative closeness RC_i = D_i⁻ / (D_i⁺ + D_i⁻); descending rank. Formül: RC_{i} = \dfrac{D_{i}^{-}}{D_{i}^{+}+D_{i}^{-}},\ 0 \le RC_{i} \le 1;\ \ \text{rank} = \text{argsort}_{\text{desc}}(RC_{i}) Anchor: Report §3.1 Formula 8: relative closeness

Commonly paired with

• •n_a + L2T-TOPSIS (common)

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