L2T-VIKOR: Linguistic extension of L2T-VIKOR

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 VIKOR directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •If C1 fails, return the maximum prefix A^(1)..A^(M) for which Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.

Common pitfalls

• •Hatalı: 'L2T-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Linguistic 2-Tuple numbers/tuples
• •Hatalı: 'L2T-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'L2T-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: L2T-VIKOR'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: L2T-VIKOR'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: Construct the 2-tuple linguistic decision matrix; each entry is (s_ij, α_ij). Determine best (s_j*, α_j*) and worst (s_j⁻, α_j⁻) per criterion (max for benefit, min for cost using Δ⁻¹). Formül: (s_{j}^{*}, \alpha_{j}^{*}) = \max_{i}(s_{ij}, \alpha_{ij})\ \text{by}\ \Delta^{-1};\ \ (s_{j}^{-}, \alpha_{j}^{-}) = \min_{i}(s_{ij}, \alpha_{ij})\ \text{by}\ \Delta^{-1} Anchor: Report §3.2 Steps 1-2; Ju 2013 2TL-VIKOR
2. 2.Adım 2 (F2): Step 2: Utility measure S_i and regret measure R_i using 2-tuple distance d via Δ⁻¹. Formül: S_{i} = \sum_{j=1}^{n} w_{j}\,\dfrac{d((s_{j}^{*},\alpha_{j}^{*}),(s_{ij},\alpha_{ij}))}{d((s_{j}^{*},\alpha_{j}^{*}),(s_{j}^{-},\alpha_{j}^{-}))},\ R_{i} = \max_{j}\left[w_{j}\,\dfrac{d((s_{j}^{*},\alpha_{j}^{*}),(s_{ij},\alpha_{ij}))}{d((s_{j}^{*},\alpha_{j}^{*}),(s_{j}^{-},\alpha_{j}^{-}))}\right] Anchor: Report §3.2 Formulas 2-3: utility & regret
3. 3.Adım 3 (F3): Step 3: VIKOR index Q_i = v·(S_i − S*)/(S⁻ − S*) + (1−v)·(R_i − R*)/(R⁻ − R*); ascending sort over S, R, Q. Formül: Q_{i} = v\,\dfrac{S_{i}-S^{*}}{S^{-}-S^{*}} + (1-v)\,\dfrac{R_{i}-R^{*}}{R^{-}-R^{*}},\ S^{*}=\min_{i}S_{i},\ S^{-}=\max_{i}S_{i},\ R^{*}=\min_{i}R_{i},\ R^{-}=\max_{i}R_{i} Anchor: Report §3.2 Formula 4: VIKOR index
4. 4.Adım 4 (F4): Step 4: Propose A^(1) (the lowest-Q alternative) as compromise solution iff both C1 (acceptable advantage) and C2 (acceptable stability) hold. If C1 fails, return the maximum prefix A^(1)..A^(M) for which Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}. Formül: DQ = \dfrac{1}{m-1};\quad C1: Q(A^{(2)}) - Q(A^{(1)}) \ge DQ;\quad C2: A^{(1)} \text{ is best in } S \text{ or in } R Anchor: Opricovic & Tzeng 2004, §2 Eqs.(6)-(7)

Commonly paired with

• •n_a + L2T-VIKOR (common)

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