PL-VIKOR: Probabilistic Linguistic extension of VIKOR

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 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: 'PL-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Probabilistic Linguistic numbers/tuples
• •Hatalı: 'PL-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PL-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PL-VIKOR'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PL-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: Identify the best and worst PLTS values per criterion using expected value E(L(p))=Σ_k r(L^(k))·p^(k): L_j^* = arg max_i E(L_ij(p_ij)) (benefit) or arg min_i (cost); L_j^- = arg min_i (benefit) or arg max_i (cost). Formül: E(L(p)) = \sum_{k=1}^{\#L} r(L^{(k)})\cdot p^{(k)};\ \ L_{j}^{*}(p_{j}^{*}) = \arg\max/\min_{i} E(L_{ij}(p_{ij})),\ L_{j}^{-}(p_{j}^{-}) = \arg\min/\max_{i} E(L_{ij}(p_{ij})) Anchor: Report §4.2 Formula 1; Li 2021 PL-VIKOR
2. 2.Adım 2 (F2): Step 2: Compute utility S_i and regret R_i in PLTS form using normalised PLTS distance d(L_j^*(p_j^*), L_ij(p_ij)) / d(L_j^*(p_j^*), L_j^-(p_j^-)) weighted by w_j. Formül: S_{i} = \sum_{j=1}^{n} w_{j}\,\dfrac{d(L_{j}^{*}(p_{j}^{*}), L_{ij}(p_{ij}))}{d(L_{j}^{*}(p_{j}^{*}), L_{j}^{-}(p_{j}^{-}))},\quad R_{i} = \max_{j}\left[w_{j}\,\dfrac{d(L_{j}^{*}(p_{j}^{*}), L_{ij}(p_{ij}))}{d(L_{j}^{*}(p_{j}^{*}), L_{j}^{-}(p_{j}^{-}))}\right] Anchor: Report §4.2 Formulas 2-3: utility and regret
3. 3.Adım 3 (F3): Step 3: Compute the VIKOR index Q_i as a convex combination of normalised S and R, weighted by the compromise coefficient v (classical VIKOR aggregation; identical structure to crisp VIKOR with PLTS-derived S, R). 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 §4.2 Formula 4: VIKOR index (classical)
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 + PL-VIKOR (common)

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