PL-MARCOS: Probabilistic Linguistic extension of MARCOS

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

Edge cases

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

Common pitfalls

• •Hatalı: 'PL-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Probabilistic Linguistic numbers/tuples
• •Hatalı: 'PL-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PL-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PL-MARCOS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PL-MARCOS'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 PLTS decision matrix L_ij(p_ij)={L_ij^(k)(p_ij^(k))} on linguistic term set S={s_0,…,s_g} with Σ_k p_ij^(k)=1; define criterion weights w_j and direction sets J^+ (benefit) / J^- (cost). Formül: L_{ij}(p_{ij}) = \{L_{ij}^{(k)}(p_{ij}^{(k)})\},\ \sum_{k} p_{ij}^{(k)} = 1;\ \ J^{+}\cup J^{-}=\{1,\dots,n\} Anchor: Report §4.4 Step 1; PL-MARCOS
2. 2.Adım 2 (F2): Step 2: Define PL anti-ideal L_AI(p) and ideal L_I(p) solutions per criterion direction using expected value E(L(p))=Σ_k r(L^(k))·p^(k): L_AI,j = arg min_i E(L_ij) (benefit) / arg max_i (cost); L_I,j = arg max_i (benefit) / arg min_i (cost). Append L_AI and L_I as extra rows to the matrix. Formül: E(L(p)) = \sum_{k} r(L^{(k)})\cdot p^{(k)};\ \ L_{AI,j} = \arg\min/\max_{i} E(L_{ij}),\ L_{I,j} = \arg\max/\min_{i} E(L_{ij}) Anchor: Report §4.4 Formulas 1-2: anti-ideal and ideal solutions
3. 3.Adım 3 (F3): Step 3: Build the weighted PLTS matrix v_ij(p) by scaling each row's PLTS by criterion weight w_j (probabilistic-linguistic weighted aggregation). Formül: v_{ij}(p) = w_{j}\otimes L_{ij}(p_{ij}) Anchor: Report §4.4 Step 4: weighted PL matrix
4. 4.Adım 4 (F4): Step 4: Weighted sum per alternative S_i = Σ_j E(v_ij(p)); compute S_AI and S_I for the appended rows. Formül: S_{i} = \sum_{j=1}^{n} E(v_{ij}(p));\ \ S_{AI} = \sum_{j} E(v_{AI,j}(p)),\ S_{I} = \sum_{j} E(v_{I,j}(p)) Anchor: Report §4.4 Step 4: weighted sum
5. 5.Adım 5 (F5): Step 5: Utility degrees K_i^- = S_i/S_AI (vs anti-ideal) and K_i^+ = S_i/S_I (vs ideal). Formül: K_{i}^{-} = \dfrac{S_{i}}{S_{AI}},\quad K_{i}^{+} = \dfrac{S_{i}}{S_{I}} Anchor: Report §4.4 Formulas 3-4: utility degrees
6. 6.Adım 6 (F6): Step 6: Utility functions f(K_i^+)=K_i^-/(K_i^+ + K_i^-) and f(K_i^-)=K_i^+/(K_i^+ + K_i^-). Formül: f(K_{i}^{+}) = \dfrac{K_{i}^{-}}{K_{i}^{+}+K_{i}^{-}},\quad f(K_{i}^{-}) = \dfrac{K_{i}^{+}}{K_{i}^{+}+K_{i}^{-}} Anchor: Report §4.4: utility functions
7. 7.Adım 7 (F7): Step 7: Final utility function f(K_i) and descending ranking. Formül: f(K_{i}) = \dfrac{K_{i}^{+}+K_{i}^{-}}{1+\dfrac{1-f(K_{i}^{+})}{f(K_{i}^{+})}+\dfrac{1-f(K_{i}^{-})}{f(K_{i}^{-})}};\ \ \text{rank} = \text{argsort}_{\text{desc}}(f(K_{i})) Anchor: Report §4.4 Formula 5: final utility

Commonly paired with

• •n_a + PL-MARCOS (common)

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