PL-MARCOS - Probabilistic Linguistic extension of MARCOS

Overview

pl-marcos extends MARCOS to handle Probabilistic Linguistic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Probabilistic Linguistic Term Set (PLTS: {L_k|p_k}) algebra. The final scores are defuzzified via expected linguistic value E = Σ p_k · index(L_k) before ranking.

Output

utility, higher is better

Data

Probabilistic Linguistic, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Probabilistic Linguistic MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 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).

   $$L_{ij}(p_{ij})=\{L_{ij}^{(k)}(p_{ij}^{(k)})\},\sum _k{p}_{ij}^{(k)}=1;J^{+}\cup J^{-}=\{1,\dots ,n\}$$

   Report §4.4 Step 1; PL-MARCOS

2. 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.

   $$E(L(p))=\sum _{k}r(L^{(k)})\cdot p^{(k)};L_{AI,j}=\arg \min /\underset{i}{\max }E(L_{ij}),L_{I,j}=\arg \max /\underset{i}{\min }E(L_{ij})$$

   Report §4.4 Formulas 1-2 - anti-ideal and ideal solutions

3. 3

   Build the weighted PLTS matrix v_ij(p) by scaling each row's PLTS by criterion weight w_j (probabilistic-linguistic weighted aggregation).

   $$v_{ij}(p)=w_j\otimes L_{ij}(p_{ij})$$

   Report §4.4 Step 4 - weighted PL matrix

4. 4

   Weighted sum per alternative S_i = Σ_j E(v_ij(p)); compute S_AI and S_I for the appended rows.

   $$S_i=\sum _{j=1}^{n}E(v_{ij}(p));S_{AI}=\sum _{j}E(v_{AI,j}(p)),S_I=\sum _{j}E(v_{I,j}(p))$$

   Report §4.4 Step 4 - weighted sum

5. 5

   Utility degrees K_i^- = S_i/S_AI (vs anti-ideal) and K_i^+ = S_i/S_I (vs ideal).

   $$K_i^{-}=\frac{S_i}{S_{AI}},K_i^{+}=\frac{S_i}{S_I}$$

   Report §4.4 Formulas 3-4 - utility degrees

6. 6

   Utility functions f(K_i^+)=K_i^-/(K_i^+ + K_i^-) and f(K_i^-)=K_i^+/(K_i^+ + K_i^-).

   $$f(K_i^{+})=\frac{K_i^{-}}{K_i^{+}+K_i^{-}},f(K_i^{-})=\frac{K_i^{+}}{K_i^{+}+K_i^{-}}$$

   Report §4.4 - utility functions

7. 7

   Final utility function f(K_i) and descending ranking.

   $$f(K_i)=\frac{K_i^{+}+K_i^{-}}{1+\frac{1-f(K_i^{+})}{f(K_i^{+})}+\frac{1-f(K_i^{-})}{f(K_i^{-})}};\text{rank}={\text{argsort}}_{\text{desc}}(f(K_i))$$

   Report §4.4 Formula 5 - final utility

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy PLTS: {L_k|p_k} where L_k is linguistic label, Σ p_k ≤ 1 before computation.

• Defuzzification method affects ranking: expected linguistic value E = Σ p_k · index(L_k) is the canonical choice but alternatives exist.

Works with

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