PiF-MARCOS - Picture extension of MARCOS

Overview

pif-marcos extends MARCOS to handle Picture uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Picture Fuzzy Set (PiFS: μ, η, ν; μ+η+ν ≤ 1) algebra. The final scores are defuzzified via score function S = μ − ν before ranking.

Output

utility, higher is better

Data

Picture Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Picture Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Form Picture Fuzzy decision matrix X = [r_ij] with r_ij = (μ_ij, η_ij, ν_ij), μ_ij+η_ij+ν_ij ≤ 1.

   $$X=[r_{ij}{]}_{m\times n},r_{ij}=({\mu }_{ij},{\eta }_{ij},{\nu }_{ij}),{\mu }_{ij}+{\eta }_{ij}+{\nu }_{ij}\le 1$$

   Cuong 2013, Definition 1

2. 2

   Cost-complement normalize: for cost criteria, (μ,η,ν) → (ν,η,μ); benefit criteria unchanged. Result: R = [r̂_ij] all in benefit direction.

   $${\hat{r}}_{ij}=({\nu }_{ij},{\eta }_{ij},{\mu }_{ij})ifj\in Cost;{\hat{r}}_{ij}=({\mu }_{ij},{\eta }_{ij},{\nu }_{ij})ifj\in Benefit$$

   Tatar & Ayvaz 2024, §3.3 step 2; Cuong 2013 complement

3. 3

   Extend R with anti-ideal AAI and ideal AI rows: AAI_j = argmin_i s(r̂_ij), AI_j = argmax_i s(r̂_ij), where s(α) = (1+μ−ν)/2 is the score function.

   $$AA{I}_j={\hat{r}}_{i\ast j}\text{ where }i\ast =\arg \underset{i}{\min }s({\hat{r}}_{ij});A{I}_j={\hat{r}}_{i\ast \ast j}\text{ where }i\ast \ast =\arg \underset{i}{\max }s({\hat{r}}_{ij});s(\alpha )=(1+\mu -\nu )/2$$

   Stević 2020 Eq.(1); Tatar & Ayvaz 2024 §3.3 step 3

4. 4

   Per-row PFWA weighted aggregation: S_i = PFWA(r̂_i1,…,r̂_in; w). Apply to all m alternative rows plus AAI and AI rows.

   $$S_i=\text{PFWA}({\hat{r}}_{i1},\dots ,{\hat{r}}_{in};w)=\left(1-\prod _{j=1}^n(1-{\mu }_{ij}{)}^{w_j},\prod _{j=1}^n{\eta }_{ij}^{w_j},\prod _{j=1}^n{\nu }_{ij}^{w_j}\right)$$

   Wei 2017 Eq.(8); Tatar & Ayvaz 2024 §3.3 step 4

5. 5

   Defuzzify via score function: s_i = s(S_i) = (1+μ_i−ν_i)/2 for each alternative and for AAI, AI rows.

   $$s_i=(1+{\mu }_i-{\nu }_i)/2;s_{AAI}=(1+{\mu }_{AAI}-{\nu }_{AAI})/2;s_{AI}=(1+{\mu }_{AI}-{\nu }_{AI})/2$$

   Cuong 2013 score function; Stević 2020 Eq.(2)-(4) crisp analogue

6. 6

   Utility ratios vs ideal and anti-ideal: K_i^+ = s_i / s_AI, K_i^- = s_i / s_AAI.

   $$K_i^{+}=s_i/s_{AI};K_i^{-}=s_i/s_{AAI}$$

   Stević 2020 Eqs.(5)-(6); Tatar & Ayvaz 2024 §3.3 step 5

7. 7

   Utility functions: f(A_i^+) = K_i^- / (K_i^+ + K_i^-), f(A_i^-) = K_i^+ / (K_i^+ + K_i^-).

   $$f(A_i^{+})=K_i^{-}/(K_i^{+}+K_i^{-});f(A_i^{-})=K_i^{+}/(K_i^{+}+K_i^{-})$$

   Stević 2020 Eqs.(7)-(8); Tatar & Ayvaz 2024 Table 2 confirms constants f(A^-)=0.4634, f(A^+)=0.5366 in their example

8. 8

   Final utility f(K_i) = (K_i^+ + K_i^-) / (1 + (1−f(A_i^+))/f(A_i^+) + (1−f(A_i^-))/f(A_i^-)); rank alternatives in descending order of f(K_i).

   $$f(K_i)=\frac{K_i^{+}+K_i^{-}}{1+\frac{1-f(A_i^{+})}{f(A_i^{+})}+\frac{1-f(A_i^{-})}{f(A_i^{-})}}$$

   Stević 2020 Eq.(9); Tatar & Ayvaz 2024 Table 2

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Picture Fuzzy 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 PiFS: μ (membership), η (neutral), ν (non-membership); μ+η+ν ≤ 1 before computation.

• Defuzzification method affects ranking: score function S = μ − ν is the canonical choice but alternatives exist.

• Bu manifestin B.extensions bloğunda kaynak gösterilen Wei (2017), 'Picture fuzzy aggregation operators...' (DOI 10.3233/JIFS-161798) GERİ ÇEKİLMİŞTİR (RETRACTED); bu kaynağı PFWA/agregasyon gerekçesi olarak KULLANMAYIN.

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