P-PROMETHEE: Plithogenic extension of P-PROMETHEE

Overview

Plithogenic outranking/ranking: Plithogenic Set (PltS: attribute values with contradiction degrees). Output typically preference_flow (higher value = preferred).

Strengths

• •Method-specific: Plithogenic outranking/ranking: Plithogenic Set (PltS: attribute values with contradiction degrees)
• •Preserves plithogenic 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: Thresholds (q, p, v) can be expressed in Plithogenic scale
• •Assumes: Veto + concordance semantics adapted to fuzzy arithmetic

Assumptions to verify

• •Thresholds (q, p, v) can be expressed in Plithogenic scale
• •Veto + concordance semantics adapted to fuzzy arithmetic

When not to use

• •Small dataset (m<3): outranking machinery underutilised

Edge cases

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

Common pitfalls

• •Hatalı: 'P-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Thresholds (q, p, v) can be expressed in Plithogenic scale
• •Hatalı: 'P-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Veto + concordance semantics adapted to fuzzy arithmetic
• •Hatalı: P-PROMETHEE'yi 'Small dataset (m<3)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Contradiction adjustment; cost complement. Formül: \hat{p}_{ij}=(T+c(1-T),I(1-c),F(1-c));\;s(\hat{p})=(1+\hat{T}-2\hat{I}-\hat{F})/2 Anchor: Smarandache 2018 Plithogeny; 2019 NSS 21
2. 2.Adım 2 (F2): Signed Plithogenic differences. Formül: d_j(i,k)=\mathrm{sign}(s(\hat{p}_{ij})-s(\hat{p}_{kj}))\cdot d(\hat{p}_{ij},\hat{p}_{kj});\;d(p_1,p_2)=\sqrt{[(T_1-T_2)^2+(I_1-I_2)^2+(F_1-F_2)^2]/3} Anchor: Brans & Vincke 1985 MS 31(6)
3. 3.Adım 3 (F3): Type V linear preference function; aggregated preference index. Formül: P_j(i,k)=0\;(d_j\leq 0);\;d_j/p_j\;(0<d_j\leq p_j);\;1\;(d_j>p_j);\;\pi(A_i,A_k)=\sum_j w_j P_j(i,k) Anchor: Brans & Vincke 1985 Sec.3
4. 4.Adım 4 (F4): Positive and negative outranking flows. Formül: \phi^+(A_i)=\tfrac{1}{m-1}\sum_{k\neq i}\pi(A_i,A_k);\;\phi^-(A_i)=\tfrac{1}{m-1}\sum_{k\neq i}\pi(A_k,A_i) Anchor: Brans & Vincke 1985 Sec.4
5. 5.Adım 5 (F5): Net flow PROMETHEE II; rank descending. Formül: \phi(A_i)=\phi^+(A_i)-\phi^-(A_i);\quad\text{rank descending (PROMETHEE II)} Anchor: Brans & Vincke 1985

Commonly paired with

• •n_a + P-PROMETHEE (common)

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