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Outranking

P-PROMETHEE - Plithogenic extension of P-PROMETHEE

Plithogenic outranking/ranking - Plithogenic Set (PltS: attribute values with contradiction degrees)

Smarandache, F.2019

Overview

p-promethee extends P-PROMETHEE to handle Plithogenic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Plithogenic Set (PltS: attribute values with contradiction degrees) algebra. The final scores are defuzzified via dominant value defuzzification; approach varies by appurtenance function before ranking.

Output: preference flow, higher is better
Data: Plithogenic, uncertainty tuples complete
Weights: Needs a weight source
Size: 2+ alternatives, 3-10 criteria works best
Used for: Plithogenic MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1
Contradiction adjustment; cost complement.
$\overset{p}{^}_{ij} = (T + c (1 - T), I (1 - c), F (1 - c)); s (\overset{p}{^}) = (1 + \hat{T} - 2 \hat{I} - \hat{F}) /2$
Smarandache 2018 Plithogeny; 2019 NSS 21
2
Signed Plithogenic differences.
$d_{j} (i, k) = sign (s (\overset{p}{^}_{ij}) - s (\overset{p}{^}_{k j})) \cdot d (\overset{p}{^}_{ij}, \overset{p}{^}_{k j}); d (p_{1}, p_{2}) = [(T_{1} - T_{2})^{2} + (I_{1} - I_{2})^{2} + (F_{1} - F_{2})^{2}] /3$
Brans & Vincke 1985 MS 31(6)
3
Type V linear preference function; aggregated preference index.
$P_{j} (i, k) = 0 (d_{j} \leq 0); d_{j} / p_{j} (0 < d_{j} \leq p_{j}); 1 (d_{j} > p_{j}); π (A_{i}, A_{k}) = j \sum w_{j} P_{j} (i, k)$
Brans & Vincke 1985 Sec.3
4
Positive and negative outranking flows.
$ϕ^{+} (A_{i}) = \frac{1}{m - 1} k \neq = i \sum π (A_{i}, A_{k}); ϕ^{-} (A_{i}) = \frac{1}{m - 1} k \neq = i \sum π (A_{k}, A_{i})$
Brans & Vincke 1985 Sec.4
5
Net flow PROMETHEE II; rank descending.
$ϕ (A_{i}) = ϕ^{+} (A_{i}) - ϕ^{-} (A_{i}); rank descending (PROMETHEE II)$
Brans & Vincke 1985

Fits when / Look elsewhere when

Fits when

•Preserves plithogenic uncertainty through the pipeline rather than premature crispification at elicitation
•Native group-decision support (multi-DM aggregation built into the pipeline)

Look elsewhere when

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

Assumptions to verify

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

Limitations

•Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)

Edge cases and pitfalls

Value-space violation: ensure all entries satisfy Plithogenic: each attribute has a set of values V with appurtenance degrees and contradiction degrees c(v,D) before computation.
Defuzzification method affects ranking: dominant value defuzzification; approach varies by appurtenance function is the canonical choice but alternatives exist.

Works with

Commonly takes its weights from

n_a

How to cite

Smarandache, F. (2019). Plithogenic MCDM methods. Infinite Study.

BibTeX RIS APA (.txt)

System ID, as it appears in reports and the API

P-PROMETHEE

Overview

Output

preference flow, higher is better

Data

Plithogenic, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

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

How it works

Contradiction adjustment; cost complement.

\overset{p}{^}_{ij} = (T + c (1 - T), I (1 - c), F (1 - c)); s (\overset{p}{^}) = (1 + \hat{T} - 2 \hat{I} - \hat{F}) /2

Smarandache 2018 Plithogeny; 2019 NSS 21

Signed Plithogenic differences.

d_{j} (i, k) = sign (s (\overset{p}{^}_{ij}) - s (\overset{p}{^}_{k j})) \cdot d (\overset{p}{^}_{ij}, \overset{p}{^}_{k j}); d (p_{1}, p_{2}) = [(T_{1} - T_{2})^{2} + (I_{1} - I_{2})^{2} + (F_{1} - F_{2})^{2}] /3

Brans & Vincke 1985 MS 31(6)

Type V linear preference function; aggregated preference index.

P_{j} (i, k) = 0 (d_{j} \leq 0); d_{j} / p_{j} (0 < d_{j} \leq p_{j}); 1 (d_{j} > p_{j}); π (A_{i}, A_{k}) = j \sum w_{j} P_{j} (i, k)

Brans & Vincke 1985 Sec.3

Positive and negative outranking flows.

ϕ^{+} (A_{i}) = \frac{1}{m - 1} k \neq = i \sum π (A_{i}, A_{k}); ϕ^{-} (A_{i}) = \frac{1}{m - 1} k \neq = i \sum π (A_{k}, A_{i})

Brans & Vincke 1985 Sec.4

Net flow PROMETHEE II; rank descending.

ϕ (A_{i}) = ϕ^{+} (A_{i}) - ϕ^{-} (A_{i}); rank descending (PROMETHEE II)

Brans & Vincke 1985

Edge cases and pitfalls

Value-space violation: ensure all entries satisfy Plithogenic: each attribute has a set of values V with appurtenance degrees and contradiction degrees c(v,D) before computation.

Defuzzification method affects ranking: dominant value defuzzification; approach varies by appurtenance function is the canonical choice but alternatives exist.