PROMETHEE Method

Preference Ranking Organization Method for Enrichment Evaluation

1986 Brans, Vincke and Mareschal 5 Steps

PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluations) is an outranking-based MCDM method used to compare and rank alternatives across multiple conflicting criteria. It converts pairwise differences into preference degrees using criterion-specific preference functions, aggregates them with criterion weights, and then uses positive and negative preference flows for ranking.

Video Demonstration

Step-by-Step Algorithm

Calculated cell

Source cells in formula

Form the decision matrix

Create the decision matrix with $m$ alternatives and $n$ criteria :

\[ X=\begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1n}\\ x_{21} & x_{22} & \cdots & x_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ x_{m1} & x_{m2} & \cdots & x_{mn} \end{bmatrix} \]

$x_{ij}$ presents the performance value of ith alternative on jth criterion, m and n are the numbers of alternatives and criteria respectively.

Alternative	$C_1$	$C_2$	$C_3$
$A_1$	10	20	30
$A_2$	15	25	35
$A_3$	25	15	35

Criterion type: C₁ Benefit C₂ Cost C₃ Benefit

Weights: $w_1=0.7279$ · $w_2=0.2009$ · $w_3=0.0712$ (Entropy method)

Preference index

Formula

For every pair of alternatives $A_i$ and $A_k$, calculate the difference for each criterion.

$$ d_j(A_i,A_{k})=x_{ij}-x_{kj} \qquad i,k \in \{1,\ldots,m\} $$

The value of the weak preference function is calculated based on the function :

\[ P_j(A_i,A_{k})=f_j\left[d_j(A_i,A_{k})\right], \qquad 0\leq P_j(A_i,A_{k})\leq1 \]

The six commonly used preference functions are :

1. Usual criterion (Type I). No parameter

$$ P(d)=\begin{cases} 0, & d\leq0\\ 1, & d>0 \end{cases} $$

Quasi-criterion (Type II). Parameter : l

$$ P(d)=\begin{cases} 0, & d\leq l\\ 1, & d>l \end{cases} $$

3. V-shape criterion (Type III). Parameter : m

$$ P(d)=\begin{cases} 0, & d\leq0\\[4pt] \dfrac{d}{m}, & 0 < d \leq m\\[8pt] 1, & d>m \end{cases} $$

4. Level criterion (Type IV). Parameters : q , p

$$ P(d)=\begin{cases} 0, & d\leq q\\[4pt] \frac12, & q < d \leq p\\[4pt] 1, & d>p \end{cases} $$

5. Linear criterion (Type V). Parameters : s , r

$$ P(d)=\begin{cases} 0, & d\leq s\\[6pt] \dfrac{d-s}{r-s}, & s < d \leq r\\[10pt] 1, & d>r \end{cases} $$

6. Gaussian criterion (Type VI). Parameter : u

$$ P(d)=\begin{cases} 0, & d\leq0\\[6pt] 1-e^{-\frac{d^2}{2u^2}}, & d>0 \end{cases} $$

Usual criterion (Type I)

Type I — No threshold

P (d )=0 if d ≤ 0 ; P (d )=1 if d > 0

Quasi-criterion (Type II)

Type II — Indifference threshold

P (d )=0 if d ≤ l ; P (d )=1 if d >l

V-shape criterion (Type III)

Type III — Preference threshold

P (d )=d /m if 0 <d ≤ m ; 1 if d > m

Level criterion (Type IV)

Type IV — Indifference & preference thresholds

0 if d ≤ q ; ½ if q < d ≤p ; 1 if d >p

Linear criterion (Type V)

Type V — Linear ramp after indifference

0 if d ≤s ; (d −s )/(r −s ) if s < d ≤ r ; 1 if d > r

Gaussian criterion (Type VI)

Type VI — Smooth S-curve

P (d )=1 − exp (−d ² / 2u ²) for d > 0

Calculate the Weak Preference Index :

$$ \pi(A_i,A_k)=\sum_{j=1}^{n} w_jP_j(A_i,A_k) $$

where $w_j$ is the weight of criterion $C_j$.

Using the Usual criterion (Type I) preference function. For cost criteria, the preference is reversed: $P=1$ when $d<0$ (lower cost is better).

$\pi(A_i,A_k)$	$A_1$	$A_2$	$A_3$
$A_1$	—	0.2009	0.0000
$A_2$	0.7991	—	0.0000
$A_3$	1.0000	0.9288	—

Entering and leaving flows

Formula

The leaving flow measures how strongly alternative $A_i$ prefers the other alternatives.

$$ \phi^{+}(A_i)=\frac{1}{m-1}\sum_{\substack{k=1\\k\neq i}}^{m}\pi(A_i,A_k) $$

A larger leaving flow is better.

The entering flow measures how strongly the other alternatives prefer $A_i$.

$$ \phi^{-}(A_i)=\frac{1}{m-1}\sum_{\substack{k=1\\k\neq i}}^{m}\pi(A_k,A_i) $$

A smaller entering flow is better.

Alternative	Leaving Flow $\phi^{+}$	Entering Flow $\phi^{-}$
$A_1$	0.1005	0.8996
$A_2$	0.3996	0.5649
$A_3$	0.9644	0.0000

Ranking PROMETHEE I

Formula

The priority of alternatives to each other is determined based on the leaving and entering flows.

$\,P\,$ Preference : $A_i\,P\,A_k$ means $A_i$ is better than $A_k$.

\[ \begin{aligned} P \quad if \quad \begin{cases}\phi^{+}(A_i)>\phi^{+}(A_k)\quad and \quad \phi^{-}(A_i)<\phi^{-}(A_k)\\\phi^{+}(A_i)=\phi^{+}(A_k)\quad and \quad \phi^{-}(A_i)<\phi^{-}(A_k)\\\phi^{+}(A_i)>\phi^{+}(A_k)\quad and \quad \phi^{-}(A_i)=\phi^{-}(A_k)\end{cases}\end{aligned} \]

$\,I\,$ Indifference : $A_i\,I\,A_k$ means means neither alternative is preferred over the other.

\[ I \quad if \quad \begin{cases}\phi^{+}(A_i)=\phi^{+}(A_k)\quad and \quad \phi^{-}(A_i)=\phi^{-}(A_k)\end{cases} \]

$\,R\,$ Incomparability : $A_i\,R\,A_k$ means no clear preference can be established.

\[ \begin{aligned} R \quad if \quad \begin{cases}\phi^{+}(A_i)>\phi^{+}(A_k)\quad and \quad \phi^{-}(A_i)>\phi^{-}(A_k)\\\phi^{+}(A_i)<\phi^{+}(A_k)\quad and \quad \phi^{-}(A_i)<\phi^{-}(A_k)\end{cases}\end{aligned} \]

Pair	Condition Check	Relation
$A_2$ vs $A_1$	$P$	$A_2 \succ A_1$
$A_3$ vs $A_1$	$P$	$A_3 \succ A_1$
$A_3$ vs $A_2$	$P$	$A_3 \succ A_2$
Reverse pairs	—	—

PROMETHEE I Final ranking: $A_3 \succ A_2 \succ A_1$

Ranking PROMETHEE II

Formula

Calculate the net flow:

The net flow combines the leaving and entering flows.

$$ \phi(A_i)=\phi^{+}(A_i)-\phi^{-}(A_i) $$

A larger $\phi(A_i)$ indicates a better alternative.

Ranking:

Sort the alternatives according to their net flow values.

$$ \phi(A_i)>\phi(A_k)\quad\Longrightarrow\quad A_i >A_k $$

Alternative	Net Flow $\phi(A_i)$
$A_1$	−0.7991
$A_2$	−0.1653
$A_3$	0.9644

Net flows: $\phi(A_3)=0.9644 >\phi(A_2)=-0.1653 >\phi(A_1)=-0.7991$
PROMETHEE II Final ranking: $A_3 \succ A_2 \succ A_1$

POMETHEE Algorithm Flowchart

START

Form the decision matrix

Preference index

Entering and leaving flows

Ranking PROMETHEE I

Ranking PROMETHEE II

END