PROMETHEE

This is a method from the family of outranking (i.e. based on pair-wise comparison) MCDA methods.

PROMETHEE stands for Preference Ranking Organization METHod for Enrichment Evaluation

Notation:

  • $A = \{a_1, ..., a_n\}$ - set of alternatives
  • $F = \{f_1, ..., f_q\}$ - set of criteria (assume we maximize everything)

Note that since we perform pair-wise comparisons, it may become computationally very expensive


There are four steps:

  1. Compute Uni-Criterion Preferences
  2. Compute Preference Matrix
  3. Compute Net-Flow Scores
  4. Rankings
    • Complete: PROMETHEE II
    • Partial: PROMETHEE I


Parameters for a problem (a decision maker needs to specify them)

  • the type of preference function $P_k$ used for each criterion $f_k$
  • weight $w_k$ for each criterion $f_k$


Uni-Criterion Preferences

we calculate the differences in all criteria for all pairs of alternatives:

  • $\forall a_i, a_j \in A: d_k(a_i, a_j) = f_k(a_i) - f_k(a_j)$
  • since we assume that we maximize everything, the bigger the difference, the better the first alternative is

The resulting value should be normalized:

  • we apply a (pre-defined for each criteria $f_k$) preference function $P_k$ to each difference
  • $\pi_k = P_k \big[ d_k(a_i, a_j) \big]$
  • this transforms the difference into the preference degree

Preference Functions

There are 6 preference functions:

promethee-pref-functions.png

Basic Shapes

  • (a) usual preference function
    • as soon as see any difference, the preference is 1
    • no need to specify any parameters
    • this function is very sensitive
  • (b) $u$-shape function
    • difference needs to exceed a certain threshold $q$
    • (a) and (b) are typically used for qualitative scales
  • (c) $v$-shape
    • with strong preference threshold $p$
  • (d) level
  • (e) linear
    • same as (c) but with sensitivity threshold $q$
    • promethee-linear-pref.png
  • (f) $v$-shape

Choosing a shape is already a decision


Preference Matrix

At this step we compute global preference degrees

  • that is, for each pair $a_i, a_j$ compute:
  • $\pi(a_i, a_j) = \sum_{k=1}^q w_k \cdot \pi_k (a_i, a_j)$
  • global preference degree is a weighted sum of uni-criteria preferences


Consequences:

  • $\pi(a_i, a_i) = 0$
    • the preference of element $a$ with itself is always 0
    • $\pi(a_i, a_i) = \sum_{k=1}^q w_k \cdot \pi_k (a_i, a_i) = \sum_{k=1}^q w_k \cdot P_k \big[ \underbrace{d_k(a_i, a_i)}_{= \ 0} \big] = 0$
  • $\pi(a_i, a_j) \geqslant 0$
    • the preference degree is always positive
    • $\pi(a_i, a_j)$ is a weighed sum of $\pi_k$ which are always positive
    • this is because the preference functions always return positive values
  • $\pi(a_i, a_j) + \pi(a_j, a_i) \leqslant 1$
    • todo


Net Flow Scores

For each $a_i, a_j \in A$ we compute net flow scores

  • $\Phi^+(a_i) = \cfrac{1}{n-1} \sum_{a_j \in A} \pi(a_i, a_j)$
    • positive outranking net flow
    • "strengths" of an alternative, how an alternative outranks others
    • want to maximize it
  • $\Phi^-(a_i) = \cfrac{1}{n-1} \sum_{a_j \in A} \pi(a_j, a_a)$
    • negatives outranking net flow
    • "weaknesses" of an alternative, how an alternative is outranked by others
    • want to minimize it

promethee-netflowscore.png

The values are normalized:

  • $\Phi^+(a_i) \in [0, 1]$ and
  • $\Phi^-(a_i) \in [0, 1]$

The flow score is computed as difference between the positive flow and negative flow

  • $\Phi(a_i) = \Phi^+(a_i) - \Phi^-(a_i) \in [-1, 1]$ and

Uni-criteria flow

  • flow computed with respect to only one criteria $f_k$
  • $\Phi_k(a) = \cfrac{1}{n-1} \sum_{b \in A} [ P_k(a, b) - P_k(b, a) ]$



Ranking

PROMETHEE I

Partial Order Ranking:

  • ranking based only on $\Phi^+(a_i)$ and $\Phi^-(a_i)$ (not on the aggregated flow score)
  • let $P^+$ denote the preference of $\Phi^+$
    • $a_i \ P^+ \ a_j \iff \Phi^+(a_i) > \Phi^+(a_j)$ (want to maximize $\Phi^+$)
  • let $P^-$ denote the preference of $\Phi^-(a_i)$
    • $a_i \ P^- \ a_j \iff \Phi^+(a_i) < \Phi^+(a_j)$ (want to maximize $\Phi^-$)
  • $a_i \ P \ a_j \iff a_i \ P^+ \ a_j \land a_i \ P^- \ a_j$
    • i.e. $a_i$ is preferred to $a_j$ only in case of Unanimity between $\Phi^+$ and $\Phi^-$
  • if there is no unanimity, $a \ J \ b$
    • $a$ is not comparable to $b$
    • in this case it's up to the decision maker to decide what to do with these alternatives


PROMETHEE II

This gives a completely ordered preference structure

  • i.e. there's no $J$ relation


Define:

  • preference as $a_i \ P \ a_j \iff \Phi(a_i) > \Phi(a_j)$
  • indifference as $a_i \ I \ a_j \iff \Phi(a_i) = \Phi(a_j)$

These relations are transitive and complete


Properties

Justification

The PROMETHEE property
  • the netflow score $\Phi(a_i)$ is a centered score $s_i$ ($\forall i$) that minimizes the following $Q$:
  • $Q = \sum_{i=1}^n \sum_{j=1}^n \big[ (s_i - s_j) - (\pi_{ij} - \pi_{ji}) \big]^2 $
  • i.e. $Q$ is the sum of squared deviation and we want to minimize it
  • proof: PROMETHEE/Properties#The PROMETHEE Property


Property 1

$\Phi(a_i) \in [-1, 1]$

Easy to see that by the way we construct $\Phi(a_i)$


Property 2

Want to show that $\sum_{i = 1}^N \Phi(a_i) = 0$


Preferential Independence

PROMETHEE respects the Preferential Independence hypothesis


Arrow's Impossibility Theorem

Recall that according to the theorem all 5 conditions cannot be satisfied at the same time.


Rank Reversal

A rank reversal happens if:

  • $\pi_{ij} \geqslant \pi_{ji}$ but in spite of that $\Phi(a_i) \leqslant \Phi(a_j)$
  • the main article: PROMETHEE/Rank Reversal


GAIA

PROMETHEE gives you scores and you can compute complete and partial rankings

  • GAIA is a tool for visualizing, complimentary to PROMETHEE
  • GAIA = Geometrical Analysis for Interactive Assistance

We can represent each alternative as a vector:

  • we can represent an alternative as a vector of $q$ components:
    • $[f_1(a_i), ..., f_q(a_i)]$
    • but in this space we have different scales - and things are not easy to compare
  • we can use the netflow score, which is a value in $[-1, 1]$
    • $\Phi(a_i) = \sum_{k=1}^{q} w_k \cdot \phi_k(a_i)$
    • where $\Phi_k(a_i) = \cfrac{1}{n-1} \sum_{a_j \ne a_i} \big[ \pi_k(a_i, a_j) - \pi_k(a_j, a_i) \big] $
  • so we can represent every alternative $a_i$ as a vector $[\Phi_1(a_i), ..., \Phi_k(a_i)]$ in $q$ dimensional space.


But $q$ dimensional space is really hard to visualize

$\delta$

  • some information is lost during the projection, and the $\delta$ coefficient shows how much information is retained on the plane
  • $\delta$ is the ratio between the projected variance and the initial variance
  • $\delta \geqslant 70\%$ is good


Example

Consider a "Buying a new car" example (from D-Sight [1])

gaia-new-car.png

  • points are alternatives
  • lines with points at the end are criteria
  • the red stick is a decision stick - the projection of weights to the plain


Based on a GAIA plane can make some observations:

  • there are conflicting criteria: ones that point in the opposite directions
  • there are also criteria that are in synergy: they point in the same direction
  • gaia-new-car2.png
  • some criteria are close to each other in 2-dim space - they also must be close in the $q$-dim space, so they should have similar profiles
  • gaia-new-car3.png
  • it is also interesting to see the position of alternatives with respect to criteria: to project each as see which one is the best, worst, etc
  • gaia-new-car4.png
  • and finally, the projection on the decision stick vector shows what are the best alternatives
  • gaia-new-car5.png


Example: Plant Location Problem

Evaluation Table

First we establish the following evaluation table:

# engineers power cost maintenance village security
Italy 75 90 600 5.4 8 5
Belgium 65 58 200 9.7 1 1
Germany 83 60 400 7.2 4 7
Sweden 40 80 1000 7.5 7 10
Austria 52 72 600 2 3 8
France 94 96 3.6 5 6
min max min min max max
  • red color - worst value in column
  • blue color - best value in column

Pair-wise Comparison

  • start with 2 alternatives and consider only 1 criteria at a time
  • for example, $G$ermany and $A$ustria:
  • cost is better in $G$: 400 vs 600
    • what does it mean?
    • need to normalize this difference to [0, 1] scale - to be able to qualify the difference
Unic. pref 0.25
-200
Germany 83 60 400 7.2 4 7
Engineers Power Cost Maint. Village Security
Austria 52 72 600 2 3 8
-31 12 -5.2 -1 1
Unic. pref 1 0.75 1 0.3 0.63


Global Preference Degree

Next step: compute global preference degree for all pairs of alternatives

  • assuming a decision maker provides the weights
  • eng: 0.1, pow: 0.2, cost: 0.2, maint: 0.1, village: 0.15, security: 0.15 (sum = 1)
  • $\pi(A, G) = 1 \cdot 0.1 + 0.75 \cdot 02 + 1 \cdot 0.1 + 0.3 \cdot 0.15 + 0.63 \cdot 0.15 = 0.498$
  • $\pi(G, A) = 0.25 \cdot 0.4 = 0.05$
  • the closest the $\Pi$ value to 1, the move preferred one alternative to another is

We do this for all pairs and build a preference matrix

Italy Belgium Germany Sweden Austria France
Italy 0 0.28 0.23 0.24 0.09 0.22
Belgium 0.27 0 0.4 0.3 0.27 0.5
Germany 0.22 0.19 0 0.3 0.05 0.43
Sweden 0.43 0.55 0.33 0 0.25 0.25
Austria 0.46 0.55 0.49 0.34 0 0.46
France 0.23 0.4 0.26 0.39 0.12 0


Net Flow Scores

Next we compute the positive and negative net flow scores:

promethee-plant-loc.png

  • positive flow: strengths, has to be maximized
  • negative flow: weaknesses, has to be minimized


Then we calculate if on average $G$ is better than the rest:

  • $\Phi^+(G) = 0.238$ (avg of all $\Pi(G, X)$)
  • $\Phi^-(G) = 0.334$ (avg of all $\Pi(X, G)$)
  • the total score: $\Phi(G) = \Phi^+(G) - \Phi^-(G) = -0.1 \in [-1, 1]$


PROMETHEE II ranking

ranking the alternatives based on their netflow scores
rank alternative $\Phi(G)$ $\Phi^+(G)$ $\Phi^-(G)$
1 Austria 0.302 0.458 0.156
2 Sweden 0.049 0.363 0.314
3 Belgium -0.041 0.347 0.397
4 Germany -0.096 0.238 0.334
5 France -0.099 0.274 0.373
6 Italy -0.115 2.11 0.326


promethee-plant-loc-barplot.png

Note that this is a complete ranking:

  • you can always say which alternative is better


PROMETHEE I ranking

$\Phi^+$ $\Phi^-$
$a$ 0.8 0.1
$b$ 0.5 0.05
$c$ 0.1 0.8

for both $\Phi^+$ and $\Phi^-$ (applying the Unanimity principle)

  • $a \ P \ c$
  • $b \ P \ c$

Cannot say anything about $a$ and $c$

  • for $\Phi^+$: $a \ P \ b$
  • for $\Phi^-$: $b \ P \ a$

promethee-2-ex.png


Sources

  • Decision Engineering (ULB)
  • An Introduction to Multicriteria Decision Aid: The PROMETHEE and GAIA Methods, Yves De Smet, Karim Lidouh, 2013
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