Outranking methods: ELECTRE and PROMETHEE

From my doctoral course

This note comes from the doctoral course “Multi-Criteria Decision Making”. I rewrote the explanations in my own words and recomputed the numerical examples. Main source of the course: Asgharpour, Multi-Criteria Decision Making (University of Tehran).

How do they differ from the rest?

Instead of building one overall score, outranking methods compare alternatives two by two and ask: “does $A$ outrank $B$?” Outranking means we have enough reasons to regard $A$ as at least as good as $B$. The result can be a partial ranking: sometimes two alternatives are not comparable, which is more honest than a definite number. For this reason they are also known as the “French school”.

ELECTRE I (Roy, 1968)

For each pair $(a,b)$ there are two indices:

• Concordance index $C(a,b)$: the sum of the weights of the attributes on which $a$ is at least as good as $b$. (How much do we agree?)
• Discordance index $D(a,b)$: the largest relative difference on attributes where $b$ is better than $a$. (How strong is the worst objection?)

$a$ outranks $b$ if $C(a,b)\ge \bar{c}$ and $D(a,b)\le \bar{d}$ (the thresholds are given by the decision maker, say 0.6 and 0.4). This is a partial veto: even with many in favour, one attribute with a sharp difference can reject the outranking.

At the end the outranking graph is drawn and the kernel (alternatives not outranked by any other) is the answer set. Later versions: ELECTRE II to IV, III and TRI.

PROMETHEE (Brans, 1982)

For each attribute $j$ and each pair $(a,b)$, a preference function $P_j(a,b)\in [0,1]$ measures the intensity of $a$‘s superiority from the difference of values. There are six types of function: usual, U-shape (quasi-criterion), V-shape (linear), level, linear with an indifference area, and Gaussian; each has thresholds ($q$ for indifference, $p$ for preference).

Steps:

1. Differences of values on each attribute.
2. Compute $P_j(a,b)$ with the preference function.
3. Overall preference index: $\pi (a,b)={\sum }_j{w}_j{P}_j(a,b)$.
4. Positive flow ${\varphi }^{+}(a)=\frac{1}{m-1}{\sum }_b\pi (a,b)$ (the strength of a), negative flow ${\varphi }^{-}(a)=\frac{1}{m-1}{\sum }_b\pi (b,a)$ (the weakness of a), net flow $\varphi (a)={\varphi }^{+}-{\varphi }^{-}$.
5. PROMETHEE I: a partial ranking from comparing ${\varphi }^{+}$ and ${\varphi }^{-}$; PROMETHEE II: a complete ranking from the net $\varphi$.

Example: the same four laptops

Weights $(0.40,0.35,0.25)$ and both methods:

ELECTRE I (concordance threshold 0.6, discordance threshold 0.4): the outrankings that hold are A over C and B over D. “A” and “B” are not comparable with each other (A is cheaper and B stronger), and the “kernel” contains A and B.

PROMETHEE II with the “usual” function:

	${\varphi }^{+}$	${\varphi }^{-}$	net $\varphi$	Rank
A	0.467	0.533	−0.067	3
B	0.617	0.383	0.233	1
C	0.400	0.600	−0.200	4
D	0.517	0.483	0.033	2

Here “B” comes first, while SAW had made “A” the winner. Why? Because PROMETHEE with the usual function ignores the magnitude of a difference and counts only “being better”; “B” is ahead of “A” on two attributes, however slightly. An important reminder: the choice of method changes the ranking and none is “more right”; they only hold different assumptions about value.

Software

Visual PROMETHEE and Decision Lab have ready software for PROMETHEE and ELECTRE; useful since they also give sensitivity analysis and the GAIA plot.

Critique

The main advantage: partial ranking and veto (a severe weakness is taken as uncompensable). The drawback: setting thresholds and choosing a preference function is subjective, and the result is harder to explain to a manager.

Related: Compromise methods: TOPSIS and VIKOR and Non-compensatory methods.

ELECTREPROMETHEEmulti_criteria_decision_making