This page is published in English.

Outranking

EXPROM I - Extended PROMETHEE (partial)

Outranking with weak+strong preferences

Diakoulaki, D., Koumoutsos, N.1991doi:10.1016/0377-2217(91)90067-6 ↗

Overview

Apply the F.steps in order; the final score determines the descending ranking.

Output: preference flow, higher is better
Data: Crisp, complete numeric matrix
Weights: Needs a weight source
Size: 2+ alternatives, 3-12 criteria works best
Used for: Complex multi-stakeholder decisions

How it works

1
Pairwise differences d_j(A_i,A_l) and weak preference function P^w_j(d).
$d_{j} (A_{i}, A_{l}) = C_{j} (A_{i}) - C_{j} (A_{l}), P_{j}^{w} (d) \in [0, 1]$
Diakoulaki-Koumoutsos 1991, p.4 Eq.(1)
2
Weak preference index π^w(A_i,A_l) = Σ w_j P^w_j(A_i,A_l).
$π^{w} (A_{i}, A_{l}) = j = 1 \sum n P_{j}^{w} (A_{i}, A_{l}) w_{j}$
Diakoulaki-Koumoutsos 1991, p.5 Eq.(2)
3
Strong preference function P^s_j(d) capturing decisive differences.
$P_{j}^{s} (d) = f_{j}^{s} (d_{j} (A_{i}, A_{l})) \in [0, 1]$
Diakoulaki-Koumoutsos 1991, p.5 Eq.(3)
4
Strong preference index π^s(A_i,A_l) = Σ w_j P^s_j(A_i,A_l).
$π^{s} (A_{i}, A_{l}) = j = 1 \sum n P_{j}^{s} (A_{i}, A_{l}) w_{j}$
Diakoulaki-Koumoutsos 1991, p.5 Eq.(4)
5
Combined flows φ^+(A_i) and φ^-(A_i) summing weak and strong preferences.
$ϕ^{+} (A_{i}) = \frac{1}{m - 1} l \neq = i \sum [π^{w} (A_{i}, A_{l}) + π^{s} (A_{i}, A_{l})]$
Diakoulaki-Koumoutsos 1991, p.6 Eq.(5)
6
Net flow φ(A_i) = φ^+(A_i) − φ^-(A_i).
$ϕ (A_{i}) = ϕ^{+} (A_{i}) - ϕ^{-} (A_{i})$
Diakoulaki-Koumoutsos 1991, p.6 Eq.(6)
7
EXPROM I partial ranking from (φ^+, φ^-); EXPROM II full ranking from φ.
$EXPROM I: partial; EXPROM II: rank by ϕ (A_{i}) descending$
Diakoulaki-Koumoutsos 1991, p.6 Sec.4

Look elsewhere when

•Small dataset (m<3). Outranking benefit minimal.
•Compensatory preferences acceptable. Simpler ranking method.

Assumptions to verify

Decision-maker can specify preference (p), indifference (q), and veto (v) thresholds
Non-compensatory preference structure

Limitations

•Rank reversal known on alternative-set changes (ref: general MCDM literature)

Edge cases and pitfalls

Applying EXPROM-I without verifying this assumption.
Requirement: Decision-maker can specify preference (p), indifference (q), and veto (v) thresholds
Applying EXPROM-I without verifying this assumption.
Requirement: Non-compensatory preference structure
Using EXPROM-I when: Small dataset (m<3) → outranking benefit minimal.
An alternative method is recommended in this situation.
Using EXPROM-I when: Compensatory preferences acceptable → simpler ranking method.
An alternative method is recommended in this situation.

Works with

Commonly takes its weights from

AHP BWM ENTROPY CRITIC SWARA

How to cite

Diakoulaki, D.; Koumoutsos, N. (1991). Cardinal ranking of alternative actions: Extension of the PROMETHEE method. European Journal of Operational Research. https://doi.org/10.1016/0377-2217(91)90067-6

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

EXPROM-I

How it works

Pairwise differences d_j(A_i,A_l) and weak preference function P^w_j(d).

d_{j} (A_{i}, A_{l}) = C_{j} (A_{i}) - C_{j} (A_{l}), P_{j}^{w} (d) \in [0, 1]

Diakoulaki-Koumoutsos 1991, p.4 Eq.(1)

Weak preference index π^w(A_i,A_l) = Σ w_j P^w_j(A_i,A_l).

π^{w} (A_{i}, A_{l}) = j = 1 \sum n P_{j}^{w} (A_{i}, A_{l}) w_{j}

Diakoulaki-Koumoutsos 1991, p.5 Eq.(2)

Strong preference function P^s_j(d) capturing decisive differences.

P_{j}^{s} (d) = f_{j}^{s} (d_{j} (A_{i}, A_{l})) \in [0, 1]

Diakoulaki-Koumoutsos 1991, p.5 Eq.(3)

Strong preference index π^s(A_i,A_l) = Σ w_j P^s_j(A_i,A_l).

π^{s} (A_{i}, A_{l}) = j = 1 \sum n P_{j}^{s} (A_{i}, A_{l}) w_{j}

Diakoulaki-Koumoutsos 1991, p.5 Eq.(4)

Combined flows φ^+(A_i) and φ^-(A_i) summing weak and strong preferences.

ϕ^{+} (A_{i}) = \frac{1}{m - 1} l \neq = i \sum [π^{w} (A_{i}, A_{l}) + π^{s} (A_{i}, A_{l})]

Diakoulaki-Koumoutsos 1991, p.6 Eq.(5)

Net flow φ(A_i) = φ^+(A_i) − φ^-(A_i).

ϕ (A_{i}) = ϕ^{+} (A_{i}) - ϕ^{-} (A_{i})

Diakoulaki-Koumoutsos 1991, p.6 Eq.(6)

EXPROM I partial ranking from (φ^+, φ^-); EXPROM II full ranking from φ.

EXPROM I: partial; EXPROM II: rank by ϕ (A_{i}) descending

Diakoulaki-Koumoutsos 1991, p.6 Sec.4

Edge cases and pitfalls

Applying EXPROM-I without verifying this assumption.

Requirement: Decision-maker can specify preference (p), indifference (q), and veto (v) thresholds

Applying EXPROM-I without verifying this assumption.

Requirement: Non-compensatory preference structure

Using EXPROM-I when: Small dataset (m<3) → outranking benefit minimal.

An alternative method is recommended in this situation.

Using EXPROM-I when: Compensatory preferences acceptable → simpler ranking method.

An alternative method is recommended in this situation.