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Ranking

TODIM - TOmada de Decisão Interativa e Multicritério (Interactive and Multicriteria Decision Making)

Prospect theory (loss aversion pairwise dominance)

Gomes, L. F. A. M., Lima, M. M. P. P.1992

Overview

ξ_i ∈ [0,1]. Higher ξ means greater overall dominance. TODIM is grounded in Prospect Theory - gains and losses are not symmetric: a loss of magnitude x is felt more intensely than a gain of the same magnitude (controlled by θ). θ=1 means standard (symmetric) case. θ>1 amplifies loss aversion; θ→0 reduces it. The reference criterion r is the one with the highest weight.

Output: utility, higher is better
Data: Crisp, complete numeric matrix
Weights: Needs a weight source
Size: 2+ alternatives, 3-12 criteria works best
Used for: Alternative selection, Supplier evaluation

How it works

1
Linear-sum normalisation per criterion.
$p_{ij} = \frac{x _{ij}}{\sum _{k = 1}^{m} x _{k j}}$
Gomes-Lima 1992, p.116 Eq.(1)
2
Reference criterion (max weight) and relative weights w_jr = w_j / w_r.
$r = ar g j max w_{j}, w_{j r} = \frac{w _{j}}{w _{r}}$
Gomes-Lima 1992, p.116 Eq.(2)
3
Pairwise dominance Φ_j(A_i,A_k) with prospect-theory loss aversion θ.
$Φ_{j} (A_{i}, A_{k}) = ⎩ ⎨ ⎧ \frac{w _{j r} ( p _{ij} - p _{k j} )}{\sum w _{k r}} 0 - \frac{1}{θ} \frac{( \sum w _{k r} ) ( p _{k j} - p _{ij} )}{w _{j r}} p_{ij} > p_{k j} p_{ij} = p_{k j} p_{ij} < p_{k j}$
Gomes-Lima 1992, p.117 Eq.(3)
4
Aggregated dominance δ(A_i,A_k) = Σ_j Φ_j.
$δ (A_{i}, A_{k}) = j = 1 \sum n Φ_{j} (A_{i}, A_{k})$
Gomes-Lima 1992, p.117 Eq.(4)
5
Global value ξ_i normalised in [0,1] and descending ranking.
$ξ_{i} = \frac{\sum _{k} δ ( A _{i} , A _{k} ) - min _{i^{'}} \sum _{k} δ ( A _{i^{'}} , A _{k} )}{max _{i^{'}} \sum _{k} δ ( A _{i^{'}} , A _{k} ) - min _{i^{'}} \sum _{k} δ ( A _{i^{'}} , A _{k} )}$
Gomes-Lima 1992, p.117 Eq.(5)

Look elsewhere when

•Criteria strongly correlated. Consider DEMATEL/ANP for interdependence.
•Non-compensatory preferences. Consider outranking (ELECTRE/PROMETHEE).

Assumptions to verify

Criteria preferences are independent (no synergistic interactions)
Compensation is acceptable: high score on one criterion can offset low on another
Decision matrix is complete (no missing values)

Edge cases and pitfalls

θ sensitivity: rankings can reverse as θ varies; always run sensitivity on θ.
Cost criterion direction: the gain/loss formula must be inverted for cost criteria (higher x_ic is worse).

Works with

Commonly takes its weights from

AHP BWM ENTROPY CRITIC SWARA

How to cite

Gomes, L. F. A. M.; Lima, M. M. P. P. (1992). TODIM: Basics and application to multicriteria ranking of projects with environmental impacts. Foundations of Computing and Decision Sciences.

BibTeX RIS APA (.txt)

System ID, as it appears in reports and the API

TODIM

Overview

Output

utility, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Alternative selection, Supplier evaluation

How it works

Linear-sum normalisation per criterion.

p_{ij} = \frac{x _{ij}}{\sum _{k = 1}^{m} x _{k j}}

Gomes-Lima 1992, p.116 Eq.(1)

Reference criterion (max weight) and relative weights w_jr = w_j / w_r.

r = ar g j max w_{j}, w_{j r} = \frac{w _{j}}{w _{r}}

Gomes-Lima 1992, p.116 Eq.(2)

Pairwise dominance Φ_j(A_i,A_k) with prospect-theory loss aversion θ.

Φ_{j} (A_{i}, A_{k}) = ⎩ ⎨ ⎧ \frac{w _{j r} ( p _{ij} - p _{k j} )}{\sum w _{k r}} 0 - \frac{1}{θ} \frac{( \sum w _{k r} ) ( p _{k j} - p _{ij} )}{w _{j r}} p_{ij} > p_{k j} p_{ij} = p_{k j} p_{ij} < p_{k j}

Gomes-Lima 1992, p.117 Eq.(3)

Aggregated dominance δ(A_i,A_k) = Σ_j Φ_j.

δ (A_{i}, A_{k}) = j = 1 \sum n Φ_{j} (A_{i}, A_{k})

Gomes-Lima 1992, p.117 Eq.(4)

Global value ξ_i normalised in [0,1] and descending ranking.

ξ_{i} = \frac{\sum _{k} δ ( A _{i} , A _{k} ) - min _{i^{'}} \sum _{k} δ ( A _{i^{'}} , A _{k} )}{max _{i^{'}} \sum _{k} δ ( A _{i^{'}} , A _{k} ) - min _{i^{'}} \sum _{k} δ ( A _{i^{'}} , A _{k} )}

Gomes-Lima 1992, p.117 Eq.(5)