Approximating Promethee II's net flow scores by piecewise linear value functions

Promethee II is a prominent method for multicriteria decision aid (MCDA) that builds a complete ranking on a set of potential actions by assigning each of them a so-called net flow score. However, to calculate these scores, each pair of actions has to be compared, causing the computational load to increase quadratically with the number of actions, eventually leading to prohibitive execution times for large decision problems. For some problems, however, a trade-off between the ranking’s accuracy and the required evaluation time may be acceptable. Therefore, we propose a piecewise linear model that approximates Promethee II’s net flow scores and reduces the computational complexity (with respect to the number of actions) from quadratic to linear at the cost of some misranked actions. Simulations on artificial problem instances allow us to quantify this time/quality trade-off and to provide probabilistic bounds on the problem size above which our model satisfyingly approximates Promethee II’s rankings. They show, for instance, that for decision problems as small as 10 actions evaluated on 3 criteria, our model ranks 9 actions accurately with a probability of 90%. Beyond its immediate applicability on large decision problems, our model also provides some insight into how (far) Promethee II’s outranking method is different from a much simpler weighted sum.