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

Promethee II is a prominent method for multi-criteria 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 wrongly ranked 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 of 10,000 actions evaluated on 7 criteria, the Pearson correlation coefficient between the original scores and our approximation is of at least 0.97. When put in balance with computation times that are more than 7000 times faster than for the Promethee II model, the proposed approximation model represents an interesting alternative for large problem instances.