On the sigma-mu stochastic multicriteria analysis: Exact solutions for common particular cases

The sigma-mu approach is one of the recent innovations in the field of multicriteria decision aiding and composite indicators, extending the stochastic multicriteria acceptability analysis (SMAA) toolbox. The initial stage of this method involves computing the mean (mu) and standard deviation (sigma) for the composite value of the units under evaluation, considering a stochastic distribution on a set of admissible weights. This work develops closed-form formulas to obtain exact values for mu and sigma without needing approximations via Monte-Carlo simulations, which can be applied in some cases that are quite common. In terms of aggregation, these cases are characterized by an additive model, such as a weighted sum, a multiattribute value function, or PROMETHEE II. In terms of stochastic distributions, these cases include uniformly distributed unconstrained vectors of weights, rank-ordered vectors of weights, or lower-bounded weights. The developed formulas are applied to a didactic example and some open problems for future research are suggested.