On the sigma-mu stochastic multicriteria analysis: Exact solutions for common particular cases
本文为sigma-mu方法推导出精确公式，无需蒙特卡洛模拟即可计算均值和标准差，适用于加权和、多属性价值函数等加性模型及无约束、排序或下限权重分布，对多准则决策和复合指标研究者有用。
• Closed-form formulas to obtain exact sigma and mu values for the sigma-mu approach. • Avoids the need for simulations in common cases for MCDA and composite indicators. • Addresses unconstrained, rank-ordered, or lower-bounded weights. • Addresses additive models (MAVT, PROMETHEE, OWA, etc.). 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.