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.

Luis C. Dias. On the sigma-mu stochastic multicriteria analysis: Exact solutions for common particular cases. Omega 2024, 127 .

Luis C. Dias. On the sigma-mu stochastic multicriteria analysis: Exact solutions for common particular cases. Omega. 2024; 127 ():.

Luis C. Dias. 2024. "On the sigma-mu stochastic multicriteria analysis: Exact solutions for common particular cases." Omega 127, no. : .

J.A. Ferreira; G. Pena. FDM/FEM for nonlinear convection–diffusion–reaction equations with Neumann boundary conditions—Convergence analysis for smooth and nonsmooth solutions. Journal of Computational and Applied Mathematics 2024, 446 .

J.A. Ferreira, G. Pena. FDM/FEM for nonlinear convection–diffusion–reaction equations with Neumann boundary conditions—Convergence analysis for smooth and nonsmooth solutions. Journal of Computational and Applied Mathematics. 2024; 446 ():.

J.A. Ferreira; G. Pena. 2024. "FDM/FEM for nonlinear convection–diffusion–reaction equations with Neumann boundary conditions—Convergence analysis for smooth and nonsmooth solutions." Journal of Computational and Applied Mathematics 446, no. : .

Building upon the exact methods presented in our earlier work (2022) [5], we introduce a heuristic approach for the star discrepancy subset selection problem. The heuristic gradually improves the current-best subset by replacing one of its elements at a time. While it does not necessarily return an optimal solution, we obtain promising results for all tested dimensions. For example, for moderate sizes $30\le n\le 240$, we obtain point sets in dimension 6 with $L_{\infty }$ star discrepancy up to 35% better than that of the first n points of the Sobol' sequence. Our heuristic works in all dimensions, the main limitation being the precision of the discrepancy calculation algorithms. We provide a comparison with an energy functional introduced by Steinerberger (2019) [31], showing that our heuristic performs better on all tested instances. Finally, our results give further empirical information on inverse star discrepancy conjectures.

François Clément; Carola Doerr; Luís Paquete. Heuristic approaches to obtain low-discrepancy point sets via subset selection. Journal of Complexity 2024, 83 .

François Clément, Carola Doerr, Luís Paquete. Heuristic approaches to obtain low-discrepancy point sets via subset selection. Journal of Complexity. 2024; 83 ():.

François Clément; Carola Doerr; Luís Paquete. 2024. "Heuristic approaches to obtain low-discrepancy point sets via subset selection." Journal of Complexity 83, no. : .