Sigma-Mu efficiency analysis: A methodology for evaluating units through composite indicators

We propose a methodology to employ composite indicators for performance analysis of units of interest using and extending the family of Stochastic Multiattribute Acceptability Analysis. We start evaluating each unit by means of weighted sums of their elementary indicators in the whole set of admissible weights. For each unit, we compute the mean, μ, and the standard deviation, σ, of its evaluations. Clearly, the former has to be maximized, while the latter has to be minimized as it denotes instability in the evaluations with respect to the variability of weights. We consider a unit to be Pareto–Koopmans efficient with respect to μ and σ if there is no convex combination of μ and σ of the rest of the units with a value of μ that is not smaller, and a value of σ that is not greater, with at least one strict inequality. The set of all Pareto–Koopmans efficient units constitutes the first Pareto–Koopmans frontier. In the spirit of context-dependent Data Envelopment Analysis, we assign each unit to one of the sequence of Pareto–Koopmans frontiers. We measure the local efficiency of each unit with respect to each frontier, but also its global efficiency taking into account all frontiers in the σ−μ plane, thus enhancing the explicative power of the proposed approach. To illustrate its potential, we present a case study of ‘world happiness’ based on the data of the homonymous report that is annually produced by the United Nations’ Sustainable Development Solutions Network.