Malmquist and Hicks–Moorsteen Productivity Indexes for Clusters Performance Evaluation

Measuring the performance of clusters characterized by the unbalancedeness and units with no correspondence in other clusters (“uncorrespondencedeness”) has not achieved the desired attention in the literature. Particularly, the operational research has been almost exclusively focused on performance evolution over time, where clusters are generally balanced and the units repeat themselves over these groups. Such analysis has been based on the Malmquist and the Hicks–Moorsteen indexes (MI and HMI), which are solely based on Shephard’s radial distance functions and do not account for all inefficiency sources. Making use of the so-called geometric distance functions (GDFs) and the GDF-based MI, we propose a generalization of the Hicks–Moorsteen index (HMI), based on targets instead of distances to the efficient frontier, allowing the introduction of all inefficiency sources in the productivity model. Moreover, we propose a Monte Carlo-based framework to achieve the pseudo-corresponding units for general cluster performance analysis. This framework is then a generalization of the conventional performance evolution over time. Then, we show that the HMI can be decomposed into economically meaningful indexes and can be rewritten as the geometric mean of the input and the output-oriented MIs. Given these conclusions and our proposed framework, the employment of the HMI to the general clusters analysis is straightforward. Other economically meaningful conclusions are also obtained in this paper.