Efficiency Evaluation of Very Large-Scale Samples : Data Envelopment Analysis with Angle-Index Synthesis

In the era of big data, the sizes of samples to be evaluated are expanding, but this situation also hinders the use of data envelopment analysis models. Furthermore, only efficient decision-making units (DMUs) can be used to evaluate the efficiency of DMUs. Thus, finding these efficient DMUs has become a key consideration in large-scale evaluation. The parallel building hull method and the framework method offer good performance in terms of finding efficient DMUs. However, the sample sizes to be solved by these two methods are still not big enough, and both of them require a reasonable parameter design to achieve the optimal solution. In this study, we provide a new algorithm for solving the aforementioned issue. First, we propose a prescoring method for DMUs by synthesizing the angle and the index, hereafter called the angle-index synthesis method. Then, we combine this proposed method with the framework method to obtain all efficient DMUs. Numerical experiments and a real application to the bankruptcy data of Polish companies show the significant advantages of our algorithm in terms of computational time in the large-scale sampling context, and optimal parameters can be ignored. Finally, in the Monte Carlo simulation of very large-scale samples, we show that our algorithm has good validity even in 1 billion DMUs, which previous methods can hardly address under the assumption that data are normally or lognormally distributed. Results indicate an approximately linear increasing trend for the computational time required by our algorithm to obtain efficient DMUs with respect to the number of evaluated DMUs