Conventional data envelopment analysis (DEA) models require that inputs and outputs are given as crisp values. Very often, however, some of inputs and outputs are given as imprecise data where they are only known to lie within bounded intervals. While a typical approach to addressing this situation for optimization models such as DEA is to conduct sensitivity analysis, it provides only a limited ex-post measure against the data imprecision. Robust optimization provides a more effective ex-ante measure where the data imprecision is directly incorporated into the model. This study aims to apply robust optimization approach to DEA models with imprecise data. Based upon a recently developed robust optimization framework which allows a flexible adjustment of the level of conservatism, we propose two robust optimization DEA model formulations with imprecise data; multiplier and envelopment models. We demonstrate that the two models consider different risks regarding imprecise efficiency scores, and that the existing DEA models with imprecise data are special cases of the proposed models. We show that the robust optimization for the multiplier DEA model considers the risk that estimated efficiency scores exceed true values, while the one for the envelopment DEA model deals with the risk that estimated efficiency scores fall short of true values. We also show that efficiency scores stratified in terms of probabilistic bounds of constraint violations can be obtained from the proposed models. We finally illustrate the proposed approach using a sample data set and show how the results can be used for ranking DMUs.

When it comes to designing a product or a process, the robust parameter design (RPD) methodology devised by Taguchi is recognized as a way to produce products or processes with less variability, while the specifications of products or processes are met, so that the ratio of nonconforming products can be as minimized as possible. Nevertheless, as a matter of fact, there have been many pros and cons concerning the RPD method, mainly because of the use of signal to noise ratio as the measure of quality characteristic. Meanwhile, the variability analysis method has been highly recommended as an alternative of the RPD in the literature. In this paper, it is demonstrated that RPD can also be implemented within the framework of the variability analysis, which is sounder in the statistical sense. In light of an example of RPD approach, the modeling and estimation procedure is discussed in some detail with a view to a comparison of the two methods.