Returns to scale and most productive scale size in DEA with negative data

• A new directional DEA model is developed in presence of negative data.
• This model is used to determine level efficiency, MPSS and RTS.
• Our static model is adapted to dynamic technology for estimating GE and RTG.
• Two examples are presented for the illustration of our proposed approach.

Non-parametric evaluation of returns to scale of production units in standard DEA models becomes problematic when their underlying technologies involve negative data. The methodology recently offered by Allahyar and Rostamy-Malkhalifeh (2015) (hence after called ARM model) is of some help to deal with this issue. However, there are two shortcomings underlying the ARM model. First, it may not be capable of locating all the production units exhibiting constant returns to scale; and second, it is also not able to determine most productive scale size. In order to deal with these two shortcomings, the current paper contributes to the DEA literature in two ways. First, it makes a unifying attempt to propose a general non-radial DEA model to determine both the most productive scale size and the returns to scale characterizations of production units in the presence of negative data. Second, the proposed model can be adapted in a dynamic DEA technology setting to determine growth efficiency and returns to growth behavior of production units facing hyper competition in a new economy.