This research explores the minimum distance inefficiency measure for the Data Envelopment Analysis (DEA) model. A critical issue is that this measure does not satisfy monotonicity, i.e., the measure may provide a better evaluation score to an inferior decision making unit (DMU) than to a superior one. To overcome this, a variant called the extended facet approach has been introduced. This approach, however, requires a certain regularity condition to be met. We discuss several special classes of the DEA model, and show that for these models, the minimum distance inefficiency measure satisfies the monotonicity property without the regularity condition. Moreover, we conducted computational experiments using real-world data sets from these special classes, and demonstrated that the extended facet approach may overestimate the performance of a DMU.