some conditions for characterizing minimum face in non-radial dea models with undesirable outputs

the problem of utilizing undesirable (bad) outputs in dea models often need replacing the assumption of free disposability of outputs by weak disposability of outputs. the kuosmanen technology is the only correct representation of the fully convex technology exhibiting weak disposability of bad and good outputs. also, there are some specific features of non-radial data envelopment analysis (dea) models for obtaining all projections of a decision making unit (dmu) on the boundary of production possibility set (pps) or efficient frontier. production technologies in dea are modeled by polyhedral sets that envelop the observed dmus. because the efficient frontiers of dea technologies are generally non-smooth and are characterized by different faces, thus, all projections of a dmu on efficient frontier can not belong to different faces that do not have common points. the rationale behind abovementioned statement is as follows: if all projections of a dmu belong to different faces then the interior points of pps will become efficient that contradicts the principles of optimality conditions in linear programming models. therefore all projections would belong to a unique face that is called minimum face. in this paper we propose a procedure to find minimum face and so all projections of a dmu on efficient frontiers in non-radial dea models with undesirable outputs. this leads us to an interesting algorithm to obtain minimum face.