evaluation of decision-making units using directional distance function with weak disposability in the presence of undesirable outputs

This study evaluates decision-making units (dmus) that produce  unavoidable undesirable outputs,  which negatively affect performance. previous studies by researchers have explored   approaches for calculating the minimum undesirable outputs, presenting the abatement range for undesirable outputs and assessing dmu efficiency, yet challenges remain.  to address this, we  introduce  a new method based on the directional distance function  model with individual-proportion weak disposability. the proposed method calculates the minimum unavoidable undesirable outputs, evaluates dmu  efficiency  for both undesirable and minimum undesirable outputs, separates the effects  of desirable and undesirable outputs on inefficiency, and resolves the limitations of earlier methods. finally, the proposed method is applied to practical examples to demonstrate its superiority over existing approaches.