A modified slacks-based measure model for data envelopment analysis with 'natural' negative outputs and inputs

This paper is concerned with Data Envelopment Analysis of systems with Natural Negative Inputs, and Natural Negative Outputs, i.e. inputs and outputs that, by their nature less than zero. Examples of situations with only positive inputs and positive and natural negative outputs are given and of situations in which both natural negative inputs and natural negative outputs occur. More attention has been paid to the former type of problem, in the literature, however, most available DEA software does not solve either type of problem. An exception to this is the DEA-Solver package (Cooper et al, 2000). However, most of the models of that package have the drawback that they cope with negative outputs and possibly negative inputs by assigning zero weights to them. The Slacks-Based Measure (SBM) model proposed by Cooper et al (2000) appears to be an attractive way to overcome this problem. It is shown that that model copes with the situation of natural negative outputs only without modification. Once natural negative inputs are incorporated, however, the SBM model no longer applies directly since efficiencies are no longer confined to the region (0,1). Two approaches are suggested to overcome this difficulty. The first generates a modified efficiency measure based on the solution to an Additive DEA model. The second replaces the Efficiency measure by a Cost Benefit Ratio confined to the region (1,0). This can be solved by a fractional planning approach. The solution to this fractional programming problem is equivalent to regarding negative outputs as pseudo-inputs and negative inputs as pseudo-outputs.