Data envelopment analysis with a single decision making unit: A graphic projected-gradient approach

Data Envelopment Analysis (DEA) gradient lines are constructed in a two-dimensional section of the production set, for the case of constant returns to scale. Differential equations are obtained and solved. The gradient lines – showing the optimal direction of partial improvement in efficiency – are derived and are represented graphically. It is shown that for proportional change of inputs and outputs: (i) both input minimization and output maximization problems have one and the same set of gradient lines, (ii) gradient lines are invariant with respect to weights of inputs and outputs, and (iii) if inputs and outputs are measured in relative units, the gradient line starting from an actual input–output point may be derived without solution of the optimizing DEA problem, so that it becomes possible to carry out a partial DEA analysis with only a single Decision Making Unit. The formula for the optimal proportional change of inputs and outputs is obtained. A numerical example is provided.