Data envelopment analysis with a single DMU: 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 inputoutput 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 (DMU). The formula for the optimal proportional change of inputs and outputs is obtained. A numerical example is provided.
data envelopment analysis decision support systems optimal behavior