Directional Distance Functions in DEA with Optimal Endogenous Directions

This paper is concerned with optimal directions in the directional distance function in data envelopment analysis. It is shown that the vector pointing in the direction that minimizes the Euclidean distance between the input–output vector ( X o , Y o ) and the efficient frontier, the input isoquant reflecting output Y o , or the output isoquant reflecting input X o is optimal, because the corresponding vector of virtual multipliers defines the relative prices that maximize profit, cost, or revenue efficiency. The associated efficiency indicator is a value measure of technical efficiency in difference form with the Euclidean distance between ( X o , Y o ) and the efficient frontier, the Y o input isoquant, or the X o output isoquant as an equivalent directional distance quantity indicator. A linear combinatorial optimization program for computing the relevant value indicators of technical efficiency in multiplier space or the equivalent quantity indicators in terms of the relevant Euclidean distances is developed. A nonlinear and nonconvex optimization model for an estimation of the relevant value indicator of efficiency in multiplier space is also developed. Preliminary computational results are reported.