Defining a new graph inefficiency measure for the proportional directional distance function and introducing a new Malmquist productivity index

• Published in: European journal of operational research Vol. 281; no. 1; pp. 222 - 230
• Main Authors: Pastor, Jesus T., Lovell, C.A. Knox, Aparicio, Juan
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 16-02-2020
• Subjects: Data envelopment analysis; Efficiency measure; Malmquist productivity index; Proportional directional distance function; Proportional directional distance function; Efficiency measure; Malmquist productivity index; Data envelopment analysis
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2019.08.021

•An efficiency measure for the proportional directional distance function is derived.•It is related to the output-oriented radial efficiency measurement model.•The standard Malmquist is written as the ratio of output and input quantity indices.
A natural multiplicative efficiency measure for the Constant Returns to Scale proportional directional distance function (pDDF) is derived, relating its associated linear program to that of the well-known output-oriented radial efficiency measurement model. Based on this relationship, a traditional CCD (Caves, Christensen and Diewert) Malmquist index is introduced to show that, when it is based on the new efficiency measure associated with the pDDF, rather than on a radial efficiency measure associated with an oriented distance function, it becomes a Total Factor Productivity (TFP) index. This constitutes a new result, because heretofore the traditional CCD Malmquist index has not been considered a TFP index. Additionally, a new decomposition of the CCD Malmquist index is proposed that expresses productivity change as the ratio of two components, productivity change due to output change in the numerator and productivity change due to input change in the denominator. In an Appendix the efficiency measure is extended to include any returns to scale pDDF.