A non-parametric dynamic model for measuring efficiency


  • Kelly Patricia Murillo Departamento de Matemáticas, Universidade de Aveiro, Aveiro, Portugal.




Dynamic model, multidireccional efficiency analysis, optimization on data analysis


Conventional efficiency evaluation systems present little diversity in the type of variables considered. This aspect generally leads to errors in the application of the models and in the corresponding interpretation of results. This study proposes a dynamic model to measure the efficiency of decision-making units, based on non-parametric Multidirectional Efficiency Analysis (MEA). The model presented here considers a complete structure, which includes intertemporal variables (desirable/undesirable intermediate inputs and outputs), discretionary/non-discretionary inputs; desirable/undesirable inputs and outputs. Dynamic score is defined first in a particular version and then in a more general version, considering two optimization problems.

The directional contribution of each variable is examined, showing excess inputs and deficit outputs. In addition, a dynamic inefficiency index to measure the number of times each input was used inefficiently, is presented for both desirable and undesirable cases.


Charnes A, Cooper W, Rhodes E. Measuring the efficiency of decision making units. Eur. J. of Operational Research. 1978; 2(6):429–444.

Asmild M, Paradi JC, Kulkarni A. Using Data Envelopment Analysis in Software Development Productivity Measurement. Software Process Improvement and Practice. 2006; 11(6): 561–72.

Kaffash S, Marra M. Data Envelopment Analysis in Financial Services: A Citations Network Analysis of Banks, Insurance Companies and Money Market Funds. Annals of Operations Research. 2017; 253(1): 307–344.

Bhat ZUH, Sultana D, Dar QF. A Comprehensive Review of Data Envelopment Analysis (DEA) in Sports. J. of Sports Economics and Management. 2019; 9(2): 82–109.

Wen H, Lim B, Lisa H. Measuring E-commerce Efficiency: A Data Envelopment Analysis (DEA) Approach. Industrial Management and Data Systems. 2003; 103(9): 703–710.

Liu J, Lu L, Lu W, Lin B. A survey of DEA applications.Omega (United Kingdom). 2013; 41(5): 893–902.

Fare R, Grosskopf S. Intertemporal Production Frontiers: With Dynamic DEA. Kluwer Academic Publishers, Boston; 1996.

Nemoto J, Goto M. Dynamic data envelopment analysis: modeling intertemporal behavior of a firm in the presence of productive inefficiencies. Economics Letters. 1999; 64(1): 51–56.

Tone K, Tsutsui M. Dynamic DEA: a slacks-based measure approach. Omega. 2010; 38(3–4): 145–156.

Mariz F, Almeida M, Aloise D. A review of dynamic data envelopment analysis: State of the art and applications, Int. Transactions in Operational Research. 2018; 25(2): 469–505.

Bogetoft P, Hougaard J. Efficiency Evaluations Based on Potential (Non-proportional) Improvements. J. of Productivity Analysis. 1999; 12(3): 233–247.

Gongbing B, Pingchun W, Feng Y, Liang L. Energy and Environmental Efficiency of China’s Transportation Sector: A Multidirectional Analysis Approach. Mathematical Problems in Engineering. 2014; 1–12.

Wang K,Wei Y, Zhang X. Energy and Emissions Efficiency Patterns of Chinese Regions: A Multi-directional Efficiency Analysis. Applied Energy. 2013; 104: 105–116

Murillo K, Rocha E. The Portuguese Manufacturing Sector During 2013- 2016 After the Troika Austerity Measures. World J. of Applied Economics. 2018; 4(1): 21–38.

Bogetoft P, Hougaard J. Super efficiency evaluations based on potential slack, Eur. J. of Operational Research. 2004; 152(1):14–21.

Asmild M, Pastor J. Slack free MEA and RDM with comprehensive efficiency measures, Omega. 2010; 38(6): 475–483.

Banker R, Charnes A, CooperW. Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis. Management Science. 1984; 30(9): 1078–92.



How to Cite

Murillo, K. P. (2023). A non-parametric dynamic model for measuring efficiency. Selecciones Matemáticas, 10(02), 324 - 332. https://doi.org/10.17268/sel.mat.2023.02.08