A Ratio-Based Efficiency Measurement for Ranking Multi-Stage Production Systems in DEA

Document Type: Original Manuscript


1 PhD, Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

2 Associate Professor, Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.



Conventional data envelopment analysis (DEA) models are used to measure efficiency score of production systems when they are considered as black boxes and their internal relationship is ignored. This paper deals with a common special case of network systems which is called multi-stage production system and can be generalized to many organizations. A multi stage production system has some stages in which the outputs of each stage are used as the inputs of the next stage to produce the final outputs of the system. Most of the approaches handling multi-stage systems in DEA evaluate efficiency measure of a production system considering the interrelationship between its stages; however, they do not present their ranking or impact of each stage in ranking of a special multi-stage system through comparison with the others. In this paper, considering the series internal structure of the multi-stage systems and their efficiency measure, we propose some new ratio-based DEA models to determine the best and the worst rank of the multi-stage systems over all sets of feasible weights. In order to improve the performance of the whole system, the proposed models are used to recognize the stages with the most important role in the system’s inefficiency. Some numerical examples are presented to illustrate the approach.


  • For the first time, we determine the best and the worst ranks of DMUs with multi-stage structure.
  • We determine the stages which have more effect in inefficiency of the corresponding system in comparison with another system.
  • A numerical example shows the applicability and efficiency of the proposed model in agriculture industry.