Journal of Optimization Theory and Applications

, Volume 166, Issue 3, pp 1002–1028 | Cite as

How to Deal with Non-Convex Frontiers in Data Envelopment Analysis

  • K. Tone
  • M. Tsutsui


In data envelopment analysis, we are often puzzled by the large difference between the constant-returns-scale and variable returns-to-scale scores, and by the convexity production set syndrome in spite of the S-shaped curve, often observed in many real data sets. In this paper, we propose a solution to these problems. Initially, we evaluate the constant-returns-scale and variable returns-to-scale scores for all decision-making units by means of conventional methods. We obtain the scale-efficiency for each decision-making unit. Using the scale-efficiency, we decompose the constant-returns-scale slacks for each decision-making unit into scale-independent and scale-dependent parts. Following this, we eliminate scale-dependent slacks from the data set, and thus obtain a scale-independent data set. Next, we classify decision-making units into several clusters, depending either on the degree of scale-efficiency or on some other predetermined characteristics. We evaluate slacks of scale-independent decision-making units within the same cluster using the constant-returns-scale model, and obtain the in-cluster slacks. By summing the scale-dependent and the in-cluster slacks, we define the total slacks for each decision-making unit. Following this, we evaluate the efficiency score of the decision-making unit and project it onto the efficient frontiers, which are no longer guaranteed to be convex and are usually non-convex. Finally, we define the scale-dependent data set by which we can find the scale elasticity of each decision-making unit. We apply this model to a data set of Japanese universities’ research activities.


Data envelopment analysis S-shaped curve Constant returns-to-scale Variable returns-to-scale Scale elasticity 

Mathematic Subject Classification 

90C05 90B50 91B06 91B38 



We are grateful to two reviewers for their comments and suggestions on the previous version of the manuscript. This research was supported by JSPS KAKENHI Grant Number 25282090.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.National Graduate Institute for Policy StudiesTokyoJapan
  2. 2.Central Research Institute of Electric Power IndustryTokyoJapan

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