# American Institute of Mathematical Sciences

January  2017, 13(1): 237-249. doi: 10.3934/jimo.2016014

## Efficiency measures in fuzzy data envelopment analysis with common weights

 1 Department of International Business, Kao Yuan University, Kaohsiung, 82151, Taiwan 2 Department of Mechanical and Automation Engineering, Ⅰ-Shou University, Kaohsiung, 84001, Taiwan 3 Department of Applied Mathematics, National Chiayi University, Chiayi, 60004, Taiwan

Received  January 2015 Revised  September 2015 Published  March 2016

This work considers providing a common base for measuring the relative efficiency for all the decision-making units (DMUs) with multiple fuzzy inputs and outputs under the fuzzy data envelopment analysis (DEA) framework. It is shown that the fuzzy DEA model with common weights can be reduced into an auxiliary bi-objective fuzzy optimization problem by considering the most and the least favorable conditions simultaneously. An algorithm with the implementation issue for finding the compromise solution of the fuzzy DEA program is developed. A numerical example is included for illustration and comparison purpose. Our results show that the proposed approach is able to provide decision makers the flexibility in measuring the relative efficiency for DMUs with fuzzy inputs and outputs, which not only differentiates efficient units on a common base but also detects some abnormal efficiency scores calculated from other existing methods.

Citation: Cheng-Kai Hu, Fung-Bao Liu, Cheng-Feng Hu. Efficiency measures in fuzzy data envelopment analysis with common weights. Journal of Industrial & Management Optimization, 2017, 13 (1) : 237-249. doi: 10.3934/jimo.2016014
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##### References:
Input and output data in [2]
 DMU Input Output j Exact Imprecise Exact Ordinal x1j x2jL x2jU y1j y2j 1 100 0.6 0.7 2000 4 2 150 0.8 0.9 1000 2 3 150 1 1 1200 5 4 200 0.7 0.8 900 1 5 200 1 1 600 3
 DMU Input Output j Exact Imprecise Exact Ordinal x1j x2jL x2jU y1j y2j 1 100 0.6 0.7 2000 4 2 150 0.8 0.9 1000 2 3 150 1 1 1200 5 4 200 0.7 0.8 900 1 5 200 1 1 600 3
The positive ideal solution $(E^{\ast}_j)$ and the negative ideal solution $(E^{-}_j)$
 DMUj Ej* Ej− 1 1 2.001 * 10−7 2 0.875 9.01 * 10−8 3 1 1.201 * 10−7 4 1 9.01 * 10−8 5 0.7 6.01 * 10−8
 DMUj Ej* Ej− 1 1 2.001 * 10−7 2 0.875 9.01 * 10−8 3 1 1.201 * 10−7 4 1 9.01 * 10−8 5 0.7 6.01 * 10−8
Efficiency scores and the associated rankings (in parentheses) calculated from different methods
 DMUj Fuzy CCR IDEA approach Model (6) in [3] p = 1 p = 2 p = ∞ 1 1(1) 1(1) 1(1) 1(1) 1(1) 1(1) 2 0.875(4) 0.875(4) 0.875(4) 0.875(4) 0.7825(4) 0.7563(4) 3 1(1) 1(1) 1(1) 1(1) 0.9233(2) 0.9062(2) 4 1(1) 1(1) 1(1) 1(1) 0.9083(3) 0.8943(3) 5 0.7(5) 0.7(5) 0.7(5) 0.7(5) 0.6239(5) 0.6022(5)
 DMUj Fuzy CCR IDEA approach Model (6) in [3] p = 1 p = 2 p = ∞ 1 1(1) 1(1) 1(1) 1(1) 1(1) 1(1) 2 0.875(4) 0.875(4) 0.875(4) 0.875(4) 0.7825(4) 0.7563(4) 3 1(1) 1(1) 1(1) 1(1) 0.9233(2) 0.9062(2) 4 1(1) 1(1) 1(1) 1(1) 0.9083(3) 0.8943(3) 5 0.7(5) 0.7(5) 0.7(5) 0.7(5) 0.6239(5) 0.6022(5)
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