Efficiency measures in fuzzy data envelopment analysis with common weights

Cheng-Kai Hu; Fung-Bao Liu; Cheng-Feng Hu

doi:10.3934/jimo.2016014

Article Contents

2017, Volume 13, Issue 1: 237-249. Doi: 10.3934/jimo.2016014

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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: December 31, 2014

Revised: August 31, 2015

Published: August 2017

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 90C70, 90B50.

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Table 1. Input and output data in [2]

DMU	Input			Output
j	Exact	Imprecise		Exact	Ordinal
j	x_1j	x_2j^L	x_2j^U	y_1j	y_2j
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

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Table 2. The positive ideal solution $(E^{\ast}_j) $ and the negative ideal solution $(E^{-}_j)$

DMU j	E_j^*	E_j⁻
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

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Table 3. Efficiency scores and the associated rankings (in parentheses) calculated from different methods

DMU j	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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