Network data envelopment analysis with fuzzy non-discretionary factors

Cheng-Kai Hu; Fung-Bao Liu; Hong-Ming Chen; Cheng-Feng Hu

doi:10.3934/jimo.2020046

Article Contents

2021, Volume 17, Issue 4: 1795-1807. Doi: 10.3934/jimo.2020046

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Network data envelopment analysis with fuzzy non-discretionary factors

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, Tunghai University, Taichung 40704, Taiwan
4.
Department of Applied Mathematics, National Chiayi University, Chiayi, 60004, Taiwan

^* Corresponding author: C.-F. Hu
^* Corresponding author: C.-F. Hu

Received: December 31, 2018

Revised: August 31, 2019

Early access: March 2020

Published: July 2021

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Abstract

Network data envelopment analysis (DEA) concerns using the DEA technique to measure the relative efficiency of a system, taking into account its internal structure. The results are more meaningful and informative than those obtained from the conventional DEA models. This work proposed a new network DEA model based on the fuzzy concept even though the inputs and outputs data are crisp numbers. The model is then extended to investigate the network DEA with fuzzy non-discretionary variables. An illustrative application assessing the impact of information technology (IT) on firm performance is included. The results reveal that modeling the IT budget as a fuzzy non-discretionary factor improves the system performance of firms in a banking industry.

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Mathematics Subject Classification: Primary: 90B50, 90C08; Secondary: 91B06.

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Figure 1. General network systems [12]

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Figure 2. Network system discussed in [18]

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Table 1. Data set for assessing IT impact on firm performance

DMU j	IT	Fixed	No. of	Deposits	Profit	Fraction
	$ \rm {budget}$	$ {\mbox{assets}} $	$ {\mbox{employees }}$			of loans
	$({＄ \ \mbox{billions})}$	$({＄ \ \mbox{billions})} $	$ ({＄ \ \mbox{billions})} $	$({＄ \ \mbox{billions})} $	$ ({＄ \ \mbox{billions})} $	${\mbox{recovered}}$
	$ X_1 $	$ X_2 $	$ X_3 $	$ Z $	$ Y_1$	$ Y_2 $
1	$ 0.150 $	$ 0.713 $	$ 13.3 $	$ 14.478 $	$ 0.232 $	$ 0.986 $
2	$ 0.170 $	$ 1.071 $	$ 16.9 $	$ 19.502 $	$ 0.340 $	$ 0.986 $
3	$ 0.235 $	$ 1.224 $	$ 24.0 $	$ 20.952 $	$ 0.363 $	$ 0.986 $
4	$ 0.211 $	$ 0.363 $	$ 15.6 $	$ 13.902 $	$ 0.211 $	$ 0.982 $
5	$ 0.133 $	$ 0.409 $	$ 18.485 $	$ 15.206 $	$ 0.237 $	$ 0.984 $
6	$ 0.497 $	$ 5.846 $	$ 56.42 $	$ 81.186 $	$ 1.103 $	$ 0.955 $
7	$ 0.060 $	$ 0.918 $	$ 56.42 $	$ 81.186 $	$ 1.103 $	$ 0.986 $
8	$ 0.071 $	$ 1.235 $	$ 12.0 $	$ 11.441 $	$ 0.199 $	$ 0.985 $
9	$ 1.500 $	$ 18.120 $	$ 89.51 $	$ 124.072 $	$ 1.858 $	$ 0.972 $
10	$ 0.120 $	$ 1.821 $	$ 19.8 $	$ 17.425 $	$ 0.274 $	$ 0.983 $
11	$ 0.120 $	$ 1.915 $	$ 19.8 $	$ 17.425 $	$ 0.274 $	$ 0.983 $
12	$ 0.050 $	$ 0.874 $	$ 13.1 $	$ 14.342 $	$ 0.177 $	$ 0.985 $
13	$ 0.370 $	$ 6.918 $	$ 12.5 $	$ 32.491 $	$ 0.648 $	$ 0.945 $
14	$ 0.440 $	$ 4.432 $	$ 41.9 $	$ 47.653 $	$ 0.639 $	$ 0.979 $
15	$ 0.431 $	$ 4.504 $	$ 41.1 $	$ 52.63 $	$ 0.741 $	$ 0.981 $
16	$ 0.110 $	$ 1.241 $	$ 14.4 $	$ 17.493 $	$ 0.243 $	$ 0.988 $
17	$ 0.053 $	$ 0.450 $	$ 7.6 $	$ 9.512 $	$ 0.067 $	$ 0.980 $
18	$ 0.345 $	$ 5.892 $	$ 15.5 $	$ 42.469 $	$ 1.002 $	$ 0.948 $
19	$ 0.128 $	$ 0.973 $	$ 12.6 $	$ 18.987 $	$ 0.243 $	$ 0.985 $
20	$ 0.055 $	$ 0.444 $	$ 5.9 $	$ 7.546 $	$ 0.153 $	$ 0.987 $
21	$ 0.057 $	$ 0.508 $	$ 5.7 $	$ 7.595 $	$ 0.123 $	$ 0.987 $
22	$ 0.098 $	$ 0.370 $	$ 14.1 $	$ 16.906 $	$ 0.233 $	$ 0.981 $
23	$ 0.104 $	$ 0.395 $	$ 14.6 $	$ 17.264 $	$ 0.263 $	$ 0.983 $
24	$ 0.206 $	$ 2.680 $	$ 19.6 $	$ 36.430 $	$ 0.601 $	$ 0.982 $
25	$ 0.067 $	$ 0.781 $	$ 10.5 $	$ 11.581 $	$ 0.120 $	$ 0.987 $
26	$ 0.100 $	$ 0.872 $	$ 12.1 $	$ 22.207 $	$ 0.248 $	$ 0.972 $
27	$ 0.0106 $	$ 1.757 $	$ 12.7 $	$ 20.670 $	$ 0.253 $	$ 0.988 $

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Table 2. The system efficiency, $ \theta_p^{\ast}, $ and the membership degree, $ \alpha_p, p = 1, 2, \cdots, 27. $

DMU j	Model (2) $ \theta^{\ast} $	${ \text{Model (6)}}$		DMU j	Model (2) $ \theta^{\ast} $	$ { \text{Model (6)}}$
DMU j	Model (2) $ \theta^{\ast} $	$\alpha^{\ast}$	$ 1-\alpha^{\ast}$	DMU j	Model (2) $ \theta^{\ast} $	$ \alpha^{\ast} $	$ 1-\alpha^{\ast} $
$ 1 $	$ 0.6388 $	$ 0.3612 $	$ 0.6388 $	$ 15 $	$ 0.6582 $	$ 0.3418 $	$ 0.6582 $
$ 2 $	$ 0.6507 $	$ 0.3493 $	$ 0.6507 $	$ 16 $	$ 0.6646 $	$ 0.3354 $	$ 0.6646 $
$ 3 $	$ 0.5179 $	$ 0.4821 $	$ 0.5179 $	$ 17 $	$ 0.7177 $	$ 0.2823 $	$ 0.7177 $
$ 4 $	$ 0.5986 $	$ 0.4014 $	$ 0.5986 $	$ 18 $	$ 1.0000 $	$ 0.0000 $	$ 1.0000 $
$ 5 $	$ 0.5556 $	$ 0.4444 $	$ 0.5556 $	$ 19 $	$ 0.8144 $	$ 0.1856 $	$ 0.8144 $
$ 6 $	$ 0.7599 $	$ 0.2401 $	$ 0.7599 $	$ 20 $	$ 0.6940 $	$ 0.3060 $	$ 0.6940 $
$ 7 $	$ 1.0000 $	$ 0.0000 $	$ 1.0000 $	$ 21 $	$ 0.7067 $	$ 0.2933 $	$ 0.7067 $
$ 8 $	$ 0.5352 $	$ 0.4648 $	$ 0.5352 $	$ 22 $	$ 0.7942 $	$ 0.2058 $	$ 0.7942 $
$ 9 $	$ 0.6249 $	$ 0.3751 $	$ 0.6249 $	$ 23 $	$ 0.7802 $	$ 0.2198 $	$ 0.7802 $
$ 10 $	$ 0.4961 $	$ 0.5039 $	$ 0.4961 $	$ 24 $	$ 0.9300 $	$ 0.0700 $	$ 0.9300 $
$ 11 $	$ 0.4945 $	$ 0.5055 $	$ 0.4945 $	$ 25 $	$ 0.6270 $	$ 0.3730 $	$ 0.6270 $
$ 12 $	$ 0.6685 $	$ 0.3315 $	$ 0.6685 $	$ 26 $	$ 1.0000 $	$ 0.0000 $	$ 1.0000 $
$ 13 $	$ 0.9487 $	$ 0.0513 $	$ 0.9487 $	$ 27 $	$ 1.0000 $	$ 0.0000 $	$ 1.0000 $
$ 14 $	$ 0.5880 $	$ 0.4120 $	$ 0.5880 $

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Table 3. The results of solving the proposed fuzzy non-discretionary Model (14)

$ \begin{array}{c} \mbox{DMU}\\ j \end{array} $	Fuzzy non-discretionary input
$ \begin{array}{c} \mbox{DMU}\\ j \end{array} $	$ \bar{X}_{1j}^{\ast} $	$ \bar{X}_{2j}^{\ast} $	$ \bar{X}_{3j}^{\ast} $	$ \alpha^{\ast} $	$ 1-\alpha^{\ast} $	Rank
1	0.1102	0.5236	9.6335	0.2654	0.7346	18
2	0.1260	0.7723	12.4259	0.2589	0.7411	17
3	0.1586	0.8079	16.1328	0.3253	0.6747	25
4	0.1506	0.2564	10.9013	0.2864	0.7136	21
5	0.0921	0.2793	11.2165	0.3077	0.6923	23
6	0.4008	4.6342	45.4289	0.1936	0.8064	10
7	0.0600	0.9180	56.4200	0.0000	1.0000	1
8	0.0485	0.7677	8.0988	0.3173	0.6827	24
9	1.0908	13.1471	64.8529	0.2728	0.7272	20
10	0.0798	1.0715	12.8295	0.3351	0.6649	26
11	0.0797	1.0997	12.7416	0.3358	0.6642	27
12	0.0376	0.6544	9.7883	0.2490	0.7510	14
13	0.3519	5.3291	11.8900	0.0488	0.9512	5
14	0.3116	3.1047	29.5929	0.2918	0.7082	22
15	0.3212	3.2772	30.4969	0.2547	0.7453	16
16	0.0824	0.8943	10.7729	0.2512	0.7488	15
17	0.0413	0.3509	5.8871	0.2202	0.7798	11
18	0.3450	5.8920	15.5000	0.0000	1.0000	1
19	0.1080	0.8154	10.6151	0.1565	0.8435	7
20	0.0421	0.3349	4.4948	0.2343	0.7657	13
21	0.0441	0.3904	4.3686	0.2268	0.7732	12
22	0.0813	0.3043	11.6775	0.1707	0.8293	8
23	0.0853	0.3216	11.9554	0.1802	0.8198	9
24	0.1925	2.4125	18.3176	0.0654	0.9346	6
25	0.0488	0.5448	7.5342	0.2717	0.7283	19
26	0.1000	0.8720	12.1000	0.0000	1.0000	1
27	0.0106	1.7570	12.7000	0.0000	1.0000	1

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