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July  2021, 17(4): 1795-1807. doi: 10.3934/jimo.2020046

## 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

Received  January 2019 Revised  September 2019 Published  July 2021 Early access  March 2020

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.

Citation: Cheng-Kai Hu, Fung-Bao Liu, Hong-Ming Chen, Cheng-Feng Hu. Network data envelopment analysis with fuzzy non-discretionary factors. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1795-1807. doi: 10.3934/jimo.2020046
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##### References:
]">Figure 1.  General network systems [12]
]">Figure 2.  Network system discussed in [18]
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 $0.0106$ $1.757$ $12.7$ $20.670$ $0.253$ $0.988$
 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 $0.0106$ $1.757$ $12.7$ $20.670$ $0.253$ $0.988$
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)$\theta^{\ast}$ ${ \text{Model (6)}}$ $\alpha^{\ast}$ $1-\alpha^{\ast}$ $\alpha^{\ast}$ $1-\alpha^{\ast}$ $1$ $0.6388$ $0.3612$ $0.6388$ $15$ $0.6582$ $0.3418$ $0.6582$ $2$ $0.6507$ $0.3493$ $0.6507$ $16$ $0.6646$ $0.3354$ $0.6646$ $3$ $0.5179$ $0.4821$ $0.5179$ $17$ $0.7177$ $0.2823$ $0.7177$ $4$ $0.5986$ $0.4014$ $0.5986$ $18$ $1.0000$ $0.0000$ $1.0000$ $5$ $0.5556$ $0.4444$ $0.5556$ $19$ $0.8144$ $0.1856$ $0.8144$ $6$ $0.7599$ $0.2401$ $0.7599$ $20$ $0.6940$ $0.3060$ $0.6940$ $7$ $1.0000$ $0.0000$ $1.0000$ $21$ $0.7067$ $0.2933$ $0.7067$ $8$ $0.5352$ $0.4648$ $0.5352$ $22$ $0.7942$ $0.2058$ $0.7942$ $9$ $0.6249$ $0.3751$ $0.6249$ $23$ $0.7802$ $0.2198$ $0.7802$ $10$ $0.4961$ $0.5039$ $0.4961$ $24$ $0.9300$ $0.0700$ $0.9300$ $11$ $0.4945$ $0.5055$ $0.4945$ $25$ $0.6270$ $0.3730$ $0.6270$ $12$ $0.6685$ $0.3315$ $0.6685$ $26$ $1.0000$ $0.0000$ $1.0000$ $13$ $0.9487$ $0.0513$ $0.9487$ $27$ $1.0000$ $0.0000$ $1.0000$ $14$ $0.5880$ $0.4120$ $0.5880$
 DMU j Model (2) $\theta^{\ast}$ ${ \text{Model (6)}}$ DMU j Model (2)$\theta^{\ast}$ ${ \text{Model (6)}}$ $\alpha^{\ast}$ $1-\alpha^{\ast}$ $\alpha^{\ast}$ $1-\alpha^{\ast}$ $1$ $0.6388$ $0.3612$ $0.6388$ $15$ $0.6582$ $0.3418$ $0.6582$ $2$ $0.6507$ $0.3493$ $0.6507$ $16$ $0.6646$ $0.3354$ $0.6646$ $3$ $0.5179$ $0.4821$ $0.5179$ $17$ $0.7177$ $0.2823$ $0.7177$ $4$ $0.5986$ $0.4014$ $0.5986$ $18$ $1.0000$ $0.0000$ $1.0000$ $5$ $0.5556$ $0.4444$ $0.5556$ $19$ $0.8144$ $0.1856$ $0.8144$ $6$ $0.7599$ $0.2401$ $0.7599$ $20$ $0.6940$ $0.3060$ $0.6940$ $7$ $1.0000$ $0.0000$ $1.0000$ $21$ $0.7067$ $0.2933$ $0.7067$ $8$ $0.5352$ $0.4648$ $0.5352$ $22$ $0.7942$ $0.2058$ $0.7942$ $9$ $0.6249$ $0.3751$ $0.6249$ $23$ $0.7802$ $0.2198$ $0.7802$ $10$ $0.4961$ $0.5039$ $0.4961$ $24$ $0.9300$ $0.0700$ $0.9300$ $11$ $0.4945$ $0.5055$ $0.4945$ $25$ $0.6270$ $0.3730$ $0.6270$ $12$ $0.6685$ $0.3315$ $0.6685$ $26$ $1.0000$ $0.0000$ $1.0000$ $13$ $0.9487$ $0.0513$ $0.9487$ $27$ $1.0000$ $0.0000$ $1.0000$ $14$ $0.5880$ $0.4120$ $0.5880$
The results of solving the proposed fuzzy non-discretionary Model (14)
 $\begin{array}{c} \mbox{DMU}\\ j \end{array}$ Fuzzy non-discretionary input $\bar{X}_{1j}^{\ast}$ $\bar{X}_{2j}^{\ast}$ $\bar{X}_{3j}^{\ast}$ $\alpha^{\ast}$ $1-\alpha^{\ast}$ 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
 $\begin{array}{c} \mbox{DMU}\\ j \end{array}$ Fuzzy non-discretionary input $\bar{X}_{1j}^{\ast}$ $\bar{X}_{2j}^{\ast}$ $\bar{X}_{3j}^{\ast}$ $\alpha^{\ast}$ $1-\alpha^{\ast}$ 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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