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doi: 10.3934/jimo.2020046

Network data envelopment analysis with fuzzy non-discretionary factors

Cheng-Kai Hu 1, , Fung-Bao Liu 2, , Hong-Ming Chen 3, and  Cheng-Feng Hu 4,,

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  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.

Keywords: Network DEA, non-discretionary, fuzzy decision making.
Mathematics Subject Classification: Primary: 90B50, 90C08; Secondary: 91B06.
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, doi: 10.3934/jimo.2020046
References:
[1]

R. D. Banker and R. Morey, Efficiency analysis for exogenously fixed inputs and outputs, Oper. Res., 34 (1986), 501-653.  doi: 10.1287/opre.34.4.513.  Google Scholar

[2]

M. Barat, G. Tohidi and M. Sanei, DEA for nonhomogeneous mixed networks, Asia Pac. Manag. Rev., 24 (2018), 161-166.  doi: 10.1016/j.apmrv.2018.02.003.  Google Scholar

[3]

R. E. Bellman and L. A. Zadeh, Decision making in a fuzzy environment, Manag. Sci., 17 (1970), B141–B164. doi: 10.1287/mnsc.17.4.B141.  Google Scholar

[4]

L. Castelli, R. Pesenti and W. Ukovich, DEA-like models for the efficiency evaluation of hierarchically structured units, Eur. J. Oper. Res., 154 (2004), 465-476.  doi: 10.1016/S0377-2217(03)00182-6.  Google Scholar

[5]

J. Zhu, Data Envelopment Analysis: A Handbook of Modeling Internal Structures and Networks, International Series in Operations Research & Management Science, 238. Springer, New York, 2016. doi: 10.1007/978-1-4899-7684-0.  Google Scholar

[6]

J. M. Cordero-Ferrera, F. Pedraja-Chaparro and D. Santín-González, Enhancing the inclusion of non-discretionary inputs in DEA, J. Oper. Res. Soc., 61 (2010), 574-584.  doi: 10.1057/jors.2008.189.  Google Scholar

[7]

R. Färe and S. Grosskopf, Intertemporal Production Frontiers: With Dynamic DEA, Boston: Kluwer Academic Publishers, 1996. Google Scholar

[8]

R. Färe and S. Grosskopf, Network DEA, Socio. Econ. Plann. Sci., 4 (2000), 35-49.   Google Scholar

[9]

D. U. A. Galagedera, Modelling social responsibility in mutual fund performance appraisal: A two-stage data envelopment analysis model with non-discretionary first stage output, Eur. J. Oper. Res., 273 (2019), 376-389.  doi: 10.1016/j.ejor.2018.08.011.  Google Scholar

[10]

B. Golany and Y. Roll, Some extensions of techniques to handle non-discretionary factors in data envelopment analysis, J. Prod. Anal., 4 (1993), 419-432.  doi: 10.1007/BF01073549.  Google Scholar

[11]

C. Kao, Network data envelopment analysis: A review, Eur. J. Oper. Res., 239 (2014), 1-16.  doi: 10.1016/j.ejor.2014.02.039.  Google Scholar

[12]

C. Kao, Efficiency decomposition and aggregation in network data envelopment analysis, Eur. J. Oper. Res., 255 (2016), 778-786.  doi: 10.1016/j.ejor.2016.05.019.  Google Scholar

[13]

C. Kao and S.-N. Hwang, Efficiency measurement for network systems: IT impact on firm performance, Decis. Support Syst., 48 (2010), 437-446.  doi: 10.1016/j.dss.2009.06.002.  Google Scholar

[14]

R. J. Kauffman and P. Weill, An evaluative framework for research on the performance effects of information technology investment, Proceedings of the 10th International Conference on Information Systems, (1989), 377–388. doi: 10.1145/75034.75066.  Google Scholar

[15]

M. A. Muniz, J. Paradi, J. Ruggiero and Z. Yang, Evaluating alternative DEA models used to control for non-discretionary inputs, Comput. Oper. Res., 33 (2006), 1173-1183.   Google Scholar

[16]

L. Simar and P. W. Wilson, Estimation and inference in two-stage, semi-parametric models of production processes, J. Econom., 136 (1997), 31-64.  doi: 10.1016/j.jeconom.2005.07.009.  Google Scholar

[17]

M. Taleb, R. Ramli and R. Khalid, Developing a two-stage approach of super efficiency slack-based measure in the presence of non-discretionary factors and mixed integer-valued data envelopment analysis, Expert. Syst. Appl., 103 (2018), 14-24.  doi: 10.1016/j.eswa.2018.02.037.  Google Scholar

[18]

C. H. Wang, R. Gopal and S. Zionts, Use of data envelopment analysis in assessing information technology impact on firm performance, Ann. Oper. Res., 73 (1997), 191-213.   Google Scholar

[19]

M. Zerafat Angiz L and A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl.-Based Syst., 49 (2013), 145-151.   Google Scholar

show all references

References:
[1]

R. D. Banker and R. Morey, Efficiency analysis for exogenously fixed inputs and outputs, Oper. Res., 34 (1986), 501-653.  doi: 10.1287/opre.34.4.513.  Google Scholar

[2]

M. Barat, G. Tohidi and M. Sanei, DEA for nonhomogeneous mixed networks, Asia Pac. Manag. Rev., 24 (2018), 161-166.  doi: 10.1016/j.apmrv.2018.02.003.  Google Scholar

[3]

R. E. Bellman and L. A. Zadeh, Decision making in a fuzzy environment, Manag. Sci., 17 (1970), B141–B164. doi: 10.1287/mnsc.17.4.B141.  Google Scholar

[4]

L. Castelli, R. Pesenti and W. Ukovich, DEA-like models for the efficiency evaluation of hierarchically structured units, Eur. J. Oper. Res., 154 (2004), 465-476.  doi: 10.1016/S0377-2217(03)00182-6.  Google Scholar

[5]

J. Zhu, Data Envelopment Analysis: A Handbook of Modeling Internal Structures and Networks, International Series in Operations Research & Management Science, 238. Springer, New York, 2016. doi: 10.1007/978-1-4899-7684-0.  Google Scholar

[6]

J. M. Cordero-Ferrera, F. Pedraja-Chaparro and D. Santín-González, Enhancing the inclusion of non-discretionary inputs in DEA, J. Oper. Res. Soc., 61 (2010), 574-584.  doi: 10.1057/jors.2008.189.  Google Scholar

[7]

R. Färe and S. Grosskopf, Intertemporal Production Frontiers: With Dynamic DEA, Boston: Kluwer Academic Publishers, 1996. Google Scholar

[8]

R. Färe and S. Grosskopf, Network DEA, Socio. Econ. Plann. Sci., 4 (2000), 35-49.   Google Scholar

[9]

D. U. A. Galagedera, Modelling social responsibility in mutual fund performance appraisal: A two-stage data envelopment analysis model with non-discretionary first stage output, Eur. J. Oper. Res., 273 (2019), 376-389.  doi: 10.1016/j.ejor.2018.08.011.  Google Scholar

[10]

B. Golany and Y. Roll, Some extensions of techniques to handle non-discretionary factors in data envelopment analysis, J. Prod. Anal., 4 (1993), 419-432.  doi: 10.1007/BF01073549.  Google Scholar

[11]

C. Kao, Network data envelopment analysis: A review, Eur. J. Oper. Res., 239 (2014), 1-16.  doi: 10.1016/j.ejor.2014.02.039.  Google Scholar

[12]

C. Kao, Efficiency decomposition and aggregation in network data envelopment analysis, Eur. J. Oper. Res., 255 (2016), 778-786.  doi: 10.1016/j.ejor.2016.05.019.  Google Scholar

[13]

C. Kao and S.-N. Hwang, Efficiency measurement for network systems: IT impact on firm performance, Decis. Support Syst., 48 (2010), 437-446.  doi: 10.1016/j.dss.2009.06.002.  Google Scholar

[14]

R. J. Kauffman and P. Weill, An evaluative framework for research on the performance effects of information technology investment, Proceedings of the 10th International Conference on Information Systems, (1989), 377–388. doi: 10.1145/75034.75066.  Google Scholar

[15]

M. A. Muniz, J. Paradi, J. Ruggiero and Z. Yang, Evaluating alternative DEA models used to control for non-discretionary inputs, Comput. Oper. Res., 33 (2006), 1173-1183.   Google Scholar

[16]

L. Simar and P. W. Wilson, Estimation and inference in two-stage, semi-parametric models of production processes, J. Econom., 136 (1997), 31-64.  doi: 10.1016/j.jeconom.2005.07.009.  Google Scholar

[17]

M. Taleb, R. Ramli and R. Khalid, Developing a two-stage approach of super efficiency slack-based measure in the presence of non-discretionary factors and mixed integer-valued data envelopment analysis, Expert. Syst. Appl., 103 (2018), 14-24.  doi: 10.1016/j.eswa.2018.02.037.  Google Scholar

[18]

C. H. Wang, R. Gopal and S. Zionts, Use of data envelopment analysis in assessing information technology impact on firm performance, Ann. Oper. Res., 73 (1997), 191-213.   Google Scholar

[19]

M. Zerafat Angiz L and A. Mustafa, Fuzzy interpretation of efficiency in data envelopment analysis and its application in a non-discretionary model, Knowl.-Based Syst., 49 (2013), 145-151.   Google Scholar

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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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 $
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)}}$
$\alpha^{\ast}$ $ 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 $$ 0.5880 $
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DMU
j		 Model (2)
$ \theta^{\ast} $		 ${ \text{Model (6)}}$ DMU
j		 Model (2)
$ \theta^{\ast} $		 $ { \text{Model (6)}}$
$\alpha^{\ast}$ $ 1-\alpha^{\ast}$ $ \alpha^{\ast} $ $ 1-\alpha^{\ast} $
$ 1 $ $ 0.6388 $ $ 0.3612 $$ 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 $
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
$ \bar{X}_{1j}^{\ast} $ $ \bar{X}_{2j}^{\ast} $ $ \bar{X}_{3j}^{\ast} $ $ \alpha^{\ast} $ $ 1-\alpha^{\ast} $Rank
10.11020.52369.63350.26540.734618
20.12600.772312.42590.25890.741117
30.15860.807916.13280.32530.674725
40.15060.256410.90130.28640.713621
50.09210.279311.21650.30770.692323
60.40084.634245.42890.19360.806410
70.06000.918056.42000.00001.00001
80.04850.76778.09880.31730.682724
91.090813.147164.85290.27280.727220
100.07981.071512.82950.33510.664926
110.07971.099712.74160.33580.664227
120.03760.65449.78830.24900.751014
130.35195.329111.89000.04880.95125
140.31163.104729.59290.29180.708222
150.32123.277230.49690.25470.745316
160.08240.894310.77290.25120.748815
170.04130.35095.88710.22020.779811
180.34505.892015.50000.00001.00001
190.10800.815410.61510.15650.84357
200.04210.33494.49480.23430.765713
210.04410.39044.36860.22680.773212
220.08130.304311.67750.17070.82938
230.08530.321611.95540.18020.81989
240.19252.412518.31760.06540.93466
250.04880.54487.53420.27170.728319
260.10000.872012.10000.00001.00001
270.01061.757012.70000.00001.00001
Table Options

Download as excel

$ \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
10.11020.52369.63350.26540.734618
20.12600.772312.42590.25890.741117
30.15860.807916.13280.32530.674725
40.15060.256410.90130.28640.713621
50.09210.279311.21650.30770.692323
60.40084.634245.42890.19360.806410
70.06000.918056.42000.00001.00001
80.04850.76778.09880.31730.682724
91.090813.147164.85290.27280.727220
100.07981.071512.82950.33510.664926
110.07971.099712.74160.33580.664227
120.03760.65449.78830.24900.751014
130.35195.329111.89000.04880.95125
140.31163.104729.59290.29180.708222
150.32123.277230.49690.25470.745316
160.08240.894310.77290.25120.748815
170.04130.35095.88710.22020.779811
180.34505.892015.50000.00001.00001
190.10800.815410.61510.15650.84357
200.04210.33494.49480.23430.765713
210.04410.39044.36860.22680.773212
220.08130.304311.67750.17070.82938
230.08530.321611.95540.18020.81989
240.19252.412518.31760.06540.93466
250.04880.54487.53420.27170.728319
260.10000.872012.10000.00001.00001
270.01061.757012.70000.00001.00001
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