TWO-DIMENSIONAL REPRESENTATION FOR TWO-STAGE NETWORK DEA MODELS

Torres, Bruno Guimarães; Reis, Juliana de Castro; Mello, João Carlos C. B. Soares de

doi:10.1590/0101-7438.2022.042.00258055

Bruno Guimarães Torres ^**Corresponding author

Universidade Federal Fluminense, Department of Production Engineering (PDE), Rua Dom Silvério 135, 25650-050 Petrópolis, Rio de Janeiro, Brazil - E-mail: brunotorres@id.uff.br

https://orcid.org/0000-0003-2095-7937

Juliana de Castro Reis

Universidade Federal Fluminense, Rua Passo da Pátria 156, 24210-240 Niterói, Rio de Janeiro, Brazil - E-mail: juliana.dcreis@gmail.com

https://orcid.org/0000-0003-2843-2211

João Carlos C. B. Soares de Mello

Universidade Federal Fluminense, Department of Production Engineering (TEP), Rua Passo da Pátria 156, 24210-240 Niterói, Rio de Janeiro, Brazil - E-mail: joaocsmello@gmail.com

https://orcid.org/0000-0002-6507-2721

*Corresponding author

DMU	Input	Int. Var.	Output	Eff. Stage 1	Eff. Stage 2	Overall Eff.
A	2	2	2	0,625	0,800	0,500
B	2,5	4	5	1,000	1,000	1,000
C	4	3	2,5	0,469	0,667	0,312
D	6	5	5,5	0,521	0,880	0,458

DMUs	Exogenous Input	Intermediate Variable	Exogenous Output	Overall Efficiency	First Stage Efficiency	Second Stage Efficiency	v1	w1	u1
A	2	2	2	0,250	0,500	0,500	1,000	0,500	0,250
B	2	4	8	1,000	1,000	1,000	0,500	0,250	0,125
C	4	4	8	0,500	0,500	1,000	0,500	0,250	0,125
D	6	12	5,5	0,229	1,000	0,229	0,167	0,083	0,042
E	3	4,5	8	0,667	0,750	0,889	0,444	0,222	0,111
F	2,5	4,5	6	0,600	0,900	0,667	0,444	0,222	0,111

DMU	Virtual input	Virtual output	$E_{0}^{1}$	$E_{0}^{2}$	$E_{0}^{1} * E_{0}^{2} = E_{0}$
1	1.015435	0.7054	0.9848	0.7054	0.6947
2	1.002908	0.626	0.9971	0.626	0.6242
3	1.449275	1	0.69	1	0.69
4	1.392564	0.4202	0.7181	0.4202	0.3018
5	1.248284	0.9457	0.8011	0.9457	0.7577
6	1.039609	0.4037	0.9619	0.4037	0.3883
7	1.464772	0.4026	0.6827	0.4026	0.2748
8	1.481921	0.4076	0.6748	0.4076	0.275
9	1.059659	0.2323	0.9437	0.2323	0.2192
10	1.274697	0.5597	0.7845	0.5597	0.4391
11	1.449485	0.2276	0.6899	0.2276	0.157
12	1	0.7596	1	0.7596	0.7596
13	1.471887	0.3052	0.6794	0.3052	0.2073
14	1.475579	0.4222	0.6777	0.4222	0.2861
15	1.067122	0.6376	0.9371	0.6376	0.5976
16	1.127269	0.3597	0.8871	0.3597	0.3191
17	1.764291	0.6183	0.5668	0.6183	0.3504
18	1.300221	0.3335	0.7691	0.3335	0.2565
19	1.003814	0.412	0.9962	0.412	0.4104
20	1.29668	0.6763	0.7712	0.6763	0.5216
21	1.345171	0.2668	0.7434	0.2668	0.1984
22	1.696353	1	0.5895	1	0.5895
23	1.22835	0.5079	0.8141	0.5079	0.4135
24	1.209629	0.1255	0.8267	0.1255	0.1037

U	First Stage			Second Stage			Overall Process
U	Virtual input	Virtual output	$E_{0}^{1}$	Virtual input	Virtual output	$E_{0}^{2}$	Virtual input	Virtual output	E ₀
1	803991.1	798021.2	0.993	0.992	0.699	0.704	803991.1	562178.4	0.699
2	1381822	1381822	0.998	1	0.625	0.626	1381822	864139.7	0.625
3	743793.2	513232.3	0.690	0.690	0.690	1	743793.2	513232.3	0.690
4	601320	437820.3	0.724	0.728	0.305	0.420	601320	183736.6	0.304
5	4349626	3613079	0.830	0.830	0.766	0.923	4349626	3336088	0.767
6	698879.6	671358.7	0.961	0.960	0.389	0.406	698879.6	272341.1	0.390
7	1572159	1054353	0.671	0.670	0.276	0.412	1572159	434822.1	0.277
8	1873530	1048827	0.663	0.559	0.276	0.415	1873530	518226.3	0.275
9	1109857	1109857	1	1	0.223	0.223	1109857	247814.3	0.223
10	1303249	1132234	0.862	0.868	0.467	0.541	1303249	609185.7	0.466
11	672414	434913.6	0.647	0.646	0.163	0.253	672414	110224.7	0.164
12	650952	650952	1	1	0.759	0.760	650952	494450.4	0.760
13	1368802	816983.9	0.672	0.596	0.215	0.309	1368802	295102.4	0.208
14	1094028	732913	0.670	0.669	0.288	0.431	1094028	315779.5	0.289
15	771871.8	771871.8	1	1	0.613	0.614	771871.8	473797.4	0.614
16	477814.9	415065.5	0.886	0.868	0.332	0.361	477814.9	158850.6	0.320
17	1180258	740736.7	0.628	0.627	0.360	0.574	1180258	424906.9	0.360
18	602106.2	477796	0.794	0.793	0.258	0.326	602106.2	155851.8	0.259
19	159422	159422	1	1	0.411	0.411	159422	65553.82	0.411
20	53518	49943.93	0.934	0.933	0.546	0.586	53518	29250.05	0.547
21	30787.92	22539.84	0.732	0.732	0.200	0.274	30787.92	6181.63	0.200
22	10934.47	6446.133	0.589	0.589	0.589	1	10934.47	6446.133	0.589
23	35196.25	29655.01	0.843	0.842	0.420	0.499	35196.25	14794.56	0.420
24	163297	70003.68	0.429	0.428	0.134	0.314	163297	22014.27	0.135

\begin{matrix} E_{0}^{1} = m i n \frac{Σ_{i = 1}^{m} v_{i} x_{i 0}}{Σ_{d = 1}^{D} w_{d} z_{d 0}} \\ s . t . \frac{Σ_{i = 1}^{m} v_{i} x_{i j}}{Σ_{d = 1}^{D} w_{d} z_{d j}} \geq 1, j = 1,2, \dots, n \\ v_{i}, w_{d} \geq 0; i = 1,2, \dots, m; d = 1,2, \dots, D \end{matrix}

\begin{matrix} E_{0}^{2} = m a x \frac{Σ_{r = 1}^{s} u_{r} y_{r 0}}{Σ_{d = 1}^{D} w_{d} z_{d 0}} \\ s . t . \frac{Σ_{r = 1}^{s} u_{r} y_{r j}}{Σ_{d = 1}^{D} w_{d} z_{d j}} \leq 1, j = 1,2, \dots, n \\ u_{r}, w_{d} \geq 0; r = 1,2, \dots, s; d = 1,2, \dots, D \end{matrix}

\begin{array}{l} E_{0}^{1} = m i n \sum_{i = 1}^{m} v_{i} x_{i 0} \\ E_{0}^{2} = m a x \sum_{r = 1}^{s} u_{r} y_{r 0} \\ s . t . \sum_{d = 1}^{D} w_{d} z_{d 0} = 1 \\ \sum_{d = 1}^{D} w_{d} z_{d j} - \sum_{i = 1}^{m} v_{i} x_{i j} \leq 0, j = 1,2, \dots, n \\ \sum_{r = 1}^{s} u_{r} y_{r j} - \sum_{d = 1}^{D} w_{d} z_{d j} \leq 0, j = 1,2, \dots, n \\ u_{r}, v_{i}, w_{d} \geq 0, i = 1,2, \dots, m, r = 1,2, \dots, s, d = 1,2, \dots, D \end{array}

\begin{array}{l} m i n δ \\ s . t . \sum_{i = 1}^{m} v_{i} x_{i 0} - δ \leq E_{j o}^{1} \\ \sum_{r = 1}^{s} u_{r} y_{r 0} + δ \geq E_{j o}^{2} \\ \sum_{d = 1}^{D} w_{d} z_{d 0} = 1 \\ \sum_{d = 1}^{D} w_{d} z_{d j} - \sum_{i = 1}^{m} v_{i} x_{i j} \leq 0, j = 1,2, \dots, n \\ \sum_{r = 1}^{s} u_{r} y_{r j} - \sum_{d = 1}^{D} w_{d} z_{d j} \leq 0, j = 1,2, \dots, n \\ u_{r}, v_{i}, w_{d}, δ \geq 0, i = 1,2, \dots, m, r = 1,2, \dots, s, d = 1,2, \dots, D \end{array}

E_{0} = E_{0}^{1} * E_{0}^{2} = \frac{1}{Σ_{i = 1}^{m} v_{i} x_{i 0}} * \sum_{r = 1}^{s} u_{r} y_{r 0} = \frac{Σ_{r = 1}^{s} u_{r} y_{r 0}}{Σ_{i = 1}^{m} v_{i} x_{i 0}}

\begin{array}{l} E_{o} = m a x \sum_{r = 1}^{s} u_{r} y_{r 0} \\ s . t . \sum_{i = 1}^{m} v_{i} x_{i 0} = 1 \\ \sum_{d = 1}^{D} w_{d} z_{d j} - \sum_{i = 1}^{m} v_{i} x_{i j} \leq 0, j = 1,2, \dots, n \\ \sum_{r = 1}^{s} u_{r} y_{r j} - \sum_{d = 1}^{D} w_{d} z_{d j} \leq 0, j = 1,2, \dots, n \\ u_{r}, v_{i}, w_{d} \geq 0, i = 1,2, \dots, m, r = 1,2, \dots, s, d = 1,2, \dots, D \end{array}

[1] *Corresponding author

Brasil

Brasil

TWO-DIMENSIONAL REPRESENTATION FOR TWO-STAGE NETWORK DEA MODELS

ABSTRACT