doi: 10.3934/naco.2019043

Resource allocation and target setting based on virtual profit improvement

1. 

Department of Mathematics, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran

2. 

Faculty of Industrial Engineering and Management Sciences, Shahrood University of Technology, Shahrood, Iran

3. 

Operations and Information Management Department, Aston Business School, Aston University, Birmingham, B4 7ET, UK

* Corresponding author: E-mail addresses: std_j.sadeghi@khu.ac.ir

Received  April 2018 Revised  July 2019 Published  September 2019

One application of Data Envelopment Analysis (DEA) is the resource allocation and target setting among homogeneous Decision Making Units (DMUs). In this paper, we assume that all units are under the supervision and control of a central decision making unit, for instance chain stores, banks, schools, etc. The aim is to allocate available resources among units in a way that the so-called organisational overall "virtual profit" is maximized. Our method is highly flexible in decision making to achieve the goals of the Decision Maker (DM). The resulting production plans maintain the following characteristics: (1) the virtual profit of each unit is calculated with a common set of weights; (2) the selected weights for calculating the virtual profit prevent the virtual profit of the system from getting worse; (3) the virtual profits of less profitable units are improved as much as possible. The proposed method is illustrated with a simple numerical example and a real life application.

Citation: Jafar Sadeghi, Mojtaba Ghiyasi, Akram Dehnokhalaji. Resource allocation and target setting based on virtual profit improvement. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2019043
References:
[1]

A. D. Athanassopoulos, Goal programming & data envelopment analysis (godea) for target-based multi-level planning: allocating central grants to the greek local authorities, European Journal of Operational Research, 87 (1995), 535-550.   Google Scholar

[2]

J. E. Beasley, Allocating fixed costs and resources via data envelopment analysis, European Journal of Operational Research, 147 (2003), 198-216.   Google Scholar

[3]

A. Charnes and W. W. Cooper, Programming with linear fractional functionals, Naval Research Logistics Quarterly, 9 (1962), 181-186.  doi: 10.1002/nav.3800090303.  Google Scholar

[4]

A. CharnesW. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8.  Google Scholar

[5]

A. DehnokhalajiM. Ghiyasi and P. Korhonen, Resource allocation based on cost efficiency, Journal of the Operational Research Society, 68 (2017), 1279-1289.   Google Scholar

[6]

J. DuL. LiangY. Chen and G. B. Bi, Dea-based production planning, Omega, 38 (2010), 105-112.   Google Scholar

[7]

A. Emrouznejad and K. De Witte, Cooper-framework: A unified process for non-parametric projects, European Journal of Operational Research, 207 (2010), 1573-1586.   Google Scholar

[8]

L. Fang and H. Li, Centralized resource allocation based on the cost–revenue analysis, Computers & Industrial Engineering, 85 (2015), 395-401.   Google Scholar

[9]

M. J. Farrell, The measurement of productive efficiency, Journal of the Royal Statistical Society, Series A (General), 253–290. Google Scholar

[10]

S. GattoufiG. R. Amin and A. Emrouznejad, A new inverse dea method for merging banks, IMA Journal of Management Mathematics, 25 (2014), 73-87.   Google Scholar

[11]

B. GolanyF. Phillips and J. Rousseau, Models for improved effectiveness based on dea efficiency results, IIE Transactions, 25 (1993), 2-10.   Google Scholar

[12]

B. Golany and E. Tamir, Evaluating efficiency-effectiveness-equality trade-offs: A data envelopment analysis approach, Management Science, 41 (1995), 1172-1184.   Google Scholar

[13]

G. R. JahanshahlooJ. Sadeghi and M. Khodabakhshi, Fair ranking of the decision making units using optimistic and pessimistic weights in data envelopment analysis, RAIRO-Operations Research, 51 (2017), 253-260.  doi: 10.1051/ro/2016023.  Google Scholar

[14]

G. R. JahanshahlooJ. Sadeghi and M. Khodabakhshi, Proposing a method for fixed cost allocation using dea based on the efficiency invariance and common set of weights principles, Mathematical Methods of Operations Research, 85 (2007), 1-18.  doi: 10.1007/s00186-016-0563-z.  Google Scholar

[15]

P. Korhonen and M. Syrjänen, Resource allocation based on efficiency analysis, Management Science, 50 (2004), 1134-1144.   Google Scholar

[16]

F. LiQ. Zhu and L. Liang, Allocating a fixed cost based on a dea-game cross efficiency approach, Expert Systems with Applications, 96 (2018), 196-207.  doi: 10.1007/s11424-015-4211-0.  Google Scholar

[17]

F. Li, Q. Zhu and L. Liang, A new data envelopment analysis based approach for fixed cost allocation, Annals of Operations Research, 1–26. Google Scholar

[18]

F. H. LotfiA. Hatami-MarbiniP. J. AgrellN. Aghayi and K. Gholami, Allocating fixed resources and setting targets using a common-weights dea approach, Computers & Industrial Engineering, 64 (2013), 631-640.   Google Scholar

[19]

S. LozanoG. Villa and B. Adenso-Dıaz, Centralised target setting for regional recycling operations using dea, Omega, 32 (2004), 101-110.   Google Scholar

[20]

S. Lozano and G. Villa, Centralized resource allocation using data envelopment analysis, Journal of Productivity Analysis, 22 (2004), 143-161.   Google Scholar

[21]

N. NasrabadiA. DehnokhalajiN. A. KianiP. J. Korhonen and J. Wallenius, Resource allocation for performance improvement, Annals of Operations Research, 196 (2012), 459-468.  doi: 10.1007/s10479-011-1016-y.  Google Scholar

[22]

M. PervinG. C. Mahata and S. K. Roy, An inventory model with declining demand market for deteriorating items under a trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251.   Google Scholar

[23]

M. PervinS. K. Roy and G. W. Weber, A two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control & Optimization, 7 (2017), 21-50.  doi: 10.3934/naco.2017002.  Google Scholar

[24]

M. PervinS. K. Roy and G.-W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460.  doi: 10.1007/s10479-016-2355-5.  Google Scholar

[25]

M. PervinS. K. Roy and G. W. Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numerical Algebra, Control & Optimization, 8 (2018), 169-191.   Google Scholar

[26]

M. PervinS. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price-and stock-dependent demand: A trade-credit policy, Journal of Industrial & Management Optimization, 15 (2019), 1345-1373.  doi: 10.3934/jimo.2018098.  Google Scholar

[27]

J. Sadeghi and A. Dehnokhalaji, A comprehensive method for the centralized resource allocation in dea, Computers & Industrial Engineering, 127 (2019), 344-352.   Google Scholar

[28]

K. Tone, A slacks-based measure of efficiency in data envelopment analysis, European Journal of Operational Research, 130 (2001), 498-509.  doi: 10.1016/S0377-2217(99)00407-5.  Google Scholar

[29]

P. WankeC. Barros and A. Emrouznejad, A comparison between stochastic dea and fuzzy dea approaches: revisiting efficiency in angolan banks, RAIRO-Operations Research, 52 (2018), 285-303.  doi: 10.1051/ro/2016065.  Google Scholar

[30]

Q. WeiJ. Zhang and X. Zhang, An inverse dea model for inputs/outputs estimate, European Journal of Operational Research, 121 (2000), 151-163.   Google Scholar

[31]

J. WuQ. AnS. Ali and L. Liang, Dea based resource allocation considering environmental factors, Mathematical and Computer Modelling, 58 (2013), 1128-1137.   Google Scholar

[32]

L. Xiaoya and C. Jinchuan, A comprehensive dea approach for the resource allocation problem based on scale economies classification, Journal of Systems Science and Complexity, 21 (2008), 540-557.  doi: 10.1007/s11424-008-9134-6.  Google Scholar

[33]

H. YanQ. Wei and G. Hao, Dea models for resource reallocation and production input/output estimation, European Journal of Operational Research, 136 (2002), 19-31.  doi: 10.1016/S0377-2217(01)00046-7.  Google Scholar

[34]

M. Zahedi-SereshtG.-R. JahanshahlooJ. Jablonsky and S. Asghariniya, A new monte carlo based procedure for complete ranking efficient units in dea models, Numerical Algebra, Control & Optimization, 7 (2017), 403-416.  doi: 10.3934/naco.2017025.  Google Scholar

show all references

References:
[1]

A. D. Athanassopoulos, Goal programming & data envelopment analysis (godea) for target-based multi-level planning: allocating central grants to the greek local authorities, European Journal of Operational Research, 87 (1995), 535-550.   Google Scholar

[2]

J. E. Beasley, Allocating fixed costs and resources via data envelopment analysis, European Journal of Operational Research, 147 (2003), 198-216.   Google Scholar

[3]

A. Charnes and W. W. Cooper, Programming with linear fractional functionals, Naval Research Logistics Quarterly, 9 (1962), 181-186.  doi: 10.1002/nav.3800090303.  Google Scholar

[4]

A. CharnesW. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8.  Google Scholar

[5]

A. DehnokhalajiM. Ghiyasi and P. Korhonen, Resource allocation based on cost efficiency, Journal of the Operational Research Society, 68 (2017), 1279-1289.   Google Scholar

[6]

J. DuL. LiangY. Chen and G. B. Bi, Dea-based production planning, Omega, 38 (2010), 105-112.   Google Scholar

[7]

A. Emrouznejad and K. De Witte, Cooper-framework: A unified process for non-parametric projects, European Journal of Operational Research, 207 (2010), 1573-1586.   Google Scholar

[8]

L. Fang and H. Li, Centralized resource allocation based on the cost–revenue analysis, Computers & Industrial Engineering, 85 (2015), 395-401.   Google Scholar

[9]

M. J. Farrell, The measurement of productive efficiency, Journal of the Royal Statistical Society, Series A (General), 253–290. Google Scholar

[10]

S. GattoufiG. R. Amin and A. Emrouznejad, A new inverse dea method for merging banks, IMA Journal of Management Mathematics, 25 (2014), 73-87.   Google Scholar

[11]

B. GolanyF. Phillips and J. Rousseau, Models for improved effectiveness based on dea efficiency results, IIE Transactions, 25 (1993), 2-10.   Google Scholar

[12]

B. Golany and E. Tamir, Evaluating efficiency-effectiveness-equality trade-offs: A data envelopment analysis approach, Management Science, 41 (1995), 1172-1184.   Google Scholar

[13]

G. R. JahanshahlooJ. Sadeghi and M. Khodabakhshi, Fair ranking of the decision making units using optimistic and pessimistic weights in data envelopment analysis, RAIRO-Operations Research, 51 (2017), 253-260.  doi: 10.1051/ro/2016023.  Google Scholar

[14]

G. R. JahanshahlooJ. Sadeghi and M. Khodabakhshi, Proposing a method for fixed cost allocation using dea based on the efficiency invariance and common set of weights principles, Mathematical Methods of Operations Research, 85 (2007), 1-18.  doi: 10.1007/s00186-016-0563-z.  Google Scholar

[15]

P. Korhonen and M. Syrjänen, Resource allocation based on efficiency analysis, Management Science, 50 (2004), 1134-1144.   Google Scholar

[16]

F. LiQ. Zhu and L. Liang, Allocating a fixed cost based on a dea-game cross efficiency approach, Expert Systems with Applications, 96 (2018), 196-207.  doi: 10.1007/s11424-015-4211-0.  Google Scholar

[17]

F. Li, Q. Zhu and L. Liang, A new data envelopment analysis based approach for fixed cost allocation, Annals of Operations Research, 1–26. Google Scholar

[18]

F. H. LotfiA. Hatami-MarbiniP. J. AgrellN. Aghayi and K. Gholami, Allocating fixed resources and setting targets using a common-weights dea approach, Computers & Industrial Engineering, 64 (2013), 631-640.   Google Scholar

[19]

S. LozanoG. Villa and B. Adenso-Dıaz, Centralised target setting for regional recycling operations using dea, Omega, 32 (2004), 101-110.   Google Scholar

[20]

S. Lozano and G. Villa, Centralized resource allocation using data envelopment analysis, Journal of Productivity Analysis, 22 (2004), 143-161.   Google Scholar

[21]

N. NasrabadiA. DehnokhalajiN. A. KianiP. J. Korhonen and J. Wallenius, Resource allocation for performance improvement, Annals of Operations Research, 196 (2012), 459-468.  doi: 10.1007/s10479-011-1016-y.  Google Scholar

[22]

M. PervinG. C. Mahata and S. K. Roy, An inventory model with declining demand market for deteriorating items under a trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251.   Google Scholar

[23]

M. PervinS. K. Roy and G. W. Weber, A two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control & Optimization, 7 (2017), 21-50.  doi: 10.3934/naco.2017002.  Google Scholar

[24]

M. PervinS. K. Roy and G.-W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460.  doi: 10.1007/s10479-016-2355-5.  Google Scholar

[25]

M. PervinS. K. Roy and G. W. Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numerical Algebra, Control & Optimization, 8 (2018), 169-191.   Google Scholar

[26]

M. PervinS. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price-and stock-dependent demand: A trade-credit policy, Journal of Industrial & Management Optimization, 15 (2019), 1345-1373.  doi: 10.3934/jimo.2018098.  Google Scholar

[27]

J. Sadeghi and A. Dehnokhalaji, A comprehensive method for the centralized resource allocation in dea, Computers & Industrial Engineering, 127 (2019), 344-352.   Google Scholar

[28]

K. Tone, A slacks-based measure of efficiency in data envelopment analysis, European Journal of Operational Research, 130 (2001), 498-509.  doi: 10.1016/S0377-2217(99)00407-5.  Google Scholar

[29]

P. WankeC. Barros and A. Emrouznejad, A comparison between stochastic dea and fuzzy dea approaches: revisiting efficiency in angolan banks, RAIRO-Operations Research, 52 (2018), 285-303.  doi: 10.1051/ro/2016065.  Google Scholar

[30]

Q. WeiJ. Zhang and X. Zhang, An inverse dea model for inputs/outputs estimate, European Journal of Operational Research, 121 (2000), 151-163.   Google Scholar

[31]

J. WuQ. AnS. Ali and L. Liang, Dea based resource allocation considering environmental factors, Mathematical and Computer Modelling, 58 (2013), 1128-1137.   Google Scholar

[32]

L. Xiaoya and C. Jinchuan, A comprehensive dea approach for the resource allocation problem based on scale economies classification, Journal of Systems Science and Complexity, 21 (2008), 540-557.  doi: 10.1007/s11424-008-9134-6.  Google Scholar

[33]

H. YanQ. Wei and G. Hao, Dea models for resource reallocation and production input/output estimation, European Journal of Operational Research, 136 (2002), 19-31.  doi: 10.1016/S0377-2217(01)00046-7.  Google Scholar

[34]

M. Zahedi-SereshtG.-R. JahanshahlooJ. Jablonsky and S. Asghariniya, A new monte carlo based procedure for complete ranking efficient units in dea models, Numerical Algebra, Control & Optimization, 7 (2017), 403-416.  doi: 10.3934/naco.2017025.  Google Scholar

Figure 1.  Farell's frontier before resource allocation
Figure 2.  Farell's frontier after resource allocation in cases Ⅰ and Ⅱ
Table 1.  Data set and results of numerical example
Case Ⅰ Case Ⅱ
$ DMU $ $ I_1 $ $ I_2 $ $ O $ $ I_1 $ $ I_2 $ $ O $ $ I_1 $ $ I_2 $ $ O $
A 8 24 4 6.4 24 4.1 6.4 21.32 4.2
B 27 27 9 32.4 27 8.55 32.4 32.4 9
C 56 8 8 54.3 8 7.6 67.2 9.6 8
D 8 14 2 6.4 14 2.1 6.4 11.2 2.1
E 42 24 6 33.6 24 6.3 33.6 20.28 6.3
F 32 8 4 25.6 8 4.2 30.18 9.6 4.2
G 30 3 3 24 3 3.15 26.82 3.6 3.15
Central 203 108 36 182.7 108 36 203 108 36.95
Case Ⅰ Case Ⅱ
$ DMU $ $ I_1 $ $ I_2 $ $ O $ $ I_1 $ $ I_2 $ $ O $ $ I_1 $ $ I_2 $ $ O $
A 8 24 4 6.4 24 4.1 6.4 21.32 4.2
B 27 27 9 32.4 27 8.55 32.4 32.4 9
C 56 8 8 54.3 8 7.6 67.2 9.6 8
D 8 14 2 6.4 14 2.1 6.4 11.2 2.1
E 42 24 6 33.6 24 6.3 33.6 20.28 6.3
F 32 8 4 25.6 8 4.2 30.18 9.6 4.2
G 30 3 3 24 3 3.15 26.82 3.6 3.15
Central 203 108 36 182.7 108 36 203 108 36.95
Table 2.  Results of numerical example
Case Ⅰ Case Ⅱ
$ DMU $ $ p_j $ $ p_j^\prime $ $ \theta_j $ $ \theta_j^\prime $ $ p_j $ $ p_j^\prime $ $ \theta_j $ $ \theta_j^\prime $
A -0.0457 -0.0239 1 1 -0.0457 -0.0091 1 1
B 0 0 1 1 0 0 1 1
C -0.0047 0 1 1 -0.0047 0 1 1
D -0.042 -0.0287 0.63 0.7 -0.042 -0.0151 0.59 0.73
E -0.0869 -0.0459 0.59 0.77 -0.0869 -0.0151 0.59 0.91
F -0.0307 -0.0037 0.75 0.97 -0.0307 -0.0099 0.69 0.9
G -0.0239 -0.0028 0.85 1 -0.0239 0 0.85 1
Central -0.234 -0.1049 0.77 0.9 -0.234 -0.0491 0.77 0.95
Case Ⅰ Case Ⅱ
$ DMU $ $ p_j $ $ p_j^\prime $ $ \theta_j $ $ \theta_j^\prime $ $ p_j $ $ p_j^\prime $ $ \theta_j $ $ \theta_j^\prime $
A -0.0457 -0.0239 1 1 -0.0457 -0.0091 1 1
B 0 0 1 1 0 0 1 1
C -0.0047 0 1 1 -0.0047 0 1 1
D -0.042 -0.0287 0.63 0.7 -0.042 -0.0151 0.59 0.73
E -0.0869 -0.0459 0.59 0.77 -0.0869 -0.0151 0.59 0.91
F -0.0307 -0.0037 0.75 0.97 -0.0307 -0.0099 0.69 0.9
G -0.0239 -0.0028 0.85 1 -0.0239 0 0.85 1
Central -0.234 -0.1049 0.77 0.9 -0.234 -0.0491 0.77 0.95
Table 3.  Data set and results of numerical example
Case Ⅰ Case Ⅱ
$ DMU $ $ I_1 $ $ I_2 $ $ O_1 $ $ O_2 $ $ I_1 $ $ I_2 $ $ O_1 $ $ O_2 $ $ I_1 $ $ I_2 $ $ O_1 $ $ O_2 $
1 79.1 4.99 115.3 1.71 87.01 4.99 121.06 1.8 71.19 4.49 121.06 1.8
2 60.1 3.3 75.2 1.81 66.11 3.3 78.96 1.9 54.09 2.97 78.96 1.9
3 126.7 8.12 225.5 10.39 139.37 8.12 214.22 9.87 139.37 8.93 225.5 10.91
4 153.9 6.7 185.6 10.42 169.29 6.7 194.88 10.94 138.51 7.37 194.88 10.94
5 65.7 4.74 84.5 2.36 72.27 4.74 88.73 2.48 59.13 4.27 88.73 2.48
6 76.8 4.08 103.3 4.35 84.48 4.08 108.46 4.57 69.12 4.24 108.46 4.57
7 50.2 2.53 78.8 0.16 55.22 2.53 82.74 0.17 55.22 2.78 82.74 0.17
8 44.8 2.47 59.3 1.3 49.28 2.47 62.27 1.37 40.32 2.72 62.27 1.37
9 48.1 2.32 65.7 1.49 52.91 2.32 68.99 1.56 43.29 2.55 68.99 1.56
10 89.7 4.91 163.2 6.26 98.67 4.91 155.04 5.95 98.67 5.4 163.2 6.26
11 56.9 2.24 70.7 2.8 62.59 2.24 74.23 2.94 51.21 2.46 74.24 2.94
12 112.6 5.42 142.6 2.75 123.86 5.42 149.73 2.89 101.34 4.88 149.73 2.89
13 106.9 6.28 127.8 2.7 117.59 6.28 134.19 2.84 96.21 5.65 134.19 2.84
14 54.9 3.14 62.4 1.42 60.39 3.14 65.52 1.49 60.39 3.45 65.52 1.49
15 48.8 4.43 55.2 1.38 53.68 4.43 57.96 1.45 53.68 3.99 57.96 1.45
16 59.2 3.98 95.9 0.74 65.12 3.98 100.7 0.78 65.12 4.38 100.7 0.78
17 74.5 5.32 121.6 3.06 81.95 5.32 127.68 3.21 67.05 5.85 127.68 3.21
18 94.6 3.69 107 2.98 104.06 3.69 112.35 3.13 102.17 3.32 112.35 3.13
19 47 3 65.4 0.62 51.7 3 68.67 0.65 42.3 2.7 68.67 0.65
20 54.6 3.87 71 0.01 60.06 3.87 74.55 0.01 57.72 3.48 74.55 0.01
21 90.1 3.31 81.2 5.12 99.11 3.31 85.26 5.38 99.11 2.98 85.26 5.38
22 95.2 4.25 128.3 3.89 104.72 4.25 134.72 4.08 104.72 3.83 134.72 4.08
23 80.1 3.79 135 4.73 88.11 3.79 135.39 4.97 87.47 4.17 141.75 4.97
24 68.7 2.99 98.9 1.86 75.57 2.99 103.85 1.95 75.57 2.69 103.85 1.95
25 62.3 3.1 66.7 7.41 68.53 3.1 63.37 7.04 68.53 3.41 66.7 7.41
Central 1901.5 102.97 2586.1 81.72 2091.65 102.97 2663.52 83.42 1901.5 102.96 2692.66 85.14
Case Ⅰ Case Ⅱ
$ DMU $ $ I_1 $ $ I_2 $ $ O_1 $ $ O_2 $ $ I_1 $ $ I_2 $ $ O_1 $ $ O_2 $ $ I_1 $ $ I_2 $ $ O_1 $ $ O_2 $
1 79.1 4.99 115.3 1.71 87.01 4.99 121.06 1.8 71.19 4.49 121.06 1.8
2 60.1 3.3 75.2 1.81 66.11 3.3 78.96 1.9 54.09 2.97 78.96 1.9
3 126.7 8.12 225.5 10.39 139.37 8.12 214.22 9.87 139.37 8.93 225.5 10.91
4 153.9 6.7 185.6 10.42 169.29 6.7 194.88 10.94 138.51 7.37 194.88 10.94
5 65.7 4.74 84.5 2.36 72.27 4.74 88.73 2.48 59.13 4.27 88.73 2.48
6 76.8 4.08 103.3 4.35 84.48 4.08 108.46 4.57 69.12 4.24 108.46 4.57
7 50.2 2.53 78.8 0.16 55.22 2.53 82.74 0.17 55.22 2.78 82.74 0.17
8 44.8 2.47 59.3 1.3 49.28 2.47 62.27 1.37 40.32 2.72 62.27 1.37
9 48.1 2.32 65.7 1.49 52.91 2.32 68.99 1.56 43.29 2.55 68.99 1.56
10 89.7 4.91 163.2 6.26 98.67 4.91 155.04 5.95 98.67 5.4 163.2 6.26
11 56.9 2.24 70.7 2.8 62.59 2.24 74.23 2.94 51.21 2.46 74.24 2.94
12 112.6 5.42 142.6 2.75 123.86 5.42 149.73 2.89 101.34 4.88 149.73 2.89
13 106.9 6.28 127.8 2.7 117.59 6.28 134.19 2.84 96.21 5.65 134.19 2.84
14 54.9 3.14 62.4 1.42 60.39 3.14 65.52 1.49 60.39 3.45 65.52 1.49
15 48.8 4.43 55.2 1.38 53.68 4.43 57.96 1.45 53.68 3.99 57.96 1.45
16 59.2 3.98 95.9 0.74 65.12 3.98 100.7 0.78 65.12 4.38 100.7 0.78
17 74.5 5.32 121.6 3.06 81.95 5.32 127.68 3.21 67.05 5.85 127.68 3.21
18 94.6 3.69 107 2.98 104.06 3.69 112.35 3.13 102.17 3.32 112.35 3.13
19 47 3 65.4 0.62 51.7 3 68.67 0.65 42.3 2.7 68.67 0.65
20 54.6 3.87 71 0.01 60.06 3.87 74.55 0.01 57.72 3.48 74.55 0.01
21 90.1 3.31 81.2 5.12 99.11 3.31 85.26 5.38 99.11 2.98 85.26 5.38
22 95.2 4.25 128.3 3.89 104.72 4.25 134.72 4.08 104.72 3.83 134.72 4.08
23 80.1 3.79 135 4.73 88.11 3.79 135.39 4.97 87.47 4.17 141.75 4.97
24 68.7 2.99 98.9 1.86 75.57 2.99 103.85 1.95 75.57 2.69 103.85 1.95
25 62.3 3.1 66.7 7.41 68.53 3.1 63.37 7.04 68.53 3.41 66.7 7.41
Central 1901.5 102.97 2586.1 81.72 2091.65 102.97 2663.52 83.42 1901.5 102.96 2692.66 85.14
Table 4.  Results of numerical example
Case Ⅰ Case Ⅱ
$ DMU $ $ p_j $ $ p_j^\prime $ $ \theta_j $ $ \theta_j^\prime $ $ p_j $ $ p_j^\prime $ $ \theta_j $ $ \theta_j^\prime $
1 -0.0237 -0.0205 0.359 0.418 -0.0246 -0.0136 0.343 0.575
2 -0.0176 -0.0154 0.431 0.494 -0.0178 -0.0098 0.423 0.62
3 0 0 1 1 -0.0015 0 1 1
4 -0.0196 -0.0109 0.849 0.923 -0.0181 -0.0006 0.849 0.992
5 -0.0213 -0.0188 0.5 0.566 -0.023 -0.0138 0.436 0.618
6 -0.0127 -0.0085 0.712 0.809 -0.0128 -0.0036 0.709 0.915
7 -0.0146 -0.0128 0.078 0.095 -0.0143 -0.0154 0.078 0.085
8 -0.0125 -0.0107 0.431 0.502 -0.0126 -0.0092 0.427 0.566
9 -0.0113 -0.0093 0.516 0.57 -0.011 -0.0068 0.516 0.659
10 0 0 1 1 0 0 1 1
11 -0.0097 -0.0069 0.805 0.86 -0.0088 -0.0027 0.805 0.902
12 -0.0321 -0.0281 0.401 0.451 -0.0315 -0.0169 0.401 0.59
13 -0.0362 -0.0328 0.367 0.418 -0.0371 -0.0237 0.346 0.502
14 -0.019 -0.0174 0.366 0.414 -0.0194 -0.0214 0.346 0.37
15 -0.0225 -0.0211 0.393 0.444 -0.0248 -0.0236 0.306 0.353
16 -0.018 -0.0156 0.243 0.3 -0.019 -0.0207 0.232 0.247
17 -0.0159 -0.0118 0.628 0.757 -0.0175 -0.011 0.583 1
18 -0.0247 -0.0214 0.562 0.617 -0.0233 -0.0196 0.562 0.739
19 -0.0164 -0.0149 0.229 0.267 -0.017 -0.011 0.215 0.332
20 -0.0243 -0.0232 0.003 0.004 -0.0255 -0.0237 0.003 0.004
21 -0.02 -0.0164 0.729 0.799 -0.0185 -0.0153 0.729 1
22 -0.0182 -0.0137 0.677 0.735 -0.0173 -0.0125 0.677 0.914
23 -0.0038 0 1 1 -0.0031 0 1 1
24 -0.0146 -0.0117 0.573 0.617 -0.0138 -0.0107 0.573 1
25 0 0 1 1 0 0 1 1
central -0.4088 -0.3419 0.591 0.658 -0.4123 -0.2856 0.588 0.714
Case Ⅰ Case Ⅱ
$ DMU $ $ p_j $ $ p_j^\prime $ $ \theta_j $ $ \theta_j^\prime $ $ p_j $ $ p_j^\prime $ $ \theta_j $ $ \theta_j^\prime $
1 -0.0237 -0.0205 0.359 0.418 -0.0246 -0.0136 0.343 0.575
2 -0.0176 -0.0154 0.431 0.494 -0.0178 -0.0098 0.423 0.62
3 0 0 1 1 -0.0015 0 1 1
4 -0.0196 -0.0109 0.849 0.923 -0.0181 -0.0006 0.849 0.992
5 -0.0213 -0.0188 0.5 0.566 -0.023 -0.0138 0.436 0.618
6 -0.0127 -0.0085 0.712 0.809 -0.0128 -0.0036 0.709 0.915
7 -0.0146 -0.0128 0.078 0.095 -0.0143 -0.0154 0.078 0.085
8 -0.0125 -0.0107 0.431 0.502 -0.0126 -0.0092 0.427 0.566
9 -0.0113 -0.0093 0.516 0.57 -0.011 -0.0068 0.516 0.659
10 0 0 1 1 0 0 1 1
11 -0.0097 -0.0069 0.805 0.86 -0.0088 -0.0027 0.805 0.902
12 -0.0321 -0.0281 0.401 0.451 -0.0315 -0.0169 0.401 0.59
13 -0.0362 -0.0328 0.367 0.418 -0.0371 -0.0237 0.346 0.502
14 -0.019 -0.0174 0.366 0.414 -0.0194 -0.0214 0.346 0.37
15 -0.0225 -0.0211 0.393 0.444 -0.0248 -0.0236 0.306 0.353
16 -0.018 -0.0156 0.243 0.3 -0.019 -0.0207 0.232 0.247
17 -0.0159 -0.0118 0.628 0.757 -0.0175 -0.011 0.583 1
18 -0.0247 -0.0214 0.562 0.617 -0.0233 -0.0196 0.562 0.739
19 -0.0164 -0.0149 0.229 0.267 -0.017 -0.011 0.215 0.332
20 -0.0243 -0.0232 0.003 0.004 -0.0255 -0.0237 0.003 0.004
21 -0.02 -0.0164 0.729 0.799 -0.0185 -0.0153 0.729 1
22 -0.0182 -0.0137 0.677 0.735 -0.0173 -0.0125 0.677 0.914
23 -0.0038 0 1 1 -0.0031 0 1 1
24 -0.0146 -0.0117 0.573 0.617 -0.0138 -0.0107 0.573 1
25 0 0 1 1 0 0 1 1
central -0.4088 -0.3419 0.591 0.658 -0.4123 -0.2856 0.588 0.714
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