June  2020, 10(2): 127-142. 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

Fund Project: We thank four reviewers of this journal for their most constructive comments

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 and Optimization, 2020, 10 (2) : 127-142. 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. 

[2]

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

[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.

[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.

[5]

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

[6]

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

[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. 

[8]

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

[9]

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

[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. 

[11]

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

[12]

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

[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.

[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.

[15]

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

[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.

[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.

[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. 

[19]

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

[20]

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

[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.

[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. 

[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.

[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.

[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. 

[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.

[27]

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

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[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.

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. 

[2]

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

[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.

[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.

[5]

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

[6]

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

[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. 

[8]

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

[9]

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

[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. 

[11]

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

[12]

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

[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.

[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.

[15]

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

[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.

[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.

[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. 

[19]

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

[20]

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

[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.

[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. 

[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.

[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.

[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. 

[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.

[27]

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

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[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.

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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