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doi: 10.3934/naco.2021032
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Second order cone programming formulation of the fixed cost allocation in DEA based on Nash bargaining game

Narges Torabi Golsefid 1, and  Maziar Salahi 2,,

1. 

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
2. 

Department of Applied Mathematics, Faculty of Mathematical Sciences, and Center of Excellence for Mathematical Modeling, Optimization and Combinatorial Computing (MMOCC), University of Guilan, Rasht, Iran

* Corresponding author: Maziar Salahi

Received  March 2020 Revised  June 2021 Early access August 2021

Fund Project: The authors would like to thank Center of Excellence for Mathematical Modeling, Optimization and Combinatorial Computing (MMOCC), University of Guilan for supporting this research

A vital issue in many organizations is the fair allocation of fixed cost among its subsets. In this paper, using data envelopment analysis, first we study fixed cost allocation based on both additive and multiplicative efficiency decompositions in the cooperative context for a two-stage structure in the presence of exogenous inputs and outputs. A conic relaxation formulation of multiplicative decomposition is given. Then, fixed cost allocation based on the leader-follower paradigm are presented. In the sequel, for allocating a fair fixed cost between the stages, using the results of the leader-follower model, we present the nonlinear Nash bargaining game model that independent of the efficiency score of each unit, allocates fixed cost to the stages. The nonlinear model is reformulated as a second order cone program which is an imporvement over the parametric linear models in the literature. Finally, two examples are used to illustrate the proposed models and compare their results with the existing models.

Keywords: Cost allocation, Data envelopment analysis, Network, Nash bargaining game, Second order cone program.
Mathematics Subject Classification: Primary: 90B50; Secondary: 90C25.
Citation: Narges Torabi Golsefid, Maziar Salahi. Second order cone programming formulation of the fixed cost allocation in DEA based on Nash bargaining game. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021032
References:
[1]

A. Amirteimoori, A DEA two-stage decision processes with shared resources, Central European Journal of Operations Research, 21 (2013), 141-151.  doi: 10.1007/s10100-011-0218-3.  Google Scholar

[2]

Q. An, P. Wang, A. Emroznejad and J. Hu, Fixed cost allocationbased on the principle of efficincy invariance in two-stage systems, European Journal of Operational Research, 283 (2020), 662-675.  doi: 10.1016/j.ejor.2019.11.031.  Google Scholar

[3]

Q. An, Y. Wen, T. Ding and Y. Li, Resource sharing and payoff allocation in a three-stage system: integrating network DEA with the Shaplley value method, Omega, 85 (2018), 16-25.   Google Scholar

[4]

J. E. Beasley, Allocating fixed costs and resources via data envelopment analysis, European Journal of Operational Research, 147 (2003), 198-216.  doi: 10.1016/S0377-2217(02)00244-8.  Google Scholar

[5] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge: Cambridge University Press, 2004.  doi: 10.1017/CBO9780511804441.  Google Scholar
[6]

A. Charnes and W. W. Cooper, Programming with linear fractional functions, Naval Research Logistics Quarterly, 9 (1962), 181-185.  doi: 10.1002/nav.3800090303.  Google Scholar

[7]

A. Charnes, W. 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

[8]

C. M. Chen and M. A. Delmas, Measuring eco-inefficiency: a new frontier approach, Operations Research, 60 (2012), 1064-1079.  doi: 10.1287/opre.1120.1094.  Google Scholar

[9]

Y. Chen, J. Du, H. D. Sherman and J. Zhu, DEA model with shared resources and efficiency decomposition, European Journal of Operational Research, 207 (2010), 339-349.  doi: 10.1016/j.ejor.2010.03.031.  Google Scholar

[10]

L. Chen, F. Lai, Y. M. Wang, Y. Huang and F. M. Wu, A two-stage network data envelopment analysis approach for measuring and decomposing environmental efficiency, Computers and Industrial Engineering, 119 (2018), 388-403.  doi: 10.1016/j.cie.2018.04.011.  Google Scholar

[11]

Y. Chen, W. D. Cook, N. Li and J. Zhu, Additive efficiency decomposition in two-stage DEA, European Journal of Operational Research, 196 (2009), 1170-1176.  doi: 10.1016/j.ejor.2008.05.011.  Google Scholar

[12]

K. Chen and J. Zhu, Second order cone programming approach to two-stage network data envelopment analysis, European Journal of Operational Research, 262 (2017), 231-238.  doi: 10.1016/j.ejor.2017.03.074.  Google Scholar

[13]

K. Chen, W. D. Cook and J. Zhu, A conic relaxation model for searching for the global optimum of network data envelopment analysis, European Journal of Operational Research, 280 (2020), 242-253.  doi: 10.1016/j.ejor.2019.07.012.  Google Scholar

[14]

J. Chu, J. Wu, C. Chu and T. Zhang, DEA-based fixed cost allocation in two-stage systems: leader-follower and satisfaction degree bargaining game approaches, Omega, 94 (2020), ID: 102054. doi: 10.1016/j.omega.2019.03.012.  Google Scholar

[15]

W. D. Cook and M. Kress, Characterizing an equitable allocation of shared costs: A DEA approach, European Journal of Operational Research, 119 (1999), 652-661.  doi: 10.1016/S0377-2217(98)00337-3.  Google Scholar

[16]

D. K. Despotis, G. Koronakos and D. Sotiros, Composition versus decomposition in two-stage newwork DEA: A reverse approach, Journal of Productivity Analysis, 45 (2014), 71-87.   Google Scholar

[17]

T. Ding, Q. Zhu, B. Zhang and L. Liang, Centralized fixed cost allocation for generalized two-stage network DEA, INFOR: Information Systems and Operational Research, 57 (2019), 123-140. doi: 10.1080/03155986.2017.1397897.  Google Scholar

[18]

J. Du, W. D. Cook, L. Liang and J. Zhu, Fixed cost and resource allocation based on DEA cross-efficiency, European Journal of Operational Research, 235 (2014), 206-214.  doi: 10.1016/j.ejor.2013.10.002.  Google Scholar

[19]

L. Fang, Centralized resource allocation based on efficiency analysis for step-by step improvement paths, Omega, 51 (2015), 24-28.  doi: 10.1016/j.omega.2014.09.003.  Google Scholar

[20]

C. Feng, F. Chu, J. Ding, G. Bi and L. Liang, Carbon emissions abatement (cea) allocation and compensation schemes based on DEA, Omega, 53 (2015), 78-89.  doi: 10.1016/j.omega.2014.12.005.  Google Scholar

[21]

C. Guo, F. Wei and Y. Chen, A note on second order cone programming approach to two-stage network data envelopment analysis, European Journal of Operational Research, 263 (2017), 733-735.  doi: 10.1016/j.ejor.2017.06.011.  Google Scholar

[22]

Z. Y. Hua, Y. Bian and L. Liang, Eco-efficiency analysis of paper mills along the huai river: an extended DEA approach, Omega, 35 (2007), 578-587.  doi: 10.1016/j.omega.2005.11.001.  Google Scholar

[23]

C. Kao and S. N. Hwang, Efficiency decomposition in two-stage data envelopment analysis: an application to non-life insurance companies in Taiwan, European Journal of Operational Research, 185 (2008), 418-429.  doi: 10.1016/j.ejor.2006.11.041.  Google Scholar

[24]

F. Li, Q. Zhu and Z. Chen, Allocating a fixed cost across the decision making units with two-stage network structures, Omega, 83 (2019), 139-154.  doi: 10.1016/j.omega.2018.02.009.  Google Scholar

[25]

Y. Li, M. Yang, Y. Chen, Q. Dai and L. Liang, Allocating a fixed cost based on data envelopment analysis and satisfaction degree, Omega, 41 (2013), 55-60.  doi: 10.1016/j.omega.2011.02.008.  Google Scholar

[26]

Y. Li, F. Li, A. Emrouznejad, L. Liang and Q. Xie, Allocating the fixed cost: an approach based on data envelopment analysis and cooperative game, Annals of Operations Reaserch, 274 (2018), 373-394.  doi: 10.1007/s10479-018-2860-9.  Google Scholar

[27]

R. Lin, Z. Chen and Z. Li, A new approach for allocating fixed costs among decision making units, Journal of Industrial and Management Optimization, 12 (2016), 211-228.  doi: 10.3934/jimo.2016.12.211.  Google Scholar

[28]

R. Lotfi, G. W. Weber, S. M. Sajadifar and N. Mardani, Interdependent demand in the two-period newsvendor problem, Journal of Industrial and Management Optimization, 16 (2018), 117-140.  doi: 10.3934/jimo.2018143.  Google Scholar

[29]

J. F. Nash, The bargaining problem, Econometrica; Journal of Econometric Society, 18 (1950), 155-162.  doi: 10.2307/1907266.  Google Scholar

[30]

J. Nash, Two-person cooperative games, Econometrica: Journal of Econometric Society, 21 (1953), 128-140.  doi: 10.2307/1906951.  Google Scholar

[31]

J. Sadeghi, M. Ghiyasi and A. Dehnokhalaji, Resource allocation and target setting based on virtual profit improvement, Numerical Algebra, Control and Optimization, 10 (2020), 127-142.  doi: 10.3934/naco.2019043.  Google Scholar

[32]

Y. Sho, G. Bi, F. Yang and Q. Xia, Resource allocation for branch network system with considering heterogeneity based on DEA method, Central European Journal of Operations Research, 26 (2018), 1005-1025.  doi: 10.1007/s10100-018-0563-6.  Google Scholar

[33]

J. Sun, J. Wu, L. Liang, R. Y. Zhong and G. Q. Huang, Allocation of emission permits using DEA: centralised and individual points of view, International Journal of Production Research, 52 (2014), 419-435.   Google Scholar

[34]

K. Wang, W. Huang, J. Wu and Y. N. Liu, Efficiency measures of the Chinese commerical banking system using an additive two-stage DEA, Omega, 44 (2014), 5-20.  doi: 10.1016/j.omega.2013.09.005.  Google Scholar

[35]

J. Wu, Q. Zhu, X. Ji, J. Chu and L. Liang, Two-stage network processes with shared resources and resources recovered from undesirable outputs, European Journal of Operational Research, 251 (2016), 182-197.  doi: 10.1016/j.ejor.2015.10.049.  Google Scholar

[36]

G. L. Yang, Y. Y. Song, D. L. Xu and J. B. Yang, Overall efficiency and its decomposision in a two-stage network DEA model, Journal of Managment Science and Engineering, 2 (2017), 161-192.  doi: 10.1016/j.ejor.2016.08.002.  Google Scholar

[37]

M. M. Yu, L. H. Chen and H. Bo, A fixed cost allocation based on the two-stage network data envelopment approach, Journal of Business Research, 69 (2016), 1817-1822.   Google Scholar

[38]

Q. Zhang, D. Koutmos, K. Chen and J. Zhu, Using operational and stock analytics to measure airline perfoemance: A network DEA approach, Decision Sciences, 52 (2021), 720-748.  doi: 10.1111/deci.12363.  Google Scholar

[39]

W. Zhu, Q. Zhang and H. Wang, Fixed costs and shared resources allocation in two-stage network DEA, Annals of Operations Research, 278 (2019), 177-194.  doi: 10.1007/s10479-017-2599-8.  Google Scholar

show all references

References:
[1]

A. Amirteimoori, A DEA two-stage decision processes with shared resources, Central European Journal of Operations Research, 21 (2013), 141-151.  doi: 10.1007/s10100-011-0218-3.  Google Scholar

[2]

Q. An, P. Wang, A. Emroznejad and J. Hu, Fixed cost allocationbased on the principle of efficincy invariance in two-stage systems, European Journal of Operational Research, 283 (2020), 662-675.  doi: 10.1016/j.ejor.2019.11.031.  Google Scholar

[3]

Q. An, Y. Wen, T. Ding and Y. Li, Resource sharing and payoff allocation in a three-stage system: integrating network DEA with the Shaplley value method, Omega, 85 (2018), 16-25.   Google Scholar

[4]

J. E. Beasley, Allocating fixed costs and resources via data envelopment analysis, European Journal of Operational Research, 147 (2003), 198-216.  doi: 10.1016/S0377-2217(02)00244-8.  Google Scholar

[5] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge: Cambridge University Press, 2004.  doi: 10.1017/CBO9780511804441.  Google Scholar
[6]

A. Charnes and W. W. Cooper, Programming with linear fractional functions, Naval Research Logistics Quarterly, 9 (1962), 181-185.  doi: 10.1002/nav.3800090303.  Google Scholar

[7]

A. Charnes, W. 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

[8]

C. M. Chen and M. A. Delmas, Measuring eco-inefficiency: a new frontier approach, Operations Research, 60 (2012), 1064-1079.  doi: 10.1287/opre.1120.1094.  Google Scholar

[9]

Y. Chen, J. Du, H. D. Sherman and J. Zhu, DEA model with shared resources and efficiency decomposition, European Journal of Operational Research, 207 (2010), 339-349.  doi: 10.1016/j.ejor.2010.03.031.  Google Scholar

[10]

L. Chen, F. Lai, Y. M. Wang, Y. Huang and F. M. Wu, A two-stage network data envelopment analysis approach for measuring and decomposing environmental efficiency, Computers and Industrial Engineering, 119 (2018), 388-403.  doi: 10.1016/j.cie.2018.04.011.  Google Scholar

[11]

Y. Chen, W. D. Cook, N. Li and J. Zhu, Additive efficiency decomposition in two-stage DEA, European Journal of Operational Research, 196 (2009), 1170-1176.  doi: 10.1016/j.ejor.2008.05.011.  Google Scholar

[12]

K. Chen and J. Zhu, Second order cone programming approach to two-stage network data envelopment analysis, European Journal of Operational Research, 262 (2017), 231-238.  doi: 10.1016/j.ejor.2017.03.074.  Google Scholar

[13]

K. Chen, W. D. Cook and J. Zhu, A conic relaxation model for searching for the global optimum of network data envelopment analysis, European Journal of Operational Research, 280 (2020), 242-253.  doi: 10.1016/j.ejor.2019.07.012.  Google Scholar

[14]

J. Chu, J. Wu, C. Chu and T. Zhang, DEA-based fixed cost allocation in two-stage systems: leader-follower and satisfaction degree bargaining game approaches, Omega, 94 (2020), ID: 102054. doi: 10.1016/j.omega.2019.03.012.  Google Scholar

[15]

W. D. Cook and M. Kress, Characterizing an equitable allocation of shared costs: A DEA approach, European Journal of Operational Research, 119 (1999), 652-661.  doi: 10.1016/S0377-2217(98)00337-3.  Google Scholar

[16]

D. K. Despotis, G. Koronakos and D. Sotiros, Composition versus decomposition in two-stage newwork DEA: A reverse approach, Journal of Productivity Analysis, 45 (2014), 71-87.   Google Scholar

[17]

T. Ding, Q. Zhu, B. Zhang and L. Liang, Centralized fixed cost allocation for generalized two-stage network DEA, INFOR: Information Systems and Operational Research, 57 (2019), 123-140. doi: 10.1080/03155986.2017.1397897.  Google Scholar

[18]

J. Du, W. D. Cook, L. Liang and J. Zhu, Fixed cost and resource allocation based on DEA cross-efficiency, European Journal of Operational Research, 235 (2014), 206-214.  doi: 10.1016/j.ejor.2013.10.002.  Google Scholar

[19]

L. Fang, Centralized resource allocation based on efficiency analysis for step-by step improvement paths, Omega, 51 (2015), 24-28.  doi: 10.1016/j.omega.2014.09.003.  Google Scholar

[20]

C. Feng, F. Chu, J. Ding, G. Bi and L. Liang, Carbon emissions abatement (cea) allocation and compensation schemes based on DEA, Omega, 53 (2015), 78-89.  doi: 10.1016/j.omega.2014.12.005.  Google Scholar

[21]

C. Guo, F. Wei and Y. Chen, A note on second order cone programming approach to two-stage network data envelopment analysis, European Journal of Operational Research, 263 (2017), 733-735.  doi: 10.1016/j.ejor.2017.06.011.  Google Scholar

[22]

Z. Y. Hua, Y. Bian and L. Liang, Eco-efficiency analysis of paper mills along the huai river: an extended DEA approach, Omega, 35 (2007), 578-587.  doi: 10.1016/j.omega.2005.11.001.  Google Scholar

[23]

C. Kao and S. N. Hwang, Efficiency decomposition in two-stage data envelopment analysis: an application to non-life insurance companies in Taiwan, European Journal of Operational Research, 185 (2008), 418-429.  doi: 10.1016/j.ejor.2006.11.041.  Google Scholar

[24]

F. Li, Q. Zhu and Z. Chen, Allocating a fixed cost across the decision making units with two-stage network structures, Omega, 83 (2019), 139-154.  doi: 10.1016/j.omega.2018.02.009.  Google Scholar

[25]

Y. Li, M. Yang, Y. Chen, Q. Dai and L. Liang, Allocating a fixed cost based on data envelopment analysis and satisfaction degree, Omega, 41 (2013), 55-60.  doi: 10.1016/j.omega.2011.02.008.  Google Scholar

[26]

Y. Li, F. Li, A. Emrouznejad, L. Liang and Q. Xie, Allocating the fixed cost: an approach based on data envelopment analysis and cooperative game, Annals of Operations Reaserch, 274 (2018), 373-394.  doi: 10.1007/s10479-018-2860-9.  Google Scholar

[27]

R. Lin, Z. Chen and Z. Li, A new approach for allocating fixed costs among decision making units, Journal of Industrial and Management Optimization, 12 (2016), 211-228.  doi: 10.3934/jimo.2016.12.211.  Google Scholar

[28]

R. Lotfi, G. W. Weber, S. M. Sajadifar and N. Mardani, Interdependent demand in the two-period newsvendor problem, Journal of Industrial and Management Optimization, 16 (2018), 117-140.  doi: 10.3934/jimo.2018143.  Google Scholar

[29]

J. F. Nash, The bargaining problem, Econometrica; Journal of Econometric Society, 18 (1950), 155-162.  doi: 10.2307/1907266.  Google Scholar

[30]

J. Nash, Two-person cooperative games, Econometrica: Journal of Econometric Society, 21 (1953), 128-140.  doi: 10.2307/1906951.  Google Scholar

[31]

J. Sadeghi, M. Ghiyasi and A. Dehnokhalaji, Resource allocation and target setting based on virtual profit improvement, Numerical Algebra, Control and Optimization, 10 (2020), 127-142.  doi: 10.3934/naco.2019043.  Google Scholar

[32]

Y. Sho, G. Bi, F. Yang and Q. Xia, Resource allocation for branch network system with considering heterogeneity based on DEA method, Central European Journal of Operations Research, 26 (2018), 1005-1025.  doi: 10.1007/s10100-018-0563-6.  Google Scholar

[33]

J. Sun, J. Wu, L. Liang, R. Y. Zhong and G. Q. Huang, Allocation of emission permits using DEA: centralised and individual points of view, International Journal of Production Research, 52 (2014), 419-435.   Google Scholar

[34]

K. Wang, W. Huang, J. Wu and Y. N. Liu, Efficiency measures of the Chinese commerical banking system using an additive two-stage DEA, Omega, 44 (2014), 5-20.  doi: 10.1016/j.omega.2013.09.005.  Google Scholar

[35]

J. Wu, Q. Zhu, X. Ji, J. Chu and L. Liang, Two-stage network processes with shared resources and resources recovered from undesirable outputs, European Journal of Operational Research, 251 (2016), 182-197.  doi: 10.1016/j.ejor.2015.10.049.  Google Scholar

[36]

G. L. Yang, Y. Y. Song, D. L. Xu and J. B. Yang, Overall efficiency and its decomposision in a two-stage network DEA model, Journal of Managment Science and Engineering, 2 (2017), 161-192.  doi: 10.1016/j.ejor.2016.08.002.  Google Scholar

[37]

M. M. Yu, L. H. Chen and H. Bo, A fixed cost allocation based on the two-stage network data envelopment approach, Journal of Business Research, 69 (2016), 1817-1822.   Google Scholar

[38]

Q. Zhang, D. Koutmos, K. Chen and J. Zhu, Using operational and stock analytics to measure airline perfoemance: A network DEA approach, Decision Sciences, 52 (2021), 720-748.  doi: 10.1111/deci.12363.  Google Scholar

[39]

W. Zhu, Q. Zhang and H. Wang, Fixed costs and shared resources allocation in two-stage network DEA, Annals of Operations Research, 278 (2019), 177-194.  doi: 10.1007/s10479-017-2599-8.  Google Scholar

Figure 1.  Two-stage system with fixed costss
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14] (b) model (23) and Li et al. [24]">Figure 2.  The allocated cost to stages: (a) model (23) and Chu et al. [14] (b) model (23) and Li et al. [24]
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Table 1.  Classification of the literature
Reference exogenous input or output efficiency decomposition system approach of fair cost allocation allocation principles
Li et al. [25] $ \times $ - single-stage Satisfaction degree Efficiency maximization
Du et al. [18] $ \times $ - single-stage Cross efficiency Efficiency maximization
Yu et al. [37] $ \times $ AED two-stage Cross efficiency Effficiency maximization
Zhu et al. [39] $ \checkmark $ AED two-stage Based on different objectives in reality Effficiency maximization
Ding et al. [17] $ \checkmark $ AED two-stage Maximal average satisfaction degree -
An et al. [3] $ \checkmark $ - three-stage The Shapley value -
Li et al. [24] $ \times $ AED two-stage By repeatedly minimizing the maximum deviation between the efficient and size allocation Efficiency invariance & efficiency maximization
Li et al.[26] $ \times $ - single-stage Cooperative game -
Chu et al. [14] $ \times $ AED two-stage Leader-follower & the Nash bargaining -
The present study $ \checkmark $ AED and MED two-stage Cooperative, leader-follower & the Nash bargaining Independent from efficiency rank and size
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Reference exogenous input or output efficiency decomposition system approach of fair cost allocation allocation principles
Li et al. [25] $ \times $ - single-stage Satisfaction degree Efficiency maximization
Du et al. [18] $ \times $ - single-stage Cross efficiency Efficiency maximization
Yu et al. [37] $ \times $ AED two-stage Cross efficiency Effficiency maximization
Zhu et al. [39] $ \checkmark $ AED two-stage Based on different objectives in reality Effficiency maximization
Ding et al. [17] $ \checkmark $ AED two-stage Maximal average satisfaction degree -
An et al. [3] $ \checkmark $ - three-stage The Shapley value -
Li et al. [24] $ \times $ AED two-stage By repeatedly minimizing the maximum deviation between the efficient and size allocation Efficiency invariance & efficiency maximization
Li et al.[26] $ \times $ - single-stage Cooperative game -
Chu et al. [14] $ \times $ AED two-stage Leader-follower & the Nash bargaining -
The present study $ \checkmark $ AED and MED two-stage Cooperative, leader-follower & the Nash bargaining Independent from efficiency rank and size
Table 2.  Notations
Notation Description
$ n $ number of DMUs
$ R $ allocated total cost to system
$ m_1 $ number of exogenous inputs of stage 1
$ m_2 $ number of exogenous inputs of stage 2
$ g_1 $ number of exogenous outputs of stage 1
$ g_2 $ number of exogenous outputs of stage 2
$ l $ number of intermediate products
$ x_{ij}^1 $ $ i $th input of DMU$ j $ of stage 1
$ x_{hj}^2 $ $ h $th input of DMU$ j $ of stage 2
$ r_{j}^1 $ allocated cost to DMU$ j $ of stage 1
$ r_{j}^2 $ allocated cost to DMU$ j $ of stage 2
$ y_{pj}^1 $ $ p $th output of DMU$ j $ of stage 1
$ y_{fj}^2 $ $ f $th output of DMU$ j $ of stage 2
$ z_{dj} $ $ d $th intermediate product of DMU$ j $
$ v_i $ weight assigned to the $ i $th input of stage 1
$ q_h $ weight assigned to the $ h $th input of stage 2
$ \mu_p $ weight assigned to the $ p $th output of stage 1
$ u_f $ weight assigned to the $ f $th output of stage 2
$ \gamma_d^1 $ weight assigned to the $ d $th intermediate product of stage 1
$ \gamma_d^2 $ weight assigned to the $ d $th intermediate product of stage 2
Table Options

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Notation Description
$ n $ number of DMUs
$ R $ allocated total cost to system
$ m_1 $ number of exogenous inputs of stage 1
$ m_2 $ number of exogenous inputs of stage 2
$ g_1 $ number of exogenous outputs of stage 1
$ g_2 $ number of exogenous outputs of stage 2
$ l $ number of intermediate products
$ x_{ij}^1 $ $ i $th input of DMU$ j $ of stage 1
$ x_{hj}^2 $ $ h $th input of DMU$ j $ of stage 2
$ r_{j}^1 $ allocated cost to DMU$ j $ of stage 1
$ r_{j}^2 $ allocated cost to DMU$ j $ of stage 2
$ y_{pj}^1 $ $ p $th output of DMU$ j $ of stage 1
$ y_{fj}^2 $ $ f $th output of DMU$ j $ of stage 2
$ z_{dj} $ $ d $th intermediate product of DMU$ j $
$ v_i $ weight assigned to the $ i $th input of stage 1
$ q_h $ weight assigned to the $ h $th input of stage 2
$ \mu_p $ weight assigned to the $ p $th output of stage 1
$ u_f $ weight assigned to the $ f $th output of stage 2
$ \gamma_d^1 $ weight assigned to the $ d $th intermediate product of stage 1
$ \gamma_d^2 $ weight assigned to the $ d $th intermediate product of stage 2
Table 3.  Data of the 27 bank branches
branches Inputs Exogenous input Intermediate products Exogenous output Outputs
1 25 619 538 854 77237 34224 2101 2947 913 224
2 27 419 489 125 88031 56559 1023 3138 478 516
3 40 1670 1459 120 164053 62776 1440 5494 1242 877
4 42 2931 1497 86 145369 65226 2458 3144 870 1138
5 52 2587 797 133 166424 85886 2202 6705 854 618
6 45 2181 697 149 215695 30179 1653 8487 1023 2096
7 33 989 1217 144 114043 43447 1919 4996 767 713
8 107 6277 2189 735 727699 294126 2486 21265 6282 6287
9 88 3197 949 101 186642 53223 648 8574 1537 1739
10 146 6222 1824 399 614241 121784 2007 21937 8008 3261
11 57 1532 2248 83 241794 83634 626 8351 1530 2011
12 42 1194 1604 447 150707 57875 1538 5594 858 1203
13 132 5608 1731 141 416754 168798 1263 15271 4442 2743
14 77 2136 906 145 276379 38763 2686 10070 2445 1487
15 43 1534 438 750 133359 48239 538 4842 1172 1355
16 43 1711 1069 106 157275 27004 2419 6505 1469 1217
17 59 3686 820 119 150827 60244 2927 6552 1209 1082
18 33 1479 2347 41 215012 78253 2975 8624 894 2228
19 38 1822 1577 232 192746 76284 2472 9422 967 1367
20 162 5922 2330 148 533273 163816 1597 18700 4249 6545
21 60 2158 1153 180 252568 77887 1745 10573 1611 2210
22 56 2666 2683 469 269402 158835 1035 10678 1589 1834
23 71 2969 1521 65 197684 100321 2108 8563 905 1316
24 117 5527 2369 175 406475 106073 1300 15545 2359 2717
25 78 3219 2738 661 371847 125323 2900 14681 3477 3134
26 51 2431 741 164 190055 142422 2316 7964 1318 1158
27 48 2924 1561 183 332641 94933 1529 11756 2779 1398
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branches Inputs Exogenous input Intermediate products Exogenous output Outputs
1 25 619 538 854 77237 34224 2101 2947 913 224
2 27 419 489 125 88031 56559 1023 3138 478 516
3 40 1670 1459 120 164053 62776 1440 5494 1242 877
4 42 2931 1497 86 145369 65226 2458 3144 870 1138
5 52 2587 797 133 166424 85886 2202 6705 854 618
6 45 2181 697 149 215695 30179 1653 8487 1023 2096
7 33 989 1217 144 114043 43447 1919 4996 767 713
8 107 6277 2189 735 727699 294126 2486 21265 6282 6287
9 88 3197 949 101 186642 53223 648 8574 1537 1739
10 146 6222 1824 399 614241 121784 2007 21937 8008 3261
11 57 1532 2248 83 241794 83634 626 8351 1530 2011
12 42 1194 1604 447 150707 57875 1538 5594 858 1203
13 132 5608 1731 141 416754 168798 1263 15271 4442 2743
14 77 2136 906 145 276379 38763 2686 10070 2445 1487
15 43 1534 438 750 133359 48239 538 4842 1172 1355
16 43 1711 1069 106 157275 27004 2419 6505 1469 1217
17 59 3686 820 119 150827 60244 2927 6552 1209 1082
18 33 1479 2347 41 215012 78253 2975 8624 894 2228
19 38 1822 1577 232 192746 76284 2472 9422 967 1367
20 162 5922 2330 148 533273 163816 1597 18700 4249 6545
21 60 2158 1153 180 252568 77887 1745 10573 1611 2210
22 56 2666 2683 469 269402 158835 1035 10678 1589 1834
23 71 2969 1521 65 197684 100321 2108 8563 905 1316
24 117 5527 2369 175 406475 106073 1300 15545 2359 2717
25 78 3219 2738 661 371847 125323 2900 14681 3477 3134
26 51 2431 741 164 190055 142422 2316 7964 1318 1158
27 48 2924 1561 183 332641 94933 1529 11756 2779 1398
Table 4.  Cost allocation results based on cooperative models
Model (3) Model (6) Model (3) Model (6)
branches $ E $ $ E $ $ r_j^1 $ $ r_j^2 $ $ r_j^1 + r_j^2 $ $ r_j^1 $ $ r_j^2 $ $ r_j^1+r_j^2 $
1 1(1) 1(1) 246.9783 229.0669 476.0452(22) 249.9640 241.3908 491.3548(26)
2 0.8798(14) 0.8152(14) 247.3826 256.3008 503.6833(12) 250.6172 248.6579 499.2751(14)
3 0.7632(23) 0.5949(22) 245.3606 263.9743 509.3349(8) 249.0929 248.1273 497.2202(24)
4 0.7382(24) 0.5849(24) 244.5442 261.6055 506.1497(11) 249.7255 248.2464 497.97198(20)
5 0.8589(18) 0.7690(16) 244.4590 256.2162 500.6752(16) 251.6313 248.5930 500.2244(10)
6 1(1) 1(1) 248.9866 274.3444 523.3309(4) 250.5248 250.8240 501.3489(7)
7 0.8644(16) 0.7747(15) 245.7330 256.5588 502.2918(14) 249.5825 248.5230 498.1055(19)
8 0.9617(4) 0.9546(4) 226.5395 244.7763 471.3158(25) 255.3597 252.1110 507.4707(1)
9 0.7943(21) 0.5885(23) 245.7459 255.0998 500.8457(15) 250.7254 248.7602 499.4857(12)
10 0.9336(8) 0.8670(10) 235.6404 295.7199 531.3602(3) 255.4656 249.2222 504.6878(4)
11 0.8978(12) 0.8599(11) 244.5437 232.8622 477.4059(20) 248.6555 248.8462 497.5017(21)
12 0.8010(20) 0.69733(19) 245.1240 230.6952 475.8192(23) 249.2772 248.0615 497.3387(23)
13 0.8624(17) 0.7247(18) 237.5553 280.0552 517.6105(5) 252.4906 248.5284 501.0190(8)
14 1(1) 1(1) 249.5513 265.0496 514.6009(6) 256.0044 248.5349 504.5393(5)
15 0.9587(5) 0.9166(7) 246.8357 228.7804 475.6061(24) 248.9389 243.5603 492.4992(25)
16 0.9172(11) 0.8314(13) 249.2900 260.6817 509.9717(7) 251.1524 248.2845 499.4369(13)
17 0.9685(2) 0.9580(3) 244.1696 264.5992 508.7689(9) 250.3736 248.2632 498.6368(17)
18 1(1) 1(1) 245.3942 215.7705 461.1648(26) 249.4011 250.3788 499.7799(11)
19 0.9638(3) 0.9277(5) 244.6776 253.1032 497.7808(17) 249.8272 248.9734 498.8005(16)
20 0.8664(15) 0.7326(17) 236.1256 304.6363 540.7618(2) 251.2982 252.9057 504.2039(6)
21 0.8936(13) 0.8452(12) 244.6708 263.000 507.6708(10) 249.7751 248.5778 498.3529(18)
22 0.9210(10) 0.9234(6) 242.6463 245.2438 487.8901(18) 255.0194 251.9952 507.0145(2)
23 0.8016(19) 0.6033(21) 243.8905 259.6044 503.4949(13) 252.5173 248.3064 500.8237(9)
24 0.7912(22) 0.6329(20) 240.8019 300.7017 541.5036(1) 250.0492 248.9657 499.0150(15)
25 0.9220(9) 0.8732(8) 243.2701 232.7953 476.0654(21) 250.1477 247.1931 497.3408(22)
26 0.9511(6) 0.9983(2) 244.5997 233.8111 478.4109(19) 256.7994 249.7160 506.5153(3)
27 0.9353(7) 0.8705(9) 0.2413 0.2289 0.4702(27) 0.0355 0.0015 0.0370(27)
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Model (3) Model (6) Model (3) Model (6)
branches $ E $ $ E $ $ r_j^1 $ $ r_j^2 $ $ r_j^1 + r_j^2 $ $ r_j^1 $ $ r_j^2 $ $ r_j^1+r_j^2 $
1 1(1) 1(1) 246.9783 229.0669 476.0452(22) 249.9640 241.3908 491.3548(26)
2 0.8798(14) 0.8152(14) 247.3826 256.3008 503.6833(12) 250.6172 248.6579 499.2751(14)
3 0.7632(23) 0.5949(22) 245.3606 263.9743 509.3349(8) 249.0929 248.1273 497.2202(24)
4 0.7382(24) 0.5849(24) 244.5442 261.6055 506.1497(11) 249.7255 248.2464 497.97198(20)
5 0.8589(18) 0.7690(16) 244.4590 256.2162 500.6752(16) 251.6313 248.5930 500.2244(10)
6 1(1) 1(1) 248.9866 274.3444 523.3309(4) 250.5248 250.8240 501.3489(7)
7 0.8644(16) 0.7747(15) 245.7330 256.5588 502.2918(14) 249.5825 248.5230 498.1055(19)
8 0.9617(4) 0.9546(4) 226.5395 244.7763 471.3158(25) 255.3597 252.1110 507.4707(1)
9 0.7943(21) 0.5885(23) 245.7459 255.0998 500.8457(15) 250.7254 248.7602 499.4857(12)
10 0.9336(8) 0.8670(10) 235.6404 295.7199 531.3602(3) 255.4656 249.2222 504.6878(4)
11 0.8978(12) 0.8599(11) 244.5437 232.8622 477.4059(20) 248.6555 248.8462 497.5017(21)
12 0.8010(20) 0.69733(19) 245.1240 230.6952 475.8192(23) 249.2772 248.0615 497.3387(23)
13 0.8624(17) 0.7247(18) 237.5553 280.0552 517.6105(5) 252.4906 248.5284 501.0190(8)
14 1(1) 1(1) 249.5513 265.0496 514.6009(6) 256.0044 248.5349 504.5393(5)
15 0.9587(5) 0.9166(7) 246.8357 228.7804 475.6061(24) 248.9389 243.5603 492.4992(25)
16 0.9172(11) 0.8314(13) 249.2900 260.6817 509.9717(7) 251.1524 248.2845 499.4369(13)
17 0.9685(2) 0.9580(3) 244.1696 264.5992 508.7689(9) 250.3736 248.2632 498.6368(17)
18 1(1) 1(1) 245.3942 215.7705 461.1648(26) 249.4011 250.3788 499.7799(11)
19 0.9638(3) 0.9277(5) 244.6776 253.1032 497.7808(17) 249.8272 248.9734 498.8005(16)
20 0.8664(15) 0.7326(17) 236.1256 304.6363 540.7618(2) 251.2982 252.9057 504.2039(6)
21 0.8936(13) 0.8452(12) 244.6708 263.000 507.6708(10) 249.7751 248.5778 498.3529(18)
22 0.9210(10) 0.9234(6) 242.6463 245.2438 487.8901(18) 255.0194 251.9952 507.0145(2)
23 0.8016(19) 0.6033(21) 243.8905 259.6044 503.4949(13) 252.5173 248.3064 500.8237(9)
24 0.7912(22) 0.6329(20) 240.8019 300.7017 541.5036(1) 250.0492 248.9657 499.0150(15)
25 0.9220(9) 0.8732(8) 243.2701 232.7953 476.0654(21) 250.1477 247.1931 497.3408(22)
26 0.9511(6) 0.9983(2) 244.5997 233.8111 478.4109(19) 256.7994 249.7160 506.5153(3)
27 0.9353(7) 0.8705(9) 0.2413 0.2289 0.4702(27) 0.0355 0.0015 0.0370(27)
Table 5.  Cost allocation results arising from leader-follower and Nash game theory
Stage 1 as the leader Stage 1 as the follower Model (23)
branches $ r_j^1 $ $ r_j^2 $ $ r_j^1 $ $ r_j^2 $ $ r_j^1 $ $ r_j^2 $ $ r_j^1+r_j^2 $
1 0 135.7885 521.7512 0 260.8756 67.8943 328.7698 (21)
2 0 140.1356 260.8284 0 130.4142 70.0678 200.4820 (27)
3 0 214.8582 357.783 0 178.8915 107.4291 286.3206(24)
4 0 0.0000 602.2915 0 301.1457 0.0000 301.1457(23)
5 0 341.3245 562.1265 0 281.0632 170.6623 451.7255(17)
6 0 476.9271 441.9432 0 220.9715 238.4636 459.4352(15)
7 0 295.2004 469.1469 0 234.5734 147.6002 382.1736(19)
8 0 687.6763 735.7962 0 367.8980 344.8382 712.7362(4)
9 0 556.8296 181.5706 0 90.7853 278.4148 369.2001(20)
10 0 967.3349 599.2162 0 299.6080 483.6675 783.2756(1)
11 0 379.7576 157.2898 0 78.6449 189.8788 268.5237(25)
12 0 281.0507 374.9296 0 187.4647 140.5254 327.9901 (22)
13 0 726.8723 369.9592 0 184.9796 363.4362 548.4158(10)
14 0 460.0660 705.4757 0 352.7378 230.0330 582.7708(7)
15 0 248.9499 154.1661 0 77.0830 124.4750 201.5580(26)
16 0 371.5729 606.3652 0 303.1825 185.7865 488.9690(14)
17 0 390.0909 735.7962 0 367.8980 195.0455 562.9435(9)
18 0 501.3772 725.2381 0 362.6190 250.6887 613.3076(6)
19 0 623.6072 615.4395 0 307.7197 311.8037 619.5234(5)
20 0 967.3349 465.8196 0 232.9097 483.6675 716.5773(3)
21 0 623.5219 462.2460 0 231.1230 311.7610 542.8840(11)
22 0 569.3522 253.7010 0 126.8505 284.6762 411.5266(18)
23 0 503.6960 528.6428 0 264.3214 251.8480 516.1694(13)
24 0 788.8750 360.5408 0 180.2704 394.4376 574.7079(8)
25 0 806.3711 735.7962 0 367.8980 403.1856 771.0837 (2)
26 0 447.1736 597.3741 0 298.6870 223.5869 522.2738(12)
27 0 492.2555 418.7666 0 209.3832 246.1278 455.5110(16)
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Stage 1 as the leader Stage 1 as the follower Model (23)
branches $ r_j^1 $ $ r_j^2 $ $ r_j^1 $ $ r_j^2 $ $ r_j^1 $ $ r_j^2 $ $ r_j^1+r_j^2 $
1 0 135.7885 521.7512 0 260.8756 67.8943 328.7698 (21)
2 0 140.1356 260.8284 0 130.4142 70.0678 200.4820 (27)
3 0 214.8582 357.783 0 178.8915 107.4291 286.3206(24)
4 0 0.0000 602.2915 0 301.1457 0.0000 301.1457(23)
5 0 341.3245 562.1265 0 281.0632 170.6623 451.7255(17)
6 0 476.9271 441.9432 0 220.9715 238.4636 459.4352(15)
7 0 295.2004 469.1469 0 234.5734 147.6002 382.1736(19)
8 0 687.6763 735.7962 0 367.8980 344.8382 712.7362(4)
9 0 556.8296 181.5706 0 90.7853 278.4148 369.2001(20)
10 0 967.3349 599.2162 0 299.6080 483.6675 783.2756(1)
11 0 379.7576 157.2898 0 78.6449 189.8788 268.5237(25)
12 0 281.0507 374.9296 0 187.4647 140.5254 327.9901 (22)
13 0 726.8723 369.9592 0 184.9796 363.4362 548.4158(10)
14 0 460.0660 705.4757 0 352.7378 230.0330 582.7708(7)
15 0 248.9499 154.1661 0 77.0830 124.4750 201.5580(26)
16 0 371.5729 606.3652 0 303.1825 185.7865 488.9690(14)
17 0 390.0909 735.7962 0 367.8980 195.0455 562.9435(9)
18 0 501.3772 725.2381 0 362.6190 250.6887 613.3076(6)
19 0 623.6072 615.4395 0 307.7197 311.8037 619.5234(5)
20 0 967.3349 465.8196 0 232.9097 483.6675 716.5773(3)
21 0 623.5219 462.2460 0 231.1230 311.7610 542.8840(11)
22 0 569.3522 253.7010 0 126.8505 284.6762 411.5266(18)
23 0 503.6960 528.6428 0 264.3214 251.8480 516.1694(13)
24 0 788.8750 360.5408 0 180.2704 394.4376 574.7079(8)
25 0 806.3711 735.7962 0 367.8980 403.1856 771.0837 (2)
26 0 447.1736 597.3741 0 298.6870 223.5869 522.2738(12)
27 0 492.2555 418.7666 0 209.3832 246.1278 455.5110(16)
Table 6.  Evaluaition results with AED of [14] and models (3) and (6)
Stage 1 of (3) Stage 2 of (3) AED of [14] Model (3) Model (6) Model(3) Model(6)
branches $ E^1 $ $ E^2 $ $ E $ $ E $ $ E $ $ r^1_j+r^2_j $ $ r^1_j+r^2_j $
1 0.7597 1 0.8634(13) 0.8799(14) 0.7884(16) 306.9362 306.6713
2 1 0.7595 0.8798(11) 0.8798(13) 0.8375(12) 299.0361 307.4420
3 0.7204 0.7839 0.7470(21) 0.7521(24) 0.5810(23) 305.4121 306.6072
4 0.5595 0.7034 0.5726(27) 0.6314(27) 0.4292(27) 300.6593 306.9801
5 0.6884 0.8412 0.7102(24) 0.7848(22) 0.5998(21) 298.6402 307.7304
6 0.9994 0.9996 0.9304(5) 0.9995(2) 0.9979(4) 295.3327 308.3022
7 0.7285 0.9343 0.8152(17) 0.8314(19) 0.7136(19) 308.9644 306.5066
8 0.9983 0.9565 0.9762(2) 0.9774(4) 0.9544(5) 294.8752 310.4954
9 0.5872 1 0.7399(22) 0.7936(21) 0.5872(22) 306.4590 307.0616
10 1 0.8625 0.9225(8) 0.9313(8) 0.8624(9) 453.1254 309.8544
11 0.9380 0.8120 0.8770(12) 0.8750(15) 0.8297(13) 307.0382 306.8330
12 0.7732 0.8288 0.7975(19) 0.8010(20) 0.6945(20) 307.1054 306.7443
13 0.7248 1 0.8402(15) 0.8624(17) 0.7247(18) 299.7402 308.6713
14 1 1 0.9499(4) 1(1) 1(1) 289.1324 308.8396
15 0.9136 0.9942 0.9513(3) 0.9538(5) 0.9128(7) 304.4242 306.7419
16 0.6846 1 0.8023(18) 0.8423(18) 0.8124(14) 311.8755 306.8844
17 0.5531 0.9534 0.6934(25) 0.7533(23) 0.5380(25) 302.9726 306.6634
18 1 0.9752 0.9876(1) 0.9876(3) 0.9997(2) 307.5180 306.7759
19 0.8078 1 0.9036(10) 0.9039(11) 0.8090(15) 308.7589 306.9259
20 0.7328 1 0.8458(14) 0.8664(16) 0.7328(17) 309.7008 307.5808
21 0.8525 0.9232 0.8850(10) 0.8878(12) 0.8454(11) 306.4648 306.8614
22 1 0.8421 0.7935(20) 0.9210(10) 0.9253(6) 286.0109 309.9431
23 0.5549 0.8946 0.6407(26) 0.7247(26) 0.5231(26) 298.3907 308.0263
24 0.5918 0.9036 0.7177(23) 0.7477(25) 0.5791(24) 300.7911 307.3677
25 0.8710 0.9938 0.9239(7) 0.9324(7) 0.8731(7) 308.9031 307.0074
26 1 0.9022 0.8276(16) 0.9511(6) 0.9989(3) 281.1311 310.4377
27 1 0.8505 0.9241(6) 0.9252(9) 0.8504(10) 0.6013 0.0448
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Stage 1 of (3) Stage 2 of (3) AED of [14] Model (3) Model (6) Model(3) Model(6)
branches $ E^1 $ $ E^2 $ $ E $ $ E $ $ E $ $ r^1_j+r^2_j $ $ r^1_j+r^2_j $
1 0.7597 1 0.8634(13) 0.8799(14) 0.7884(16) 306.9362 306.6713
2 1 0.7595 0.8798(11) 0.8798(13) 0.8375(12) 299.0361 307.4420
3 0.7204 0.7839 0.7470(21) 0.7521(24) 0.5810(23) 305.4121 306.6072
4 0.5595 0.7034 0.5726(27) 0.6314(27) 0.4292(27) 300.6593 306.9801
5 0.6884 0.8412 0.7102(24) 0.7848(22) 0.5998(21) 298.6402 307.7304
6 0.9994 0.9996 0.9304(5) 0.9995(2) 0.9979(4) 295.3327 308.3022
7 0.7285 0.9343 0.8152(17) 0.8314(19) 0.7136(19) 308.9644 306.5066
8 0.9983 0.9565 0.9762(2) 0.9774(4) 0.9544(5) 294.8752 310.4954
9 0.5872 1 0.7399(22) 0.7936(21) 0.5872(22) 306.4590 307.0616
10 1 0.8625 0.9225(8) 0.9313(8) 0.8624(9) 453.1254 309.8544
11 0.9380 0.8120 0.8770(12) 0.8750(15) 0.8297(13) 307.0382 306.8330
12 0.7732 0.8288 0.7975(19) 0.8010(20) 0.6945(20) 307.1054 306.7443
13 0.7248 1 0.8402(15) 0.8624(17) 0.7247(18) 299.7402 308.6713
14 1 1 0.9499(4) 1(1) 1(1) 289.1324 308.8396
15 0.9136 0.9942 0.9513(3) 0.9538(5) 0.9128(7) 304.4242 306.7419
16 0.6846 1 0.8023(18) 0.8423(18) 0.8124(14) 311.8755 306.8844
17 0.5531 0.9534 0.6934(25) 0.7533(23) 0.5380(25) 302.9726 306.6634
18 1 0.9752 0.9876(1) 0.9876(3) 0.9997(2) 307.5180 306.7759
19 0.8078 1 0.9036(10) 0.9039(11) 0.8090(15) 308.7589 306.9259
20 0.7328 1 0.8458(14) 0.8664(16) 0.7328(17) 309.7008 307.5808
21 0.8525 0.9232 0.8850(10) 0.8878(12) 0.8454(11) 306.4648 306.8614
22 1 0.8421 0.7935(20) 0.9210(10) 0.9253(6) 286.0109 309.9431
23 0.5549 0.8946 0.6407(26) 0.7247(26) 0.5231(26) 298.3907 308.0263
24 0.5918 0.9036 0.7177(23) 0.7477(25) 0.5791(24) 300.7911 307.3677
25 0.8710 0.9938 0.9239(7) 0.9324(7) 0.8731(7) 308.9031 307.0074
26 1 0.9022 0.8276(16) 0.9511(6) 0.9989(3) 281.1311 310.4377
27 1 0.8505 0.9241(6) 0.9252(9) 0.8504(10) 0.6013 0.0448
Table 7.  Comparison results
Model (23) Chu et al. [14] Li et al. [24]
branches $ r_j^1 $ $ r_j^2 $ $ r_j^1+r_j^2 $ $ r_j^1 $ $ r_j^2 $ $ r_j^1+r_j^2 $ $ r_j^1 $ $ r_j^2 $ $ r_j^1+r_j^2 $
1 44.2099 41.7809 86.2678 29.2523 54.8286 84.0809 34.8394 99.5911 134.4306
2 50.3884 43.1185 93.5068 33.3404 61.0391 94.3795 57.6112 4.1483 61.7595
3 93.9028 56.1099 160.0128 62.1326 102.6452 164.7778 66.9629 83.9139 150.8768
4 83.2082 0.0000 83.2082 55.0563 50.3562 105.4125 42.5563 5.3306 47.8869
5 95.2600 105.0225 200.2825 63.0305 130.1635 193.1941 105.6896 19.9776 125.6671
6 123.4624 146.7462 270.2086 81.6912 185.8070 267.4982 149.2000 46.4915 195.6915
7 65.2774 90.8305 156.1080 43.1920 105.1728 148.3648 27.0534 58.2580 85.3113
8 416.5300 212.2072 628.7372 275.6048 412.7297 688.3345 662.6315 455.6247 1118.2562
9 106.8326 171.3315 278.1641 70.6878 192.9383 263.626 55.8416 158.4308 214.2724
10 351.5874 297.6403 649.2277 232.6343 421.0993 653.7337 428.1954 448.9607 877.1561
11 138.4013 116.8480 255.2492 91.5758 170.7374 262.3132 91.0412 79.0747 170.1160
12 86.2637 86.4768 172.7404 57.0780 116.1171 173.1951 38.3474 35.4254 73.7728
13 238.5472 223.6521 462.1993 157.8392 305.5089 463.3481 257.0286 451.538 708.5666
14 158.1975 141.5582 299.7556 104.6743 195.2432 299.9174 164.9817 245.7357 410.7174
15 76.3338 76.5996 152.9334 50.5077 104.8872 155.3949 85.7256 103.1206 188.8462
16 90.0231 114.3296 204.3528 59.5655 138.6559 198.2214 62.9564 161.9881 224.9445
17 86.3324 120.0275 206.3598 57.1234 140.0173 197.1407 65.8529 108.8606 174.7134
18 123.0714 154.2693 277.3407 81.4325 192.0992 273.5317 84.4604 4.1483 88.6087
19 110.3265 191.8783 302.2048 72.9996 207.1934 280.193 100.2016 53.5750 153.7766
20 305.2419 297.6403 602.8821 201.9689 421.0993 623.0683 306.3559 340.4241 646.7800
21 144.5682 191.8521 336.4203 95.6562 230.5790 326.2353 164.1695 115.4121 279.5816
22 154.2039 175.1846 329.3884 102.0319 220.4889 322.5207 131.0596 56.5463 187.6059
23 113.1530 154.9827 268.1357 74.8698 181.0649 255.9346 76.9843 13.9235 90.9078
24 232.6635 242.7298 475.3933 153.9462 316.4011 470.3472 196.4414 133.2967 329.7382
25 212.8427 248.1131 460.9559 140.8313 313.3001 454.1314 195.7660 344.9362 540.7023
26 108.7862 137.5913 246.3775 71.9804 164.8987 236.8791 163.0222 66.0493 229.0715
27 190.4015 151.4626 341.8641 125.9827 218.2432 344.2259 250.7998 239.4424 490.2422
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Model (23) Chu et al. [14] Li et al. [24]
branches $ r_j^1 $ $ r_j^2 $ $ r_j^1+r_j^2 $ $ r_j^1 $ $ r_j^2 $ $ r_j^1+r_j^2 $ $ r_j^1 $ $ r_j^2 $ $ r_j^1+r_j^2 $
1 44.2099 41.7809 86.2678 29.2523 54.8286 84.0809 34.8394 99.5911 134.4306
2 50.3884 43.1185 93.5068 33.3404 61.0391 94.3795 57.6112 4.1483 61.7595
3 93.9028 56.1099 160.0128 62.1326 102.6452 164.7778 66.9629 83.9139 150.8768
4 83.2082 0.0000 83.2082 55.0563 50.3562 105.4125 42.5563 5.3306 47.8869
5 95.2600 105.0225 200.2825 63.0305 130.1635 193.1941 105.6896 19.9776 125.6671
6 123.4624 146.7462 270.2086 81.6912 185.8070 267.4982 149.2000 46.4915 195.6915
7 65.2774 90.8305 156.1080 43.1920 105.1728 148.3648 27.0534 58.2580 85.3113
8 416.5300 212.2072 628.7372 275.6048 412.7297 688.3345 662.6315 455.6247 1118.2562
9 106.8326 171.3315 278.1641 70.6878 192.9383 263.626 55.8416 158.4308 214.2724
10 351.5874 297.6403 649.2277 232.6343 421.0993 653.7337 428.1954 448.9607 877.1561
11 138.4013 116.8480 255.2492 91.5758 170.7374 262.3132 91.0412 79.0747 170.1160
12 86.2637 86.4768 172.7404 57.0780 116.1171 173.1951 38.3474 35.4254 73.7728
13 238.5472 223.6521 462.1993 157.8392 305.5089 463.3481 257.0286 451.538 708.5666
14 158.1975 141.5582 299.7556 104.6743 195.2432 299.9174 164.9817 245.7357 410.7174
15 76.3338 76.5996 152.9334 50.5077 104.8872 155.3949 85.7256 103.1206 188.8462
16 90.0231 114.3296 204.3528 59.5655 138.6559 198.2214 62.9564 161.9881 224.9445
17 86.3324 120.0275 206.3598 57.1234 140.0173 197.1407 65.8529 108.8606 174.7134
18 123.0714 154.2693 277.3407 81.4325 192.0992 273.5317 84.4604 4.1483 88.6087
19 110.3265 191.8783 302.2048 72.9996 207.1934 280.193 100.2016 53.5750 153.7766
20 305.2419 297.6403 602.8821 201.9689 421.0993 623.0683 306.3559 340.4241 646.7800
21 144.5682 191.8521 336.4203 95.6562 230.5790 326.2353 164.1695 115.4121 279.5816
22 154.2039 175.1846 329.3884 102.0319 220.4889 322.5207 131.0596 56.5463 187.6059
23 113.1530 154.9827 268.1357 74.8698 181.0649 255.9346 76.9843 13.9235 90.9078
24 232.6635 242.7298 475.3933 153.9462 316.4011 470.3472 196.4414 133.2967 329.7382
25 212.8427 248.1131 460.9559 140.8313 313.3001 454.1314 195.7660 344.9362 540.7023
26 108.7862 137.5913 246.3775 71.9804 164.8987 236.8791 163.0222 66.0493 229.0715
27 190.4015 151.4626 341.8641 125.9827 218.2432 344.2259 250.7998 239.4424 490.2422
Table 8.  The characteristics of two different techniques
Model Model(23) Li et al. [24]
Technique Bargaining game Goal programming
Max-Cost(Efficiency rank) DMU 10 (7) DMU 8 (4)
Min-Cost(Efficiency rank) DMU 4 (27) DMU 4 (27)
Mean cost (stage1, stage2) (148.1488,148.1475) (150.5813,145.712)
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Model Model(23) Li et al. [24]
Technique Bargaining game Goal programming
Max-Cost(Efficiency rank) DMU 10 (7) DMU 8 (4)
Min-Cost(Efficiency rank) DMU 4 (27) DMU 4 (27)
Mean cost (stage1, stage2) (148.1488,148.1475) (150.5813,145.712)
Table 9.  CPU time of SOCP model (23) and Chu et al. [14] model
Model(23) Chu et al. [14] Chu et al. [14] Chu et al. [14]
Model $ (\gamma_d^1=\gamma_d^2) $ (100 iterations) (500 iterations) (1000 iterations)
Time (s) 1.7 109.7 515.7 1000
$ s_1 $ 0.52 0.52 0.52 0.52
$ s_2 $ 0.51 0.54 0.53 0.53
Mean cost
(stage1, stage2) (97.9866,198.3097) (98.6419,197.6544) (97.8199,198.4764) (98.0254,198.2709)
Table Options

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Model(23) Chu et al. [14] Chu et al. [14] Chu et al. [14]
Model $ (\gamma_d^1=\gamma_d^2) $ (100 iterations) (500 iterations) (1000 iterations)
Time (s) 1.7 109.7 515.7 1000
$ s_1 $ 0.52 0.52 0.52 0.52
$ s_2 $ 0.51 0.54 0.53 0.53
Mean cost
(stage1, stage2) (97.9866,198.3097) (98.6419,197.6544) (97.8199,198.4764) (98.0254,198.2709)
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