2017, 7(4): 379-401. doi: 10.3934/naco.2017024

Performance evaluation of four-stage blood supply chain with feedback variables using NDEA cross-efficiency and entropy measures under IER uncertainty

Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran

* Corresponding author: Abolfazl Mirzazadeh

Received  February 2017 Revised  July 2017 Published  October 2017

Fund Project: The reviewing process of this paper was handled by Associate Editors A. (Nima) Mirzazadeh, Kharazmi University, Tehran, Iran, and Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey. This paper was for the occasion of The 12th International Conference on Industrial Engineering (ICIE 2016), which was held in Tehran, Iran during 25-26 January, 2016

Blood supply chain management has been considered by many managers in recent years, as one of the major challenges in health systems. In order to ensure the optimal performance of the supply chain, enable continuous improvement and create competitive advantage, establishment of a performance evaluation system is essential. For this purpose, the current study proposes a Network Data Envelopment Analysis (NDEA) model for measuring efficiency of four-stage serial network of blood supply chain in presence of feedback variables by identifying comprehensive and balanced criteria as evaluation variables. Since criteria values are obtained from subjective judgment of individuals, are uncertain. Interval Evidential Reasoning (IER) approach that deals with a variety of uncertainties such as ignorance and vagueness, has been used to control the uncertainty and provide reliable evaluation. In order to rank the units a new cross-efficiency based model is presented as a remedy for the issue of non-uniqueness of optimal weights in cross efficiency. Then, Gibbs entropy is utilized to measure the uncertainty of obtained interval cross efficiency. Finally, a numerical example is provided to illustrate the proposed model.

Citation: Shiva Moslemi, Abolfazl Mirzazadeh. Performance evaluation of four-stage blood supply chain with feedback variables using NDEA cross-efficiency and entropy measures under IER uncertainty. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 379-401. doi: 10.3934/naco.2017024
References:
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J. C. Baez, T. Fritz, T. Leinster, A characterization of entropy in terms of information loss, Entropy, 13 (2011), 1945-1957. doi: 10.3390/e13111945.

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J. Beliën, H. Forcé, Supply chain management of blood products: A literature review, European Journal of Operational Research, 217 (2012), 1-16. doi: 10.1016/j.ejor.2011.05.026.

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C. Chen, H. Yan, Network DEA model for supply chain performance evaluation, European Journal of Operational Research, 213 (2011), 147-155. doi: 10.1016/j.ejor.2011.03.010.

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K. S. Chin, Y. M. Wang, J. B. Yang, K. K. G. Poon, An evidential reasoning based approach for quality function deployment under uncertainty, Expert Systems with Applications, 36 (2009), 5684-5694.

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M. A. Cohen, W. P. Pierskalla, Management policies for a regional blood bank, Transfusion, 15 (1975), 58-67.

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W. W. Cooper, K. S. Park, G. Yu, IDEA and AR-IDEA: Models for dealing with imprecise data in DEA, Management science, 45 (1999), 597-607.

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M. Dotoli, N. Epicoco, M. Falagario, A technique for supply chain network design under uncertainty using cross-efficiency fuzzy data envelopment analysis, IFAC-PapersOnLine, 48 (2015), 634-639.

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J. Doyle, R. Green, Efficiency and cross-efficiency in DEA: Derivations, meanings and uses, Journal of the Operational Research Society, 45 (1994), 567-578.

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M. Guo, J. B. Yang, K. S. Chin, H. W. Wang, X. B. Liu, Evidential reasoning approach for multiattribute decision analysis under both fuzzy and interval uncertainty, IEEE Transactions on Fuzzy Systems, 17 (2009), 683-697.

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A. Hatami-Marbini, P. J. Agrell, M. Tavana, P. Khoshnevis, A flexible cross-efficiency fuzzy data envelopment analysis model for sustainable sourcing, Journal of Cleaner Production, 142 (2017), 2761-2779.

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G. R. Jahanshahloo, M. Khodabakhshi, F. H. Lotfi, M. M. Goudarzi, A cross-efficiency model based on super-efficiency for ranking units through the TOPSIS approach and its extension to the interval case, Mathematical and Computer Modelling, 53 (2011), 1946-1955. doi: 10.1016/j.mcm.2008.07.009.

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C. Kao, Efficiency decomposition for general multi-stage systems in data envelopment analysis, European Journal of Operational Research, 232 (2014), 117-124. doi: 10.1016/j.ejor.2013.07.012.

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K. Katsaliaki, Cost-effective practices in the blood service sector, Health policy, 86 (2008), 276-287.

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S. Keikha-Javan, M. Rostamy-Malkhalifeh, Efficiency measurement of NDEA with interval data, International Journal of Mathematical Modelling and Computations, 6 (2016), 199-210.

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K. Khalili-Damghani, M. Taghavifard, A three-stage fuzzy DEA approach to measure performance of a serial process including JIT practices, agility indices, and goals in supply chains, International Journal of Services and Operations Management, 13 (2012), 147-188.

[16]

K. Khalili-Damghani, M. Taghavi-Fard, A. R. Abtahi, A fuzzy two-stage DEA approach for performance measurement: real case of agility performance in dairy supply chains, International Journal of Applied Decision Sciences, 5 (2012), 293-317.

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L. Liang, Z. Q. Li, W. D. Cook, J. Zhu, Data envelopment analysis efficiency in two-stage networks with feedback, IIE Transactions, 43 (2011), 309-322.

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C. lo Storto, Ecological efficiency based ranking of cities: A combined DEA cross-efficiency and Shannon's entropy method Sustainability, 8 (2016), 124.

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F. H. Lotfi, M. Navabakhs, A. Tehranian, M. Rostamy-Malkhalifeh, R. Shahverdi, Ranking bank branches with interval data-The application of DEA, In International Mathematical Forum, 2 (2007), 429-440. doi: 10.12988/imf.2007.07039.

[20]

T. Lu and S. T. Liu, Ranking DMUs by comparing DEA cross-efficiency intervals using entropy measures Entropy, 18 (2016), 452.

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A. M. Mathai, H. J. Haubold, On a generalized entropy measure leading to the pathway model with a preliminary application to solar neutrino data, Entropy, 15 (2013), 4011-4025. doi: 10.3390/e15104011.

[22]

S. M. Mirhedayatian, M. Azadi, R. F. Saen, A novel network data envelopment analysis model for evaluating green supply chain management, International Journal of Production Economics, 147 (2014), 544-554.

[23]

K. H. Mistry, J. H. Lienhard, An economics-based second law efficiency, Entropy, 15 (2013), 2736-2765. doi: 10.3390/e15062046.

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A. F. Osorio, S. C. Brailsford, H. K. Smith, A structured review of quantitative models in the blood supply chain: a taxonomic framework for decision-making, International Journal of Production Research, 53 (2015), 7191-7212.

[25]

A. Pereira, Economies of scale in blood banking: a study based on data envelopment analysis, Vox Sanguinis, 90 (2006), 308-315.

[26]

C. Pitocco, T. R. Sexton, Alleviating blood shortages in a resource-constrained environment, Transfusion, 45 (2005), 1118-1126.

[27]

G. B. Schreiber, K. S. Schlumpf, S. A. Glynn, D. J. Wright, Y. Tu, M. R. King, M.J. Higgins, D. Kessler, R. Gilcher, C. C. Nass, A. M. Guiltinan, Convenience, the bane of our existence, and other barriers to donating, Transfusion, 46 (2006), 545-553.

[28]

T. R. Sexton, R. H. Silkman, A. J. Hogan, Data envelopment analysis: Critique and extensions, New Directions for Evaluation, 32 (1986), 73-105.

[29]

G. Shafer, A Mathematical Theory of Evidence Princeton: Princeton University Press, 1976.

[30]

Y. S. Shao, D. Brooks, ISA-independent workload characterization and its implications for specialized architectures, ,in Performance Analysis of Systems and Software (ISPASS), 2013 IEEE International Symposium on., IEEE, (2013), 245-255.

[31]

M. Tavana, H. Mirzagoltabar, S. M. Mirhedayatian, R. F. Saen, M. Azadi, A new network epsilon-based DEA model for supply chain performance evaluation, Computers and Industrial Engineering, 66 (2013), 501-513.

[32]

Y. M. Wang, K. S. Chin, A neutral DEA model for cross-efficiency evaluation and its extension, Expert Systems with Applications, 37 (2010a), 3666-3675.

[33]

Y. M. Wang, K. S. Chin, Some alternative models for DEA cross-efficiency evaluation, International Journal of Production Economics, 128 (2010b), 332-338.

[34]

L. Wang, L. Li and N. Hong, Entropy cross-efficiency model for decision making units with interval data Entropy, 18 (2016), 358.

[35]

B. Y. Wong, J. B. Yang, R. Greatbanks, Using DEA and the ER approach for performance measurement of UK retail banks, MCDM, (2004), 6-11.

[36]

J. Wu, L. Liang, F. Yang, Determination of the weights for the ultimate cross efficiency using Shapley value in cooperative game, Expert Systems with Applications, 36 (2009), 872-876.

[37]

J. Wu, J. S. Sun, L. A. Liang, Y. C. Zha, Determination of weights for ultimate cross efficiency using Shannon entropy, Expert Syst. Appl., 38 (2011), 5162-5165.

[38]

J. Wu, J. S. Sun, L. Liang, DEA cross-efficiency aggregation method based upon Shannon entropy, Int. J. Prod. Res., 50 (2012), 6726-6736.

[39]

F. Yang, S. Ang, Q. Xia, C. Yang, Ranking DMUs by using interval DEA cross efficiency matrix with acceptability analysis, European Journal of Operational Research, 223 (2012), 483-488. doi: 10.1016/j.ejor.2012.07.001.

[40]

J. B. Yang, M. G. Singh, An evidential reasoning approach for multiple-attribute decision making with uncertainty, IEEE Transactions on systems, Man, and Cybernetics, 24 (1994), 1-18.

[41]

J. B. Yang, Y. M. Wang, D. L. Xu, K. S. Chin, The evidential reasoning approach for MADA under both probabilistic and fuzzy uncertainties, European journal of operational research, 171 (2006), 309-343. doi: 10.1016/j.ejor.2004.09.017.

[42]

G. L. Yang, J. B. Yang, W. B. Liu, X. X. Li, Cross-efficiency aggregation in DEA models using the evidential-reasoning approach, European Journal of Operational Research, 231 (2013), 393-404. doi: 10.1016/j.ejor.2013.05.017.

[43]

Q. Yu, F. Hou, A cross evaluation-based measure of super efficiency in DEA with interval data, Kybernetes, 45 (2016), 666-679. doi: 10.1108/K-05-2014-0089.

[44]

Y. Zha, X. Ding, L. Liang, Z. Huang, A two-stage DEA approach with feedback for team performance evaluation, In Applications of Management Science. Emerald Group Publishing Limited, (2012), 3-18.

show all references

References:
[1]

J. C. Baez, T. Fritz, T. Leinster, A characterization of entropy in terms of information loss, Entropy, 13 (2011), 1945-1957. doi: 10.3390/e13111945.

[2]

J. Beliën, H. Forcé, Supply chain management of blood products: A literature review, European Journal of Operational Research, 217 (2012), 1-16. doi: 10.1016/j.ejor.2011.05.026.

[3]

C. Chen, H. Yan, Network DEA model for supply chain performance evaluation, European Journal of Operational Research, 213 (2011), 147-155. doi: 10.1016/j.ejor.2011.03.010.

[4]

K. S. Chin, Y. M. Wang, J. B. Yang, K. K. G. Poon, An evidential reasoning based approach for quality function deployment under uncertainty, Expert Systems with Applications, 36 (2009), 5684-5694.

[5]

M. A. Cohen, W. P. Pierskalla, Management policies for a regional blood bank, Transfusion, 15 (1975), 58-67.

[6]

W. W. Cooper, K. S. Park, G. Yu, IDEA and AR-IDEA: Models for dealing with imprecise data in DEA, Management science, 45 (1999), 597-607.

[7]

M. Dotoli, N. Epicoco, M. Falagario, A technique for supply chain network design under uncertainty using cross-efficiency fuzzy data envelopment analysis, IFAC-PapersOnLine, 48 (2015), 634-639.

[8]

J. Doyle, R. Green, Efficiency and cross-efficiency in DEA: Derivations, meanings and uses, Journal of the Operational Research Society, 45 (1994), 567-578.

[9]

M. Guo, J. B. Yang, K. S. Chin, H. W. Wang, X. B. Liu, Evidential reasoning approach for multiattribute decision analysis under both fuzzy and interval uncertainty, IEEE Transactions on Fuzzy Systems, 17 (2009), 683-697.

[10]

A. Hatami-Marbini, P. J. Agrell, M. Tavana, P. Khoshnevis, A flexible cross-efficiency fuzzy data envelopment analysis model for sustainable sourcing, Journal of Cleaner Production, 142 (2017), 2761-2779.

[11]

G. R. Jahanshahloo, M. Khodabakhshi, F. H. Lotfi, M. M. Goudarzi, A cross-efficiency model based on super-efficiency for ranking units through the TOPSIS approach and its extension to the interval case, Mathematical and Computer Modelling, 53 (2011), 1946-1955. doi: 10.1016/j.mcm.2008.07.009.

[12]

C. Kao, Efficiency decomposition for general multi-stage systems in data envelopment analysis, European Journal of Operational Research, 232 (2014), 117-124. doi: 10.1016/j.ejor.2013.07.012.

[13]

K. Katsaliaki, Cost-effective practices in the blood service sector, Health policy, 86 (2008), 276-287.

[14]

S. Keikha-Javan, M. Rostamy-Malkhalifeh, Efficiency measurement of NDEA with interval data, International Journal of Mathematical Modelling and Computations, 6 (2016), 199-210.

[15]

K. Khalili-Damghani, M. Taghavifard, A three-stage fuzzy DEA approach to measure performance of a serial process including JIT practices, agility indices, and goals in supply chains, International Journal of Services and Operations Management, 13 (2012), 147-188.

[16]

K. Khalili-Damghani, M. Taghavi-Fard, A. R. Abtahi, A fuzzy two-stage DEA approach for performance measurement: real case of agility performance in dairy supply chains, International Journal of Applied Decision Sciences, 5 (2012), 293-317.

[17]

L. Liang, Z. Q. Li, W. D. Cook, J. Zhu, Data envelopment analysis efficiency in two-stage networks with feedback, IIE Transactions, 43 (2011), 309-322.

[18]

C. lo Storto, Ecological efficiency based ranking of cities: A combined DEA cross-efficiency and Shannon's entropy method Sustainability, 8 (2016), 124.

[19]

F. H. Lotfi, M. Navabakhs, A. Tehranian, M. Rostamy-Malkhalifeh, R. Shahverdi, Ranking bank branches with interval data-The application of DEA, In International Mathematical Forum, 2 (2007), 429-440. doi: 10.12988/imf.2007.07039.

[20]

T. Lu and S. T. Liu, Ranking DMUs by comparing DEA cross-efficiency intervals using entropy measures Entropy, 18 (2016), 452.

[21]

A. M. Mathai, H. J. Haubold, On a generalized entropy measure leading to the pathway model with a preliminary application to solar neutrino data, Entropy, 15 (2013), 4011-4025. doi: 10.3390/e15104011.

[22]

S. M. Mirhedayatian, M. Azadi, R. F. Saen, A novel network data envelopment analysis model for evaluating green supply chain management, International Journal of Production Economics, 147 (2014), 544-554.

[23]

K. H. Mistry, J. H. Lienhard, An economics-based second law efficiency, Entropy, 15 (2013), 2736-2765. doi: 10.3390/e15062046.

[24]

A. F. Osorio, S. C. Brailsford, H. K. Smith, A structured review of quantitative models in the blood supply chain: a taxonomic framework for decision-making, International Journal of Production Research, 53 (2015), 7191-7212.

[25]

A. Pereira, Economies of scale in blood banking: a study based on data envelopment analysis, Vox Sanguinis, 90 (2006), 308-315.

[26]

C. Pitocco, T. R. Sexton, Alleviating blood shortages in a resource-constrained environment, Transfusion, 45 (2005), 1118-1126.

[27]

G. B. Schreiber, K. S. Schlumpf, S. A. Glynn, D. J. Wright, Y. Tu, M. R. King, M.J. Higgins, D. Kessler, R. Gilcher, C. C. Nass, A. M. Guiltinan, Convenience, the bane of our existence, and other barriers to donating, Transfusion, 46 (2006), 545-553.

[28]

T. R. Sexton, R. H. Silkman, A. J. Hogan, Data envelopment analysis: Critique and extensions, New Directions for Evaluation, 32 (1986), 73-105.

[29]

G. Shafer, A Mathematical Theory of Evidence Princeton: Princeton University Press, 1976.

[30]

Y. S. Shao, D. Brooks, ISA-independent workload characterization and its implications for specialized architectures, ,in Performance Analysis of Systems and Software (ISPASS), 2013 IEEE International Symposium on., IEEE, (2013), 245-255.

[31]

M. Tavana, H. Mirzagoltabar, S. M. Mirhedayatian, R. F. Saen, M. Azadi, A new network epsilon-based DEA model for supply chain performance evaluation, Computers and Industrial Engineering, 66 (2013), 501-513.

[32]

Y. M. Wang, K. S. Chin, A neutral DEA model for cross-efficiency evaluation and its extension, Expert Systems with Applications, 37 (2010a), 3666-3675.

[33]

Y. M. Wang, K. S. Chin, Some alternative models for DEA cross-efficiency evaluation, International Journal of Production Economics, 128 (2010b), 332-338.

[34]

L. Wang, L. Li and N. Hong, Entropy cross-efficiency model for decision making units with interval data Entropy, 18 (2016), 358.

[35]

B. Y. Wong, J. B. Yang, R. Greatbanks, Using DEA and the ER approach for performance measurement of UK retail banks, MCDM, (2004), 6-11.

[36]

J. Wu, L. Liang, F. Yang, Determination of the weights for the ultimate cross efficiency using Shapley value in cooperative game, Expert Systems with Applications, 36 (2009), 872-876.

[37]

J. Wu, J. S. Sun, L. A. Liang, Y. C. Zha, Determination of weights for ultimate cross efficiency using Shannon entropy, Expert Syst. Appl., 38 (2011), 5162-5165.

[38]

J. Wu, J. S. Sun, L. Liang, DEA cross-efficiency aggregation method based upon Shannon entropy, Int. J. Prod. Res., 50 (2012), 6726-6736.

[39]

F. Yang, S. Ang, Q. Xia, C. Yang, Ranking DMUs by using interval DEA cross efficiency matrix with acceptability analysis, European Journal of Operational Research, 223 (2012), 483-488. doi: 10.1016/j.ejor.2012.07.001.

[40]

J. B. Yang, M. G. Singh, An evidential reasoning approach for multiple-attribute decision making with uncertainty, IEEE Transactions on systems, Man, and Cybernetics, 24 (1994), 1-18.

[41]

J. B. Yang, Y. M. Wang, D. L. Xu, K. S. Chin, The evidential reasoning approach for MADA under both probabilistic and fuzzy uncertainties, European journal of operational research, 171 (2006), 309-343. doi: 10.1016/j.ejor.2004.09.017.

[42]

G. L. Yang, J. B. Yang, W. B. Liu, X. X. Li, Cross-efficiency aggregation in DEA models using the evidential-reasoning approach, European Journal of Operational Research, 231 (2013), 393-404. doi: 10.1016/j.ejor.2013.05.017.

[43]

Q. Yu, F. Hou, A cross evaluation-based measure of super efficiency in DEA with interval data, Kybernetes, 45 (2016), 666-679. doi: 10.1108/K-05-2014-0089.

[44]

Y. Zha, X. Ding, L. Liang, Z. Huang, A two-stage DEA approach with feedback for team performance evaluation, In Applications of Management Science. Emerald Group Publishing Limited, (2012), 3-18.

Figure 1.  Network model of $j^{th}$ blood supply chain
Table 1.  Verified performance criteria and sub-criteria of $ j^{th}$ supply chain network and nature of them in NDEA model
Variable Criteria and Sub-criteria Nature
$x_{1j}^{(1)}$ The process of attracting donor:
-Make culture of blood donation in society
-Inform about the benefits of blood donation and the need to donate blood
Donor's input
$x_{1j}^{(2)}$ Intellectual capital of blood donation center:
-Employee competence in terms of scientific and practical
-The number of trained employee
-give suggestions to improve doing tasks
-Timely presence
First input of blood donation center
$x_{2j}^{(2)}$ Space and Facilities of blood donation center:
-Space, light, ventilation, cleanliness, temperature
-Having the necessary equipment
Second input of blood donation center
$x_{3j}^{(2)}$ Blood donation center costsThird input of blood donation center
$x_{1j}^{(3)}$ Intellectual capital of blood bank:
-Employee competence in terms of scientific and practical
-The number of trained employee
-give suggestions to improve doing tasks
-Presence timely
First blood bank input
$x_{2j}^{(3)}$ Space and Facilities of blood bank:
-Space, light, ventilation, cleanliness, temperature
-Having the necessary equipment (Special refrigerators and freezers, conventional refrigerator to store blood samples, incubator shaker, sero-fuge, etc.
Second blood bank input
$x_{3j}^{(3)}$ Blood bank costsThird blood bank input
$x_{1j}^{(4)}$ Intellectual capital of hospital:
-Employees competence in terms of scientific and practical
-The number of trained employee
-give suggestions to improve doing tasks
-Timely presence
First hospital input
$x_{2j}^{(4)}$Space and Facilities of hospital:
-Space, light, ventilation, cleanliness, temperature
-Having the necessary equipment
Second hospital input
$x_{3j}^{(4)}$ Hospital costsThird hospital input
$y_{1j}^{(2)}$ Environmental actions of blood donation centers:
-Proper disposal of waste
First output of blood donation center
$y_{2j}^{(2)}$ Management of financial resources (supportive budget) in the blood donation centerSecond output of blood donation center
$y_{3j}^{(2)}$ Waste management in blood donation center
*Waste due to expiration, lack of demand
Third output of blood donation center
$y_{4j}^{(2)}$ Loss rate in blood donation center
*Waste due to improper blood storage
Fourth output of blood donation center
$y_{5j}^{(2)}$Social actions of blood donation center
-Employee satisfaction
-Health and safety of employees
Fifth output of blood donation center
$y_{1j}^{(3)}$ Environmental actions of blood bank:
-Proper disposal of waste
First blood bank output
$y_{2j}^{(3)}$Management of financial resources in blood bankSecond blood bank output
$y_{3j}^{(3)}$ Waste management in blood bank
*Waste due to expiration, lack of demand, negative result of cross-match test
Third blood bank output
$y_{4j}^{(3)}$ Loss rate in blood bank
*Waste due to expiration, improper storage, lack of demand, negative result of test (Cross-match, ...)
Fourth blood bank output
$y_{5j}^{(3)}$Social actions of blood bank
-Employee satisfaction
-Health and safety of employees
Fifth blood bank output
$y_{1j}^{(4)}$ Environmental actions of hospital:
-Proper disposal of waste
First hospital output
$y_{2j}^{(4)}$Hospital revenuesSecond hospital output
$y_{3j}^{(4)} $Waste management in hospital
Additional and Durable blood or blood near to meet expiration date
Third hospital output
$y_{4j}^{(4))}$ Loss rate in hospital
Cancellation of surgery, Cross match/Transfusion, expiration, etc.
Fourth hospital output
$y_{5j}^{(4)}$ Social actions of hospital
-Employee satisfaction
-Health and safety of employees
-Crisis management (Sudden hazards such as earthquake, possible complications of blood transfusion (e.g. severe reactions, allergies, fever, hypotension, bleeding
-Inform patients about possible reactions of blood transfusions
-Controlling blood bags at the time of) receiving it from blood bank and before blood transfusion
-Controlling patient's clinical and laboratory signs before and after blood transfusion
-Controlling patient profile before blood transfusion and matching it with blood bag
Fifth hospital output
$y_{6j}^{(4)}$Recording and archiving in blood donation center
-Completeness
-accuracy and validity
Sixth hospital output
$y_{7j}^{(4)}$ Patient satisfaction from hospital
-How to do sampling from patient blood
-How to do blood transfusion
-Responsiveness to expectations and complaints
-Availability of doctors during complication
Seventh hospital output
$y_{8j}^{(4)}$ Quality in hospital:
-Hygiene
-Health, quality and freshness of transfused blood
-Implementation of hemovigilance program
-Blood transportation in hospital ward
-Blood storage until transfusion (temperature, special refrigerators)
-Timely blood transfusion (a maximum of 20 minutes after receiving)
Eighth hospital output
$y_{9j}^{(4)}$ Inventory management in hospital
-Management of blood shortages (unavailability of required blood group, etc.)
-Proper selection of blood for transfusion considering condition (freshness, near to meet the expiration date, ...)
-Ordering policies
Ninth hospital output
$z_{1j}^{(1))}$ The number of donors (Amount of donated blood)First intermediate variable between donor and blood donation center
$z_{1j}^{(2)}$ Recording and archiving in blood donation center
-Completeness
-accuracy and validity
First intermediate variable between blood donation center and blood bank
$z_{4j}^{(2)}$Other Social actions of blood donation center:
-Doing Medical examination (checking donor medical record, interval time between two blood donations, Blood donation eligibility criteria (age, weight, physical and mental conditions, etc.))
-Crisis management (Sudden hazards such as earthquake, possible complications of blood donation such as Inflammation of the veins, localized tenderness, a collection of blood under the skin, bruising, etc.)
Fourth intermediate variable between blood donation center and blood bank
$z_{5j}^{(2)}$ Partnership and cooperation between blood donation center and blood bank:
-Sharing information
Fifth intermediate variable between blood donation center and blood bank
$z_{6j}^{(2)}$Inventory management in blood donation center:
-Management of blood shortages
-Proper selection of blood to send considering condition (freshness, near to meet expiration date, ...)
Sixth intermediate variable between blood donation center and blood bank
$z_{1j}^{(3)}$ Recording and archiving in blood bank
-Completeness -Accuracy and validity
First intermediate variable between blood bank and hospital
$z_{2j}^{(3)}$ Hospital satisfaction from blood bank:
-Storage of blood and blood products in terms of temperature and storage place (after receiving from the blood donation center, doing tests, reservations / not)
-Timely delivery of blood to the hospitals
-Transportation
-Responsiveness
-Freshness of received blood
Second intermediate variable between blood bank and hospital
$z_{3j}^{(3)}$ Quality in blood bank:
-Hygiene
-How to do tests on donated blood (HIV, hepatitis, etc.) and blood samples, and cross-match tests
-Accuracy of test results
-How to do separation (analysis) Blood
Third intermediate variable between blood bank and hospital
$z_{4j}^{(3)}$ Other Social actions of blood bank:
-Crisis management (Sudden hazards such as earthquake, etc.)
-Controlling blood bags during (at the time of) delivery to hospital (hemolysis, clots. discoloration, etc.)
Fourth intermediate variable between blood bank and hospital
$z_{5j}^{(3)}$ Partnership and cooperation between blood bank and hospital:
-Sharing information
Fifth intermediate variable between blood bank and hospital
$z_{6j}^{(3)}$Inventory management in blood bank
-Management of blood shortages (unavailability of required blood group, etc.)
-Proper selection of blood to send considering condition (freshness, close to the expiration date, ...)
-Cross-matching policies
Sixth intermediate variable between blood bank and hospital
$f_{1j}^{(2)}$Donor satisfaction from blood donation center
-Employee attitude
-Responsiveness
-How to get blood from a donor -Consulting for hepatitis, AIDS, thalassemia, etc.
First feedback variable of blood donation center
$f_{1j}^{(3)}$ Satisfaction of blood donation center from blood bank
-Accuracy of results of test on donated blood (HIV, hepatitis, etc.) provided by blood bank
First feedback variable of blood bank
$f_{1j}^{(4)} $ Blood bank satisfaction from hospital
-Storage and transportation of blood samples in hospital for sending to blood bank
First feedback variable of hospital
Variable Criteria and Sub-criteria Nature
$x_{1j}^{(1)}$ The process of attracting donor:
-Make culture of blood donation in society
-Inform about the benefits of blood donation and the need to donate blood
Donor's input
$x_{1j}^{(2)}$ Intellectual capital of blood donation center:
-Employee competence in terms of scientific and practical
-The number of trained employee
-give suggestions to improve doing tasks
-Timely presence
First input of blood donation center
$x_{2j}^{(2)}$ Space and Facilities of blood donation center:
-Space, light, ventilation, cleanliness, temperature
-Having the necessary equipment
Second input of blood donation center
$x_{3j}^{(2)}$ Blood donation center costsThird input of blood donation center
$x_{1j}^{(3)}$ Intellectual capital of blood bank:
-Employee competence in terms of scientific and practical
-The number of trained employee
-give suggestions to improve doing tasks
-Presence timely
First blood bank input
$x_{2j}^{(3)}$ Space and Facilities of blood bank:
-Space, light, ventilation, cleanliness, temperature
-Having the necessary equipment (Special refrigerators and freezers, conventional refrigerator to store blood samples, incubator shaker, sero-fuge, etc.
Second blood bank input
$x_{3j}^{(3)}$ Blood bank costsThird blood bank input
$x_{1j}^{(4)}$ Intellectual capital of hospital:
-Employees competence in terms of scientific and practical
-The number of trained employee
-give suggestions to improve doing tasks
-Timely presence
First hospital input
$x_{2j}^{(4)}$Space and Facilities of hospital:
-Space, light, ventilation, cleanliness, temperature
-Having the necessary equipment
Second hospital input
$x_{3j}^{(4)}$ Hospital costsThird hospital input
$y_{1j}^{(2)}$ Environmental actions of blood donation centers:
-Proper disposal of waste
First output of blood donation center
$y_{2j}^{(2)}$ Management of financial resources (supportive budget) in the blood donation centerSecond output of blood donation center
$y_{3j}^{(2)}$ Waste management in blood donation center
*Waste due to expiration, lack of demand
Third output of blood donation center
$y_{4j}^{(2)}$ Loss rate in blood donation center
*Waste due to improper blood storage
Fourth output of blood donation center
$y_{5j}^{(2)}$Social actions of blood donation center
-Employee satisfaction
-Health and safety of employees
Fifth output of blood donation center
$y_{1j}^{(3)}$ Environmental actions of blood bank:
-Proper disposal of waste
First blood bank output
$y_{2j}^{(3)}$Management of financial resources in blood bankSecond blood bank output
$y_{3j}^{(3)}$ Waste management in blood bank
*Waste due to expiration, lack of demand, negative result of cross-match test
Third blood bank output
$y_{4j}^{(3)}$ Loss rate in blood bank
*Waste due to expiration, improper storage, lack of demand, negative result of test (Cross-match, ...)
Fourth blood bank output
$y_{5j}^{(3)}$Social actions of blood bank
-Employee satisfaction
-Health and safety of employees
Fifth blood bank output
$y_{1j}^{(4)}$ Environmental actions of hospital:
-Proper disposal of waste
First hospital output
$y_{2j}^{(4)}$Hospital revenuesSecond hospital output
$y_{3j}^{(4)} $Waste management in hospital
Additional and Durable blood or blood near to meet expiration date
Third hospital output
$y_{4j}^{(4))}$ Loss rate in hospital
Cancellation of surgery, Cross match/Transfusion, expiration, etc.
Fourth hospital output
$y_{5j}^{(4)}$ Social actions of hospital
-Employee satisfaction
-Health and safety of employees
-Crisis management (Sudden hazards such as earthquake, possible complications of blood transfusion (e.g. severe reactions, allergies, fever, hypotension, bleeding
-Inform patients about possible reactions of blood transfusions
-Controlling blood bags at the time of) receiving it from blood bank and before blood transfusion
-Controlling patient's clinical and laboratory signs before and after blood transfusion
-Controlling patient profile before blood transfusion and matching it with blood bag
Fifth hospital output
$y_{6j}^{(4)}$Recording and archiving in blood donation center
-Completeness
-accuracy and validity
Sixth hospital output
$y_{7j}^{(4)}$ Patient satisfaction from hospital
-How to do sampling from patient blood
-How to do blood transfusion
-Responsiveness to expectations and complaints
-Availability of doctors during complication
Seventh hospital output
$y_{8j}^{(4)}$ Quality in hospital:
-Hygiene
-Health, quality and freshness of transfused blood
-Implementation of hemovigilance program
-Blood transportation in hospital ward
-Blood storage until transfusion (temperature, special refrigerators)
-Timely blood transfusion (a maximum of 20 minutes after receiving)
Eighth hospital output
$y_{9j}^{(4)}$ Inventory management in hospital
-Management of blood shortages (unavailability of required blood group, etc.)
-Proper selection of blood for transfusion considering condition (freshness, near to meet the expiration date, ...)
-Ordering policies
Ninth hospital output
$z_{1j}^{(1))}$ The number of donors (Amount of donated blood)First intermediate variable between donor and blood donation center
$z_{1j}^{(2)}$ Recording and archiving in blood donation center
-Completeness
-accuracy and validity
First intermediate variable between blood donation center and blood bank
$z_{4j}^{(2)}$Other Social actions of blood donation center:
-Doing Medical examination (checking donor medical record, interval time between two blood donations, Blood donation eligibility criteria (age, weight, physical and mental conditions, etc.))
-Crisis management (Sudden hazards such as earthquake, possible complications of blood donation such as Inflammation of the veins, localized tenderness, a collection of blood under the skin, bruising, etc.)
Fourth intermediate variable between blood donation center and blood bank
$z_{5j}^{(2)}$ Partnership and cooperation between blood donation center and blood bank:
-Sharing information
Fifth intermediate variable between blood donation center and blood bank
$z_{6j}^{(2)}$Inventory management in blood donation center:
-Management of blood shortages
-Proper selection of blood to send considering condition (freshness, near to meet expiration date, ...)
Sixth intermediate variable between blood donation center and blood bank
$z_{1j}^{(3)}$ Recording and archiving in blood bank
-Completeness -Accuracy and validity
First intermediate variable between blood bank and hospital
$z_{2j}^{(3)}$ Hospital satisfaction from blood bank:
-Storage of blood and blood products in terms of temperature and storage place (after receiving from the blood donation center, doing tests, reservations / not)
-Timely delivery of blood to the hospitals
-Transportation
-Responsiveness
-Freshness of received blood
Second intermediate variable between blood bank and hospital
$z_{3j}^{(3)}$ Quality in blood bank:
-Hygiene
-How to do tests on donated blood (HIV, hepatitis, etc.) and blood samples, and cross-match tests
-Accuracy of test results
-How to do separation (analysis) Blood
Third intermediate variable between blood bank and hospital
$z_{4j}^{(3)}$ Other Social actions of blood bank:
-Crisis management (Sudden hazards such as earthquake, etc.)
-Controlling blood bags during (at the time of) delivery to hospital (hemolysis, clots. discoloration, etc.)
Fourth intermediate variable between blood bank and hospital
$z_{5j}^{(3)}$ Partnership and cooperation between blood bank and hospital:
-Sharing information
Fifth intermediate variable between blood bank and hospital
$z_{6j}^{(3)}$Inventory management in blood bank
-Management of blood shortages (unavailability of required blood group, etc.)
-Proper selection of blood to send considering condition (freshness, close to the expiration date, ...)
-Cross-matching policies
Sixth intermediate variable between blood bank and hospital
$f_{1j}^{(2)}$Donor satisfaction from blood donation center
-Employee attitude
-Responsiveness
-How to get blood from a donor -Consulting for hepatitis, AIDS, thalassemia, etc.
First feedback variable of blood donation center
$f_{1j}^{(3)}$ Satisfaction of blood donation center from blood bank
-Accuracy of results of test on donated blood (HIV, hepatitis, etc.) provided by blood bank
First feedback variable of blood bank
$f_{1j}^{(4)} $ Blood bank satisfaction from hospital
-Storage and transportation of blood samples in hospital for sending to blood bank
First feedback variable of hospital
Table 2.  Efficiency of DMUs in each group
GroupEfficiency of the $DMU_o$ in the group
1 $e_o^L=e_o^U=1$
2 $e_o^L < 1, e_o^U=1$
3 $e_o^L < 1, e_o^U < 1$
GroupEfficiency of the $DMU_o$ in the group
1 $e_o^L=e_o^U=1$
2 $e_o^L < 1, e_o^U=1$
3 $e_o^L < 1, e_o^U < 1$
Table 3.  Typical cross-efficiency matrix for DMUs (Jahanshahloo et al, 2011)
$DMU_1$ $DMU_2$... $DMU_n$
$DMU_1$[$e_{11}^L, e_{11}^U$][$e_{12}^L, e_{12}^U$]...[$e_{1n}^L, e_{1n}^U$]
$DMU_2$[$e_{21}^L, e_{21}^U$][$e_{22}^L, e_{22}^U$] ... [$e_{2n}^L, e_{2n}^U$]
$.$ $.$ $.$ $.$ $.$
$.$ $.$ $.$ $.$ $.$
.....
$DMU_n$[$e_{n1}^L, e_{n1}^U$][$e_{n2}^L, e_{n2}^U$]...[$e_{nn}^L, e_{nn}^U$]
$DMU_1$ $DMU_2$... $DMU_n$
$DMU_1$[$e_{11}^L, e_{11}^U$][$e_{12}^L, e_{12}^U$]...[$e_{1n}^L, e_{1n}^U$]
$DMU_2$[$e_{21}^L, e_{21}^U$][$e_{22}^L, e_{22}^U$] ... [$e_{2n}^L, e_{2n}^U$]
$.$ $.$ $.$ $.$ $.$
$.$ $.$ $.$ $.$ $.$
.....
$DMU_n$[$e_{n1}^L, e_{n1}^U$][$e_{n2}^L, e_{n2}^U$]...[$e_{nn}^L, e_{nn}^U$]
Table 4.  Final interval data of each supply chain based on experts' beliefs
$\bf DMU_j$
$\bf Data$ 1 2 3 4 5 6 7 8
$\bf x^{(1)}_{1j}$ [2.5- 6.1] [4.3-7.8] [2.4-7.16] [4.52-8.4] [2.82-6.3] [1.8-6.6] [4.5-8.4] [2.5-7]
$\bf x^{(2)}_{1j}$ [2.6-7.4] [1-7.4] [1-9] [1.8-7.4] [2.6-9] [1-5.8] [4.2-5.8] [3.4-9]
$\bf x^{(2)}_{2j}$ [3.4- 6.6] [1-6.6] [3.8- 7.6] [3.4-6.6] [5-9] [4.2-5.8] [2-7] [7.8- 7.8]
$\bf x^{(2)}_{3j}$ [2-7] [1-2.6] [1-8.2] [2.6-9] [4-9] [1.6-7.8] [3.4-6.6] [5.2-9]
$\bf x^{(3)}_{1j}$ [4.2-5.8] [1- 6] [2.4- 7.6] [2- 8] [1- 9] [4.6- 6.2] [2.2- 4.2] [4.6-7.8]
$\bf x^{(3)}_{2j}$ [3.4- 6.6] [2.2- 5.4] [5- 7] [6- 6] [5- 9] [1-7.4] [2- 8] [5.2- 9]
$\bf x^{(3)}_{3j}$ [4.6- 7.8] [1- 6.6] [3- 8] [3.4- 7.8] [3- 8] [2.2- 5.4] [2.6-7.4] [3-8]
$\bf x^{(4)}_{1j}$ [3.8-7.6] [3- 8] [5.8-5.8] [3.4- 6.6] [4.2- 7.4] [1.8-6.6] [1- 8] [2.6-5.8]
$\bf x^{(4)}_{2j}$ [3.4- 9] [3.8-6.2] [4- 9] [2- 7] [6.2- 6.2] [1.4-1.4] [4-8] [3.4-6.6]
$\bf x^{(4)}_{3j}$ [4-8] [2-8] [3-9] [5.8-6.6] [4-8] [1-6] [3.6- 6.2] [2-7]
$\bf y^{(2)}_{1j}$ [5.2-9] [1-7.6] [7.4-9] [4.2-7.4] [3.4-6.6] [1-4.2] [3.8-6.2] [2.4-7.6]
$\bf y^{(2)}_{2j}$ [2.4-6.2] [2-6] [1-7.6] [2-8] [1-5] [4.6-7.8] [4-4] [4-9]
$\bf y^{(2)}_{3j}$ [1.8-8.2] [1-5.4] [1-8.2] [2-9] [1-4.8] [2.6-7.4] [2.4-6.2] [4.4-7]
$\bf y^{(2)}_{4j}$ [4.6-7.8] [1-4.8] [1-6] [1-9] [1-5.8] [3.4-9] [1-9] [7.4-9]
$\bf y^{(2)}_{5j}$ [1-4.8] [4.2-9] [2.9-5.6] [4-6] [1-6.6] [2.3-5.1] [1-9] [2.6-9]
$\bf y^{(3)}_{1j}$ [3-8] [2.6-4.2] [3.6-5.2] [6.6-9] [1-4.2] [2.4-4.8] [3.4-6.6] [5.2-9]
$\bf y^{(3)}_{2j}$ [3-7] [3.4-8.2] [1-7.4] [2.2-9] [1-5] [3-7] [2-6] [4.6-7.8]
$\bf y^{(3)}_{3j}$ [2.4-6.2] [1.8-8.2] [5-8.4] [4.2-6.6] [8.2-8.2] [2-2.8] [6.6-8.2] [3-7.8]
$\bf y^{(3)}_{4j}$ [3.8-9] [3-7] [3-7] [6.2-8.2] [6.6-8.2] [2.8-6.2] [6.2-8.6] [2.2-7/4]
$\bf y^{(3)}_{5j}$ [2.2-5.4] [1-6.6] [2.6-9] [2.6-7] [4.8-6.4] [6.6-8.2] [1-9] [3.4- 5]
$\bf y^{(4)}_{1j}$ [1-6.6] [4.2-4.2] [1.8-7.4] [6.2-6.2] [3-6.2] [2.6-6.2] [3-7.8] [4.6-6.2]
$\bf y^{(4)}_{2j}$ [2.6-3.8] [1-9] [1-4.8] [5-8.2] [6.6-8.6] [4.2-8.2] [5.8-5.8] [2.2-6.2]
$\bf y^{(4)}_{3j}$ [2.2-6.04] [6-8.3] [4.2-7.8] [2.4-8.6] [3.6-7.4] [3.7-7.8] [3.7-7.8] [5.4-8.08]
$\bf y^{(4)}_{4j}$ [6.2-7.9] [2.8-5.4] [4.2-6.4] [2.6-6.8] [3.8-6.3] [3.3-6.4] [5.04-7.7] [3.1-6.9]
$\bf y^{(4)}_{5j}$ [3.7-8.08] [3.1-8.4] [2.8-7.6] [3.8-8.1][ 2.2-6.9] [5.1-8.3] [2.04-] [1.8-7.1]
$\bf y^{(4)}_{6j}$ [2.5-5.4] [4.6-8.1] [4.5-8.4] [2.8-7.2] [3.7-8.2] [3.7-7.8] [2.6-6.5] [2.2-6.1]
$\bf y^{(4)}_{7j}$ [1.2-5.9] [3.1-6.6] [2.3-6.8] [4.9-8.4] [3.08-5.3] [1.2-5.6] [3.2-6.8] [2.9-5.6]
$\bf y^{(4)}_{8j}$ [4.03-8.6] [2.5-6.3] [2.1-6.6] [4.2-8.04] [3.8-8.2] [1.28-6] [3.4-7.8] [3.8-7.1]
$\bf y^{(4)}_{9j}$ [4.2-5.8] [5.2-7.6] [3.6-7.6] [3.4-6.2] [1-6.6] [2.6-9] [1-9] [5.8-7.4]
$\bf z^{(1)}_{1j}$ [2.3-5.1] [1-9] [4-9] [1-7] [2.6-8.2] [2.5-6] [1-6.2] [2.3-7.7]
$\bf z^{(2)}_{1j}$ [2.6-7.4][1-6.6] [1-8.2] [2.6-9] [4.2-5.8] [5.2-9] [4.2-5.8] [2.4-6.2]
$\bf z^{(2)}_{2j}$ [3-6.2] [3.6-5.8] [1.4-8.2] [3.6-7.6] [7.2-8] [1-6.2] [1.8-7.4] [1-7]
$\bf z^{(2)}_{3j}$ [1-9] [4.2-5.8] [3-7.8] [3.4-9] [3.6-7.6] [3-6.2] [3-7] [5-8.2]
$\bf z^{(2)}_{4j}$ [8.4-8.4] [2.4-6.2] [4.6-7.8] [1-7] [3.8-7.6] [2.3-5.1] [4.2-7.4] [2.6-9]
$\bf z^{(2)}_{5j}$ [3.7-8.08] [3.2-6.8] [2.6-8.5] [1-7.8] [4.4-7.2] [3-8.2] [2-5.7] [1-9]
$\bf z^{(2)}_{6j}$ [5.6-5.6] [4.9-7.7] [6.6-9] [3-4] [7.4-9] [3.4-6.6] [4.8-8.2] [4.2-9]
$\bf z^{(3)}_{1j}$ [4-4] [5-8.2] [3.9-5.1] [4.4-7.2] [1-5] [5.2-9] [1.8-7.4] [2.6-5.8]
$\bf z^{(3)}_{2j}$ [1.2-5.6] [4.9-8.4] [2.8-7.2] [4.6-8.1] [1-4.9] [3.5-7.6] [2.6-6.4] [1-4.4]
$\bf z^{(3)}_{3j}$ [1-6.6] [2.2-5.4] [4.2-5.8] [5.8-9] [2.4-4.8] [2-7] [4.2-9] [2.4-6.2]
$\bf z^{(3)}_{4j}$ [4.9-9] [1-6] [4.6-9] [2.3-7.7] [4.6-7.8] [3-8] [5.8-7.4] [2.4-9]
$\bf z^{(3)}_{5j}$ [4.2-5.8] [4.2-5.8] [2.4-6.2] [2.2-6.6] [1-6] [5.2-9] [4-9] [3-7]
$\bf z^{(3)}_{6j}$ [3-9] [4.4-7.4] [3.4-6.6] [3-8] [2.4-4.8] [3.4-7.8] [1-9] [3.8-7.6]
$\bf f^{(2)}_{1j}$ [5.8-9] [6.6-6.6] [3-8] [2-6] [4.6-7.8] [3-9] [5.8-7.4] [4-9]
$\bf f^{(3)}_{1j}$ [3.4-6.6] [4.6-7.8] [4.6-9] [3-7] [2.2-7.8] [6.2-8.4] [1-7] [2-7]
$\bf f^{(4)}_{1j}$ [3-9] [1-4.2] [5-9] [4.2-5.8] [2.6-4.2] [3-9] [1.4-7.8] [1.8-8.2]
$\bf DMU_j$
$\bf Data$ 1 2 3 4 5 6 7 8
$\bf x^{(1)}_{1j}$ [2.5- 6.1] [4.3-7.8] [2.4-7.16] [4.52-8.4] [2.82-6.3] [1.8-6.6] [4.5-8.4] [2.5-7]
$\bf x^{(2)}_{1j}$ [2.6-7.4] [1-7.4] [1-9] [1.8-7.4] [2.6-9] [1-5.8] [4.2-5.8] [3.4-9]
$\bf x^{(2)}_{2j}$ [3.4- 6.6] [1-6.6] [3.8- 7.6] [3.4-6.6] [5-9] [4.2-5.8] [2-7] [7.8- 7.8]
$\bf x^{(2)}_{3j}$ [2-7] [1-2.6] [1-8.2] [2.6-9] [4-9] [1.6-7.8] [3.4-6.6] [5.2-9]
$\bf x^{(3)}_{1j}$ [4.2-5.8] [1- 6] [2.4- 7.6] [2- 8] [1- 9] [4.6- 6.2] [2.2- 4.2] [4.6-7.8]
$\bf x^{(3)}_{2j}$ [3.4- 6.6] [2.2- 5.4] [5- 7] [6- 6] [5- 9] [1-7.4] [2- 8] [5.2- 9]
$\bf x^{(3)}_{3j}$ [4.6- 7.8] [1- 6.6] [3- 8] [3.4- 7.8] [3- 8] [2.2- 5.4] [2.6-7.4] [3-8]
$\bf x^{(4)}_{1j}$ [3.8-7.6] [3- 8] [5.8-5.8] [3.4- 6.6] [4.2- 7.4] [1.8-6.6] [1- 8] [2.6-5.8]
$\bf x^{(4)}_{2j}$ [3.4- 9] [3.8-6.2] [4- 9] [2- 7] [6.2- 6.2] [1.4-1.4] [4-8] [3.4-6.6]
$\bf x^{(4)}_{3j}$ [4-8] [2-8] [3-9] [5.8-6.6] [4-8] [1-6] [3.6- 6.2] [2-7]
$\bf y^{(2)}_{1j}$ [5.2-9] [1-7.6] [7.4-9] [4.2-7.4] [3.4-6.6] [1-4.2] [3.8-6.2] [2.4-7.6]
$\bf y^{(2)}_{2j}$ [2.4-6.2] [2-6] [1-7.6] [2-8] [1-5] [4.6-7.8] [4-4] [4-9]
$\bf y^{(2)}_{3j}$ [1.8-8.2] [1-5.4] [1-8.2] [2-9] [1-4.8] [2.6-7.4] [2.4-6.2] [4.4-7]
$\bf y^{(2)}_{4j}$ [4.6-7.8] [1-4.8] [1-6] [1-9] [1-5.8] [3.4-9] [1-9] [7.4-9]
$\bf y^{(2)}_{5j}$ [1-4.8] [4.2-9] [2.9-5.6] [4-6] [1-6.6] [2.3-5.1] [1-9] [2.6-9]
$\bf y^{(3)}_{1j}$ [3-8] [2.6-4.2] [3.6-5.2] [6.6-9] [1-4.2] [2.4-4.8] [3.4-6.6] [5.2-9]
$\bf y^{(3)}_{2j}$ [3-7] [3.4-8.2] [1-7.4] [2.2-9] [1-5] [3-7] [2-6] [4.6-7.8]
$\bf y^{(3)}_{3j}$ [2.4-6.2] [1.8-8.2] [5-8.4] [4.2-6.6] [8.2-8.2] [2-2.8] [6.6-8.2] [3-7.8]
$\bf y^{(3)}_{4j}$ [3.8-9] [3-7] [3-7] [6.2-8.2] [6.6-8.2] [2.8-6.2] [6.2-8.6] [2.2-7/4]
$\bf y^{(3)}_{5j}$ [2.2-5.4] [1-6.6] [2.6-9] [2.6-7] [4.8-6.4] [6.6-8.2] [1-9] [3.4- 5]
$\bf y^{(4)}_{1j}$ [1-6.6] [4.2-4.2] [1.8-7.4] [6.2-6.2] [3-6.2] [2.6-6.2] [3-7.8] [4.6-6.2]
$\bf y^{(4)}_{2j}$ [2.6-3.8] [1-9] [1-4.8] [5-8.2] [6.6-8.6] [4.2-8.2] [5.8-5.8] [2.2-6.2]
$\bf y^{(4)}_{3j}$ [2.2-6.04] [6-8.3] [4.2-7.8] [2.4-8.6] [3.6-7.4] [3.7-7.8] [3.7-7.8] [5.4-8.08]
$\bf y^{(4)}_{4j}$ [6.2-7.9] [2.8-5.4] [4.2-6.4] [2.6-6.8] [3.8-6.3] [3.3-6.4] [5.04-7.7] [3.1-6.9]
$\bf y^{(4)}_{5j}$ [3.7-8.08] [3.1-8.4] [2.8-7.6] [3.8-8.1][ 2.2-6.9] [5.1-8.3] [2.04-] [1.8-7.1]
$\bf y^{(4)}_{6j}$ [2.5-5.4] [4.6-8.1] [4.5-8.4] [2.8-7.2] [3.7-8.2] [3.7-7.8] [2.6-6.5] [2.2-6.1]
$\bf y^{(4)}_{7j}$ [1.2-5.9] [3.1-6.6] [2.3-6.8] [4.9-8.4] [3.08-5.3] [1.2-5.6] [3.2-6.8] [2.9-5.6]
$\bf y^{(4)}_{8j}$ [4.03-8.6] [2.5-6.3] [2.1-6.6] [4.2-8.04] [3.8-8.2] [1.28-6] [3.4-7.8] [3.8-7.1]
$\bf y^{(4)}_{9j}$ [4.2-5.8] [5.2-7.6] [3.6-7.6] [3.4-6.2] [1-6.6] [2.6-9] [1-9] [5.8-7.4]
$\bf z^{(1)}_{1j}$ [2.3-5.1] [1-9] [4-9] [1-7] [2.6-8.2] [2.5-6] [1-6.2] [2.3-7.7]
$\bf z^{(2)}_{1j}$ [2.6-7.4][1-6.6] [1-8.2] [2.6-9] [4.2-5.8] [5.2-9] [4.2-5.8] [2.4-6.2]
$\bf z^{(2)}_{2j}$ [3-6.2] [3.6-5.8] [1.4-8.2] [3.6-7.6] [7.2-8] [1-6.2] [1.8-7.4] [1-7]
$\bf z^{(2)}_{3j}$ [1-9] [4.2-5.8] [3-7.8] [3.4-9] [3.6-7.6] [3-6.2] [3-7] [5-8.2]
$\bf z^{(2)}_{4j}$ [8.4-8.4] [2.4-6.2] [4.6-7.8] [1-7] [3.8-7.6] [2.3-5.1] [4.2-7.4] [2.6-9]
$\bf z^{(2)}_{5j}$ [3.7-8.08] [3.2-6.8] [2.6-8.5] [1-7.8] [4.4-7.2] [3-8.2] [2-5.7] [1-9]
$\bf z^{(2)}_{6j}$ [5.6-5.6] [4.9-7.7] [6.6-9] [3-4] [7.4-9] [3.4-6.6] [4.8-8.2] [4.2-9]
$\bf z^{(3)}_{1j}$ [4-4] [5-8.2] [3.9-5.1] [4.4-7.2] [1-5] [5.2-9] [1.8-7.4] [2.6-5.8]
$\bf z^{(3)}_{2j}$ [1.2-5.6] [4.9-8.4] [2.8-7.2] [4.6-8.1] [1-4.9] [3.5-7.6] [2.6-6.4] [1-4.4]
$\bf z^{(3)}_{3j}$ [1-6.6] [2.2-5.4] [4.2-5.8] [5.8-9] [2.4-4.8] [2-7] [4.2-9] [2.4-6.2]
$\bf z^{(3)}_{4j}$ [4.9-9] [1-6] [4.6-9] [2.3-7.7] [4.6-7.8] [3-8] [5.8-7.4] [2.4-9]
$\bf z^{(3)}_{5j}$ [4.2-5.8] [4.2-5.8] [2.4-6.2] [2.2-6.6] [1-6] [5.2-9] [4-9] [3-7]
$\bf z^{(3)}_{6j}$ [3-9] [4.4-7.4] [3.4-6.6] [3-8] [2.4-4.8] [3.4-7.8] [1-9] [3.8-7.6]
$\bf f^{(2)}_{1j}$ [5.8-9] [6.6-6.6] [3-8] [2-6] [4.6-7.8] [3-9] [5.8-7.4] [4-9]
$\bf f^{(3)}_{1j}$ [3.4-6.6] [4.6-7.8] [4.6-9] [3-7] [2.2-7.8] [6.2-8.4] [1-7] [2-7]
$\bf f^{(4)}_{1j}$ [3-9] [1-4.2] [5-9] [4.2-5.8] [2.6-4.2] [3-9] [1.4-7.8] [1.8-8.2]
Table 5.  Interval efficiency values of DMUs and sub-DMUs obtained from model (Ⅱ) and (Ⅲ)
$\bf DMU_o$ $\bf (e_o^{1})^L$ $\bf (e_o^{1})^U$ $\bf (e_o^{2})^L$ $\bf (e_o^{2})^U$ $\bf (e_o^{3})^L$ $\bf (e_o^{3})^U$ $\bf (e_o^{4})^L$ $\bf (e_o^{4})^U$ $\bf e_o^L$ $\bf e_o^U$
10.590.8650.6340.9760.6170.9710.74310.1710.82
20.620.8680.7910.6120.940.6450.9820.1930.801
30.8370.9860.58710.6840.9840.7210.2410.97
40.65110.69510.7510.85510.291
50.57610.51110.63510.7410.1381
60.660.8160.760.970.90.9690.8570.8870.3860.68
70.6650.940.8250.9760.73310.8890.9860.3570.904
80.6230.780.7110.8360.920.6760.7930.250.569
$\bf DMU_o$ $\bf (e_o^{1})^L$ $\bf (e_o^{1})^U$ $\bf (e_o^{2})^L$ $\bf (e_o^{2})^U$ $\bf (e_o^{3})^L$ $\bf (e_o^{3})^U$ $\bf (e_o^{4})^L$ $\bf (e_o^{4})^U$ $\bf e_o^L$ $\bf e_o^U$
10.590.8650.6340.9760.6170.9710.74310.1710.82
20.620.8680.7910.6120.940.6450.9820.1930.801
30.8370.9860.58710.6840.9840.7210.2410.97
40.65110.69510.7510.85510.291
50.57610.51110.63510.7410.1381
60.660.8160.760.970.90.9690.8570.8870.3860.68
70.6650.940.8250.9760.73310.8890.9860.3570.904
80.6230.780.7110.8360.920.6760.7930.250.569
Table 6.  cross-efficiency matrix for DMUs obtained from model (Ⅳ) and (Ⅴ)
$DMU_1$ $DMU_2$ $DMU_3$ $DMU_4$ $DMU_5$ $DMU_6$ $DMU_7$ $DMU_8$
$DMU_1$[0.06-1][0.12-0.99][0.13-0.9][0.09-1.23][0.2-0.74][0.37-0.68][0.2-0.83][0.23-0.49]
$DMU_2$[0.12-1][0.17-1][0.13-0.94][0.14-1.12][0.2-0.88][0.34-0.71][0.19-0.8][0.17-0.55]
$DMU_3$[0.13-0.98][0.17-1.1][0.13-1][0.15-1][0.3-0.92][0.18-0.84][0.27-0.89][0.18-0.69]
$DMU_4$[0.22-1][0.19-0.95][0.15-1.12][0.19-1][0.15-0.87][0.25-0.89][0.26-0.75][0.18-0.93]
$DMU_5$[0.14-1.1][0.18-1][0.16-0.9][0.16-0.99][0.14-0.87][0.28-0.83][0.14-0.9][0.21-0.92]
$DMU_6$[0.22-0.78][0.2-1][0.34-0.65][0.37-1][0.25-0.79][0.33-0.62][0.25-0.75][0.33-0.63]
$DMU_7$[0.18-0.8][0.17-0.92][0.26-0.95][0.19-0.86][0.28-0.79][0.37-0.8][0.19-0.77][0.22-0.84]
$DMU_8$[0.23-0.64][0.18-0.82][0.16-0.65][0.19-0.74][0.24-0.7][0.18-0.68][0.23-0.72][0.17-0.56]
$DMU_1$ $DMU_2$ $DMU_3$ $DMU_4$ $DMU_5$ $DMU_6$ $DMU_7$ $DMU_8$
$DMU_1$[0.06-1][0.12-0.99][0.13-0.9][0.09-1.23][0.2-0.74][0.37-0.68][0.2-0.83][0.23-0.49]
$DMU_2$[0.12-1][0.17-1][0.13-0.94][0.14-1.12][0.2-0.88][0.34-0.71][0.19-0.8][0.17-0.55]
$DMU_3$[0.13-0.98][0.17-1.1][0.13-1][0.15-1][0.3-0.92][0.18-0.84][0.27-0.89][0.18-0.69]
$DMU_4$[0.22-1][0.19-0.95][0.15-1.12][0.19-1][0.15-0.87][0.25-0.89][0.26-0.75][0.18-0.93]
$DMU_5$[0.14-1.1][0.18-1][0.16-0.9][0.16-0.99][0.14-0.87][0.28-0.83][0.14-0.9][0.21-0.92]
$DMU_6$[0.22-0.78][0.2-1][0.34-0.65][0.37-1][0.25-0.79][0.33-0.62][0.25-0.75][0.33-0.63]
$DMU_7$[0.18-0.8][0.17-0.92][0.26-0.95][0.19-0.86][0.28-0.79][0.37-0.8][0.19-0.77][0.22-0.84]
$DMU_8$[0.23-0.64][0.18-0.82][0.16-0.65][0.19-0.74][0.24-0.7][0.18-0.68][0.23-0.72][0.17-0.56]
Table 7.  Entropy values and ranking results of DMUs
$\bf DMU_j$$\bf \bar {e_{jo}}^L$ $\bf \bar {e_{jo}}^U$ $\bf K_j$ $\bf H_j$Rank
10.170.850.510.8667
20.180.850.5150.8986
30.180.920.550.9445
40.190.940.560.9763
50.150.930.540.9744
60.280.770.520.9852
70.230.840.531.011
80.190.680.430.88
$\bf DMU_j$$\bf \bar {e_{jo}}^L$ $\bf \bar {e_{jo}}^U$ $\bf K_j$ $\bf H_j$Rank
10.170.850.510.8667
20.180.850.5150.8986
30.180.920.550.9445
40.190.940.560.9763
50.150.930.540.9744
60.280.770.520.9852
70.230.840.531.011
80.190.680.430.88
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