# American Institute of Mathematical Sciences

December  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

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
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##### References:
Network model of $j^{th}$ blood supply chain
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 costs Third 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 costs Third 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 costs Third 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 center Second 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 bank Second 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 revenues Second 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 costs Third 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 costs Third 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 costs Third 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 center Second 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 bank Second 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 revenues Second 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
Efficiency of DMUs in each group
 Group Efficiency 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 $e_o^L < 1, e_o^U < 1$
 Group Efficiency 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 $e_o^L < 1, e_o^U < 1$
Typical cross-efficiency matrix for DMUs (Jahanshahloo et al, 2011)
 $DMU_1$ $DMU_2$ ... $DMU_n$ $DMU_1$ [$e_{11}^L, e_{11}^U$] [$e_{12}^L, e_{12}^U$] ... [$e_{1n}^L, e_{1n}^U$] $DMU_2$ [$e_{21}^L, e_{21}^U$] [$e_{22}^L, e_{22}^U$] ... [$e_{2n}^L, e_{2n}^U$] $.$ $.$ $.$ $.$ $.$ $.$ $.$ $.$ $.$ $.$ . . . . . $DMU_n$ [$e_{n1}^L, e_{n1}^U$] [$e_{n2}^L, e_{n2}^U$] ... [$e_{nn}^L, e_{nn}^U$]
 $DMU_1$ $DMU_2$ ... $DMU_n$ $DMU_1$ [$e_{11}^L, e_{11}^U$] [$e_{12}^L, e_{12}^U$] ... [$e_{1n}^L, e_{1n}^U$] $DMU_2$ [$e_{21}^L, e_{21}^U$] [$e_{22}^L, e_{22}^U$] ... [$e_{2n}^L, e_{2n}^U$] $.$ $.$ $.$ $.$ $.$ $.$ $.$ $.$ $.$ $.$ . . . . . $DMU_n$ [$e_{n1}^L, e_{n1}^U$] [$e_{n2}^L, e_{n2}^U$] ... [$e_{nn}^L, e_{nn}^U$]
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]
Interval efficiency values of DMUs and sub-DMUs obtained from model (Ⅱ) and (Ⅲ)
 $\bf DMU_o$ $\bf (e_o^{1})^L$ $\bf (e_o^{1})^U$ $\bf (e_o^{2})^L$ $\bf (e_o^{2})^U$ $\bf (e_o^{3})^L$ $\bf (e_o^{3})^U$ $\bf (e_o^{4})^L$ $\bf (e_o^{4})^U$ $\bf e_o^L$ $\bf e_o^U$ 1 0.59 0.865 0.634 0.976 0.617 0.971 0.743 1 0.171 0.82 2 0.62 0.868 0.79 1 0.612 0.94 0.645 0.982 0.193 0.801 3 0.837 0.986 0.587 1 0.684 0.984 0.72 1 0.241 0.97 4 0.651 1 0.695 1 0.75 1 0.855 1 0.29 1 5 0.576 1 0.511 1 0.635 1 0.74 1 0.138 1 6 0.66 0.816 0.76 0.97 0.9 0.969 0.857 0.887 0.386 0.68 7 0.665 0.94 0.825 0.976 0.733 1 0.889 0.986 0.357 0.904 8 0.623 0.78 0.71 1 0.836 0.92 0.676 0.793 0.25 0.569
 $\bf DMU_o$ $\bf (e_o^{1})^L$ $\bf (e_o^{1})^U$ $\bf (e_o^{2})^L$ $\bf (e_o^{2})^U$ $\bf (e_o^{3})^L$ $\bf (e_o^{3})^U$ $\bf (e_o^{4})^L$ $\bf (e_o^{4})^U$ $\bf e_o^L$ $\bf e_o^U$ 1 0.59 0.865 0.634 0.976 0.617 0.971 0.743 1 0.171 0.82 2 0.62 0.868 0.79 1 0.612 0.94 0.645 0.982 0.193 0.801 3 0.837 0.986 0.587 1 0.684 0.984 0.72 1 0.241 0.97 4 0.651 1 0.695 1 0.75 1 0.855 1 0.29 1 5 0.576 1 0.511 1 0.635 1 0.74 1 0.138 1 6 0.66 0.816 0.76 0.97 0.9 0.969 0.857 0.887 0.386 0.68 7 0.665 0.94 0.825 0.976 0.733 1 0.889 0.986 0.357 0.904 8 0.623 0.78 0.71 1 0.836 0.92 0.676 0.793 0.25 0.569
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]
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 1 0.17 0.85 0.51 0.866 7 2 0.18 0.85 0.515 0.898 6 3 0.18 0.92 0.55 0.944 5 4 0.19 0.94 0.56 0.976 3 5 0.15 0.93 0.54 0.974 4 6 0.28 0.77 0.52 0.985 2 7 0.23 0.84 0.53 1.01 1 8 0.19 0.68 0.43 0.8 8
 $\bf DMU_j$ $\bf \bar {e_{jo}}^L$ $\bf \bar {e_{jo}}^U$ $\bf K_j$ $\bf H_j$ Rank 1 0.17 0.85 0.51 0.866 7 2 0.18 0.85 0.515 0.898 6 3 0.18 0.92 0.55 0.944 5 4 0.19 0.94 0.56 0.976 3 5 0.15 0.93 0.54 0.974 4 6 0.28 0.77 0.52 0.985 2 7 0.23 0.84 0.53 1.01 1 8 0.19 0.68 0.43 0.8 8
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