American Institute of Mathematical Sciences

Advanced
May  2022, 18(3): 2237-2253. doi: 10.3934/jimo.2021065

A novel risk ranking method based on the single valued neutrosophic set

Kuei-Hu Chang ,

Department of Management Sciences, R.O.C. Military Academy, Kaohsiung 830, Taiwan, Institute of Innovation and Circular Economy, Asia University, Taichung 413, Taiwan

* Corresponding author: Kuei-Hu Chang

Received  February 2020 Revised  January 2021 Published  May 2022 Early access  April 2021

Fund Project: The research is supported by the Ministry of Science and Technology, Taiwan(Grant No. MOST 107-2410-H-145-001, MOST 108-2410-H-145-001, and MOST 109-2410-H-145-002)

Risk assessment is a key issue in the process of product design and manufacturing. Traditionally risk assessment uses the risk priority number (RPN) method to rank the extent of a threat. However, this simultaneously includes quantitative and qualitative evaluation factors in the process of risk assessment. Moreover, the information provided by different experts for evaluation factors contain ambiguous, incomplete and inconsistent information. These problems lead to more difficulty for risk assessment, and cannot be effectively solved by the traditional RPN method. To solve some limits of the traditional risk analysis method, this paper integrates the single valued neutrosophic set and subsethood measure method to rank the extent of the threat. For missing or incomplete information in the information aggregation process, the minimum, averaging and maximum operators are used to perform data imputation to avoid the distortion of decision results. Finally, a numerical example of high-dose-rate (HDR) brachytherapy treatments is provided to demonstrate the effectiveness and feasibility of the proposed method, and a comparative analysis with some other existing methods is given.

Keywords: Risk assessment, neutrosophic set, single valued neutrosophic set, multi-criteria decision-making.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Kuei-Hu Chang. A novel risk ranking method based on the single valued neutrosophic set. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2237-2253. doi: 10.3934/jimo.2021065
References:
[1]

K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87-96.  doi: 10.1016/S0165-0114(86)80034-3.

[2]

P. Biswas, S. Pramanik and B. C. Giri, TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment, Neural Comput. Appl., 27 (2016), 727-737.  doi: 10.1007/s00521-015-1891-2.

[3]

British Standards Institute, Reliability of Systems, Equipment and Components, Guide to Failure Modes, Effects and Criticality Analysis (FMEA and FMECA), Vol. BS 5760-5, British Standards Institute, United Kingdom, 1991.

[4]

K. H. Chang, Evaluate the orderings of risk for failure problems using a more general RPN methodology, Microelectron. Reliab., 49 (2009), 1586-1596.  doi: 10.1016/j.microrel.2009.07.057.

[5]

K. H. Chang, A novel reliability calculation method under neutrosophic environments, Ann. Oper. Res., (2021) in press. doi: 10.1007/s10479-020-03890-4.

[6]

K. H. Chang, A more general risk assessment methodology using soft sets based ranking technique, Soft Comput., 18 (2014), 169-183.  doi: 10.1007/s00500-013-1045-3.

[7]

K. H. Chang, Y. C. Chang and P. T. Lai, Applying the concept of exponential approach to enhance the assessment capability of FMEA, J. Intell. Manuf., 25 (2014), 1413-1427.  doi: 10.1007/s10845-013-0747-9.

[8]

Y. C. Chang, K. H. Chang and C. Y. Chen, Risk assessment by quantifying and prioritizing 5S activity for semiconductor manufacturing, Proc. Inst. Mech. Eng. Part B-J. Eng. Manuf., 227 (2013), 1874-1887.  doi: 10.1177/0954405413493901.

[9]

N. Chanamool and T. Naenna, Fuzzy FMEA application to improve decision-making process in an emergency department, Appl. Soft. Comput., 43 (2016), 441-453.  doi: 10.1016/j.asoc.2016.01.007.

[10]

K. H. Chang, Generalized multi-attribute failure mode analysis, Neurocomputing, 175 (2016), 90-100.  doi: 10.1016/j.neucom.2015.10.039.

[11]

D.C. US Department of Defense Washington, Procedures for Performing a Failure Mode Effects and Criticality Analysis, US MIL-STD-1629A, 1980.

[12]

M. Giardina, F. Castiglia and E. Tomarchio, Risk assessment of component failure modes and human errors using a new FMECA approach: Application in the safety analysis of HDR brachytherapy, J. Radiol. Prot., 34 (2014), 891-914.  doi: 10.1088/0952-4746/34/4/891.

[13]

Y. H. Guo and A. Sengur, A novel 3D skeleton algorithm based on neutrosophic cost function, Appl. Soft. Comput., 36 (2015), 210-217.  doi: 10.1016/j.asoc.2015.07.025.

[14]

International Electrotechnical Commission, Analysis Techniques for System Reliability- Procedures for Failure Mode and Effect Analysis, Geneva, IEC 60812, 1985.

[15]

H. A. Khorshidi, I. Gunawan and M. Y. Ibrahim, Applying UGF concept to enhance the assessment capability of FMEA, Qual. Reliab. Eng. Int., 32 (2016), 1085-1093.  doi: 10.1002/qre.1817.

[16]

P. Kraipeerapun and C. C. Fung, Binary classification using ensemble neural networks and interval neutrosophic sets, Neurocomputing, 72 (2009), 2845-2856.  doi: 10.1016/j.neucom.2008.07.017.

[17]

S. Li and W. Zeng, Risk analysis for the supplier selection problem using failure modes and effects analysis (FMEA), J. Intell. Manuf., 27 (2016), 1309-1321.  doi: 10.1007/s10845-014-0953-0.

[18]

H. C. Liu, J. X. You, X. J. Fan and Q. L. Lin, Failure mode and effects analysis using D numbers and grey relational projection method, Expert Syst. Appl., 41 (2014), 4670-4679.  doi: 10.1016/j.eswa.2014.01.031.

[19]

O. Mohsen and N. Fereshteh, An extended VIKOR method based on entropy measure for the failure modes risk assessment - A case study of the geothermal power plant (GPP), Saf. Sci., 92 (2017), 160-172.  doi: 10.1016/j.ssci.2016.10.006.

[20]

H. Safari, Z. Faraji and S. Majidian, Identifying and evaluating enterprise architecture risks using FMEA and fuzzy VIKOR differentiables, J. Intell. Manuf., 27 (2016), 475-486.  doi: 10.1007/s10845-014-0880-0.

[21]

R. Sahin and P. D. Liu, Maximizing deviation method for neutrosophic multiple attribute decision making with incomplete weight information, Neural Comput. Appl., 27 (2016), 2017-2029.  doi: 10.1007/s10700-006-0022-z.

[22]

R. Sahin and A. Kucuk, Subsethood measure for single valued neutrosophic sets, J. Intell. Fuzzy Syst., 29 (2015), 525-530.  doi: 10.3233/ifs-141304.

[23]

F. Smarandache, A unifying field in logics, neutrosophy: Neutrosophic probability, set and logic, preprint, arXiv: 0101228.

[24]

Z. P. Tian, H. Y. Zhang, J. Wang, J. Q. Wang and X. H. Chen, Multi-criteria decision-making method based on a cross-entropy with interval neutrosophic sets, Int. J. Syst. Sci., 47 (2016), 3598-3608.  doi: 10.1080/00207721.2015.1102359.

[25]

H. Wang, F. Smarandache, Y. Q. Zhang and R. Sunderraman, Single valued neutrosophic sets, Multispace Multistructure, 4 (2014), 410-413.  doi: 10.1007/s00357-017-9225-y.

[26]

J. Ye, A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets, J. Intell. Fuzzy Syst., 26 (2014), 2459-2466.  doi: 10.3233/IFS-130916.

[27]

J. Ye, Similarity measures between interval neutrosophic sets and their applications in multicriteria decision-making, J. Intell. Fuzzy Syst., 26 (2014), 165-172.  doi: 10.3233/IFS-130916.

[28]

J. Ye, Multicriteria decision-making method using the correlation coefficient under single-value neutrosophic environment, J. Intell. Fuzzy Syst., 42 (2013), 386-394.  doi: 10.1080/03081079.2012.761609.

[29]

L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.

[30]

E. K. Zavadskas, R. Bausys and M. Lazauskas, Sustainable assessment of alternative sites for the construction of a waste incineration plant by applying WASPAS method with single-valued neutrosophic set, Sustainability, 7 (2015), 15923-15936.  doi: 10.3390/su71215792.

[31]

H. Y. Zhang, J. Q. Wang and X. H. Chen, Interval neutrosophic sets and their application in multicriteria decision making problems, Sci. World J., 2014 (2014), Article ID 645953. doi: 10.1155/2014/645953.

[32]

J. H. Zhao, X. Wang, H. M. Zhang, J. Hu and X. M. Jian, Side scan sonar image segmentation based on neutrosophic set and quantum-behaved particle swarm optimization algorithm, Mar. Geophys. Res., 37 (2016), 229-241.  doi: 10.1007/s11001-016-9276-1.

show all references

References:
[1]

K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87-96.  doi: 10.1016/S0165-0114(86)80034-3.

[2]

P. Biswas, S. Pramanik and B. C. Giri, TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment, Neural Comput. Appl., 27 (2016), 727-737.  doi: 10.1007/s00521-015-1891-2.

[3]

British Standards Institute, Reliability of Systems, Equipment and Components, Guide to Failure Modes, Effects and Criticality Analysis (FMEA and FMECA), Vol. BS 5760-5, British Standards Institute, United Kingdom, 1991.

[4]

K. H. Chang, Evaluate the orderings of risk for failure problems using a more general RPN methodology, Microelectron. Reliab., 49 (2009), 1586-1596.  doi: 10.1016/j.microrel.2009.07.057.

[5]

K. H. Chang, A novel reliability calculation method under neutrosophic environments, Ann. Oper. Res., (2021) in press. doi: 10.1007/s10479-020-03890-4.

[6]

K. H. Chang, A more general risk assessment methodology using soft sets based ranking technique, Soft Comput., 18 (2014), 169-183.  doi: 10.1007/s00500-013-1045-3.

[7]

K. H. Chang, Y. C. Chang and P. T. Lai, Applying the concept of exponential approach to enhance the assessment capability of FMEA, J. Intell. Manuf., 25 (2014), 1413-1427.  doi: 10.1007/s10845-013-0747-9.

[8]

Y. C. Chang, K. H. Chang and C. Y. Chen, Risk assessment by quantifying and prioritizing 5S activity for semiconductor manufacturing, Proc. Inst. Mech. Eng. Part B-J. Eng. Manuf., 227 (2013), 1874-1887.  doi: 10.1177/0954405413493901.

[9]

N. Chanamool and T. Naenna, Fuzzy FMEA application to improve decision-making process in an emergency department, Appl. Soft. Comput., 43 (2016), 441-453.  doi: 10.1016/j.asoc.2016.01.007.

[10]

K. H. Chang, Generalized multi-attribute failure mode analysis, Neurocomputing, 175 (2016), 90-100.  doi: 10.1016/j.neucom.2015.10.039.

[11]

D.C. US Department of Defense Washington, Procedures for Performing a Failure Mode Effects and Criticality Analysis, US MIL-STD-1629A, 1980.

[12]

M. Giardina, F. Castiglia and E. Tomarchio, Risk assessment of component failure modes and human errors using a new FMECA approach: Application in the safety analysis of HDR brachytherapy, J. Radiol. Prot., 34 (2014), 891-914.  doi: 10.1088/0952-4746/34/4/891.

[13]

Y. H. Guo and A. Sengur, A novel 3D skeleton algorithm based on neutrosophic cost function, Appl. Soft. Comput., 36 (2015), 210-217.  doi: 10.1016/j.asoc.2015.07.025.

[14]

International Electrotechnical Commission, Analysis Techniques for System Reliability- Procedures for Failure Mode and Effect Analysis, Geneva, IEC 60812, 1985.

[15]

H. A. Khorshidi, I. Gunawan and M. Y. Ibrahim, Applying UGF concept to enhance the assessment capability of FMEA, Qual. Reliab. Eng. Int., 32 (2016), 1085-1093.  doi: 10.1002/qre.1817.

[16]

P. Kraipeerapun and C. C. Fung, Binary classification using ensemble neural networks and interval neutrosophic sets, Neurocomputing, 72 (2009), 2845-2856.  doi: 10.1016/j.neucom.2008.07.017.

[17]

S. Li and W. Zeng, Risk analysis for the supplier selection problem using failure modes and effects analysis (FMEA), J. Intell. Manuf., 27 (2016), 1309-1321.  doi: 10.1007/s10845-014-0953-0.

[18]

H. C. Liu, J. X. You, X. J. Fan and Q. L. Lin, Failure mode and effects analysis using D numbers and grey relational projection method, Expert Syst. Appl., 41 (2014), 4670-4679.  doi: 10.1016/j.eswa.2014.01.031.

[19]

O. Mohsen and N. Fereshteh, An extended VIKOR method based on entropy measure for the failure modes risk assessment - A case study of the geothermal power plant (GPP), Saf. Sci., 92 (2017), 160-172.  doi: 10.1016/j.ssci.2016.10.006.

[20]

H. Safari, Z. Faraji and S. Majidian, Identifying and evaluating enterprise architecture risks using FMEA and fuzzy VIKOR differentiables, J. Intell. Manuf., 27 (2016), 475-486.  doi: 10.1007/s10845-014-0880-0.

[21]

R. Sahin and P. D. Liu, Maximizing deviation method for neutrosophic multiple attribute decision making with incomplete weight information, Neural Comput. Appl., 27 (2016), 2017-2029.  doi: 10.1007/s10700-006-0022-z.

[22]

R. Sahin and A. Kucuk, Subsethood measure for single valued neutrosophic sets, J. Intell. Fuzzy Syst., 29 (2015), 525-530.  doi: 10.3233/ifs-141304.

[23]

F. Smarandache, A unifying field in logics, neutrosophy: Neutrosophic probability, set and logic, preprint, arXiv: 0101228.

[24]

Z. P. Tian, H. Y. Zhang, J. Wang, J. Q. Wang and X. H. Chen, Multi-criteria decision-making method based on a cross-entropy with interval neutrosophic sets, Int. J. Syst. Sci., 47 (2016), 3598-3608.  doi: 10.1080/00207721.2015.1102359.

[25]

H. Wang, F. Smarandache, Y. Q. Zhang and R. Sunderraman, Single valued neutrosophic sets, Multispace Multistructure, 4 (2014), 410-413.  doi: 10.1007/s00357-017-9225-y.

[26]

J. Ye, A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets, J. Intell. Fuzzy Syst., 26 (2014), 2459-2466.  doi: 10.3233/IFS-130916.

[27]

J. Ye, Similarity measures between interval neutrosophic sets and their applications in multicriteria decision-making, J. Intell. Fuzzy Syst., 26 (2014), 165-172.  doi: 10.3233/IFS-130916.

[28]

J. Ye, Multicriteria decision-making method using the correlation coefficient under single-value neutrosophic environment, J. Intell. Fuzzy Syst., 42 (2013), 386-394.  doi: 10.1080/03081079.2012.761609.

[29]

L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.

[30]

E. K. Zavadskas, R. Bausys and M. Lazauskas, Sustainable assessment of alternative sites for the construction of a waste incineration plant by applying WASPAS method with single-valued neutrosophic set, Sustainability, 7 (2015), 15923-15936.  doi: 10.3390/su71215792.

[31]

H. Y. Zhang, J. Q. Wang and X. H. Chen, Interval neutrosophic sets and their application in multicriteria decision making problems, Sci. World J., 2014 (2014), Article ID 645953. doi: 10.1155/2014/645953.

[32]

J. H. Zhao, X. Wang, H. M. Zhang, J. Hu and X. M. Jian, Side scan sonar image segmentation based on neutrosophic set and quantum-behaved particle swarm optimization algorithm, Mar. Geophys. Res., 37 (2016), 229-241.  doi: 10.1007/s11001-016-9276-1.

Table 1.  Traditional RPN method scale for severity [18,19]
Rating Effect Severity of effect
10 Hazardous without warning Highest severity ranking of a failure mode, occurring without warning and consequence is hazardous
9 Hazardous with warning Higher severity ranking of a failure mode occurring with warning, consequence is hazardous
8 Extreme Operation of system or product is broken down without compromising safe
7 Major Operation of system or product may be continued but performance of system or product is affected
6 Significant Operation of system or product is continued and performance of system or product is degraded
5 Moderate Performance of system or product is affected seriously and the maintenance is needed
4 Low Performance of system or product is small affected and the maintenance may not be needed
3 Minor System performance and satisfaction with minor effect
2 Very minor System performance and satisfaction with slight effect
1 None No effect
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Rating Effect Severity of effect
10 Hazardous without warning Highest severity ranking of a failure mode, occurring without warning and consequence is hazardous
9 Hazardous with warning Higher severity ranking of a failure mode occurring with warning, consequence is hazardous
8 Extreme Operation of system or product is broken down without compromising safe
7 Major Operation of system or product may be continued but performance of system or product is affected
6 Significant Operation of system or product is continued and performance of system or product is degraded
5 Moderate Performance of system or product is affected seriously and the maintenance is needed
4 Low Performance of system or product is small affected and the maintenance may not be needed
3 Minor System performance and satisfaction with minor effect
2 Very minor System performance and satisfaction with slight effect
1 None No effect
Table 2.  Traditional RPN method scale for occurrence [18,19]
Rating Probability of failure Possible failure rates
10 Extremely high: failure almost inevitable $ \geq $in 2
9 Very high 1 in 3
8 Repeated failures 1 in 8
7 High 1 in 20
6 Moderately high 1 in 80
5 Moderate 1 in 400
4 Relatively low 1 in 2000
3 Low 1 in 15,000
2 Remote 1 in 150,000
1 Nearly impossible $ \leq $1 in 1,500,000
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Rating Probability of failure Possible failure rates
10 Extremely high: failure almost inevitable $ \geq $in 2
9 Very high 1 in 3
8 Repeated failures 1 in 8
7 High 1 in 20
6 Moderately high 1 in 80
5 Moderate 1 in 400
4 Relatively low 1 in 2000
3 Low 1 in 15,000
2 Remote 1 in 150,000
1 Nearly impossible $ \leq $1 in 1,500,000
Table 3.  Traditional RPN method scale for detection [18,19]
Rating Detection Likelihood of detection by design control
10 Absolute uncertainty Potential occurring of failure mode cannot be detected in concept, design and process failure mode and effects analysis (FMEA)/mechanism and subsequent failure mode
9 Very remote The possibility of detecting the potential occurring of failure mode is very remote/mechanism and subsequent failure mode
8 Remote The possibility of detecting the potential occurring of failure mode is remote/mechanism and subsequent failure mode
7 Very low The possibility of detecting the potential occurring of failure mode is very low/mechanism and subsequent failure mode
6 Low The possibility of detecting the potential occurring of failure mode is low/mechanism and subsequent failure mode
5 Moderate The possibility of detecting the potential occurring of failure mode is moderate/mechanism and subsequent failure mode
4 Moderately high The possibility of detecting the potential occurring of failure mode is moderately high/mechanism and subsequent failure mode
3 High The possibility of detecting the potential occurring of failure mode is high/mechanism and subsequent failure mode
2 Very high The possibility of detecting the potential occurring of failure mode is very high/mechanism and subsequent failure mode
1 Almost certainThe potential occurring of failure mode will be detect/ mechanism and subsequent failure mode
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Rating Detection Likelihood of detection by design control
10 Absolute uncertainty Potential occurring of failure mode cannot be detected in concept, design and process failure mode and effects analysis (FMEA)/mechanism and subsequent failure mode
9 Very remote The possibility of detecting the potential occurring of failure mode is very remote/mechanism and subsequent failure mode
8 Remote The possibility of detecting the potential occurring of failure mode is remote/mechanism and subsequent failure mode
7 Very low The possibility of detecting the potential occurring of failure mode is very low/mechanism and subsequent failure mode
6 Low The possibility of detecting the potential occurring of failure mode is low/mechanism and subsequent failure mode
5 Moderate The possibility of detecting the potential occurring of failure mode is moderate/mechanism and subsequent failure mode
4 Moderately high The possibility of detecting the potential occurring of failure mode is moderately high/mechanism and subsequent failure mode
3 High The possibility of detecting the potential occurring of failure mode is high/mechanism and subsequent failure mode
2 Very high The possibility of detecting the potential occurring of failure mode is very high/mechanism and subsequent failure mode
1 Almost certainThe potential occurring of failure mode will be detect/ mechanism and subsequent failure mode
Table 4.  The FMEA of the HDR brachytherapy treatments [12]
Identification number (ID) Component Failure mode Failure effect
1 Stepping motor Electrical blackout High-dose-rate (HDR) unit is stopped and dc motor withdraws the source to the safe
2 Direct current safety motor Loss of power Operator goes into the treatment room (TR) to manually return the source to the safe
3 Dwell position distance control device Stepper motor failure Source position not correct
4 Secondary timer Electronic fault Incorrect check of the primary timer
5 Backup battery Power-off Direct current motor fault
6 Backup battery Operator forgets to charge the battery Direct current motor fault
7 Software Power-off Safety and control system fault
8 Stop button on the console Contact fault During treatment, the stop button on the console did not retract the wire source
9 Physicist Dose calculation errors during treatment planning system (TPS) Incorrect HDR treatment
10 Therapist Data insertion errors during TPS Incorrect HDR treatment
11 Medical operator Incorrect patient identification Incorrect data are used during treatment control system (TCS)
12 Medical operator Incorrect medical application of the catheter or applicator Incorrect HDR treatment
13 Therapist Error in loading patient information (from the database) Incorrect data are used during TCS
14 Therapist Error in the data entry for dwell time or dwell position programmingIncorrect data are used during TCS
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Identification number (ID) Component Failure mode Failure effect
1 Stepping motor Electrical blackout High-dose-rate (HDR) unit is stopped and dc motor withdraws the source to the safe
2 Direct current safety motor Loss of power Operator goes into the treatment room (TR) to manually return the source to the safe
3 Dwell position distance control device Stepper motor failure Source position not correct
4 Secondary timer Electronic fault Incorrect check of the primary timer
5 Backup battery Power-off Direct current motor fault
6 Backup battery Operator forgets to charge the battery Direct current motor fault
7 Software Power-off Safety and control system fault
8 Stop button on the console Contact fault During treatment, the stop button on the console did not retract the wire source
9 Physicist Dose calculation errors during treatment planning system (TPS) Incorrect HDR treatment
10 Therapist Data insertion errors during TPS Incorrect HDR treatment
11 Medical operator Incorrect patient identification Incorrect data are used during treatment control system (TCS)
12 Medical operator Incorrect medical application of the catheter or applicator Incorrect HDR treatment
13 Therapist Error in loading patient information (from the database) Incorrect data are used during TCS
14 Therapist Error in the data entry for dwell time or dwell position programmingIncorrect data are used during TCS
Table 5.  Single valued neutrosophic number conversion for S, O and D factors (adapted from [30])
Level S O D Single valued neutrosophic
10 Hazardous Extremely high Absolute uncertainty (1.00, 0.00, 0.00)
9 Serious Very high Very remote (0.90, 0.10, 0.10)
8 Extreme Repeated failures Remote (0.80, 0.15, 0.20)
7 Major High Very low (0.70, 0.25, 0.30)
6 Significant Moderately high Low (0.60, 0.35, 0.40)
5 Moderate Moderate Moderate (0.50, 0.50, 0.50)
4 Low Relatively low Moderately high (0.40, 0.65, 0.60)
3 Minor Low High (0.30, 0.75, 0.70)
2 Very minor Remote Very high (0.20, 0.85, 0.80)
1 None Nearly impossible Almost certain (0.10, 0.90, 0.90)
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Level S O D Single valued neutrosophic
10 Hazardous Extremely high Absolute uncertainty (1.00, 0.00, 0.00)
9 Serious Very high Very remote (0.90, 0.10, 0.10)
8 Extreme Repeated failures Remote (0.80, 0.15, 0.20)
7 Major High Very low (0.70, 0.25, 0.30)
6 Significant Moderately high Low (0.60, 0.35, 0.40)
5 Moderate Moderate Moderate (0.50, 0.50, 0.50)
4 Low Relatively low Moderately high (0.40, 0.65, 0.60)
3 Minor Low High (0.30, 0.75, 0.70)
2 Very minor Remote Very high (0.20, 0.85, 0.80)
1 None Nearly impossible Almost certain (0.10, 0.90, 0.90)
Table 6.  The $ S $, $ O $ and $ D $ factors of the possible range of linguistic rating
ID S O D
TM1 TM2 TM3 TM4 TM1 TM2 TM3 TM4 TM1 TM2 TM3 TM4
1 2 2 1 2 1 1 2 1 10 10 9 10
2 1 1 1 1 10 10 9 9 2 2 1 2
3 1 2 1 1 8 7 8 8 3 2 3 4
4 3 2 3 3 7 7 8 6 2 2 2 3
5 4 4 3 3 9 8 9 9 2 1 2 2
6 3 2 2 3 9 9 9 9 2 2 2 2
7 1 1 2 1 9 8 8 9 9 8 9 9
8 1 1 1 2 10 9 10 10 9 9 9 10
9 4 3 3 5 9 8 8 9 3 2 3 3
10 5 6 4 5 9 9 9 10 2 3 2 2
11 5 5 * 6 9 8 * 9 3 4 * 3
12 2 3 * 2 1 1 * 2 10 9 * 10
13 5 5 4 6 9 10 10 8 2 2 2 3
14 4 4 5 4 9 9 10 9 2 2 1 2
* Missing or incomplete information
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ID S O D
TM1 TM2 TM3 TM4 TM1 TM2 TM3 TM4 TM1 TM2 TM3 TM4
1 2 2 1 2 1 1 2 1 10 10 9 10
2 1 1 1 1 10 10 9 9 2 2 1 2
3 1 2 1 1 8 7 8 8 3 2 3 4
4 3 2 3 3 7 7 8 6 2 2 2 3
5 4 4 3 3 9 8 9 9 2 1 2 2
6 3 2 2 3 9 9 9 9 2 2 2 2
7 1 1 2 1 9 8 8 9 9 8 9 9
8 1 1 1 2 10 9 10 10 9 9 9 10
9 4 3 3 5 9 8 8 9 3 2 3 3
10 5 6 4 5 9 9 9 10 2 3 2 2
11 5 5 * 6 9 8 * 9 3 4 * 3
12 2 3 * 2 1 1 * 2 10 9 * 10
13 5 5 4 6 9 10 10 8 2 2 2 3
14 4 4 5 4 9 9 10 9 2 2 1 2
* Missing or incomplete information
Table 7.  The $ S $, $ O $ and $ D $ factors by single valued neutrosophic numbers
ID S O D
1 (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (1.00, 0.00, 0.00)
2 (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (0.20, 0.85, 0.80)
3 (0.10, 0.90, 0.90) (0.80, 0.15, 0.20) (0.30, 0.75, 0.70)
4 (0.30, 0.75, 0.70) (0.70, 0.25, 0.30) (0.20, 0.85, 0.80)
5 (0.40, 0.65, 0.60) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
6 (0.30, 0.75, 0.70) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
7 (0.10, 0.90, 0.90) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10)
8 (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10)
9 (0.40, 0.65, 0.60) (0.90, 0.10, 0.10) (0.30, 0.75, 0.70)
10 (0.50, 0.50, 0.50) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
11 (0.50, 0.50, 0.50) (0.90, 0.10, 0.10) (0.30, 0.75, 0.70)
12 (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (1.00, 0.00, 0.00)
13 (0.50, 0.50, 0.50) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
14 (0.40, 0.65, 0.60) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
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ID S O D
1 (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (1.00, 0.00, 0.00)
2 (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (0.20, 0.85, 0.80)
3 (0.10, 0.90, 0.90) (0.80, 0.15, 0.20) (0.30, 0.75, 0.70)
4 (0.30, 0.75, 0.70) (0.70, 0.25, 0.30) (0.20, 0.85, 0.80)
5 (0.40, 0.65, 0.60) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
6 (0.30, 0.75, 0.70) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
7 (0.10, 0.90, 0.90) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10)
8 (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10)
9 (0.40, 0.65, 0.60) (0.90, 0.10, 0.10) (0.30, 0.75, 0.70)
10 (0.50, 0.50, 0.50) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
11 (0.50, 0.50, 0.50) (0.90, 0.10, 0.10) (0.30, 0.75, 0.70)
12 (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (1.00, 0.00, 0.00)
13 (0.50, 0.50, 0.50) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
14 (0.40, 0.65, 0.60) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
Table 8.  The value of $ d\left(A^{*}, A^{*} \cap A_{i}\right) $ by subsethood measure method
ID S O D
1 0.267 0.283 0.267 0.300 0.300 0.300 0.000 0.000 0.000
2 0.300 0.300 0.300 0.000 0.000 0.000 0.267 0.283 0.267
3 0.300 0.300 0.300 0.067 0.050 0.067 0.233 0.250 0.233
4 0.233 0.250 0.233 0.100 0.083 0.100 0.267 0.283 0.267
5 0.200 0.217 0.200 0.033 0.033 0.033 0.267 0.283 0.267
6 0.233 0.250 0.233 0.033 0.033 0.033 0.267 0.283 0.267
7 0.300 0.300 0.300 0.033 0.033 0.033 0.033 0.033 0.033
8 0.300 0.300 0.300 0.000 0.000 0.000 0.033 0.033 0.033
9 0.200 0.217 0.200 0.033 0.033 0.033 0.233 0.250 0.233
10 0.167 0.167 0.167 0.033 0.033 0.033 0.267 0.283 0.267
11 0.167 0.167 0.167 0.033 0.033 0.033 0.233 0.250 0.233
12 0.267 0.283 0.267 0.300 0.300 0.300 0.000 0.000 0.000
13 0.167 0.167 0.167 0.033 0.033 0.033 0.267 0.283 0.267
14 0.200 0.217 0.200 0.033 0.033 0.033 0.267 0.283 0.267
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ID S O D
1 0.267 0.283 0.267 0.300 0.300 0.300 0.000 0.000 0.000
2 0.300 0.300 0.300 0.000 0.000 0.000 0.267 0.283 0.267
3 0.300 0.300 0.300 0.067 0.050 0.067 0.233 0.250 0.233
4 0.233 0.250 0.233 0.100 0.083 0.100 0.267 0.283 0.267
5 0.200 0.217 0.200 0.033 0.033 0.033 0.267 0.283 0.267
6 0.233 0.250 0.233 0.033 0.033 0.033 0.267 0.283 0.267
7 0.300 0.300 0.300 0.033 0.033 0.033 0.033 0.033 0.033
8 0.300 0.300 0.300 0.000 0.000 0.000 0.033 0.033 0.033
9 0.200 0.217 0.200 0.033 0.033 0.033 0.233 0.250 0.233
10 0.167 0.167 0.167 0.033 0.033 0.033 0.267 0.283 0.267
11 0.167 0.167 0.167 0.033 0.033 0.033 0.233 0.250 0.233
12 0.267 0.283 0.267 0.300 0.300 0.300 0.000 0.000 0.000
13 0.167 0.167 0.167 0.033 0.033 0.033 0.267 0.283 0.267
14 0.200 0.217 0.200 0.033 0.033 0.033 0.267 0.283 0.267
Table 9.  The $ S $, $ O $ and $ D $ factors of the possible range by single valued neutrosophic number
ID S O D
TM1 TM2 TM3 TM4 TM1 TM2 TM3 TM4 TM1 TM2 TM3 TM4
1 (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10) (1.00, 0.00, 0.00)
2 (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80)
3 (0.10, 0.90, 0.90) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.80, 0.15, 0.20) (0.70, 0.25, 0.30) (0.80, 0.15, 0.20) (0.80, 0.15, 0.20) (0.30, 0.75, 0.70) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70) (0.40, 0.65, 0.60)
4 (0.30, 0.75, 0.70) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70) (0.30, 0.75, 0.70) (0.70, 0.25, 0.30) (0.70, 0.25, 0.30) (0.80, 0.15, 0.20) (0.60, 0.35, 0.40) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70)
5 (0.40, 0.65, 0.60) (0.40, 0.65, 0.60) (0.30, 0.75, 0.70) (0.30, 0.75, 0.70) (0.90, 0.10, 0.10) (0.80, 0.15, 0.20) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80)
6 (0.30, 0.75, 0.70) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80)
7 (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.90, 0.10, 0.10) (0.80, 0.15, 0.20) (0.80, 0.15, 0.20) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.80, 0.15, 0.20) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10)
8 (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (1.00, 0.00, 0.00)
9 (0.40, 0.65, 0.60) (0.30, 0.75, 0.70) (0.30, 0.75, 0.70) (0.50, 0.50, 0.50) (0.90, 0.10, 0.10) (0.80, 0.15, 0.20) (0.80, 0.15, 0.20) (0.90, 0.10, 0.10) (0.30, 0.75, 0.70) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70) (0.30, 0.75, 0.70)
10 (0.50, 0.50, 0.50) (0.60, 0.35, 0.40) (0.40, 0.65, 0.60) (0.50, 0.50, 0.50) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (1.00, 0.00, 0.00) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80)
11 (0.50, 0.50, 0.50) (0.50, 0.50, 0.50) (*, *, *) (0.60, 0.35, 0.40) (0.90, 0.10, 0.10) (0.80, 0.15, 0.20) (*, *, *) (0.90, 0.10, 0.10) (0.30, 0.75, 0.70) (0.40, 0.65, 0.60) (*, *, *) (0.30, 0.75, 0.70)
12 (0.20, 0.85, 0.80) (0.30, 0.75, 0.70) (*, *, *) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (*, *, *) (0.20, 0.85, 0.80) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10) (*, *, *) (1.00, 0.00, 0.00)
13 (0.50, 0.50, 0.50) (0.50, 0.50, 0.50) (0.40, 0.65, 0.60) (0.60, 0.35, 0.40) (0.90, 0.10, 0.10) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00) (0.80, 0.15, 0.20) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70)
14 (0.40, 0.65, 0.60) (0.40, 0.65, 0.60) (0.50, 0.50, 0.50) (0.40, 0.65, 0.60) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80)
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ID S O D
TM1 TM2 TM3 TM4 TM1 TM2 TM3 TM4 TM1 TM2 TM3 TM4
1 (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10) (1.00, 0.00, 0.00)
2 (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80)
3 (0.10, 0.90, 0.90) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.80, 0.15, 0.20) (0.70, 0.25, 0.30) (0.80, 0.15, 0.20) (0.80, 0.15, 0.20) (0.30, 0.75, 0.70) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70) (0.40, 0.65, 0.60)
4 (0.30, 0.75, 0.70) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70) (0.30, 0.75, 0.70) (0.70, 0.25, 0.30) (0.70, 0.25, 0.30) (0.80, 0.15, 0.20) (0.60, 0.35, 0.40) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70)
5 (0.40, 0.65, 0.60) (0.40, 0.65, 0.60) (0.30, 0.75, 0.70) (0.30, 0.75, 0.70) (0.90, 0.10, 0.10) (0.80, 0.15, 0.20) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80)
6 (0.30, 0.75, 0.70) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80)
7 (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.90, 0.10, 0.10) (0.80, 0.15, 0.20) (0.80, 0.15, 0.20) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.80, 0.15, 0.20) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10)
8 (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (1.00, 0.00, 0.00)
9 (0.40, 0.65, 0.60) (0.30, 0.75, 0.70) (0.30, 0.75, 0.70) (0.50, 0.50, 0.50) (0.90, 0.10, 0.10) (0.80, 0.15, 0.20) (0.80, 0.15, 0.20) (0.90, 0.10, 0.10) (0.30, 0.75, 0.70) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70) (0.30, 0.75, 0.70)
10 (0.50, 0.50, 0.50) (0.60, 0.35, 0.40) (0.40, 0.65, 0.60) (0.50, 0.50, 0.50) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (1.00, 0.00, 0.00) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80)
11 (0.50, 0.50, 0.50) (0.50, 0.50, 0.50) (*, *, *) (0.60, 0.35, 0.40) (0.90, 0.10, 0.10) (0.80, 0.15, 0.20) (*, *, *) (0.90, 0.10, 0.10) (0.30, 0.75, 0.70) (0.40, 0.65, 0.60) (*, *, *) (0.30, 0.75, 0.70)
12 (0.20, 0.85, 0.80) (0.30, 0.75, 0.70) (*, *, *) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.10, 0.90, 0.90) (*, *, *) (0.20, 0.85, 0.80) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10) (*, *, *) (1.00, 0.00, 0.00)
13 (0.50, 0.50, 0.50) (0.50, 0.50, 0.50) (0.40, 0.65, 0.60) (0.60, 0.35, 0.40) (0.90, 0.10, 0.10) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00) (0.80, 0.15, 0.20) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.30, 0.75, 0.70)
14 (0.40, 0.65, 0.60) (0.40, 0.65, 0.60) (0.50, 0.50, 0.50) (0.40, 0.65, 0.60) (0.90, 0.10, 0.10) (0.90, 0.10, 0.10) (1.00, 0.00, 0.00) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80) (0.20, 0.85, 0.80) (0.10, 0.90, 0.90) (0.20, 0.85, 0.80)
Table 10.  Data imputation by minimum, averaging and maximum operators
ID Minimum operator
S O D
11 0.50 0.50 0.50 0.80 0.15 0.20 0.30 0.75 0.70
12 0.20 0.85 0.80 0.10 0.90 0.90 0.90 0.10 0.10
Averaging operator
11 0.536 0.444 0.464 0.874 0.114 0.126 0.335 0.715 0.665
12 0.235 0.815 0.765 0.135 0.883 0.865 1.000 0.000 0.000
Maximum Operator
11 0.60 0.35 0.40 0.90 0.10 0.10 0.40 0.65 0.60
12 0.30 0.75 0.70 0.20 0.85 0.80 1.00 0.00 0.00
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ID Minimum operator
S O D
11 0.50 0.50 0.50 0.80 0.15 0.20 0.30 0.75 0.70
12 0.20 0.85 0.80 0.10 0.90 0.90 0.90 0.10 0.10
Averaging operator
11 0.536 0.444 0.464 0.874 0.114 0.126 0.335 0.715 0.665
12 0.235 0.815 0.765 0.135 0.883 0.865 1.000 0.000 0.000
Maximum Operator
11 0.60 0.35 0.40 0.90 0.10 0.10 0.40 0.65 0.60
12 0.30 0.75 0.70 0.20 0.85 0.80 1.00 0.00 0.00
Table 11.  Aggregated $ S $, $ O $ and $ D $ factors by minimum operator
ID S O D
1 (0.18, 0.86, 0.82) (0.13, 0.89, 0.87) (1.00, 0.00, 0.00)
2 (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (0.18, 0.86, 0.82)
3 (0.13, 0.89, 0.87) (0.78, 0.17, 0.22) (0.30, 0.75, 0.70)
4 (0.28, 0.77, 0.72) (0.71, 0.24, 0.29) (0.23, 0.82, 0.77)
5 (0.35, 0.70, 0.65) (0.88, 0.11, 0.12) (0.18, 0.86, 0.82)
6 (0.25, 0.80, 0.75) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
7 (0.13, 0.89, 0.87) (0.86, 0.12, 0.14) (0.88, 0.11, 0.12)
8 (0.13, 0.89, 0.87) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00)
9 (0.38, 0.65, 0.62) (0.86, 0.12, 0.14) (0.28, 0.77, 0.72)
10 (0.51, 0.49, 0.49) (1.00, 0.00, 0.00) (0.23, 0.82, 0.77)
11 (0.53, 0.46, 0.47) (0.86, 0.12, 0.14) (0.33, 0.72, 0.67)
12 (0.23, 0.82, 0.77) (0.13, 0.89, 0.87) (1.00, 0.00, 0.00)
13 (0.51, 0.49, 0.49) (1.00, 0.00, 0.00) (0.23, 0.82, 0.77)
14 (0.43, 0.61, 0.57) (1.00, 0.00, 0.00) (0.18, 0.86, 0.82)
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ID S O D
1 (0.18, 0.86, 0.82) (0.13, 0.89, 0.87) (1.00, 0.00, 0.00)
2 (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (0.18, 0.86, 0.82)
3 (0.13, 0.89, 0.87) (0.78, 0.17, 0.22) (0.30, 0.75, 0.70)
4 (0.28, 0.77, 0.72) (0.71, 0.24, 0.29) (0.23, 0.82, 0.77)
5 (0.35, 0.70, 0.65) (0.88, 0.11, 0.12) (0.18, 0.86, 0.82)
6 (0.25, 0.80, 0.75) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
7 (0.13, 0.89, 0.87) (0.86, 0.12, 0.14) (0.88, 0.11, 0.12)
8 (0.13, 0.89, 0.87) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00)
9 (0.38, 0.65, 0.62) (0.86, 0.12, 0.14) (0.28, 0.77, 0.72)
10 (0.51, 0.49, 0.49) (1.00, 0.00, 0.00) (0.23, 0.82, 0.77)
11 (0.53, 0.46, 0.47) (0.86, 0.12, 0.14) (0.33, 0.72, 0.67)
12 (0.23, 0.82, 0.77) (0.13, 0.89, 0.87) (1.00, 0.00, 0.00)
13 (0.51, 0.49, 0.49) (1.00, 0.00, 0.00) (0.23, 0.82, 0.77)
14 (0.43, 0.61, 0.57) (1.00, 0.00, 0.00) (0.18, 0.86, 0.82)
Table 12.  Aggregated $ S $, $ O $ and $ D $ factors by averaging operator
ID S O D
1 (0.18, 0.86, 0.82) (0.13, 0.89, 0.87) (1.00, 0.00, 0.00)
2 (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (0.18, 0.86, 0.82)
3 (0.13, 0.89, 0.87) (0.78, 0.17, 0.22) (0.30, 0.75, 0.70)
4 (0.28, 0.77, 0.72) (0.71, 0.24, 0.29) (0.23, 0.82, 0.77)
5 (0.35, 0.70, 0.65) (0.88, 0.11, 0.12) (0.18, 0.86, 0.82)
6 (0.25, 0.80, 0.75) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
7 (0.13, 0.89, 0.87) (0.86, 0.12, 0.14) (0.88, 0.11, 0.12)
8 (0.13, 0.89, 0.87) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00)
9 (0.38, 0.65, 0.62) (0.86, 0.12, 0.14) (0.28, 0.77, 0.72)
10 (0.51, 0.49, 0.49) (1.00, 0.00, 0.00) (0.23, 0.82, 0.77)
11 (0.54, 0.44, 0.46) (0.87, 0.11, 0.13) (0.34, 0.72, 0.66)
12 (0.23, 0.82, 0.77) (0.13, 0.88, 0.87) (1.00, 0.00, 0.00)
13 (0.51, 0.49, 0.49) (1.00, 0.00, 0.00) (0.23, 0.82, 0.77)
14 (0.43, 0.61, 0.57) (1.00, 0.00, 0.00) (0.18, 0.86, 0.82)
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ID S O D
1 (0.18, 0.86, 0.82) (0.13, 0.89, 0.87) (1.00, 0.00, 0.00)
2 (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (0.18, 0.86, 0.82)
3 (0.13, 0.89, 0.87) (0.78, 0.17, 0.22) (0.30, 0.75, 0.70)
4 (0.28, 0.77, 0.72) (0.71, 0.24, 0.29) (0.23, 0.82, 0.77)
5 (0.35, 0.70, 0.65) (0.88, 0.11, 0.12) (0.18, 0.86, 0.82)
6 (0.25, 0.80, 0.75) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
7 (0.13, 0.89, 0.87) (0.86, 0.12, 0.14) (0.88, 0.11, 0.12)
8 (0.13, 0.89, 0.87) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00)
9 (0.38, 0.65, 0.62) (0.86, 0.12, 0.14) (0.28, 0.77, 0.72)
10 (0.51, 0.49, 0.49) (1.00, 0.00, 0.00) (0.23, 0.82, 0.77)
11 (0.54, 0.44, 0.46) (0.87, 0.11, 0.13) (0.34, 0.72, 0.66)
12 (0.23, 0.82, 0.77) (0.13, 0.88, 0.87) (1.00, 0.00, 0.00)
13 (0.51, 0.49, 0.49) (1.00, 0.00, 0.00) (0.23, 0.82, 0.77)
14 (0.43, 0.61, 0.57) (1.00, 0.00, 0.00) (0.18, 0.86, 0.82)
Table 13.  Aggregated $ S $, $ O $ and $ D $ factors by maximum operator
ID S O D
1 (0.18, 0.86, 0.82) (0.13, 0.89, 0.87) (1.00, 0.00, 0.00)
2 (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (0.18, 0.86, 0.82)
3 (0.13, 0.89, 0.87) (0.78, 0.17, 0.22) (0.30, 0.75, 0.70)
4 (0.28, 0.77, 0.72) (0.71, 0.24, 0.29) (0.23, 0.82, 0.77)
5 (0.35, 0.70, 0.65) (0.88, 0.11, 0.12) (0.18, 0.86, 0.82)
6 (0.25, 0.80, 0.75) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
7 (0.13, 0.89, 0.87) (0.86, 0.12, 0.14) (0.88, 0.11, 0.12)
8 (0.13, 0.89, 0.87) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00)
9 (0.38, 0.65, 0.62) (0.86, 0.12, 0.14) (0.28, 0.77, 0.72)
10 (0.51, 0.49, 0.49) (1.00, 0.00, 0.00) (0.23, 0.82, 0.77)
11 (0.55, 0.42, 0.45) (0.88, 0.11, 0.12) (0.35, 0.70, 0.65)
12 (0.25, 0.80, 0.75) (0.15, 0.87, 0.85) (1.00, 0.00, 0.00)
13 (0.51, 0.49, 0.49) (1.00, 0.00, 0.00) (0.23, 0.82, 0.77)
14 (0.43, 0.61, 0.57) (1.00, 0.00, 0.00) (0.18, 0.86, 0.82)
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ID S O D
1 (0.18, 0.86, 0.82) (0.13, 0.89, 0.87) (1.00, 0.00, 0.00)
2 (0.10, 0.90, 0.90) (1.00, 0.00, 0.00) (0.18, 0.86, 0.82)
3 (0.13, 0.89, 0.87) (0.78, 0.17, 0.22) (0.30, 0.75, 0.70)
4 (0.28, 0.77, 0.72) (0.71, 0.24, 0.29) (0.23, 0.82, 0.77)
5 (0.35, 0.70, 0.65) (0.88, 0.11, 0.12) (0.18, 0.86, 0.82)
6 (0.25, 0.80, 0.75) (0.90, 0.10, 0.10) (0.20, 0.85, 0.80)
7 (0.13, 0.89, 0.87) (0.86, 0.12, 0.14) (0.88, 0.11, 0.12)
8 (0.13, 0.89, 0.87) (1.00, 0.00, 0.00) (1.00, 0.00, 0.00)
9 (0.38, 0.65, 0.62) (0.86, 0.12, 0.14) (0.28, 0.77, 0.72)
10 (0.51, 0.49, 0.49) (1.00, 0.00, 0.00) (0.23, 0.82, 0.77)
11 (0.55, 0.42, 0.45) (0.88, 0.11, 0.12) (0.35, 0.70, 0.65)
12 (0.25, 0.80, 0.75) (0.15, 0.87, 0.85) (1.00, 0.00, 0.00)
13 (0.51, 0.49, 0.49) (1.00, 0.00, 0.00) (0.23, 0.82, 0.77)
14 (0.43, 0.61, 0.57) (1.00, 0.00, 0.00) (0.18, 0.86, 0.82)
Table 14.  The value of $ d\left(A^{*}, A^{*} \cap A_{i}\right) $ by minimum operator
ID S O D
1 0.275 0.287 0.275 0.291 0.296 0.291 0.000 0.000 0.000
2 0.300 0.300 0.300 0.000 0.000 0.000 0.275 0.287 0.275
3 0.291 0.296 0.291 0.074 0.057 0.074 0.232 0.249 0.232
4 0.241 0.258 0.241 0.097 0.080 0.097 0.258 0.275 0.258
5 0.216 0.233 0.216 0.040 0.037 0.040 0.275 0.287 0.275
6 0.249 0.266 0.249 0.033 0.033 0.033 0.267 0.283 0.267
7 0.291 0.296 0.291 0.047 0.041 0.047 0.040 0.037 0.040
8 0.291 0.296 0.291 0.000 0.000 0.000 0.000 0.000 0.000
9 0.206 0.218 0.206 0.047 0.041 0.047 0.241 0.258 0.241
10 0.165 0.163 0.165 0.000 0.000 0.000 0.258 0.275 0.258
11 0.158 0.152 0.158 0.047 0.041 0.047 0.225 0.241 0.225
12 0.258 0.275 0.258 0.291 0.296 0.291 0.000 0.000 0.000
13 0.165 0.163 0.165 0.000 0.000 0.000 0.258 0.275 0.258
14 0.191 0.203 0.191 0.000 0.000 0.000 0.275 0.287 0.275
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ID S O D
1 0.275 0.287 0.275 0.291 0.296 0.291 0.000 0.000 0.000
2 0.300 0.300 0.300 0.000 0.000 0.000 0.275 0.287 0.275
3 0.291 0.296 0.291 0.074 0.057 0.074 0.232 0.249 0.232
4 0.241 0.258 0.241 0.097 0.080 0.097 0.258 0.275 0.258
5 0.216 0.233 0.216 0.040 0.037 0.040 0.275 0.287 0.275
6 0.249 0.266 0.249 0.033 0.033 0.033 0.267 0.283 0.267
7 0.291 0.296 0.291 0.047 0.041 0.047 0.040 0.037 0.040
8 0.291 0.296 0.291 0.000 0.000 0.000 0.000 0.000 0.000
9 0.206 0.218 0.206 0.047 0.041 0.047 0.241 0.258 0.241
10 0.165 0.163 0.165 0.000 0.000 0.000 0.258 0.275 0.258
11 0.158 0.152 0.158 0.047 0.041 0.047 0.225 0.241 0.225
12 0.258 0.275 0.258 0.291 0.296 0.291 0.000 0.000 0.000
13 0.165 0.163 0.165 0.000 0.000 0.000 0.258 0.275 0.258
14 0.191 0.203 0.191 0.000 0.000 0.000 0.275 0.287 0.275
Table 15.  The value of $ d\left(A^{*}, A^{*} \cap A_{i}\right) $ by averaging operator
ID S O D
1 0.275 0.287 0.275 0.291 0.296 0.291 0.000 0.000 0.000
2 0.300 0.300 0.300 0.000 0.000 0.000 0.275 0.287 0.275
3 0.291 0.296 0.291 0.074 0.057 0.074 0.232 0.249 0.232
4 0.241 0.258 0.241 0.097 0.080 0.097 0.258 0.275 0.258
5 0.216 0.233 0.216 0.040 0.037 0.040 0.275 0.287 0.275
6 0.249 0.266 0.249 0.033 0.033 0.033 0.267 0.283 0.267
7 0.291 0.296 0.291 0.047 0.041 0.047 0.040 0.037 0.040
8 0.291 0.296 0.291 0.000 0.000 0.000 0.000 0.000 0.000
9 0.206 0.218 0.206 0.047 0.041 0.047 0.241 0.258 0.241
10 0.165 0.163 0.165 0.000 0.000 0.000 0.258 0.275 0.258
11 0.155 0.148 0.155 0.042 0.038 0.042 0.222 0.238 0.222
12 0.255 0.272 0.255 0.288 0.294 0.288 0.000 0.000 0.000
13 0.165 0.163 0.165 0.000 0.000 0.000 0.258 0.275 0.258
14 0.191 0.203 0.191 0.000 0.000 0.000 0.275 0.287 0.275
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ID S O D
1 0.275 0.287 0.275 0.291 0.296 0.291 0.000 0.000 0.000
2 0.300 0.300 0.300 0.000 0.000 0.000 0.275 0.287 0.275
3 0.291 0.296 0.291 0.074 0.057 0.074 0.232 0.249 0.232
4 0.241 0.258 0.241 0.097 0.080 0.097 0.258 0.275 0.258
5 0.216 0.233 0.216 0.040 0.037 0.040 0.275 0.287 0.275
6 0.249 0.266 0.249 0.033 0.033 0.033 0.267 0.283 0.267
7 0.291 0.296 0.291 0.047 0.041 0.047 0.040 0.037 0.040
8 0.291 0.296 0.291 0.000 0.000 0.000 0.000 0.000 0.000
9 0.206 0.218 0.206 0.047 0.041 0.047 0.241 0.258 0.241
10 0.165 0.163 0.165 0.000 0.000 0.000 0.258 0.275 0.258
11 0.155 0.148 0.155 0.042 0.038 0.042 0.222 0.238 0.222
12 0.255 0.272 0.255 0.288 0.294 0.288 0.000 0.000 0.000
13 0.165 0.163 0.165 0.000 0.000 0.000 0.258 0.275 0.258
14 0.191 0.203 0.191 0.000 0.000 0.000 0.275 0.287 0.275
Table 16.  The value of $ d\left(A^{*}, A^{*} \cap A_{i}\right) $ by maximum operator
ID S O D
1 0.275 0.287 0.275 0.291 0.296 0.291 0.000 0.000 0.000
2 0.300 0.300 0.300 0.000 0.000 0.000 0.275 0.287 0.275
3 0.291 0.296 0.291 0.074 0.057 0.074 0.232 0.249 0.232
4 0.241 0.258 0.241 0.097 0.080 0.097 0.258 0.275 0.258
5 0.216 0.233 0.216 0.040 0.037 0.040 0.275 0.287 0.275
6 0.249 0.266 0.249 0.033 0.033 0.033 0.267 0.283 0.267
7 0.291 0.296 0.291 0.047 0.041 0.047 0.040 0.037 0.040
8 0.291 0.296 0.291 0.000 0.000 0.000 0.000 0.000 0.000
9 0.206 0.218 0.206 0.047 0.041 0.047 0.241 0.258 0.241
10 0.165 0.163 0.165 0.000 0.000 0.000 0.258 0.275 0.258
11 0.149 0.139 0.149 0.040 0.037 0.040 0.216 0.233 0.216
12 0.249 0.266 0.249 0.283 0.292 0.283 0.000 0.000 0.000
13 0.165 0.163 0.165 0.000 0.000 0.000 0.258 0.275 0.258
14 0.191 0.203 0.191 0.000 0.000 0.000 0.275 0.287 0.275
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ID S O D
1 0.275 0.287 0.275 0.291 0.296 0.291 0.000 0.000 0.000
2 0.300 0.300 0.300 0.000 0.000 0.000 0.275 0.287 0.275
3 0.291 0.296 0.291 0.074 0.057 0.074 0.232 0.249 0.232
4 0.241 0.258 0.241 0.097 0.080 0.097 0.258 0.275 0.258
5 0.216 0.233 0.216 0.040 0.037 0.040 0.275 0.287 0.275
6 0.249 0.266 0.249 0.033 0.033 0.033 0.267 0.283 0.267
7 0.291 0.296 0.291 0.047 0.041 0.047 0.040 0.037 0.040
8 0.291 0.296 0.291 0.000 0.000 0.000 0.000 0.000 0.000
9 0.206 0.218 0.206 0.047 0.041 0.047 0.241 0.258 0.241
10 0.165 0.163 0.165 0.000 0.000 0.000 0.258 0.275 0.258
11 0.149 0.139 0.149 0.040 0.037 0.040 0.216 0.233 0.216
12 0.249 0.266 0.249 0.283 0.292 0.283 0.000 0.000 0.000
13 0.165 0.163 0.165 0.000 0.000 0.000 0.258 0.275 0.258
14 0.191 0.203 0.191 0.000 0.000 0.000 0.275 0.287 0.275
Table 17.  Comparison of different ranking methods
ID S O D RPN [4,12] Ranking RPN [4,12] $S\left(A^{*}, A_{i}\right)$ [22] Ranking subsethood measure [22] $S_{1}(A, B)$ [27] Ranking $S_1(A, B)$ [27] $S_2(A, B)$ [27] Ranking $S_2(A, B)$ [27] $d_H(A, B)$ [27] Ranking $d_H(A, B)$ [27] $d_E(A, B)$ [27] Ranking $d_E(A, B)$ [27]
1 2 1 10 20 12 0.806 10 0.865 10 0.449 13 0.572 10 0.702 12
2 1 10 2 20 12 0.806 10 0.865 10 0.475 12 0.572 10 0.702 12
3 1 8 3 24 11 0.794 13 0.856 13 0.514 9 0.600 13 0.673 11
4 3 7 2 42 10 0.789 14 0.854 14 0.485 11 0.606 14 0.649 10
5 4 9 2 72 7 0.822 7 0.885 7 0.520 7 0.511 7 0.594 7
6 3 9 2 54 9 0.811 9 0.874 9 0.508 10 0.544 9 0.630 9
7 1 9 9 81 6 0.878 2 0.933 2 0.755 2 0.367 2 0.526 3
8 1 10 9 90 3 0.889 1 0.944 1 0.762 1 0.333 1 0.523 2
9 4 9 3 108 2 0.833 6 0.896 6 0.576 4 0.478 6 0.549 4
10 5 9 2 90 3 0.839 4 0.898 4 0.525 5 0.472 4 0.556 5
11 5 9 3 135 1 0.850 3 0.909 3 0.582 3 0.439 3 0.508 1
12 2 1 10 20 12 0.806 10 0.865 10 0.449 13 0.572 10 0.702 12
13 5 9 2 90 3 0.839 4 0.898 4 0.525 5 0.472 4 0.556 5
14 4 9 2 72 7 0.822 7 0.885 7 0.520 7 0.511 7 0.594 7
ID S O D $C(A, B)$ [28] Ranking $C(A, B)$ [28] Proposed method
Minimum operator Ranking minimum operator Averaging operator Ranking averaging operators Maximum Operator Ranking maximum operator
1 2 1 10 0.86 12 0.806 10 0.806 10 0.806 10
2 1 10 2 0.86 12 0.804 12 0.804 12 0.804 12
3 1 8 3 1.05 11 0.794 13 0.794 13 0.794 13
4 3 7 2 1.18 10 0.790 14 0.790 14 0.790 14
5 4 9 2 1.43 7 0.813 8 0.813 8 0.813 9
6 3 9 2 1.21 9 0.806 10 0.806 10 0.806 10
7 1 9 9 1.79 2 0.872 2 0.872 2 0.872 2
8 1 10 9 1.72 3 0.901 1 0.901 1 0.901 1
9 4 9 3 1.69 6 0.826 7 0.826 7 0.826 7
10 5 9 2 1.71 4 0.853 3 0.853 4 0.853 4
11 5 9 3 2.02 1 0.851 5 0.855 3 0.860 3
12 2 1 10 0.86 12 0.810 9 0.811 9 0.814 8
13 5 9 2 1.71 4 0.853 3 0.853 4 0.853 4
14 4 9 2 1.43 7 0.837 6 0.837 6 0.837 6
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ID S O D RPN [4,12] Ranking RPN [4,12] $S\left(A^{*}, A_{i}\right)$ [22] Ranking subsethood measure [22] $S_{1}(A, B)$ [27] Ranking $S_1(A, B)$ [27] $S_2(A, B)$ [27] Ranking $S_2(A, B)$ [27] $d_H(A, B)$ [27] Ranking $d_H(A, B)$ [27] $d_E(A, B)$ [27] Ranking $d_E(A, B)$ [27]
1 2 1 10 20 12 0.806 10 0.865 10 0.449 13 0.572 10 0.702 12
2 1 10 2 20 12 0.806 10 0.865 10 0.475 12 0.572 10 0.702 12
3 1 8 3 24 11 0.794 13 0.856 13 0.514 9 0.600 13 0.673 11
4 3 7 2 42 10 0.789 14 0.854 14 0.485 11 0.606 14 0.649 10
5 4 9 2 72 7 0.822 7 0.885 7 0.520 7 0.511 7 0.594 7
6 3 9 2 54 9 0.811 9 0.874 9 0.508 10 0.544 9 0.630 9
7 1 9 9 81 6 0.878 2 0.933 2 0.755 2 0.367 2 0.526 3
8 1 10 9 90 3 0.889 1 0.944 1 0.762 1 0.333 1 0.523 2
9 4 9 3 108 2 0.833 6 0.896 6 0.576 4 0.478 6 0.549 4
10 5 9 2 90 3 0.839 4 0.898 4 0.525 5 0.472 4 0.556 5
11 5 9 3 135 1 0.850 3 0.909 3 0.582 3 0.439 3 0.508 1
12 2 1 10 20 12 0.806 10 0.865 10 0.449 13 0.572 10 0.702 12
13 5 9 2 90 3 0.839 4 0.898 4 0.525 5 0.472 4 0.556 5
14 4 9 2 72 7 0.822 7 0.885 7 0.520 7 0.511 7 0.594 7
ID S O D $C(A, B)$ [28] Ranking $C(A, B)$ [28] Proposed method
Minimum operator Ranking minimum operator Averaging operator Ranking averaging operators Maximum Operator Ranking maximum operator
1 2 1 10 0.86 12 0.806 10 0.806 10 0.806 10
2 1 10 2 0.86 12 0.804 12 0.804 12 0.804 12
3 1 8 3 1.05 11 0.794 13 0.794 13 0.794 13
4 3 7 2 1.18 10 0.790 14 0.790 14 0.790 14
5 4 9 2 1.43 7 0.813 8 0.813 8 0.813 9
6 3 9 2 1.21 9 0.806 10 0.806 10 0.806 10
7 1 9 9 1.79 2 0.872 2 0.872 2 0.872 2
8 1 10 9 1.72 3 0.901 1 0.901 1 0.901 1
9 4 9 3 1.69 6 0.826 7 0.826 7 0.826 7
10 5 9 2 1.71 4 0.853 3 0.853 4 0.853 4
11 5 9 3 2.02 1 0.851 5 0.855 3 0.860 3
12 2 1 10 0.86 12 0.810 9 0.811 9 0.814 8
13 5 9 2 1.71 4 0.853 3 0.853 4 0.853 4
14 4 9 2 1.43 7 0.837 6 0.837 6 0.837 6
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