Abstract
Multi-criteria decision-making (MCDM) methods deeply assist decision-makers in evaluating and analyzing input data as accurately as possible in the area of safety and reliability. The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) can be used to rank a set of alternatives based on the different types of criteria. However, this method still needs to be improved; fuzzy set theory and its corresponding extensions are bold in the line of TOPSIS improvement. In this study, the original form of TOPSIS method is extended using Pythagorean fuzzy-based Chebyshev distance measures. Pythagorean fuzzy set has better reflection compared to the conventional fuzzy set theory and Chebyshev distance measures can provide more accurate results, which are used to compute the separation measures in the TOPSIS method. Therefore, the novel improved TOPSIS methods have been developed underlying the ideal of Pythagorean fuzzy-based Chebyshev distance measures. The developed method is applied on a humanitarian case-study to show the efficiency of the approach in an emergency response situation. Results show that the proposed method is efficiently applicable in real-life decision-making situations.
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References
Abdel-Basset M, Mohamed R (2020) A novel plithogenic TOPSIS-CRITIC model for sustainable supply chain risk management. J Clean Prod 247:119586. https://doi.org/10.1016/j.jclepro.2019.119586
Adesina KA, Nedjati A, Yazdi M (2020) A Short communication improving marine safety management system by addressing common safety program. Res Mar Sci 5:671–680
Al-Qerem A, Alauthman M, Almomani A, Gupta BB (2020) IoT transaction processing through cooperative concurrency control on fog–cloud computing environment. Soft Comput 24:5695–5711. https://doi.org/10.1007/s00500-019-04220-y
Alsmirat MA, Al-Alem F, Al-Ayyoub M, Jararweh Y, Gupta B (2019) Impact of digital fingerprint image quality on the fingerprint recognition accuracy. Multimed Tools Appl 78:3649–3688. https://doi.org/10.1007/s11042-017-5537-5
Andreopoulou Z, Koliouska C, Galariotis E, Zopounidis C (2018) Renewable energy sources: using PROMETHEE II for ranking websites to support market opportunities. Technol Forecast Soc Change 131:31–37. https://doi.org/10.1016/j.techfore.2017.06.007
Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96. https://doi.org/10.1016/S0165-0114(86)80034-3
Behzadian M, Khanmohammadi Otaghsara S, Yazdani M, Ignatius J (2012) A state-of the-art survey of TOPSIS applications. Expert Syst Appl 39:13051–13069. https://doi.org/10.1016/j.eswa.2012.05.056
Bian T, Zheng H, Yin L, Deng Y (2018) Failure mode and effects analysis based on D numbers and TOPSIS. Qual Reliab Eng Int 34:501–515. https://doi.org/10.1002/qre.2268
Bolturk E (2018) Pythagorean fuzzy CODAS and its application to supplier selection in a manufacturing firm. J Enterp Inf Manag 31:550–564. https://doi.org/10.1108/JEIM-01-2018-0020
Chang YH, Shao PC, Chen HJ (2015) Performance evaluation of airport safety management systems in Taiwan. Saf Sci 75:72–86. https://doi.org/10.1016/j.ssci.2014.12.006
Chatterjee K, Zavadskas EK, Tamošaitiene J, Adhikary K, Kar S (2018) A hybrid MCDM technique for risk management in construction projects. Symmetry (Basel). https://doi.org/10.3390/sym10020046
Chen TY (2018) Remoteness index-based Pythagorean fuzzy VIKOR methods with a generalized distance measure for multiple criteria decision analysis. Inf Fusion 41:129–150. https://doi.org/10.1016/j.inffus.2017.09.003
Chen T (2019) Multiple criteria group decision making using a parametric linear programming technique for multidimensional analysis of preference under uncertainty of pythagorean fuzziness. IEEE Access 7:174108–174128. https://doi.org/10.1109/ACCESS.2019.2957161
Chen TY (2020) New Chebyshev distance measures for Pythagorean fuzzy sets with applications to multiple criteria decision analysis using an extended ELECTRE approach. Expert Syst Appl 147:113164. https://doi.org/10.1016/j.eswa.2019.113164
Chen YC, Lien HP, Tzeng GH (2010) Measures and evaluation for environment watershed plans using a novel hybrid MCDM model. Expert Syst Appl 37:926–938. https://doi.org/10.1016/j.eswa.2009.04.068
Chen F, Wang J, Deng Y (2015) Road safety risk evaluation by means of improved entropy TOPSIS-RSR. Saf Sci 79:39–54. https://doi.org/10.1016/j.ssci.2015.05.006
Chen TL, Hsu HM, Pan SY, Chiang PC (2019) Advances and challenges of implementing carbon offset mechanism for a low carbon economy: the Taiwanese experience. J Clean Prod 239:117860. https://doi.org/10.1016/j.jclepro.2019.117860
Chu Y, Sun L, Li L (2019) Lightweight scheme selection for automotive safety structures using a quantifiable multi-objective approach. J Clean Prod 241:118316. https://doi.org/10.1016/j.jclepro.2019.118316
Daneshvar S, Yazdi M, Adesina KA (2020) Fuzzy smart failure modes and effects analysis to improve safety performance of system: case study of an aircraft landing system. Qual Reliab Eng Int. https://doi.org/10.1002/qre.2607
Ding XF, Liu HC (2019) A new approach for emergency decision-making based on zero-sum game with Pythagorean fuzzy uncertain linguistic variables. Int J Intell Syst 34:1667–1684. https://doi.org/10.1002/int.22113
dos Santos BM, Godoy LP, Campos LMS (2019) Performance evaluation of green suppliers using entropy-TOPSIS-F. J Clean Prod 207:498–509. https://doi.org/10.1016/j.jclepro.2018.09.235
El-Latif AAA, Abd-El-Atty B, Hossain MS, Rahman MA, Alamri A, Gupta BB (2018) Efficient quantum information hiding for remote medical image sharing. IEEE Access 6:21075–21083. https://doi.org/10.1109/ACCESS.2018.2820603
Golilarz NA, Gao H, Pirasteh S, Yazdi M, Zhou J, Fu Y (2021) Satellite multispectral and hyperspectral image de-noising with enhanced adaptive generalized gaussian distribution threshold in the wavelet domain. Remote Sens. https://doi.org/10.3390/rs13010101
Hsu TK, Tsai YF, Wu HH (2009) The preference analysis for tourist choice of destination: a case study of Taiwan. Tour Manag 30:288–297. https://doi.org/10.1016/j.tourman.2008.07.011
Hwang C-L, Yoon K (1981) Multiple attribute decision making. Springer. https://doi.org/10.1007/978-3-642-48318-9
Inamdar PM, Sharma AK, Cook S, Perera BJC (2018) Evaluation of stormwater harvesting sites using multi criteria decision methodology. J Hydrol 562:181–192. https://doi.org/10.1016/j.jhydrol.2018.04.066
Jato-Espino D, Castillo-Lopez E, Rodriguez-Hernandez J, Canteras-Jordana JC (2014) A review of application of multi-criteria decision making methods in construction. Autom Constr 45:151–162. https://doi.org/10.1016/j.autcon.2014.05.013
Jozaghi A, Alizadeh B, Hatami M, Flood I, Khorrami M, Khodaei N, Tousi EG (2018) A comparative study of the AHP and TOPSIS techniques for dam site selection using GIS: a case study of Sistan and Baluchestan Province. Iran Geosci 8:1–23. https://doi.org/10.3390/geosciences8120494
Kabir S, Yazdi M, Aizpurua JI, Papadopoulos Y (2018) Uncertainty-aware dynamic reliability analysis framework for complex systems. IEEE Access. https://doi.org/10.1109/ACCESS.2018.2843166
Kabir S, Geok TANKIM, Kumar M, Yazdi M, Hossain F (2020) A method for temporal fault tree analysis using intuitionistic fuzzy set and expert elicitation. IEEE Access 8:980–996. https://doi.org/10.1109/ACCESS.2019.2961953
Kaya R, Yet B (2019) Building Bayesian networks based on DEMATEL for multiple criteria decision problems: a supplier selection case study. Expert Syst Appl 134:234–248. https://doi.org/10.1016/j.eswa.2019.05.053
Khalili-Damghani K, Abtahi AR, Tavana M (2014) A decision support system for solving multi-objective redundancy allocation problems. Qual Reliab Eng Int 30:1249–1262. https://doi.org/10.1002/qre.1545
Khan MW, Ali Y, De Felice F, Petrillo A (2019) Occupational health and safety in construction industry in Pakistan using modified-SIRA method. Saf Sci 118:109–118. https://doi.org/10.1016/j.ssci.2019.05.001
Li M, Cao P (2019) Computers and industrial engineering extended TODIM method for multi-attribute risk decision making problems in emergency response. Comput Ind Eng 135:1286–1293. https://doi.org/10.1016/j.cie.2018.06.027
Li D, Zeng W (2018) Distance measure of pythagorean fuzzy sets. Int J Intell Syst 33:348–361. https://doi.org/10.1002/int.21934
Li YL, Ying CS, Chin KS, Yang HT, Xu J (2018) Third-party reverse logistics provider selection approach based on hybrid-information MCDM and cumulative prospect theory. J Clean Prod 195:573–584. https://doi.org/10.1016/j.jclepro.2018.05.213
Li J, Fang H, Song W (2019) Sustainable supplier selection based on SSCM practices: a rough cloud TOPSIS approach. J Clean Prod 222:606–621. https://doi.org/10.1016/j.jclepro.2019.03.070
Liu J, Wei Q (2018) Risk evaluation of electric vehicle charging infrastructure public-private partnership projects in China using fuzzy TOPSIS. J Clean Prod 189:211–222. https://doi.org/10.1016/j.jclepro.2018.04.103
Liu HC, Ren ML, Wu J, Lin QL (2014) An interval 2-tuple linguistic MCDM method for robot evaluation and selection. Int J Prod Res 52:2867–2880. https://doi.org/10.1080/00207543.2013.854939
Liu Q, Chen Y, Tian C, Zheng XQ, Li JF (2016) Strategic deliberation on development of low-carbon energy system in China. Adv Clim Chang Res 7:26–34. https://doi.org/10.1016/j.accre.2016.04.002
Liu J, Zhao HK, Li ZB, Liu SF (2017) Decision process in MCDM with large number of criteria and heterogeneous risk preferences. Oper Res Perspect 4:106–112. https://doi.org/10.1016/j.orp.2017.07.001
Liu A, Ji X, Lu H, Liu H (2019a) The selection of 3PRLs on self-service mobile recycling machine: Interval-valued pythagorean hesitant fuzzy best-worst multi-criteria group decision-making. J Clean Prod 230:734–750. https://doi.org/10.1016/j.jclepro.2019.04.257
Liu X, Zhou X, Zhu B, He K, Wang P (2019b) Measuring the maturity of carbon market in China: an entropy-based TOPSIS approach. J Clean Prod 229:94–103. https://doi.org/10.1016/j.jclepro.2019.04.380
Mamta D, Gupta B (2021) An attribute-based keyword search for m-Health networks. J Comput Virol Hacking Tech 17:21–36. https://doi.org/10.1007/s11416-020-00361-z
Marzouk M, Daour IA (2018) Planning labor evacuation for construction sites using BIM and agent-based simulation. Saf Sci 109:174–185. https://doi.org/10.1016/j.ssci.2018.04.023
May Tzuc O, Bassam A, Ricalde LJ, Jaramillo OA, Flota-Bañuelos M, Escalante Soberanis MA (2020) Environmental-economic optimization for implementation of parabolic collectors in the industrial process heat generation: case study of Mexico. J Clean Prod. https://doi.org/10.1016/j.jclepro.2019.118538
Mehrabadi ZK, Boyaghchi FA (2019) Thermodynamic, economic and environmental impact studies on various distillation units integrated with gasification-based multi-generation system: comparative study and optimization. J Clean Prod 241:118333. https://doi.org/10.1016/j.jclepro.2019.118333
Meriam JL, James L, Kraige LG, Bolton JN, Jeff N (2009) Engineering mechanics
Nawaz F, Asadabadi MR, Janjua NK, Hussain OK, Chang E, Saberi M (2018) An MCDM method for cloud service selection using a Markov chain and the best-worst method. Knowl-Based Syst 159:120–131. https://doi.org/10.1016/j.knosys.2018.06.010
Nedjati A, Vizvari B, Izbirak G (2016) Post-earthquake response by small UAV helicopters. Nat Hazards 80:1669–1688. https://doi.org/10.1007/s11069-015-2046-6
Nguyen S (2018) Development of an MCDM framework to facilitate low carbon shipping technology application. Asian J Shipp Logist 34:317–327. https://doi.org/10.1016/j.ajsl.2018.12.005
Nie W, Liu W, Wu Z, Chen B, Wu L (2019) Failure mode and effects analysis by integrating Bayesian fuzzy assessment number and extended gray relational analysis-technique for order preference by similarity to ideal solution method. Qual Reliab Eng Int 35:1676–1697. https://doi.org/10.1002/qre.2468
Opricovic S, Tzeng G-H (2004a) Compromise solution by MCDM methods: a comparative analysis of VIKOR and TOPSIS. Eur J Oper Res 156:445–455. https://doi.org/10.1016/S0377-2217(03)00020-1
Peng X, Selvachandran G (2019) Pythagorean fuzzy set: state of the art and future directions. Artif Intell Rev 52:1873–1927. https://doi.org/10.1007/s10462-017-9596-9
Peng X, Yang Y (2014) Some results for pythagorean fuzzy sets. Int J Intell Syst 29:495–524
Rausand M, Hoyland A (2004) System reliability theory: models. Stat Methods Appl. https://doi.org/10.1109/WESCON.1996.554026
Ren P, Xu Z, Gou X (2016) Pythagorean fuzzy TODIM approach to multi-criteria decision making. Appl Soft Comput J 42:246–259. https://doi.org/10.1016/j.asoc.2015.12.020
Rezaei J (2015) Best-worst multi-criteria decision-making method. Omega (United Kingdom) 53:49–57. https://doi.org/10.1016/j.omega.2014.11.009
Rostamzadeh R, Ghorabaee MK, Govindan K, Esmaeili A, Nobar HBK (2018) Evaluation of sustainable supply chain risk management using an integrated fuzzy TOPSIS-CRITIC approach. J Clean Prod 175:651–669. https://doi.org/10.1016/j.jclepro.2017.12.071
Selim H, Yunusoglu MG, Yilmaz Balaman Ş (2016) A dynamic maintenance planning framework based on fuzzy TOPSIS and FMEA: application in an international food company. Qual Reliab Eng Int 32:795–804. https://doi.org/10.1002/qre.1791
Şimşek B, Iç YT (2014) Multi-response simulation optimization approach for the performance optimization of an Alarm Monitoring Center. Saf Sci 66:61–74. https://doi.org/10.1016/j.ssci.2014.02.001
Song W, Ming X, Wu Z, Zhu B (2014) A rough TOPSIS approach for failure mode and effects analysis in uncertain environments. Qual Reliab Eng Int 30:473–486. https://doi.org/10.1002/qre.1500
Srdjevic B, Medeiros YDP, Faria AS (2004) An objective multi-criteria evaluation of water management scenarios. Water Resour Manag 18:35–54. https://doi.org/10.1023/B:WARM.0000015348.88832.52
Taylan O, Bafail AO, Abdulaal RMS, Kabli MR (2014) Construction projects selection and risk assessment by fuzzy AHP and fuzzy TOPSIS methodologies. Appl Soft Comput J 17:105–116. https://doi.org/10.1016/j.asoc.2014.01.003
Tsaura SH, Chang TY, Yen CH (2002) The evaluation of airline service quality by fuzzy MCDM. Tour Manag 23:107–115. https://doi.org/10.1016/S0261-5177(01)00050-4
Tzeng GH, Chiang CH, Li CW (2007) Evaluating intertwined effects in e-learning programs: a novel hybrid MCDM model based on factor analysis and DEMATEL. Expert Syst Appl 32:1028–1044. https://doi.org/10.1016/j.eswa.2006.02.004
Wang YM, Elhag TMS (2006) Fuzzy TOPSIS method based on alpha level sets with an application to bridge risk assessment. Expert Syst Appl 31:309–319. https://doi.org/10.1016/j.eswa.2005.09.040
Wang CN, Huang YF, Cheng IF, Nguyen VT (2018) A multi-criteria decision-making (MCDM) approach using hybrid SCOR metrics, AHP, and TOPSIS for supplier evaluation and selection in the gas and oil industry. Processes. https://doi.org/10.3390/pr6120252
Wang B, Xie HL, Ren HY, Li X, Chen L, Wu BC (2019a) Application of AHP, TOPSIS, and TFNs to plant selection for phytoremediation of petroleum-contaminated soils in shale gas and oil fields. J Clean Prod 233:13–22. https://doi.org/10.1016/j.jclepro.2019.05.301
Wang L, Hu YP, Liu HC, Shi H (2019b) A linguistic risk prioritization approach for failure mode and effects analysis: a case study of medical product development. Qual Reliab Eng Int 35:1735–1752. https://doi.org/10.1002/qre.2472
Wu WW (2008) Choosing knowledge management strategies by using a combined ANP and DEMATEL approach. Expert Syst Appl 35:828–835. https://doi.org/10.1016/j.eswa.2007.07.025
Wu B, Yan X, Wang Y, Guedes Soares C (2016) Selection of maritime safety control options for NUC ships using a hybrid group decision-making approach. Saf Sci 88:108–122. https://doi.org/10.1016/j.ssci.2016.04.026
Wu Y, Xu C, Zhang T (2018) Evaluation of renewable power sources using a fuzzy MCDM based on cumulative prospect theory: a case in China. Energy 147:1227–1239. https://doi.org/10.1016/j.energy.2018.01.115
Xu Z, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 35:417–433. https://doi.org/10.1080/03081070600574353
Xu X, Du Z, Chen X (2015) Consensus model for multi-criteria large-group emergency decision making considering non-cooperative behaviors and minority opinions. Decis Support Syst 79:150–160. https://doi.org/10.1016/j.dss.2015.08.009
Yager RR (2013) Pythagorean fuzzy subsets. In: Proc. 2013 Jt. IFSA World Congr. NAFIPS Annu. Meet. IFSA/NAFIPS, vol 2, pp 57–61. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375
Yager RR (2014) Pythagorean membership grades in multicriteria decision making. IEEE Trans Fuzzy Syst 22:958–965. https://doi.org/10.1109/TFUZZ.2013.2278989
Yang JJ, Chuang YC, Lo HW, Lee TI (2020) A two-stage MCDM model for exploring the influential relationships of sustainable sports tourism criteria in Taichung City. Int J Environ Res Public Health. https://doi.org/10.3390/ijerph17072319
Yaseen Q, Aldwairi M, Jararweh Y, Al-Ayyoub M, Gupta B (2018) Collusion attacks mitigation in internet of things: a fog based model. Multimed Tools Appl 77:18249–18268. https://doi.org/10.1007/s11042-017-5288-3
Yazdi M (2018) Improving failure mode and effect analysis (FMEA) with consideration of uncertainty handling as an interactive approach. Int J Interact Des Manuf. https://doi.org/10.1007/s12008-018-0496-2
Yazdi M (2018b) Footprint of knowledge acquisition improvement in failure diagnosis analysis. Qual Reliab Eng Int. https://doi.org/10.1002/qre.2408
Yazdi M (2019a) A review paper to examine the validity of Bayesian network to build rational consensus in subjective probabilistic failure analysis. Int J Syst Assur Eng Manag 10:1–18. https://doi.org/10.1007/s13198-018-00757-7
Yazdi M (2019b) Ignorance-aware safety and reliability analysis: a heuristic approach. Qual Reliab Eng Int 36:652–674. https://doi.org/10.1002/qre.2597
Yazdi M (2019c) A perceptual computing-based method to prioritize intervention actions in the probabilistic risk assessment techniques. Qual Reliab Eng Int. https://doi.org/10.1002/qre.2566
Yazdi M (2019d) Introducing a heuristic approach to enhance the reliability of system safety assessment. Qual Reliab Eng Int. https://doi.org/10.1002/qre.2545
Yazdi M, Kabir S (2017) A fuzzy Bayesian network approach for risk analysis in process industries. Process Saf Environ Prot 111:507–519. https://doi.org/10.1016/j.psep.2017.08.015
Yazdi M, Kabir S (2018) Fuzzy evidence theory and Bayesian networks for process systems risk analysis. Hum Ecol Risk Assess. https://doi.org/10.1080/10807039.2018.1493679
Yazdi M, Nedjati A, Abbassi R (2019) Fuzzy dynamic risk-based maintenance investment optimization for offshore process facilities. J Loss Prev Process Ind. https://doi.org/10.1016/j.jlp.2018.11.014
Yazdi M, Adesina KA, Korhan O, Nikfar F (2019a) Learning from fire accident at Bouali sina petrochemical complex plant. J Fail Anal Prev. https://doi.org/10.1007/s11668-019-00769-w
Yazdi M, Hafezi P, Abbassi R (2019b) A methodology for enhancing the reliability of expert system applications in probabilistic risk assessment. J Loss Prev Process Ind. https://doi.org/10.1016/j.jlp.2019.02.001
Yazdi M, Kabir S, Walker M (2019c) Uncertainty handling in fault tree based risk assessment: state of the art and future perspectives. Process Saf Environ Prot 131:89–104. https://doi.org/10.1016/j.psep.2019.09.003
Yazdi M, Korhan O, Daneshvar S (2020) Application of fuzzy fault tree analysis based on modified fuzzy AHP and fuzzy TOPSIS for fire and explosion in the process industry. Int J Occup Saf Ergon 26:319–335
Yazdi M, Nedjati A, Zarei E, Abbassi R (2020) A novel extension of DEMATEL approach for probabilistic safety analysis in process systems. Saf Sci. https://doi.org/10.1016/j.ssci.2019.09.006
Yazdi M, Khan F, Abbassi R, Rusli R (2020) Improved DEMATEL methodology for effective safety management decision-making. Saf Sci 127:104705. https://doi.org/10.1016/j.ssci.2020.104705
Yazdi M, Golilarz NA, Nedjati A, Adesina KA (2021) An improved lasso regression model for evaluating the efficiency of intervention actions in a system reliability analysis. Neural Comput Appl. https://doi.org/10.1007/s00521-020-05537-8
Yousefzadeh S, Yaghmaeian K, Mahvi AH, Nasseri S, Alavi N, Nabizadeh R (2020) Comparative analysis of hydrometallurgical methods for the recovery of Cu from circuit boards: Optimization using response surface and selection of the best technique by two-step fuzzy AHP-TOPSIS method. J Clean Prod 249:119401. https://doi.org/10.1016/j.jclepro.2019.119401
Yu X, Xu Z (2013) Prioritized intuitionistic fuzzy aggregation operators. Inf Fusion 14:108–116. https://doi.org/10.1016/j.inffus.2012.01.011
Zadeh L (1965) Fuzzy sets. Inf Control 8:338–353
Zadeh LA (2015) Fuzzy logic—a personal perspective. Fuzzy Sets Syst 281:4–20. https://doi.org/10.1016/j.fss.2015.05.009
Zarbakhshnia N, Soleimani H, Ghaderi H (2018) Sustainable third-party reverse logistics provider evaluation and selection using fuzzy SWARA and developed fuzzy COPRAS in the presence of risk criteria. Appl Soft Comput J 65:307–319. https://doi.org/10.1016/j.asoc.2018.01.023
Zare M, Pahl C, Rahnama H, Nilashi M, Mardani A, Ibrahim O, Ahmadi H (2016) Multi-criteria decision making approach in E-learning: a systematic review and classification. Appl Soft Comput J 45:108–128. https://doi.org/10.1016/j.asoc.2016.04.020
Zarei E, Khakzad N, Cozzani V, Reniers G (2019) Safety analysis of process systems using Fuzzy Bayesian Network (FBN). J Loss Prev Process Ind. https://doi.org/10.1016/j.jlp.2018.10.011
Zavadskas EK, Turskis Z, Tamošaitiene J (2010) Risk assessment of construction projects. J Civ Eng Manag 16:33–46. https://doi.org/10.3846/jcem.2010.03
Zeng W, Li D, Yin Q (2018) Distance and similarity measures of Pythagorean fuzzy sets and their applications to multiple criteria group decision making. Int J Intell Syst 33:2236–2254. https://doi.org/10.1002/int.22027
Zhang X (2016) Multicriteria Pythagorean fuzzy decision analysis: a hierarchical QUALIFLEX approach with the closeness index-based ranking methods. Inf Sci (NY) 330:104–124. https://doi.org/10.1016/j.ins.2015.10.012
Zhang X, Xu Z (2014) Extension of TOPSIS to multiple criteria decision making with pythagorean fuzzy sets. Int J Intell Syst 29:1061–1078. https://doi.org/10.1002/int.21676
Zhang L, Wang Y, Zhao X (2018) Knowledge-Based systems a new emergency decision support methodology based on multi-source knowledge in 2-tuple linguistic model. Knowl-Based Syst 144:77–87. https://doi.org/10.1016/j.knosys.2017.12.026
Zhao H, Guo S, Zhao H (2019) Comprehensive assessment for battery energy storage systems based on fuzzy-MCDM considering risk preferences. Energy 168:450–461. https://doi.org/10.1016/j.energy.2018.11.129
Zhou L, Wu X, Xu Z, Fujita H (2018) Emergency decision making for natural disasters: An overview. Int J Disaster Risk Reduct 27:567–576. https://doi.org/10.1016/j.ijdrr.2017.09.037
Acknowledgements
This work was partially supported by National Natural Science Foundation of China under the contract number 71761030 and 51965051, Natural Science Foundation of Inner Mongolia under the contract number 2019LH07003.
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Guang-Jun Jiang, Hong-Xia Chen and Hong-Hua Sun contributed to methodology, software, and writing—original draft; ; Mohammad Yazdi contributed to conceptualization, methodology, software, validation, and writing—original draft; Arman Nedjati was involved in writing—original draft and investigation; Kehinde A. Adesina contributed to writing—original draft and investigation.
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Appendices
Appendix 1
Proof of Theorem 5 (T.4.A)
According to the definition 3 and definition 7, we can simply means that \({\left({\fancyscript{u}}_{\fancyscript{p}_{t }}\right)}^{2}\), \({\left({\fancyscript{v}}_{\fancyscript{p}_{t }}\right)}^{2}\), \({\left({\dot{\pi }}_{\fancyscript{p}_{t }}\right)}^{2}\), \({\left({r}_{\fancyscript{p}_{t }}\right)}^{2}\), and \({d}_{\fancyscript{p}_{t}}\) fit into the interval range between one and zero [0,1] for all \(t\). The theorem shows that \({-1\le \left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\le 1\), \(\le {\left({\fancyscript{v}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{v}}_{\fancyscript{p}_{2 }}\right)}^{2 }\le 1\), \(-1{\le \left({\dot{\pi }}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\dot{\pi }}_{\fancyscript{p}_{2 }}\right)}^{2 }\le 1\), \({1\le \left({r}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({r}_{\fancyscript{p}_{2 }}\right)}^{2 }\le 1\), and \(-1\le {d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{2 }}\le 1\). Therefore, the property of \({0\le D}_{\mathrm{ Chebyshev }}^{ 5\text{-terms }}\left({\fancyscript{p}}_{1 },{\fancyscript{p}}_{2 }\right)\le 1\) is acetated and T.4.A is valid.
Agree that the T.4.B and T.4.C are valid.□
Proof of Theorem 4 (T.4.D)
In order to meet the property of Separability, \({D}_{\mathrm{ Chebyshev }}^{ 5\text{-terms }}\left({\fancyscript{p}}_{1 },{\fancyscript{p}}_{2 }\right)=0\) is assumed, and the following circumstances must be fulfilled: \({\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }={\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2}\), \({\left({\fancyscript{v}}_{\fancyscript{p}_{1 }}\right)}^{2 }={\left({\fancyscript{v}}_{\fancyscript{p}_{2 }}\right)}^{2}\), \({\le \left({\dot{\pi }}_{\fancyscript{p}_{1 }}\right)}^{2 }={\left({\dot{\pi }}_{\fancyscript{p}_{2 }}\right)}^{2}\), \({\left({r}_{\fancyscript{p}_{1 }}\right)}^{2 }={\left({r}_{\fancyscript{p}_{2 }}\right)}^{2}\), and \({d}_{\fancyscript{p}_{1 }}={d}_{\fancyscript{p}_{2}}\). T.4.D proves that, \({\fancyscript{u}}_{\fancyscript{p}_{1 }}={\fancyscript{u}}_{\fancyscript{p}_{2}}\), \({\fancyscript{v}}_{\fancyscript{p}_{1 }}={\fancyscript{v}}_{\fancyscript{p}_{2}}\), and \({r}_{\fancyscript{p}_{1 }}={r}_{\fancyscript{p}_{2}}\). One can say that once \({\fancyscript{p}}_{1 }={\fancyscript{p}}_{2}\), which is requirement of Separability, then \({D}_{\mathrm{ Chebyshev }}^{ 5\text{-terms }}\left({\fancyscript{p}}_{1 },{\fancyscript{p}}_{2 }\right)=0\). Therefore, T.4.D is viable and valid. □
Proof of Theorem 4 (T.4.E)
The triangular inequality can be represented in the form of different functions. Let use the membership function as an example, i.e. \(\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|\le \left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\). To prove the property of triangular inequality, four assumptions that must be taken into account: the relationships between three membership functions \({\fancyscript{u}}_{\fancyscript{p}_{1 }},{\fancyscript{u}}_{\fancyscript{p}_{2}}\), and \({\fancyscript{u}}_{\fancyscript{p}_{3}}\) as: (1) \({\fancyscript{u}}_{\fancyscript{p}_{1 }}\le \) \({\fancyscript{u}}_{\fancyscript{p}_{2}}{\le \fancyscript{u}}_{\fancyscript{p}_{3}}\), (2) \({\fancyscript{u}}_{\fancyscript{p}_{1 }}\ge {\fancyscript{u}}_{\fancyscript{p}_{2}}{\ge \fancyscript{u}}_{\fancyscript{p}_{3}}\), (3) \({\fancyscript{u}}_{\fancyscript{p}_{2 }}\le {\mathrm{min}}\{{\fancyscript{u}}_{\fancyscript{p}_{1 }},{\fancyscript{u}}_{\fancyscript{p}_{3 }}\}\), and (4) \({\fancyscript{u}}_{\fancyscript{p}_{2 }}\ge \mathrm{max}\left\{{\fancyscript{u}}_{\fancyscript{p}_{1 }},{\fancyscript{u}}_{\fancyscript{p}_{3 }}\right\}.\) An inference can be concluded based on assumption (1) and (2), \(\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|\le \left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\), that according to the assumption (3), the two conditions as \({\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\ge 0\) and \({\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2}\) underlies the ideal of this assumption. In addition, the following results can also be concluded:
Therefore, \(\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3}}\right)}^{2}\right|\le \left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\) underlies the ideal of assumption (3). In addition, assumption (4) denotes that \({\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }\ge 0\) and \({\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }\ge 0\) and accordingly the following conclusion can be obtained:
Up to this point, by evaluating all possible four assumptions, the triangular inequality, \(\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|\le \left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\), is fully satisfied. Analogously, the following relationships between the square difference of the non-membership function, the indeterminacy, and the commitment strength can be achieved: \(\left|{\left({\fancyscript{v}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{v}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|\le \left|{\left({\fancyscript{v}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{v}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\fancyscript{v}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{v}}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\), \(\left|{\left({\dot{\pi }}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\dot{\pi }}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|\le \left|{\left({\dot{\pi }}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\dot{\pi }}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\dot{\pi }}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\dot{\pi }}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\), and \(\left|{\left({r}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({r}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|\le \left|{\left({r}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({r}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({r}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({r}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\). In addition, assuming that relationships between the difference of commitment direction exists, it is proposed that \(\left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{3 }}\right|\le \left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{2} }\right|+\left|{d}_{\fancyscript{p}_{2 }}-{d}_{\fancyscript{p}_{3}}\right|\) satisfies \({d}_{\fancyscript{p}_{1 }}\le {d}_{\fancyscript{p}_{2 }}\le {d}_{\fancyscript{p}_{3}}\) and \({d}_{\fancyscript{p}_{1 }}\ge {d}_{\fancyscript{p}_{2 }}\ge {d}_{\fancyscript{p}_{3}}\). According to the aforementioned assumption, \({d}_{\fancyscript{p}_{2 }}\le {\mathrm{min}}\{{ d}_{\fancyscript{p}_{1 }}\,\mathrm{and}\,\,{ d}_{\fancyscript{p}_{3 }}\}\), the following can be concluded that:
Finally, the relationship \(\left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{3 }}\right|\le \left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{2 }}\right|+\left|{d}_{\fancyscript{p}_{2 }}-{d}_{\fancyscript{p}_{3}}\right|\) is fulfilled. Therefore, combination of the aforementioned outputs, with consideration of a maximization operator, the following result can be derived:
According to the aforementioned above results, the property of triangular inequality \({D}_{\mathrm{ Chebyshev }}^{ 5\text{-terms }}\left({\fancyscript{p}}_{1 },{\fancyscript{p}}_{3 }\right)\le {D}_{\mathrm{ Chebyshev }}^{ 5\text{-terms }}\left({\fancyscript{p}}_{1 },{\fancyscript{p}}_{2 }\right)+{D}_{\mathrm{ Chebyshev }}^{ 5\text{-terms }}\left({\fancyscript{p}}_{2 },{\fancyscript{p}}_{3}\right)\) is proved. Therefore, T.4.E is viable and valid. □
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Jiang, GJ., Chen, HX., Sun, HH. et al. An improved multi-criteria emergency decision-making method in environmental disasters. Soft Comput 25, 10351–10379 (2021). https://doi.org/10.1007/s00500-021-05826-x
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DOI: https://doi.org/10.1007/s00500-021-05826-x