Abstract
Bipolar complex fuzzy set (BCFS) is an enlargement of the bipolar fuzzy set and complex fuzzy set. A bipolar complex fuzzy number (BCFN) is used to represent both the bipolarity of ambiguity and bipolarity of periodicity semantics. This paper develops an integrated BCF–CRITIC–WASPAS methodology to elucidate the multi-criteria decision-making (MCDM) problems with bipolar complex fuzzy (BCF) data. Firstly, we propose the idea of cross-entropy of BCFSs followed by the introduction of some Archimedean operations between the BCFNs. Secondly, we discuss some BCF-Archimedean weighted aggregation operators with the proposed Archimedean aggregation operators that are generalizations of many renowned operators’ namely algebraic operator, Einstein operator, Hamachar operator and Frank operator to combine the BCF information. Thirdly, a procedure is given to estimate the decision expert’s weights using proposed cross-entropy measure of BCFSs. Since cross-entropy measure quantifies the degree of uncertain information and hence it reflects the influence of each expert. Fourthly, criteria weights are estimated through CRITIC (criteria importance through inter-criteria correlation) method which is a well-known objective method based on aggregated score values of options, intensity contrast of every criteria and conflict among attributes. Further, a green supplier selection (GSS) problem is taken to show the applicability of the developed methodology in real decision problems. We also exhibit a sensitivity investigation with diverse parameter values and criteria weights sets to examine the stability of proposed approach. Finally, we draw attention towards a comparison between the proposed approach and the extant methods to show its superiority.
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Abareshi A, Molla A (2013) Greening logistics and its impact on environmental performance: an absorptive capacity perspective. Int J of Logis Res Appl 16(3):209–226
Abdulkareem KH, Arbaiy N, Zaidan AA, Zaidan BB, Albahri OS, Alsalem MA, Salih MM (2020) A novel multi-perspective benchmarking framework for selecting image dehazing intelligent algorithms based on BWM and group VIKOR techniques. Int J Inform Tech Decis Mak 19(3):909–957
Adali EA, Tus A (2019) Hospital site selection with distance-based multi-criteria decision-making methods. Int J Healthc Manag. https://doi.org/10.1080/20479700.2019.1674005
Akram M, Arshad M (2019) A novel trapezoidal bipolar fuzzy TOPSIS method for group decision-making. Group Decis Negot 28:565–584
Akram M, Shumaiza Al-Kenani AN (2020) Multi-criteria group decision-making for selection of green suppliers under bipolar fuzzy PROMETHEE process. Symmetry 12:1–25
Alghamdi MA, Alshehri NO, Akram M (2018) Multi-criteria decision-making methods in bipolar fuzzy environment. Int J Fuzzy Syst 20:2057–2064
Al-Husban A, Amourah A, Jaber JJ (2020) Bipolar complex fuzzy sets and their properties. Ital J Pure Appl Maths 43:754–761
Alkouri AUMJ, Massa’dehb MO, Ali M (2020) On bipolar complex fuzzy sets and its application. J Intell Fuzzy Syst. https://doi.org/10.3233/JIFS-191350
Alrasheedi M, Mardani A, Mishra AR, Rani P, Loganathan N (2021) An extended framework to evaluate sustainable suppliers in manufacturing companies using a new Pythagorean fuzzy entropy-SWARA-WASPAS decision-making approach. J Enterp Inf Manag. https://doi.org/10.1108/JEIM-07-2020-0263
Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96
Awasthi A, Kannan G (2015) Green suppliers development program selection using NGT and VIKOR under fuzzy environment. Comput Ind Eng. https://doi.org/10.1016/j.cie.2015.11.011
Badalpur M, Nurbakhsh E (2019) An application of WASPAS method in risk qualitative analysis: a case study of a road construction project in Iran. Int J Constr Manag. https://doi.org/10.1080/15623599.2019.1595354
Baidya J, Garg H, Saha A, Mishra AR, Rani P, Dutta D (2021) Selection of third party reverses logistic providers: an approach of BCF–CRITIC-MULTIMOORA using Archimedean power aggregation operators. Complex Intell Syst. https://doi.org/10.1007/s40747-021-00413-x
Bakeshlou EA, Khamseh AA, Goudarzian Asl MA, Sadeghi J, Abbaszadeh M (2017) Evaluating a green supplier selection problem using a hybrid MODM algorithm J. Intell Manuf 28:913–927
Banaeian N, Bobil H, Neilsen IE, Omid M (2015) Criteria definition and approaches in green supplier selection- a case study for raw material and packaging of food industry. Prod Manuf Res 3(1):149–168
Banaeian N, Mobli H, Fahimnia B, Nielsen IE, Omid M (2018) Green supplier selection using fuzzy group decision making methods: a case study from the agri-food industry. Comput Oper Res 89:337–347
Bid S, Siddique G (2019) Human risk assessment of Panchet Dam in India using TOPSIS and WASPAS multi-criteria decision-making (MCDM) methods. Heliyon. https://doi.org/10.1016/j.heliyon.2019.e01956)
Cao G (2020) A multi-criteria picture fuzzy decision-making model for green supplier selection based on fractional programming. Int J Comput Commun Cont 15(1):1–14
Chen HMW, Chou SY, Luu QD, Yu THK (2016) A fuzzy MCDM approach for green supplier selection from the economic and the environmental aspects. Math Prob Eng. https://doi.org/10.1155/2016/8097386
Chen KS, Wang CH, Tan KH (2019) Developing a fuzzy green supplier selection model using six sigma quality indices. Int J Prod Econ 212:1–7
Chen L, Duan D, Mishra AR, Alrasheedi M (2021) Sustainable third-party reverse logistics provider selection to promote circular economy using new uncertain interval-valued intuitionistic fuzzy-projection model. J Enterp Inf Manag. https://doi.org/10.1108/JEIM-02-2021-0066
Chin TA, Tat HH, Sulaiman Z (2015) Green supply chain management, environmental collaboration and sustainability performance. Procedia CIRP 26:695–699
Cousens MDG (2000) Consensus eating. North Atlantic Books, Berkeley, California. ISBN:1-55643-285-2
Davoudabadi R, Mousavi SM, Mohagheghi V (2020) A new last aggregation method of multi-attributes group decision-making based on concepts of TODIM, WASPAS and TOPSIS under interval-valued intuitionistic fuzzy uncertainty. Knowl Inf Syst 62:1371–1391
Deveci M, Canıtez F, Gokasar I (2018) WASPAS and TOPSIS based interval type-2 fuzzy MCDM method for a selection of a car sharing station. Sustain Cities Soc 41:777–791
Diakoulaki D, Mavrotas G, Papayannakis L (1995) Determining objective weights in multiple criteria problems: the CRITIC method. Comput Oper Res 22:763–770
Dorfeshan Y, Mousavi SM (2020) A novel interval type-2 fuzzy decision model based on two new versions of relative preference relation-based MABAC and WASPAS methods (with an application in aircraft maintenance planning). Neural Comput Appl 32:3367–3385
Dweiri F, Kumar S, Khan SA, Jain V (2016) Designing an integrated AHP based decision support system for supplier selection in automotive industry. Expert Syst Appl 62:273–283
Fallahpour A, Olugu E, Musa S, Khezrimotlagh D, Wong K (2015) An integrated model for green supplier selection under fuzzy environment: application of data envelopment analysis and genetic programming approach. Neural Comput Appl. https://doi.org/10.1007/s00521-015-1890-3
Fan J, Liu X, Wu M, Wang Z (2019) Green supplier selection with undesirable outputs DEA under Pythagorean fuzzy environment. J Intell Fuzzy Syst 37(2):2443–2452
Fang C, Zhang J (2018) Performance of green supply chain management: a systematic review and meta analysis. J Clean Prod 183:1064–1081
Ghorabaee MK, Zavadskas EK, Amiri M, Esmaeili A (2016) Multi-criteria evaluation of green suppliers using an extended WASPAS method with interval type-2 fuzzy sets. J Clean Prod 137:213–229
Ghorabaee MK, Amiri M, Zavadskas EK, Antuchevicience J (2017) Assessment of third-party logistics providers using a CRITIC–WASPAS approach with interval type-2 fuzzy sets. Transport 32(1):66–78
Ghorabaee MK, Govindan K, Amiri M, Zavadskas EK, Antucheviciene J (2019) An integrated type-2 fuzzy decision model based on WASPAS and SECA for evaluation of sustainable manufacturing strategies. J Environ Eng Landsc Manag 27(4):187–200
Govindan K, Khodaverdi R, Jafarian A (2013) A fuzzy multi criteria approach for measuring sustainability performance of a supplier based on triple bottom line approach. J Clean Prod 47:345–354
Gundogdu FK, Kahraman C (2019) Extension of WASPAS with spherical fuzzy sets. Informatica 30(2):269–292
Gupta S, Soni U, Kumar G (2019) Green supplier selection using multi-criterion decision-making under fuzzy environment: a case study in automotive industry. Comput Ind Eng 136:663–680
Gurel O, Acar AZ, Onden I, Gumus I (2015) Determinants of the GREEN SUPPLIER SELECTION. Proc Soc Behavi Sci 181:131–139
Haeri SAS, Rezaei J (2019) A grey-based green supplier selection model for uncertain environments. J Clean Prod 221:768–784
Han Y, Shi P, Chen S (2015) Bipolar-valued rough fuzzy set and its applications to decision information system. IEEE Trans Fuzzy Syst 23(6):2358–2370
Hsu CW, Kuo TC, Chen SH, Hu AH (2013) Using DEMATEL to develop a carbon management model of supplier selection in green supply chain management. J Clean Prod 56:164–172
Javad MOM, Darvishi M, Javad AOM (2020) Green supplier selection for the steel industry using BWM and fuzzy TOPSIS: a case study of Khouzestan steel company. Sust Future. https://doi.org/10.1016/j.sftr.2020.100012
Jiang P, Hu YC, Yen GF, Tsao SJ (2018) Green supplier selection for sustainable development of the automotive industry using grey decision-making. Sustain Dev 26(6):890–903
Kahraman C, Onar SC, Oztaysi B, Ilbahar E (2019) Selection among GSM operators using Pythagorean fuzzy WASPAS method. J Multi-Valued Logic Soft Comput 33:459–469
Kannan D, Khodaverdi R, Olfat L, Jafarian A, Diabat A (2013) Integrated fuzzy multi criteria decision making method and multi-objective programming approach for supplier selection and order allocation in a green supply chain. J Clean Prod 47:355–367
Klement EP, Mesiar R (2005) Logical, algebraic, analytic and probabilistic aspects of triangular norms. Elsevier, New York
Klir G, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and applications. Prentice Hall, Upper Saddle River
Koksalmis E, Kabak O (2018) Deriving decision makers weights in group decision making: an overview of objective methods. Inf Fus. https://doi.org/10.1016/J.INFFUS.2018.11.009
Krishankumar R, Subrajaa LS, Ravichandran KS, Kar S, Saeid AB (2019) A framework for multi-attribute group decision-making using double hierarchy hesitant fuzzy linguistic term set. Int J Fuzzy Syst 21:1130–1143
Krishankumar R, Gowtham Y, Ahmed I, Ravichandran KS, Kar S (2020) Solving green supplier selection problem using q-rung orthopair fuzzy-based decision framework with unknown weight information. Appl Soft Comput. https://doi.org/10.1016/j.asoc.2020.106431
Kumari R, Mishra AR (2020) Multi-criteria COPRAS method based on parametric measures for intuitionistic fuzzy sets: application of green supplier selection. Iran J Sci Tech Trans Electr Eng. https://doi.org/10.1007/s40998-020-00312-w
Laari S, Töyli J, Solakivi T, Ojala L (2016) Firm performance and customer-driven green supply chain management. J Clean Prod 112(1):1960–1970
Laosirihongthong T, Adebanjo D, Tan KC (2013) Green supply chain management practices and performance. Ind Manag Data Syst 113(8):1088–1109
Liang Y (2020) An EDAS Method for multiple attribute group decision-making under intuitionistic fuzzy environment and its application for evaluating green building energy-saving design projects. Symmetry 12:1–12. https://doi.org/10.3390/sym12030484
Liu P, Gao H, Ma J (2019) Novel green supplier selection method by combining quality function deployment with partitioned Bonferroni mean operator in interval type-2 fuzzy environment. Inf Sci 490:292–316
Liu P, Ali Z, Mahmood T (2020) The distance measures and cross-entropy based on complex fuzzy sets and their applications in decision making. J Intell Fuzzy Syst. https://doi.org/10.3233/JIFS-191718
Ma W, Lei W, Sun B (2020) Three-way group decisions under hesitant fuzzy linguistic environment for green supplier selection. Kybernetes. https://doi.org/10.1108/K-09-2019-0602
Mardani A, Nilashi M, Zakuan N, Loganathan N, Soheilirad S, Saman MZ, Ibrahim O (2017) A systematic review and meta-analysis of SWARA and WASPAS methods: theory and applications with recent fuzzy developments. Appl Soft Comput 57:265–292
Min H, Kim I (2012) Green supply chain research: past, present, and future. Logis Res 4:39–47
Mishra AR, Rani P (2018) Interval-valued intuitionistic fuzzy WASPAS method: application in reservoir flood control management policy. Group Decis Negot 27:1047–1078
Mishra AR, Rani P (2021) Assessment of sustainable third party reverse logistic provider using the single-valued neutrosophic combined compromise solution framework. Clean Resp Consum. https://doi.org/10.1016/j.clrc.2021.100011
Mishra AR, Singh RK, Motwani D (2019a) Multi-criteria assessment of cellular mobile telephone service providers using intuitionistic fuzzy WASPAS method with similarity measures. Granul Comput 4:511–529
Mishra AR, Rani P, Pardasani KR, Mardani A (2019b) A novel hesitant fuzzy WASPAS method for assessment of green supplier problem based on exponential information measures. J Clean Prod. https://doi.org/10.1016/j.jclepro.2019.117901
Mishra AR, Rani P, Pandey K (2021a) Fermatean fuzzy CRITIC-EDAS approach for the selection of sustainable third-party reverse logistics providers using improved generalized score function. J Ambient Intell Humaniz Comput. https://doi.org/10.1007/s12652-021-02902-w
Mishra AR, Rani P, Bharti S (2021b) Assessment of agriculture crop selection using Pythagorean fuzzy CRITIC–VIKOR decision-making framework. In: Garg H (ed) Pythagorean fuzzy sets. Springer, Singapore. https://doi.org/10.1007/978-981-16-1989-2_7
Mishra AR, Rani P, Krishankumar R, Zavadskas EK, Cavallaro F, Ravichandran KS (2021c) A hesitant fuzzy combined compromise solution framework-based on discrimination measure for ranking sustainable third-party reverse logistic providers. Sustainability 13:2064. https://doi.org/10.3390/su13042064
Mohagheghi V, Mousavi SM (2020) D-WASPAS: addressing social cognition in uncertain decision-making with an application to a sustainable project portfolio problem. Cognit Comput 12:619–641
Mohammed MA, Abdulkareem KH, Al-Waisy AS, Mostafa SA, Al-Fahdami S, Dihar AM, Alhakami W, Abdullah B, Al-Mhiaani MN, Alhakimi H, Arbaiy N, Maashi MS, Multag AA, Garcia-Zapirain B, De La Tare DI (2020) Benchmarking methodology for selection of optimal COVID-19 diagnostic model based on entropy and TOPSIS methods. IEEE Access 8:99115–99131
Mohammed RT, Zaidan AA, Sharef NM, Abdullah RH, Zaidan BB, Albabri OS, Abdulkareem KH (2021) Determining importance of many-objective optimization competitive algorithms evaluation criteria based on a novel fuzzy-weighted zero-inconsistency method. Int J Inform Tech Decis Mak. https://doi.org/10.1142/s0219622021500140
Nguyen HT, Walker EA (1997) A first course in fuzzy logic. CRC Press, Boca Raton
Pamucar D, Sremac S, Stevic Z, Cirovic G, Tomic D (2019) New multi-criteria LNN WASPAS model for evaluating the work of advisors in the transport of hazardous goods. Neural Comput Appl. https://doi.org/10.1007/s00521-018-03997
Peng X (2019) New multi-parametric similarity measure and distance measure for interval neutrosophic set with IoT industry evaluation. IEEE Access 7:28258–28280
Peng X, Huang H (2020) Fuzzy decision making method based on CoCoSo with CRITIC for financial risk evaluation. Tech Econ Dev Econ. https://doi.org/10.3846/tede.2020.11920
Peng X, Zhang X, Luo Z (2020) Pythagorean fuzzy MCDM method based on CoCoSo and CRITIC with score function for 5G industry evaluation. Artific Intell Rev 53:3813–3847
Popovic M, Kuzmanovic M, Savic G (2018) A comparative empirical study of analytic hierarchy process and conjoint analysis: literature review. Decis Mak Appl Manag Engg 1(2):153–163
Qin J, Liu PW (2016) An extended TODIM multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment. Eur J Oper Res. https://doi.org/10.1016/j.ejor.2016.09.059
Quan J, Zeng B, Liu D (2018) Green supplier selection for process industries using weighted grey incidence decision model. Complexity 2018:4631670. https://doi.org/10.1155/2018/4631670
Ramot D, Milo R, Friedman M, Kandel A (2002) Complex Fuzzy Sets. IEEE Trans Fuzzy Syst 10(2):171–186
Ramot D, Friedman M, Langholz G, Kandel A (2003) Complex fuzzy logic. IEEE Trans on Fuzzy Syst 11:450–461
Rani P, Mishra AR (2020a) Pardasani KR (2020) A novel WASPAS approach for multi-criteria physician selection problem with intuitionistic fuzzy type-2 sets. Soft Comput 24:2355–2367
Rani P, Mishra AR (2020b) Multi-criteria weighted aggregated sum product assessment framework for fuel technology selection using q-Rung orthopair fuzzy sets. Sustain Prod Consum. https://doi.org/10.1016/j.spc.2020.06.015
Rani P, Mishra AR, Krishankumar R, Mardani A, Cavallaro F, Ravichandran KS, Balasubramanian K (2020) Hesitant fuzzy SWARA-complex proportional assessment approach for sustainable supplier selection (HF-SWARA-COPRAS). Symmetry 12(7):1152. https://doi.org/10.3390/sym12071152
Rani P, Mishra AR, Krishankumar R, Mardani A, Ravichandran KS, Kar S (2021) Multi-criteria food waste treatment method selection using single-valued neutrosophic-CRITIC-MULTIMOORA framework. Appl Soft Comput 111:107657. https://doi.org/10.1016/j.asoc.2021.107657
Schitea D, Deveci M, Iordache M, Bilgili K, Akyurt IZ, Iordache I (2019) Hydrogen mobility roll-up site selection using intuitionistic fuzzy sets based WASPAS, COPRAS and EDAS. Int J Hydrog Energy 44(16):8585–8600
Sharma R, Pradhan S (2020) Investigation of machinability criteria during micro-abrasive finishing of SUS-304L steel using fuzzy combined with WASPAS approach. J Braz Soc Mech Sci Eng. https://doi.org/10.1007/s40430-020-2198-5
Shaw K, Shankar R, Yadav SS, Thakur LS (2012) Supplier selection using fuzzy AHP and fuzzy multi-objective linear programming for developing low carbon supply chain. Expert Syst Appl 39(9):8182–8192
Shumaiza AM, Al-Kenani AN (2019) Multiple-attribute decision making ELECTRE II method under bipolar fuzzy model. Algorithms 12(11):1–24
Singh PK (2020) Bipolar δ-equal complex fuzzy concept lattice with its application. Neural Comput Appl 32:2405–2422
Stanujkic D, Karabasevic D (2018) An extension of the WASPAS method for decision-making problems with intuitionistic fuzzy numbers: a case of website evaluation. Oper Res Eng Sci Theory Appl 1(1):29–39
Tamir DE, Lin J, Kandel A (2011) A new interpretation of complex membership grade. Int J Intell Syst 26:285–312
Tang SL (2017) Green supplier selection model with hesitant fuzzy information. J Intell Fuzzy Syst 32:189–195
Tsai SB, Chien MF, Xue Y, Li L, Jiang X, Chen Q, Zhou J, Wang L (2015) Using the fuzzy DEMATEL to determine environmental performance: a case of printed circuit board industry in Taiwan. PLoS ONE. https://doi.org/10.1371/journal.pone.0129153
Turskis Z, Zavadskas EK, Antucheviciene J (2015) A hybrid model based on fuzzy AHP and fuzzy WASPAS for construction site selection. Int J Comput Commun Control 10:873–888
Tus A, Adali EA (2019) The new combination with CRITIC and WASPAS methods for the time and attendance software selection problem. Opsearch 56:528–538
Wei G, Alsaadi FE, Hayat T, Alsaedi A (2017) Bipolar fuzzy Hamacher aggregation operators in multiple-attribute decision making. Int J Fuzzy Syst. https://doi.org/10.1007/s40815-017-0338-6
Wei G, Lei F, Lin R, Wang R, Wei Y, Wu J, Wei C (2020) Algorithms for probabilistic uncertain linguistic multiple-attribute group decision-making based on the GRA and CRITIC method: application to location planning of electric vehicle charging stations. Econ Res Ekonomska Istrazivanja 33(1):828–846
Wu MQ, Zhang CH, Liu XN, Fan J (2019) Green supplier selection based on DEA model in interval-valued Pythagorean fuzzy environment. IEEE Access 7:108001–108013
Younis H, Sundarakani B, O’Mahony B (2019) Green supply chain management and corporate performance: developing a roadmap for future research using a mixed method approach. I IMB Manag Rev. https://doi.org/10.1016/j.iimb.2019.10.011
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
Zavadskas EK, Turskis Z, Antucheviciene J, Zakarevicius A (2012) Optimization of weighted aggregated sum product assessment. Electr Electron Eng 6:3–6
Zhang WR (1994) Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multi agent decision analysis. In: The workshop on fuzzy inf. Proce. society bi. Conf., pp 305–309
Zhang WR, Zhang L (2004) Bipolar logic and bipolar fuzzy logic. Inf Sci 165:265–287
Zhang WR, Pandurangi KA, Peace KE, Zhang Y, Zhao Z (2011) Mental squares—a generic bipolar support vector machine for psychiatric disorder classification, diagnostic analysis and neurobiological data mining. Int J Data Min Bioinf 5(5):532–572
Zolfani SH, Yazdani M, Torkayesh AE, Derakhti A (2010) Application of a gray-based decision support framework for location selection of a temporary hospital during COVID-19 pandemic. Symmetry 12:1–15
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Liu, P., Saha, A., Mishra, A.R. et al. A BCF–CRITIC–WASPAS method for green supplier selection with cross-entropy and Archimedean aggregation operators. J Ambient Intell Human Comput 14, 11909–11933 (2023). https://doi.org/10.1007/s12652-022-03745-9
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DOI: https://doi.org/10.1007/s12652-022-03745-9