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Improved CoCoSo Method Based on Frank Softmax Aggregation Operators for T-Spherical Fuzzy Multiple Attribute Group Decision-Making

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Abstract

In this article, a novel CoCoSo (Combined compromise solution) method based on Frank operational laws and softmax function is investigated to handle multiple attribute group decision-making problems for T-spherical fuzzy sets. We extend Frank operations in T-spherical fuzzy environment and develop a series of aggregation operators, including T-spherical fuzzy Frank softmax (T-SFFS) average and geometric operators, and their weighted forms, i.e., T-SFFS weighted averaging (T-SFFSWA) and T-SFFS weighted geometric (T-SFFSWG) operators. Some of their basic properties and particular cases are discussed. Meanwhile, the monotonicity of proposed operators is also analyzed, and it is discussed that how they indicate the decision-makers’ optimistic and pessimistic decision attitudes with risk preference. Furthermore, a novel CoCoSo method based on Hamming distance measure is proposed, which considers both decision-maker’s decision attitude and attribute priority, and a multiple attribute group decision-making framework with two independent and parallel T-spherical fuzzy information processing processes are designed. Lastly, a real case of spent power battery recycling technology (SPBRT) selection is presented to show the practicability of the proposed method. Also sensitivity and comparative analyses are carried out to prove the reliability, effectiveness, and superiority of our proposed method.

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Data availability

Some or all data that support this study's findings are available from the corresponding author upon reasonable request.

Abbreviations

3PRL:

Third-party reverse logistic

AD:

Abstinence degree

AO:

Aggregation operator

AOLs:

Algebraic operational laws

CFS:

Classical fuzzy set

CoCoSo:

Combined compromise solution

DEMATEL:

Decision-making trial and evaluation laboratory

DM:

Decision-maker

DOLs:

Dombi operational laws

FNs:

Fuzzy numbers

FOLs:

Frank operational laws

GMIR:

Graded mean integration representation

HFEs:

Hesitant fuzzy elements

HFWA:

Hesitant fuzzy weighted averaging

HFLEs:

Hesitant fuzzy linguistic elements

IFS:

Intuitionistic fuzzy set

IFWA:

Intuitionistic fuzzy weighted averaging

IFWG:

Intuitionistic fuzzy weighted geometric

IVIFNs:

Interval-valued intuitionistic fuzzy numbers

IVIFWA:

Interval-valued intuitionistic fuzzy weighted averaging

IVNNs:

Interval valued neutrosophic numbers

INNWAA:

Interval valued neutrosophic weighted arithmetic averaging

MAGDM:

Multi-attribute group decision-making

MD:

Membership degree

MULTIMOORA:

Multi-objective optimization based on the ratio analysis with the full multiplicative form

ND:

Non-membership degree

NIS:

Negative ideal solution

OLs:

Operational laws

PAO:

Prioritized averaging operator

PFS:

Picture fuzzy set

PFNs:

Picture fuzzy numbers

PFWA:

Picture fuzzy weighted averaging

PFWG:

Picture fuzzy weighted geometric

PIS:

Positive ideal solution

PLEs:

Probabilistic linguistic elements

PyFS:

Pythagorean fuzzy set

PyFNs:

Pythagorean fuzzy numbers

PyFWA:

Pythagorean fuzzy weighted averaging

PyFWG:

Pythagorean fuzzy weighted geometric

PyFWPA:

Pythagorean fuzzy weighted power averaging

q-ROFS:

q-Rung orthopair fuzzy set

q-ROFWA:

q-Rung orthopair fuzzy weighted averaging

q-ROFWG:

q-Rung orthopair fuzzy weighted geometric

RN:

Rough number

RNDWAA:

RN Dombi weighted arithmetic averaging

RNDWGA:

RN Dombi weighted geometric averaging

SFS:

Spherical fuzzy set

SFWA:

Spherical fuzzy weighted averaging

SFWG:

Spherical fuzzy weighted geometric

SIFWA:

Softmax intuitionistic fuzzy weight averaging

SIFWG:

Softmax intuitionistic fuzzy weight geometric

SPBRT:

Spent power battery recycling technology

SVNNs:

Single-valued Neutrosophic numbers

SVNWA:

Single-valued Neutrosophic weighted averaging

TODIM:

Portuguese acronym meaning Interactive Multi-Criteria Decision Making

TOPSIS:

Technique for Order Preference by Similarity to an Ideal Solution

T-SF:

T-spherical fuzzy

T-SFDM:

T-SF decision matrix

T-SFDPWA:

T-SF Dombi prioritized weighted arithmetic

T-SFDPWG:

T-SF Dombi prioritized weighted geometric

T-SFDRM:

T-SF direct relation matrix

T-SFFS:

T-spherical fuzzy Frank softmax

T-SFN:

T-SF number

T-SFS:

T-spherical fuzzy set

T-SFFWA:

T-spherical Frank weighted averaging

T-SFFWG:

T-spherical Frank weighted geometric

T-SFWA:

T-spherical fuzzy weighted averaging

T-SFWG:

T-spherical fuzzy weighted geometric

T-SFFSA:

T-spherical fuzzy Frank softmax average

T-SFFSWA:

T-spherical fuzzy Frank softmax weighted average

T-SFFSG:

T-spherical fuzzy Frank softmax geometric

T-SFFSWG:

T-spherical fuzzy Frank softmax weighted geometric

T-SFWAI:

T-SF weighted average interaction

T-SFWGI:

T-SF weighted geometric interaction

T-SFWGMSM:

T-SF weighted generalized Maclarurin symmetric mean

VIKOR:

VlseKriterijumska Optimizacija I Kompromisno Resenje

WHM:

Weighted Heronian mean

WPM:

Weighted product model

WSM:

Weighted sum model

a, b, η 1, η 2, η :

Non-negative real numbers

ac(δ):

Accuracy function of T-SFN δ

δ :

T-SFN of ℑ

D t :

Individual T-SFDM by the t-th expert

D H(δ 1, δ 2):

Hamming distance between two T-SFNs

d ij t :

Initial T-SF evaluation value of alternative i w.r.t. attribute j by expert t

(d ij t)c :

Complement set of dijt

∂i (1), ∂i (2) :

Closeness degree of alternative i with optimistic (ϒ = 1) and pessimistic decision type (ϒ = 2)

E :

Expert set

e t :

The t-th expert

EM ( ϒ ) :

Extended group T-SFDM with decision type ϒ

f(θ):

Score value of T-SFFSWA representing a function with respect to parameter θ

ϕ i κ :

Softmax function with parameter κ

Φi κ :

Weighted softmax function with parameter κ

℘i (ϒ), ℘Θ (ϒ) :

Performance values of alternatives i, PIS and NIS with decision type ϒ(ϒ = 1,2)

g j (ϒ) NIS, g j (ϒ) PIS :

NIS and PIS w.r.t. attribute j with decision type ϒ

H :

Attribute set

h j :

The j-th attribute

i,j :

Index of number

φ :

Adjustment parameter in combined weight

κ :

Modulation parameter in softmax function

K i :

Comprehensive utility value of alternative i

K ia :

Additive normalization of ∂i(1) and ∂i(2)

K ib :

Sum of the relative relations of ∂i(1) and ∂i(2)

K ic :

Tradeoff of ∂i(1) and ∂i(2)

λ :

Weight vector of expert set E

λ t :

Weight value of expert et

ℵt :

Individual initial T-SFDRM by the t-th expert

n, m :

Number of evaluation objects

Θ:

Index of “PIS” and “NIS”

θ :

Base number in Frank t-norms

ρ :

Compromise coefficient in Kic

q :

Power of MD,AD and ND of T-SFN

ϒ:

Index of optimistic decision type (ϒ = 1) and pessimistic decision type (ϒ = 2)

R t :

Normalized T-SFDM by the t-th expert

r ij t :

Normalized T-SF evaluation value of alternative i w.r.t. attribute j by expert t

γ jl t :

Initial T-SF evaluation value between two attributes j, l by expert t

ℑ :

T-spherical fuzzy set

S :

Alternative set

s i :

The i-th alternative

sc(δ):

Score function of T-SFN δ

sc(A), sc(G):

Score functions of T-SFFSWA and T-SFFSWG operators

S(δ 1, δ 2):

Similarity measure between two T-SFNs

t(a, b), s(a, b):

Frank product and Frank sum

T i :

Sum of i-1values in softmax function

T (ϒ) :

Total relation matrix with decision type ϒ

t :

Index of expert

t jl (ϒ) :

Total relative T-SF value with decision type ϒ

Γl (ϒ), Λj (ϒ) :

Row sum and column sum in total relation matrix T(ϒ)

τ, ψ, ϑ :

MD, AD, ND of T-SFN

ϖ jl t, ϖ ij t :

Priority weight value of expert t for T-SFDRM and T-SFDM

w i :

Weight of the i-th T-SFN δi

w oj (ϒ), w sj (ϒ), w cj (ϒ) :

Objective, subjective and combined weight value of attribute j with decision type ϒ

ω ij (ϒ), ω Θj (ϒ) :

Priority weight of attribute j w.r.t. alternative i, PIS and NIS with decision type ϒ

X M ( ϒ ), X A ( ϒ ), X N ( ϒ ) :

Normalized MD, AD, ND sub-matrix of group initial T-SFDRM ℵ(ϒ)

X :

Universe

Ψ1, Ψ2 :

Benefit and cost attribute type

z :

Number of experts

References

  1. Xu, Z.S.: An interactive approach to multiple attribute group decision making with multigranular uncertain linguistic information. Group Decis Negot. 18(2), 119–145 (2009)

    Google Scholar 

  2. Zadeh, L.A.: Fuzzy sets. Inf Contr. 8(3), 338–353 (1965)

    MATH  Google Scholar 

  3. Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 31, 343–349 (1986)

    MATH  Google Scholar 

  4. Yager, R.R.: Pythagorean membership grades in multicriteria decision making. IEEE Trans Fuzzy Syst. 22(4), 958–965 (2014)

    Google Scholar 

  5. Yager, R.R.: Generalized orthopair fuzzy sets. IEEE Trans Fuzzy Syst. 25(5), 1222–1230 (2017)

    Google Scholar 

  6. Cuong, B.C.: Picture fuzzy sets. J Comput Sci Cyb. 30(4), 409–420 (2015)

    Google Scholar 

  7. Mahmood, T., Ullah, K., Khan, Q., Jan, N.: An approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy sets. Neural Comput Appl. 31(11), 7041–7053 (2019)

    Google Scholar 

  8. Gul, M., Lo, H.W., Yucesan, M.: Fermatean fuzzy TOPSIS-based approach for occupational risk assessment in manufacturing. Complex Intell Syst. 7(5), 2635–2653 (2021)

    Google Scholar 

  9. Opricovic, S., Tzeng, G.H.: Extended VIKOR method in comparison with outranking methods. Eur J Oper Res. 178(2), 514–529 (2007)

    MATH  Google Scholar 

  10. Mahmood, T., Warraich, M.S., Ali, Z., Pamucar, D.: Generalized MULTIMOORA method and Dombi prioritized weighted aggregation operators based on T-spherical fuzzy sets and their applications. Int J Intell Syst. 36(9), 4659–4692 (2021)

    Google Scholar 

  11. Ju, Y.B., Liang, Y.Y., Luo, C., Dong, P.W., Gonzale, E.D.R.S., Wang, A.H.: T-spherical fuzzy TODIM method for multi-criteria group decision-making problem with incomplete weight information. Soft Comput. 25, 2981–3001 (2021)

    MATH  Google Scholar 

  12. Yazdani, M., Zarate, P., Zavadskas, E.K., Turskis, Z.: A combined compromise solution (CoCoSo) method for multi-criteria decision-making problems. Manage Decis. 57(9), 2501–2519 (2019)

    Google Scholar 

  13. Liu, P.D., Zhu, B.Y., Wang, P.: A multi-attribute decision-making approach based on spherical fuzzy sets for Yunnan Baiyao’s R&D project selection problem. Int J Fuzzy Systems. 21(7), 2168–2191 (2019)

    Google Scholar 

  14. Liu, P.D., Khan, Q., Mahmood, T., Hassan, N.: T-spherical fuzzy power Muirhead mean operator based on novel operational laws and their application in multi-attribute group decision making. IEEE Access 7, 22613–22632 (2019)

    Google Scholar 

  15. Grag, H., Ullah, K., Mahmood, T., Hassan, N., Jan, N.: T-spherical fuzzy power aggregation operators and their applications in multi-attribute decision making. J Ambient Intell Human Comput. 12, 9067–9080 (2021)

    Google Scholar 

  16. Munir, M., Kalsoom, H., Ullah, K., Mahmood, T., Chu, Y.M.: T-spherical fuzzy Einstein hybrid aggregation operators and their applications in multi-attribute decision making problems. Symmetry. 12, 365 (2020)

    Google Scholar 

  17. Ullah, K., Mahmood, T., Garg, H.: Evaluation of the performance of search and rescue robots using T-spherical fuzzy Hamacher aggregation operators. Int J Fuzzy Syst. 22(2), 570–582 (2020)

    Google Scholar 

  18. Zeng, S.Z., Garg, H., Munir, M., Mahmood, T., Hussain, A.: A multi-attribute decision making process with immediate probabilistic interactive averaging aggregation operators of t-spherical fuzzy sets and its application in the selection of solar cells. Energies 12(23), 4436 (2019)

    Google Scholar 

  19. Garg, H., Munir, M., Ullah, K., Mahmood, T., Jan, N.: Algorithm for T-spherical fuzzy multi-attribute decision making based on improved interactive aggregation operators. Symmetry. 10, 670 (2018)

    Google Scholar 

  20. Munir, M., Mahmood, T., Hussain, A.: Algorithm for T-spherical fuzzy MADM based on associated immediate probability interactive geometric aggregation operators. Artif Intell Rev. 54, 6033–6061 (2021)

    Google Scholar 

  21. Ullah, K., Ali, Z., Mahmood, T., Garg, H., Chinram, R.: Methods for multi-attribute decision making, pattern recognition and clustering based on T-spherical fuzzy information measures. J Intell Fuzzy Syst. 2021, 1–21 (2022). https://doi.org/10.3233/JIFS-210402

    Article  Google Scholar 

  22. Frank, M.J.: On the simultaneous associativity of F(x, y) and x+y-F(x, y). Aequationes Math. 19, 194–226 (1979)

    MathSciNet  MATH  Google Scholar 

  23. Zhang, Z.M.: Interval-valued intuitionistic fuzzy Frank aggregation operators and their applications to multiple attribute group decision making. Neural Comput Appl. 28(6), 1471–1501 (2017)

    Google Scholar 

  24. Qin, J.D., Liu, X.W., Pedrycz, W.: Frank aggregation operators and their application to hesitant fuzzy multiple attribute decision making. Appl Soft Comput. 41, 428–452 (2016)

    Google Scholar 

  25. Xing, Y.P., Zhang, R.T., Wang, J., Zhu, X.M.: Some new Pythagorean fuzzy Choquet-Frank aggregation operators for multi-attribute decision making. Int J Intell Syst. 33(11), 2189–2215 (2018)

    Google Scholar 

  26. Xing, Y.P.: q-Rung orthopair fuzzy Frank power point aggregation operators with new multi-parametric distance measures. J Intell Fuzzy Syst. 41(6), 7275–7297 (2021)

    MathSciNet  Google Scholar 

  27. Mahmood, T., Waqas, H.M., Ali, Z., Ullah, K., Pamucar, D.: Frank aggregation operators and analytic hierarchy process based on interval-valued picture fuzzy sets and their applications. Int J Intell Syst. 36(12), 7925–7962 (2021)

    Google Scholar 

  28. Ji, P., Wang, J.Q., Zhang, H.Y.: Frank prioritized Bonferroni mean operator with single-value neutrosophic sets and its application in selecting third-party longistics providers. Neural Comput Appl. 30(3), 799–823 (2018)

    Google Scholar 

  29. Yager, R.R.: Prioritized OWA aggregation. Fuzzy Optim Decis Ma. 8(3), 245–262 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Fahmi, A., Ul, A.N.: Group decision-making based on bipolar neutrosophic fuzzy prioritized Muirhead mean weighted averaging operator. Soft Comput. 25(15), 10019–10036 (2021)

    Google Scholar 

  31. Wei, G.W.: Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowl Based Syst. 31, 176–182 (2012)

    Google Scholar 

  32. Lu, B.Q., Xu, Z.S.: Prioritized aggregation operators based on the priority degrees in multicriteria decision-making. Int J Intell Syst. 34(9), 1985–2018 (2019)

    MathSciNet  Google Scholar 

  33. Gao, H.: Pythagorean fuzzy Hamacher prioritized aggregation operators in multiple attribute decision making. J Intell Fuzzy Syst. 35(2), 2229–2245 (2018)

    Google Scholar 

  34. Zhu, D.Y., Lu, S.Y., Wang, M.Q., Lin, J., Wang, Z.F.: Efficient precision-adjustable architecture for softmax function in deep learning. IEEE T Circuits-II. 67(12), 3382–3386 (2020)

    Google Scholar 

  35. Yu, D.J.: Softmax function based intuitionistic fuzzy multi-criteria decision making and applications. Oper Res. 16, 327–348 (2016)

    Google Scholar 

  36. Torres, R., Salas, R., Astudillo, H.: Time-based hesitant fuzzy information aggregation approach for decision making problems. Int J Intell Syst. 29(6), 579–595 (2014)

    Google Scholar 

  37. Ecer, F., Pamucar, D.: Sustainable supplier selection: a novel integrated fuzzy best worst method (F-BWM) and fuzzy CoCoSo with Bonferroni (CoCoSo’B) multi-criteria model. J Clean Prod. 266, 121981 (2020)

    Google Scholar 

  38. Peng, X.D., Zhang, X., Luo, Z.G.: Pythagorean fuzzy MCDM method based on CoCoSo and CRITIC with score function for 5G industry evaluation. Artif Intell Rev. 53(5), 3813–3847 (2020)

    Google Scholar 

  39. Liao, H.C., Qin, R., Wu, D., Yazdani, M., Zavadskas, E.K.: Pythagorean fuzzy combined compromise solution method integrating the cumulative prospect theory and combined weights for cold chain logistics distribution center selection. Int J Intell Syst. 35(12), 2009–2031 (2020)

    Google Scholar 

  40. Peng, X.D., Huang, H.H.: Fuzzy decision making method based on CoCoSo with CRITIC for financial risk evaluation. Technol Econ Dev Eco. 26(4), 695–724 (2020)

    Google Scholar 

  41. Yazdani, M., Chatterjee, P., Pamucar, D., Chakraborty, S.: Development of an integrated decision making model for location selection of logistics centers in the Spanish autonomous communities. Expert Syst Appl. 148, 113208 (2020)

    Google Scholar 

  42. Svadlenka, L., Simic, V., Dobrodolac, M., Lazarevic, D., Todorovic, G.: Picture fuzzy decision-making approach for sustainable last-mile delivery. IEEE Access. 8, 209393 (2020)

    Google Scholar 

  43. Peng, X., Li, W.: Spherical fuzzy decision making method based on combined compromise solution for IIoT industry evaluation. Artif Intell Rev. 55, 1857–1886 (2022)

    Google Scholar 

  44. Tavana, M., Shaabani, A., Di Caprio, D., Bonyani, A.: A novel interval type-2 fuzzy best-worst method and combined compromise solution for evaluating eco-friendly packaging alternatives. Expert Syst Appl. 200, 117188 (2022)

    Google Scholar 

  45. Ullah, K., Mahmood, T., Jan, N.: Similarity measures for T-spherical fuzzy sets with applications in pattern recognition. Symmetry 10, 193 (2018)

    MATH  Google Scholar 

  46. Xu, Z.S.: Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst. 15(6), 1179–1187 (2007)

    Google Scholar 

  47. Zeng, S.Z., Chen, J.P., Li, X.S.: A hybrid method for Pythagorean fuzzy multiple-criteria decision making. Int J Inf Tech Decis. 15(2), 403–422 (2016)

    Google Scholar 

  48. Liu, P.D., Wang, P.: Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making. Int J Intell Syst. 33(2), 259–280 (2018)

    Google Scholar 

  49. Wei, G.W.: Picture fuzzy aggregation operators and their application to multiple attribute decision making. J Intell Fuzzy Syst. 33(2), 713–724 (2017)

    MATH  Google Scholar 

  50. Ashraf, S., Abdullah, S.: Spherical aggregation operators and their application in multiattribute group decision-making. Int J Intell Syst. 34, 493–523 (2019)

    Google Scholar 

  51. Ashraf, S., Abdullah, S., Mahmood, T., Ghani, F., Mahmood, T.: Spherical fuzzy sets and their applications in multi-attribute decision making problems J. Intell Fuzzy Syst. 36, 2829–2844 (2019)

    Google Scholar 

  52. Ullah, K., Hassan, N., Mahmood, T., Jan, N., Hassan, M.: Evaluation of investment policy based on multi-attribute decision-making using interval valued T-spherical fuzzy aggregation operators. Symmetry. 11, 357 (2019)

    Google Scholar 

  53. Mahnaz, S., Ali, J., Malik, M.G., Bashir, Z.: T-spherical fuzzy Frank aggregation operators and their application to decision making with unknown weight information. IEEE Access 10, 7408–7438 (2022)

    Google Scholar 

  54. Gül, S.: Spherical fuzzy extension of DEMATEL (SF-DEMATEL). Int J Intell Syst. 35, 1329–1353 (2020)

    Google Scholar 

  55. Harper, G., Sommerville, R., Kendrick, E., et al.: Recycling lithium-ion batteries from electric vehicles. Nature 575(7781), 75–86 (2019)

    Google Scholar 

  56. Zhao, S.Q., Li, G.M., He, W.Z., Huang, J.W.: Recovery methods and regulation status of waste lithium-ion batteries in China: a mini review. Waste Manage Res. 37(11), 1142–1152 (2019)

    Google Scholar 

  57. Stević, Ž, Pamučar, D., Puška, A., Chatterjee, P.: Sustainable supplier selection in healthcare industries using a new MCDM method: Measurement of alternatives and ranking according to compromise solution (MARCOS). Comput Ind Eng. 140, 106231 (2020)

    Google Scholar 

  58. Zavadskas, E.K., Turskis, Z., Antucheviciene, J., Zakarevicius, A.: Optimization of weighted aggregated sum product assessment. Elekton Elektrotech. 122(6), 3–6 (2012)

    Google Scholar 

  59. Ali, Z., Mahmood, T., Yang, M.S.: Complex T-spherical fuzzy aggregation operators with application to multi-attribute decision making. Symmetry. 12(8), 1311 (2020)

    Google Scholar 

  60. Wen, Z., Liao, H.C., Zavadskas, E.K., Al-Barakati, A.: Selection third-party logistics service providers in supply chain finance by a hesitant fuzzy linguistic combined compromise solution method. Econ Res-Ekon Istraz. 32(1), 4033–4058 (2019)

    Google Scholar 

  61. Wen, Z., Liao, H.C., Ren, R.X., Bai, C.G., et al.: Cold chain logistics management of medicine with an integrated multi-criteria decision-making method. Int J Env Res Pub He. 16, 4843 (2019)

    Google Scholar 

  62. Zhang, Z.Y., Liao, H.C., Al-Barakati, A., Zavadskas, E.K., Antucheviciene, J.: Supplier selection for housing development by an integrated method with interval rough boundaries. Int J Strateg Prop M. 24(4), 269–284 (2020)

    Google Scholar 

  63. Deveci, M., Pamucar, D., Gokasar, I.: Fuzzy power Heronian function based CoCoSo method for the advantage prioritization of autonomous vehicles in real-time traffic management. Sustain Cities Soc. 69, 102846 (2021)

    Google Scholar 

  64. Mishra, A.R., Rani, P., Krishankumar, R., Zavadskas, E.K., Cavallaro, F., Ravichandran, K.S.: A hesitant fuzzy combined compromise solution framework-based on discrimination measure for ranking sustainable third-party reverse logistic providers. Sustainability. 13, 2064 (2021)

    Google Scholar 

  65. Alrasheedi, M., Mardani, A., Mishra, A.R., Streimikiiene, D., Liao, H.C., Al-nefaie, A.H.: Evaluating the green growth indicators to achieve sustainable development: a novel extended interval-valued intuitionistic fuzzy-combined compromise solution approach. Sustain Dev. 29(1), 120–142 (2021)

    Google Scholar 

  66. Cui, Y.F., Liu, W., Rani, P., Alrasheedi, M.: Internet of things (IoT) adoption barriers for the circular economy using pythagorean fuzzy SWARA-CoCoSo decision-making approach in the manufacturing sector. Technol Forecast Soc Change. 171, 120951 (2021)

    Google Scholar 

  67. Rani, P., Ali, J., Krishankumar, R., Mishra, A.R., Cavallaro, F., Ravichandran, K.S.: An integrated single-valued Neutrosophic combined compromise solution methodology for renewable energy resource selection problem. Energies 14, 5494 (2021)

    Google Scholar 

  68. Liu, P.D., Rani, P., Mishra, A.R.: A novel Pythagorean fuzzy combined compromise solution framework for the assessment of medical waste treatment technology. J Clean Prod. 292, 126047 (2021)

    Google Scholar 

  69. Yazdani, M., Torkayesh, A.E., Stevic, Z., Chatterjee, P., Ahari, S.A., Hernandez, V.D.: An interval valued neutrosophic decision-making structure for sustainable supplier selection. Expert Syst Appl. 183, 115354 (2021)

    Google Scholar 

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Funding

This study was supported by the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China, 19YJC630164 and the Postdoctoral Science Foundation of Jiangxi Province, 2019KY14 to Haolun Wang.

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Authors and Affiliations

  1. School of Economics and Management, Nanchang Hangkong University, Nanchang, China

    Haolun Wang

  2. Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad, Pakistan

    Tahir Mahmood

  3. Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University, Lahore, Pakistan

    Kifayat Ullah

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  1. Haolun Wang
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Correspondence to Haolun Wang.

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Wang, H., Mahmood, T. & Ullah, K. Improved CoCoSo Method Based on Frank Softmax Aggregation Operators for T-Spherical Fuzzy Multiple Attribute Group Decision-Making. Int. J. Fuzzy Syst. 25, 1275–1310 (2023). https://doi.org/10.1007/s40815-022-01442-5

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