Document Type : Article

**Authors**

School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan Shandong 250014, China

**Abstract**

Linguistic Z-numbers (LZNs), as a more rational extension of linguistic description, not only consider the fuzzy restriction of assessment information but also take the reliability of the information into account. Maclaurin symmetric mean (MSM) operator has the advantage which can take account of interrelationship of different attributes and there are a lot of research results on it. However, it has not been used to handle multi-attribute decision-making (MADM) problems expressed by LZNs. To sum up the advantages of LZNs and MSM, in this paper, we present the linguistic Z-Numbers MSM (LZMSM) and linguistic Z-Numbers weight MSM (LZWMSM) operators, respectively, and several characters and several special cases of them are explored. Moreover, we propose an approach to handle some MADM problems by using LZWMSM operator. In the end, an example is given to illustrate the effectiveness and superiority of this new presented approach by comparing with several existing approaches.

**Keywords**

[1] Zadeh, L.A. “Fuzzy sets”, Information and Control, 8 (3), pp. 338-356 (1965).

[2] Atanassov, K.T. “Intuitionistic fuzzy sets”, Fuzzy Sets and Systems, 20 (1), pp. 87-96 (1986).

[3] DEschrijver, G., Kerre, E.E. “On the position of intuitionistic fuzzy set theory in the framework of theories modelling imprecision”, Information

Sciences, 177(8), pp. 1860-1866 (2007).

[4] Zadeh, L.A. “A note on Z-numbers”, Information Sciences, 181, pp. 2923–32 (2011).

[5] Akhbari, M., Sadi-Nezhad, S. “Equilibrium solution of noncooperative bimatrix game of Z-numbers,” Bulletin of the Georgian National Academy of

Sciences, 9(1), pp. 33–47 (2015).

[6] Yaakob, A.M., Gegov, A. “Interactive TOPSIS based group decision making methodology using Z-numbers”, International Journal of Computational

Intelligence Systems, 9, pp. 311–24 (2016).

[7] Kang, B.Y., Zhang, P.D., Gao, Z.Y., Chhipi-Shrestha, G., Hewage, K., Sadiq, R. “Environmental assessment under uncertainty using Dempster–Shafer

theory and Z-numbers”, Journal of Ambient Intelligence and Humanized Computing, https://doi.org/10.1007/s12652-019-01228-y (2019).

[8] Aliev, R.A., Alizadeh, A.V., Huseynov, O.H. “An introduction to the arithmetic of Z-numbers by using horizontal membership functions”, Procedia

computer science, 120, pp. 349-356 (2017).

[9] Kang, B.Y., Wei, D., Li, Y. “Decision making using Z-numbers under uncertain environment”, Journal of Computational Information Systems, 8(7), pp.

2807-2814 (2012).

[10] Aliev, R.A., Alizadeh, A.V., Huseynov, O.H. “The arithmetic of discrete Z-numbers”, Information Sciences, 290, pp.134–55 (2015).

[11] Kang, B.Y., Wei, D., Li, Y. “A method of converting Z-number to classical fuzzy number”, Journal of Information and Computational Science, 9(3),

pp.703-709 (2012).

[12] Peng, H.G., Wang, X.K., Wang, T.L., Wang, J.Q. “Multi-criteria game model based on the pairwise comparisons of strategies with Z-numbers”, Applied

Soft Computing, 74, pp. 451-465 (2019).

[13] Saravi, N.A., Yazdanparast, R., Momeni, O., Heydarian, D., Jolai, F. “Location optimization of agricultural residues-based biomass plant using

Z-number DEA”, Journal of Industrial and Systems Engineering, 12(1), pp. 39-65 (2019).

[14] Zadeh, L.A. “The concept of a linguistic variable and its application to approximate reasoning—I”, Information Sciences, 8(3), pp. 199-249 (1975).

[15] Zadeh, L.A. “The concept of a linguistic variable and its application to approximate reasoning—II”, Information Sciences, 8(4), pp. 301-357 (1975).

[16] Zadeh, L.A. “The concept of a linguistic variable and its application to approximate reasoning—III”, Information Sciences, 9 (1), pp. 43-80 (1975).

[17] Xu, Z.S. “Induced uncertain linguistic OWA operators applied to group decision making”, Information Fusion, 7 (2), pp. 231-238 (2006).

[18] Liu, P.D., Liu, J.L. “Partitioned Bonferroni mean based on two-dimensional uncertain linguistic variables for multiattribute group decision making”,

International Journal of Intelligent Systems, 34(2), pp. 155–187 (2019).

[19] Meng, F.Y., Tan, C.Q., Zhang, Q. “An approach to multi-attribute group decision making under uncertain linguistic environment based on the Choquet

aggregation operators”, Journal of Intelligent & Fuzzy Systems, 26(2), pp. 769-780 (2014).

[20] Zhao, N., Xu, Z.S., Ren, Z.L. “Hesitant fuzzy linguistic prioritized superiority and inferiority ranking method and its application in sustainable energy

technology evaluation”, Information Sciences, 478, pp. 239-257 (2019).

[21] Yu, X.H., Xu, Z.S., Liu, S.S., Chen, Q. “Multi-criteria decision making with 2-dimension linguistic aggregation techniques”, International Journal of

Intelligent Systems, 27 (6), pp. 539–562 (2012).

[22] Wang, P., Xu, X.H., Wang, J.Q., Cai, C.G. “Interval-valued intuitionistic linguistic multi-criteria group decision-making method based on the interval

2-tuple linguistic information”, Journal of Intelligent & Fuzzy Systems, 33(2), pp. 985-994 (2017).

[23] Wan, S.P. Dong, J.Y. “Multi-attribute group decision making with trapezoidal intuitionistic fuzzy numbers and application to stock selection”,

Informatica, 25(4), pp. 663-697 (2014).

[24] Nehi, H.M. “A new ranking method for intuitionistic fuzzy numbers”, International Journal of Fuzzy Systems, 12 (1), pp. 80-86 (2010).

[25] Chen, Z.C., Liu, P.H. “An approach to multiple attribute group decision making based on linguistic intuitionistic fuzzy numbers”, International Journal

16

of Computational Intelligence Systems, 8(4), pp. 747-760 (2015).

[26] Liu, P.D., Liu, J.L., Merigó, J. M. “Partitioned Heronian means based on linguistic intuitionistic fuzzy numbers for dealing with multi-attribute group

decision making”, Applied Soft Computing, 65, pp. 395-422 (2017).

[27] Meng, F.Y., Tang, J., Fujita, H. “Linguistic intuitionistic fuzzy preference relations and their application to multi-criteria decision making”, Information

Fusion, 46, pp. 77-90 (2019).

[28] Wang, J.Q., Cao, Y.X., Zhang, H.Y. “Multi-criteria decision-making method based on distance measure and choquet integral for linguistic Z-numbers”,

Cognitive Computation, 9, pp. 827–842 (2017).

[29] Bhanu, M.S., Velammal, G. “Operations on Zadeh’s Z-number”, IOSR Journal of Mathematics, 11(3), pp. 88–94 (2015).

[30] Xu, Z.S. “Intuitionistic fuzzy aggregation operators”, IEEE Transactions on Fuzzy Systems, 15 (6), pp. 1179-1187 (2007).

[31] Bonferroni, C. “Sulle medie multiple di potenze”, Bolletino Matematica Italiana, 5, pp. 267-270 (1950).

[32] Beliakov, G., Pradera, A., Calvo, T. “Aggregation functions: A guide for practitioners”, Springer, 2007.

[33] Liang, D.C., Zhang, Y.R.J., Xu, Z.S. “Pythagorean fuzzy Bonferroni mean aggregation operator and its accelerative calculating algorithm with the

multithreading”, International Journal of Intelligent Systems, 33 (3), pp. 615-633 (2018).

[34] Yang, W., Pang, Y.F. “New q-rung orthopair fuzzy partitioned Bonferroni mean operators and their application in multiple attribute decision making”,

International Journal of Intelligent Systems, 34 (3), pp. 439-476 (2019).

[35] Wei, G.W., Lu, M., Gao, H. “Picture fuzzy heronian mean aggregation operators in multiple attribute decision making”, International Journal of

Knowledge-based and Intelligent Engineering Systems, 22 (3), pp. 167-175 (2018).

[36] Maclaurin, C. “A second letter to Martin Folkes, Esq.; concerning the roots of equations, with the demonstartion of other rules in algebra”,

Philosophical Transactions of the Royal Society of London Series A, 36, pp. 59–96 (1729).

[37] Detemple, D.W., Robertson, J.M. “On generalized symmetric means of two variables”, Publikacije ElektrotehniË‡ckog fakulteta. Serija Matematika i

fizika, 634/677, pp. 236–238 (1979).

[38] Wei, G.W., Wei, C., Wang, J., Gao, H., Wei, Y. “Some q-rung orthopair fuzzy maclaurin symmetric mean operators and their applications to potential

evaluation of emerging technology commercialization”, International Journal of Intelligent Systems, 34 (1), pp. 50-81 (2019).

[39] Yang, W., Pang, Y.F. “New Pythagorean Fuzzy Interaction Maclaurin Symmetric Mean Operators and Their Application in Multiple Attribute

Decision Making”, IEEE ACCESS, 6, pp. 39241-39260 (2018).

[40] Peng, X.D. “Neutrosophic reducible weighted Maclaurin symmetric mean for undergraduate teaching audit and evaluation”, IEEE Access, 7, pp.

18634-18648 (2019).

[41] Zhang, Z.M. “Maclaurin symmetric means of dual hesitant fuzzy information and their use in multi-criteria decision making”, Granular Computing,

https://doi.org/10.1007/s41066-018-00152-4 (2019).

[42] Wang, J.Q., Wu, J.T., Wang, J., Zhang, H.Y., Chen, X.H. “Multi-criteria decision-making methods based on the hausdorff distance of hesitant fuzzy

linguistic numbers”, Soft Computing, 20(4), pp. 1621–1633 (2016).

[43] Qin, J.D., Liu, X.W. “An approach to intuitionistic fuzzy multipleattribute decision making based on Maclaurin symmetric mean operators”, Journal of

Intelligent & Fuzzy Systems, 27(5), pp. 2177–2190 (2014).

[44] Qiao, D., Shen, K.W., Wang, J.Q., Wang, T.L. “Multi-criteria PROMETHEE method based on possibility degree with Z-numbers under uncertain

linguistic environment”, Journal of Ambient Intelligence and Humanized Computing, (2019) 1-15.

[45] Liu, P., Wang, P. “Multiple-Attribute Decision Making based on Archimedean Bonferroni Operators of q-Rung Orthopair Fuzzy Numbers”, IEEE

Transactions on Fuzzy systems, 27(5), pp. 834-848 (2019).

[46] Liu, P. “Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to Group Decision

Making”, IEEE Transactions on Fuzzy systems, 22(1), pp. 83 – 97 (2014).

[47] Liu, P., Chen, S.M., Wang, P. “Multiple-Attribute Group Decision-Making Based on q-Rung Orthopair Fuzzy Power Maclaurin Symmetric Mean

Operators”, IEEE Transactions on Systems, Man, and Cybernetics: Systems, DOI: 10.1109/TSMC.2018.2852948, In Press.(2020).

[48] Liu, P., Chen, S.M., Tang, G. “Multicriteria Decision Making With Incomplete Weights Based on 2-D Uncertain Linguistic Choquet Integral Operators”,

IEEE Transactions on Cybernetics, DOI: 10.1109/TCYB.2019.2913639, In Press. (2020).

Transactions on Industrial Engineering (E)

September and October 2021Pages 2910-2925