Abstract
With respect to multiple attribute group decision making (MAGDM) problems in which the attributes are dependent and the attribute values take the forms of intuitionistic linguistic numbers and intuitionistic uncertain linguistic numbers, this paper investigates two novel MAGDM methods based on Maclaurin symmetric mean (MSM) aggregation operators. First, the Maclaurin symmetric mean is extended to intuitionistic linguistic environment and two new aggregation operators are developed for aggregating the intuitionistic linguistic information, such as the intuitionistic linguistic Maclaurin symmetric mean (ILMSM) operator and the weighted intuitionistic linguistic Maclaurin symmetric mean (WILMSM) operator. Then, some desirable properties and special cases of these operators are discussed in detail. Furthermore, this paper also develops two new Maclaurin symmetric mean operators for aggregating the intuitionistic uncertain linguistic information, including the intuitionistic uncertain linguistic Maclaurin symmetric mean (IULMSM) operator and the weighted intuitionistic uncertain linguistic Maclaurin symmetric mean (WIULMSM) operator. Based on the WILMSM and WIULMSM operators, two approaches to MAGDM are proposed under intuitionistic linguistic environment and intuitionistic uncertain linguistic environment, respectively. Finally, two practical examples of investment alternative evaluation are given to illustrate the applications of the proposed methods.
Access this article
Rent this article via DeepDyve
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96
Detemple D, Robertson J (1979) On generalized symmetric means of two variables. Univ Beograd Publ Elektrotehn Fak Ser Mat Fiz No 634(677):236–238
Dong YC, Xu XF, Yu S (2009) Computing the numerical scale of the linguistic term set for the 2-tuple fuzzy linguistic representation model. IEEE Trans Fuzzy Syst 17:1366–1378
Herrera F, Alonso S, Chiclana F, Herrera-Viedma E (2009) Computing with words in decision making: foundations, trends and prospects. Fuzzy Optim Decis Mak 8(4):337–364
Herrera F, Martínez L (2000a) An approach for combining linguistic and numerical information based on 2-tuple fuzzy representation model in decision making. Int J Uncertain Fuzz Knowl -Based Syst 8(5):539–562
Herrera F, Martínez L (2000b) A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans Fuzzy Syst 8(6):746–752
Ju YB (2014) A new method for multiple criteria group decision making with incomplete weight information under linguistic environment. Appl Math Model 38(21–22):5256–5268
Ju YB, Yang SH (2014) A new method for multiple attribute group decision-making with intuitionistic trapezoid fuzzy linguistic information. Soft Comput. doi:10.1007/s00500-014-1403-9
Li DF (2010) Multiattribute decision making method based on generalized OWA operators with intuitionistic fuzzy sets. Expert Syst Appl 37:8673–8678
Liu PD, Jin F (2012) Methods for aggregating intuitionistic uncertain linguistic variables and their application to group decision making. Inf Sci 205:58–71
Liu PD (2013) Some generalized dependent aggregation operators with intuitionistic linguistic numbers and their application to group decision making. J Comput Syst Sci 79:131–143
Liu PD, Chen YB, Chu YC (2014a) Intuitionistic uncertain linguistic weighted Bonferroni OWA operator and its application to multiple attribute decision making. Cybern Syst 45:418–438
Liu PD, Liu ZM, Zhang X (2014b) Some intuitionistic uncertain linguistic Heronian mean operators and their application to group decision making. Appl Math Comput 230:570–586
Liu PD, Rong LL, Chu YC, Li YW (2014c) Intuitionistic linguistic weighted Bonferroni mean operator and its application to multiple attribute decision making. Sci World J 2014:1–13
Liu XY, Ju YB, Yang SH (2014d) Some generalized interval-valued hesitant uncertain linguistic aggregation operators and their applications to multiple attribute group decision making. Soft Comput. doi:10.1007/s00500-014-1518-z
Liu PD, Wang YM (2014) Multiple attribute group decision making methods based on intuitionistic linguistic power generalized aggregation operators. Appl Soft Comput 17:90–104
Maclaurin C (1729) A second letter to Martin Folkes, Esq concerning the roots of equations, with demonstration of other rules of algebra. Philos Trans Roy Soc London Ser A 36:59–96
Mendel JM (2009) Historical reflections and new positions on perceptual computing. Fuzzy Optim Decis Mak 8(4):325–335
Meng FY, Zhang Q, Cheng H (2013) Approaches to multiple-criteria group decision making based on interval-valued intuitionistic fuzzy Choquet integral with respect to the generalized \(\lambda \)-Shapley index. Knowl-Based Syst 37:237–249
Merigó JM, Gil-Lafuente AM (2013a) Induced 2-tuple linguistic generalized aggregation operators and their application in decision-making. Inf Sci 236:1–16
Merigó JM, Gil-Lafuente AM (2013b) A method for decision making based on generalized aggregation operators. Int J Intel Syst 26:453–473
Pecaric J, Wen JJ, Wang WL, Lu T (2005) A generalization of Maclaurin’s inequalities and its applications. Math Inequal Appl 8:583–598
Qin JD, Liu XW (2014) An approach to intuitionistic fuzzy multiple attribute decision making based on Maclaurin symmetric mean operators. J Intell Fuzzy Syst 27(5):2177–2190
Torra V (2010) Hesitant fuzzy sets. Int J Intel Syst 25:529–539
Wang JQ, Li JJ (2009) The multi-criteria group decision making method based on multi-granularity intuitionistic two semantics. Sci Technol Inf 33:8–9 (In Chinese)
Wei GW, Zhao XF (2012) Some dependent aggregation operators with 2-tuple linguistic information and their application to multiple attribute group decision making. Expert Syst Appl 39:5881–5886
Wei GW, Zhao XF, Lin R, Wang HJ (2013) Uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making. Appl Math Model 37:5277–5285
Xu ZS (2004a) A method based on linguistic aggregation operators for group decision making with linguistic preference relations. Inf Sci 166(1):19–30
Xu ZS (2004b) Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment. Inf Sci 168:171–184
Xu ZS (2006a) A note on linguistic hybrid arithmetic averaging operator in multiple attribute group decision making with linguistic information. Group Decis Negot 15(6):593–604
Xu ZS (2006b) An approach based on the uncertain LOWG and induced uncertain LOWG operators to group decision making with uncertain multiplicative linguistic preference relations. Decis Support Syst 41:488–499
Xu ZS, Zhang XL (2013) Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information. Knowl Based Syst 52:53–64
Yager RR (2004) Generalized OWA aggregation operators. Fuzzy Optim Decis Mak 3:93–107
Yue ZL (2011) An extended TOPSIS for determining weights of decision makers with interval numbers. Knowl Based Syst 24:146–153
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353
Zhang N, Wei GW (2013) Extension of VIKOR method for decision making problem based on hesitant fuzzy set. Appl Math Model 37:4938–4947
Zhang ZH, Wei FJ, Zhou SH (2014) Approaches to comprehensive evaluation with 2-tuple linguistic information. J Intell Fuzzy Syst 28(1):469–475
Zhou LG, Chen HY (2012) A generalization of the power aggregation operators for linguistic environment and its application in group decision making. Knowl Based Syst 26:216–224
Acknowledgments
This research is supported by Program for New Century Excellent Talents in University (NCET-13-0037), Humanities and Social Sciences Foundation of Ministry of Education of China (14YJA630019).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. Loia.
Appendices
Appendix A: Proof of Theorem 2
According to the operational laws in Definition 2, we have
and
then we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs00500-015-1761-y/MediaObjects/500_2015_1761_Equ106_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs00500-015-1761-y/MediaObjects/500_2015_1761_Equ107_HTML.gif)
In addition, since
it follows that
Similarly, we have
which completes the proof of Theorem 2.
Appendix B: Proof of Theorem 3
-
(1)
Since \(a_{i}(i\)=1,2,...,n) are equal, i.e., \(a_i =\langle s_{\theta (a)} ,(\mu (a), v(a))\rangle \) for all i=1,2,...,n, according to Theorem 2, we have
-
(2)
Since \(s_{\theta (a_i )} \le s_{\theta (a_i^{'})} \) for any i, we have \(\theta (a_{i_j } )\le \theta (a_{i_j }^{'} )\). Based on the assumpation condition, for all j=1,2,...,k, we have
$$\begin{aligned} \begin{array}{l} \displaystyle s_{\prod \limits _{j=1}^k {\theta (a_{i_j } )} } \le s_{\prod \limits _{j=1}^k {\theta (a_{i_j }^{'} )} } \\ \displaystyle \Rightarrow s_{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta (a_{i_j } )} } } \le s_{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta (a_{i_j }^{'} )} } } \\ \displaystyle \Rightarrow s_{\frac{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta (a_{i_j } )} } }{C_n^k }} \le s_{\frac{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta (a_{i_j }^{'} )} } }{C_n^k }.} \\ \end{array} \end{aligned}$$Therefore, we have
$$\begin{aligned} s_{\left( {\frac{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta (a_{i_j } )} } }{C_n^k }}\right) ^{\frac{1}{k}}} \le s_{\left( {\frac{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta (a_{i_j }^{'} )} } }{C_n^k }}\right) ^{\frac{1}{k}}}. \end{aligned}$$(47)In addition, since \(\mu (a_i )\le \mu (a_i^{'} )\) and \(v(a_i )\ge v(a_i^{'} )\) for all i, we have
$$\begin{aligned} \begin{array}{l} \prod \limits _{j=1}^k {\mu (a_{i_j } )} \le \prod \limits _{j=1}^k {\mu (a_{i_j }^{'} )} \\ \Rightarrow 1-\prod \limits _{j=1}^k {\mu (a_{i_j } )} \ge 1-\prod \limits _{j=1}^k {\mu (a_{i_j }^{'} )} \\ \Rightarrow \prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu (a_{i_j } )} }\right) } \\ \quad \ge \prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu (a_{i_j }^{'} )} }\right) }. \end{array} \end{aligned}$$Therefore, we can obtain
$$\begin{aligned}&\left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu \left( a_{i_j } \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k} \nonumber \\&\quad \le \left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu \left( a_{i_j }^{'} \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}.\nonumber \\ \end{aligned}$$(48)Similarly, we have
$$\begin{aligned}&\displaystyle 1-\left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\left( 1-v\left( a_{i_j } \right) \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k} \nonumber \\&\quad \displaystyle \quad \ge 1-\left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\left( 1-v\left( a_{i_j }^{'} \right) \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}. \nonumber \\ \end{aligned}$$(49)Therefore, based on Eqs.(48) and (49), we can obtain
$$\begin{aligned} \begin{array}{l} \displaystyle \left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu \left( a_{i_j } \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}+ \\ \displaystyle 1-\left( {1-\left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\left( 1-v\left( a_{i_j } \right) \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}}\right) \le \\ \displaystyle \left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu \left( a_{i_j }^{'} \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}+ \\ \displaystyle 1-\left( {1-\left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\left( 1-v\left( a_{i_j }^{'} \right) \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}}\right) . \end{array} \end{aligned}$$(50)Let \(a\mathrm{=ILMSM(}a_\mathrm{1}, a_\mathrm{2} \mathrm{,}\ldots \mathrm{,}a_n )\) and \(a^{'}=\mathrm{ILMSM(}a_1^{'} \mathrm{,}a_2^{'} \mathrm{,}\ldots \mathrm{,}a_n^{'} )\); based on the Eqs. (47) and (50), using the score function of ILN in Eq. (6), we can easily obtain that \(S(a)\le S(a^{'})\). Therefore, we should discuss the following two cases:
-
(1)
If \(S(a)<S(a^{'})\), according to Theorem 1, we can obtain
$$\begin{aligned} \mathrm{ILMSM}^{(k)}(a_1 ,a_2 ,\ldots ,a_n )<\mathrm{ILMSM}^{(k)}(a_1^{'} ,a_2^{'} ,\ldots ,a_n^{'} ). \end{aligned}$$ -
(2)
If \(S(a)=S(a^{'})\), then
$$\begin{aligned} \begin{array}{l} \displaystyle \left( \left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu \left( a_{i_j } \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}+\right. \displaystyle \left. 1-\left( {1-\left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\left( 1-v\left( a_{i_j } \right) \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}}\right) \right) \,\\ \displaystyle \quad \quad \times \left( {\frac{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta \left( a_{i_j } \right) } } }{C_n^k }}\right) ^{\frac{1}{k}} \\ \displaystyle \quad =\left( \left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu \left( a_{i_j }^{'} \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}\right. \\ \displaystyle +\quad \left. 1-\left( {1-\left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\left( 1-v\left( a_{i_j }^{'} \right) \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}}\right) \right) \displaystyle \times \left( {\frac{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta \left( a_{i_j }^{'} \right) } } }{C_n^k }}\right) ^{\frac{1}{k}}. \end{array}\nonumber \\ \end{aligned}$$(51)Based on the Eqs. (47) and (48), we can get the following inequality:
$$\begin{aligned}&\left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu \left( a_{i_j } \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}\\&\quad =\left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu \left( a_{i_j }^{'} \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k} \end{aligned}$$and
$$\begin{aligned}&\left( {\frac{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta \left( a_{i_j } \right) } } }{C_n^k }}\right) ^{\frac{1}{k}}\\&\quad =\left( {\frac{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta \left( a_{i_j }^{'} \right) } } }{C_n^k }}\right) ^{\frac{1}{k}}, \end{aligned}$$i.e.,
$$\begin{aligned}&\left( {2\times \left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu \left( a_{i_j } \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}+1}\right) \nonumber \\&\quad \quad \times \left( {\frac{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta \left( a_{i_j } \right) } } }{C_n^k }}\right) ^{\frac{1}{k}} \nonumber \\&\quad =\left( {2\times \left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu \left( a_{i_j }^{'} \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}+1}\right) \nonumber \\&\quad \quad \times \left( {\frac{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta \left( a_{i_j }^{'} \right) } } }{C_n^k }}\right) ^{\frac{1}{k}}, \end{aligned}$$(52)and then subtracting Eq. (51) from Eq. (52), we can obtain that
$$\begin{aligned}&\left( \left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu \left( a_{i_j } \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}\right. \\&\quad \left. +\left( {1-\left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\left( 1-v\left( a_{i_j } \right) \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}}\right) \right) \, \\&\quad \times \left( {\frac{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta \left( a_{i_j } \right) } } }{C_n^k }}\right) ^{\frac{1}{k}} \\&\quad =\left( \left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\mu \left( a_{i_j }^{'} \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}\right. \\&\quad \left. +\left( {1-\left( {1-\left( {\prod \limits _{1\le i_1 <\cdots <i_k \le n} {\left( {1-\prod \limits _{j=1}^k {\left( 1-v\left( a_{i_j }^{'} \right) \right) } }\right) } }\right) ^{\frac{1}{C_n^k }}}\right) ^{1/k}}\right) \right) \, \\&\quad \times \left( {\frac{\sum \limits _{1\le i_1 <\cdots <i_k \le n} {\prod \limits _{j=1}^k {\theta \left( a_{i_j }^{'} \right) } } }{C_n^k }}\right) ^{\frac{1}{k}}, \end{aligned}$$that is,
$$\begin{aligned} H(a)=H(a^{'}). \end{aligned}$$Therefore, according to Theorem 1, we can obtain
$$\begin{aligned}&\mathrm{ILMSM}^{(k)}(a_1 ,a_2 ,\ldots ,a_n )\\&\quad \le \mathrm{ILMSM}^{(k)}(a_1^{'} ,a_2^{'} ,\ldots ,a_n^{'} ). \end{aligned}$$ -
(3)
Since \(s_{\theta ^-} =\mathop {\min }\limits _{1\le i\le n} \{s_{\theta (a_i )} \}\),\(s_{\theta ^+} =\mathop {\max }\limits _{1\le i\le n} \{s_{\theta (a_i )} \}\), \(\mu ^-=\mathop {\min }\limits _{1\le i\le n} \{\mu (a_i )\}\),\(\mu ^+=\mathop {\max }\limits _{1\le i\le n} \{\mu (a_i )\}\), \(v^-=\mathop {\min }\limits _{1\le i\le n} \{v(a_i )\}\) and \(v^+=\mathop {\max }\limits _{1\le i\le n} \{v(a_i )\}\), then we have \(s_{\theta ^-} \le s_{\theta (a_i )} \le s_{\theta ^+} , \quad \mu ^-\le \mu (a_i )\le \mu ^+,\) and \(v^-\le v(a_i )\le v^+\) for all i=1,2,...,n. Therefore, according to the monotonicity and idempotency of Theorem 3, we can obtain
$$\begin{aligned}&\langle s_{\theta ^-} ,(\mu ^-,v^+)\rangle \le \mathrm{ILMSM}^{(k)}(a_1 ,a_2 ,...,a_n )\\&\quad \le \langle s_{\theta ^+} ,(\mu ^+,v^-)\rangle . \end{aligned}$$ -
(4)
Since \(a_i^{'} =\langle s_{\theta (a_i^{'} )} ,(\mu (a_i^{'} ),v(a_i^{'} ))\rangle \) is any permutation of \(a_i =\langle s_{\theta (a_i )} ,(\mu (a_i ),v(a_i ))\rangle \) (i=1,2,...,n), then
$$\begin{aligned}&\left( {\frac{\mathop \oplus \limits _{1\le i_1 <...<i_k \le n} (\mathop \otimes \limits _{j=1}^n a_{i_j } )}{C_n^k }}\right) ^{1/k}\\&\quad =\left( {\frac{\mathop \oplus \limits _{1\le i_1 <...<i_k \le n} (\mathop \otimes \limits _{j=1}^n a_{i_j }^{'} )}{C_n^k }}\right) ^{1/k}. \end{aligned}$$Therefore, according to Theorem 2, we can obtain
$$\begin{aligned} \mathrm{ILMSM}^{(k)}(a_1 ,a_2 ,\ldots ,a_n )=\mathrm{ILMSM}^{(k)}(a_1^{'} ,a_2^{'} ,\ldots ,a_n^{'} ). \end{aligned}$$
Appendix C: Proof of Theorem 4
For \(a_i =\langle s_{\theta (a_i )} ,(\mu (a_i ),v(a_i ))\rangle (i\)=1,2,...,n), based on Theorem 2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs00500-015-1761-y/MediaObjects/500_2015_1761_Equ109_HTML.gif)
Let
Based on Lemma 1, we can see that the function \(f(k)_{ }\)is monotonically decreasing with respect to the parameter k. Moreover, based on Lemma 2, we can see that the functions p(k) and q(k) are monotonically decreasing and increasing with respect to the parameter k, respectively.
According to the score function in Eq. (6), we have
For any positive integer \(k\in [1,n-1]\), we can obtain
i.e., \(S(\mathrm{ILMSM}^{(k+1)}(a_1 ,a_2 ,\ldots ,a_n ))\le S(\mathrm{ILMSM}^{(k)} (a_1 ,a_2 ,\ldots ,a_n ))\) for any integer \(k\in [1,n-1]\), which completes the proof of Theorem 4.
Rights and permissions
About this article
Cite this article
Ju, Y., Liu, X. & Ju, D. Some new intuitionistic linguistic aggregation operators based on Maclaurin symmetric mean and their applications to multiple attribute group decision making. Soft Comput 20, 4521–4548 (2016). https://doi.org/10.1007/s00500-015-1761-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-015-1761-y