doi: 10.3934/jimo.2020086

Approach to the consistency and consensus of Pythagorean fuzzy preference relations based on their partial orders in group decision making

1. 

School of Mathematics and Statistics, Linyi University, Linyi, 276005, China

2. 

Business School, Sichuan University, Chengdu, 610064, China

* Corresponding author: Ze Shui Xu and Zhen Ming Ma

Received  October 2019 Revised  January 2020 Published  April 2020

Fund Project: The first author is supported by NSF of Shandong Province grant: ZR2017MG027

Although intuitionistic fuzzy preference relations have become powerful techniques to express the decision makers' preference information over alternatives or criteria in group decision making, some limitations of them are pointed out in this paper, then they are overcame by developed the group decision making with Pythagorean fuzzy preference relations (PFPRs). Specially, we provide a partial order on the set of all the PFPRs, based on which, a deviation measure is defined. Then, we check and reach the acceptably multiplicative consistency and consensus of PFPRs associated with the partial order and mathematical programming. Concretely, acceptably multiplicative consistent PRPRs are defined by the deviation between a given PFPR and a multiplicative consistent PFPR constructed by a normal Pythagorean fuzzy priority vector. Then acceptable consensus of a collection of PFPRs is defined by the deviation of each PFPR and the aggregated result from symmetrical Pythagorean fuzzy aggregation operators. Based on which, a method which can simultaneously modify the unacceptable consistency and consensus of PFPRs in a stepwise way is provided. Particularly, we also prove that the collective PFPR obtained by aggregating several individual acceptably consistent PFPRs with various symmetric aggregation operators is still acceptably consistent. Then, a procedure is provided to solve group decision making with PRPRs and a numerical example is given to illustrate the effectiveness of our method.

Citation: Zhen Ming Ma, Ze Shui Xu, Wei Yang. Approach to the consistency and consensus of Pythagorean fuzzy preference relations based on their partial orders in group decision making. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020086
References:
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E. BarrenecheaJ. FernandezM. PagolaF. Chiclana and H. Bustince, Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations, Application to Decision Making, Knowledge Based Systems, 58 (2014), 33-44.   Google Scholar

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F. J. CabrerizoJ. M. MorenoI. J. Perez and E. Herrera-Viedma, Analyzing consensus approaches in fuzzy group decision making: Advantages and drawbacks, Soft Computing, 14 (2010), 451-463.  doi: 10.1007/s00500-009-0453-x.  Google Scholar

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F. ChiclanaE. Herrera-ViedmaF. Herrera and S. Alonso, Induced ordered weighted geometric operators and their use in the aggregation of multiplicative preference relations, International Journal of Intelligent Systems, 19 (2004), 233-255.  doi: 10.1002/int.10172.  Google Scholar

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Z. M. Ma and Z. S. Xu, Symmetric Pythagorean fuzzy weighted geometric/averaging operators and their application in multicriteria decision-making problems, International Journal of Intelligent Systems, 31 (2016), 1198-1219.  doi: 10.1002/int.21823.  Google Scholar

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[23]

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X. Peng and Y. Yang, Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators, International Journal of Intelligent Systems, 31 (2016), 444-487.   Google Scholar

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T. Tanino, Fuzzy preference orderings in group decision making, Fuzzy Sets and Systems, 12 (1984), 117-131.  doi: 10.1016/0165-0114(84)90032-0.  Google Scholar

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show all references

References:
[1]

K. Atanassov, Intuitionistic fuzzy set, Fuzzy Sets and Systems, 20 (1986), 87-96.  doi: 10.1016/S0165-0114(86)80034-3.  Google Scholar

[2]

E. BarrenecheaJ. FernandezM. PagolaF. Chiclana and H. Bustince, Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations, Application to Decision Making, Knowledge Based Systems, 58 (2014), 33-44.   Google Scholar

[3]

H. Behret, Group decision making with intuitionistic fuzzy preference relations, Knowledge-Based Systems, 70 (2014), 33-43.  doi: 10.1016/j.knosys.2014.04.001.  Google Scholar

[4]

J. ChuX. LiuY. Wang and K. Chin, A group decision making model considering both the additive consistency and group consensus of intuitionistic fuzzy preference relations, Computers and Industrial Engineering, 101 (2016), 227-242.   Google Scholar

[5]

F. J. CabrerizoR. UrenaW. Pedrycz and E. Herrera-Viedma, Building consensus in group decision making with an allocation of information granularity, Fuzzy Sets and Systems, 255 (2014), 115-127.  doi: 10.1016/j.fss.2014.03.016.  Google Scholar

[6]

F. J. CabrerizoJ. M. MorenoI. J. Perez and E. Herrera-Viedma, Analyzing consensus approaches in fuzzy group decision making: Advantages and drawbacks, Soft Computing, 14 (2010), 451-463.  doi: 10.1007/s00500-009-0453-x.  Google Scholar

[7]

F. ChiclanaF. Herrera and E. Herrera-Viedma, Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations, Fuzzy Sets and Systems, 97 (1998), 33-48.  doi: 10.1016/S0165-0114(96)00339-9.  Google Scholar

[8]

F. ChiclanaE. Herrera-ViedmaF. Herrera and S. Alonso, Induced ordered weighted geometric operators and their use in the aggregation of multiplicative preference relations, International Journal of Intelligent Systems, 19 (2004), 233-255.  doi: 10.1002/int.10172.  Google Scholar

[9]

Y. DongX. Chen and F. Herrera, Minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group decision making, Information Sciences, 297 (2015), 95-117.  doi: 10.1016/j.ins.2014.11.011.  Google Scholar

[10]

Y. DongJ. XiaoH. Zhang and T. Wang, Managing consensus and weights in iterative multiple-attribute group decision making, Applied Soft Computing, 48 (2016), 80-90.  doi: 10.1016/j.asoc.2016.06.029.  Google Scholar

[11]

Z. W. GongL. S. LiJ. Forrest and Y. Zhao, The optimal priority models of the intuitionistic fuzzy preference relation and their application in selecting industries with higher meteorological sensitivity, Expert Systems with Applications, 38 (2011), 4394-4402.  doi: 10.1016/j.eswa.2010.09.109.  Google Scholar

[12]

H. Garg, New Logarithmic operational laws and their aggregation operators for Pythagorean fuzzy set and their applications, International Journal of Intelligent Systems, 34 (2019), 82-106.  doi: 10.1002/int.22043.  Google Scholar

[13]

E. Herrera-ViedmaF. HerreraF. Chiclana and M. Luque, Some issues on consistency of fuzzy preference relations, European Journal of Operational Research, 154 (2004), 98-109.  doi: 10.1016/S0377-2217(02)00725-7.  Google Scholar

[14]

F. JinZ. NiH. Chen and Y. Li, Approaches to group decision making with intuitionistic fuzzy preference relations based on multiplicative consistency, Knowledge-Based Systems, 97 (2016), 48-59.  doi: 10.1016/j.knosys.2016.01.017.  Google Scholar

[15]

J. B. LanM. M. HuX. M. Ye and S. Q. Sun, Deriving interval weights from an interval multiplicative consistent fuzzy preference relation, Knowledge-Based Systems, 26 (2012), 128-134.  doi: 10.1016/j.knosys.2011.07.014.  Google Scholar

[16]

H. C. Liao and Z. S. Xu, Consistency and consensus of intuitionistic fuzzy preference relations in group decision making, imprecision and uncertainty in information representation and processing, Studies in Fuzziness and Soft Computing, 332 (2016), 189-206.  doi: 10.1007/978-3-319-26302-1_13.  Google Scholar

[17]

H. C. Liao and Z. S. Xu, Priorities of intuitionistic fuzzy preference relation based on multiplicative consistency, IEEE Transactions on Fuzzy Systems, 22 (2014), 1669-1681.   Google Scholar

[18]

H. C. LiaoZ. S. XuX. J. Zeng and J. M. Merigo, Framework of group decision making with intuitionistic fuzzy preference information, IEEE Transactions on Fuzzy Systems, 23 (2014), 1211-1227.   Google Scholar

[19]

H. C. Liao and Z. S. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process, Applied Soft Computing, 35 (2015), 812-826.  doi: 10.1016/j.asoc.2015.04.015.  Google Scholar

[20]

H. C. LiaoZ. S. Xu and X. J. Zeng, An enhanced consensus reaching process in group decision making with intuitionistic fuzzy preference relations, Information Sciences, 329 (2016), 274-286.   Google Scholar

[21]

Z. M. Ma and Z. S. Xu, Symmetric Pythagorean fuzzy weighted geometric/averaging operators and their application in multicriteria decision-making problems, International Journal of Intelligent Systems, 31 (2016), 1198-1219.  doi: 10.1002/int.21823.  Google Scholar

[22]

Z. M. Ma and Z. S. Xu, Hyperbolic scales involving appetites-based intuitionistic multiplicative preference relations for group decision making, Information Sciences, 451/452 (2018), 310-325.  doi: 10.1016/j.ins.2018.04.040.  Google Scholar

[23]

F. Mata, L. G. Perez, S. M. Zhou and F. Chiclana, Type-1 OWA methodology to consensus reaching processes in multi-granular linguistic contexts,Knowledge-Based Systems, 58 (2014) 11–22. doi: 10.1016/j.knosys.2013.09.017.  Google Scholar

[24]

S. A. Orlovsky, Decision-making with a fuzzy preference relation, Fuzzy Sets and Systems, 1 (1978), 155-167.  doi: 10.1016/0165-0114(78)90001-5.  Google Scholar

[25]

X. Peng and Y. Yang, Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators, International Journal of Intelligent Systems, 31 (2016), 444-487.   Google Scholar

[26]

X. Peng and G. Selvachandran, Pythagorean fuzzy set: State of the art and future directions, Artificial Intelligence Review, 52 (2019), 1873-1927.  doi: 10.1007/s10462-017-9596-9.  Google Scholar

[27]

X. Peng, New operations for interval-valued Pythagorean fuzzy set, Scientia Iranica, 26 (2019), 1049-1076.  doi: 10.24200/sci.2018.5142.1119.  Google Scholar

[28]

T. L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, 1980.  Google Scholar

[29]

E. Szmidt and J. Kacprzyk, A consensus-reaching process under intuitionistic fuzzy preference relations, International Journal of Intelligent Systems, 18 (2003), 837-852.  doi: 10.1002/int.10119.  Google Scholar

[30]

E. Szmidt and J. Kacprzyk, A new concept of a similarity measure for intuitionistic fuzzy sets and its use in group decision making, Lecture Notes in Computer Science, 3558 (2005), 272-282.   Google Scholar

[31]

T. Tanino, Fuzzy preference orderings in group decision making, Fuzzy Sets and Systems, 12 (1984), 117-131.  doi: 10.1016/0165-0114(84)90032-0.  Google Scholar

[32]

R. UrenaF. ChiclanaJ. A. Morente-Molinera and E. Herrera-Viedma, Managing incomplete preference relations in decision making: A review and future trends, Information Sciences, 302 (2015), 14-32.  doi: 10.1016/j.ins.2014.12.061.  Google Scholar

[33]

Z. S. Xu, A method based on linguistic aggregation operators for group decision making with linguistic preference relations, Information Sciences, 166 (2004), 19-30.  doi: 10.1016/j.ins.2003.10.006.  Google Scholar

[34]

Z. S. Xu, On compatibility of interval fuzzy preference relations, Fuzzy Optimization and Decision Making, 3 (2004), 217-225.  doi: 10.1023/B:FODM.0000036864.33950.1b.  Google Scholar

[35]

Z. S. Xu, Deviation measures of linguistic preference relations in group decision making, Omega, 33 (2005), 249-254.   Google Scholar

[36]

Z. S. Xu, Incomplete linguistic preference relations and their fusion, Information Fusion, 7 (2006), 331-337.  doi: 10.1016/j.inffus.2005.01.003.  Google Scholar

[37]

Z. S. Xu, Intuitionistic preference relations and their application in group decision making, Information Sciences, 177 (2007), 2363-2379.  doi: 10.1016/j.ins.2006.12.019.  Google Scholar

[38]

Z. S. XuX. Q. Cai and E. Szmidt, Algorithms for estimating missing elements of incomplete intuitionistic preference relations, International Journal of Intelligent Systems, 26 (2011), 787-813.  doi: 10.1002/int.20494.  Google Scholar

[39]

Z. S. Xu and H. Liao, Intuitionistic fuzzy analytic hierarchy process, IEEE Transactions on Fuzzy Systems, 22 (2014), 749-761.   Google Scholar

[40]

Z. S. Xu and H. Liao, A survey of approaches to decision making with intuitionistic fuzzy preference relations, Knowledge-Based Systems, 80 (2015), 131-142.  doi: 10.1016/j.knosys.2014.12.034.  Google Scholar

[41]

G. XuS. WanF. WangJ. Dong and Y. Zeng, Mathematical programming methods for consistency and consensus in group decision making with intuitionistic fuzzy preference relations, Knowledge-Based Systems, 98 (2016), 30-43.  doi: 10.1016/j.knosys.2015.12.007.  Google Scholar

[42]

Y. J. XuK. W. Li and H. M. Wang, Distance-based consensus models for fuzzy and multiplicative preference relations, Information Sciences, 253 (2013), 56-73.  doi: 10.1016/j.ins.2013.08.029.  Google Scholar

[43]

Z. J. Wang, Derivation of intuitionistic fuzzy weights based on intuitionistic fuzzy preference relations, Applied Mathematical Modelling, 37 (2013), 6377-6388.  doi: 10.1016/j.apm.2013.01.021.  Google Scholar

[44]

S. P. WanF. WangL. L. Lin and J. Y. Dong, An intuitionistic fuzzy linear programming method for logistics outsourcing provider selection, Knowledge-Based Systems, 82 (2015), 80-94.  doi: 10.1016/j.knosys.2015.02.027.  Google Scholar

[45]

S. P. WanF. Wang and J. Dong, A preference degree for intuitionistic fuzzy values and application to multi-attribute group decision making, Information Sciences, 370/371 (2016), 127-146.   Google Scholar

[46]

S. P. WanZ. Jin and J. Y. Dong, Pythagorean fuzzy mathematical programming method for multi-attribute group decision making with Pythagorean fuzzy truth degrees, Knowledge and Information Systems, 55 (2018), 437-466.  doi: 10.1007/s10115-017-1085-6.  Google Scholar

[47]

S. P. WanZ. Jin and J. Y. Dong, A three-phase method for Pythagorean fuzzy multi-attribute group decision making and application to haze management, Computers & Industrial Engineering, 123 (2018), 348-363.   Google Scholar

[48]

S. P. WanZ. Jin and J. Y. Dong, A new order relation for Pythagorean fuzzy numbers and application to multi-attribute group decision making, Knowledge and Information Systems, 62 (2020), 751-785.  doi: 10.1007/s10115-019-01369-8.  Google Scholar

[49]

J. Wu and F. Chiclana, Multiplicative consistency of intuitionistic reciprocal preference relations and its application to missing values estimation and consensus building, Knowledge-Based Systems, 71 (2014), 187-200.  doi: 10.1016/j.knosys.2014.07.024.  Google Scholar

[50]

S. P. WanG. Xu and J. Dong, A novel method for group decision making with interval-valued Atanassov intuitionistic fuzzy preference relations, Information Sciences, 372 (2016), 53-71.   Google Scholar

[51]

R. R. Yager and A. M. Abbasov, Pythagorean membership grades, complex numbers, and decision making, International Journal of Intelligent Systems, 28 (2013), 436-452.  doi: 10.1002/int.21584.  Google Scholar

[52]

R. R. Yager, Pythagorean membership grades in multicriteria decision making, IEEE Transactions on Fuzzy Systems, 22 (2014), 958-965.  doi: 10.1109/TFUZZ.2013.2278989.  Google Scholar

[53]

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Table 1.  The modified PFPRs $ {P}^{(h)} $ and their corresponding multiplicatively consistent PFPRs $ \widetilde{P}^{(h)}, h = 0, 1, 2 $
$ {P}^{(0)}$ $(0.7071, 0.7071)$ $(0.469, 0.8216)$ $(0.7969, 0.3674)$ $(0.9301, 0.2646)$
$(0.8216, 0.469)$ $(0.7071, 0.7071)$ $(0.8689, 0.2121)$ $(0.9618, 0.1449)$
$(0.3674, 0.7969)$ $(0.2121, 0.8689)$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$
$(0.2646, 0.9301)$ $(0.1449, 0.9618)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$
$ {P}^{(1)}$ $(0.7071, 0.7071)$ $(0.469, 0.8216)$ $\underline{(0.8559, 0.3462)}$ $(0.9301, 0.2646)$
$(0.8216, 0.469)$ $(0.7071, 0.7071)$ $(0.8689, 0.2121)$ $(0.9618, 0.1449)$
$\underline{(0.3462, 0.8559)}$ $(0.2121, 0.8689)$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$
$(0.2646, 0.9301)$ $(0.1449, 0.9618)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$
$ {P}^{(2)}$ $(0.7071, 0.7071)$ $(0.469, 0.8216)$ $(0.8559, 0.3462)$ $(0.9301, 0.2646)$
$(0.8216, 0.469)$ $(0.7071, 0.7071)$ $\underline{(0.9178, 0.2121)}$ $(0.9618, 0.1449)$
$(0.3462, 0.8559)$ $\underline{(0.2121, 0.9178)}$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$
$(0.2646, 0.9301)$ $(0.1449, 0.9618)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$
$\widetilde{P}^{(2)}$ $(0.7071, 0.7071)$ $(0.469, 0.8209)$ $(0.856, 0.3462)$ $(0.9451, 0.1952)$
$(0.8209, 0.469)$ $(0.7071, 0.7071)$ $(0.9179, 0.2121)$ $(0.9627, 0.1136)$
$(0.3462, 0.856)$ $(0.2121, 0.9179)$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$
$(0.1952, 0.9451)$ $(0.1136, 0.9627)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$
$\widetilde{P}^{(1)}$ $(0.7071, 0.7071)$ $(0.4691, 0.8209)$ $(0.8561, 0.3462)$ $(0.9452, 0.1952)$
$(0.8209, 0.4691)$ $(0.7071, 0.7071)$ $(0.9179, 0.2121)$ $(0.9627, 0.1136)$
$(0.3462, 0.8561)$ $(0.2121, 0.9179)$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$
$(0.1952, 0.9452)$ $(0.1136, 0.9627)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$
$\widetilde{P}^{(0)}$ $(0.7071, 0.7071)$ $(0.47, 0.8206)$ $(0.8587, 0.3443)$ $(0.9452, 0.1922)$
$(0.8206, 0.47)$ $(0.7071, 0.7071)$ $(0.9192, 0.2111)$ $(0.9628, 0.1122)$
$(0.3443, 0.8587)$ $(0.2111, 0.9192)$ $(0.7071, 0.7071)$ $(0.7593, 0.3851)$
$(0.1922, 0.9452)$ $(0.1122, 0.9628)$ $(0.3851, 0.7593)$ $(0.7071, 0.7071)$
$ {P}^{(0)}$ $(0.7071, 0.7071)$ $(0.469, 0.8216)$ $(0.7969, 0.3674)$ $(0.9301, 0.2646)$
$(0.8216, 0.469)$ $(0.7071, 0.7071)$ $(0.8689, 0.2121)$ $(0.9618, 0.1449)$
$(0.3674, 0.7969)$ $(0.2121, 0.8689)$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$
$(0.2646, 0.9301)$ $(0.1449, 0.9618)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$
$ {P}^{(1)}$ $(0.7071, 0.7071)$ $(0.469, 0.8216)$ $\underline{(0.8559, 0.3462)}$ $(0.9301, 0.2646)$
$(0.8216, 0.469)$ $(0.7071, 0.7071)$ $(0.8689, 0.2121)$ $(0.9618, 0.1449)$
$\underline{(0.3462, 0.8559)}$ $(0.2121, 0.8689)$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$
$(0.2646, 0.9301)$ $(0.1449, 0.9618)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$
$ {P}^{(2)}$ $(0.7071, 0.7071)$ $(0.469, 0.8216)$ $(0.8559, 0.3462)$ $(0.9301, 0.2646)$
$(0.8216, 0.469)$ $(0.7071, 0.7071)$ $\underline{(0.9178, 0.2121)}$ $(0.9618, 0.1449)$
$(0.3462, 0.8559)$ $\underline{(0.2121, 0.9178)}$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$
$(0.2646, 0.9301)$ $(0.1449, 0.9618)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$
$\widetilde{P}^{(2)}$ $(0.7071, 0.7071)$ $(0.469, 0.8209)$ $(0.856, 0.3462)$ $(0.9451, 0.1952)$
$(0.8209, 0.469)$ $(0.7071, 0.7071)$ $(0.9179, 0.2121)$ $(0.9627, 0.1136)$
$(0.3462, 0.856)$ $(0.2121, 0.9179)$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$
$(0.1952, 0.9451)$ $(0.1136, 0.9627)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$
$\widetilde{P}^{(1)}$ $(0.7071, 0.7071)$ $(0.4691, 0.8209)$ $(0.8561, 0.3462)$ $(0.9452, 0.1952)$
$(0.8209, 0.4691)$ $(0.7071, 0.7071)$ $(0.9179, 0.2121)$ $(0.9627, 0.1136)$
$(0.3462, 0.8561)$ $(0.2121, 0.9179)$ $(0.7071, 0.7071)$ $(0.7583, 0.3873)$
$(0.1952, 0.9452)$ $(0.1136, 0.9627)$ $(0.3873, 0.7583)$ $(0.7071, 0.7071)$
$\widetilde{P}^{(0)}$ $(0.7071, 0.7071)$ $(0.47, 0.8206)$ $(0.8587, 0.3443)$ $(0.9452, 0.1922)$
$(0.8206, 0.47)$ $(0.7071, 0.7071)$ $(0.9192, 0.2111)$ $(0.9628, 0.1122)$
$(0.3443, 0.8587)$ $(0.2111, 0.9192)$ $(0.7071, 0.7071)$ $(0.7593, 0.3851)$
$(0.1922, 0.9452)$ $(0.1122, 0.9628)$ $(0.3851, 0.7593)$ $(0.7071, 0.7071)$
Table 2.  Individual preference information from three decision makers
$P^{(1)}$ $(0.7071, 0.7071)$ $(0.9487, 0.1)$ $(0.8367, 0.3162)$
$(0.1, 0.9487)$ $(0.7071, 0.7071)$ $(0.7746, 0.4472)$
$(0.3162, 0.8367)$ $(0.4472, 0.7746)$ $(0.7071, 0.7071)$
$P^{(2)}$ $(0.7071, 0.7071)$ $(0.7746, 0.3162)$ $(0.8367, 0.4472)$
$(0.3162, 0.7746)$ $(0.7071, 0.7071)$ $(0.7071, 0.3162)$
$(0.4472, 0.8367)$ $(0.3162, 0.7071)$ $(0.7071, 0.7071)$
$P^{(3)}$ $(0.7071, 0.7071)$ $(0.7746, 0.4472)$ $(0.8944, 0.3162)$
$(0.4472, 0.7746)$ $(0.7071, 0.7071)$ $(0.7746, 0.3162)$
$(0.3162, 0.8944)$ $(0.3162, 0.7746)$ $(0.7071, 0.7071)$
$P^{(1)}$ $(0.7071, 0.7071)$ $(0.9487, 0.1)$ $(0.8367, 0.3162)$
$(0.1, 0.9487)$ $(0.7071, 0.7071)$ $(0.7746, 0.4472)$
$(0.3162, 0.8367)$ $(0.4472, 0.7746)$ $(0.7071, 0.7071)$
$P^{(2)}$ $(0.7071, 0.7071)$ $(0.7746, 0.3162)$ $(0.8367, 0.4472)$
$(0.3162, 0.7746)$ $(0.7071, 0.7071)$ $(0.7071, 0.3162)$
$(0.4472, 0.8367)$ $(0.3162, 0.7071)$ $(0.7071, 0.7071)$
$P^{(3)}$ $(0.7071, 0.7071)$ $(0.7746, 0.4472)$ $(0.8944, 0.3162)$
$(0.4472, 0.7746)$ $(0.7071, 0.7071)$ $(0.7746, 0.3162)$
$(0.3162, 0.8944)$ $(0.3162, 0.7746)$ $(0.7071, 0.7071)$
Table 3.  Iterative process of the proposed method for the acceptable concensus
$h$ $(D(P^{(1h)}, P_{Agg}^{(h)}), D(P^{(2h)}, P_{Agg}^{(h)}), D(P^{(3h)}, P_{Agg}^{(h)}))$ $l_0$ $i_0, j_0$
0 $(0.3589, 0.2902, 0.2578)$ 1 $(1, 2)$
1 $(0.271, 0.2774, 0.2244)$ 1 $(2, 3)$
2 $(0.2205, 0.2606, 0.2291)$ 2 $(1, 3)$
3 $(0.2175, 0.2288, 0.2203)$ 2 $(2, 3)$
4 $(0.2037, 0.1762, 0.1986)$ 2 $(1, 2)$
5 $(0.1943, 0.1377, 0.1875)$ 1 $(1, 3)$
6 $(0.1493, 0.1540, 0.1598)$ 3 $(1, 2)$
7 $(0.1188, 0.1355, 0.1076)$ 2 $(1, 3)$
8 $(0.1017, 0.0787, 0.0814)$ 1 $(1, 2)$
9 $(0.0780, 0.0628, 0.0757)$
$h$ $(D(P^{(1h)}, P_{Agg}^{(h)}), D(P^{(2h)}, P_{Agg}^{(h)}), D(P^{(3h)}, P_{Agg}^{(h)}))$ $l_0$ $i_0, j_0$
0 $(0.3589, 0.2902, 0.2578)$ 1 $(1, 2)$
1 $(0.271, 0.2774, 0.2244)$ 1 $(2, 3)$
2 $(0.2205, 0.2606, 0.2291)$ 2 $(1, 3)$
3 $(0.2175, 0.2288, 0.2203)$ 2 $(2, 3)$
4 $(0.2037, 0.1762, 0.1986)$ 2 $(1, 2)$
5 $(0.1943, 0.1377, 0.1875)$ 1 $(1, 3)$
6 $(0.1493, 0.1540, 0.1598)$ 3 $(1, 2)$
7 $(0.1188, 0.1355, 0.1076)$ 2 $(1, 3)$
8 $(0.1017, 0.0787, 0.0814)$ 1 $(1, 2)$
9 $(0.0780, 0.0628, 0.0757)$
Table 4.  Individual preference information from three decision makers
$P^{(1)}$ $(0.707, 0.707)$ $(0.707, 0.447)$ $(0.837, 0.316)$ $(0.707, 0.548)$
$(0.447, 0.707)$ $(0.707, 0.707)$ $(0.775, 0.447)$ $(0.548, 0.775)$
$(0.316, 0.837)$ $(0.447, 0.775)$ $(0.707, 0.707)$ $(0.548, 0.775)$
$(0.548, 0.707)$ $(0.775, 0.548)$ $(0.775, 0.548)$ $(0.707, 0.707)$
$P^{(2)}$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.894, 0.447)$ $(0.775, 0.548)$
$(0.316, 0.775)$ $(0.707, 0.707)$ $(0.707, 0.316)$ $(0.548, 0.837)$
$(0.447, 0.894)$ $(0.316, 0.707)$ $(0.707, 0.707)$ $(0.632, 0.775)$
$(0.548, 0.775)$ $(0.837, 0.548)$ $(0.775, 0.632)$ $(0.707, 0.707)$
$P^{(3)}$ $(0.707, 0.707)$ $(0.775, 0.447)$ $(0.894, 0.316)$ $(0.837, 0.447)$
$(0.447, 0.775)$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.447, 0.837)$
$(0.316, 0.894)$ $(0.316, 0.775)$ $(0.707, 0.707)$ $(0.447, 0.548)$
$(0.447, 0.837)$ $(0.837, 0.447)$ $(0.548, 0.447)$ $(0.707, 0.707)$
$P^{(1)}$ $(0.707, 0.707)$ $(0.707, 0.447)$ $(0.837, 0.316)$ $(0.707, 0.548)$
$(0.447, 0.707)$ $(0.707, 0.707)$ $(0.775, 0.447)$ $(0.548, 0.775)$
$(0.316, 0.837)$ $(0.447, 0.775)$ $(0.707, 0.707)$ $(0.548, 0.775)$
$(0.548, 0.707)$ $(0.775, 0.548)$ $(0.775, 0.548)$ $(0.707, 0.707)$
$P^{(2)}$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.894, 0.447)$ $(0.775, 0.548)$
$(0.316, 0.775)$ $(0.707, 0.707)$ $(0.707, 0.316)$ $(0.548, 0.837)$
$(0.447, 0.894)$ $(0.316, 0.707)$ $(0.707, 0.707)$ $(0.632, 0.775)$
$(0.548, 0.775)$ $(0.837, 0.548)$ $(0.775, 0.632)$ $(0.707, 0.707)$
$P^{(3)}$ $(0.707, 0.707)$ $(0.775, 0.447)$ $(0.894, 0.316)$ $(0.837, 0.447)$
$(0.447, 0.775)$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.447, 0.837)$
$(0.316, 0.894)$ $(0.316, 0.775)$ $(0.707, 0.707)$ $(0.447, 0.548)$
$(0.447, 0.837)$ $(0.837, 0.447)$ $(0.548, 0.447)$ $(0.707, 0.707)$
Table 5.  Modified individual preference information from three decision makers
$P^{(1, 44, 0)}$ $(0.707, 0.707)$ $(0.767, 0.341)$ $(0.89, 0.149)$ $(0.88, 0.136)$
$(0.341, 0.767)$ $(0.707, 0.707)$ $(0.769, 0.316)$ $(0.748, 0.271)$
$(0.149, 0.89)$ $(0.316, 0.769)$ $(0.707, 0.707)$ $(0.626, 0.549)$
$(0.136, 0.88)$ $(0.271, 0.748)$ $(0.549, 0.626)$ $(0.707, 0.707)$
$P^{(2, 44, 0)}$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.894, 0.154)$ $(0.883, 0.128)$
$(0.316, 0.775)$ $(0.707, 0.707)$ $(0.772, 0.322)$ $(0.76, 0.271)$
$(0.154, 0.894)$ $(0.322, 0.772)$ $(0.707, 0.707)$ $(0.624, 0.549)$
$(0.128, 0.883)$ $(0.271, 0.76)$ $(0.549, 0.624)$ $(0.707, 0.707)$
$P^{(3, 44, 0)}$ $(0.707, 0.707)$ $(0.764, 0.327)$ $(0.894, 0.155)$ $(0.874, 0.13)$
$(0.327, 0.764)$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.756, 0.274)$
$(0.155, 0.894)$ $(0.316, 0.775)$ $(0.707, 0.707)$ $(0.628, 0.548)$
$(0.13, 0.874)$ $(0.274, 0.756)$ $(0.548, 0.628)$ $(0.707, 0.707)$
$\widetilde{N}^{(44)}$ $(0.707, 0.707)$ $(0.75, 0.316)$ $(0.894, 0.149)$ $(0.883, 0.128)$
$(0.316, 0.775)$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.76, 0.271)$
$(0.149, 0.894)$ $(0.316, 0.775)$ $(0.707, 0.707)$ $(0.628, 0.548)$
$(0.128, 0.883)$ $(0.271, 0.76)$ $(0.548, 0.628)$ $(0.707, 0.707)$
$P^{(44, 0)}$ $(0.707, 0.707)$ $(0.769, 0.328)$ $(0.893, 0.153)$ $(0.879, 0.131)$
$(0.328, 0.769)$ $(0.707, 0.707)$ $(0.772, 0.318)$ $(0.755, 0.272)$
$(0.153, 0.893)$ $(0.318, 0.772)$ $(0.707, 0.707)$ $(0.626, 0.549)$
$(0.131, 0.879)$ $(0.272, 0.755)$ $(0.549, 0.626)$ $(0.707, 0.707)$
$P^{(1, 44, 0)}$ $(0.707, 0.707)$ $(0.767, 0.341)$ $(0.89, 0.149)$ $(0.88, 0.136)$
$(0.341, 0.767)$ $(0.707, 0.707)$ $(0.769, 0.316)$ $(0.748, 0.271)$
$(0.149, 0.89)$ $(0.316, 0.769)$ $(0.707, 0.707)$ $(0.626, 0.549)$
$(0.136, 0.88)$ $(0.271, 0.748)$ $(0.549, 0.626)$ $(0.707, 0.707)$
$P^{(2, 44, 0)}$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.894, 0.154)$ $(0.883, 0.128)$
$(0.316, 0.775)$ $(0.707, 0.707)$ $(0.772, 0.322)$ $(0.76, 0.271)$
$(0.154, 0.894)$ $(0.322, 0.772)$ $(0.707, 0.707)$ $(0.624, 0.549)$
$(0.128, 0.883)$ $(0.271, 0.76)$ $(0.549, 0.624)$ $(0.707, 0.707)$
$P^{(3, 44, 0)}$ $(0.707, 0.707)$ $(0.764, 0.327)$ $(0.894, 0.155)$ $(0.874, 0.13)$
$(0.327, 0.764)$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.756, 0.274)$
$(0.155, 0.894)$ $(0.316, 0.775)$ $(0.707, 0.707)$ $(0.628, 0.548)$
$(0.13, 0.874)$ $(0.274, 0.756)$ $(0.548, 0.628)$ $(0.707, 0.707)$
$\widetilde{N}^{(44)}$ $(0.707, 0.707)$ $(0.75, 0.316)$ $(0.894, 0.149)$ $(0.883, 0.128)$
$(0.316, 0.775)$ $(0.707, 0.707)$ $(0.775, 0.316)$ $(0.76, 0.271)$
$(0.149, 0.894)$ $(0.316, 0.775)$ $(0.707, 0.707)$ $(0.628, 0.548)$
$(0.128, 0.883)$ $(0.271, 0.76)$ $(0.548, 0.628)$ $(0.707, 0.707)$
$P^{(44, 0)}$ $(0.707, 0.707)$ $(0.769, 0.328)$ $(0.893, 0.153)$ $(0.879, 0.131)$
$(0.328, 0.769)$ $(0.707, 0.707)$ $(0.772, 0.318)$ $(0.755, 0.272)$
$(0.153, 0.893)$ $(0.318, 0.772)$ $(0.707, 0.707)$ $(0.626, 0.549)$
$(0.131, 0.879)$ $(0.272, 0.755)$ $(0.549, 0.626)$ $(0.707, 0.707)$
Table 6.  The modified individual preference information from three decision makers by Xu's method [41]
$\widetilde{P}^{(1)}$ $(0.707, 0.707)$ $(0.71, 0.445)$ $(0.01, 0.056)$ $(0.011, 0.553)$
$(0.445, 0.71)$ $(0.707, 0.707)$ $(0.047, 0.013)$ $(0.512, 0.802)$
$(0.056, 0.01)$ $(0.013, 0.047)$ $(0.707, 0.707)$ $(0.529, 0.619)$
$(0.553, 0.011)$ $(0.802, 0.512)$ $(0.619, 0.529)$ $(0.707, 0.707)$
$\widetilde{P}^{(2)}$ $(0.707, 0.707)$ $(0.71, 0.445)$ $(0.01, 0.056)$ $(0.011, 0.553)$
$(0.445, 0.71)$ $(0.707, 0.707)$ $(0.047, 0.014)$ $(0.512, 0.802)$
$(0.056, 0.01)$ $(0.014, 0.047)$ $(0.707, 0.707)$ $(0.529, 0.619)$
$(0.553, 0.011)$ $(0.802, 0.512)$ $(0.619, 0.529)$ $(0.707, 0.707)$
$\widetilde{P}^{(3)}$ $(0.707, 0.707)$ $(0.728, 0.447)$ $(0.011, 0.132)$ $(0.011, 0.553)$
$(0.447, 0.728)$ $(0.707, 0.707)$ $(0.046, 0.03)$ $(0.509, 0.804)$
$(0.132, 0.011)$ $(0.03, 0.046)$ $(0.707, 0.707)$ $(0.5, 0.573)$
$(0.553, 0.011)$ $(0.804, 0.509)$ $(0.573, 0.5)$ $(0.707, 0.707)$
$\widetilde{P}$ $(0.707, 0.707)$ $(0.716, 0.446)$ $(0.01, 0.074)$ $(0.011, 0.553)$
$(0.446, 0.716)$ $(0.707, 0.707)$ $(0.047, 0.018)$ $(0.511, 0.803)$
$(0.074, 0.01)$ $(0.018, 0.047)$ $(0.707, 0.707)$ $(0.519, 0.603)$
$(0.553, 0.011)$ $(0.803, 0.511)$ $(0.603, 0.519)$ $(0.707, 0.707)$
$\widetilde{P}^{(1)}$ $(0.707, 0.707)$ $(0.71, 0.445)$ $(0.01, 0.056)$ $(0.011, 0.553)$
$(0.445, 0.71)$ $(0.707, 0.707)$ $(0.047, 0.013)$ $(0.512, 0.802)$
$(0.056, 0.01)$ $(0.013, 0.047)$ $(0.707, 0.707)$ $(0.529, 0.619)$
$(0.553, 0.011)$ $(0.802, 0.512)$ $(0.619, 0.529)$ $(0.707, 0.707)$
$\widetilde{P}^{(2)}$ $(0.707, 0.707)$ $(0.71, 0.445)$ $(0.01, 0.056)$ $(0.011, 0.553)$
$(0.445, 0.71)$ $(0.707, 0.707)$ $(0.047, 0.014)$ $(0.512, 0.802)$
$(0.056, 0.01)$ $(0.014, 0.047)$ $(0.707, 0.707)$ $(0.529, 0.619)$
$(0.553, 0.011)$ $(0.802, 0.512)$ $(0.619, 0.529)$ $(0.707, 0.707)$
$\widetilde{P}^{(3)}$ $(0.707, 0.707)$ $(0.728, 0.447)$ $(0.011, 0.132)$ $(0.011, 0.553)$
$(0.447, 0.728)$ $(0.707, 0.707)$ $(0.046, 0.03)$ $(0.509, 0.804)$
$(0.132, 0.011)$ $(0.03, 0.046)$ $(0.707, 0.707)$ $(0.5, 0.573)$
$(0.553, 0.011)$ $(0.804, 0.509)$ $(0.573, 0.5)$ $(0.707, 0.707)$
$\widetilde{P}$ $(0.707, 0.707)$ $(0.716, 0.446)$ $(0.01, 0.074)$ $(0.011, 0.553)$
$(0.446, 0.716)$ $(0.707, 0.707)$ $(0.047, 0.018)$ $(0.511, 0.803)$
$(0.074, 0.01)$ $(0.018, 0.047)$ $(0.707, 0.707)$ $(0.519, 0.603)$
$(0.553, 0.011)$ $(0.803, 0.511)$ $(0.603, 0.519)$ $(0.707, 0.707)$
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