Pythagorean fuzzy Muirhead mean operators in multiple attribute decision making for evaluating of emerging technology commercialization

Figures & data

Table 1. PFN decision matrix.

	C1	C2	C3	C4
O1	(0.50,0.80)	(0.60,0.50)	(0.30,0.60)	(0.60,0.70)
O2	(0.70,0.50)	(0.70,0.40)	(0.60,0.20)	(0.40,0.60)
O3	(0.70,0.50)	(0.50,0.70)	(0.50,0.30)	(0.60,0.20)
O4	(0.80,0.20)	(0.60,0.30)	(0.40,0.50)	(0.60,0.60)
O5	(0.60,0.40)	(0.40,0.70)	(0.70,0.50)	(0.60,0.80)

Table 2. The aggregating Result of PFMM, PFWMM, PFDMM and PFWDMM operators.

	O1	O2	O3	O4	O5
PFMM	(0.4827,0.6742)	(0.5862,0.4590)	(0.5695,0.4882)	(0.5833,0.4411)	(0.5639,0.6448)
PFWMM	(0.4556,0.708)	(0.5620,0.5247)	(0.5362,0.6065)	(0.5542,0.4671)	(0.5262,0.6842)
PFDMM	(0.522,0.6407)	(0.6229,0.3947)	(0.5874,0.382)	(0.637,0.3673)	(0.5936,0.5794)
PFWDMM	(0.5960,0.6007)	(0.6926,0.3695)	(0.6400,0.3673)	(0.6929,0.3416)	(0.6280,0.5410)

Table 3. The rank and score of emerging technology enterprises by using PFMM, PFWMM, PFDMM and PFWDMM operators.

	O1	O2	O3	O4	O5	Order
PFMM	0.3892	0.5665	0.543	0.5728	0.4511	O4> O2> O3> O5> O1
PFWMM	0.3531	0.5203	0.4599	0.5445	0.4043	O4> O2> O3> O5> O1
PFDMM	0.4310	0.6161	0.5996	0.6354	0.5084	O4> O2> O3> O5> O1
PFDWMM	0.4972	0.6716	0.6373	0.6817	0.5508	O4> O2> O3> O5> O1

Table 4. Ranking results by utilizing different parameter vector R in the PFWMM operator.

$\left({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\right)$		Scores				Order
	O1	O2	O3	O4	O5
(1,0,0,0)	0.4126	0.5869	0.6269	0.6000	0.5147	O3>O4>O2>O5>O1
(1,1,0,0)	0.3793	0.5401	0.5521	0.5708	0.4696	O4>O3>O2>O5>O1
(2,0,0,0)	0.4389	0.5962	0.6440	0.6166	0.5412	O3>O4>O2>O5>O1
(3,0,0,0)	0.4640	0.6052	0.6574	0.6309	0.5611	O3>O4>O2>O5>O1
(1,1,1,0)	0.3638	0.5273	0.4993	0.5548	0.4374	O4>O2>O3>O5>O1
(1,1,1,1)	0.3524	0.5197	0.4576	0.5438	0.4026	O4>O2>O3>O5>O1

Table 5. Ranking results by utilizing different parameter vector R in the PFWDMM operator.

$\left({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\right)$		Scores				Order
	O1	O2	O3	O4	O5
(1,0,0,0)	0.3750	0.5261	0.5875	0.5353	0.4572	O3>O4>O2>O5>O1
(1,1,0,0)	0.4317	0.6145	0.6137	0.6081	0.5234	O3>O2>O4>O5>O1
(2,0,0,0)	0.3535	0.4827	0.5734	0.4913	0.4233	O3>O4>O2>O5>O1
(3,0,0,0)	0.3368	0.4486	0.5609	0.4603	0.3939	O3>O4>O2>O5>O1
(1,1,1,0)	0.4647	0.6502	0.6285	0.6582	0.5401	O4>O2>O3>O5>O1
(1,1,1,1)	0.4991	0.6730	0.6379	0.6832	0.5515	O4>O2>O3>O5>O1

Table 6. The Comparison of the different aggregation operators under PFNs.

Aggregation operators	Whether the operator can capture the interrelationship between any two or three PFNs	Whether the operator can capture the interrelationship among all PFNs	Whether a parameter vector exists to manipulate the ranking results
PFWA and PFWG (Yager, 2014)	No	No	No
SPFWA and SPFWG (Ma & Xu, 2016)	No	No	No
PFOWAWAD (Zeng et al., 2016)	No	No	No
PFEWA, PFEOWA (Garg, 2016a)	No	No	No
PFHWA and PFHWG (Wu & Wei, 2017)	No	No	No
PFHOWA, PFHOWG (Wu & Wei, 2017)	No	No	No
PFHHA, PFHHG (Wu & Wei, 2017)	No	No	No
DHPFHWA, DHPFHWG (Wei & Lu, 2017a)	No	No	No
DHPFHOWA,DHPFHOWG (Wei & Lu, 2017a)	No	No	No
DHPFHHA, DHPFHHG (Wei & Lu, 2017a)	No	No	No
HPFHWA, HPFHWG (Lu et al., 2017)	No	No	No
HPFHOWA, HPFHOWG (Lu et al., 2017)	No	No	No
HPFHHA, HPFHHG (Lu et al., 2017)	No	No	No
PFCIA and PFCIG (Peng & Yang, 2016)	Yes	No	No
PFPWA, PFPWG (Wei & Lu, 2018b)	Yes	No	No
PFIWA, PFIWG (Wei, 2017a)	Yes	No	No
PFIOWA, PFIOWG (Wei, 2017a)	Yes	No	No
PFIHA, PFIHG (Wei, 2017a)	Yes	No	No
PFMSM,PFWMSM (Wei & Lu, 2018a)	Yes	Yes	No
GPFWBM,GPFWBGM (Zhang, Wang, Zhu, Xia, & Yu, 2017)	Yes	Yes	Yes
PFMM, PFDMM	Yes	Yes	Yes
PFWMM, PFWDMM	Yes	Yes	Yes

Table 7. Ordering of the emerging technology enterprises.

	Ordering
PFWA operator (Yager, 2014)	A3﹥4﹥2﹥5﹥1
PFWG operator (Yager, 2014)	A3﹥4﹥2﹥5﹥1
SPFWA operator (Ma & Xu, 2016)	A3﹥4﹥2﹥5﹥1
SPFWG operator (Ma & Xu, 2016)	A3﹥4﹥2﹥5﹥1

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