Picture Hesitant Fuzzy Set and Its Application to Multiple Criteria DecisionMaking
Abstract
:1. Introduction
2. Preliminaries
2.1. PFS
2.2. HFS
2.3. The PA Operator
3. PHFS
 (1)
 If$s\left({\tilde{n}}_{1}\right)>s\left({\tilde{n}}_{2}\right)$, then${\tilde{n}}_{1}>{\tilde{n}}_{2}$;
 (2)
 If$s\left({\tilde{n}}_{1}\right)=s\left({\tilde{n}}_{2}\right)$, then
 a.
 If$h\left({\tilde{n}}_{1}\right)>h\left({\tilde{n}}_{2}\right)$, then${\tilde{n}}_{1}>{\tilde{n}}_{2}$;
 b.
 If$h\left({\tilde{n}}_{1}\right)=h\left({\tilde{n}}_{2}\right)$, then${\tilde{n}}_{1}={\tilde{n}}_{2}$;
 c.
 If$h\left({\tilde{n}}_{1}\right)<h\left({\tilde{n}}_{2}\right)$, then${\tilde{n}}_{1}<{\tilde{n}}_{2}$
 (1)
 ${\tilde{n}}_{1}{}^{c}=\left\{\left\{0.2,0.3\right\},\left\{0.2\right\},\left\{0.3,0.4\right\}\right\}$, ${\tilde{n}}_{2}{}^{c}=\left\{\left\{0.1,0.2\right\},\left\{0.2,0.3\right\},\left\{0.3\right\}\right\}$;
 (2)
 ${\tilde{n}}_{1}\oplus {\tilde{n}}_{2}=\left\{\left\{0.51,0.58\right\},\left\{0.04,0.06\right\},\left\{0.02,0.03,0.04,0.06\right\}\right\};$
 (3)
 ${\tilde{n}}_{1}\otimes {\tilde{n}}_{2}=\left\{\left\{0.09,0.12\right\},\left\{0.36,0.44\right\},\left\{0.28,0.36,0.37,0.44\right\}\right\};$
 (4)
 $\lambda {\tilde{n}}_{1}=\left\{\left\{0.51,0.64\right\},\left\{0.04\right\},\left\{0.04,0.09\right\}\right\},$$\lambda {\tilde{n}}_{2}=\left\{\left\{0.51\right\},\left\{0.04,0.09\right\},\left\{0.01,0.04\right\}\right\};$
 (5)
 ${\tilde{n}}_{1}{}^{\lambda}=\left\{\left\{0.09,0.16\right\},\left\{0.36\right\},\left\{0.36,0.51\right\}\right\}$, ${\tilde{n}}_{2}{}^{\lambda}=\left\{\left\{0.09\right\},\left\{0.36,0.51\right\},\left\{0.19,0.36\right\}\right\}.$
 (1)
 ${\tilde{n}}_{1}\oplus {\tilde{n}}_{2}={\tilde{n}}_{2}\oplus {\tilde{n}}_{1};$
 (2)
 ${\tilde{n}}_{1}\otimes {\tilde{n}}_{2}={\tilde{n}}_{2}\otimes {\tilde{n}}_{1};$
 (3)
 $\lambda \left({\tilde{n}}_{1}\oplus {\tilde{n}}_{2}\right)=\lambda {\tilde{n}}_{1}\oplus \lambda {\tilde{n}}_{2};$
 (4)
 ${\left({\tilde{n}}_{1}\otimes {\tilde{n}}_{2}\right)}^{\lambda}={\tilde{n}}_{1}{}^{\lambda}\otimes {\tilde{n}}_{2}{}^{\lambda};$
 (5)
 ${\lambda}_{1}\tilde{n}\oplus {\lambda}_{2}\tilde{n}=\left({\lambda}_{1}+{\lambda}_{2}\right)\tilde{n};$
 (6)
 ${\tilde{n}}^{{\lambda}_{1}}\otimes {\tilde{n}}^{{\lambda}_{2}}={\tilde{n}}^{\left({\lambda}_{1}+{\lambda}_{2}\right)};$
 (7)
 ${\left({\tilde{n}}^{{\lambda}_{1}}\right)}^{{\lambda}_{2}}={\tilde{n}}^{{\lambda}_{1}{\lambda}_{2}}.$
4. Generalized Picture Hesitant Fuzzy Aggregation Operators
4.1. The GPHFWA Operator
4.2. The GPHFWG Operator
4.3. The GPHFPWA Operator
4.4. The GPHFPWG Operator
5. MCDM Methods under PHF Environment
Algorithm 1. MCDM method based on the GPHFWA or the GPHFWG operator. 
1: Normalize the PHF evaluation matrix $N$ to obtain the standardized PHF evaluation matrix $\overline{N}$ combined with Equation (51). 
2: Utilize the GPHFWA operator
$$GPHFW{A}_{\lambda}\left({\overline{n}}_{i1},{\overline{n}}_{i2},\dots ,{\overline{n}}_{in}\right)={\left({w}_{1}{\overline{n}}_{i1}{}^{\lambda}\oplus {w}_{2}{\overline{n}}_{i2}{}^{\lambda}\oplus \cdots \oplus {w}_{n}{\overline{n}}_{in}{}^{\lambda}\right)}^{1/\lambda}=\underset{j=1}{\overset{n}{\oplus}}{\left({w}_{j}{\overline{n}}_{ij}{}^{\lambda}\right)}^{1/\lambda}\phantom{\rule{0ex}{0ex}}={\displaystyle \underset{{\alpha}_{i1}\in {\tilde{\mu}}_{i1},{\alpha}_{i2}\in {\tilde{\mu}}_{i2},\dots ,{\alpha}_{in}\in {\tilde{\mu}}_{in},{\beta}_{i1}\in {\tilde{\eta}}_{i1},{\beta}_{i2}\in {\tilde{\eta}}_{i2},\dots ,{\beta}_{in}\in {\tilde{\eta}}_{in},{\gamma}_{i1}\in {\tilde{v}}_{i1},{\gamma}_{i2}\in {\tilde{v}}_{i2},\dots ,{\gamma}_{in}\in {\tilde{v}}_{in}}{\cup}\{\left\{{\left(1{\displaystyle \prod _{j=1}^{n}{\left(1{\overline{\alpha}}_{ij}{}^{\lambda}\right)}^{{w}_{j}}}\right)}^{1/\lambda}\right\},}\left\{1{\left(1{\displaystyle \prod _{j=1}^{n}{\left(1{\left(1{\overline{\beta}}_{ij}\right)}^{\lambda}\right)}^{{w}_{j}}}\right)}^{1/\lambda}\right\},\phantom{\rule{0ex}{0ex}}\left\{1{\left(1{\displaystyle \prod _{j=1}^{n}{\left(1{\left(1{\overline{\gamma}}_{ij}\right)}^{\lambda}\right)}^{{w}_{j}}}\right)}^{1/\lambda}\right\}\}$$
$$GPHFW{G}_{\lambda}\left({\overline{n}}_{i1},{\overline{n}}_{i2},\dots ,{\overline{n}}_{in}\right)=\frac{1}{\lambda}\left(\lambda {\overline{n}}_{i1}{}^{{w}_{1}}\otimes \lambda {\overline{n}}_{i2}{}^{{w}_{2}}\otimes \cdots \otimes \lambda {\overline{n}}_{in}{}^{{w}_{n}}\right)=\frac{1}{\lambda}\underset{j=1}{\overset{n}{\otimes}}\left(\lambda {\overline{n}}_{ij}{}^{{w}_{j}}\right)\phantom{\rule{0ex}{0ex}}={\displaystyle \underset{{\alpha}_{i1}\in {\tilde{\mu}}_{i1},{\alpha}_{i2}\in {\tilde{\mu}}_{i2},\dots ,{\alpha}_{in}\in {\tilde{\mu}}_{in},{\beta}_{i1}\in {\tilde{\eta}}_{i1},{\beta}_{i2}\in {\tilde{\eta}}_{i2},\dots ,{\beta}_{in}\in {\tilde{\eta}}_{in},{\gamma}_{i1}\in {\tilde{v}}_{i1},{\gamma}_{i2}\in {\tilde{v}}_{i2},\dots ,{\gamma}_{in}\in {\tilde{v}}_{in}}{\cup}\{\left\{1{\left(1{\displaystyle \prod _{j=1}^{n}{\left(1{\left(1{\overline{\alpha}}_{ij}\right)}^{\lambda}\right)}^{{w}_{j}}}\right)}^{1/\lambda}\right\},\left\{{\left(1{\displaystyle \prod _{j=1}^{n}{\left(1{\overline{\beta}}_{ij}{}^{\lambda}\right)}^{{w}_{j}}}\right)}^{1/\lambda}\right\},}\phantom{\rule{0ex}{0ex}}\left\{{\left(1{\displaystyle \prod _{j=1}^{n}{\left(1{\overline{\gamma}}_{ij}{}^{\lambda}\right)}^{{w}_{j}}}\right)}^{1/\lambda}\right\}$$

3: Compute the score and accuracy values of each alternative using Equation (14) and (15). 
4: Based on the comparison method of PHFEs, rank the alternatives. 
Algorithm 2. MCDM method based on the GPHFPWA or the GPHFPWG operator. 
1: Normalize the PHF evaluation matrix $N$ to obtain the standardized PHF evaluation matrix combined with Equation (51). 
2: Compute the values of ${T}_{ij}$ using the equations as
$${T}_{ij}={\displaystyle \prod _{k=1}^{j1}s\left({\tilde{n}}_{ik}\right)},{T}_{i1}=1.$$

3: Utilize the GPHFPWA operator
$$GPHFPW{A}_{\lambda}\left({\overline{n}}_{i1},{\overline{n}}_{i2},\dots ,{\overline{n}}_{in}\right)={\left(\frac{{T}_{i1}}{{\displaystyle {\sum}_{j=1}^{n}{T}_{ij}}}{\overline{n}}_{i1}{}^{\lambda}\oplus \frac{{T}_{i2}}{{\displaystyle {\sum}_{j=1}^{n}{T}_{ij}}}{\overline{n}}_{i2}{}^{\lambda}\oplus \cdots \oplus \frac{{T}_{in}}{{\displaystyle {\sum}_{j=1}^{n}{T}_{ij}}}{\overline{n}}_{in}{}^{\lambda}\right)}^{1/\lambda}\phantom{\rule{0ex}{0ex}}={\displaystyle \underset{{\alpha}_{i1}\in {\tilde{\mu}}_{i1},{\alpha}_{i2}\in {\tilde{\mu}}_{i2},\dots ,{\alpha}_{in}\in {\tilde{\mu}}_{in},{\beta}_{i1}\in {\tilde{\eta}}_{i1},{\beta}_{i2}\in {\tilde{\eta}}_{i2},\dots ,{\beta}_{in}\in {\tilde{\eta}}_{in},{\gamma}_{i1}\in {\tilde{v}}_{i1},{\gamma}_{i2}\in {\tilde{v}}_{i2},\dots ,{\gamma}_{in}\in {\tilde{v}}_{in}}{\cup}\{\left\{{\left(1{\displaystyle \prod _{j=1}^{n}{\left(1{\overline{\alpha}}_{ij}{}^{\lambda}\right)}^{\frac{{T}_{ij}}{{\displaystyle {\sum}_{j=1}^{n}{T}_{ij}}}}}\right)}^{1/\lambda}\right\},}\left\{1{\left(1{\displaystyle \prod _{j=1}^{n}{\left(1{\left(1{\overline{\beta}}_{ij}\right)}^{\lambda}\right)}^{\frac{{T}_{ij}}{{\displaystyle {\sum}_{j=1}^{n}{T}_{ij}}}}}\right)}^{1/\lambda}\right\},\phantom{\rule{0ex}{0ex}}\left\{1{\left(1{\displaystyle \prod _{j=1}^{n}{\left(1{\left(1{\overline{\gamma}}_{ij}\right)}^{\lambda}\right)}^{\frac{{T}_{ij}}{{\displaystyle {\sum}_{j=1}^{n}{T}_{ij}}}}}\right)}^{1/\lambda}\right\}\}$$
$$GPHFPW{G}_{\lambda}\left({\overline{n}}_{i1},{\overline{n}}_{i2},\dots ,{\overline{n}}_{in}\right)=\frac{1}{\lambda}\left({\left(\lambda {\overline{n}}_{i1}\right)}^{\frac{{T}_{i1}}{{\displaystyle {\sum}_{j=1}^{n}{T}_{ij}}}}\otimes {\left(\lambda {\overline{n}}_{i2}\right)}^{\frac{{T}_{i2}}{{\displaystyle {\sum}_{j=1}^{n}{T}_{ij}}}}\otimes \cdots \otimes {\left(\lambda {\overline{n}}_{in}\right)}^{\frac{{T}_{in}}{{\displaystyle {\sum}_{j=1}^{n}{T}_{ij}}}}\right)\phantom{\rule{0ex}{0ex}}={\displaystyle \underset{{\alpha}_{i1}\in {\tilde{\mu}}_{i1},{\alpha}_{i2}\in {\tilde{\mu}}_{i2},\dots ,{\alpha}_{in}\in {\tilde{\mu}}_{in},{\beta}_{i1}\in {\tilde{\eta}}_{i1},{\beta}_{i2}\in {\tilde{\eta}}_{i2},\dots ,{\beta}_{in}\in {\tilde{\eta}}_{in},{\gamma}_{i1}\in {\tilde{v}}_{i1},{\gamma}_{i2}\in {\tilde{v}}_{i2},\dots ,{\gamma}_{in}\in {\tilde{v}}_{in}}{\cup}\{\left\{1{\left(1{\displaystyle \prod _{j=1}^{n}{\left(1{\left(1{\overline{\alpha}}_{ij}\right)}^{\lambda}\right)}^{\frac{{T}_{ij}}{{\displaystyle {\sum}_{j=1}^{n}{T}_{ij}}}}}\right)}^{1/\lambda}\right\},\left\{{\left(1{\displaystyle \prod _{j=1}^{n}{\left(1{\overline{\beta}}_{ij}{}^{\lambda}\right)}^{\frac{{T}_{ij}}{{\displaystyle {\sum}_{j=1}^{n}{T}_{ij}}}}}\right)}^{1/\lambda}\right\},}\phantom{\rule{0ex}{0ex}}\left\{{\left(1{\displaystyle \prod _{j=1}^{n}{\left(1{\overline{\gamma}}_{ij}{}^{\lambda}\right)}^{\frac{{T}_{ij}}{{\displaystyle {\sum}_{j=1}^{n}{T}_{ij}}}}}\right)}^{1/\lambda}\right\}$$

4: Compute the score and accuracy values of each alternative using Equation (14) and (15). 
5: Based on the comparison method of PHFEs, rank the alternatives. 
6. Numerical Examples
6.1. Implementation
 Step 1: Because of all the criteria are the benefit type, the standardized PHF evaluation matrix $\overline{N}$ is as same as the PHF evaluation matrix $N$.
 Step 2: Use the GPHFWA ($\lambda =1$) operator to aggregate the standardized PHF evaluation matrix $\overline{N}$, and the collective evaluation information of each alternative is obtained as$${\tilde{n}}_{1}=\{\left\{0.3636,0.3877,0.4113,0.4336\right\},\left\{0.3862,0.4048,0.4136,0.4335\right\},\left\{0.0444,0.0482,0.0502,0.0544\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{2}=\{\left\{0.4061,0.4401,0.4684,0.5023,0.5308,0.5545\right\},\left\{0.2563,0.2612,0.2612,0.2659,0.2662,0.2709,0.2709,0.2761\right\},\left\{0.0891,0.0942,0.0964,0.1019\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{3}=\{\left\{0.4014,0.4789,0.5180,0.5230,0.5804,0.6159\right\},\{0.1627,0.1677,0.1725,0.1763,0.1779,0.1870,0.1884,0.1943,0.1999,0.2042,0.2061,0.2166\},\left\{0.0448,0.0551\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{4}=\{\left\{0.3549,0.3738,0.4077,0.4251\right\},\left\{0.3834,0.3989,0.4018,0.4181,0.4248,0.4420,0.4452,0.5632\right\},\left\{0.0576\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{5}=\{\left\{0.3468,0.4038,0.4756\right\},\left\{0.3321,0.3444,0.3516,0.3647\right\},\left\{0.0612,0.0642\right\}\}.$$
 Step 3: Compute the score values of each alternative combined with Equation (14):$$s\left({\tilde{n}}_{1}\right)=0.4701,s\left({\tilde{n}}_{2}\right)=0.5611,s\left({\tilde{n}}_{3}\right)=0.6409,s\left({\tilde{n}}_{4}\right)=0.4553,s\left({\tilde{n}}_{5}\right)=0.4989.$$
 Step 4: According to the score values, the ranking result of the five ERP systems is determined as ${A}_{3}\succ {A}_{2}\succ {A}_{5}\succ {A}_{1}\succ {A}_{4}$.
 Step 1′: See Step 1.
 Step 2′: Use the GPHFWG ($\lambda =1$) operator to aggregate the standardized PHF evaluation matrix $\overline{N}$, and the collective evaluation information of each alternative is obtained as$${\tilde{n}}_{1}=\{\left\{0.2307,0.2343,0.2405,0.2443\right\},\left\{0.5365,0.6663,0.5380,0.6673\right\},\left\{0.0600,0.0797,0.0661,0.0856\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{2}=\{\left\{0.2017,0.2101,0.2151,0.2212,0.2304,0.2358\right\},\left\{0.4526,0.4550,0.4670,0.4693,0.4914,0.4936,0.5047,0.5070\right\},\left\{0.0901,0.0993,0.0999,0.1090\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{3}=\{\left\{0.1986,0.2057,0.2081,0.2342,0.2427,0.2454\right\},\{0.3334,0.3363,0.3602,0.3630,0.3766,0.3792,0.4016,0.4042,0.4830,0.4852,0.5038,0.5059\},\left\{0.0481,0.0634\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{4}=\{\left\{0.1024,0.1037,0.1044,0.1058\right\},\left\{0.5949,0.5977,0.6709,0.6732,0.7038,0.7058,0.7594,0.7611\right\},\left\{0.0591\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{5}=\{\left\{0.1613,0.1660,0.1696\right\},\left\{0.5850,0.6372,0.6503,0.6943\right\},\left\{0.0716,0.0805\right\}\}.$$
 Step 3′: Compute the score values of each alternative combined with Equation (14):$$s\left({\tilde{n}}_{1}\right)=0.2813,s\left({\tilde{n}}_{2}\right)=0.3197,s\left({\tilde{n}}_{3}\right)=0.3778,s\left({\tilde{n}}_{4}\right)=0.1808,s\left({\tilde{n}}_{5}\right)=0.2240.$$
 Step 4′: According to the score values, the ranking result of the five ERP systems is determined as ${A}_{3}\succ {A}_{2}\succ {A}_{1}\succ {A}_{5}\succ {A}_{4}$.
 Step 1: Because of all the criteria are the benefit type, the standardized PHF evaluation matrix $\overline{N}$ is as same as the PHF evaluation matrix $N$.
 Step 2: Compute the values of ${T}_{ij}$ using the Equation (54)$${T}_{ij}=\left[\begin{array}{cccc}1.000& 0.6667& 0.4783& 0.3157\\ 1.000& 0.7600& 0.5225& 0.4311\\ 1.000& 0.7475& 0.3526& 0.1992\\ 1.000& 0.7100& 0.3905& 0.3417\\ 1.000& 0.7000& 0.4620& 0.3719\end{array}\right].$$
 Step 3: Use the GPHFPWA ($\lambda =1$) operator to aggregate the standardized PHF evaluation matrix $\overline{N}$, and the collective evaluation information of each alternative is obtained as$${\tilde{n}}_{1}=\{\left\{0.5003,0.5177,0.5360,0.5382,0.5522,0.6230,0.6361,0.6516,0.6516\right\},\left\{0.0495,0.0521,0.0562,0.0592\right\},\left\{0.1176,0.1345,0.1558,0.1783\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{2}=\{\left\{0.6290,0.6396,0.6446,0.6548,0.6723,0.6816,0.6861,0.6950\right\},\left\{0.0460,0.0559,0.0594,0.0722\right\},\left\{0.1084,0.1175,0.1186,0.1287,0.1316,0.1427,0.1441,0.1562\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{3}=\{\left\{0.5335,0.5533,0.5537,0.5727,0.5996,0.6169\right\},\{0.0494,0.0550,0.0617,0.0686\},\left\{0.1692,0.1732,0.1857,0.1902,0.2018,0.2066,0.2216,0.2269\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{4}=\{\left\{0.5593,0.5820,0.5978,0.6185,0.6425,0.6609\right\},\left\{0.0921,0.1015,0.1055,0.1162\right\},\left\{0.0947,0.1044,0.1159,0.1277\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{5}=\{\left\{0.5888,0.6181,0.6429,0.6683\right\},\left\{0.0605,0.0670,0.0796,0.0881\right\},\left\{0.1670,0.1742,0.1871,0.1951\right\}\}.$$
 Step 4: Compute the score values of each alternative combined with Equation (14)$$s\left({\tilde{n}}_{1}\right)=0.6888,s\left({\tilde{n}}_{2}\right)=0.7368,s\left({\tilde{n}}_{3}\right)=0.6580,s\left({\tilde{n}}_{4}\right)=0.6978,s\left({\tilde{n}}_{5}\right)=0.6874.$$
 Step 5: According to the score values, the ranking result of the five foreign professors is determined as ${A}_{2}\succ {A}_{4}\succ {A}_{1}\succ {A}_{5}\succ {A}_{3}$.
 Step 1′: See Step 1.
 Step 2′: See Step 2.
 Step 3′: Use the GPHFPWG ($\lambda =1$) operator to aggregate standardized the PHF evaluation matrix $\overline{N}$, and the collective evaluation information of each alternative is obtained as$${\tilde{n}}_{1}=\{\left\{0.4753,0.4963,0.5142,0.5204,0.5434,0.5966,0.6231,0.6455,0.6455\right\},\left\{0.0527,0.0597,0.0609,0.0678\right\},\left\{0.1203,0.1402,0.1614,0.1804\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{2}=\{\left\{0.6049,0.6124,0.6267,0.6345,0.6376,0.6456,0.6606,0.6689\right\},\left\{0.0668,0.0739,0.0809,0.0878\right\},\left\{0.1176,0.1230,0.1350,0.1403,0.1462,0.1515,0.1630,0.1682\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{3}=\{\left\{0.4297,0.4424,0.4902,0.5047,0.5788,0.5959\right\},\{0.0526,0.0571,0.0703,0.0748\},\left\{0.2020,0.2110,0.2216,0.2304,0.2410,0.2495,0.2596,0.2680\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{4}=\{\left\{0.5026,0.5363,0.5416,0.5769,0.5779,0.6155\right\},\left\{0.1119,0.1178,0.1264,0.1322\right\},\left\{0.0994,0.1236,0.1297,0.1531\right\}\};\phantom{\rule{0ex}{0ex}}{\tilde{n}}_{5}=\{\left\{0.5596,0.5733,0.6141,0.6292\right\},\left\{0.0640,0.0801,0.0838,0.0995\right\},\left\{0.1791,0.1975,0.1985,0.2164\right\}\}.$$
 Step 4′: Compute the score values of each alternative combined with Equation (14)$$s\left({\tilde{n}}_{1}\right)=0.6757,s\left({\tilde{n}}_{2}\right)=0.7080,s\left({\tilde{n}}_{3}\right)=0.6040,s\left({\tilde{n}}_{4}\right)=0.6550,s\left({\tilde{n}}_{5}\right)=0.6572.$$
 Step 5′: According to the score values, the ranking result of the five foreign professors is obtained as ${A}_{2}\succ {A}_{1}\succ {A}_{5}\succ {A}_{4}\succ {A}_{3}$.
6.2. Sensitivity Analysis
 (1)
 In Example 1, the score values of each alternative obtained by the GPHFWA operator are bigger than those obtained by the GPHFWG operator, and the difference between them increases along with the increasing of $\lambda $. It means that the GPHFWA operator is more suitable to aggregate the PHFEs of optimistic decision makers, while the GPHFWG operator can reflect the opinion of pessimistic decision makers. Furthermore, the level of optimism and pessimism are greater with the bigger value of $\lambda $.
 (2)
 In Example 2, the score values of each alternative obtained by the GPHFPWA and GPHFPWG operators are relatively stable when the different values of $\lambda $ are used; the parameter $\lambda $ cannot reflect the attitude of decision makers. In addition, the best alternative varies when the value of $\lambda $ is relatively high, while the best alternative is always the same in Example 1. It means that the rankings obtained by the GPHFPWA and GPHFPWG operators are more affected by the parameter $\lambda $ than those obtained by the GPHFWA and GPHFWG operators.
6.3. Comparative Analysis
6.4. Application of Web Service Selection
 Step 1: According to the Definition 3, normalize the PF evaluation matrix $A=\left({a}_{ij}\right)$ to the standardized PF evaluation matrix $\overline{A}=\left({\overline{a}}_{ij}\right)$ as$${\overline{a}}_{ij}=\{\begin{array}{ll}{a}_{ij},& \mathrm{for}\text{}\mathrm{the}\text{}\mathrm{benefit}\text{}\mathrm{criteria};\\ {\left({a}_{ij}\right)}^{c},& \mathrm{for}\text{}\mathrm{the}\text{}\mathrm{cost}\text{}\mathrm{criteria}.\end{array}$$
 Step 2: Utilize the GPFWA ($\lambda =1$) operator$$GPFW{A}_{\lambda =1}\left({\overline{a}}_{i1},{\overline{a}}_{i2},\dots ,{\overline{a}}_{i9}\right)={a}_{i}=\left(1{\displaystyle \prod _{j=1}^{9}{\left(1{\overline{\mu}}_{ij}\right)}^{{w}_{j}}},{\displaystyle \prod _{j=1}^{9}{\left({\overline{\eta}}_{ij}\right)}^{{w}_{j}}},{\displaystyle \prod _{j=1}^{9}{\left({\overline{v}}_{ij}\right)}^{{w}_{j}}}\right)$$
 Step 3: Compute the score values of each web service using the equation$$s\left({a}_{i}\right)=\left(1+{\mu}_{i}{\eta}_{i}{v}_{i}\right)/2.$$
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
Appendix B
Appendix C
References
 Mardani, A.; Jusoh, A.; Zavadskas, E.K. Fuzzy multiple criteria decisionmaking techniques and applications–two decades review from 1994 to 2014. Expert Syst. Appl. 2015, 42, 4126–4148. [Google Scholar] [CrossRef]
 MorenteMolinera, J.A.; Kou, G.; GonzálezCrespo, R.; Corchado, J.M.; HerreraViedma, E. Solving multicriteria group decision making problems under environments with a high number of alternatives using fuzzy ontologies and multigranular linguistic modelling methods. Knowl. Syst. 2017, 137, 54–64. [Google Scholar] [CrossRef]
 Hernández, F.L.; Ory, E.G.D.; Aguilar, S.R.; GonzálezCrespo, R. Residue properties for the arithmetical estimation of the image quantization table. Appl. Soft Comput. 2018, 67, 309–321. [Google Scholar] [CrossRef]
 Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef] [Green Version]
 Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
 Atanassov, K.T. Gargov, G. Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 1989, 31, 343–349. [Google Scholar] [CrossRef]
 Cường, B.C. Picture fuzzy sets. J. Comput. Sci.Cybern. 2015, 30, 409–420. [Google Scholar] [CrossRef]
 Torra, V. Hesitant fuzzy sets. Int. J. Intell. Syst. 2010, 25, 529–539. [Google Scholar] [CrossRef]
 Ye, J. Trapezoidal neutrosophic set and its application to multiple attribute decisionmaking. Neural Comput. Appl. 2015, 26, 1157–1166. [Google Scholar] [CrossRef]
 Zhu, B.; Xu, Z.S.; Xia, M.M. Dual hesitant fuzzy sets. J. Appl. Math. 2012, 2012, 2607–2645. [Google Scholar] [CrossRef]
 Liu, F.; Yuan, X.H. Fuzzy number intuitionistic fuzzy set. Fuzzy Syst. Math. 2007, 21, 88–91. [Google Scholar]
 Garg, H. Some picture fuzzy aggregation operators and their applications to multicriteria decisionmaking. Arab. J. Sci. Eng. 2017, 42, 1–16. [Google Scholar] [CrossRef]
 Cuong, B. Picture fuzzy setsfirst results. Part 1. In Proceedings of the Third World Congress on Information and Communication WICT’2013, Hanoi, Vietnam, 15–18 December 2013; pp. 1–6. [Google Scholar]
 Singh, P. Correlation coefficients for picture fuzzy sets. J. Intell. Fuzzy Syst. 2015, 28, 1–12. [Google Scholar]
 Son, L.H. Generalized picture distance measure and applications to picture fuzzy clustering. Appl. Soft Comput. 2016, 46, 284–295. [Google Scholar] [CrossRef]
 Wei, G.W. Picture fuzzy crossentropy for multiple attribute decision making problems. J. Bus. Econ. Manag. 2016, 17, 491–502. [Google Scholar] [CrossRef]
 Torra, V.; Narukawa, Y. On hesitant fuzzy sets and decision. In Proceedings of the IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 20–24 August 2009; pp. 1378–1382. [Google Scholar]
 Chen, N.; Xu, Z.S.; Xia, M.M. Intervalvalued hesitant preference relations and their applications to group decision making. Knowl. Syst. 2013, 37, 528–540. [Google Scholar] [CrossRef]
 Farhadinia, B. Correlation for dual hesitant fuzzy sets and dual intervalvalued hesitant fuzzy sets. Int. J. Intell. Syst. 2013, 29, 184–205. [Google Scholar] [CrossRef]
 Xu, Z.S.; Yager, R.R. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Intell. Syst. 2006, 35, 417–433. [Google Scholar] [CrossRef]
 Xu, Z.S. Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst. 2007, 15, 1179–1187. [Google Scholar]
 Xu, Z.S. Methods for aggregating intervalvalued intuitionistic fuzzy information and their application to decision making. Control Decis. 2007, 22, 215–219. [Google Scholar]
 Xu, Z.S.; Chen, J. Approach to group decision making based on intervalvalued intuitionistic judgment matrices. Syst. Eng. Theory Pract. 2007, 27, 126–133. [Google Scholar] [CrossRef]
 Xu, Z.S.; Chen, J. On geometric aggregation over intervalvalued intuitionistic fuzzy information. In Proceedings of the International Conference on Fuzzy Systems and Knowledge Discovery, Haikou, China, 24–27 August 2007; pp. 466–471. [Google Scholar]
 Wei, G.W. Picture fuzzy aggregation operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 2017, 33, 713–724. [Google Scholar] [CrossRef]
 Wang, C.; Zhou, X.; Tu, H.; Tao, S. Some geometric aggregation operators based on picture fuzzy sets and their application in multiple attribute decision making. Ital. J. Pure Appl. Math. 2017, 37, 477–492. [Google Scholar]
 Xia, M.M.; Xu, Z.S. Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason. 2011, 52, 395–407. [Google Scholar] [CrossRef] [Green Version]
 Yu, D.J.; Zhang, W.Y.; Huang, G. Dual hesitant fuzzy aggregation operators. Technol. Econ. Develop. Econ. 2016, 22, 194–209. [Google Scholar] [CrossRef]
 Wang, H.J.; Zhao, X.F.; Wei, G.W. Dual hesitant fuzzy aggregation operators in multiple attribute decision making. J. Intell. Fuzzy Syst. Appl. Eng. Technol. 2014, 26, 2281–2290. [Google Scholar]
 Zhang, W.K.; Li, X.; Ju, Y.B. Some aggregation operators based on Einstein operations under intervalvalued dual hesitant fuzzy setting and their application. Math. Probl. Eng. 2014. [Google Scholar] [CrossRef]
 Yager, R.R. Prioritized aggregation operators. Int. J. Approx. Reason. 2008, 48, 263–274. [Google Scholar] [CrossRef]
 Yu, D.J. Intuitionistic fuzzy prioritized operators and their application in multicriteria group decision making. Technol. Econ. Develop. Econ. 2013, 19, 1–21. [Google Scholar] [CrossRef]
 Yu, D.J.; Wu, Y.Y.; Lu, T. Intervalvalued intuitionistic fuzzy prioritized operators and their application in group decision making. Knowl. Syst. 2012, 30, 57–66. [Google Scholar] [CrossRef]
 Wei, G.W. Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowl. Syst. 2012, 31, 176–182. [Google Scholar] [CrossRef]
 Bedregal, B.; Reiser, R.; Bustince, H.; LopezMolina, C.; Torra, V. Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms. Infor. Sci. 2014, 255, 82–99. [Google Scholar] [CrossRef] [Green Version]
 Alcantud, J.C.R.; Torra, V. Decomposition theorems and extension principles for hesitant fuzzysets. Infor. Fusion. 2018, 41, 48–56. [Google Scholar] [CrossRef]
 Bagga, P.; Joshi, A.; Hans, R. QoS based web service selection and multicriteria decision making methods. Int. J. Int. Multimed. Artif. Intell. 2017. [Google Scholar] [CrossRef]
 Zhao, H.; You, J.X.; Liu, H.C. Failure mode and effect analysis using MULTIMOORA method with continuous weighted entropy under intervalvalued intuitionistic fuzzy environment. Soft Comput. 2017, 12, 5355–5367. [Google Scholar] [CrossRef]
 Kou, G.; Lu, Y.Q.; Peng, Y.; Shi, Y. Evaluation of classification algorithms using MCDM and rank correlation. Int. J. Inf. Technol. Decis. Mak. 2012, 11, 197–225. [Google Scholar] [CrossRef]
Alternatives  ${\mathit{C}}_{1}$  ${\mathit{C}}_{2}$  ${\mathit{C}}_{3}$  ${\mathit{C}}_{4}$ 

${A}_{1}$  {{0.43,0.53},{0.33}, {0.06,0.09}}  {{0.76,0.89}, {0.05,0.08},{0.03}}  {{0.42},{0.35}, {0.12,0.18}}  {{0.08},{0.75,0.89}, {0.02}} 
${A}_{2}$  {{0.53,0.65,0.73}, {0.10,0.12},{0.08}}  {{0.13},{0.53,0.64}, {0.12,0.21}}  {{0.03},{0.77,0.82}, {0.10,0.13}}  {{0.58,0.73},{0.15}, {0.08}} 
${A}_{3}$  {{0.72,0.86,0.91}, {0.03},{0.02}}  {{0.07},{0.05,0.09}, {0.05}}  {{0.04},{0.65,0.72,0.85},{0.05,0.10}}  {{0.45,0.68},{0.18,0.26},{0.06}} 
${A}_{4}$  {{0.77,0.85},{0.09}, {0.05}}  {{0.65,0.74},{0.10,0.16},{0.10}}  {{0.02},{0.78,0.89},{0.05}}  {{0.08},{0.65,0.84}, {0.06}} 
${A}_{5}$  {{0.70,0.81,0.90}, {0.05},{0.02}}  {{0.68},{0.08}, {0.13,0.21}}  {{0.05},{0.77,0.87},{0.06}}  {{0.13},{0.65,0.75}, {0.09}} 
Alternatives  ${\mathit{C}}_{1}$  ${\mathit{C}}_{2}$  ${\mathit{C}}_{3}$  ${\mathit{C}}_{4}$ 

${A}_{1}$  {{0.40,0.50,0.70},{0.05},{0.10,0.20}}  {{0.65},{0.05,0.08}, {0.15}}  {{0.40,0.50,0.60},{0.03},{0.10,0.20}}  {{0.55},{0.10,0.15}, {0.15}} 
${A}_{2}$  {{0.65,0.75},{0.02,0.04},{0.15}}  {{0.60},{0.05,0.10}, {0.10,0.20}}  {{0.75,0.80},{0.06}, {0.05,0.08}}  {{0.40,0.50},{0.20}, {0.15,0.25}} 
${A}_{3}$  {{0.70},{0.06,0.10}, {0.10,0.15}}  {{0.20,0.30,0.50},{0.04},{0.30,0.40}}  {{0.50},{0.03,0.06}, {0.30,0.35}}  {{0.50,0.70},{0.10}, {0.10}} 
${A}_{4}$  {{0.50,0.60,0.70},{0.08},{0.10}}  {{0.40,0.50},{0.20}, {0.10,0.20}}  {{0.85},{0.03,0.07}, {0.05}}  {{0.45},{0.10,0.20}, {0.15,0.30}} 
${A}_{5}$  {{0.65},{0.05,0.10}, {0.15,0.20}}  {{0.50,0.70},{0.08}, {0.20}}  {{0.70,0.80},{0.04}, {0.10}}  {{0.35},{0.10,0.20},{0.30,0.40}} 
Values  $\mathit{s}\left({\tilde{\mathit{n}}}_{1}\right)$  $\mathit{s}\left({\tilde{\mathit{n}}}_{2}\right)$  $\mathit{s}\left({\tilde{\mathit{n}}}_{3}\right)$  $\mathit{s}\left({\tilde{\mathit{n}}}_{4}\right)$  $\mathit{s}\left({\tilde{\mathit{n}}}_{5}\right)$  Ranking 

$\lambda =0.001$  0.4117  0.4988  0.5787  0.3388  0.4058  ${A}_{3}\succ {A}_{2}\succ {A}_{1}\succ {A}_{5}\succ {A}_{4}$ 
$\lambda =0.5$  0.4412  0.5328  0.6124  0.3977  0.4526  ${A}_{3}\succ {A}_{2}\succ {A}_{5}\succ {A}_{1}\succ {A}_{4}$ 
$\lambda =1$  0.4701  0.5611  0.6409  0.4553  0.4989  ${A}_{3}\succ {A}_{2}\succ {A}_{5}\succ {A}_{1}\succ {A}_{4}$ 
$\lambda =2$  0.5197  0.6001  0.6812  0.5426  0.5743  ${A}_{3}\succ {A}_{2}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}$ 
$\lambda =3$  0.5578  0.6242  0.7074  0.5981  0.6255  ${A}_{3}\succ {A}_{5}\succ {A}_{2}\succ {A}_{4}\succ {A}_{1}$ 
$\lambda =5$  0.6133  0.6522  0.7406  0.6613  0.6852  ${A}_{3}\succ {A}_{5}\succ {A}_{4}\succ {A}_{2}\succ {A}_{1}$ 
$\lambda =10$  0.6965  0.6831  0.7837  0.7274  0.7481  ${A}_{3}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ 
$\lambda =20$  0.7679  0.7062  0.8197  0.7722  0.7916  ${A}_{3}\succ {A}_{5}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ 
$\lambda =50$  0.8243  0.7274  0.8527  0.8063  0.8280  ${A}_{3}\succ {A}_{5}\succ {A}_{1}\succ {A}_{4}\succ {A}_{2}$ 
Values  $\mathit{s}\left({\tilde{\mathit{n}}}_{1}\right)$  $\mathit{s}\left({\tilde{\mathit{n}}}_{2}\right)$  $\mathit{s}\left({\tilde{\mathit{n}}}_{3}\right)$  $\mathit{s}\left({\tilde{\mathit{n}}}_{4}\right)$  $\mathit{s}\left({\tilde{\mathit{n}}}_{5}\right)$  Ranking 

$\lambda =0.001$  0.3227  0.3791  0.4486  0.2142  0.2720  ${A}_{3}\succ {A}_{2}\succ {A}_{1}\succ {A}_{5}\succ {A}_{4}$ 
$\lambda =0.5$  0.3019  0.3492  0.4120  0.1960  0.2457  ${A}_{3}\succ {A}_{2}\succ {A}_{1}\succ {A}_{5}\succ {A}_{4}$ 
$\lambda =1$  0.2813  0.3197  0.3778 