Aiming at the mixed multiattribute group decision-making problem of interval Pythagorean fuzzy numbers, a weighted average (WA) operator model based on interval Pythagorean fuzzy sets is constructed. Furthermore, a decision-making method based on the technique for order preference by similarity to ideal solution (TOPSIS) method with interval Pythagorean fuzzy numbers is proposed. First, based on the completely unknown weights of decision-makers and attributes, interval Pythagorean fuzzy numbers are applied to TOPSIS group decision-making. Second, the interval Pythagorean fuzzy number WA operator is used to synthesize the evaluation matrices of multiple decision-makers into a comprehensive evaluation matrix, and the relative closeness of each scheme is calculated based on the TOPSIS decision-making method. Finally, an example is given to illustrate the rationality and effectiveness of the proposed method.

In a real decision-making process, the information obtained by decision-makers is often fuzzy and uncertain. Therefore, decision-making research based on fuzzy information is essential. Zadeh [

In the existing decision-making processes, it is often difficult for decision-makers to use an accurate value to evaluate the advantages and disadvantages of the scheme. Methods using interval Pythagorean fuzzy values are more conducive to expressing fuzzy information. Peng [

The technique for order preference by similarity to ideal solution (TOPSIS) decision-making method is a widely used multiattribute decision-making method, and it is one the most frequently used techniques to deal with multicriteria group decision-making (MCGDM) conflicts. Umer et al. [

In recent years, experts and scholars have conducted multidomain, in-depth research in the field of Pythagorean fuzzy numbers, extended Pythagorean fuzzy numbers to interval Pythagorean fuzzy numbers, and studied various operators and decision models in the environment of interval Pythagorean fuzzy numbers. However, the application of the TOPSIS decision method in an interval Pythagorean fuzzy environment is rare. Therefore, the weighted average (WA) operator of interval Pythagorean fuzzy numbers is defined, and a TOPSIS hybrid multiattribute decision-making method based on interval Pythagorean fuzzy numbers is presented. The advantages of this method are as follows: (1) it alleviates the loss of information in the decision-making process, (2) it effectively solves different types of attribute information problems and fully utilizes the attribute advantages of each decision-maker, and (3) it enlarges the gap between the closeness of each scheme and the advantages and disadvantages, making the decision-making results more convincing.

Suppose

For the Pythagorean fuzzy number

For two Pythagorean fuzzy numbers

Suppose

Suppose

For two Pythagorean fuzzy numbers

In decision-making, scholars have proposed the concept and algorithms of interval Pythagorean fuzzy numbers. An interval Pythagorean fuzzy number adds the upper and lower limits of the membership and nonmembership based on the Pythagorean fuzzy number.

Suppose

For the interval Pythagorean fuzzy number

For two interval Pythagorean fuzzy numbers

Suppose

Suppose

For two interval Pythagorean fuzzy numbers

For two interval Pythagorean fuzzy numbers

There is a mapping WA:

For the interval Pythagorean fuzzy number

If

Because

This paper presents a new TOPSIS decision-making method based on interval Pythagorean fuzzy numbers.

Suppose there is a mixed multiattribute decision-making problem. The scheme set is

The TOPSIS decision-making method is used to analyze interval Pythagorean fuzzy problems. First, the positive ideal matrix and negative ideal matrix of the interval Pythagorean fuzzy sets are defined, and multiple decision matrices are integrated into a comprehensive decision matrix.

According to the evaluated values of the all decision-makers, the positive ideal matrix

According to the positive and negative ideal matrices, the Hamming distance

The closeness of each evaluated value is calculated according to the valuated values of the positive and negative ideal matrices:

The weights are determined as follows:

The weighted average operator (IPVFWA) of the interval Pythagorean fuzzy number

The positive and negative ideal solutions of the interval Pythagorean fuzzy comprehensive decision matrix

Positive ideal solution:

Negative ideal solution:

The closeness coefficient is determined as follows:

The greater the closeness coefficient of the scheme is, the closer the scheme is to being optimal.

In this example, it is assumed that when an epidemic occurs in an area, there are three affected areas (

Evaluation matrix of three decision-makers.

([0.7,0.8], [0.2,0.3]) | ([0.7,0.8], [0.4,0.5]) | ([0.4,0.6], [0.2,0.4]) | ||

([0.5,0.6], [0.3,0.4]) | ([0.5,0.6], [0.2,0.3]) | ([0.4,0.5], [0.1,0.2]) | ||

([0.6,0.8], [0.4,0.5]) | ([0.7,0.9], [0.3,0.5]) | ([0.6,0.7], [0.2,0.3]) | ||

([0.4,0.5], [0.2,0.3]) | ([0.5,0.7], [0.3,0.4]) | ([0.7,0.8], [0.4,0.5]) | ||

([0.5,0.6], [0.4,0.5]) | ([0.6,0.7], [0.4,0.5]) | ([0.5,0.6], [0.2,0.3]) | ||

([0.4,0.6], [0.1,0.3]) | ([0.5,0.7], [0.3,0.5]) | ([0.6,0.7], [0.2,0.4]) | ||

([0.5,0.6], [0.3,0.4]) | ([0.6,0.7], [0.4,0.5]) | ([0.7,0.8], [0.2,0.4]) | ||

([0.4,0.5], [0.2,0.3]) | ([0.7,0.8], [0.2,0.4]) | ([0.4,0.5], [0.2,0.3]) | ||

([0.7,0.8], [0.2,0.4]) | ([0.6,0.8], [0.4,0.5]) | ([0.6,0.7], [0.4,0.5]) |

According to the evaluated values of all the decision-makers, the positive ideal matrix

Positive and negative ideal matrices.

([0.5333, 0.6333], [0.2333, 0.3333]) | ([0.6000, 0.7333], [0.3667, 0.4667]) | ([0.6000, 0.7333], [0.2667, 0.4333]) | ||

([0.4667, 0.5667], [0.3000, 0.4000]) | ([0.6000, 0.7000], [0.2667, 0.4000]) | ([0.4333, 0.5333], [0.1667, 0.2667]) | ||

([0.5667, 0.7333], [0.2333, 0.4000]) | ([0.6000, 0.8000], [0.3333, 0.5000]) | ([0.6000, 0.7000], [0.2667, 0.4000]) | ||

([0.4,0.5], [0.3,0.4]) | ([0.5,0.7], [0.4,0.5]) | ([0.4,0.6], [0.4,0.5]) | ||

([0.4,0.5], [0.4,0.5]) | ([0.5,0.6], [0.4,0.5]) | ([0.4,0.5], [0.2,0.3]) | ||

([0.4,0.6], [0.4,0.5]) | ([0.5,0.7], [0.4,0.5]) | ([0.6,0.7], [0.4,0.5]) | ||

([0.5,0.6], [0.2,0.3]) | ([0.6,0.7], [0.3,0.4]) | ([0.7,0.8], [0.2,0.4]) | ||

([0.5,0.6], [0.2,0.3]) | ([0.7,0.8], [0.2,0.4]) | ([0.5,0.6], [0.2,0.3]) | ||

([0.7,0.8], [0.1,0.3]) | ([0.6,0.8], [0.3,0.5]) | ([0.6,0.7], [0.2,0.4]) |

According to the positive and negative ideal matrices, the Hamming distance

Hamming distance

0.1000 | 0.0583 | 0.1083 | |||

0.0167 | 0.0917 | 0.0500 | |||

0.0917 | 0.0583 | 0.0417 | |||

0.2000 | 0.0750 | 0.0750 | |||

0.1000 | 0.1000 | 0.0500 | |||

0.1000 | 0.1250 | 0.1000 | |||

0.1000 | 0.1000 | 0.1250 | |||

0.0500 | 0.1250 | 0.1000 | |||

0.1500 | 0.0500 | 0.0250 | |||

0.0833 | 0.0667 | 0.0917 | |||

0.0667 | 0.0583 | 0.0500 | |||

0.1333 | 0.0583 | 0.0167 | |||

0.0500 | 0.0500 | 0.1250 | |||

0.0500 | 0.0500 | 0.0500 | |||

0.1250 | 0.0250 | 0.0750 | |||

0.0500 | 0.0250 | 0.0750 | |||

0.1000 | 0.1250 | 0.0000 | |||

0.1250 | 0.0500 | 0.0000 | |||

0.0500 | 0.0250 | 0.0667 | |||

0.0833 | 0.0667 | 0.0333 | |||

0.0583 | 0.0167 | 0.0583 | |||

0.0500 | 0.0250 | 0.2000 | |||

0.1000 | 0.1750 | 0.0000 | |||

0.2000 | 0.0500 | 0.0000 | |||

0.0500 | 0.0500 | 0.0000 | |||

0.0500 | 0.0000 | 0.0500 | |||

0.0500 | 0.0250 | 0.0750 |

The closeness

Closeness of each evaluated value.

0.7500 | 0.7500 | 0.6486 | ||

0.9000 | 0.7105 | 0.7500 | ||

0.7317 | 0.7500 | 0.7500 | ||

0.5455 | 0.5294 | 0.6857 | ||

0.6923 | 0.7500 | 0.5000 | ||

0.6522 | 0.5625 | 0.8182 | ||

0.6667 | 0.7500 | 0.7500 | ||

0.6429 | 0.7241 | 0.6000 | ||

0.8108 | 0.8182 | 0.5625 |

The weight

Weight of each evaluated value.

0.3822 | 0.3696 | 0.3112 | ||

0.4027 | 0.3252 | 0.4054 | ||

0.3334 | 0.3520 | 0.3520 | ||

0.2780 | 0.2609 | 0.3290 | ||

0.3097 | 0.3433 | 0.2703 | ||

0.2972 | 0.2640 | 0.3840 | ||

0.3398 | 0.3696 | 0.3598 | ||

0.2876 | 0.3315 | 0.3243 | ||

0.3694 | 0.3840 | 0.2640 |

At this time, the interval Pythagorean fuzzy number WA operator (IPVFWA) can be used to synthesize the evaluated value weight of each decision-maker obtained in Step

Comprehensive evaluation matrix.

([0.5763 ,0.6806], [0.2295 ,0.3308]) | ([0.6218, 0.7427], [0.3711, 0.4717]) | ([0.6359,0.7546], [0.2512,0.4305]) | |

([0.5618,0.5746], [0.2919,0.3946]) | ([0.6124, 0.7147], [0.2537,0.3933]) | ([0.4305,0.5305], [0.1510,0.2545]) | |

([0.7633, 0.7569], [0.2051, 0.3956]) | ([0.6196, 0.8275], [0.3350,0.5000]) | ( 0.6000,0.7000], [0.2402,0.3834]) |

The positive and negative ideal solutions of the interval Pythagorean fuzzy comprehensive decision matrix, the Hamming distance between each evaluated value and the positive and negative ideal evaluated values, and the closeness degree and weight of each evaluated value were determined. The results are shown in Table

Positive and negative ideal matrices of the comprehensive matrix.

([0.7633,0.7569], [0.2919,0.3956]) | ([0.6218, 0.8275], [0.3711, 0.5000]) | ([0.6359, 0.7546], [0.2512, 0.4305]) | ||

([0.5618,0.5746], [0.2051,0.3308]) | ([0.6124, 0.7147], [0.2537, 0.3933]) | ([0.4305,0.5305], [0.1510,0.2545]) | ||

0.0976 | 0.0283 | 0.0000 | ||

0.0962 | 0.0866 | 0.1764 | ||

0.0217 | 0.0096 | 0.0371 | ||

0.0362 | 0.0583 | 0.1764 | ||

0.0376 | 0.0000 | 0.0000 | ||

0.1122 | 0.0770 | 0.1392 | ||

0.2707 | 0.6734 | 1.0000 | ||

0.2812 | 0.0000 | 0.0000 | ||

0.8379 | 0.8895 | 0.7894 | ||

0.1392 | 0.3464 | 0.5144 | ||

1.0000 | 0.0000 | 0.0000 | ||

0.3329 | 0.3534 | 0.3136 |

The weighted distance and closeness between the evaluated value of each scheme and the positive and negative ideal solutions were determined, and the results are shown in Table

According to the results in Table

Therefore, the TOPSIS hybrid multiattribute decision-making method based on interval Pythagorean fuzzy numbers has three advantages:

In the TOPSIS hybrid multiattribute decision-making method based on interval Pythagorean fuzzy numbers, the decision-making method was used to provide the upper and lower limits of the membership and nonmembership and then decisions were made through the TOPSIS decision-making method. This method alleviates the loss of information in the decision-making process to a great extent, making the decision-making process more accurate and scientific.

For mixed decision-making involving multiple decision-makers, when the decision-maker weight and attribute weight are unknown, the corresponding weight can be obtained from the decision matrix by using a TOPSIS hybrid multiattribute decision-making method based on interval Pythagorean fuzzy numbers. Compared with the case where the weights of the decision-makers are known, this method can effectively solve the problem of different types of attribute information and utilize the attribute advantages of each decision-maker to the greatest extent.

Through the interval Pythagorean fuzzy TOPSIS hybrid multiattribute decision-making method, the decision matrices of multiple decision-makers are collected into a comprehensive evaluation matrix, and then, the TOPSIS decision-making method is used to solve the optimal scheme. In the whole decision-making process, the closeness values of each scheme calculated by the TOPSIS decision-making method twice would be significantly different. This would allow decision-makers to more easily select the optimal scheme, making the decision result more convincing.

Weighted distance and closeness of the comprehensive matrix.

0.0234 | 0.0962 | 0.0223 | |

0.1160 | 0.0376 | 0.1082 | |

0.1678 | 0.7188 | 0.1705 | |

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In this paper, a mixed multiattribute decision-making problem was studied in which the decision-maker’s weight and attribute weight are completely unknown and the attribute value is an interval Pythagorean fuzzy number. A TOPSIS mixed multiattribute decision-making method based on interval Pythagorean fuzzy numbers is proposed. The advantages of this method are as follows: First, in the decision-making process, by giving the upper and lower limits of the membership and nonmembership of the evaluated value, the information loss of decision-makers in the decision-making process is alleviated to the greatest extent. Second, the decision-makers are no longer given a decision weight directly in decision-making, fully utilizing the attribute advantages of each decision-maker and effectively solving the problems of different types of attribute information. Third, compared with other decision-making methods, the TOPSIS hybrid multiattribute decision-making method using interval Pythagorean fuzzy numbers increases the gap between the advantages and disadvantages of each scheme. As a result, the decision-making results are more convincing and it is easier for decision-makers to select the optimal scheme. The TOPSIS hybrid multiattribute decision-making method using interval Pythagorean fuzzy numbers not only reduces the subjectivity of the weight determination to a great extent but also uses interval Pythagorean fuzzy numbers to improve the risk impact of the deviation between membership and nonmembership on the decision-making results.

The data analyzed by the example in this paper are interval values randomly given in line with the assumptions in this paper. The data are real and can be used as a reference for readers.

The authors declare that they have no conflicts of interest.

This work was supported by the General Project of National Natural Science Foundation of China, “Price Distortion of Energy Factors: Causes, Measurement and Appropriate Correction Strategy” (71874073); the project supported by the National Social Science Foundation of China, “Research on Knowledge Transfer Game and Innovation Performance of Industry University Research Alliance” (19fglb029); and NSFC, “Research on Coupling Mechanism and Promotion Strategy of Knowledge Subject Collaborative Behavior and Value Creation in Innovation Ecosystem” (71771161).