In recent years, the multi-attribute group decision making (MAGDM) problem has received extensive attention and research, and it plays an increasingly important role in our daily life. Fuzzy environment provides a more accurate decision-making environment for decision makers, so the research on MAGDM problem under fuzzy environment sets (SFSs) has become popular. Taxonomy method has become an effective method to solve the problem of MAGDM. It also plays an important role in solving the problem of MAGDM combined with other environments. In this paper, a new method for MAGDM is proposed by combining Taxonomy method with SFSs (SF-Taxonomy). In addition, we use entropy weight method to calculate the objective weight of attributes, so that more objective results can be produced when solving MAGDM problems.

In order to improve the accuracy of decision-making, Zadeh (

Taxonomy was proposed in 1763 and subsequently extended by a Polish mathematical group, and introduced as a means of classifying and determining levels of development (Jurkowska,

According to the existing literature on the study of SFSs, we have not found a method to use Taxonomy to solve the problem of MAGDM in the background of SFSs. Therefore, it is necessary to combine SFSs and Taxonomy to solve the MAGDM problem in this paper, which will provide a new method to solve the MAGDM problem in SFSs. This paper uses case analysis to carry on the concrete calculation, and also makes the relative comparison with the other methods which have been proved in this environment to confirm the practicability of this method. To this end, this paper has the following research ideas: (1) Use SFSs to express the decision maker’s (DM) overall evaluation of the method. (2) Combine Taxonomy method with SFSs, and present the specific calculation process. (3) Take car rental as an example to present the actual operation method of the algorithm. (4) Compare and verify the method in this paper with the existing method in this environment.

This paper is structured as follows: Firstly, SFSs and Taxonomy methods are briefly introduced and their applications are introduced. Secondly, in order to make readers better understand the method, we listed the formulas and calculation steps related to SFS and Taxonomy in this part. Later, we used the example to carry out specific operations. In order to verify the correctness of this method, we used the existing SF-VIKOR and SF-TOPSIS methods for verification. Finally, we compare and summarize the methods.

The definition of an SFSs, each

Some basic operations about SFSs.

Add operation

The multiplication

Multiplication by a scalar

For any set of fuzzy numbers

Spherical Weighted Arithmetic Mean (SWAM) and Spherical Weighted Geometric Mean (SWGM).

The calculation formula of the score function and the accuracy function is given below

The score function is used to compare the size of two fuzzy numbers. If the scoring functions are equal, then compare the calculations and compare the accuracy functions.

Note that

Euclidean distance formula:

Taxonomy was proposed in 1763, subsequently extended by a Polish mathematical group, and introduced as a means of classifying and determining levels of development (Jurkowska,

In this calculation,

If every row has a value that doesn’t fall within this range, it will not work, and the mean and standard deviation of each row will need to be calculated again.

Then, calculate the final progression order using the following formula:

In this section, we combine Taxonomy method with SFS (SF-Taxonomy) method to solve the problem of MAGDM. Let

The scoring function of the standard matrix is calculated, and the matrix obtained is normalized by the following formula:

Calculate the degree of entropy

Calculate the rate of degree of entropy (

A company needs to rent a car for a major event, and there are four types of car rental companies that can offer this service.

Decision matrix by DM_{1}.

Decision matrix by DM_{2}.

Decision matrix by DM_{3}.

Decision matrix by DM_{1}.

Decision matrix by DM_{2}.

Decision matrix by DM_{3}.

The overall decision matrix.

The

– | 0.1058 | 0.1396 | 0.1487 | |

0.10583 | – | 0.1181 | 0.1234 | |

0.1396 | 0.1181 | – | 0.1190 | |

0.1487 | 0.1234 | 0.1190 | – |

The SFPIS of each alternative.

Similarly, the upper limit of

The SFDA.

0.7669 | 0.4757 | 0.8867 | 0.5433 |

From the final value of

In order to verify the correctness of the SF-Taxonomy method, we adopted the examples and original data previously given in the paper, and adopted the SF-TOPSIS (Kutlu Gündoğdu and Kahraman,

The overall weight matrix.

The score function of the overall weight matrix.

−0.233 | −0.299 | −0.136 | 0.132 | |

−0.071 | −0.159 | −0.264 | 0.147 | |

−0.083 | −0.145 | −0.213 | 0.101 | |

−0.149 | −0.240 | −0.360 | 0.204 |

The SFPIS and SFNIS.

The distance between the overall weight matrix and the SFPIS and SFNIS.

0.033 | 0.078 | |

0.078 | 0.081 | |

0.068 | 0.074 | |

0.074 | 0.080 |

The closeness ratio of each alternative (

Closeness ratio | |

0.2947 | |

0.4893 | |

0.4789 | |

0.4807 |

According to the above calculation results of SF-TOPSIS method with the same data, we can get the final decision ranking of the scheme is

As above, we will also directly show the calculation results of SF-VIKOR method here.

The SFPIS and SFNIS.

(0.24,0.36,0.34) | (0.33,0.41,0.45) | (0.27,0.43,0.45) | (0.40,0.45,0.49) | |

(0.43,0.18,0.23) | (0.39,0.32,0.26) | (0.45,0.28,0.36) | (0.47,0.39,0.25) |

The weight distance

0.1977 | 0.3382 | 0.0100 | 0.0141 | |

0.1877 | 0.0558 | 0.0168 | 0.0523 | |

0.2805 | 0.4049 | 0.0117 | 0.0483 | |

0.2836 | 0.3740 | 0.0229 | 0.0160 |

The separation measures

0.3382 | 0.1877 | 0.4049 | 0.3740 | |

0.5599 | 0.3127 | 0.7455 | 0.6966 |

The

0.1877 | 0.4049 | ||

0.3127 | 0.7455 |

The

0.6320 | 0.0000 | 1.0000 | 0.8723 |

According to

In order to more clearly and intuitively see the results of these two methods and the SF-Taxonomy method, the results are shown in Table

The comparative analysis result.

Methods | Consequences |

SF-TAXONOMY | |

SF-TOPSIS | |

SF-VIKOR |

In order to improve the accuracy of comparison, we used the same case above to conduct a comparative study on the SF-TOPSIS method and the SF-VIKOR method, and found that the SF-Taxonomy method formed by applying the Taxonomy method in the SFS environment in this paper was objective and effective. The optimal solution is consistent when the optimal decision is made. There was little difference in the rankings for the rest. In the research of SF-Taxonomy method, entropy weight method is introduced to calculate the objective weight because the attribute weight is unknown, so as to make the result more accurate and objective.

Through the study of SFSs by scholars and the application of Taxonomy method in other backgrounds, this paper combines Taxonomy method with SFSs to form a new method to solve the multi-attribute decision problem in SFSs environment. In this paper, the concrete steps of SF-Taxonomy method are given. In order to make readers understand the method more clearly, the paper also gives the relevant calculation example analysis. In order to verify the correctness of such methods, the SF-TOPSIS method and the SF-VIKOR method, which have been confirmed by scholars, were compared in the following part of the paper, and relevant comparative analysis was made. The optimal scheme obtained by them in comparison is consistent, which confirms the correctness of this method. In the future, this approach could also have important applications in other contexts.