Due to the complexity of decision-making problems and the subjectivity of decision-makers in practical application, it is necessary to adopt different forms of information expression according to the actual situation of specific decision-making problems and choose the best method to solve them. Multi-valued neutrosophic set, as an extension of neutrosophic set, can more effectively and accurately describe incomplete, uncertain or inconsistent information. TODIM and TOPSIS methods are two commonly used multi-attribute decision-making methods, each of which has its advantages and disadvantages. This paper proposes a new method based on TODIM and TOPSIS to solve multi-attribute decision-making problems under multi-valued neutrosophic environment. After introducing the related theory of multi-valued neutrosophic set and the traditional TODIM and TOPSIS methods, the new method based on a combination of TODIM and TOPSIS methods is described. And then, two illustrative examples proved the feasibility and validity of the proposed method. Finally, the result has been compared with some existing methods under the same examples and the proposed method's superiority has been proved. This paper studies this kind of decision-making problem from algorithm idea, algorithm steps and decision-making influencing factors.

Multi-attribute decision-making (MADM) problem refers to how to rank alternatives and select the optimal one when the decision-making problem contains multiple attributes. As an important part of modern decision-making science, it has made significant progress and wide application since the early 1960s. Its methodology and theory have been applied to many fields, such as investment decision-making [

A complete MADM process mainly consists of two core parts: representation of decision information and integration of decision information. The representation of decision information refers to using correct language to express decision information correctly. Due to the complexity of the objective problems and the vagueness of the subjective thinking of decision makers in practical applications, it is difficult for decision makers to give accurate evaluation values for MADM problems with inaccurate, uncertain or incomplete information. In this case, fuzzy language is considered to be the best tool to express fuzzy information. Therefore, there emerge a lot of sets to describe uncertain decision information. Such as fuzzy set, soft set, neutrosophic set (NS) [

The second core part of MADM is integrating decision information and the ranking or selection of alternatives. At present, commonly used decision-making methods mainly contain TODIM (TOmada de Decisao Iterativa Multicriterio) [

As a most used MADM method, TODIM method takes psychological behavior of decision makers into consideration which is more in line with actual decision-making situations. Therefore, it has been developed rapidly and widely used since it was proposed by Gomes et al. [

TOPSIS method is also a popular MADM method for its efficient and straightforward calculation. Rıdvan et al. [

Based on different methodologies, each MADM method has its advantages, disadvantages and scope of application. For this, with the increasing number of decision-making methods, more and more researchers begin to consider combining the existing methods to propose some improved methods. Some studies of combining TODIM or TOPSIS with other methods are proposed. Ji et al. [

Based on the above analysis, this paper proposes a method by combining the TODIM and TOPSIS methods under the MVNS, considering the bounded rationality of decision makers. The main motivations behind this paper are as follows:

TODIM method fully considers the decision makers’ risk aversion attitude on the basis of prospect theory, and can reflect the decision makers’ risk preference by adjusting the parameters, which is more in line with the actual decision-making requirements. However, any two alternatives need to be compared in TODIM method and also lead to high computational complexity. It is meaningful to find an improvement method that not only reflects the bounded rationality of decision makers, but also overcome its computational complexity.

TOPSIS method sorts the alternatives according to their closeness to the ideal solution (PIS and NIS) rather than making a pairwise comparison of any two alternatives thereby reduces the number of comparisons between alternatives and makes the calculation simple. But it only considers the closeness between each alternatives and the PIS (NIS), and does not taking the decision makers’ risk aversion attitude into account. Therefore, finding an improvement method based on TOPSIS method to reflect the bounded rationality of decision makers is significant.

MVNS is a good tool to describe the incomplete, indeterminate and inconsistent information accurately. Therefore it is necessary to propose a hybrid method under MVNS environment that not only has concise calculation process and scientific results but considers the decision makers’ risk attitude.

In summary, inspired from these two methods, this paper proposed a new method based on TOPSIS and TODIM methods under MVNS environment. This hybrid method not only has concise calculation like TOPSIS method but also takes the decision makers’ aversion attitude into account, which makes it has good scientificity, accuracy and maneuverability.

The reminder of this paper unfolds as

Section | Contents |
---|---|

Section 2 | Some basic notions of MVNS and its related definitions are listed. |

Section 3 | The traditional TODIM and TOPSIS methods are reviewed. |

Section 4 | A hybrid MADM method based on TODIM and TOPSIS methods under MVNS environment is proposed. |

Section 5 | Two illustrate cases about personnel selection and teaching evaluation are used to verify the feasibility of the proposed method. |

Section 6 | The sensitivity and comparative analysis of the two cases are given to verify the validity and superiority. |

Section 7 | The conclusions drawn are presented. |

For a better understanding the content of this paper, this section introduces some basic concepts of NSs and MVNSs and their related definitions.

So,

Especially,

if

if

if

if

For convenience, suppose that

The traditional TODIM method is a MADM method proposed by Gomes et al. [

Suppose that there are

Step 1: Standardize the decision information. That is, normalizing

Step 2: Figure out

Step 3: Figure out the dominance degree of

Step 4: Work out the overall dominance of

Step 5: Ranking all alternatives according to the value of

It can be seen that TODIM method fully considers the decision makers’ risk aversion attitude on the basis of prospect theory, and can reflect the decision makers’ risk preference by adjusting the parameters, thereby resulting in a more accurate decision-making process. However, decision-making information needs to be standardized firstly. Besides, any two alternatives also need to be compared in TODIM method in the case of more decision-making data, which leads to high computational complexity and large amount of calculation (such as distance and expectation values).

TOPSIS is a MADM method based on geometric thinking, which ranks candidate alternatives relatively far from ideal solutions and negative ideal solutions. On the basis of the TOPSIS method with IHFS in paper [

Let alternatives are

Step 1: Get the decision matric and the corresponding weight.

Step 2: Figure out the weighted normalization matrix.

Step 3: Determine the positive ideal solution (PIS)

Step 4: Calculate the distance between

Step 5: Figure out the relative approximation coefficients of each alternative according to

Step 6: Ranking the alternative according to the value of

TOPSIS method puts forward PIS (NIS) as reference point and only needs to compare each alternative with the ideal solution, which reduces the number of comparisons between alternatives when facing more decision-making data. So the calculation of TOPSIS method is simpler than TODIM method. However, from the formula of relative approximation coefficients, it can be seen that this method only uses distance measure to judge the quality of alternatives, and does not take the decision makers’ risk aversion attitude into account, which lacks subjectivity.

When faced with a large amount of decision-making data, to consider the risk attitude of decision makers and simplify the calculation process, a hybrid MADM method based on TOPSIS and TODIM is proposed in this section.

Suppose in a MADM problem, there are

and if

where

The parameter

It is obvious that the proposed method does not need to standardize the overall decision-making data according to attribute classification and it does not need to calculate the distance between any two alternatives and the expectation value of each alternative. In this sense, the proposed method is simpler than TODIM method in calculation. Besides, the proposed method is also based on TODIM method, taking the bounded rationality of decision makers into account, thus making up for the shortcoming of TOPSIS method. Therefore, facing to large number of alternatives, the proposed method can reduce the computational complexity and load more than TODIM or TOPSIS, which lead to a more simple decision-making process and a more reasonable result.

In this section, two illustrate examples of the personnel selection problem and teaching evaluation problem adapted from Wang et al. [

Personnel selection is an important part of modern human resource management and also an important part of the company's operation. It is directly related to the company's output quality and efficiency. Good talents are the guarantee of good products. In the personnel selection process, the interviewer needs to evaluate candidates based on their performance in various aspects, which can be regarded as an MADM process. Thinking about the ambiguity and complexity of people's thinking and language, MVMS can be used to express the evaluation information of the interviewer.

Suppose that a food company wants to hire a product manager. After preliminary screening of dozens of resumes, there are four candidates in the talent market that meet the recruitment requirements, which are denoted by

The original decision matrix is obtained as shown in

According to the attributes characteristics,

According to

According to

0.200 | 0.250 | 0.550 | 0.364 | 0.455 | 1.00 | 1.818 |

Let the parameter

Finally, the overall dominances are shown in

Ranking | Optimal choice | ||||
---|---|---|---|---|---|

0.395 | 0.677 | 0.000 | 1.000 |

Teaching evaluation is an important means for education administrative departments to supervise the teaching quality of colleges and universities. The purpose is to improve the quality of education and teaching. It is not only an important part of the teaching process but the basis of effective teaching and successful teaching. Specifically, according to certain teaching goals and standards, such as teaching completion, teaching concepts, teaching quality, etc., it systematically detects teachers’ teaching and evaluates its value, advantages and disadvantages, in order to improve. This can also be regarded as a MADM problem. Due to the complexity of the reality, its attribute values are often not directly represented by crisp set. In this condition, MVNS has become a good tool for expressing attribute values.

Assume that an education administration department in a certain place wants to evaluate the teaching of four local universities which are denoted by

The original decision matrix is obtained as shown in

According to the description,

Then, the distance

Next, according to

0.250 | 0.350 | 0.400 | 0.625 | 0.875 | 1.000 | 2.500 |

Assume the parameter

Finally, the overall dominances are shown in

Ranking | Optimal choice | ||||
---|---|---|---|---|---|

0.413 | 1.000 | 0.000 | 0.846 |

Since the proposed method is based on TODIM method, the parameter

As mentioned in Section 4, parameter

Ranking | Optimal choice | |
---|---|---|

0.01–11.12 | ||

11.13–15.00 |

Ranking | Optimal choice | |
---|---|---|

0.01–2.44 | ||

2.45–5.00 |

In summary, when the parameter

In order to verify the effectiveness, feasibility and superiority of the proposed methodology, we compared the results of the two cases applied in Section 5 with some existing methods. Details are as follows:

As shown in

Method | Author | Year | Ranking | Optimal choice |
---|---|---|---|---|

TODIM ( |
Wang et al. [ |
2015 | ||

MVNWBM ( |
Liu et al. [ |
2016 | ||

MVNWGBM ( |
Liu et al. [ |
2016 | ||

VIKOR | Liu et al. [ |
2017 | ||

TOPSIS | Giri [ |
2020 | ||

The proposed method ( |

Method | Author | Year | Ranking | Optimal choice |
---|---|---|---|---|

SVNHFWA | Ye [ |
2014 | ||

SVNHFWG | Ye [ |
2014 | ||

TODIM ( |
Wang et al. [ |
2015 | ||

Hamming distance | Şahin [ |
2015 | ||

Correlation coefficient | Şahin et al. [ |
2017 | ||

SVNHFCOA | Li et al. [ |
2018 | ||

SVNHFCOG | Li et al. [ |
2018 | ||

The proposed method ( |
||||

The proposed method ( |

From

In summary, among the above methods, only the proposed method and TODIM method take the decision makers bounded rationality into account, which can make the result more accurate. But when the alternatives or attributes increase, the calculation amount of the proposed method is much smaller than that of the traditional TODIM method. In this sense, the proposed method is superior to the TODIM method. Thus, the advantages of the proposed method are as follows:

Based on the TODIM method, it considers the bounded rationality of decision makers and reflect the risk preference of decision makers by adjusting the loss attenuation coefficient

The proposed method does not need to standardize the original matrix according to different attributes, which simplifies the calculation process to a certain extent.

The proposed method uses PIS and NIS in the TOPSIS method as reference points, and uses the distance between the alternatives and the reference point to replace the pairwise comparison between alternatives, which makes the calculation simple. When faced with a large amount of decision-making data, the proposed method can use relatively simple calculations to obtain more reasonable result.

As a subset of neutrosophic set, MVNS is an excellent tool to describe uncertainty, incomplete and imprecise information. The elements in the truth-membership, the indeterminacy-membership and the falsity-membership in MVNS are extended to finite sets of discrete values, which enriched the expression of fuzzy information on the basis of fuzzy languages such as SVNS and INS. Therefore, based on the traditional TODIM and TOPSIS methods and the related research achievements of MVNS, this paper proposed a new method to handle the MCDM problems under MVNS environment. First, the traditional TODIM and TOPSIS methods are reviewed. And then, the specific steps of the hybrid method are introduced. In the new method, the distance between the alternatives and the ideal solutions is calculated by standardized Hamming distance of MVNS. Subsequently, the proposed method is used to solve two MADM problems about company personnel selection and university teaching evaluation under MVNS environment. These two cases all illustrate the feasibility and applicability of the proposed method. Moreover, by adjusting the loss attenuation coefficient

There are several directions for future research. Firstly, in addition to personnel selection and teaching evaluation, the proposed method should be used in more fields, such as venture capital selection, medical diagnoses, supplier selection, and so on. In addition, the proposed method assumes that the weight of the attribute is given by the decision maker, and does not discuss the situation with unknown attribute weight. Therefore, in the future, we will devote ourselves to improving the method to cover this deficiency and applying it to natural and complex decision-making processes.

Thanks to the co-authors for their help in the research process, for providing language assistance, writing assistance and proofreading for this article.