The interval-valued intuitionistic fuzzy sets (IVIFSs), based on the intuitionistic fuzzy sets (IFSs), combine the classical decision method and its research and application is attracting attention. After a comparative analysis, it becomes clear that multiple classical methods with IVIFSs’ information have been applied to many practical issues. In this paper, we extended the classical EDAS method based on the Cumulative Prospect Theory (CPT) considering the decision experts (DEs)’ psychological factors under IVIFSs. Taking the fuzzy and uncertain character of the IVIFSs and the psychological preference into consideration, an original EDAS method, based on the CPT under IVIFSs (IVIF-CPT-EDAS) method, is created for multiple-attribute group decision making (MAGDM) issues. Meanwhile, the information entropy method is used to evaluate the attribute weight. Finally, a numerical example for Green Technology Venture Capital (GTVC) project selection is given, some comparisons are used to illustrate the advantages of the IVIF-CPT-EDAS method and a sensitivity analysis is applied to prove the effectiveness and stability of this new method.

During the long history of humanity, conflicts between environmental issues and economic growth have always been one of the key issues (Yang

On the other hand, multiple criteria decision-making (MCDM) is usually classified (Hashemi-Tabatabaei

That being the case, it is necessary to design a suitable and advanced MAGDM model in a fuzzy environment to select the optimal GTVC project. To enhance the readability of the proposed research work, we have sorted out some important abbreviations, as shown in Table

Description of abbreviations.

Abbreviations | Description |

DE | Decision expert |

DM | Decision making |

FS | Fuzzy set |

IFS | Intuitionistic fuzzy set |

IVIFS | Interval valued intuitionistic fuzzy set |

IVIFN | Interval valued intuitionistic fuzzy number |

IVIFWA | Interval valued intuitionistic fuzzy weighted averaging |

IVIFWG | Interval valued intuitionistic fuzzy geometric |

SF | Score function |

AF | Accuracy function |

CPT | Cumulative prospect theory |

GTVC | Green technology venture capital |

PDA(NDA) | Positive (Negative) distance from average |

MAGDM | Multiple attribute group decision making |

CRITIC | CRiteria importance through intercriteria correlation |

EDAS | Evaluation based on distance from average solution |

TOPSIS | Technique for order preference by similarity to ideal solution |

TODIM | An acronym in Portuguese for interactive and multicriteria decision making |

MABAC | Muti-attributive border approximation area comparison |

The motivations of this study are given below:

With the increasing complexity of DM problems and the ambiguity of informative data, DEs find it is difficult to accurately quantify their evaluations using single numerals, but their preferences can be expressed more completely in natural language. Linguistic IVIFS theory is one of the most effective generalizations in FS theory, which can describe the assessment information of DEs in even more detail.

Compared with other evaluation methods, the EDAS method is a highly effective tool to execute classification and decision for contradictory attributes in MAGDM. Classical EDAS algorithms assume that all decision makers are perfectly rational, but people have different views on the same level of risks and benefits in practical environments. They are more conservative when facing equal risks than when facing returns. Therefore, it is necessary to develop a technology that simulates the real DM environment to model IVIF information and evaluate the weights of attributes. From this perspective, it is necessary to integrate the CPT, which takes the DEs’ risk preference into consideration, and the IVIF-EDAS method.

Due to the fact that the entropy method determines attribute weights based on the degree of the indicator confusion, the higher the degree of indicator information confusion, the greater the entropy value, and the smaller the assigned weights. Combined with the advantages of the EDAS method in solving DM problems with conflicting attributes, it is necessary to extend the information entropy method to handle qualitative information in the IVIF environment to guarantee the stability of the entire DM system.

As a key research issue in the MAGDM field, the selection of high-quality GTVC projects has a great significance for environmental protection and sustainable economic development. In this regard, considering the advantages of the EDAS method, the CPT’s characteristics and the IVIFSs which may contain more information, the aim of this paper is to extend the EDAS method based on CPT for MAGDM under IVIFSs and apply it to the GTVC project selection.

The contributions of this paper are as follows:

The integration of CPT and the EDAS method in IVIFSs, named as the IVIF-CPT-EDAS method, is discussed to determine which GTVC project is suitable. The attribute weights are determined by using IVIF-entropy method and alternatives are ranked by using CPT-EDAS method in a IVIF context. Therefore, our proposed method combines DEs evaluation values which makes it more profitable to use and the decision results more precise.

The IVIF-CPT-EDAS method, which considers not only the relatively simple and reasonable classical method, but also the psychological state of DEs, which is more realistic, has been constructed.

We implement the constructed approach to a numerical example for GTVC project selection to demonstrate the applicability of our proposed methodology.

We compare the constructed approach with the existing methods to illustrate the advantages of the IVIF-CPT-EDAS method. Furthermore, comparative analysis and sensitivity analysis are used to illustrate the effectiveness and authenticity of the developed approach.

In order to do so, the overall structure of the article is as follows: the basic knowledge of IVIFSs is briefly introduced in Section

Full text framework diagram.

In most practical decisions, the DEs’ evaluation of an index cannot be simply expressed by real numbers (Ye,

The EDAS method (Huang and Lin,

Characteristic table of different approaches.

Year | Approach | Linguistic data | Fuzzy information form | Application |

2012 | IVIF-TOPSIS (Izadikhah, |
✓ | Intuitionistic fuzzy number | Supplier selection |

2018 | EHFLTS-EDAS (Feng |
✓ | Hesitant fuzzy linguistic term set | Company project selection |

2019 | IVIF-VIKOR (Wu |
✕ | Interval-valued intuitionistic fuzzy set | Financing risk assessment |

2020 | IVIF-EDAS (Li and Wang, |
✕ | Intuitionistic fuzzy number | Computer network system assessment |

2022 | F-SECA (Keshavarz-Ghorabaee |
✓ | Triangular fuzzy numbers | E-waste scenario evaluation |

2022 | F-SBWM (Amiri |
✓ | Triangular fuzzy numbers | Warehouse location and medical selection |

2022 | PDHFWEPGMSM (Ning |
✕ | Probabilistic dual hesitant fuzzy set | Sustainable supplier selection |

2022 | IVIF-GRA (Zhang and Wang, |
✕ | Interval-valued intuitionistic fuzzy set | The service quality evaluation of agricultural e-commerce |

2023 | SF-CPT-TODIM (Zhang |
✕ | Spherical fuzzy set | Commercial insurance selection |

2023 | PHF-CPT-EDAS (Liao |
✕ | Probabilistic hesitant fuzzy set | Commercial vehicles and green supplier selection |

Current study | ✓ | Interval-valued intuitionistic fuzzy set | Green technology venture capital selection |

According to the literature review, it is evident that while some scholars have successfully utilized the CPT-EDAS method in various fuzzy environments (Zhang

In the entire process of GTVC, project screening and evaluation are crucial steps. However, since GTVC primarily focuses on emerging technology projects related to environmental protection and improvement, the evaluation of such projects often involves numerous qualitative indicators and cognitive limitations, which can also cause significant ambiguity. To address this issue, it is important to construct a scientific evaluation index system for green finance risk capital projects and establish an appropriate evaluation model. This will bridge the gap and help venture investors make informed and optimal investment decisions related to these projects.

After reviewing the existing research on GTVC projects, we can state that there is a lack of emphasis on the psychological well-being. However, the psychological state of DM is an inevitable objective presence in influencing their investment decisions. In order to address this gap, the improved EDAS algorithm has been applied to GTVC, offering new methods and tools for venture capitalists to effectively screen green investment projects.

Overall, it is meaningful to use the IVIF-CPT-EDAS method for the GTVC project selection. This study aims to promote the future development of the venture capital field, provide solutions to guide practical cases, and also provide references for research on DM methods and theories.

In order to better understand the content of this article, in this section, we will review some basic concepts of FSs and aggregation operators of IVIFSs.

Set

Set

Set

Usually, the interval-valued intuitionistic fuzzy number (IVIFN) (Xu and Chen,

Especially when

Suppose any three IVIFNs

Commutative laws:

Associative laws:

Distributive laws:

Exponential operation laws:

Let

If

According to Definition

If

If

If

If

If

If

Let a set of

Especially when

Let a set of

Especially, when

Suppose two IVIFNs

Keshavarz-Ghorabaee

We assume that the set of decision-makers is

As previously mentioned, the IVIF-MAGDM matrices (

The flowchart of IVIF-CPT-EDAS.

Then, the IVIF-CPT-EDAS detailed steps are as follows (Fig.

Integrate the IVIF decision matrix

Transform the IVIF decision matrix

The specific steps of IVIF-entropy method are shown as follows:

Determine the IVIF negative ideal point (IVIF-NIP):

Calculate the hybrid distance matrix

The normalized distance matrix

Calculate the entropy degree

Figure out the original attribute weight

Development is a timeless topic. However, due to the limitations and irreversibility of resources, it has become a new demand to reduce pollution, save resources, and take the path of sustainable development by developing a green economy. From an investment perspective, GTVC, as one of the five key green investment areas, has a broad space to promote the development of the circular economy, such as comprehensive utilization of resource technology, environmental protection technology, ecological agriculture technology, comprehensive utilization of resource technology, new energy technology, etc. At the same time, evaluating investment projects that can yield significant benefits scientifically and quickly is particularly important for a decision-maker. Therefore, this paper takes the GTVC project selection as an example to verify the effectiveness and applicability of the enhanced EDAS method.

We invite five experts

Here, we will refer to the IVIF linguistic evaluation scale proposed by Peng and Luo (

IVIF linguistic evaluation scale.

Linguistic scale | IVIFN |

Extremely terrible (ET) | |

Very terrible (VT) | |

Terrible (T) | |

Medium terrible (MT) | |

Medium (M) | |

Medium good (MG) | |

Good (G) | |

Very good (VG) | |

Extremely good (EG) | |

Perfectly good (PG) |

The linguistic assessment matrices by five experts.

DEs | Alternatives | Attributes | |||||

MG | G | M | G | MG | MG | ||

VG | M | EG | M | EG | G | ||

MG | VG | M | MG | G | M | ||

M | G | MT | M | MG | T | ||

G | VT | M | M | G | MG | ||

G | MG | MG | G | M | MG | ||

EG | MT | EG | M | EG | G | ||

G | VG | M | M | MG | MT | ||

MT | G | M | M | G | T | ||

MG | MT | MT | MT | MG | M | ||

MG | G | G | MG | G | M | ||

EG | M | VG | G | EG | G | ||

MG | G | MG | MG | M | MT | ||

M | G | M | MG | G | MT | ||

G | M | VT | M | MG | MG | ||

G | G | M | G | G | MG | ||

EG | M | EG | M | VG | VG | ||

M | EG | MG | MG | MG | M | ||

MT | G | MG | M | G | MT | ||

G | MT | MT | M | MG | M | ||

VG | MG | G | G | MG | G | ||

EG | MT | EG | MG | VG | VG | ||

MG | VG | G | MG | G | M | ||

M | G | MG | M | VG | MT | ||

VG | M | MT | M | M | MG |

Next, the IVIF-CPT-EDAS technique is developed for selecting the best GTVC project.

The normalized IVIF decision matrix

Using the decision information in

Calculate the distance between each alternative solution and IVIF-NIS by employing Eq. (

Figure out the entropy degree

Compute the original weight

IVIF-NIS

IVIF-NIP | |||

IVIF-NIP | |||

The normalized hybrid distance matrix

Hybris distance | ||||||

0.209 | 0.138 | 0.217 | 0.000 | 0.022 | 0.260 | |

0.364 | 0.306 | 0.466 | 0.182 | 0.605 | 0.393 | |

0.131 | 0.000 | 0.193 | 0.176 | 0.144 | 0.122 | |

0.000 | 0.120 | 0.124 | 0.337 | 0.229 | 0.000 | |

0.296 | 0.436 | 0.000 | 0.354 | 0.000 | 0.225 |

The entropy

0.821 | 0.778 | 0.785 | 0.839 | 0.625 | 0.814 | |

0.179 | 0.222 | 0.215 | 0.161 | 0.375 | 0.186 |

The original attribute weight

Original attribute weight | ||||||

0.134 | 0.166 | 0.160 | 0.121 | 0.280 | 0.139 |

The average solution

Average solution | |||

Average solution | |||

The relative attribute matrix

0.202 | 0.230 | 0.226 | 0.190 | 0.314 | 0.219 | |

0.215 | 0.238 | 0.234 | 0.190 | 0.308 | 0.219 | |

0.202 | 0.230 | 0.226 | 0.190 | 0.314 | 0.207 | |

0.202 | 0.230 | 0.226 | 0.204 | 0.314 | 0.207 | |

0.215 | 0.238 | 0.226 | 0.204 | 0.314 | 0.219 |

The positive distance

0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.387 | |

1.436 | 0.849 | 2.131 | 0.000 | 1.412 | 1.444 | |

0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

0.000 | 0.000 | 0.000 | 0.508 | 0.000 | 0.000 | |

0.770 | 1.840 | 0.000 | 0.847 | 0.000 | 0.120 |

The negative distance

0.740 | 1.824 | 0.775 | 2.566 | 2.068 | 0.000 | |

0.000 | 0.000 | 0.000 | 0.418 | 0.000 | 0.000 | |

2.464 | 4.251 | 1.333 | 0.491 | 1.023 | 2.170 | |

5.126 | 2.151 | 2.799 | 0.000 | 0.384 | 4.124 | |

0.000 | 0.000 | 5.184 | 0.000 | 2.240 | 0.000 |

The normalized weighted distance.

The normalized weighted distance | |||||

0.048 | 1.000 | 0.000 | 0.059 | 0.456 | |

0.400 | 0.975 | 0.158 | 0.000 | 0.403 |

The overall assessment score

Overall assessment score | |||||

0.224 | 0.987 | 0.079 | 0.029 | 0.429 |

It follows from the above:

In this section, we incorporate the effects of CPT on the model and refer to the parameter values given by Tversky and Kahneman (

We assign the parameter

The calculation results with different value of

Ranking results | The optimal solution | ||||||

0.61 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

0.05 | 0.228 | 0.987 | 0.079 | 0.039 | 0.458 | ||

0.15 | 0.227 | 0.987 | 0.079 | 0.037 | 0.453 | ||

0.25 | 0.227 | 0.987 | 0.079 | 0.035 | 0.448 | ||

0.35 | 0.226 | 0.987 | 0.079 | 0.034 | 0.443 | ||

0.45 | 0.225 | 0.987 | 0.079 | 0.032 | 0.438 | ||

0.55 | 0.224 | 0.987 | 0.079 | 0.030 | 0.433 | ||

0.65 | 0.224 | 0.987 | 0.079 | 0.029 | 0.427 | ||

0.75 | 0.223 | 0.987 | 0.079 | 0.027 | 0.422 | ||

0.85 | 0.222 | 0.987 | 0.079 | 0.026 | 0.417 | ||

0.95 | 0.222 | 0.987 | 0.079 | 0.025 | 0.411 |

According to Table

We assign the parameter

The calculation results of different value of parameter

Ranking results | The optimal solution | ||||||

0.69 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

0.05 | 0.259 | 0.985 | 0.106 | 0.029 | 0.492 | ||

0.15 | 0.254 | 0.985 | 0.102 | 0.029 | 0.484 | ||

0.25 | 0.251 | 0.986 | 0.098 | 0.029 | 0.475 | ||

0.35 | 0.246 | 0.986 | 0.094 | 0.029 | 0.465 | ||

0.45 | 0.240 | 0.986 | 0.090 | 0.029 | 0.455 | ||

0.55 | 0.238 | 0.987 | 0.086 | 0.029 | 0.445 | ||

0.65 | 0.227 | 0.987 | 0.081 | 0.029 | 0.434 | ||

0.75 | 0.219 | 0.988 | 0.076 | 0.029 | 0.423 | ||

0.85 | 0.211 | 0.988 | 0.071 | 0.029 | 0.411 | ||

0.95 | 0.202 | 0.988 | 0.066 | 0.029 | 0.398 |

According to Table

We assign the parameter

The calculation results with different value of parameter

Ranking results | The optimal solution | ||||||

0.88 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

0.05 | 0.275 | 0.987 | 0. 079 | 0. 079 | 0.539 | ||

0.15 | 0.265 | 0.987 | 0. 079 | 0. 070 | 0.515 | ||

0.25 | 0.257 | 0.987 | 0. 079 | 0.062 | 0.495 | ||

0.35 | 0.250 | 0.987 | 0. 079 | 0.056 | 0.479 | ||

0.45 | 0.244 | 0.987 | 0. 079 | 0.049 | 0.466 | ||

0.55 | 0.238 | 0.987 | 0. 079 | 0.044 | 0.455 | ||

0.65 | 0.233 | 0.987 | 0. 079 | 0.039 | 0.446 | ||

0.75 | 0.229 | 0.987 | 0. 079 | 0.035 | 0.437 | ||

0.85 | 0.225 | 0.987 | 0. 079 | 0.031 | 0.431 | ||

0.95 | 0.222 | 0.987 | 0. 079 | 0.027 | 0.426 |

According to Table

We assign the parameter

The calculation result with different value of parameter

Ranking results | The optimal solution | ||||||

0.88 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

0.05 | 0.094 | 0.936 | 0. 000 | 0. 094 | 0.515 | ||

0.15 | 0.099 | 0.945 | 0. 000 | 0. 083 | 0.498 | ||

0.25 | 0.106 | 0.953 | 0. 000 | 0.069 | 0.482 | ||

0.35 | 0.114 | 0.960 | 0. 000 | 0.053 | 0.465 | ||

0.45 | 0.122 | 0.967 | 0. 000 | 0.034 | 0.448 | ||

0.55 | 0.144 | 0.973 | 0. 016 | 0.029 | 0.440 | ||

0.65 | 0.169 | 0.978 | 0. 036 | 0.029 | 0.436 | ||

0.75 | 0.194 | 0.983 | 0. 055 | 0.029 | 0.433 | ||

0.85 | 0.217 | 0.986 | 0. 074 | 0.029 | 0.430 | ||

0.95 | 0.240 | 0.989 | 0. 091 | 0.029 | 0.428 |

According to Table

We assign the parameter

The calculation results with different value of parameter

Ranking results | The optimal solution | ||||||

2.25 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

1.55 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

2.55 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

3.55 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

4.55 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

5.55 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

6.55 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

7.55 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

8.55 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

9.55 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 | ||

10.00 | 0.224 | 0.987 | 0.079 | 0.029 | 0.429 |

According to Table

From the simulation analysis of the above five parameters, it can be seen that despite the slight fluctuations in the model calculation and ranking results caused by changing the parameters in the weight and value functions, the optimal and sub-optimal schemes remain unchanged, demonstrating the IVIF-CPT-EDAS model’s robustness and effectiveness.

In this subsection, we continue to explore the effectiveness of the enhanced EDAS approach based on the same original matrices, the expert weighting vector

The original matrices of this article are substituted by Eq. (

The calculation results of IVIFWA operator.

Alternative | IVIFWA operator | Ranking results | ||

0.176 | 0.809 | 3 | ||

0.588 | 0.863 | 1 | ||

0.126 | 0.812 | 4 | ||

0.099 | 0.803 | 5 | ||

0.240 | 0.816 | 2 |

The calculation results of IVIF-TOPSIS.

Alternative | Ranking results | |||

0.129 | 0.061 | 0.320 | 3 | |

0.013 | 0.177 | 0.932 | 1 | |

0.150 | 0.040 | 0.212 | 5 | |

0.145 | 0.045 | 0.238 | 4 | |

0.104 | 0.085 | 0.449 | 2 |

We substitute the initial data of this article by the IVIF-TOPSIS method (Izadikhah,

The DM process is as follows:

We substitute the initial data of this article by the IVIF-Taxonomy method (Xiao

The calculation results of IVIF-Taxonomy.

Alternative | Ranking results | |

0.617 | 3 | |

0.095 | 1 | |

0.708 | 4 | |

0.741 | 5 | |

0.496 | 2 |

The calculation results of IVIF-TODIM.

Alternative | Ranking results | |

0.180 | 3 | |

1.000 | 1 | |

0.018 | 4 | |

0.000 | 5 | |

0.623 | 2 |

The initial data of this article are substituted by the IVIF-TODIM method (Krohling and Pacheco,

The ranking results of different DM methods.

Methods | Order | The best choice | The worst choice |

IVIFWA (Xu and Chen, |
|||

IVIF-TOPSIS (Izadikhah, |
|||

IVIF-TOXONOMY (Xiao |
|||

IVIF-TODIM (Krohling and Pacheco, |
|||

IVIF-CPT-EDAS |

The score value of different DM methods.

The ranking results of different DM methods.

After conducting the validity analysis above and analysing the research results presented in this article, we can conclude that

In this article, we modified the traditional EDAS method for settling the MAGDM issues better by combining IVIFs. First of all, we briefly review the fundamental definition of IVIFSs, aggregation operator and distance formula. We are aware that the EDAS method is highly effective in resolving DM issues involving contradictory attributes. Therefore, the prominent advantage of introducing the enhanced EDAS technology to IVIFSs in order to address MAGDM problems due to the influence of the actual environment is that it not only lets the average scheme reduce the impact of extreme data, but also takes into account the proximity of the distance between each attribute and the average scheme. Simultaneously, the implementation of the entropy weight method also guarantees the stability of the DM system. Finally, we verified the feasibility of the developed technique by applying the IVIF-CPT-EDAS method to a numerical example of GTVC. Furthermore, sensitivity analysis and several contrast analyses are conducted to validate the rationality and feasibility of MAGDM problems in the practical application process, respectively. Therefore, the main advantages of this paper can be shown as follows: (1) The attribute weights via exploiting information entropy method in IVIFS are constructed. (2) The IVIF-CPT-EDAS approach is designed for addressing MAGDM issues. (3) The presented method of this study not only provides DEs with a broader space for information expression, but also fully considers the psychological characteristics of DEs when facing gains and losses, providing inspiration for subsequent research on DM methods under the framework of bounded rationality.

Nonetheless, the proposed method also has some shortages. On the one hand, the technology developed in this paper solely takes into account the scenario where evaluation information is provided in IVIFs, and real-world evaluation might encounter diverse DM environments. On the other hand, we should thoroughly take into account the genuine psychological alteration trend of decision-makers. Therefore, in future research, we will concentrate on creating novel models and functions for determining attribute weights, enabling them to vary dynamically based on the data. Simultaneously, with the advancement of network technology, numerous DM methods in network form have arisen, including microblog, WeChat, and other online voting forms. In theory, when confronted with such complex DM, we can also extend the findings of this paper to the realm of complex DM that evolves with time.

This article does not contain any studies with human participants or animals performed by any of the authors.

The authors declare that they have no conflicts of interest.

The data used to support the findings of this study are included within the article.

The work was supported by the Sichuan Province Social Development Key R&D Projects under Grant No. 2023YFS0375 and Natural Science Foundation of Sichuan Province under Grant No. 2022NSFSC1821.