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Double-Level Multi-attribute Group Decision-making Method Based on Intuitionistic Fuzzy Theory and Evidence Reasoning

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Abstract

Intuitionistic fuzzy (IF) theory and evidence reasoning (ER) are two commonly used tools in the decision-making process. They have their own advantages and disadvantages in information expression and information aggregation. How to use these two methods’ strengths to make up for each other’s shortcomings in the decision-making process is a very interesting topic. We refine IF expressions of decision attributes and develop a conversion formula about membership degree and belief degree, which unifies IF theory and ER. Then, we propose a double-level multi-attribute group decision-making method based on IF theory and ER. In the new method, we study the setting and aggregation of weights under double-level attributes, and optimize the process of information aggregation. The proposed method is verified by a detailed case study, which shows its validity and stability. It is expected that this method, which combines IF theory and ER, will become a useful tool for multi-attribute group decision-making.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Funding

The work was supported by the National Natural Science Foundation of China (No. 72071135).

Author information

Authors and Affiliations

  1. Business School, Sichuan University, Chengdu, 610064, Sichuan, China

    Xuecheng Fan & Zeshui Xu

Authors
  1. Xuecheng Fan
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  2. Zeshui Xu
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Corresponding authors

Correspondence to Xuecheng Fan or Zeshui Xu.

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Appendices

Appendix 1

The expert \({V}_{2}\)’s IF evaluations

  1. (1)

    Comprehensive index \({\mathrm{y}}_{1}\)  

Basic Index

\(\boldsymbol{{e}_{1}}\)  

\(\boldsymbol{{e}_{2}}\)  

\(\boldsymbol{{a}_{1}}\)  

u

(\({H}_{3},0.7\))(\({H}_{4},0.8\))(\({H}_{5},0.5\))

(\({H}_{3},0.3\))(\({H}_{4},0.2\))(\({H}_{5},0.2\))

(\({H}_{4},0.7\))(\({H}_{5},0.5\))

(\({H}_{4},0.1\))(\({H}_{5},0.3\))

v

\(\boldsymbol{{a}_{2}}\)  

u

(\({H}_{3},0.6\))(\({H}_{4},0.8\))

(\({H}_{3},0.3\))(\({H}_{4},0.2\))

(\({H}_{3},0.8\))(\({H}_{4},0.6\))

(\({H}_{3},0.2\))(\({H}_{4},0.1\))

v

\(\boldsymbol{{a}_{3}}\)

u

(\({H}_{2},0.6\))(\({H}_{3},0.4\))

(\({H}_{2},0.2\))(\({H}_{3},0.4\))

(\({H}_{2},0.7\))(\({H}_{3},0.6\))

(\({H}_{2},0.2\))(\({H}_{3},0.3\))

v

  1. (2)

    Comprehensive index \({\mathrm{y}}_{2}\)  

Basic Index

\(\boldsymbol{{e}_{3}}\)  

\(\boldsymbol{{e}_{4}}\)  

\(\boldsymbol{{e}_{5}}\)  

\(\boldsymbol{{a}_{1}}\)  

u

(\({H}_{3},0.5\))(\({H}_{4},0.8\))(\({H}_{5},0.7\))

(\({H}_{3},0.2\))(\({H}_{4},0.2\))(\({H}_{5},0.1\))

(\({H}_{2},0.8\))(\({H}_{3},0.5\))

(\({H}_{2},0.1\))(\({H}_{3},0.5\))

\(({H}_{3},0.6\))(\({H}_{4},0.8\))

v

\(({H}_{3},0.3\))(\({H}_{4},0.1\))

\(\boldsymbol{{a}_{2}}\)  

u

(\({H}_{3},0.8\))(\({H}_{4},0.7\))(\({H}_{5},0.4\))

(\({H}_{3},0.2\))(\({H}_{4},0.3\))(\({H}_{5},0.3\))

(\({H}_{2},0.9\))(\({H}_{3},0.7\))

(\({H}_{2},0.1\))(\({H}_{3},0.3\))

(\({H}_{2},0.6\))(\({H}_{3},0.7\))

v

(\({H}_{2},0.1\))(\({H}_{3},0.2\))

\(\boldsymbol{{a}_{3}}\)  

u

(\({H}_{3},0.6\))(\({H}_{4},0.7\))

(\({H}_{3},0.4\))(\({H}_{4},0.2\))

(\({H}_{3},0.8\))(\({H}_{4},0.8\))

(\({H}_{3},0.2\))(\({H}_{4},0.2\))

(\({H}_{3},0.6\))(\({H}_{4},0.8\))

v

(\({H}_{3},0.1\))(\({H}_{4},0.2\))

  1. (3)

    Comprehensive index \({y}_{3}\)

Basic Index

\(\boldsymbol{{e}_{6}}\)  

\(\boldsymbol{{e}_{7}}\)  

\(\boldsymbol{{e}_{8}}\)  

\(\boldsymbol{{a}_{1}}\)  

u

(\({H}_{1},0.5\))(\({H}_{2},0.6\))

(\({H}_{1},0.4\))(\({H}_{2},0.3\))

(\({H}_{2},0.6\))(\({H}_{3},0.7\))

(\({H}_{2},0.2\))(\({H}_{3},0.1\))

\(({H}_{1},0.4\))(\({H}_{2},0.7\))(\({H}_{3},0.7\))

v

\(({H}_{1},0.5\))(\({H}_{2},0.3\)) (\({H}_{3},0.2\))

\(\boldsymbol{{a}_{2}}\)  

u

(\({H}_{2},0.5\))(\({H}_{3},0.7\))

(\({H}_{2},0.5\))(\({H}_{3},0.2\))

(\({H}_{2},0.8\))(\({H}_{3},0.6\))

(\({H}_{2},0.1\))(\({H}_{3},0.3\))

(\({H}_{2},0.6\))(\({H}_{3},0.5\))

v

(\({H}_{2},0.2\))(\({H}_{3},0.3\))

\(\boldsymbol{{a}_{3}}\)  

u

(\({H}_{4},0.6\))(\({H}_{5},0.7\))

(\({H}_{4},0.2\))(\({H}_{5},0.1\))

(\({H}_{3},0.6\))(\({H}_{4},0.8\))

(\({H}_{3},0.3\))(\({H}_{4},0.1\))

(\({H}_{3},0.2\))(\({H}_{4},0.7\))

v

(\({H}_{3},0.7\))(\({H}_{4},0.2\))

  1. (4)

    Comprehensive index \({y}_{4}\)

Basic Index

\(\boldsymbol{{e}_{9}}\)  

\(\boldsymbol{{e}_{10}}\)  

\(\boldsymbol{{a}_{1}}\)  

u

(\({H}_{3},0.6\))(\({H}_{4},0.7\))

(\({H}_{3},0.1\))(\({H}_{4},0.2\))

(\({H}_{2},0.8\))(\({H}_{3},0.8\))

(\({H}_{2},0.1\))(\({H}_{3},0.2\))

v

\(\boldsymbol{{a}_{2}}\)  

u

(\({H}_{3},0.7\))(\({H}_{4},0.5\))

(\({H}_{3},0.1\))(\({H}_{3},0.3\))

(\({H}_{3},0.5\))(\({H}_{4},0.7\))

(\({H}_{2},0.4\))(\({H}_{3},0.3\))

v

\({a}_{3}\)

u

\(({H}_{1},0.8\))(\({H}_{2},0.7\))(\({H}_{3},0.6\))

\(({H}_{1},0.1\))(\({H}_{2},0.2\))(\({H}_{3},0.3\))

(\({H}_{1},0.7\))(\({H}_{2},0.6\))

(\({H}_{1},0.1\))(\({H}_{2},0.3\))

v

The expert \({V}_{3}\)’s IF evaluations

  1. (1)

    Comprehensive index \({\mathrm{y}}_{1}\)

Basic Index

\(\boldsymbol{{e}_{1}}\)

\(\boldsymbol{{e}_{2}}\)  

\(\boldsymbol{{a}_{1}}\)  

u

(\({H}_{3},0.7\))(\({H}_{4},0.8\))(\({H}_{5},0.5\))

(\({H}_{3},0.3\))(\({H}_{4},0.2\))(\({H}_{5},0.2\))

(\({H}_{4},0.7\))(\({H}_{5},0.5\))

(\({H}_{4},0.1\))(\({H}_{5},0.3\))

v

\(\boldsymbol{{a}_{2}}\)  

u

(\({H}_{3},0.6\))(\({H}_{4},0.8\))

(\({H}_{3},0.3\))(\({H}_{4},0.2\))

(\({H}_{3},0.8\))(\({H}_{4},0.6\))

(\({H}_{3},0.2\))(\({H}_{4},0.1\))

v

\(\boldsymbol{{a}_{3}}\)  

u

(\({H}_{2},0.6\))(\({H}_{3},0.4\))

(\({H}_{2},0.2\))(\({H}_{3},0.4\))

(\({H}_{2},0.7\))(\({H}_{3},0.6\))

(\({H}_{2},0.2\))(\({H}_{3},0.3\))

v

  1. (2)

    Comprehensive index \({y}_{2}\)

Basic Index

\(\boldsymbol{{e}_{3}}\)  

\(\boldsymbol{{e}_{4}}\)  

\(\boldsymbol{{e}_{5}}\)  

\(\boldsymbol{{a}_{1}}\)  

u

(\({H}_{3},0.5\))(\({H}_{4},0.8\))(\({H}_{5},0.7\))

(\({H}_{3},0.2\))(\({H}_{4},0.2\))(\({H}_{5},0.1\))

(\({H}_{2},0.8\))(\({H}_{3},0.5\))

(\({H}_{2},0.1\))(\({H}_{3},0.5\))

\(({H}_{3},0.6\))(\({H}_{4},0.8\))

v

\(({H}_{3},0.3\))(\({H}_{4},0.1\))

\(\boldsymbol{{a}_{2}}\)  

u

(\({H}_{3},0.8\))(\({H}_{4},0.7\))(\({H}_{5},0.4\))

(\({H}_{3},0.2\))(\({H}_{4},0.3\))(\({H}_{5},0.3\))

(\({H}_{2},0.9\))(\({H}_{3},0.7\))

(\({H}_{2},0.1\))(\({H}_{3},0.3\))

(\({H}_{2},0.6\))(\({H}_{3},0.7\))

v

(\({H}_{2},0.1\))(\({H}_{3},0.2\))

\(\boldsymbol{{a}_{3}}\)  

u

(\({H}_{3},0.6\))(\({H}_{4},0.7\))

(\({H}_{3},0.4\))(\({H}_{4},0.2\))

(\({H}_{3},0.8\))(\({H}_{4},0.8\))

(\({H}_{3},0.2\))(\({H}_{4},0.2\))

(\({H}_{3},0.6\))(\({H}_{4},0.8\))

v

(\({H}_{3},0.1\))(\({H}_{4},0.2\))

  1. (3)

    Comprehensive index \({y}_{3}\)

Basic Index

\(\boldsymbol{{e}_{6}}\)  

\(\boldsymbol{{e}_{7}}\)  

\(\boldsymbol{{e}_{8}}\)  

\(\boldsymbol{{a}_{1}}\)  

u

(\({H}_{1},0.5\))(\({H}_{2},0.6\))

(\({H}_{1},0.4\))(\({H}_{2},0.3\))

(\({H}_{2},0.6\))(\({H}_{3},0.5\))

(\({H}_{2},0.4\))(\({H}_{3},0.3\))

(\({H}_{2},0.7\))(\({H}_{3},0.3\))

v

(\({H}_{2},0.2\)) (\({H}_{3},0.6\))

\(\boldsymbol{{a}_{2}}\)  

u

(\({H}_{2},0.5\))(\({H}_{3},0.7\))

(\({H}_{2},0.5\))(\({H}_{3},0.2\))

(\({H}_{2},0.5\))(\({H}_{3},0.7\))

(\({H}_{2},0.3\))(\({H}_{3},0.2\))

(\({H}_{2},0.6\))(\({H}_{3},0.8\))

v

(\({H}_{2},0.2\))(\({H}_{3},0.1\))

\(\boldsymbol{{a}_{3}}\)  

u

(\({H}_{3},0.6\))(\({H}_{4},0.4\))(\({H}_{5},0.7\))

(\({H}_{3},0.3\))(\({H}_{4},0.2\))(\({H}_{5},0.1\))

(\({H}_{4},0.7\))(\({H}_{5},0.6\))

(\({H}_{4},0.3\))(\({H}_{5},0.3\))

(\({H}_{3},0.5\))(\({H}_{4},0.7\))

v

(\({H}_{3},0.4\))(\({H}_{4},0.1\))

  1. (4)

    Comprehensive index \({y}_{4}\)

Basic Index

\(\boldsymbol{{e}_{9}}\)  

\(\boldsymbol{{e}_{10}}\)  

\(\boldsymbol{{a}_{1}}\)  

u

(\({H}_{3},0.6\))(\({H}_{4},0.5\)) (\({H}_{5},0.5\))

(\({H}_{3},0.1\))(\({H}_{4},0.2\)) (\({H}_{5},0.3\))

(\({H}_{2},0.5\))(\({H}_{3},0.7\))

(\({H}_{2},0.5\))(\({H}_{3},0.2\))

v

\(\boldsymbol{{a}_{2}}\)

u

(\({H}_{3},0.4\))(\({H}_{4},0.6\))

(\({H}_{3},0.3\))(\({H}_{3},0.1\))

(\({H}_{3},0.8\))(\({H}_{4},0.3\))

(\({H}_{3},0.1\))(\({H}_{4},0.6\))

v

\(\boldsymbol{{a}_{3}}\)  

u

\(({H}_{1},0.8\))(\({H}_{2},0.5\))(\({H}_{3},0.4\))

\(({H}_{1},0.3\))(\({H}_{2},0.2\))(\({H}_{3},0.1\))

(\({H}_{4},0.7\))(\({H}_{5},0.6\))

(\({H}_{4},0.1\))(\({H}_{5},0.3\))

v

The expert \({V}_{4}\)’s IF evaluations

  1. (1)

    Comprehensive index \({\mathrm{y}}_{1}\)

Basic Index

\(\boldsymbol{{e}_{1}}\)  

\(\boldsymbol{{e}_{2}}\)  

\(\boldsymbol{{a}_{1}}\)  

u

(\({H}_{3},0.5\))(\({H}_{4},0.7\))(\({H}_{5},0.6\))

(\({H}_{3},0.3\))(\({H}_{4},0.2\))(\({H}_{5},0.2\))

(\({H}_{3},0.6\))(\({H}_{4},0.7\))

(\({H}_{3},0.3\))(\({H}_{4},0.1\))

v

\(\boldsymbol{{a}_{2}}\)  

u

(\({H}_{3},0.4\))(\({H}_{4},0.8\))(\({H}_{5},0.7\))

(\({H}_{3},0.3\))(\({H}_{4},0.2\))(\({H}_{5},0.1\))

(\({H}_{3},0.8\))(\({H}_{4},0.9\))(\({H}_{5},1\))

(\({H}_{3},0.2\))(\({H}_{4},0.1\))(\({H}_{5},0\))

v

\(\boldsymbol{{a}_{3}}\)  

u

(\({H}_{2},0.7\))(\({H}_{3},0.8\))

(\({H}_{2},0.2\))(\({H}_{3},0.1\))

(\({H}_{2},0.7\))(\({H}_{3},0.6\))

(\({H}_{2},0.2\))(\({H}_{3},0.2\))

v

  1. (2)

    Comprehensive Index \({y}_{2}\)

Basic Index

\(\boldsymbol{{e}_{3}}\)  

\(\boldsymbol{{e}_{4}}\)  

\(\boldsymbol{{e}_{5}}\)  

\(\boldsymbol{{a}_{1}}\)  

u

(\({H}_{3},0.5\))(\({H}_{4},0.5\))(\({H}_{5},0.7\))

(\({H}_{3},0.2\))(\({H}_{4},0.3\))(\({H}_{5},0.1\))

(\({H}_{1},0.7\))(\({H}_{2},0.8\))(\({H}_{3},0.3\))

(\({H}_{1},0.2\))(\({H}_{2},0.1\))(\({H}_{3},0.6\))

\(({H}_{2},0.7\))(\({H}_{3},0.6\))

v

\(({H}_{2},0.2\))(\({H}_{3},0.3\))

\(\boldsymbol{{a}_{2}}\)  

u

(\({H}_{3},0.6\))(\({H}_{4},0.6\))

(\({H}_{3},0.4\))(\({H}_{4},0.3\))

(\({H}_{1},0.5\))(\({H}_{2},0.7\))(\({H}_{3},0.4\))

(\({H}_{1},0.3\))(\({H}_{2},0.1\))(\({H}_{3},0.\) 3)

(\({H}_{1},0.5\))(\({H}_{2},0.6\))(\({H}_{3},0.5\))

v

(\({H}_{1},0.4\))(\({H}_{2},0.1\))(\({H}_{3},0.2\))

\(\boldsymbol{{a}_{3}}\)  

u

(\({H}_{3},0.4\))(\({H}_{4},0.7\))(\({H}_{5},0.8\))

(\({H}_{3},0.4\))(\({H}_{4},0.2\))(\({H}_{5},0.2\))

(\({H}_{3},0.7\))(\({H}_{4},0.8\))(\({H}_{5},1\))

(\({H}_{3},0.2\))(\({H}_{4},0.2\))(\({H}_{5},0\))

(\({H}_{3},0.7\))(\({H}_{4},0.8\))

v

(\({H}_{3},0.3\))(\({H}_{4},0.1\))

  1. (3)

    Comprehensive Index \({y}_{3}\)

Basic Index

\(\boldsymbol{{e}_{6}}\)  

\(\boldsymbol{{e}_{7}}\)  

\(\boldsymbol{{e}_{8}}\)  

\(\boldsymbol{{a}_{1}}\)  

u

(\({H}_{2},0.7\))(\({H}_{3},0.5\))

(\({H}_{2},0.1\))(\({H}_{3},0.2\))

(\({H}_{2},0.6\))(\({H}_{3},0.7\))

(\({H}_{2},0.2\))(\({H}_{3},0.1\))

(\({H}_{1},0.4\))(\({H}_{2},0.7\))(\({H}_{3},0.\) 7)

v

(\({H}_{1},0.5\))(\({H}_{2},0.3\))(\({H}_{3},0.\) 2)

\(\boldsymbol{{a}_{2}}\)  

u

(\({H}_{2},0.8\))(\({H}_{3},0.6\))

(\({H}_{2},0.1\))(\({H}_{3},0.2\))

(\({H}_{2},0.8\))(\({H}_{3},0.6\))

(\({H}_{2},0.1\))(\({H}_{3},0.3\))

(\({H}_{2},0.6\))(\({H}_{3},0.5\))

v

(\({H}_{2},0.2\))(\({H}_{3},0.3\))

\(\boldsymbol{{a}_{3}}\)  

u

(\({H}_{3},0.5\))(\({H}_{4},0.6\))(\({H}_{5},0.6\))

(\({H}_{3},0.4\))(\({H}_{4},0.2\))(\({H}_{5},0.3\))

(\({H}_{3},0.6\))(\({H}_{4},0.8\))

(\({H}_{3},0.3\))(\({H}_{4},0.1\))

(\({H}_{3},0.7\))(\({H}_{4},0.7\))

v

(\({H}_{3},0.1\))(\({H}_{4},0.2\))

  1. (4)

    Comprehensive index \({y}_{4}\)

Basic Index

\(\boldsymbol{{e}_{9}}\)  

\(\boldsymbol{{e}_{10}}\)  

\(\boldsymbol{{a}_{1}}\)  

u

(\({H}_{3},0.6\))(\({H}_{4},0.5\))

(\({H}_{3},0.1\))(\({H}_{4},0.2\))

(\({H}_{2},0.8\))(\({H}_{3},0.8\))

(\({H}_{2},0.1\))(\({H}_{3},0.2\))

v

\(\boldsymbol{{a}_{2}}\)  

u

(\({H}_{3},0.7\))(\({H}_{4},0.5\))

(\({H}_{3},0.1\))(\({H}_{3},0.3\))

(\({H}_{2},0.5\))(\({H}_{3},0.7\))

(\({H}_{2},0.4\))(\({H}_{3},0.3\))

v

\(\boldsymbol{{a}_{3}}\)

u

\(({H}_{1},0.8\))(\({H}_{2},0.7\))(\({H}_{3},0.6\))

\(({H}_{1},0.1\))(\({H}_{2},0.2\))(\({H}_{3},0.3\))

(\({H}_{2},0.6\))(\({H}_{3},0.7\))

(\({H}_{2},0.3\))(\({H}_{3},0.1\))

Appendix 2

The total IF evaluation of the comprehensive attribute \({y}_{2}\)

  

\(\boldsymbol{{H}_{1}}\)  

\(\boldsymbol{{H}_{2}}\)  

\(\boldsymbol{{H}_{3}}\)  

\(\boldsymbol{{H}_{4}}\)  

\(\boldsymbol{{H}_{5}}\)  

  

\(u\)

\(v\)

\(u\)

\(v\)

\(u\)

\(v\)

\(u\)

\(v\)

\(u\)

\(v\)

\(\boldsymbol{{e}_{3}}\)  

\({a}_{1}\)

0

0

0

0

0.5049

0.2214

0.6836

0.2450

0.7

0.1

\({a}_{2}\)

0

0

0

0

0.7143

0.2418

0.6522

0.3

0.2197

0

\({a}_{3}\)

0

0

0

0

0.5062

0.4

0.6605

0.2

0.3598

0

\(\boldsymbol{{e}_{4}}\)  

\({a}_{1}\)

0.4523

0

0.8

0.1

0.4083

0.5478

0

0

0

0

\({a}_{2}\)

0.2999

0

0.8240

0.1

0.5715

0.3

0

0

0

0

\({a}_{3}\)

0

0

0

0

0.7531

0.2

0.8

0.2

1

0

\(\boldsymbol{{e}_{5}}\)  

\({a}_{1}\)

0

0

0.4527

0

0.6

0.3

0.5523

0

0

0

\({a}_{2}\)

0.2999

0.6

0.6098

0.2

0.2932

0

0

0

0

0

\({a}_{3}\)

0

0

0

0

0.6555

0.1769

0.8

0.1395

0

0

The total IF evaluation of the comprehensive attribute \({y}_{3}\)

  

\(\boldsymbol{{H}_{1}}\)  

\(\boldsymbol{{H}_{2}}\)  

\(\boldsymbol{{H}_{3}}\)  

\(\boldsymbol{{H}_{4}}\)  

\(\boldsymbol{{H}_{5}}\)  

  

\(u\)  

\(v\)

\(u\)  

\(v\)

\(u\)

\(v\)

\(u\)

\(v\)

\(u\)

\(v\)

\(\boldsymbol{{e}_{6}}\)  

\({a}_{1}\)

0.2926

0

0.6337

0.1731

0.2932

0

0

0

0

0

\({a}_{2}\)

0

0

0.6879

0.2185

0.6522

0.2

0

0

0

0

\({a}_{3}\)

0

0

0

0

0.4405

0

0.5589

0.2

0.6517

0.1769

\(\boldsymbol{{e}_{7}}\)  

\({a}_{1}\)

0

0

0.6

0.2827

0.6128

0.1731

0

0

0

0

\({a}_{2}\)

0

0

0.6879

0.1705

0.6522

0.2464

0

0

0

0

\({a}_{3}\)

0

0

0

0

0.3772

0

0.7567

0.1700

0.3577

0

\(\boldsymbol{{e}_{8}}\)  

\({a}_{1}\)

0.2256

0

0.7

0.2450

0.5419

0.3462

0

0

0

0

\({a}_{2}\)

0

0

0.2450

0.6

0.3462

0.6796

0

0

0

0

\({a}_{3}\)

0

0

0

0

0.5142

0.3115

0.7

0.1431

0

0

The total IF evaluation of the comprehensive attribute \({y}_{4}\)

  

\(\boldsymbol{{H}_{1}}\)  

\(\boldsymbol{{H}_{2}}\)  

\(\boldsymbol{{H}_{3}}\)  

\(\boldsymbol{{H}_{4}}\)  

\(\boldsymbol{{H}_{5}}\)  

  

\(u\)

\(v\)

\(u\)

\(v\)

\(u\)

\(v\)

\(u\)

\(v\)

\(u\)

\(v\)

\(\boldsymbol{{e}_{9}}\)  

\({a}_{1}\)

0

0

0

0

0.6

0.1

0.5600

0.2

0.2926

0

\({a}_{2}\)

0

0

0

0

0.5799

0.1705

0.5819

0.2303

0

0

\({a}_{3}\)

0.6509

0

0.6160

0.2

0.5134

0.1764

0

0

0

0

\(\boldsymbol{{e}_{10}}\)  

\({a}_{1}\)

0

0

0.6839

0.2234

0.7511

0.2

0

0

0

0

\({a}_{2}\)

0

0

0.2999

0

0.7536

0.1760

0.1590

0

0

0

\({a}_{3}\)

0.2507

0

0.5012

0

0.4649

0

0.2518

0

0.1981

0

Appendix 3

The belief degrees of the basic attributes under the comprehensive attribute \({y}_{2}\)

  

\(\boldsymbol{{H}_{1}}\)  

\(\boldsymbol{{H}_{2}}\)  

\(\boldsymbol{{H}_{3}}\)  

\(\boldsymbol{{H}_{4}}\)  

\(\boldsymbol{{H}_{5}}\)  

\(\boldsymbol{{e}_{3}}\)

\({a}_{1}\)

0

0

0.1141

0.1504

0.1614

\({a}_{2}\)

0

0

0.1566

0.1418

0.0573

\({a}_{3}\)

0

0

0.1013

0.1488

0.0968

\(\boldsymbol{{e}_{4}}\)  

\({a}_{1}\)

0.1272

0.1836

0.0707

0

0

\({a}_{2}\)

0.0776

0.1709

0.1199

0

0

\({a}_{3}\)

0

0

0.1749

0.1816

0.1985

\(\boldsymbol{{e}_{5}}\)  

\({a}_{1}\)

0

0.1047

0.1129

0.1238

0

\({a}_{2}\)

0.0708

0.1274

0.1229

0

0

\({a}_{3}\)

0

0

0.1442

0.1667

0

The belief degrees of the basic attributes under the comprehensive attribute \({y}_{3}\)

  

\(\boldsymbol{{H}_{1}}\)  

\(\boldsymbol{{H}_{2}}\)  

\(\boldsymbol{{H}_{3}}\)

\(\boldsymbol{{H}_{4}}\)  

\(\boldsymbol{{H}_{5}}\)  

\(\boldsymbol{{e}_{6}}\)  

\({a}_{1}\)

0.0656

0.1254

0.0657

0

0

\({a}_{2}\)

0

0.1489

0.1436

0

0

\({a}_{3}\)

0

0

0.1065

0.1161

0.1346

\(\boldsymbol{{e}_{7}}\)  

\({a}_{1}\)

0

0.1285

0.1384

0

0

\({a}_{2}\)

0

0.1516

0.1408

0

0

\({a}_{3}\)

0

0

0.0862

0.1409

0.0817

\(\boldsymbol{{e}_{8}}\)  

\({a}_{1}\)

0.0601

0.1567

0.1162

0

0

\({a}_{2}\)

0

0.1304

0.1463

0

0

\({a}_{3}\)

0

0

0.1089

0.1581

0

The belief degrees of the basic attributes under the comprehensive attribute \({y}_{4}\)

  

\(\boldsymbol{{H}_{1}}\)  

\(\boldsymbol{{H}_{2}}\)  

\(\boldsymbol{{H}_{3}}\)  

\(\boldsymbol{{H}_{4}}\)  

\(\boldsymbol{{H}_{5}}\)  

\(\boldsymbol{{e}_{9}}\)  

\({a}_{1}\)

0

0

0.1271

0.1132

0.0688

\({a}_{2}\)

0

0

0.1282

0.1244

0

\({a}_{3}\)

0.1469

0.1286

0.1097

0

0

\(\boldsymbol{{e}_{10}}\)  

\({a}_{1}\)

0

0.1509

0.1635

0

0

\({a}_{2}\)

0

0.0707

0.1457

0.0349

0

\({a}_{3}\)

0.0502

0.1002

0.0939

0.0504

0.0386

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Fan, X., Xu, Z. Double-Level Multi-attribute Group Decision-making Method Based on Intuitionistic Fuzzy Theory and Evidence Reasoning. Cogn Comput 15, 838–855 (2023). https://doi.org/10.1007/s12559-023-10109-8

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