Attribute Normalization Approaches to Group Decision-making and Application to Software Reliability Assessment

Abstract

A group decision-making (GDM) process is a social cognition process, which is a sub-topic of cognitive computation. The normalization of attribute values plays an important role in multi-attribute decision-making (MADM) and GDM problems. However, this research finds that the existing normalization methods are not always reasonable for GDM problems. To solve the problem of attribute normalization in GDM systems, some new normalization models are developed in this paper. An integrative study contributes to cognitive MADM and GDM systems. In existing normalization models, there are some bounds, such as \(\text {Max}(u_{j}), \text {Min}(u_{j}),\sum (u_{j}),\text {and} \sqrt {\sum (u_{j})^{2}}\). They are limited to a single attribute vector uj. The bound of new normalization method proposed in this work is related to one or more attribute vectors, in which the attribute values are graded in the same measure system. These related attribute vectors may be distributed to all decision matrices graded by this decision system. That is, the new bound in developed normalization model is an uniform bound, which is related to a decision system. For example, this uniform bound can be written as one of \(\text {Max}(.), \text {Min}(.), \sum (.),\sqrt {\sum (.)^{2}}\). Some illustrative examples are provided. A practical application to the evaluation of software reliability is introduced in order to illustrate the feasibility and practicability of methods introduced in this paper. Some experimental and computational comparisons are provided. The results show that new normalization methods are feasibility and practicability, and they are superior to the classical normalization methods. This work has provided some new normalization models. These new methods can adapt to all decision problems, including MADM and GDM problems. Some important limitations and future research are introduced.

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Acknowledgements

The author would like to thank the editors and the anonymous reviewers for their insightful and constructive comments and suggestions that have led to this improved version of the paper.

Funding

This study was funded by the Young Creative Talents Project from Department of Education of Guangdong Province (No. 2016KQNCX064), the Project of Enhancing School with Innovation of Guangdong Ocean University (No. GDOU2017052802), the Project of Professional Core Course from College of Mathematics and Computer Science, Guangdong Ocean University (No. 571119134), and the Project of Teaching Innovation in 2019 from Guangdong Ocean University (No. 570219088).

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Appendix. Some related tables in the section “Comparison with Other Normalization Methods”

Appendix. Some related tables in the section “Comparison with Other Normalization Methods”

Table 26 Weighted decisions of four software products based on the classical Eqs. 2 and 3
Table 27 Ideal decisions of four software products based on the classical Eqs. 2 and 3
Table 28 Weighted decisions of four software products based on the classical Eqs. 7 and 8
Table 29 Ideal decisions of four software products based on the classical Eqs. 7 and 8
Table 30 Weighted decisions of four software products based on Eqs. 9 and 10 proposed in this work
Table 31 Ideal decisions of four software products based on Eqs. 9 and 10 proposed in this work
Table 32 Weighted decisions of four software products based on the classical Eqs. 11 and 12
Table 33 Ideal decisions of four software products based on the classical Eqs. 11 and 12
Table 34 Weighted decisions of four software products based on Eqs. 13 and 14 proposed in this work
Table 35 Ideal decisions of four software products based on Eqs. 13 and 14 proposed in this work
Table 36 Weighted decisions of four software products based on the classical Eqs. 15 and 16
Table 37 Ideal decisions of four software products based on the classical Eqs. 15 and 16
Table 38 Weighted decisions of four software products based on Eqs. 17 and 18 proposed in this work
Table 39 Ideal decisions of four software products based on Eqs. 17 and 18 proposed in this work

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Yue, C. Attribute Normalization Approaches to Group Decision-making and Application to Software Reliability Assessment. Cogn Comput 13, 139–163 (2021). https://doi.org/10.1007/s12559-019-09707-2

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Keywords

  • Normalization
  • Multi-attribute decision-making
  • Group decision-making
  • Normalized projection
  • Software reliability assessment