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

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 Max(uj),Min(uj),∑(uj),and∑(uj)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\text {Max}(u_{j}), \text {Min}(u_{j}),\sum (u_{j}),\text {and} \sqrt {\sum (u_{j})^{2}}$\end{document}. 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 Max(.),Min(.),∑(.),∑(.)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\text {Max}(.), \text {Min}(.), \sum (.),\sqrt {\sum (.)^{2}}$\end{document}. 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.