Study on credibility level of trustworthy software development process based on grey nonlinear cluster

1 Introduction

With software functional requirements and increasing level of improvement, software systems become increasingly complex and large. Therefore, what are bound to increase are the loopholes and flaws. In the development process, the developer will bring some risk of loss, in the course, users can also assume that failure to bring the risk of loss. The international software design and use of defects as a result of accidents and loss are common. Thus, credibility in use and the process of software development have become a widespread international concern. Analysis of credibility in the process of software development and use is particularly important.

There are many researchers who have researched risk in software. Chittister and Haimes (1993) studied software development risk using holographic modeling from function, source and process three dimensions. Bannerman (2008) studied risk and risk management in software from a reassessment point of view. Huang et al. (2008) presented a software reliability index model based on Bayesian network. Shi et al. (2010) proposed a fuzzy AHP and fuzzy multi‐attribute decision making model for assessment of the model. Lang et al. (2010) established software credible evidence reference model. Cheng et al. (2010) analyzed China's credibility of software industry and puts forward strategies for promoting China's software industry development. Houmb et al. (2010) studied a quantifying method to security risk level from CVSS estimates of frequency and impact.

Grey cluster is an important part of grey system theory, and it is used in assessment and decision analysis commonly. Grey whiting weight function cluster, which is used widely, is a kind of grey cluster. Determination of the whitening weight function is mainly based on expertise's experience, and there is a little existing research on whitening weight function. Wang (1987) proposed a new grey relational analysis mode (B‐mode). Tang (1996) put forward experience value method and mean value method, and examples ware used to prove that average method is superior to experience. Experience value method's downside is it can be influenced by human factors. Zhang and Liu (2007) examined the turning point in albino choice of weight function. Dong et al. (2010) researched the determination of the whitening weight function and gave the general steps. Mi et al. (2007) studied grey cluster with two‐tuple linguistic assessment method.

In this paper, nonlinear whitening weight function is built by nonlinear function. Grey clustering method steps are given and are applied into software development process.

2 Nonlinear grey whitening weight function

Let n Clustering objects, m clustering indexes, s grey types. According to the observation data x_ij(i=1, 2, … , n, j=1, 2, … , m) about jth (j=1, 2, … , m) index of ith (i=1, 2, … , n) object, classify ith object into kth (k∈{1, 2, … , s}) grey type. This is called grey clustering.

Definition 1

Classify jth (j=1, 2, … , m) index of n clustering objects into s grey types. This is subclasses of jth (j=1, 2, … , m) index. Whitening weight function for jth index belonging into kth (k∈{1, 2, … , s}) grey type is f_j^k(x).

Definition 2. (typical nonlinear grey whitening weight function)

Let whitening weight function for jth index belonging into kth (k∈{1, 2, … , s}) grey type is f_j^k(x) which is typical nonlinear grey whitening weight function (Figure 1).

x_j^k(1), x_j^k(2), x_j^k(3), x_j^k(4) are turning points of f_j^k(x). Typical nonlinear grey whitening weight function notes for f_j^k[x_j^k(1),x_j^k(2),x_j^k(3),x_j^k(4)], whose formula is: Equation 1

Definition 3.

• •

  If nonlinear grey whitening weight function f_j^k(x) does not have the first and second turning points x_j^k(1), x_j^k(2) (Figure 2), then f_j^k(x) is called nonlinear lower measure whitening weight function and notes for f_j^k[–,–,x_j^k(3),x_j^k(4)].

• •

  If nonlinear grey whitening weight function f_j^k(x)'s second and third turning points x_j^k(2), x_j^k(3) are same (Figure 3), then f_j^k(x) is called nonlinear moderate measure whitening weight function and notes for f_j^k[x_j^k(1),x_j^k(2),−,x_j^k(4)].

• •

  If nonlinear grey whitening weight function f_j^k(x) does not have the third and forth turning points x_j^k(3), x_j^k(4) (Figure 4), then f_j^k(x) is called nonlinear upper measure whitening weight function and notes for f_j^k[x_j^k(1),x_j^k(2),–,–].

Proposition 1

Formula of nonlinear lower measure whitening weight function is: Equation 2 Formula of nonlinear moderate measure whitening weight function is: Equation 3 Formula of nonlinear upper measure whitening weight function is: Equation 4

Proposition 2

Nonlinear grey whitening weight function is derivable except endpoint.

3 Grey clustering method given the right based on nonlinear whitening weight function

x_ij (i=1, 2, … , n, j=1, 2, … , m) are the observation date about jth index of ith (i=1, 2, … , n) object. f_j^k(x) is the whitening weight function for jth index belonging into kth (k∈{1, 2, … , s}) grey type η_j^k is the weight for jth index belonging into kth (k∈{1, 2, … , s}) grey type and it is nothing with k. Then η_j^k=η_j(j=1,2, … ,m).

Clustering coefficient for i object belongs to kth grey type is σ_i^k=∑_j=1^mf_j^k(x_ij) · η_j.

Definition 4

Classify the object according to clustering coefficient based on nonlinear whitening weight function is call grey clustering method given the right based on nonlinear whitening weight function.

The steps of grey clustering method given the right based on nonlinear whitening weight function are:

1. 1.

   Determine f_j^k(x) which is the whitening weight function for jth index belonging into kth (k∈{1, 2, … , s}) grey type.

2. 2.

   Determine η_j(j=1, 2, … , m) which is the weight for jth index belonging into kth (k∈{1, 2, … , s}) grey type.

3. 3.

   Calculate σ_i^k=∑_j=1^mf_j^k(x_ij) · η_j.

4. 4.

   Determine i (i=1, 2, … , n) object belonging to k* grey type according to _max^1≤k≤s{σ_i^k}=σ_i^k*.

4 Application for determining the credible risk of software development process

5 Conclusion

Determination on credibility level in the software development process is dynamic, as credibility of the results may be different with different time and project requirements. Qualitative methods are not entirely convincing, and total quantitative methods are not flexible enough. Grey cluster based on nonlinear grey whitening weight function put forward in this paper is both convincing and flexible. Finally eight projects from ISBSG database are used for empirical analysis, which confirms that the method put forward in this paper is available and credible.