Optimal release time determination via fuzzy goal programming approach for SDE-based software reliability growth model

Abstract

Faults in software can be of varied severity and so is the priority with which they are fixed. Hence, these two factors impact the reliability of the software to a large extent and is worthy of being taken into consideration while framing the software reliability growth model (SRGM). In this paper, a stochastic differential equation-based SRGM has been formulated where detection of faults has been associated with severity of fault occurrence and the process of correcting them have been knit with priority of defect fixing. Another matter of concern is the determination of the optimal time to release the software in the market with high reliability figures keeping the development cost margin minimum. Release time determination in the literature is done mainly using crisp or type-1 fuzzy system approach wherein the cost constraint in the optimization framework is considered to be of deterministic nature which is something highly unrealistic. In this study, the cost constraint is considered to comprise of a randomized cost budget with variable coefficients being interval type-2 fuzzy numbers which can represent the dynamicity associated with changes in specification during software development process in a much more realistic fashion. This encourages the release time determination problem to be framed as a multi-objective chance constrained optimization problem, which is thereby solved using fuzzy goal programming approach based on Taylor series method. The proposed SRGM and the release time approach have been validated on real time data, which show much promising results than many of those available in the literature.

Appendices

Appendix A: Deterministic formulation of chance constraint

$C_{\text{B}}$ is obtained to be a FRV following Johnson SB distribution having probability density function (pdf) given by,

$$ f(C_{\text{B}} ,\gamma ,\delta ,\lambda ,\xi ) = \frac{\delta }{{\lambda \sqrt {2\pi } z(1 - z)}}\exp \left( { - \frac{1}{2}\left( {\gamma^{'} + \delta^{'} \ln \left( {\frac{z}{1 - z}} \right)} \right)^{2} } \right) $$

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where $z = \frac{{C_{\text{B}} - \xi^{'} }}{{\lambda^{'} }}$; $\gamma^{'} \in \,\,\gamma [\alpha ],\,\,\,\delta^{'} \in \delta [\alpha ]$

such that $C_{\text{B}} \ge \xi$

In expression above, $\gamma^{'} \in \,\tilde{\gamma }[\alpha ]\,,\,\delta^{'\,} \in \,\tilde{\delta }[\alpha ]$, $\xi^{'} \in \xi [\alpha ]$, $\lambda^{'} \in \,\lambda [\alpha ]$ are defined the $\alpha \,$ cut of fuzzy numbers $\tilde{\gamma }\,,\,\tilde{\delta }\,,\,\tilde{\xi },\,\tilde{\lambda }$. The support of $\delta \,,\,\lambda$ are set of positive real numbers and the support of $\xi ,\,\,\gamma$ are set of real numbers.

The chance constraint expression is given by, $\Pr \left( {\tilde{C}(t) \le \tilde{C}_{B} } \right) \ge 1 - p$ whereby $C_{\text{B}}$ follows Johnson SB distribution. Hence, from the expression for pdf it may be well stated that the above inequality takes the form,

$$ \int\limits_{{h_{i} }}^{\infty } {\frac{{\lambda^{'} \delta^{'} }}{{\sqrt {2\pi } \left( {b - \xi } \right)\left( {\lambda^{'} - b + \xi^{'} } \right)}}\exp \left( { - \frac{1}{2}\left( {\gamma^{'} + \delta^{'} \ln \left( {\frac{b - \xi }{{\lambda^{'} - b}}} \right)} \right)^{2} } \right)} \,db\, \ge 1 - p $$

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$h_{i} \, \in \,\tilde{C}(t)$ and $h_{i} \, \ge \,\xi^{'}$ Implies that,

$$ h_{i} \, \le \,\frac{{\lambda^{'} \exp \left( {\frac{{\omega - \gamma^{'} }}{{\delta^{'} }}} \right)}}{{\exp \left( {\frac{{\omega - \gamma^{'} }}{{\delta^{'} }}} \right)\, + 1}} + \xi^{'} \,\,\,\,\,\,\,\,;\,\,\,\omega = erf^{ - 1} (2p) $$

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Since, the above inequality is true for all $\alpha \in (0,1]$, the expression may be reframed in terms of $\alpha$-cut as,

$$ \tilde{C}(t)[\alpha ] \le \frac{{\lambda [\alpha ]\exp \left( {\frac{\omega - \gamma [\alpha ]}{\delta [\alpha ]}} \right)}}{{\exp \left( {\frac{\omega - \gamma [\alpha ]}{\delta [\alpha ]}} \right) + 1}} + \xi [\alpha ] $$

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and $\tilde{C}(t)[\alpha ] \ge \xi [\alpha ]$

Using the first decomposition theorem the above expression takes the form,

$$ \tilde{C}(t) \le \frac{{\tilde{\lambda }\exp \left( {\frac{{\omega - \tilde{\gamma }}}{{\tilde{\delta }}}} \right)}}{{\exp \left( {\frac{{\omega - \tilde{\gamma }}}{{\tilde{\delta }}}} \right) + 1}} + \xi \,\,\,\,\,\,\,\,\,{\text{and}}\,\,\,\,\,\,\,\,\,\tilde{C}(t) \ge \tilde{\xi } $$

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By fitting the cost data corresponding to Release 1 (as shown in “Appendix B”) to the Easy fit Software it was observed that, $\gamma = - 0.50168\,,\,\delta = \,0.51181,\,\lambda = 6.1499 \times 10^{5} ,\,\xi = - 16470.0$

$C(t)$ denoting the cost function it always assumes a value greater than zero. Hence, the constraint $\tilde{C}(t) \ge \xi$ becomes a redundant constraint.

Thereby, $\tilde{C}(t) \le \tilde{C}_{n}$ whereby $\tilde{C}_{n} = \frac{{\tilde{\lambda }\exp \left( {\frac{{\omega - \tilde{\gamma }}}{{\tilde{\delta }}}} \right)}}{{\exp \left( {\frac{{\omega - \tilde{\gamma }}}{{\tilde{\delta }}}} \right) + 1}} + \tilde{\xi }$ and hence $\tilde{C}_{{B^{'} }} = \left( \begin{aligned} \left( {459833.9,506933.818,526933.75,584975.80;0.97,0.99} \right), \hfill \\ \left( {506933.818,537875.69,558734.88,564537.23;0.87,0.98} \right) \hfill \\ \end{aligned} \right)$

In the above case value of p is considered equal to 0.2089.

Appendix B: Dataset based on which cost calculation has been carried out

The data set below was obtained from a large IT firm. The data comprises of specification of the Resources, Effort Man days, Effort Man Hours, Total Hours, Rate and Price. Further it contains the weekly cost breakup corresponding to each resource. The Effort Man Week column represents the number of weeks the particular resource had worked in course of building the project. The column corresponding to Effort Man Days corresponds to number of days the resource had put in the development process. It has been considered that the firm supports five working days a week with 8 to 9 working hours per day. The column corresponding to Rate illustrates the amount paid to the each resource per week. The column corresponding to price defines the total amount spend in regard to the corresponding resource. The remaining part of the dataset defines the proportion of the total cost that was spent per resource on a weekly basis. The above data set was used for estimating the distribution corresponding to the cost budget in formulation of the cost constraint in the optimization problem.

Cost dataset: