combining two attributes like cost and reliability, cost and
failure intensity etc. X. Li, et alproposed open source software release based on release indicator and reliability
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Software Release Time Determination: A Combination of Three Attributes
Dr. Subhadra Rajpoot
—————————— ——————————
With growing penetration of information technology, computer software is playing an important role in our lives.
testing and releasing.
In late years, due to the significance of software application, professional testing of software becomes an increasingly
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Software reliability becomes a problem that can’t be
overlooked. Many models (generally called software reliability growth models, SRGM) have been proposed to describe the software testing processes. A software reliability growth model (SRGM) can be considered to be a mathematical expression which fits the experimental data. G-O, Yamada, K-G models are most well-known models among these models [1, 8, 12, 17, 18, 19, 23]. GO model assumes that the number of defects detected up to time t follows a non-homogenous Poisson process (NHPP) with constant detection rate, Yamada and K-G assume that the total number of defects at the start of testing is a known constant, detection rate is function of time and the failure rate at any time is proportional to the number of defects remaining at that moment [2,3].
Software release is one of the most prominent issues involved in software development to decide upon the most appropriate software release plans. The problem of determining when we should stop testing emerges. If we stop testing too early, there may be too many defects in the software, which will result in too many failures during operation, and lead to significant losses due to the failure penalty or user dissatisfaction. On the other side, spending too much time in testing may result in a high testing cost and delay the introduction of the product into the market place. Therefore, there is a trade-off between software
important task. Once all detected faults are removed,
project managers can begin to determine when to stop testing. Software reliability has important relations with many aspects of software, including the structure, the operational environment, and the amount of testing. Actually, software reliability analysis is a key factor of software quality and can be used for planning and controlling the testing resources during development [2,3]. Debian: in recent years, the project has faced increasingly delayed and unpredictable releases. Most notably, the release process of Debian 3.1 was characterized by major delays. Initially announced for December 1, 2003, the software was finally released in June 2005 – a delay of one and a half years. By the time the new version was released, the previous stable release was largely considered out of date and did not run on modern hardware [].
GNU tools: despite their popularity and importance, development has been slow in recent years and there is a long interval between releases. Version1.13 of tar came out in August 1999, followed by version 1.14 at the end of 2004. The compression utility gzip saw a new version in December 2006, more than a decade after the last stable release in 1993. As a consequence of these long delays between stable releases, several vendors started shipping pre-releases. Thus there is a trade-off, and the issue is to find an optimal point at which costs justify the stop
decision [].
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Yang, et. al worked on the approach taken is to minimize the expected total cost (ETC) of the software project, or further consider the software reliability requirement[24]. Yun et al. [25] study the optimal software release problem based on software reliability growth models with random life–cycle length. Yamada et al. [19,20] present an optimal software release problem based on a dual constraint of minimizing a total average software cost and satisfying a reliability requirement. Brown [2] describes a cost model to determine the optimal number of test cases.
This stream of research is closely related to the broader software reliability literature, a good summary of which is provided by Pham [6,11] and Kapur [3,4,5,7,31,32]. In all these studies, the optimal release time is determined from the cost and reliability viewpoint. Testing should continue until the gain from the improved reliability cannot justify the cost of continued testing. An implicit assumption made in these studies is that software testing stops completely after release. Recently many researchers have used Multi attribute utility theory to find the optimal release time of
new version of software by combining different attributes
In Sec 2, we have discussed basic modeling of SRGM. In Section 3 we have described the theme of our discussion i.e. problem of releasing software in the market and later we have used MAUT as an evaluation approach to formulate the problem followed by numerical illustration in Sec 4. Conclusion and Acknowledgement are given in section 5 and 6 respectively.
SRGM is mathematical model. It shows how software reliability improves as faults are detected and repaired. SRGM can be used to predict when a particular level of reliability is likely to be attained. Software reliability models which assume that software failures display the behaviour of a non-homogeneous Poisson process (NHPP).The parameter of the stochastic process, which denotes the failure intensity of the software at time , is time dependent. Let denote the cumulative number of faults detected by time , and denote its expectation.
Then, and the failure intensity is related as follows:
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like cost, reliability, and failure intensity. Several
researchers have done work on multi attribute utility theory to determine the release time of software by
t
m(t ) = ∫0 λ (s)ds
dm(t )
combining two attributes like cost and reliability, cost and
failure intensity etc. X. Li, et alproposed open source software release based on release indicator and reliability
and,
= λ (t )
dt
using MAUT [21]. Lately Kapur et al. solved a release problem using cost and failure intensity as attribute [].
Recently Kapur et al. defined a scenario of release time
N (t) is known to have a Poisson probability mass function with
parameter m(t ) , that is:
m t n ⋅ −m t
using different structure of multi attribute utility function combining just two attribute reliability and cost [5]. These studies by different researcher are bound for two attribute
only.
Pr {N (t ) = n} =
( ) exp (
n!
( ) )
,
n = 0,1, 2,...
In this paper, we address the problem faced by most software managers, namely, time to stop testing or time of releasing software using three different attribute. This is a problem of decision-making under uncertainly and involves a trade-off among reliability; cost and detection rate indicator (rate of change of detection rate).
We will further investigate the modelling of fault removal process using logistic distribution functions. Specifically, the multi-attribute utility theory (MAUT) is adopted for determining the optimal time for release, where three important strategies are considered simultaneously: reliability of software, rate of change of detection rate and the acceptable level cost. The paper is outlined as follows:
Various time dependent models have appeared in the
literature which describes the stochastic failure process by an NHPP. These models differ in their failure intensity function and hence . Let ‘ ’ denote the expected number of faults that would be detected given infinite testing time in case of finite failure NHPP models. Then, the mean value function of the finite failure NHPP models can also be written as [1,6,8,9,10,11,12].
(1)
where is a distribution function.
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In our model, NHPP is used to describe the time-dependent nature of the cumulative number of faults detected up to a specific testing time.
The differential equation for describing the removal phenomenon can be given by:
dm(t ) = b(t ) [ a − m (t ) ]
dt
when should be done? Where? Other managerial functions, such as organizing, implementing, and controlling rely heavily on decision making.
Multi-Attribute Utility Theory (MAUT) is a structured methodology designed to handle the tradeoffs among multiple objectives. One of the first applications of MAUT involved a study of alternative locations for a new airport in Mexico City in the early 1970s. The factors that were
considered included cost, capacity, access time to the
= b
(1 + β e−bt )
.[ a − m (t ) ]
airport, safety, social disruption and noise pollution. Utility theory is a systematic approach for quantifying an individual's preferences. It is used to rescale a numerical
value on some measure of interest onto a 0-1 scale with 0
Solving the above differential equation (1), under initial condition, we get mean value function as[6, 7]:
−bt
representing the worst preference and 1 the best
[24,26,27,28,29,30].
To tackle these two conflicting factors simultaneously,
multi attribute utility theory (MAUT) is adopted in our
m (t ) = a.F (t ) = a.
1 − e
decision model. The application of MAUT can obtain a one-
1 + β e−bt
Logistic Distribution model proposed by Kapur and Garg is
dimensional multi-attribute utility function, which is the measure of the attractiveness of the conjoint outcome of attributes given a specified alternative.
S-shape in nature. We use S-Ishape cJurve softwarSe reliability ER
model to capture all possible behavior of data set. The
analysis can be similarly carried out using any other model. Model assessment evaluates how well a data-set conforms to a chosen model. An important objective of model-based software reliability analysis is to guide decisions by
Let x1 , x2 , x3 ............xn , n ≥ 2 , be a set of attributes
associated with the consequences of a decision problem. The utility of a consequence ( x1 , x2 , x3 ............xn ) can be determined from
Decomposed assessment: estimate n conditional utilities
providing accurate future predictions. Thus, a model that
fits the observed data well, but makes poor predictions,
Ui ( xi )
for the given values of the n attributes; and compute
raises serious doubts regarding its practical utility. The R2 is a complimentary measure of a model's statistical adequacy [13,14,15,16,17,18]. The value of the estimated parameters of all the models and value of R2 are given in Table-1.
Parameter Estimation
SRGMs | Parameters | |||
K- G(Logistic) | a | b | β | R2 |
K- G(Logistic) | 113 | 0.12 | 3.22 | 0.989 |
Decision-making is a process of choosing among alternative courses of action in order to attain goals and objectives. Release time of software, for example, involves deciding
U ( x1 , x2 , x3 ............xn ) by combining the Ui ( xi ) of all
attributes:
U ( x1 , x2 , x3 ............xn ) = f (U1 ( x1 ),U 2 ( x2 ),U3 ( x3 )..........U n ( xn ))
, i = 1....n
The attributes we are seeking should be the most important ones deemed relevant to the final decision. They should preferably be mutually exclusive: the attributes should be viewed as independent entities among which appropriate trade-offs may later be made. Most importantly, the chosen attributes should be measurable in a meaningful and practical way, for each of the proposed alternatives.
When considering the attributes for optimal release of any software the main objective of software industry is to prepare software which is much reliable and satisfy the customer needs. Software reliability represents a customer oriented view of software quality. Therefore maximizing software reliability is also a major concern of management. A simple index to measure
the reliability is the ratio of the number of cumulative detected
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faults at time t to the mean value of initial faults in the software.
( ) ( ) ( )
Hence, the attribute reliability can be represented by it should be maximized.
m(t )
a
and
C T
Where,
= C1m T
+ C2 a − m T
+ C3T
Maximize R = m(t )
a
(5)
C1 be the cost of fixing a fault during testing phase.
C2 be the cost of fixing a fault during operational phase.
Where the approximated reliability R is attribute in MAUT. Since the reliability is an increasing function of time, it reaches its maximum when time goes to infinity.
Many software reliability models (SRMs) have been proposed to explain the software fault detection process and to quantify the different metrics that are related to system reliability [13,14,15,16,17,18]. These SRMs follow a wide variety of fault detection rates. A majority of these SRMs first choose a well- known mathematical distribution to characterize the software fault detection rate, and then interpret the parameters of the chosen distribution in the context of the testing process. For example, an early SRM developed by Goel and Okumoto assumed a constant fault detection rate. Yamada used time dependent detection rate. Common sense, however, suggests that one must first identify the characteristics or behaviour of the detection rate that drives the fault detection process followed by
cumulative number of faults i.e. how detection rate is changing as
C3 is the testing cost per unit testing time.
m(T ) is the expected number of faults removed during testing phase.
C (T ) is the total cost.
A firm never wants to spend more than its capacity, therefore the next attribute that we consider is:
minC = C (T )
CB
Where, CB is the total budget allocated to the firm.
The single utility function for each attribute represents management’s satisfaction level towards the performance of each
attribute. It is usually assessed by a few particular points on the
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testing process goes on. To measure the rate of change of
utility curve [24,26,27]. More specifically, suppose that the single
detection rate we differentiate the detection rate equation (2).
b(t ) used in
utility function for cost is to be determined, the worst and best
values of cost are selected first as C + and C − . These values are of great importance because C + and C − represent its lowest
b' (t ) =
b2 β e
−bt
(6)
cost and its highest cost expectation respectively. At these
boundary points, we have u(C + ) = 1 and u(C − ) = 0 . Finally,
We use
(1 + β e−bt )2
b' (t ) as an attribute to measure the behaviour of
to determine functional form of utility functions either an additive or exponential form needs to examine through interviews, surveys
detection rate. It is reasonable to assume that the b' (t ) follows a hump-shaped curve. It is worth noting here that b' (t ) reaches its
2
or lottery. It may be noted that we use lottery when there is a
preference or indifference between two lotteries. If they are equal to each other, management is risk neutral and the linear form should be used. Otherwise, if management is not risk neutral then
maximum value b' (t )
= b at
4
t = 1 log β . This new
b
the exponential form will be selected.
attribute is called Detection Rate Indicator (DRI) and defined as which is to maximize.
b' (t )
MaximizeD = (7)
b' (t )
On the other hand the software performance during the field is dependent on the reliability level achieved during testing. In general, it is observed the longer the testing phase, the better the performance. Better system performance also ensures less number of faults required to be fixed during operational phase. On the other hand prolonged software testing unduly delays the software release. Here we wish to determine the optimal testing time of software so that the total expected costs of the software can be minimized. For this purpose we construct a cost model for the software by assuming that there are three type of cost
u(C ) = l + m.C or u(C ) = l + m exp(k.C )
Where, l , m , k are constants. The single utility functions u(R)
and u(D) for the reliability can be obtained as well [26,28].
In this section we have discussed about estimation of weight parameters wc , wD and wR . There are two common methods to assess the scaling constants: certainty scaling and probabilistic scaling [28,29,30]. Given that the number of attributes considered in our problem is only three and this is a small number, the probabilistic scaling technique is recommended for use.
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In probabilistic scaling, management is asked to compare the choices. Let (R+ , D+ , C + ) and (R− , D− , C− ) denote the best and worst possible consequence respectively. There is a certain
joint outcome (R+ , D+ , C − ) comprised three attribute R , D
faults in the software. The number of faults removed up to time t
, m(t ) reaches its maximum value at tmax = ∞ .
Similarly for the second attribute given by equation (7) is the ratio of change in detection rate to maximum change in detection rate.
and C at the best and worst level with probability
p, q and
For other attribute i.e. cost we use the cost model as discussed
(1 − p − q) , respectively. In these situations, the weight for
earlier in Section (3).We set parameters C1 = 15 , C2 = 18 ,
attribute R equals p , where p is the indifference probability
C3 = 5 and CB = 20000
as parameters of cost function. The
among them, [29,30]. At indifference,
q and (1 − p − q) is
cost function is then calculated using the value of estimated
equal to the weight parameters for the detection rate indicator and cost. Since the sum of weight parameters must be equal to one, the other weight parameter wD wR can be obtained with ease.
Based on the previously estimated single utility functions and scaling constants, choosing the structure of the multi-attribute utility function is important.
The additive linear form of the MAUF is given as [26,27,29,30]:
U (R, D, C ) = wR × u(R) + wD × u(D) + wC × u(C )
wR + wD + wC = 1
Where wR , wD and wc are the weight parameters for attribute R
parameters as given in the Table.2.
The single utility function for each attribute is elicited based on the management’s own scenarios. Since these management scenarios are subjective assessments from management, they may not be precise. Suppose that management scenarios in our application example are as follows:
For Reliability, management has verified that at least 70% of software faults should be detected; its highest expected value is 100%.
For Detection Rate Indicator, management has verified that
70% of change in detection rate show minimum, while its
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, D and C respectively. u(R) , u(D)
and u(C )
are the single
highest expected value is 100%.
utility function for each attribute i.e. for reliability, detection rate indicator and cost respectively. From the manager’s point of view R and D are to be maximized while cost attribute C is to be minimized. To synchronize the two utility together, we convert minimization of cost utility by multiplying “–‘sign before cost
Under minimization cost strategy, management indicates that
at least 60% of budget must be consumed.
According to the above management decision, some important points on the utility curve are obtained. In particular, for third
utility. By maximizing multi-attribute utility functions, the
optimal time to release, T * will be obtained.
release the lowest reliability requirement is R0 = 0.7
1
and the
maximum reliability expectation is R =1. The lowest change in
detection rate is
D0 = 0.7
and the highest change is D1 = 1 .
0
Data [22] comprises of four successive releases. The proposed decision model has been validated for its first release. We will
The lowest cost requirement is
1
C = 0.6
and the highest cost
find the release time of software for using first release data. The first version of software is released after 20 weeks.
For management, it is of utmost importance to predict the optimal
expectation C = 1 . Additionally, based on management’s risk
neutral attitude towards these attributes, three form of the single utility function should be used. Specifically, we have
release time. The Multi Attribute Utility Theory approach is used
to determine the release time. Multi-Attribute Utility Theory
(MAUT) is a label for a family of methods. These methods are a
u(C ) = − 5 + 5 C
4 2
u(D) = − 1 + 5
4 2
D and
means to analyze situations and create an evaluation process. The
objective of MAUT is to attain a conjoint measure of the
attractiveness (utility) of each outcome of a set of alternatives.
u(R) = − 7 + 10
3 3
R .It is worth noting here that although the
As discussed, reliability R D and cost C are three important factors for management to determine the optimal release. Based on the failure data shown in Table 1, the model parameters can be estimated as shown in Table 2. Then, these three attributes are quantitatively measured by (5) (7) and (8).
The reliability attribute defined in equation (6) is the ratio of
linear form is simple for evaluating the utility for the attributes
[21,26,27].
The weight parameter wC is estimated by comparing the two choices by lottery approach [29,30]. Management has claimed that it is indifferent among these choices when p is equal to 0.3;
number of faults removed up to time t to the total number of
hence wC
= 0.3 i.e. the weight related to cost parameter is 0.3.
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It is easy to calculate
wR and wD
based on the sum of weight
parameters is equal to one, therefore wR , wD
and .3.
are respectively .4
Finally, based on the estimated single utility functions and the weight parameter, the multi-attribute utility function is evaluated using three attributes.
The Multi Attribute Utility Function is maximized by using of Maple package Software and the optimal time to release the first version of software. The optimal Release time is given in table-4.
According to Tandem data failure, real time to release thefirstversion of software is 20weeks. Based on optimal result, we can say that software in first release must release after this time.
Figure-1 shows the multi attribute utility function for first release
of software. Figure-2 represents the behavior of the cost function for first release.
Figure-2: Cost function graph
Optimal release time can be determined by maximizing the multi- attribute function. However, since most parameters in the MAUT
are obtained based on the subjective assessments from
management, the optimal introduction time received may not be
precise. The weight attached to each attribute is purely management decision. Different combination of attributes can be used by attaching different weights to them. In numerical example we have find the release time with fixed weight .3,.4,.3, for cost, reliability and detection rate indicator respectively. Accordingly for choice of different weight for attributes, sensitivity analysis is needed. Sensitivity analysis is generally done by changing one parameter and setting the other parameters at their fixed values. We choose different combination of weight for three attributes and find optimal release time as we have done in our numerical example section. Sensitivity for weight parameter and respective release time are summarized in following table.
Sensitivity on weight
Weights Optimal
Values
Single Attributes Utility
Figure-1: Utility function graph
Comb inatio
n No.
wc wR
w T *
M ult i
Uti
lit y
C = C Bu
R = m D = b
b
1 .3 .4 .3
0 0 0
2 .3 .3 .4
0 0 0
20 .75 .094 .720 .650
15 .95 .096 .730 .890
3 .2 .2 .4
9 9 2
14 .98 .091 .540 .910
.9
7
4 .2 .4 .4
0 0 0
17 .81 .095 .613 .810
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3. P. K. Kapur, H. Pham, Anu G. Aggarwal, Gurjeet Kaur, "Two-dimensional multi-release software reliability
modelling and optimal release planning" IEEE Trans on
Reliability, Vol. 61 (3), pp. 758-768, 2012
Optimal release times for different choice of weight for three attributes are given in above table. The final utility values along with single utility value for three attributes are given in last four columns in above table. Different combination of weight gives different release time which shows its sensitivity. Management has several choices for release time according to their preference in attributes. One thing is very clear from the table that changing weight for cost parameter does not affect its own utility value. While weight parameter is very sensitive for rest two attributes. If we increase or decrease the weight of reliability its own utility follows same. Similarly for detection rate indicator is happening (Combination No 5, 1, 2, 3). This shows the importance of reliability and detection rate indicator as an attribute. Less weight to detection rate indicator gives bad release time (Combination No 5).
The strategy behind the developing and releasing software is not an easy thing. Most of organizations are struggling to find the
4. P. K. Kapur, A. tondon, G. Kaur, "Multi up- gradation software reliability model," 2nd international conference on reliability, safety & hazard (ICRESH-2010), pp-468-
474, Mumbai, 2010.
5. P K Kapur, V B Singh, Ompal Singh, Jyotish N P Singh, software release time based on differentmulti- attribute utility functions” International Journal of Reliability, Quality and Safety Engineering, Vol. 20, No. 4 (2013) 1350012 (15 pages).
6. H. Pham and X. Zhang, "Software release policies with gain in reliability justifying costs," Annals of Software Engineering, vol. 8, 1999.
7. P. K. Kapur and R. B. Garg, "Cost reliability optimum release policies for a software system with testing effort," Operations Research, vol. 27, pp. 109-116,
1990.
8. H. Pham, Software Reliability. Singapore: Springer,
2000.
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exact release time of software. From a management point of view
it is important to understand the balance between reliability of software and cost. A vital decision problem that the software developer encounters is to determine when to stop testing and release the software system to the user. If the release of the software is unduly delayed, the manufacturer (software developer) may suffer in terms of penalties and revenue loss, while a premature release may cost heavily in terms of fixes (removals of faults) to be done after release, which consequently might harm the manufacturer’s reputation. On one hand, when there is limited cost budget for testing, the software is expected to be tested in such a manner that it costs reasonable.In order to make a judicious decision on the optimal time of release of software, a decision model based on MAUTis proposed. We maximize the Multi attribute utility function using cost, reliability and detection rate indicator and obtained optimal release time.We conduct sensitivity analysis for weight parameter. We find different optimal release time for different combination of weight for attributes.
1. K. Okumoto and A. L. Goel, "Optimal release time for computer software," IEEE Transactions on Software Engineering, vol. 9, pp. 323-327, 1983.
2. D. B. Brown, S. Maghsoodloo, and W. H. Deason “A cost model for determining the optimal number of test cases” IEEE Trans. on Software Engineering,
15(2):218–221, February 1989.
9. Kapur, P. K., Pham, H., Gupta, A. and Jha, P.C. (2011),
“Software Reliability Assessment with OR
Applications”, Springer –UK.
10. H. Ohtera and S. Yamada., “Optimum software-release time considering an error detection phenomenon during operation” IEEE Trans. on Reliability, 39(5):596–
599, December 1990.
11. H. Pham and X. Zhang., “A software cost model with warranty and risk costs”. IEEE Trans. on Computers,
48(1):71–75, January 1999.
12. R. S. Pressman. Software Engineering: A Practitioner’s
Approach. McGraw Hill, 1997.
13. S. M. Ross. “Software reliability: The stopping rule problem”. IEEE Trans. on Software Engineering, SE-
11(12):1472–1476, December 1985.
14. Musa JD, Iannino A, Okumoto K. “Software reliability: Measurement, Prediction, Applications” 1987; New York: Mc Graw Hill.
15. L. Goel and K. Okumoto, "Time-dependent error- detection rate model for software reliability and other performance measures," IEEE Trans. on Reliab., vol.
28, pp. 206–211, 1979.
16. M. Obha, "Software reliability analysis models," IBM Journal of Research and Development, vol. 28, pp. 428–
443, 1984.
IJSER © 2013 http://www.ijser.org
International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December-2013 1769
ISSN 2229-5518
17. M. Xie, Software Reliability ModelingWorld Scientific
Publishing, 1991.
18. H. Pham, System Software Reliability. U. K.: Springer- Verlag, 2006.
19. Yamada, S., Ohba, M. and Osaki, S. (1984), “S-shaped software reliability growth models and their applications”, IEEE Trans. on Reliability, Vol. 33 No.
4, pp. 289–292.
20. Yamada, S., Narihisa, H., Osaki, S., (1984), “Optimum release policies for a software system with a cheduled software delivery time”, International Journal of Systems Science, Vol. 15 No. 8, pp. 905–914.
21. Li, X., Li, Y.F., Xie, Min. and Ng, S.H. (2011), “Reliability analysis and optimal version-updating for open source software”, Information and Software Technology, Vol. 53, pp. 929–936.
22. Wood,, "Predicting software reliability," IEEE Computer, vol. 9, pp. 69-77, 1996.
model” Recherche Operationnelle – Operations
Research 1994; 28: 165-180.
32. Kapur PK, Garg RB. “Optimal release policies for software systems with testing effort” Int. Journal System Science, 1990; 22(9), 1563-1571
23. Kapur PK, Garg RB, Kumar S. “Contributions to hardware and software reliability” 1999; Singapore:
World Scientific PublIishing CJo. Ltd. SER
24. M. C. K. Yang and A. Chao. “Reliability-estimation and
stopping-rules for software testing based on repeated
appearance of bugs”. IEEE Trans. on Reliability,
44(2):315–321, June 1995.
25. W. Y. Yun and D. S. Bai., “Optimum software release policy with random life cycle” IEEE Trans. on Reliability, 39(2):167–170, June 1990.
26. Ferreira, R.J.P.,Almeida, A.T., Cavalcante, C.A.V., “A multi-criteria decision modelto determine inspection intervals of condition monitoring based on delay timeanalysis,” Reliability Engineering and System Safety 94 (2009) 905–912.
27. Fishburn, C P, Utility Theory for Decision Making, Wiley, New York, 1970.
28. Keeney, R.L. and Raiffa, H. (1976), “Decisions with Multiple Objectives: Preferences and Value Tradeoffs, Wiley, New York.
29. Web reference, http://wiki.ece.cmu.edu/ddl/index.php/Multiattribute_uti lity_theory.
30. Winterfeldt, D. and Edwards, W. (1986), “Decision Analysis and Behavioral Research”, Cambridge University Press, Cambridge, UK
31. Kapur PK, Agarwala S, Garg RB. “Bicriterion release policy for exponential software reliability growth
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