Abstract---Software Testing is the process of implementing a program with the definite intent of finding errors former to delivery to the end user. Due to the increase and the complexity of the software system the problem is how to optimally assign the narrow testing resource during the testing phase has become more important and difficult. Traditional Optimal Testing Resource Allocation Problems (OTRAPs) includesin afinest allocation of a limited amount of testing resources with respect to reliability, cost etc. To solve the OTRAPs with Multi-Objective Algorithms called as Hierarchy Particle Swarm Optimization Algorithm (HPSO) is suggested. Especially, organize OTRAPs for two types of multi-objective problems. First one is reliability of the system and the testing cost of the system as two objectives. Second, the total testing resource consumed is also taken into account as the third objective, sensitivity is the fourth objective. The existing algorithms require more time and the both the evolutionary and the NSGA algorithm having more drawbacks. To overcome the drawbacks of the existing algorithm, the proposed HPSO algorithm which is used in this paper, satisfy allthe four objective of this research. Experimental results show that the proposed algorithm is more efficient than the existing algorithms.
Keywords---Particle Swarm Optimization algorithm, Parallel Series Modular Software System, Software Engineering, Software Reliability, Software Testing, Star-Structure Modular Software System
I. INTRODUCTION
ONTEMPORARY complex software-systems are
developed with machinery supplied by contractors under various environments. Particularly component-based software engineering [1] [2] has drainedfabulous attention in mounting cost-effective and reliable applications to meet short time-to-market requirements. The systems incorporated with such modules, the system-testing problem can be created as a combination optimization problem of the system components. The system-reliability problem of this type is the series-parallel redundancy allocation problem, weather the system reliability is maximized or total system testing cost is reduced. The formations generally involve in system-level constraints on suitable cost, effort, and reduce the system-reliability levels.Series parallel redundancy allocation is a problem used for hardware systems with dynamic programming [3], [4],
M. Pavithra, Student, CIET, India.
DOI: 10.9756/BIJSESC.10101
integer programming [5], nonlinear optimization [6], and heuristic techniques [7] [8].The reliability of positive distribution is considered as a multiple objective process using fuzzy techniques [9].The reliability growth model [10] is applied for hardware components to achieve overall system reliability. Software reliability allocation problem is in [11].To exploit the user’s service, subject to cost and technical constraints predicts how dependable software model and programs are used.For multiple software programs, various techniques for reliability of software allocation problems using redundanciesare discussed in [12].Even though, still do not consider the testing-time of the components of the software and their reliabilitydevelopment.Allowance of testing times of the software is based on a software reliability model on a particular software system is discussedin [13].However it considers as a single application in the system and the reliability-growth model is restricted to the Hyper-Geometric Distribution (S-shaped) Model [14].
A software system is in general comprised of a number of modules.Each and everyelement needs to be assigned for appropriate testing resources before the testing phase. Consequently, a question may arise how to allocate the testing resources to the model so that the consistency of a software system is increased.Such a problem was formally defined as the Optimal Testing Resource Allocation Problems (OTRAPs) [15].Even thoughthe testing resources can be allocated in simple ways is discussed in[16], it has been proved that an optimal allocation scheme couldpossibly direct to considerable improvement of the reliability of a software system. In other words, it is valued optimizing the allocation scheme,solving OTRAPs is a non-trivial task.More effort has been dedicated to this topic since the 1990s [17] and development has been made in the way of implementing more accurate/practical formulations of OTRAPs or problem-solving techniques. Byrevisingthe literature from the previous type of work, an OTRAP is naturally anxious with three factors like reliability, cost, and testing resources. Clearly create an OTRAP made aconnection between these termswants to be defined exactly. In the prose, the relationship between reliability and testing resources was generallydevised by Software Reliability Growth Models (SRGMs), where the reliability is usually have some metric of the failure data, such as the number of failures, time of occurrence, failure severity, or the interval between two consecutive failures [18]. The software system’s reliability growth during software improvement processes is explained by SRGM and it can be viewed as devising the reliability of a software system as a purpose of the testing resource allocation scheme. Obviously, early works completely have a goal toincrease the reliability with a known budget of source by means of reducing the remaining errors
M. Pavithra
Optimal Testing Resource Allocation Problems in
Software System using Heuristic Algorithm
[19], minimizing the number of software faults detected [20] or directly maximizing a function that quantifies the system reliability [21].In recent times, testing cost attracts more attention.Instinctively speaking, testing cost measures the cost required for attaining a given level of reliability.It is essentially a function of reliability, and thus also a function of testing resources [22][23].Current work on OTRAPs in additiondesignedto reduce the testing cost instead of maximizing reliability or includedother constraints on the testing cost [24].
From the past decade, the OTRAPs were frequently handled as optimization problems with a single objective that is the testing resource was billed with the only purpose to maximize the system reliability or minimize the testing cost.However, both reliability and testing cost are important for software development and it is impractical to fail to notice either of them.Unfortunately, by giving a budget of testing resources, more testing cost is usually predictable if want to improve the reliability of a software system.Hence, it is not possibleso as to a single solution is optimal in terms of both reliability and cost. Instead of that it may route to matching between reliability and cost, and looking for a good trade-off. Although some preceding researchers did effort to do so [22],they still solved the problem in a single-objective optimization manner, where the resource allocation scheme is optimized or devised with respect to a weighted sum of reliability and testing cost. This strengthwould be unsuitable because the reliability and cost are generally of different scales.Summing them up does not really provide meaningful information about the quality of solutions, and might cause difficulties in practice as it is hard to determine the appropriate values of weights.
Multi-Objective Evolutionary Algorithms (MOEAs) is designed to solve OTRAPsand it is discussed in [25]. Exclusively, the formulated OTRAPs as two types of multi-objective problem in whichfirst consider the reliability of the system, and the testing cost as two separate objectives.Second, the total testing resource consumed was taken into account as the third goal and the sensitivity is taken as a fourth objective. The advantages of MOEAs over existing approaches were evaluated by applying two MOEAs, namely Nondominated Sorting Genetic Algorithm II (NSGA-II) [26] and multi-objective differential evolution, on two simple parallel-series modular software systems.Sensitivity analysis is used to determine how “sensitive” a model is to changes in the value of the parameters of the model and to changes in the structure of the model.
In this paper proposed an algorithm called Hierarchy particle swarm optimization [27], which require less time for execution and it overcomes the drawbacks of both the Multi-Objective Evolutionary Algorithms and Nondominated Sorting Genetic Algorithm II.
Advantages of the HPSO algorithm can handle the multi level-objective optimization problem successfully, and has the advantages of good convergence property and simplicity.
II. RELATED WORKS
As apromising area of evolutionary algorithm, memeticalgorithm combines global search strategies with local search heuristics [28] and thus searches more efficiently than conventional genetic algorithms [29]. The achievement of memetic algorithm has been established on a selection of single objective optimization problems.In [30] the author discussed about a memetic algorithm which is used to solve single objective redundancy allocation in multi-level system.
Compared to the existing single-objective approaches, which typicallygive one single optimal solution at a time, MOEAs can offer a set of solutions that are well known as non-dominated solutions.For this reason, MOEAs can provide the designers a lot of choices with different level of tradeoff between reliability and cost and the total testing-resource consumed.Competence of this approaches are established on software reliability growth model(SRGMs) [31], which is a main type of software system models which is based on Nonhomogeneous Poisson process (NHPP).
MOEAs, such as Strength Pareto Evolutionary Algorithm II (SPEA2) [32], Pareto Archived Evolution Strategy (PAES) [33], Nondominated Sorting Genetic Algorithm II (NSGA2) [34] and Multi-Objective Differential Evolution Algorithms (MODEs) [35] [36], are the evolutionary approaches to multi-objective optimization problems.They have been demonstrated to perform well on lots of real-world problems.The MOEAs have been never applied to the TRAP before. Consider two scenarios, one is the reliability of the software and the second one is the testing cost. So the goal is to find a good trade-off between the reliability and the cost.Then, the total testing-resource consumed (i.e. the total time) is brought in as the third objective.
Hongfeng Wang [37] discussed about a particle swarm optimization based on memetic algorithm for dynamic optimization problems. T. Warren Liao [38] employed random walk with direction utilization method as the local search operator in the single objective memetic algorithm.The application of memetic algorithm in multi-objective optimization has not drawn a large amountof attention.Slim Bechikh [39] developed a new multi-objective memetic algorithm (PHC-NSGA-II) for continuous optimization of hybridization of the NSGA-II algorithm with polynomial mutation as local search procedure and the efficiency of this local method could be improved.
III. PROBLEMS FORMULATION
There are two types of formation of the problem on the software system is discussed below
A. Problem Formulations on Parallel-Series Modular Software Systems
Due to the increase of the size of the software systems, a system generally comprised of many modules in the designing and testing phase.Figure 1 shows the basic structure of a general parallel-series modular software system with n groups of parallel modules and m serial modules.
Figure 1: Basic Structure of a Parallel-Series Modular Software System
According to the software reliability of the software system text, the following models are adopted to enumerate the reliability of a parallel series modular system and the testing cost.A summary of the classification are presented in Table I.
Table 1 :The Nomenclature of the Software System
T Total testing resources (or time) Testing resource allocated on module i Optimal allocation resource on module i C the total testing cost
cost function of module i according to the reliability of module i
m(t) mean value function in NHPP
R system reliability
reliability function of module i, after a testing period Ti
For every modular in this system, the failure intensity of module iis and it can be calculated by:
Where ai and bi are constants.ai is the mean value of the total errors in modular i and bi describes the rate of detected errors in modular i. Ti is the testing time allocated to module i.
Thus the reliability of module i is
After the Ti unit time of testing, the probability of no fault occurring in the interval (Ti, Ti + x] is Ri(x|Ti). The reliability of modular iis exponentially increasing to the resource allocated to the modular iis find from the equation (1) and (2) is explained in [40].
Based on the above two equations, the reliability of this parallel-series modular software system are calculated as following equation:
Where is the reliability of the lth
parallel group. There are n groups of parallel modules, thus the total reliability of these n groups of parallel modules is
. is the total
reliability of the m serial modules.
The testing cost for each module is defined as
where, , and are constants that control the increment speed of the testing cost corresponding to the reliability in module . Correspondingly, the testing cost for the whole parallel-series modular software system is
Where, is the total testing cost for the n groups of parallel modules, and is the total testing cost for the series modules.
The aim is by knowing an OTRAP, maximizing the reliability and minimizing the testing cost. In the meantime, the consumed testing resource should not exceed a pre-defined budget or (ideally) be minimized.However, eqn (4) shows that the testing cost rapidly increases with reliability. Therefore, the goal of maximizing reliability and minimizing the testing cost will may clash with each other.In scenario, it has been searched under the name of multi-objective optimization. Mathematically, a multi-objective problem with conflicting minimizing objectives can be formulated as (6).
where x, X , y and Y are called decision vector, decision space, objective vector, and objective space, respectively. is the ith objective function of the problem. Because different objective functions frequently conflict with each other, there hardly ever exists a unique solution that is optimal in terms of all objective functions. For this cause, the widespread approach to a multi-objective optimization problem is to look for a set of Pareto optimal solutions.In other words, each solution should not be low-grade to any other solution on all objectives. Such type of solutions is referred to as nondominated solutions.
Table I represents the mathematical formulation of the two multi-objective OTRAPs on parallel-series modular software systems.
Table 2 :Formulations of Two Multi-Objective Problems on Parallel-Series Modular Software Systems (Maximum Testing
Resource to be Allocated Is T) 1. Bi-objective problem
2.Tri- Objective Problem
B. Problem Formulations on Star-Structure Modular Software Systems
The Star-structure modular software systems are consisting of two types of units. They are central, and non-central. The central unit which is acts as a server and can be comprised of parallel, serial or parallel-series modules. A non-central unit consists of some connected modules, each of which is an input of the whole system.
Figure 2 illustrates a star-structure modular software system with non-central modules, and central modules.
Figure 2: Basic Structure of a Star-Structure Modular Software System
Equations (1), (2), and (4) can be readily utilized to quantify the reliability and cost of every single module in a star-structure modular software system. On the basis of these equations, the reliability of a star-structure modular software system can be calculated as
Where, is the reliability of the ith module in the noncentral unit, and is the reliability of the central unit (i.e., total reliability of the m central modules). Similarly, the cost of testing a star-structure modular software system can be calculated by (8).
Where, is the cost of the ith module in the non-central unit, and stands for the cost of the jth module of the central unit. Similar to parallel-series modular software systems, the bi-objective, and tri-objective OTRAPs on star-structure modular software systems are mathematically formulated in Table II.
TABLE II
Formulations of two multi-objective problems on star-structure modular software systems (maximum testing
resource to be Allocated is T) 1. Bi-objective problem
2. Tri objective Problem Input NC2 NC2 NC2 NC2 Input Input Input CM Output Non-Central Modules Central Modules
Hence to solve the OTRAPsthe evolutionary algorithm this is used as a multi objective optimization technique. During the earlier period, MOEAs which are used to solvethe multi-objective optimization problems based on evolutionary algorithms.It has been systematically investigated mainly because of the fact that they can be suitably applied to find multiple Pareto-optimal solutions in one single simulation run. PSO has been effectively used for the continuous nonlinear and discrete single-objective optimization problems and it seems to be particularly suitable for multiobjective optimization problems.Since, it presents the high speed of convergence for single-objective optimization. However, basic PSO is not appropriate to handling multi-objective problems in that there is no absolute global optimum in functions, which leads to complexity in defining a single gBest or pBest during each generation. So to achieve all the objective in this paper, a new particle swarm optimization algorithm solving hierarchy multi-objective optimization problems is proposed.
IV. IMPLEMENTATION OF HIERARCHY PARTICLE SWARM
OPTIMIZATION ALGORITHM (HPSO) FOR THE PROBLEM
FORMULATION
In this paper, a new PSO technique is used to solve hierarchy multi-objective multiple-objective operation optimizations, called Hierarchy particle swarm optimization algorithm (HPSO), is presented.
The main idea of HPSO is as follows:
Ever sincein multiple-objective operation optimization of Optimal Testing Resource Allocation Problems has understandable hierarchy and priority according to this feature, the updating strategies ofparticle pBest and gBest are as follows:
1. First, calculatethe objective function vector of j-th particle, then fromhigh to low sort according to the objective function valueof particle pBest and gBest, According to the sortingresults to determine whether update the pBest and gBestof j-th particle.
2. Which adopts an objective-based fitness assignment approach; these three methods assign fitness based on the concept of domination.
3. Given a solution that a point in the objective space, the crowding distance is calculated as the 1-norm distance between the two nearest neighbors of the
solution. In a multi-objective optimization problem, it is usually expected that the final solutions cover the whole objective space well. Hence, in the problem-solving process, the solutions with larger crowding distances are preferable.
4. It is very difficult to solve multi-objective optimalregulation of the problem due to complexconstraints or parameters. Based on the characteristics of a cascadesystem, change the constraints to thefeasible region of the crowding distance,and then the evolution of the particles of the swarm isrestricted to this region. Thereby, the constrainedoptimization problem is changed to unconstrainedoptimization problems.
The structure of HPSO is shown as follows: Step 1: Initial algorithm parameters;
Step 2: Generate initial population in thefeasible region of particles;
Step 3: Generate the next generation population: FOR j = 1 TO POP_SIZE
FOR i= 1 TO M
Compute the feasible region of particles a.
Compute the velocity limited region b.
C. Compute the position in t + 1generation; NEXT
D. Compute the objective function vector ofj–th
particle
E. Update gBestof population and pBestofj-th particle in accordance with thepreference among the objective function;
F. Mutate the i-th particle; NEXT
Step 4: Judgment over the convergence of HPSO; Step 5: Output the gBestparticle.
where, POP_SIZE is population size. Accelerating genetic operator of HPSO
The particle's velocity in decision space is one of themost important parameters to control the rate of convergenceof the algorithm and it restricts the searching step and directionof the particles during the evolutionary process. If thesearching step is too small, the algorithm may entrap inlocal optima or elseit may cause particle oscillationaround a position [41]. If the domain knowledge ofoptimum object can be obtained, it can guide the particlesto the global optimal solution, so as to improve theconverging speed evidently.
An accelerating genetic operator based on theuniqueness of peaking operation of the software system is designed, which can change the searchingdirection of particles by the particle's position during theevolutionary process. Particularly, if the particles arelocated on the peak load of residual load
process,the decision variable will be decreased or increase with agreat probability, and the closer the peak or valley load, the greater the probability. The accelerating geneticoperator is implemented by the following:
First of all, the decision variable of particles arenormalized to [-1, 1] (let's call decision variable as p).Assuming the absolute value of p is greater a thresholdvalue (such as 0.85), the particles are located on the peakload (p > 0) or valley load (p < 0), the maximal velocityof particles are set to 0 for the former, and the minimalvelocity of particles are set to 0 for the latter. The greaterof p of particles, the more the variation range of velocityis. So the searching step and direction are changed bylimiting variation range of velocity.The realization ofalgorithm is as shown in Fig.1. where, isthe average value of decision variable of j-th particle, R israndom number between [0,1], sgn(·) is sign function, αis real number between [0,1]. The algorithm of velocity inHPSO degenerated to general PSO algorithm.
V. EXPERIMENTAL RESULTS
Experimental studies have been carried out to evaluate the effectiveness of the proposed approach. Here, to prove the effectiveness of the proposed Hierarchy Particle Swarm Optimization Algorithm technique this method is compared with the other existing algorithm like NRGA, NSGA etc.
5.1. Performance Analysis of the Optimization Algorithms
The performance is evaluated based on the parameters like Convergence behavior
Processing Time
a. Convergence Behavior
Table 3 shows the performance comparison of the optimization techniques such as Genetic Algorithm, Memtic Algorithm, PSO, GDPSO and the proposed HPSABC.
The Proposed HPSO algorithm outperforms the NRGA Algorithm, NSGA Algorithm and HaD-MOEA algorithm in attaining the reliability in terms of convergence behavior.
Table 3: Performance Comparison of the Optimization Techniques using Convergence Behavior
Optimization Algorithms Convergence Behavior
NRGA 60
NSGA 55
HaD-MOEA 45
Proposed HPSO 30
Figure 3: Comparison of Convergence Behavior Figure 3 shows the comparison of the convergence behavior of the NSGA, NRGA HaD-MOEA and the proposed HPSO approach. It is observed from the figure that the HPSO converges in lesser iterations (i.e. 30 iterations) when compared with the other optimization techniques. Thus the proposed HPOS technique is very significant when compared with the other optimization approaches taken for consideration.
b. Processing Time
Table 4 shows the performance comparison of the optimization techniques such as Genetic Algorithm, Memtic Algorithm, PSO, GDPSO and the proposed HPSABC.
The Proposed HPSO algorithm outperforms the NRGA Algorithm, NSGA Algorithm and HaD-MOEA algorithm in attaining the reliability in terms of the processing time.
Table 4: Performance Comparison of the Optimization Techniques using Processing Time
Optimization Algorithms Processing Time (sec) NRGA 16 NSGA 21 HaD-MOEA 15 Proposed HPSO 8
Figure 4: Comparison of Processing Time 0 10 20 30 40 50 60 70
NRGA NSGA HaD-MOEAProposed HPSO
Convergence Behavior Converge… 0 5 10 15 20 25
NRGA NSGA HaD-MOEAProposed HPSO
Processing Time (sec)
It is observed from the figure 4, that the proposed HPSO optimization technique takes processing time of 8 seconds, where as the other optimization techniques such as NRGA, NSGA and HaD-MOEA takes longer processing time such as 16, 21 and 15 seconds respectively.
5.2. Performance Comparison of the Proposed Approach
The experiments are evaluated based on the parameters like reliability, testing cost and the run time. In this section, experimentally compare the proposed multiobjectiveapproach with traditional single-objective approaches on two examples. Three state-of-the-art single objective methods are chosen for comparison.
A. Performance Evaluation for Parallel-Series Modular Software System
Table 5 shows that the comparison of various approaches for parallel-series modular software system. From the table, it can be shows that the proposed approach is more efficient in solving the multiobjective problems than the other existing approaches.
Table 5: Comparison of Various Approaches for Parallel-Series Modular Software System
Approaches Reliability Testing cost (units) Total Testing Resource consumed Sensitivity NSGA-II and HaD-MOEA 0.8957 9.4 15.13 0.3 Proposed HPSO Approach 0.944 8.1 13.01 0.10
Figure 5: Comparison of Various Approaches for Parallel-Series Software Modular Structure
Figure 5 shows the reliability versus cost graph. From the figure, it can be shows that the proposed approach is more efficient than the other existing approaches. Hence the proposed approach is well suited for solving the multiobjective problems.
Figure 6: Comparison of Various Approaches for Parallel-Series Software Modular Structure
Figure 6, shows the time versus sensitivity graph. From the figure, it can be shown that the proposed approach is more efficient than the other existing approaches. Hence the proposed approach is well suited for solving the multiobjective problems. Hence the proposed approach consists of better sensitivity.
B. Performance Evaluation for a Star-Structure Modular Software System
Table 6 shows that the comparison of various approaches. From the table, it can be shows that the proposed approach is more efficient in solving the multiobjective problems than the other existing approaches.
Table 6: Comparison of Various Approaches for Star-Structure Modular Software Star-Structure
Approaches Reliability Testing cost (units) Total Testing Resource consumed Sensitivity NSGA-II and HaD-MOEA 0.89 9.456 14.92 0.6 Proposed HPSO Approach 0.961 7.3 11.04 0.17 0 2 4 6 8 10 12 14 0.4 0.5 0.6 0.7 0.8 0.9 1 Cost Reliability proposed HPSO 0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4 5 Sensitivity Time (msec) proposed HPSO 0 2 4 6 8 10 12 14 0.4 0.5 0.6 0.7 0.8 0.9 1 Cost Reliability proposed HPSO
Figure 7: Comparison of Various Approaches for Star-Structure Modular Software Star-Structure
Figure 5 shows the reliability versus cost graph for star-structure modular software star-structure. From the figure 7, it can be shows that the proposed approach is more efficient than the other existing approaches. Hence the proposed approach is well suited for solving the multiobjective problems.
Figure 8: Comparison of Various Approaches for Star-Structure Modular Software Star-Structure
Figure 8, shows the time versus sensitivity graph. From the figure, it can be shown that the proposed approach is more efficient than the other existing approaches. Hence the proposed approach is well suited for solving the multiobjective problems. Hence the proposed approach consists of better sensitivity.
Hence, from the experimental results the proposed HPSO algorithm performs better to solve the multi objective problems also that it performs better on star structure modular software system than the parallel series software modular structure.
VI. CONCLUSION
Nowadays the software systems have become very large and compound. In spite of the decrease of the hardware costs, the costs of software systems have increased rapidly during the past several decades. In modern computer systems,software takes larger portion of the system cost than the hardware. In this paper, a new particle swarm optimization algorithm called Hierarchy particle swarm optimization algorithm solving multi-objective optimization problems is proposed.The proposed technique is used to organize OTRAPs for three types of multi-objective problems like reliability of the system, the testing cost of the system, the total testing resource consumed and the sensitivity is also taken into account. The existing algorithms require more time and the both the evolutionary and the NSGA algorithm having more drawbacks. Experimental results show that the proposed algorithm is more efficient than the existing algorithms.
REFERENCES
[1] X. Cai, M. R. Lyu, K. F.Wong, and R. Ko, “Component-based software engineering: Technologies, development frameworks, and quality assurance schemes,” in Proc. Asia-Pacific Software Engineering Conf., Dec. 2000, Pp. 372–379.
[2] W. Kozaczynski and G. Booch, “Component-based software engineering,” IEEE Software, Vol. 155, Pp. 34–36, Sep./Oct. 1998. [3] E. Fyffe, W. W. Hines, and N. K. Lee, “System reliability allocation and a computational algorithm,” IEEE Trans. Reliability, Vol. R-17, Pp. 64–69, 1968.
[4] Y. Nakagawa and S. Miyazaki, “Surrogate constraints algorithm for reliability optimization problems with two constraints,” IEEE Trans. Reliability, Vol. R-30, Pp. 175–181, 1981.
[5] R. L. Bulfin and C. Y. Liu, “Optimal allocation of redundant components for large systems,” IEEE Trans. Reliability, Vol. R-34, Pp. 241–247, 1985.
[6] F. A. Tillman, C. L. Hwang, and W. Kuo, “Determining component reliability and redundancy for optimum system reliability,” IEEE Trans. Reliability, Vol. R-26, Pp. 162–165, 1977.
[7] W. Coit and A. E. Smith, “Reliability optimization of series-parallel systems using a genetic algorithm,” IEEE Trans. Reliability, Vol. 45, 1996.
[8] L. Painton and J. Campbell, “Genetic algorithms in optimization of system reliability,” IEEE Trans. Reliability, Vol. 44, Pp. 172–178, 1995.
[9] K. Dhingra, “Optimal apportionment of reliability and redundancy in series systems under multiple objectives,” IEEE Trans. Reliability, Vol. 41, Pp. 576–582, Dec. 1992.
[10] W. Coit, “Economic allocation of test times for subsystem-level reliability growth testing,” IIE Trans., Vol. 30, No. 12, Pp. 1143– 1151, Dec. 1998.
[11] Zahedi and N. Ashrafi, “Software reliability allocation based on structure, utility, price, and cost,” IEEE Trans. Software Eng., Vol. 17, Pp. 345–355, Apr. 1991.
[12] O. Berman and N. Ashrafi, “Optimization models for reliability of modular software systems,” IEEE Trans. Software Reliability, Vol. 19, Pp. 1119–1123, Nov. 1993.
[13] R.-H. Hou, S.-Y. Kuo, and Y.-P. Chang, “Efficient allocation of testing resources for software module testing based on the hyper-geometric distribution software reliability growth model,” in Proc. 7th Int. Symp. Software Reliability Eng., Oct./Nov. 1996, Pp. 289– 298.
[14] Y. Tohma, K. Tokunaga, S. Nagase, and Y. Murata, “Structural approach to the estimation of the number of residual software faults based on the hyper-geometric distribution,” IEEE Trans. Software Eng., Vol. 15, Pp.345–355, Mar. 1989.
[15] Ohtera and S. Yamada, “Optimal allocation and control problems for software-testing resource,” IEEE Transactions on Reliablity, Vol. 39, No. 2, Pp. 171–176, 1990.
[16] R. H. Huo, S. Y. Kuo, and Y. P. Chang, “Efficient allocation of testing resources for software module testing based on the hyper-geometric distribution software reliability growth model,” IEEE Transcations on Reliablity, Vol. 45, No. 4, Pp. 541–549, 1996. [17] P. Y. Yin and J. Y. Wang, “A particle swarm optimization
approach to the nonlinear resource allocation problem,” Applied Mathematics and Computation, Vol. 183, Pp. 232–242, 2006. [18] M. R. Lyu, S. Rangarajan, and A. P. A. V. Moorsel, “Optimal
allocation of testing resources for software reliability growth modeling in component-based software development,” IEEE Transactions on Reliability, Vol. 51, No. 2, Pp. 183–192, 2002. [19] S. Yamada, T. Lehimori, and M. Nishiwaki, “Optimal allocation
policies for testing resource based on a software reliability growth model,” Mathematical and Computer Modelling, Vol. 22, Pp. 295– 301, 1995.
[20] R. H. Huo, S. Y. Kuo, and Y. P. Chang, “Efficient allocation of testing resources for software module testing based on the hyper-geometric distribution software reliability growth model,” IEEE Transactions on Reliability, Vol. 45, No. 4, Pp. 541–549, 1996. [21] W. Coit, “Economic allocation of test times for subsystem-level
reliability growth testing,” IIE Transcations on Business, Vol. 30, No. 12,Pp. 1143–1151, 1998. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 1 2 3 4 5 6 Sens itivity Time (msec) proposed HPSO
[22] B. Yang and M. Xie, “Optimal testing-time allocation for modular systems,” International Journal of Quality and Reliability Management, Vol. 18, No. 8, Pp. 854–863, 2001.
[23] Y. Huang and M. R. Lyu, “Optimal testing resource allocation, and sensitivity analysis in software development,” IEEE Transactions on Reliability, Vol. 54, No. 4, Pp. 592–603, 2005.
[24] Y. S. Dai, M. Xie, K. L. Poh, and B. Yang, “Optimal testing-resource allocation with genetic algorithm for modular software systems,” The Journal of Systems and Software, Vol. 66, Pp. 47– 55, 2003.
[25] Z. Wang, K. Tang, and X. Yao, “A multi-objective approach to testing resource allocation in modular software systems,” in
Proceedings of the 2008 IEEE Congress on Evolutionary Computation (CEC2008), Hongkong, China, 2008, Pp. 1148–1153. [26] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” IEEE Transactions on Evolutionary Computation, Vol. 6, No. 2, pp. 182–197, 2002.
[27] Junjie Yang, “A New Particle Swarm Optimization Algorithm to Hierarchy Multi-objective Optimization Problems and Its Application in Optimal Operation of Hydropower Stations”, JOURNAL OF COMPUTERS, Vol. 7, No. 8, AUGUST 2012. [28] P. Moscato, On Evolution, Search, Optimization, Genetic
Algorithms and Martial Arts: Towards Memetic Algorithm, CalTech Concurrent Computation Program Report 826. Pasadena, CA: CalTech, 1989.
[29] N. Krasnogor, and J. Smith, “A tutorial for competent memetic algorithms: model, taxonomy, and design issues,” IEEE Trans. Evolutionary Computation, Vol. 9, Pp. 474– 488, Oct. 2005. [30] Z. Wang, K. Tang, and X. Yao, “A memetic algorithm for
multi-level redundancy allocation,” IEEE Trans.Reliability, Vol. 59, Pp. 754-765, Dec. 2010.
[31] M. R. Lyu(1996). Handbook of Software Reliability Engineering, McGraw Hill.
[32] Zitzler, M. Laumanns, and L. Thiele, “SPEA2: Improving the Strength Pareto Evolutionary Algorithm,” In K. Giannakoglou et al., editor, EUROGEN 2001. Evolutionary Methods for Design, Optimization and Control with Applications to Industrial Problems, Pp. 95-100, Athens, Greece, 2002.
[33] J. D. Knowles and D. W. Corne, “Approximating the Nondominated Front Using the Pareto Archived Evolution Strategy,” In Evolutionary Computation, Vol. 8, No. 2, Pp. 149-172, 2000.
[34] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II,” In IEEE Transactions on Evolutionary Computation, Vol. 6, No. 2, Pp. 182-197, April 2002.
[35] C. S. Chang, D. Y. Xu. and H. B. Quek. “Pareto-Optimal Set Based Multiobjective Tuning of Fuzzy Automatic Train Operation for Mass Transit System,” In IEE Proceedings on Electric Power Applications, September 1999.
[36] V. L. Huang, P. N. Suganthan, A. K. Qin, and S. Baskar, “Multi- Objective Differential Evolution with External Archive and Harmonic Distance-Based Diversity Measure,” Nanyang Technological University,Singapore, Tech. Rep. TR-07-01, 2006. [37] H.F. Wang, S.X. Yang, W.H. Ip and D.W. Wang, “A particle
swarm optimization based memetic algorithm for dynamic optimization problems,” Nat. Comput., Vol.9, Pp. 703-725, Sep. 2010.
[38] T.W. Liao, “Two hybrid differential evolution algorithms for engineering design optimization,” Applied Soft Computing, Vol.10, Pp. 1188–1199, Sep. 2010.
[39] S. Bechikh, N. Belgasmi, L.B. Said, and K.Ghédira, “PHCNSGA- II: a novel multi-objective memetic algorithm for continuous optimization,” in Proc. 20th IEEE International Conference on Tools with Artificial Intelligence, Dayton,OH USA, Nov. 2008, Pp.180-189.
[40] B. Yang, M. Xie, 2000. “A Study of Operational and Testing Reliability in Software Reliability Analysis,” Reliable. Eng. Syst. Safety 70, Pp. 323- 329.
[41] Y. P. Chen, P. Jiang, “Analysis of particle interaction inparticle swarm optimization,” Theoretical Computer Science, Vol. 411, Pp. 2101-2115, 2010.
