Intelligent Algorithm of Optimal Allocation of Test Resource Based on Imperfect Debugging and Non-homogeneous Poisson Process

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1 Intelligent Algorithm of Optiml Alloction of Test Resource Bsed on Imperfect Debugging nd Non-homogeneous Poisson Process Xiong Wei 1, 2, Guo Bing * 1, Shen Yn 3, Wenli Zhng 1, 4 1 Computer Science College, Sichun University, Chengdu , Chin 2 Leshn Voctionl & Technicl College, Leshn , Chin 3 School of Control Engineering, Chengdu University of Informtion Technology, Chengdu , Chin 4 Chengdu Technologicl University, Chengdu , Chin Abstrct Bsed on non-homogeneous Poisson Process (NHPP), by combintion of imperfect debugging in the testing stge nd introducing wrong model, the optiml distribution expression nd constrint conditions of test resources re derived in this pper, nd due to the NP-hrd nture of this problem, so we introduce the intelligent lgorithm. By combintion of bsic intelligent lgorithms of GA, SA nd CS, the lgorithms of GA-CSA nd GA-SA re put forwrd in this pper, which re lso compred with ech other concerning the efficiency of solving the problem. In terms of optiml degree nd convergence rte of solutions, the effect of GA-CSA is the best nd the effect of GA-SA is the second, the effect of GA is equivlent to tht of SA, the effect of the ltter two lgorithms is lower thn tht of the former two lgorithms. The experiment result hs proved tht our optiml mthemticl deduction of test resources is correct nd the intelligent lgorithm is effective. Key words: NHPP, TCM, GA-CSA, GA-SA 1. INTRODUCTION As n importnt sfegurding mesure of relibility of softwre, the softwre test hs received universl ttention. How to crry out resonble softwre test under limited conditions nd relize the blnce of softwre test cost nd relibility is n importnt subject of softwre development. The lloction problem of softwre test resources refers to the fct tht under the premise of limited test resources, how to llocte the test resource in resonble nd optiml wy in order to reduce the cost of softwre testing while the relibility of the softwre system is up to the stndrds t the sme time. Literture (Fiondell nd Gokhle, 2012) pplies softwre rchitecture theory to optimize lloction of test resources, in order to reduce the cost of test; literture (Wng et l., 2010) discusses multi-objective evolutionry lgorithm to solve optiml lloction of test resource; literture (Hung nd Lo, 2006) discusses relibility of modulrized softwre nd lloction of optiml test resource of cost; literture (Lyu et l., 2002) considers the optiml lloction of test resource under the growth model of softwre relibility; literture (Aggrwl et l., 2012; Bermn nd Culter, 2004; Di et l., 2013; Morli nd Soyer, 2003; Pietrntuono et l., 2010; Zhng nd Phm, 1998) discusses the optiml resource lloction of type of softwre test from multiple ngles nd methods. The reltion between the relibility of softwre nd the test resources cn be indicted by Relibility Growth Model (RGM). The reltion between cost of softwre test nd resources cn be indicted by Test Cost Model (TCM). In the following, RGM nd TCM will be introduced respectively. Over the pst thirty yers, vrious softwre models hve been put forwrd (Goel nd Okumoto, 1979; Lyu, 1996; Mus et l., 2010), in which G-O model nd NHPP model re the most commonly used nd the most well-known. 79

2 2. MATHEMATICAL MODEL We used Non-Homogeneous Poisson Process for modeling of softwre relibility. According to the theories of NHPP, suppose the number of defects emerging during the time of (0, t) is N(t), for ny t 0, x 0, we cn get: k mtxmt m t x m t PN t x N t k e (1) k! m(t) is the men function of NHPP. As for N(t), the distribution function is: (2) In order to build up RGM, we hve mde the following hypothesis: 1) For ny model i, the wrong removing process cn be indicted by NHPP with the men vlue of m i (t) 2) In the wrong removing phse, the wrong content would be removed with probbility of p; 3) In the wrong removing phse, new mistkes my emerge; Wrong remove rte is S-type non-decresing logistic function. 4) Bsed on the hypotheses bove, for the testing of some module, the process of removing mistkes cn be described with the following expression; S(t) indictes the test resources of cumultive consumption during the time of (0, t], (t) nd b(t) cn be indicted s: (3) (4) b is the wrong detection rte, is the number of mistkes existing in the system t the strt of the testing of the softwre, is the rtio of production of new mistkes, β is constnt. (4) nd (5) re put into (3), the vlue of m(t) is: (5) Prove: (4) nd (5) re put into (3), which cn be simplified into: (6) 1 dm t pb ds t pb ds t m bst bst dt 1e dt 1e dt (7) Let: 80

3 1 pb ds t Pt bst 1 e dt The following first order liner differentil eqution cn be obtined: dm t The generl solution of the first order liner differentil eqution is: dt Ptmt Pt 1 (8) Ptdt Ptdt mt e Pt e dt C 1 (9) which cn be simplified into P t dt m t C e (10) 1 P t dt 1 pb ds t dt bst 1 e dt bst 1 1 pbs t p 1 log 1 e C (11) (11) is put into (10) to get: bst 1 pbstp1 log 1e C 1 mt C e 1 bst 1 pbst p1 log1 e C e e e 1 bst 1 pbst p1 log1 e C2 e e 1 bst e C2 1 1 p1 bst e (12) As m(0)=s(0)=0, (12) cn be put into the formul to obtin the vlue of C2: p 1 C 2 1 (13) 1 is put into (12), finlly the vlue of m(t) cn be obtined: bst e mt C2 bs 1 t 1 e p1 e bs 1 1 t 1 e bst p 1 1 p1 bst 1 e 1 bs 1 t 1 e p1 C1 81

4 The whole process of demonstrtion is complete. Suppose there re N modules in the system, the totl testing time is T, during the time of T, the test resources consumed by ny module is S i, then the formul (6) cn be rewritten into the form bsed on the test resources, indicted s followed: In the formul, b i is the wrong detection rte of the i module, i is the number of mistkes for the i module when the softwre strted, α i is the rtio of new mistkes in the i module, β i is the constnt, p i is the probbility of removing mistke in the i module. Kpur et l., (1999) put forwrd generl TCM, which cn be described in the following expression: (14) With combintion of the formul (14) nd (15), the relibility of ny module i during the testing time T cn be defined s: (15) (16) 3. COST MODEL OF SOFTWARE TEST Kpur et l., (1999) put forwrd generl TCM, which cn be described in the following expression: The formul (17) is chnged into the model between test cost nd test resources, which is indicted in the following expression: (17) In the expression, S indictes the resources consumed by softwre test, C(S) indictes the cost of softwre test, the first prt indictes the cost of looking for nd revising mistkes in the softwre test, the second prt indictes the cost of looking for nd revising mistkes fter the softwre is published, the lst prt indictes the other cost before the softwre is published, such s the pyment of the testing personnel. C 1 indictes the cost of looking for nd revising softwre mistke during the testing period, C 2 indictes the cost of looking for nd revising softwre mistke during the opertion of the softwre, C 3 indictes other cost of resources consumed in unit testing. In the softwre testing, every module is tested independently, so the totl cost of softwre system is the sum of test cost of every module, which cn be indicted s followed: (18) During the period of softwre test, every module of the softwre is tested independently, nd the importnce of every module is different, suppose w i is the weight fctor of the i model, the formul (14) nd (18) re put into the formul (19), we cn get the totl cost of the test of the system: (19) 82

5 In the expression, C 1i nd C 2i represent C 1 nd C 2 in the corresponding formul (1-11) of the i module. (20) 4. OPTIMIZING THE OBJECT In the optiml lloction of the test resources of the softwre, our optiml objective is to minimize the test cost, independent vrible is the test resource consumed by every model. The optiml objective cn be indicted s: Minimize: 5. CONSTRAINT PROGRAMMING Suppose the cost sum of test resource of every model is S ce, the sum of llocted test resource of every module does not exceed some fixed vlue U Sce, so the constrint progrmming of the test resources cn be indicted s: (21) The relibility of every module is not less thn some fixed vlue L Re, then the relible constrint cn be indicted s: (22) 6. SYSTEM OPTIMIZATION ALGORITHM The whole optimiztion issue is combintion optimiztion issue, which is lso Non-deterministic Polynomil issue of difficulty. Therefore, we need heuristic serching method to chieve the optiml nswer of issues. 6.1 The system optimiztion lgorithm bsed on GA nd CSA In the heuristic lgorithm, Genetic Algorithm (GA) hs possessed strong nd globl serching bility, which is often used to solve NP issue of difficulty, but the locl serching bility is reltively poor. On the contrry, CUCKOO Serch Algorithm (CSA) (Yng nd Deb, 2009; Yng nd Deb, 2010; Yng nd Deb, 2011) hs possessed good locl serching bility, but its globl serch bility is reltively poor. Therefore, we considered combining GA with CSA, we put Levy flight of CSA into the use of GA to improve its locl serch bility. We nmed the improved optimized lgorithm s GA-CSA, we used S=(S 1, S 2,, S N) to encode individuls of group. Suppose p t iindictes the i individul in the genertion of t, the corresponding p t ij indictes the j component, then the method of updting individuls by Levy flight (Pvlyukevich, 2007) is s followed: (23) p p s p gbest t 1 t t ij ij ij j (24) In the expression, gbest indictes the globl optimized sttus of the current popultion, s indictes the step length of Levy flight: s=α*μ/ v 1/β, the prmeter μ~n(0, σ 2 μ ), v~n(0, 1), mong which: 83

6 1 sin / 2 1 /2 1 / 22 1/ 2 (25) Generlly we set α=0.01, β=1.5. Detiled procedure of GA-CSA lgorithm is s followed: Step 1 Initilized popultion P 0, the number of itertions mxgen, the scle of popultion sizepop; Step 2 The historicl optiml of initilized individuls pbest i, the globlly optiml of the popultion gbest; Step 3 Initiliztion t=0; Step 4 If t = mxgen, stop the implementtion of procedures; Step 5 Crry out the selection, cross-opertion nd muttion of, producing the offspring; Step 6 As for p P t+1, updte individul p ccording to Levy flight; Step 7 Judge whether p is more optiml thn pbest, if it is yes, then updte pbest; Step 8 Judge whether pbest is more optiml thn gbest, if it is yes, then updte gbest; Step 9 Updte current number of itertions: t= t System optimiztion lgorithm bsed on GA nd SA Besides the mentioned fct tht CSA hs very good locl serch bility, in the heuristic lgorithm, Simulted Anneling (SA) lso hs very good locl serch bility, but globl serch bility is poor, nd the convergence rte is slow. In this section, we considered combining GA with SA to form new optimiztion lgorithm. In the following sections, we will mke comprison with the former GA-CSA through the experiment simultion. We introduced Boltzmnn mechnism of SA into GA to improve the locl serch bility of GA nd nmed it s GA-SA. We used S=(S 1, S 2,, S N) to encode the individuls of popultion, C(S) indictes the corresponding objective function, detiled procedure of GA-SA is s followed: Step 1 Initilized strting temperture T=T 0, the termintion temperture T end, the cooling coefficient μ, the number of internl recycle Loop; Step 2 Initilized prent popultion P, the historicl optiml of individuls pbest, the globlly optiml of popultion gbest; Step 3 If T<T end, the lgorithm is terminted; Step 4 Initiliztion i=1; Step 5 If i>loop, then T=μ*T, turn to Step 3; Step 6 Initilized offspring popultion Q=Φ; Step 7 Crry out the selection, overlpping nd muttion of the popultion P, producing new popultion P new; Step 8 For ech individul P new P new, P P in P new nd P, dopt Boltzmnn mechnism of SA, judge the individuls needed to be stored P sve=boltzmnn(p new, P), which should be stored into the offspring popultion Q; Step 9 Judge whether P sve is more optiml thn pbest, if it is yes, then updte pbest; 84

7 Step 10 Judge whether pbest is more optiml thn gbest, if it is yes, then updte gbest; Step 11 Crry out i = i + 1, turn to the procedure Step 5. In Step 8 of GA-SA, judge whether we should store P new or p, Boltzmnn mechnism is introduced, the detiled reliztion method is indicted s followed: P sve=boltzmnn(p new, P){ 1. If P new meets the constrint conditions, nd does not stisfy the constrints then 2. P sve=p new 3. else if Pnew does not meet the conditions of constrint then 4. P sve=p 5. else 6. ΔObj=Obj(P new)-obj(p) 7. if ΔObj 0 then 8. P sve=p new 9. else 10. Pr = exp(-δobj/t), r = Rndom[0, 1] 11. if Pr>r then P sve=p new end if 12. end if 13. end if 14. } 7. EXPERIMENT SIMULATION Through the effectiveness nd robustness of verifiction lgorithm GA-CSA nd GA-SA in the experiment simultion system, we selected GA nd SA s comprtive lgorithms. The comprison of lgorithm performnce is minly mde by the dvntge nd disdvntge of test cost of the softwre. In the experiment, 6 different test exmples re selected in totl, their nmes re EX1, EX2, EX3, EX4, EX5 nd EX6 respectively; the corresponding number of test model is 10, 12, 14, 16, 18 nd 20. The prmeter settings of EX1, EX2, EX3, EX4, EX5 nd EX6 re s indicted in the tble 1. Tble 1 Prmeter setting of test exmples Test cse EX1 EX2 EX3 EX4 EX5 EX6 Number of modules Test resource constrints (USce) Relibility constrint (LRe) Test phse correction cost (C1) Runtime modifiction cost (C2)

8 Unit test resource cost (C3) Error removl probbility (P) [0.9, 1] [0.9, 1] [0.9, 1] [0.9, 1] [0.9, 1] [0.9, 1] Number of errors before the strt of the [0, 100] [0, 100] [0, 100] [0, 100] [0, 100] [0, 100] test() Error detection rte (b) [0, 0.1] [0, 0.1] [0, 0.1] [0, 0.1] [0, 0.1] [0, 0.1] Generte new error probbility (α) [0, 0.2] [0, 0.2] [0, 0.2] [0, 0.2] [0, 0.2] [0, 0.2] Constnt(β) [0.2, 1] [0.2, 1] [0.2, 1] [0.2, 1] [0.2, 1] [0.2, 1] 7.1 Algorithm prmeter setting The prmeter setting of GA-CSA lgorithm is s followed in the tble 2. Tble 2 Prmeters Of GA-CSA Popultion size (n) 50 Itertion number (G) 100 Crossover-probbility (PC) 0.7 Muttion-probbility (PM) 0.3 lph (α) 0.01 bet (β) 1.5 The prmeter setting of n lgorithm is indicted s followed in the tble 3: Tble 3 Prmeters Of GA-SA Initil temperture (T0) 500 Termintion temperture (Tend) 1 Cooling coefficient (MU) 0.5 Internl-cycle-times (Loop) 20 Crossover-probbility (PC) 0.7 Muttion-probbility (PM) 0.3 Popultion size (n) 50 The prmeter setting of GA, SA lgorithm is s followed in the tble 4, tble 5: 7.2 Performnce comprison Tble 4 Prmeters Of GA Crossover-probbility (PC) 0.7 Muttion-probbility (PM) 0.3 Popultion size (n) 50 Itertion number (G) 100 Tble 5 Prmeters Of SA Initil temperture (T0) 1000 Termintion temperture (Tend) 0.1 Cooling coefficient (μ) 0.95 Internl-cycle-times (Loop) 10 The experiment is fulfilled in the encoding environment of AMD Athlon(tm) II P340 Dul-Core 2.20GHZ CPU, 4GB RAM, Mtlb R2008. In order to reduce the effect of the rndom dt over the experiment results, the lgorithm is run independently for ten times in 6 test exmples, then the verge vlue of the test cost C is clculted, the men vlue replcement lgorithm is used in every test exmple to obtin the miniml test cost. 86

9 Figure 1 is the miniml test cost solved through the lgorithm in 6 test exmples. From the figure, it cn be seen tht the miniml test cost solved by GA-SA is smller thn tht by GA-CSA, GA nd SA, which indictes tht the test cost solved by GA-SA is optiml, while the test cost solved by GA-CSA is smller thn tht by GA nd SA, which indictes tht the test cost solved by GA-CSA is more optiml thn tht by GA nd SA, the test costs of GA nd SA re very close. Figure 1. Algorithm for the minimum test cost Tble 6 show the solutions in detil, the verge test cost solved by GA-CSA, GA-SA, GA nd SA is 12946, 12268, nd respectively, the solution of GA-SA is the miniml, in terms of the optimiztion result: GA-SA > GA-CSA > GA SA. Tble 6 Miniml test cost comprison lgorithm EX1 EX2 EX3 EX4 EX5 EX6 GA-CSA GA-SA GA SA CONCLUSION Bsed on NHPP, this pper derives the optiml lloction progrm under limittion of test resources from the combintion of imperfect debugging, introducing mistkes in debugging nd the hypothesis tht the wrong remove rte is S-type non-decresing logic function; on this bsis, pply GA-CSA, GA-SA, GA nd SA lgorithms for solutions, nd mke comprison of four intelligent lgorithms. Among the four lgorithms, the solution of GA-SA is the best, the solution of GA-CSA is the second best, the solution of GA nd the solution of SA re equivlent. The experiment result hs shown tht the ssumptions, mthemticl derivtion nd intelligent lgorithm re resonble. ACKNOWLEDGMENTS This work ws supported in prt by the Stte Key Progrm of Ntionl Nturl Science Foundtion of Chin under Grnt No ; The Ntionl Nturl Science Foundtion of Chin under Grnt No nd ; the Science nd Technology Plnning Project of Sichun Province under Grnt No. 2014JY0257, 2015GZ0103 nd 2014-HM SF. REFERENCES Aggrwl A.G., Kpur P.K., Kur G. (2015). Genetic Algorithm Bsed Optiml Testing Effort Alloction Problem for Modulr Softwre, Bvicms Interntionl Journl of Informtion Technology, 12(5). Bermn O., Cutler M. (2004). Resource lloction during tests for optimlly relible softwre, Computers & Opertions Reserch, 31(11), Di Y.S., Xie M., Poh K.L. (2003). Optiml testing-resource lloction with genetic lgorithm for modulr softwre systems, Journl of Systems & Softwre, 66(1),

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