A Hybrd ACO Algorthm for the Next Release Problem He Jang School of Software Dalan Unversty of Technology Dalan 116621, Chna janghe@dlut.edu.cn Jngyuan Zhang School of Software Dalan Unversty of Technology Dalan 116621, Chna zhangjy019@hotmal.com Jfeng Xuan School of Mathematcal Scences Dalan Unversty of Technology Dalan 116024, Chna xuan@mal.dlut.edu.cn Zhle Ren School of Mathematcal Scences Dalan Unversty of Technology Dalan 116024, Chna ren@mal.dlut.edu.cn Yan Hu School of Software Dalan Unversty of Technology Dalan 116621, Chna huyan@dlut.edu.cn Abstract In ths paper, we propose a Hybrd Ant Colony Optmzaton algorthm (HACO) for Next Release Problem (NRP). NRP, a NP-hard problem n requrement engneerng, s to balance customer requests, resource constrants, and requrement dependences by requrement selecton. Inspred by the successes of Ant Colony Optmzaton algorthms (ACO) for solvng NP-hard problems, we desgn our HACO to approxmately solve NRP. Smlar to tradtonal ACO algorthms, multple artfcal ants are employed to construct new solutons. Durng the soluton constructon phase, both pheromone trals and neghborhood nformaton wll be taen to determne the choces of every ant. In addton, a local search (frst found hll clmbng) s ncorporated nto HACO to mprove the soluton qualty. Extensvely wde experments on typcal NRP test nstances show that HACO outperforms the exstng algorthms (GRASP and smulated annealng) n terms of both soluton qualty and runnng tme. Keywords-next release problem (NRP); ant colony optmzaton; local search; requrment engneerng I. INTRODUCTION As a well-nown problem arsng n software requrement engneerng, Next Release Problem (NRP) sees to maxmze the customer benefts from a set of nterdependent requrements, under the constrant of a predefned budget bound. NRP was frstly formulated as an optmzaton problem n software evoluton [1] by Bagnall n 2001 [2]. To optmze software requrements, NRP and ts varants have attracted much attenton from the research communty n recent years [3-5], such as Component Selecton and Prortzaton [6], Farness Analyss [7], Mult-Objectve Next Release Problem (MONRP) [7, 8], and Release Plannng [1, 9]. Snce NRP has been proved as NP-hard, no exact algorthm exsts to fnd optmal solutons n polynomal tme unless P=NP [10]. Therefore, many heurstc algorthms have been proposed for NRP to obtan near optmal solutons n reasonable tme, ncludng greedy algorthms, hll clmbng, smulated annealng (SA) [2, 6], and genetc algorthms [3, 8]. Among these algorthms, LMSA (a Smulated Annealng algorthm by Lundy and Mees) can wor effcently on some problem nstances [2]. Ant Colony Optmzaton (ACO) s one of the new technologes n approxmately solvng NP-hard problems snce 1991 [11]. Wth a colony of artfcal ants, ACO can acheve good solutons for numerous hard dscrete optmzaton problems. To mprove the performance of ACO, some hybrd ACO algorthms (.e. ACO wth local search) have also been proposed for solvng classcal optmzaton problems, ncludng Travelng Salesman Problem (TSP) [11], Quadratc Assgnment Problem (QAP) [12], 3-Dmensonal Assgnment Problem (AP3) [13], etc. Motvated by the great successes of ACO n taclng NPhard problems, we propose a new algorthm named Hybrd ACO (HACO) for solvng large NRP nstances n ths paper. HACO updates the solutons under the gudelne of artfcal ants and pheromone trals. At every teraton, HACO constructs the solutons and updates the pheromones for the next teraton. In contrast to tradtonal ACO algorthms, a frst found hll clmbng (FHC) operator s ncorporated nto HACO to mprove the soluton qualty. Ths operator always updates solutons when the frst better soluton s found. Snce t s a challengng and tme consumng tas to tune approprate parameters for ACO algorthms, we employ CALIBRA procedure (an automatc tunng procedure by Adenso-Daz and Laguna n 2006) [14] to effectvely determne those nterrelated parameter settngs. Expermental results on typcal NRP test nstances show that our new algorthm outperforms those exstng algorthms n both soluton qualty and runnng tme. The remander of ths paper s organzed as follows. Secton II gves out the related defntons of NRP. Secton III ntroduces ACO and HACO for NRP. Secton IV descrbes the Ths wor s partally supported by the Natural Scence Foundaton of Chna under Grant No. 60805024 and the Natonal Research Foundaton for the Doctoral Program of Hgher Educaton of Chna under Grant No. 20070141020. 166
expermental results on NRP nstances. Secton V dscusses some related wor and Secton VI concludes ths paper. II. PRELIMINARIES Ths secton gves the defntons of NRP and some notatons whch wll be used n the followng part of ths paper. In a software system, let R={r 1, r 2,, r n } denotes all the possble software requrements. Each requrement r R s assocated wth cost c Z +. A drected acyclc graph G=(R,E) denotes the dependency of requrements, where ( rr, ') E ndcates that r must be satsfed before r'. A set of requrements parent(r) contans all requrements whch must be satsfed before r. Let S={1, 2,, m} denotes all the customers related to the requrements. Each customer s satsfed, f and only f a set of requrements R R s satsfed. A proft w Z + denotes the prorty of customer. To satsfy a customer, both the requrements of R and the ones satsfed before them parent( R ) = parent( r) must be satsfed. Let r R ' R = R parent( R) denotes all the requrements whch have ' to be developed to satsfy customer. Let cost ( R ) = rj R c ' j be the cost of satsfyng customer. The cost of a subset S' S ' s defned as cost( S ') = cost( S' R) and the overall proft obtaned s defned as ω( S') = Sw. ' Based on above defntons, the goal of NRP s to fnd a subset S' S to maxmze ω ( S '), subject to cost( S ') B, where B s the predefned development budget bound [2]. Gven a NRP nstance (denoted as NRP(S,R,W)), a feasble soluton s a subset S' S subject to cost( S ') B. III. A HYBRID ACO ALGORITHM In ths secton, we present the HACO algorthm for solvng NRP n detal. In contrast to tradtonal ACO algorthms, a local search operator called FHC s ncorporated nto HACO to mprove the qualty of solutons. We present the framewor of ths HACO n Part A and the local search operator n Part B, respectvely. A. HACO Ant Colony Optmzaton (ACO) s a meta-heurstc algorthm based on communcaton of a colony of smple agents (artfcal ants), whch are drected by artfcal pheromone trals. The ants use pheromone trals and heurstc nformaton to probablstcally construct solutons. In addton, the ants also use these pheromone trals and heurstc nformaton durng the algorthm s executon to reflect ther search experences [11, 12]. In ths subsecton, we propose HACO algorthm to solve NRP. The man framewor of HACO s presented n Algorthm 1. After the ntalzaton, a seres of teratons are conducted to fnd solutons. At every teraton, HACO employs those ants to construct solutons by a probablstc acton choce rule and ncorporates a local search operator L to further mprove those constructed solutons (see Step (2.1)). Then, HACO updates the pheromone trals for the next teraton, ncludng the pheromone evaporaton (see Step (2.2)) and deposton (see Step (2.3)). Durng the teratons, when a better soluton s found, our best soluton wll be updated. Fnally, the best soluton wll be returned. Algorthm 1: HACO Input: NRP(S,R,W), local search operator L, teraton tmes t, ant number h Output: Soluton S Begn B(1) S =, ω( S ) = 0 ; B(2) for =1 to t do BB(2.1) for =1 to h do BBBBBBB // SolutonConstructon phase BBBBBBB ant construct solutons S ; BBBBBBB // Local search phase BBBBBBB call L to optmze S ; BB//Pheromone update phase BB(2.2) evaporate pheromone on all customers, by (2); BB(2.3) for =1 to h do BBBBBBB depost pheromone on customers n S, by (4); BBBBBBB f ω( S )<ω(s ) then =S ; S B(3) return S ; End More detals related to HACO are dscussed as follows: 1) Pheromone tral and heurstc nformaton: The pheromone trals τ for NRP refers to the desrablty of addng a customer to the current partal soluton. We defne τ = θ w, where θ s a parameter assocated wth customer profts w to mae sure that the ntal pheromone value vares between 0 and 1. We also defne the heurstc nformaton η =w /cost(r '). Obvously, the hgher proft and the lower requrements cost a customer has, the hgher η s. 2) Soluton Constructon: Intally, there re no customers n ants for NRP. At each constructon step, ant apples a probablstc acton choce rule (random proportonal rule [11]) to decde whch customer to add next. In partcular, the probablty wth whch ant chooses customer s gven by: p α β [ τ] [ η] =, f N. (1) α β [ τ ] [ η ] l N l l where α and β are two parameters whch determne the relatve nfluences of the pheromone tral and the heurstc nformaton, and N s the set of customers that ant has not chosen yet. The pseudo-code for the ant soluton constructon s gven n Algorthm 2. Ths procedure wors as follows. Frstly, there s no customer n ant (see Step (1)). Secondly, the next customer s chosen probablstcally by the roulette wheel selecton procedure of evolutonary computaton, proposed by 167
Goldberg n 1989 [15]. The roulette wheel s dvded nto slces proportonal to weghts of customers that ant has not chosen yet (see Step (2.3.2)). Fnally, the wheel s rotated and the next customer chosen by ant s the one whch the marer ponts to (see Step (2.3.3)). Algorthm 2: SolutonConstructon Input: NRP(S,R,W), budget B, ant Output: Soluton S Begn B(1) S =, ω( S ) = 0 ; B(2) whle (cost( S ) B) BB(2.1) sumprob = 0; BB(2.2) let r be a value randomly generated n [0, 1]; BB(2.3) for each customer do BBB(2.3.1) f ( has been added to S ) BBBBBBBB p =0.0; BBB(2.3.2) else BBBBBBBB p s calculated by (1); BBBBBBBB sumprob += p ; BBB(2.3.3) f (sumprob r) BBBBBBBB add customer to S, brea; B(3) return S ; End 3) Update of pheromone trals: The pheromone trals are updated after all the ants have constructed ther solutons. Frstly, we decrease the pheromone value of every customer by a constant factor, and then add pheromone to those customers that the ants have chosen n ther solutons. Pheromone evaporaton s mplemented by τ (1 ρ) τ. (2) where 0< ρ 1 s the pheromone evaporaton rate, whch enables the algorthm to forget bad decsons prevously taen and have more opportuntes to choose other customers. All ants depost pheromone on the customers whch they have chosen n ther solutons after evaporaton: τ τ + Δ h τ. (3) = 1 where Δτ s the amount of pheromone that ant deposts on those chosen customers. It s defned as follows: γw, s selected Δ τ = 0, otherwse where W, the qualty of the soluton S bult by ant, s computed as the sum of customer profts n S. The parameter γ s used to tune Δ. τ (4) B. FHC To enhance the performance of HACO, we ncorporate a local search operator L named FHC nto HACO. In ths secton, we ntroduce the framewor of FHC (see Algorthm 3). Algorthm 3: FHC Input: NRP(S,R,W), budget B, teraton tmes t Output: Soluton S Begn B(1) S =, ω( S ) = 0 ; B(2) for =1 to t do BB(2.1) let be a random feasble soluton; S 0 BB(2.2) flag = true; BB(2.3) whle (flag = true) do BBB(2.3.1) randomly choose j S \ S0 ; BBB(2.3.2) let S1 = S0 {} j ; BBB // when S 1 s feasble BBB(2.3.3) f cost(s 1)<B then S0 = S1; BBBBBBB else // swap a customer n S 0 wth j BBBB(2.3.3.1) flag = false; BBBB(2.3.3.2) S1 = S0; BBBB(2.3.3.3) for every customer l S 0 do BBBBBBBBBBB f cost( S0 {}\{} j l ) < B and BB ω( S0 { j}\{}) l > ω( S1) BBBBBBBBBBB then S1 = S0 {}\{} j l, flag = true; BBBB(2.3.3.4) S0 = S1; * * BB(2.4) f ω( S0) > ω( S ) then S = S 0 ; B(3) return S ; End The hll clmbng algorthm s a classc local search technology, whch can fnd local optmal solutons n the soluton neghborhood. FHC always updates the soluton when a frst local optmal soluton s found. FHC manly conssts of a seres of teratons. At every teraton, a feasble soluton wll be randomly generated as the current soluton (see Step (2.1)). After that, ths current soluton S 0 wll be further mproved by local search as follows (see Step (2.3)). Frstly, an unselected customer j S \ S0 wll be arbtrarly chosen out (see Step (2.3.1)). A new soluton can be acheved by addng customer j to S0 (see Step (2.3.2)). If such acton wll result n a new feasble soluton, then the current soluton S 0 wll be updated wth the new generated soluton and the local search process s further conducted to mprove t (see Step (2.3.3)). Otherwse, we try to replace a customer l S 0 wth j such that the resultng soluton s feasble and ts proft s maxmzed. If succeeded, the current soluton wll be updated wth the resultng soluton and the local search s further conducted to mprove t (see Step (2.3.3.1)-(2.3.3.4)). After every teraton, when a better soluton s obtaned, the best soluton wll be updated (see Step (2.4)). Fnally, the best soluton s returned (see Step (3)). 168
TABLE I. THE DETAILS OF INSTANCE GENERATION Instance NRP-1 NRP-2 NRP-3 NRP-4 NRP-5 Number 20/40/80 20/40/80/160/320 250/500/750 250/500/750/1000/750 500/500/500 Cost 1~5/2~8/5~10 1~5/2~7/3~9/4~10/5~15 1~5/2~8/5~10 1~5/2~7/3~9/4~10/5~15 1~3/2/3~5 Max. 8/2/0 8/6/4/2/0 8/2/0 8/6/4/2/0 4/4/0 Customer 100 500 500 750 1000 Request 1~5 1~5 1~5 1~5 1 Proft 1~30 1~30 1~30 1~30 1~30 IV. EXPERIMENTAL RESULTS In ths secton, we descrbe the expermental results to evaluate the performance of HACO and exstng algorthms. A. Instances generaton Snce the customer requrements are usually prvate data of a company, no publc NRP nstance s avalable. We generate NRP test nstances by followng the methods n the classc paper of NRP [2]. These nstances consst of fve randomly generated problems, each of whch contans some mult-level requrements. Each requrement n a level s wth the constrant of numbers, dependency, and cost. For each problem, the predefned development budget bound s defned as 30%, 50%, and 70% of the total cost of requrements. But the ranges of customer profts have not been mentoned before. We assume each customer proft s selected randomly from 1 to 30. Table I gves the detals of nstance generaton. The 6 rows of ths table show the nstance names, number of requrement n each level, cost of requrements, the maxmum number of requrement dependency, number of customers, request of each customer, and the customer profts. B. Parameters for HACO The performance of ACO algorthms usually depends on the confguraton of fve parameters. In ths secton, we use CALIBRA procedure [14], proposed by Adenso-Daz and Laguna n 2006, to determne the nterrelated HACO parameter settngs that mprove the algorthm defaults. CALIBRA s an automated tool for fndng performanceoptmzng parameter settngs, whch lberates algorthm desgners from the tedous tas of manually searchng the parameter space. It uses Taguch s fractonal factoral expermental desgns coupled wth local search to fnely tune algorthm parameters wth a wde range of possble values. The beneft of usng CALIBRA s evdent n stuatons where the algorthm beng fne-tuned has parameters whose values have sgnfcant mpact on performance. When lnng the onlne supplement and usng the current verson of CALIBRA 1, 5 nstances are selected randomly for testng, one for each group sze, to ensure proper representaton of problem nstances from all szes. We pc lower and upper bounds for parameters and dscretze the ntervals unformly. For example, the contnuous pheromone nfluence parameter α n the range 0.1 to 4.0 wth an accuracy of one decmal place s nternally handled as an nteger varable wth a lower bound of 1 and an upper bound of 40. The other four parameters are handled n the same way. Table II shows the parameters, the orgnal range consdered before dscretzaton and the values found by CALIBRA. Wth these parameter values, HACO can acheve good solutons on all problem nstances. TABLE II. HACO PARAMETER RANGES CONSIDERED FOR TUNING AND RESULTS OF RUNNING CALIBRA Parameters Ranges Values by CALIBRA α [0.1, 4.0] 1.1 β [0.1, 6.0] 1.5 γ [0.010, 0.050] 0.020 ρ [0.01, 0.99] 0.13 h [1, 30] 10 C. Results and analyss In ths secton, we show the expermental results of heurstc algorthms, ncludng GRASP, SA [2], FHC, ACO, and HACO, where ACO s referred as the HACO verson usng no local search. All the algorthms are mplemented n C++ and run on a personal computer wth Mcrosoft Wndows XP, Intel Core 2.53GHz CPU and 4GB memory. For those heurstc algorthms, we set ther parameters as follows. Both n GRASP and n FHC, each soluton qualty s the best found after 100 restartng tmes. For GRASP, the constructon phase s mplemented accordng to [2] and the length of Restrcted Canddate Lst (RCL) s set to 10. The local search phase s smlar to FHC. For SA algorthm, the Lundy and Mees coolng schedule n [2] s conducted wth temperature control parameter to be 10-8. In ACO and HACO, we set ts fve parameters to 1.1, 1.5, 0.020, 0.13, and 10 as suggested by CALIBRA. For the HACO, 10 teratons are executed and 10 ants apply local search at each teraton for comparson wth FHC wthn smlar runnng tme. Table III llustrates ACO and HACO outperform other algorthms n terms of soluton qualty and runnng tme. The frst column shows the nstance names, and the followng 5 columns show the 5 algorthms n the experments. The 2 subcolumns of each algorthm present the soluton qualty and runnng tme, respectvely. 1 http://or.pubs.nforms.org/org/pages.collect.html 169
TABLE III. EXPERIMENTAL RESULTS OF FIVE ALGORITHMS ON NRP INSTANCES Instance GRASP SA FHC ACO HACO soluton tme(s) soluton tme(s) soluton tme(s) soluton tme(s) soluton tme(s) NRP-1-0.3 810 1.406 827 14.453 817 0.968 829 1.703 835 0.734 NRP-1-0.5 1364 1.281 1397 27.312 1353 1.281 1416 1.671 1418 0.89 NRP-1-0.7 1945 1.359 1960 32.921 1939 0.812 1963 1.656 1968 0.859 NRP-2-0.3 4084 144.328 3902 401.031 3802 334.062 4139 42.593 4262 187.218 NRP-2-0.5 6616 333.187 6629 655.687 6352 401.64 6712 43.125 6786 337.484 NRP-2-0.7 9284 351.437 9602 801.859 9436 349.937 9628 42.5 9732 274.593 NRP-3-0.3 6061 140.25 5556 820.812 5423 678.39 6063 48.984 6067 397.531 NRP-3-0.5 9131 300.14 8836 940.687 8647 666.921 9136 48.812 9196 464.312 NRP-3-0.7 11723 102.078 11648 352.312 11572 191.25 11726 47.187 11728 79.812 NRP-4-0.3 9161 1329.59 8498 5763.55 7935 4729.84 9165 137.156 9183 2810.8 NRP-4-0.5 13731 1220.11 13279 8827.31 12822 12208.4 13751 136.968 13810 3059.88 NRP-4-0.7 17999 273.312 17932 1809.71 17771 1433.06 18022 131.156 18029 424.234 NRP-5-0.3 14940 465.14 14808 2535.55 13322 506.296 15592 158.515 15616 282.76 NRP-5-0.5 20030 686.718 19991 2682.19 19608 1461.38 20784 156.531 20946 1036.47 NRP-5-0.7 24508 138.203 24439 375.656 24410 245.781 24568 155.578 24570 163.312 On the small nstances as NPR-1, the runnng tme of ACO s nearly the same as the exstng ones, but on large NRP nstances, ACO algorthm can get better solutons n much less computng tme than other algorthms. However, HACO can acheve better solutons than ACO and need less runnng tme to obtan a local optmal soluton than FHC. Fg. 1 shows the soluton qualty comparson of FHC, ACO and HACO algorthms n a more ntutvely way. ACO and HACO prove to be much more suffcent than FHC on all nstances, especally on large ones le NRP-4 and NRP-5. HACO can obtan the best soluton qualty on all nstances among these 3 algorthms and can be mplemented as a general approach for NRP problems. Soluton Qualty 25700 20700 15700 10700 5700 700 1-0.3 1-0.5 1-0.7 2-0.3 2-0.5 2-0.7 3-0.3 3-0.5 3-0.7 4-0.3 4-0.5 4-0.7 5-0.3 5-0.5 5-0.7 NRP Instances FHC ACO HACO Fgure 1. Soluton qualtes obtaned by FHC, ACO and HACO algorthms on all nstances. Fg. 2 llustrates the runnng tme of FHC, ACO and HACO algorthms. On all the NRP nstances, ACO uses the shortest computng tme. The runnng tme of FHC on larger nstances le NRP-4 s much longer than that of HACO. It mples that when we ncorporate local search operator FHC nto HACO, those hgh-qualty solutons constructed by ACO can reduce the tme of searchng for local optma to some extent. Runnng Tme (s) 10000 8000 6000 4000 2000 0-2000 1-0.31-0.51-0.72-0.32-0.52-0.73-0.33-0.53-0.74-0.34-0.54-0.75-0.35-0.55-0.7 NRP Instances FHC ACO HACO Fgure 2. An llustraton of runnng tme by FHC, ACO and HACO algorthms on all nstances. V. RELATED WORK There already exsted several heurstc approaches for the NRP problem. Among those algorthms, the hll clmbng (HC) and smulated annealng (SA) were mplemented n [2, 6]. When usng greedy algorthm n [2] as the constructon phase, GRASP can fnd better solutons than SA on some larger nstances wthn only modest amounts of computng tme (as shown n Table III n Secton IV). Recently, an approxmate bacbone based multlevel algorthm (ABMA) was proposed n [17] to solve large scale NRP nstances. ACO s a nd of meta-heurstc approach and Ant System (AS) [18] was the frst ACO algorthm, usng Travelng Salesman Problem (TSP) as an example applcaton. Then a number of drect extensons of AS were provded [19-21], 170
beng dfferent n the way the pheromone updated, as well as the management detals of the pheromone trals. When usng ACO algorthms, automatc tunng procedures could effectvely fnd better parameter settngs. There already exsted several approaches for automatc algorthm parameters tunng. Boyan and Moore [22] proposed a tunng algorthm based on machne learnng technques. But there was no emprcal analyss of ths algorthm when appled to parameters that have wde range of possble values. Audet and Orban [23] ntroduced a mesh adaptve drect search that used surrogate models for algorthmc tunng. Nevertheless, ths approach has never been used for tunng local search algorthms. Adenso- Daz and Laguna [14] desgned an algorthm called CALIBRA for fne tunng algorthms. It can narrow down choces for parameter values when facng a wde range of possble values and search for the best possble parameter settngs. VI. CONCLUSION AND FUTURE WORK In ths paper, we propose HACO for NRP, whch ncorporates FHC nto ACO to solve NRP. We compare the performance of standard ACO and HACO wth exstng heurstc algorthms on some typcal NRP nstances. The expermental results llustrate that ACO outperforms the exstng algorthms n soluton qualty and runnng tme. When coupled wth FHC, HACO can acheve better soluton qualty than ACO. In future wor, we wll nvestgate how to further mprove the performance of HACO and extend HACO for multobjectve NRP. ACKNOWLEDGMENT We would le to than Prof. Belarmno Adenso-Daz and Prof. Manuel Laguna for provdng the latest onlne verson of CALIBRA. Prof. Adenso-Daz s wth Computer Engneerng School and Industral Engneerng School, Unversty of Ovedo and Prof. Laguna s wth Leeds School of Busness, Unversty of Colorado. REFERENCES [1] K. H. Bennett and V. T. 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