Ant Colony Algorithm Based Scheduling for Handling Software Project Delay

Transcription

1 At Coloy Algorith Based Schedulig for Hadlig Software Project Delay Wei Zhag 1,2, Yu Yag 3, Juchao Xiao 4, Xiao Liu 5, Muhaad Ali Babar 6 1 School of Coputer Sciece ad Techology, Ahui Uiversity, Hefei, Ahui, Chia School of software, East Chia Istitute of Techology, Nachag, Jiagxi, Chia School of Software ad Electrical Egieerig, Swibure Uiversity of Techology, Melboure, Australia Istitute of Software, Chiese Acadey of Scieces, Beijig, Chia Software Egieerig Istitute, East Chia Noral Uiversity, Shaghai, Chia School of Coputer Sciece, The Uiversity of Adelaide, Adelaide, Australia 5005 ABSTRACT Delay o a critical path ay cause the failure i eetig the software project deadlie. By addig extra eployees with siilar skills for help, the delay is expected to be eliiated or reduced. However, the origially scheduled activities ay be suspeded due to reallocatio of eployees, which ay lead to the proble of delay propagatio. So how to iiize ad eve eliiate the delay without delay propagatio is worth ivestigatio. I this paper, we first use a siple sceario to deostrate the proble of eployee schedulig which shows that i the schedulig process, oe activity ca have ay ways for selectig eployees fro aother project. I fact, the searchig path i a ulti-brach tree ad its coplete traversal is a NP hard proble. Furtherore whe the scale of the proble becoes large, it is ipractical to geerate a search tree for ipleetatio. Therefore, we propose a at coloy algorith to address such a proble. Both case studies ad iitial siulatio results deostrate that our proposed algorith ca obtai feasible solutios uder differet circustaces. Categories ad Subject Descriptors D.2.9 [Software]: Maageet Keywords Schedulig, Software Process, Project Delay Hadlig 1. INTRODUCTION Curretly, ay software copaies provide dedicated solutios for specific fields by desigig uiversal busiess flows. So, the fial custoized software products for differet custoers have siilar structures ad fuctios. For cocurret custoer s orders, the copay eeds to orgaize correspodig project teas accordigly with eployees havig siilar skills. Custoers expect that their orders ca be delivered before their deadlies. But because of various reasos, delays ofte occur. Serious delay ca ake custoers very usatisfied, that ca daage a copay s reputatio ad profit. Whilst delays are Perissio to ake digital or hard copies of all or part of this work for persoal or classroo Perissio useto is ake grateddigital withoutor feehard provided copies that of copies all or are part otof ade this orwork distributed for for persoal profit or or coercial classroo advatage use is grated ad thatwithout copies bear fee this provided otice ad that thecopies full citatio are o ot theade first page. or distributed Copyrights for forprofit copoets or coercial of this work advatage owed byad others that tha copies ACM ust be hoored. Abstractig with credit is peritted. To copy otherwise, or republish, tobear postthis o servers otice or ad tothe redistribute full citatio to lists, o requires the first prior page. specific To copy perissio otherwise, ad/or to a fee. republish, Requestto perissios post servers fro Perissios@ac.org. or to redistribute to lists, requires prior specific ICSSP 15, perissio August ad/or a 24 26, fee. 2015, Talli, Estoia c ICSSP'15, 2015 ACM. August /15/08...$ , 2015, Talli, Estoia. Copyright 2015 ACM /15/08... $ soeties ievitable, it is iportat to fid out how to eliiate or iiize delays. Accordig to oe of the software egieerig priciples [2], addig a iexperieced perso would cause further delay due to the learig curve ad couicatio overhead. This research ais at reallocatig skilled eployees for solvig such a issue. I geeral, resource schedulig is a proisig approach for hadlig delays. Most project plaig odels are based o CPM (critical path ethod) or PERT (progra evaluatio review techique) [6]. Ay delay to ay activity o the critical path ay postpoe the copletio of the project. I acadeia, uch attetio is paid o how to copress the duratios of activities o critical path. For exaple, authors of [11] proposed a K-step algorith to copress the critical path. Authors of [7] discussed the feature of the logest path beyod critical path ad the aout of copressio. Those strategies aily focus o the copressio of the critical path. I fact, reasoable aout of copressio would ot oly reduce the delay, but also esure that the critical path reais uchaged. To ipleet copressio, oe solutio is to request the eployees to work overtie. This is ot ecessarily a good solutio as workig overtie ofte results i coplaits fro the eployees, i additio to huge extra costs ad lower quality of products. Aother solutio is related to resource schedulig. I resource schedulig, ay scholars pay attetio to project schedulig proble (PSP). As explaied i [12], project schedulig ivolves separatig the total work ivolved i a project ito separate activities ad judgig the tie required to coplete these activities. Usually, soe of these activities are carried out i parallel. Project schedulers ust coordiate these parallel activities ad orgaize the work so that the workforce is used optially. Authors of [13] preseted a value-based hua resource schedulig ethod aog ultiple software projects by usig a geetic algorith. Authors of [5] forulated the PSP as the proble of eployee allocatio ad sequece that workpackages are selected to be hadled i a queue syste. Differet search algoriths based o sigle ad ulti-objective evolutioary algorith (EA) were discussed. Authors of [10] proposed a ew desig to solve the PSP proble based o proble foratio proposed i [1]. It icludes oralizatios of dedicatio values, a tailored utatio operator, ad fitess fuctios with a strog gradiet towards feasible solutios. Noralizatio reoves the proble of overwork ad allows EA to focus o the solutio quality. Authors of [8] discussed estiated schedules ad project schedulig. They proposed a two-stage probabilistic schedulig strategy, which ais to 52

2 decrease schedule overrus. Authors of [3] propose a ew schedulig strategy based o evet-based scheduler ad at coloy optiizatio algorith is applied to search feasible solutio for a sigle project. However, our research is to hadle delays ivolvig ultiple cocurret software projects. Sice eployees ca work o several tasks siultaeously, existig work aily discusses o how to allocate optial hua resources i order to save tie ad oey for software copaies at the plaig stage. However, the proble is that delay ofte occurs at the rutie stage. How to hadle rutie delays has ot bee well addressed i these papers. Authors of [9] itroduced a achie learig algorith to staff assiget based o workflow evet log at rutie stage. I our research, we propose that a eployee with the sae/siilar skills ca be reallocated to help for the activity where delay occurs i order to eliiate or iiize the delay. It is ofte the case that oe activity ca be partitioed ito several sub-activities so that extra eployees ca be itroduced to help for sub-activities as explaied i Sectio 2. For the sceario addressed at the very begiig of this sectio, if eployees with the sae/siilar skills are available, they ca be reallocated across the projects without uch learig curve ad couicatio overhead. However, due to eployee reallocatio, other projects ivolved ay be ipacted. It is ot desirable that reducig the delay of oe project causes the delay of other projects. So i this paper, we ivestigate a ovel at coloy algorith based schedulig ethod, aiig to reallocate eployees for cocurret projects i order to eliiate or iiize the delays of the projects i overall ters. The reaider of this paper is orgaized as follows. Sectio 2 describes the proble forulatio. Sectio 3 proposes our solutio. Sectio 4 presets the evaluatio. Sectio 5 cocludes our cotributios ad poits out future work. 2. PROBLEM FORMULATION Our proble forulatio is based o the existig work [1]. Assue that we are give a set of projects P 1.P with critical paths cp 1.cp respectively; a set of eployees e 1 i.e qi (i idicates the eployee coig fro the i th project) ad a set of skills skill 1.skill p respectively; a set of activities a 1.a s; a activity precedece graph (APG) a directed graph with activities as odes ad activities precedece as edges where i APG, fro start to ed there are ay paths path 1.path g. Here our graph is slightly differet to its couterpart i [1]. I order to coveietly reallocate the eployees ad oitor the tie chage of activities, we further break dow the task defied i [1]: oe activity requires oly oe skill ad is coducted by oly oe eployee. I order to siplify the proble without losig geerality, we itroduce soe costraits: (1) whe oe activity is started, it caot be iterrupted; ad (2) if oe activity ca be further broke dow which is ofte the case, it will be partitioed ito sub-activities whe eeded. For exaple, a desig activity for the iterface betwee iteral ad exteral systes ca be divided ito sub-iterfaces for such as hardware, software ad couicatio. Such activity ca be partitioed ito three subactivities accordigly. However, i order to reduce the coplexity of this research, we assue that a activity is helped by oe extra eployee at ost. These costraits will be relaxed i our future work. We use a case to illustrate the proble. Suppose there are two siilar projects P ad P, cp is the critical path of project P. cp is the critical path of project P which icludes activities of the sae fuctios but differet duratios with P.cp. Suppose that the delay will occur at P.cp.a 1 whilst P.cp is o schedule. We illustrate how to schedule eployees fro those activities betwee the two paths i order to reduce the delay of cp as uch as possible without jeopardizig cp. Of course, we assue that soe activities ca be partitioed. I fact, if we oly reallocate a eployee fro cp to work o cp, it ay reduce the delay of cp but cp ay get delayed seriously. Obviously we expect that delay o cp is reduced or eve eliiated while cp is ot ipacted. Table 1. Duratio of each activity at two paths (cp & cp ) cp Scheduled start tie ad ed tie Duratio (plaed) Duratio (helped) a1 Fro the 5 th day to the 10 th day 6 5 a2 Fro the 11 th day to the 16 th day 6 3 cp Plaed start tie ad ed tie Duratio (plaed) a1 Fro the 7 th day to the 14 th day 8 6 a2 Fro the 15 th day to the 18 th day 4 2 Duratio (helped) Based o the paraeters listed i Table 1, Figure 1(a) shows the plaed schedule where delay occurs at activity a 1 of cp which is 6 days. O oe had, the plaed tie cp.a 1 is fro the 5 th day to the 10 th day. Hece its duratio is 6 days. If eployee e 1 with skill 1 fro cp..a 1 is reallocated to help the activity which requires skill 1, the duratio of cp.a 1, say, ca be reduced to 5 days. Siilarly, the plaed tie of cp.a 2 is fro the 11 th day to the 16 th day. Hece its duratio is 6 days. If eployee e 2 with skill 2 fro cp.a 2 is reallocated to help the activity which requires skill 2, the duratio of cp.a 2 ca be reduced to 3 days. O the other had, i cp, the plaed tie of cp.a 1 is fro the 7 th day to the 14 th day. Hece its duratio is 8 days. If eployee e 1 fro cp.a 1 with skill 1 is scheduled to help the activity, which requires skill 1, the duratio of cp.a 1, say, ca be reduced to 6 days. Siilarly, the plaed tie of cp.a 2 is fro the 15 th day to the 18 th day. Hece its duratio is 4 days. If eployee e 2 fro cp.a 2 with skill 2 is scheduled to help the activity, which requires skill 2, the duratio of cp.a 2, say, ca be reduced to 2 days. Now suppose the start tie of cp.a 1 is postpoed to the 8 th day. We illustrate two strategies below. cp (b) First strategy 8 12 a1:5 cp a2:3 a1:6 (a) Plaed schedule 5 10 cp a1: a2:6 cp a2:2 a1:8 15 a2:4 18 (c) Secod strategy 8 13 a1: a1: Figure 1. Two differet schedulig strategies cp cp a2: a2:2 53

3 Firstly we provide oe siple strategy as show i Figure 1(b). We allocate e 1 fro cp.a 1 to help cp.a 1. Accordig to the paraeter settig, with help, duratio of the activity ca be reduced to 5 days. The ed tie of cp.a 1 would be the 12 th day. The delay is reduced but ot eliiated which is propagated to cp.a 2. Furtherore, sice cp.a 1.e 1 is trasferred to cp.a 1, cp.a 1 ust wait util cp.a 1 is fiished. So the delay will also occur at cp.a 1. Its start tie will be postpoed to the 13 th day. Based o the above aalysis, eve if cp.a 1 is helped, its ed tie will still be delayed to the 18 th day without cp.a 2 helped. However, with the help of cp.a 2.e 2, the duratio of cp.a 2 is reduced to 3 days. Hece the ed tie of cp.a 2 is the 16th day, i.e., delay eliiated for cp. Siilarly, duratios of cp.a 1 ad cp.a 2 ca be reduced to 6 ad 2 days with help of e 1 ad e 2 respectively fro cp eve if they are available. Its ed tie is the 20 th day, i.e., the 2-day delay of cp is itroduced. Our goal is that the delay of cp is reduced or eve eliiated without cp ipacted which ca be see ext. Secodly, we chage the strategy as show i Figure1(c). Suppose cp.a 1 rus without ay help which eas that the delay is fully propagated to cp.a 2. Due to the delay, the start tie of cp.a 2 is chaged to the 14 th day. With the help of e 2 fro cp.a 2, its duratio is ow 3 days. So the ed tie of cp.a 2 becoes the 16 th day, i.e. delay eliiated. For cp, ow cp.a 2 eeds to wait util cp.a 2 is fiished. So it starts fro the 17 th day. With the help of e 2 fro cp.a 2, its duratio becoes 2 days. Hece the fial ed tie of the activity is the 18 th day, i.e. o delay itroduced. Clearly this is a better schedulig for reallocatio. Fro the above siple case, we ca coclude that ot all delayed activities eed to ivolve schedulig. I our case, whe we schedule e 2 fro cp.a 2 to help cp.a 2 firstly, the start tie of cp.a 2 will be delayed to the 17 th day. Hece o atter whether cp.a 2 is scheduled or ot, the delay will be itroduced. Therefore, soeties we ay eed to propagate the delay to the ext activity where feasible solutio ca be foud. I additio, the order of schedulig eployees ay also affect the fial result. 3. FEASIBLE SOLUTION 3.1 Search Tree Based o the aalysis above, the activities ivolved i the schedulig ad the order of schedulig eployees will deterie whether a feasible solutio ca be obtaied. The questio is how to fid the feasible solutio. This is siilar to the cobiatio optiizatio proble. Firstly, we start with the discussio of two paths. I such a circustace, for oe activity, there are two possible scearios. Oe is that the activity is executed aloe by the origial eployee. Aother is that the activity is executed together by the origial eployee ad the help eployee who is a eployee borrowed fro aother project. So we have differet cobiatios. I the case preseted i Sectio 2, whe a 1 fro cp is selected firstly, it eeds to lock i the help eployee first. Assue that the activity chooses to be executed without ay help, ad the a 1 fro cp chooses to be executed with help, we defie this for as <(cp.e 1 ) (cp.a 1), (cp.e 1, cp.e 1 ) ( cp.a 1)>. Here (cp.e 1 ) (cp.a 1) idicates eployee e 1 fro cp will work o activity cp.a 1 while (cp.e 1, cp.e 1 ) (cp.a 1) idicates eployee cp.e 1 ad eployee cp.e 1 will work o activity cp.a 1 together. cp.a 1 precedes cp.a 1. If activity cp.a 1 selects cp.e 1 first, cp.a 1 ca oly select cp.e 1 later. The the resource schedulig order is cp.a 1 ad cp.a 1. Whe two activities choose the sae eployee, the eployee will work o activity cp.a 1 first. Because the activity caot be iterrupted, the correspodet activity fro oe path will be suspeded whe the activity at aother path is helped. So there are seve differet cobiatios: (1) <(cp.e 1 ) (cp.a 1), (cp.e 1, cp.e 1 ) (cp.a 1)>, (2) <(cp.e 1, cp.e 1 ) (cp.a 1), (cp.e 1, cp.e 1 ) (cp.a 1)>, (3) <(cp.e 1, cp.e 1 ) (cp.a 1), (cp.e 1 ) (cp..a 1)>, (4) <(cp.e 1 ) (cp.a 1), (cp.e 1 ) (cp.a 1)>, (5) <(cp.e 1, cp.e 1 ) (cp.a 1), (cp.e 1 ) (cp.a 1)>, (6) <(cp.e 1, cp.e 1 ) (cp.a 1), (cp.e 1, cp.e 1 ) (cp.a 1)> ad (7) <(cp.e 1 ) (cp.a 1), (cp.e 1, cp.e 1 ) (cp.a 1)>. The proble is that which cobiatio at oe activity is chose so as to achieve our goal. Fro activity a 1 to activity a, if all possible cobiatios are liked together, a seve-fork tree is produced accordig to data structure i [4]. A feasible solutio ay exist i the tree. How do we fid the feasible solutio? Oe siple ethod is to traverse all possible paths i the tree. However, whe the uber of activities icreases, the uber of cobiatio odes i the tree icreases expoetially. Obviously this is a NP hard proble. So we eed to ivestigate a itelliget algorith such as at coloy algorith to solve the proble. I reality, the scale of the proble ca have ( >2) paths. I such a circustace, there are three possible scearios for each activity give the assuptio of that a activity is helped by oe extra eployee at ost. The first is that the activity is executed etirely by the origial eployee aloe. The secod is that the activity is executed together by the origial eployee ad a help eployee fro aother path if the activity ca be partitioed. The third is that the activity is executed etirely by a help eployee fro aother path. Likewise, there exists a ulti-way tree. Due to paths, all possible cobiatios caot be euerated i a reasoable tie. 3.2 Schedulig Strategy Here we propose a feasible solutio based o at coloy algorith. Note that whe the scale of the proble icreases, buildig a ulti-brach tree becoes ipractical sice a at does ot kow which ode to select ext. Whilst the detailed cotet for the SelectNode fuctio will be described i Sectio 3.3, here we itroduce the overall procedure of the schedulig strategy based o at coloy algorith. (1) Iitializatio (2) For oe geeratio, each at k (k=1, 2, 3.) selects the cobiatio ode accordig to the SelectNode fuctio (3) Calculate the actual ed tie of the fial activities of all paths ad deterie whether those critical paths are delayed. (4) Whe oe at fiishes its tour, set the delay of path i as a iiu value if the fial delay of path i is reduced ad all other paths are ot ipacted. If other ats ca obtai better result, the iiu value will be updated, ad fially record the correspodet at searchig path of the iiu value. (5) Whe all ats i oe geeratio have fiished the tour, the correspodet pherooe o the at searchig path of the iiu value will be updated. If there is o iiu value, pherooe will ot be updated. (6) Icrease the geeratio uber by 1, if exceedig the axiu geeratio uber set, the algorith teriates with the fial result as output, otherwise, go to step 2. 54

4 3.3 SelectNode Fuctio This fuctio deteries which cobiatio ode will be visited by a at. I our algorith, i order to siplify the proble, the activity is helped by oe extra eployee fro other paths at ost if oe activity ca be partitioed. The geeral cotrol flow of the fuctio is show i Figure 2. Firstly at activity a i, executio order is rado ad is stored i array path[]. The order deteries the path priority,,i.e., which path will select help eployee firstly. The if a i-1 of all paths is fiished o tie, algorith teriates ad outputs the cobiatio result. Otherwise, Accordig to actual ed tie of a i-1 of path j, the actual start tie of a i of path path j is calculated. Subsequetly, it starts to select the cobiatio ode. If the ed tie of ew eployee cobiatio workig at the activity is earlier tha the ed tie of origial eployee workig aloe at the activity, the ed tie of the activity will be updated. I additio, the available tie of all eployees fro the cobiatio will be updated. Otherwise, the cobiatio will be selected agai. The process of selectig cobiatio is detailed as follows. To deterie whether the origial eployee is ready for the activity, we copare the available tie of the eployee with the actual start tie of the activity. Whe the origial eployee is ready, there are two choices. The first is that the origial eployee executes the activity aloe. The secod is that if the activity ca be partitioed, a help eployee is selected radoly fro aother path who has the sae skills so that the group of two eployees works together o the activity. If the ed tie of the group workig o the activity is greater tha or equal to that of the origial eployee workig o the activity aloe, a ew eployee cobiatio is selected agai. If the origial eployee is ot ready, there are three choices. The first is that the activity will wait for the origial eployee. The secod is that if the activity ca be partitioed, the activity will wait for the group that is coprised of the origial eployee ad help eployee fro aother path. If the ed tie of the group workig o the activity is less tha that of the origial eployee workig o the activity aloe, a ew eployee cobiatio is selected agai. The third is to radoly select oe help eployee fro aother path. The help eployee will execute the activity aloe. If the ed tie of the help eployee workig o the activity is greater tha or equal to that of the origial eployee workig o the activity aloe, a ew eployee cobiatio is selected agai. Activity ai is chose 4. EVALUATION 4.1 Case Studies Case Study 1 Here all paraeters of cp 1, cp 2 are geerated radoly. I order to illustrate the proble better, we choose two siilar critical paths. I cp 1 ad cp 2, activity a 2 caot be partitioed. Other activities ca be partitioed. All paraeters are listed i Table 2. Assue that the delay occurs at activity a 1 of cp 1 ad the delay is 8 days, ad cp 2 rus orally. If we do othig at path cp 1, the fial ed tie of cp 1 is delayed to the 102 th day. By usig our schedulig algorith described i Sectio 3, we ca obtai a feasible cobiatio path as follows: <(cp 1.e 1 1,cp 2.e 1 2 )(cp 1.a 1),(cp 2.e 1 2,cp 1.e 1 1 )(cp 2.a 1)> <(cp 2.e 2 2 )(cp 2.a 2),(cp 1.e 2 1 )(cp 1.a 2)> <(cp 1.e 3 1,cp 2.e 3 2 )(cp 1.a 3),(cp 2.e 3 2,cp 1.e 3 1 )(cp 2.a 3)> <(cp 1.e 4 1 )( cp 1.a 4),(cp 2.e 4 2 )(cp 2.a 4)> Table 2. Paraeters of activities at cp 1, cp 2 cp 1 a 1 a 2 a 3 a 4 Plaed start tie 6 th day 27 th day 38 th day 67 th day Duratio : cp 1.e 21 days 11 days 29 days 28 days Duratio : cp 2.e 19 days 9 days 26 days 27 days Duratio:(cp 1.e, cp 2.e) 13 days N/A 19 days 20 days (a) cp 1 with a deadlie of the 94 th day cp 2 a 1 a 2 a 3 a 4 Plaed start tie 12 th day 31 th day 53 th day 76 th day Duratio : cp 2.e 19 days 22 days 23 days 22days Duratio : cp 1.e 17 days 22 days 19 days 18 days Duratio:(cp 1.e, cp 2.e) 12 days N/A 15 days 14 days (b) cp 2 with a deadlie of the 97 th day The correspodig fial ed ties of cp 1 ad cp 2 are the 94 th day ad the 97 th day. I copariso to the delay before schedulig, the delay is eliiated after schedulig as show i Table 3. Table 3. Copariso betwee before ad after schedulig Delay of cp 1 Delay of cp 2 Before 8 days 0 days After 0 days 0 days Algorith teriates ad Yes output result Store the cobiatio ad select ext path fro path[] Ed tie of ai ad free tie of all workers i the Yes Cobiatio will be updated executio serial which is produced radoly :path[] ai-1 of all paths is fiished o tie? No coputer actual start tie of ai of pathj, Select ew cobiatio radoly Ed tie of ew worker cobiatio at ai is earlier tha that of origial worker at the ai Case Study 2 Here all paraeters of cp 1, cp 2, cp 3 are geerated radoly. There are three siilar critical paths. I cp 1, cp 2, cp 3, activity a 3 caot be partitioed. Other activities ca be partitioed. All paraeters are listed i Table 4. Assue that the delay occurs at activity a 1 of cp 1 ad the delay is 10 days, ad cp 2 ad cp 3 ru orally. If we do othig at path cp 1, the fial ed tie of cp 1 is delayed to the 87 th day. By usig our schedulig algorith described i Sectio 3, we ca obtai a feasible cobiatio path as follows: No Select agai Figure 2. Geeral cotrol flow of SelectNode fuctio <(cp 3.e 1 3,cp 1.e 1 1 )(cp 3.a 1),(cp 2.e 1 2 )(cp 2.a 1),(cp 1.e 1 1,cp 3.e 1 3 )(cp 1.a 1)> <(cp 1.e 2 1,cp 2.e 2 2 )(cp 1.a 2),(cp 2.e 2 2,cp 1.e 2 1 )(cp 2.a 2),(cp 3.e 2 3 )(cp 3.a 2)> <(cp 3.e 3 3 )(cp 3.a 3 ),(cp 2.e 3 2 )(cp 2.a 3),(cp 1.e 3 1 )(cp 1.a 3)> <(cp 1.e 4 1 )(cp 1.a 4),(cp 3.e 4 3 )(cp 3.a 4),(cp 2. e 4 2 )(cp 2.a 4)> 55

5 Table 4. Paraeters of activities at cp 1, cp 2, cp 3 cp1 a 1 a 2 a 3 a 4 Plaed start tie 6 th day 23 th day 36 th day 55 th day Duratio : cp1.e 17 days 13 days 19 days 23 days Duratio : cp2.e 20 days 8 days 18 days 13 days Duratio : cp3.e 19 days 13 days 19 days 12 days Duratio:(cp1.e, cp2.e) 14 days 6 days N/A 17 days Duratio:(cp1.e, cp3.e) 10 days 10 days N/A 16 days (a) cp 1 with a deadlie of the 77 th day cp2 a 1 a 2 a 3 a 4 Plaed start tie 12 th day 38 th day 47 th day 76 th day Duratio : cp2.e 26 days 9 days 29 days 25 days Duratio : cp1.e 19 days 9 days 16 days 16 days Duratio : cp3.e 25 days 14 days 24 days 26 days Duratio:(cp2.e, cp1.e) 17 days 6 days N/A 16 days Duratio:(cp2.e, cp3.e) 24 days 5 days N/A 22 days (b) cp 2 with a deadlie of the 100 th day cp3 a 1 a 2 a 3 a 4 Plaed start tie 6 th day 35 th day 49 th day 73 th day Duratio : cp3.e 29 days 14 days 24 days 13 days Duratio : cp1.e 27 days 15 days 16 days 8 days Duratio : cp2.e 20 days 18 days 22 days 16 days Duratio:(cp3.e, cp1.e) 16 days 9 days N/A 17 days Duratio:(cp3.e, cp2.e) 18 days 7 days N/A 15 days (c) cp 3 with a deadlie of the 85 th day The correspodig fial ed ties of cp 1, cp 2 ad cp 3 are the79 th day, the 100 th day, ad the 85 th day. I copariso to the delay before schedulig, the delay is reduced sigificatly after schedulig as show i Table 5. Table 5. Copariso betwee before ad after schedulig Delay of cp 1 Delay of cp 2 Delay of cp 3 Before 10 days 0 days 0 days After 2 days 0 days 0 days 4.2 Siulatio Experiets Figure 3. Hit cout uder differet ubers of paths & activities Here we test the hit couts (i.e. the uber of feasible solutios foud) for 2, 4, 6, ad 8 paths whe the ubers of activities are 10, 20, ad 30 respectively. The uber of geeratios of the at coloy algorith is set as 100 ad the algorith is ru for 20 ties each. Coputatio took less tha 1 iute for all tests. The results are depicted i Figure 3. Fro the results, the hit cout is high ad stable which illustrates that our algorith ca easily fid feasible solutios. 5. CONCLUSION AND FUTURE WORK I this paper, we have discussed priarily o how to eliiate or reduce the delay of oe path i a software project without ipactig o the other paths. To achieve this goal, a ovel at coloy based searchig algorith has bee proposed. The case studies have deostrated the effectiveess of eliiatig or reducig the delays. The siulatio experiets have also show that our proposed algorith ca obtai stable hit rates uder differet circustaces, which ea that feasible solutios ca be easily foud for hadlig software project delay. To siplify the scearios ad focus o the ai ideas, the work preseted i this paper is based o soe costraits for eployees ad activities. Therefore, our future work is to relax these costraits i order to ake our algorith ore geeral. 6. REFERENCES [1] E. Alba ad J. F. Chicao.2007.Software Project Maageet with Gas. Iforatio Scieces, 177, [2] F. Brooks The Mythical Ma-Moth: Essays o Software Egieerig (Aiversary Editio). Addiso- Wesley. [3] W.-N. Che ad J. Zhag At Coloy Optiizatio for Software Project Schedulig ad Staffig with a Evet- Based Scheduler. IEEE Trasactios o Software Egieerig, 39(1), [4] T. H. Core.2001.Itroductio to Algoriths (Secod Editio).The MIT Press. [5] M. Di Peta, M. Hara, ad G. Atoiol The Use of Search Based Optiizatio Techiques to Schedule ad Staff Software Projects: A Approach ad a Epirical Study. Software: Practice ad Experiece, 41, [6] B. Hughes Software Project Maageet (Fifth Editio).McGraw-Hill. [7] X. Li. ad J. Qi Free Float Theore ad Algorith of Seekig the k-th Order Critical Path. Joural of Maageet Scieces i Chia, 12, [8] X. Liu ad Y. Yag, Achievig O-Tie Delivery: A Two- Stage Probabilistic Schedulig Strategy for Software Projects. I 2009 Iteratioal Coferece o Software Process, (Vacouver, Caada, 2009), Lecture Notes i Coputer Sciece, [9] Y. Liu, J. Wag, Y. Yag ad J. Su A Sei- Autoatic Approach for Workflow Staff Assiget. Coputers i Idustry, 59 (5), [10] L. L. Miku, D. Sudholt, ad X. Yao Iproved Evolutioary Algorith Desig for the Project Schedulig Proble Based o Rutie Aalysis. IEEE Trasactios o Software Egieerig, 40(1), [11] J. Qi,S. Yi The Max Efficiet Solutio of Cotractio: the Liit of the Project with i Cost. Joural of North Chia Istitute of Electric Power, 2, [12] I. Soerville Software Egieerig (Eighth Editio). Addiso-Wesley. [13] J. Xiao ad Q. Wag, Value-based Multiple Software Projects Schedulig with Geetic Algorith. I 2009 Iteratioal Coferece o Software Process, (Vacouver, Caada, 2009), Lecture Notes i Coputer Sciece,