An Approach for Project Scheduling Using PERT/CPM and Petri Nets (PNs) Tools

Transcription

1 Inernaional Journal of Modern Engineering Research (IJMER) Vol., Issue. 5, Se - Oc ISSN: n roach for Projec Scheduling Using PERT/CPM and Peri Nes (PNs) Tools mer. M. oushaala (Dearmen of Indusrial and Manufacuring Sysems Engineering / Universiy of enghazi, Libya) STRCT: The Peri Nes are more owerful models as comared o he PERT/CPM chars. The Peri Nes show he same ye of scheduling consrains as he PERT/CPM chars does. We can easily conver a PERT/CPM neworks o a Peri Ne model.this work shows how o use Peri ne s for modeling and verifying rojec managemen neworks. Sofware for a funcional analysis of a Peri Ne model is develoed. The develoed sofware enables analyzing of a sysem for is various roeries irresecive of ime. The sofware analysis is based on some aroaches used o deermine he various roeries of PN model: livens, safeness, reach-abiliy and conservaiveness. This aer rooses a PN based modeling aroach o rovide a formal way o verify ha all he aciviies are well conneced in he rojec nework. This PN model can hen be analyzed hrough (PN based) scheduling echniques, used o find he criical ah, used as a basis o develo algorihms for resource-consrained rojec managemen, and used for oher decisions relaed o rojec managemen. The roosed PN based aroach enables he rojec manager o consider no only resources bu also all differen yes of variables/consrains relaed o a rojec, e.g. coss of aciviies. Keywords: CPM, PERT, Peri Ne s, Verificaions I. INTRODUCTION CPM/PERT are major ools for rojec managemen. They are neworks ha show recedence relaions of aciviies of a rojec wih he aciviy imes. They are used o find rojec comleion ime, which is he longes ah in he nework. They are also used o find oher relaed informaion abou he aciviies, e.g., slack ime, earlies sar and end ime. ll hese analyses are made based on he infinie resource assumion. Then, resource allocaion of he rojec is carried ou. Tha is, finie resources are allocaed o he rojec aciviies whose ime requiremens were calculaed based on he infinie resource assumion. However, insead of his wo-se aroach o he resource-consrained rojec managemen, a onese aroach, hrough Peri nes (PNs), is ossible. Furhermore, CPM/PERT; do no rovide a formal way o verify ha all he aciviies are well conneced in he rojec nework. Liang, Chen and Wang [] inroduced a rojec model, called SPREM, which exends CPM/PERT's noaion o four yes of verices o exress he non-deerminisic and ieraive behaviors of sofware engineering rojecs. They comared i wih PNs, and discussed is analysis and behavioral roeries. These roeries are verified hrough several algorihms hey roosed. However, as discussed in secion 5, he well known lace (ransiion) invarians hel verify some of hese roeries wihou any addiional algorihms. Desrochers and l-jaar [2] give some advanages of PNs: PNs caure he recedence relaions; conflics and buffer sizes can be modeled easily; PN models reresen a hierarchical modeling ool wih a well-develoed mahemaical and racical foundaion. These advanages hel model and verify (resource-consrained) rojec neworks. Several PN based aroaches o rojec managemen were roosed in he lieraure. Jeeendra e al. [] evaluae convenional rojec managemen ools, sae heir inadequacies and lis advanages of using PNs for rojec managemen. Chang and Chrisensen [] roose PM-Ne for sofware develomen, which ados he basic conces of PNs wih exensions o reresen boh decisions and arifacs. shok Kumar and Ganesh [5] also describe rojec managemen models wih heir inadequacies, and roose a new model based on PNs and demonsrae is usefulness for real-ime aciviy scheduling in a resource-consrained rojec environmen. However, hese aroaches need o modify and exend basic PN semanics and add new ools o analyze he rojec nework. For examle, Jeeendra e al. [] change PN semanics hrough a recedence marix hey develoed. Conflicing ransiions in radiional PNs behave concurrenly in heir definiion. This marix is hen used o hel deermine floas and he criical ah and o find deadlocks. shok Kumar and Ganesh [5] also add several feaures o basic PN semanics for rojec managemen. The roosed PN based aroach faciliaes modeling (resource-consrained) rojecs, and verifying some roeries of he rojecs neworks, exloiing he well known lace (ransiion) invarians, hrough he basic PN definiions. Tha is, conrary o some aroaches in he lieraure, here is no need o modify he basic PN semanics o model aciviies and decisions relaed o a rojec. This increases he caabiliy of he basic PNs, wihou any exension in he heory, o model rojecs. Transiions are used o model all he recondiions, for an aciviy o sar, which is modeled via a lace. The lace invarians are hen used o verify some roeries of he rojecs, as discussed in secion 5. This model can hen be analyzed hrough (PN based) scheduling echniques [, 7], and used o find he criical ah [], and used as a basis o develo algorihms for resource-consrained rojec managemen [9, ]. These consideraions are beyond he scoe of his aer. The aim of his aer is o show how o model rojec neworks wih PNs wihou any modificaion or exension in he basic PN semanics, and how o comue he criical ah and verify some rojec roeries hrough he well known lace invarians. 2 Page

2 Inernaional Journal of Modern Engineering Research (IJMER) Vol., Issue. 5, Se - Oc ISSN: II. PERT/CPM CTIVITY NETWORKS The foundaion of he aroach came from he Secial Projecs Office of he US Navy in 95. I develoed a echnique for evaluaing he erformance of large develomen rojecs, which became known as PERT - Projec Evaluaion and Review Technique. Oher variaions of he same aroach are known as he Criical Pah Mehod (CPM) or Criical Pah nalysis (CP) [, 2]. The hear of any PERT char is a nework of asks needed o comlee a rojec, showing he order in which he asks need o be comleed and he deendencies beween hem. This is reresened grahically as shown in Fig.. The diagram consiss of a number of circles, reresening evens wihin he develomen lifecycle, such as he sar or comleion of a ask, and lines, which reresen he asks hemselves. Each ask is addiionally labeled by is ime duraion. Thus he ask beween evens and 5 is lanned o ake ime unis. The rimary benefi is he idenificaion of he criical ah. The criical ah = oal ime for aciviies on his ah is greaer han any oher ah hrough he nework (delay in any ask on he criical ah leads o a delay in he rojec). Diagram Symbols illusraed in Fig. 2. Projec nework comlexiy is ofen recognized way, bu no comleely undersood by everyone. oushaala [] roosed a new measure of rojec comlexiy. The develoed measure showed more sensiiviy o he changes in he nework daa and give accurae quanified resuls in evaluaing he rojec comlexiy where he criical aciviies, he criical ahs, number of criical aciviies o he oal number of rojec aciviies, he lengh of criical ah, he resource yes and heir availabiliy are considered. Fig. Examle of an aciviy nework Fig. 2 Symbols used in aciviy neworks III. MODELING WITH PETRI NET S Peri ne (PN) is formally defined as a four-ule C= (P,T, I,O) where P is a finie se of laces, T is a finie se of ransiions, I is a maing from ransiions o a subse of laces (bag of inu laces) such ha I() reresens inu laces for ransiion, and O is a maing from ransiions o a subse of laces (a bag of ouu laces) such ha O() reresens ouu laces for ransiion. Mulile occurrences of each in he inu and ouu bags are allowed [, 5]. PN can also be described by a biarie direced grah wih wo yes of nodes: circles for laces and bars for ransiions. Direced arcs conneced laces and ransiions. Le (, ) and F (, ) be, resecively, he number of occurrences of laces P in he inu and ouu bags of ransiion. Then, F and D = F -, resecively, define he backward, forward and incidence marices of he PN. These marices define he oology of he PN. The dynamics of he PN are defined by marking of he PN; is a sae vecor wih () is he number of okens in lace. The dynamics of he PN are conrolled by he execuion of ha PN. PN execues by firing is ransiions. ransiion fires by removing okens 2 Page

3 Inernaional Journal of Modern Engineering Research (IJMER) Vol., Issue. 5, Se - Oc ISSN: from is inu laces and deosiing okens a is ouu laces. ransiion may fire if i is enabled. ransiion is enabled in marking if.f where f = (,,...,,,..., ) wih corresonding o ransiion. If is a new marking afer firing ransiion, hen = + D.f defines he dynamics of he PN. For a sequence of n ransiions, he dynamics equaion becomes n = + D.f where f = f, is a se of n ransiions and is he iniial marking; f is called he firing vecor of he sequence. Each marking defines a sae. Firing a ransiion may resul in a new sae. ll he ossible saes define he sae sace of he PN. From an analyical ersecive, i is quie imoran o deermine all he reachable saes. I is also imoran o deermine wheher or no he PN is live or dead-lock free, bounded (number of okens in any lace is finie in any marking), conservaive (he weighed number of okens in any marking is fixed and finie) and consisen (here is a firing vecor wih all osiive elemens). live and consisen PN is cyclic, which is yical roery of manufacuring sysems. One may also be ineresed in oher feaures of a PN as a conroller, such as recoverabiliy and fairness. Some of hese roeries can be mahemaically analysed hrough he P-and T-invarians of he PN [5, ].. Marked Grah marked grah is a PN in which each lace is an inu for exacly one ransiion and an ouu for exacly one ransiion. lernaively, we can say ha each lace exacly one inu and one ouu [, 7]. Definiion: marked grah is a PN C = ( P,T,I,O ) such ha for each i P,I(i) = { j / i O( j )} = and O(i) = { j / i I(j)} =. Marked grahs can model concurrence and synchronizaion bu canno model conflic or daa-deenden decisions. The roeries which have been invesigaed for marked grahs have been Livens, safeness, and reach-abiliy. Marked grahs can model concurrence and synchronizaion bu canno model conflic or daa-deenden decisions. The roeries which have been invesigaed for marked grahs have been Livens, safeness, and reach-abiliy. In he invesigaion of hese roeries, he major srucural ars of a marked grah of ineres are is cycles. cycle in a marked grah is a sequence of ransiions j j2... jk such ha for each jr and jr+ in he sequence here is a lace ir he ir O( jr ) and ir I( jr+ ) and j = jk. cycle is such a closed ah from a ransiion back o ha same ransiion. If P` P is a subse of laces ha comose a cycle, hen he characerisic vecor U= ( u i i = {, 2,..., n} such ha crd i U = for i P`, is a - invarian of PN. If U' and U'' are P- invarians of PN, hen U = U' + U'' +... is also a P-invarian of PN, where crd i U = u i for U=(u,u 2,...,u i,...,u r ) []..2 P Invarian U is said o be a P-invarian of a PN if and only if U=(u,u 2,...,u n ) is a vecor such ha D*U= and u i for I =,2,...,n and D is he incidence marix. The following heorem rovides he condiion allowing solving he reachabiliy roblem saes. Given a Peri ne C =( P,T,I,O,µo ) wih marking µ ( R ( C,µo ) ) and marking µ` ( R ( C,µo ) ). The roblem is if µ` is reachable from µ, i.e. µ`( R ( C,µ ) ) [,]. Theorem (): Le µ be an iniial marking and le µ ( R ( C,µ ) ). If U is an P-invarian, hen µ U T = µ U T The above heorem rovides he condiion allowing solving he so called reachabiliy roblem. The roblem can be saed as following. Given a PN C = ( P,T,I,O,µo ) wih marking µ ( R ( C,µ ) ) and marking µ`( R ( C,µ ) ). Is µ` reachable from µ, i.e. µ`( R (C,µ) )?.s shown in Fig. [9].. Sofware for a Funcional nalysis of a Peri Ne Model The develoed sofware enables analyzing of a sysem for is various roeries irresecive of ime. Funcional analysis has been used in he Peri ne o refer o his ye of analysis. The sofware analysis is based on some aroaches used o deermine he various roeries of PN model: livens, safeness, reach-abiliy and conservaiveness. Fig. shows he flow char of comuing P-invarians. Inu / Ouu Marices Comue Loos U = u + u Is U * D +? No Deadlock Yes M' is reached from M Yes Is M * U = M' * U No M' is no reached from M Fig. : Flow char of comuing P-invarians 22 Page

4 Inernaional Journal of Modern Engineering Research (IJMER) Vol., Issue. 5, Se - Oc ISSN: IV. HEDINGS CONVERSION OF PERT/CPM CHRTS INTO EQUIVLENT PETRI NET MODEL The Peri Nes show he same ye of scheduling consrains as he PERT/CPM chars does. We can easily conver a PERT/CPM neworks o a Peri Ne model. Each aciviy in he PERT char is reresened by a lace, while he recedence consrains are reresened by he ransiions. In CPM char each even is reresened by lace (ransiion), while he aciviy is reresened by ransiion (lace). The Peri Ne is excellen vehicle o reresen he concurrency and recedence consrains of he PERT char. In modelling he PERT char wih he Peri Ne, aciviies ha are reresened by laces and ransiions are reserved o model mulile resources of limied amouns. Fig. demonsraes he conversion rocess of PERT/CPM chars ino Peri ne char. civiy On Node civiy on rc 2 a ) PERT Char b ) Equivalen CPM Char for PERT Char Transiion P P P P2 P Equivalen PN diagram for he PERT diagram OR 2 P P C d ) Equivalen PN diagram for he CPM diagram P P 2 e ) CPM char 5 PC f ) Equivalen PN diagram for e ) CPM char Fig. : Conversion of PERT/CPM chars ino Peri ne char The Peri Nes are more owerful models as comared o he PERT/CPM chars. The following reasons may be given for i: The reeaed erformance of aciviies, if necessary, can be modelled by he PN. Required resources er aciviy aear exlicily as okens in reresenaion. Non-deerminisic asecs can be deal wih. For examle: he order in which a aricular resource erforms some asks may no be oally secified. V. MODELING PROJECT NETWORKS THROUGH PETRI NETS y definiion of a rojec, he rojec is comleed when all he aciviies are rocessed exacly once. In PN erminology, his means ha all he ransiions mus fire exacly once, i.e. U.D = for U =, where is a column vecor wih all enries being one. This ransiion invarian verifies ha he recedence relaions among he aciviies are well esablished (he nodes are srongly conneced), i.e. he rojec can be comleed. Noe ha because he PN model of a rojec is a marked grah, he ransiion invarian is also a sufficien condiion. D. = means ha he sum of each and every row of D should be zero. Hence, o verify ha he rojec can be comleed, i is sufficien o show ha he incidence marix of he PN model of he rojec has his roery and ha he PN is a marked grah. The following demonsraes his hrough an illusraive examle. Consider a rojec consiss of aciviies,, C, D, E, F, G, and H wih heir imes of comleion as shown in Table []. The rojec will be finished afer 2 weeks, which is he CPM nework conains wo of criical ahs. The firs criical bah consiss of aciviies,, E, and H, and he second criical bah consiss of 5 aciviies,, D, F, and H. The aciviies D and D2 were dummies wih zero duraion. Fig. 5 shows CPM rojec nework for Table wih comleion ime for he rojec. Table : The sequence of he rojec aciviies civiy Precedecessor(s) Duraion ( Weeks ) C D E F G H , C, D E, F 2 Page

5 Inernaional Journal of Modern Engineering Research (IJMER) Vol., Issue. 5, Se - Oc ISSN: Page C D E F G H D D Fig. 5: CPM rojec ne work for able wih comleion ime for he rojec [] P P P5 P9 P P P P7 P9 P2 P P P PC PD PE PD PD2 PF PG PH Fig. : Peri ne (PN) model of figure 5 Fig. shows he PN model of he rojec nework in Fig. 5. For examle aciviy (P) needs days o be comleed. civiy imes are denoed wihin he circles (laces), which model he imed laces. The bars denoe immediae ransiions, which model logical condiions (ransiions and 2 were re and os condiions of P) and ake zero ime o fire. The same as for he oher aciviies from hrough H including dummy aciviies D and D2 have re and os condiions reresened in ransiions. ll oher laces P, P, P,, and P2 were connecion laces beween all he rojec aciviies, for examle P is he connecion beween he wo aciviies P and P hrough 2 which is os condiion of P and which is recondiion of P, as well as for he oher remaining laces ( e.g. P, P5,, P2 ) and needs zero ime o be comleed. The incidence marix of Fig. is shown below. The sum of each and every row of he incidence marix should be zero and each and every column should have a leas one - and + enry o verify ha he corresonding PN models he rojec correcly, i.e. all he aciviies, hence he rojec, can be comleed H F G E D D D C -u + u 7 = u u 2 = u u 7 =

6 Inernaional Journal of Modern Engineering Research (IJMER) Vol., Issue. 5, Se - Oc ISSN: In order o illusrae he alicaion of he aroach roosed, le consider he examle rovided in he revious examle of Fig.. n even grah (like PN in general) said o be srongly conneced if here is a direced ah joining any node o any node of he grah. The even grah resened in Fig. srongly conneced. We also define an elemenary circui in a srongly conneced even as a direced ah goes from one node, i.e. a lace or ransiion, back o he same node, which any oher node is no reeaed. For insance, Fig. exoses five elemenary circuis, namely: = (, ;,2;,;,;9,;G,7; ) 2 = (,;,2;,;,;,9;E,;9,;H,7; ) = (,;,2;,;,;,7;D,2;7,;F,5;2,;H,7; ) = (,;,2;,5;C,2;7,;F,5;2,;H,7; ) 5 = (,;,2;5,;D,;,;D2,2;7,;F,5;2,;H,7; ) If he number of okens in a marking remains fixed for all markings in he reachabiliy se, he Peri ne is hen said o be sricly conservaive. n immediae consequence of conservaiveness is boundedness of he Peri ne. The boundedness roery imlies ha he number of okens in any lace does no increase beyond a limi. This in urn guaranees a finie reachabiliy se for he Peri ne. If here exis an U wih all osiive elemens he Peri ne is hen said o be conservaive. If U = (,...,) hen he Peri ne is sricly conservaive. The oal number of okens in i ( i=,2,5 is hen: n ( ) = μ ( ) + μ ( ) + μ ( ) + μ ( ) + μ ( 9 ) + μ ( E ) + = = n 2 ( 2 ) = μ ( ) + μ ( ) + μ ( ) + μ ( ) + μ ( ) + μ ( G ) + μ ( 9 ) + μ ( H ) = = n ( ) = μ ( ) + μ ( ) + μ ( ) + μ ( ) + μ ( ) + μ ( D ) + μ ( 7 ) + μ ( F ) + μ ( 2 ) + μ ( H ) = = n ( ) = μ ( ) + μ ( ) + μ ( ) + μ ( C) + μ ( 7 ) + μ ( F ) + μ ( 2 ) + μ ( H ) = = N 5 ( 5 ) = μ ( ) + μ ( ) + μ ( 5 ) + μ ( D) + μ ( ) + μ ( D2 ) + μ ( 7 ) + μ ( F ) + μ ( 2 ) + μ ( H ) = = Therefore, he oal P invarians of all loos U = (5, 5,,,,,,,,,,,,,,,, ). y alying he equaions : U * D = (,,,,,,,,,,,,,,,, ), his means ha he sysem is deadlock free. To comue he oal ime for each loo o find he criical ah(s) as follows: = (,,,,9,G ) = = weeks 2 = (,,,,,E,9,H ) = = 2 weeks = (,,,,,D,7,F,2,H, ) = = 2 weeks = (,,,C,7,F,2,H ) = = 9 weeks 5 = (,,5,D,,D2,7,F,2,H ) = = weeks Then here is wo criical ahs γ 2 and γ, as he same as of he CPM nework Figure 5. VI. CONCLUSIONS ND REMRKS The roosed PN based aroach o rojec managemen faciliaed modeling (resource-consrained) rojecs, and verifying some roeries of he rojecs neworks, exloiing he well known lace ( ransiion ) invarians, hrough he basic PN semanics. Tha is, conrary o some aroaches in he lieraure, here is no need o modify he basic PN semanics o model aciviies and decisions relaed o a rojec. This increases he caabiliy of he basic PNs, wihou any exension in he heory, o model rojecs. Places were used o model all he recondiions, including resources, for an aciviy o sar, which was modeled via a lace (ransiion). The lace (ransiion) invarians were hen used o verify some roeries of he rojecs. This PN model can hen be analyzed hrough (PN based) scheduling echniques, used o find he criical ah, used as a basis o develo algorihms for resource-consrained rojec managemen, and used for oher decisions relaed o rojec managemen. The ransiion invarian means ha he sum of each and every row of he incidence marix should be zero. Hence, o verify ha he rojec can be comleed, i is sufficien o show ha he sum of each and every row of he incidence marix is zero, and ha each and every column has a leas one - and + enry. The roosed PN based aroach enables he rojec manager o consider no only resources bu also all differen yes of variables/consrains relaed o a rojec, e.g. coss of aciviies. ny of hese variables can be considered fuzzy as well. In his case, fuzzy arc weighs cause he firing rule o be modified. These are ossible fuure research direcions. REFERENCES [].S., Liang, J.N., Chen, and F.J., Wang, rojec model for sofware develomen, Journal of Informaion Science and Engineering,, 2, 2. [2].., Desrochers, and R.Y, l-jaar, licaion of Peri nes in manufacuring sysems: modeling, conrol, and erformance analysis (New York: IEEE Press, 995). [] V.., Jeeendra, O.V., Chey, and J., Reddy, Peri nes for rojec managemen and resource leveling, Inernaional Journal of dvanced Manufacuring Technology,, 2, [] C.K.,Chang and M., Chrisensen, ne racice for sofware rojec managemen, IEEE Sofware, November/ December, 999, -. [5] V.K., Kumar, and L.S., Ganesh, Use of Peri nes for resource allocaion in rojecs, IEEE Transacions on Engineering Managemen, 5(), 99, 9 5. [] W.M.P., Van der als, Peri ne based scheduling, OR Secrum,, 99, Page

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