An Enhaned Critial Path Method for Multiple Resoure Constraints Chang-Pin Lin, Hung-Lin Tai, and Shih-Yan Hu Abstrat Traditional Critial Path Method onsiders only logial dependenies between related ativities that may have to share ommon resoures. The method has then been expanded to allow for the identifiation of resoure-onstrained ativities and the size of those resoures. They are usually alled ativity-based resoure assignments and leveling. However, one the number of resoures inreases, it beomes more and more diffiult to draw the orresponding graph/network and explore the resoure- onstrained ritial path. Therefore, in order to solve a multiple-resoure-onstrained projet, the Petri Nets theory is introdued for modeling those resoure sharing proesses. Beause Petri nets is ommonly used for modeling the dynami behavior of disrete systems, it is intended in this researh to apply Petri net theory in modeling and analyzing ompliated projets with multiple resoure onstraints. Furthermore, analysis on the size of eah resoure for performane enhanement an then be done by hanging the numbers of tokens in eah of the resoure plaes that were added for resoure onstraints. Cost analysis has to be reonsidered sine the additions of resoures onstraints will alter the original shedule of eah ativities and inrease the duration of the projet. Ativity rashing may have to be done that shifts more resoures towards required ativity and results in dereased projet duration. I. INTRODUCTION URRENTLY for the design and management of omplex Cprojets, Critial Path Method (CPM) and Program Evaluation and Review Tehnique (PERT) are the two most ommonly used tools. CPM and PERT an be used with all kinds of projets, inluding manufaturing, onstrution, software development, researh projets, produt development, engineering, and plant maintenane, among others. Any projet with interdependent ativities an apply these methods of mathematial analysis. The original CPM onsidered only logial dependenies between related ativities that may have to share ommon resoures. The method has then been expanded to allow for the identifiation of resoure-onstrained ativities and the size of those resoures. They are usually defined through proesses alled ativity-based resoure assignments and resoure leveling. A resoure-leveled shedule may inlude delays due to resoure bottleneks and may ause a previously shorter path to beome the longest or the "ritial" path. Manusript reeived August 30, 2013. This work was supported in part by the National Siene Counil under Grant NSC 101-2221-E-019-015. C. P. Lin is with the National Taiwan Oean University, Keelung, TAIWAN 20224 (phone: 2-2462-2192 ext.3243; fax: 2-2462-0836; e-mail: b0195@mail.ntou.edu.tw). H. L. Tai is with the National Taiwan Oean University, Keelung, TAIWAN 20224 (e-mail: 19972069@mail.ntou.edu.tw). S. Y, Hu is with the National Taiwan Oean University, Keelung, TAIWAN 20224 (e-mail: 10072054@mail.ntou.edu.tw). In Kelley and Walker [1], a large engineering projet involves almost all the engineering and raft skills as well as the funtions represented by researh, development, design, prourement, onstrution, vendors, fabriators and the ustomer. Management must devise plans whih will tell with as muh auray as possible how the efforts of the people representing these funtions should be direted toward the projet's ompletion.[2] However, one the size of eah resoure inrease, it beomes more and more diffiult to draw a graph/network and explore the resoure-onstrained ritial path. Therefore, in order to solve a omplex resoure-onstrained projet, the Petri Nets (PN) theory is then introdued for modeling and analysis. Beause Petri nets is ommonly used for modeling the dynami behavior of disrete systems, it is intended in this researh to apply Petri net theory in modeling and analyze large resoure-onstrained projets. Peterson[3], [4] has developed properties, onepts, and tehniques of Petri nets for a searh of natural, simple, and powerful methods in desribing and analyzing the flow of information and ontrol in systems, partiularly systems that may exhibit asynhronous and onurrent ativities. The major use of Petri nets has been the modeling of systems of events in whih it is possible for some events to our onurrently but there are onstraints on the onurrene, preedene, or frequeny of these ourrenes.[5], [6] A.G. Colombo and A. Saiz de Bustamante [7] have defined Timed PN (TPN) as a marked PN in whih a set of speifiations are provided and a set of rules are defined suh that to eah legal exeution sequene, a timed exeution sequene an be univoally assoiated.[8] Time is represented as a state of the system. Beause the number of different shedules grows very quikly with the number of mahines as well as the length of the shedule, olored Petri nets were proposed by Zuberek [9] as a uniform representation of entire lasses of shedules. Verifiation of real-time systems is a tough problem: time, unertainty, omplex synhronizations, preemptive sheduler, distribution. [10]-[13] Kumar and Ganesh [14] have presented oversights that may be due to the inadequay of the traditional models to represent these fators effetively. The later models suh as venture (V)ERT, graphial (G)ERT, ontrolled ativity networks, and tools like time/ost tradeoff urves and influene matrix approahes also are not apable of inorporating these fators. In ontrast, Petri nets an inorporate these fators and thus provide a powerful methodology for resoure alloation in projets. In the ontext of researh by Chetty and Reddy [15], the benefits of PNs were presented. Extensions to PNs were
proposed to suit this area of appliation. The use of a P-matrix for token movements was proposed. Chang-Pin Lin and MuDer Jeng [16] have developed unique methodology in performing integration at today s manufaturing environment by reating a ommon, modular, flexible, and integrated objet model that unifies an advaned objet-oriented arhiteture onept and heterogeneous manufaturing appliation development in an open and multi-supplier manufaturing environment. In Pantouvakis and Manoliadis [17], leveling generally onsists of two stages: The first one performs projet sheduling without imposing resoure onstraints. Early/late start and finish dates with ativity floats are then alulated. The seond stage attempts to reshedule the ativities of the projet, preferably within their float time, so that the projet an be exeuted with a limited number of resoures and with or without time delay. Penzek and Polrola [18] onsider two main models of real-time systems: Petri nets with time and timed automata. They have defined Petri nets, disuss their time extensions, and provide a definition of time Petri nets. Some work by Didier Lime [19] has been devoted to analytial results of shedulability, giving worst-ase response-time for tasks. Time management in Pedro and Silva [20] is a ritial omponent of workflow-based proess. Important aspets of time management inlude: the planning for the workflow proess exeution on time, estimating workflow exeution duration, avoiding deadline violations, and satisfying all time onstraints speified. Gonzalez del Foyo and Reinaldo Silva [21] apply Time Petri Nets to model the temporal behavior for workflow systems using TINA as a tool to support the verifiations of the ativities deadlines. The proess is based on the onstrution of a unique Timed Graph where verifiation of TCTL formulas is made avoiding the neessity of the onstrution of a Timed Graph for eah formula. Latif Salum [22] presented Timed PNs to model event duration that improves the urrent analysis tehniques and overomes these diffiulties. Ismail, Rashid and Hilo [23] have indiated that a traditional CPM shedule is not realisti sine it assumes unlimited resoures, some of whih are highly limited in pratie. Although traditional RCS an onsider resoure limitations, they do not provide the orret floats and ritial path, as the CPM does. An enhaned projet management methodology has been developed in order to handle multiple limited resoures whih have beome more and more important in ompliated large projets. The existing and newly developed analysis tools for Petri net an be applied to ahieve this goal by onverting CPM network to its orresponding Petri net model in whih the resoure onstraints ould be added into the projet model. One the Petri net model was built, the reahability analysis an be arried out to obtain the optimal resoure sharing sequene whih will ahieve the shortest projet duration with the existing limited resoures. However, the addition of resoure onstraints will inrease the projet duration, it is then neessary to apply ost analysis by inreasing the number of eah resoure type and by rashing the duration of eah ativity. Analysis on the size of eah resoure for performane enhanement an be done by inreasing the numbers of tokens in eah of the resoure plaes being added for resoure onstraints. Corresponding ost analysis should be done to determine the most ost effetive resoure. Ativity rash may have to be determined that is ahieved by shifting more resoures towards the ompletion of that ativity, resulting in dereased time spent and often a redued quality of work, as the premium is set on speed. II. RELATED THEORY A. Critial Path Method CPM is a projet modeling tehnique developed in the late 1950s by Morgan R. Walker of DuPont and James E. Kelley, Jr. of Remington Rand. At about the same time Booz Allen Hamilton and the U.S. Navy have developed the Program Evaluation and Review Tehnique (PERT) whih is very similar to the basis and exeution of the CPM. The essential tehnique for using CPM is to onstrut a model of the projet that inludes the following: 1. A list of all ativities required to omplete the projet, 2. The duration that eah ativity will take to ompletion, and 3. The preedene relationships between all ativities. Using these values, CPM alulates the longest path of planned ativities to the end of the projet, and the earliest and latest that eah ativity an start and finish without making the projet longer. This proess determines whih ativities are "ritial" (i.e., on the longest path) and whih have "total float" (i.e., an be delayed without making the projet longer). In projet management, a ritial path is the sequene of projet network ativities whih add up to the longest overall duration. This determines the shortest time possible to omplete the projet. Any delay of an ativity on the ritial path diretly impats the planned projet ompletion date (i.e. there is no float on the ritial path). A projet an have several, parallel, near ritial paths. An additional parallel path through the network with the total durations shorter than the ritial path is alled a sub-ritial or non-ritial path. Sine projet shedules hange regularly, CPM allows ontinuous monitoring of the shedule, whih allows the projet manager to trak the ritial ativities. In additions, beause it is possible that non-ritial ativities may be delayed beyond their total float, the CPM network will reate a new ritial path and delay projet ompletion. B. Petri Net Theory Petri net theory onsists of a graphial tool for modeling systems dynami behavioral and a set of mathematial theorems to analyze their behavioral and strutural properties. Definition 1: A Petri net is a 5-tuple PN= (N, M o ) = (P, T, I +, I -, M o ), where (1) P={p 1,p 2,, p m } denotes a finite set of plaes, (2) T={t 1,t 2,, t n } denotes a finite set of transitions,
(3) P T andp T, (4) I - is the input inidene funtion defined on P T, I + is the output inidene funtion defined on T P, and (I -, I + ) denote the set of ar links between plaes and transitions, (5) p Pt T I pt 0I pt 0 and t Tp P I pt 0I pt 0, (6) M o is a set of token values defined on P and is alled the initial marking. Definition 2: A Petri net graph ontains three basi entities: Cirles represent plaes (onditions), Bars represent transitions (events), and Ars represent input and output inidene funtions. A plae represents one and only one proposition and a transition represents a transformation funtion. Marking is represented by Tokens whih are shown as dots in plaes. Definition 3: Petri net transition firing rule is defined as follows: (1) Transition t j is enabled and ready to be fired if: M(p) I ( p, t j ), p t j (where t j represent the set of all input plaes of t j ). (2) An enabled transition an be fired aording to the ourrene of the atual event on the transition. (3) After fired the transition, I ( p, t j ) tokens are removed from eah of the assoiated input plaes and I ( p, t j ) tokens are added into eah of its orresponding output plaes. Definition 4: A Petri net an be represented in a mathematial form by its inidene matrix C. The inidene matrix C an be represented as in Eq. 1: C mn C C 11 21.. m1 12 m2 where = I - (p i,t j ) and ij 1n 11. 21.... mn m1 ij = I + (p i,t j ) 12 m2 1n... mn The mathematial formulation of transforming an initial marking M o into a new marking M by firing a set of enabled transitions is shown as in Eq. 2, M = M o + C y [2] where y is a transition firing vetor. Definition 5: A Petri net graph onsists of 4 basi omponents as: Plae: The input data or signals, data buffer onditions Transition: Aggregation, tasks or ativities Token: Dynami Ar: Diretion or path Timed Petri Net (TPN) is a time extension of the lassial Petri net model that allows for expliit treatment of real-time, whih is assoiated usually with the transition in the net and represents the time intervals to be used in order to fire the respetive transition. By ooperating time variables into Petri nets, timed Petri [1] nets further allow us to derive prodution yle time, identify bottlenek workstations, verify timing onstraints, and so on. They an also be used to obtain prodution rates, throughput, average delays, ritial resoure utilization, reliability measures, and so on when the time variables are random. III. RESEARCH METHODOLOGY The main objetive of researh is to develop a methodology in order to extend the traditional projet management tool for multiple resoure onstrained projets that are the quite ommon in today's projet analysis, espeially in the areas of onstrution management, manufaturing proess planning, and et. Steps in the proposed methodology is presented as follows. Step 1.Apply traditional CPM methodology to draw the projet network, ompute all the basi timely fators, inluding ES, EF, LS, LF, TF, and find the ritial path of the projet network with all the ritial ativities Step 2.In order to onsider multiple resoure onstraints in managing the projet, the CPM network will firstly be onverted into orresponding Petri net by the rules of onversion. Furthermore, the resulting Petri net has to be verified by applying strutural analysis algorithm and the omputation of make span in Timed Petri net. Step 3.Next, the required resoure onstraints should be added into the Petri net whih will then reate one onflit in the net for eah resoure onstraint. Repeat this step for eah of the resoure onstraint. Step 4.Reahability analysis with timely onsideration will be arried out to obtain the best sequene for eah of the onflit in the net. The optimal sequene ombination with the lowest make span of the omplete projet an then be found as the best pratial projet shedule. Step 5. In order to find the atual ritial path of the projet, it is then neessary to onvert the Petri net model bak to the CPM model with the onsiderations of sequene for onflit ativities. One onverted, the CPM network an be reanalyzed to disover the atual ritial path and the assoiated ritial ativities. Step 6. Finally, in order to arry out neessary redutions on the projet make span, the proess to rash ritial ativity one by one based on their limited rashing time has to be done. the ost analysis for ativity rashing will be exeuted to obtain the possible rashing time and its orresponding rashing ost for the projet management, and the results will be quite different from the traditional CPM ost analysis without onsiderations of resoure onstraints. A. Conversion between CPM network and Petri net The onversion rules to transform CPM network into Petri net as shown in Fig. 1 whih inlude : 1. Burst node whih reates several transitions, one for the burst purpose from one plae into multiple plaes, and the others represent the underlying ativities; 2. Merge node whih also reates several transitions, one for the merge purpose from multiple plaes into one plae, and the others represent the underlying
ativities. Fig. 2. CPM network of the sample ase. Table I PROJECT DETAILS Ativity Duration Preedene Resoures A 2 - B 3 - C 5 - D 4 B E 4 A F 3 B G 6 CD First, the traditional CPM that does not onsider resoure sharing is applied to ompute and analyze the sample ase projet. The Early Start(ES) Early Finish(EF) Late Start (LS) Late Finish(LF) Total Float(TF) times are omputed through forward and bakward proesses as shown in Table II. The irtial path an then be determined to be B-D-G with ritial ativities of zero TF. Fig. 1. Conversion rules from CPM to Petri Net. B. Representation of resoure onstraints in Petri net model Petri nets inorporate Resoure onstraints for the designated projet by adding resoure plaes into the graph of CPM. The number of eah resoure type an then be represented by the number of tokens in eah of the resoure plaes. The duration of eah ativity is presented by the time onsumption for the transition whih is onverted from the ativity in the orresponding CPM. The resoure sharing is represented by the onneted ars from and to the resoure plae with the transitions that share the same resoure. IV. CASE STUDY For the demonstration purpose, one simple ase was used to present the feasibility of the proposed methodology and the importane of the results for deision making. The projet being shown as in Fig. 2 is a ase whih involve 7 (A, B, C, D, E, F, G) ativities and the required preedene relationships between them. In additions, 2 types of limited resoures are introdued to represent the possible sharing of required resoure among ativities as listed in Table I. Table II PROJECT COMPUTATIONAL RESULTS Ativity ES EF LS LF TF A 0 2 3 5 3 B 0 2 0 2 0 C 0 3 2 5 2 D 2 5 2 5 0 E 2 8 5 11 3 F 2 8 5 11 3 G 5 11 5 11 0 Existing method for resoure sharing an be applied for single resoure onstraint, suh as, Woodworth s shedule after forward resoure sheduling pass to adjust the shedules of ativities that share the same resoure. Fig. 3 shows the move of shedule due to the resoure sharing among ativities C, D, and E that inreases the total projet duration from 11 days for none resoure onstraint to 12 days for single resoure onstraint. However, one the types of limited resoures used in the projet inrease, it is very diffiult to reshedule the ativities due to the ompliated resoure sharing proesses by the existing resoure leveling methods. The proposed methodology for handling multiple resoure onstraints will then be presented in the following setions. Fig. 3. Resheduling for limited resoure.
In order to onsider multiple resoure onstraints in managing the projet, the CPM network will firstly be onverted into orresponding Petri net by the rules of onversion. Furthermore, the resulting Petri net has to be verified by applying strutural analysis algorithm and the omputation of make span in Timed Petri net. Burst node will reate several transitions, one for the burst purpose from one plae into multiple plaes, and the others represent the underlying ativitie. Merge node will also reates several transitions, one for the merge purpose from multiple plaes into one plae, and the others represent the underlying ativities. The onverted Petri net from the CPM in Fig. 2 is shown as in Fig. 4. duration and the ost effetiveness in the inreasing number of resoures. Aordingly, the number of eah resoure an not be higher than the number of ativities (transitions) sharing this type of resoure, otherwise, it beomes no resoure onstraint. In order to show the hanges by inreasing the number of eah resoure type, the reahable markings of the Petri net model is listed in Table III by the exeuting time. P0 T0 P1 s P0 T1 s T0 P4 P3 P2 T4(2) T3(3) T2(2) P1 T1 P7 P6 P5 P16 T6(6) T5 R1 P4 P3 P2 P17 R2 P9 P8 T4(2) T3(3) T2(2) T8(3) T7(6) P7 P6 P5 T6(6) T5 P11 T9 P9 P8 P13 T8(3) T7(6) T10(6) P11 P12 P14 P10 T9 T11 P13 P15 T10(6) Fig. 5. Petri Net with resoure onstraints. P12 P14 T11 P15 P10 Fig. 4. CPM onverted to Petri Net. The use of Fig. 5 after the onversion inreases Resoure onstraint graph for the example, this example of onneting three Transition Resoure onstraint were Transition3, Transition6, Transition8, in the absene of restritions under what onditions there are many paths to go. A path Transition3 Transition8 Transition6 were obtained for the shortest projet duration. Studies in this sample ase onsider only one unit for eah resoure type, therefore the ativities sharing the same resoure have to be exeuted sequentially instead of in parallel. In additions, the resoure sizes for eah resoure type should be inreased to find their effets on the projet Table III PROJECT DURATIONS FOR DIFFERENT RESOURCE SIZES Resoure 0/0 1/1 1/2 1/3 2/1 2/2 2/3 sizes(r1/r2) Time(day) 11 23 14 12 23 14 11 In order to justify the effetiveness of inreasing numbers of token in eah resoure type, it is neessary to arry out ost analysis. The expeted benefit for the sample projet is 1 million with no resoure onstraint. By inreasing the number of resoure onstraints, the benefit will derease aordingly as shown in Table IV. The ost for one type one resoure is 50 thousands and for type two resoure is 80 thousands. The ost is 15 thousands for eah day whih was delayed. The onsiderations of ost inlude inventory, raw material, equipment maintenane and et. The best ombination of resoure numbers between resoure
type one and two is 1/2 whih will provide the highest benefit for the projet management. Table IV PROJECT PROFITS FOR DIFFERENT RESOURCE SIZES Resoure Duration Resoure Duration Profit onstraint Cost Cost 0/0 11 0 0 10000000 1/1 23 130000 180000 690000 1/2 14 210000 45000 745000 1/3 12 290000 15000 695000 2/1 23 180000 180000 640000 2/2 14 260000 45000 695000 2/3 11 340000 0 660000 V. CONCLUSIONS A new projet management methodology based on Critial Path Method and Petri net theory was developed in order to handle multiple limited resoures whih have beome more and more important in ompliated large projets. Conversion rules between CPM and Petri net were developed to reate the Petri net model for the additions of resoure onstraints into the projet model. Petri net strutural properties verifiation analysis an then be applied to verify the orretness of the network model. One the Petri net model was built, the reahability analysis an be arried out to obtain the optimal resoure sharing sequene whih will ahieve the shortest projet duration with the existing limited resoures. Furthermore, analysis on the numbers of eah resoure types for performane enhanement an be done by inreasing the numbers of tokens in eah of the resoure plaes being added for resoure onstraints. Corresponding ost analysis should be done to determine the most ost effetive resoure. The ritial path for the projet by onsidering the resoure onstraints an then be found by onverting the Petri net model bak to CPM network with additional preedene ativities and apply the neessary redution rules. The modified CPM network an then be used to find the right ritial path and ritial ativities whih require more attentions for projet management. Results through the ase study show the feasibility of the proposed methodology and the value of the Petri net analysis algorithm. 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