Critical Path Segments Scheduling TechniqueTarek Hegazy, M.ASCE1; and Wail Menesi, S.M.ASCE2

Abstract: While the critical path method CPM has been useful for scheduling construction projects, years of practice and research havehighlighted serious drawbacks that hinder its use as a decision support tool. This paper argues that many of CPM drawbacks stem fromthe rough level of detail at which the analysis is conducted, where activities durations are considered as continuous blocks of time. Thepaper thus proposes a new critical path segments CPS mechanism with a finer level of granularity by decomposing the duration of eachactivity into separate time segments. Three cases are used to prove the benefits of using separate time segments in avoiding complexnetwork relationships, accurately identifying all critical path fluctuation, better allocation of limited resources, avoiding multiple-calendarproblems, and accurate analysis of project delays. The paper discusses the proposed CPS mechanism and comments on several issuesrelated to its calculation complexity, its impact on existing procedures, and future extensions. This research is more beneficial toresearchers and has the potential to revolutionize scheduling computations to resolve CPM drawbacks.

DOI: 10.1061/ASCECO.1943-7862.0000212

CE Database subject headings: Construction management; Scheduling; Critical path method; Project management; Computation.Author keywords: Construction; Scheduling; Critical path method; Float; Project management.

Introduction

Scheduling the construction process using critical path methodCPM is essential so that projects can be completed profitablyand on time. Because of its benefits and the significant advance-ments that have been made in both computer hardware and sched-uling software, the use of the CPM and its precedence diagrammethod PDM variation in all industries, including construction,has dramatically increased in the last three decades Galloway2006; Liberatore et al. 2001. For the purpose of this paper, CPMwill be used to indicate both CPM and PDM.

While the CPM calculations are simple and straightforward,CPM-based scheduling is a challenging process. At the planningstage before construction, the CPM network may contain complexrelationships that complicate the scheduling process. In addition,the CPM algorithm has no formulation to account for the multipleconstraints in a project such as deadline and resource limit. Whileresearchers have introduced remediational techniques such astime-cost trade-off analysis and resource leveling Hegazy 2002,it is often difficult to produce a realistic schedule since a solutionto one constraint e.g., resource limits may interfere with thesolution to another e.g., deadline. This difficulty adds to theperception that CPM and existing software are useful for organi-zational and reporting purposes but not for decision support toreflect and react to reality Kuhn 2006.

The lack of CPM-based decision support is even more vividonce a project has started. While the schedule acts as a baselinefor measuring progress, it is difficult to use it to initiate appropri-ate corrective actions for recovering delays and overruns. Further-more, CPM has an important role in the analysis of the finalas-built schedules in order to determine the responsibility of thedifferent parties for any delays experienced during construction.The boards of contract appeals and the courts have shown theirwillingness to utilize CPM network analysis to identify the sourceof delays in construction projects Ostrowski 2006. CPM sched-ules, however, are difficult to analyze due to many well-documented factors that impact calculation accuracy andrepeatability. The following list shows the most important criticalviews of the CPM:1. Problems with multiple-complex relationships:

Networks with multiple relations finish-to-finish, FF, andstart-to-start, SS are complex to analyze and cause partsof an activity, not the whole, to be critical, which is un-detectable by existing software Lowsley and Linnett2006; Lu and Lam 2009.

Non-finish-to-start relationships with lags complicate totalfloat determination and interpretation, potentially affect-ing critical path identification Lu and Lam 2009.

SS and FF relationships use time, but not work-amount,lags OBrien and Plotnick 2006.

2. Inaccurate schedule calculations: Floats and the critical path can be inaccurate due to the

extensive use of leads and lags Wickwire and Ockman2000.

Multiple calendars make it harder to analyze the criticalpath and floats Scavino 2003.

Primavera software can produce inaccurate dates whenresource calendars are used Kim and de la Garza 2005;Lu and Lam 2008.

Unrealistic activity durations can result from wrong cal-culations of remaining durations Street 2000; Wickwireand Ockman 2000.

1Professor, Civil and Environmental Engineering Dept., Univ. of Wa-terloo, Waterloo ON, Canada N2L 3G1 corresponding author. E-mail:tarek@uwaterloo.ca

2Ph.D. Student, Civil and Environmental Engineering Dept., Univ. ofWaterloo, Waterloo ON, Canada N2L 3G1. E-mail: wmenesi@engmail.uwaterloo.ca

Note. This manuscript was submitted on March 2, 2009; approved onMarch 2, 2010; published online on September 15, 2010. Discussionperiod open until March 1, 2011; separate discussions must be submittedfor individual papers. This paper is part of the Journal of ConstructionEngineering and Management, Vol. 136, No. 10, October 1, 2010.ASCE, ISSN 0733-9364/2010/10-10781085/$25.00.

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3. Difficult schedule analysis during and after execution: Out-of-sequence progress e.g., activities starting prior to

completion of their predecessors makes CPM schedulesdifficult to analyze Herold 2004.

Existing software are sometimes abused to produce badlyflawed schedules that look good but are unpractical toimplement. The result is confusion, delays, and lawsuitsKorman and Daniels 2003.

Schedule analysis is not a straightforward task under mul-tiple baseline updates Livengood and Anderson 2006.

The lack of clear representation of site events in schedul-ing software makes it difficult to visualize the actionsmade by the various parties and accordingly analyze theschedule Baweja 2006; Menesi and Hegazy 2008.

Several alternatives to the CPM have been introduced to dealwith special cases of projects. While CPM can effectively handlesequential and parallel activities, Fayez et al. 2003, for example,criticized the CPMs inability to handle iterations, which is acharacteristic of design projects. They then introduced a designstructure matrix approach to represent information flows in addi-tion to workflows in a project. Several other researchers have alsoextensively used discrete event simulation as a scheduling tool toovercome deficiencies in the network-based methods Sawhney etal. 2003. Simulation-based systems can consider project risksand provide means to optimize resources for projects that involvecyclic operations such as earth-moving and tunneling, etc.

Many other efforts in the literature have discussed ways toimprove CPM scheduling and avoid some of the calculation mis-takes listed earlier. In an effort to improve planning and avoid theproblem of dealing with complex relationships, Ponce de Leon2008 presented a logic diagramming method, which uses anactivity notation that resembles arrow diagramming, albeit on atime scale. Interestingly, SS, FF, and SF logics are permitted byinserting embedded nodes on, or between, the activity start andfinish nodes.

Lu and Lam 2009 proved through a PDM network examplecontaining non-finish-to-start FS relationships that non-FS rela-tionships complicate total float determination and interpretation.They proposed generic transformation schemes to transformnon-FS relationships in a project network into equivalent FS re-lationships with zero lag, which provides a better understandingof the scheduling results and paves the way for conducting furthersophisticated schedule analysis.

Plotnick 2006 also focused on a better understanding of therelationships between activities. He discussed the confusion ofwhether the lag duration is measuring the passage of time oractual progress. To represent various practical situations, Plotnickintroduced a new system called the relationship diagrammingmethod, a variant of the CPM, which records additional informa-tion about the relationships, such as the purpose of the relation-ship. He also introduced additional relationship types such asbegin-to-start, progressed-to-start, remaining-to-start, end-to-finish, finish-to-remainder, and finish-to-progressed. All these re-lationships, however, require some effort to use it efficiently.

From the project control point of view, Herold 2004 pointedout that the industry and software vendors need to develop en-hanced PDM scheduling software that not only can perform com-plex calculations, but that also has the ability to present scheduleinformation clearly and concisely. In an attempt to improveschedule representation, Herold suggested an interesting approachof sorting and ranking various activity paths for legible viewwithin a bar chart.

Basu 2008 investigated how CPM scheduling software

handles business rules such as resource allocation, cost tracking,and claim management. He stressed the need to establish math-ematical basis for schedule development and use. To that end, hesuggested that software functionalities be validated so that theCPM calculations yield repeatable and consistent results.

All the above efforts, and many others, introduced incrementalenhancements to the traditional CPM analysis. Yet, however, theanalysis is still done at an activity by activity level, which is arough level of detail that produces calculation errors and is notsuited to detailed progress analysis. This research, therefore, aimsat introducing a finer detailed, yet simple, approach to overcomethe current CPM drawbacks.

New Critical Path Segments Mechanism

To overcome the CPM drawbacks discussed earlier, the proposedcritical path segments CPS mechanism has three innovativefronts: 1 representing activity duration using separate time seg-ments, 2 better representation of activity progress, and 3mechanism to incorporate project constraints into the CPS analy-sis. These aspects are discussed in detail in the following sections,followed by three cases to prove that CPS offers less complexrepresentation of activity relationships, thus leading to betteridentification of critical path fluctuation, and better ability to ana-lyze the schedule and mitigate delays.

Representing Activity Duration as Separate TimeSegmentsSince the representation of activities and their durations is thebasis for schedule calculations, improving the representation ofthe activities would solve many of the problems mentioned ear-lier. As opposed to the traditional representation of activity dura-tion as a continuous block of time that spans the activity duration,the CPS represents each activity as a number of separate consecu-tive time segments that add up to the total duration of the activity.For example, an activity with duration of three days is representedby three time segments Fig. 1, where each time segment is oneday in this case. This representation has major advantages, asfollows: It permits direct conversion of any complex logical relation-

ship SS, FF into a simple FS relationship three cases shownin Fig. 1, without the lag times which cause float calculationproblems in the traditional CPM. In this representation, eachactivity is bound by start and finish milestones, as indicated bythe dark solid lines in Fig. 1. For example, the SS0 relation-ship was converted to a FS0 relationship from the milestonepreceding the first activity to the milestone preceding the sec-ond activity.

It provides more flexible options to better represent the intentof the relationships among the activities. The CPS can definethe relationship between activities not only as time-based, butalso as production-based. For instance, instead of indicatingthat steel reinforcement work can start two days after theformwork begins, the CPS would enable the project managerto specify that each 20% of the formwork completed is fol-lowed by 20% of the steel reinforcement. This kind of rela-tionship is illustrated in Fig. 2. Another example of preservingthe relationship intent is shown in Cases 4 and 5 of Fig. 1where relation segments may need to be added.

By reporting the daily percentages on the time segments as inFigs. 1 and 2, it is possible to clearly convey information

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related to speed of construction e.g., actual versus plannedand the use of different resource calendars in different days orin different activities.

It offers a more granular level of analysis that uses a time-segmentbytime-segment analysis, rather than traditionalactivity-by-activity analysis. This also facilitates the trackingof resources and allows more flexible resource allocationoptions.

New Representation of Activity ProgressIn the CPS, progress is clearly represented, in consistency withthe separate time segment approach, so that schedule analysis canbe carried out accurately with less disagreement among parties. Inthe proposed representation, work progress in percentage isshown on the associated time segment the progress percentagesdo not mandate daily inputs, rather, can be averaged over a num-ber of days/weeks. Also, additional time segments are inserted torepresent known events that happened on specific dates andcaused by the owner O, the contractor C, and/or neither partyN e.g., weather.

Fig. 3 shows the proposed representation of an activity withfive-day baseline duration. During construction progress, the con-tractor started the activity one-day late. After completing 5% inthe next day slower than baseline, the work was stopped for oneday due to owner interruption and then resumed with a 15%progress in day 4. Thus, the contractor had only one day to com-plete the activity as planned. Accordingly, the contractor decidedto use a faster and more expensive construction method to accel-erate the activity and finish the remaining work in two days, eachallocated 40%. Such a generic activity representation clearlyshows the evolution of all activity events, including the effect ofdecisions such as acceleration and resource allocation. This rep-resentation is therefore general enough for progress recording.Accordingly, the CPS automatically calculates the remaining du-ration of each activity, based on the progress entered to-date asshown on the left-side of the actual bar in Fig. 3, as follows:

Activity duration = Actual duration to-date

+ activity remaining duration 1

Fig. 1. Sample CPS representation to transform relationships into FS with no lag time

Fig. 2. Time-based and production-based options in the CPS Fig. 3. Proposed representation of progress events

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Activity remaining duration

= 1-% complete to-date Baseline duration 2

It is noted that the remaining duration can also consider expectedfuture events such as accelerations or delays as shown on theright side of the actual bar in Fig. 3.

Mechanism to Incorporate Project Constraintsinto the CPSFor traditional CPM networks with continuous activity durations,a simple approach to facilitate the resolution of multiple-projectconstraints was proposed by Hegazy 2006 and can be adapted tothe CPS with separate time segments. To demonstrate the process,a small case study Fig. 4 is considered. The optional estimatesfor the four activities of the case study are shown in the figure,with the project having a strict 10-day deadline, a late penalty of$2,000 per day, a $100 per day indirect cost, and a strict resourcelimit of two per day. It should be noted that the optional estimatesrepresent practical options that vary from cheap and slow to fastand expensive. These estimates can be different subcontractorquotes, or crews with different skill levels, different equipment, orsimply working overtime hours.

A quick look at the project network reveals that activitiestrench 1 and trench 2 run in parallel and will require fourresources limit is two per day. In addition, using the cheapestoption estimate 1 for each activity, project duration becomes 13days three days beyond the deadline with a total cost of $14,300$7,000 direct cost+$1,300 indirect cost+$6,000 penalty.To meet the deadline and resource-limit constraints, it is possibleto experiment with varying decisions. Given the sequence amongthe activities and the various construction options, it is possible tocome up with a least-expensive plan that meets both the deadlineand the resource limit. The solution in Fig. 5, for example, showsa plan with 10-day project duration, which meets the deadline andin which all the activities are scheduled so that the resource limitis not exceeded.

It is important to note that the solution in Fig. 5 shows the twoquantitative decisions that need to be taken: a an index to themethod that makes a good trade-off between the duration and costof the activity i.e., time-cost tradeoff analysis and b the start-delay time applies to the start of the activity that resolves re-source over-allocations by preventing many activities to run inparallel. The formulation of the key activity variables decisionsin Fig. 5, as such, simplifies their direct incorporation into themathematics of the CPM algorithm, not only for scheduling be-fore construction but also for corrective actions during construc-

tion. If a project is delayed, for example, then a suitablecorrective action is to decide on modified values for the two de-cisions, which will affect the remaining portion of the schedule.More details on this formulation of project constraints and the useof genetic algorithms optimization to address the large combina-torial nature of this problem can be found in Hegazy 2002 andwas implemented in the Easyplan computer prototype discussedin Hegazy 2006. Since Easyplan is prototype developed onExcel, it has its limitations; however, interested readers candownload it www.civil.uwaterloo.ca/tarek/easyplan.html andexperiment with its optimization features.

This approach will be even more powerful in facilitating deci-sions when it is reformulated for the CPS to consider separatetime segments on-going work by the authors. As such, the CPSwill allow each time segment of an activity to be independent andflexible. A generic representation of the revised project decisionsin the CPS is shown in Fig. 6, where the method-index decisionapplies to each activity, while the start-delay decision applies toeach individual time segment of the activity. As such, the CPSformulation enables resource allocation to produce more practicaland realistic schedules since it will enable all individual timesegments to be stopped and restarted, as necessary, so that thelimited resources are not exceeded.

Proof of Concept

To demonstrate the ability of the CPS to provide better analysisthan traditional CPM, three simple case studies are used to showits ability to: 1 simplify network relationships and accuratelycalculate floats and the critical path; 2 achieve a better resource

Fig. 4. Case study activities and their optional estimates and resourceneeds Fig. 5. Details of a solution that meets all constraints

Fig. 6. Representation of activity variables in the CPS

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allocation and facilitate accurate delay analysis; and 3 overcomesome of the problems associated with multiple calendars.

Case 1As mentioned earlier, complex relationships such as FF, SS, andSF, complicate the CPM and can lead to situations in which theactivities might be partially critical Lowsley and Linnett 2006.Such a situation is not detectable by available software systemsmainly because of the assumption that each activity is a singleundivided block of a given duration. This results in errors in thefloat and critical path calculations. Fig. 7 illustrates a simple casestudy discussed in Lowsley and Linnett 2006. The figure showsa network in which each activity is linked by both a SS and a FFrelationship. The network calculations in this case Fig. 7a re-veal that the start dates are critical for all activities but, because ofthe overlap created by the SS and FF relationships, the finishdates for the first three activities contain float. Such a situation iscomplex to analyze, particularly under cases involving resourcelimits and/or schedule crashing, let alone progress evaluation anddelay analysis. As shown in Fig. 7b, Primavera P3 and Mi-crosoft Project software, which use continuous activity durations,show all the activities as critical.

For comparison purposes, the CPS representation has beensimulated on Microsoft Project software, with each time segmentbeing simulated by a separate activity with a one-day duration, asshown in Fig. 7c. Although the software is not readily suited forthe CPS representation, Fig. 7c clearly shows that only the firsttwo days of activities B and C are critical, not the whole activity.Fig. 7c also shows that the CPS used only FS relationships with

zero lag times. As such, Case 1 clearly illustrates the ability of theCPS to simplify network representation and accurately calculatethe floats and the critical path.

Case 2In Case 2, a small project is considered and is intended to showthat CPS is more flexible in terms of resource allocation andallows detailed schedule analysis of project delays. Fig. 8ashows the as-planned schedule of a seven-activity project, withactivities B, C, and D, each requiring one R1 resource limit=2 R1 /day. The 13-day as-planned schedule of Fig. 8a, there-fore, meets the resource limit.

During the course of construction, the owner caused a delay inactivity B on day 3 Fig. 8b. Although the delay did not affectthe critical path, it made the initial resource allocation for theremaining work impractical on day 8 Fig. 8b. To resolve thisresource over allocation, the contractor would be forced to delaythe project one day Fig. 8c to become 14 days. Accordingly,regular schedule analysis would indicate that the contractor mayclaim a one-day extension due to the resource over-allocationresulting from the owner delay, which is an issue discussed byseveral researchers e.g., Hegazy and Menesi 2008; Ibbs andNguyen 2007.

Using the CPS approach, the resource-leveling solution forthis case results in a 13-day schedule, thus causing no projectextension Fig. 8d. The solution was achieved by having a onetime segment delay inserted before the fifth time segment of theactivity D. It is important to note that although existing softwaresystems includes an option for splitting the activities during theresource-leveling process, it does not permit activities that havebeen already started to be split Son and Mattila 2004. Since theCPS does not suffer from this limitation, it offers more flexible

Fig. 7. Case 1

Fig. 8. Case 2

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resource leveling solutions, particularly for project updates andcorrective action plans. The result of this case illustrates the flex-ibility of the CPS in resource leveling and schedule analysis.

Case 3In Case 3, a small project involving the use of multiple calendarsis used. The example was used by Kim and de la Garza 2005 toprove that Primavera P3 software generates incorrect dates for theactivities when multiple calendars are used. As shown in the toppart of Fig. 9, activities A and B have a FS relationship with 1lag. Accordingly, forward pass calculation in CPM determinesthat the early start time EST of B is day 8. However, since theFS relationship with 1 lag means that the successor can startwhenever the remaining duration of the predecessor is one day,other options exist for the EST of activity B. Because of thedifference in the calendars, the bottom part of Fig. 9 shows thatactivity B can start either on day 4 or day 5. As such, day 4 is theEST of activity B, not day 8, which is not detectable by CPMcalculations and existing software systems.

Both Primavera P3 and Microsoft Project software specify day8 as the EST of activity B Fig. 10a, without taking advantageof the other possible EST times for activity B. Kim and de laGarza 2005, therefore, recommended not to utilize negative lagswith multiple calendars. It is noted that because existing softwareshows only one calendar on the bar chart, Fig. 10a wronglyshows that activity A extends over the nonworking days of calen-dar 2 Th. and Fr., which does not give a correct indication of theactivity duration.

Using the CPS, Case 3 was simulated using separate activitieson Microsoft Project, as shown in Fig. 10b. The FS 1 rela-tionship was converted into a simple FS between the end of seg-ment 3 of activity A and the start of segment 1 of activity B,

without lag relationship is highlighted on Fig. 10b. The figureclearly shows that no work will be performed for activity A on itsnonworking days. More important, it illustrates how the CPS iscapable of taking advantage of possible earlier times to finish theproject in 8 days, rather than 10.

Step-by-Step CPS

As shown in the above examples, the steps followed to employthe CPS approach using existing scheduling software are as fol-lows:1. Generate a regular CPM baseline schedule using the soft-

ware;2. Generate a CPS schedule from the CPM schedule by manu-

ally converting each activity into time segments and the re-lationships into FS relationships, as explained before;

3. Use the CPS schedule to determine the float for each activitysegment;

4. During actual progress, additional time segments are insertedto represent daily events progress, delays, etc. and theschedule is adjusted accordingly;

5. Repeat step 4 for the duration of the project until a full as-built schedule is generated.

It is important to note that the above steps represent a manualprocess that can possibly be used for small exercises for valida-tion purposes. Part of the ongoing research, therefore, is to fullyautomate the process as an add-on module to existing schedulingsoftware.

Discussion of the CPS Approach

Based on the results of the three cases presented earlier, CPSproved to be a good basis for improving the scheduling process.One of the benefits of using the CPS in background computationsof a schedule is the fact that it offers little changes to the mannerby which scheduling basics are taught. Segmenting activity dura-tion simply into days also adds the necessary level of detail thatis consistent with the findings of other research. Al-Gahtani2009, for example, presented an approach to divide the float andallocate it on a daily basis among the project parties according tothe levels of risk they assume. A day-to-day system was thenproposed to monitor the dynamics of float management. Severalother studies also discussed the granularity of construction activi-

Fig. 9. Case 3: possible early-start times under multiple calendars

Fig. 10. Simulating CPS for Case 3

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ties from the process or product perspectives e.g., Song and Chua2007; Chua and Shen 2005; Morad and Beliveau 1994; Winstan-ley et al. 1993; Hendrickson et al. 1987. These important effortsmainly focus on planning and can generate generic baselineschedules by deriving precedence relationships from productmodels. Moreover, research on the concepts of Lean Constructionsuch as short-interval planning that goes into crew-level detailsproved to tremendously improve productivity Kim 2002. Ac-cording to Kim, even limited implementation of short-intervalplanning which the CPS is capable of doing can be far moreeffective than typical planning efforts employed in the construc-tion industry. It is important to note that the proposed CPS tech-nique is a detailed scheduling technique that is most advantageousfor documenting and analyzing as-built schedules. The CPS tech-nique, therefore, can work nicely in collaboration with these ef-forts to provide the lower level of granularity that allows efficientproject control through better recording of site events, resourcemanagement, delay analysis, and corrective actions. As such, thedifference between CPM and CPS is similar to the differencebetween design drawings and the more detailed shop drawingsthat are used during construction.

There are several issues related to the CPS that further re-search can improve upon. For example, segmenting activity du-rations may seem to add more computational burden to thecritical path analysis. However, this may not be true since all therelationships have been simplified in the CPS. One other issue iswhether the activities separate time segments cause unnecessarysplits in the activities, particularly when trying to resolve resourceover allocations. It is important, therefore, in resource allocationand schedule optimization to add a constraint to minimize activitydisruption. Other possibilities for further research include revisingcurrent techniques for resource leveling, constrained-resource al-location, time-cost trade-off, cash flow analysis, and schedule op-timization to work smoothly with the CPS. Future contributionsby other researchers could also generalize the CPS formulationfor repetitive scheduling and investigate the use of varying timesegments in different activities. For example, activities with cy-clic nature such as earth-moving operations can be modeled usinga time segment that is a second or a minute, and as such, asimulation model of this operation can be integrated within theCPS network.

Conclusions

This paper presented a CPS approach for microlevel critical pathanalysis, particularly suited to progress documentation, as-builtschedule analysis, and corrective action optimization. CPS con-siders activity duration as a chain of separate time segments thatcorrespond to a desired level of analysis detail. It facilitates ac-curate schedule analysis by simplifying complex relationships andavoiding the use of leads and lags. It also provides a flexiblerepresentation of project variables that offer a wide range of pos-sible solutions to project constraints. A prototype of the CPS iscurrently being developed and will be followed by extensive test-ing on real life projects. CPS is a good basis that can be extendedby other researchers and practitioners and has the potential torevolutionize the way schedules are generated and managed. Ul-timately, CPS is expected to assist project managers in preparingreliable schedules that better reflect reality and offer better sup-port for planning, corrective action, and schedule analysis deci-sions.

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