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Formalizing A Path-Float-Based Approach to Determine and Interpret Total

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DOI: 10.1080/15623599.2016.1207366

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 International Journal of Construction Management

ISSN: 1562-3599 (Print) 2331-2327 (Online) Journal homepage: http://www.tandfonline.com/loi/tjcm20

Formalizing a path-float-based approach to

determine and interpret total float in project
scheduling analysis

Ming Lu, Jing Liu & Wenying Ji

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approach to determine and interpret total float in project scheduling analysis, International
Journal of Construction Management, DOI: 10.1080/15623599.2016.1207366

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 International Journal of Construction Management, 2016
http://dx.doi.org/10.1080/15623599.2016.1207366

Formalizing a path-float-based approach to determine and interpret total float in project

scheduling analysis
Ming Lu*, Jing Liu and Wenying Ji

Construction Engineering and Management, Department of Civil and Environmental Engineering, University of Alberta, Edmonton, AB,
Canada T6G2R3

The classic critical path method (CPM) determines total float (TF) for each individual activity by performing forward pass
and backward pass analyses. A comprehensive literature review has shown that TF is the most crucial attribute of a
scheduled activity, and plays a fundamental part in advanced scheduling research. This research proposes a simplified
version of CPM, called path-float-based critical path method (PFCPM), which determines TF based on identification of
path float (PF) instead of entailing a backward pass analysis in the classic CPM. Analytical proof is provided and step-by-
step application procedures are generalized. Then, PFCPM application examples are given based on two demonstration
projects represented in activity-on-node (AON) and precedence diagram method (PDM) networking formats, respectively.
Downloaded by [University of Alberta] at 20:37 25 July 2016

Results are compared with the classic CPM for cross-validation. The newly proposed PFCPM enhances CPM-based
scheduling through circumventing the backward pass analysis in deriving TF based on PF; helping researchers and
practitioners interpret the TF ownership issue and account for changes on TF as a result of activity delay by relating TF
with PF; and laying a theoretical foundation for further research into advanced construction planning methods such as
resource loading, time
cost tradeoff and risk analysis.
Keywords: project management; scheduling; critical path method; construction planning

Problem statement
Critical path method (CPM) has been widely implemented in both practice and research for construction project schedul-
ing ever since it was formalized by Kelley and Walker (1959). Over the past five decades, although the classic CPM has
its own limitations in project scheduling subject to uncertainty, finite resources and time
cost tradeoff, it does provide a
solid methodological foundation for developing advanced scheduling methods (e.g. project/programme evaluation and
review techniques (PERT), resource-constrained CPM, time
cost tradeoff, etc.).
Total float (TF), as the most important attribute of the classic CPM, is defined as the maximum time that one activity
can be delayed without increasing the total project duration. The path linking the activities with zero TF establishes the
critical path for a given project. In practice, TF is recognized as a ‘treasured asset’ by both owners and contractors (de la
Garza et al. 2007). Contractors treat TF as an effective metric to prioritize the execution of activities in resource levelling,
resource allocation and time
cost tradeoff analyses, while owners mainly use TF to process claims (Ammar 2003; Bayr-
aktar et al. 2011). In the majority of CPM-related scheduling research, forward pass and backward pass analyses need to
be performed in order to determine earliest start time (ES), earliest finish time (EF), latest start time (LS) and latest finish
time (LF) for each activity prior to determining TF. In particular, the backward pass analysis is necessary to decide LS and
LF in relation to keeping the total project duration.
Researchers have endeavoured to develop new techniques and algorithms aimed to elucidate on TF while facilitating
advanced CPM scheduling analyses. Lu and Lam (2008) proposed a new iterative method to accurately determine the TF
of each activity subject to resource calendar constraints based on the definition of TF. This approach requires forward pass
analysis only in order to arrive at the total project duration. However, it needs an iterative simulation process on each activ-
ity so as to determine TF. Zhong and Zhang (2003) proposed a new method for calculating path float (PF) in order to cope
with uncertainties on construction projects. Moreover, researchers have noticed that TF should be shared by all the activi-
ties on the same path instead of being owned by an individual activity (Callahan et al. 1992; Ammar 2003; Hegazy &
Menesi 2010). TF consumption on a noncritical activity would decrease the amount of float time of other activities on the
same path. Many utilized the TF of a path (identical to PF as defined in this paper) to address the TF ownership issue in
€
the litigation of delay claims (Okmen €
& Oztaş 2008; Al-Gahtani 2009). For instance, Al-Gahtani (2009) proposed a total
risk approach to allocate TF and clarify its ownership among different stakeholders. However, no research has been done
in regard to analytically connecting TF with PF and directly deriving TF from PF in a simple and valid way.

*Corresponding author. Email: mlu6@ualberta.ca

Ó 2016 Informa UK Limited, trading as Taylor & Francis Group

 2 M. Lu et al.

This paper aims to develop a new algorithm, which can be used to determine the TF of each individual activity based
on PF through an elegant procedure which is executed at the end of a straightforward forward pass analysis to identify
paths and calculate PF for each on a project network. Taking TF as a path attribute can shed light on the implication of TF
ownership and how a particular activity delay or constrained milestone causes TF to dynamically change on a project net-
work. Meanwhile, the proposed algorithm can circumvent the backward pass analysis as needed to determine LS and LF in
the classic CPM. The proposed new method is analytically proved and ready for further applications such as resource load-
ing or time
cost tradeoff based on CPM scheduling.
In the remaining sections of this paper, previous research efforts related to CPM, PERT and resource-constrained
scheduling are reviewed first to illuminate the importance of TF in CPM scheduling research. Then, the concepts of path
and path float are formally defined. Next, the path-float-based critical path method (PFCPM) is proposed to determine TF
based on PF. PFCPM is then analytically proven by relating it to the classic CPM algorithm. A detailed manual calculation
procedure is presented so as to facilitate application. Following the procedure, a simple activity-on-node (AON) network
example and a precedence diagram method (PDM) case containing non-finish-to-start relationships are used to illustrate
the application of PFCPM.

CPM-based scheduling research

The three principal research objectives in project scheduling are (1) minimizing project total duration, (2) minimizing vari-
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ation in resource use profile, and (3) minimizing project total cost (Nudtasomboon & Randhawa 1997). Classic CPM plays
a pivotal role in the development of advanced CPM scheduling techniques, such as PERT, resource-constrained CPM and
time
cost tradeoff.
In the classic CPM, the assumption that activity duration remains deterministic may not be realistic in the real world.
As an extension to CPM, PERT incorporates uncertainties by using probabilistic activity duration estimates in order to
approximate the probability that a project can be completed by a given date (Lu & AbouRizk 2000; Halpin 2006). In
PERT, three times are estimated for each activity: the optimistic time, the most likely time and the pessimistic time. It is
noteworthy that PERT determines the critical path in the same way as classic CPM by entailing a backward pass to fix
activity TF and linking all activities with zero TF.
Similarly, the assumption of unlimited resources (e.g. labour, equipment and material) in classic CPM may not be justi-
fied in many real world circumstances since only a finite amount of resources are available or the cost of acquiring addi-
tional resources is high (Hegazy 2002). In such cases, resource constraints can significantly affect the schedule. This issue
has inspired resource-constrained CPM scheduling research. Heuristic methods based on activity prioritization rules

such as minimum slack time (MST), earlier LS, or minimum current float (CF)
are commonly utilized in tackling
resource-constrained CPM scheduling (Harris 1990; Lu & Li 2003; Christodoulou et al. 2009; Hegazy & Menesi 2011). It
is worth mentioning that TF, ES and LS as calculated by classic CPM analyses usually serve as part of the criteria to define
particular activity-priority rules (Lu & Li 2003).
Another limitation of the original CPM lies in the difficulty of confining a schedule to a specified duration in a cost-
effective way. In general, there is a tradeoff between the project duration and the direct cost of completing a project: the
less expensive the resources, the lower the direct cost, and the longer it takes to complete a project (Hegazy 2002).
Time
cost tradeoff analysis based on the results of CPM analysis overcomes this drawback and can be used to optimize
the project’s schedule under time and cost constraints.
To better understand the determination of TF in previous research, the authors further looked into CPM-related schedul-
ing research from 2000 to 2015 by reviewing major journals in construction engineering and project management, they are:
(1) Journal of Construction Engineering and Management (JCEM), ASCE; (2) Journal of Management in Engineering
(JME), ASCE; (3) Construction Management and Economics (CME); and (4) International Journal of Project Management
(IJPM). The procedures of searching for CPM-related papers are summarized as follows. The titles and keywords of the
papers were first scanned using the following keywords: ‘CPM’, ‘Critical Path Method’, ‘Project Evaluation and Review
Technique’, ‘PERT’, ‘resource-constrained scheduling’, ‘resource levelling’ and ‘resource allocation’. Their abstracts were
then checked to determine whether they are genuinely CPM-related research. In Tables 1 and 2, the relevant papers are
categorized by (1) CPM/PERT and (2) resource-constrained scheduling, respectively. Note time
cost tradeoff-related papers
are excluded, in that their main focuses are on path length calculation rather than determining the TF of individual activities.
In 22 of the 36 papers listed in Table 1 and Table 2, the calculation of TF of each individual activity is required; among
the collection of 22 papers, 10 CPM/PERT-related papers and 10 resource-constrained scheduling-related papers are based
on application of the classic CPM to derive TF (the forward pass and backward pass calculation algorithms). The literature
review has provided us the motivation to develop an elegant, more efficient, yet valid alternative for the determination
of TF.
 International Journal of Construction Management 3

Table 1. CPM/PERT-related papers.

TF calculation TF calculation
Authors Journal Title involved method

Lu and AbouRizk (2000) JCEM Simplified CPM/PERT simulation model No

Pontrandolfo (2000) IJPM Project duration in stochastic networks by No

the PERT-path technique
Kuchta (2001) IJPM Use of fuzzy numbers in project risk Yes FP-BP
(criticality) assessment
Lu (2002) JCEM Enhancing project evaluation and review No

technique simulation through artificial
neural network-based input modeling
Zhang et al. (2002) CME Simulation-based methodology for project Yes FP-BP
scheduling
Zhong and Zhang (2003) JCEM New method for calculating path float in No

program evaluation and review
technique (PERT)
Zhang (2005) JCEM Financial viability analysis and capital No

structure optimization in privatized
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public infrastructure projects

Lee and Arditi (2006) JCEM Automated statistical analysis in stochastic Yes FP-BP
project scheduling simulation
de la Garza et al. (2007) JCEM Preallocation of total float in the Yes FP-BP
application of a critical path method
based construction contract
€
Okmen €
and Oztaş (2008) JCEM Construction project network evaluation Yes FP-BP
with correlated schedule risk analysis
model
Zhang (2009) JCEM Win
win concession period determination Yes FP-BP
methodology
Lu and Lam (2009) JCEM Transform schemes applied on non-finish- Yes FP-BP
to-start logical relationships in project
network diagrams
Al-Gahtani (2009) JCEM Float allocation using the total risk Yes FP-BP
approach
Zammori et al. (2009) IJPM A fuzzy multi-criteria approach for critical Yes FP-BP
path definition
Hegazy and Menesi (2010) JCEM Critical path segments scheduling Yes Gant chart (visual based)
technique
Trietsch and Baker (2012) IJPM PERT 21: Fitting PERT/CPM for use in the No

21st century
Shi and Blomquist (2012) IJPM A new approach for project scheduling No

using fuzzy dependency structure matrix
San Cristobal (2012) JME Critical path definition using multicriteria No

decision making: PROMETHEE
method
Wang et al. (2014) JCEM Incorporation of alternatives and Yes FP-BP
importance levels in scheduling
complex construction programs
Acebes et al. (2014) IJPM A new approach for project control under No

uncertainty: going back to the basics
Colin and Vanhoucke (2015) IJPM Developing a framework for statistical No

process control approaches in project
management

Note: FP-BP: Forward pass and backward pass analysis.

 4 M. Lu et al.

Table 2. Resource-constrained scheduling related papers.

Authors Journal Title TF calculation TF calculation

involved method

Shi and Deng (2000) IJPM Object-oriented resource-based planning Yes FP-BP
method (ORPM) for construction
Hiyassat (2000) JCEM Modification of minimum moment No

approach in resource leveling
Abeyasinghe et al. (2001) IJPM An efficient method for scheduling Yes FP-BP
construction projects with resource
constraints
Hiyassat (2001) JCEM Applying modified minimum moment No

method to multiple resource leveling
Wei et al. (2002) IJPM Resource-constrained project management Yes FP-BP
using enhanced theory of constraint
Leu and Hung (2002) CME A genetic algorithm-based optimal Yes FP-BP
resource- constrained scheduling
simulation model
Ammar and Mohieldin (2002) CME Resource constrained project scheduling No

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using simulation
Kim and de la Garza (2003) JCEM Phantom float Yes FP-BP
Lu and Li (2003) JCEM Resource-activity critical-path method for Yes FP-BP
construction planning
Bonnal et al. (2005) CME A linear
discrete scheduling model for the Yes FP-BP
resource
constrained project
scheduling problem
Lu and Lam (2008) JCEM Critical path scheduling under resource Yes Iterative forward pass analysis
calendar constraints
El-Rayes and Jun (2009) JCEM Optimizing resource leveling in Yes FP-BP
construction projects
Christodoulou et al. (2009) JCEM Minimum moment method for resource Yes FP-BP
leveling using entropy maximization
Montoya-Torres et al. (2010) IJPM Project scheduling with limited resources No

using a genetic algorithm
Kim (2013) CME Genetic algorithm stopping criteria for Yes FP-BP
optimization of construction resource
scheduling problems

Note: FP-BP: Forward pass and backward pass analysis.

Path-float based critical path method (PFCPM)

Path and path float
Compared to an activity-on-arrow (AOA) network, an AON network does not require the use of any dummy activities to
represent precedence relationships for executing a project. In general, an AON network simply consists of finish-to-start
(FS) logical relationships between activities. Regarded as a more complex form of AON, a PDM project network can have
four types of precedence relationships with lag times, namely: finish-to-start, start-to-start (SS), finish-to-finish (FF) and
start-to-finish (SF). Additionally, the AON network is suitable for the object-oriented programming scheme which is
implemented in most project management software (Lu & Lam 2009). As AON is more straightforward to communicate
and interpret than PDM, generic transform schemes for converting a PDM network into an equivalent AON network were
developed (Lu & Lam 2009). As such, the concepts of path and path float are formally defined based on the AON network
in this paper.
In the classic CPM, a path in a simple AON network diagram is a continuous chain of activities linked with FS relation-
ships from project start to project end. The longest continuous path defines the shortest time duration required to complete
the whole project (i.e. total project duration), which is identified as the critical path. As shown in Figure 1 (a), path Pi is
assumed as the critical path linking dummy start activity (ST), Activity Aik (k D 1,…, n), and dummy finish activity (FN).
Similarly, Path Pj is the noncritical path linking ST, Activity Aik (k D 1,…, m) and FN.
 International Journal of Construction Management 5
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Figure 1. Illustration of path and path float.

PF for each path can be defined as the difference in length between a path and the critical path. PF is nonnegative: a
zero PF denotes a critical path; otherwise, a positive PF denotes a noncritical path. As shown in Figure 1 (b), the path
length of a critical path Pi is Tcp, equal to the summation of time duration Dik on Activity Aik (k D 1,…, n). The path length
of a noncritical path Pj is Tj, which is the summation of time duration Djk on Activity Ajk (k D 1,…, m). A zero PFi indi-
cates that Pi is the critical path. For a noncritical path, PFj for Pj is Tcp
Tj. The value of PF is correlated with the criticality
ranking of a path: the longer the path is, the higher criticality and the less PF it has. Meanwhile, PF can be related to TF of
individual activities residing on the same path. The connection between TF and PF can be mathematically expressed as in
Equation (1).

PFj if the activity only belongs to path j
TF D (1)
min1jm PFj if the activity belongs to m paths

As per Equation (1), the relationship between TF and PF can be interpreted as: if one activity belongs to one path only,
then its TF is equal to the PF of the path; if one activity resides on more than one path, then the TF is equal to the minimum
PF value factoring in all the relevant paths. The analytical proof is presented in the next section.

Analytical proof of PFCPM

Activity i belongs to Path j only

When an activity only belongs to one path in the project network, i.e. Activity i belongs to Path j only define its duration as
Di, earliest start time as ESi, earliest finish time as EFi, latest start time as LSi, latest finish time as LFi, and total float as
TFi. The activity immediately preceding Activity i is denoted as Activity i¡1. According to the classic CPM, EFi-1 is equal
to ESi due to the finish
start precedence constraint between the two activities. Conversely, the activity immediately suc-
ceeding Activity i is denoted as Activity iC1; LSiC1 is equal to LFi due to the finish
start precedence constraint between
the two activities. And for Activity i, EFi is equal to the summation of ESi and Di; LSiC1 is equal to the difference between
LFiC1 and DiC1. The relationship between EFi and EFiC1 is shown in Equation (2). The relationship between LFi and
 6 M. Lu et al.

LFiC1 is shown in Equation (3).

EFi D ESi C Di D EFi
1 C Di (2)

LFi D LSi C 1 D LFi C 1
Di C 1 (3)

According to the definition of TF, TFi is equal to the difference between LFi and EFi, as given in Equation (4).

TFi D LFi
EFi (4)

Substituting Equation (2) and Equation (3) in Equation (4), Equation (5) is derived,

TFi D LFi C 1
Di C 1
ðEFi
1 C Di Þ

(5)
D LFi C 1
EFi
1
ðDi C Di C 1 Þ

By continuing the recursive process and summarizing duration of relevant activities, TF of Activity i can be expressed
as a function of (1) LFFN (the latest finish time of the ‘project finish activity’, which is equal to the critical path duration
Tcp), (2) EFST (the earliest finish time of the ‘project start activity’, which is equal to zero), and (3) the summation of time
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duration of relevant activities on this path (which is Tj), as shown in Equation (6) .

TFi D LFi C 2
Di C 2
EFi
2
Di
1
ðDi C Di C 1 Þ

D LFi C 2
EFi
2
ðDi
1 C Di C Di C 1 C Di C 2 Þ
D 
D LFFN
EFST
ðD1 C    C Di C    C Dn Þ
Xn (6)
D LFFN
EFST
Di
iD1

D Tcp
0
Tj
D Tcp
Tj

As per path float definition, the path float of a noncritical path equals to the difference between the critical path dura-
tion Tcp and the duration of the current path Tj. Therefore, for path j,

PFj D Tcp
Tj (7)

Equations (6) and (7) lead to Equation (8), which means that when an activity belongs to one path only, its TF equals to
PF of the path.

TFi D PFj (8)

Activity i belongs to multiple paths

For one activity residing on more than one path, assume Activity i is one of these activities and belongs to m paths (m > 1).
The total duration of each path can be expressed as:

Tj D Tcp
PFj ; j D 1;    ; m (9)

By TF definition, TFi is the maximum time that Activity i can be delayed without increasing the total project duration.
Therefore, if Activity i is delayed by TFi, the path with the longest duration among the m paths becomes a critical path,
that is,
 
Tcp D max Tj C TFi (10)
 International Journal of Construction Management 7

By PF definition, Equation (9) is given, namely, Tj D Tcp
PFj . Expanding Tj in Equation (10) results in Equation
(11) and Equation (12), which indicate that when an activity belongs to multiple paths, its TF equals to the minimum PF
value, factoring in all the relevant paths.
 
Tcp D max Tcp
PFj C TFi
  (11)
D Tcp
min PFj C TFi
TFi D minðPFj Þ (12)

From the above analytical proof, the TF of an individual activity can be determined based on the PF relevant to the cur-
rent activity. Thus, PFCPM provides an alternative method for more efficient CPM analysis, since it requires a forward
pass analysis only to identify the path and arrive at PF. In addition, PFCPM is conducive to the interpretation and the utili-
zation of TF, which should be taken as an attribute of a path, analogous to PF. Thus, any consumption of TF on a particular
activity will reduce PF, thereby decreasing TF values on other activities along the path.

Step-by-step PFCPM
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In this section, the procedures of manually employing PFCPM to calculate the TF of each activity are formalized, includ-
ing five steps as follows:

(1) Represent the given project network in AON and determine the duration of individual activities.
(2) Identify all the paths in the AON network.
(3) Calculate the length of each path by adding up the duration of all the activities making up the path. The path/paths
with the longest duration is/are critical path(s).
(4) Compute the PF of each path by subtracting the path duration from the critical path duration (i.e. total project
duration).
(5) Determine the TF for each activity. If the activity belongs to one path only, then its TF equals the PF of the path; if
the activity resides on more than one path, then its TF is the minimum PF value of all the relevant paths.

By applying the above procedures, PFCPM can be easily performed by hand calculations.

AON case
In this section, the PFCPM application is illustrated by an example taken from the textbook written by Hegazy (2002). The
AON network is shown in Figure 2. There are three paths in the example AON network (A-B-E, A-C-E and A-D-E).
Activity duration (in days) is also shown in Figure 2. The classic CPM and the proposed PFCPM are applied respectively
in order to determine TF for Activities A, B, C, D and E. For the classic CPM, forward pass and backward pass calcula-
tions are performed to fix ES, EF, LS and LF of each activity. The calculation results are given in Figure 3. And then TF is
calculated by subtracting ES from LS or subtracting EF from LF, as shown in Table 4.
For PFCPM, following the formalized procedure, calculation steps are listed as follows:

(1) AON network and durations of all activities are shown in Figure 2.
(2) Identify three paths in the AON network. They are A-B-E, A-C-E and A-D-E.
(3) Calculate the path length and identify the critical path, shown in Table 3. Path A-D-E is the critical path.

Figure 2. AON case project network.

 8 M. Lu et al.

Figure 3. Forward and backward pass calculation of AON case.

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(4) Compute the PF of each path by subtracting the path duration from the critical path duration, shown in Table 3.
(5) Determine the TF for each activity by following Equation (1), shown in Table 4.

The results generated by the two methods are contrasted in Table 4. The newly proposed PFCPM can arrive at the same
results as the classic CPM (forward pass and backward pass calculations). This example clearly demonstrates the capabil-
ity of PFCPM to streamline TF determination and derive valid TF for each activity.

PDM case
PDM is more compact and sophisticated to represent precedence relationships for executing construction activities (Lu &
Lam 2009). The proposed PF-based TF calculation equations are also applicable for PDM with start
start, finish
finish,
start
finish relationships, and lag time. An alternative approach to cope with non-finish-to-start relationships with lag
times is to directly represent special events and relationships in PDM as special activities linked with finish
start

Table 3. Path float analysis on AON case.

Path No. Path Path length (day) Path float (day)

1 A-B-E 11 3
2 A-C-E 12 2
3 A-D-E 14 0

Table 4. Results comparison between PFCPM and classic CPM for AON case.

Forward and backward pass calculation results PFCPM results

Activity EST (day) LST (day) TF (day) TF (day)

A 0 0 0 Min(3,2,0) D 0
B 3 6 3 3
C 3 5 2 2
D 3 3 0 0
E 9 9 0 Min(3,2,0) D 0
 International Journal of Construction Management 9

Figure 4. PDM case project network.

relationships. As such, the start end, the finish end and the relationship with lag can be treated as special ‘activities’ on a
path; while the duration can be equal to zero for an activity denoting a start or finish event or set as the lag time on an activ-
ity denoting a relationship. Then the same calculation equations can be used to fix TF based on PF for all the activities in
the network. In this paper, the network transformation approach proposed by Lu and Lam (2009) is advocated. For this
case, a typical PDM project network containing FS, SS and FF relationships with positive lag time is taken from Lu and
Lam (2009) in order to illustrate the proposed PFCPM application. Figure 4 shows the PDM network along with activity/
Downloaded by [University of Alberta] at 20:37 25 July 2016

lag time duration. The generic transform schemes proposed in Lu and Lam (2009) are first adopted to convert the network
with start
start, finish
finish and start
finish relationships and lag time into an equivalent AON network that only
involves finish
start relationships. Then, the equations provided for the general AON case are applied to determine PT
and hence calculate TF for each activity.

Base case scenario

As per precedence relationships and activity duration (in days) shown in Figure 4, the following steps are taken to perform
PFCPM analysis.

(1) According to the transform scheme (Lu & Lam 2009), the PDM network is first converted into an equivalent AON
network shown in Figure 5.
(2) Five paths in the AON network are identified, shown in Table 5.
(3) Calculate the path length and determine the critical path. The results are listed in Table 5.
(4) PF of each path is computed, as shown in Table 5.
(5) TF of each activity is calculated accordingly and listed in Table 6.

Table 6 also gives the TF results from the classic CPM. The results of this case also demonstrate that PFCPM can arrive
at the same results as the classic CPM analysis on a PDM network, but involves forward pass calculation only.

Activity-delay scenario
To better elaborate the impact on TF due to activity delay, a two-day delay is assumed to occur on Activity B2 in Figure 4.
Updated PF analysis results can be found in Table 7. PF for Paths 1, 2 and 3 are each reduced by two days. Comparison of
activity TF is given in Table 8. As a result of B2 being delayed by two days, TF of Activity B2 itself is used up and B2
turns critical. Meanwhile, the two-day delay on Activity B2 also takes away two-day TF from Activity D1, because Activ-
ity D1 and Activity B2 fall on the same path.

Figure 5. Equivalent AON network for PDM case.

 10 M. Lu et al.

Table 5. Path float analysis on PDM case: base case.

Path No. Path Path length (day) Path float (day)

1 ST-A-B1-B2-D1-D2-FN 8 4
2 ST-A-B1-B2-C2-E1-D2-FN 8 4
3 ST-A-B1-B2-C2-E1-E2-FN 10 2
4 ST-A-B1-C1-C2-E1-D2-FN 10 2
5 ST-A-B1-C1-C2-E1-E2-FN 12 0

Table 6. Results comparison between PFCPM and classic CPM for PDM case: base case.

Activity PFCPM results Classic CPM results

TF (day) TF (day)

A Min(4, 4, 2, 2, 0) D 0 0
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B1 Min(4, 4, 2, 2, 0) D 0 0
B2 Min(4, 4, 2) D 2 2
C1 Min(2, 0) D 0 0
C2 Min(4, 2, 2, 0) D 0 0
D1 4 4
D2 Min(4, 4, 2) D 2 2
E1 Min(4, 2, 2, 0) D 0 0
E2 Min(2, 0) D 0 0

Table 7. Path float analysis on PDM case: activity-delay scenario.

Path No. Path Path length (day) Path float (day)

1 ST-A-B1-B2-D1-D2-FN 10 2
2 ST-A-B1-B2-C2-E1-D2-FN 10 2
3 ST-A-B1-B2-C2-E1-E2-FN 12 0
4 ST-A-B1-C1-C2-E1-D2-FN 10 2
5 ST-A-B1-C1-C2-E1-E2-FN 12 0

Table 8. Results Comparison between activity-delay scenario and base scenario for PDM case.

Activity Results (Two-day delay for activity B2) Results (Base scenario)
TF (day) TF (day)

A Min(2, 2, 0, 2, 0) D 0 0
B1 Min(2, 2, 0, 2, 0) D 0 0
B2 Min(2, 2, 0) D 0 2
C1 Min(2, 0) D 0 0
C2 Min(2, 0, 2, 0) D 0 0
D1 2 4
D2 Min(2, 2, 2) D 2 2
E1 Min(2, 0, 2, 0) D 0 0
E2 Min(0, 0) D 0 0
 International Journal of Construction Management 11

Table 9. Path float analysis of PDM case: constrained project milestone (15 days) scenario.

Path No. Path Path length (day) Permitted longest duration (day) Path float (day)

1 ST-A-B1-B2-D1-D2-FN 8 15 7
2 ST-A-B1-B2-C2-E1-D2-FN 8 15 7
3 ST-A-B1-B2-C2-E1-E2-FN 10 15 5
4 ST-A-B1-C1-C2-E1-D2-FN 10 15 5
5 ST-A-B1-C1-C2-E1-E2-FN 12 15 3

Constrained milestone scenario

For the above two scenarios, it is noteworthy that no milestone is imposed on the project schedule. According to Project
Management Institute’s specification (2011), a milestone represents
the start or completion of a portion or deliverable of the project and may also be associated with external constraints, such as the
Downloaded by [University of Alberta] at 20:37 25 July 2016

delivery of specific required permissions or equipment. Each project should have a start milestone and a finish milestone.

As the schedule model is created, a list of milestones can be imposed as additional constraints originating from the
client, team members or other stakeholders on the project. Simply put, the milestone is an event to mark a specific point in
time along a project timeline. It can be a constraint on the start or finish of a specific activity or the whole project. The
proposed approach is focused on the determination of PF and TF, both of which are relevant to the total project completion
time. In this paper, the milestone of project completion is considered to illustrate the effect of inserting milestones into the
analytical results.
In this scenario, the project is assumed to be constrained by two different milestones on project completion: (1) the
total project duration is set to 15 days, (2) the total project duration is set to 11 days.
The algorithm being applied remains the same except for substituting the milestone time (‘permitted longest duration’)
for the length of the longest path prior to calculating PF. PF and TF in respective milestone settings are derived and shown
in Tables 9, 10 and 11.
For the scenario with the 11-day project completion milestone, several activities end up with negative TF (
1), which
means the milestone is impractical and the scheduled project completion time will be one day later than the project com-
pletion milestone (11 days).
The results of case studies show that TF should be interpreted as a path attribute instead of an activity attribute. The PF
definition makes it clear that the float time is a path attribute. Therefore, the concept of PF can be instrumental in clarifying
the issue of float time ownership. Meanwhile, it is straightforward to dynamically update the TF of each activity when any
activity on one path consumes certain float time. This potentially would enable project managers to better allocate the float
time to different parties in the project in response to various constraints such as key activity or project milestones and par-
ticular activity delays. This also provides the solid basis to guide more sophisticated CPM-based analyses that generally
rely on the values of TF for making decisions in intermediate steps (such as risk analysis, resource allocation and levelling,
and time
cost tradeoff analysis).

Table 10. Path float analysis of PDM case: constrained project milestone (11 days) scenario.

Path No. Path Path length (day) Permitted longest duration (day) Path float (day)

1 ST-A-B1-B2-D1-D2-FN 8 11 3
2 ST-A-B1-B2-C2-E1-D2-FN 8 11 3
3 ST-A-B1-B2-C2-E1-E2-FN 10 11 1
4 ST-A-B1-C1-C2-E1-D2-FN 10 11 1
5 ST-A-B1-C1-C2-E1-E2-FN 12 11 -1
 12 M. Lu et al.

Table 11. TF results comparison among different milestone constraints scenarios and base case scenario.

TF (days)

Activity Activity milestone scenario Project milestone (15 days) scenario Project milestone (11 days) scenario Base case

A Min(4, 3, 1) D 1 Min(7, 7, 5, 5, 3) D 3 Min(3, 3, 1, 1,
1) D
1 0

B1 Min(4, 3, 1) D 1 Min(7, 7, 5, 5, 3) D 3 Min(3, 3, 1, 1,
1) D
1 0
B2 Min(4, 3) D 3 Min(7, 7, 5) D 5 Min(3, 3, 1) D 1 2
C1 1 Min(5, 3) D 3 Min(1,
1) D
1 0
C2 Min(3, 1) D 1 Min(7, 5, 5, 3) D 3 Min(3, 1, 1,
1) D
1 0
D1 4 7 3 4
D2 Min(4, 2) D 2 Min(7, 7, 5) D 5 Min(3, 3, 1) D 1 2
E1 Min(2, 0) D 0 Min(7, 5, 5, 3) D 3 Min(3, 1, 1,
1) D
1 0
E2 0 Min(5, 3) D 3 Min(1,
1) D
1 0

Conclusions
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CPM provides the fundamental knowledge and the analytical engine to construction planning and project scheduling.
Advanced scheduling techniques and methods such as PERT, resource-constrained scheduling and time
cost tradeoff have
been developed based on the classic CPM. Meanwhile, simulation based on random sampling concepts and computer appli-
cations adds to project scheduling and facilitates real-world decision-making processes. On the other hand, CPM is crucial to
contract administration, claim and dispute resolution. Thus, an elegant CPM algorithm along with an unambiguous explana-
tion of TF resulting from CPM analysis is vitally important to advance academic research and improve industry practice.
In an attempt to clarify TF interpretation and streamline TF determination, this paper proposes a new version of CPM,
called path-float-based critical path method (PFCPM) for path identification and PF determination so as to arrive at TF
based on PF. A comprehensive literature review has helped the authors gain insight of TF determination and interpretation
in CPM-related scheduling research. This research has made academic contributions in regard to formalizing PFCPM and
analytically proving PFCPM. To facilitate practical application, a manual calculation procedure is formalized in detailed
steps. PFCPM application examples are given based on small-scale projects represented in AON and PDM networking for-
mats and results are compared with the classic CPM for cross-validation purpose.
PFCPM is conducive to clarifying two critical issues revolving around CPM scheduling, they are: (1) interpreting TF as
a path attribute instead of an activity attribute, (2) accounting for TF ownership in a transparent, unambiguous fashion by
relating TF to PF.
As the schedule model is created, a list of milestones can be imposed as additional constraints originating from the client,
team members or other stakeholders on the project. Simply put, the milestone is an event to mark a specific point in time
along a project timeline. In this paper, the milestone of project completion is considered to illustrate the effect of inserting
milestones upon the analytical results. As for the milestones to be inserted in the middle of the project, their definitions are
different from the objective of defining TF or PF in the first place. For example, certain fixed date constraints are imposed as
such that by those dates particular activities must complete or start, instead of confining the start or finish on the total project.
The float time available in connection with those milestones and implications upon how the total project completion is
affected are out of the scope of the current work and can be addressed in future research. Other related future research may
include: (1) incorporating PFCPM into existing CPM-based scheduling methods, including CPM/PERT simulation for risk
analysis, resource allocation and levelling analyses; and time
cost tradeoff analyses; (2) automating PFCPM as an add-on
module to existing scheduling software to enhance the accuracy of CPM results when project schedules are revised or
updated when activity delay occurs or complex calendar constraints or resource availability constraints are imposed.

Disclosure statement
No potential conflict of interest was reported by the authors.

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