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Advances in Social Sciences Research Journal – Vol.8, No.1
Publication Date: January 25, 2021
DOI:10.14738/assrj.81.9568.
Copertari, L. F. (2021). Comprehensive Project Risk Management Methodology. Advances in Social Sciences Research Journal, 8(1) 94-
115.
94
Comprehensive Project Risk Management Methodology
Luis F. Copertari
Computer Engineering Program
Autonomous University of Zacatecas (UAZ). Zacatecas, México.
ABSTRACT
The objective of this paper is to introduce and discuss the basics of a
methodology called the Probabilistic Critical Path Method (PCPM) for
managing the previously identified risks (uncertainty) of the three
project management dimensions: time, cost and return (performance).
An interactive Graphic User Interface (GUI) has been designed for
visualizing the tradeoffs among these three dimensions as well as their
uncertainties on a flat computer screen. The user can choose to visualize
the probability of failure (exceeding some user given due date, budget
or not exceeding a given Minimally Attractive Rate of Return – MARR) or
the probability of success (not exceeding the due date and the budget
and exceeding the MARR). PCPM allows for comprehensive project risk
management and it constitutes a new integrative project risk
management framework. This paper shows that it is possible to
integrate all three project management dimensions (time, cost and
return) and show their known risks as well as determining the optimal
cost and the associated time and return for such optimal cost. Finally, it
is possible to interactively show all this multidimensional information
on a flat computer screen.
Keywords Risk management; project management; decision support
systems.
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or
not-for-profit sectors.
INTRODUCTION
One of the most important tasks in project management, particularly in technology and information- based organizations, is to successfully manage risk. Thus, project risk management becomes critical
for successful project management (Buganová & Šimíčková, 2019; Khameneh, Taheri, & Ershadi,
2016; Odimabo & Oduoza, 2018; Serpella, Ferrada, Howard, & Rubio, 2014; Szymański, 2017).
The methodology introduced in this paper, called the Probabilistic Critical Path Method (PCPM)
allows to manage the three project management dimensions: time, cost and performance (the latter
being measured by the Internal Rate of Return, or simply return, of the project) and meaningfully
link them to the identified risks of each dimension in the form of the probability of failure or,
alternatively, the probability of success. It integrates these dimensions by allowing to visually see
their tradeoffs as well as the probability of failure, that is, of exceeding the due date and the budget
and the probability of not exceeding the Minimally Attractive Rate of Return (MARR), or
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Copertari, L. F. (2021). Comprehensive Project Risk Management Methodology. Advances in Social Sciences Research Journal, 8(1) 94- 115.
URL: http://dx.doi.org/10.14738/assrj.81.9568
alternatively, to visualize the probability of succeeding by not exceeding the due date or the budget
and exceeding the MARR. A specially designed and interactive Graphic User Interface (GUI) for
successful project risk management decision making was created and even the optimal cost can be
calculated given some specific indirect project cost slope (m). This methodology allows for
comprehensive project risk management. This paper explains the basics of PCPM. Calculations have
been automated by creating a Decision Support System (DSS) called Schedule1, which can be
downloaded for free from www.copertari.net/schedule.
Managing Risk And Uncertainty
A great deal of project management involves good risk management. Risk management in a project
can be defined as the consequence of the existence of significative uncertainty concerning the level of
project achievement (Chapman & Ward, 1997). Tight time, cost or performance goals increase time,
cost and performance risks, respectively. A risky situation is often considered as the existence of
potentially high and unacceptable costs due to events considered more or less likely to happen. This
negative approach to risk leads to the idea that risk management essentially deals with reducing or
removing the possibility of under-achievement. Risk analysis is not a ‘throwing the dice’ situation,
but rather an area of study where a proactive, creative and intelligent prior planning approach must
be used, instead of entrenching in a defensive position (Adams, 2001; Dey, 2001; McManus, 2001;
Schimmoller, 2001; Walker, 2001).
Within this context, it is important to distinguish between risk and uncertainty. Risk is the
possibility or probability of failure, whereas uncertainty is the variability of the relevant outcomes
for a given risk or eventuality. Brealey and Myers (1991) define risk saying that more things can
happen (at present time) that will happen (in the future). Uncertainty, on the other hand, is the
degree in which an identified threat or risk (at present time, after previous analysis) will
(presumably, based on experience, historical data or assumptions) vary. Uncertainty is an identified
(and quantified) risk. Even then, the degree in which such identified risk will vary is unknown.
Uncertainty constitutes the ‘known’ unknowns because although a specific risk has been identified,
its exact impact is still unknown. Unidentified risks are ‘unknown’ unknowns because, generally
speaking, a risk is unquantified uncertainty about something not yet being considered possible as a
future eventuality. It will be assumed throughout this paper that risk identification has been
successfully and completely carried out and will focus attention on the risks due to the uncertainty
for the most relevant variables previously identified by decision makers. Risk sources are any
factors that may affect the project dimensions (time, cost and performance). Setting a tight time
target such as an optimistic due date increases time risk. Likewise, an irrationally low budget
increases cost risk and setting a minimum Internal Rate of Return (IRR) increases performance risk.
On the other hand, setting slack times, emergency budget allocations or a lower IRR reduces time,
cost and performance risks, respectively (Dawson, 1998; Farrell, 1996; Lefley, 1997; Tavares,
1998).
1 The heuristic specially designed for probabilistic crashing is proprietary software technology developed for Schedule.
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Systemic Relationships Between The Project Dimensions: Time, Cost And Return
(Performance)
Although the relations among the project management dimensions vary from time to time and from
project to project, a systemic approach can be used to elucidate the nature of the underlying
balances (Icmeli, 1996; Johnson & Schou, 1990; Sunde & Lichtenberg, 1995).
Figure 1a illustrates the systemic relationships between time and cost using influence diagrams. If
the project is delayed (it takes longer) will cost more, so that there is a positive correlation between
time and cost. But if in order to deliver the project on time, additional resources are used for critical
activities, maintaining resources to a minimum for non-critical activities (which is called crashing)
there is a negative correlation between cost and time (Winston, 1994). The existence of both a
positive and a negative correlation between time and cost implies the existence of a balance point
in which an optimal project completion can be achieved at a minimum cost. Figure 1b illustrates
how the time/cost balancing is additionally influenced by performance. Improving the quality of the
product requires investing more resources, which will increase cost and increase time if those
resources are limited. But if more resources are invested and it takes longer to complete the project,
it costs more, so that the Internal Rate of Return (IRR) of the project measuring its profitability is
reduced. As a consequence, there must also be an optimal balance between time/cost achieving an
optimal performance as measured according to the project’s IRR.
Figure 1. Balances among time, cost and performance.
Project Scope And Time, Cost And Performance Risks
The first and most important task in the management of a project in particular is determining its
scope, that is, exactly what is intended to achieve with the project and exactly in what measure or
proportion. From the scope determination, the different aspects of the project are planned. A
procedure called Work Breakdown Structure (WBS) is used in order to identify the tasks and sub- tasks (or simply activities, being the total task the project itself) as well as activity precedence, that
is, which activity should go before which other or if it has no precedence and it can begin at the start
of the project. Two very important dimensions of the project are time and cost, which are closely
linked by a procedure called crashing. An additional dimension is performance, although some
speak about quality instead of performance. From the point of view of this paper, quality will be the
degree of success in the performance of the project, and the performance will be measured using
the Internal Rate of Return (IRR) of the project, for which the target is set based on the Minimally
Attractive Rate of Return or MARR (Copertari, 2014).
a. Time-cost balance
Time Cost
+
Delay
-
Crashing
b. Time/cost-performance
balance
Time/Cost Performance
-
Profit
+
Quality
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Copertari, L. F. (2021). Comprehensive Project Risk Management Methodology. Advances in Social Sciences Research Journal, 8(1) 94- 115.
URL: http://dx.doi.org/10.14738/assrj.81.9568
The Probabilistic Critical Path Method (PCPM) considers the project and its dimensions from a
probabilistic point of view. Thus, if the project is intended to be finished by a certain due date, the
probability of success in time and the risk of not fulfilling the due date are illustrated in Figure 2a.
Also, if the goal is to finish the project within a certain budget, project success in the cost dimension
and cost risk are shown in Figure 2b. Finally, if the project goal is to reach a minimum MARR,
performance success as measured by the project’s IRR as well as the risk of not reaching the MARR
are pictured in Figure 2c.
Figure 2. Project success and risk based on the time dimension (set by the due date), cost (set by the
budget) and performance (set by the MARR and measuring performance according to the project’s
IRR).
The Targets Of Project Management
When we talk about project management, the most important aspects to consider in this area of
study come up: project scope, time, cost and quality. Time and cost are clearly two of the most
important dimensions of all projects and they are intimately connected. The quality of the project is
something more difficult to measure. Generally speaking, it could be said that quality is satisfying
the customer’s expectations. However, who is the customer of a project? The project team members
could be considered the customers, or the people receiving the project deliverables, or maybe the
decision makers wishing the project to be successful so that they can innovate in their products or
services or even the company stockholders wishing a successful project in order to receive a
satisfactory return over their investment. It is precisely the latter point of view of quality
measurement in a project, the one of the stockholders, being considered in order to define the way
in which the project performance is measured. In this case, it is the Internal Rate of Return (IRR), or
simply return, which is the rate of return over the investment the project yields, that is, that rate at
which the sum of all cash flows is equal to zero, being those either positive (the final income to
Due date Time
(days)
Time risk
Probability of
delivering on
time
a. Time dimension
Budget Cost
($)
Cost risk
Probability of
delivering on
budget
b. Cost dimension
MARR Performance
(IRR)
Performance
risk
Probability of
exceeding the
MARR
c. Performance dimension
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obtain in a project) or negative (the different costs associated to each activity, being the activity
critical or not and whether it is decided to crash it or not).
Figure 3. The dimensions of project management as an ideal balancing among the time, cost
and performance dimensions.
In this way, for the purposes of this paper, there are three dimensions of the project to be measured:
time, cost and performance. In PCPM time is measured probabilistically and the due date is used as
a basis to indicate the project’s time risk (see Figure 2a). The cost is also measured probabilistically
and the budget is used as an evaluation point in order to know the probability of exceeding the
budget, which is the cost risk. Finally, there is performance, measured as a dimension to exceed
(while time and cost are dimensions not to exceed), having the IRR as indicator of performance and
the MARR as the indicator of the threshold for success or failure. Figure 3 illustrates the three
targets of project management and the way in which they can be visualized in a probabilistic three- dimensional chart.
THE SHAPE PARAMETERS EQUATION FOR THE BETA DISTRIBUTION
In general, we have that the mean () and the variance (2) can be calculated in practice according
to equations (1) and (2), respectively, where the minimum, most like and maximum duration times
are given as a, m, and b, respectively (Meredith & Mantel, 1995).
Performance
(IRR)
Cost
(Money)
Time
(Duration)
Due date
Budget
MARR
Performance
risk
Cost risk
Time risk
Target
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Copertari, L. F. (2021). Comprehensive Project Risk Management Methodology. Advances in Social Sciences Research Journal, 8(1) 94- 115.
URL: http://dx.doi.org/10.14738/assrj.81.9568
μ =
a+4m+b
6
(1)
σ
2 = (
b−a
6
)
2
(2)
Also, the general formula for the mean and the variance of a beta probability density function with
range (a and b) and shape ( and ) parameters (Hastings & Peacock, 1975) are given in equations
(3) and (4), respectively.
μ =
aβ+bα
α+β
(3)
σ
2 = (b − a)
2 αβ
(α+β)
2(α+β+1)
(4)
Since the values for and 2 are known from equations (1) and (2) based on PERT’s three point
estimates (a, m and b), equations (3) and (4) constitute a non-linear system of two equations with
two unknowns ( and ), which can be solved. Equations (5) and (6) provide the solution to this
system of non-linear equations, which yield the values for and , respectively.
α = (μ − a) (
(μ−a)(b−μ)−σ
2
(b−a)σ2
) , b > a (5)
β = (b − μ) (
(μ−a)(b−μ)−σ
2
(b−a)σ2
) , b > a (6)
Equations (5) and (6) are novel and specific findings of the Probabilistic Critical Path Method
(PCPM). What is the practical relevance of equations (5) and (6)? Now, it is possible to correctly
calculate the shape parameters ( and ) of any activity or any variable described using a beta
probability density function based on the minimum (a), maximum (b), mean () and variance (2)
values. In turn, the mean () and the variance (2) values can be calculated using equation (1) and
(2), respectively, which are based on three-point estimates for minimum (a), most likely (m) and
maximum (b) values related to the variable being modelled using the beta probability density
function.
THE STARTING OF ACTIVITIES IN A PROJECT MANAGEMENT NETWORK
Let k be any activity and N be the total number of activities in a project management network. In
order to calculate the projects’ IRR (or simply return) it is necessary to know when to apply activity
costs. The problem with CPM is that there is no specific starting or finishing activity completion
times if they have a slack greater than zero. Let dk, ESk, EFk, LSk, LFk and Sk be a specific duration,
early start, early finish, late start, late finish and slack for any activity k, where k = 1, 2., ..., N. Also,
let Ck = {1,0} indicate with a 1 if the activity is critical and with a 0 if it is not. A critical activity delays
the entire project, so they have a slack of zero. However, for non-critical activities, the slack is
greater than zero. The problem is how to specify when non-critical activities start. Let Startk (or
simply Stk) be the starting of activity k, where k = 1, 2, ..., N. The user decides how optimistic (0 ≤ O
≤ 1) or pessimistic (0 ≤ P ≤ 1) wants to be. A completely optimistic user (O = 1) would start the
activity at its early starting time (ESk), whereas a completely pessimistic user (P = 1) would start
the activity at its latest starting time (LSk). Clearly, O+P = 1 and combinations of O and P different
than one and zero are possible. Equation (7) indicate how to calculate the starting time (Startk or
simply Stk). Notice that equation (8) is also satisfied.
Startk = O × ESk + P × LSk (7)
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Copertari, L. F. (2021). Comprehensive Project Risk Management Methodology. Advances in Social Sciences Research Journal, 8(1) 94- 115.
URL: http://dx.doi.org/10.14738/assrj.81.9568
Figure 5. Minimum activity completion time calculations for the testing example.
Figure 6. Maximum activity completion time calculations for the testing example.
4 0 4.5
9 | I 1
4 0.5 4.5
5 | E 0
1 0.5 1.5
2 2 2.5
0.5 0.5 1.0
0 0.5 0.5
1 | A 0
0 0 1
2 | B 1
0 1 1
1 1 4
6 | F 1
1 3 4
3.5 1 4.5
8 | H 0
2.5 1 3.5
2.5 1 4.5
7 | G 0
2 3.5 s
1.5
Finish 10
0 4.5
Start 0
1.5 1 3.5
3 | C
0.5 2 2.5
0
2 1.5 2.5
4 | D
0.5 0.5 1
0
11 19
9 8 17
7 | G 0
2
4 5 9
5 | E 0
6 2 11
1 1 4
0 3 3
1 | A 0
0 4 4
2 | B 1
0 0 4
4 9 13
6 | F 1
4 0p
13
1 19 s
12
11 7 18
8 | H 0
13 6 19
9 | I 1
13 0 19
Finish 10
19
Start 0
0
3 8
4 1 12
11
3 | C 0
7 4
3 4 7
4 | D 0
11
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Figure 7. Mean activity completion time calculations for the testing example.
In Figures 5, 6 and 7, the critical path resulting in each case is highlighted using bold lines. The only
thing missing in order to be able to calculate beta distributed completion times is the variance (2)
in order to be able to successfully apply equations (5) and (6). However, we are not going to use the
PERT approach based on assuming the stochastic completion times of the activities to be
represented using the mean activity duration time () and CPM calculations. This is because we
want more than just a probabilistic project completion PERT time, which assumes a normally
distributed completion time that tends to underestimate the actual completion time, because we
also want the probabilistic beta distributed completion times of all the activities.
Criticality and Project Completion Variance
Now we need a new concept, called criticality. Criticality in a PCPM network is the probability for
each activity to be critical. Thus, Ck is no longer either zero or one, but a number between zero and
one, since activities now have a given probability of being critical. For that, we are going to use
simulation. Let s be the total number of simulation runs, Ck,i be the criticality obtained for activity k
(k = 1, 2, ..., N) in iteration i (i = 1, 2, ..., s), 2 the variance in the completion time for the whole
project, k
2 the variance of the completion time for activity k, path p be the set of all the activities
(non-repetitive) of all the alternative project network paths in which activity k exists. Then,
equations (9), (10) and (11) apply.
Ck =
1
s
∑ Ck,i
s
i=1
, k = 1, 2, ... ,N, where Ck,i = {
1, if the activity is critical
0, if the activity is not critical (9)
σ
2 = ∑ Ckσk
N 2
k=1
, 0 ≤ Ck ≤ 1 (10)
σk
2 = ∑ Chσh
2
h⊂p , 0 ≤ Ck ≤ 1 (11)
For the testing example shown in Table 1, a total of 1,000 runs and 10 re-runs where carried out.
Table 2 shows the results. Notice that a slack is zero if it is between a zero value of ±0.0001.
9 16
7 7 14
7 | G 0
2
3 4 7
5 | E 0
5 2 9
1 1 3
0 2 2
1 | A 0
0 3 3
2 | B 1
0 0 3
3 8 11
6 | F 1
3 0p
11
1 16 s
10
9 6 15
8 | H 0
11 5 16
9 | I 1
11 0 16
Finish 10
16
Start 0
0
2 7
3 1 10
9
3 | C 0
6 4
2 3 5
4 | D 0
9
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Copertari, L. F. (2021). Comprehensive Project Risk Management Methodology. Advances in Social Sciences Research Journal, 8(1) 94- 115.
URL: http://dx.doi.org/10.14738/assrj.81.9568
Table 2. Criticality example using simulation (s = 10,000, r = 10) where the activity is critical if
the slack is approximately zero.
k = 1 2 Block 1 3 4 5 6 Block 2 7 8 9 Block 3
r C1 C2 C1+C2 C3 C4 C5 C6
C3+C4+
C5+C6 C7 C8 C9
C7+C8+
C9
1 32.5% 67.5% 100.0% 29.1% 0.0% 6.5% 64.5% 100.0% 6.5% 29.1% 64.5% 100.0%
2 31.9% 68.1% 100.0% 28.6% 0.1% 6.5% 64.9% 100.0% 6.5% 28.6% 64.9% 100.0%
3 32.7% 67.3% 100.0% 28.7% 0.1% 5.8% 65.4% 100.0% 5.9% 28.7% 65.4% 100.0%
4 32.4% 67.6% 100.0% 28.9% 0.0% 6.0% 65.1% 100.0% 6.0% 28.9% 65.1% 100.0%
5 32.9% 67.1% 100.0% 29.4% 0.0% 6.3% 64.3% 100.0% 6.3% 29.4% 64.3% 100.0%
6 32.3% 67.7% 100.0% 28.7% 0.0% 5.8% 65.4% 100.0% 5.9% 28.7% 65.4% 100.0%
7 32.4% 67.6% 100.0% 28.8% 0.0% 6.1% 65.1% 100.0% 6.1% 28.8% 65.1% 100.0%
8 32.4% 67.6% 100.0% 28.9% 0.0% 6.2% 64.9% 100.0% 6.3% 28.9% 64.9% 100.0%
9 32.8% 67.2% 100.0% 29.4% 0.0% 6.9% 63.6% 100.0% 7.0% 29.4% 63.6% 100.0%
10 31.7% 68.3% 100.0% 28.2% 0.0% 6.5% 65.3% 100.0% 6.5% 28.2% 65.3% 100.0%
Ck 32.4% 67.6% 100.0% 28.9% 0.0% 6.3% 64.9% 100.0% 6.3% 28.9% 64.9% 100.0%
The advantage of using criticality is that now we have beta distributed probabilistic completion
times for selected activities marked as milestones in the project and not only a normally distributed
probabilistic completion time.
THE OPTIMAL COST WITH UNLIMITED RESOURCES
Let the project completion time be denoted as T. The result of applying the probabilistic crashing
heuristic is a series of points beginning with the maximum project completion (T = b) having the
minimum overall cost (CA or U) and proceeding in a series of iterations until the minimum project
completion (a) is reached having a maximum project cost (CB or V). For simplicity, crashing can be
assumed to be an inversely proportional relationship between time and cost (Babu & Surech, 1996;
Foldes & Soumis, 1993). Notice that crashing refers to a CPM technique, whereas probabilistic
crashing refers to a PCPM heuristic technique specially designed for the PCPM methodology.
Generally speaking, however, it can be said that there are two kinds of costs: direct and indirect
costs. Direct costs, which are inversely proportional to the project completion time (T) include the
cost of material, equipment, and direct labor required to perform an activity, although in this case
we will consider the entire project. Indirect costs include in addition to supervision and other
overhead costs, interest charges on the cumulative project investment and overdue penalty costs
(Auguston, 1993; Clark, 1992; Meredith & Mantel, 1995; Rivenbank, 2000; Tattersall, 1990).
Let D denote the direct costs of the whole project. Then, direct cost is inversely proportional to time
(T), as indicated in equation (12).
D ∝
1
T
(12)
In order to transform such proportionality relationship into an equality, proportionality constants
are added, as indicated in equation (13).
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D =
χ
T
+ φ (13)
In this case, there are for practical purposes, two direct costs: V, which is associated to time T = a,
and U, which is associated with time T = b. Thus, equations (14) and (15) are created.
V =
χ
a
+ φ (14)
U =
χ
b
+ φ (15)
These system of two equations have four known values (after probabilistic crashing and time
calculations): V, U, a and b and two unknowns: and . Thus, it can easily be solved. The solution
for the unknown variables ( and ) are indicated in equations (16) and (17), respectively.
χ =
ab(V−U)
b−a
, b > a (16)
φ =
Ub−Va
b−a
, b > a (17)
Thus, substituting from equations (16) and (17) into equation (13) results in the direct cost
expression, shown in equation (18).
D =
ab
T
(V−U)
(b−a)
+
Ub−Va
b−a
(18)
The project indirect cost, I, is directly proportional to the project completion time (T), as indicated
in equation (19).
I ∝ T (19)
The proportionality expression in equation (19) can be transformed into an equality adding
proportionality constants. Let O denote the minimum overhead (indirect) cost at time T = 0, and m
the increments (slope) of the indirect cost as a function of time. Then, equation (20) indicates the
relationship between time and indirect cost.
I = O + m T (20)
The total cost (C) is the result of adding the indirect (I) and the direct (D) costs, as indicated in
equation (21).
C = I + D (21)
Substituting I from equation (20) and D from equation (18) into equation (21) yields equation (22),
which is the sum of the indirect and direct costs, or total cost.
C = (O + m T) + (
ab
T
(V−U)
(b−a)
+
Ub−Va
b−a
) (22)
Taking the derivative of the total cost (C) with respect to time (T) from equation (22) and equating
that to zero yields the optimal project completion time (T*). The derivative is shown in equation
(23).
δC
δT
= m −
ab
T2
(V−U)
(b−a)
= 0 (23)
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URL: http://dx.doi.org/10.14738/assrj.81.9568
Solving for T from equation (23) yields the optimal completion time (T*) as indicated in equation
(24).
T
∗ = √
ab
m
(V−U)
(b−a)
(24)
Figure 8a illustrates the direct (D) and indirect (I) costs, whereas Figure 8b shows the total cost and
it highlights the fact that the derivative, when equating to zero, would result in the optimal project
completion time (T*) for the optimal minimal total cost (C*).
a. Direct and indirect project costs. b. Optimal project cost.
Figure 8. Time-cost tradeoffs.
TIME-COST TRADEOFF
When considering the total cost of the project, a concept that is central not only to CPM but also to
PCPM is the time-cost tradeoff. When considering there are unlimited resources, it is possible in
CPM as well as in PCPM to calculate a tradeoff between time and cost.
Generally speaking, the longer it takes to complete the project because fewer resources are used,
the lower the cost of such doing is. If we use additional resources, it should be possible to reduce
the time it takes to complete the project at the expense of incurring in additional costs. By following
an optimal algorithm, a curve similar to the one corresponding to the direct costs in Figure 8a is
generated. However, because there are too many activities in realistic projects, it is not possible to
consider all alternatives of activity crashing. A project with 10 activities would have 210 = 1,024
crashing combinations to consider. A project with 20 activities would have 220 = 1’048,576 crashing
combinations. Schedule allows to have a maximum of 254 activities. Considering 2254 crashing
combinations is out of any practical algorithm that could be devised. Even a relatively realistic
Time
(T)
Cost
(C)
a b
U
O
V
O+mb
m
D
I
Time
(T)
Cost
(C)
a b
C*
O+ma+V
T*
O+mb+U
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project with 50 activities (250 combinations) would mean disaster for a combinatorial algorithm to
find the optical time-cost tradeoff.
In any case, the theory holds that if such optimal time-cost crashing calculations could be performed,
the result should look like the direct cost curve in Figure 8a. Let iteration i = 0 correspond to the
case where there is no activity crashing at all having the maximum project completion time (b), and
the maximum number of iterations (I) be the case where all activities are crashed. Clearly, iteration
i = I is reached only after the minimum project completion time (a) has been obtained in the
previous iteration (that is, iteration i = I-1).
Let CAk be the normal activity duration cost (corresponding to the maximum activity duration time
bk) and CBk be the crashed activity duration cost (corresponding to the minimum activity duration
time ak) for activities k = 1, 2, ..., N. Also, let time ti be the completion time of the project for iteration
i and cost ci be the corresponding cost to such project completion time.
Figure 9 shows the ideal (optimal) time-cost tradeoff sequence of I iterations. For this case, the
minimum project cost (CA) is calculated according to equation (25) and the maximum project cost
(CB) is given according to equation (26).
Figure 9. Ideal (optimal) time-cost tradeoff crashing sequence of I iterations.
CA = ∑ CAk
N
k=1
, k = 1, 2, . . . ,N (25)
CB = ∑ CBk
N
k=1
, k = 1, 2, . . . ,N (26)
The variance in the cost of activity k (Ck
2) is given according to equation (27).
0
1
2
3
i-1
i
i+1
I
I-2
I-1
Project time
Project cost
a b
CA
CB
(t0,c0)
(t1
,c1
)
(t2
,c2
)
(t3
,c3
)
(ti-1
,ci-1
)
(ti
,ci
)
(ti+1,ci+1)
(tI-2
,cI-2
)
(tI-1
,cI-1
)
(tI
,cI
)
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URL: http://dx.doi.org/10.14738/assrj.81.9568
Cσk
2 = (
CBk−CAK
6
)
2
, k = 1, 2, ... ,N (27)
The only information we need to be able to apply the beta distribution shape parameters equations
is the mean cost. For that, it is necessary to interpolate the mean cost (C) corresponding to the
mean project completion time ().
Now, we need to consider the case where there is a time t corresponding to a cost c between
iterations i and i-1. Equation (28) calculate the interpolated cost c corresponding to an interpolation
time t, whereas equation (29) yield the interpolated time t corresponding to an interpolation cost c.
Figure 10 illustrates the situation.
Figure 10. Ideal (optimal) time-cost tradeoff crashing sequence of I iterations for a given
interpolation time and cost (t,c).
c = ci −
(ti−1−t)(ci−ci−1
)
ti−ti−1
, ci ≥ c > ci−1 and ti ≤ t < ti−1 (28)
t = ti +
(ci−c)(ti−1−ti
)
ci−ci−1
,ti ≤ t < ti−1 and ci ≥ c > ci−1 (29)
Thus, having a mean interpolation time , it is possible to calculate the interpolated cost C
according to equation (30).
Cμ = ci −
(ti−1−μ)(ci−ci−1
)
ti−ti−1
, ci ≥ Cμ > ci−1 and ti ≤ μ < ti−1 (30)
Project cost
i
1 0
2
3
i-1
(t,c)
i+1
I-2
I-1
I
a b Project time
CA
CB
t
c
(ti,ci)
(ti-1
,ci-1
)
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Advances in Social Sciences Research Journal (ASSRJ) Vol.8, Issue 1, January-2021
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Now, having the minimum project cost (CA), maximum project cost (CB), the mean project cost (C)
and the variance project cost (C2), we can calculate the shape parameters (C and C) for the
corresponding beta distributed cost probability density function as given by equations (31) and
(32).
Cα = (Cμ − CA) (
(Cμ−CA)(CA−Cμ)−Cσ
2
(CB−CA)Cσ2
) , CB > CA (31)
Cβ = (CB − Cμ) (
(Cμ−CA)(CB−Cμ)−Cσ
2
(CB−CA)Cσ2
) , CB > CA (32)
THE RETURN (PERFORMANCE) DIMENSION
The Net Present Value (NPV) and the Internal Rate of Return (IRR) or simply return
Let T be the project duration for certain iteration i (i = 0, 1, 2, ..., I) and the Startk (Stk as a shortcut)
the beginnings of each activity for k = 1, 2, ..., N. It is assumed a certain cost k (ck) occurs at the
beginning of each activity (at time Stk). Also, suppose there is an income, which is assumed to be
collected at the end of the project (at time T for each time-cost tradeoff iteration). The costs of each
activity are the ones collected for such iteration. Finally, such income (Income) is fixed and should
in principle (but not necessarily) be greater than the maximum project cost (CB). It is initially
assumed that the Internal Rate of Return (IRR) is, as a consequence, positive. In order to simplify
calculations, let X denote the value for 1+IRR, as indicated in equation (33).
X = 1 + IRR (33)
Then, the IRR is the rate at which the Net Present Value (NPV) equals zero. The equation that
explains the NPV calculation when it is zero is shown in equation (34).
NPV = 0 = Income − ∑ ckX
N T−Stk
k=1
(34)
Equation (34) obtains the value of X when the NPV = 0. From that X, based on equation (33), we can
calculate the value of the IRR. It is assumed there is one IRR for the project, regardless of the
duration units (days, weeks, months or years). This is because calculating the annualized rate of
return brings additional complications. Thus, we have a function of X, f(X), indicated in equation
(35).
f(X) = Income − ∑ ckX
N T−Stk
k=1
(35)
We need to find the value (and there should be only one because there is one sign change only in
the cash flows) for which f(X) = 0. For that, we rely on the Newton-Raphson method of assured
convergence (Burden & Faires, 1985). Beginning with any given value for X (such as 1.5, for
example), we proceed from such initial value (X0 = 1.5 for j = 0) and continue to the next iteration.
The process for each new iteration j is indicated in equation (36).
Xj+1 = Xj −
f(Xj)
f
′(Xj)
(36)
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We can see that equation (36) requires us to calculate the derivative of the function f(Xj), that is
f’(Xj). That can easily be done by derivation of equation (35), since X is the unknown variable, which
is shown in equation (37).
f
′
(X) = − ∑ ck
(T − Stk
)X
N T−Stk−1
k=1
(37)
The rate of return obtained (IRR) does not necessarily increases as the time-cost tradeoff iterations
proceed, because the result obtained according to Schedule’s heuristic is not necessarily convex.
Thus, we take the minimum such rate of return and call it RA as well as the maximum rate of return
and call it RB. Let RR be the rate of return corresponding to some given time completion value t.
Thus, we can relate the rate of return RR to a given time t and calculate a given time t for a given
rate of return RR, as indicated in equations (38) and (39), where a is the minimum (all crashed)
project completion time and b the maximum (normal) project completion time.
RR = RA +
(t−a)(RB−RA)
b−a
, b > a (38)
t = a +
(RR−RA)(b−a)
RB−RA
, RB > RA (39)
The mean rate of return (R) is assumed to be the rate of return at which the mean project
completion time () occurs. Thus, we have equation (40).
Rμ = RA +
(μ−a)(RB−RA)
b−a
, b > a (40)
The return variance (R2) is calculated according to the original PERT equation, which is shown in
equation (41).
Rσ
2 = (
RB−RA
6
)
2
(41)
Since we have the minimum (RA), maximum (RB), mean (R) and variance (R2) of the IRR or
simply return, we can calculate the beta shape parameters (R and R) according to equations (42)
and (43).
Rα = (Rμ − RA) (
(Rμ−RA)(RA−Rμ)−Rσ
2
(RB−RA)Rσ2
) , RB > RA (42)
Rβ = (RB − Rμ) (
(Rμ−RA)(RB−Rμ)−Rσ
2
(RB−RA)Rσ2
) , RB > RA (43)
Testing The Theory With The Example From Table 1.
Table 3 shows the time (ti), cost (ci) and return (ri) for each iteration (where I = 15+1) calculated
for the example from Table 1 as given by Schedule.
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Advances in Social Sciences Research Journal (ASSRJ) Vol.8, Issue 1, January-2021
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Table 3. Time, cost and return calculations from Schedule for the realistic enough example shown in
Table 1 where Income = $2,500.
I Time (ti) Cost (ci) Return (ri)
0 19 $ 1,245.0000 5.36516914%
1 18 $ 1,253.3333 5.52892210%
2 17 $ 1,278.3333 5.64622149%
3 15 $ 1,348.3333 6.03133936%
4 13 $ 1,418.3333 6.59978418%
5 12 $ 1,458.6364 6.99933285%
6 11 $ 1,498.9394 7.67486409%
7 10 $ 1,692.8788 6.50907407%
8 9 $ 1,732.8788 6.18065098%
9 8 $ 1,839.5455 5.23776127%
10 7.5 $ 1,892.8788 4.91881442%
11 7 $ 1,926.2121 4.87586633%
12 6.5 $ 1,959.5455 4.81307255%
13 6 $ 2,029.7835 4.47847154%
14 4.5 $ 2,260.9524 3.03750096%
15 4.5 $ 2,430.0000 0.94333776%
Figure 11 shows the time-cost tradeoffs and Figure 12 shows the return for the time/cost-return
calculations from Table 3.
Notice in Figure 11 that the time-cost tradeoff obtained according to Schedule’s proprietary
probabilistic crashing heuristic is not a convex function as shown in Figure 8a, Figure 9 and Figure
10. However, as the completion time decreases, the cost always increases.
Also, notice that in Figure 12 the returns obtained are not always increasing as the completion time
increases. This is because of the complexity of the functions in equations (35) and (37).
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URL: http://dx.doi.org/10.14738/assrj.81.9568
Figure 11. Time-cost tradeoff for the realistic enough example from Table 1.
Figure 12. Time-return calculations for the realistic enough example from Table 1.
THE INTERACTIVE GRAPHIC USER INTERFACE (GUI) FOR SCHEDULE
As we have previously seen, it is possible to calculate beta distributed time, cost and return
probability density functions, which give us the “risk” of failure (and alternatively success) of each
dimension. Furthermore, these dimensions are related through the time-cost tradeoff curve. Figure
13 shows a scheme of this interactive user interface.
$1200.0
$1300.0
$1400.0
$1500.0
$1600.0
$1700.0
$1800.0
$1900.0
$2000.0
$2100.0
$2200.0
$2300.0
$2400.0
$2500.0
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
17
17.5
18
18.5
19
Total cost
Time
1 0
2
3
4
5
6
7
8
9
10 11 12
13
14
15
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
17
17.5
18
18.5
19
Rate of return (IRR)
Project completion (Time)
0
2 1
3
4
5
6
7
8
9
11 10
12
13
14
15
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Advances in Social Sciences Research Journal (ASSRJ) Vol.8, Issue 1, January-2021
112
Figure 13. Interactive GUI for Schedule.
These dimensions are related according to equations (28), (29) and (38). If we change the due date
(t), it immediately changes the budget (c) and the return (RR) and if we change the budget (c) it
changes the due date (t) and the return (RR). To avoid a cyclical user interaction, we can change the
MARR at will and see the effect on the bubble.
DISCUSSION AND CONCLUSION
CPM presents a considerable constraint: activity durations must be deterministic. Besides, there is
the inherent problem with the CPM that for the non-critical activities, they can begin at any time
between the Early Start (ES) of the activity and such ES plus the slack (S). Also, activities can finish
at any time between the Early Finish (EF) of the activity and such EF plus the slack (S). In all these
cases, when do the actual beginning and end of the non-critical activities actually occurs?
We require, then, a modification to CPM. Using the optimistic percentage (O) and its corresponding
pessimistic one (P = 1- O), it is possible to specify values given the starting and finishing times of
the activities to know when it is they occur in each iteration of the simulation required to calculate
the criticality of each activity.
c
t
RR = MARR
RB
Return
a b
CA
CB
RA
0% 100%
100% 100%
Cost
Time
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Copertari, L. F. (2021). Comprehensive Project Risk Management Methodology. Advances in Social Sciences Research Journal, 8(1) 94- 115.
URL: http://dx.doi.org/10.14738/assrj.81.9568
The CPM has, nevertheless, something very important, which is the crashing method, which can be
extended to PCPM (calling it probabilistic crashing using the proprietary heuristic for the
computations) to obtain a time and cost relationship and also a time-cost and return relationship,
which allows to link all variables among them, even probabilistically, so that the risk element for
each dimension is added.
PERT, on the other hand, offers the idea of minimum, most likely and maximum times for the
duration of each activity, which is used in PCPM to calculate the mean and the variance of each
activity according to the PERT textbook formula. However, PERT tends to lead to optimistic
estimates of the duration of the activities by assuming a normal distribution (and not a beta
distribution, as with PCPM) representing the project duration. In principle, this is theoretically
wrong, because there is an absolute minimum project duration (called a in PCPM) and an absolute
maximum project duration (called b in PCPM). The normal distribution (unlike the beta
distribution) does not have minimum nor maximum duration, but rather a duration time that goes
from - to +, despite the fact that the probabilities of having these extreme durations are
infinitesimal.
There is, nonetheless, an additional limitation to PERT: what is the project duration variance when
there is more than one critical path? We could assume that in this case the sum of the variances of
the activities in each and every critical path are used and choose whichever is the highest of all
possible critical paths. However, PERT does not provide such consideration for assuming it unlikely
to happen, which constitutes a serious limitation of PERT. On the other hand, PCPM considers the
variance of the project based on the criticalities and it does not assume a normal distribution for
such duration, but rather a beta distribution, so that it does not lead to optimistic (nor pessimistic)
assumptions concerning the project duration. Instead, it leads to a precise probabilistic assessment.
Furthermore, it allows to calculate the beta distributed completion times of all the activities
considered to be milestones.
Notice that in this case the return calculation (IRR) was consistent with the theory, that is, as time
increases, the related cost is reduced and the return increases. It is possible that the return
calculations are not always following this precise trend. It is possible to have fluctuations such that
with time, return not always increases. In any case, the lowest project duration (a) is associated
with the minimum return (RA) and the largest project duration (b) is associated with the highest
return obtained (RB). However, the actual results can be consulted to find out the correct return for
any given time and cost, even for the optimal time and the optimal cost.
Finally, PCPM is the only method leading to a project duration, cost and return described by a beta
distribution, so that it is possible to estimate the time, cost and return risks for given due dates,
budgets or MARR, respectively, and consequently, define probabilistic tradeoffs among these
dimensions. In conclusion, PCPM not only emphasizes the risk element to the project, but it also
offers a comprehensive approach to Project Management (Project Risk Management to be more
specific) by considering the three dimensions and their inherent tradeoffs: time, cost and return
(the latter one measuring project performance).
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