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doi:10.1016/j.ejor.2017.12.039 •www.stefancreemers.be •info@stefancreemers.be
Moments and distribution of the net present value of a
serial project
Stefan Creemers
Abstract -We study the Net Present Value (NPV) of a project with multiple
stages that are executed in sequence. A cash flow (positive or negative) may be
incurred at the start of each stage, and a payoff is obtained at the end of the
project. The duration of a stage is a random variable with a general distribution
function. For such projects, we obtain exact, closed-form expressions for the
moments of the NPV, and develop a highly accurate closed-form approximation
of the NPV distribution itself. In addition, we show that the optimal sequence
of stages (that maximizes the expected NPV) can be obtained efficiently, and
demonstrate that the problem of finding this optimal sequence is equivalent to
the least cost fault detection problem. We also illustrate how our results can be
applied to a general project scheduling problem where stages are not necessarily
executed in series. Lastly, we prove two limit theorems that allow to approximate
the NPV distribution. Our work has direct applications in the fields of project
selection, project portfolio management, and project valuation.
Keywords -project scheduling, project management, net present value, NPV
distribution, least cost fault detection problem
1 Introduction
We consider a project with multiple stages that are executed in sequence. Each stage of the
project has a random duration with general distribution function. At the start of a stage,
a deterministic cash flow (positive or negative) may be incurred, and a deterministic payoff
is obtained upon completion of the project. Continuous compounding is used to determine
the Net Present Value (NPV) of the project (i.e., the sum of the discounted cash flows that
are incurred during the project lifetime; the convolution of the NPV distributions of the
individual cash flows). We develop exact, closed-form expressions to obtain the moments of
the project NPV distribution. In addition, we provide a highly accurate approximation of the
NPV distribution itself. The approximation uses a three-parameter lognormal distribution to
match the first three moments of the NPV distribution. A lognormal distribution was chosen
because: (1) the moment-matching procedure uses closed-form expressions, and (2) we show
that the NPV of a cash flow converges to a (reflected) lognormal distribution if the cash
flow is not incurred during the early stages of the project. We also show that, if a sufficient
number of cash flows are incurred, the project NPV converges to a normal distribution. In
addition, we show that the sequence of stages that maximizes the expected NPV (eNPV)
over all possible sequences can be found efficiently, and that the problem of finding this
optimal sequence is equivalent to the Least Cost Fault Detection Problem (LCFDP). Lastly,
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doi:10.1016/j.ejor.2017.12.039 •www.stefancreemers.be •info@stefancreemers.be
if stages are not executed in sequence, we demonstrate that our approach can still be used
to approximate the moments and the distribution of the project NPV. We use examples to
illustrate our results, and to show that our approach can easily be implemented.
Our work has direct applications in the fields of project selection, project portfolio man-
agement, and project valuation. In these fields, the detailed scheduling of activities is often
not considered, and it is assumed that: (1) a project is a sequence of stages with cash flows
that are incurred at the start of a stage, and (2) a (uncertain) payoff is obtained upon com-
pletion of the project. Such projects are not only prevalent in the real world, but also in
the literature. For instance, Huchzermeier and Loch (2001) consider an R&D project that
is divided in sequential stages, and develop a dynamic program to determine the expected
value of the project. Santiago and Vakili (2005) build on the work of Huchzermeier and
Loch, and also consider an R&D project with sequential stages. De Reyck and Leus (2008)
discuss the literature on R&D project scheduling, and conclude that most of the literature
is limited to sequential R&D stages only. Girotra et al. (2007) determine the eNPV of a
drug development project where stages have been defined by a regulator. They also mention
that a stage-gate development process is prevalent in most industries. Chao et al. (2014)
also investigate the use of a state-gate processes to manage NPD projects. They argue that
decisions based on eNPV alone are dangerous, and that risk should be taken into account
when making project selection/investment decisions. Often, the risk of a project is modeled
using the variance of the NPV (Van Horne 1966). Other measures of risk are the skewness
and/or kurtosis of the NPV, and the probability to have a negative NPV. Until now, how-
ever, Monte Carlo simulation was the only technique available to obtain higher moments
and/or the NPV distribution itself. In this article, we develop a closed-form characterization
of the NPV distribution of a project that is a valid alternative to Monte Carlo simulation,
and that can be directly applied to evaluate project selection/investment decisions.
On the tactical/strategical level, projects are often seen as a sequence of stages. Oper-
ational factors, however, may also result in the serial execution of a project. For instance,
a bottleneck resource may force activities to be executed in series. A bottleneck resource
has been considered, among others, by Kaviadias and Loch (2003), who study NPD projects
that compete for a scarce resource, and that are divided into stages. In addition, some in-
dustries are more likely to have a serial project execution due to an abundance of technical
precedence relationships (e.g., the construction industry).
In the (more operational) field of project scheduling, our work is related to CPM/PERT
in the sense that we also focus on a single sequence of stages, and that we also use normal
(lognormal) approximations. The study of CPM/PERT dates back to the work of Kelley and
Walker (1959) and Malcolmn et al. (1959), and still continues today (refer to Demeulemeester
and Herroelen (2002) and Trietsch and Baker (2012) for an overview of the literature).
Whereas CPM/PERT deals with the project completion time, we focus on the NPV. In a
recent survey, Wiesemann and Kuhn (2015) not only highlight the importance of NPV over
project completion time, but also stress the importance of stochastic project scheduling. In
stochastic project scheduling, stage durations and/or cash flows are random variables, and as
a result the project NPV is a random variable as well. Although most of the literature deals
with minimizing the expected completion time of a project (Herroelen 2005; Ballest´ın and
Leus 2009), some research has already been devoted to maximizing the eNPV of a project
(Vanhoucke et al. 2001; Szmerekovsky 2005). Higher moments of the NPV distribution,
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doi:10.1016/j.ejor.2017.12.039 •www.stefancreemers.be •info@stefancreemers.be
and/or the NPV distribution of a project itself, have never been studied before. In general,
it is considered to be impossible to efficiently determine the NPV distribution of a project
(Wiesemann and Kuhn 2015). In fact, for the completion time of a project, Hagstr¨om
(1988) has shown that it is #P-complete to determine even a single point of the Cumulative
Distribution Function (CDF). Even for serial projects, Kamburowski (1986) has shown that
the result of Hagstr¨om holds.
The remainder of this article is structured as follows. Section 2 develops exact, closed-
form expressions for the moments and the distribution of the NPV of a cash flow that is
obtained after a single stage. Multiple stages are considered in Section 3. In Section 3, we
also show that the NPV of a single cash flow converges to a (reflected) lognormal distribution
if the cash flow is not incurred during the early stages of the project. Section 4 introduces the
lognormal approximation that can be used to model the NPV distributions of both individual
cash flows as well as projects. In Section 5, we develop exact, closed-form expressions for
the moments of the NPV distribution of a multi-stage project with intermediate cash flows.
In addition, we also show that the NPV of a project converges to a normal distribution,
and assess the accuracy of the lognormal and normal approximations of the project NPV
distribution. In Section 6, we show that: (1) the problem of finding the optimal sequence
of stages is equivalent to the LCFDP, (2) if stages are not precedence related, a well-known
result from the literature on the LCFDP can be used to obtain the optimal sequence in
polynomial time, and (3) efficient methods exist to obtain the optimal sequence of stages if
they are precedence related. Section 7 illustrates how our results can be used to approximate
the moments and the distribution of the NPV of a general project where stages are scheduled
using a scheduling policy. Section 8 discusses a number of model extensions, and Section 9
concludes and provides directions for future research.
2 NPV of a cash flow obtained after a single stage
In this section, we investigate the basic case where a cash flow cis incurred after a single
stage. Under continuous compounding, the NPV of a cash flow cis given by:
v=ce−rt,(1)
where ris the discount rate, and tis the time at which cash flow cis incurred. If tis a
realization of T, and if Tis a random variable with probability function f(t), the eNPV of
cash flow cis given by:
µ=
∞
Z
0
f(t)ce−rtdt.
Lemma 1. Consider a cash flow cthat is incurred at time T, where Tis a random variable
with probability function f(t). Given a discount rate r, the eNPV of cis given by:
µ=cMT(−r),
where MT(u)is the Moment Generating Function (MGF) of T.
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For notational convenience, let φ(r)≡MT(−r) such that:
µ=cMT(−r) = cφ(r).(2)
φ(r) can be interpreted as the eNPV of a cash flow c= 1 that is obtained at time Tif discount
rate rapplies. For most distributions, the MGF (and hence φ(r)) is readily available. There
are some distributions, however, for which the MGF does not have a closed-form expression
(e.g., the Weibull distribution), or for which the MGF is undefined (e.g., the lognormal
distribution). For those distributions, φ(r) has to be approximated. In addition, note that
φ(r) is not always defined for all values of r. For instance, if Tis exponentially distributed,
its MGF is given by MT(u) = λ(λ−u)−1. Hence, if r=−λ, the MGF about −ris undefined,
and µcannot be determined. In practice, however, this is rarely an issue.
We use an example to illustrate Lemma 1. Consider a cash flow c= 1,000 that is
incurred at time T, where Tfollows a gamma distribution with shape parameter k= 5 and
scale parameter τ= 1. The MGF of the gamma distribution is MT(u) = (1 −τu)−k. As a
result, φ(r) = (1 + τ r)−k, and the eNPV of cash flow cis µ=cφ(r) = 620.92 for discount
rate r= 0.1.
Theorem 1. Consider a cash flow cthat is incurred at time T, where Tis a random variable
with probability function f(t). Given a discount rate r, the mean, variance, skewness, and
kurtosis of the NPV of care given by:
µ=cφ(r),
σ2=c2(φ(2r)−φ2(r)),(3)
γ=c3φ(3r)−3φ(2r)φ(r)+2φ3(r)σ−3,
θ=φ(4r)−4φ(3r)φ(r)+6φ(2r)φ2(r)−3φ4(r)φ(2r)−φ2(r)−2.
If we revisit the previous example, the moments of the NPV distribution of cash flow c
are: µ= 620.92, σ2= 16,334, γ=−0.2347, and θ= 2.7064 for discount rate r= 0.1.
Theorem 2. Consider a cash flow cthat is incurred at time T, where Tis a random
variable with probability function f(t). Given a discount rate r, the CDF and Probability
Density Function (PDF) of the NPV of cash flow care given by:
G(v)=1−Fln c
vr−1,
g(v) = fln c
vr−1
|r|v,
where F(t)is the CDF of T. Note that: (1) if r > 0, then vhas range 0≤v < c, (2) if
r= 0, then v=c, and (3) if r < 0, then vhas range c<v≤ ∞.
We illustrate Theorem 2 by means of an example. In the example, a cash flow cis
incurred at time T, where Tfollows an exponential distribution with rate parameter λ. For
a given discount rate r, the CDF of the NPV of cash flow cis:
G(v) = c
v−λr−1
.
Similar results can be obtained for other probability functions.
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3 NPV of a cash flow obtained after multiple stages
In this section, we consider the NPV of a cash flow that is incurred after multiple stages.
Below, we use payoff pto demonstrate our results (as payoff pis obtained at the end of the
project; after all stages have been completed). Note, however, that the results in this section
hold for any cash flow that is incurred during the lifetime of the project.
Lemma 2. Consider a project with multiple stages w:w∈N={1,2, . . . , n}that are exe-
cuted in sequence. Each stage w:w∈Nhas duration distribution fw(t)and corresponding
factor φw(r)that is obtained using Eq. (2). If the durations of the individual stages are
independent, the duration of the project itself has factor:
φ1,n(r) = Y
w∈N
φw(r).
We can combine Theorem 1 with Lemma 2 to determine the moments of the NPV of a
cash flow that is incurred after multiple stages. For instance, consider the NPV of a payoff
pthat is obtained upon completion of a project with three stages. The stages have factors
φ1(r), φ2(r), and φ3(r), respectively. The mean and variance of the NPV of payoff pare
given by:
µ=pφ1(r)φ2(r)φ3(r) = pφ1,3(r),
σ2=p2(φ1(2r)φ2(2r)φ3(2r)−φ2
1(r)φ2
2(r)φ2
3(r)) = p2(φ1,3(2r)−φ2
1,3(r)).
The skewness, kurtosis, and higher-order moments are obtained in the same way.
Lemma 3. Consider a project with multiple stages w:w∈N={1,2, . . . , n}that are
executed in sequence. Each stage w:w∈Nhas a duration distribution fw(t)with mean
dwand variance s2
w. If the durations of the individual stages are independent, the mean and
variance of the project duration are given by:
dN=X
w∈N
dw,
s2
N=X
w∈N
s2
w.
If nis sufficiently large, and if no stage dominates the others, the duration of the project will
converge to a normal distribution with mean dNand standard deviation sN.
Lemma 3 is a well-known result in the literature (Malcolm et al. 1959; Van Slyke 1963;
Moder and Phillips 1970), and allows to predict the completion time of a project. We will use
Lemma 3 to show that the NPV of a payoff pconverges to a (reflected) lognormal distribution
if nis sufficiently large.
Theorem 3. Consider a project with multiple stages w:w∈N={1,2, . . . , n}that are
executed in sequence. If the durations of the individual stages are independent, and if nis
sufficiently large, the NPV of payoff pconverges to a (reflected) lognormal distribution g(v)
with location parameter α= ln(p)−rdNand scale parameter β=rsN.
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Exact NPV distribution
n1 5 10 25 50 100
µ666.67 620.92 613.91 609.53 608.04 607.29
σ255,556 16,334 8,654 3,589 1,817 914
γ-0.566 -0.235 -0.163 -0.101 -0.071 -0.050
θ2.4000 2.7060 2.8300 2.9252 2.9613 2.9803
LNapproximation
n1 5 10 25 50 100
µLN687.29 621.89 614.16 609.57 608.05 607.29
σ2
LN134,164 19,829 9,549 3,734 1,853 923
γLN1.7500 0.6909 0.4814 0.3018 0.2128 0.1502
θLN8.8980 3.8606 3.4148 3.1623 3.0806 3.0401
K–S 0.1587 0.0596 0.0421 0.0266 0.0188 0.0133
Table 1: Accuracy of the LNapproximation for various number of stages
In order to illustrate Theorem 3, consider a project with nstages that are executed in
sequence, and that have i.i.d. exponential durations with rate parameter λ(i.e., the project
duration follows an Erlang distribution with parameters nand λ). A payoff pis obtained
upon completion of the project. After applying Theorem 2, we obtain the PDF of the NPV
of payoff p:
g(v) = λp
v−λr−1ln p
vλr−1n−1
|r|v(n−1)! .
The approximate lognormal distribution has location parameter α= ln(p)−rnλ−1and scale
parameter β=r√nλ−1, and is denoted by LN. Given a payoff p= 1,000, and a rate
parameter λ= 1, Fig. 1 shows the exact and the approximate PDF of the distribution of
the NPV of payoff pfor various values of n. The discount rate ris set equal to 0.5n−1.
Table 1 reports the mean, variance, skewness, kurtosis, and Kolmogorov–Smirnov (K–S)
test statistic (i.e., the maximum absolute difference in cumulative probability; the maximum
absolute difference between G(v) and the CDF of LN). We observe that, if nis small,
Lemma 3 (and hence Theorem 3) does not hold, and the approximation performs poorly. If,
on the other hand, nis large, the approximation is fairly accurate, and the NPV of a payoff
pmay be approximated by a lognormal distribution.
4 A lognormal approximation of the NPV distribution
Theorem 3 only holds for cash flows that are incurred after a sufficient number of stages.
Hence, the NPV of a cash flow does not always follow a lognormal distribution. Often, it
is impossible to characterize the exact NPV distribution of a cash flow, however, we can
use Theorem 1 to obtain its moments. A moment-matching procedure can then be used to
define a distribution that approximates the true NPV distribution.
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0 250 500 750 1000
g(v)
×10-3
0
1
2
3
n=1
Exact
LN
0 250 500 750 1000
×10-3
0
1
2
3
4
n=5
0 250 500 750 1000
g(v)
×10-3
0
1
2
3
4
5
n=10
0 250 500 750 1000
×10-3
0
2
4
6
n=25
v
0 250 500 750 1000
g(v)
0
0.002
0.004
0.006
0.008
0.01
n=50
v
0 250 500 750 1000
0
0.005
0.01
n=100
Figure 1: PDF of the exact NPV and the LNapproximation for various number of stages
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Moment-matching procedures can be evaluated along three lines: (1) the number of
moments matched, (2) the computational efficiency, and (3) the generality of the solution.
Ideally, a moment-matching procedure uses closed-form expressions to match as many mo-
ments as possible under general conditions. Most of the literature on moment matching
has focussed on the use of phase-type (PH) distributions (Osogami 2005). Using PH dis-
tributions, up to three moments can be matched using closed-form expressions (Osogami
and Harchol-Balter 2006). In this article, we do not adopt PH distributions, however, we
use a lognormal approximation of the NPV distribution of a cash flow c. Not only does
the lognormal distribution allow us to develop closed-form expressions to match up to three
moments of any real-valued distribution with non-zero skew, it is also a logical choice as the
NPV distribution of a cash flow cconverges to a (reflected) lognormal distribution if it is
incurred after a sufficient number of stages (see also Theorem 3).
In what follows, we define two moment-matching procedures. In a first procedure, we
match the first two moments of the NPV distribution. A second procedure matches the first
three moments. We use L2and L3to denote both approximations, respectively.
Proposition 1. We can approximate the NPV distribution by matching its first two moments
using a (reflected) lognormal distribution with scale and location parameters:
β=pln (1 + η2),
α= ln(µ)−0.5β2,
where µand η=σ2µ−2are the mean and Squared Coefficient of Variation (SCV) of the
NPV distribution, respectively.
In order to match three moments, we use a three-parameter (or bounded) lognormal
distribution (Aitchison and Brown 1957) with location, shape, and threshold parameters α,
β, and κ, respectively. The threshold parameter can be used to bound the support of the
distribution, and can either serve as a lower or as an upper bound (for matching distributions
with positive/negative skew, respectively). The mean, variance, skewness, kurtosis, PDF,
and CDF of the three-parameter lognormal distribution are given by:
µL3=κ+δeα+0.5β2,(4)
σ2
L3=eβ2−1e2α+β2,(5)
γL3=δ2 + eβ2peβ2−1,(6)
θL3=e2β23 + eβ22 + eβ2−3,
gL3(v) = 1
δ(v−κ)β√2πe(ln(δ(v−κ))−α)2
2β2,
GL3(v) = 1
2−δ
2Erf α−ln (δ(v−κ))
β√2,
where δ=−1 if the distribution has negative skew, and δ= 1 otherwise.
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L2approximation
n1 5 10 25 50 100
µL2666.67 620.92 613.91 609.53 608.04 607.29
σ2
L255,556 16,334 8,654 3,589 1,817 914
γL21.1049 0.6262 0.4581 0.2958 0.2106 0.1495
θL25.2463 3.7053 3.3754 3.1560 3.0790 3.0397
K–S 0.1357 0.0597 0.0421 0.0266 0.0188 0.0133
L3approximation
n1 5 10 25 50 100
µL3666.67 620.92 613.91 609.53 608.04 607.29
σ2
L355,556 16,334 8,654 3,589 1,817 914
γL3-0.566 -0.235 -0.163 -0.101 -0.071 -0.050
θL33.5743 3.0981 3.0470 3.0182 3.0090 3.0045
K–S 0.0590 0.0118 0.0059 0.0023 0.0011 0.0006
Table 2: Accuracy of the L2and L3approximations for various number of stages
Proposition 2. We can approximate the NPV distribution by matching its first three mo-
ments using a bounded lognormal distribution with parameters:
β=v
u
u
u
u
tln
21
/3
2+γ2+p4γ2+γ41
/3+2+γ2+p4γ2+γ41
/3
21
/3−1
,
α=0.5ln σ2
eβ2−1−β2,
κ=µ−δeα+0.5β2,
where µ,σ2, and γare the mean, variance, and skewness of the NPV distribution.
In order to illustrate the accuracy of the lognormal approximations, we revisit the last
example of Section 3. Fig. 2 shows the exact and the approximate PDF of the NPV distri-
bution for various values of n. Table 2 reports the mean, variance, skewness, kurtosis, and
Kolmogorov-Smirnov test statistic. We observe that the L3approximation is almost always
very accurate, whereas the L2approximation has more or less the same accuracy as the LN
approximation. This latter observation is no surprise. If nis small, neither Lemma 3 nor
Theorem 3 hold, and the approximations fail to achieve a good accuracy. In addition, the
L2and LNapproximations only take into account the first two moments. As a result, they
are always dominated by the L3approximation.
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0 250 500 750 1000
g(v)
×10-3
0
1
2
3
4
n=5
Exact
L2
0 250 500 750 1000
g(v)
×10-3
0
1
2
3
4
5
n=10
v
0 250 500 750 1000
g(v)
×10-3
0
2
4
6
n=25
0 250 500 750 1000
×10-3
0
1
2
3
4
n=5
Exact
L3
0 250 500 750 1000
×10-3
0
1
2
3
4
5
n=10
v
0 250 500 750 1000
×10-3
0
2
4
6
n=25
Figure 2: PDF of the exact NPV, the L2, and the L3approximation for various number of
stages
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5 NPV of a project with multiple stages and interme-
diate cash flows
In this section, we consider a project with multiple stages w:w∈N={1,2, . . . , n}, and
assume that a cash flow cwis incurred at the start of stage w. A payoff pis obtained
upon completion of the project. For notational convenience, we let cn+1 ≡p. Let c=
{c1, c2, . . . , cn, cn+1}denote the set of cash flows that are incurred during the lifetime of the
project. In addition, define Vw, the random variable that represents the NPV of cash flow
cw, and let Vc=Pn+1
w=1 Vwdenote the random variable that captures the NPV of the project.
Because the NPV of a cash flow cxdepends on the NPV of an earlier cash flow cw,Vxdepends
on Vwfor all x, w : 1 ≤w < x ≤n+ 1. Hence, Vcis the sum of a number of dependent
random variables whose distribution converges to a (reflected) lognormal distribution if their
associated cash flow is not incurred during the early stages of the project.
Determining the distribution of Vcis closely related to finding the distribution of the
lognormal sum (i.e., the sum of a number of random variables that follow a lognormal
distribution). Even though the lognormal sum has received considerable attention in the
literature, few exact results are available (Yan et al. 2016). In what follows, we first develop
exact, closed-form expressions for the moments of the distribution of Vc. We then use the
lognormal approximation developed in Section 4 to approximate the NPV distribution and
illustrate its accuracy by means of an example. Next, we show that Vcis normally distributed
if the number of cash flows is sufficiently large, and propose a new approximation based on
the normal distribution. Again, we illustrate the accuracy of this approximation by means
of an example.
Theorem 4. Consider a project with multiple stages w:w∈N, and let c=
{c1, c2, . . . , cn, cn+1}denote the set of cash flows that are incurred at the start of each stage
(where cn+1 ≡pis the payoff that is obtained upon project completion). In addition, Vw
denotes the random variable that represents the NPV of cash flow cw, and Vc=Pn+1
w=1 Vwis
the random variable that captures the NPV of the project. The moments of the distribution
of Vcare:
µc=
n+1
X
w=1
µw,
σ2
c=eΣce,
γc= (eΓce)σ−3
c,
θc= (eΘce)σ−4
c,
where eis a vector of ones, and Σc,Γc, and Θcare the central covariance, coskewness, and
cokurtosis matrices, respectively. Σc,Γc, and Θccapture the covariance, coskewness, and
cokurtosis of the NPV of the cash flows in c. Table 3 provides a summary of the closed-form
expressions that allow to calculate the entries of these cross-moment matrices.
In order to illustrate Theorem 4, we use an example project with 3 stages. In the example,
cash outflows are incurred at the start of the project, and at the start of the third stage. Cash
inflows, on the other hand, are received at the start of the second stage, and upon completion
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Mean µ
µw=cwa1
Covariance matrix Σc
Σc(w, w) = σ2
w=c2
w(a2−a2)
Σc(w, x) = cwcxb1(a2−a2) = c−1
wcxb1Σc(w, w)
Central coskewness matrix Γc
Γc(w, w, w) = γwσ3
w=c3
w(a3−3a2a1+ 2a3)
Γc(w, w, x) = c−1
wcxb1Γc(w, w, w)
Γc(w, x, x) = cwc2
x(a3b2−a2a1(2b2+b2)+2a3b2)
Γc(w, x, y) = c−1
xcyh1Γc(w, x, x)
Central cokurtosis matrix Θc
Θc(w,w,w,w)= θwσ4
w=c4
w(a4−4a3a1+ 6a2a2−3a4)
Θc(w,w,w,x) = c−1
wcxb1Θc(w,w,w,w)
Θc(w,w,x,x) = c2
wc2
x(a4b2−2a3a1(b2+b2)+a2a2(b2+5b2)−3a4b2)
Θc(w,x,x,x) = cwc3
x(a4b3−a3a1(b3+3b2b1)+3a2a2(b2b1+b3)−3a4b3)
Θc(w,w,x,y) = c−1
xcyh1Θc(w,w,x,x)
Θc(w,x,x,y) = c−1
xcyh1Θc(w,x,x,x)
Θc(w,x,y,y) = cwcxc2
y((a4−a3a1)b3h2−(h2+2h2) ((a3a1−a2a2)b2b1)+(a2a2−a4) 3b3h2)
Θc(w,x,y,z) = c−1
yczo1(r)Θc(w,x,y,y)
ai=φ1,w−1(ir)bi=φw,x−1(ir)hi=φx,y−1(ir)oi=φy,z−1(ir)
ai=φi
1,w−1(r)bi=φi
w,x−1(r)hi=φi
x,y−1(r)
Table 3: Summary of closed-form expressions that allow to calculate the moments of the
NPV distribution of a project
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w cwfw(t)kwτwdws2
w
1 -300 gamma 1.5 1.0 1.5 1.5
2 250 gamma 2.5 1.0 2.5 2.5
3 -750 gamma 0.5 1.0 0.5 0.5
p1000
r0.05
Table 4: Data of the example project with three stages
Exact Simulation L2L3L3without
cross moments
µ168.21 168.21 168.21 168.21 168.21
σ21,533 1,533 1,533 1,533 10,276
γ-1.035 -1.035 0.1006 -1.035 -2.620
θ4.7421 4.7420 3.0180 4.9631 17.269
G(0) NA 0.0105 0.0008 0.0105 0.1018
K–S NA NA 0.0734 0.0055 0.1018
Table 5: Accuracy of the L2and L3approximations of the NPV distribution of a project
with intermediate cash flows
of the project. Each stage whas a duration that follows a gamma distribution with shape
and scale parameters kwand τw, respectively. The gamma distribution was chosen because it
is a general distribution (e.g., the exponential, Erlang, and chi-squared distributions are all
special cases of the gamma distribution) that has many practical applications. We assume a
discount rate r= 0.05. The data of the example project are summarized in Table 4. Fig. 3
shows the L2and L3approximations, as well as the simulated PDF of the project NPV
(note that we have to resort to simulation as it is no longer an easy task to determine the
exact NPV distribution). It is clear that, in this example, the L2approximation performs
very poorly. The L3approximation, however, is once more very accurate. Table 5 reports
on the moments of the NPV distribution, the probability to have a negative project NPV,
and the Kolmogorov–Smirnov test statistic. The exact moments have been obtained using
Theorem 4. We observe that the simulation (with 1 billion replications) almost perfectly
matches the exact moments, which supports the claim that the simulated PDF is close to
the true PDF of the project NPV. As was also shown by Fig. 3, the L3approximation yields
excellent accuracy. If, however, cross moments are ignored (i.e., if we assume that the NPVs
of the cash flows are independent), the accuracy is abysmal. This is also reflected in the
probability to have a negative project NPV. Only the L3approximation is able to provide
an accurate estimate.
Theorem 5. Consider a project with multiple stages w:w∈N={1,2, . . . , n}that are
executed in sequence. At the start of each stage w:w∈N, a cash flow cwis incurred, and
a payoff p≡cn+1 is obtained upon completion of the project. Let Vwdenote the random
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-100 -50 0 50 100 150 200 250 300
g(v)
0
0.005
0.01
0.015
Simulation
L2
v
-100 -50 0 50 100 150 200 250 300
g(v)
0
0.005
0.01
0.015
Simulation
L3
Figure 3: PDF of the simulated NPV, and the L2and L3approximations for a project with
intermediate cash flows
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variable that represents the NPV of cash flow cw, and let Vc=Pn+1
w=1 Vwdenote the random
variable that captures the NPV of the project. If r > 0, and if s2
w>0for all w∈N,
the project NPV converges to a normal distribution, with mean µcand variance σ2
c, as the
number of stages increases.
Note that Theorem 5 also applies in a more general context where stages are not neces-
sarily executed in sequence. In fact, Theorem 5 holds as long as a sufficient number of cash
flows are incurred during the lifetime of a project.
We use an example to illustrate Theorem 5. The example project has nstages with
gamma-distributed durations with shape and scale parameters kiand τi, respectively. Cash
outflows are incurred at the start of odd stages. Cash inflows, on the other hand, are obtained
at the start of even stages, and upon completion of the project. The discount rate requals
0.1n−1. Table 6 summarizes the data of the example project. Fig. 4 shows the simulated
and the approximate PDF of the distribution of the project NPV. Next to the lognormal L3
approximation, we now also include a normal approximation that has mean µcand variance
σ2
c, and that is denoted by N. We observe that, as nincreases, the project NPV converges
to a normal distribution, and the accuracy of the Napproximation improves. Even so, the
L3approximation still performs better due to the extra moment matched. These findings
are confirmed by Table 7 that reports on the moments of the NPV distribution, and on
the Kolmogorov–Smirnov test statistic. For reference, we have also included the CPU time
required to run the simulation (with 1 billion replications) and to calculate the moments
using the closed-form expressions provided in Table 3. Both the simulation as well as the
exact approach were implemented in Visual Studio C++. Although the simulation model
yields good accuracy (also for a lower number of replications), it can hardly compete with
an exact, closed-form approach that requires less than a second of CPU time when 100
stages are considered. In addition, most of the computation time is spent on calculating
the cokurtosis matrix. Our approach, however, only requires that the first three moments
are specified (i.e., there is no need to determine the kurtosis). If only three moments are
calculated, the required CPU time drops to 0.046 seconds (for n= 100). Even if, for very
large n, the computation of the coskewness matrix becomes too time consuming, it suffices
to calculate only the first two moments as the Napproximation becomes more accurate as
nincreases (i.e., for large n, it is no longer necessary to calculate the coskewness matrix).
6 Optimal sequence of stages
The problem of finding the optimal sequence (that maximizes the eNPV over all possible se-
quences) can be seen as a special case of the stochastic NPV maximization problem (SNPV).
The SNPV tries to maximize the eNPV of a project with nstages that do not necessarily
have to be scheduled in series. A solution to the SNPV is a policy that schedules stages such
that the eNPV of the project (i.e., the expected sum of the discounted cash flows that are
incurred during the lifetime of the project) is maximized. The SNPV has been considered
by, among others, Sobel et al. (2009), Creemers et al. (2010), and Wiesemann et al. (2010).
For a review of the literature on the SNPV, refer to Wiesemann and Kuhn (2015).
Another related problem is the LCFDP; a variant of the Sequential Testing Problem
(STP) where nprecedence-related tests have to be scheduled such that the expected cost
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cw=250 if wis even
−250 if wis odd
kw=
0.5 if w∈ {1,6,11, . . .}
1.0 if w∈ {2,7,12, . . .}
1.5 if w∈ {3,8,13, . . .}
2.0 if w∈ {4,9,14, . . .}
2.5 if w∈ {5,10,15, . . .}
fw(t) =
gamma if w∈ {1,6,11, . . .}
exponential if w∈ {2,7,12, . . .}
gamma if w∈ {3,8,13, . . .}
Erlang if w∈ {4,9,14, . . .}
gamma if w∈ {5,10,15, . . .}
τw=2.0 if wis even
1.0 if wis odd
p= 1000
r= 0.1n−1
Table 6: Data of the example project with nstages and intermediate cash flows
n= 10 n= 30 n= 100
Sim N L3Sim N L3Sim N L3
µ783.04 783.04 783.04 782.16 782.16 782.16 781.86 781.86 781.86
σ22,584 2,584 2,584 875 875 875 264 264 264
γ-0.361 0.0 -0.361 -0.211 0.0 -0.211 -0.117 0.0 -0.116
θ3.1159 3.0 3.1162 3.0415 3.0 3.0402 3.0769 3.0 3.0122
K–S NA 0.0297 0.0032 NA 0.0155 0.0001 NA 0.0080 0.0003
CPU (s) 2,250 0.000 11,279 0.015 90,955 0.967
Table 7: Accuracy of the Nand L3approximations for the NPV of a project with interme-
diate cash flows and nstages
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600 800 1000
g(v)
0
0.002
0.004
0.006
0.008
0.01
n= 10
600 800 1000
g(v)
0
0.005
0.01
0.015
n= 30
v
600 800 1000
g(v)
0
0.005
0.01
0.015
0.02
0.025
n= 100
Sim
N
600 800 1000
0
0.002
0.004
0.006
0.008
0.01
n= 10
600 800 1000
0
0.005
0.01
0.015
n= 30
v
600 800 1000
0
0.005
0.01
0.015
0.02
0.025
n= 100
Sim
L3
Figure 4: PDF of the simulated NPV, and the Nand L3approximations for various number
of stages
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of the diagnosis of a system is minimized. Each test w:w∈N={1, . . . , n}has a known
cost cwand a failure probability qw. In this article, we consider the setting where a single
test results in the failure of the system (i.e., we study so-called n-out-of-nor serial systems).
For such a setting, it can be shown that there exists a full order sequence of tests in Nthat
is globally optimal. The LCFDP is related to the R&D project scheduling problem studied
in De Reyck and Leus (2008), who show that their problem is NP-hard. It follows that the
LCFDP is also NP-hard if tests are precedence-related (Wei et al., 2013). The LCFDP arises
in many practical contexts, such as the inspection of containers arriving at a port (Madigan
et al., 2011) and the identification of toxic chemicals (Gowtham et al., 2012). A literature
review on the STP in general, and on the LCFDP in particular, may be found with ¨
Unl¨uyurt
(2004), Wei et al. (2013), and Coolen et al. (2014).
We define the serial SNPV as the problem to find the optimal sequence of stages that
maximizes the eNPV over all possible sequences. For serial projects, the serial SNPV is
equivalent to the SNPV. In what follows, we show that: (1) the LCFDP is equivalent to
the serial SNPV, (2) a well-known result from the literature on the LCFDP may be used
to obtain the optimal solution to the serial SNPV if stages are not precedence related,
and (3) methods for solving the SNPV can also be used to solve the LCFDP. In addition,
we perform a computational experiment that shows that the state-of-the-art procedure for
solving the SNPV (a more general problem where stages are allowed to be executed in
parallel) outperforms the state-of-the-art procedure for solving the LCFDP.
6.1 Equivalence of the serial SNPV and the LCFDP
Let s={s1, . . . , sn}denote a sequence of nstages, where swis the stage at position win
the sequence. As shown in Section 5, the eNPV of a sequence sis given by:
cs1+
n+1
X
w=2
φ1,(w−1)(r)csw,
where φ1,w is the discount factor for a sequence of stages {s1, . . . , sw}. The objective of the
serial SNPV is to find a sequence that maximizes:
cs1+ n
X
x=2
csx
x−1
Y
w=1
φsw(r)!+ cn+1
n
Y
w=1
φw(r)!,
where the latter term is a constant that does not depend on the sequence of stages (i.e.,
the latter term may be ignored when making sequencing decisions), and hence the objective
reduces to:
max
scs1+ n
X
x=2
csx
x−1
Y
w=1
φsw(r)!.(7)
The objective of the LCFDP, on the other hand, is to find a sequence of tests that
minimizes the cost of the sequential diagnosis of a system, and is given by:
max
scs1+ n
X
x=2
csx
x−1
Y
w=1
psw!,(8)
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w cwλwφwcw(1 −φw)−1sw
1 -10 1
/20.833 -60 5
2 10 1
/40.714 35 2
3 -15 1
/60.625 -40 3
4 20 1
/80.556 45 1
5 -36 1
/20 0.250 -48 4
p100
r0.1
Table 8: Data of the example project
where pw= 1 −qwis the success probability of test w. Eq. (7) and Eq. (8) are equivalent
if φw(r)≡pwfor all w:w∈N. We conclude that the LCFDP is equivalent to the serial
SNPV, which in turn is a special case of the SNPV.
6.2 Optimal sequence
In the absence of precedence relationships, Boothroyd (1960) has shown that the optimal
solution to the LCFDP is a sequence that arranges tests in (increasing) order of their ratio
of cost over failure probability. Therefore, for the LCFDP, the optimal sequence can be
determined in polynomial time, and if:
cs1
qs1≤cs2
qs2≤. . . ≤csn
qsn
,
then s={s1, s2, . . . , sn}is optimal.
The above result can also be used to determine the optimal sequence that maximizes the
eNPV of a project where stages are not precedence related. More precisely, in the absence
of precedence relationships, sequence s={s1, s2, . . . , sn}is optimal if:
cs1
1−φs1(r)≤cs2
1−φs2(r)≤. . . ≤csn
1−φsn(r).
To illustrate this finding, we use an example project with 5 stages that have exponentially-
distributed durations with rate parameter λw:w∈N={1,2,3,4,5}. The data of the
example project are summarized in Table 8. Note that φwis the moment-generating function
of fwabout −r, and is given by:
φw(r) = λw
λw+r(9)
for an exponentially-distributed duration with rate parameter λw. The optimal sequence
executes stages 4, 2, 3, 5, and 1 in series, and yields an eNPV of 15.22.
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6.3 Solving the LCFDP
We solve the LCFDP using the state-of-the-art procedure of Creemers (2017) that was de-
signed to solve the SNPV. Creemers (2017) assumes that stage durations are exponentially
distributed, and uses a Continuous-Time Markov Chain (CTMC) to model the state space.
A backward Stochastic Dynamic Program (SDP) is used to obtain the globally optimal pol-
icy that maximizes the eNPV of a project (note that a solution to the SNPV is a scheduling
policy rather than a sequence). In contrast to most of the literature on the scheduling of
Markovian PERT networks (i.e., PERT networks where stages have exponentially-distributed
durations), Creemers (2017) does not use Uniformly Directed Cuts (UDCs) to structure the
state space, nor does he represent the state of the system using sets of idle, ongoing, and
finished stages (see e.g., Creemers et al., 2010). Instead, Creemers (2017) uses arrays to
store states that are defined only by the set of finished stages. The cardinality of a state
(i.e., the number of finished stages) determines the array in which the state is stored (there
is one array for each number of finished stages). Because states with cardinality (i+ 1)
are only accessible from states with cardinality i, at most two arrays need to be stored in
memory (i.e., after calculating all value functions of states with cardinality i, states with
cardinality (i+ 1) are no longer needed, and they can be removed from memory). Together
with a stricter definition of the state space (by only using the set of finished stages), this
more efficient structuring of the state space results in a significant reduction of memory and
computational requirements (when compared to other methods that solve the SNPV).
In order to solve an instance of the LCFDP by means of a procedure for solving the SNPV,
tests first need to be “transformed” into stages. As explained in Section 6.1, the serial SNPV
and the LCFDP are equivalent if φw(r)≡pwfor all w:w∈N. In the procedure of Creemers
(2017), stages are assumed to have exponentially-distributed durations, and therefore, the
discount factor of a stage wis given by Eq. (9). As such, a test wwith cost cwand failure
probability qwcan be transformed into a stage wwith cost cwand rate parameter:
λw=qwr
1−qw
.
After transforming all tests into stages, the procedure of Creemers (2017) can be used to
solve an instance of the LCFDP. Note that, in order to make sure that stages are executed in
a sequence, we impose a resource constraint (i.e., each stage requires one unit of a renewable
resource that has unit availability).
We compare the performance of the above approach with the state-of-the-art procedure
of Wei et al. (2013). Wei et al. (2013) propose both a Branch-and-Bound (B&B) as well as
an SDP procedure to solve the k-out-of-nSTP (i.e., at least kout of ncomponents should
be functional, otherwise the system is down). The SDP procedure significantly outperforms
the B&B, and in what follows, we will compare its performance with that of the procedure of
Creemers (2017). Note that, if we let k=n, the k-out-of-nSTP corresponds to the LCFDP
as defined by Boothroyd (1960). Similar to the procedures of Creemers et al. (2010) and
Coolen et al. (2014), The SDP procedure of Wei et al. (2013) uses UDCs to structure the
state space. Once the states of a UDC are no longer required, the UDC is discarded, and
the memory is freed.
We use the instances of Wei et al. (2013) to compare the performance of both SDP
procedures. Wei et al. (2013) use RanGen (Demeulemeester et al., 2003) to generate three
20
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Creemers (2017) Wei et al. (2013)
nOS = 0.8 OS = 0.6 OS = 0.4 OS = 0.8 OS = 0.6 OS = 0.4
10 10 10 10 10 10 10
20 10 10 10 10 10 10
30 10 10 10 10 10 10
40 10 10 10 10 10 10
50 10 10 10 10 10 10
60 10 10 10 10 10 10
70 10 10 10 10 10 9
80 10 10 10 10 10 0
90 10 10 0 10 10 0
100 10 10 0 10 10 0
110 10 10 0 10 7 0
120 10 9 0 10 0 0
Table 9: Number of instances solved (out of 10) by the procedures of Creemers (2017) and
Wei et al. (2013)
data sets that each contain 10 instances for each value of n:n∈ {10,20,...,120}and for
each value of OS : OS ∈ {0.4,0.6,0.8}(where OS is the Order Strength; a measure of the
density of the project network). Each data set has different failure probabilities. Because
failure probabilities do not impact the computational performance of the SDP procedures,
we select the data set that has the lowest failure probabilities (i.e., failure probabilities are
drawn from a uniform distribution with a minimum of 0.8 and a maximum of 1). Both
procedures are tested on an Intel 3.3 GHz desktop computer with 16GB of RAM.
Table 9 reports on the number of instances solved by each approach, and shows that the
procedure of Creemers (2017) outperforms the procedure of Wei et al. (2013). This can
be explained by the more efficient memory-management techniques adopted by Creemers
(2017). When comparing average computation times (in seconds) on instances that could
be solved by Wei et al., however, Table 10 shows that the procedure of Creemers (2017) is
somewhat slower (26.5% on average). Since the procedure of Creemers (2017) was designed
to solve the SNPV (a more general problem where stages are allowed to be executed in
parallel), this does not come as a surprise. In addition, it is clear that memory requirements
rather than CPU times are the bottleneck for the problem at hand. Even for larger problems,
Table 11 shows that the procedure of Creemers (2017) is able to solve instances within a
reasonable time frame. We conclude that the procedure of Creemers (2017) is the current
state-of-the-art for solving the LCFDP/serial SNPV.
7 NPV of a general project with multiple stages that
are scheduled using a scheduling policy
In this section, we illustrate how to determine the NPV of a general project with multiple
stages that are scheduled using a scheduling policy. To this end, we use an example project
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Creemers (2017) Wei et al. (2013)
nOS = 0.8 OS = 0.6 OS = 0.4 OS = 0.8 OS = 0.6 OS = 0.4
10 0 0 0 0 0 0
20 0 0 0 0 0 0
30 0 0 0.01 0 0 0.01
40 0 0.01 0.17 0 0.01 0.13
50 0 0.04 1.99 0 0.02 1.44
60 0.01 0.19 25.4 0 0.11 19.3
70 0.01 0.94 178 0.01 0.58 156
80 0.03 4.00 – 0.01 2.40 –
90 0.05 15.0 – 0.02 9.48 –
100 0.11 77.1 – 0.05 45 –
110 0.24 223 – 0.10 151 –
120 0.56 – – 0.24 – –
Table 10: Comparison of average computation time (in seconds) for the instances that could
be solved by Wei et al. (2013)
nOS = 0.8 OS = 0.6 OS = 0.4
10 0 0 0
20 0 0 0
30 0 0 0.01
40 0 0.01 0.17
50 0 0.04 1.99
60 0.01 0.19 25.4
70 0.01 0.94 205
80 0.03 4.00 2,013
90 0.05 15.0 –
100 0.11 77.1 –
110 0.24 323 –
120 0.56 1,009 –
Table 11: Average computation time (in seconds) for the procedure of Creemers (2017)
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w cwdwλwPredecessor
1 -50 1 1.0 ∅
2 -20 2 0.5 ∅
3 -10 2 0.5 {2}
p200
r0.1
Table 12: Data of the example project with general structure
with three stages. The first stage can be executed together with any of the other two stages.
The third stage, however, can only start after the second stage has finished. At the start of
each stage w, a cash flow cwis incurred. Upon completion of the project, a payoff p= 200 is
obtained. The discount rate r= 0.1. Each stage whas an exponentially-distributed duration
with rate parameter λw. The data of the example project are summarized in Table 12.
There are several policies that allow to schedule the stages during the execution of the
project. We discuss two: the Early-Start (ES) policy and the optimal policy. The ES policy
first starts stages 1 and 2, and upon completion of stage 2, stage 3 is started (note that stage
1 can still be ongoing at that time). The optimal policy maximizes the eNPV of the project,
and starts stages 1 and 3 only upon completion of stage 2. In what follows, we first discuss
how to determine the NPV of the ES policy.
In the ES policy, cash flows c1and c2are incurred at the start of the project (i.e., no
discounting is required). Cash flow c3is incurred upon completion of stage 2 (i.e., when stage
3 starts). Hence, the NPV of cash flow c3can be determined using factor φ2(r) = λ2(λ2+r)−1.
The NPV of payoff p, on the other hand, is only obtained if both stage 1 and the series of
stage 2 and 3 are finished. In other words, the distribution of time until we obtain payoff
pis the distribution of the maximum of an exponential distribution with rate parameter λ1
and an Erlang distribution with two phases that both have rate parameter λ2=λ3. The
corresponding factor can be determined as follows:
φp(r) = λ2
λ2+r2
−rλ2
λ1+λ22
λ1+r.
The eNPV of the ES policy can then be computed using Theorem 4, and amounts to 58.78.
Note that Theorem 4 can always be used to calculate the exact eNPV of any project/policy
as the eNPV does not require to calculate cross moments. For higher moments of the NPV
distribution, however, cross moments may need to be approximated if the sequence of cash
flows is not fixed (i.e., if the sequence of cash flows is probabilistic). In the ES policy, for
instance, the NPV of payoff pcan be impacted by stage 1 (if it takes long enough) as well as
by stages 2 and 3. In other words, there are two possible sequences that impact the NPV of
payoff p. Theorem 4 assumes that there is only a single sequence, and as a result, we have to
approximate the cross moments between payoff pand the cash flows of the preceding stages.
In our example, we can follow three different approaches:
•We can assume that stage 1 was finished before stage 2. In this case, the higher
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Exact Simulation Stage 1 finished Stage 1 ongoing Weighted approach
µ58.780 58.780 58.780 58.780 58.780
σ2971.08 971.10 967.25 969.50 968.00
γ-0.580 -0.581 -0.579 -0.580 -0.580
θ2.8549 2.8552 2.8361 2.8695 2.8457
Table 13: Accuracy of different approaches to determine the NPV of the ES policy
moments of the NPV of payoff pare determined solely by stages 2 and 3 (i.e., the
sequence consists of stages 2 and 3).
•We can assume that stage 1 was still ongoing after completion of stage 2. In this
case, the higher moments of the NPV of payoff pare determined by stage 2 and the
maximum of stages 1 and 3 (i.e., the sequence consists of stage 2 and the maximum of
stages 1 and 3).
•We can adopt a weighted approach where both of the previous approaches are weighed
depending on their probability of occurrence.
The moments corresponding to each of the above approaches are given in Table 13. Table 13
also reports the exact and simulated moments. We observe that the error is relatively small,
however, further research is required to assess the accuracy for larger projects.
Next, we observe the optimal policy. In the optimal policy, there is only one possible
sequence (stage 2 is followed by stages 1 and 3 that are executed in parallel), and as result,
the moments of the project NPV can be determined in an exact manner using Theorem 4.
Whereas cash flow c2is incurred at the start of the project, cash flows c1and c3are incurred
upon completion of stage 2 (i.e., factor φ2(r) applies). Payoff pis obtained after both stage
1 and stage 3 are finished (i.e., after the maximum of two exponential durations). The
corresponding factor is given by:
φp(r) = λ1λ3(λ1+λ3+r)
(λ1+r)(λ3+r)(λ1+λ3+r).
The moments of the NPV distribution of the optimal policy are µ= 64.154, σ2= 698.43,
γ=−0.5342, and θ= 2.9649.
We also assess the accuracy of the L3approximation. Table 14 summarizes the results,
and Fig. 5 shows the simulated and the approximate PDF of the NPV distribution of the ES
and the optimal policy. We conclude that the L3approximation is once more very accurate.
8 Model extensions
In this section, we discuss two model extensions. A first extension allows stages (and hence
projects) to fail. Stage/project failure is common in the literature on R&D projects (Sommer
2004; De Reyck and Leus 2008; Creemers et al. 2015), and can easily be incorporated in our
24
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ES policy Optimal policy
Simulation L3Simulation L3
µ58.780 58.780 64.154 64.154
σ2971.10 971.08 698.45 698.43
γ-0.581 -0.580 -0.534 -0.534
θ2.8552 3.6048 2.9654 3.5116
K–S NA 0.0267 NA 0.0190
Table 14: Accuracy of the L3approximations to model the NPV of the ES policy and the
optimal policy
-50 0 50 100 150
g(v)
0
0.005
0.01
0.015
0.02 ES policy
Simulation
L3
v
-50 0 50 100 150
g(v)
0
0.005
0.01
0.015
0.02 Optimal policy
Simulation
L3
Figure 5: PDF of the simulated NPV and the L3approximation of the ES policy and the
optimal policy
25
doi:10.1016/j.ejor.2017.12.039 •www.stefancreemers.be •info@stefancreemers.be
approach. We need only to redefine factor φw(r):
φw(r) = pwφ∗
w(r),
where pwis the probability of success of stage w, and φ∗
w(r) is the factor given by Eq. (2)
(i.e., the factor that does not take into account stage/project failure).
A second model extension allows for different discount rates to be applied during different
stages of the project. This extension requires a redefinition of factor φw,x(r):
φw,x(r) =
x
Y
y=w
φy(ry),
where ryis the discount rate that applies for stage y, and r={w, w + 1, . . . , x}is the vector
of discount rates that apply to stages y:w≤y≤x.
9 Conclusions
In this article, we considered projects with multiple stages w:w∈N={1,2, . . . , n}
that are executed in sequence. Each stage w:w∈Nhas a random duration Tequipped
with probability function f(t). A cash flow cw(positive or negative) may be incurred upon
the start of stage w, and a payoff pis obtained at the completion of the project. We use
continuous compounding and a discount rate rto determine the NPV of a project.
Our main contributions can be summarized as follows: (1) we obtain exact, closed-form
expressions for the moments of the NPV of a project, (2) we develop a highly accurate
closed-form approximation of the distribution of the project NPV, (3) we show that the
NPV of a single cash flow converges to a (reflected) lognormal distribution if the cash flow
is not incurred during the early stages of the project, (4) we show that the NPV of a project
converges to a normal distribution if a sufficient number of cash flows are incurred during
the lifetime of the project, (5) we show that the problem of finding the optimal sequence
of stages is equivalent to the LCFDP, (6) we show that the optimal sequence of stages
can be obtained in polynomial time if stages are not precedence related, (7) we perform a
computational experiment to identify the state-of-the-art procedure to determine the optimal
sequence of stages if they are precedence related, and (8) we illustrate how our approach
can be used to determine the moments and the NPV of a general project where stages are
scheduled using a scheduling policy.
Our work can be directly applied in the fields of project selection, project portfolio
management, and project valuation. In these fields, a project is often seen as a sequence of
stages with intermediate cash flows (including a payoff that is obtained upon the successful
completion of the project). Project selection/investment decisions can be made based on the
eNPV and the risk of a project. The risk of a project/an investment is often modeled using
the variance of the NPV. Other measures of risk include the skewness and/or kurtosis of
the NPV, and the probability to have a negative NPV. Until now, Monte Carlo simulation
was the only tool available to obtain estimates for these measures. Our work, however,
offers a valid alternative to Monte Carlo simulation, and allows to obtain a highly accurate
approximation of the NPV distribution, and an exact characterization of its moments.
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In the (more operational) field of project scheduling, our work is related to CPM/PERT
in the sense that we also focus on a single sequence of stages, and also use normal (log-
normal) approximations. As a result, the limitations of our work are similar to those of
CPM/PERT. Therefore, future research should further investigate methods that have been
used to generalize CPM/PERT, and that may also be applied here (e.g., network transforma-
tions/reductions and bounding procedures). In addition, the scheduling of stages of a general
project is also an important direction for future research. Determining the optimal release
dates to maximize the eNPV of a project is also a topic worthy of study. Other directions
for extending our results are the inclusion of time-dependent cash flows and interdependent
stage durations.
Appendix. Proofs
Proof of Lemma 1.The proof follows from the definition of the MGF:
MT(u) =
∞
Z
0
f(t)eutdt, if Tis continuous,
=
∞
X
t=0
f(t)eut,if Tis discrete.
Proof of Theorem 1.Let Vdenote the random variable that represents the NPV of a
cash flow cthat is incurred at time T. The MGF of Vis:
MV(u) =
∞
X
i=0
uimi
i!,
where miis the ith raw moment of the NPV distribution:
mi=
∞
Z
0
f(t)(e−rt)idt=φ(ir),if Tis continuous,
=
∞
X
t=0
f(t)(e−rt)i=φ(ir),if Tis discrete.
Using these raw moments, we can obtain the mean, variance, skewness, kurtosis, and even
higher-order moments of the NPV of cash flow c.
Proof of Theorem 2.If we solve Eq. (1) for t, we obtain:
tv= ln c
vr−1.
where tvis the time at which cash flow cneeds to be incurred in order to obtain NPV vfor
a given discount rate r. As a result, F(tv) not only represents the probability to have a time
t≤tv, but it also represents the probability to have an NPV ≥v.
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Proof of Lemma 2.Factor φ1,n(r) can be obtained as follows:
φ1,n(r) =
∞
Z
0
···
∞
Z
0
f1(t1)e−rt1···fn(tn)e−rtndt1···dtn,
=φ1(r)
∞
Z
0
···
∞
Z
0
f2(t2)e−rt2···fn(tn)e−rtndt2···dtn,
···
=Y
w∈N
φw(r).
In general, let φw,x(r) denote the factor for stages wto x, where x≥w:
φw,x(r) =
x
Y
y=w
φy(r).
Proof of Lemma 3.The proof follows from the Central Limit Theorem (CLT).
Proof of Theorem 3.The proof is a direct application of Theorem 2 and Lemma 3. If
nis sufficiently large, the duration of the project is normally distributed, and if F(t) is a
normal distribution function, G(v) can be expressed as follows:
G(v) = 1
2+1
2Erf ln(v)−(ln(p)−rdN)
√2rsN.
When substituting ln(p)−rdNby αand rsNby β, we get:
G(v) = 1
2+1
2Erf ln(v)−α
√2β,
which is the CDF of the lognormal distribution with location parameter α= ln(p)−rdNand
scale parameter β=rsN. Note that, if pis negative, the NPV distribution of pconverges to
a reflected lognormal distribution.
Proof of Proposition 1.Consider a (reflected) lognormal distribution with location and
scale parameters αand β, respectively. The mean and SCV of that distribution are given
by:
µL2=eα+0.5β2,(10)
η2
L2=e(2α+β2)eβ2−1e−2(α+0.5β2).(11)
The unique solution for βcan easily be obtained by solving Eq. (11):
β=qln(1 + η2
L2).
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Note that, as long as σ2
L2>0, then η2
L2>0, and therefore β > 0. Given β, the unique
solution for αcan easily be obtained by solving Eq. (10):
α= ln(µL2)−0.5β2.
Note that, if µis negative, a reflected lognormal distribution can be used to match the NPV
distribution. We conclude that αand βare the unique solution to Eqs. (10–11), and that
they are well defined for all µL2, σ2
L2∈R, and for σ2
L2>0.
Proof of Proposition 2.First, we obtain the unique solution for βfrom Eq. (6). We
have:
γL3=δ(2 + eβ2)peβ2−1,
γ2
L3=e3β2+ 3e2β2−4.
If we let q=eβ2and l= 4 + γ2
L3, we obtain the following cubic equation:
q3+ 3q2−l= 0.
The discriminant of q3+ 3q2−lis ∆ = −27(l−4)l, and is always negative if γL36= 0. For
cubic equations, if ∆ <0, the equation has one unique real root and two non-real complex
conjugate roots. The unique real root of q3+q2−lis:
q=21
/3
−2 + l+√−4l+l21
/3+−2 + l+√−4l+l21
/3
21
/3−1.
After substituting q=eβ2and l= 4 + S2
L3, we obtain the unique, real solution for β:
β=v
u
u
u
u
tln
21
/3
2 + γ2
L3+p4γ2
L3+γ4
L31
/3+2 + γ2
L3+p4γ2
L3+γ4
L31
/3
21
/3−1
.
Given β, the unique solution for αcan easily be obtained by solving Eq. (5):
α= 0.5ln σ2
L3
eβ2−1−β2.
Given αand β, the unique solution for κcan easily be obtained by solving Eq. (4):
κ=µL3−δeα+0.5β2.
We conclude that α,β, and κare the unique solution to Eqs. (4–6), and that they are well
defined for all µL3, σ2
L3, γL3∈R,σ2
L3>0, and for γL36= 0.
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Proof of Theorem 4.The covariance between the NPV of cash flow cxand the NPV of
an earlier cash flow cwis given by:
Σc(w, x) =
∞
Z
0
...
∞
Z
0
x−1
Y
y=1
fy(ty)
cwe
−r w−1
P
y=1
ty!−µw
cxe
−r x−1
P
y=1
ty!−µx
dt1...dtx−1.
Which can be rewritten as a sum of 4 parts:
Σc(w, x) =
∞
Z
0
···
∞
Z
0
x−1
Y
y=1
fy(ty)
cwe
−r w−1
P
y=1
ty!cxe
−r x−1
P
y=1
ty!
dt1···dtx−1
−
∞
Z
0
···
∞
Z
0
x−1
Y
y=1
fy(ty)
µxcwe
−r w−1
P
y=1
ty!
dt1···dtx−1
−
∞
Z
0
···
∞
Z
0
x−1
Y
y=1
fy(ty)
µwcxe
−r x−1
P
y=1
ty!
dt1···dtx−1
+
∞
Z
0
···
∞
Z
0
x−1
Y
y=1
fy(ty) (µwµx) dt1···dtx−1.
After application of Lemma 2, we get:
Σc(w, x) =
cwcxφ1,w−1(2r)φw,x−1(r)
−µxcwφ1,w−1(r)
−µwcxφ1,w−1(r)φw,x−1(r)
+µwµx.
From Theorem 1, we have that µw=cwφ1,w−1(r) and µx=cxφ1,x−1(r), and therefore:
Σc(w, x) =
cwcxφ1,w−1(2r)φw,x−1(r)
−cwcxφ1,w−1(r)φ1,w−1(r)φw,x−1(r)
−cwcxφ1,w−1(r)φ1,w−1(r)φw,x−1(r)
+cwcxφ1,w−1(r)φ1,w−1(r)φw,x−1(r).
Which, finally, can be simplified to:
Σc(w, x) = cwcxφw,x−1(r)φ1,w−1(2r)−φ2
1,w−1(r).(12)
The same approach can be used to determine the coskewness, the cokurtosis, and even the
higher-order cross moments of the NPV of the cash flows that are incurred during the lifetime
of the project.
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Proof of Theorem 5.Let (V) = {V1, V2, . . . , Vn, Vn+1}denote the non-stationary se-
quence of dependent random variables Vw: 1 ≤w≤n+ 1. For such a sequence, Bradley
and Tone (2015) have shown that a CLT holds if:
•the sequence is strongly mixing,
•the sequence has a maximum correlation that is strictly smaller than 1 for some Vw
and Vw+1 in (V),
•the Lindeberg condition holds.
Several mixing conditions have been defined in the literature (for an overview, refer to
Bradley (2005)). In this proof, we will show that sequence (V) is ρ-mixing (which automati-
cally implies that (V) is strongly mixing). A sequence is said to be ρ-mixing if the maximum
correlation between two random variables Vw, Vx∈(V) tends to zero for some wand xthat
are “far apart”. We use Eq. (12) to obtain the expression for the correlation between two
random variables Vwand Vx:
Corr(w, x) = φw,x−1(r)φ1,w−1(2r)−φ2
1,w−1(r)
qφ1,w−1(2r)−φ2
1,w−1(r)qφ1,x−1(2r)−φ2
1,x−1(r)
,
=φw,x−1(r)sφ1,w−1(2r)−φ2
1,w−1(r)
φ1,x−1(2r)−φ2
1,x−1(r).
It is easy to verify that Corr(w, x)→0 if φw,x−1(r)→0, or if φ1,w−1(2r) = φ2
1,w−1(r). If
cw>0, and if at least one stage z: 1 ≤z < w has s2
z>0, then σw>0, and it follows from
Eq. (3) that φ1,w−1(2r)> φ2
1,w−1(r). Therefore, we say that Corr(w, x)→0 if and only if
φw,x−1(r)→0. From Lemma 2 we know that φw,x−1(r) = Qx−1
y=wφy(r). In addition, if r > 0,
and if s2
y>0, then φy(r)<1, and φw,x−1(r)→0 if s2
y>0 for sufficient y:w≤y < x, and
for x−w→ ∞. In other words, if r > 0, and if s2
y>0 for sufficient y∈N, then sequence
(V) is ρ-mixing as n→ ∞.
In order to show that sequence (V) satisfies the second condition, we observe the corre-
lation between random variables Vwand Vw+1:
Corr(w, w + 1) = φw(r)sφ1,w−1(2r)−φ2
1,w−1(r)
φ1,w(2r)−φ2
1,w(r),
=φ1,w−1(2r)φ2
w(r)−φ2
1,w(r)
φ1,w−1(2r)φw(2r)−φ2
1,w(r).
A perfect correlation is achieved if φ1,w−1(2r)→0, or if φw(2r) = φ2
w(r). If s2
w>0, then
φw(2r)> φ2
w(r), and as a result, Corr(w, w + 1) →1 if and only if φ1,w−1(2r)→0. If
w→ ∞,φ1,w−1(2r)→0, however, because φ1,w−1(2r)> φ2
1,w−1(r), φ2
1,w−1(r) goes to zero
even faster. In addition, φw(2r)> φ2
w(r), and therefore, the maximum correlation between
random variables Vwand Vw+1 is always strictly smaller than 1.
To complete the proof, we still need to show that the Lindeberg condition holds. Instead
of verifying the Lindeberg condition itself, we show that sequence (V) satisfies the more strict
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doi:10.1016/j.ejor.2017.12.039 •www.stefancreemers.be •info@stefancreemers.be
Lyapunov condition. The Lyapunov condition requires that all random variables Vw∈(V)
have finite mean, variance, and at least one finite higher-order moment (Greene 2003). In
our case, the ith moment of Vxis finite if φ1,x−1(ir) is finite; if the MGF about −ir is defined
for all duration distributions fw(t):1≤w < x (see also Lemma 1). In general, the MGF
is defined for most duration distributions, and for most values of r. Therefore, we conclude
that, in general, the Lyapunov condition (and hence the Lindeberg condition) holds.
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