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Quantifying the structural complexity of projects with
first-order joint binary entropy
__________________________________________________________________________
Philippe Ruiz *
IESEG School of Management, 3 Rue de la Digue, 59000 Lille, France
* Phone: +33 3 20 54 58 92
_________________________________________________________________________
Abstract: The present research develops a new measure of projects‟ complexity: It analyzes
the structural sophistication of PERT networks through Information Theory and utilizes
Shannon‟s first-order joint binary entropy. Higher entropy means that a project is more
complex. A complex project is a project which is neither sure nor impossible. Unsurprisingly,
project entropy (or structural complexity) increases with time (i.e., longest path or number of
arrows that follow each other), together with the number of connections among the tasks. The
main discovery of this research is that later occurring risks increase project entropy
significantly more than equivalent risks occurring earlier.
Keywords: project complexity, entropy, PERT, Gantt chart, network diagram, project risk
APA reference for quoting this article:
Ruiz, P. (2013). Quantifying the structural complexity of projects with first-order joint binary
entropy. In P. Ruiz (Ed.), La gestion des projets complexes (pp. 101 – 115). Lille, France:
C2BR Press.
© 2013 C2BR Press. All rights reserved.
_________________________________________________________________________
1- Gantt and PERT-charts
Project management is a developing field of academic study in management, of considerable
diversity and richness (Turner, Anbari, & Bredillet, 2013).
The Gantt chart is a central project management instrument; it offers a graphic
timetable for the planning and controlling of various tasks, and recording advancement
towards completion of a project (Baker & Trietsch, 2009). The chart has a contemporary
variant, called the Program Evaluation and Review Technique.
The Program Evaluation and Review Technique – usually shortened to PERT – is a
model for project management discovered by the Department of Defense‟s US Navy Special
Projects Office in 1958 as part of the Polaris submarine-launched ballistic missile
development. This project was a clear reaction to the Sputnik emergency (Sapolsky, 1972).
PERT is essentially a way to scrutinize the tasks concerned in finishing a certain
project, in particular the time required to finish each task, and discovering the minimum time
required to close the whole project. It was created largely to make simpler the planning and
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scheduling of big and intricate projects. PERT was able to integrate uncertainty and it became
feasible to plan a project not knowing accurately the minutiae and lengths of all the activities.
This project representation was the first of its kind, a resurgence of scientific
management, founded in Fordism and Taylorism. Though every business now has its own
"project model" of some kind, they all look like PERT in some respect. Only DuPont
Corporation‟s critical path method was conceived at approximately the same time as PERT.
The most well-known part of PERT is the "PERT Networks", charts of timelines that
interrelate. PERT is proposed for one-time, very large-scale, intricate, non-routine projects.
Basic PERT rules:
- A PERT chart is an instrument that eases decision making; a PERT chart does not
take decisions.
- A PERT chart shows interrelated events (each of which is an essential objective),
typically represented by arrows between circles in the examples below (1 to 14).
- The first sketch of a PERT chart may number its events successively in 10s (10, 20,
30, etc.) to permit the later inclusion of supplementary events.
- Two successive events in a PERT chart are connected by activities, which are
traditionally symbolized by arrows in the examples below (1 to 14).
- The events are shown in a logical string and no activity can begin until its directly
preceding event is finished.
- The planner decides which objectives should be PERT events and also decides their
“proper” succession.
- A PERT chart may have numerous pages with lots of sub-tasks.
- A PERT event is a point that indicates the start or finish of one (or more) tasks. It
uses no time, and consumes no resources.
- A PERT event that indicates the close of one (or more) tasks is not “reached” until all
of the activities leading to that event have been finished.
- A predecessor event: an event (or events) that directly precedes some other event
without any other events prevailing. It may be the outcome or the result of more than
one activity.
- A successor event: an event (or events) that instantly follows some other event
without any other events interfering. It may be the outcome of more than one activity.
- A PERT activity is the concrete execution of a task. It consumes time, it requires
resources (such as labor, materials, space, machinery), and it can be seen as
symbolizing the time, effort, and resources necessary to move from one event to
another.
- A PERT activity cannot be accomplished until the event preceding it has taken place.
2. Risks of projects and probabilities of events
Risk management is one of the most important areas of project management that must be
considered in connection with the PERT-chart. PMI takes the systems approach to risk
management (Project Management Institute, 2013). The risk process is divided into six major
processes: risk management planning, risk identification, risk quantification, risk response
planning, and risk monitoring and control.
All the work to be done on a project or all the events of a PERT Network are risky
because they have a probability of greater than zero but less than 100% of occurring. Risks
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can be divided into known and unknown risks. Known risks are those risks that can be
identified. Unknown risks are those that cannot be well identified (Williams, 2002). Even
though unknown risks are fuzzily identified at best, we can sometimes recognize the effect of
these unknown risks and we can plan for them (“known unknown” vs. “unknown unknown”).
This planning can be accomplished by looking at expert opinion and observations of similar
projects, evaluating the risks that occurred there, and adjusting schedules and budgets
accordingly.
There are many ways to discover and identify risks. We will discuss several of them
here (Heagney, 2011):
- Documentation reviews
- Brainstorming
- Delphi technique
- Nominal group technique
- Crawford slip
- Expert interviews
- Checklists
- Scenario-based Risk Identification
Documentation reviews
Documentation reviews comprise reviewing all of the project materials that have been
generated up to the date of this risk review. This includes reviewing lessons learned and risk
management plans from previous projects, contract obligations, or project baselines.
Brainstorming
Brainstorming is probably the most popular technique for identifying risk. It is useful in
generating any kind of list by mining the ideas of the participants. To use the technique, a
meeting is called to make a comprehensive list of risks. It is important that the purpose of the
meeting be explained clearly to the participants and the meeting should have between ten and
fifteen participants. If there are fewer than ten, there is not enough interaction between the
participants; if there are more than fifteen people, the meeting tends to be difficult to control
and keep focused. The meeting should take less than two hours. In larger projects, it may be
necessary to have several meetings.
Delphi technique
The Delphi technique is similar to brainstorming, but the participants do not know one
another. This technique is useful if the participants are some distance away. The Delphi
technique is much more efficient and useful today than in the past because of the use of e-mail
or chat-rooms as a medium for conducting the exercise. Because the participants in this
technique are anonymous, there is little to inhibit the flow of ideas. Peer pressure and the risk
of embarrassment from putting forth a silly idea are avoided. Where the participants are not
anonymous, there is a tendency for one or more people to dominate the meeting. If one of the
participants is a higher-level manager than the others in the meeting, many of the participants
will be inhibited or will try to show off in front of the upper level manager. All of this is
avoided in the Delphi technique.
Nominal group technique
In the nominal group technique, the idea is to eliminate the problems associated with person‟s
inhibitions and reluctance to participate. A group size of seven to ten people is used. The
facilitator instructs each of the participants to privately and silently list his or her ideas on a
piece of paper. When this is completed, the facilitator takes each piece of paper and lists the
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ideas on a flip chart or blackboard. At this time no discussion takes place. Once all of the
ideas are listed on the flip chart, the group discusses each idea. Each member of the group
now ranks the ideas in order of importance, again in secret. The result is an ordered list of the
risks in order of their importance. This process not only identifies risk but also does a
preliminary evaluation of them.
Crawford slip
The Crawford slip process has become popular recently. The Crawford slip process does not
require as strong a facilitator as the other techniques, and it produces a lot of ideas very
quickly – it can take place in less than 30 minutes. The usual number of seven to ten
participants is used. The facilitator begins by instructing the group that he will ask ten
questions, one at a time. Each participant must answer each question with a different answer.
The same answer cannot be used for more than one question. The participants are to write
their answer to each question on a separate piece of paper. The facilitator tells the participants
that they will have one minute to answer each question. When all the participants are ready,
the facilitator begins by asking a question such as, “What is the most important risk to this
project?” The participants write down the answer. After one minute, the facilitator repeats the
question. This is repeated nine times. The effect is that the participants are forced the think of
ten separate risks in the project. Even with duplicates among the members, the number of
risks identified is formidable.
Expert interviews
Experts or people with experience on this type of project or problem can be of great help in
avoiding solving the same problems over and over again. Caution must be exercised
whenever using expert opinions. If an expert is trusted implicitly and his or her advice is taken
without question, the project can head off in the wrong direction. The use of experts can be
costly too. Before the expert interview is conducted, the input information must be given to
the expert and the goals of the interview must be clearly understood. During the interview, the
information from the expert must be recorded. If more than one expert is used, the output
information from the interviews should be consolidated and circulated to the other experts.
Checklists
Checklists have gained popularity in recent years because of the ease of communications
through computers and the ease of sharing information through databases. There are many
commercially available databases, and there are many checklists that are generated locally for
specific companies and applications. In their basic form, these checklists are simply
predetermined lists of risks that are possible for given projects. In their specific form, they are
risks that have occurred in the particular types of projects that a company has worked on in
the past.
Scenario-based Risk Identification
In scenario analysis different scenarios are created. The scenarios may be the alternative ways
to achieve an objective, or an analysis of the interaction of forces in, for example, a market or
battle. Any event that triggers an undesired scenario alternative is identified as risk.
Since all risks have a probability of greater than zero and less than 100 percent, the
probability of a risk occurring is essential to the assessment of the risk. Any risk event that
has a probability of zero cannot occur and need not be considered as a risk. A risk event that
has a probability of 100 percent is not a risk either. It has a certainty of occurring and must be
planned for in the project plan (Ho & Pike, 1998).
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Williams (2002) further advocates the use of a Project Risk Register (PRR) as a
central analysis tool, and he asserts the need to combine the effects of time, cost and technical
risk. An important point he makes is that two separate sets of uncertainty need to be
combined: the ordinary engineering aleatory uncertainties, and the epistemic risks – the extent
of some risks will not be known until the development is actually tried. He further considers
both risk reduction actions to reduce the probability of the risk occurring, and contingency
plans reducing the consequences of the risk in case it does occur.
It is thus possible to give a probability value of success pi (s, ti) at the beginning of a
project (Chapman & Ward, 1997) to all the interconnected PERT-chart events (each of which
is an important milestone), conventionally represented as numbered circles linked by arrows
in a diagram – where pi (s, ti) is the likelihood of success of event i at time ti (the time when
event i begins). Success is here defined in terms of the minimum scope (within a given cost
and time frame) that will allow the following events on the PERT-chart to take place. pi (s, ti)
is mutually exclusive to the negative risk, also called the probability of failure pi (f, ti) of this
event. We write that for every event i of the PERT-chart:
pi (s, ti) + pi (f, ti) = 1 (1)
By definition, the events of a PERT-chart are presented in a logical sequence and no activity
can commence until its immediately preceding event is completed. In other words, the
completion of a particular event is a necessary condition for the completion of all the future
events linked (via an arrow path) to this completed event on the same PERT-chart. This is a
typical case of conditional probability. This case is similar to a basic card problem, which
consists in calculating the probability of drawing without replacement two hearts from a well-
mixed deck of 52 cards. There are 13 hearts at first, so the probability of obtaining one heart
is p(H1) = 13/52 or 1/4. Afterwards, the probability of obtaining a second heart knowing that
the first card was a heart is p(H2|H1) = 12/51. And the likelihood to draw two hearts in a row
is therefore:
p(H1 ∩ H2) = p(H1) * p(H2|H1) = 13/52 * 12/51 = 0.06 (this is also known as Bayes‟ law).
The product rule is often used with stochastic processes, which are processes where
the outcome of an experiment depends on the outcomes of previous experiments (a Markov
chain is a typical example of a stochastic process).
For the sake of simplicity we write that the probability of success of both events i and j
at time i, with event j immediately consecutive to event i on a one-dimensional PERT-chart
(or single path) and with both events having probabilities of success pi (s, ti) and pj (s, tj)
respectively, is:
p(i, j) (s, ti) = pi (s, ti) * pj (s, tj) = pj (s, ti) (2)
Before the beginning of a project (t0), we can further write that the probability of occurrence
of a particular event pn (s, t0) on a one-dimensional PERT-chart is the product of the
probabilities of all the preceding events. In the case of a two-event project (2 following 1),
we write:
p2 (s, t0) = p1 (s, t1)* p2 (s, t2) (3)
And for any event n and the immediately preceding event n-1 on a one-dimensional PERT-
chart:
pn (s, t0) = pn-1 (s, tn-1) * pn (s, tn) (4)
This is also the likelihood of the project to reach stage or event n at t0.
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In the case of several preceding events that do not follow each other (multi-dimensional
PERT-chart), the multiplication rule still applies. For instance, if event j is preceded by two
events that are not on the same path – there are two arrows pointing at event j – and if these
events are named event i1 and event i2 respectively, then:
p(i1,i2,j) (s, t0) = pi1 (s, ti1) * pi2 (s, ti2)* pj (s, tj) (5)
3. Entropy and Information Theory
“Information theory is a discipline in applied mathematics involving the quantification of data
with the goal of enabling as much data as possible to be reliably stored on a medium and/or
communicated over a channel” (Jing, 2005).
Information theory is narrowly connected with a group of pure and applied fields that
have been explored and shrunk to engineering applications throughout the world over the past
half century or more: informatics, machine learning, adaptive systems, artificial intelligence,
complex systems, complexity science, cybernetics, along with systems sciences (Georgescu-
Roegen, 1971).
Information theory is a wide and profound statistical theory, with similarly large and
deep applications, among which is the crucial field of coding theory.
In information theory, entropy is the value of the quantity of information that is not
there before reception. This kind of entropy is called “information entropy” whereas
thermodynamic entropy is an energetic measure of irreversibility. Information entropy can be
defined too as “measure of the amount of information in a message” (Oxford English
Dictionary). Information entropy is sometimes used as well in information theory as a
measure of complexity.
The degree of uncertainty in data, known as information entropy, is generally
equivalent to the average number of bits required for storage or communication. For instance,
if there are 16 potential messages and everyone is similarly prone to be sent, a total of 4 bits
are needed to identify a given message and the entropy will be 4 bits. But if half of these
messages are in no way sent, (i.e. have a probability of zero) then one would only need 3 bits.
The problem of the relation between information entropy and thermodynamic entropy
is a passionately disputed subject. Many authors say that there is a relationship between the
two (Jing, 2005) whereas others will state that they have not much to do with one another
(Shu-Kun, 1999).
The beginning of this expression can be traced back to a discussion between John von
Neumann and Claude Shannon. In the 1940s, as he was working at Bell Telephone
Laboratories, Shannon was also trying to mathematically calculate the statistical nature of lost
information in phone-line signals. After Shannon had been developing his equations for a
while, he visited John von Neumann. During their discussion, it was recommended that
Shannon name his probabilistic measure of uncertainty (or attenuation in phone-line signals)
entropy. Explicitly, according to Tribus and McIrvine (1971):
My greatest concern was what to call it. I thought of calling it „information‟, but the
word was overly used, so I decided to call it „uncertainty‟. When I discussed it with
John von Neumann, he had a better idea. Von Neumann told me, „You should call it
entropy, for two reasons. In the first place your uncertainty function has been used in
statistical mechanics under that name, so it already has a name. In the second place,
and more important, nobody knows what entropy really is, so in a debate you will
always have the advantage.
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Pursuing this suggestion, in 1948 Shannon published his prominent paper “A Mathematical
Theory of Communication”, in which he wrote a part on what he calls Choice, Uncertainty,
and Entropy (Shannon, 1948). In this part, Shannon introduces an “H function” of the
following variety:
k
iipip
KH 1)(log)(
where K is a positive constant. Shannon then says that “any quantity of this form, where K
merely amounts to a choice of a unit of measurement, plays a central role in information
theory as measures of information, choice, and uncertainty.” Then, as an illustration of how
this expression works in various fields, he references R. C. Tolman‟s 1938 Principles of
Statistical Mechanics, affirming that “the form of H will be recognized as that of entropy as
defined in certain formulations of statistical mechanics where pi is the probability of a system
being in cell i of its phase space…”
The entropy, H, of a discrete random variable M is a measure of the level of
uncertainty one has about the value of M. It is here that the characterization of bit used is
important. For instance, suppose one sends 1000 bits in the usual sense (0s and 1s). If these
bits are known before transmission (to be a given value with fixed probability), then no
information has been conveyed. If, conversely, each is uniformly and independently likely to
be 0 or 1, 1000 bits (in the information-theory sense) have been sent. Between these two
boundaries, information can be computed as follows: If M is the set of all messages m and
p(m) = Pr(M = m), then M has
bits of entropy. A key property of entropy is that it is maximized when all the messages in the
message space have the same probability – i.e., most unpredictable – in which case H(M) =
log | M | .
Sometimes the function H is expressed in terms of the probabilities of the distribution:
where
An essential particular case of this is the binary entropy function:
The joint entropy of two discrete (not necessarily independent) random variables X and Y is
simply the entropy of their pairing, (X,Y). For example, if (X,Y) represents the position of a
chess piece – X the row and Y the column – then the joint entropy of the row of the piece and
the column of the piece will be the entropy of the position of the piece. Precisely,
(6)
(7)
(8)
(9)
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4. Entropy in projects until now (December 2013)
It is possible to apply the concept of entropy to projects as demonstrated by Bushuyev &
Sochnev (1999) who have introduced an entropy model for estimating and managing the
uncertainty of projects. (This is the only direct attempt to apply entropy that I have found in
the entire literature on projects.) This work by Bushuyev & Sochnev was recently refined by
Tamvakis & Xenidis (2013). Entropy is proposed there as a unified measure of uncertainty
which allows the comparison of various types and sources of risk. They show that managers
can consider different categories of entropy such as budget contingency, schedule
contingency or quality contingency. A computational example for simultaneous usage of
Monte Carlo and a proposed entropy model is given as well.
Approaches to the project control and risk management from the viewpoint of
information theory and entropy measurements help them to deal with project uncertainties in a
simple intuitive manner. According to the authors in the previous paragraph: “entropy in a
project is similar to the kind of resource which is assigned to the activities. The consumption
of this resource in a broad sense means managerial efforts for eliminating uncertainty.” Their
model considers that future progress in the project depends on the realization of uncertainties,
as well as on the future managerial actions, and entropy measurement is therefore one of the
possible alternative approaches to risk analyses and project control.
On the other hand, in spite of their pioneering initiative, one may question some of the
assumptions of Bushuyev & Sochnev (1999), in particular the notion that the individual
entropies are independent from one another. To them, the total entropy of a project is simply
the sum of all the individual entropies calculated at the beginning of each activity. But there is
a limit to the notion that all entropies can be added. This limit is conditional probability or the
fact that some events are undeniably linked. Besides, a clear time frame must be established
when calculating entropy. The probability of success of a particular event at the beginning of
a project is conditioned by the probability of success of all the preceding events (see equation
5). For example, if at the beginning of a project, the conditional probability of an event j is .9
and if there is only one event i with a probability of .8 preceding it on the PERT-chart, then
the real probability of success of event j at the beginning of the project is not .9 but rather .8 *
.9 = .78.
The Axiomatic Method was developed by Suh (1990) as a basic background theory
and a key part of his design-build planning and control methodology. In general terms,
Axioms are defined as generic principles and inherent parameters that characterize and govern
the design or construction processes (Ruiz, 2011). The Axiomatic Method comprises of two
fundamental Axioms, the first is the Independence Axiom, and the second is the Information
Content Axiom.
In order to compare and choose from a number of candidate/potential design or
construction solutions, each solution is examined separately for satisfaction of the two
Axioms in two discrete steps. The Axiomatic Method has been tested by some academics
(Albano, 1992) and several practitioners in various areas of application, primarily in the
manufacturing industry.
Axiom 2, or the information Axiom, which measures the information content for the
solutions transferred from (accepted by) Axiom 1, and selects/recommends the best (most
effective) solution is related to the present research because it utilizes a formula akin to
entropy in a project management context, even if significantly different nonetheless.
(10)
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Nobody so far and to our knowledge, therefore, has tried to systematically apply the
concept of entropy to PERT-charts in order to evaluate the complexity of projects.
5. Quantifying the entropy of projects through PERT-charts
All the diagrams used in this section are in fact Network Diagrams – although the actual name
is not really the important thing. What is important is the distinction between CPM/ Network
Analysis and the true meaning of PERT – rather than the meaning given to it by common use.
People tend to indiscriminately use the term PERT when they actually mean a Network
Diagram. The distinction is the use of a range of time estimates and probabilities for each
event which is specific to PERT. Given that I use probabilities for events and that time
estimates are implied, the use of “PERT” is fully justified.
Entropy is defined here as a measure of the informational degree of organization of a
project and is not directly related to the thermodynamic property used in physics (although
they do possess a common mathematical ancestor as previously discussed).
Information can be defined in the „statistical‟ sense (Shannon, 1948), as the degree to
which data are non-compressible (e.g., the word „meet‟ can be compressed by „e‟ * 2; the
word „meat‟, however, cannot be compressed without losing information. Thus, there is more
statistical or entropic information in „meat‟ than „meet‟).
Consequently, to avoid any confusion, we define „information‟ in the present study
using Shannon‟s original use in information theory, that of statistical information, also known
as entropy. The first-order entropy takes into account the different probability of occurrence
of each event (see formula 7).
We will define the total first-order entropy of a project at a given time t as the joint binary
entropy Hbt(Q) (see formula 10), where Q is the set of all the events that remain to be
completed on the PERT-chart at time t.
If there are q events of the project left to be completed at time t, we write:
Hbt (Q) =
q
i1
Hbt (pi) bits/project left (11)
where Hbt (pi) is the binary entropy function (see 9) of each event i at time t, and pi the
probability of event i at time t as explained in (5). Base 2 is used because the „bit‟ can be
considered the most familiar logarithmic form for an information measure (but any other base
would do just as well). A bit (binary digit) is a digit in the binary numeral system, which
consists of base 2; it is the smallest unit of information on a machine. John Tukey, a leading
statistician, first used the term in 1946. A single bit can hold only one of two values: 0 or 1.
More meaningful information is obtained by combining consecutive bits into larger units. For
example, a byte is composed of 8 consecutive bits.
In order to understand better the meaning of formula 11 and what it means for the complexity
of real projects, let‟s take several simulated examples based on diverse projects each made of
six events or subparts, and with each event having the same arbitrary conditional probability
of success pi (s, ti) = .9.
As a starter, let‟s calculate the zero-order joint binary entropy (H0b) of all such projects at t0 :
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H0b = 6 * log2 2 = 6 bits (for one entire project).
Let‟s notice that entropy usually decreases from one order to the next (Shannon, 1948). This
means that all the first-order entropies of the fourteen projects below will be lower than 6 bits.
(We will not compute any entropy of the second order or higher because they would be
meaningless in this context.)
Let‟s now calculate the first-order joint binary entropies of several six-event projects at t0 :
Example 1: First-order joint binary entropy of this project at t0 = 4.92 bits
[Detailed calculations for Ex. 1: Hbt0 = (0.9 * log2 0.9 + 0.1 * log2 0.1) + (0.81 * log2 0.81 +
0.19 * log2 0.19) + (0.73 * log2 0.73 + 0.27 * log2 0.27) + (0.66 * log2 0.66 + 0.34 * log2 0.34)
+ (0.59 * log2 0.59 + 0.41 * log2 0.41) + (0.53 * log2 0.53 + 0.47 * log2 0.47) = 4.92.]
Example 2: First-order joint binary entropy of this project at t0 = 2.81 bits
Example 3: First-order joint binary entropy of this project at t0 = 4.09 bits
Example 4: First-order joint binary entropy of this project at t0 = 4.49 bits
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Example 5: First-order joint binary entropy of this project at t0 = 4.03 bits
Example 6: First-order joint binary entropy of this project at t0 = 3.51 bits
Example 7: First-order joint binary entropy of this project at t0 = 4.23 bits
Example 8: First-order joint binary entropy of this project at t0 = 4.66 bits
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Example 9: First-order joint binary entropy of this project at t0 = 4.62 bits
Example 10: First-order joint binary entropy of this project at t0 = 4.19 bits
Example 11: First-order joint binary entropy of this project at t0 = 3.65 bits
Example 12: First-order joint binary entropy of this project at t0 = 3.42 bits
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Example 13: First-order joint binary entropy of this project at t0 = 4.80 bits
Example 14: First-order joint binary entropy of this project at t0 = 4.26 bits
6. Discussion
Let‟s first insist on the reliability and general usefulness of this method: both are high because
it does not matter so much whether the true probability of success of a single event is really
.90, .99 or .80 (and therefore accurate risk assessment is not the main issue here). What
matters most is how the events (equal to six in all the examples above) are interconnected.
Higher entropy simply means that a project is more complex. A complex project is a
project which is neither sure nor impossible, and low entropy consequently means that a
project is either a certainty or an impossibility.
We did not make the number of events vary systematically in the present study given
that entropy is obviously linked to the number of events (entropy is higher in two well-mixed
decks of cards than in one deck). We are more interested in entropy as a measure of the
relational complexity – or structural sophistication – of a project represented by its typical
PERT-chart.
What appears clearly in the examples above is that the entropy (or structural complexity) of
projects increases – the number of events and their conditional probabilities being kept
constant – with :
1° time (longest path or number of arrows that follow each other) and,
2° the number of connections among the tasks.
The projects with the highest levels of entropy are projects 1 and 13, and there are also the
two projects with the longest paths (five arrows and three respectively) or, so to speak, where
time is most critical.
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Then come projects 8 and 9 with a total number of eight arrows each, but with longest
paths of only two arrows. Let‟s notice that project 10, even though it has nine arrows, is
considered less complex than projects 8 and 9. This is because time (the number of arrows
that follow each other in this case) is at least as important as sheer connectivity (e.g., total
number of arrows): for project 10, the longest path is simply one arrow, whereas it is two
arrows for projects 8 and 9.
Nevertheless, even though time is essential, simple connectivity is very important for
entropy: project 12 has a longest path of two arrows compared to only one arrow for the
longest path of project 10, but project 10 is nonetheless more complex because it contains a
total of nine arrows versus only two for project 12.
Finally, it is worth noticing that the difference in entropy between projects 3 and 4 is
not due to the number of arrows nor to their longest path lengths because all are equal in the
two projects: this is genuinely new information that is not obvious. Starting with three events
and finishing with one is less complex than to start with one event and to finish with three.
This makes sense if we do consider, again, the link between time and entropy. To start with
three events is to take a bigger risk now, and to finish with one is to take a smaller risk later;
whereas to start with one event is to take a smaller risk now, and to finish with three events is
to take a bigger risk later. Therefore, the later the risk, the higher the entropy.
Entropy increase is also often explained using the image of a pack of cards being
shuffled. Let‟s suppose for example, that the pack is arranged so that all the red cards occur
together on top and all the black cards below – an “ordered” state. Now let‟s shuffle them
repeatedly. We would expect them to end up mixed together, randomly, in a “disordered”
state, so shuffling increases the amount of disorder in the pack. Analogously, let‟s suppose
that at some moment all the oxygen molecules in a room are concentrated at one end and all
the nitrogen molecules are at the other. This is an ordered thermodynamic state. Collisions,
however, will mix all the molecules together, more or less uniformly throughout the room,
depending on their frequency.
This idea does seem to apply also to the project examples above: the more shuffling or
collisions (if we compare events to gas molecules), that is the more arrows or interactions
among the events, the higher the entropy, and the more complex the project.
Also, shuffling a deck of card several times will result in a more random deck if more
thorough shuffling is made later than sooner (assuming of course than no single shuffling
results in a totally random deck).
It is therefore possible to quantify the general structural complexity of projects based
on their PERT-charts.
This indicator is the first-order joint binary entropy of all the events of a project also
called Hbt(Q). The method is useful because, until now, there was no simple/single indicator
to compare the complexity of projects which were either completely different or apparently
similar (made of the same number of events for instance). Hbt(Q) takes into account not only
the number of events and their likelihood of success, but mostly the number of connections
among the events and how such connections evolve with time.
It is hoped that the first-order joint binary entropy will be used fruitfully in the future
to assess the structural complexity of projects before or during their execution and to evaluate
more accurately their inherent risk.
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