Schneider J., Kuchta D., Michalski R. (2023). A vector visualization of uncertainty complementing the traditional fuzzy approach with applications in project management, Applied Soft Computing, 137, 110155. Cited (JCR): 0, Other Cites: 0 IF:8.7 5yIF: Pt:200
$$\mu\mathbf{=}\frac{\mathbf{l}\mathbf{+}\mathbf{r}}{\mathbf{2}}$$
Abstract
In view of recent technological developments and new computational and graphical possibilities, scientists and practitioners have become increasingly interested in studying how data and information should be presented. For instance, in project management, it is now recommended to employ dashboards instead of traditional reports. It is also believed that the usage of vectors, resembling the hands of a clock, may increase the efficiency and effectiveness of data presentation and information processing. In light of this, we propose a novel approach to visualization of uncertainty as defined by triangular fuzzy numbers. This new representation is based on vectors whose length represents the range of possible values of an uncertain parameter, while the slope reflects tendencies within possible scenarios. The mathematical foundations and definitions along with the basic properties of this approach are demonstrated in detail. In particular, we show how to transform triangular representations into vector ones and vice versa. The arithmetic operations of addition and multiplication by a crisp number on these vectors are demonstrated as well. Possible applications of the new vector visualization to project uncertainty representation and in project management are described. We also discuss both the advantages and disadvantages of our approach in relation to the traditional visualization as graphs of membership functions. Our proposal is complementary to the traditional one, and they should be used in combination. The new graphical representation of triangular fuzzy numbers expands the available toolbox for visualizing uncertainty not only in project management but in any other area.
Keywords:
Decision making; Membership function; Project management; Triangular fuzzy number; Uncertainty visualization; Vector representation
1. Introduction
Uncertainty, defined on the basic level as “lack of
certainty” [1 p. 3], is present in everyday life both of
individuals and organisations. It is classified in various ways, of
which the following stand out [1 p. 3334,2 p. 33]:
Ambiguity or knowledge uncertainty, defined as lack or
incompleteness of information. This type of uncertainty depends on the
quantity, quality, and relevance of the data and on the reliability and
relevance of the models and assumptions.
Inherent variability uncertainty refers to true differences in
attributes due to heterogeneity or diversity. It cannot be reduced by
further measurement or study, it can only be better
characterised.
Both types of uncertainty can take on different forms [2 p. 36], that
is, the scenario, model, and the parameter/input uncertainty. Here, we
concentrate on the latter: the parameter uncertainty involved in the
specification of numerical values, like cost, duration, etc.
Other authors differentiate between statistical and nonstatistical
uncertainty [3]. Statistical uncertainty is the one where probability
distributions can be used for quantitative uncertainty analysis. The
focus of our research is on nonstatistical uncertainty, where
probability distributions are not available. Several approaches have
been developed over the years to manage this type of uncertainty. The
most popular mathematical models are based on fuzzy [4,5], rough [6],
soft [7] and grey sets [8–10]. They are studied and compared to each
other in several papers, e.g., [11–13].
In this paper, we focus on triangular fuzzy numbers that can be
applied to represent uncertain parameters in various contexts. Apart
from uncertainty, the most important problem that lies at the root of
the present paper is that of information visualization. Designing
appropriate and effective visualizations is far from simple and
effortless. As stressed in [14], it cannot be equated with only
presenting data on a generated graph, especially if a suitable and
insightful graphical representation is crucial for the decisionmaking
process.
Traditionally, fuzzy numbers are represented as graphs of membership
functions. In theory, membership functions of any shape can be
generated, and many of them may be adequate for specific applications
[15]. However, in practice, the vast majority of realworld
applications, especially in social, economic, and management sciences,
rely on triangular fuzzy numbers. They have been used to model parameter
uncertainty in a number of different areas, for example, in supply chain
management [16], inventory [17], production planning [18], service
quality [19], product positioning [20], decision making [21], project
risk estimation [22], project selection [23] or project network analysis
[24].
A theoretical explanation for this predominance of triangular fuzzy
numbers over all other shapes is given in [25]. The following two
arguments may be put forward:
Simple definition. It requires the estimation of the only three
parameters which are used in computing algorithms while intermittent
values are usually not involved. This makes triangular fuzzy numbers
akin to an enhanced interval number.
Mathematical tractability. Due to the low complexity of addition
and multiplication by a crisp number on the closed set of triangular
fuzzy numbers, it is relatively easy to transfer a great number of
classical optimization algorithms or procedures from real numbers
R into triangular fuzzy numbers.
Triangular fuzzy number membership functions take the shape of
triangles. However, this representation, as simple as it may seem, does
not have to be the most appropriate for everyone. For many, the link
between the uncertain information and the graphics of triangles may not
be obvious. On top of that, certain types of parameter changes in time
will be easier to track if they are illustrated by shifts of the tip of
a vector, resembling the fluctuations of a clock hand, instead of
changes in a triangle shape.
This is especially true in the context of modern project management
that, according to both researchers and practitioners, should be based
on dashboards instead of traditional reports [26,27]. Dashboards [28]
are data visualization and analysis tools that show on one screen all
the basic information necessary to make decisions. The idea has resulted
in some software applications (e.g., https://thedigitalprojectmanager.com/tools/projectdashboardsoftware/).
Having the appropriate technologies at our disposal, researchers are
obliged to investigate modern ways of visualization.
For this reason, in the present work, we propose an alternative
vectorbased graphical representation of uncertainty as quantified by
triangular fuzzy numbers and show its application to modelling data in
the context of project management. Because the correspondence between
traditional and vectorbased representations of triangular fuzzy numbers
is unique (one to one), the new representation may be used
interchangeably with the traditional one. Since we deal with two
alternative representations of the same uncertainty, for the sake of
simplicity, we will sometimes call the vectors “another
representation of triangular fuzzy numbers”.
The rest of the paper is organized as follows. In the next section,
we briefly provide some specifics about uncertainty modelling and
picturing metrics in the context of project management. The following
sections include a detailed description of our proposal, with its
mathematical foundations and selected properties. We also give some
practical examples in the context of project management. The article
ends with a discussion of our proposal, conclusions, and some future
research prospects.
For convenience, the list of mathematical symbols and abbreviations
used in this paper is provided in (Table 1).
Table 1. List of mathematical symbols and abbreviations.
ξ 
An uncertain parameter, a fuzzy number 
Page 6 
l 
The smallest possible the uncertain parameter ξ can
attain 
Page 6 
m 
The possible value the uncertain parameter ξ can
attain 
Page 6 
r 
The largest possible value the uncertain parameter ξ can
attain 
Page 7 
ξ(x) 
Membership function of a fuzzy number 
Page 7 
tr^{*}(l, m, r) 
A triangular fuzzy number in traditional representation 
Page 7 
μ 
The arithmetic mean of the smallest possible and the largest possible value:
$$\mu\mathbf{=}\frac{\mathbf{l}\mathbf{+}\mathbf{r}}{\mathbf{2}}$$

Page 7 
$$\overrightarrow{vc}(\mu,\s,\gamma)$$

A triangular fuzzy number in vector representation 
Page 8 
TV 
Transform taking the traditional representation of a triangular
fuzzy number into the vector one 
Page 9 
s 
Length of the vector $\overrightarrow{vc}(\mu,\ s,\gamma)$: s = r − l 
Page 9 
γ 
Angle of deviation from symmetry: γ = arctan (m − μ) 
Page 9 
Tail 
Cartesian coordinates of the vector initial point in vector
representation 
Page 10 
Tip 
Cartesian coordinates of the vector terminal point in vector
representation 
Page 10 
γ_{△min} 
Smallest possible angle of deviation from symmetry in vector
representation 
Page 11 
γ_{△max} 
Largest possible angle of deviation from symmetry in vector
representation 
Page 11 
$$\underset{\text{\{} \bigtriangleup
\text{\}}}{\oplus}$$ 
Addition operator of triangular fuzzy numbers in traditional
representation 
Page 15 
$$\underset{\text{\{} \rightarrow
\text{\}}}{\oplus}$$ 
Addition operator of triangular fuzzy numbers in vector
representation 
Page 15 
$$\underset{\text{\{}\sphericalangle\text{\}}}{\oplus}$$ 
Addition operator of angles in the vector representation of
triangular fuzzy numbers 
Page 15 
$\$\$\backslash underset\{\backslash text\{\backslash \{\}\; \backslash bigtriangleup\; \backslash text\{\backslash \}\}\}\{\backslash odot\}\$\$$ 
Multiplication operator of triangular fuzzy numbers in traditional
representation 
Page 19 
$$\underset{\text{\{} \rightarrow
\text{\}}}{\odot}$$ 
Multiplication operator of triangular fuzzy numbers in vector
representation 
Page 19 
VT 
Transform taking the vector representation of a triangular fuzzy
number into the traditional one 
Page 20 
2. Project management context
Projects can be defined as “temporary endeavours undertaken to
create a unique product, service or result” [29]. Uniqueness,
inherent in projects [30] together with the turbulent project
environment and the limited and biased human perception [31], imply that
uncertainty is omnipresent in project management. This is particularly
true for parameter uncertainty [1 p. 3334]. Therefore, our research is
focused on this uncertainty form, especially in the context of the
duration and costs of the project activities.
The uncertainty in projects has to be monitored and controlled, like
any other project feature [29]. The knowledge uncertainty is usually the
highest at the beginning of projects and diminishes with their
advancement. The variability uncertainty, in turn, may be described
better and better as the project progresses.
Similarly to other areas, uncertainty modelling in a project can
involve fuzzy numbers [32–34]. They have often been used in the project
planning stage [35] to model, known only to a certain extent, project
task duration times or cost values, as well as other quantitative
project parameters. As mentioned in the Introduction, triangular fuzzy
numbers are the form of fuzzy numbers that is used most frequently, also
in the project management context. Triangular fuzzy numbers express the
actual knowledge of experts and their subjective opinion on the
pessimistic, most possible, and optimistic value of the respective
parameters. This approach has certain advantages over the classical
3point PERT method of project planning, based on probability theory.
Triangular fuzzy numbers do not assume any probability distribution, but
refer to the subjective and mathematically simpler possibility theory
[5]. This feature makes their application potentially easier for
nonmathematicians and nonengineers. Projects are today omnipresent,
and they appear not only in companies but also in notprofit
organisations, such as nongovernmental or public institutions. This
makes the problem of understanding and processing information by project
participants of all possible backgrounds of utmost importance.
Moreover, projects have been becoming more and more complex [27]. For
this reason, project management is nowadays a difficult metaprocess
where visualization plays an important role in supporting project
managers in their role ([36] [37] [38]). Visualization should accelerate
the decisionmaking process and replace text reports. However, an
incorrect graphical presentation can lead to improper decisions [39].
That is why it is important to select an adequate way of visualizing
uncertain information, adapted to each individual recipient.
The significance of appropriate visualization techniques in conveying
information has been recognized in the scientific literature for
decades. Some basic rules of human visual processing of graphical data
were identified by Gestalt psychologists as early as the beginning of
twentieth century [40,41]. They proposed a set of Gestalt laws regarding
perceptual grouping, e.g., similarity, closure, proximity, or
continuation [42,43]. More recently, among the first approaches to
provide guidelines in this regard based on theoretical foundations
supported by experimental data, were the works of Cleveland and McGill
[44,45]. The authors identified ten elementary perceptual tasks and
determined their hierarchy based on the accuracy of the properties
compared. They suggested to use those graphical encodings that are as
high as possible in the following ordering: position on a common scale
and nonaligned scales, length, direction, angle, area, volume,
curvature, shading, and colour saturation. In the context of project
management, Kerzner [27] provides practical indications concerning the
artwork (image, icon) used in the project dashboards, the positioning,
accuracy, colour, size, texture, etc. He claims that the selection of
graphics used to convey information is extremely important in project
management.
Accordingly, our new vector representation of triangular fuzzy
numbers extends the set of available techniques for visualizing
uncertain information. Therefore, it seems to be potentially interesting
and attractive for various stakeholders in project management. The new
approach may have some advantages over the traditional triangle
representation. The preliminary study related to the usability of vector
and membership functionbased representations of uncertain data [46]
shows its potential usefulness for visualization purposes in the context
of project management. In particular, the authors performed an
experimentallybased research designed to compare vector and membership
functionbased representations of uncertainty with respect to their
effectiveness, efficiency, and participant satisfaction. They recorded
the accuracy of proper recognition of uncertain information presented by
both visualizations relative to their textual depiction to objectively
assess the effectiveness. The efficiency and user satisfaction
measurements were subjectively evaluated by asking about ease of
interpretation and attractiveness, respectively. Based on the results
from 76 subjects, they found that, overall, both representations appear
to be effective to a similar degree in conveying uncertain information.
Some significant differences between men and women have been observed.
The females performed better if vectors were used, while the males were
more effective in the case of traditional representations. Similar
patterns were identified for ease of use and attractiveness. Generally,
women favoured vector representations, while men preferred traditional
representations.
On the whole, the vector representation of fuzzy numbers proposed in
[46] was found acceptable and supportive by a substantial number of
users. This result prompted us to investigate theoretical aspects
regarding soft computing for the vector representation (arithmetical
operations), the problem of formal equivalence between the traditional
representation of fuzzy numbers and the vectorbased one, as well as
several mathematical properties of the vector representation. The
results of this research are presented here. Also, we focus here
specifically on project management context.
A project resembles a car journey. The car driver is heading
a destination and controls time and other parameters by means of the
dashboard with several meters, that change their indications
dynamically. The same is true for a project manager. Here, we discuss,
especially in the examples, the dynamic changes in the project situation
and compare the respective visualizations (traditional and vectorbased
one) from the point of view of their userfriendliness. We investigate
whether a vectorbased dashboard could be, at least for some users, more
helpful than the traditional one in decision making during the usually
changeable and unstable project implementation process.
No research has been conducted on the difference in the perception of
dynamic changes or tendencies in uncertain parameter values represented
by triangles and vectors. Here, the abovementioned clock hand
similarity of vectors may be potentially important for facilitating the
project monitoring and control for certain project stakeholders.
3. Triangular fuzzy numbers and their graphical representations
Triangular fuzzy numbers are a special case of fuzzy sets [5], which
are the basis of the possibility theory [5,47]. In economic and social
sciences, triangular and trapezoidal fuzzy numbers are by far the most
commonly used to represent a quantity whose exact value is not known at
the given time [48] and the probability distribution cannot be
determined. In this context, the more subjective possibility degree
of occurrence is used instead of probability [49]. The
possibility degree is given by experts. The uncertain value of the
parameter ξ can be represented
by three crisp numbers l, m and r such that l ≤ m ≤ r. These
numbers correspond to the smallest possible value considered (l
as left), the largest possible value (r as right) and the value
assumed to be the most possible, i.e. m (middle). The most
possible value m has a degree
of possibility of 1, ξ(m) = 1, whereas the lower
and upper bound degrees of possibility are 0, i.e. ξ(l) = ξ(r) = 0.
A degree of possibility for each x: l ≤ x ≤ r
is linearly increasing from ξ(l) = 0 to ξ(m) = 1, and linearly
decreasing from ξ(m) = 1 to ξ(r) = 0. This procedure
formally defines a socalled membership function ξ(x) that, for each real
x, assigns a degree of
possibility according to expert knowledge (1).
Notation 1. The shorthand notation for (1)
used in this paper is presented in (2):
The support of ξ,
as defined in (1), is the closed interval [l, r] [50]. Values between
l and r (excluding l i r) are
considered to be possible to a positive degree. Values beyond this
interval are seen as impossible. The broader the interval [l, r], the less
information we have about the quantity in question. Its width gives us
some information about the indeterminacy degree, which is defined in the
Collins dictionary [51] as the quality of being uncertain or vague, and
is linked with fuzziness in [52]. The relative position of the value
m, whose possibility degree is
1, indicates the value which is assumed
to be the most possible. Therefore, m shows the skewness of the
fuzzy numbers, that is, the inclination of the possibility degree of the
values from support [l, r] towards lower (close
to l), medium (around the middle
$\mu\mathbf{=}\frac{\mathbf{l}\mathbf{+}\mathbf{r}}{\mathbf{2}}$),
or higher (close to r) values.
Let us assume that a neutral situation occurs when the middle value
μ is at the same time the most
possible value m = μ.
Then, if m ≠ μ, the
experts have a more pessimistic or optimistic opinion on the possibility
distribution. The choice of the adjective depends on the position of
μ with respect to m and on the nature of the value
modelled by the fuzzy number. For example, if (m > μ) and the
estimated parameter refers to benefits, the expert opinion will be
qualified as optimistic. If m > μ and the parameter
refers to cost, the opinion represented by the same fuzzy number would
be pessimistic.
3.1. Traditional representation of triangular fuzzy numbers
The hitherto only and generally accepted way to visualize triangular
fuzzy numbers is by drawing their membership function ξ(x). Any triangular fuzzy
number given in a form tr^{*}(l, m, r)
uniquely determines its membership function (1). For instance, the
triangular fuzzy number with m = 4 and a scope of indeterminacy
[2, 5], denoted as tr^{*}(2, 4, 5)
corresponds to (3) and is visualized by means of the membership function
from Figure 1.
Figure 1. Triangular fuzzy number tr^{*}(2, 4, 5)
represented by a triangle.
3.2. A new, vector representation of triangular fuzzy numbers
In this paper, we give an alternative representation of the
uncertainty defined by triangular fuzzy numbers as vectors pointing
upwards from the value $\mu\mathbf{=}\frac{\mathbf{l}\mathbf{+}\mathbf{r}}{\mathbf{2}}$
and leaning to the left or right side of the vertical line x=μ. For
example, the triangular fuzzy number tr^{*}(2, 4, 5) can
be represented as the vector $\overrightarrow{vc}(\mu,\ s,\gamma) =
\overrightarrow{vc}\left( 3.5,\ 3,\,{26.55}^{\circ} \right)$
defined in polartype coordinates, where s is vector’s length,
γ is the angle between the
vector and a vertical line x=μ, and
μ is vector’s tail as shown in
Figure 2.
Figure 2. Visualization of triangular fuzzy number
tr^{*}(2, 4, 5) by
the vector $\overrightarrow{vc}\left( 3.5,\
3,{26.55}^{\circ} \right)$
attached to $x = \frac{7}{2}$, of
length 3, and angle γ = 26.55^{∘}.
In Figure 2 and subsequent figures a blue dot is placed at $x = \mu = \frac{r + l}{2}$. Vector length
s represents the scope of
indeterminacy and is equivalent to the length of fuzzy triangle number’s
support [l, r]. The
angle γ, by which the vector
is inclined to the left or right, is determined by the relative position
of the middle value μ and the
most possible value m. The
angle indicates that the value m is less than μ by leaning to the left or greater
than μ – leaning to the right.
Thus, the position of the pointer shows clearly the inclination of the
represented expert opinion, its pessimistic or optimistic touch.
As a mathematical formula, our new vector representation is defined
by the following equivalence transform (TV, Triangle → Vector),
which takes the three defining parameters of a triangular fuzzy number:
l, m, and r into the alternative three
parameters μ, s, γ (4).
where:
μ is the arithmetic
mean of l and $r,\ \mu = \frac{r + l}{2},\ $.
s is the vector’s
length equal to the triangle’s support length s = r − l,
γ is the angle of
inclination to the left or right from the vertical line x = μ, which is is given by
(5).
Thus, the uncertain quantity ξ may be interchangeably and
equivalently represented by either the triple (l, m, r) or (μ, s, γ). In the
latter case, the notation we adapt is given in (6).
Notation 2.
In compact form the transform TV of one defining triple to
the other is given by (7):
The geometric meaning and provenance of μ, s, and γ is shown in Figure 3, where we can
also compare both representations. The vector coefficients (parameters)
can be identified graphically in the traditional representation as is
presented in the graph of its membership function.
Figure 3. A triangular fuzzy number in its
traditional representation in upper graph (orange) and in a vector form
in down graph (light blue).
The vector representation of triangular fuzzy numbers defined in
polar coordinates, can also be expressed by Cartesian coordinates, which
may, at times, be more convenient to use. The vector’s tip and tail
coordinates of the given triple (μ, s, γ) can be
computed according to (8), (9), and (10).
whereby
and
Example 1. Tip and tail vector representation of
a triangular fuzzy number.
The triangular fuzzy number tr^{*}(1, 4, 5) may
be written in a vector representation as (11).
or as tip and tail coordinates by (12).
as shown in Figure 4.
Figure 4. The traditional and vector representations
corresponding to the triangular fuzzy number tr^{*}(1, 4, 5)
given in Cartesian coordinates with indicated tail and tip
coordinates.
3.3. Border cases of the vector representation
Theoretically, in the general case, γ can take values from −90^{∘} and +90^{∘} as given in (13), (14),
and (15) and schematically presented in Figure 5.
that is to say that:
Figure 5. Visualization of γ maximal and minimal values in
general, and in triangular fuzzy numbers for an exemplary s = 8, and γ_{△min} = −75.96^{∘} < γ < + 75.96^{∘} = γ_{△max} ,
marked with the green dot.
However, the triangular fuzzy number fulfils in fact the condition
γ ∈ [γ_{△min }, γ_{△max }],
where the interval of possible γ values is a proper subset of the
interval (−90^{∘}, + 90^{∘}). Since
μ is the middle point
of the support of the triangle tr^{*}(l, m, r),
(m − μ) measures the
degree of deviation from symmetry $\text{meant
as}\text{ }m = \mu = \frac{r + l}{2}$. One may single out three
border cases that determine the value range of γ. For symmetric triangular fuzzy
numbers, we have (16):
The other two cases refer to the socalled degenerate triangle
numbers of type tr^{*}(m, m, r)
or tr^{*}(l, m, m).
This time the most possible value m coincides with one of the
endpoints of the support [l, r], and then goes to
infinity with the left or right endpoint. Therefore, according to(13)
and (14), one receives an angle of γ = −90^{∘} or +90^{∘}. Maximal and minimal values
of γ depend on maximal and
minimal values of (m − μ) and can be computed
by (17) and (18) respectively.
Thus, for degenerate triangular fuzzy numbers, the (m − μ) value ranges from
$ \frac{r  l}{2}$ to $+ \frac{r  l}{2}$ and extreme values of
γ depend only on the
indeterminacy degree s. One can also observe that the higher
the indeterminacy measured by s, the higher maximal possible
absolute values of γ are. For
example, for any triangular fuzzy numbers such that s = r − l= 4 and
r ≥ m ≥ l,
γ ∈ (−63.43^{∘}, + 63.43^{∘}).
Sample cases of degenerative triangular fuzzy numbers with s = 4, i.e. tr^{*}(2, 2, 6), tr^{*}(2, 4, 6), and tr^{*}(2, 6, 6)
are shown in Figure 6. The light blue area in the vector representations
shows the possible range of γ
values for these examples.
Figure 6. Symmetricity and degeneracies in
traditional and vector representations of the uncertainty defined by
(1).
3.4. Arithmetical operations in traditional and vector representations
In this section, we present the addition of two triangular fuzzy
numbers and the multiplication of a triangular fuzzy number by positive
crisp numbers in both representations. These operations are sufficient
for most project management applications, e.g., the determination of
project budgets, schedules, or risk appraisal.
Notation 3.
In this paper, we denote addition in traditional representation
by $\underset{\text{\{} \bigtriangleup
\text{\}}}{\oplus}$ and addition in the vector representation
by $\underset{\text{\{} \rightarrow
\text{\}}}{\oplus}$.
Before defining the addition of triangular fuzzy numbers in their
vector representation, we provide a reminder of the addition definition
for membership functions. For two triangular fuzzy numbers given in
their traditional representation tr^{*}(l_{1}, m_{1}, r_{1})
and tr^{*}(l_{2}, m_{2}, r_{2})
one has by definition (19) [4].
The addition operation in the vector representation for two vectors
$\overrightarrow{vc}\,\left(
\mu_{1},s_{1},\gamma_{1} \right)$ and $\overrightarrow{vc}\,\left(
\mu_{2},s_{2},\gamma_{2} \right)$ must be mathematically
consistent with (19), which necessarily entails definitions (20) and
(21). For easier readability, we first introduce the following addition
operation on angles.
Notation 4.
In this paper, we denote the addition of angles in the vector
representation of triangular fuzzy numbers by $\underset{\text{\{}\sphericalangle\text{\}}}{\oplus}\,$.
Definition 2.
Definition 3.
Remark 1.
The vectors given in (8) cannot be added in the
usual ℝ^{2} sense
because of (22) and (23):
For instance (24):
Example 2. Adding two vector representations of
triangular fuzzy numbers.
as depicted in Figure 7.
Figure 7. Addition of two triangular fuzzy numbers
in traditional and vector representations.
Some more additions of sample vector representations are given in
Example 3 (25). They serve mainly to give a feeling for the
TVaddition of angles, as the first and second components of
the triples added in (21) are straightforward.
Example 3. Additions of sample vectors.

$$\overrightarrow{vc}\,\left( 3,\, 2, 
30^{\circ} \right)\underset{\text{\{} \rightarrow
\text{\}}}{\oplus}\,\overrightarrow{vc}\,\left( 5,\, 1,  10^{\circ}
\right) = \overrightarrow{vc}\,\left( 8,\, 3,  {37.00}^{\circ}
\right)$$ 

$$\overrightarrow{vc}\,\left( 3,\,
2,50^{\circ} \right)\underset{\text{\{} \rightarrow
\text{\}}}{\oplus}\,\overrightarrow{vc}\,\left( 5,\, 1,  60^{\circ}
\right) = \overrightarrow{vc}\,\left( 8,\, 3,  {28.38}^{\circ}
\right)$$ 
Remark 2. In general, we have, by
construction, (26):
which can be schematically seen in Figure 5. For
consistency, Definition 2 (20) must yield (27):
Assuming (28):
we observe that (29):
Let us add, for instance, two triangular fuzzy numbers with a large
indeterminacy of s = 115, e.g., tr^{*}(0, 115, 115).
The maximal vector angle for these individual cases is γ_{△max} = 89^{∘}.
The maximal γ_{△max}
for the resulting vector tr^{*}(0, 230, 230)
is 89.5^{∘} (30).
This specific borderline example shows that the sensitivity of
changing the vector direction is smaller for larger absolute values of
γ, that is while deviating
farther from the vertical line x = μ.
Notation 5.
In this paper, we denote the product of a positive crisp real
number and a triangular fuzzy number in the traditional representation
by $\underset{\text{\{} \bigtriangleup
\text{\}}}{\odot}$ and the equivalent for the vector
representation by $\underset{\text{\{}
\rightarrow \text{\}}}{\odot}.$
For any real positive crisp value c and a triangular fuzzy number with
nonnegative l, m, r, given in its
traditional representation tr^{*}(l, m, r)
one has by definition (31).
The equivalent formula of (31) in the vector representation of the
triangular fuzzy number is given in (32).
Formula (32) results from transformation TV (7) in the
following way (33).
For example, $2\underset{\text{\{}
\bigtriangleup \text{\}}}{\odot}\, tr^{*}(0,\ 2,\ 3) = tr^{*}(0,\ 4,\
6)$ is equivalent with $2\underset{\text{\{} \rightarrow
\text{\}}}{\odot}\,\overrightarrow{vc}(1.5,\ 3,\ 26.57{^\circ}) =
\overrightarrow{vc}(3,\ 6,\ 45{^\circ})$. A graphical
illustration of this example is shown in Figure 8. It should be noted
that the multiplication of the triangular fuzzy number by a crisp
positive real value does not change the sign of the γ angle in its vector
representation.
Figure 8. Multiplication of the triangular fuzzy
number by a crisp positive real value in traditional $\left( 2\underset{\text{\{} \bigtriangleup
\text{\}}}{\odot}\, tr^{*}(0,\ 2,\ 3) = tr^{*}(0,\ 4,\ 6)
\right)$, and vector $\left(
2\underset{\text{\{} \rightarrow
\text{\}}}{\odot}\,\overrightarrow{vc}(1.5,\ 3,\ 26.57{^\circ}) =
\overrightarrow{vc}(3,\ 6,\ 45{^\circ}) \right)\
$representations.
3.5. Backtransforming from vectors to triangles
With (7) showing the transformation from traditional representation
to vectors, it is necessary to be able to go back. That is, given vector
representation $\overrightarrow{vc}\,(\mu,s,\gamma)$,
compute the corresponding traditional representation tr^{*}(l, m, r).
Obviously, $l = \mu  \frac{s}{2}$, and
$r = \mu + \frac{s}{2}$. The m parameter can be obtained from the
fact that tan(γ) = m − μ
(see (5)), which gives us m = tan (γ) + μ.
Thus, this backtransformation TV^{−1} = VT
from vector to traditional representation is given in (34):
It may be desirable to perform computations in one representation
(triangles) but display results in the other (vectors). Therefore, it
needs to be ensured that at least additive operations carry over from
one representation to the other and back. That is, we need to show
(35):
This is shown by performing the following calculations (36).
4. Examples of applications to project management
In this section, we illustrate the usage of both representations of
the uncertainty represented by triangular fuzzy numbers in the context
of project management. In project management, one distinguishes 10
project management knowledge areas and 49 project management process
groups [53]. Our examples relate mainly to the knowledge area of Project
Schedule Management and to two process groups linked to this knowledge
area: Estimate Activity Durations and Control Schedule, as well as to
the knowledge area of Project Risk Management.
The example fuzzy numbers which are displayed in Figures 9, 10, and
11 represent durations of not yet started project activities, which are
characterized by nonstatistical uncertainty.
A possible scenario served by the examples is the following. There
are three potential suppliers of a service which is indispensable for
the activity (for instance, providing a number of hoists and equipment
handling at some stage of a construction project) whose time duration is
being estimated. The first supplier’s service is superior to other
suppliers’ service, resulting in a maximally shortened activity
performance time. The activity duration corresponding to this supplier
is represented by the number l. Due to the supplier’s high
financial demands, it is hardly possible that this firm will be
contracted. Then there is another supplier, supplier 3, whose service is
inferior, resulting in a prolonged duration of the activity, denoted as
r. Finally, there is a compromise supplier 2 – the time we
estimate they will need for the job is m. Most possibly, this
supplier will be contracted; therefore, the degree of possibility is
1.
A change in the internal situation of each of these suppliers
(financial trouble, unexpected sick leaves, defaults of other clients,
or on the contrary, a sudden increase in productivity) can affect each
of the three estimates of l, m, r
independently – as the project proceeds, each of the estimates may have
to be moved in one or other direction.
A different scenario serving the same example Figures 9, 10, and 11
could be a single contractor for the same activity who has a varying
(maximal, minimal, in between) workforce.
In the context of time duration of activities in a project the
optimistic inclination would mean that the most possible value
m is closer to the minimal possible value l and
γ negative, the pessimistic inclination (γ positive)
would occur in the opposite case. The fuzzy estimate of an activity
duration directly influences the predicted entire project duration, if
the task belongs to the critical path [39].
The selection of experts and the entire Estimate Activity Durations
process are described in detail in [54]. The process is usually based on
interviews with people who are knowledgeable about the project or its
selected aspects. They are called “subject matter experts” and
are most often members of the project team and its management,
contractors, or advisors. The type and structure of interviews must be
adapted to the interviewee and the information required. They have to
take into account various types of biases that deteriorate the
estimation quality. Sometime estimates are developed in workshops with
2025 participants rather than interviews with a single or a few
persons. Threepoint estimates are especially popular. In our examples,
we assumed arbitrary triangular fuzzy values that were selected to
represent some of the typical scenarios in the project management
practice.
The estimations of activity durations are performed in the following
moments:
Before the project starts with respect to all project activities
or tasks – the two notions will be treated as synonyms. Various duration
estimation methods mainly based on expert knowledge can be used here
[53].
Repeatedly at constant intervals during the whole project course
with respect to yet nonstarted project activities. Here, updated expert
knowledge and experience gathered in the project so far, is
used.
In the first two examples (Examples 4 and 5) we consider three
consecutive moments of the project and one and the same activity, which
has not been started up to the latest of those moments. The activity
duration is expressed, in each of the considered moments, by means of
triangular fuzzy numbers provided by experts. The first moment t_{0} is the project start,
the two other moments occur during project execution.
Example 4. Duration estimation of a task updated
by experts in consecutive moments. From a rather optimistic assessment
to a gradually more and more pessimistic assessment.
Here (Table 2) the experts are convinced at all the three moments of
time t_{i}, i = 0…2
that the task duration x would
be included somewhere in the interval x ∈ [1, 6]. However, with the
project advancing, and additional information available, they change
their mind about the inclination (skewness) of the fuzzy estimation,
i.e. the relative position of the most possible value with respect to
the support centre. At the beginning (t_{0}), they think the most
possible value of the task duration is about m = 3, thus, they assign to the
fuzzy estimation an optimistic inclination. Later, at time t_{1}, it is reevaluated to
m = 4 changing the estimation
inclination in the pessimistic direction, and at moment t_{2}, the experts’ opinion
shifts to m = 5, thus the
pessimistic inclination increases. This means that as the project
progresses, the estimation of the most possible value of the task
duration turns out to be more pessimistic than it was judged at the
project beginning, and this tendency is persevering. Furthermore, at a
certain moment in time the optimism of experts with respect to the
duration of the task in question turns into pessimism. Such pieces of
information are crucial for the Control Schedule and Project Risk
Management processes. The project manager and the project team should
address the problem as soon as possible, preferably before the task in
question starts. The sooner they react, the more chances they have to
solve the issue: either by reducing the increasing most possible task
duration or by making sure that the longer most possible duration will
be acceptable from the point of view of the final project evaluation.
The tendency observed in the task estimation may also indicate some
serious problems in the project, and the sooner they are identified, the
higher the project success chances.
Table 2. The task duration uncertainty estimated at
three moments in time for Example 4, given in traditional and vector
representations.
t_{0} 
t_{1} 
t_{2} 
Triangular 
tr^{*}(1, 3, 6) 
tr^{*}(1, 4, 6) 
tr^{*}(1, 5, 6) 
Vector 
$$\overrightarrow{vc}\left( 3.5,\ 5, 
\,{26.57}^{\circ} \right)$$ 
$$\overrightarrow{vc}\left( 3.5,\ 5, +
\,{26.57}^{\circ} \right)$$ 
$$\overrightarrow{vc}\left( 3.5,\ 5, +
\,{56.31}^{\circ} \right)$$ 
Figure 9 shows a graphical confrontation of the traditional and
vector representations of this example. Let us compare these
visualizations with respect to the basic information they should convey,
that is, the increasing most possible duration of the task in question
and the switch from an optimistic opinion on this value to a more and
more pessimistic one. This is shown by the shift of the most possible
duration value of the activity in question from below the middle value
of the interval [1, 6] to farther and
farther to the right. The visualization should be as striking as
possible to attract the attention of the, usually very busy, authorised
members of the team, at the earliest possible moment. In the authors’
opinion, the vector representation works in this respect better than the
traditional one. The traversal of the middle point of interval [1, 6] is much more clearly visible in the
case of vectors, as the pointer completely changes the direction.
Also, if we consider the Stype shape of the function in Figure 5, we
can see that the tendency which started before moment t_{1} (the gradual passage
from optimistic inclination in moment t_{0}to a deepening pessimism) will be
most visible shortly before and right after the traversal of the middle
point. In this area, small changes of m − μ influence the most
the value of γ. This means
that the vector representation is most appealing (the clock hand moves
the most) around the neutral point m − μ: shortly before it
and right after it. Although the changes in the triangles’ shapes show
the same moves of inclination from optimism to pessimism, they do so by
far less clearly, thus, seem to be less useful from the point of view of
effective project management.
Figure 9. Estimation of the task duration changing
over time in Example 4. From rather optimistic assessment to gradually
more and more pessimistic.
Example 5. Duration estimation of a task updated
by experts in consecutive moments. From rather optimistic assessment to
a pessimistic one, and back to more optimistic that the original
one.
In this example, we take a look at a similar situation as in Example
4 but the tendency in the task duration estimations is different. Here,
the triangular fuzzy numbers are defined as in Table 3.
Table 3. The task duration uncertainty estimated at
three moments in time for Example 5, given in traditional and vector
representations.
t_{0} 
t_{1} 
t_{2} 
Triangular 
tr^{*}(2, 4, 7) 
tr^{*}(2, 5, 7) 
tr^{*}(2, 3, 7) 
Vector 
$$\overrightarrow{vc}\left( 4.5,\ 5, 
\,{26.57}^{\circ} \right)$$ 
$$\overrightarrow{vc}\left( 4.5,\ 5, +
\,{26.57}^{\circ} \right)$$ 
$$\overrightarrow{vc}\left( 4.5,\ 5, 
\,{56.31}^{\circ} \right)$$ 
Graphical visualizations of both representations are shown in Figure
10. The changes, as in the previous example, regard the most possible
values. However, here, after the negative shift from below the middle
point of the interval [2, 7] (t_{0}) to a value over the
middle point in moment t_{1}, the estimation of the
most possible value in moment t_{2} returns below the
middle value and is better (lower) than in moment t_{0}. Thus, we have a shift
from optimism to pessimism and then back to even higher optimism than
originally. This could be achieved thanks to taking some decisive
measures addressing the initially negative trend, after the warning
received in moment t_{1}. Again, the
interpretation of the three estimations of parameter uncertainty of this
trend and the switch between optimism and pessimism appears to be
clearer in the vector representation compared to the traditional
approach. The changes in the positions of clock hands are more
appealing than the changes in the shape of the triangles.
Figure 10. Estimation of task duration changing over
time in Example 5. From positive assessment to negative and again to
even more positive.
In Example 6 we consider one single moment, before the project
starts, in which we estimate the duration of a task. In this moment, two
different execution modes are possible, e.g., executed in two distinct
technologies or by two separate teams. We should choose one mode for
actual execution.
Example 6. Comparison of two different task
modes at the same time, before the project starts.
Let us now consider a single moment t_{0} before the project
starts, and a project task with duration estimations of task execution
modes A and B, specified as in Table 4.
Table 4. Estimations of the duration of a project
task in two modes A and B for Example 6, given in traditional and vector
representations.
Mode A 
Mode B 
Triangular 
tr^{*}(3, 5, 8) 
tr^{*}(1, 5, 8) 
Vector 
$$\overrightarrow{vc}\left( 5.5,\ 5, 
\,{26.57}^{\circ} \right)$$ 
$$\overrightarrow{vc}\left( 4.5,\ 7, +
\,{26.57}^{\circ} \right)$$ 
Here we are facing a situation where the duration estimations for
both modes have the same most possible value (m = 5), but its position with
respect to the middle value of the supports along with the lengths of
the supports are different. These representations are visualized in
Figure 11.
Figure 11. Two different tasks at the same time in
traditional and vector representations.
In this example, the duration of the task in both modes has the same
most possible value. Thus, the selection of the mode has to be performed
using other criteria. The natural proposals are as follows:
the minimal possible value – preferably as small as
possible,
the maximal possible value – preferably as small as
possible,
the indeterminacy degree – preferably as small as
possible,
the optimism degree – preferable as big as possible.
The decision maker has to select the criteria and their relative
importance to make the final decision. For the evaluation of the modes
under criteria I and II, the decision maker will probably be better
supported by the traditional representation. However, for criteria III
and IV, the vector representation appears to be more efficient and
effective. The vector position immediately shows the inclination, either
pessimistic or optimistic, whereas the vector length permits one to
evaluate easily the indeterminacy degree. Although the same information
can be extracted from the traditional representation, this would require
from the decision maker much more mental effort.
Similar images as in Figs. 911 could be displayed on the project
dashboard for the total duration of the project, estimated in the given
moment and calculated as the sum of activity durations from the critical
path, thus the longest project network path. Since in the fuzzy case the
longest path is not always uniquely determined (see, e.g., [55–58]), the
length estimates and the consecutive changes for several paths might
have to be displayed on one dashboard. This means that, depending on the
representation chosen, the decision maker would be presented either with
several triangles or with several vectors. The vectorbased display that
uses “clocksimilar” symbols can be, in our opinion, more userfriendly
than a display based on triangles.
5. Discussion and conclusions
In this research, we proposed a novel method for visualizing
uncertainty. In this new approach, we use vectors to graphically show
uncertain information about a given parameter, which is traditionally
represented by triangular membership functions. Our vectors are defined
by three crisp numbers (μ, s, γ). The
traditional representation also uses three parameters (l, m, r) that
define the triangular membership function. However, the interpretation
of the parameters is different in both representations. The vector
anchor μ is equivalent to the
middle point of the support [l, r] of the triangular
membership function. The vector length s corresponds to the
support length, which represents the indeterminacy of the parameter
represented. The vector direction is specified by γ, which is defined as the angle
between vertical line x = μ and the line going
through points (μ, 0) and
(m, 1), where m denotes the most possible value of the
triangular fuzzy number. The vector representation may also be specified
by Cartesian coordinates of its tail and tip points.
We showed that the vector representation is mathematically equivalent
to the traditional one and derived formulas for transforming one into
the other. In this context, we analysed border cases to illustrate
properties of the new representation. We also gave formulas for the
addition of two fuzzy parameters and multiplication of a parameter by a
crisp number in the vector representation.
Although both representations are defined by three crisp numbers, the
difference in their shapes (vector versus triangle) is considerable and
may have a high influence on the perception of the underlying
information. Triangles, apart from moving along the abscissa, may change
the lengths of their three sides and the size of three angles, which
does not correspond to any phenomena occurring in everyday life.
Accordingly, a change in a single uncertain parameter involves
simultaneous changes of seven intertangled features. In a vector only
three features may change: fixing point, length, and inclination. Moving
points and changing lengths are natural for the human eye to detect.
Changes in inclination correspond to the movements of the hands of
clocks or meters on the car dashboards, which humans are very accustomed
to and comfortable with. For convenience, we have put together the main
properties of our proposal and compared them with the corresponding
features of the traditional approach in Table 5.
Table 5. Properties Comparison of traditional and
our representations of triangular fuzzy numbers.
Graphical form: 
Triangle △ 
Vector → 
Numerical form: 
Three crisp numbers (l, m, r):
$l = \mu  \frac{s}{2}$ (smallest
value),
m = tan (γ) + μ
(most possible value),
$r = \mu + \frac{s}{2}$ (largest
value).

Three crisp numbers (μ, s, γ):
$\mu = \frac{l + r}{2}$ (middle
value),
s= r − l
(indeterminacy degree),
γ = arctan(m − μ)
(pessimistic or
optimistic tendency).
In Cartesian coordinates:
Tip (μ, 0)
,
Tail (μ + s ⋅ sin(γ), s ⋅ cos(γ)).

Graphical features influenced by a change in an
uncertain parameter value: 
Location along the abscissa,
Lengths of three triangle sides,
Sizes of three triangle angles.


Seven components may change, thus making visual processing more
challenging and requiring more cognitive effort. 
Only three components change, thus simpler and faster visual
processing takes place and less cognitive effort is required. 
Difference between pessimistic or optimistic tendency
(m − μ): 
Is measured by distance, which is scaledependent. 
Is measured by angle, which is scaleinvariant. 
Change in pessimistic or optimistic tendency (m − μ): 
The change in the shape of a triangle is not obvious and is visually
more difficult to detect. 
It is easier to spot the change by observing the vector direction.
This may facilitate early identification of forthcoming problems and
speed up corrective decisions. 
Dynamic visualization of uncertain parameter changes in
time: 
Less obvious interpretation since more graphical features
change. 
Better suited as they resemble clock hands movements or car
meters. 
Visualization of multiple uncertain parameters: 
Processing many triangles is more difficult and graphically awkward
as compared to processing multiple vectors. 
Displaying many vectors is less visually cluttered and easier to
interpret than displaying multiple triangles. They resemble vector
fields. 
Computer implementations and visualizations: 
More troublesome compared to vectors from the user interface design
point of view. 
Easier, since vectors are graphically more compact and take up less
space in the graphical user interface. 
By providing illustrative examples, we show how this new approach may
be applied to project management. We considered and compared
visualizations of both representations for the duration estimation of a
task whose assessment changed from rather optimistic to more and more
pessimistic (Example 4) as well as from optimistic to more pessimistic
and back to optimistic (Example 5). The underlying problem was the ease
of perception of the respective phenomenon by the project manager. It is
especially important in the midst of project execution, when they are
dealing with dozens of project tasks and have to identify as quickly as
possible the problematic ones. It appears that the vector representation
is more natural and appealing and, thus, would provide a more efficient
(from the point of view of the project manager) visualization of the
current situation of individual tasks. The last example (Example 6)
concerned the estimation of the project task duration in two possible
implementation modes. The underlying problem here was the need to choose
one mode for actual project implementation. This is a multicriterial
problem. Here the two representations turned out to be complementary:
the vector representation visualizes better the difference between the
two modes according to some criteria, and the traditional one –
according to other ones.
An important soft computing problem needs to be solved if we want to
identify a critical path when fuzzy numbers are used to represent
project parameters (we mentioned it after Example 6). In such a case, it
is necessary to rank (or find the maximum of) fuzzy numbers representing
the lengths of several possible paths. This problem is complex and has
been subject to extensive research for the traditional representation of
fuzzy numbers (e.g., [59,60]). It becomes even more difficult to rank
project network paths with fuzzy activity durations (e.g., [55–58]).
Often no unique solution exists, and decision makers have to decide
arbitrarily which ranking they prefer. The existing ranking methods of
fuzzy numbers are strongly determined by their traditional triangular
representation – they use notions like centroids, are based on fuzzy
number levels, or areas under a selected section of the membership
function.
Our new approach for representing fuzzy numbers as vectors provides a
completely new perspective for the problem of ranking fuzzy numbers,
which should be investigated in the future. We hypothesize that the
vector representation may give rise to new fuzzy numbers ranking methods
that can be potentially more appropriate in some cases and better
reflect the decision maker preferences. It would be interesting to
compare the rankings provided by various decision makers for both
representations. In the context of project management, this could
substantially increase the spectrum of project uncertainty management
methods.
In the presented context, vector representations seem to be better
and less cognitively demanding than triangles, although in some
applications they should be accompanied by the traditional
representation. As was demonstrated, vector direction changes are clear
and evident at first glance, especially when the angle’s sign reverses,
which is not the case for changes in the shapes of triangles.
Our previous study on the usability of using vectors as a graphical
representation of uncertainty [46] showed the potential usefulness of
this approach for visualization purposes in a managerial context. In
particular, it appears to be more adapted to the needs of project
managers in some practical cases. This preliminary investigation
involved scenarios in which static information was presented. It
indicated the overall equivalence of both representations in terms of
efficiency and effectiveness. This paper’s sample visualizations of
uncertainty changes over time (Example 4 and 5) in a project management
context suggests that the vector representation may be better suited for
presenting the dynamics of uncertainty than triangles. Example 6 proves
that a combined approach (triangle, vector) would provide more
information for project planning than the traditional representation
alone. Moreover, as mentioned in the introduction, some earlier studies
on cognitive aspects of information visualization allow us to think that
the proposal may have some psychologically based advantages over the
traditional representation by triangles [44,45].
In various optimization problems such as scheduling, supply chain
management, transportation, it is necessary to perform mathematical
operations on fuzzy parameters. In most of those problems, addition and
multiplication by a crisp positive number are sufficient. In Section
3.4, we provide appropriate definitions, formulas, and examples of
performing these operations in both representations. In this regard, the
vector representation is clearly inferior to the traditional one.
Although arithmetic of vectors as given by (21) and (32) is not
particularly difficult, it involves taking both the tangent and its
inverse. This computational complexity provides much more room for
rounding error as compared to the traditional version. The classic
arithmetic performed on triangular fuzzy numbers is more
straightforward, since it involves either three additions of crisp
numbers (19) or three multiplications of crisp numbers (31). Thus, it
appears advantageous to perform mathematical manipulations on triangular
fuzzy numbers using their traditional representations, regardless of how
uncertainty data were gathered or graphically presented. We provide the
necessary direct formulae for the passage from one representation to the
other.
Furthermore, the vector representation also has some specific
features that could prevent it from being suitable in some situations.
Among them, there is the property of nonlinear changes in the γ angle with the increase of the
distance between the support middle point μ and the most possible value m. This feature will be very useful
in project management when the neutrality (neither optimism nor
pessimism) of duration estimation means that the middle value of the
support is at the same time the most possible value. However, it is not
always the case. Therefore, in practical applications, it seems to be
reasonable to use both representations. Combining their advantages will
provide a fuller picture of parameter uncertainty and its changes over
time. We have summarized the key advantages and weaknesses of our
proposal in Table 6.
Table 6. Advantages and weaknesses of our vector representation.
Extends the arsenal of visually representing uncertainty. May be
better suited for some groups of users in practical applications
[46]. 
The new approach, which scientists and practitioners are not
familiar with. The need to implement it in existing software that is
used for uncertainty visualisations. 
Strict equivalence with the traditional triangular representation,
which allows for interchangeable or simultaneous use. Easy
transformations between these representations. 
Mathematical operations may be subject to a higher rounding error
compared to the traditional version. 
May be better suited for representing some type of problem, for
instance, those in which the detection of the change in the tendency
(optimistic/pessimistic) is of the greatest importance. 
The maximal absolute value of γ angle is smaller than 90^{∘} in practical applications.
This could be misleading when interpreting the magnitude of an
optimistic or pessimistic tendency. 
Facilitates new research directions. Possible future applications
relate to fuzzy number ranking methods. Current solutions are strongly
based on geometrical features of the triangular representation. 
The change in the γ gamma
angle is not linearly dependent on (m − μ). The vector
direction is more sensitive closer to the vertical line x = μ. This may be
misleading and makes interpretations more complex. 
Despite the concerns presented, this research approach provides a new
perspective on how to visualize the uncertainty of various parameters.
Although we focus in this paper on the project management context,
exactly the same representations can be used in any other area. Our
proposal may be particularly useful when uncertainty needs to be
presented graphically and explored by experts or decisionmakers without
a formal mathematical background. Extending the arsenal of visualization
techniques will allow the decisionmakers to define, present, and
interpret the uncertain information more accurately. This, in turn,
should result in the achievement of specific goals more effectively and
efficiently.
The idea of using vectors to visualize uncertainty defined by
triangular fuzzy numbers provides further possibilities for scientific
exploration. For instance, one may try to search for vector
specifications different from those presented in this paper. The
presented idea is applicable only to triangular fuzzy numbers, but
future research can also be directed to develop other vector
representations for different fuzzy number types, e.g., trapezoidal,
LR, intuitionistic, or type2. It may also be possible to discover
ideas for vectorlike representations pertaining to other forms of
uncertainty than the ones treated in this paper.
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