During making
decisions in various fields, managers of different levels often have to take
into account a large number of various factors, as well as taking into account
external conditions. Often these decisions are made intuitively and are based
mainly on the experience and knowledge of the decision maker (DM). However,
this is not the only way to make decisions. In modern science, a wide variety
of decision making methods based on specialized approaches and algorithms are
widely used [1]. At the same time, the solution of multicriteria decision making
problems by means of these specialized approaches and algorithms can be not
very effective when there are dozens and hundreds of alternatives, and they all
have more than a dozen criteria. These situations are quite common when, for
example, the source of alternatives is multisensory systems [2], or when
initially many alternatives are formed by means of specialized systems in the
course of multiple simulation [3, 4].
Therefore, in such
cases, the initial choice set is firstly filtered, and then decision making
methods are used already on a filtered selection of alternatives.
Traditionally, statistical methods are used for filtering. Taking into account
the fact that during decision making DM activates his mental activity, the same
factor can be used to solve the problem of filtering alternatives. To do this, DM’s
mental activity can be addressed to a comparative analysis of alternative visual
images (visual representations of vector criteria) [5]. But it is necessary to
develop effective algorithms for visual filtering of alternatives, so that the
main DM’s efforts should be focused exactly on intelligent visual selection,
and not on accompanying actions or calculations.
In multicriteria
decision making problems, the criteria characterizing alternatives can be set
both as quantitative and qualitative characteristics. In order to work with
various criteria, first it is necessary to reduce them to numeric variables.
A wide range of different algorithms and methods are used for these
purposes [1]. These methods and algorithms allow to convert the initial values
of the criteria into numerical values in the form of the corresponding
functions f_{i}(k), where , K is the number of criteria.
Function f_{i}(k) is usually nonlinear and may contain
additional conditions for different intervals of k initial values of icriterion.
The dimension (i.e. range of possible values) of f_{i}(k)
function for different criteria may vary significantly. For this reason, for
further work with alternative visual images having these criteria, it is
necessary to normalize f_{i}(k) functions. One of the
traditional approaches in this case is normalization by interval [0; 1] based
on the maximum and minimum possible values of the function.
1.
, if
the maximum criterion value corresponds to the best option;
2.
, if
the minimum criterion value corresponds to the best option.
Values f_{i,min}
and f_{i,max} are defined in the given choice set , , , where N is a number of alternatives
either is from valid (anticipated) values, or is defined according to the
formulas:
,
.
Various methods
are currently used to visualize many alternatives [6]: parallel coordinates
plot (Fig. 1), bar diagrams (Fig. 2), radar diagrams ([7], Fig. 3), pie charts
(Fig. 4) and others.
Most of these
approaches focus on displaying several alternatives in a single chart (Fig.
13). This way is quite appropriate when the number of alternatives (about
310) and criteria (about 37) is not big. However, such visualization methods
are not suitable when you have to analyze dozens and hundreds of alternatives,
each of which can have more than 10 criteria, because the chart becomes too
overloaded and complex to analyze. A large number of alternatives can be
visualized, for example, by using a set of pie charts, each representing a
different alternative (Fig. 4). However, effective filtering of alternatives
requires a more holistic perception of their visual image. Pie charts do not
sufficiently provide such a perception with a large number of criteria due to
color diversity.
Fig. 1. Alternatives’ visualization in the form
of polyline criteria values
Fig. 2.
Alternatives visualization in the form of bar
diagrams
Fig. 3. Alternative
visualization in the form of a
radar diagram
Fig. 4. Alternatives
visualization in the form of pie charts
Taking into
account the considered peculiarities we formulate the main criteria of
constructing an algorithm for visualizing alternatives in multicriteria
decision making problems in order to filter them.
1.
Each alternative should
be represented as a single image.
2.
Since there may be too
many alternatives, it is necessary to provide a mechanism for focusing on a
small sample of them and the possibility of changing this focus.
3.
To highlight equivalent
criteria with a different color is inappropriate, because the color can
adversely affect DM’s alternative.
4.
Color effect is useful
when visualizing alternatives for criteria values close to optimal in order to
further focus DM’s attention on them.
Based on these
criteria we proposed an appropriate algorithm. Within this algorithm, two main
aspects can be distinguished.
1. Visualization
of one alternative.
2. Allocation
of alternatives and focusing method on their subset (focus subset).
The construction
of a visual image is based on pie charts and radar diagrams (Fig. 3, 4). Their
common feature is that the value of the alternative by a separate criterion is
located on a separate beam (radar diagram) or a sector of the circle (pie
chart). However, moving away from a traditional representation of radar diagrams,
we will place each alternative on a separate circle (as in pie charts).
For an alternative
to be represented as a single image, it is advisable to use a single filling
style for all criteria: for the pie chart, the corresponding sectors are
filled, and for the radar diagram, the corresponding polygon is filled. Taking
into account that the sector radius in the pie chart and the position of
polygon points is determined by the proximity of the normalized criterion value
of the corresponding alternative to 1 (the closer to one, the better), it is
advisable to use a gradient radial fill: in the center of the circle the color
is neutral, and close to the border it is contrast (for example, red).
In addition to the
color effect in this approach, an additional source of focusing and choice
preferences among alternatives is the area of the corresponding figure.
1. For a pie chart, the shape area is defined
as a sum of sector areas:
,
where in
the case, if all the sectors have the same angle (this is usually the case
where all criteria are equal) and radius of sector is equal to normalized value
of corresponding criterion (algorithm "Sectors (radius)"). If,
however, there is some preference concerning criteria, then a sector with a
larger angle can be specified for the more preferred criteria. Let us denote
this angle as (radian), then
.
The value of angle can be defined on the basis of
ranking algorithms or weighting criteria used in decision making methods [1].
This approach has a peculiarity that the sector area is proportional to the square
of the alternative value by the criterion, which may unnecessarily draw
attention to the alternatives that have the maximum value of one of the
criteria. To reduce this degree of influence, it is possible to establish not a
quadratic, but a linear dependence between the sector area and the alternative value
according to the corresponding criterion: . Then the radius of the corresponding
sector will be defined by the formula: (algorithm
"Sectors (radius root)").
2. For a radar diagram, the area of a shape is
defined as the sum of the areas of triangles:
,
where , .
The main
peculiarity of calculating the area of the polygon according to this formula is
that its final value is in addition affected by the order of criteria placement,
because when placing nearby criteria with large values , then the total area is larger, which
means that such alternatives are more focused on themselves attention of DM than
others. On the other hand, the focusing factor in this case may also be a large
number of sharp angles in the polygon. Therefore, for this type of diagrams let
us consider two modifications of exchanging criteria:
• grouping a
number of criteria with higher values for the displayed alternatives;
• alternation of
criteria with higher and lower values for the displayed alternatives.
In both versions,
we first calculate the mean value of m_{i} for each criterion in
the set of displayed alternatives , where , T is the number of displayed alternatives:
.
Next, we sort
descending m_{i} with remembering the initial i position.
Let us present the result as a sequence
,
where .
For the first
modification (grouping), p_{1 }exchange is defined as follows (radar
(p_{1 }exchange)):
For the second
modification (grouping), p_{2 }exchange is defined as follows (radar
(p_{1 }exchange)):
For example, suppose
we have two alternatives A_{1 }= {1.0; 0.4; 0.8; 1.0} and A_{2}
= {0.5; 1.0; 1.0; 0.2}. Average values of the criteria will be {0.75; 0.7; 0.9;
0.6}. Then p_{0} = {3; 1; 2; 4} (i.e. maximum value we have for
the 3rd criterion, and minimal value for the 4th). The permutation p_{1}
will be: p_{1}={3; 2; 4; 1}, and p_{2}={3; 2; 1;
4}. Those for p_{1}: A_{1}(p_{1})={0.8;
0.4; 1.0; 1.0}, A_{2}(p_{1})={1.0; 1.0; 0.2;
0.5}, and for p_{2}: A_{1}(p_{2})={0.8;
0.4; 1.0; 1.0}, A_{2}(p_{2})={1.0; 1.0; 0.5;
0.2}.
Building a visual
image of alternatives in the case of using radar diagrams starts with the use
of one of the two considered exchanges of criteria: p_{1} or p_{2}.
When visual images
of alternatives is placed on the screen, we follow these principles:
1.
It is necessary to
visualize all alternatives in a simplified form on a smaller part of the screen
within a rectangle (simplified display). In this part of the screen you need to
place the selection area of alternatives (it is also advisable to do this in
the form of a rectangle). This area must be movable.
2.
When you change the
position of the selection area on the simplified display, you need to define a
list of alternatives completely being inside it. These alternatives are
displayed in the focus area, which is also a rectangular portion of the screen
that takes up significantly more space than the simplified display area.
The position of alternatives
in these areas is determined by the grid consisting of rows and columns (Fig.
5). The alternatives themselves are placed at the nodes of this grid. In this case,
for a more uniform distribution of alternatives, the grid is not orthogonal,
but with an offset in even rows by the circle radius in which the alternative
is visualized.
Fig. 5. Grid
of alternatives layout
The algorithms
described above were implemented in a special program “AlternativesVisualizer”. The developed software allows to load from the
table view a list of alternatives with a numerical representation of values
according to the relevant criteria, as well as to filter them in the rendering
mode (Fig. 6).
For filtering the
program provides two approaches – manual filtering by hiding or displaying
alternatives (the central part of the form in Fig. 6), as well as filtering
based on the threshold values of the criteria (the right part of the form in
Fig. 6) – all alternatives that do not meet the thresholds are hidden.
At the bottom of
the form there is a simplified area of alternatives visualization where the
user can select a range of alternatives to be displayed in the main area by
moving the corresponding rectangular block. Using the toolbar buttons, the user
can enter the hide or show alternatives mode. Hiding or displaying is done by
clicking the mouse button on the corresponding alternative on the main
visualization area.
The program “AlternativesVisualizer” supports all four visualization options
considered in the paper:
• sectors with
radii proportional to the criteria for the alternative;
• sectors with
radii proportional to the roots of the criteria values for the alternative;
• radar diagram
with p_{1 }criteria exchange;
• radar diagram
with p_{2 }criteria exchange.
Working with the
program, the user can perform several experiments on filtering alternatives by
means of different ways of their visualization. The results are summarized in a
table (Fig. 7).
In addition, on
the basis of the table obtained, the program “AlternativesVisualizer” analyzes the results and displays the numbers of
alternatives that the user has chosen in all experiments in a separate list.
This approach allows one to reduce the resulting set of filtered alternatives
further, leaving only those that the DM has preferred for all visualization
methods.
With the help of
the developed program “AlternativesVisualizer”, an experiment was conducted. A choice set
(200 alternatives) with 15 criteria was randomly generated. An alternative with
maximum values for all criteria out of all 200 generated was added to this set
(in order to verify that this alternative will not be filtered). Thus, there
were 201 alternatives in total. The problem of filtering alternatives was
solved 5 times using different methods: 4 methods of different visualization
options and one method – setting low threshold values for all criteria in order
to reduce the number of displayed alternatives by an order of magnitude. The
results of the experiments are presented in table 1.
Fig. 6. Interface
of software for visual filtering of alternatives “AlternativesVisualizer”
Table 1. Experiment results of filtering alternatives
by using various methods.
¹

Filtering type

Number of alternatives

Alternatives list

1

Sectors (radius)

41

4, 5, 7, 8, 12, 14, 18, 27, 29, 34, 41, 42, 46, 52,
53, 55, 56, 62, 76, 79, 82, 83, 89, 97, 103, 106, 110, 120, 127, 130, 146,
147, 155, 156, 158, 159, 161, 170, 184, 197, 201

2

Sectors (radius root)

50

4, 5, 7, 14, 17, 18, 19, 21, 27, 29, 33, 34, 36, 41, 42, 46, 53, 56, 62, 63, 76, 79, 81, 82, 83, 84, 87, 90, 97, 103, 120, 122, 127, 130, 132, 133, 138, 139, 141, 146, 147, 154, 155, 156, 165, 170, 173, 177, 184, 201

3

Radar (p_{1 }exchange)

45

4,
5, 7, 14,
17, 18, 21, 27, 31, 33, 34, 36, 42, 52, 56, 62, 63, 76, 79, 83, 84, 89, 90, 97, 103, 106, 118, 120, 121, 130, 141, 142, 146, 147, 154, 155, 156, 158, 161, 173, 177, 178, 182, 184, 201

4

Radar (p_{2 }exchange)

36

4, 5, 7, 14, 17, 18, 19, 21, 27, 29, 33, 34, 36, 52, 53, 55, 62, 76, 79, 83, 84, 90, 103, 104, 106, 120, 127, 130, 138, 139, 146, 147, 154, 155, 177, 201

5

Threshold

27

4, 5, 27, 36, 39, 42, 43, 50, 52, 62, 64, 83, 84, 90, 92, 98, 100, 104, 106, 110, 149, 154, 155, 175, 177, 190, 201

Using the program (on
the “Results” tab) we find the intersection of choice sets obtained in 14
experiments, and we get 18 alternatives (Fig. 7). When determining the
intersection of choice sets obtained in all experiments, we obtain only 7
alternatives: 4, 5, 27, 62, 83, 155, 201. In the case of combining choice sets (this
feature is also available in the "Results" tab of the program) obtained
in experiments 14, we will get 68 alternatives. And if we combine choice sets obtained
in all five experiments, we get 78 alternatives.
Fig. 7. Interface
of «Results» tab
Analyzing the
results obtained, we can conclude that the greatest effect of filtering alternatives
is achieved by finding the intersection of sets obtained in experiments with
different methods of visualization in combination with threshold filtering alternatives.
Further work on selecting the optimal alternative should be carried out with
this subset using other methods of decision making, designed and applicable on a
small number of alternatives.
With the help of
the developed software “AlternativesVisualizer”, there was also an experiment conducted
for real alternatives. We got estimations according to 144 criteria for norms
L1 and L2 for the computational problem of evaluating the accuracy of the
calculations of inviscid flow around a cone by means of several OpenFoam
solvers (rhoCentralFoam, pisoCentralFoam, sonicFoam, rhoPimpleFoam, QGDFoam) [8,
9].
After preliminary
processing of the initial data according to norm L1, only 88 criteria were left
(as for the rest of the criteria, the data were incomplete). Having constructed
and analyzed visual images for different algorithms, we determined that the alternative
corresponding to pisoCentralFoam solver is almost always occupies a large area
and is more contrast by using the red fill on the border of the corresponding
visual image, so we can conclude that this algorithm is the most preferable
(Fig. 811).
Fig. 8. Alternatives
visualization using method “Sectors (radius)” for criteria on L1 norm
Fig. 9. Alternatives
visualization using method “Sectors (radius root)” for criteria on L1 norm
Fig. 10. Alternatives
visualization using method “Radar (p_{1} exchange)” for
criteria on L1 norm
Fig. 11. Alternatives
visualization using method “Radar (p_{1} exchange)” for
criteria on L1 norm
176 criteria were
selected when comparing the alternatives according to two norms, and one of the
alternatives was excluded from consideration, as there were no its data on L2
norm. As a result of visual filtering, it was determined that from four
alternatives the two alternatives have a large area, as well as more contrast
(due to the use of red color at the borders of the visual image) – this visual
images are determinate rhoCentralFoam and pisoCentralFoam solvers (Fig. 1215).
However, preference can again be given to pisoCentralFoam solver, because in
almost all images it visually occupies a slightly larger area compared of
rhoCentralFoam solver.
Fig. 12. Alternatives
visualization using method “Sectors (radius)” for criteria on L1 and L2 norms
Fig. 13. Alternatives
visualization using method “Sectors (radius root)” for criteria on L1 and L2
norms
Fig. 14. Alternatives
visualization using method “Radar (p_{1} exchange)” for
criteria on L1 and L2 norms
Fig. 15. Alternatives
visualization using method “Radar (p_{2} exchange)” for
criteria on L1 and L2 norms
The approaches to
placement and focusing alternatives and algorithms of constructing visual
images for multicriteria alternatives (four algorithms) considered in the
paper have shown that visual filtering can be quite an effective method in
decreasing the initial choice set. As a result of several experiments on
filtration by means of the developed software an initial set of 201 alternatives
has been reduced to 2750 alternatives. Identifying a subset from the results
of different experiments, which is common for the results of all experiments,
allowed us to decrease this set to only 7 alternatives, i.e., to reduce the
number by about 28 times.
Four ways of
constructing visual images are considered in the paper. However, it is possible
to expand these options further through the use of additional visualization
techniques: 3D visualization, other types of diagrams, other types of defining
criteria exchange, etc.
Also, an
experiment was conducted on the visual selection of the best alternative
(solver) according to the known criteria characterizing the accuracy of
calculations. Comparison was made for 5 alternatives and the number of criteria
was 176. According to the comparative visual analysis it was clearly seen that
pisoCentralFoam solver gives more accurate calculation results.
An additional
feature of the developed mathematical support and software is that it is
suitable for using by a group of experts, each of whom can conduct a series of
experiments with different types of constructing visual images, and then the
results can be summarized.
The work was
supported by Russian Science Foundation grant ¹ 181100215.
[1]
Figuera J., Greco S.
and Ehrgott M. (Eds). Multiple Criteria Decision Analysis: State of the Art
Surveys. – New York: Springer, 2005. – DOI: 10.1007/b100605.
[2]
Zakharova A., Vekhter E.,
Shklyar A., Pak A. Visual modeling in an analysis of multidimensional data //
Journal of Physics: Conference Series. – 944. – 012127, 2018. DOI: 10.1088/17426596/944/1/012127.
[3]
Isaev R.A.,
Podvesovskii A.G. Generalized Model of Pulse Process for Dynamic Analysis of
Sylov’s Fuzzy Cognitive Maps // CEUR Workshop Proceedings of the Mathematical
Modeling Session at the International Conference Information Technology and
Nanotechnology (MMITNT 2017), Vol. 1904. – P. 5763. – DOI:
10.18287/16130073201719045763
[4]
Podvesovskii A.G.,
Isaev R.A. Visualization Metaphors for Fuzzy Cognitive Maps // Scientific
Visualization, 2018, Vol. 10, Num. 4, P. 1329. – DOI: 10.26583/sv.10.4.02
[5]
Zakharova, A.A.,
Shklyar, A.V. Informative features of data visualization tasks // Scientific
Visualization, 2015, Vol. 7, Issue 2, P. 7380.
[6]
Pomerol JC., Romero S.
Multicriterion Decision in Management: Principles and Practice. – Kluwer
Academic Publishers: Boston, 2000. – DOI: 10.1007/9781461544593.
[7]
Morris M.F. Kiviat
graphs: Conventions and “figures of merit” // ACM SIGMETRICS Performance
Evaluation Review. – V. 3, N. 3. – P. 28. – New York: ACM, 1974. – DOI:
10.1145/1041691.1041692.
[8]
Bondarev A.E.,
Kuvshinnikov A.E. Analysis of the Accuracy of OpenFOAM Solvers for the Problem
of Supersonic Flow Around a Cone // ICCS 2018, Lecture Notes in Computer
Science (LNCS) 10862. – P. 221–230, 2018. – DOI:10.1007/9783319937137_18.
[9]
Bondarev A.,
Kuvshinnikov A. Comparative Estimation of QGDFoam Solver Accuracy for Inviscid
Flow Around a Cone // IEEE The Proceedings of the 2018 Ivannikov ISPRAS Open
Conference (ISPRAS2018). – P. 8287, 2018. – DOI: 10.1109/ISPRAS.2018.00019.