=Paper=
{{Paper
|id=Vol-2744/paper23
|storemode=property
|title=Methods for Construction and Research Stereometric Visual Images of Multi-criteria Alternatives
|pdfUrl=https://ceur-ws.org/Vol-2744/paper23.pdf
|volume=Vol-2744
|authors=Alena Zakharova,Dmitry Korostelyov
}}
==Methods for Construction and Research Stereometric Visual Images of Multi-criteria Alternatives==
https://ceur-ws.org/Vol-2744/paper23.pdf
Methods for Construction and Research Stereometric
Visual Images of Multi-criteria Alternatives*
Alena Zakharova1,2[0000-0003-4221-7710] and Dmitriy Korostelyov1,2[0000-0002-0853-7940]
1 Keldysh Institute of Applied Mathematics Russian Academy of Sciences, Moscow, Russia
2 Bryansk State Technical University, Bryansk, Russia
zaa@tu-bryansk.ru, nigm85@mail.ru
Abstract. The article discusses methods for constructing and research stereomet-
ric visual images of multi-criteria alternatives. The difficulty of applying tradi-
tional decision-making methods when choosing the best alternative in the case of
decision-making tasks with a large dimension is noted. Methods are proposed for
reducing the dimension of a decision-making problem by constructing stereomet-
ric visual images that describe choice set and criteria set. The tasks of visual an-
alytics are highlighted in the course of reducing the dimension of the decision-
making task: grouping and ranking criteria, pairwise comparison and filtering of
alternatives. A description of the methods for constructing stereometric visual
images of multi-criteria alternatives based on histograms of alternatives and cri-
teria is given. Also describes a method for constructing surface of the criteria's
correlation. Using the developed method of alternatives visualization as a single
image enables their visual filtering. The application method of the criteria visu-
alization as a single image in combination with the construction and analysis of
surface of the criteria's correlation allows for the grouping and ranking of criteria.
The experiment is described on the application of the proposed methods for con-
structing and studying stereometric visual images of multi-criteria alternatives
using the example of the problem of choosing the best OpenFoam solver, with
the help of which estimates of the accuracy of calculating the inviscid flow
around the cone were obtained. It is shown how, due to the above visualization
methods, it is possible to reduce the count of criteria by several times.
Keywords: Alternatives Visualization, Multi-criteria Alternatives, Decision-
making Tasks Dimension Decrease, Criteria's Convolution, Alternatives Visual
Filtering.
1 Introduction
In various subject areas, managers and researchers are often faced with the task of
choosing the best option from the many possible alternatives. There may be only a few
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
* The work was supported by Russian Science Foundation grant № 18-11-00215
2 A. Zakharova, D. Korostelyov
such options, or there may be hundreds (or thousands). Moreover, each option, in turn,
can also be characterized by both just a few criteria, and dozens or even hundreds.
Usually, this choice problem is considered as a decision-making problem [1]. When the
alternatives and criteria a bit, then the decision-maker (DM) can successfully use the
well-known methods of decision making: SMART, PROMETHEE, AHP, etc. [2].
In the case when we are faced with a large count of alternatives or criteria that char-
acterize them, the use of traditional methods already becomes difficult. In such a situa-
tion, one usually resorts to methods of filtering alternatives and the method of decreas-
ing a dimension of them. However, this is not always effective, since decision-making
is a subjective and creative process, and if the above methods are applied, we can lose
some important alternatives and criteria, without even noticing it. Therefore, it is im-
portant to also involve decision-makers at the stage of compressing the available set of
alternatives and criteria.
The decision-maker is often unable to perceive large volumes of multi-criteria alter-
natives without using special tools. The decision-maker can effectively and, which is
also important, holistically perceive large volumes of information only with the use of
specialized methods of data visualization [3-6].
If the decision-maker is involved in the procedure of filtering alternatives and reduc-
ing their count and dimension, he faces the following tasks: criteria's grouping, criteria's
ranking, pairwise comparison of alternatives [7, 8]. Effective 3D visualization methods
can be used to solve these problems. The development and improvement of such meth-
ods will allow us to study a choice set and set of criteria in order to select the most
informative ones for achieving visualization goals. Consider several methods to con-
struct and research such stereometric visual images of alternatives to filter and reduce
the dimension.
2 Methods for Constructing and Researching
Stereometric Images of Multi-criteria Alternatives
To increase the information content of stereometric images of multi-criteria alterna-
tives, it is proposed to solve the following tasks:
1. Ranking of criteria's.
2. Grouping of criteria's.
3. Filtering of alternatives.
Suppose we have the choice set A = {A1, A2, …, AN}, where N – count of alternatives.
Each alternative is specified by a vector of normalized criteria values: Ai = {ai,j}, where
1 ≤ j ≤ K, K – count of criteria, 0 ≤ ai,j ≤ 1. Under criterion commonly, understood
parameter (quantitative or qualitative) that characterized the alternative (can be ob-
tained by calculation, empirically, or synthetically). In the case of using qualitative cri-
teria, an additional procedure for their normalization using specialized scales is required
[8]. At the same time, higher values of the criteria ai,j corresponds to a more priority
alternative.
Methods for Construction and Research Stereometric Visual Images of Multi-criteria … 3
Consider several methods for visualizing multi-criteria alternatives that can be used
in the course of visual analytics in solving the above problems.
2.1 Methods for Visualizing Multi-criteria Alternatives
There are various methods for 2D imaging of single multi-criteria alternative [9]: His-
tograms, 2D plots, radar plots, radial plots, etc. The simplest method for visualizing
many alternatives in three-dimensional space is to place two-dimensional images of
alternatives along the 3rd coordinate axis (Fig. 1).
Fig. 1. Stereometric visualization of three multi-criteria alternatives using histograms
The use of this method can be effective in the case when the count of alternatives is
small and the count of criteria is large, and the task is to filter out alternatives. Based
on the obtained stereometric image, it is possible to visually determine the “strong” and
“weak” alternatives due to the prevalence of columns with large values. However, in
the case of similar characteristics of the initial data, it would be more appropriate to try
to reduce the count of criteria. This would allow the use of traditional decision-making
methods. However, this visualization method will not allow you to do this, because it
is not obvious how the count of criteria can be reduced. Therefore, we will consider
other methods of visualizing stereometric images of multi-criteria alternatives.
One possible method is to swap the criteria and alternatives for the previous visualiza-
tion method - i.e. each row visualizes the values of a single criterion for all alternatives
(Fig. 2). In this case, a reduction in the number of criteria can be achieved by excluding
criteria that have a similar histogram with other criteria. However, the effectiveness of
4 A. Zakharova, D. Korostelyov
this procedure depends on the relative position of the criteria along the axis of the cri-
teria (Fig. 2).
Another visualization method for searching similar criteria at the given choice set
can be basis on calculating correlation coefficients of the criteria values. To do this,
you must first calculate the matrix of correlation coefficients between all pairs of crite-
ria: M={mi,j}, where 1 ≤ i, j ≤ K, mi,j – correlation coefficient between the criteria's
values of the alternative. As a coefficient in the simplest case, you can use the Pearson
correlation coefficient, calculated by the formula [10]:
∑𝑁
𝑡=1(𝑎𝑡,𝑖 −𝑎𝑖 )∙(𝑎𝑡,𝑗 −𝑎𝑗 )
𝑚𝑖,𝑗 = , where (1)
2 2
√∑𝑁 𝑁
𝑡=1(𝑎𝑡,𝑖 −𝑎𝑖 ) ∙∑𝑡=1(𝑎𝑡,𝑗 −𝑎𝑗 )
1
𝑎𝑖 = ∑𝑁
𝑡=1 𝑎𝑡,𝑖 . (2)
𝑁
Fig. 2. Stereometric visualization of criteria for three alternatives using histograms
The thus obtained matrix M can be visualized as a surface (Fig. 3). A larger value of
the correlation coefficient means that the criteria are very similar and one of them can
be removed from consideration. As seen in Fig. 3, areas with a high cross-correlation
coefficient may form on the surface, containing groups of criteria. The detection of such
zones may indicate the advisability of replacing the group of criteria that form it with a
single criterion.
It is worth noting that the given method (Fig. 3) can also be applied in cases where
the count of alternatives and criteria is large. However, it can only be used to group and
rank criteria, but it does not provide filtering of alternatives. It is also worth noting that
an additional advantage of this method is that when the group of criteria is collapsed,
the rank of this group can also be obtained, which can be used in classical decision-
making methods.
Methods for Construction and Research Stereometric Visual Images of Multi-criteria … 5
The rank of the group (rx) can be determined, for example, in the following ways:
1. rx = count of elements in a group x.
2. The sum of the correlation coefficients between the criteria that make up the group
and the criterion g, which will replace the entire group:
𝑟𝑥 = ∑𝑖∈𝑥
𝑖 𝑚𝑖,𝑔 . (3)
Fig. 3. Stereometric visualization of the surface of the criteria's correlation
Thus, as a result of applying this method, we get a new decision-making problem, char-
acterized by the same choice set, but having a smaller number of criteria and supple-
mented by the ranks of these criteria: A'={a'i}, where a'i = {(ai,j; rj)}, 1 ≤ j ≤ K' ≤ K, K'
– new count of criteria.
2.2 Visual Analytics of Stereometric Images of Multi-criteria
Alternatives
The effectiveness of research and analysis of stereometric images depends significantly
on a foreshortening of view. Therefore, for this problem, it is important for the ability
6 A. Zakharova, D. Korostelyov
to change the foreshortening of the constructed stereometric image. At the same time,
the analyst should have a convenient tool for making a quick change of foreshortening.
The best way to do this is to construct a stereo model of the scene containing the cor-
responding visual images, and control the foreshortening with the camera, as is done in
modern environments and 3D modeling tools.
Appropriate tools that ensure the implementation of the following functions should
accompany visual analytics of stereometric images of multi-criteria alternatives:
• hide and show alternative;
• hide and show criterion;
• alternatives permutation;
• criteria permutation;
• selection of a specific alternative as a focused one (when using the method of visu-
alizing alternatives as a single image);
• selection of a specific criterion as a focusable one (when using the method of visu-
alizing criteria as a single images);
• in the case of visualizing alternatives with as a single image (Fig. 1), when focusing
on a specific alternative, automatically change a color of remaining alternatives
based on a level of similarity with the selected one (for example, by applying a gra-
dient);
• in the case of visualization of criteria as a single image (Fig. 2) when choosing a
criterion, automatically change a color of remaining criteria based on a level of sim-
ilarity with the selected one;
• automatic reordering of alternatives depending on a level of similarity with the fo-
cusable alternative;
• automatic reordering of criteria depending on a level of similarity with the focusable
criterion;
• in case of using the stereometric visualization method of criteria's correlation, pro-
vide an ability to customize a scales of a level of similarity for automatic grouping
of criteria.
It is also possible to improve the quality of visual analytics of images obtained by the
methods discussed above by using autostereoscopic displays [11].
3 Experiment Description
Consider the operation of the described methods for the problem of choosing the opti-
mal solver OpenFoam [12, 13]. Alternatively, consider three solvers – rhoCentralFoam,
pisoCentralFoam, sonicFoam. The criteria are the results of calculating the deviation
norm L1 of the numerical solution from the analytical one for the three-dimensional
problem of inviscid flow around a cone. The computational problem was solved for
following variants of initial parameters:
1. Mach number: 3, 5, 7.
2. Cone half-opening angle (in degrees): 10, 15, 20.
Methods for Construction and Research Stereometric Visual Images of Multi-criteria … 7
3. Angle of attack (in degrees): 0, 5, 10.
In total, 24 values were calculated (for the combination of half-opening angle = 10° and
angle of attack = 10°, no calculations were performed).
To begin with, constructing a correlation surface and making criteria's permutation
so that we clearly have groups (Fig. 4).
In this figure, the following groups (as the basic was using criterion with the maxi-
mum average correlation with other criteria of the group):
Fig. 4. The surface of the criteria's correlation
1. Criteria 1-12 (basic criterion – 7th).
2. Criteria 13-18 (basic criterion – 18th).
3. Criteria 19-20 (basic criterion – 19th).
4. Criterion 21.
5. Criteria 22-24 (basic criterion – 24th).
The ranks of the new criteria are as follows: {11.9966; 5.9983; 2; 1; 2.9995}. Visualize
all three alternatives only for these remaining five criteria, sorted in descending order
of rank (Fig. 5). As you can see in this figure, the far alternative (corresponds to the
8 A. Zakharova, D. Korostelyov
sonicFoam solver) is significantly worse than the other two (all columns below), i.e. it
can be filtered.
4 Discussion of Experimental Results
The use of the developed methods for constructing and researching stereometric images
of multi-criteria alternatives made it possible to reduce count of alternatives to two, and
count of criteria to five. Thus, it was obtained the ranks of these criteria; one of the
criteria (№ 21) has a substantially different correlation to the others, which may indicate
that the corresponding computer experiment was performed incorrectly.
Fig. 5. Stereometric visualization of three alternatives after grouping criteria
Between the remaining two alternatives (Fig. 5) there is no longer such an unambiguous
predominance, but traditional decision-making methods can already be applied to the
choice of the best one. We solve this problem with four different decision-making meth-
ods - SMART, REGIME, PROMETHEE, TAXONOMY [2], and in all cases, we find
that the solver rhoCentralFoam is the preferred alternative.
5 Conclusions
The developed methods for constructing and researching stereometric images of multi-
criteria alternatives allow solving the problems of reducing the set of alternatives, re-
ducing the criteria by grouping them, as well as ranking the criteria. The use of the
Methods for Construction and Research Stereometric Visual Images of Multi-criteria … 9
developed method for visualizing alternatives as single images allows for their visual
filtering, and the use of the criteria visualization method as a single image in combina-
tion with the construction and analysis of the surface of the criteria's correlation allows
grouping and ranking them. The experiment has demonstrated the effectiveness of the
developed methods, since using visual analytics, we get a rationale for reducing the
count of alternatives and criteria.
The development of methods for visualizing alternatives and criteria, as well as
methods for their visual analytics and research, makes it possible to improve the quality
and validity of the decisions obtained when choosing preferred alternatives and reduces
the likelihood of errors when using traditional methods of decision-making by reducing
the dimension of the original problem. It is possible to increase the efficiency of the
described methods through the development and use of specialized software tools for
visual analytics, which is one of the promising areas for the development of this topic.
6 Acknowledgments
The work was supported by Russian Science Foundation grant № 18-11-00215.
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