INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, 10(4):508-519, August, 2015.

A Visualization Technique for Accessing Solution Pool in
Interactive Methods of Multiobjective Optimization

E. Filatovas, D. Podkopaev, O. Kurasova

Ernestas Filatovas*, Olga Kurasova
Vilnius University
Institute of Mathematics and Informatics,
Akademijos str. 4, LT–08663 Vilnius, Lithuania
Ernest.Filatov@gmail.com, Olga.Kurasova@mii.vu.lt
*Corresponding author: Ernest.Filatov@gmail.com

Dmitry Podkopaev
1. Systems Research Institute
Polish Academy of Sciences
Newelska str. 6, 01-447 Warsaw, Poland
2. University of Jyväskylä
Departament of Mathematical Information Technology
P.O. Box 35 (Agora), FI-40014 University of Jyväskylä, Finland
Dmitry.Podkopaev@gmail.com

Abstract: Interactive methods of multiobjective optimization repetitively derive
Pareto optimal solutions based on decision maker’s preference information and present
the obtained solutions for his/her consideration. Some interactive methods save the
obtained solutions into a solution pool and, at each iteration, allow the decision
maker considering any of solutions obtained earlier. This feature contributes to the
flexibility of exploring the Pareto optimal set and learning about the optimization
problem. However, in the case of many objective functions, the accumulation of
derived solutions makes accessing the solution pool cognitively difficult for the decision
maker. We propose to enhance interactive methods with visualization of the set
of solution outcomes using dimensionality reduction and interactive mechanisms for
exploration of the solution pool. We describe a proposed visualization technique and
demonstrate its usage with an example problem solved using the interactive method
NIMBUS.
Keywords: Multiobjective optimization, interactive methods, Pareto front visual-
ization, dimensionality reduction, multidimensional scaling.

1 Introduction

Many real-world optimization problems are multiobjective in their nature. Existence of a
solution optimizing all objectives simultaneously is unlikely in practice. Solving a multiobjective
problem is usually understood as finding a Pareto optimal solution which is the most preferred
for a decision maker (DM) [16]. Therefore most of the multiobjective optimization methods
rely on preference information obtained from the DM. We concentrate on interactive methods
consisting of iterations, where at each iteration the DM provides some preference information
and receives solution(s) derived based on this information as feedback [3].

In many interactive methods, a solution obtained at each iteration serves as the starting
point for the next iteration. The DM considers this solution and expresses preference information
aiming at its improvement. It is useful to allow the DM selecting for consideration the solution
obtained not only during the last iteration but any previous iteration. Few interactive methods
include this possibility (see e.g. [8, 22]) and to our knowledge, none of the papers describe the
selection process in detail. The issue to be addressed here is high cognitive load placed on the

Copyright © 2006-2015 by CCC Publications



A Visualization Technique for Accessing Solution Pool in
Interactive Methods of Multiobjective Optimization 509

DM when dealing with many-objective problem and a large number of accumulated solutions.
Even though the number of iterations in an interactive solution process can be moderate [7],
many interactive methods such as Tchebycheff method [33], NIMBUS [22] and others generate
several Pareto optimal solutions at each iteration, which makes the solution set grow fast.

In this paper, we address the problem of assisting the DM in selecting one among many
Pareto optimal solutions at each iteration of an interactive method in the case of a large number
of objectives. We propose to enhance interactive methods with a technique which visualizes a set
of the obtained solution outcomes and provides interactive mechanisms of its exploration. This
technique is based on multidimensional scaling method [2] which maps the solution outcomes on
a plane while trying to preserve distances between them. Despite significant lost of information,
such dimensionality reduction creates a holistic view on the outcome set helping the DM to
develop a cognitive map. This method has been successfully applied earlier for visualizing solution
outcomes in multiobjective evolutionary algorithms [10], but never in interactive methods. Its
integration into interactive methods requires some modification for handling inclusion of new
solutions into the solution pool.

This paper is organized as follows. In Section 2 we present background information and
studies underlying our research, namely basic notions of multiobjective optimization, a general
structure of interactive methods and the employed approach to dimensionality reduction. In
Section 3 we propose a visualization technique which can be integrated in many interactive
methods. In order to demonstrate this technique in action, in Section 4 we integrate it to the
interactive method NIMBUS, and use the technique for solving an example problem. Section 5
presents conclusions.

2 Background

2.1 The problem of multiobjective optimization

The multiobjective optimization problem is formulated as follows [16]:

min
x∈S

f(x), (1)

where x is a decision (variable) vector ; S ⊆ Rn is a set of feasible solutions;
f(x) = (f1(x),f2(x), . . . ,fk(x))T, fi : Rn → R, i ∈ {1,2, . . . ,k} are objective functions, k ≥ 2.
For each decision vector x ∈ S, f(x) is called its outcome.

A decision vector x∗ is called Pareto optimal solution of problem (1) (or Pareto optimum for
short), if there does not exist another vector x ∈ S such that fi(x) ≤ fi(x∗) for all i ∈{1,2, . . . ,k}
and fj(x) < fj(x∗) for at least one j. The set of Pareto optimal solutions is called Pareto optimal
set, and the set of outcomes of all Pareto optimal solutions is called Pareto front.

Solving the problem (1) is commonly understood as finding a Pareto optimum x∗ whose
outcome f(x∗) is the most preferred for the DM (see e.g. [3, 16]). Therefore, most of multi-
objective optimization methods use preference information provided by the DM. In this paper,
we concentrate on interactive methods for they are believed to be most promising methods of
multiobjective optimization because of numerous advantages [25]. An important advantage of
interactive methods is the possibility for the DM to learn about the problem during the solution
process, which makes him/her more confident in the final choice [1].

2.2 Interactive methods

In interactive methods [16,18], the DM provides preference information progressively during
the solution process and obtains Pareto optima derived based on this information as feedback.



510 E. Filatovas, D. Podkopaev, O. Kurasova

Many interactive methods have the underlying idea of exploring the Pareto optimal set by shifting
DM’s attention from one solution to another. The notion of current solution is introduced as the
Pareto optimum considered by the DM in the current iteration. The DM expresses preferences
aiming (explicitly or implicitly) at improving the current solution, i.e. finding a more preferred
Pareto optimum. A Pareto optimum derived based on these preferences can be considered as the
current solution in the next iteration. This process of producing a sequence of Pareto optima
can be viewed as moving from one Pareto optimum to another, where preference information
determines the direction of movement.

In the case of a cognitively complex model underlying a multiobjective optimization prob-
lem, the DM cannot be sure that his/her preference information always leads to a more preferred
solution. Besides that, sequential improvement of Pareto optimal solutions based on DM’s pref-
erences cannot guarantee the convergence to the most preferred solution among the whole Pareto
optimal set, by analogy with global optimization problems where a series of local improvements
of a feasible solution does not necessary lead to a global optimum. Therefore it is reasonable to
give the DM possibility of exploring different parts of the Pareto optimal set by going back to
solutions derived earlier and starting moving from them in different directions. This possibility
can be implemented by saving all the obtained Pareto optima in the solution pool and enabling
the DM in each iteration selecting any element of the pool as the current solution for the next
iteration. Thus instead of a sequence of Pareto optima one can have a tree where each branch
relates to such a move back.

A general structure of interactive methods including the possibility of moving back is de-
scribed below step-by-step (Algorithm 1).

Algorithm 1

Step 0. Initialize. Set initial variable values and derive Pareto optimal solution(s) using an
initial preference model (based on some initial DM’s preference information or a rule of
thumb).

Step 1. Update. Add the obtained solution(s) into the pool.

Step 2. Select. Ask the DM to select a solution from the pool.

Step 3. Preference expression. If the DM regards the selected solution the most preferred,
then the method stops with the selected solution as the final one. Otherwise set the selected
solution as the current solution of the next iteration and ask the DM to express preferences
aiming at its improvement.

Step 4. Solution derivation. Build the preference model based on DM’s preference informa-
tion and derive new Pareto optimal solution(s) using this model. Go to Step 1.

Some authors mention the possibility of selecting the current solution from the solution pool
in interactive methods (see e.g. [6, 8, 22, 30]). Many interactive methods can be easily enhanced
by adding such possibility and thereby be fit into the structure described above. However to our
knowledge, no papers present any details about implementing the procedure of solution selection.

As the size of the solution pool grows, it becomes cognitively more difficult for the DM to keep
and operate with it in his/her mind. Indeed, comparing and choosing among a number of objects
exceeding the mental capacities obviously leads to mistakes and inconsistencies as illustrated e.g.
in [32]. Therefore the DM can benefit from a graphical tool visualizing the solution pool and
assisting with solution selection.



A Visualization Technique for Accessing Solution Pool in
Interactive Methods of Multiobjective Optimization 511

2.3 Graphical representation of solution outcomes

It must be noted that most of the multiobjective optimization methods rely on the assumption
that the DM compares different solutions and chooses the most preferred one only based on
their outcomes. Therefore most of existing visualization techniques (see e.g. [10, 12, 13, 17])
concentrate on presenting the set of the Pareto optimal outcomes in the so-called objective space
Rk. Visualizing a set of Pareto optimal outcomes is a non-trivial task in the case k > 3. If
the solution set is relatively small, then all components of all outcomes can be presented in a
diagram such as scatter plot, bar charts, spider-web charts and others (see comprehensive surveys
in [9, 17]). For bigger solution sets, such representations enforce the DM to deal with abundance
of information. Then dimensionality reduction methods can be helpful.

As dimensionality reduction involves loss of information, the problem of extracting informa-
tion which remains can be approached in different ways depending on the aim the visualization
serves. For example, in self-organizing maps [36], artificial neural networks are trained to preserve
the topological properties of the input space. Principal component analysis (PCA) [15] tries to
preserve variances of data. Interactive decision maps [4, 12] are based on two-dimensional pro-
jections, they are also used for the visualization of Pareto outcomes.

Multidimensional scaling method (MDS) [2, 5] aims at finding points v1, v2, . . . , vm in the
low-dimensional space Rs, s < k, such that the distances between them are as close as possible to
the distances between the original points z1, z2, . . . , zm in the multidimensional space Rk. This
is achieved by minimizing the stress function

EMDS =
∑
i<j

(
d(vi, vj)−d(zi, zj)

)2
. (2)

Here d(·, ·) is the distance between two points in the corresponding space. The procedure of
finding points v1, v2, . . . , vm by minimizing (2) is called mapping. We propose to use MDS for
visualizing the solution pool in interactive methods because it detects the similarity between
Pareto optimal outcomes, and allows the DM distinguishing areas by how much they have been
explored. This helps in focusing the DM’s attention on solutions located in preferred parts of
the Pareto front.

When using MDS during an interactive solution process, a problem of mapping new solution
outcomes occurs. After adding them to the solution pool and recalculating low-dimensional
points by minimizing (2), the points corresponding to already mapped solution outcomes will
also change. In order to avoid this changing, so-called relative mapping [27] can be applied,
where only coordinates of the newly added points are calculated. The stress function of relative
mapping is following:

ER =
m+m̂∑

i,j=m+1,i<j

(
d(v̂i, v̂j)−d(ẑi, ẑj)

)2
+

m+m̂∑
i=m+1

m∑
j=1

(
d(v̂i,vj)−d(ẑi,zj)

)2
, (3)

where z1,z2, . . . ,zm are the previously mapped multidimensional points and v1, v2, . . . , vm are
their corresponding lower-dimensional points which are fixed, m̂ is the number of new points,
ẑm+1, ẑm+2, . . . , ẑm+m̂ and v̂m+1, v̂m+2, . . . , v̂m+m̂ are the new points and their corresponding
lower-dimensional points, respectively.

However, after applying the relative mapping, the value of the stress function EMDS cal-
culated for all the solution outcomes by formula (2) increases, which indicates deterioration of
mapping quality. Therefore, after visualizing a large number of the new outcomes using relative
mapping, all the mapped points must be recalculated by the MDS method again, i.e. remapping
must be performed.



512 E. Filatovas, D. Podkopaev, O. Kurasova

Our previous research was devoted to usage MDS for visualizing the Pareto front in multiob-
jective evolutionary algorithms [10]. A similar approach was developed in [34] for reviewing the
set of obtained Pareto optimal solutions when solving the problem (1). However, the possibility
of integrating MDS into interactive methods has not been studied before. In the next section, we
describe the proposed visualization technique based on MDS and its integration into interactive
methods.

3 Enhancement of interactive methods with visualization

The general structure of interactive methods described in Subsection 2.2 is modified and
presented as a scheme in Figure 1. The visualization tool is used in Step 2 for assisting the DM
in exploring the solution pool and selecting from it a Pareto optimal solution to be considered as
the most preferred solution or the current solution in the next iteration. In order to enable using
the visualization, a visual representation of the solution pool is constructed or updated in Step 1.
We propose to allow the DM choosing, whether to use the visualization technique or select the
solution in a traditional way implemented in the method. Thus, adding the visualization in the
method does not imply any additional efforts from the DM. If the DM finds the visualization
useless, he/she can ignore it which means that its integration into the method does not deteriorate
DM’s experience.

Figure 1: A general enhanced scheme of interactive methods.

The proposed visualization technique consists of mapping of solution outcomes on a plane
using the MDS method and interacting with the DM for exploring the outcome set and selecting a
solution. Our approach does not depend on the interactive method used, therefore the technique
can be implemented for a wide range of interactive methods. We propose to visualize outcomes
from the solution pool as a scatter plot of two-dimensional points mapped using the MDS method
or relative mapping, that can be implemented in visualization tool (see Figure 1). The following



A Visualization Technique for Accessing Solution Pool in
Interactive Methods of Multiobjective Optimization 513

two operations will be used for constructing and updating this visualization:

• map all solution outcomes – given outcomes z1, z2, . . . , zm of all Pareto optimal so-
lutions from the solution pool, corresponding two-dimensional points v1, v2, . . . , vm are
calculated by minimizing the MDS stress function (2) as described in Subsection 2.3. Here
m is the number of solution outcomes, zj ∈ Rk and vj ∈ R2 for all j = 1, . . . ,m.

• map new solution outcomes – given outcomes z1, z2, . . . , zm of Pareto optimal solu-
tions from the solution pool mapped earlier, their corresponding two-dimensional points
v1, v2, . . . , vm, and the outcomes of newly derived Pareto optimal solutions
ẑm+1, ẑm+2, . . . , ẑm+m̂, the two-dimensional points v̂m+1, v̂m+2, . . . , v̂m+m̂ corresponding
to the new solutions are calculated by minimizing the relative mapping stress function (3)
as described in Subsection 2.3.

In order to evaluate similarities between the obtained solution outcomes the MDS method
requires several points. Therefore the visualization starts being constructed after a predefined
number of solutions m has been collected in the solution pool. In the case of a small number
of solutions, it can be easier for the DM to select one of them by straightforward comparison
of the solution outcomes rather than studying the two-dimensional visualization. Therefore we
propose to set initial value of m not smaller than 7.

Once new Pareto optimal solutions are derived in Step 4 (Figure 1) and added to the solution
pool, the visual representation is updated in Step 1 either by mapping all solutions or only new
solutions. In the first case, the locations of the previously mapped solutions will also change
which may distort the cognitive map of the solution pool the DM has developed. On the other
hand, mapping only new solutions may affect the accuracy of correspondence between distances
in the objective space and two-dimensional space. Therefore, we propose to use a compromise
strategy where new solutions are mapped during most of iterations, but the map of all solutions
is recalculated on DM’s demand or after accumulation of a large number of new solutions. When
the number of the added solutions l reaches the predefined threshold limit mp, the map of all
Pareto optimal outcomes is recalculated by applying the MDS method again. We propose to set
the limit mp = 0.5m, where m is the number of outcomes already mapped by MDS. Hence, the
higher value m is, the less frequent remapping by MDS is required. The detailed procedure of
constructing the map of the obtained outcomes is presented as Algorithm 2.

Algorithm 2 Constructing the map of the solution outcomes
Input: The solution outcomes z1, . . . , zm, obtained by using an interactive method, where m is

the number of outcomes.
Output: The 2D map (scatter plot).
Step 1: Set the number of solution outcomes mapped by relative MDS to l = 0.
Step 2: The dimensionality of the solution outcomes z1, . . . , zm is reduced to 2 by the MDS

method, and the two-dimensional points v1, . . . , vm are displayed in a 2D scatter plot and
presented to the DM. Set the value of threshold limit mp = 0.5m.

Step 3: When m̂ new solution outcomes ẑm+1, . . . , ẑm+m̂ are obtained by the interactive
method, two-dimensional points v̂m+1, . . . , v̂m+m̂ are calculated by the relative mapping, they
are added to the 2D scatter plot and we set m = m+m̂, l = l +m̂. If the DM is satisfied with
the obtained solution, then go to Step 5.

Step 4: Check if l > mp or the DM wants to remap the solution outcomes. If Yes then go to
Step 1, else go to Step 3.

Step 5: Stop.



514 E. Filatovas, D. Podkopaev, O. Kurasova

The visualization should be accompanied with graphical tools and interactive mechanisms
based on graphical user interface, allowing the DM:

• to see additional information related to the way solution were obtained, e.g. information
about the extent at which solutions satisfy preferences based on which they were derived,
relations between solutions related to the sequence in which they were obtained;

• to perform operations on visualized outcomes, e.g. hide selected outcomes or outcomes
satisfying a given property in order to focus on the most interesting outcomes, recalculate
mapping on demand;

• to get full information about the solution corresponding to any selected outcome.

The implementation of the above enhancements of the visualization might depend on the
interactive method used as well as the graphical user interface of the method implementation.

4 Example implementation

In this section, we demonstrate the proposed visualization technique by solving an example
problem using a popular interactive multiobjective optimization method NIMBUS [16, 19]. It is
implemented in the IND-NIMBUS software platform ( [28, 29]) designed for solving industrial
problems, as well as in a web-based application WWW-NIMBUS available via the Internet [21].
The NIMBUS method has been applied to a number of engineering problems [11, 22, 24].

In the NIMBUS method, new Pareto optimal solutions are derived by solving scalarized
problems based on preference information provided by the DM in the form of a classification. In
each iteration of this method, the DM considers the current Pareto optimal solution xc ∈ S and
corresponding outcome and expresses preferences by classifying the objective functions fi(xc),
i = 1,2, . . . ,k into up to the five classes related to five types of desirable or acceptable changes
of the function values [22]: I< – to be improved (i.e. decreased) as much as possible; I≤ – to
be improved till some desirable aspiration level z̄i < fi(xc); I= – the current function values are
acceptable; I≥ – may be impaired (i.e. increased) till some upper bound ai > fi(xc); I⋄ – the
function values are allowed to change freely. This classification must be done in such a way that
at least one objective function value should be improved, and at least one is allowed to impair.
The classification is used to build scalarizing functions for deriving new Pareto optimal solutions.
In synchronous NIMBUS [22], four different scalarization functions are used in order to present to
the DM diverse solutions related to expressed preferences, thereby up to four different solutions
can be obtained according the same preference information.

4.1 Numerical example

As an example for demonstrating the proposed visualization technique in use, we chose the
five-objective river pollution problem which was implemented in IND-NIMBUS. The problem was
originally presented in [26] and modified in [20], where the fifth (nonsmooth) objective function
was included. The mathematical formulation of the problem is as follows:



A Visualization Technique for Accessing Solution Pool in
Interactive Methods of Multiobjective Optimization 515

maximize f1(x) = 4.07 + 2.27x1

maximize f2(x) = 2.60 + 0.03x1 + 0.02x2 +
0.01

1.39−x21
+

0.30

1.39−x22
maximize f3(x) = 8.21−

0.71

1.09−x21
minimize f4(x) = −0.96 +

0.96

1.09−x22
minimize f5(x) = max[|x1 −0.65|, |x2 −0.65|]
subject to 0.3 ≤ x1,x2 ≤ 1.0.

(4)

It describes a (hypothetical) pollution problem of a river, where a fishery and a city are
polluting water. The decision variables x1 and x2 represent the proportional amounts of bio-
chemical oxygen demanding material removed from water in two treatment plants located after
the fishery and after the city [23]. Here f1 and f2 describe the quality of water after the fishery
and after the city, respectively; f3 and f4 show the percent return on investment at the fishery
and the addition to the tax rate in the city, respectively; and f5 describes the functionality of
the treatment plants. Hence, the first three objective functions must be maximized, while the
fourth and fifth objective functions are to be minimized.

The DM’s strategy and solution process when solving the original four-objective river pollu-
tion problem [26] using NIMBUS method was described in [14]. Below we describe the process of
solving the problem (4) using synchronous NIMBUS implemented in IND-NIMBUS, where the
DM gets additional support from our visualization technique. When addressing multiobjective
optimization problems the DM can have different priorities across objectives in mind. Further
we assume that the DM has higher priority to improving the quality of the water after the fishery
and the city. The solution process described below is divided into several stages according to the
DM’s actions related to the usage of the visualization technique:

Stage 1. When starting solving the problem by using the IND-NIMBUS, at first a so-
called neutral compromise solution is obtained using an initial preference model and presented
to the DM. Naturally the DM is not satisfied with the obtained solution, therefore he/she tries
to improve it by making several different classifications of the objective functions. During the
solution process the DM selects the most preferred solution from the obtained four solutions as
the current solution and tries to improve it in the next iteration. This process is repetitive, and
the solution pool is constantly updated with the new derived solutions.

Stage 2. After a predefined number of the solutions is collected in the solution pool, the
visualization technique is started to be used by the DM (see Figure 2a). Here the points of the
map correspond to the obtained solution outcomes. We omit drawing axes of the two-dimensional
coordinate system due to the fact that the coordinates do not have individual meaning, as the
only important aspect of mapping is preserving distance between points. The points of the scatter
plot are coloured using RdYlGn colour scheme, where colour of each mapped point represents its
distance to the reference point that express the current DM’s preferences (red colour corresponds
to large distance, green – to small distance). The numbers near the points mean the order in
which the corresponding solutions were obtained. The presented map of obtained outcomes is
constructed by using the MDS method, when the 8th solution has been selected as the current
one. Now the DM can explore the map, see the relations between the outcomes and think
about the further directions of the solution process. It can be noticed from the map that the
10th solution is located furthest from the area of the objective space where the DM started the
solution process. As the DM wants to explore the objective space in direction further away from
first solution, he decides to select the 10th solution as the current one.



516 E. Filatovas, D. Podkopaev, O. Kurasova

(a) (b) (c)

Figure 2: Two-dimensional map of the Pareto optimal solution outcomes, constructed by the
visualization technique.

Stage 3. The DM tries to improve the 10th solution by using a different classification of
objective functions. Then the solution pool is updated by new outcomes and they are added to
the map by using the relative mapping (see Figure 2b). The points mapped using MDS have
circular shape while the new points are presented as squares. Here the graph of the solution
process is also displayed (the arrows are drawn between the points corresponding to obtained
outcomes which once served as current solutions). The location of new points confirms that
the selection of 10th solution as the current solution and provision of the preference information
allowed the DM to advance in exploring the objective space, e.g. obtain Pareto optimal solutions
which demonstrate bigger differences from earlier obtained solutions. Afterwards, the DM wants
to remap all the obtained outcomes by MDS in order to see relations between them more precisely,
and the updated map is presented (see Figure 2c). After remapping, the places of the points
on the map have been changed slightly comparing with the previous ones, which does not affect
the cognitive map developed in the DM’s mind. However, after remapping the furthest location
of 15th solution outcome from earlier solutions is more expressed. Here the arrows are drawn
between all the solutions from the solution pool in sequence order in which the solutions were
obtained (see Figure 2c). Next, the DM can select other solution from the solution pool as the
current solution for further improvement, change the solution strategy by giving higher priority
to other objectives, and search for more preferable solutions in unexplored parts of the decision
space with the support of the MDS-based visualization technique, or using only the original
IND-NIMBUS user interface. Hence, the obtained map helps the DM to keep in mind a holistic
image of the current state of the solution pool, as well as the history of the solution process.

It must be pointed out that the purpose of this example is not to describe the whole solution
process of the problem, but to demonstrate the performance and flexibility of the proposed
visualization technique. Hence, the three presented stages cover the main aspects of its usage
during the solution process.

5 Conclusions

The core idea of interactive methods is using a Pareto optimum derived during each iteration
(so-called current solution) as the starting point for the next iteration. The possibility of selecting
Pareto optima derived in any of previous iterations as the current solution contributes to the
flexibility and freedom of Pareto optimal set exploration. This aspect of implementing interactive
methods has not received any attention in the literature.



A Visualization Technique for Accessing Solution Pool in
Interactive Methods of Multiobjective Optimization 517

In this paper we have proposed to enhance interactive methods with a technique visualizing
the outcomes from the solution pool and assisting the DM in its exploration and solution selection.
The resulting visualization is presented in the form of an interactive 2D scatter plot obtained
using MDS method, which helps the DM to detect similarity between outcomes and focus on
more preferred parts of the Pareto front. This visualization facilitates the DM in developing a
cognitive map of the set of solution outcomes. We have described an general approach of creating
and integrating the visualization into interactive methods, and also illustrated it with a simple
example. However the visualization should be implemented using a graphical user interface and,
depending on the method, enhanced with graphical and interactive elements improving DM’s
experience.

Acknowledgment

Ernestas Filatovas is supported by the postdoctoral fellowship funded by European Union
Structural Funds project “Postdoctoral Fellowship Implementation in Lithuania”.

Bibliography

[1] Belton, V., Branke, J., Eskelinen, P., Greco, S., Molina, J., Ruiz, F., Słowiński, R. (2008);
Interactive multiobjective optimization from a learning perspective, in Branke, J., Deb, K.,
Miettinen, K., Słowiński, R., editors, Multiobjective Optimization (Lecture Notes in Computer
Science, 5252), Springer, ISBN 978-3-540-88907-6, 405–433.

[2] Borg, I., Groenen, P. J. (2005); Modern multidimensional scaling: Theory and applications,
Springer, ISBN 978-0-387-25150-9.

[3] Branke, J., Deb, K., Miettinen, K., Słowiński, R. (2008); Multiobjective optimization: In-
teractive and evolutionary approaches (Lecture Notes in Computer Science 5252), Springer,
ISBN 978-3-540-88908-3.

[4] Castelletti, A., Lotov, A. V., Soncini-Sessa, R. (2010); Visualization-based multi-objective
improvement of environmental decision-making using linearization of response surfaces, En-
vironmental Modelling & Software, ISSN 1364-8152, 25(12): 1552–1564.

[5] Dzemyda, G., Kurasova, O., Žilinskas, J. (2013); Multidimensional Data Visualization: Meth-
ods and Applications (Springer Optimization and Its Applications, 75), Springer, ISBN 978-
1-4419-0235-1.

[6] Filatovas, E., Kurasova, O. (2011); A decision support system for solving multiple criteria
optimization problems. Informatics in Education, ISSN 1648-5831, 10(2): 213–224.

[7] Gardiner, L., Vanderpooten, D. (1997); Interactive multiple criteria procedures: Some re-
flections. In Climaco, J., editor, Multicriteria Analysis, Springer, ISBN 978-3-642-64500-6,
290–301.

[8] Jaszkiewicz, A., Słowiński, R. (1999). The "light beam search" approach - an overview of
methodology and applications. European Journal of Operational Research, ISSN 0377-2217,
113(2): 300–314.



518 E. Filatovas, D. Podkopaev, O. Kurasova

[9] Korhonen, P., Wallenius, J. (2008); Visualization in the multiple objective decision-making
framework, in Branke, J., Deb, K., Miettinen, K., Słowiński, R., editors, Multiobjective Op-
timization (Lecture Notes in Computer Science, 5252), Springer, ISBN 978-3-540-88907-6,
195–212.

[10] Kurasova, O., Petkus, T., Filatovas, E. (2013); Visualization of pareto front points
when solving multi-objective optimization problems. Information Technology And Control,
ISSN 1392-124, 42(4): 353–361.

[11] Laukkanen, T., Tveit, T.-M., Ojalehto, V., Miettinen, K., Fogelholm, C.-J. (2010); An inter-
active multi-objective approach to heat exchanger network synthesis. Computers & Chemical
Engineering, ISSN 0098-1354, 34(6): 943–952.

[12] Lotov, A., Bushenkov, V., Kamenev, G. (2004); Interactive Decision Maps, Approximation
and Visualization of Pareto Frontier (Applied Optimization, 89), Springer, ISBN 978-1-4419-
8851-5.

[13] Lotov, A. V., Miettinen, K. (2008); Visualizing the Pareto frontier, in Branke, J., Deb, K.,
Miettinen, K., Słowiński, R., editors, Multiobjective Optimization (Lecture Notes in Computer
Science, 5252), Springer, ISBN 978-3-540-88907-6, 5252: 213–243.

[14] Luque, M., Ruiz, F., Miettinen, K. (2011); Global formulation for interactive multiobjective
optimization. OR Spectrum, ISSN 0171-6468, 33(1): 27–48.

[15] Mareschal, B., De Smet, Y. (2009); Visual PROMETHEE: Developments of the
PROMETHEE & GAIA multicriteria decision aid methods, in Proceedings of IEEE In-
ternational Conference on Industrial Engineering and Engineering Management, 2009, e-
ISBN 978-1-4244-4870-8, 1646–1649.

[16] Miettinen, K. (1999); Nonlinear multiobjective optimization (International Series in Opera-
tions Research & Management Science, 12), Springer, ISBN 978-1-4615-5563-6.

[17] Miettinen, K. (2014); Survey of methods to visualize alternatives in multiple criteria decision
making problems. OR Spectrum, ISSN 0171-6468, 36(1): 3–37.

[18] Miettinen, K., Hakanen, J., Po (2014a); Interactive nonlinear multiobjective optimization
methods. In Ehrgott, M., Figueia, J., and Greco, S., editors, Multiple Criteria Decision
Analysis: State of the Art Surveys, Springer, (to appear).

[19] Miettinen, K., Mäkelä, M. (1995); Interactive bundle-based method for nondifferentiable
multiobjective optimization: NIMBUS. Optimization, ISSN 0233-1934, 34(3): 231–246.

[20] Miettinen, K., Mäkelä, M. M. (1997); Interactive method NIMBUS for nondifferentiable
multiobjective optimization problems, in Climaco, J., editor, Multicriteria Analysis, Springer,
ISBN 978-3-642-64500-6, 310–319.

[21] Miettinen, K., Mäkelä, M. M. (2000); Interactive multiobjective optimization system
WWW-NIMBUS on the Internet. Computers & Operations Research, ISSN 0305-0548,
27(7): 709–723.

[22] Miettinen, K., Mäkelä, M. M. (2006); Synchronous approach in interactive multiobjective
optimization. European Journal of Operational Research, ISSN 0377-2217, 170(3): 909–922.



A Visualization Technique for Accessing Solution Pool in
Interactive Methods of Multiobjective Optimization 519

[23] Miettinen, K., Mäkelä, M. M., Kaario, K. (2006); Experiments with classification-based
scalarizing functions in interactive multiobjective optimization. European Journal of Opera-
tional Research, ISSN 0377-2217, 175(2): 931–947.

[24] Miettinen, K., Mustajoki, J., Stewart, T. J. (2014b); Interactive multiobjective optimiza-
tion with NIMBUS for decision making under uncertainty. OR Spectrum, ISSN 0171-6468,
36(1): 39–56.

[25] Miettinen, K., Ruiz, F., Wierzbicki, A. (2008); Introduction to multiobjective optimization:
Interactive approaches, in Branke, J., Deb, K., Miettinen, K., Słowiński, R., editors, Multiob-
jective Optimization: Interactive and Evolutionary Approaches (Lecture Notes in Computer
Science, 5252), Springer, ISBN 978-3-540-88908-3, 27–57.

[26] Narula, S. C., Weistroffer, H. (1989); A flexible method for nonlinear multicriteria decision-
making problems. IEEE Transactions on Systems, Man and Cybernetics, ISSN 1083-4419,
19(4): 883–887.

[27] Naud, A., Duch, W. (2000); Interactive data exploration using MDS mapping, in Proceedings
of the Fifth Conference: Neural Networks and Soft Computing, ISBN 83-908587-2-X, 255–260.

[28] Ojalehto, V., Miettinen, K., Laukkanen, T. (2014); Implementation aspects of interactive
multiobjective optimization for modeling environments: the case of GAMS-NIMBUS. Com-
putational Optimization and Applications, ISSN 0926-6003, 58(3): 757–779.

[29] Ojalehto, V., Miettinen, K., Mäkelä, M. (2007); Interactive software for multiobjective
optimization: IND-NIMBUS, WSEAS Transactions on Computers, ISSN 2224-2872, 6(1): 87–
94.

[30] Petkus, T., Filatovas, E., Kurasova, O. (2009); Investigation of human factors while solving
multiple criteria optimization problems in computer network. Technological and Economic
Development of Economy, ISSN 2029-4913, 15(3): 464–479.

[31] Pohlheim, H. (2006); Multidimensional scaling for evolutionary algorithms – visualization
of the path through search space and solution space using Sammon mapping. Artificial Life,
ISSN 1064-5462, 12(2): 203–209.

[32] Saaty, T., Ozdemir, M. (2003); Why the magic number seven plus or minus two, Mathemat-
ical and Computer Modelling, ISSN 0895-7177, 38(3–4): 33–244.

[33] Steuer, R. E. (1989); The Tchebycheff procedure of interactive multiple objective program-
ming, in Karpak, B., Zionts, S., editors, Multiple Criteria Decision Making and Risk Analysis
Using Microcomputers (NATO ASI Series, 56), Springer, ISBN 978-3-642-74919-3, 235–249.

[34] Valdés, J. J., Barton, A. J. (2007); Visualizing high dimensional objective spaces for multi-
objective optimization: A virtual reality approach, in Proceedings of the IEEE Congress on
Evolutionary Computation(CEC 2007), IEEE, ISBN 978-1-4244-1339-3, 4199–4206.

[35] Walker, D. J., Everson, R. M., Fieldsend, J. E. (2013); Visualizing mutually nondominating
solution sets in many-objective optimization. IEEE Transactions on Evolutionary Computa-
tion, ISSN 1089-778X, 17(2): 165–184.

[36] Yoshimi, M., Kuhara, T., Nishimoto, K., Miki, M., Hiroyasu, T. (2012). Visualization of
pareto solutions by spherical self-organizing map and it’s acceleration on a GPU. Journal of
Software Engineering & Applications, ISSN 1945-3116, 5(3): 129–137.