Towards Enhanced Decision Making: Integrating Weighted Expected

Solution Points in Multi-Criteria Analysis

Andrii Shekhovtsov

1,2 a

, Bartłomiej Kizielewicz

1,2 b

and Wojciech Sałabun

1,2 c

West Pomeranian Univ. of Technology

Zołnierska 49, 71-210 Szczecin, Poland

National Institute of Telecommunications Szachowa 1, 04-894 Warsaw, Poland

Keywords:

MCDA, Multi-Criterion Decision-Making, Fuzzy Logic, Expected Solution, COMET.

Abstract:

Multi-Criteria Decision Analysis (MCDA) addresses complex problems across various domains by consider-

ing multiple decision criteria. This interdisciplinary ﬁeld offers a systematic approach to decision-making,

accommodating contradictory criteria and non-linear factors. Reference points are crucial in MCDA, facilitat-

ing a nuanced understanding of decision interrelationships and outcomes. While classic MCDA methods rely

on static reference points, recent advances introduce manual allocation mechanisms, such as the Stable Pref-

erence Ordering Toward Ideal Solution (SPOTIS) and Characteristic Objects Method (COMET). However,

incorporating reference points alone may overlook the signiﬁcance of individual criteria, leading to the para-

dox of equal evaluations. To address this issue, an extension of the COMET method, Expected Solution Point

(ESP-COMET), introduces weighted considerations to accurately reﬂect experts’ preferences. This paper pro-

poses a methodology to integrate weights into ESP-COMET, enhancing its efﬁcacy in decision modeling. We

applied the proposed approach in the case study focused on the evaluation of hydrogen-fueled vehicles. Iden-

tifying the decision model and considering both the expected solution point and the relevance of the criteria to

it, we demonstrated the utility of weighted ESP in improving decision-making processes.

1 INTRODUCTION

Multi-Criteria Decision Analysis (MCDA) is an in-

terdisciplinary ﬁeld that analyzes decisions in the

context of multiple criteria. It is used when faced

with complex problems, considering various factors.

These criteria may frequently conﬂict with one an-

other, as they embody various priorities like efﬁ-

ciency, proﬁt, or other considerations. An essential

element of MCDA is the consideration of non-linear

criteria that may be relevant to the analysis. MCDA is

used in various areas, from energy to medicine (Saraji

et al., 2023; Kizielewicz et al., 2020), logistics to sus-

tainability issues (Moslem, 2023; Wi˛eckowski et al.,

2024).

MCDA, using a variety of mechanisms, is an ef-

fective tool to solve such problems. Approaches re-

lying on pairwise comparisons prove advantageous

when experts aim to identify the model based on their

knowledge. Alternatively, methods dependent on the

https://orcid.org/0000-0002-0834-2019

https://orcid.org/0000-0001-5736-4014

https://orcid.org/0000-0001-7076-2519

coefﬁcients of individual components facilitate adapt-

able decision making by permitting dynamic adjust-

ments to these coefﬁcient values. In addition, meth-

ods that use reference points efﬁciently represent the

decision grid by focusing solely on these reference

points.

Using reference points makes it possible to better

understand the relationships between different deci-

sion criteria and assess how changes in these criteria

affect the outcome of a decision. Reference points

form the basis of many classic MCDA methods, such

as the Technique for Order of Preference by Similar-

ity to Ideal Solution (TOPSIS) (Aldino et al., 2023)

and the Viekriterijumsko Kompromisno Rangiranje

(VIKOR) (Nath et al., 2023). In addition, the logic of

reference points is also reﬂected in newer approaches

such as Compromise Ranking of Alternatives from

Distance to Ideal Solution (CRADIS) (Chakraborty

et al., 2024).

However, the methods above operate on statically

deﬁned reference points based on a decision matrix.

With reference points, decision experts can better

reﬂect their preferences, which has resulted in the de-

velopment of methods that allow experts to allocate

272

Shekhovtsov, A., Kizielewicz, B. and Sałabun, W.

Towards Enhanced Decision Making: Integrating Weighted Expected Solution Points in Multi-Criteria Analysis.

DOI: 10.5220/0013120100003890

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 17th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2025) - Volume 3, pages 272-279

ISBN: 978-989-758-737-5; ISSN: 2184-433X

these points manually. The logic behind such man-

ual mechanisms assumes that the decision expert for

a given problem assigns his ideal values to speciﬁc

criteria. Thus, to normalize the decision space, these

very objects are used. In addition, using ﬁxed val-

ues for these objects with ﬁxed boundaries of the de-

cision problem allows the method to be immune to

the rank-reversal paradox. Examples of these meth-

ods are methods that will allow reference points to

be used in this way include Stable Preference Order-

ing Toward Ideal Solution (SPOTIS) (Dezert et al.,

2020) and Characteristic Objects Method (COMET)

(Shekhovtsov et al., 2023).

In the SPOTIS approach methods, an expert de-

termines a single point that forms the basis for a de-

cision grid. The COMET approach, on the other

hand, is based on fuzzy logic, speciﬁcally the Mam-

dani model, where characteristic objects are created

based on speciﬁc characteristic values. This raises a

dimensionality problem, where the expert is forced

to compare characteristic objects among themselves

to get the right preference. To solve this prob-

lem, (Shekhovtsov et al., 2023) proposed an exten-

sion of the COMET method with the Expected So-

lution Point (ESP-COMET). With this approach, an

expected point is assigned based on which the pref-

erence values of the characteristic objects are deter-

mined. This eliminates the difﬁculty of an expert

evaluating a high-dimensional problem and facilitates

comparing objects.

However, with this solution, there is another prob-

lem related to the relevance of the criteria. Using ESP

alone allows for grid modeling to evaluate alternatives

according to a declared reference point. However, it

does not consider the relevance of individual criteria

to that point. Therefore, this paper will focus on a pro-

posal to add weight to ESP-COMET to better reﬂect

the preferences of experts. In the remainder of the pa-

per, we will present the process of identifying such

a model and its practical implementation, using the

example of evaluating hydrogen-powered vehicles.

The remainder of the paper is organized as fol-

lows. Section 2 presents related work in relation to

the topic of decision making. Section 3 presents the

methodology, with the initial assumptions related to

the COMET method, the expanded COMET method

with ESP and correlation coefﬁcients. Section 3.2 dis-

cusses the proposed approach. Section 4 presents a

simple example related to weighting the expected so-

lution point. Section 5 presents research on a practical

problem related to hydrogen-powered cars. Section 6

presents conclusions and future research propositions.

2 RELATED WORKS

Methods based on reference objects have gained sig-

niﬁcant attention and are widely applied across vari-

ous ﬁelds due to their robust ability to handle complex

multi-criteria decision-making processes. For exam-

ple, (Awodi et al., 2023) developed a fuzzy TOPSIS-

based risk assessment model to effectively man-

age the risks associated with nuclear decommission-

ing. This approach allowed for a more nuanced as-

sessment by incorporating uncertainty through fuzzy

logic. Similarly, (Aldino et al., 2023) used the TOP-

SIS method with an alternative weighting procedure

to identify the highest performing graduates, demon-

strating the versatility of the method in educational

evaluation.

In the context of sustainable technologies,

(Wi˛eckowski et al., 2024) applied the RANking

COMparison (RANCOM) method in combination

with ESP-SPOTIS to optimize decision making for

the selection of electric vehicles, highlighting the ﬂex-

ibility of SPOTIS in handling adaptive systems.

Moreover, (Nath et al., 2023) proposed a VIKOR

framework for biodiesel production using heteroge-

neous agricultural waste-based catalysts. Their work

underscores the adaptability of VIKOR in sustainable

energy research. (Saraji et al., 2023) utilized the Fer-

matean CRITIC-VIKOR approach to assess the chal-

lenges in implementing renewable energy technolo-

gies in rural areas, demonstrating the applicability of

the method to assess complex technological adoption

scenarios.

On the other hand, pairwise comparison methods,

such as the Analytic Hierarchy Process (AHP) and

ELimination Et Choix Traduisant la REalité (ELEC-

TRE), compare alternatives in pairs, judging which

of the two performs better relative to speciﬁc crite-

ria. These techniques are instrumental when the di-

rect comparison between multiple criteria is complex,

allowing for a more gradual and structured evaluation

process.

In his work, (Romero-Ramos et al., 2023) inte-

grated a GIS-AHP approach to assess the potential of

solar energy to meet the demand for heat in indus-

trial areas in the south-eastern part of Spain. This

study demonstrates the effectiveness of AHP in spa-

tial decision-making, combining geographical data

with multi-criteria evaluation. Similarly, (Ahadi et al.,

2023) used the AHP method to determine the opti-

mal site for a solar power plant in Iran, highlighting

the utility of AHP in planning energy infrastructure,

mainly when multiple conﬂicting criteria such as land

use, environmental impact and cost are involved.

Towards Enhanced Decision Making: Integrating Weighted Expected Solution Points in Multi-Criteria Analysis

273

While the discussed methods provide valuable

frameworks for multi-criteria evaluations, they often

lack a comprehensive approach that enables decision-

makers or experts to express their preferences and in-

sights in a clear and transparent manner. Address-

ing this limitation we propose a novel weighted ESP-

COMET approach. This method aims to offer a more

holistic solution, enhancing the ability of decision-

makers to provide their judgments while maintaining

the robustness of the decision-making process.

3 METHODOLOGY

In this paper, we focus on our proposed weighting

method within the ESP-COMET approach introduced

by (Shekhovtsov et al., 2023). To this end, we devel-

oped a framework consisting of ﬁve stages. The ﬁrst

stage involved creating research data to evaluate the

new approach. For this purpose, we utilized an en-

vironment created in Python, and the data was gener-

ated from a uniform distribution. In the next stage, we

implement the proposed weighted ESP-COMET ap-

proach. After its implementation, we moved on to the

third stage, where we compared the unweighted ESP-

COMET approach with our proposal. Once the dif-

ferences between these approaches were determined,

a simulation was created in which ESP-based ap-

proaches with weighting options, such as SPOTIS and

our proposed ESP-COMET, were applied. In the ﬁ-

nal ﬁfth stage, we analyze the approaches using the

weighted Spearman correlation coefﬁcient (r

). The

entire framework of the work is presented in Figure 1,

while the techniques used and the details of our pro-

posed approach are discussed in subsequent sections.

Data generation

Proposal for an ESP weighting

approach in COMET

Comparative research

Simulation studies

Analysis of results

Figure 1: The framework of the proposed approach.

3.1 Preliminaries

3.1.1 Characteristic Objects Method

The Characteristic Objects Method (COMET) is

a distinctive Multiple Criteria Decision Analysis

(MCDA) approach that is completely resistant to

the ranking reversal phenomenon. This method en-

sures consistent and unequivocal results regardless of

changes in the set of alternatives (Kizielewicz et al.,

2021). The main steps of the COMET algorithm are

as follows:

Step 1. Identify the criteria of the decision problem

and represent each criterion using fuzzy numbers.

Step 2. Create a set of Characteristic Objects by ap-

plying the Cartesian product of fuzzy numbers, repre-

senting all potential combinations.

Step 3. Conduct pairwise comparisons of the Charac-

teristic Objects based on expert judgment, summarize

these judgments, and compute preference values.

Step 4. Transform each characteristic object and its

associated preference value into a fuzzy rule.

Step 5. Utilize the fuzzy rule base along with Mam-

dani’s inference method to assess and rank alterna-

tives, where higher preference values indicate better

alternatives.

The complete algorithm of the COMET method

can be found in (Kizielewicz et al., 2021), while

the implementation of this method is presented

(Kizielewicz et al., 2023).

3.1.2 Expected Solution Point COMET

Expected Solution Point COMET was created as an

answer to the dimensionality problem in the standard

COMET procedure (Shekhovtsov et al., 2023). This

problem appears when the Matrix of Expert Judge-

ments (MEJ) should be identiﬁed by an expert. Sup-

pose that we have t characteristic objects, therefore,

the identiﬁcation of the MEJ matrix of size t ×t will

require

t(t−1)

pairwise comparisons. This makes it

difﬁcult to identify the decision model due to the large

number of pairwise comparisons required.

To answer that, ESP-COMET was proposed, cre-

ating the alternative way to identify the pairwise com-

parison matrix. To use it, ﬁrst, the expert or decision

maker should deﬁne the n Expected Solution Points

based on their preferences and domain knowledge.

Each of the ESP vectors consists of r values, where

r is a number of criteria in the decision problem (1).

ESP =



esp

i j



n×r

(1)

ESP-COMET uses Equation (2), which incorpo-

rates the function f

ESP

. This function computes an

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

274

aggregated normalized distance from predeﬁned Ex-

pected Solution Points concerning a characteristic ob-

ject. A smaller distance indicates a preferable charac-

teristic object.

i j







1.0, f

ESP

(CO

) < f

ESP

(CO

)

0.5, f

ESP

(CO

) = f

ESP

(CO

)

0.0, f

ESP

(CO

) > f

ESP

(CO

)

, (2)

The function f

ESP

(CO

) is deﬁned as (3).

ESP

(X) = min

∑

j=1

′

− esp

′

i j

)

(3)

In Equation (3), X represents the abstract Char-

acteristic Object, comprising the values x

, where

j ∈ 1, 2, . . . , r, and esp

i j

denotes the expected value

i for the criterion j. The values x

′

and esp

′

i j

represent

the normalized counterparts of x

and esp

i j

, respec-

tively, computed using Equation (4). This normal-

ization process relies on the values c

(min)

and c

(max)

which denote the smallest and largest characteristic

values for criterion j. The same normalization proce-

dure outlined in Equation (4) is applied to each ESP.

′

− c

(min)

(max)

− c

(min)

, (4)

where c

(min)

and c

(max)

denote the minimum and

maximum characteristic values within criterion C

3.1.3 Similarity Coefﬁcients

In this work, we use two similarity coefﬁcients to

measure the differences between the rankings and

weights.

The Weighted Spearman’s correlation coefﬁcient

(Dancelli et al., 2013) extends the traditional Spear-

man coefﬁcient by incorporating weights, emphasiz-

ing changes at the top of the rankings. It evaluates the

correlation between two rankings, with values rang-

ing from -1 (reversed rankings) to 1 (identical rank-

ings), and 0 indicating no correlation.

The Weights Similarity Coefﬁcient (Shekhovtsov,

2023) provides a robust measure for comparing differ-

ences in sets of criteria weights. It uses the Manhat-

tan distance, normalized for cross-set comparability,

to quantify dissimilarities, offering a similarity value

scaled between 0 and 1.

3.2 Proposed Approach

The paper which introduced the ESP-COMET ap-

proach focuses mainly on the idea of the approach

and the inﬂuence of the selection of characteristic ob-

jects on the identiﬁcation of the preference function

and therefore the accuracy of the model (Shekhovtsov

et al., 2023). However, this paper missed the impor-

tant point of weighting the distances to ESP during

the aggregation process. This paper addresses this is-

sue by introducing the weighting mechanism to the

ESP-COMET method and investigates it’s properties.

To introduce the weights into the ESP-COMET

method, we modify Equation (3), adding the weight

vector to it (5):

ESP

(X) = min

∑

j=1

· (x

′

− esp

′

i j

)

, (5)

Where w

is the importance weight for the jth cri-

terion. Such modiﬁcation of the algorithm allows us

to simply manipulate the importance of the weights

by the expert if needed. This paper focuses on this

version. However, it is possible to extend the ap-

proach further, introducing weight sets for different

ESP, and also providing ability to weight ESP points.

4 SIMPLE EXAMPLE

To demonstrate our proposed approach, we ﬁrst want

to apply it to a simple two-criterion problem. This

simple example contains two criteria, each in the

range [0, 10], with chosen ESP {3, 3} as shown on the

leftmost part of Figure 3. All of those values were ar-

bitrary selected, as they will best illustrate the changes

of the preference function.

In this example we compare unweighted ver-

sion of the ESP-COMET (which can be considered

weighted with equal weights) with weighted ESP with

the weights vector w

ESP

= {0.01, 0.99}.

Such extreme change in the weights will in our

opinion show how big can be change in the preference

function when weights are introduced.

The comparison results are presented in Table 1,

which contains the indices of the characteristic ob-

jects (COs) evaluatedCO

as well as the criteria values

for these characteristic objects (C

and C

). Next, we

include preferences for these COs for normal P

and

weighted P

algorithms, as well as respective rank-

ings of characteristic objects R

and R

Analyzing Table 1 one can notice that in the case

of the unweighted method there are several links, such

as the link between CO

and CO

, CO

and CO

and

others. This can be easily explained by the fact that

those characteristic objects have the same distance to

the ESP and are therefore treated equally. In case of

weighted ESP however, we can see that weights work

Towards Enhanced Decision Making: Integrating Weighted Expected Solution Points in Multi-Criteria Analysis

275

correctly, introducing the change in the ranking. CO

which has the same criteria values as ESP is placed

ﬁrst in both rankings, and COs such CO

and CO

that are most distant from ESP are ranked worst.

Table 1: Preferences and rankings of unweighted (P

, R

)

and weighted (P

, R

) ESP models.

0 0 0.6000 0.5000 4 5

0 3 0.8000 0.8750 2.5 2

0 10 0.2000 0.1250 7.5 8

3 0 0.8000 0.6250 2.5 4

3 3 1.0000 1.0000 1 1

3 10 0.4000 0.2500 5.5 7

10 0 0.2000 0.3750 7.5 6

10 3 0.4000 0.7500 5.5 3

10 10 0.0000 0.0000 9 9

In addition, we calculate several numerical co-

efﬁcients to show how different those rankings and

preferences are. First, the value of the coefﬁcient

W SC

between equal weights and the vector w =

{0.01, 0.99} used in this example is 0.5100, which is

the maximum value of W SC

which can be obtained

by measuring the difference with equal weights. Next,

we evaluate preferences and CO ranking, obtain-

ing Weighted Spearman’s correlation r

0.8752 and

Mean Absolute Error (MAE) 0.1222. Both these val-

ues point that these rankings and preferences are sim-

ilar, and of order of the characteristic objects in rank-

ing are generally preserved, however, weighted ver-

sion has no ties.

In Figure 2 we present two identiﬁed MEJ ma-

trices, respectively, for unweighted, weighted ESP-

COMET models and the absolute difference between

those MEJ matrices. As can be seen, the introduction

of weights mostly inﬂuences ties, but there are also

changes (i.e., in the 8 th row, for CO

), which explain

the change in the preference and ranking position for

Next, in Figure 3, the preference functions are

presented for both the unweighted and the weighted

ESP-COMET models. We also present the differ-

ence between them, calculated as values of the pref-

erence function for the unweighted model reduced by

the corresponding values of the weighted model. In

case of chosen weights, it can be seen that a darker re-

gion with higher preference values is stretched among

values, instead of concentrating around the ESP as

was in case of unweighted ESP. From the difference,

it can be seen that the preferences for CO

= {10, 3}

and for the region adjusted to it change the most.

The preference value of CO

changed from 0.400 to

0.750 after weighting was introduced, implying such

a large change in the preference function. The Char-

acteristic Objects for CO

= {3, 0} and CO

= {3, 10}

are removed after the weights are introduced, which

can also be seen in the visualization of the differ-

ences. Only the most distant characteristic object

= {10, 10} does not change its preference value,

which remains 0 for both weighted and unweighted

models.

5 CASE STUDY

However, since a simple example does not present

real data, we decided to present our approach to the

real-world data collected for the purpose of this re-

search. With the increasing popularity of alternative

fuel for cars, we present a case study of choosing the

most suitable hydrogen car, based on the data col-

lected from the manufacturers’ web pages.

The cars are evaluated using the criteria presented

in Table 2. The minimum and maximum values are ar-

bitrarily chosen based on the collected data to build a

model that ﬁts all the alternatives chosen. The weights

of the criteria were calculated using the RANCOM

method (Wi˛eckowski et al., 2023) and the ESP was

arbitrary chosen by the decision maker, based on their

needs and expectations for the car. According to the

input of the decision maker, the most important cri-

terion for them is price, followed by range and tank

capacity. The least important criterion is year, which

is expected, because the range for years is quite small.

Table 2: Criteria description, as well as the weights deter-

mined using the RANCOM method.

Crit. names Units w

ESP Min Max

Year – 0.012 2023 2015 2024

Est. Price $k 0.210 30 30 100

Range km 0.173 800 400 800

Power output KW 0.111 200 100 400

Tank cap. kg 0.173 6 4.4 6.5

Horse power – 0.136 130 130 550

Max torque Nm 0.062 300 260 410

0 to 100 time s 0.050 8 4.5 12.5

Max speed km/h 0.074 150 130 250

Data for the hydrogen cars chosen for the evalua-

tion are presented in Table 3. It can be seen that most

of the cars are actually new, with only A

being an

outlier in the year criterion. The price and other cri-

terion values are different, and there is no simple way

to decide which one is will ﬁt the decision-maker’s

expectations best. The data of the alternatives was

collected from the manufacturers’ web pages.

We evaluated the data presented in Table 3 with

both ESP-COMET and weighted ESP-COMET using

the weights shown in Table 2. The results of the com-

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

276

Figure 2: Identiﬁed MEJ for unweighted (left) and weighted (center) ESP-COMET and difference between these two functions

(right), characteristic objects are represented by the black dots.

Figure 3: Contour representation of the preference functions for unweighted (left), weighted (center) ESP-COMET and

difference between these two functions (right), characteristic objects are represented by the black dots.

Table 3: Data of the alternatives.

2023 50.0 646 128 5.6 182 406 9.2 161

2023 60.0 612 135 6.3 161 394 9.2 179

2021 59.0 579 103 5.5 174 299 9.0 165

2019 45.0 437 155 4.4 211 364 5.8 160

2022 30.0 400 150 4.4 134 260 7.8 130

2015 66.5 594 100 5.6 136 406 12.5 160

2024 90.0 504 295 6.0 401 347 6.0 180

2023 45.0 590 134 5.6 182 300 7.8 170

2022 68.0 800 220 5.4 300 347 6.5 200

2022 100.0 800 400 5.4 550 347 4.5 250

putations are presented in Table 4. We ﬁrst present

preferences for unweighted (P

) and weighted (P

)

models, as well as respective rankings R

and R

Change in weights introduces several changes in the

ranking, however, alternative A

, which has the best

preference, and alternatives A

and A

which per-

form the worst preserve their positions in the rank-

ings. Three alternatives worsen their position in the

ranking, and four alternatives improved. For exam-

ple, alternatives A

and A

moved from third and ﬁfth

positions to second and third, because they have a bet-

ter price to other criteria ratio.

The differences in both rankings are clearly visible

in the visualization in Figure 5. In this visualization,

every dot represents an alternative, and the x coordi-

nate of the dot represents its position in the R

rank-

Table 4: Case study result.

0.8367 0.8275 3 2

0.8111 0.7974 5 3

0.8193 0.7767 4 6

0.5798 0.6218 7 8

0.8493 0.7942 2 4

0.5404 0.7182 8 7

0.5271 0.4408 9 9

0.9054 0.8556 1 1

0.7803 0.7855 6 5

0.2907 0.3294 10 10

ing and the y coordinate represents the position in the

ranking. It can be seen that deviations from the di-

agonal are not big, implying that rankings are rather

similar. This is also conﬁrmed by the Weighted Spear-

man correlation coefﬁcient, which is equal to 0.9012

for these two rankings. Similarly, W SC

value be-

tween equal weights and weights computed using the

RANCOM methods is 0.7531, which show that these

weight vectors are similar. For characteristic objects,

we provide only the value r

between their classiﬁca-

tions, which is 0.8843. The number of characteristic

objects for this problem is 729, and, therefore, it can

be represented or visualized in readable form.

Results obtained in this case study shows, that in

this particular problem weights does not strongly in-

Towards Enhanced Decision Making: Integrating Weighted Expected Solution Points in Multi-Criteria Analysis

277

Figure 4: Visual difference of the rankings for the weighted

and unweighted ESP-COMET models.

ﬂuence the ﬁnal ranking, however this needs to be in-

vestigated further.

5.1 Simulation Experiment

In order to investigate how strong the inﬂuence of

the weights is in the ESP-COMET approach, we de-

sign a simple simulation experiment. This experi-

ment is based on the data derived from the case study

of hydrogen cars previously presented. We use the

same criteria and ESP as provided in Table 2, and

then change the weights to check how much different

results we can obtain (compared to the unweighted

ESP-COMET).

We also include the Stable Preference Ordering

Towards Ideal Solution (SPOTIS) method in the sim-

ulation, as this method is ESP-capable and therefore

can be compared with the COMET method. The sim-

ulation experiment is described with Algorithm 1. For

this experiment, we ﬁrst need to deﬁne the decision

problem consists of characteristic values cv and the

decision matrix X, as well as ESP and the number of

iterations to make in the simulation. Next, we calcu-

late the rankings R

for alternatives from X and the

rankings of the characteristic objects using the stan-

dard ESP-COMET procedure (line 1). Additionally,

in line 2 we calculate the ranking of the alternatives

using the SPOTIS method and deﬁned ESP. Next,

in lines 3-10 we repeat n = 10000 calculations of

these rankings, but with randomly generated weights

. The sum of random weights w

is ensured to be

equal to 1. Next, with this weight, we calculate the

ranking of alternatives R

and the ranking of charac-

teristic objects R

using the weighted ESP-COMET

algorithm. Next, we also calculate the ranking R

using the SPOTIS approach and the same random

weight vector. Finally, we memorize all the results

for further analysis.

Algorithm 1: Algorithm of the simulation research.

Input: Number of iterations n ← 10000

Input: Decision matrix X

Input: Expected Solution Point ESP

Input: Characteristic values cv

1: R

, R

← ESP_COMET (X, ESP)

2: R

← SPOT IS(X , ESP, w

)

3: for i ← 1, 2, . . . , n do

4: w

← random_weights()

5: R

, R

← wESP_COMET (X, ESP, w

)

6: R

← SPOT IS(X , ESP, w

)

7: Write r

between R

and R

8: Write r

between R

and R

9: Write r

between R

and R

10: end for

Output: Collected r

values for different rankings.

The results of the simulation are presented in Fig-

ure 5 in the form of box plots. This visualization

presents the distributions of the r

values between

the classiﬁcation calculated with equal weights and

the weights generated randomly. It can be seen that

the values of both methods r

(named COMET and

SPOTIS in the visualization) have a similar range

[0.5, 1.0], however, the quantiles and the average val-

ues are different. When weighted ESP-COMET is

used, the average r

values between the classiﬁca-

tion build with equal weights and random weights is

≈ 0.92, however, in the case of the SPOTIS method

the average r

is ≈ 0.85, which is lower. The aver-

age value of r

is also higher in case of ranking of

the Characteristic objects (”COs” label). Such good

correlations can be explained with the fact that alter-

natives in the COMET methods are evaluated based

on COs and not based on the weights, and because

the distances between characteristic objects are usu-

ally larger than between alternatives, it is less pos-

sible to have changes in order of alternatives, when

weights and order of characteristic objects have some

small changes. This implies that ESP-COMET is a

more robust and stable approach.

6 CONCLUSIONS

This paper presents an essential aspect of the ESP-

COMET algorithm, introducing the weighting mech-

anism and examining changes in the preference func-

tion and the method’s stability. We present the ap-

plication of the method to a simple example, con-

sisting of two criteria, where we observe changes in

the adaptation of the decision map and the effect of

ESP weights on its formation. In addition, the paper

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

278

Figure 5: Results of the comparison of 10000 models with

random weights and unweighted ESP-COMET on alterna-

tives and characteristic objects rankings.

presents a practical example related to the evaluation

of hydrogen cars. Based on the study, the proposed

approach shows high application potential. Moreover,

comparing it with the SPOTIS method, it turns out

that the inﬂuence of the weights on the ﬁnal ranking

is more limited, which translates into obtaining stable

rankings, resistant to slight deviations in the weights.

However, this method needs future investigations.

One of the feature of this methods is a possibility for

providing several ESP points, which can be used to

introduce more complex weighting algorithms. Such

algorithms are useful in group decision making and

other complex decision scenarios. In addition, con-

sideration should be given to integrating this tool for

possible re-identiﬁcation of MCDA models, where

the research direction may be a Stochastic Identi-

ﬁcation of Weights (SITW) - ESP-COMET hybrid

(Kizielewicz et al., 2024).

ACKNOWLEDGMENTS

The work was supported by the National Science Cen-

tre 2021/41/B/HS4/01296.

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