EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6049
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Petroleum Well Site Selection Using MCRAT
Integrated with Rough Set Theory

Shaaban M. Shaaban1, Anas M. El-Sherif2, Ahmed A. Hammam3,
Husham M. Attaalfadeel4,∗, Yehya I. Mesalam5,6

1 Center for Scientific Research and Entrepreneurship, Northern Border University,
Arar 73213, Saudi Arabia
2 Engineering College, Northern Border University, Arar 1321, Saudi Arabia
3 General Courses Department, Northern Border University, Arar 1321, Saudi Arabia
4 Department of Mathematics, College of Science, Northern Border University, Arar 91431,
Saudi Arabia
5 Industrial Engineering Department, Faculty of Engineering, Zagazig University, Zagazig
44519, Egypt
6 Department of Industrial Engineering, College of Engineering, Northern Border
University, Arar 1321, Saudi Arabia

Abstract. Shale Oil (SO) has emerged as an attractive additional supply of conventional crude oil
throughout the world in recent years. Shale oil quality evaluation includes a variety of geochemical
parameters. In this research, we propose a novel Integrated shale oil evaluation approach. This
method firstly determines the relative weights of parameters using rough set theory. Finally,
Multiple Criteria Ranking by Alternative Trace (MCRAT) technique is utilized to calculate the
rank of Shale oil wells. Twenty-seven samples of shale oil were collected from various distinct well
sites. Twelve different geochemical parameters were examined in the gathered samples in order
to determine the grade of the shale oil. The results reveal that shale oil sample 17 geochemical
parameters, with shale oil grade I is the best collection parameters concentrations. The suggested
approach is compared to five Multi-Criteria Decision Making (MCDM) approaches to demonstrate
its effectiveness. The obtained results gained by the MCRAT Integrated with Rough Set Theory
(RST) is clear in ideas and easy in computation, which may be effectively applied to address a
variety of problems, both similar and different.

2020 Mathematics Subject Classifications: 90B50, 91B06, 62C99, 03E72

Key Words and Phrases: Multi criteria decision making (MCDM), shale oil, multiple criteria
ranking by alternative trace (MCRAT), rough set theory (RST)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6049

Email addresses: hushamalhassan@nbu.edu.sa (H. M. Attaalfadeel),
ymesalam@yahoo.com (Y. I. Mesalam)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



S. M. Shaaban et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6049 2 of 21

1. Introduction

Exploration in the petroleum sector is primarily concerned with locating commercially
recoverable reserves. Exploring for petroleum is a complicated and uncertain process that
is expensive and complex. An important step in petroleum exploration is the modelling
of the petroleum location. The potential modelling is important step to decide where the
well will be drilled to avoid the high cost needed in petroleum location because of uncer-
tainty and complexity process. The evaluation of the petroleum location for undiscovered
petroleum accumulations involves integrating surface and subsurface datasets, including
information on geology, geochemistry, geophysics, petrophysics, and geography.

Numerous data-based approaches have been created and effectively used in mineral
potential mapping (MPM) during the last few decades as Decision-tree analysis (DTA)
using a geographic information system (GIS) [1], certainty factor C-F model [2], Wild-
cat modelling [3], Preference Ranking Organization Method for Enrichment Evaluation
(PROMETHEE II) [4], Elimination and Choice Translation Reality III (ELECTRE III)
technique [5], structural surface-restoration and logistic regression (LR) analysis [6], Boost-
WofE is new algorithm which using a weighted training sample to implement weights [7],
Bayesian Network classifier using Naive Bayes (NB) [8], Random Forests (RF) algorithm
[9], index overlay integration method [10], Data Envelopment Analysis (DEA) technique
[11], Fuzzy Logic Analysis [12], Analytic Hierarchy Process - Technique for Order Prefer-
ence by Similarity to Ideal Solution (AHP–TOPSIS) algorithm [13], TOPSIS method [14–
16], fuzzy comprehensive assessment model with entropy weights (FCAEW) method [17],
extreme learning machine (ELM) [18], maximum entropy (MaxEnt) model [19], weights-
of-evidence (WofE) method [20], artificial neural networks (ANN) and random forest(RF)
[21], isolation forest model based on data-mining algorithm [22], Fuzzy logic, an outrank-
ing technique, and the Artificial Bee Colony (ABC) optimization algorithm are combined
to create the optimized fuzzy ELECTRE (OFE) methodology [23], Evaluation based on
Distance from Average Solution (EDAS) method [24, 25], Combinative Distance based
Assessment (CODAS) method [26, 27], Weighted Aggregated Sum Product Assessment
(WASPAS) method [28].

However, a geographical analysis of identified petroleum reservoirs within an area of
interest is not required by knowledge-based approaches. Some examples of these methods
include Dempster-Shafer belief theory, Boolean logic, outranking techniques, fuzzy logic,
and index overlay. Expert knowledge is applied to select the parameters of models [29, 30].
The research [31] provides a knowledge-driven method based on digital maps (GIS) of ge-
ology, heat flow, young faults, and young volcanism, to evaluate geothermal systems. Also,
[32] provides data-driven methods in order to model petroleum potential in a geograph-
ical context [33] provides a hybrid fuzzy weights-of-evidence (WofE) approach that uses
information-based fuzzy membership values and data-based conditional probabilities to
produce fuzzy predictor patterns. In a geographic information system (GIS) environment
[34] integrated primary geological control factors on petroleum occurrence to determine
which regions have the most petroleum potential for further investigation using weights of
evidence.



S. M. Shaaban et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6049 3 of 21

Numerous scholars have used a variety of strategies in an effort to evaluate petroleum
potential. over the past few years. In [35] Tounsi created an expert system called approxi-
mation fuzzy assessment (AFA) based on the strength of fuzzy set and possibility theories
for evaluating oil potential in Algeria’s Hassi Messaoud areas. The petroleum potential
was evaluated in a non-spatial framework, despite the system’s ability to manage data with
ambiguity. Chen and Osadetz [36] suggested a model-based simulation strategy employ-
ing the Fourier transform algorithm to construct a petroleum accumulation map. In [37]
the author developed a GIS-based fuzzy multi criteria evaluation model for determining
petroleum potential. A fractal model was used by Chen and Osadetz in [38] to replicate
how unknown oil and gas deposits would be distributed geographically.

Multi-criteria decision-making (MCDM) techniques have been applied more often in
recent years to address a wide range of issues in several scientific domains [39, 40]. One of
the scientific domains where the use of MCDM [41, 42] techniques is essential for developing
a trustworthy model in the decision-making process is engineering, medical, and business
[43, 44]. Accordingly, a vast array of MCDM techniques have been created to address these
kinds of challenging issues. As new MCDM techniques advanced quickly, techniques for
determining the weights of criteria were also created. This indicates that a considerable
portion of the best answer is determined by the weights of the criterion. Criteria weight
evaluation is a major issue in MCDM approaches, as it significantly impacts decision-
making outcomes. In this paper rough set theory (RST) is used for evaluating the weights
of attributes. Professor Pawlak put up a theory of rough sets in 1982 [45], which offers a
structured method for handling vague or insufficient data.

Even though previous researchers have conducted a significant amount of research
on petroleum well site selection using various MCDM techniques, a straightforward and
methodical mathematical approach is still required to help the decision maker choose the
right site for a given engineering context.

this study developed an integrated MCRAT Rough Set Theory approach for shale oil
quality classification. The integrated model overcomes the limitations of single algorithms
and maintains good predictive performance, providing a reliable and practical solution for
shale oil classification in engineering context.

The rest of the article is structured as follows: Section 2 explains the process method-
ology of the proposed approach, while results, validations and discussions are presented
in Section 3. Lastly, the final section gives the research results, and the paper concludes
with a list of references.

2. Methods

2.1. Preliminaries of rough Set Theory

RST is a mathematical data processing technology presented by Polish academic
Pawlak. Incomplete, imprecise, and erroneous data may be processed and mined efficiently
using this strategy [45]. RST aids in eliminating redundant features from high-dimensional
datasets. RST provides an objective description of uncertain problems without requiring



S. M. Shaaban et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6049 4 of 21

additional information beyond the present dataset. Expert subjectivity and empirical
knowledge may be avoided by using a weighting approach based on the knowledge granu-
larity and attribute significance of rough sets. It calculates the weight after analyzing the
association between data and indicators objectively. We’ll go over the RST related ideas
in this part.

2.2. Decision table

In RST, a data set is represented as a table, with each row representing a process, an
instance, or a simple object. The observable properties of an item are represented in each
of the table’s columns [45]. This table is referred to as a decision table (DT). Any 4-tuple
DT = (U,A, Va, Fa) represents a decision table in more formal terms. Where U represents
a nonempty finite set of objects, process or cases (data from experiments) called universe,
A represents a set of primitive features for objects; A = C ∪ D, where C represents a
condition attribute set (input) and D represents a decision attribute set (output), for each
a ∈ A, the set Va contains all possible values of attribute a, Fa : U → Va is called the
decision function.

2.3. Indiscernibility relation

Let DT = (U,A, Va, Fa) be a decision table, (B ⊆ A). The binary relation IND(B)
called indiscernibility relation, It gathers indistinguishable objects, defined here as those
having the same a (characteristic values) with regard to R. IND(B) is defined by

IND(B) =
{
(x, y) ∈ U2 : ∀a ∈ R, a(x) = a(y)

}
(1)

So, IND(B) is an equivalent relation and IND(B) = ∩a∈BIND(a).

2.4. Criteria importance degree

This section introduces the basic definition of criteria importance [46, 47]. To determine
the importance of each criterion, we use the attribute reduction method of RST. By
minimizing attributes, we can identify a core attribute, remove unimportant attributes,
and establish a relationship between attributes. As an alternative, we determine the
attribute contribution degree by measuring the variation size in system structure after
reducing one attribute. The criteria weight increases as the variation size increases.

In a DT, we define SigR(c) as criteria c important for subset c:

SigR(c) = 1− Card(R ∪ {c})
card(R)

(2)

Where: card(R) = card(IND(R))
In reality, we could run across the following scenario while calculating the contribu-

tion degree using formula (2): certain attribute contribution degrees are equal to zero or



S. M. Shaaban et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6049 5 of 21

have values that are inconsistent with the facts. To overcome this issue, we express the
enhancement importance relation as:

SigR (ci) =

∣∣∣∣∣∣SigR (ci)−
1

n− 1

n∑
j=1(j ̸=i)

[
SigR (ci,, cj,)− SigR (ci)

2

]∣∣∣∣∣∣ (3)

2.5. Multiple Criteria Ranking by Alternative Trace (MCRAT) Approach

Katarina et al. (2021), developed multiple criteria ranking by alternative trace (MCRAT)
[48]. This approach consists of two stages, first stage involves normalizing, weighting, and
determination of a component’s ”magnitude” after getting an optimal alternative and
decomposition of alternatives. The MCRAT techniques steps are defined as follow [44]:

Step 1: define the main parameters and determine the alternatives.
Step 2: Build the decision matrix of X

X = [Xij ]mn =


x11x12 . . . x1n
x21x22 . . . x2n

...
...

. . .
...

xm1xm2 . . . xmn

 (4)

Where n is the number of parameters and m is the number of selections, and Xij is
the outcomes value of ith alternative on jth parameters.

Step 3: Use the formula below to normalize the choice matrix, removing dimensions
from various criterion,

rij =

{
Xij−minXij

maxXij−minXij
if j ∈ B

maxXij−Xij

maxXij−minXij
if j ∈ C

(5)

Where B represents the collection of maximized characteristics, and C represents the
subset of minimized characteristics.

Step 4: For each alternative, prepare a weighted normalised matrix U as following:

U = [Uij ]mn =


u11u12...u1n
u21u22...u2n

...
...

. . .
...

um1um2...umn

 (6)

Step 5: Calculation the optimal alternative

qj = max (uij ⌈1 ≤ j ≤ n⌉ ) , ∀i ∈ [1, 2, 3, . . . ,m] (7)

Then the obtained optimal alternative is calculated using the following formula.

Q = { q1, q2, . . . , qj } , j = 1, 2, . . . , n (8)



S. M. Shaaban et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6049 6 of 21

Step 6: Decomposition step
This phase entails decomposing the ideal choice into two subgroups or components.

The set Q may be represented as the union of the two subsets:

Q = Qmax ∪Qmin (9)

Q = { q1, q2, . . . , qk } ∪ { q1, q2, . . . , qh } ; k + h = j (10)

where k indicates the total number of factors to be maximized, and h denotes the total
number of parameters to be decreased.

Step 7: Decomposition of the alternative
Similarly, to Step 6 we perform decomposition of each alternative:

U = Umax ∪ Umin, ∀ i ∈ [1, 2, . . . , m] (11)

Ui = { ui1, ui2, . . . , uik } ∪ { ui1, ui2, . . . , uih } , ∀ i ∈ [1, 2, . . . , m] (12)

Step 8: Magnitude of component
Using the following formula to calculate the magnitude

Qk =
√

q21 + q22 + . . . ..q2k(13)

Qh =
√

q21 + q22 + . . . ..q2h(14)

Every preference is treated using the same methodology.

Uik =
√
u2i1 + u2i2 + . . . ..u2ik ∀ i ∈ [1, 2, .., m](15)

Uih =
√

u2i1 + u2i2 + . . . ..u2ih ∀ i ∈ [1, 2, .., m](16)

Step 9: Ranking Alternative using (MCRAT) approach
Construct the matrix F using the best substitute elements:

F =

[
Qk 0
0 Qh

]
(17)

Additionally, construct the matrix Gi using various elements:

Gi =

[
Uik 0
0 Uih

]
∀ i ∈ [1, 2, . . . , m] (18)

If matrix F and matrix Gi are multiplied to get the matrix Ti:



S. M. Shaaban et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6049 7 of 21

Ti = F ×Gi =

[
t11 0
0 t22

]
∀ i ∈ [1, 2, . . . , m] (19)

The trace of the matrix Ti is obtained by the following formula:

tr (Ti) = t11,i + t22,i, ∀ i ∈ [1, 2, . . . , m] (20)

According to the descending order of tr (Ti) the alternatives are ranked

3. Proposed Methodology

The suggested technique involves three fundamental stages:
Stage I. determining criteria to be implemented inside the framework
Stage II. RST weight calculation
Stage III. Using the Multiple Criteria Ranking by Alternative Trace technique, the

alternatives are ranked in order of preference.
The general framework of the suggested approach is shown in Fig. 1.

Figure 1: The diagrammatic layout of the Integrated MCRAT Rough Set Theory approach.



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4. Shale oil wells Assessment Using the integrated RST and MCRAT
Approach

This portion presents a real-world application to verify the effectiveness and perfor-
mance of the recommended method for evaluating the quality of shale oil wells.

4.1. Information System of Evaluating Shale oil location

Numerous chemical and physical factors are involved in the shale oil site (SOs) evalua-
tion problem. In this paper, twenty-seven shale oil samples were taken from Al-Quseir area
of the Red Sea coast of Egypt as shown in table 1 [20]. For each sample, twelve parameter
including hydrogen total organic carbon (Cr1), the amount of thermovaporized-free hydro-
carbon (Cr2), the amount of hydrocarbon compounds originating from kerogen cracking
(Cr3), an indication of residual petroleum potential of the rock (Cr4), the amount of CO2

generated through thermal heating (Cr5), Production Index (Cr6), Oxygen Index (Cr7),
Hydrogen Index (Cr8), Rolyzable carbon index (Cr9), genetic potential (Cr10), vitrinite
reflectance (Cr11), and the temperature at which the maximum release of hydrocarbons
(Cr12) were investigated.

Table 1: Shale Oil samples information system

Alternative / Cr1 Cr2 Cr3 Cr4 Cr5 Cr6 Cr7 Cr8 Cr9 Cr10 Cr11 Cr12
Criterion Max Max Max Max Min Max Min Max Max Max Max Min

SOs1 19.07 3.04 97.57 3.83 422 553.44 21.35 0.045 97.31 25.48 83.51 0.43
SOs2 22.2 2.99 107.35 5.02 422 521.08 23.83 0.045 107.04 21.38 91.58 0.43
SOs3 23.56 5.19 153.18 6.65 423 695.211 29.65 0.045 155.07 23.03 131.45 0.45
SOs4 23.82 5.38 142.97 5.36 424 642.51 23.62 0.055 145.05 26.67 123.13 0.47
SOs5 20.76 2.86 94.99 4.46 423 494.64 22.72 0.045 94.55 21.30 81.22 0.45
SOs6 19.86 3.27 99.17 4.87 422 539.61 26 0.045 99.14 20.36 85.03 0.43
SOs7 22.69 3.34 104.94 4.88 421 498.58 22.63 0.045 104.98 21.50 89.87 0.41
SOs8 18.97 4.38 101.55 4.6 423 578.72 25.78 0.055 102.63 22.08 87.92 0.45
SOs9 19.22 2.33 60.45 6.17 425 343.84 34.11 0.045 59.48 9.80 52.11 0.49
SOs10 19.03 1.93 59.92 5.62 424 344.29 31.39 0.045 58.55 10.66 51.34 0.47
SOs11 12.38 1.26 22.63 3.2 424 202.67 28.44 0.045 20.59 7.07 19.83 0.47
SOs12 12.09 2.83 68.04 7.1 423 621.75 64.78 0.045 67.57 9.58 58.82 0.45
SOs13 13.48 2.9 67.41 7.74 423 549.27 62.67 0.055 67.01 8.71 58.36 0.45
SOs14 12.14 3.37 68.66 7.24 424 624.66 65.76 0.055 68.73 9.48 59.78 0.47
SOs15 21.73 5.46 139.54 11.27 422 688.7 54.71 0.055 141.7 12.38 120.35 0.43
SOs16 21.23 5.4 140.88 11.7 422 711.93 58.21 0.055 142.98 12.04 121.41 0.43
SOs17 22.15 5.85 180.53 9.43 425 870.4 44.86 0.055 183.08 19.14 154.70 0.49
SOs18 21.21 5.2 170.53 11.35 424 860.27 56.53 0.055 172.43 15.02 145.86 0.47
SOs19 21.11 5.48 139.23 10.9 425 707.8 54.56 0.055 141.41 12.77 120.11 0.49
SOs20 21.72 2.58 91 4.85 421 453.28 23.56 0.045 90.28 18.76 77.67 0.41
SOs21 22.63 3.02 86.68 8.52 421 414.98 39.63 0.045 86.4 10.17 74.45 0.41
SOs22 21.43 3.59 89.86 9.81 423 453.81 48.33 0.055 90.15 9.16 77.56 0.45
SOs23 21.16 3.1 99.47 5.78 423 507.59 28.86 0.045 99.27 17.21 85.13 0.45
SOs24 3.03 1.13 13.05 2.28 420 582.07 120 0.045 10.88 5.72 11.77 0.4
SOs25 2.83 1.05 8.91 0.36 422 404.3 21.17 0.045 6.66 24.75 8.27 0.43
SOs26 3.33 1.05 12.52 0.82 422 481.76 37.27 0.045 10.27 15.27 11.26 0.43
SOs27 2.93 1.29 8.52 0.27 420 361.45 15 0.045 6.51 31.56 8.14 0.4



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4.2. Weights calculation of the assessment parameters by RST

The weights of parameters for Shel oil site evaluation are computed using RST. Shale
oil information system need to be in discretized form before apply the proposed approach,
consequently, equal width technique prior to analysis is used. Where the attribute signifi-
cant is calculated through calculating the equivalence relation of each parameter using Eq.
2. Finally, using Eq. 3 is applied to calculate the degree of importance of each criterion.
Table 2 displays the respective weights of the evaluation criteria.

Table 2: The criteria weight calculation

Cr1 Cr2 Cr3 Cr4 Cr5 Cr6 Cr7 Cr8 Cr9 Cr10 Cr11 Cr12
0.0859 0.0618 0.0928 0.0756 0.0941 0.0943 0.0908 0.0682 0.0929 0.0855 0.0925 0.0655

4.3. Assessment of the available locations of Shale oil wells by MCRAT

Initially, the attributes for shale oil assessment are transformed into dimensionless
values using the linear normalization procedure in the MCRAT method. This enables
the comparison of all parameters. Table 3 shows shale oil normalized decision matrix.
The corresponding weighted normalized matrix is subsequently constructed using Eq. 5,
as illustrated in Table 4. After that the optimal alternatives for the given criteria using
formula 7 & 8 are obtained. Table 5 shows the Optimal alternative and decomposition of
each alternative. Then, the decomposition calculation for each alternative to obtain the
optimum solution is executed as shown in table 6. Consequently, the computation of the
magnitude for the optimal alternatives and other components using formula section 2.6.
is calculated as shown in table 7. Finally, the calculation of the alternative rank using the
formula 17-20 is performed. Table 8 shows the alternatives rank.



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Table 3: Normalized decision matrix

Alternatives Cr1 Cr2 Cr3 Cr4 Cr5 Cr6 Cr7 Cr8 Cr9 Cr10 Cr11 Cr12
SOs1 0.8006 0.5197 0.5405 0.3274 0.9953 0.6358 0.7026 0.8182 0.5315 0.8073 0.5398 0.9302
SOs2 0.9320 0.5111 0.5946 0.4291 0.9953 0.5987 0.6295 0.8182 0.5847 0.6777 0.5920 0.9302
SOs3 0.9891 0.8872 0.8485 0.5684 0.9929 0.7987 0.5059 0.8182 0.8470 0.7300 0.8497 0.8889
SOs4 1.0000 0.9197 0.7919 0.4581 0.9906 0.7382 0.6351 1.0000 0.7923 0.8453 0.7960 0.8511
SOs5 0.8715 0.4889 0.5262 0.3812 0.9929 0.5683 0.6602 0.8182 0.5164 0.6749 0.5250 0.8889
SOs6 0.8338 0.5590 0.5493 0.4162 0.9953 0.6200 0.5769 0.8182 0.5415 0.6453 0.5496 0.9302
SOs7 0.9526 0.5709 0.5813 0.4171 0.9976 0.5728 0.6628 0.8182 0.5734 0.6815 0.5810 0.9756
SOs8 0.7964 0.7487 0.5625 0.3932 0.9929 0.6649 0.5818 1.0000 0.5606 0.6996 0.5684 0.8889
SOs9 0.8069 0.3983 0.3348 0.5274 0.9882 0.3950 0.4398 0.8182 0.3249 0.3105 0.3368 0.8163
SOs10 0.7989 0.3299 0.3319 0.4803 0.9906 0.3956 0.4779 0.8182 0.3198 0.3379 0.3318 0.8511
SOs11 0.5197 0.2154 0.1254 0.2735 0.9906 0.2328 0.5274 0.8182 0.1125 0.2241 0.1282 0.8511
SOs12 0.5076 0.4838 0.3769 0.6068 0.9929 0.7143 0.2316 0.8182 0.3691 0.3037 0.3802 0.8889
SOs13 0.5659 0.4957 0.3734 0.6615 0.9929 0.6311 0.2393 1.0000 0.3660 0.2760 0.3772 0.8889
SOs14 0.5097 0.5761 0.3803 0.6188 0.9906 0.7177 0.2281 1.0000 0.3754 0.3005 0.3865 0.8511
SOs15 0.9123 0.9333 0.7729 0.9632 0.9953 0.7912 0.2742 1.0000 0.7740 0.3924 0.7780 0.9302
SOs16 0.8913 0.9231 0.7804 1.0000 0.9953 0.8179 0.2577 1.0000 0.7810 0.3816 0.7848 0.9302
SOs17 0.9299 1.0000 1.0000 0.8060 0.9882 1.0000 0.3344 1.0000 1.0000 0.6067 1.0000 0.8163
SOs18 0.8904 0.8889 0.9446 0.9701 0.9906 0.9884 0.2653 1.0000 0.9418 0.4761 0.9429 0.8511
SOs19 0.8862 0.9368 0.7712 0.9316 0.9882 0.8132 0.2749 1.0000 0.7724 0.4048 0.7764 0.8163
SOs20 0.9118 0.4410 0.5041 0.4145 0.9976 0.5208 0.6367 0.8182 0.4931 0.5946 0.5021 0.9756
SOs21 0.9500 0.5162 0.4801 0.7282 0.9976 0.4768 0.3785 0.8182 0.4719 0.3224 0.4813 0.9756
SOs22 0.8997 0.6137 0.4978 0.8385 0.9929 0.5214 0.3104 1.0000 0.4924 0.2903 0.5014 0.8889
SOs23 0.8883 0.5299 0.5510 0.4940 0.9929 0.5832 0.5198 0.8182 0.5422 0.5454 0.5503 0.8889
SOs24 0.1272 0.1932 0.0723 0.1949 1.0000 0.6687 0.1250 0.8182 0.0594 0.1814 0.0761 1.0000
SOs25 0.1188 0.1795 0.0494 0.0308 0.9953 0.4645 0.7085 0.8182 0.0364 0.7843 0.0534 0.9302
SOs26 0.1398 0.1795 0.0694 0.0701 0.9953 0.5535 0.4025 0.8182 0.0561 0.4839 0.0728 0.9302
SOs27 0.1230 0.2205 0.0472 0.0231 1.0000 0.4153 1.0000 0.8182 0.0356 1.0000 0.0526 1.0000



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Table 4: Normalize DM

Alternatives Cr1 Cr2 Cr3 Cr4 Cr5 Cr6 Cr7 Cr8 Cr9 Cr10 Cr11 Cr12
SOs1 0.0688 0.0321 0.0502 0.0247 0.093654 0.0600 0.0638 0.0558 0.0494 0.0690 0.0499 0.0609
SOs2 0.0801 0.0316 0.0552 0.0324 0.093654 0.0565 0.0572 0.0558 0.0543 0.0579 0.0548 0.0609
SOs3 0.0850 0.0548 0.0787 0.0430 0.093433 0.0753 0.0459 0.0558 0.0787 0.0624 0.0786 0.0582
SOs4 0.0859 0.0568 0.0735 0.0346 0.093212 0.0696 0.0577 0.0682 0.0736 0.0723 0.0736 0.0557
SOs5 0.0749 0.0302 0.0488 0.0288 0.093433 0.0536 0.0599 0.0558 0.0480 0.0577 0.0486 0.0582
SOs6 0.0716 0.0345 0.0510 0.0315 0.093654 0.0585 0.0524 0.0558 0.0503 0.0552 0.0508 0.0609
SOs7 0.0818 0.0353 0.0539 0.0315 0.093876 0.0540 0.0602 0.0558 0.0533 0.0583 0.0537 0.0639
SOs8 0.0684 0.0463 0.0522 0.0297 0.093433 0.0627 0.0528 0.0682 0.0521 0.0598 0.0526 0.0582
SOs9 0.0693 0.0246 0.0311 0.0399 0.092993 0.0373 0.0399 0.0558 0.0302 0.0265 0.0312 0.0535
SOs10 0.0686 0.0204 0.0308 0.0363 0.093212 0.0373 0.0434 0.0558 0.0297 0.0289 0.0307 0.0557
SOs11 0.0446 0.0133 0.0116 0.0207 0.093212 0.0220 0.0479 0.0558 0.0104 0.0192 0.0119 0.0557
SOs12 0.0436 0.0299 0.0350 0.0459 0.093433 0.0674 0.0210 0.0558 0.0343 0.0260 0.0352 0.0582
SOs13 0.0486 0.0306 0.0347 0.0500 0.093433 0.0595 0.0217 0.0682 0.0340 0.0236 0.0349 0.0582
SOs14 0.0438 0.0356 0.0353 0.0468 0.093212 0.0677 0.0207 0.0682 0.0349 0.0257 0.0357 0.0557
SOs15 0.0784 0.0577 0.0717 0.0728 0.093654 0.0746 0.0249 0.0682 0.0719 0.0335 0.0720 0.0609
SOs16 0.0766 0.0570 0.0724 0.0756 0.093654 0.0771 0.0234 0.0682 0.0726 0.0326 0.0726 0.0609
SOs17 0.0799 0.0618 0.0928 0.0609 0.092993 0.0943 0.0304 0.0682 0.0929 0.0519 0.0925 0.0535
SOs18 0.0765 0.0549 0.0877 0.0733 0.093212 0.0932 0.0241 0.0682 0.0875 0.0407 0.0872 0.0557
SOs19 0.0761 0.0579 0.0716 0.0704 0.092993 0.0767 0.0250 0.0682 0.0718 0.0346 0.0718 0.0535
SOs20 0.0783 0.0273 0.0468 0.0313 0.093876 0.0491 0.0578 0.0558 0.0458 0.0508 0.0464 0.0639
SOs21 0.0816 0.0319 0.0446 0.0551 0.093876 0.0450 0.0344 0.0558 0.0438 0.0276 0.0445 0.0639
SOs22 0.0773 0.0379 0.0462 0.0634 0.093433 0.0492 0.0282 0.0682 0.0457 0.0248 0.0464 0.0582
SOs23 0.0763 0.0327 0.0511 0.0373 0.093433 0.0550 0.0472 0.0558 0.0504 0.0466 0.0509 0.0582
SOs24 0.0109 0.0119 0.0067 0.0147 0.094100 0.0631 0.0114 0.0558 0.0055 0.0155 0.0070 0.0655
SOs25 0.0102 0.0111 0.0046 0.0023 0.093654 0.0438 0.0643 0.0558 0.0034 0.0671 0.0049 0.0609
SOs26 0.0120 0.0111 0.0064 0.0053 0.093654 0.0522 0.0365 0.0558 0.0052 0.0414 0.0067 0.0609
SOs27 0.0106 0.0136 0.0044 0.0017 0.094100 0.0392 0.0908 0.0558 0.0033 0.0855 0.0049 0.0655

Table 5: Optimal alternative Decomposition

Alternative Max Min Max Min Max Min
Cr1 Cr2 Cr3 Cr4 Cr5 Cr6 Cr7 Cr8 Cr9 Cr10 Cr11 Cr12

q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11 q12

Q Max 0.0859 0.0618 0.0928 0.0756 0.0943 0.0682 0.0929 0.0855 0.0925

Q Min 0.0941 0.0908 0.0655



S. M. Shaaban et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6049 12 of 21

Table 6: Decomposition of alternatives.

Alternative / Max Max Max Max Min Max Min Max Max Max Max Min
Criterion Cr1 Cr2 Cr3 Cr4 Cr5 Cr6 Cr7 Cr8 Cr9 Cr10 Cr11 Cr12

U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12

SOs1 UMax 0.0688 0.0321 0.0502 0.0247 0.0600 0.0558 0.0494 0.0690 0.0499
SOs1 UMin 0.093654 0.0638 0.0609
SOs2 UMax 0.0801 0.0316 0.0552 0.0324 0.0565 0.0558 0.0543 0.0579 0.0548
SOs2 UMin 0.093654 0.0572 0.0609
SOs3 UMax 0.0850 0.0548 0.0787 0.0430 0.0753 0.0558 0.0787 0.0624 0.0786
SOs3 UMin 0.093433 0.0459 0.0582
SOs4 UMax 0.0859 0.0568 0.0735 0.0346 0.0696 0.0682 0.0736 0.0723 0.0736
SOs4 UMin 0.093212 0.0577 0.0557
SOs5 UMax 0.0749 0.0302 0.0488 0.0288 0.0536 0.0558 0.0480 0.0577 0.0486
SOs5 UMin 0.093433 0.0599 0.0582
SOs6 UMax 0.0716 0.0345 0.0510 0.0315 0.0585 0.0558 0.0503 0.0552 0.0508
SOs6 UMin 0.093654 0.0524 0.0609
SOs7 UMax 0.0818 0.0353 0.0539 0.0315 0.0540 0.0558 0.0533 0.0583 0.0537
SOs7 UMin 0.093876 0.0602 0.0639
SOs8 UMax 0.0684 0.0463 0.0522 0.0297 0.0627 0.0682 0.0521 0.0598 0.0526
SOs8 UMin 0.093433 0.0528 0.0582
SOs9 UMax 0.0693 0.0246 0.0311 0.0399 0.0373 0.0558 0.0302 0.0265 0.0312
SOs9 UMin 0.092993 0.0399 0.0535
SOs10 UMax 0.0686 0.0204 0.0308 0.0363 0.0373 0.0558 0.0297 0.0289 0.0307
SOs10 UMin 0.093212 0.0434 0.0557
SOs11 UMax 0.0446 0.0133 0.0116 0.0207 0.0220 0.0558 0.0104 0.0192 0.0119
SOs11 UMin 0.093212 0.0479 0.0557
SOs12 UMax 0.0436 0.0299 0.0350 0.0459 0.0674 0.0558 0.0343 0.0260 0.0352
SOs12 UMin 0.093433 0.0210 0.0582
SOs13 UMax 0.0486 0.0306 0.0347 0.0500 0.0595 0.0682 0.0340 0.0236 0.0349
SOs13 UMin 0.093433 0.0217 0.0582
SOs14 UMax 0.0438 0.0356 0.0353 0.0468 0.0677 0.0682 0.0349 0.0257 0.0357
SOs14 UMin 0.093212 0.0207 0.0557
SOs15 UMax 0.0784 0.0577 0.0717 0.0728 0.0746 0.0682 0.0719 0.0335 0.0720
SOs16 UMin 0.093654 0.0249 0.0609
SOs16 UMax 0.0766 0.0570 0.0724 0.0756 0.0771 0.0682 0.0726 0.0326 0.0726
SOs16 UMin 0.093654 0.0234 0.0609
SOs17 UMax 0.0799 0.0618 0.0928 0.0609 0.0943 0.0682 0.0929 0.0519 0.0925
SOs17 UMin 0.092993 0.0304 0.0535
SOs18 UMax 0.0765 0.0549 0.0877 0.0733 0.0932 0.0682 0.0875 0.0407 0.0872
SOs18 UMin 0.093212 0.0241 0.0557
SOs19 UMax 0.0761 0.0579 0.0716 0.0704 0.0767 0.0682 0.0718 0.0346 0.0718
SOs19 UMin 0.092993 0.0250 0.0535
SOs20 UMax 0.0783 0.0273 0.0468 0.0313 0.0491 0.0558 0.0458 0.0508 0.0464
SOs20 UMin 0.093876 0.0578 0.0639
SOs21 UMax 0.0816 0.0319 0.0446 0.0551 0.0450 0.0558 0.0438 0.0276 0.0445
SOs21 UMin 0.093876 0.0344 0.0639
SOs22 UMax 0.0773 0.0379 0.0462 0.0634 0.0492 0.0682 0.0457 0.0248 0.0464
SOs22 UMin 0.093433 0.0282 0.0582
SOs23 UMax 0.0763 0.0327 0.0511 0.0373 0.0550 0.0558 0.0504 0.0466 0.0509
SOs23 UMin 0.093433 0.0472 0.0582
SOs24 UMax 0.0109 0.0119 0.0067 0.0147 0.0631 0.0558 0.0055 0.0155 0.0070
SOs24 UMin 0.094100 0.0114 0.0655
SOs25 UMax 0.0102 0.0111 0.0046 0.0023 0.0438 0.0558 0.0034 0.0671 0.0049
SOs25 UMin 0.093654 0.0643 0.0609
SOs26 UMax 0.0120 0.0111 0.0064 0.0053 0.0522 0.0558 0.0052 0.0414 0.0067
SOs26 UMin 0.093654 0.0365 0.0609
SOs27 UMax 0.0106 0.0136 0.0044 0.0017 0.0392 0.0558 0.0033 0.0855 0.0049
SOs27 UMin 0.094100 0.0908 0.0655



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Table 7: Magnitude of optimal alternative

Alternatives Max Min
Qk Qh
Uik Uik

Q 0.252122 0.146255
SOs1 0.1590 0.1287
SOs2 0.1647 0.1255
SOs3 0.2081 0.1193
SOs4 0.2068 0.1230
SOs5 0.1540 0.1254
SOs6 0.1569 0.1234
SOs7 0.1643 0.1285
SOs8 0.1675 0.1221
SOs9 0.1227 0.1145
SOs10 0.1207 0.1170
SOs11 0.0833 0.1187
SOs12 0.1298 0.1121
SOs13 0.1346 0.1122
SOs14 0.1379 0.1106
SOs15 0.2040 0.1145
SOs16 0.2056 0.1142
SOs17 0.2365 0.1115
SOs18 0.2284 0.1112
SOs19 0.2031 0.1101
SOs20 0.1497 0.1274
SOs21 0.1500 0.1186
SOs22 0.1597 0.1136
SOs23 0.1560 0.1198
SOs24 0.0891 0.1152
SOs25 0.0991 0.1289
SOs26 0.0892 0.1176
SOs27 0.1110 0.1463



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Table 8: Alternative Ranking

Alternatives Total value Rank

SOs1 0.0589 11
SOs2 0.0599 10
SOs3 0.0699 4
SOs4 0.0701 3
SOs5 0.0572 13
SOs6 0.0576 12
SOs7 0.0602 8
SOs8 0.0601 9
SOs9 0.0477 22
SOs10 0.0475 23
SOs11 0.0384 27
SOs12 0.0491 21
SOs13 0.0504 19
SOs14 0.0509 18
SOs15 0.0682 6
SOs16 0.0685 5
SOs17 0.0759 1
SOs18 0.0739 2
SOs19 0.0673 7
SOs20 0.0564 16
SOs21 0.0552 17
SOs22 0.0569 14
SOs23 0.0569 15
SOs24 0.0393 26
SOs25 0.0438 24
SOs26 0.0397 25
SOs27 0.0494 20

4.4. Evaluation the suggested approach in comparison to other compre-
hensive assessment methods

In order to assess the efficacy and reliability of the proposed method, we also compared
the shale oil ranking outcomes to those of known optimization techniques like: TOPSIS
[14, 39], EDAS [24, 24], Codas [26, 27], and WASPAS [28]. The results of rankings of
various approaches are presented in table 9, and Fig.2.

Based on a comparison of ranking results, it appears that ranking outcomes are gener-
ally aligned. Based on the slight variations in rankings between established and proposed
methods, the proposed method may be a good decision-making tool.



S. M. Shaaban et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6049 15 of 21

Table 9: Results of shale Oil Ranking

SO. No. Ranked based on different methods
Proposed CODAS EDAS TOPSIS WASPAS

SOs1 11 12 14 14 14
SOs2 10 8 10 10 10
SOs3 4 3 6 6 6
SOs4 3 7 7 7 7
SOs5 13 16 17 17 15
SOs6 12 14 13 13 13
SOs7 8 9 11 11 11
SOs8 9 10 8 9 8
SOs9 22 22 21 22 21
SOs10 23 23 22 23 22
SOs11 27 27 24 27 24
SOs12 21 20 19 18 20
SOs13 19 19 18 19 19
SOs14 18 18 15 16 17
SOs15 6 5 5 5 5
SOs16 5 4 3 3 4
SOs17 1 1 1 2 1
SOs18 2 2 2 1 2
SOs19 7 6 4 4 3
SOs20 16 17 20 20 18
SOs21 17 15 16 15 16
SOs22 14 11 9 8 9
SOs23 15 13 12 12 12
SOs24 26 21 23 21 23
SOs25 24 25 27 25 26
SOs26 25 26 26 26 25
SOs27 20 24 25 24 27

The Comparison of the rankings shows that the highest-ranked alternatives (SOs17
and SOs18) remain consistent across all MCDM methods, highlighting the reliability of
the proposed approach. Moreover, the mid-ranked and lower-ranked alternatives display
only minor variations, suggesting that the methodology follows a decision-making pattern
similar to traditional MCDM techniques. Although slight discrepancies are observed,
particularly in alternatives such as SOs3, SOs4, and SOs22, the overall ranking trend
remains largely stable.

Based on the results, it appears that the proposed approach is in line with other
established methods. Furthermore, this method effectively handles datasets with multiple
input variables, providing an enhanced decision-making process.



S. M. Shaaban et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6049 16 of 21

5. Conclusions

The evaluation of shale Oil quality is one of the most important aspects in crude oil
administration.

This study introduces a new two-stage method to evaluate shale oil quality. It combines
Rough Set Theory (RST) with (MCRAT). The approach identifies key geochemical factors
and ranks shale oil samples based on quality.

The obtained results are compared with established decision-making methods like CO-
DAS, EDAS, TOPSIS, and WASPAS shows that the proposed method produces similar
rankings. The top-ranked samples remain stable across different techniques, confirming
the model’s reliability. Although some mid- and lower-ranked samples show slight differ-
ences, the overall trend aligns with conventional methods.

This approach has several benefits. It reduces computational complexity, handles
large datasets efficiently, and is easy to implement. RST helps simplify the evaluation by
eliminating unnecessary parameters while preserving accuracy.

Overall, the suggested approach is a practical and reliable tool for assessing shale oil
quality. Future studies can apply it to other decision-making problems in the petroleum
industry, integrate machine learning techniques, or expand the dataset for better predic-
tions.



S. M. Shaaban et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6049 17 of 21

Figure 2: Comparative rankings of suggested method with other MCDM techniques.

Acknowledgements

The authors extend their appreciation to Northern Border University, Saudi Arabia,
for supporting this work through project number (NBU-CRP-2025-289).

Conflicts of Interest

The authors declare no conflict of interest.

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