EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6578
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

1

A New Method for Multi-Criteria Decision-Making:2

Adaptive Ranking with Ideal Evaluation (ARIE)3

Nur Fathiah Fatin Mohamad Fauzi1, Zahari Md Rodzi1,2,∗,4

Zaifilla Farrina Zainuddin3,∗, Norzieha Mustapha4, Faisal Al-Sharqi55

1 Fakulti Sains Komputer dan Matematik, Universiti Teknologi MARA (UiTM) Cawangan6

Negeri Sembilan Kampus Seremban, Negeri Sembilan, Malaysia7

2 Fakulti Sains dan Teknologi, Universiti Kebangsaan Malaysia, Selangor, Malaysia8

3 Manufacturing and Management Technology Section, Universiti Kuala Lumpur Malaysia9

Italy Design Institute (UniKL MIDI), Malaysia10

4 Fakulti Sains Komputer dan Matematik, Universiti Teknologi MARA (UiTM) Cawangan11

Kelantan Kampus Machang, Kelantan, Malaysia12

5 Department of Mathematics, Faculty of Education for Pure Sciences, University of Anbar,13

Ramadi, Iraq14

5 College of Pharmacy, National University of Science and Technology, Dhi Qar, Iraq15

16

Abstract. Decision makers frequently confront complex criteria, some requiring maximization,
others minimization, and still others precise target attainment, yet classical Multi-Criteria Decision-
Making (MCDM) methods (e.g., TOPSIS, VIKOR and SAW) offer limited flexibility to handle such
mixed preference directions, often producing inconsistent or opaque rankings. To address these
challenges, this study proposed the Adaptive Ranking with Ideal Evaluation (ARIE), a
fully flexible similarity-based framework that unifies benefit, cost, and target-type normalization
under one roof. ARIE leverages a novel dual-parameter score function with a sensitivity exponent
γ to control the nonlinearity of deviations and a balancing coefficient κ to tailor the trade-off
between aspiration toward the ideal and avoidance of the anti-ideal to convert weighted normal-
ized ratios into a single, interpretable closeness measure. We demonstrate ARIE in a case study
of halal supplier selection, perform a sensitivity analysis between γ and κ values, and carry out
comparative and simulation-based analyzes using MATLAB-generated weight scenarios against
seven benchmark methods (CRADIS, MABAC, ARAS, MOORA, VIKOR, TOPSIS and SAW).
The results show that ARIE’s new scoring technique consistently yields more stable, transparent,
and decision-maker-aligned rankings in diverse decision contexts.

2020 Mathematics Subject Classifications: 90B50, 68T37, 62C0517

Key Words and Phrases: Multi-criteria decision-making (MCDM), arie method, adaptive rank-18

ing, ideal evaluation, performance analysis, decision modeling19

20

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6578

Email addresses: zahari@uitm.edu.my (Z. M. Rodzi), zaifilla@unikl.edu.my (Z. F. Zainuddin)

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



N. F. F. M. Fauzi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6578 2 of 24

1. Introduction21

Decision-making has become an increasingly complex and critical process across diverse22

domains such as business [1, 2], engineering [3, 4], healthcare [5, 6], and public policy [7],23

[8]. In the context of globalization, technological advancement, and increasingly intercon-24

nected challenges, both organizations and individuals face the need to evaluate alternatives25

under multiple, often conflicting, criteria [9, 10]. Multi-Criteria Decision-Making (MCDM)26

methods provide structured, logical, and repeatable tools to navigate such complexity [11].27

These methods support the comparison of alternatives, weighting of criteria, and synthesis28

of preferences into consistent and interpretable rankings [12, 13].29

MCDM has found widespread application in areas such as supplier selection [14],30

project prioritization [15], and sustainability evaluations [16] where decision-makers of-31

ten need to manage trade-offs between cost, quality, risk, and environmental impact.32

Despite these advancements, current methods still face several limitations. Many tra-33

ditional models assume criteria are either to be maximized or minimized. However, in34

real-world decision problems, some criteria are best evaluated by how closely they match35

a specific target value — a scenario poorly addressed in methods such as TOPSIS, VIKOR,36

and SAW. Furthermore, classical methods typically use static preference models and lack37

adjustable parameters for sensitivity control, making them less robust in dynamic envi-38

ronments where decision-maker attitudes or input data may vary [17, 18].39

To address these gaps, this paper introduces a novel method: Adaptive Ranking40

with Ideal Evaluation (ARIE). ARIE extends classical MCDM by integrating three41

key innovations:42

• A unified normalization framework that incorporates benefit, cost, and a newly43

introduced target-type criterion — where the optimal value lies within, rather44

than at the edge of, the acceptable range.45

• A dual-parameter scoring function governed by the sensitivity parameter γ and bal-46

ancing coefficient κ, which together modulate the influence of deviations and the47

trade-off between closeness to the ideal and distance from the anti-ideal.48

• A comprehensive evaluation structure validated through sensitivity analysis, compar-49

ative benchmarking against seven classical MCDM methods, and simulation-based50

robustness checks using real-world inspired data.51

ARIE is designed to offer greater flexibility, stability, and adaptability compared to52

existing methods. It responds effectively to criteria directionality (max, min, or target),53

allows decision-makers to tune parameters based on risk preferences or aspiration levels,54

and maintains consistency across diverse decision environments.55

The remainder of this paper is organized as follows. Section 2 presents a critical review56

of MCDM literature, with emphasis on current methodological gaps and the motivation57

for ARIE. Section 3 explains the ARIE methodology in detail, including its mathematical58

formulation and computational steps. Section 4 demonstrates the method’s performance59

through a halal supplier selection case study, sensitivity analysis, comparative study, and60



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simulation-based evaluation. Finally, Section 5 concludes the study and suggests future61

recommendations.62

2. Literature Review63

Nowadays, MCDM techniques are widely used to guide important decisions in engi-64

neering, environmental planning, education and finance. They are designed to weigh dif-65

ferent conflicting factors and direct decision-makers to the best solutions. There are many66

MCDM methods that differ in their basic concepts, strengths and weaknesses. Here, the67

most prominent MCDM approaches are studied to compare their outcomes and underline68

the need to build more flexible strategies. One of the most prominent MCDM techniques69

is the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), intro-70

duced by Hwang and Yoon [19]. TOPSIS ranks alternatives based on their proximity to71

the Positive Ideal Solution (PIS) and distance from the Negative Ideal Solution (NIS). Its72

has been applied in supplier selection [20, 21], project prioritization [22, 23], and environ-73

mental management [24, 25]. On the other hand, the use of fixed points and the same74

significance for distances rise difficulty to handle real life challenges [26, 27].75

The Simple Additive Weighting (SAW) or Weighted Sum Model (WSM) [28] is another76

foundational method that aggregates weighted criteria scores. While SAW is easy to use77

and understand [29], it struggles to handle cases where different choices affect each other78

[30]. For the Additive Ratio Assessment (ARAS) method [31], alternatives are judged79

against an optimal solution to determine their performance. It is an efficient and effective80

process when considering both cost and benefit factors [32, 33]. The approach is still81

sensitive to changes in data and fails to offer reliable results [34, 35].82

Moreover, the Compromise Ranking of Alternatives from Distance to Ideal Solution83

(CRADIS) [36] extends TOPSIS and ARAS by integrating multiple evaluation perspec-84

tives. CRADIS effectively incorporates the deviation of each alternative from ideal and85

anti-ideal points [37, 38]. Nevertheless, it still depends on static benchmarks and may86

face scalability issues with large datasets. The Multi-Attributive Border Approximation87

Area Comparison (MABAC) method [39] proposes a shift from static ideals to a border88

approximation area for comparison. The method is found effective in dealing with nega-89

tive values and less responsive to outliers [40, 41] which is good for making decisions in90

unpredictable situations. Even so, MABAC can be complex to compute when there are91

numerous criteria and its interpretation may not be clear.92

Next, the MOORA method (Multi-Objective Optimization by Ratio Analysis) uses93

normalizing through ratios to ensure simplicity and fast computing, as seen in Brauers94

(2010, 2006) and Chakraborty (2011) [42]. It may result in unfair rankings when criteria95

are not valued equally [43, 44]. VIKOR (VlseKriterijumska Optimizacija I Kompromisno96

Resenje) [45] aims to find a compromise by minimizing regret and maximizing the overall97

group utility [46, 47]. Sometimes, the decisions made from the model are unclear as they98

depend on the personal views and opinions of the participants [48, 49].99

A range of comparative analysis study between MCDM methods has been conducted100

to discuss the benefits and weaknesses of every MCDM method. Li et al. [50] proposed101



N. F. F. M. Fauzi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6578 4 of 24

and compared extended TOPSIS, MOORA, ARAS, SAW and other methods. The pro-102

posed extended MCDM methods are found robust and effective in solving decision-making103

problems. Abacıoğlu et al. [51] investigates the evolving landscape of green universities104

using CRADIS, MABAC and other methods. The study analyzes how the significance of105

the six sustainability criteria changes when different MCDM weighting methods are ap-106

plied. Some weighting techniques affect the final rankings as they may emphasize certain107

criteria more than others. In order to improve sustainability through drilling machine108

efficiency, Ramdani et al. [52] compared TOPSIS with VIKOR. The TOPSIS approach109

is found to exhibit a strong association with reference ranking, particularly with regard110

to specific energy. On the other hand, the VIKOR coefficient shows a moderate degree111

of similarity. Anic [53] compared VIKOR, MABAC, and other methods in tower geodetic112

micro-network application. It was shown that change was reflected in a different ranking113

list compared to the corresponding ranking lists provided using the MABAC, and other114

methods.115

In other applications, Susilo and Wahyuni [54] compared SAW and TOPSIS methods116

in decision support system in contraceptive methods application. The results show that117

SAW more accurately reflects expert opinions and more effective than TOPSIS. George et118

al. [55] compared TOPSIS, VIKOR, and MOORA methods for vendor selection in man-119

ufacturing industry. Muni et al. [56] compared MABAC, MOORA and other methods120

to select the best egg supplier. The results of the sensitivity test show that MABAC has121

the highest value of 4.4274 percent, then MOORA with a value of 2.3442 percent and122

the other method with a value of 0.4573 percent. Hendrawan [57] employed MOORA,123

ARAS, and other methods for location development priorities. The study’s results are124

expected to reveal how different MCDM methods rank industrial area development loca-125

tions. Mete [58] utilized VIKOR, ARAS, SAW, and other methods to analyze the Turkish126

insurance companies’ financial performance traded. According to the result, there is a127

strong relationship between ARAS and SAW with Proximity Indexed Value (PIV), while128

the relationship with VIKOR is moderate. Kumar et al. [59] used MOORA, TOPSIS,129

ARAS, and other methods to solve milling process optimization problems. The study130

finds 255 rpm, 82 mm/min feed rate, and 0.75 mm depth of cut as the optimal machin-131

ing parameters, with six MCDM methods agreeing on this result. In these studies, it is132

shown that there is no single MCDM method that works best in every case, so it remains133

important to choose methods that are suited to each unique problem.134

These findings further highlight the limitations of existing methods when applied across135

varying contexts and underscore the need for more adaptive and versatile approaches.136

CRADIS and MABAC approaches can fix many problems with traditional ways, yet they137

usually struggle with complexity, variation or do not adapt in real time. Although SAW,138

MOORA and ARAS are simple, they lack the robustness required for many applications.139

Even though VIKOR and TOPSIS find middle choices, they do this by sticking to fixed140

standards and assumptions. Alternatively, the proposed ARIE method addresses these141

issues by constantly changing the references and the way values are decided in real time.142

ARIE is not meant to take over from present models, but improves how reliable decisions143

can be by including flexibility, visibility and advanced computation. The next part outlines144



N. F. F. M. Fauzi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6578 5 of 24

the ARIE methodology and compare it with current MCDM techniques.145

3. Methodology146

ARIE method delivers a comprehensive framework to evaluate and prioritize alterna-147

tives in MCDM issues. This method is proposed as an improvement to the classical TOP-148

SIS, introducing a similarity-based approach with greater flexibility through risk-sensitive149

tuning and multi-type normalization. It is designed to handle criteria of different nature150

(benefit, cost, target) in MCDM problems, and each phase is designed to bolster the ro-151

bustness, flexibility, and clarity of the resulting rankings. The step-by-step flowchart of152

the ARIE procedure is presented in Figure 1.153

Figure 1: ARIE Method Flowchart.

The following is a detailed explanation of the ARIE method procedure.154

Step 1: Construct the Decision Matrix155

In the MCDM behavior, there must be the number of criteria, j = 1, 2, . . . , n, and alter-156

natives, i = 1, 2, . . . ,m to create a decision matrix form as below:157

X = [xij ]m×n (1)

The raw decision matrix X = xij underpins the entire evaluation process where each158

element xij denotes the performance score of the ith alternative on the jth criterion.159

To ensure comparability across diverse criteria, practitioners typically agree on consis-160

tent scoring scales, conduct calibration workshops to align interpretation, and perform161

data-quality checks such as outlier detection and handling of missing values before pro-162

ceeding to ARIE’s multi-type normalization step. This rigorous grounding of xij values163

in validated since context-specific metrics is essential for generating credible similarity164

assessments and, ultimately, robust and interpretable rankings.165



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Step 2: Normalize the Decision Matrix166

To ensure dimensionless and comparable criteria values, the decision matrix is normalized.167

This normalized process eliminates the unit measurement in the decision matrix including168

price, percentage, ratio, and points. In this method, three types of normalization are169

introduced:170

• Max-type (benefit) criterion where higher values are recommended:171

rij =
xij
xmax
j

(2)

• Min-type (cost) criterion where lower values are preferable:172

rij =
xmin
j

xij
(3)

• Target-type (goal): For criteria with a desired target value xTj :173

rij = 1 −
|xij − xTj |

max
(
|xmax

j − xTj |, |xmin
j − xTj |

) (4)

xij is the original value of the ith alternative with respect to the jth criterion, while174

xmax
j is the maxmimum value of the jth criterion across al alternatives in step 1. ARIE175

method uses xTj as the benchmark for each criterion whether set by expert consensus, reg-176

ulatory standards, historical averages, or strategic goals and normalizes each alternative’s177

score xij by its distance from this target so that those closest to the benchmark score178

highest. By applying benefit, cost, and target-based normalization, the method avoids179

distortions from inappropriate scaling, accurately reflects each criterion’s true preference180

direction, and with its innovative inclusion of target-based transformation outperforms181

classical models in ensuring consistent, fair, and balanced comparisons across mixed cri-182

teria.183

Step 3: Weighted Normalized Decision Matrix184

The normalized values are multiplied by the corresponding criteria weights, wj , where185 ∑n
j=1wj = 1 given by:186

vij = wj · rij (5)

The weights wj capture each criterion’s relative importance and may be established187

through subjective techniques, like AHP and BWM methods, objective methods, such as,188

Entropy and MEREC methods, or decision-makers judgment [60, 61]. Once applied to189



N. F. F. M. Fauzi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6578 7 of 24

the normalized scores, they produce the matrix V = [vij ], in which higher-priority cri-190

teria exert a proportionally greater influence and variables with wider numerical ranges191

are balanced. Incorporating weights at this stage both embeds stakeholder preferences192

(or data-driven priorities) and sets the foundation for meaningful similarity calculations.193

Keeping
∑

j wj = 1 further guarantees that comparisons across alternatives remain trans-194

parent and interpretable.195

Step 4: Compute Similarity to Ideal and Anti-Ideal Solutions196

To evaluate how close each alternative is to the best (ideal) and worst (anti-ideal) alter-197

natives by using a similarity-based approach. Let:198

vmax
j = max

i
vij , vmin

j = min
i

vij

A linear similarity model is used by setting the sensitivity parameter γ = 1. Then:199

Simbest
i =

n∑
j=1

(
vij
vmax
j

)γ

(6)

Simworst
i =

n∑
j=1

(
vmin
j

vij

)γ

(7)

Here, vmax
j and vmin

j denote the highest and lowest weighted normalized scores for200

criterion j. The sensitivity parameter γ controls how sharp deviations from these bench-201

marks are penalized. Larger values γ intensify penalties, while smaller ones temper them,202

thus capturing both the proximity of an alternative to the ideal and its distance from203

the nadir. By framing this assessment as a similarity measure rather than a raw dis-204

tance, interpretability is enhanced and the approach seamlessly supports both benefit and205

cost criteria. The exponentiation governed by γ also accommodates non-linear preference206

patterns to model risk-averse behavior when γ > 1 or risk-seeking tendencies when γ < 1.207

Step 5: Compute Relative Closeness and Ranking208

A balancing parameter κ ∈ [0, 1] allows decision makers to control the emphasis be-209

tween being close to the ideal and far from the worst. The formula is defined as:210

RCi =
κ · Simbest

i

κ · Simbest
i + (1 − κ) · Simworst

i

(8)

This improvement address the classical RC formula used in TOPSIS. When κ = 0.5,211

the method behaves like a neutral model where equal importance is given to similarity212

to the best and separation from the worst. A value of κ > 0.5 places more emphasis on213

closeness to the ideal, which is suitable for optimistic or goal-driven decision strategies.214

On the other hand, κ < 0.5 focuses more on avoiding poor alternatives which may align215

with conservative or risk-averse thinking. The RC score remains within the range [0, 1]216

where each alternative is ranked in descending order of RCi. The top-ranked alternatives217

is considered the most suitable under the given preferences.218



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4. Computational Analyses219

This section presents four sub-sections: a numerical example demonstrating the appli-220

cation of the ARIE framework in halal supplier selection, a sensitivity analyses, as well221

as comparative analysis comparing the performance of ARIE and other MCDM methods222

and a simulation-based analysis. The numerical example was computed in Microsoft Ex-223

cel, while all subsequent analyses were performed in MATLAB software. The numerical224

data presented in this section was created by the authors for illustrative purposes in this225

study. The goal of this section is to evaluate the validity and stability of the ARIE method226

through these computational analyses.227

4.1. Numerical Example228

In the halal food production industry, the selection of an appropriate supplier is a229

critical decision that significantly affects the company’s operations, compliance with halal230

standards, and overall company’s reputation. The decision-making process must carefully231

balance multiple criteria, such as halal certification compliance, cost efficiency, delivery232

performance, product quality, and supplier reputation. This example demonstrates the233

implementation of the Adaptive Ranking with Ideal Evaluation (ARIE) method234

to evaluate and rank five suppliers (S1, S2, S3, S4, S5) based on their performance across235

these criteria. The objective is to determine the most suitable supplier by applying the236

ARIE method.237

Problem Context and Criteria238

The company has shortlisted five suppliers for an evaluation based on the following criteria:239

• Halal Compliance, C1: The ability of the supplier to meet stringent halal certification240

standards (Benefit criterion).241

• Cost, C2: The price per unit of raw materials supplied (Cost criterion).242

• Delivery Time, C3: The time taken to deliver the goods, measured in days (Target-243

type criterion).244

• Product Quality, C4: Consistency and adherence to quality standards required by245

the manufacturer (Benefit criterion).246

• Supplier Reputation, C5: The supplier’s standing in the market based on past per-247

formance and reliability (Benefit criterion).248

The weights assigned to the criteria were determined through consultations with stake-249

holders, prioritizing halal compliance and product quality, while considering cost and de-250

livery time. Table 1 summarizes the criteria and their corresponding weights.251

Initial Data and Decision Matrix252



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Table 1: Criteria and Weights for Halal Supplier Selection.

Criterion Description Type Weight (wj)

C1 Ability to meet halal certification standards Benefit 0.3
C2 Price per unit of raw materials (MYR) Cost 0.2
C3 Time taken for delivery (in days) Goal 0.15
C4 Consistency and adherence to quality standards Benefit 0.25
C5 Market perception and reliability Benefit 0.10

The decision matrix includes raw data provided by the suppliers during the selection253

process.254

Step 1: Decision Matrix Construction255

Each entry xij denotes the performance score of supplier i on criterion j, each row rep-256

resents one of the five suppliers, and each column corresponds to one of the five evaluation257

criteria.258

X = [xij ] (i = 1, . . . , 5; j = 1, . . . , 5),

Table 2 presents the 5×5 decision matrix.259

Table 2: Decision Matrix for Halal Supplier Selection.

Supplier C1 C2 C3 C4 C5

S1 8 25,000 15 9 8
S2 9 28,000 20 7 9
S3 6 22,000 25 8 7
S4 7 30,000 10 10 9
S5 10 24,000 12 9 10

Step 2: Normalization Decision Matrix260

The decision matrix is normalized using equations (2), (3), and (4) shown in Table 3.261

Table 3: Normalized Decision Matrix.

Supplier C1 C2 C3 C4 C5

S1 0.8000 0.8800 1.0000 0.9000 0.8000
S2 0.9000 0.7857 0.5000 0.7000 0.9000
S3 0.6000 1.0000 0.0000 0.8000 0.7000
S4 0.7000 0.7333 0.5000 1.0000 0.9000
S5 1.0000 0.9167 0.7000 0.9000 1.0000



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Normalization ensures that all raw scores each originally in different units and ranges262

are converted into a common, dimensionless scale so no single criterion can dominate the263

results.264

xmax
1 = 10, xmin

2 = 22, 000, xmax
3 = 25, xmin

3 = 10, xT3 = 15, xmax
4 = 10, xmax

5 = 10

The target type value, xTj , can be adjust by decision makers’ preferences like 8, 14 or265

even 21. Each raw score xij is transformed using the normalization formula that matches266

its criterion’s orientation wether benefit, cost, or target so that each values reflects whether267

higher, lower, or benchmark-proximity values are preferable.268

C1 (Max-type):

r11 =
8

10
= 0.8000, r21 =

9

10
= 0.9000,

C2 (Min-type):

r12 =
22 000

25 000
= 0.8800, r22 =

22 000

28 000
= 0.7857,

C3 (Target-type):

r13 = 1 − |15 − 15|
max{25 − 15, |10 − 15|}

= 1.0000,

r23 = 1 − |20 − 15|
max{25 − 15, |10 − 15|}

= 0.5000.

Step 3: Weighted Normalization269

The weighted-normalized matrix is given by:270

V =


0.2400 0.1760 0.1500 0.2250 0.0800
0.2700 0.1571 0.0750 0.1750 0.0900
0.1800 0.2000 0.0000 0.2000 0.0700
0.2100 0.1467 0.0750 0.2500 0.0900
0.3000 0.1833 0.1050 0.2250 0.1000


Each alternative is calculated as follows:271

For v11 = 0.30 · 0.8000 = 0.2400,

For v23 = 0.15 · 0.5000 = 0.0750

Step 4: Similarity to Ideal and Anti-Ideal Solutions272

In this step, we quantify each alternative’s proximity to both the ideal (best) and273

anti-ideal (worst) solutions using a similarity-based approach:274



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Table 4: Similarity Computation.

Alternatives Sbest Sworst

S1 4.3800 3.2361
S2 3.7857 3.3778
S3 3.1000 3.6083
S4 3.8333 3.3349
S5 4.5167 2.8778

With275

vj(max) = [0.3000, 0.2000, 0.1500, 0.2500, 0.1000],

vj(min) = [0.1800, 0.1467, 0.0000, 0.1750, 0.0700]

Similarity values computed by letting γ = 1,276

Simbest
1 =

0.2400

0.3000
+

0.1760

0.2000
+

0.1500

0.1500
+

0.2250

0.2500
+

0.0800

0.1000
= 4.3800

Simworst
1 =

0.1800

0.2400
+

0.1467

0.1760
+

0.0000

0.1500
+

0.1750

0.2250
+

0.0700

0.0800
= 3.2361

Step 5: Relative Closeness and Ranking277

Let κ = 0.5,278

RC1 =
0.5 · 4.3800

0.5 · 4.3800 + (1 − 0.5) · 3.2361
= 0.5751,

RC2 =
0.5 · 3.7857

0.5 · 3.7857 + (1 − 0.5) · 3.3778
= 0.5285

The final ranks and scores with S5 ≺ S1 ≺ S4 ≺ S2 ≺ S3 are shown in Figure 2.279

Figure 2: Final Score using ARIE Method.

Based on the figure, supplier S5 is the best option as it excels in halal compliance,280

product quality, and reputation while keeping costs and delivery schedules under control.281



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Supplier S3, on the other hand, is placed last among the alternatives indicate that it282

does not satisfy any of the standard requirements of supplier selection. By combining283

dynamic normalization with ideal/anti-ideal similarity measures and adaptive scoring,284

ARIE captures subtle performance nuances and directly reflects stakeholder priorities.285

This thorough example demonstrates ARIE’s resilience and suitability for choosing halal286

suppliers. ARIE offers a valuable tool for intricate decision-making situations requiring287

the evaluation and balancing of numerous parameters. In the following section, we explore288

how the two key ARIE parameters shape the ranking outcomes.289

4.2. Sensitivity Analysis290

This sub-section examines the sensitivity of two parameters in the ARIE method: the291

γ parameter and the κ parameter. The same data in numerical example has been used in292

this section by using MATLAB. Each sensitivity analysis focused on a different objective,293

for example, Figure 3 below focuses on the γ parameter, with the κ parameter fixed at294

0.5, and vice versa for Figure 4.295

Figure 3: Radar Plot for Sensitivity Analysis of γ using ARIE Method.

Based on the figure, the radar chart consists of radial axes representing the γ parameter,296

which ranges from 0.2 to 2. The distance from the center represents the score RCi of an297

alternative. Each line represents one alternative, which is supplier, where line that close298

to the outer edge represents higher scores which perform better than others. While line299

that close to the center edge represents lower scores which perform worse. S5 ranks first300

across all the parameter values while S3 being the last rank with different parameter301

values. S3 maintains robust and consistently perform well since it remains close to the302

outer edge. All supplier sensitive under lower γ values as the line moves inward as the303

parameter γ decreases. In the radar plot, when γ < 1 all the supplier values bunch up304

into a nearly circular shape reflecting a risk-seeking stance that softens performance gaps305

and treats most alternatives as similarly acceptable. As γ grows above 1, the supplier306

plot becomes increasingly spiky and irregular, since exponentiation amplifies even small307

differences typical of risk-averse behavior that punishes deviation from the ideal. Figure308

4 shows the impact of sensitivity parameter κ on the final score by implementing ARIE309

method.310



N. F. F. M. Fauzi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6578 13 of 24

Figure 4: Line Chart for Sensitivity Analysis of κ using ARIE Method.

Based on the figure, the κ sensitivity plot shows that all alternatives start at RC = 0311

when κ = 0 since only the anti-ideal term remains and converge at RC = 1 when κ = 1312

because the formula reduces to pure similarity to the ideal. As κ grows from 0 to 1, each RC313

curve rises smoothly, with steeper slopes for suppliers whose best-vs-worst similarity gap314

is larger. Once κ exceeds 1, the denominator’s anti-ideal component becomes negative,315

causing RC to jump above 1 and the curves to fan out: suppliers with the greatest316

performance contrast (e.g., S3 soar highest, while those with smaller contrasts (e.g., S5317

climb more modestly. This behavior underscores why κ is typically bounded in [0,1] to318

maintain RC scores within [0,1] and retain a balanced trade-off between closeness to the319

ideal and distance from the worst. Next sub-section discusses the relationship of ARIE320

method between various MCDM methods.321

4.3. Comparative Analysis322

This section analyses the ranking order between ARIE, and CRADIS, MABAC, ARAS,323

MOORA, VIKOR, TOPSIS, SAW methods using MATLAB software. The data illustrated324

in this section are based on the data in numerical example. Figure 5 illustrates a compar-325

ative analysis between the ARIE and other existing MCDM methods.326

Figure 5: Ranking Score Comparison of MCDM Methods.

Based on Figure 5, the bubble chart represents x-axis as the supplier 1 until 5, and327

y-axis as the score values of MCDM methods. Five suppliers are ranked in the figure328



N. F. F. M. Fauzi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6578 14 of 24

utilizing a variety of decision-making techniques. Each scatter shows how suppliers are329

assessed using various ranking methods. Since some points overlap because of identical330

ranking values, MATLAB’s jitter function is utilized to make sure all scatter plots are331

viewable. Although VIKOR typically ranks alternatives in ascending order, it here follows332

the descending order used by the other methods to enhance figure readability and inter-333

pretation. As the top-performing supplier, supplier 5 continuously receives the highest334

scores across the majority of approaches, while supplier 3 is still at the bottom, having335

the lowest scores across all techniques. Other MCDM systems establish similar ranking336

patterns when compared to the evaluation produced by the ARIE method. Supplier 5337

shows excellent performance since it reaches one of the highest positions in the ARIE338

rankings. Supplier 3 maintains a low ranking position as it consistently performs poorly339

in different assessments. The supplier ranking approach in ARIE establishes a more mod-340

erate grading structure than ARAS which gives steep scores to the top supplier candidate.341

ARIE maintains consistent scoring patterns which enables the method to separate superior342

performing suppliers from subpar ones without creating artificial advantages.343

For SAW, TOPSIS, CRADIS, MOORA, and MABAC methods exhibit a similar pat-344

tern to ARIE. MABAC allows negative values because it calculates the deviation of al-345

ternatives from the border approximation area. Values below the reference point result346

in negative scores [62] indicates weaker performance. Out of all the approaches that have347

been analyzed, supplier 5 is constantly recognized as the best option. The overall scores348

from MOORA and MABAC are comparatively lower, indicating that these approaches349

may use more rigorous evaluation criteria, even though SAW and CRADIS give it bet-350

ter marks. Notably, supplier 3 continuously earns the lowest score despite the variations351

in ranking methods, indicating that its poor performance is unaffected by the particular352

methodology used. As a result, ARIE delivers a consistent and balanced ranking. These353

results demonstrate the durability and consistency of the ranking techniques and guar-354

antee a trustworthy MCDM framework for supplier selection decision-making. Figure 6355

provides the comparison of Spearman and Pearson correlation coefficient for ARIE across356

MCDM methods.357

Figure 6: Spearman and Pearson Correlation Coefficients for ARIE Method.

This study used two different statistical analysis techniques, the Pearson and Spearman358

correlation coefficients, to determine the direction and strength of the association between359



N. F. F. M. Fauzi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6578 15 of 24

two methods. The Pearson correlation method identifies linear associations between meth-360

ods yet Spearman approaches method ordering without requiring assumptions of normal361

distribution [63]. The monotonic strength of correlation appears as continuous upward362

or downward changes according to the Spearman coefficient although Pearson produces a363

measure of both direction and strength [64]. The values of the correlation coefficient range364

from -1 to 1. Based on the above figure, the orange bars show Pearson (ranking values)365

correlations, the blue bars show Spearman (ranking order) correlations. The Spearman366

correlations have reached 1.0 for each method which exhibits closely matching strength to367

Pearson correlations that slightly lower at 0.9685, 0.9521 and 0.9510. The visual depicts368

correlation strength through value labels placed above each bar which simplifies compari-369

son between measures. The results demonstrate that ARIE shows strong correspondence370

with most ranking techniques, but does not match the correlation level of VIKOR. The371

proposed ARIE method stands out through its comparative analysis with the renowned372

existing MCDM methodologies presented in Table 5.373

Table 5: Comparison of ARIE Method with Existing MCDM Method.

Method Adaptability Outliers Proximity Weighting Complexity

ARIE High Balanced Tunable Moderate
CRADIS Medium Balanced Implicit Low
MABAC Medium Robust Approximate High
ARAS Stable Effective Implicit Low

MOORA Stable Stable Implicit Low
TOPSIS Medium Stable Fixed Low
VIKOR Medium Balanced Fixed Moderate

SAW Medium Vulnerable NA Low

Table 5 presents an extensive comparison of ARIE with other prominent MCDM tech-374

niques. Four main areas are the focus of the comparison: adaptability, outliers, proximity375

weighting, and complexity. The customizable parameters of ARIE make it unique among376

the methods that allow decision makers to modify the model for the best results in various377

situations. Because of its great degree of adaptability, ARIE is especially flexible for appli-378

cations that need dynamic ranking modifications. By combining ideal and anti-ideal refer-379

ences with structured boundary approximation, MABAC and CRADIS also demonstrate380

strong adaptability and work well in multi-objective optimization scenarios. MABAC381

demonstrates exceptional resilience in managing extremes and computational complexity382

by providing stability against outliers and extreme values. MOORA and ARAS offer a383

stable complexity for issues requiring fast computations.384

TOPSIS and CRADIS are able to maintain a compromise between ranking accuracy385

and computational efficiency by utilizing distance-based weighting approaches. The sys-386

tems’ methods to proximity weighting differ; ARIE is notable for its adaptability, whereas387

TOPSIS and VIKOR use fixed or structured weighting processes that improve stability.388

One important conclusion is that the changes in weight have a major influence on ranks in389

ARIE, CRADIS, MABAC, ARAS, and MOORA, where the sensitivity to criteria weights390



N. F. F. M. Fauzi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6578 16 of 24

is most noticeable. These methods are dependable in decision-making scenarios where391

accurate weighting is required due to their high sensitivity and excellent highlighting of392

important factors. Overall, the table shows that although ARIE offers a flexible and well-393

balanced framework for decision-making, each MCDM approach has unique advantages394

that allow it to be applied to various situations depending on the requirements for adapt-395

ability, outliers, weighting, and complexity. Next sub-section focuses on different decision396

matrix size, and various iterations or situations in random MCDM problems between397

ARIE and other MCDM approaches.398

4.4. Simulation-Based Analysis399

This sub-section presents a comprehensive simulation-based comparison of ARIE with400

other MCDM methods. To rigorously validate ARIE’s stability, MATLAB was used to401

generate four distinct sets of decision matrices, each with corresponding criteria weights402

and types. Criteria were classified as benefit or cost for all methods, with ARIE addi-403

tionally handling target-type criteria. Each technique was implemented using its native404

MATLAB commands; because VIKOR normally identifies the best alternative by the low-405

est score, its code was adjusted to produce a descending order ranking for consistency406

with the others. Category I comprised four alternatives and four criteria, Category II407

eight of each, Category III ten of each, and Category IV twelve of each. Figure 7 shows408

the resulting ranking scores for all methods across these four categories.409

Figure 7: Comparison of Score Values across MCDM Methods for Category I and II.

Significant differences in the ranking behavior as the number of alternatives increases410

are shown in Figure 8. Most of the approaches in Category I show an upward trend from411

Alternative 1 to Alternative 3 with a minor downturn at Alternative 4. Among all the412

methods, Alternative 3 has the highest ranking, while Alternative 1 has the lowest. All413

methods continue to show steady trends with consistent ranking order, Alternatives 3 ≺414

4 ≺ 2 ≺ 1. At the lowest point value, ARAS and ARIE’s scores are still slightly better415

than MABAC and MOORA. Although MABAC continues to produce negative results, it416

shows a noticeable improvement compared other methods, TOPSIS, SAW, and CRADIS.417

The majority of approaches in Category II show a trend of instability. With a discernible418

decrease at Alternative 4 and a high at Alternative 3, where all methods sharply increases419



N. F. F. M. Fauzi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6578 17 of 24

before falling once more. Similar steady trends are followed by TOPSIS, SAW, VIKOR,420

and ARAS, although ARIE and VIKOR show only slight variations. Out of all the options,421

MABAC has the lowest score as the method allows negative values. MOORA has the422

smallest vertical range around 0.20–0.35, suggesting it’s a very conservative method it423

never gives extreme highs or lows. ARIE on the other hand shows the largest variance,424

so it’s highly sensitive to small changes in the data or the way target-type criteria are425

specified.426

Figure 8: Comparison of Score Values across MCDM Methods for Category III and IV.

In Category III, VIKOR, MOORA and MABAC exhibits extreme fluctuations and427

drop sharply to its lowest value at Alternative 7. CRADIS and ARAS show a strong428

preference for these alternatives as the consistent peaks of the method in Alternatives 3429

and 8. TOPSIS and SAW maintain moderate variations, while ARIE remain relatively430

stable with smaller deviations across the alternatives. The lower score in MOORA and431

MABAC highlight significant methodological differences where Alternatives 8 becomes a432

common evaluation tendency among most methods. In Category IV, all methods start at433

the lowest values at Alternative 1. With regular peaks at Alternative 4, all method remain434

steady progressions. The pattern of ARAS varies slightly greater than the structure of435

CRADIS. Due to their consistency and comparatively smaller scores, ARIE, SAW and436

TOPSIS exhibit more stable ranking behavior. Minimal fluctuations shown by ARIE437

across all categories compared to other MCDM approaches. Therefore, ARIE is found to438

be consistently stable for all categories. The score values across different iterations and439

MCDM method are illustrated in Figure 9, 10, 11, and 12.440

Figure 9: ARIE and ARAS Score Performance.



N. F. F. M. Fauzi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6578 18 of 24

Figure 10: CRADIS and MABAC Score Performance.

Figure 11: MOORA and SAW Score Performance.

Figure 12: TOPSIS and VIKOR Score Performance.

All of the figures display color intensities that represent alternative rankings during441

multiple iterations of assessment. The ranking system utilizes bright colors to represent442

high scores while darker hues correspond to lower scores along with lower placement for al-443

ternatives across the methods. Based on the distribution of these color intensities, ARIE,444

MABAC, MOORA, TOPSIS and SAW methods share a similar ranking order pattern445

for all iterations. While ARAS, CRADIS and VIKOR have a similar color pattern espe-446

cially for the Alternative 3. The ARIE, TOPSIS, and MOORA scores heatmap display447

a balanced and structured ranking system with scores ranging. Alternatives 4 frequently448

receive the highest scores as shown by warm colors, meanwhile either Alternatives 1 or 8449

are the last ranked for the fourth iteration across all method. Alternative 3 consistently450

appears in bright tones, being a top favorable option in first until third and ninth and last451



N. F. F. M. Fauzi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6578 19 of 24

iterations and being a stable ranking behavior is seen in ARAS, CRADIS and VIKOR452

heatmap.453

The ARAS heatmap also presents a well-organized ranking structure compared to454

ARIE. While Alternatives 2 and 9 are consistently ranked lowest in second iteration, Al-455

ternatives 3, 4, and 7 typically receive the highest scores. The moderate range of scores456

indicates that ARIE offers a methodical approach to evaluation. TOPSIS, SAW and457

CRADIS maintain stable and balanced rankings while VIKOR demonstrates sudden ag-458

gressive ranking changes. VIKOR demonstrates better performance when alternatives459

have stark differences yet ARIE together with MABAC and CRADIS demonstrate supe-460

rior reliability in situations demanding consistent ranking results. Overall, the heatmaps461

reveal that ARIE offers a well-balanced ranking approach since the method avoids ex-462

treme fluctuations while maintaining adaptability. SAW and CRADIS provide stable and463

structured rankings as the methods are ideal for decision-making contexts that require con-464

trolled score variations. In contrast, VIKOR exhibit highly dynamic ranking behaviours465

which better suited for cases where strong differentiation among alternatives is essential.466

Thus, ARIE becomes a preferred MCDM method especially for scenarios requiring both467

ranking consistency and adaptability.468

One of ARIE’s key innovations is its unified multi-type normalization scheme, which469

seamlessly integrates benefit, cost, and explicit target-based scaling alongside the usual470

extreme-value (maximum and minimum) adjustments. ARIE then applies a parameterized471

similarity based scoring function driven by a sensitivity exponent γ to tune nonlinearity472

and a balancing coefficient κ to trade off closeness to the ideal against distance from the473

anti-ideal to quantify each alternative’s overall performance. The method automatically474

adjust its parameters γ and κ in real-time to adapt to changing between network depen-475

dencies, and varying criteria and datasets therefore delivering automatic resistance against476

outliers and high variability. ARIE implements a dynamic benchmarking framework which477

enables production of correct rankings derived from settings where rigid static methods478

fail to work and delivers results that adapt according to decision-making needs.479

5. Conclusion480

In challenging conditions, robust decision-making demands precise and trustworthy481

evaluation methods, and the ARIE method meets this need by combining a novel nor-482

malization technique with a similarity-to-ideal/anti-ideal solutions process to create a483

reliable, stable, and adaptable benchmarking framework that overcomes the limitations484

of traditional MCDM approaches; through comparative, sensitivity, and simulation-based485

analyses. ARIE method consistently outperforms established methods in stability and486

efficiency, confirming its practical value, yet its reliance on complete information can limit487

its effectiveness when data are missing. Future research should explore hybridizing ARIE488

with other MCDM techniques or embedding it within advanced fuzzy and neutrosophic set489

frameworks to broaden its applicability, with the method’s inherent adaptability under-490

scoring its robustness and laying the groundwork for next-generation decision frameworks491

capable of tackling today’s complex challenges.492



N. F. F. M. Fauzi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6578 20 of 24

Acknowledgements493

The authors would like to thank the anonymous referees of European Journal of Pure494

and Applied Mathematics for the comprehensive reading of this paper and their valuable495

comments and suggestions.496

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