© D. D. Trung, Published by EDP Sciences 2025

Nomenclature

MCDM: Multi-Criteria Decision-Making

VIKOR: Višekriterijumsko Kompromisno Rangiranje

TOPSIS: Techniques For Order Preference By Similarity To An Ideal Solution

WSM: Weighted sum method

WPM: Weighted Product Method

DNMs: Decision-makers

PSI: The Preference Selection Index

SAW: Simple additive weighting

AHP: Analytical Hierarchy Process

WASPAS: Weighted Aggregated Sum Product Assessment

MOORA: Multi-Objective Optimization By Ratio Analysis

ELECTRE: Elimination and Choice Translating Reality

RAPS: Ranking Alternatives by Perimeter Similarity

MCRAT: Multiple Criteria Ranking by Alternative Trace)

SPC: Symmetry point of criterion

APC: Average Pearson Correlation

ASC: Average Spearman Correlation

SD: Standard deviation

PV: Plurality Voting

MARCOS: Measurement Of Alternatives And Ranking According To A Compromise Solution

CRADIS: Compromise Ranking Of Alternatives From Distance To Ideal Solution

DNSAW: Double normalization simple additive weighting

DNMARCOS: Double normalization Measurement Of Alternatives And Ranking According To A Compromise Solution

DNCRADIS: Double normalization Compromise Ranking Of Alternatives From Distance To Ideal Solution

DNWPM: Double normalization Weighted Product Method

DNMCDM: Double normalization Multi-Criteria Decision-Making

RCI: Ranking consistency index

SPOTIS: Stochastic Expected Solution Point

MEPSI: Mutriss Enhanced Preference Selection Index

RAWEC: Ranking of Alternatives with Weights of Criterion

S: Second

Mm: Millimeter

USD: United States Dollar

rpm: Revolutions per minute

g/cm³ : Grams per cubic centimeter

BHN: Brinell Hardness Number

GPa: Gigapascal

MPa: megapascal

W/mk: Watts per meter-kelvin

μm/m-k: Micrometers per meter-kelvin

J/gC: Joules per gram degree Celsius

1 Introduction

Since its emergence in 1960, MCDM has evolved continuously and become a widely used method in various fields [1] and has attracted considerable interest from researchers in diverse fields owing to its broad applicability [2,3]. Conversely, the implementation of numerous MCDM methods typically begins with the transformation of the data in the decision matrix into a standardized unit, a process commonly known as normalization [4].

The selection of the ideal normalization technique for an MCDM approach has become a compelling research topic, particularly with the development of big data. In the past, various attempts have been made to examine the applicability of various normalization techniques [4]. The initial analyses of the effect of normalization techniques on calculation results were carried out by Peldschus and Börner in the 1980s [5,6]. Additional studies have been carried out by several researchers. [7] examined the suitability of the logarithmic normalization technique for the VIKOR and TOPSIS methods. [1] applied five various normalization procedures to the WSM, WPM, and TOPSIS methods to demonstrate the effects of various procedures. [8] studied suitable normalization techniques in combination with the PSI method, finding that four out of twelve DNMs were suitable for combination with the PSI method. [9] examined the applicability of various normalization techniques using the ROV method. Similarly, studies investigating the effectiveness of various normalization techniques for the SAW, AHP, and DMCDM methods are noteworthy [10–12]. In another study, [13] explored the applicability of different normalization procedures for the TOPSIS method. [14] investigated the change in ranking performance based on modifications in the normalization technique for the WASPAS method. [15] compared the results of the MOORA method based on various normalization techniques. [16] compared the effects of various normalization techniques in the Hellwig method. [17] investigated the effectiveness of various normalization techniques for the single-valued m-polar fuzzy (mfs) ELECTRE-I algorithm in their study on the robot selection problem. In their work, [18] developed a framework for applying the mFs ELECTRE-I algorithm and examined the impact of combining AHP and Entropy weight methods with normalization strategies [19] explored the suitability of alternative data normalization techniques for the RAPS method. The studies summarized above have examined the effectiveness of the selection of normalization techniques for many different MCDM methods. However, no study has provided a comprehensive review of the various normalization procedures for the MCRAT method.

The MCRAT method, proposed by [20], is a relatively new approach. This technique is based on the idea of perimeter similarity to establish the ranking of alternatives, and this method is employed as an effective tool in complex and multidimensional decision-making processes. Despite being a relatively new approach, it is frequently preferred by researchers across various topics [21–25]. In the literature, there is only one study that examines the suitability of different data normalization techniques for the MCRAT method. [26] tested the suitability of four different normalization techniques (max-min, vector, max linear, enhanced accuracy, z-score) for the MCRAT method solely by comparing them with different MCDM methods. They concluded that the vector and enhanced accuracy techniques are suitable for the MCRAT method. Testing the suitability of normalization techniques for MCDM methods solely through comparative analysis is a limited approach. The aim of this study is to apply various normalization techniques in conjunction with the MCRAT method and compare the effectiveness of the ranking results based on different metrics. In this regard, the efficacy of nine distinct normalization techniques has been examined using various metrics within the framework of the SPC-MCRAT model.

The key contributions of this study to literature are as follows: i) The SPC-MCRAT model has been used for the first time in literature. ii) The study provides a new perspective that fills the gap in the literature by comparing the effects of nine various normalization techniques on the results of the MCRAT method. iii) Identifying the normalization technique that most enhances the effectiveness of the MCRAT method offers a new approach to improving the efficiency of decision-making processes. iv) The suitability of normalization techniques has been examined through a comprehensive five-step framework, which includes various distance measures, allowing for more reliable and consistent evaluation of the results. v) The applicability of the double normalization technique, selected as the most effective method for the MCRAT, has been demonstrated through comparative analysis, providing valuable insights into the technique’s performance. vi) Analyses have been conducted based on four case studies with varying numbers of alternatives/criteria, benefit/cost criteria, datasets, and problem types, presenting a robust and comprehensive decision-making framework.

After the introduction, the second section presents the methods used in the study, while the third section is dedicated to the application. The fourth section introduces the framework used to evaluate the effectiveness of the normalization techniques. The fifth section discusses the results obtained, and the final section is dedicated to the conclusion.

Fig. 1 AMCD Mframe work for optimal normalization selection for the MCRAT.Source: Created by the authors.

2 Methods and techniques

This section will provide an explanation of the techniques employed in the study. The data normalization techniques, criterion weighting method, and the MCDM approach used for ranking the alternatives are summarized in the following subsections.

2.1 Data normalization techniques

This section presents the normalization techniques for the MCRAT method, along with their corresponding formulations, as shown in Table 1. The terms beneficial (B) and cost (C) refer to criteria where a higher value is desirable or not, respectively.

As demonstrated in Table 1, this study seeks to encompass a variety of normalization techniques identified in the literature. This study has adopted normalization techniques commonly utilized in literature. Furthermore, alongside traditional normalization techniques, the study also aims to evaluate the efficancy of the double normalization technique.

2.2 Weighting method

This section outlines the steps involved in the SPC method employed for criterion weighting [34].

Step 1: Create the decision matrix

Step 2: Calculate the SPC (SPC_j) using equation (1)

$$SP{C}_j=\frac{\min \left(x_{ij}\right)+\max \left(x_{ij}\right)}{2};\text{i}=1,2,\dots ,\text{m};\forall \text{j}\in [1÷\text{n}].$$(1)

Step 3: Create the matrix of absolute distances as shown in equation (2).

$$D={\left|d_{ij}\right|}_{mxn}=\left[\begin{array}{cccc}\left|x_{11}-SP{C}_1\right|& \left|x_{12}-SP{C}_2\right|& \cdots & \left|x_{1n}-SP{C}_n\right|\\ \left|x_{21}-SP{C}_1\right|& \left|x_{22}-SP{C}_2\right|& \cdots & \left|x_{2n}-SP{C}_n\right|\\ \cdots & \cdots & \cdots & \cdots \\ \left|x_{m1}-SP{C}_1\right|& \left|x_{m2}-SP{C}_2\right|& \cdots & \left|x_{mn}-SP{C}_n\right|\end{array}\right].$$(2)

Step 4: Construct the matrix of symmetry moduli using equation (3).

$$R={\left[r_{ij}\right]}_{m\times n}=\left[\begin{array}{cccc}\left|\frac{{\text{∑}}_{i=1}^m{d}_{i1}}{m\times x_{11}}\right|& \left|\frac{{\text{∑}}_{i=1}^m{d}_{i2}}{m\times x_{12}}\right|& \cdots & \left|\frac{{\text{∑}}_{i=1}^m{d}_{in}}{m\times x_{1n}}\right|\\ \left|\frac{{\text{∑}}_{i=1}^m{d}_{i1}}{m\times x_{21}}\right|& \left|\frac{{\text{∑}}_{i=1}^m{d}_{i2}}{m\times x_{22}}\right|& \cdots & \left|\frac{{\text{∑}}_{i=1}^m{d}_{in}}{m\times x_{2n}}\right|\\ \cdots & \cdots & \cdots & \cdots \\ \left|\frac{{\text{∑}}_{i=1}^m{d}_{i1}}{m\times x_{m1}}\right|& \left|\frac{{\text{∑}}_{i=1}^m{d}_{i2}}{m\times x_{m2}}\right|& \cdots & \left|\frac{{\text{∑}}_{i=1}^m{d}_{in}}{m\times x_{mm}}\right|\end{array}\right].$$(3)

Step 5: Compute the symmetry modulus for each criterion using equation (4).

$$Q={\left[q_{1j}\right]}_{1\times n}=\left[\begin{array}{cccc}\frac{{\text{∑}}_{i=1}^m{r}_{i1}}{m}& \frac{{\text{∑}}_{i=1}^m{r}_{i2}}{m}& \dots & \frac{{\text{∑}}_{i=1}^m{r}_{in}}{m}\end{array}\right].$$(4)

Step 6: Determine the weights of criteria using equation (5)

$$W={\left[w_{1j}\right]}_{1\times n}=\left[\begin{array}{cccc}\frac{q_1}{{\text{∑}}_{j=1}^n{q}_j}& \frac{q_2}{{\text{∑}}_{j=1}^n{q}_j}& \dots & \frac{q_j}{{\text{∑}}_{j=1}^n{q}_j}\end{array}\right].$$(5)

2.3 MCDM method

The steps for scoring and ranking alternatives using the MCRAT method are as follows [20]:

Step 1: Decision matrix is normalized by equations (6) and (7).

$$r_{ij}=\frac{x_{ij}}{ma{x}_i\left(x_{ij}\right)}B$$(6)

$$r_{ij}=\frac{mi{n}_i\left(x_{ij}\right)}{x_{ij}}C.$$(7)

Step 2: The weighted normalized decision matrix is constructed by equation (8).

$$u_{ij}=w_j\ast r_{ij}.$$(8)

Step 3: The optimum alternative is determined by equation (9).

$$q_i=\left(\max \left(u_{ij}\right)|1\le j\le n\right),{\forall }_i\in [1,2,\dots ,m].$$(9)

The best alternative is expressed using the following set.

$$Q=\left\{q_1,q_2,\dots ,q_j\right\},\text{j}=1,2,\dots ,\text{n}.$$

Step 4: The optimal alternative is divided into two subsets or components.

$$Q=Q^{max}\cup Q^{min}.$$(10)

The optimal alternative is expressed as:

$$\begin{array}{l}\text{Q}=\left\{{\text{q}}_1,{\text{q}}_2,\dots ,q_{\text{k}}\right\}{{\displaystyle \cup }}^{\text{}}\left\{{\text{q}}_1,{\text{q}}_2,\dots ,{\text{q}}_{\text{h}}\right\};\hfill \\ \text{k}+\text{h}=\text{j}\hfill \end{array}.$$

Step 5: Alternatives are separated by equation (11).

$$U_i=U_i^{max}\cup U_i^{min},{\forall }_i\in [1,2,\dots ,m]$$(11)

$$U_i=\left\{u_{i1},u_{i2},\dots ,u_{ik}\right\}\cup \left\{u_{i1},u_{i2},\dots ,u_{ih}\right\},{\forall }_i\in [1,2,\dots ,m]$$

Step 6: The magnitude of each component of the optimum alternative is computed with equations (12) and (13).

$$Q_k=\sqrt{q_1^2+q_2^2+\mathrm{..}+q_k^2}$$(12)

$$Q_h=\sqrt{q_1^2+q_2^2+\mathrm{..}+q_h^2}.$$(13)

Each alternative is treated with the same approach.

$$U_{ik}=\sqrt{u_{i1}^2+u_{i2}^2+\mathrm{..}+u_{ik}^2}$$(14)

$$U_{ih}=\sqrt{u_{i1}^2+u_{i2}^2+\mathrm{..}+u_{ih}^2}.$$(15)

Alternatives are ranked by considering the trace of a matrix.

Step 6.1. T_i matrix is created

$${\text{T}}_{\text{i}}={\text{FxG}}_{\text{i}}=\left[\begin{array}{cc}{t}_{11;i}& 0\\ 0& t_{22;i}\end{array}\right],{\forall }_i,\in [1,2,\dots ,m].$$(16)

Step 6.2. Alternatives are ranked

$$\mathrm{tr}\left(T_i\right)=t_{11;i}+t_{22;i},{\forall }_i,\in [1,2,\mathrm{..},m].$$(17)

The decreasing order of (T_i) is considered when ranking the alternatives.

3 Case study applications

This section presents various case studies, each with varying numbers of alternatives and criteria, as well as different proportions of beneficial and cost criteria, to assess the stability of the MCRAT method across different normalization techniques. The process of the study is presented in Figure 1.

3.1 Case study 1

In this section, the performances of MCRAT combinations based on different normalization techniques are evaluated based on the selection of 17 metal drilling alternatives [35]. Each alternative was characterized by six cost criteria: “drilling time” (C1), “entry burr height” (C2), “exit burr height” (C3), “entry burr thickness” (C4), “exit burr thickness” (C5), and “surface roughness” (C6). The data for these alternatives was compiled in Table 2. The MCRAT results are presented in Figure 2.

The results presented in Figure 2 indicate that normalization techniques have a substantial influence on the outcomes. Notably, the results obtained with the N4 technique show distinct differences from the other rankings, particularly with the ranking of alternative D1, which contrasts with the results of most other techniques. On the other hand, rankings derived from the other normalization techniques exhibit only minor variations. The highest correlation is observed between N1 and N5 (r=1), while the lowest correlation is between N4 and N6 (r=.254). These findings underscore the importance of selecting appropriate normalization techniques in MCDM approaches.

Fig. 2 Ranking of alternatives based on various normalization techniques (Case 1).

3.2 Case study 2

In this study, eight types of CNC machines, labeled CNC-L1 to CNC-L8, were ranked using the SPC-MCRAT model based on nine normalization techniques. Each machine was characterized by seven criteria, including price (C1), spindle speed (C2), maximum workpiece diameter (C3), rapid traverse rate in the X-axis (C4), rapid traverse rate in the Z-axis (C5), machine center height (C6), and maximum workpiece length (C7), respectively, where C1 is a cost criterion, and the remaining six criteria are benefit criteria. Table 3 summarizes the information about the eight CNC machines to be ranked [36]. The SPC-based MCRAT results are presented in Figure 3.

According to Figure 3, while the results derived from employing various normalization techniques remain generally consistent, the rankings generated using the N7 normalization technique show significant deviations from the others. The rankings obtained with five normalization techniques (N1, N3, N5, N8, and N9) are identical. The lowest correlation of r=.377 was observed between N6 and N7.

Fig. 3 Ranking of alternatives based on various normalization techniques (Case 2).

3.3 Case study 3

In this case, seven types of piston materials, labeled A1 to A7, were ranked using the SPC-MCRAT model based on various normalization techniques [37]. Each material was characterized by ten criteria including density (C1), hardness (C2), modulus of elasticity (C3), tensile strength (C4), tensile yield strength (C5), relative elongation (C6), thermal conductivity (C7), thermal diffusivity (C8), specific heat (C9), and fatigue strength (C10), respectively, where C6 and C8 are cost criteria, and the remaining eight criteria are benefit criteria. Table 4 summarizes the information about the alternatives and criteria. The MCRAT rankings obtained based on various normalization techniques are provided in Figure 4.

The results presented in Figure 4 shows that there are mostly minor changes between the different rankings. On the other hand, the correlation coefficient between the N1-N5 and N3-N4 normalization techniques is 1. The lowest correlation was observed between the N6 and N9 techniques (r = 0.948).

Fig. 4 Ranking of alternatives based on various normalization techniques (Case 3).

3.4 Case study 4

In this case, ten sets of numbers, labeled A1 to A10, were ranked. Each set was described by seven criteria, C1 to C7, where C1, C2, and C3 are cost criteria, and the remaining four are beneficial criteria. Table 5 summarizes the information about the alternatives and criteria [38].

The results presented in Figure 5 indicate that the rankings obtained using the N7 technique significantly differ from those of the other techniques. On the other hand, the rankings for N1 and N5 are identical. The lowest correlation is observed between the N7 and N8 techniques (r = −.969). The variations in rankings derived from different techniques highlight the importance of normalization selection, while also indicating that certain techniques can negatively impact the performance of the MCRAT method.

Fig. 5 Ranking of alternatives based on various normalization techniques (Case 4).

4 Evaluation framework for selecting the optimal combination

In the previous section, analyses were conducted based on four distinct case studies with different characteristics to determine the extent to which various normalization techniques affect the performance of the MCRAT method. According to the overall results, for Case 1 and Case 2, the N6 and N7 normalization techniques were found to reduce the performance of the MCRAT method, while in Case 3, N6 and in Case 4, N6 and N9 were identified as techniques that degrade performance. The determination that these techniques negatively impact the performance of the MCRAT method, while other techniques are relatively more suitable, was made solely based on Spearman correlation results. However, relying solely on the Spearman correlation coefficient to determine the most appropriate technique for the MCRAT method is a limited approach and insufficient. In this context, this section employs various metrics to select the normalization technique that most enhances the performance of the MCRAT method. The five-step process can be summarized as follows:

Step I: Determining the Average Pearson Correlation (APC)

The approach followed by [13] has been adopted, and in this section, the Pearson correlation for all rankings obtained based on different normalization techniques was calculated using equation (18).

$$r=\frac{{\displaystyle \sum \left(x_i-\overline{x}\right)}\left(y_i-\overline{y}\right)}{(N-1){\sigma }_x{\sigma }_y}.$$(18)

Step II: Determining the Average Spearman Correlation (ASC)

In the second step, the Spearman correlation was calculated using equation (19) [39].

$$r_s=1-6\frac{{\displaystyle {\sum }_{i=1}^m{D}_i^2}}{m\left(m^2-1\right)}.$$(19)

Step III: Calculation of SD

In the third step, to evaluate the suitability of the different rankings, the SD was calculated using equation (20) [10].

$$STD=\sqrt{\frac{{\displaystyle {\sum }_{\text{i=1}}^{\text{p}}(X_1}-\overline{\text{X}}{)}^2}{q-1}}.$$(20)

Step IV: The calculation of various distance measurements

In the final step, various distance measures were used to assess the normalization techniques. Distance measures are among the most used methods in the normalization process. In this context, the Minkowski measures (Manhattan, Euclidean, Chebyshev) and the Lorentzian distance measure, as described below, were used to evaluate the suitability of different rankings.

Manhattan

$$(p=1):d(x,y)={\displaystyle {\sum }_{i=1}^n\left|x_i-y_i\right|}$$(21)

Euclidean

$$(p=2):(x,y)=\sqrt{{{\displaystyle {\sum }_{i=1}^n(x_i-y_i)}}^2}$$(22)

Chebyshev

$$(p=\infty ):d(x,y)=\underset{i}{\max }\left(\left|x_i-y_i\right|\right)$$(23)

Lorentzian

$$\text{ d}lorantzian={\displaystyle {\sum }_{j=1}^{n}In}\left(1+\left|x_i-y_i\right|\right).$$(24)

Step V: The application of Plurality Voting (PV)

To determine the technique ranked first, the results obtained in Steps I-IV were finalized using the PV method, with equations (25)–(27) applied [40].

$$f\left(a_{ij}\right)=\{\begin{array}{l}1 if a_{ij}=1\hfill \\ 0 otherwise\hfill \end{array}$$(25)

$$A_j={\displaystyle \sum _{i=1}^{m}f}\left(a_{ij}\right)$$(26)

$$A_{j*}=\underset{j}{\text{max}}\left\{A_j\right\}.$$(27)

4.1 Application of the proposed evaluation framework for Case 1

To identify the normalization technique that most effectively improves the performance of the MCRAT method, steps I-V was sequentially applied, and the following results were obtained. The higher the values obtained with the seven approaches used, the better the outcome [12].

Based on the results obtained from the seven different metrics presented in Table 6, among the nine normalization techniques symbolized by N1-N9, the N9 (Double Normalization) technique was identified as the one that most enhances the performance of the MCRAT method. A detailed analysis of the metric results revealed that the rankings based on the Lorentzian and Manhattan distance measures were identical, as were the rankings for Euclidean and SD, while the ASC and APC results differed from the other measures. Accordingly, in the Spearman and correlation ranking, the N7 normalization technique ranked highest, while N9 ranked first according to the other measure results. Based on the result derived from the PV method, N9 was determined to be the most suitable technique, with N8 ranked second and N7 ranked last.

4.2 Application of the proposed evaluation framework for Case 2

A similar application was carried out for Case 1, and the results obtained using various metrics are presented in Table 7.

According to the results in Table 7, similar to Case 1, the N9 technique ranked first in the PV results. In the ranking based on the seven different metrics, N9 exhibited the highest performance, followed by N6. N7, on the other hand, was the technique that most reduced the performance of the MCRAT method. While the normalization technique ranked first remained consistent across all measurements, the Spearman and Pearson rankings generally differed from the results of the other metrics.

4.3 Application of the proposed evaluation framework for Case 3

A five-step process was applied based on the rankings presented in Figure 4, leading to the results shown in Table 8.

Based on the results presented in Table 8, similar to the previous two case studies, the N9 technique was identified as the most appropriate for the MCRAT method. In a similar fashion, the Spearman and Pearson rankings generally differed from the results of the other metrics. Overall, N9 ranked first, followed by N2, while N6 was identified as the technique that most reduced performance.

4.4 Application of the proposed evaluation framework for Case 4

A five-step process (Step I-V) was applied based on the rankings presented in Table 9, leading to the results shown[CE3] in Table 17.

According to the results presented in Table 9, the common outcome across all metric rankings is that the N9 technique ranks first. Based on the final integrated results obtained using the PV method, N9 emerged as the highest performing technique, with N2 ranked second. Meanwhile, N3 ranked first among the normalization techniques deemed unsuitable for the MCRAT method.

5 Discussion

MCDM is a process that requires decision-makers to choose the best alternative based on various, and sometimes conflicting, criteria. In such problems, different criteria may be expressed using various units of measurement, and in this case, normalization is necessary to ensure accurate comparisons during the decision-making process. The choice of different normalization techniques used in MCDM processes can directly affect the quality of the results [11]. In this context, the selection of a normalization technique and the process of making that selection are crucial for obtaining accurate and reliable results in the field of MCDM. In this study, a five-step process was followed based on four different case studies to determine the most suitable normalization combination for the MCRAT method. Initially, MCRAT results were obtained using nine normalization techniques included in the analysis. The common outcome across the four case studies was that the selection of the normalization technique has a considerable impact on the MCRAT ranking outcomes, with SPC-MCRAT results showing a fluctuating pattern across the nine techniques. These findings are consistent with the results presented in example studies from the literature, which highlight the impact of various normalization techniques on MCDM outcomes [9–11, 29, 41]. In Section 3, only the Spearman correlation was calculated for the obtained rankings, allowing for the observation of similarities and differences between the rankings. However, the Spearman correlation coefficient results alone are insufficient to determine which normalization technique most enhances the performance of the MCRAT method, and which technique should be definitively excluded. Therefore, as previously stated, a five-step process based on seven different metrics was followed to identify the normalization techniques that are suitable and unsuitable for the MCRAT method.

In this context, the SPC-MCRAT rankings obtained based on the nine normalization techniques were used to sequentially derive ASC, APC results, calculate the SD, and evaluate the effectiveness of the rankings using Minkowski measures and the Lorentzian distance measure. In the final stage, the PV approach was applied to integrate the rankings obtained from seven different metrics and derive a final result. The results based on five different case studies can be summarized as follows: For Case 1, the technique that most enhanced the performance of the MCRAT method was N9, followed by N8. According to the PV final result, N7 was the technique that most reduced the performance of the MCRAT method. For Case 2, N9 ranked first, with N6 in second place. N7 was the technique that should be avoided, ranking first among the unsuitable techniques. For Case 3, similar to the other results, N9 was the technique that most improved the MCRAT performance, with N2 following. N6, however, was the technique that reduced performance most. For Case 4, the technique that most enhanced performance was N9, with N2 ranking second. N3, however, was ranked first among the normalization techniques unsuitable for the MCRAT method, followed by N7. When considering the results of the four case studies, several common conclusions can be drawn. The techniques that most improved the performance of the MCRAT method included N9, N2, N8, and N4, while N7, N3, N5, and N6 were among the techniques that reduced performance. However, it is important to note that the results obtained using MCDM methods are sensitive to various factors such as data type, normalization, and weighting coefficients [42]. Overall, N9 was the technique that most improved performance, while N7 was the technique that reduced it most. The procedures followed in this study, which involve various metrics, are similar to those employed in some studies in literature that evaluate the effectiveness of normalization techniques, where successful results have been achieved [9–11, 13].

The most significant common finding in this study is that the optimal normalization technique for the MCRAT method is N9. Recently, the application of double normalization in MCDM methods has become increasingly prevalent in literature, and it has been observed that the double normalization technique enhances the performance of MCDM methods in case studies when compared to classical normalization techniques [33, 43–45]. In this study, a two-stage comparative analysis was conducted to more clearly demonstrate the effectiveness of the double normalization technique for the MCRAT method. In the first stage, the classical MCRAT method was compared with the SAW, MARCOS, CRADIS, and WPM methods, and the results are summarized in Table 10. The selection of methods was influenced by their frequent use in literature and their previous integration of double normalization into the algorithms of these methods. In the second stage, the DNMCRAT method was compared with the DNSAW, DNMARCOS, DNCRADIS, and DNWPM methods (Tab. 11).

When comparing the alternative rankings using the Spearman correlation coefficient (Tab. 10), it was shown that there are minor differences between the methods. Each MCDM technique has distinct functionality, characteristics, and applicability that directly influence the decision-making process, the evaluation method, and the final ranking of alternatives [7].

The average Spearman correlation coefficients for all methods were calculated and presented in the last column of Tables 10 and 11. Based on this, for Case 1, the average value of DNMCRAT is higher than that of the MCRAT method, while for Case 4, they are equal. In the other cases, the average values are very close to each other. The comparative results presented above demonstrate the effectiveness of the DNMCRAT method.

6 Conclusion

The anatomy of each MCDM method can be restructured through changes in the weighting process, aggregation process, and normalization technique [7]. The choice of data normalization technique significantly impacts the effectiveness of applying MCDM methods. This study aims to examine the effectiveness of different normalization techniques for the MCRAT method. In this context, after determining the criterion weights using the SPC technique, the suitability of nine known normalization techniques for the MCRAT method has been evaluated based on four different case studies. A five-step process, based on various metrics, was adhered to determine the most appropriate technique.

According to the SPC-MCRAT method results, the common conclusion based on different case studies is that normalization techniques have an impact on MCDM outcomes. To determine which technique most enhances the performance of the MCRAT method, a five-step approach was applied. Although the results obtained from different case studies varied at times, the key common finding was that N9 was the technique that most improved performance, while N7 was the technique that most reduced it. To test the applicability of the double normalization technique and demonstrate its suitability for the MCRAT method, the classical MCRAT method was first compared with the SAW, MARCOS, CRADIS, and WPM methods, revealing a high correlation between the methods. In the second stage, the DNMCRAT method was compared with the DNSAW, DNMARCOS, DNCRADIS, and DNWPM methods, and similarly, a high correlation was found between DNMCRAT and the other DNMCDM methods. This research demonstrates that the application of double normalization ensures stability in the decision-making process. In this regard, it is expected that the applicability of double normalization for other MCDM techniques will provide valuable insights for researchers.

The limitations of the study can be expressed as follows: The effectiveness of normalization techniques with the MCRAT method has been evaluated based only on four different case studies. This may limit the generalizability of the results. Furthermore, only nine normalization techniques have been considered; however, there is wider range of normalization techniques available in the literature, and evaluating these techniques could expand the scope of the study. Methodologically, while a five-step process was followed, the application of measurement metrics such as RCI could provide a broader perspective. Future studies could test this approach on new MCDM methods, such as SPOTIS, MEPSI, and RAWEC, to examine the impact of normalization techniques across different methodologies.

Funding

None.

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper

Data availability statement

Not applicable.

Author contribution statement

Conceptualization, D.D.T., T.V.D. and H.X.T.; Methodology, D.D.T., T.V.D., N.E. and H.X.T; Software, D.D.T., T.V.D.; Validation, D.D.T., N.E.; Formal Analysis, D.D.T., T.V.D., N.E. and H.X.T.; Investigation, H.X.T.; Resources, D.D.T., T.V.D.; Data Curation, D.D.T; Writing – Original Draft Preparation D.D.T., N.E; Writing – Review & Editing, D.D.T., T.V.D. and H.X.T.; Visualization, N.E..; Supervision, D.D.T.; Project Administration, T.V.D.; Funding Acquisition, H.X.T.

References

All Tables

All Figures

	Fig. 1 AMCD Mframe work for optimal normalization selection for the MCRAT.Source: Created by the authors.
In the text

	Fig. 2 Ranking of alternatives based on various normalization techniques (Case 1).
In the text

	Fig. 3 Ranking of alternatives based on various normalization techniques (Case 2).
In the text

	Fig. 4 Ranking of alternatives based on various normalization techniques (Case 3).
In the text

	Fig. 5 Ranking of alternatives based on various normalization techniques (Case 4).
In the text