An investigation to offer conclusive recommendations on suitable benefit/cost criteria-based normalization methods for TOPSIS

Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a popular multi-criteria decision-making method that ranks the available alternatives by examining the ideal-positive and ideal-negative solutions for each decision criterion. The first step of using TOPSIS is to normalize the presence of incommensurable data in the decision matrix. There are several normalization methods, and the choice of these methods does affect TOPSIS results. As such, some efforts were made in the past to compare and recommend suitable normalization methods for TOPSIS. However, such studies merely compared a limited collection of normalization methods or used a noncomprehensive procedure to evaluate each method's suitability, leading to equivocal recommendations. This study, therefore, employed an alternate, comprehensive procedure to evaluate and recommend suitable benefit/cost criteria-based normalization methods for TOPSIS (out of ten methods extracted from past literature). The procedure was devised based on three evaluation metrics: the average Spearman's rank correlation, average Pearson correlation, and standard deviation metrics, combined with the Borda count technique.• The first study examined the suitability of ten benefit/cost criteria-based normalization methods over TOPSIS.• Users should combine the sum-based method and vector method into the TOPSIS application for safer decision-making.• The maximum method (version I) or Jüttler's-Körth's method has an identical effect on TOPSIS results.


a b s t r a c t
Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a popular multi-criteria decision-making method that ranks the available alternatives by examining the ideal-positive and ideal-negative solutions for each decision criterion. The first step of using TOPSIS is to normalize the presence of incommensurable data in the decision matrix. There are several normalization methods, and the choice of these methods does affect TOPSIS results. As such, some efforts were made in the past to compare and recommend suitable normalization methods for TOPSIS. However, such studies merely compared a limited collection of normalization methods or used a noncomprehensive procedure to evaluate each method's suitability, leading to equivocal recommendations. This study, therefore, employed an alternate, comprehensive procedure to evaluate and recommend suitable benefit/cost criteria-based normalization methods for TOPSIS (out of ten methods extracted from past literature). The procedure was devised based on three evaluation metrics: the average Spearman's rank correlation, average Pearson correlation, and standard deviation metrics, combined with the Borda count technique.
• The first study examined the suitability of ten benefit/cost criteria-based normalization methods over TOPSIS. • Users should combine the sum-based method and vector method into the TOPSIS application for safer decision-making.

Introduction
A multi-criteria decision-making (MCDM) technique enables decision-makers to systematically evaluate and rank a finite group of alternatives based on various conflicting decision criteria [1] . A wide range of popular MCDM techniques are available for solving complex decision problems, including TOPSIS, the Analytic Hierarchy Process (AHP) and the Analytic Network Process (ANP).
However, the TOPSIS technique, originally developed by Hwang and Yoon in 1981 [2] , holds some notable merits compared to AHP and ANP. For example, TOPSIS is relatively easy to understand and apply since it has a straightforward computational procedure and uses a simple yet logical principle in analyzing the preference over alternatives [3] . In contrast, AHP and ANP can be more complex and require greater expertise for the appropriate application, particularly when more criteria or alternatives are present [4] . Besides, AHP and ANP require pairwise comparisons between criteria or alternatives, which can be time-consuming [5] and challenging to maintain an acceptable degree of consistency in the comparisons made [6] . TOPSIS, on the other hand, does not require pairwise comparisons and hence is less demanding in the context of data collection and processing [7] .
As such, more and more academic researchers and practitioners are magnetized to employ TOPSIS in dealing with complex decision problems. Indeed, in the past, the method was applied to MCDM problems that occurred in domains like water resources management [8] , renewable energy [9] , higher education [10] , manufacturing [11] , agriculture [12] , pharmaceutical [13] , to name a few. The use of TOPSIS has also been considered in the development of various recommender systems, such as the hotel recommender system [14] , the sports venue recommender system [15] , and the job training recommendation system [16] .
The technique, in principle, identifies the positive-ideal and negative-ideal solutions for each decision criterion and ranks the alternatives by comparing their geometric distance from the positive-ideal and negative-ideal solutions [17] . The technique ensures the alternative with the shortest distance from the positive-ideal solution and the farthest distance from the negative-ideal solution tops the rank.
On the other hand, like many other MCDM techniques, the core input for TOPSIS is the decision matrix that encapsulates the data of alternatives across different decision criteria [18] . Assume an MCDM problem in which i = { A 1 , A 2 , … , A m } represents the alternatives under consideration and = { 1 , C 2 , … , n } denotes the set of decision criteria. The decision matrix can then be expressed in a general form as (1), where denotes the data value of alternative over criterion .
In reference to (1) , the criteria in a decision matrix usually carry different units of measurement [19] . For instance, in a drone selection problem, maximum speed, camera quality, battery power, and price are measured in metres per second, megapixels, milliamperehours, and dollars, respectively. As such, the data in the decision matrix should be converted into a commensurable unit to allow the decision-makers to make a logical data comparison across different criteria [20] . Indeed, the use of many MCDM techniques, including TOPSIS, begins with a procedure for transforming the data in the decision matrix into a standard unit. This procedure is widely known as normalization. The procedure eliminates unit differences and converts the data values across all criteria to a standard scale [21] .

Literature review and motivation of the study
There are numerous normalization methods available in the literature. These methods can generally be divided into two categories, i.e., the benefit/cost criteria (BCC)-based methods and non-BCC-based methods [22] . However, the scope of this study is limited to benefit/cost criteria (BCC)-based methods. It is because these methods appear as a more preferred choice for most of the MCDM problems since they consider whether a criterion is a benefit or cost criterion before normalization -the benefit criterion implies that the higher the value, the better, whereas the cost criterion means that the lower the value, the better [ 23 , 24 ]. As such, using a BCC-based method for an MCDM problem permits a logical comparison of an alternative across different criteria.
Having said that, it is known that associating different BCC-based normalization methods with an MCDM technique can deliver deviating results [25] . Therefore, choosing the ideal normalization methods for an MCDM technique has emerged as an intriguing topic of investigation. Various efforts were made in the past to compare the suitability of a normalization method, and Table 1 presents a list of such comparative studies and the BCC-based methods compared. The MCDM technique(s) considered in each study is (are) also presented in Table 1 . The formulas for each method listed in Table 1 , for both benefit and cost criteria, are denoted by (2) - (11), where is the data value of alternative with respect to criterion , is the normalized data value of alternative over criterion , ma x ( ) is the maximum data value of criterion , and mi n ( ) is the minimum data value of criterion .
, for a benef it cr iter ion , for a cost cr iter ion (7) , for a benef it cr iter ion , for a cost cr iter ion , for a benef it cr iter ion , for a cost cr iter ion N9 ; = ⎧ ⎪ ⎨ ⎪ ⎩ MCDM technique examined for its suitable normalization methods. To be precise, around 40% of the studies in Table 1 focused on TOPSIS, and one of the earliest studies testing TOPSIS with different normalization methods was made by Milani et al. [27] . The test involved N1, N3, N4, and N6, but it only revealed that the use of different normalization methods might lead TOPSIS to deliver varying alternative ranks; the study did not apply any evaluation metrics to compare and decide the most suitable normalization methods.
In contrast to Milani et al. [27] , Chakraborty and Yeh [28] compared N1, N2, N3, and N6 using an evaluation metric called Ranking Consistency Index (RCI). With the aid of RCI, they finally recommended TOPSIS's default normalization method, i.e., N6, as the most suitable method, thanks to its ability to produce robust and reliable alternative ranks. A similar study was later conducted by Liao et al. [29] , with the same set of normalization methods and evaluation metric, only to reconfirm the recommendation made by Chakraborty and Yeh [28] . Liao et al. [29] also reported that N6 has the advantage of effectively dealing with MCDM problems with various decision matrix sizes, data ranges, and attribute types.
Likewise, the comparison carried out by Celen [30] using RCI once again recommended N6 as the best combination for TOPSIS. Interestingly, his results suggest that N2 or N3 can be a suitable substitute for N6 when applying TOPSIS. The researcher found both N2 and N3 delivered ranks consistent with N6 when he measured the financial performance of 13 deposit banks in Turkish.
In the following year, Vafaei et al. [35] did a slightly different study in which they utilized the average Spearman's rank correlation coefficient as a metric to determine the normalization method that produces the most reliable alternative ranks for TOPSIS. The switch of average Spearman's rank correlation with the common RCI metric is undoubtedly helpful because the computation involving the latter metric becomes exponentially more difficult, especially when a greater number of normalization methods are compared [53] .
Lakshmi and Venkatesan [32] , on the other hand, compared the suitability of normalization methods over TOPSIS from a rare perspective, the usability perspective. They compared how well methods like N1, N2, N3, and N6 work over the TOPSIS based on two usability metrics, i.e., time complexity and space complexity. They lastly concluded N1 is the best method since it consumes the least time and space. This conclusion goes against the findings of Liao et al. [29] , who reported N6 as the most recommendable normalization method for TOPSIS, mainly because Liao et al. [29] performed the comparison based on the result reliability perspective, not usability. The discrepancy in the final recommendations made in these two studies demonstrates that the evaluation metrics employed can influence the selection of suitable normalization methods.
As such, recent attempts have begun using more metrics to evaluate the suitability of various normalization methods over an MCDM technique, e.g. [44] . Usually, a popular voting technique, namely the plurality technique, is used to aggregate the different performance ranks obtained by a normalization method across such metrics to finalize its suitability [46] . To simplify, evaluation metrics for normalization methods can be categorized into three [54] . The first category metrics, like RCI and average Spearman's rank correlation, are used to evaluate the suitability of normalization methods based on their ability to produce reliable alternative ranks [42] . Metrics from the second category, such as the average Pearson correlation coefficient, evaluate suitability based on a method's ability to generate reliable alternative scores [41] . The third category metrics, for example, standard deviation, are used to measure a method's suitability based on its ability to spread distinctive scores across the most and least preferred alternatives [40] . However, none of the existing TOPSIS-related studies tested the suitability of the normalization methods based on the said three perspectives, thus leading to a biased conclusion where a less suitable normalization method is recommended for the TOPSIS application. In conclusion, our survey of past literature uncovers the following limitations: (a) Comparison based on a limited range of normalization methods -To date, no single study has simultaneously compared the suitability of all the ten normalization methods identified in Table 1 for an MCDM technique, including TOPSIS. The maximum number of normalization methods considered in previous TOPSIS-based studies was four, e.g. [35] . Therefore, the conclusiveness of the prior recommendations can be argued. For instance, Chatterjee and Chakraborty [33] considered N2, N6, N7, and N10 to examine the suitability of these four methods for a few MCDM techniques, including TOPSIS. Their investigation recommended N6 and N10 as the most and least preferred methods for TOPSIS, respectively. However, they could have reached a different recommendation if a more exhaustive array of normalization methods had been considered. They indeed provided numerical proof that TOPSIS is an MCDM technique that is very sensitive to different normalization methods, indicating the immediate need for conclusive recommendations on the suitable normalization methods for TOPSIS. (b) Noncomprehensive evaluation procedure -The literature review discloses that the suitability of a normalization method over an MCDM technique must be evaluated based on three essential perspectives. Those perspectives are the method's ability to produce reliable alternative ranks, generate reliable alternative scores, and spread distinctive scores across the most and least preferred alternatives. Also, the plurality technique is generally utilized to synthesize different performance ranks obtained by a normalization method across the evaluation metrics to finalize its overall suitability. Unfortunately, the available studies on TOPSIS only evaluated suitability based on the alternative ranks' or scores' reliability; evaluation based on the spread of alternative scores was absent, e.g. [35] . On the other hand, the plurality technique is known to be non-effective in capturing the relative ranks across various metrics [55] , thus may lead to biased suitability results. This limitation signals the urgency for an alternate, comprehensive procedure to evaluate and select suitable normalization methods for TOPSIS. (c) Conflicting recommendations -Literature reports mixed and sometimes conflicting recommendations on suitable normalization methods for TOPSIS. For example, Celen [30] proposed N2 as one of the normalization methods for TOPSIS, but on the contrary, Vafaei et al. [35] claimed N2 is an absolutely unsuitable choice. The crux of this issue was the lack of comprehensiveness in the context of the range of normalization methods compared and/or the procedure used to evaluate suitability. These mixed, uncertain recommendations in the literature leave the users in a great puzzle as to which is the safer choice of normalization methods for TOPSIS. As such, the room for offering concrete recommendations regarding suitable normalization methods is still available, where such recommendations will lead to safer and more confident decision-making.
Owing to the above limitations, this study aimed to use an inclusive evaluation procedure to make more conclusive recommendations on the suitable BCC-based normalization methods for TOPSIS (out of ten methods traced from previous literature). The contribution of this study is threefold: (a) Literature contribution -This study is the first inclusive endeavor in the current literature to test TOPSIS with all the ten BCC-based normalization methods listed in Table 1 . (b) Methodological contribution -The study employed an alternate, comprehensive procedure to evaluate each normalization method's suitability. The procedure was developed based on three evaluation metrics, namely the average Spearman's rank correlation, average Pearson correlation, and standard deviation metrics, combined with the Borda count technique. This interesting combination allows us to select the normalization methods with decent performance across all three metrics to be selected as the ideal ones. Table 2 clarifies the differences between our proposed evaluation procedure and other procedures reported in similar TOPSIS studies based on four characteristics. In short, our proposed procedure has the following four merits: (c) It uses the average Spearman's rank correlation metric to evaluate each normalization method's ability to produce reliable alternative ranks, (d) It uses the average Pearson correlation metric to compare each normalization method's ability to produce reliable alternative scores, (e) It uses the standard deviation metric to test each normalization method's ability to distribute distinctive scores across the most and least preferred alternatives, and (f) It mathematically captures the relative ranks attained by a normalization method across different evaluation metrics in determining the method's overall suitability. (g) Practical contribution -This study sends a clear message to the users that they should consider combining N4 and N6 into the TOPSIS application for a more reliable result. In other words, they are recommended to aggregate the ranks resulting from both methods to facilitate safer decision-making. In addition, definitive recommendations made through this study could convince users that they are employing the optimal normalization methods, allowing TOPSIS-based decisions to be executed with a better sense of confidence.

Methodology
We developed a real decision matrix involving 30 alternatives, i.e. smartphone models in the current marketplace, to serve as the primary input for our comparative study (see Table 3 ). The decision matrix compiles the data of 30 alternatives, 1 , 2 Table 3 Decision matrix of 30 smartphone models.
, …, 30 based on five smartphone criteria suggested by [56] . Those criteria are the price measured in Malaysian Ringgit ( 1 ), the screen size measured in inches ( 2 ), the pixel density measured in pixels per centimetre ( 3 ), the thickness measured in millimetres ( 4 ), and the mass measured in grams ( 5 ). The data were gleaned from various popular gadget websites, e.g. www.stuff.tv , https://www.techadvisor.com , and https://technave.com/gadget . In other words, the size of the decision matrix is sufficiently large to yield reliable results, and it has a mixture of both the benefit and cost criteria measured with different units, making it suitable for this study. Note that 1 , 4 , and 5 are cost criteria, and the remaining are benefit criteria.
All in all, the study's investigation based on the developed decision matrix can be summarized into three major stages. The stages and steps involved in each stage are outlined in the following sub-sections.
Stage 1 -Application of TOPSIS with the combination of different normalization methods Stage 1 aimed at determining the scores and ranks of the alternatives available in the decision matrix by employing TOPSIS with the combination of ten normalization methods listed in Table 1 . Following are the exact steps involved in Stage 1: (a) Step 1 -We first normalized the data in the decision matrix using the N1 method to convert them into a common scale using (2). At the end of Step 1, a normalized matrix as expressed by (12) was derived.
where is the normalized data of alternative over criterion , where = 1 , 2 , … , and = 1 , 2 , … , . (b) Step 2 -We assigned the weight for each criterion, and developed the weighted normalized matrix, [ ] as expressed in (13). Note that for this study, we used the criteria weights estimated by Krishnan et al. [56] , where these weights were estimated via the combination of various objective weighting techniques. Table 4 shows the weight of each criterion. Source: [56] .
Step 3 -We determined the positive-ideal, * and negative-ideal, − solutions based on the weighted normalized matrix, using Eqs. (14) and (15) Step 4 -We calculated the separation measure for alternatives using -dimensional Euclidean distance. The separation/distance of each alternative from the positive-ideal solution, * , can be obtained using Eq. (16) . Similarly, the separation of each alternative from the negative-ideal solution, − can be computed based on Eq. (17) . (e) Step 5 -We determined the final alternative score for each alternative, , using Eq. (18) , with = 1 , 2 , … , . The alternatives were then ranked based on , where higher indicates better preference over alternative .
(f) Step 6 -Steps 1-5 were repeated based on the other nine normalization methods by properly applying equations (3) - (11). Note that we performed all the calculations with the aid of Microsoft Excel.

Stage 2 -Evaluation of normalization methods based on three metrics
In Stage 2, the results obtained from all the different normalization methods were further evaluated based on the three metrics, namely, average Spearman's rank correlation metric, average Pearson correlation metric, and standard deviation metric, to ensure a comprehensive suitability check. Each metric's role and computational steps are available in the following explanation.
(a) Step 1 -The alternative ranks resulting from all ten methods were analyzed using Spearman's rank correlation test. The test is notably applied to determine the degree of consistency between two different sets of ranks [57] . In other words, this test was carried out to understand each method's ability to provide reliable alternative ranks. The Spearman's correlation coefficient between any two ranks was computed using Eq. (19) [ 58 , 59 ].
where = Spearman's rank correlation coefficient between the alternative ranks resulting from normalization methods, and ; = rank difference for an alternative based on the normalization methods, and ; = total alternatives; = 1 , 2 , … , . The average corresponding to each normalization was then computed, where higher implies the method's better ability to produce reliable alternative ranks [60] .
Step 2 -The Pearson correlation coefficient is a statistical measure that is widely used to study the degree of consistency between two interval or ratio-based variables [61] . As such, this study performed the Pearson correlation test to measure each normalization method's ability to generate reliable alternative scores. The test indeed has been employed by various scholars for a similar purpose. For example, Kabassi et al. [62] analyzed the Pearson correlations between the scores of four museum websites estimated by different MCDM methods, including fuzzy VIKOR and TOPSIS, in search of the most reliable method.
The Pearson correlation coefficient between every two sets of alternative scores was calculated using Eq. (20) [63] .
where = Pearson correlation coefficient of alternative scores identified using normalization method, and ; = score of an alternative identified using the normalization method, ; = score of an alternative identified using the normalization method, ; = average alternative score of normalization method, ; = average alternative score of normalization method, ; = 1 , 2 , .., . Note that a value ranges between − 1 to 1. A value closer to 1 indicates higher consistency/similarity, and vice versa if it is closer to − 1 [64] . The calculated Rs values were then averaged out according to each normalization method, where higher implies the method's better ability to estimate reliable alternative scores [65] .
(c) Step 3 -The standard deviation of the alternative scores resulting from each normalization method was computed using Eq. (21) , where a higher standard deviation indicates the method's better ability to distribute distinctive scores across the least to most preferred alternatives.

Stage 3-Finalization of suitable normalization methods
Since the normalization methods showcased different performance ranks across all three evaluation metrics, it was difficult to conclude which methods suit TOPSIS. Thus, the analysis was furthered by aggregating these ranks by applying the Borda technique. Although some earlier studies used the plurality technique to aggregate such ranks, this study decided to opt for the Borda technique. The decision was made in light of the technique's ability to control information loss by mathematically capturing the entire relative ranks across different evaluation metrics [ 66 , 67 ]. Also, previous findings hint that the technique can be wise enough to ensure only the normalization method that has acceptable decent performance across all the metrics is chosen to be the best [68] .
Following were the steps adhered to determine the final suitability rank of each normalization method using the Borda technique [ 69 , 70 ]: (a) Step 1 -Points were assigned to the normalization methods according to their rank over the first metric, i.e., the average Spearman's rank correlation metric. The method ranked first was assigned with ( − 1 ) points, ( − 2 ) points for the method ranked second, and so. (b) Step 2 -Step 1 was repeated based on the other two metrics, i.e. the average Pearson correlation metric and standard deviation metric. (c) Step 3 -The aggregated Borda score of each normalization was then obtained by summing up the points. (d) Step 4 : The normalization methods were re-ranked ascendingly based on the aggregated Borda scores, where each score implies the method's overall suitability over TOPSIS. A higher Borda score, of course, indicates that the method is more suitable for TOPSIS.
The flowchart and pseudocode illustrating the inputs, steps and outputs of all three stages are presented via Fig. 1 and Table 5 to allow easy understanding and quick conversion into a proper programming language for future use.

Results
Tables 6-15 present the normalized matrices resulting from all ten normalization methods. It is evident that each method has generated a distinct normalized data pattern, as each method transformed data based on various statistical principles or measures, such as the maximum, minimum, sum, and sum of square values. Following is the summary of the patterns we observe in the normalized data structure: (g) N6 neither transforms the most preferred value of a criterion to 1 nor the least preferred value to 0. (h) Similar to N6, N8 neither transforms the most preferred value of a criterion to 1 nor the least preferred value to 0. In addition, we can also spot a weak distinction in the data normalized by N8, especially for the cost criteria.  As expected, the difference in normalized data patterns has led to varying TOPSIS results. Tables 16-25 present the TOPSIS calculations and results derived from each normalization method, including the weighted decision matrix, the positive-and negativeideal values, the separation measure values, and the alternative scores and ranks. We further summarize the different alternative scores and ranks from all normalization methods in Table 26 to enable easy comparison.
Nonetheless, based on Fig. 2 , which labels the alternatives in the 1st and 30th ranks, it can be noticed that all methods, except N5 and N8, identified A29 as the most preferred alternative. Whereas five methods, i.e., N2, N3, N4, N6, and N10, indicated A24 as the least preferred alternative.
Meanwhile, Table 27 suggests that N1, N2, N3, N4, N6, and N10 are the more suitable methods for TOPSIS in terms of ranking reliability since these six methods scored a higher average Spearman's rank correlation, i.e., which is more than 0.6000. Whereas N5, which recorded the lowest score, i.e., 0.0002, is deemed the least suitable.
On the other hand, in the context of reliable alternative scores, based on Table 28 , it can be claimed that N6 is the most suitable method for TOPSIS, with an average score of 0.7795. N1, with an average of 0.7766, is considered the second-most suitable method. N5, with the lowest average, is again reported as the poorest combination for TOPSIS.
However, from the overall perspective, with the aid of the Borda technique, we can conclude N1 is the best fit for TOPSIS, followed Table 30 ).  Table 7 Normalized matrix based on N2.
Model/criterion   Table 9 Normalized matrix based on N4.

Discussion
As predicted, Table 26 and Fig. 2 prove that employing different normalization methods in TOPSIS results in distinct alternative scores and ranks. This finding does not only apply to TOPSIS but also to other MCDM techniques. For example, Ersoy [44] , Vafaei et al. [41] , Palczewski and Sa ł abun [39] , and Mathew et al. [37] reported varying alternative scores and ranks after testing a different combination of normalization methods with ROV, AHP, PROMETHEE, and WASPAS technique, respectively. Interestingly, it can be noticed that, in our study, all normalization methods combined with TOPSIS, except N5 and N8, identified A29 as the least preferred alternative. Based on the TOPSIS principle, this finding interprets that A29 has the shortest geometric distance from the positive-ideal solutions and the farthest distance from the negative-ideal solutions.
The difference in the alternative scores and ranks urged us to further our investigation to identify the most reliable results or, in other words, to trace the most suitable normalization methods for TOPSIS. Hence, we used three metrics to evaluate the suitability of these methods. The evaluation based on the first metric, average Spearman's rank correlation, suggests N1, N2, N3, N4, N6, and N10 as the more suitable methods for TOPSIS. It is because these six normalization methods delivered identical alternative ranks, making them record the highest average Spearman's rank correlation [71] . Jafaryeganeh et al. [42] reported a similar finding where they concluded N3 and N6 as the ideal choices for TOPSIS after detecting the presence of a strong Spearman's rank correlation in ranks produced by both methods. On the other hand, N5 was detected as the least suitable one since the ranks the method produced were far inconsistent from others, which led to the lowest average Spearman's rank correlation.
The analysis was then extended based on the second metric, hoping to bring more distinction in the suitability rank of the normalization methods. Moreover, the first metric only inspects suitability in the context of alternative ranks and not alternative scores [54] . Thus, the second metric, i.e., average Pearson correlation, was applied. N6 records the highest average Pearson correlation, and therefore it is considered the most suitable method since it delivers alternative scores that agree well with other methods. On the contrary, N5 exhibits the poorest suitability mainly because the alternative scores computed via the method show a strong nonconformity against N8 and N9, with a negative Pearson correlation value, respectively [72] . This finding conforms with the investigation carried out by Chatterjee and Chakraborty [33] , in which N6 was declared a more suitable normalization method for TOPSIS compared to N7 and N10.
All the methods were further evaluated based on the third metric, i.e., the standard deviation, to make a more inclusive judgement. This metric was used to evaluate the methods' ability to spread distinctive scores across the most and least suitable alternatives [40] . Interestingly, N8, which failed to perform well over the previous two metrics, delivered outstanding performance on the third metric,       leaving N1 and N6 in the second and third positions, respectively. The method managed to discriminate the preference over the alternatives with distinctive scores. Also, it is surprising that N1 outperforms TOPSIS's default method, N6, on the third metric, indicating that the alternative scores estimated by N6 are not distinct enough. A similar finding was also reported in the literature for other MCDM techniques, confirming its incompetency in producing distinguishing alternative scores. For example, Vafaei et al. [40] , who tested the effect of multiple normalization methods on a dynamic MCDM technique, found that N6 recorded the lowest standard deviation compared to N1 and N2.
In also interesting to witness that N3 and N10 share the same suitability rank with respect to all three metrics. Although both methods resulted in different sets of normalized data values, they had the same positive-ideal and negative-ideal solutions for each criterion, thus leading to identical alternative scores and ranks. This discovery leads to two important conclusions. Firstly, we can conclude that N3 and N10 have the same effect on the results of TOPSIS. Secondly, we can also conclude that future comparative studies on TOPSIS can merely incorporate either N3 or N10; it is pointless to include both methods in such comparative studies since both lead to identical alternative scores. This unique discovery might only apply to TOPSIS and not other MCDM techniques.
Since the normalization methods are tagged with different ranks across all three evaluation metrics, it was difficult to conclude which methods suited TOPSIS well. Thus, the analysis was continued by aggregating these ranks using the Borda technique. The computed Borda scores finally suggest N1 as the most suitable normalization method for TOPSIS. If we used the usual plurality voting method to aggregate the ranks, the result would have equally favoured N1, N6, and N8 since each exhibits top-notch performance, i.e. 1st rank, with at least one evaluation metric. It is indeed insensible to equally treat N6 or N8 with N1, mainly because of N6's and N8's relatively lower performance against the average Spearman's rank correlation metric. In addition, N8's performance against the average Pearson correlation metric is abysmal, i.e. 8th rank. However, with the aid of the Borda technique, N1, which holds a decent performance against all three metrics, is selected as the most suitable method. At the same time, it is not surprising to conclude N9 is the least suitable choice for TOPSIS due to its poor performance with respect to all three metrics. In short, the Borda technique's ability to control information loss by mathematically capturing the complete relative rank information across all three metrics has led to this compelling result [ 73 , 74 ].
Another unique finding of this study is that the methods securing the overall 1st and 2nd ranks belong to two distinct groups of normalization methods, i.e. the linear and non-linear methods. N1, the method in the 1st rank, belongs to the former, and N6, ranked 2nd, belongs to the latter. Note that both linear and non-linear methods have their own advantage. Linear normalization is usually preferred if the data are distributed evenly across the range of values, but non-linear normalization may be necessary if the data are skewed or if there are outliers. Thus, this study recommends that TOPSIS users should first analyze the distribution of the data in the

Table 30
Final ranks of normalization methods based on the aggregated Borda scores.

Conclusion
Literature reports that the choice of normalization methods can affect the results produced by an MCDM technique. Numerous studies have thus been conducted to compare and recommend suitable normalization methods for an MCDM technique, including TOPSIS. However, the existing TOPSIS-related studies either tested a limited number of normalization methods or used incomprehensive metrics to evaluate the methods' suitability. As such, the conclusiveness of the previous studies can be called into question, leaving the TOPSIS users in the dilemma of whether to adhere to the normalization methods recommended in those studies. In light of these limitations, this study decided to employ an alternate, comprehensive procedure to evaluate and recommend suitable BCCbased normalization methods for TOPSIS (out of ten methods traced from past literature). The procedure was devised using three evaluation metrics, namely the average Spearman's rank correlation, the average Pearson correlation, and the standard deviation metrics, merged with the Borda count technique.
The results suggest that TOPSIS users must analyze the data distribution in the decision matrix before choosing the suitable normalization method. If the data are evenly distributed, they can opt for the best linear normalization method identified in this study, the sum-based method. Otherwise, the vector method, the finest non-linear method, can be applied. This study, however, strongly suggests that users should consider incorporating the sum-based method and vector method into TOPSIS. They can aggregate the ranks from both methods before making decisions based on the TOPSIS results.
The study also concludes the enhanced accuracy method is the least suitable choice for TOPSIS due to its inefficiency in producing reliable alternative ranks or failure to spread distinctive scores across the least or most preferred alternatives. Also, the study discovered that applying the maximum method (version I) or Jüttler's-Körth's method does not cause any difference to the TOPSIS results. Although both methods generate different normalized data sets, they result in the same positive-ideal and negative-ideal solutions, thus leading to identical alternative scores and ranks.
From the literature viewpoint, this study can be regarded as the first to consider a total of ten BCC-based normalization methods to offer more concrete recommendations to address the ongoing arguments about the suitable normalization methods for TOPSIS. Past investigations indeed only tested a maximum of six normalization methods.
Meanwhile, from the methodological point of view, this is the first comparative study that used a combination of three different evaluation metrics with the Borda technique to guarantee comprehensiveness and minimize possible bias in selecting ideal normalization methods for TOPSIS. Using three evaluation metrics enabled us to test the suitability of all ten normalization methods based on crucial perspectives. To be exact, the methods were tested based on their ability to produce reliable alternative scores, generate reliable alternative ranks and spread distinctive scores across the most and least preferred alternatives. Whereas the use of the Borda technique ensured the normalization method that has a decent performance across all three metrics is selected as the ideal method.
On the other hand, from the practical standpoint, this study conveys solid recommendations for users to consider merging both the sum-based method and vector method into the TOPSIS application for a more reliable result. In other words, they should aggregate the ranks from both methods to enable safer decision-making. The conclusive recommendations reported in this study could also help users feel surer that they are employing the best possible normalization methods, thus enabling the decisions based on TOPSIS results to be executed with better confidence.
Nevertheless, this study has a limitation. It merely traced and compared the suitability of the BCC-based normalization methods over TOPSIS. The study's scope was narrowed to BCC-based methods on the ground that they can consider whether a criterion is a benefit or cost-oriented criterion before normalization, thus permitting a logical comparison of an alternative across different criteria. The non-BCC-based methods, e.g., the z-score normalization, the decimal normalization, reference-based normalization etc., were sidelined since they are just appropriate for a homogenous decision matrix, which does not contain a mixture of benefit or cost criteria. However, a future comparative study may also broaden its attention to recommend suitable non-BCC-based methods for TOPSIS, where such a recommendation is helpful in case the MCDM problem at hand only incorporates criteria with parallel orientation .

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Declaration of Competing 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
The data used for the investigation are provided within the manuscript.