Operational Research in Engineering Sciences: Theory and Applications First online ISSN: 2620-1607 eISSN: 2620-1747 DOI: https://doi.org/10.31181/oresta101122091d * Corresponding author. doductrung@haui.edu.vn (D.T.Do), duatv@haui.edu.vn (V.D. Tran), duongvanduc@haui.edu.vn (V.D. Duong), tungnn@haui.edu.vn (Nhu-Tung. Nguyen) INVESTIGATION OF THE APPROPRIATE DATA NORMALIZATION METHOD FOR COMBINATION WITH PREFERENCE SELECTION INDEX METHOD IN MCDM Duc Trung Do 1, Van Dua Tran 1, Van Duc Duong 1, Nhu-Tung Nguyen 2* 1 Faculty of Mechanical Engineering, Hanoi University of Industry, Vietnam 2 HaUI Institute of Technology, Hanoi University of Industry, Vietnam Received: 11 August 2022 Accepted: 06 October 2022 First online: 11 October 2022 Research Paper Abstract: Preference Selection Index (PSI) that is a Multi-Criteria Decision Making Method (MCDM) does not need to determine the weights for criteria and it has been applied in many different fields. However, using only the data normalization method (DNM) proposed by the inventor of the PSI method may narrow the application scope of this method. This study aims to expand the application range of the PSI method by identifying the appropriate DNMs in combination with the PSI method. Twelve different DNMs were used in combination with the PSI method. These twelve combinations were used in turn to solve several problems in different fields. The ranked results of solutions by these combinations were all compared with the results in the published studies. The sensitivity analysis of the ranked results of the solutions in each case also was performed. In this study, four out of twelve DNMs were found to be appropriate in combination with the PSI method. This discovery has extended the application scope of the PSI method that the previous methods have not met. Keywords: MCDM, PSI, DNM 1. Introduction Most MCDM methods perform the steps of determining the weights and normalizing the data. Therefore, the ranked results of the solutions depend significantly on the selection of the weighting method and the data normalization method. The research direction to rank solutions using the MCDM method without using the weighting method or without using the data normalization method is being studied by scientists to improve the stability of MCDM. PSI that is a MCDM method does not need to determine the weights for the criteria. The detailed steps to ranking the solutions according to this method will be presented Duc Trung Do et al./Oper. Res. Eng. Sci. Theor. Appl. First online in section three of this paper. The application of this method is also considered to be very simple with a small number of calculations (Yadav et al., 2019). This method has been applied to multi-criteria decision making in many cases, in many different fields: to evaluate the performance of machines (Sari, 2019), to propose a method for waste recovery from electrical/electronic products (Sari, 2020), to choose an automated system development method in selecting the students with enough conditions to receive the scholarship (Arifin and Saputro, 2022), for decision-making in the selection of materials for tooth restoration/beautification (Yadav, 2022), to choose the life cycle design solutions of the product system (Attri and Grover, 2015), to select the technological parameters for turning (Prasad et al., 2018), to select the parameters of Electrical Discharge Machining (Phan et al., 2022), to select the technological parameters for the grinding process (Tien et al., 2021), to rank the efficiency of production lines (Akyuz and Aka, 2015), to rank the types of materials for engineering (Maniya and Bhatt, 2010), to rank the individuals with enough conditions for credit loans in Indonesia (Sianturi et al., 2020), to choose where to sell used computers (Sahir et al., 2018) , to compare the tourism potential of some countries (Stanujkic et al., 2020), to select the machines in the manufacturing companies (Jian et al., 2015), and so on. Thus, it is seen that the PSI method has been successfully applied for MCDM in many different fields. However, the authors of this study can confirm that all applied PSI studies used linear normalization to normalize the data. Linear normalization is also the method used by the scientists who proposed the PSI method. The formulas for normalizing data in this way as well as many other ways of data normalization will be presented in the second section of this study. However, linear normalization cannot be used if some criterion is equal to zero in some solutions. In these cases, if cannot find other DNMs in combination with the PSI method, the application of the PSI method will not be possible. From this point of view, this study will combine all twelve above-mentioned DNMs with PSI method to identify the appropriate DNMs in combining with PSI method. This is the first study using all twelve DNMs in combination with one MCDM method. Those twelve combinations were used to rank the solutions from different fields. In addition to the linear normalization method, this study identified three other DNMs that were determined to be suitable for combining with the PSI one. This obtained result contributes to extend the application scope of the PSI method. The structure of the next sections of this study is presented as follows: (1) The literature review presented the importance of determining an appropriate DNM to combine with one of the MCDM methods. This section also presented the formulas for normalizing data by twelve different methods. The suitability of combining some DNMs with some MCDM methods was also confirmed in published studies as the third content in this section; (2) Summary the performed steps according to the PSI method; (3) Perform the calculations in different cases to rank the solutions in different fields using the PSI method; (4) Identify the DNMs (when combined with PSI method) that show the same best solution as in the published studies; (5) Analyze the sensitivity of the ranking results in each case by creating different scenarios to confirm the appropriate DNMs when combined with the PSI method; (6) Discuss the obtained results and draw the conclusions from this study as well as propose the research directions in the future. Investigation of the Appropriate Data Normalization Method for Combination with Preference Selection Index Method in MCDM 2. Literature review Except for some methods such as Collaborative Unbiased Rank List Integration (CURLI) and Ranking of the attributes and alternatives (R), for most of the remaining MCDM methods, data normalization is the work that needs to be conducted when apply them (Trung, 2022a). Each MCDM method that was proposed often contains at least one DNM. However, because the implementation method in MCDM methods as well as in DNMs is not the same, the ranked results of the solutions when using MCDM methods are also not the same (Zopounidis and Doumpos, 2017). Selection of the DNM has a great influence on the ranking results of the solutions (Budiman et al., 2021; Souissi and Hafdhi, 2021; Aytekin, 2021). When comparing the two methods Vlsekriterijumska optimizacijaI KOmpromisno Resenje (VIKOR) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), the authors have concluded that the ranked results of the solutions are different when using these two methods. The reason is that these two methods used different DNMs (Opricovic and Tzeng, 2004). Mhlanga and Lall (2022) used the VIKOR method to rank ten websites in combination with five different DNMs. This study has shown very different results in those combinations. A solution may rank number one when using one DNM but rank number ten (last rank) when using another DNM. Yazdani et al. (2017) used the COmplex PRoportional ASsessment of alternatives with Grey relations (COPRAS-G) method to rank the material types. The authors concluded that the suitability of a DNM when combined with an MCDM method depends on the number of solutions as well as the number of criteria. Sarraf and McGuire (2021) also concluded that with the same DNM but when combined with different MCDM methods, the ranking results can also be different. The above analysis shows that the determination of the suitable DNM for each MCDM method has a decisive influence on the ranking results of the solutions. It is a very important work to ensure the accuracy of the ranking results of the solutions. Twelve DNMs that listed below are the combined results from two studies of (Aytekin, 2021; Ersoy, 2021a). Linear normalization (N1) 𝑁𝑖𝑗 = 𝑦𝑖𝑗 π‘šπ‘Žπ‘₯ 𝑦𝑖𝑗 , if j οƒŽ B (1) 𝑁𝑖𝑗 = π‘šπ‘–π‘› 𝑦𝑖𝑗 𝑦𝑖𝑗 , if j οƒŽ C (2) Weitendorf normalization (N2) 𝑁𝑖𝑗 = π‘¦π‘–π‘—βˆ’π‘šπ‘–π‘› 𝑦𝑖𝑗 π‘šπ‘Žπ‘₯ π‘¦π‘–π‘—βˆ’π‘šπ‘–π‘› 𝑦𝑖𝑗 , if j οƒŽ B (3) 𝑁𝑖𝑗 = π‘šπ‘Žπ‘₯ π‘¦π‘–π‘—βˆ’π‘¦π‘–π‘— π‘šπ‘Žπ‘₯ π‘¦π‘–π‘—βˆ’π‘šπ‘–π‘› 𝑦𝑖𝑗 , if j οƒŽ C (4) Sum linear normalization (N3) Duc Trung Do et al./Oper. Res. Eng. Sci. Theor. Appl. First online 𝑁𝑖𝑗 = 𝑦𝑖𝑗 βˆ‘ 𝑦𝑖𝑗 π‘š 𝑖=1 , if j οƒŽ B (5) 𝑁𝑖𝑗 = 1 𝑦𝑖𝑗⁄ βˆ‘ 1 𝑦𝑖𝑗⁄ π‘š 𝑖=1 , if j οƒŽ C (6) Vector normalization (N4) 𝑁𝑖𝑗 = 𝑦𝑖𝑗 βˆšβˆ‘ (𝑦𝑖𝑗) 2π‘š 𝑖=1 , if j οƒŽ B (7) 𝑁𝑖𝑗 = 1 βˆ’ 𝑦𝑖𝑗 βˆšβˆ‘ (𝑦𝑖𝑗) 2π‘š 𝑖=1 , if j οƒŽ C (8) Logarithmic normalization (N5) 𝑁𝑖𝑗 = 𝑙𝑛𝑦𝑖𝑗 𝑙𝑛(∏ 𝑦𝑖𝑗 π‘š 𝑖=1 ) , if j οƒŽ B (9) 𝑁𝑖𝑗 = 1 βˆ’ 𝑙𝑛𝑦𝑖𝑗 𝑙𝑛(∏ 𝑦𝑖𝑗 π‘š 𝑖=1 ) , if j οƒŽ C (10) Max linear normalization (N6) 𝑁𝑖𝑗 = 𝑦𝑖𝑗 π‘šπ‘Žπ‘₯ 𝑦𝑖𝑗 , if j οƒŽ B (11) 𝑁𝑖𝑗 = 1 βˆ’ 𝑦𝑖𝑗 π‘šπ‘Žπ‘₯ 𝑦𝑖𝑗 , if j οƒŽ C (12) Min linear normalization (N7) 𝑁𝑖𝑗 = 1 βˆ’ π‘šπ‘–π‘› 𝑦𝑖𝑗 𝑦𝑖𝑗 , if j οƒŽ B (13) 𝑁𝑖𝑗 = π‘šπ‘–π‘› 𝑦𝑖𝑗 𝑦𝑖𝑗 , if j οƒŽ C (14) JΓΌttler-KΓΆrth normalization (N8) 𝑁𝑖𝑗 = 1 βˆ’ | π‘šπ‘Žπ‘₯ π‘¦π‘–π‘—βˆ’π‘¦π‘–π‘— π‘šπ‘Žπ‘₯ 𝑦𝑖𝑗 |, if j οƒŽ B (15) 𝑁𝑖𝑗 = 1 βˆ’ | π‘šπ‘–π‘› π‘¦π‘–π‘—βˆ’π‘¦π‘–π‘— π‘šπ‘Žπ‘₯ 𝑦𝑖𝑗 |, if j οƒŽ C (16) Peldschus normalization (N9) 𝑁𝑖𝑗 = ( 𝑦𝑖𝑗 π‘šπ‘Žπ‘₯ 𝑦𝑖𝑗 ) 2 , if j οƒŽ B (17) Investigation of the Appropriate Data Normalization Method for Combination with Preference Selection Index Method in MCDM 𝑁𝑖𝑗 = ( 𝑦𝑖𝑗 π‘šπ‘Žπ‘₯ 𝑦𝑖𝑗 ) 3 , if j οƒŽ C (18) Stop normalization (N10) 𝑁𝑖𝑗 = 100𝑦𝑖𝑗 π‘šπ‘Žπ‘₯ 𝑦𝑖𝑗 , if j οƒŽ B (19) 𝑁𝑖𝑗 = 100π‘šπ‘–π‘› 𝑦𝑖𝑗 𝑦𝑖𝑗 , if j οƒŽ C (20) Z-score normalization (N11) 𝑁𝑖𝑗 = π‘¦π‘–π‘—βˆ’ βˆ‘ 𝑦𝑖𝑗 π‘š 𝑖=1 π‘š βˆšβˆ‘ (π‘¦π‘–π‘—βˆ’πœ‡π‘—) 2 π‘š 𝑖=1 π‘š , if j οƒŽ B (21) 𝑁𝑖𝑗 = βˆ’ π‘¦π‘–π‘—βˆ’ βˆ‘ 𝑦𝑖𝑗 π‘š 𝑖=1 π‘š βˆšβˆ‘ (π‘¦π‘–π‘—βˆ’πœ‡π‘—) 2 π‘š 𝑖=1 π‘š , if j οƒŽ C (22) Enhanced accuracy normalization (N12) 𝑁𝑖𝑗 = 1 βˆ’ π‘šπ‘Žπ‘₯ π‘¦π‘–π‘—βˆ’π‘¦π‘–π‘— βˆ‘ (π‘šπ‘Žπ‘₯π‘¦π‘–π‘—βˆ’π‘¦π‘–π‘—) π‘š 𝑖=1 , if j οƒŽ B (23) 𝑁𝑖𝑗 = 1 βˆ’ π‘¦π‘–π‘—βˆ’π‘šπ‘–π‘›π‘¦π‘–π‘— βˆ‘ (π‘¦π‘–π‘—βˆ’π‘šπ‘–π‘›π‘¦π‘–π‘—) π‘š 𝑖=1 , if j οƒŽ C (24) In the equations from Eq. (1) to Eq. (24), yij is the value of criterion j at the solution i; Nij is the normalized value of criterion j in solution i; B describes the larger the better criterion; C describes the smaller the better criterion; m is the number of solutions; j is the mean value of the solutions of the criterion j. In addition to have to determine the appropriate DNM in combining with each MCDM method as mentioned above, even if a suitable DNM has been identified, but if only one DNM in combining with a MCDM method may narrow the application scope of that MCDM method. The analysis results from mentioned above about twelve DNMs show that, if there exists a certain criterion whose maximum value is zero, then the methods N1, N3, N5, N6, N7, N8, N9, and N10 will not be available. Or when there exists at least one value of a certain criterion is negative, the N5 method cannot be used. At that time, if an alternative DNM cannot be identified, the decision-making will be difficult, even impossible. However, even if a different DNM is chosen to instead, will the ranked results of the solutions be accurate? Because the ranked results of the solutions are heavily influenced by the used DNMs (Trung, 2022b; Aytekin, 2021; Kaplinski and Tamosaitiene, 2015; Dragisa et al., 2013). From this aspect, many studies that have been performed to combine each MCDM method with several different DNMs. The aim of these studies is determination of the Duc Trung Do et al./Oper. Res. Eng. Sci. Theor. Appl. First online suitable DNMs when combining with each MCDM method. Sanjib and Dragan (2021) simultaneously used two methods N1 and N5 to combine with COmbinative Distance- based Assessment (CODAS) method when ranking the smartphones. They found that in determining the best solution, N1 was equivalent to N5, but in terms of rank inversion, N5 was better than N1. Trung (2022b) combined the CODAS method with six methods including N1, N2, N3, N4, N5, and N6 to make a decision in choosing a robot, assessing the air quality in the working room, and evaluating the machining in lathe machine. The author showed that if only in terms of finding the best solution, the five methods including N1, N2, N3, N4, and N5 are all suitable to combine with CODAS method except for N6 method. Vafaei et al. (2022) combined the Simple Additive Weighting (SAW) method with four methods including N2, N3, N4, and N6 to make decisions in the evaluation of the PhD candidates. They showed that only N2 is suitable for combination with the SAW method. Ersoy (2021a) combined the Proximity Indexed Value (PIV) method with N2, N11, and N12 to rank the financial position of forty-five companies. He showed that only N2 is suitable to combine with the PIV method. Ersoy (2021b) combined the Range Of Value (ROV) method with eight methods including N1, N2, N3, N4, N6, N7, N9, and N12 to rank the financial performance of ten companies. He concluded that only N9 was suitable for combining with the ROV method. Vafaei et al. (2016) combined the Analytic Hierarchy Process (AHP) method with 5 methods including N2, N3, N4, N5, and N6 to rank smart parking locations. They concluded that N6 was the most suitable method to combine with AHP, whereas the combination of AHP and N3 was the worst method. Martin (2021) combined two methods Weighted Aggregates Sum Product Assessment (WASPAS) and TOPSIS with four DNMs including N1, N2, N3, and N4 to select the food processing methods. This research showed an amazing result that all those combinations determine the best solution. Mic & Antmen (2021) used simultaneously three methods including the WASPAS, TOPSIS, and Multiobjective Optimization On the basis of Ratio Analysis (MOORA) to select the location of universities in Turkey. Although the DNMs that were used in combination with the MCDM methods were different, all three cases gave a similar ranked result in all solutions. Zavadskas et al. (2022) combined the Simple Weighted Sum Product (WISP-S) method with three methods including N1, N3, and N4 to rank the solutions for a set of random numbers. The authors have confirmed that the WISP-S method is really powerful when combined with all three DNMs. All these combinations gave the same ranking results. Vafaei et al. (2018) combined the TOPSIS method with six methods including N1, N2, N3, N4, N5, and membership function to rank the drone landing solutions. They confirmed that only N3 is suitable for combination with the TOPSIS method. In another study, Vafaei et al. (2021) also combines the TOPSIS method with six methods including N1, N2, N3, N4, N5, and membership function to select the cars. In this case, the authors point out that the membership function is the best method when combined with the TOPSIS method. Baghla and Bansal (2014) combined the VIKOR method with three methods including N1, N2, and N4 to rank the wireless internet systems. They showed that combining N2 with the VIKOR method gives the best results. Alrababah and Atyeh (2019) combined the VIKOR method with four methods including N1, N2, N3, and N4 to rank the products through the customer feedback. They showed that the combination of VIKOR and N4 gives the best results. Mathew et al. (2017) combined the WASPAS method with six methods including N2, N3, N4, N5, N6, and N12 to rank the robots. The authors found that the combination of WASPAS with N2 gave the best results. Even, in several studies, when applying a certain MCDM method, people did not even use the DNMs Investigation of the Appropriate Data Normalization Method for Combination with Preference Selection Index Method in MCDM available by itself but use other DNMs. Zolfani et al. (2020) combined simultaneously N5 with TOPSIS and VIKOR methods to rank the solutions in two cases, case one is the ranking of the apartments in Madrid (Spain) and the other is the ranking of the solutions with a set of random data. It should be noted that N5 is not the DNM proposed by the authors of both TOPSIS and VIKOR methods. However, an unexpected result occurred, the ranked results when combining TOPSIS with N5 completely coincided with the case when combining VIKOR with N5. Thus, it is seen that finding the appropriate DNMs for each MCDM method has been carried out by many scientists and has also been applied in many different fields. In addition, any study that has done in this direction has attracted a lot of interest. Based on the characteristics of the PSI method as discussed in the introduction, this study was selected the PSI method to perform the research mission follow the proposed research direction. 3. PSI Method The order of the performing the ranking of solutions according to the PSI method is presented as follows (Maniya and Bhatt, 2010). - Build a decision matrix including the solutions and the criteria. - Standardized the data. + For the larger the better criterion. 𝑁𝑖𝑗 = 𝑦𝑖𝑗 π‘šπ‘Žπ‘₯𝑦𝑖𝑗 (25) + For the smaller the better criterion. 𝑁𝑖𝑗 = π‘šπ‘–π‘›π‘¦π‘–π‘— 𝑦𝑖𝑗 (26) Eq. (25) and (26) that are data normalization formulas used by the proponent of the PSI method (method N1). The application cases in the next sections of this paper will fully apply all twelve DNMs as presented in section 2. - Calculate the mean values of the standardized data (N). 𝑁 = 1 𝑛 βˆ‘ 𝑁𝑖𝑗 𝑛 𝑖=1 (27) - Determine the preference values from the mean values (οͺj). πœ‘π‘— = βˆ‘ [𝑁𝑖𝑗 βˆ’ 𝑁] 2𝑛 𝑖=1 (28) - Determine the deviation in the preference values (j). πœƒπ‘— = [1 βˆ’ πœ‘π‘— ] (29) - Determine the overall preference value (j) for the criteria. Duc Trung Do et al./Oper. Res. Eng. Sci. Theor. Appl. First online  𝒋 = πœƒπ’‹ βˆ‘ πœƒπ’‹ π’Ž 𝒋=𝟏 (30) - Calculate the PSIj of each solution, with i = 1Γ·m. 𝑃𝑆𝐼𝑗 = βˆ‘ 𝑁𝑖𝑗 . 𝑗 π‘š 𝑗=1 (31) where n is the number of criteria. - Rank the solutions according to the principle that the solution with the largest PSIj is the best one. To identify the appropriate DNMs when combined with the PSI method, this study performed ranking in several cases from the different fields. In each case, the number of criteria and the number of solutions is also different. Selecting the cases from different fields will lead to draw the most general conclusions. The selected cases were all referenced from published studies. The reason for this is: in those studies, the solutions were also ranked either by PSI method combined with N1 or by another MCDM method. The ranking results of the solutions in the published studies will be used to compare with the obtained ranking results in this study. Specific contents when ranking the solutions in each case are presented in the section 4 of this paper. 4. Results and Discussion 4.1. Application Cases In this section, a combination of the PSI method and the twelve data normalization methods as described above will be used to rank the solutions in four different cases. The data of all four cases were referenced from published studies. In those studies, the ranking of the solutions was also performed by different MCDM methods. The ranked results of the solutions when using different MCDM methods will be used to compare with those ones when using PSI method. Case 1 The data on the personnel selection solutions for a textile company in Denizli (Turkey) were used in this example (Tus and AdalΔ±, 2018). Selection of a marketing assistant from seven candidates was performed. Table 1. The data of case 1 (Tus and AdalΔ±, 2018) No. C1 C2 C3 C4 C5 A1 2 110 3 2 3 A2 5 100 5 3 3 A3 3 90 4 5 2 A4 10 80 3 4 4 A5 4 85 2 4 5 A6 8 80 3 4 4 A7 5 95 2 4 3 Investigation of the Appropriate Data Normalization Method for Combination with Preference Selection Index Method in MCDM Five criteria to evaluate the candidates include work experience (C1), foreign language ability (C2), problem-solving ability (C3), communication ability (C4), and group management ability (C5). The scores for each criterion for each candidate are presented in Table 1. In which, all five criteria are in the form of the larger the better criteria. In this study, the ranking of solutions was conducted by two methods: one is the PSI method combined with N1 and the other one is the CODAS method. The ranked results from two above methods will be used for comparison with the ranked results from this study. And next, the ranking of solutions according to the PSI method combined with different DNMs will be performed. First of all, the data normalization by the N2 method will be applied. Eq. (3) and Eq. (4) were used to normalize the data according to the N2 method, the normalized data are presented in Table 2. Table 2. The data normalization values in case 1 according to the N2 method No. C1 C2 C3 C4 C5 A1 0.0000 1.0000 0.3333 0.0000 0.3333 A2 0.3750 0.6667 1.0000 0.3333 0.3333 A3 0.1250 0.3333 0.6667 1.0000 0.0000 A4 1.0000 0.0000 0.3333 0.6667 0.6667 A5 0.2500 0.1667 0.0000 0.6667 1.0000 A6 0.7500 0.0000 0.3333 0.6667 0.6667 A7 0.3750 0.5000 0.0000 0.6667 0.3333 Eq. (27) and Eq. (28) were used to determine the preference values from the mean (οͺj). The calculated results are presented in Table 3. Table 3. Values of οͺj in case 1 when data normalization according to the N2 method C1 C2 C3 C4 C5 οͺj 0.7411 0.8175 0.7619 0.6032 0.6349 The deviation in the preference value (Ξ²j) is calculated by Eq. (29), the overall preference value (j) is determined by Eq. (30), and the calculated results are presented in Table 4. Table 4. Values of Ξ²j and j in case 1 when data normalization according to N2 method C1 C2 C3 C4 C5 Ξ²j 0.2589 0.1825 0.2381 0.3968 0.3651 j 0.1796 0.1266 0.1652 0.2753 0.2533 The PSIi is calculated according to Eq. (31), the calculated results are presented in Table 5. The ranked results of the solutions according to the values of the PSI were also stored in this table. Duc Trung Do et al./Oper. Res. Eng. Sci. Theor. Appl. First online Table 5. PSIi values in case 1 when data normalization according to the N2 method and ranked results of the solutions No. PSIi Rank A1 0.2661 7 A2 0.4931 4 A3 0.4501 5 A4 0.5871 1 A5 0.5028 3 A6 0.5422 2 A7 0.3986 6 Thus, the ranking of the solutions for case 1 when normalizing data by the N2 method was completed. The ranking of solutions using other DNMs (from N3 to N12) was also performed. Table 6 presents the ranking results of the solutions when using all DNMs. The ranked results of the solutions according to the CODAS method and PSI method combined with N1 by Tus and AdalΔ± (2018) were also included in this table. Table 6. The ranked results of solutions in case 1 No. CODAS N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 A1 7 7 7 7 1 7 7 7 7 7 7 7 7 A2 3 3 4 3 2 1 3 1 3 1 3 2 2 A3 5 5 5 5 4 4 5 6 5 3 5 5 5 A4 1 1 1 1 7 2 1 2 1 2 1 1 1 A5 4 4 3 4 5 5 4 4 4 4 4 4 4 A6 2 2 2 2 6 3 2 3 2 5 2 3 3 A7 6 6 6 6 3 6 6 5 6 6 6 6 6 From the results in Table 6. - When using eleven DNMs to combine with the PSI method, all confirmed A1 as the worst solution (except for N4). Solution A1 was also confirmed to be the worst one when using the CODAS method (Tus and AdalΔ±, 2018). From these results, a solid conclusion can be drawn that A1 is the worst solution. - Solution A4 was determined to be the best solution when using CODAS method (Tus and AdalΔ±, 2018). When using the PSI method in combination with eight DNMs including N1, N2, N3, N6, N8, N10, N11, and N12, A4 was also determined to be the best solution. However, it would be a subjective statement if only considering the results in case 1 to conclude that all eight methods including N1, N2, N3, N6, N8, N10, N11, and N12 are all suitable to be combined with the PSI method. To draw the generalized conclusions, it is necessary to perform more applications with many cases in many different fields. Furthermore, sensitivity analysis in different situations is also required to ensure the accuracy of the conclusions. Case 2 The investigated data on robots were used in this case (Keshavarz-Ghorabaee et al., 2016; Trung, 2022b). Seven types of robots were given for the ranking process. Five criteria were selected to evaluate the robots including Load capacity (C1), Maximum tip speed (C2), Memory capacity (C3), Manipulator reach (C4), and Repeatability (C5). In which C1, C2, C3, and C4 are the larger the better criteria, Investigation of the Appropriate Data Normalization Method for Combination with Preference Selection Index Method in MCDM whereas C5 is the smaller the better criterion. The investigated data is presented in Table 7. Similar to case 1, for this case, the ranking results of the solution when applying the PSI method with twelve different DNMs (N1 to N12) are presented in Table 8. The ranking results of the solutions using the CODAS method (Keshavarz-Ghorabaee et al., 2016) and the two methods R and CURLI (Trung, 2022a) are also presented in this table. Table 7. The data of case 2 (Keshavarz-Ghorabaee et al., 2016; Trung, 2022b) No. C1 C2 C3 C4 C5 A1 60 0.4 500 990 2540 A2 6.35 0.15 3000 1041 1016 A3 6.8 0.1 1500 1676 1727.2 A4 10 0.2 2000 965 1000 A5 2.5 0.1 500 915 560 A6 4.5 0.08 350 508 1016 A7 3 0.1 1000 920 1778 The obtained results in Table 8 show that A2 is the best solution when ranking by the CODAS method (Keshavarz-Ghorabaee et al., 2016) and when ranking by two methods R and CURLI (Trung, 2022a). A2 was also identified as the best solution when combining the PSI method with six DNMs including N1, N4, N5, N6, N8, and N11. Table 8. The ranked results of solutions in case 2 No. CODAS R CURLI N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 A1 3 2 2 2 3 1 3 1 3 3 3 1 1 2 3 A2 1 1 1` 1 2 2 1 5 1 2 1 3 2 1 2 A3 2 4 4 4 1 4 4 2 4 4 4 2 4 4 7 A4 5 3 3 3 4 3 2 3 2 1 2 4 3 3 1 A5 7 5 5 5 5 5 5 7 5 5 5 6 5 5 4 A6 6 7 7 7 7 7 7 4 7 7 7 7 7 7 6 A7 4 6 6 6 6 6 6 6 6 6 6 5 6 6 5 Thus, if we only consider the results in this case, it is seen that five methods N1, N4, N6, N8, and N11 are suitable methods to combine with the PSI method. However, to draw general conclusions, further applications of the ranking of these processes in other fields are still needed to perform. Case 3 The experimental data about the turning processes were used in this case (Prasad et al., 2018). In this study, nine different solutions to a turning process were implemented. Each solution is evaluated through three criteria including arithmetic average roughness height (C1), Ten-point mean roughness (C2), and material removal rate (C3). In which, C1 and C2 are the smaller the better criteria, whereas C3 is the larger the better criterion. The calculated results are presented in Table 9. Duc Trung Do et al./Oper. Res. Eng. Sci. Theor. Appl. First online Table 9. The data of case 3 (Prasad et al., 2018) No. C1 C2 C3 A1 2.11 9.04 9.21 A2 5.023 22.68 24.85 A3 9.17 36.103 32.57 A4 2.036 8.546 20.57 A5 7.16 26.94 39 A6 11.59 43.963 24.85 A7 3.35 13.263 41.14 A8 7.25 26.086 27 A9 11.75 45.376 39.85 The ranking of solutions according to the PSI method when combined with eleven different DNMs (N2 to N11) was performed similarly to case 1. The calculation results are presented in Table 10. The ranking results of the solutions when using the PSI method in combination with N1 (Prasad et al., 2018) were also summarized in this table. Table 10. The ranked results of solutions in case 3 No. N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 A1 6 1 2 3 2 8 9 8 7 1 4 9 A2 7 4 5 5 4 4 5 4 6 8 5 7 A3 5 7 8 7 7 6 6 6 3 5 7 3 A4 2 2 1 2 1 3 1 3 8 9 2 8 A5 3 6 4 4 5 2 3 2 5 4 3 4 A6 9 8 9 9 8 9 8 9 1 7 9 2 A7 1 3 3 1 3 1 2 1 9 3 1 6 A8 8 5 6 6 6 5 7 5 4 6 6 5 A9 4 9 7 8 9 7 4 7 2 2 8 1 The obtained results in Table 10 show that A7 is determined to be the best solution when using the PSI method in combination with N1 (Prasad et al., 2018). When four methods N4, N6, N8, and N11 were used in combination with the PSI method, it was also determined that A7 was the best solution. In this case, it can be concluded that the five methods N1, N4, N6, N8, and N11 are suitable methods to combine with the PSI method. Case 4 The investigated data on air condition in offices was used in this case (Keshavarz- Ghorabaee et al., 2016). Six criteria were used to evaluate the air condition in the office including the amount of air per head (C1), relative air humidity (C2), air temperature (C3), illumination during work hours (C4), rate of airflow (C5), and dew point (C6). In which, the criteria C1 to C4 are the large the better criteria, whereas C5 and C6 are the smaller the better criteria. The data about the solutions and the criteria in this case are presented in Table 11. Investigation of the Appropriate Data Normalization Method for Combination with Preference Selection Index Method in MCDM Table 11. The data of case 4 (Keshavarz-Ghorabaee et al., 2016) No. C1 C2 C3 C4 C5 C6 A1 7.6 46 18 390 0.1 11 A2 5.5 32 21 360 0.05 11 A3 5.3 32 21 290 0.05 11 A4 5.7 37 19 270 0.05 9 A5 4.2 31 19 240 0.1 8 A6 4.4 38 19 260 0.1 8 A7 3.9 42 16 270 0.1 5 A8 7.9 44 20 400 0.05 6 A9 8.1 44 20 380 0.05 6 A10 4.5 46 18 320 0.1 7 A11 5.7 48 20 320 0.05 11 A12 5.2 48 20 310 0.05 11 A13 7.1 49 19 280 0.1 12 A14 6.9 49 16 250 0.05 10 In this case, the ranking of the solutions according to the PSI method in combining with twelve different DNMs (N1 to N12) was performed similarly to case 1. The calculated results are presented in Table 12. The ranking results of the solutions when using the CODAS method (Keshavarz-Ghorabaee, et al., 2016) were also summarized in this table. Table 12. The ranked results of solutions in case 4 No. CODAS N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 A1 3 3 9 7 7 3 3 3 3 1 9 5 6 A2 6 8 6 6 6 8 8 8 8 10 7 6 5 A3 9 12 8 9 9 9 11 11 11 12 8 9 9 A4 10 10 7 8 8 6 10 9 10 14 6 7 8 A5 14 14 14 14 14 13 14 14 14 5 14 14 14 A6 13 13 13 13 13 12 13 13 13 4 13 12 12 A7 12 11 12 10 12 14 12 12 12 6 10 13 13 A8 1 1 1 1 1 2 1 1 1 9 1 1 1 A9 2 2 2 2 2 1 2 2 2 11 2 2 2 A10 11 7 11 11 10 11 6 7 6 3 11 11 11 A11 4 4 3 3 3 7 4 4 4 7 4 3 3 A12 7 5 4 4 4 10 5 5 5 8 5 4 4 A13 8 6 10 12 11 4 7 6 7 2 12 10 10 A14 5 9 5 5 5 5 9 10 9 13 3 8 7 The calculated results in Table 12 show that A8 is determined to be the best solution when using the CODAS method (Keshavarz-Ghorabaee, et al., 2016). A8 was also determined to be the best solution when using other methods N1, N2, N3, N4, N6, N8, N10, and N11 in combination with the PSI method. From the analyzed results, it is shown that, in this case, eight methods that include N1, N2, N3, N4, N6, N8, N10, and N11 are suitable methods to combine with the PSI method. 4.2. Sensitivity Analysis The combined results from the four above cases give an overview of the fit/nonconformity when combining the DNMs with the PSI method and as presented Duc Trung Do et al./Oper. Res. Eng. Sci. Theor. Appl. First online in Table 13. In which, the cells that were marked "βœ“" show the suitability of combining the DNM with the PSI method. In contrast, the blank cells represent nonconformities when combining the DNM with the PSI method. However, this suitability only considers the factors that the method of data normalization when combined with the PSI method can determine the best solution in comparing to published studies. In order to confirm that a DNM is appropriate in combination with the PSI method, it is necessary to analyze the sensitivity in ranking the solutions. Of course, the sensitivity analysis only needs to be performed for the data normalized methods that was jointly identified the best solution. With above four cases, these methods were N1, N6, N8, and N11. Table 13. Suitable normalization methods for combining with the PSI method Examples Normalization method N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 Example 1 βœ“ βœ“ βœ“ βœ“ βœ“ βœ“ βœ“ βœ“ Example 2 βœ“ βœ“ βœ“ βœ“ βœ“ Example 3 βœ“ βœ“ βœ“ βœ“ βœ“ Example 4 βœ“ βœ“ βœ“ βœ“ βœ“ βœ“ βœ“ βœ“ βœ“ The sensitivity analysis is the determination of the degree of variation in the ranking results of the solutions under the different scenarios. The scenarios that were commonly used for sensitivity analysis include changing the weight of the criteria, removing one/several solutions from the list of solutions, and changing the criterion type (Bozanic et al., 2021; Zopounidis and Doumpos, 2017). In this case, the generation of different scenarios is done by eliminating a certain solution. In each case, the eliminated solution will also be selected differently. For case 1, solution A5 was removed from the list of solutions. According to the ranking results of the solutions in case 1 (section 4.1), A5 ranked 4, A1 ranked 7, and A4 ranked 1 (when using N1, N4, N8, and N11). Therefore, if removing A5 from the list of solutions does not affect on the ranking of the solutions, then A4 is still the best solution and A1 is still the worst solution. After removing A5 from the list of solutions, the ranking results of solutions are shown in Figure 1. Figure 1. Ranked results of the solutions without A5 solution in case 1 Investigation of the Appropriate Data Normalization Method for Combination with Preference Selection Index Method in MCDM It is seen that although the rank inversion occurred in some solutions, however, A4 is still the best solution, and A1 is still the worst solution for all four different DNMs. It shows that the removal of A5 from the list of solutions does not change the best solution and the worst solution. In this case, it can be concluded that N1, N6, N8, and N11 methods are suitable methods to combine with the PSI method. For case 2, solution A6 was removed from the list of solutions. According to the ranking of the solutions in case 2 (section 4.2), A6 is the worst solution and A2 is the best solution (when using N1, N4, N8, and N11). Therefore, if removing A6 from the list of solutions does not influence on the ranking of solutions, then A2 is still the best solution. On the other hand, currently, A7 ranks 6, so if A6 is removed from the list of solutions, A7 will rank last. After removing A6 from the list of solutions, the ranking results of the solutions are shown in Figure 2. Figure 2. Ranked results of the solutions without A6 solution in case 2 It is seen that although the rank inversion was occurred in some solutions, however, A2 is still the best solution, and A7 is still the worst solution when using four different DNMs. That shows that the removal of A6 from the list of solutions was not changed the best and worst solution. In this case, it is again certainty established that methods N1, N6, N8, and N11 are suitable methods to combine with the PSI method. For case 3, once again, the worst solution is removed from the list of solutions (solution A6). According to the ranking of solutions in case 3 (section 4.3), A7 is the best solution. If removing A6 from the list of solutions does not affect on the ranking of solutions, then A7 is still the best solution. After removing A6 from the list of solutions, the results of ranking solutions are shown in Figure 3. It is seen that although the rank inversion also occurred in some solutions, however, A7 is still the best solution and ranks 2, 3, and 4 are the same those when using DNMs. In this case, we can again confirm that N1, N6, N8, and N11 are suitable methods to combine with the PSI method. Duc Trung Do et al./Oper. Res. Eng. Sci. Theor. Appl. First online Figure 3. Ranked results of the solutions without A6 solution in case 3 For case 4, the best solution was removed from the list of solutions, (solution A8). According to the ranking results of the solutions in case 4 (item 3.4), A9 ranked 2 nd, and A5 ranked last. Figure 4. Ranked results of the solutions without A8 solution in case 4 Therefore, if removing A8 from the list of solutions does not affect on the ranking of solutions, then A9 will rank 1, and A5 will still rank last. After removing A8 from the list of solutions, the ranking results of solutions are shown in Figure 4. It is seen that rank inversion also occurred in some solutions. However, A9 is always the best solution, and A5 is always the worst solution. So, the removal of A8 from the list of solutions does not change the best solution and the worst solution. Once again, we can confirm that methods N1, N6, N8, and N11 are suitable methods to combine with the PSI method. 4.3. The appropriate DNM for combination with PSI Method From the above-performed analyzed results, it is seen that in the above-mentioned twelve DNMs, there are only four DNMs including N1, N6, N8, and N11 are suitable methods to combine with the PSI method in all studied cases. These combinations not only consistently identified the same best solution, but also gave equivalent results in Investigation of the Appropriate Data Normalization Method for Combination with Preference Selection Index Method in MCDM comparing to other methods (CODAS, R, and CURLI) as analyzed in each case. The sensitivity analysis of the ranking results of the solutions was also performed with different scenarios. The results all confirmed that N1, N6, N8, and N11 are suitable methods to combine with the PSI method. These obtained results could open a wide application range for the PSI method. It can be said that because in the cases, there does not exist any value of yij equal to 0, all four methods of data normalization can be applied. However, when there exists a certain value yij = 0, then the method N1 cannot be applied, the remaining three methods (N6, N8, and N11) can still be applied. Even when there exists a value max(yij) = 0, then all three methods N1, N6, and N8 cannot be applied, there is still an alternative method (N11). This can be considered a great discovery to be able to apply the PSI method in all cases. The case that was applied immediately below will make this statement clearer. In this case, there are 3 different solutions A1, A2, and A3. Each solution is evaluated through 5 criteria C1, C2, C3, C4, and C5. In which, C1, C2, and C3 are criteria as the larger the better, whereas C4 and C5 are criteria as the smaller the better. The values of the criteria at the solutions are selected at random, in which, there are both positive values, zero values, and negative values (Table 14). It is clear that in this case, methods N1, N6, and N8 cannot be applied, but only method N11 can be applied to rank the solutions. Using the PSI method with the DNM (N11) to rank solutions, the ranking results were summarized in table 14. In addition, to verify the ranking results, R and CURLI methods were also applied with the ranked results as summarized in Table 14. Table 14. Ranked results when using PSI+N11, CURLI, and R methods No. Criteria Rank C1 C2 C3 C4 C5 PSI + N11 CURLI R A1 5 -3 10 1 0 1 1 1 A2 6 -2 8 0 2 2 2 2 A3 3 0 6 3 1 3 3 3 The calculated results in Table 14 show that when ranking the solutions by PSI method in combining with N11, the ranking results are completely consistent with those ones when using CURLI and R methods. Once again, we see that the N11 method is perfectly suited to combine with the PSI method. This combination will create more effective when other DNMs (N1, N6, and N8) cannot be applied. The identification of the appropriate DNMs when combined with a specific MCDM method is a suitable research direction in studying on the MCDM. Therefore, in this case, the first time the PSI method was selected as the research object both showing the correctness of the approach as well as the novelty of this work. This study identified four DNMs suitable to combine with the PSI method. This discovery has expanded the PSI method application scope that has not been considered in previous studies. 5. Conclusion With the simplicity of application and no need to determine the weights for the criteria, the PSI method has been widely applied for MCDM in many different fields. Duc Trung Do et al./Oper. Res. Eng. Sci. Theor. Appl. First online However, the proponent of the PSI method as well as all the studies that applied this method all normalized the data according to the N1 method. It is clear that in all mentioned cases, the author has not considered cases when a certain criterion has a value of 0 in a certain solution. In these cases, the N1 method cannot be applied, and then the PSI method also cannot be applied. To overcome this limitation, this study investigated the suitability of combining twelve different DNMs with the PSI method. All those combinations were tested in four cases in four different fields. The number of solutions, the number of criteria, and the type of criteria (the larger the better, the smaller the better) are not the same in all cases. In this study, it was determined that in all four cases, four methods including N1, N6, N8, and N11 were identified as suitable methods to combine with the PSI method. These results from this study open a wide application range for the PSI method. Specifically, when there exists yij = 0 and/or max(yij) = 0, then the N1, N6, and N8 methods cannot be applied, the N11 method can still be applied for multi-criteria decision making. However, all twelve DNMs that were mentioned in this study cannot be applied if the criteria are in the qualitative form (color, preferences, etc.). In these cases, the assignment of these qualitative criteria to the numbers is necessary to be done before performing the data normalization. In these cases, the studies that apply the PSI method for MCDM when having the qualitative criteria are the next research direction of this study. When the value of the criteria at each solution is a fuzzy set, the evaluation of the suitable degree when combining the DNMs (N1, N6, N8, and N11) with the PSI method, which is also a new research direction should be performed as soon as possible. All twelve used DNMs in this study should also be tested to determine the methods that are suitable when combined with other MCMD methods. References Akyuz, G., & Aka, S. (2015). An Alternative Approach for Manufacturing Performance Measurement: Preference Selection Index (PSI) Method. Business and Economics Research Journal, 6(1), 63-77. Alrababah, S. A.A., & Atyeh, A.J. (2019). Effect of Normalization Techniques in VIKOR Approach for Mining Product Aspects in Customer Reviews. International Journal of Computer Science and Network Security, 19(12), 112-118. Arifin, N., & Saputro, P.H. (2022). Selection Index (PSI) Method in Developing a Student Scholarship Decision Support System. International Journal of Computer and Information System, 3(1), 12-16. https://doi.org/10.29040/ijcis.v3i1.55 Attri, R., & Grover, S. (2015). Application of preference selection index method for decision making over the design stage of production system life cycle. Journal of King Saud University-Engineering Sciences, 27(2), 207-216. https://doi.org/10.1016/j.jksues.2013.06.003 Aytekin, A. (2021). Comparative analysis of the normalization techniques in the context of MCDM problems. Decision Making: Applications in Management and Engineering, 4(2), 1-25. https://doi.org/10.31181/dmame210402001a Baghla, S., & Bansal, S. (2014). Effect of normalization techniques in VIKOR method for network selection in heterogeneous networks. 2014 IEEE International Conference on https://doi.org/10.29040/ijcis.v3i1.55 https://doi.org/10.1016/j.jksues.2013.06.003 https://doi.org/10.31181/dmame210402001a Investigation of the Appropriate Data Normalization Method for Combination with Preference Selection Index Method in MCDM Computational Intelligence and Computing Research, 15435508. https://doi.org/10.1109/ICCIC.2014.7238357 Bozanic, D., Milic, A., Tesic, D., Salabun, W., Pamucar, D. (2021). D numbers–FUCOM– fuzzy RAFSI model for selecting the group of construction machines for enabling mobility. Facta Universitatis. Series: Mechanical Engineering, 19(3), 447-471. http://dx.doi.org/10.22190/FUME210318047B Budiman, E., Hairah, U., Wati, M., & Haviluddin, H. (2021). Sensitivity analysis of data normalization techniques in social assistance program decision making for online learning. Advances in Science, Technology and Engineering Systems Journal, 6(1), 49- 56. https://doi.org/10.25046/aj060106 Stanujkic, D., Djordjevic, B., & Djordjevic, M. (2013). Comparative analysis of some prominent MCDM methods: A case of ranking Serbian banks. Serbian journal of management, 8(2), 213-241. https://doi.org/10.5937/sjm8-3774 Ersoy, N. (2021). Application of the PIV method in the presence of negative data: an empirical example from a real-world case. Hitit Sosyal Bilimler Dergisi, 14(2), 318- 337. https://doi.org/10.17218/hititsbd.974522 Ersoy, N. (2021). Selecting the best normalization technique for ROV method: Towards a real life application. Gazi University Journal of Science, 34(2), 592-609. https://doi.org/10.35378/gujs.767525 Jian, S.Y., Tao, S.J., & Huang, X.R. (2015). Preference selection index method for machine selection in a flexible manufacturing cell. Advanced Materials Research, 1078, 290-293. https://doi.org/10.4028/www.scientific.net/AMR.1078.290 Kaplinski, O., & Tamosaitiene, J. (2015). Analysis of normalization methods influencing results: a review to honour Professor Friedel Peldschus on the occasion of his 75th birthday. Procedia Engineering, 122, 2-10. https://doi.org/10.1016/j.proeng.2015.10.001 Keshavarz Ghorabaee, M., Zavadskas, E.K., Turskis, Z., & Antucheviciene, J. (2016). A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making. Economic Computation & Economic Cybernetics Studies & Research, 50(3), 25-44. Maniya, K., & Bhatt, M.G. (2010). A selection of material using a novel type decision- making method: Preference selection index method. Materials & Design, 31(4), 1785- 1789. https://doi.org/10.1016/j.matdes.2009.11.020 Martin, N. (2021). Plithogenic SWARA-TOPSIS Decision Making on Food Processing Methods with Different Normalization Techniques. Advances in Decision Making. https://doi.org/10.5772/intechopen.100548 Mathew, M., Sahu, S., & Upadhyay, A.K. (2017). Effect of normalization techniques in robot selection using weighted aggregated sum product assessment. International Journal of Innovative Research and Advanced Studies, 4(2), 59-63. Mhlanga, S.T., & Lall, M. (2022). Influence of Normalization Techniques on Multi- criteria Decision-making Methods. Journal of Physics: Conference Series, 2224, 012076. https://doi.org/10.1088/1742-6596/2224/1/012076 Mic, P., & Antmen, Z.F. (2021). A Decision-Making Model Based on TOPSIS, WASPAS, and MULTIMOORA Methods for University Location Selection Problem. SAGE Open, 11(3). https://doi.org/10.1177/21582440211040115 http://dx.doi.org/10.22190/FUME210318047B https://doi.org/10.25046/aj060106 https://doi.org/10.5937/sjm8-3774 https://doi.org/10.17218/hititsbd.974522 https://doi.org/10.35378/gujs.767525 https://doi.org/10.4028/www.scientific.net/AMR.1078.290 https://doi.org/10.1016/j.proeng.2015.10.001 https://doi.org/10.1016/j.matdes.2009.11.020 https://www.intechopen.com/books/10908 https://doi.org/10.1177/21582440211040115 Duc Trung Do et al./Oper. Res. Eng. Sci. Theor. Appl. First online Opricovic, S., & Tzeng, G.H. (2004). Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. European journal of operational research, 156(2), 445-455. https://doi.org/10.1016/S0377-2217(03)00020-1 Phan, N.H., Vu, N.N., Shirguppikar, S., Ly, N.T., Tam, N.C., & Tai, B.T. (2022). Multi- criteria decision making in electrical discharge machining with nickel coated aluminium electrode for titanium alloy using preferential selection index. Manufacturing Review, 9, 13. https://doi.org/10.1051/mfreview/2022010 Prasad, R. V., Rao, C. M., & Raju, B. N. (2018). Application of preference selection index (PSI) method for the optimization of turning process parameters. International journal of modern trends in engineering and research, 5(5), 140-144. https://doi.org/10.21884/IJMTER.2018.5152.IISGD Sahir, S.H., Afriani, J., Ginting, G., Fachri, B., Siregar, D., Simbolon, R., Simarmata, J. (2018). The Preference Selection Index method in determining the location of used laptop marketing. International Journal of Engineering and Technology, 7(3.4), 260- 263. Sanjib, B., & Dragan, B. (2021). Combinative distance based assessment (CODAS) framework using logarithmic normalization for multi-criteria decision making. Serbian Journal of Management, 16(2), 321-340. https://doi.org/10.5937/sjm16- 27758 Sari, E.B. (2019). Measuring The performances of the machines via Preference Selection Index (PSI) method and comparing them with values of Overall Equipment Efficiency (OEE). Izmir journal of economics, 34(4), 573-581. https://doi.org/10.24988/ije.2019344859 Sari, E.B. (2020). Recovery alternatives decision by using fuzzy based preference selection index method. LogForum, 16(1), 171-181. https://doi.org/10.17270/J.LOG.2020.386 Sarraf, R., & McGuire, M. P. (2021). Effect of Normalization on TOPSIS and Fuzzy TOPSIS. In Proceedings of the Conference on Information Systems Applied Research, 14(5551), 1-11. Sianturi, L.T., Mesran, M., Purba, E., & Rahim, R. (2020). Implementation of Preference Selection Index Method In Determination of People’s Business Credit Receiver. Proceedings of the Third Workshop on Multidisciplinary and Its Applications. https://doi.org/10.4108/eai.11-12-2019.2290817 Souissi, M., & Hafdhi, S. (2021). On ranking by using weighted self-normalizing distance metrics in multi-attribute decision-making. Decision Science Letters, 10, 463- 470. https://doi.org/10.5267/j.dsl.2021.7.003 Stanujkic, M., Stanujkic, D., Karabasevic, D., Sava, C., & Popovic, G. (2020). Comparison Of Tourism Potentials Using Preference Selection Index Method. QUAESTUS multidisciplinary research journal, 16, 177-187. Tien, D.H., Trung, D.D., Thien, N.V., & Nguyen, N.T. (2021). Multi-objective optimization of the cylindrical grinding process of scm440 steel using preference selection index method. Journal of Machine Engineering, 21(3), 110-123. https://doi.org/10.36897/jme/141607 Trung, D.D. (2022). Comparison R and CURLI methods for multi-criteria decision making. Advanced Engineering Letters, 1(2), 46-56. https://doi.org/10.46793/adeletters.2022.1.2.3 https://doi.org/10.1016/S0377-2217(03)00020-1 https://doi.org/10.1051/mfreview/2022010 https://doi.org/10.5937/sjm16-27758 https://doi.org/10.5937/sjm16-27758 https://doi.org/10.24988/ije.2019344859 https://eudl.eu/proceedings/WMA-3/2019 https://doi.org/10.46793/adeletters.2022.1.2.3 Investigation of the Appropriate Data Normalization Method for Combination with Preference Selection Index Method in MCDM Trung, D.D. (2022). Expanding data normalization method to CODAS method for multi- criteria decision making. Applied Engineering Letters, 7(2), 54-66. https://doi.org/10.18485/aeletters.2022.7.2.2 Tus, A., & AdalΔ±, E. A. (2018). Personnel assessment with CODAS and PSI methods. Alphanumeric Journal, 6(2), 243-256. http://dx.doi.org/10.17093/alphanumeric.432843 Vafaei, N., Ribeiro, R. A., & Camarinha-Matos, L. M. (2016). Normalization techniques for multi-criteria decision making: analytical hierarchy process case study. Doctoral conference on computing, electrical and industrial systems, 140, 261-269. https://doi.org/10.1007/978-3-319-31165-4_26 Vafaei, N., Ribeiro, R.A., & Camarinha-Matos, L.M. (2018). Data normalisation techniques in decision making: case study with TOPSIS method. International journal of information and decision sciences, 10(1), 19-38. https://doi.org/10.1504/IJIDS.2018.090667 Vafaei, N., Ribeiro, R.A., & Camarinha-Matos, L.M. (2021). Assessing Normalization Techniques for TOPSIS Method. Technological Innovation for Applied AI Systems. DoCEIS 2021. IFIP Advances in Information and Communication Technology, 626, 132-141. https://doi.org/10.1007/978-3-030-78288-7_13 Vafaei, N., Ribeiro, R.A., & Camarinha-Matos, L.M. (2022). Assessing Normalization Techniques for Simple Additive Weighting Method. Procedia Computer Science, 199, 1229-1236. https://doi.org/10.1016/j.procs.2022.01.156 Yadav, R. (2022). Fabrication, characterization, and optimization selection of ceramic particulate reinforced dental restorative composite materials. Polymers and Polymer Composites, 30, 1-10. https://doi.org/10.1177/09673911211062755 Yadav, S., Pathak, V. K., & Gangwar, S. (2019). A novel hybrid TOPSIS-PSI approach for material selection in marine applications. Sādhanā, 44, 58. https://doi.org/10.1007/s12046-018-1020-x Yazdani, M., Jahan, A., & Kazimieras Zavadskas, E.K. (2017). Analysis in material selection: Influence of normalization tools on COPRAS-G. Economic Computation & Economic Cybernetics Studies & Research, 51(1), 59-74. https://EconPapers.repec.org/RePEc:cys:ecocyb:v:50:y:2017:i:1:p:59-74 Zavadskas, E. K., Stanujkic, D., Karabasevic, D., & Turskis, Z. (2022). Analysis of the Simple WISP Method Results Using Different Normalization Procedures. Studies in informatics and control, 31(1), 5-12. https://doi.org/10.24846/v31i1y202201 Zolfani, S., Yazdani, M., Pamucar, D., & Zarate, P. (2020). A VIKOR and TOPSIS focused reanalysis of the MADM methods based on logarithmic normalization. Facta Universitatis: Mechanical Engineering, 18(3), 341 – 355. https://doi.org/10.48550/arXiv.2006.08150 Zopounidis, C., & Doumpos, M. (2017). Multiple Criteria Decision Making - Applications in Management and Engineering. Springer. https://doi.org/10.1007/978-3-319- 39292-9 Β© 2022 by the authors. Submitted for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.18485/aeletters.2022.7.2.2 http://dx.doi.org/10.17093/alphanumeric.432843 https://doi.org/10.1007/978-3-319-31165-4_26 https://doi.org/10.1504/IJIDS.2018.090667 https://doi.org/10.1007/978-3-030-78288-7_13 https://doi.org/10.1016/j.procs.2022.01.156 https://doi.org/10.1177/09673911211062755 https://doi.org/10.1007/s12046-018-1020-x https://econpapers.repec.org/RePEc:cys:ecocyb:v:50:y:2017:i:1:p:59-74 https://doi.org/10.48550/arXiv.2006.08150 https://doi.org/10.1007/978-3-319-39292-9 https://doi.org/10.1007/978-3-319-39292-9 INVESTIGATION OF THE appropriate data normalization method FOR COMBINATION WITH Preference selection index METHOD IN MCDM Duc Trung Do 1, Van Dua Tran 1, Van Duc Duong 1, Nhu-Tung Nguyen 2* 1. Introduction 2. Literature review 3. PSI Method 4. Results and Discussion 4.1. Application Cases Case 1 Case 2 Case 3 Case 4 4.2. Sensitivity Analysis 4.3. The appropriate DNM for combination with PSI Method 5. Conclusion References