Operational Research in Engineering Sciences: Theory and Applications Vol. 5, Issue 3, 2022, pp. 131-152 ISSN: 2620-1607 eISSN: 2620-1747 DOI: https:// https://doi.org/10.31181/oresta051022061d * Corresponding author. doductrung@haui.edu.vn (D. T. Do) APPLICATION OF FUCA METHOD FOR MULTI-CRITERIA DECISION MAKING IN MECHANICAL MACHINING PROCESSES Duc Trung Do Hanoi University of Industry, Vietnam Received: 12 August 2022 Accepted: 29 September 2022 First online: 05 October 2022 Research paper Abstract: Multi-criteria decision making (MCDM) is a very useful tool to find the best solution among many solutions. For most MCDM methods, the data must be normalized. However, the data normalization method has a significant influence on the results of ranking solutions. Choosing the right data normalization method is sometimes complicated. In many MCDM methods, FUCA is known as the method without using normalize the data. However, the FUCA method has a small limitation. All publications that were applied this method have not mentioned this limitation. In this study, this limitation was overcome and then used for multi-criteria decision making in some cases in the mechanical processing field. The ranked results of the solutions when determined by the FUCA method are compared with those ones when using other MCDM methods. The sensitivity analysis was also performed. The results show that the FUCA method can be used for multi-criteria decision making in mechanical machining. It is also expected to be successful when applying in other fields. The works in the future were mentioned in the last section of this article as well. Keywords: MCDM, FUCA method, Mechanical machining 1. Introduction The decision to choose one of many solutions always happens in many situations in many different fields. Each solution is described by different criteria, in which, there are criteria as the larger the better such as machining productivity, tool life, and product quality, etc. Conversely, there are also criteria as the smaller the better such as cost, energy consumption, etc. In these cases, the decision making to select a solution is known as “multi-criteria decision making” (Zopounidis & Doumpos, 2017). Over the years, MCDM methods have received more and more attention from many scholars. A common feature of most MCDM methods is the need to perform the data normalization (Zopounidis & Doumpos, 2017). The criteria with different dimensions D. T. Do /Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 131-152 132 are converted to the same dimensionless form as the basis for ranking options, which is the goal of data normalization (Wen et al. 2020; Krishnan, 2022). However, the data normalization method in each MCDM method is not exactly the same, which leads to different ranking results of the MCDM methods (Aytekin, 2021; Ersoy, 2021; Palczewski & Sałabun, 2019; Lakshmi & Venkatesan, 2014). The rank inversion phenomenon can also occur if the selected data normalization method is not suitable with the MCDM method (Trung, 2022). Currently, many MCDM methods have been proposed by reseachers, it is quite difficult for decision makers to choose one of them in ranking process. FUCA is known as a multi-criteria decision making method without using data normalization (Fernando et al., 2011). Simple steps to implement decision making using this method, its limitations as well as improvements to overcome those limitations will be presented in the next sections of this paper. Baydas (2022) simultaneously used three methods including MOORA, MABAC, and FUCA to assess the rankings of companies in the period before and after the Covid 19 pandemic. The author showed that the FUCA method gives more effective than the other two methods. In another study, Baydas (2022) used two methods FUCA and WSA to evaluate the financial performance of companies. The results of this study show that the FUCA method is better than the WSA method in finding the best solution. In another study, Baydas & Pamucar (2022) also used the FUCA method to evaluate the financial performance of companies. In addition to the FUCA method, in this study, six other methods were used simultaneously including PROMETHEE, COPRAS, TOPSIS, SAW, CODAS, and MOORA. Their results showed that FUCA and PROMETHEE were equally effective in finding the best solution, and that these two methods were better than the other five ones. Baydas et al. (2022) one time again used simultaneously ten multi- criteria decision making methods including FUCA, PROMETHEE, TOPSIS, GRA, S-, WSA, SAW, COPRAS, MOORA, and LINMAP to evaluate the financial performance of twenty- three companies. The authors concluded that the two methods FUCA and PROMETHEE were equally effective and better than other eight methods. Ouattara et al. (2022) used two methods TOPSIS and FUCA to make multi-criteria decisions in the selection of chemical manufacturing processes. They confirmed that the FUCA method is better than the TOPSIS method. The analysis results from some of the above studies show that the FUCA method has been successful in ranking the solutions in the economic and chemical manufacturing fields. It has also been determined to be better or equivalent to other MCDM methods. However, the number of studies that have applied this method is very limited. This method has never been applied to multi-criteria decision making in the field of mechanical processing. The application of FUCA method in multi-criteria decision-making in mechanical processing is a novelty and is also the first reason to conduct this study. It is important to note that the FUCA method has a small limitation. This limitation has not been considered in any published studies. That limitation occurs when a certain criterion has equal value in two or more solutions. The detailed analysis of this limitation of the FUCA method as well as the improvement to overcome this limitation will be presented in section 2 of this paper. This is also the second reason for doing this study. From the above analysis, the structure of the next sections of this paper includes: (1) Discovering the limitation of the FUCA method and improving this method to overcome the limitation; (2) Apply FUCA method for multi-criteria decision making for some common mechanical machining processes. In each example, the data were https://ieeexplore.ieee.org/author/37087868293 https://www.sciencedirect.com/science/article/abs/pii/S0098135411002870?via%3Dihub#! Application of FUCA Method for Multi-Criteria Decision Making in Mechanical Machining Processes 133 referenced from published studies. The ranking results of the solutions when using FUCA method were compared to that ones when using other MCDM methods. The sensitivity analysis in each case was also performed for different scenarios; (3) discussing about the achieved results; and (4) conclusion of this study and proposal of the further studies are the closing content of this paper. 2. FUCA Method The FUCA method performs the ranking of solutions in just three simple steps as follows (Fernando et al. 2021): Step 1. Rank the solutions for each criterion (rij). Suppose there are m solutions, the best value will be ranked 1, otherwise the worst value will be ranked m. If there are n criteria, perform n ranking times for each criterion. However, at this step, we have noticed a limitation of the FUCA method that when a certain criterion has the same value in two or more solutions, how will the ranking of the solutions (for each criterion) be implemented? To clarify this issue, a simple example is presented as below. Suppose there are four solutions including A1, A2, A3, and A4, each of which is described by three criteria C1, C2, and C3, where C1 and C2 are the criteria as the larger the bettere, and C3 is the criterion as the smaller the better as shown in Table 1. Table 1. Example of a certain criterion having equal value in several solutions No. Criteria C1 C2 C3 A1 4 3 4 A2 6 5 2 A3 2 5 4 A4 8 7 4 The ranking of alternatives for each criterion will be conducted as follows. For criterion C1 (the larger the better): A4 ranked 1, A2 ranked 2, A1 ranked 3, and A2 ranked 4. For this criterion, its values in the four solutions are different. So the ranking process is performed easy. For criterion C2 (the larger the better): Because C2 at A4 is the largest, so A4 ranked 1, C2 at A1 is the smallest, so A1 ranked 4. However, C2 at A2 and A3 are equal. So, what is the ranking order of A2 and A3? A simple proposal that A2 and A3 should have the same rank, and equal to 2.5 (the average of 2 and 3). For criterion C3 (the smaller the better): because C3 at A2 is the smallest, so A2 is ranked 1. C3 has the same value in three solutions A1, A3, and A4, so all three solutions ranked 3 (the average of 2, 3, and 4). From above analyzed results, a table of the ranking results of the solutions for the data in Table 1 was presented in Table 2. https://ieeexplore.ieee.org/author/37087868293 D. T. Do /Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 131-152 134 Table 2. The ranked results of the solutions according to the data in table 1 No. Rank C1 C2 C3 A1 3 4 3 A2 2 2.5 1 A3 4 2.5 3 A4 1 1 3 Step 2. Calculate the score of each solution according to the Eq. (1). 𝑣𝑖 = ∑ 𝑟𝑖𝑗 . 𝑤𝑗 𝑛 𝑗=1 (1) where wj is the weight of the criterion j. Step 3. Rank the solutions by the value of vi. The solution with the smallest vi is the best one, and vice versa. The discovery of the limitation of the FUCA method as well as the proposed method to overcome that limitation were performed. To evaluate the effectiveness of this remedial method, in the next sections of this study, the proposed method will be applied to multi-criteria decision making in some cases in the mechanical processing field. Because the main purpose of this study is the application of the FUCA method for multi-criteria decision making in mechanical machining processes, the data are therefore all referenced from the published studies, the number of criteria in each case is not the same. Two main reasons for performing this content include: first, not spending too much effort on conducting the experiments; and second, published studies have used other MCDM methods to rank solutions. The ranking results of the solutions when using those MCDM methods are used to compare to those ones when using the FUCA method. In each case, first, the weight of the criteria that was used was the value in the published studies. Then, in each case, the sensitivity analysis was also performed for different scenarios by varying the weights of the criteria. The number of the generated scenarios in each case also varies. The implementation of examples in different mechanical processing processes, the number of criteria in different situations, the number of different scenarios aim to draw the most general conclusions. 3. Applying the FUCA method for Multi-Criteria Decision Making in Several Cases 3.1. Multi-Criteria Decision Making in Milling Process (example 1) This case used the experimental data of the milling process of Ti-6Al-4V alloy by Nguyen et al. (2021). In that study, they conducted nine experiments, each of which changed three parameters including cutting speed, feed rate, and depth of cut. Two criteria were measured in each experiment including surface roughness (C1) and material removal rate (C2). The experimental data are presented in Table 3. In which Application of FUCA Method for Multi-Criteria Decision Making in Mechanical Machining Processes 135 C1 is the smaller the better criterion, C2 is the larger the better criterion. In addition, in that study, they used the Entropy method to determine the weights for the criteria, and the determined weights of C1 and C2 were 0.2906 and 0.7094, respectively. That study also used the TOPSIS method for multi-criteria decision making with the aim of determining the solution Ai (with i = 1 ÷ 9) with simultaneously ensuring the smallest C1 and the largest C2. Table 3. Experimental data when milling process of alloy Ti-6Al-4V (Nguyen et al. 2021). No. Criteria C1 (m) C2 (cm3/min) A1 0.281 5.42 A2 0.337 1.08 A3 0.737 16.25 A4 0.328 21.67 A5 0.321 10.83 A6 0.507 2.17 A7 0.359 32.5 A8 0.412 43.33 A9 0.636 16.52 The ranking of the solutions according to the FUCA method will be performed as follows. Step 1. Rank the solutions for each criterion. In this case, both criteria C1 and C2 have different values for all solutions, so the ranking of solutions according to the FUCA method is conducted normally. The results are presented in the Table 4. Table 4. Ranking the solutions for each criterion (example 1) No. Rank (rij) C1 C2 A1 1 7 A2 4 9 A3 9 4 A4 3 3 A5 2 6 A6 7 8 A7 5 2 A8 6 1 A9 8 5 Step 2. Calculate the score of each solution according to Eq (1). First of all, the weights of the selected criteria are the same as their values in the referenced literature, i.e., the weights of C1 and C2 are 0.2906 and 0.7094, respectively (Nguyen et al. 2021). The calculated results are presented in Table 5. D. T. Do /Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 131-152 136 Table 5. The vi score of each solution (example 1) No. rij. wj vi C1 C2 A1 1 7 5.2564 A2 4 9 7.5470 A3 9 4 5.4530 A4 3 3 3.0000 A5 2 6 4.8376 A6 7 8 7.7094 A7 5 2 2.8718 A8 6 1 2.4530 A9 8 5 5.8718 Step 3. Ranking the solutions according to the value of vi, the calculated results are presented in Table 6. The ranking results of the solutions when using the TOPSIS method are also presented in this table. Table 6. Ranking the solutions for example 1 No. Rank FUCA TOPSIS A1 5 7 A2 8 9 A3 6 4 A4 3 3 A5 4 6 A6 9 8 A7 2 2 A8 1 1 A9 7 5 The calculated results from Table 6 show that when using the improved FUCA method, it was determined that A8 is the best solution. This result is also similar to the result when ranking solutions by TOPSIS method (Nguyen et al. 2021). In addition, the second ranked solution (A7) and the third ranked solution (A4) also coincide when using both improved FUCA and TOPSIS methods. Thus, in this case, it is seen that when using the same set of weight values, two methods including improved FUCA and TOPSIS are considered to be equally effectiveness. However, in order to evaluate the effectiveness of a certain MCDM method in each case, the last work that needs to be done is the sensitivity analysis (Bozanic et al. 2021; Muhammad et al. 2021). Many studies have performed the sensitivity analysis by changing the weighted values of the criteria and using Sperman's rank correlation coefficient (Bobar et al. 2020; Pamucar et al. 2021; Dimic et al. 2019; Le et al. 2022; Lamba et al. 2019). In this study, the sensitivity analysis was also performed in the same way. The Sperman's rank correlation coefficient is determined according to Eq. (2). 𝑆 = 1 − 6 ∑ 𝐷𝑖 2𝑛 𝑖=1 𝑛(𝑛2 − 1) (2) Application of FUCA Method for Multi-Criteria Decision Making in Mechanical Machining Processes 137 where Di presents the difference of the rank according to the given scenario and the rank in the corresponding scenario, and n is the number of ranked elements. Six different scenarios were created by randomly changing the weights of the criteria as presented in Table 7. In which, S4 is the scenario just implemented above. Table 7. Weight of criteria in different scenarios (example 1) Criteria Scenarios S1 S2 S2 S4 S5 S6 C1 0.2 0.22 0.25 0.2906 0.3 0.35 C2 0.8 0.78 0.75 0.7094 0.7 0.65 The ranked results solutions according to six different scenarios are presented in Table 8. We see that in all six given scenarios, A8 is still the best solution. Table 8. Ranking the solutions in different scenarios (example 1) No. Scenarios S1 S2 S3 S4 S5 S6 A1 7 7 6 5 5 5 A2 9 9 9 8 8 8 A3 4 4 5 6 6 6 A4 3 3 3 3 3 2 A5 5 5 4 4 4 4 A6 8 8 9 9 9 9 A7 2 2 2 2 2 3 A8 1 1 1 1 1 1 A9 6 6 7 7 7 7 Table 9 presents the values of the Spearman coefficients calculated according to formula (2) for comparison between scenarios as well as comparison of the initial ranking Si. Table 9. The values of Sperman’s rank correlation coefficients (example 1) Si S1 S2 S3 S4 S5 S6 Si 1 1 1.000 0.958 0.900 0.900 0.883 S1 1 1.000 0.958 0.900 0.900 0.883 S2 1 0.958 0.900 0.900 0.883 S3 1 0.975 0.975 0.958 S4 1 1.000 0.983 S5 1 0.983 S6 1 The calculed results in Table 9 show that the Sperman's rank correlation coefficient of the solution is in the range S  [0.883, 1]. It means the degree of correlation is very high. This shows that the change in rankings is not significant even though the weight of the criteria changed with a relatively large degree (the weight of C1 changed from 0.2 to 0.35, the weight of C2 changed from 0.8 to 0.65). One great thing that was achieved is that solution A8 is always determined to be the best one of all scenarios. D. T. Do /Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 131-152 138 Thus, a solid conclusion is drawn that the FUCA method was successful in solving the problem in this example. 3.2. Multi-Criteria Decision Making in Turning Process (example 2) Singh et al. (2019) conducted twenty-seven experiments when turning Ti-6Al-4V steel. In each experiment, the input parameters were adjusted in each experiment including cutting speed, feed rate, and depth of cut. The criteria that were used to evaluate each solution included tool wear (C1), surface roughness (C2), cutting heat (C3), and cutting force (C4). All four of these criteria are the smaller tha better criteria. The values of the criteria at the solutions are as presented in Table 10. Table 10. Experimental data when turning process of steel (Singh et al. 2019) No. Criteria C1 (m) C2 (m) C3 (0C) C4 (N) A1 70 0.5 405 310 A2 85 0.53 410 315 A3 95 0.55 420 323 A4 110 0.62 440 295 A5 135 0.68 445 300 A6 120 0.6 435 298 A7 195 0.76 503 290 A8 180 0.72 490 280 A9 190 0.74 495 285 A10 118 0.62 438 296 A11 125 0.66 443 295 A12 132 0.69 455 305 A13 175 0.75 485 283 A14 180 0.73 490 289 A15 190 0.75 500 292 A16 65 0.52 410 314 A17 90 0.56 415 321 A18 98 0.57 425 325 A19 168 0.73 485 288 A20 175 0.74 497 284 A21 188 0.78 501 290 A22 92 0.54 415 328 A23 100 0.55 420 320 A24 105 0.57 425 332 A25 115 0.62 448 302 A26 130 0.63 450 308 A27 140 0.65 447 310 In that study, the ranking of the solutions by TOPSIS and SAW methods was also performed. In which, the weights of C1, C2, C3, and C4 were determined by the AHP method, and those values were 0.5846, 0.2570, 0.1088, and 0.0556, respectively. The application of the FUCA method to ranking solutions is similar to the example in section 3.1. However, in this case, the value of each criterion is equal in some Application of FUCA Method for Multi-Criteria Decision Making in Mechanical Machining Processes 139 solutions. Therefore, the ranking of the solutions for each criterion will have to consider the proposed solution. The specific steps are as follows. For criterion C1, the ranks from rank 1 to rank 19 are ranked normally. Because C1 at A13 and A20 are equal to each other, both A13 and A20 ranked 20.5 (average of 20 and 21); C1 at A8 and A14 are equal each other, both A8 and A14 ranked 22.5 (average of 22 and 23); C1 at A9 and A15 are equal each other, both A9 and A15 ranked 25.5 (average of 25 and 26). For criterion C2, the ranks from rank 1 to rank 4 are ranked normally. Because C2 at A3 and A23 are equal, both A3 and A23 ranked 5.5 (average of 5 and 6); C2 at A18 and A24 are equal each other, both A18 and A24 ranked 8.5 (average of 8 and 9); ect. For the remaining criteria (C3 and C4), the ranking of solutions was performed similarly to this method. The ranking results of the solutions for each criterion are presented in Table 11. Table 11. Ranking the solutions for each criterion in example 2 No. Criteria Rank (rij) C1 C2 C3 C4 C1 C2 C3 C4 A1 70 0.5 405 310 2 1 1 18.5 A2 85 0.53 410 315 3 3 2.5 21 A3 95 0.55 420 323 6 5.5 6.5 24 A4 110 0.62 440 295 10 12 12 10.5 A5 135 0.68 445 300 17 17 14 14 A6 120 0.6 435 298 13 10 10 13 A7 195 0.76 503 290 27 26 27 7.5 A8 180 0.72 490 280 22.5 19 21.5 1 A9 190 0.74 495 285 25.5 22.5 23 4 A10 118 0.62 438 296 12 12 11 12 A11 125 0.66 443 295 14 16 13 10.5 A12 132 0.69 455 305 16 18 18 16 A13 175 0.75 485 283 20.5 24.5 19.5 2 A14 180 0.73 490 289 22.5 20.5 21.5 6 A15 190 0.75 500 292 25.5 24.5 25 9 A16 65 0.52 410 314 1 2 2.5 20 A17 90 0.56 415 321 4 7 4.5 23 A18 98 0.57 425 325 7 8.5 8.5 25 A19 168 0.73 485 288 19 20.5 19.5 5 A20 175 0.74 497 284 20.5 22.5 24 3 A21 188 0.78 501 290 24 27 26 7.5 A22 92 0.54 415 328 5 4 4.5 26 A23 100 0.55 420 320 8 5.5 6.5 22 A24 105 0.57 425 332 9 8.5 8.5 27 A25 115 0.62 448 302 11 12 16 15 A26 130 0.63 450 308 15 14 17 17 A27 140 0.65 447 310 18 15 15 18.5 D. T. Do /Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 131-152 140 After ranking the solutions for each criterion, apply Eq. (1) to calculate the value of i. First, the weights of the selected criteria are the same as their values in the references, i.e., the weights of C1, C2, C3, and C4 are 0.5846, 0.2570, 0.1088, and 0.0556, respectively (Singh et al. 2019). The ranked results of the solutions by FUCA method and two other methods (including TOPSIS and SAW) are presented in Table 12. Table 12. Ranking the solutions for example 2 No. FUCA TOPSIS SAW A1 1 1 1 A2 3 3 3 A3 6 5 6 A4 10 11 11 A5 16 17 17 A6 12 10 10 A7 27 26 26 A8 20 19 20 A9 24 23 24 A10 11 13 12 A11 14 16 15 A12 17 18 18 A13 21 24 23 A14 23 21 21 A15 26 25 25 A16 2 2 2 A17 5 7 5 A18 8 8 8 A19 19 20 19 A20 22 22 22 A21 25 27 27 A22 4 4 4 A23 7 6 7 A24 9 9 9 A25 13 12 13 A26 15 14 14 A27 18 15 16 The calculated results in Table 12 show that using the FUCA method, A1 was identified as the best solution. This result is also consistent with cases using two methods including TOPSIS and SAW. In addition, all three methods jointly identify that A16 solution ranked 2, and A2 solution ranked 3. Seven different scenarios were generated by randomly varying the weights of the criteria as presented in Table 13. Where S7 is the scenario that was performed above. Application of FUCA Method for Multi-Criteria Decision Making in Mechanical Machining Processes 141 Table 13. Weight of criteria in different scenarios (example 2) Criteria Scenarios S1 S2 S3 S4 S5 S6 S7 C1 0.1 0.2 0.3 0.3 0.3 0.4 0.5846 C2 0.2 0.1 0.2 0.1 0.3 0.4 0.2570 C3 0.3 0.4 0.1 0.3 0.1 0.1 0.1088 C4 0.4 0.3 0.4 0.3 0.3 0.1 0.0556 The ranking results of the solutions according to different scenarios are presented in Table 14. The calculated results show that in all 7 scenarios, it is always determined that A1 is the best solution, A16 ranked 2, A2 ranked 3, and A7 ranked 27. Table 14. Ranking the solutions in different scenarios (example 2) No. Scenarios S1 S2 S3 S4 S5 S6 S7 A1 1 1 1 1 1 1 1 A2 3 3 3 3 3 3 3 A3 12 9 11 7 7 6 6 A4 4 6 4 5 5 10 10 A5 16 15 19 17 18 16 16 A6 5 8 6 10 9 11 12 A7 27 27 27 27 27 27 27 A8 10 18 10 18 15 20 20 A9 20 24 21 24 24 24 24 A10 6 10 5 9 10 12 11 A11 9 11 8 11 13 14 14 A12 24 23 23 21 21 18 17 A13 13 16 15 15 19 21 21 A14 19 22 18 23 22 22 23 A15 25 26 26 26 26 25 26 A16 2 2 2 2 2 2 2 A17 7 4 7 4 4 5 5 A18 17 12 17 12 11 8 8 A19 14 17 14 16 16 19 19 A20 18 21 16 20 20 23 22 A21 26 25 25 25 25 26 25 A22 11 5 12 6 6 4 4 A23 8 7 9 8 8 7 7 A24 21 13 22 14 14 9 9 A25 15 14 13 13 12 13 13 A26 22 19 20 19 17 15 15 A27 23 20 24 22 23 17 18 Eq. (2) is used to calculate the Spearman’s rank correlation coefficients. Table 15 presents the values of the Spearman coefficients when comparing between scenarios as well as the initial rank Si. D. T. Do /Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 131-152 142 Table 15. The values of Sperman coefficients (example 2) Si S1 S2 S3 S4 S5 S6 S7 Si 1 1 0.904 0.988 0.907 0.897 0.752 0.751 S1 1 0.904 0.988 0.907 0.897 0.752 0.751 S2 1 0.886 0.991 0.980 0.947 0.946 S3 1 0.901 0.901 0.744 0.747 S4 1 0.988 0.938 0.941 S5 1 0.946 0.948 S6 1 0.998 S7 1 The calculated data in Table 15 show that the Sperman's rank correlation coefficients of the solutions is in the range S  [0.747, 1], this value represents a very high degree of correlation. This shows that the change in rankings is not significant even though the weight of the criteria changed with a relatively large degree. Specifically, although C1 changed 5.846 times, C2 and C3 changed 4 times, and C4 changed 7.19 times, the solutions ranked 1st, 2nd, 3rd, and 27th are all same to each other in all seven scenarios. Thus, for each criterion, when ranking solutions with equal value in several solutions was implemented according to the proposed method, the FUCA method was also successful in solving the problem of this example. 3.3. Multi-Criteria Decision Making in Drill Process of Magnesium AZ91 Material (example 3) Varatharajulu et al. (2021) performed the drilling process of magnesium AZ91 in seventeen different experiments. In each experiment the input parameters are changed including spindle speed and feed rate. Six criteria that were used to evaluate each experiment included drilling time (C1), entry burr height (C2), exit burr height (C3), entry burr thickness (C4), exit burr thickness (C5), and surface roughness (C6). All six of these criteria are the smaller the better criteria. The data on the criteria for the seventeen experiments is presented in Table 16. The multi-criteria decision-making that was performed to find a solution that ensures simultaneously all six criteria to be the same minimum values using TOPSIS and COPRAS methods (Varatharajulu et al. 2021). In which, the weights of C1 and C6 were chosen to be 0.3 and the weights of all the remaining four criteria were chosen to be 0.1. The application of the FUCA method to rank solutions was performed similarly to the example in section 3.1. It is note with the cases that one certain criterion is equally valid in several solutions. This process was presented follows. The values of criterion C1 are different in all seventeen solutions, so ranking of the solutions for this criterion is performed normally. For criterion C2: C2 at A15 is the smallest, so A15 ranked 1st; C2 at A8, A9, and A12 are equal to each other, so all three solutions are ranked 3 (the average of 2, 3, and 4); C2 at A4 is equal to C2 at A7, so both solutions ranked 5.5 (average of 5 and 6); C2 at A10, A11, and A16 are equal to each other, so all three solutions ranked 11 (average of 10, 11, and 12), ect. Application of FUCA Method for Multi-Criteria Decision Making in Mechanical Machining Processes 143 Table 16. Experimental data when drilling process of magnesium material (Varatharajulu et al. 2021) No. Criteria C1 (s) C2 (mm) C3 (mm) C4 (mm) C5 (mm) C6 (m) A1 14.03 0.051 0.058 0.105 0.21 0.479 A2 7.59 0.053 0.058 0.155 0.245 1.211 A3 7.34 0.035 0.06 0.165 0.215 0.916 A4 4.06 0.033 0.075 0.18 0.215 0.535 A5 5.4 0.048 0.078 0.25 0.195 0.601 A6 5.5 0.05 0.084 0.185 0.185 0.703 A7 2.81 0.033 0.058 0.185 0.185 0.466 A8 2.62 0.028 0.048 0.2 0.19 0.577 A9 2.88 0.028 0.05 0.18 0.15 0.417 A10 2.75 0.043 0.051 0.23 0.195 0.675 A11 2.84 0.043 0.055 0.165 0.205 0.418 A12 1.59 0.028 0.074 0.145 0.17 0.601 A13 1.88 0.038 0.064 0.185 0.175 0.563 A14 3.44 0.049 0.066 0.19 0.185 0.391 A15 2.04 0.023 0.059 0.16 0.18 0.493 A16 2.1 0.043 0.05 0.235 0.185 0.675 A17 1.25 0.04 0.049 0.44 0.19 0.65 Table 17. Ranking the solutions when drilling process of magnesium material No. Rank (rij) Rank C1 C2 C3 C4 C5 C6 FUCA TOPSIS COPRAS A1 17 16 8 1 14 5 13 17 17 A2 16 17 8 3 17 17 17 16 16 A3 15 7 11 5.5 15.5 16 15 15 15 A4 12 5.5 15 7.5 15.5 7 11 12 12 A5 13 13 16 16 11.5 10.5 14 13 13 A6 14 15 17 10 6.5 15 16 14 14 A7 8 5.5 8 10 6.5 4 4 5 6 A8 6 3 1 13 9.5 9 6 7 7 A9 10 3 3.5 7.5 1 2 2 2 2 A10 7 11 5 14 11.5 13.5 12 10 11 A11 9 11 6 5.5 13 3 7 6 5 A12 2 3 14 2 2 10.5 3 4 3 A13 3 8 12 10 3 8 5 3 4 A14 11 14 13 12 6.5 1 9 8 8 A15 4 1 10 4 4 6 1 1 1 A16 5 11 3.5 15 6.5 13.5 10 9 9 A17 1 9 2 17 9.5 12 8 11 10 The ranking of the remaining criteria (C3, C4, C5, C6) was also conducted in a similar way. The ranked results of the solutions for each criterion are presented in Table 17. The data in Table 17 show that the FUCA method indicates that A15 is the best solution. This result is also similar to the results when using TOPSIS and COPRAS D. T. Do /Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 131-152 144 methods. In addition, all three methods FUCA, TOPSIS, and COPRAS identified that A9 ranked 2. Eight different scenarios were also generated by randomly varying the weights of the criteria as shown in Table 18, where S5 scenario was the just analyzed above. Table 18. Weight of criteria in different scenarios (example 3) Criteria Scenarios S1 S2 S3 S4 S5 S6 S7 S8 C1 0.2 0.2 0.25 0.28 0.3 0.32 0.33 0.35 C2 0.15 0.1 0.15 0.2 0.1 0.1 0.15 0.15 C3 0.2 0.1 0.15 0.2 0.1 0.1 0.1 0.1 C4 0.2 0.2 0.15 0.2 0.1 0.1 0.15 0.1 C5 0.15 0.15 0.2 0.1 0.1 0.1 0.1 0.15 C6 0.1 0.25 0.1 0.02 0.3 0.28 0.17 0.15 The ranking results of the solutions according to the different scenarios are presented in Table 19. It is seen that in all eight scenarios, A15 is always determined to be the best solution. Table 19. Ranking the solutions in different scenarios (example 3) No. Scenarios S1 S2 S3 S4 S5 S6 S7 S8 A1 11 10 13 13 13 13 13 13 A2 15 17 16 15 17 17 15 17 A3 14 14 14 12 15 15 14 14 A4 13 12 12 11 11 11 12 12 A5 17 16 17 17 14 14 16 15 A6 16 15 15 16 16 16 17 16 A7 5 4 6 7 4 5 7 7 A8 4 7 5 4 6 6 5 5 A9 2 2 3 3 2 2 3 3 A10 10 13 10 10 12 12 11 11 A11 8 6 9 9 7 7 8 9 A12 3 3 2 2 3 3 2 2 A13 6 5 4 6 5 4 4 4 A14 12 8 11 14 9 9 10 10 A15 1 1 1 1 1 1 1 1 A16 9 11 8 8 10 10 9 8 A17 7 9 7 5 8 8 6 6 Eq. (2) is again used to calculate the Sperman coefficients. Table 20 presents the values of the Sperman coefficients when comparing between scenarios as well as the initial rank Si. The data in Table 20 show that the Sperman's rank correlation coefficients of the solutions is in the range S  [0.853, 1], which means that the correlation level in this case is very high. This shows that the change in rankings is not significant even though the weight of the criteria changed with a relatively large degree. Specifically, the weight of C1 changed from 0.2 to 0.35, the weight of four criteria C2, C3, C4, and C5 all changed from 0.1 to 0.2. In particular, the weight of C6 changed from 0.02 to 0.3. In all Application of FUCA Method for Multi-Criteria Decision Making in Mechanical Machining Processes 145 scenarios, A15 is always determined to be the best solution. One time again, the FUCA method was confirmed as a successful applied method in this example. Table 20. The values of Sperman’s rank correlation coefficients (example 3) Si S1 S2 S3 S4 S5 S6 S7 S8 Si 1 1 0.931 0.978 0.966 0.946 0.944 0.971 0.961 S1 1 0.931 0.978 0.966 0.946 0.944 0.971 0.961 S2 1 0.924 0.853 0.973 0.971 0.929 0.924 S3 1 0.968 0.953 0.958 0.985 0.988 S4 1 0.897 0.900 0.961 0.956 S5 1 0.998 0.961 0.963 S6 1 0.968 0.971 S7 1 0.990 S8 1 3.4. Multi-Criteria Decision Making with the Qualitative Criteria (example 4) The analyzed results in the three examples that were performed above confirmed that the FUCA method was successfully applied when used in each example. However, in all those examples, the the criteria are the quantitative ones. In this example, both qualitative and quantitative criteria will be considered. To implement the content of these cases, the authors of this paper were conducted the surface grinding process of SUJ2 steel with some basic parameters of the experimental system and the experimental conditions as summarized follows: The grinding machine was the APSG- 820/2A machine, grinding wheel was the WA46J7V1A-180-13-31.5, workpiece material was SUJ2 steel; workpiece dimensions (length x width x height) were 60 mm x 40 mm x 10 mm, respectively. The workpiece was heat treated to reach a hardness of 62 HRC, the coolant was 10% emulsion oil with the flow of 4.6 l/min. Eight experiments were carried out with the values of the changed cutting conditions in each experiment as listed in Table 21. Two quantitative criteria include the surface roughness (C1) and material remove rate (C2). The values of C1 and C2 at each experiment are also summarized in Table 21. In addition, in this study, another criterion is used which is the number of the grinding grains adhered in the surface of the part (C3). The number of grinding grains adhered in the surface of the part after grinding has a great influence on the workability of the part. If there are a large number of the grinding grains adhered in the surface of the part of the part, these grinding grains will scratch the surfaces when they contact with each other. It makes the level of wear happening quickly, especially in the initially wear stage. Thereby it will rapidly reduce the life of the product (Malkin & Guo, 2018; Marinescu et al. 2006). Therefore, creating a surface after grinding with a small number of the grinding grains adhered in the surface of the part is always desirable. However, it is very difficult to determine the exact number of the grinding grains adhered in the surface of the part. Instead, we can only evaluate them at the qualitative level, i.e., through the observation (using specialized equipment) to evaluate the number of the grinding grains adhered more or less in the surface of the part. It means that according to this measurement method, C3 is in the form of a qualitative criterion. The evaluation of C3 in this study was performed through the observation of workpiece surface micrographs after grinding (Figure 1). D. T. Do /Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 131-152 146 Table 21. Experimental data when surface grinding process of SUJ2 steel No. Criteria C1 (m) C2 (mm3/min) C3 (in Fig. 1) A1 0.278 325 (1) A2 0.844 1625 (2) A3 1.041 975 (3) A4 1.548 1300 (4) A5 0.502 1950 (5) A6 0.225 650 (6) A7 1.059 2925 (7) A8 1.542 3900 (8) (1) (2) (3) (4) (5) (6) (7) (8) Figure 1. The surface of workpiece in surface grinding process of SUJ2 steel Observation of Figure 1 shows that: In the photo (8) corresponding to the A8, the number of the grinding grains adhered in the surface of the part is the least, thus, C3 at A8 ranked 1. As observed the C3 at A3 and A7 is quite the same and only more that that one at A8, so, both A3 and A7 rated 2.5 (the average of 2 and 3). For the remaining solutions, C3 decrease in order A5, A6, A4, A1, and A2. Therefore, the ranks of A5, A6, A4, A1, and A2 are rank 4, rank 5, rank 6, rank 7, and rank 8, respectively. The ranked results of the solutions for all three criteria are listed in Table 22. Application of FUCA Method for Multi-Criteria Decision Making in Mechanical Machining Processes 147 Table 22. Ranking the solutions for each criterion when surface grinding of SUJ2 steel No. Criteria Rank (rij) C1 (m) C2 (mm3/min) C3 (in Figure 1) C1 C2 C3 A1 0.278 325 (1) 2 8 7 A2 0.844 650 (2) 4 4 8 A3 1.041 975 (3) 5 6 2.5 A4 1.548 1300 (4) 8 5 6 A5 0.502 1950 (5) 3 3 4 A6 0.225 650 (6) 1 7 5 A7 1.059 2925 (7) 6 2 2.5 A8 1.542 3900 (8) 7 1 1 The score of each solution was calculated according to Eq. (1) with six randomly selected different weight sets of the criteria (Table 23). The calculated results are presented in Table 24. The ranked results of the solutions according to the FUCA method as presented in Table 25. Table 23. Weight of criteria in different scenarios (example 4) Criteria Scenarios S1 S2 S3 S4 S5 S6 S7 S8 C1 0.2 0.25 0.28 0.3 0.32 1/3 0.35 0.38 C2 0.3 0.25 0.37 0.4 0.42 1/3 0.35 0.32 C3 0.5 0.4 0.35 0.3 0.26 1/3 0.3 0.3 Table 24. The vi score of each solution (example 4) No. Scenarios S1 S2 S3 S4 S5 S6 S7 S8 A1 6.300 6.100 5.970 5.900 5.820 5.667 5.600 5.420 A2 6.000 5.600 5.400 5.200 5.040 5.333 5.200 5.200 A3 4.050 4.350 4.495 4.650 4.770 4.500 4.600 4.570 A4 6.100 6.150 6.190 6.200 6.220 6.333 6.350 6.440 A5 3.500 3.400 3.350 3.300 3.260 3.333 3.300 3.300 A6 4.800 4.700 4.620 4.600 4.560 4.333 4.300 4.120 A7 3.050 3.200 3.295 3.350 3.410 3.500 3.550 3.670 A8 2.200 2.500 2.680 2.800 2.920 3.000 3.100 3.280 Table 25. Ranking the solutions according to the improved FUCA (example 4) No. Scenarios S1 S2 S3 S4 S5 S6 S7 S8 A1 8 7 7 7 7 7 7 7 A2 6 6 6 6 6 6 6 6 A3 4 4 4 5 5 5 5 5 A4 7 8 8 8 8 8 8 8 A5 3 3 3 2 2 2 2 2 A6 5 5 5 4 4 4 4 4 A7 2 2 2 3 3 3 3 3 A8 1 1 1 1 1 1 1 1 D. T. Do /Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 131-152 148 The data in Table 25 shows that solution A8 is always determined to be the best solution for all scenarios. Seven of the eight scenarios identified A4 as the worst solution (except for S1). The ranking results for all solution are the same in the five scenarios S4, S5, S6, S7, and S8. The two scenarios S2 and S3 also give the same ranking results. In addition, there is only a small difference in ranking results between scenario S1 and the rest. Eq. (2) is again used to calculate the Sperman coefficients. Table 26 presents the value of the Sperman coefficients when comparing between scenarios as well as the initial rank Si. Table 26. The values of Sperman’s rank correlation coefficients (example 4) Si S1 S2 S3 S4 S5 S6 S7 S8 Si 1 1 0.943 0.943 0.829 0.829 0.829 0.829 0.829 S1 1 0.943 0.943 0.829 0.829 0.829 0.829 0.829 S2 1 1 0.886 0.886 0.886 0.886 0.886 S3 1 0.886 0.886 0.886 0.886 0.886 S4 1 1 1 1 1 S5 1 1 1 1 S6 1 1 1 S7 1 1 S8 1 The calculated results in Table 26 show that the Sperman’s rank correlation coefficients of the solutions are in the range S  [0.886, 1]. This represents a very high degree of correlation in this case. Thus, in this example, once again the FUCA method was successfully applied. Although the four examples that were performed belonging to different machining processes (milling, turning, drilling, and grinding). The number of solutions, number of criteria, and number of scenarios that used in each case also were different. However, in each case, the obtained results confirmed the successful application of the FUCA method in multi-criteria decision making. From the obtained results, it can be concluded that the proposed method to overcome the limitations of the FUCA method is an accurate one. So, the application of FUCA method completely ensures the reliability when using for multi-criteria decision making, firstly in the mechanical processing field. 4. Conclusion Having to choose a certain MCDM method to combine with a certain data normalization method to ensure the accuracy of multi-criteria decision making is a relatively complicated work with a lot of time consumption of decision makers. FUCA is a multi-criteria decision making method without requirement of data normalization. When using this method, the first mission is ranking the solutions for each criterion. However, the case with a certain criterion having equal value in several solutions has not considered in any published studies. In that case, the decision maker will not be able to rank the solutions. This is the first study to discover that limitation and to propose a method to overcome that one. With the additional use of the proposed method, the FUCA one was used for multi-criteria decision making for four different Application of FUCA Method for Multi-Criteria Decision Making in Mechanical Machining Processes 149 cases in the mechanical processing field. In each of those cases, the number of solutions, the number of criteria, and the type of criteria (qualitative, quantitative) are also not the same. The sensitivity analysis in ranking process was also performed with different scenarios for each case. Although there are many differences in the examples, the obtained results confirm that the FUCA method was successfully applied in the mentined cases. The discovery of the limitation of the FUCA method and the improvement of this method to overcome its limitation extends the application scope of this method. It was not only successful applied in multi-criteria decision making in the field of mechanical machining as done in this study, but it also promises to be successful applied in other fields as well. The method to overcome the limitation of the FUCA one that was proposed in this study has not been presented in the form of a general mathematical formula. This limitation needs to be implemented in the next time. In addition, this study as well as the published studies that applied the FUCA method only considered the case the values of each criterion at each solution as a unique quantity. The case these values as a fuzzy set have been not considered in any studies. This gap also needs to be filled in the further studies. In this study, the weighted values of the criteria were selected according to the studies that this study references (in those references, the weights were determined by the Entropy, AHP method, ect.), or were selected according to random values without considering the importance of the criteria. The use of weighting methods considering the importance of criteria, such as the PIPRECIA method (Stanujkic et al. 2017) in combination with the FUCA method are also the contents of works to be done in the future. Refercences Aytekin, A. (2021). Comparative Analysis of the Normalization Techniques in the Context of MCDM Problems. 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T. Do /Oper. Res. Eng. Sci. Theor. Appl. 5(3)2022 131-152 152 Abbreviations MCDM: Multi-Criteria Decision Making FUCA: Faire Un Choix Adéquat (in French) - Make an Adequate Choice MOORA: Multiobjective Optimization On the basis of Ratio Analysis MABAC: Multi-Attributive Border Approximation area Comparison WSA: Weighted Sum Approach PROMETHEE: Preference Ranking Organization METHod for Enrichment of Evaluations COPRAS: COmplex PRroportional Assessment TOPSIS: Technique for Order of Preference by Similarity to Ideal Solution S-: Negative Ideal Separation SAW: Simple Additive Weighting CODAS: COmbinative Distance-based Assessment GRA: Grey Relational Analysis LINMAP: LINear programming technique for Multidimensional Analysis of Preference AHP: Analytic Hierarchy Process COPRAS: COmplex PRoportional ASsessment PIPRECIA: PIvot Pairwise RElative Criteria Importance Assessment 3.1. Multi-Criteria Decision Making in Milling Process (example 1) 3.2. Multi-Criteria Decision Making in Turning Process (example 2) 3.3. Multi-Criteria Decision Making in Drill Process of Magnesium AZ91 Material (example 3) 3.4. Multi-Criteria Decision Making with the Qualitative Criteria (example 4) Baydas, M. (2022). Comparison of the Performances of MCDM Methods under Uncertainty: An Analysis on Bist SME Industry Index. OPUS – Journal of Society Research, 19(46), 308-326. https://doi.org/10.26466/opusjsr.1064280 Bobar, Z., Bozanic, D., Djuric, K., & Pamucar, D. (2020). Ranking and Assessment of the Efficiency of Social Media using the Fuzzy AHP-Z Number Model - Fuzzy MABAC. Acta Polytechnica Hungarica, 17(3), 43-70. Bozanic, D., Milic, A., Tesic, D., Sałabun, W., & Pamucar, D. (2021). D numbers – fucom – fuzzy rafsi model for selecting the group of construction machines for enabling mobility. FACTA UNIVERSITATIS - Mechanical Engineering, 19(3), 447 – 471. https:/... Dimic Srđan, H., & Ljubojevic Srđan, D. (2019). 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Economic and environmental strategies for process design. Computers & Chemical Engineering, 36, 174-188. https://doi.org/10.1016/j.compchemeng.2011.09.016 Pamucar, D., Behzad, M., Bozanic, D., & Behzad, M. (2021). Decision making to support sustainable energy policies corresponding to agriculture sector: Case study in Iran's Caspian Sea coastline. Journal of Cleaner Production, 292, 125302. https://doi....