Modeling and optimization in turning of PA66-GF30% and PA66 using multi-criteria decision-making (PSI, MABAC, and MAIRCA) methods: a comparative study

Semi-crystalline polymers are widely used in modern industry. Indeed, they are highly demanded because of their excellent compromise between advantageous mechanical properties, high lightness, good productivity, and low cost. In this work, a modeling study of performance parameters such as (Ra), (Fz), (Pc), and (MRR) was carried out using the response surface methodology (RSM). Dry machining operations were performed on two polyamides (PA66-GF30% and PA66) following the L9 (33) orthogonal array. The results were used to perform a mono-objective optimization based on the Taguchi signal-to-noise ratio (S/N). In addition, a comparative study between three multi-objective optimization methods MCDM (PSI, MABAC, and MAIRCA) coupled with the Taguchi approach was realized. The target objective is to reduce (Ra, Fz, and Pc) and maximize (MRR) simultaneously. The results found are original and can help researchers working in the field of machining polyamides with and without reinforcement.


Introduction
Nowadays, polymers have become indispensable materials in our daily lives; they have established themselves in all industrial fields [1]. Technical polymers are generally used to manufacture various machine parts due to their lightweight and higher specific strength than metallic materials, in addition to their competitive cost. Some polymer materials have been practically developed for industrial use since only the second half of the twentieth century [2,3]. Indeed, polyamide, whether reinforced or not, is among the most frequently used thermoplastic polymers. Reinforced polymeric materials and particularly polyamide composites are widely used in various engineering fields such as aeronautics, automotive, and robotics to substitute several metals and alloys. The addition of 30% glass fiber to PA66 significantly increases the mechanical and thermal properties of polyamide (PA66) [4,5].
Polymer's machining is infrequent compared to metal machining; however, shaping by material removal is recommended when the number of pieces to be manufactured does not justify the cost of molds. Furthermore, when a product requires a costly dimensional precision, the machining of these technical polymers is required for the realization of high-precision pieces [6]. In the case of the polymer machining process, the researchers investigated some process-mastery criteria such as surface quality, cutting forces generated, power consumed, specific pressure exerted, geometry and nature of the tools used, and finally productivity.
Several research studies have been conducted on the machining of polyamides (PA66-GF30% and PA66). M. Marin [7] studied the results of the L12 design to analyze the effects of (Vc, f, and ap) on cutting force (Fz) when turning PA 66. ANOVA revealed that the largest contribution is attributed to (ap) followed by (f), while (Vc) has a very reduced contribution and (Fz) decreases with its increase. H. Cherafa et al. [8] performed a modeling study when turning polyamide (PA66-GF30%) based on L18 orthogonal design. DF approach was used in order to find an ideal cutting regime that guarantees the minimization of (Ra, Fz and Pc) and the maximization of (MRR). Fountas et al. [9] investigated the influence of machining parameters, namely (Vc, f, ap) on the components of cutting force during the turning of PA66-GF30%, using ANOVA analysis. It was found that cutting forces decrease with an increase in (Vc) and augment with the increase in (f). Silva et al. [10] made a comparison between the geometry of an uncoated carbide tool and tools with modified geometry during the precision turning of polyamide with and without reinforcement. The results indicated that the cutting forces are reduced with the radius of the tool nose and (Ks) with a higher feed rate (f). Also, the roughness is found to augment with the increase of (f) and the reduction of the nose of the tool for the two polyamides. Gaitonde et al. [11] examined the machinability of polyamides (PA66-GF30% and PA66) during turning using a metal carbide tool. RSM method was adopted to quantify the effect of (Vc and f) on the output parameters such as F machining , F cut , P cut , and K specific . J. P Davim et al. [12] compared the machinability of polyamide PA 66 with and without glass fiber reinforcement when turning using four cutting tools (CVDD, PCD, K15, K15-KF). The effects of process parameters (Vc and f) on machinability characteristics, namely (Ra, Fz, and Ks), have been studied. G. Özden et al. [13] proposed an approach using (ANN) to predict the cutting force components when turning polyamides with and without reinforcement using two cutting tools (K15 and PCD). Based on the deviation indicators (R 2 and MAPE), they concluded that the predicted results are very close to the experimental results. T. Tezel [14] compared two different manufacturing techniques for polyamides (PA/PA6G) using molding and 3D printing. He performed drilling and reaming operations to obtain holes with good surface finish based on various input parameters.
Taguchi's (S/N) ratio-based parametric optimization method is extremely used in many types of research because it saves time and resources by identifying factors that influence the experimental design process with minimal experimentation [15]. P. Quitiaquez et al. [16] and D. Lazarević et al. [17] successfully used Taguchi's method to minimize the roughness (Ra) when turning PA-6. V. N. Gaitonde et al. [18] conducted an experimental study on machining PA66-GF30% with a metal carbide tool (K10) using Taguchi's method to obtain the optimal cutting regimes of responses (Pc and Ks) individually. Y.H çelik et al. [19] applied Taguchi's approach in order to determine the optimal cutting regime of each response when turning GFRP. Likewise, other researches successfully applied Taguchi's approach to optimize the input factors when machining polymer [20,21].
Among the optimization methods, multi-criteria decision-making (MCDM) methods have seen widespread use in a variety of fields. The application and development of MCDM methods have proven their effectiveness in determining optimal machining conditions [22,23]. In the literature, there are many MCDM methods that have been proposed to solve multi-objective optimization problems [24]. We cite some examples: the ARAS, TOPSIS, MOORA, PROMETHEE, VIKOR, ELECTRE, MOOSRA, CODAS, SAW, and MAUT methods. Each of these methods proposes specific steps to follow in order to arrive to the optimal solution that satisfies desired objectives simultaneously.
The PSI (2010), MABAC (2015), and MAIRCA (2018) methods are also methods that are part of the MCDM family. Various studies have applied these three methods in different industrial sectors in order to make a decision regarding the choice of an optimal cutting regime according to the desired criteria. PSI method [25] has revealed itself to be very interesting when there is a conflict in deciding the relative importance of various output parameters. D. Petković et al. [26] applied the PSI method for two study cases that concern the machinability of materials and the selection of the most suitable cutting fluid during turning. The authors affirm that the PSI method has the advantage, that it is not obligatory to determine the weights of output parameters as in other MCDM methods. S. Chakraborty and S. Chakraborty [27] presented a review article containing 120 research papers concerning the application of MCDM methods. The authors declare that the PSI method seems to be interesting in case of conflict to decide the relative importance between different criteria. The method (PSI) was applied by N. H. Phan et al. [28] to optimize the output parameters (MRR and Ra) using the new technology (PMEDM) when machining SKD11 (AISI D2), during the evaluation of decision alternatives according to criteria. Pamučar and Ćirovic [29] developed the MABAC method, in which the basic framework of this method is reflected in the definition of the distance of a criterion function for each alternative from the approximate boundary domain. Several authors have used the MABAC method in machining. D. Lukic et al. [30] performed a comparative study between the MABAC method and 13 other MCDM methods during highspeed milling of thin-walled Al7075 aluminum alloy parts. The selection of optimal levels of machining parameters and their ranking was established for the 14 (MCDM) methods used, during (EDM) of AISI D3 steel. I. Shivakoti et al. [31] utilized the Taguchi L16 experimental design to study the relationship between output parameters and input factors. The MABAC method was used to optimize the input parameters in order to minimize surface roughness and maximize cutting speed. S.S.S.S Paramasivam et al. [32] applied the MABAC approach while drilling magnesium alloy AM60 to optimize the input parameters. The goal is to maximize (MRR) and minimize surface roughness, power requirement, feed force, burr height, and circularity error. The authors state that the MABAC approach is simple and can be applied effectively to solve practical decision-making problems. Also, the MAIRCA method was used by D. S. Pamucar et al. [33]; this approach is distinguished by its simplicity and novelty and requires less computational time, compared to several other methods (MCDM). The studies concerning the application of the MAIRCA method in literature are very limited and particularly in the field of materials processing. Some research works have been successfully used in different sectors. D. Muravev et al. [34], E. Aksoy. [35], Ayçin et al. [36], Muravev et al. [34], M. Bakir et al. [34], and S. Güler and Can [37] are among those who have contributed to this work. Table 1 summarizes some recent studies (2021-2022) using the MCDM methods (PSI, MAIRCA, and MABAC) during machining.
Based on the aforementioned literature, it is evident that MCDM methods play a vital role in solving optimization problems in a machining process with multiple responses by presenting suitable settings of process parameters, thus improving the overall machining performance. However, there are few published research works that apply MCDM methods during the process of polymer machining, especially polyamides with and without reinforcement. To the authors' knowledge, no research has been reported in the literature on the optimization process of polyamides with and without reinforcement using promising MCDM methods such as PSI, MABAC, and MAIRCA. In order to fill this gap, the present contribution focuses mainly on a statistical and modeling study of the technological performance parameters, namely, surface roughness, cutting force, cutting power, and material removal rate during the turning of two different polyamides (PA66-GF30% and PA66). On the other hand, a multi-objective optimization study was conducted in order to find the optimal cutting regimes that guarantee a minimization of (Ra, Fz, and Pc) and the maximization of (MRR) simultaneously. The Taguchi approach based on (S/N) coupled with the methods (PSI, MABAC, and MAIRCA) were used to achieve this objective. Also, a comparison of the results obtained by the methods used was made.

Experimental procedures
The polymer chosen in this study is unreinforced polyamide (PA 66) and also composite polyamide reinforced with 30 vol% of glass fibers (PA66-GF30%), supplied by Licharz. The specimens used are cylindrical bars having the same diameter (Ø) of 80 mm and length of 280 mm. The mechanical and thermal properties of both polyamides are shown in Table 2.
The machining tests were carried out on a lathe (SN40C) with SPGR120308 WC (Tungsten Carbide) inserts which were fixed in a SDPN2525 M12 tool holder. During experiments, Kistler's dynamometer (9257B) was  (1) and (2), respectively [44,45]. Figure 1 shows the experimental setup used in this study. Experiments were performed according to Taguchi's (L 9 ) orthogonal design (3 × 3) to minimize the trial number and therefore reduce the cost and time involved in the experiments. The input parameters and their levels are as follows: Vc = (80, 115, and 206) m/min, f = (0.08, 0.12, and 0.16) mm/rev, and ap = (0.5, 1, and 2) mm. The orthogonal array and the experimental layout plan of the present study are shown in Table 3.

Analysis of variance (ANOVA)
ANOVA is a statistical method usually used to analyze the impact of input factors on technological performance parameters [46]. The statistical significance of prediction models is evaluated by the probability value "P" and Fisher's value "F." If the P value is higher than 0.05 (P > 0.05), the studied parameter is considered as non-significant, and on the other side (P < 0.05), it is considered as significant [47,48]. ANOVA results for Ra, Fz, and Pc in the function of Vc, f, and ap are estimated with a 95% confidence level. Sum squares (SS), mean squares (MS), percentage of contributions (cont. %), and value of freedom degree (FD) for each output parameter are given in Table 5.
The ANOVA results of PA66-GF30%, for Ra, showed that (f and Vc) are significant with the contributions of 55.09% and 35.68%, respectively, and their P values are lower than 0.005 (P < 0.05). On the other side, (ap) is not significant with a contribution of 5.07%. The ANOVA results for Fz indicate that the (ap) is the most dominant factor on Fz with a contribution of 73.18%, which is followed by (f) with a contribution of 16.23%, while (Vc) is insignificant with a contribution of 6%. The ANOVA results for Pc confirm that both factors (ap and Vc) are significant with contributions of 49.69% and 39.31%, respectively. However, (f) factor is not significant because its P value is higher than 0.05, and its contribution is low (5.53%). For polyamide (PA66), the ANOVA of (Ra) shows that (Vc, f, and ap) are significant since (P < 0.05) with contributions of 20.32%, 65.93%, and 9.60% consecutively. The most predominant factor on (Ra) is (f) revolution per minute. The ANOVA of (Fz) indicates that (ap) is the most predominant factor on (Fz) with contributions of 72.79%, which is followed by the feed (f) with a contribution of 17.01%, whereas (Vc) is insignificant with a contribution of 5.19%. The ANOVA of (Pc) asserts that three factors (Vc, f, and ap) are significant with contributions of 39.89%, 6.09%, and 50.11% respectively. The factors of (ap) and (f) are the main factors affecting (Fz) and that (ap) and (Vc) are the main factors causing the elevation of (Pc). The comparison between output parameters for the two polymers shows that glass fiber reinforcement for polyamide (PA66 GF30%) consequences in the increase of roughness (Ra) as well as effort (Fz).
The comparison between the two machined polyamides shows that the values of surface roughness (Ra) and cutting force (Fz) for the reinforced polyamide (PA66-GF30%) are greater. This can be explained by the presence of glass fibers in the polymer matrix (PA66-GF30%), which improves the physical and mechanical characteristics of the material and thus increases the cutting forces. Furthermore, by machining this material reinforced with glass fibers, rougher machined surfaces are possible than with unreinforced polyamide. This was possibly due to fiber pull-out caused by a single-point cutting tool contact with the workpiece. Similar explanatory remarks have been given by [12,52]. Figure 2 presents Pareto graphs for the responses (Ra, Fz, and Pc) for two materials with and without reinforced (PA66-GF30% and PA66), with (95%) confidence interval (α = 0.05). Factors higher than the line (2.571) are significant, while factors lower than these values are

Modeling of performance parameters
Modeling of performance parameters is an important step to control the machining process of engineering polymers with and without reinforcement. The desired goal is to propose mathematical models that can predict outputs as a function of varying machining conditions. In our case, the correlation between performance parameters (Ra, Fz, and Pc) and input factors (Vc, f, ap) for both polymers used was obtained by linear regression equations. This correlation is illustrated by the equations (Eqs. 3 to 8) with the different coefficients of determination (R 2 ). The closer the (R 2 ) is to 1, the more accurate the model [51,52]. Surface roughness Tangential force Cutting power The Eqs. 3-8 were used to draw the probability normality plots and the 3D response surface for the output parameters (Ra, Fz, and Pc) as a function of input factors (Vc, f, ap) for unreinforced and 30% glass fiber reinforced polyamide. Probability plots for the two polyamides are presented in Figs. 3 and 4 and were used to validate the distribution of the measured response values (Ra, Fz, and Pc). The diagram contains the means of the measured responses, a standard deviation (StDev), the Anderson Darling (AD) value, which is used to validate the normality hypothesis, and the P-value, which is paramount for acceptance or rejection of the null hypothesis regarding the normal probability distribution. Figures 3 and 4 indicate that the experimental data for all responses are close to the fitted line, when the AD statistical values are low, the P value is greater, implying that the data follow a normal distribution. Three-dimensional response surface graphs for both polyamide (PA66-GF30% and PA66) are shown in Fig. 5. Figure 5a and d demonstrate the variation of (Ra) as a function of experimental factors. It can be noticed that the values of Ra were strongly influenced by f since the slopes of the graphs were high. The predominance of feed on Ra is explained by the kinematics of the turning process. Indeed, the surface generated in turning has helical grooves resulting from the shape of the tool and the helical movement of the tool and workpiece. The higher the feed, the deeper and wider the grooves, which degrade the machined surface [53,54]. In addition, the elevation of (ap) leads to a slight increase of (Ra). The trends of the curves obtained are in good agreement with those of the literature [49]. For tangential force (Fz) (Fig. 5b and e), it is clear that the slopes of (ap) are greater than that of (f) values. This implies that the (ap) is the predominant factor on (Fz). This case can be explained by increasing the chip sectional area (ap × f) which makes it more difficult to shear [55]. An increase of (Vc) makes the material easily machinable which causes a decrease for (Fz). These results found coincide with the literature findings [56,57]. Power consumption (Pc) graphs are shown in Fig. 5c and f. It is clearly visible that the increase of three input factors (Vc, f, and ap) induce an increase of (Pc). The most important slope is that of (ap), followed by (Vc), and lastly by (f), respectively. The increase of (Pc) is in direct relation with the terms of Eq. 1, which means that the increase of (Vc and Fz) leads to the rise of (Pc) [58].
The combination of minimum values of couples (f-Vc), (ap-Vc), and (f-ap) leads to the minimum values of (Pc). Also, the combination of couple's values (f min -Vc max ), (ap min -Vc max ) and (f min -ap min ) leads to minimum values of (Ra) and (Fz) respectively for both machined materials.

Optimization of cutting conditions
The optimization of cutting parameters is an important step in the machining process, which has been the subject of several research works using different techniques [59][60][61]. It makes it possible to choose the optimal cutting conditions in order to satisfy the desired objective. This has a direct impact on productivity, quality, and the total cost of machining [59]. In this work, four optimization methods have been proposed: one for singleobjective optimization, the Taguchi approach, and three for multi-objective optimization, i.e., PSI (preference selection index), MABAC (multi-attributive border approximation area comparison), and MAIRCA (multiattributive ideal-real comparative analysis). The three multi-objective optimization methods were coupled with Taguchi's approach based on the ratio (S/N) to find the optimal combination of cutting conditions in machining process for two machined polyamides. The following figure (Fig. 6) shows the multi-objective optimization implementation procedure.

Taguchi parametric optimization
Taguchi's mono-objective optimization method, based on the (S/N) ratio, is used to analyze experimental results, such that S is the signal factor that indicates the true value of the system and N is the noise factor that represents the factor not included in the experiment design [60]. According to Taguchi's method, the S/N ratio of each machining parameter is calculated by Eqs. 9 and 10 according to desired objectives, "the bigger the better" and "the smaller the better" respectively.
where n is the number of observations, y i is the observed data, and i = 1, 2, ... n.
In our study case, "Minitab software 18" was exploited to perform the Taguchi analysis based on the (S/N) ratio. Equation 10 was used for (S/N) ratio calculation to minimize the output parameters (Ra, Fz, and Pc) individually, while Eq. 9 was used to maximize the (MRR). The calculation results are given in Table 6. Table 7 summarizes mean values of S/N for monoobjective optimization for each parameter (Ra, Fz, Pc, and MRR) obtained for PA66 and PA66-GF30% work materials. The optimal regime for each response is given by the levels of the input factors corresponding to a maximum value of the mean S/N. Results presented in Table 7 show that optimal combination of cutting parameters is the same for the two considered materials, which correspond to (Vc 3 =206 m/min, f 1 =0.08 mm/rev and ap 1 =0.5  Table 8 summarizes the optimal combinations found and their corresponding response values. The comparison of the values of the different outputs obtained for both polyamides machined allows us to see that the presence of reinforcement in the polyamide (PA66-GF30%) has modified physical and mechanical characteristics of the material, which led to an increase in parameters (Ra, Pc, and Fz) of 10% to 15%. Quite similar results have been reported in refs [1,16,61].
Furthermore, the optimal combination of cutting parameters is determined by the optimization criterion used. In other words, each discovered combination satisfies only one optimization criterion, whereas in the industrial environment, these objectives must always be met concurrently. This prompted us to seek a compromise among the various technological parameters under consideration, employing multi-objective optimization methods to do so.

PSI method
PSI is a method (MCDM) that was proposed by Maniya and Bhatt in 2010 [25]. This method is easy to apply in order to solve multi-objective optimization (MOO) type problems. The execution steps of this method are as follows [62]: Step 1: Definition of the objectives.
Step 2: Create a decision matrix (X) based on available information.
where m is number of trials, and n is number of responses.
Step 3: Normalization:Normalization is a process of converting data in different units to a common scale and similar units.
where i is the line number of matrix (i = 1 to m), j is the column number of matrix (j = 1 to n), and x ij is the value of criterion in range i and column j.

For beneficial criterion
Step 4: Calculation of mean values for normalized data.
Step 5: Determination of preferred values from mean values.
Step 6: Determination of the gap to preferred value.
Step 7: Determine the overall preferred value for the criteria.
Step 8: Calculation of selection preference index (PSI) for each solution.
From the results of ratio (S/N) for two polyamides presented in Table 6, the normalized values of decision matrix and PSI (θ i ) preference selection index of each experiment were calculated according to Eqs. 11 and 17, and their values are recorded in Table 9.
The main effect plots for PSI method are exposed in Fig. 7 for two polyamides tested. The optimal cutting regimes of performance parameters (Ra, Fz, Pc, and MRR) optimized simultaneously are given in the form of a combination (Vc 3 f 1 ap 3 ) and (Vc 3 f 1 ap 2 ) which correspond to the values (Vc= 206 m/min, f= 0.08 mm/ rev, ap= 2 mm), (Vc= 206 m/min, f= 0.08 mm/rev and ap= 1 mm) for polyamide PA66-GF30% and PA66, respectively. The optimal levels for both polyamides are indicated in Fig. 7 by red dots.

MABAC method
MABAC method also counts allows methods (MCDM); it was proposed by Pamučar and Ćirović [29], and it is characterized by its simplicity and accuracy in solving multi-objective optimization problems. This method is performed in the following steps [63]: Step 1: Definition of the decision matrix (Step 2 of the PSI method).
In the second step, normalization of the decision matrix is performed to make each component dimensionless for easy comparison.
where X + i = max of the alternative, and X ij , X − i = min of the alternative X ij .
Step 3: Calculation of the weighted matrix (q).
where W j is the weighting of each criterion (where ∑ n j=1 w j = 1).
Step 4: Determination of the approximate boundary matrix (b).
where q ij represents the elements of the weighted matrix (q), and m is the number of trials.
Step 6: Calculate the sum of the values (r ij ) in each row of the difference matrix.
The normalized t ij values of decision matrix, for each of the two polyamides considered, as well as the Si index, are presented in Table 10. From the results of S-index, we plotted the main effect graphs (Fig. 8), in order to determine the optimal cutting regimes for both machined polyamides, which concern the minimization of (Ra, Fz, and Pc) as well as the maximization of (MRR) simultaneously. As the best performance corresponds to the highest value of S i , it is obvious that the optimal combination for both polyamides is the same (Vc 3 = 206 m/min, f 1 = 0.08 mm/rev, ap 1 = 0.5 mm), but the optimized output parameters that correspond to this regime are deferent. Finally, the optimal regimes for both materials are highlighted by red dots in Fig 8. (21)

MAIRCA method
MAIRCA is a new method that belongs to MCDM family, and it was first introduced by D.S. Pamucar et al. in (2018) [33]. Several research works have applied this multi-criteria decision method in various fields [33]. The steps of this method are as follows [35]: Step 1: Creating the initial matrix as step 2 in the PSI method (Eq. 11).
Step 2: Determining the priority of an alternative (P Aj ), the priority of the alternative and the decision-makers' neutrality in the selection of alternatives show that each of the proposed alternatives is of equal importance. It is an assumption of the method that the decision-maker has not assigned probability values for any alternative selection: (23) P Aj = 1 m Step 3: Calculation of the theoretical quantities tp ij according to Eq. 24.
W j is the weight of each criterion (where ∑ n j=1 w j = 1).
Step 4: Definition of the elements of the real ranking matrix (tr).
where X + i = max of the alternative X ij , and X − i = min of the alternative X ij .
Step 5: Calculation of the total deviation matrix (g). That is, the elements of this matrix are obtained by the differ- i ence between the theoretical ( t p ij ) and real ( t r ij ) quantities.
Step 6: Calculate the sum of values (g ij ) in each line for each alternative Table 11 shows the normalized values t r ij of decision matrix, for each of the two materials considered, along with the index (Qi). The best combination is that which corresponds to the smallest Qi value. For our case, the optimal regimes for the four responses optimization (Ra, Fz, Pc, and MRR) simultaneously have been determined on the basis of the main effect graphs (Fig. 9) for the two polyamides (PA66-GF30% and PA66). The analysis of the results shows that the optimal speeds are the same for both machined materials (Vc 3 = 206 m/min, f 1 = 0.08 mm/rev, ap 1 = 0.5 mm), but the optimized output parameters are different. Finally, the optimal speeds for both materials are indicated by red dots in Fig. 9.
From the optimization results, it can be seen that (Vc) and (f) remain the same for all three methods, but the values of (ap) vary for the PSI method. Indeed, obtaining a large value of (ap) by the (PSI) method led to obtaining a maximum value of MRR, (MRR PA66-GF30% = 32.95 cm 3 /min) and (MRR PA66 = 16.484 cm 3 /min). On the other hand, it led to an increase of (Fz and Pc) as indicated in the ANOVA analysis (Table 3). In addition, we find that the values of (Ra) did not change much, which is expected since the ANOVA study (Table 3) revealed that the factor (ap) has almost no effect on the criterion (Ra).

Validation tests
An experimental validation was performed to confirm the accuracy of the optimal regimes obtained by the three MCDM methods considered for the two polyamides with and without reinforcement. The calculation of the percentage error is performed using Eq. 28 [64].
The confirmatory tests were planned and performed based on the optimal levels of the optimal cutting parameters obtained by the three methods PSI, MAIRCA, and MABAC for the two polyamides with and without reinforcement (Table 12). The percentage variations between the predicted and experimental values of Ra, Fz, Pc, and MRR are shown in Table 13. It can be seen that the  maximum error reaches 3.02%, which confirms the validity of the optimal results obtained in this study. This percentage of error can be attributed to other uncontrollable factors such as machine vibration, climatic conditions, and human error.

Conclusion
The present experimental study focused on modeling and optimization during dry machining of two polyamides with and without reinforcement (PA66-GF30% and PA66) The optimal regime for each response is given by the levels of the input factors corresponding to a maximum value of the mean S/N, presented in bold in Table 7 Materials    during turning using a metal carbide cutting tool leads to the following conclusions: 1. The ANOVA of Ra, Fz, and Pc for polyamide material of PA66-GF30% shows that (f) is the most influential factor on the criterion of (Ra) with a contribution of (55.09%), followed by (Vc) with a contribution of (35.68%). On the other hand, the cutting force (Fz) is largely influenced by (ap) with a contribution of (73.18%), followed by the feed (f) with a contribution of (16.23%). Also, (ap) is the first   factor influencing (Pc) with a contribution of (49.69%), followed by (Vc) with a contribution of (39.31%). 2. For polyamide material of PA66, the variance analysis of each response (Ra, Fz, and Pc) indicates that all three factors (Vc, f, and ap) are significant for (Ra) and that (f) is the most important factor affecting on (Ra) with a contribution of (65.93%). On the other hand, the factors of (Vc) and (ap) have contribution of (20.32% and 9.60%) respectively. The ANOVA of (Fz) shows that (ap) is the predominant factor with a contribution of 72.79%; it is followed by the lead factor of (f) with a contribution of 17.01%. The ANOVA of (Pc) reveals that three the factors (Vc, f, and ap) are significant with contributions of 39.89%, 6.09%, and 50.11% respectively. 3. The mathematical models found of (Ra, Fz, and Pc) for the two polyamides are in good agreement with the experimental values, as the different (R²) are high. Indeed, (R²) varies from (94.53% to 95.83%) and (95.86% to 96.09%) for (PA66-GF30% and PA66) respectively. From an industrial point of view, these models are very useful for the prediction and optimization of cutting conditions. 4. Taguchi's mono-objective optimization was able to, in a simple way, provide the same optimal cutting regimes of the two polyamides (PA66-GF30%) and (PA66) for the four technological parameters studied individually.  min and f= 0.08 mm/rev). Furthermore, the two methods (MABAC and MAIRCA) proposed a factor p of 0.5 mm for the two polyamides (PA66-GF30%) and (PA66), and the PSI method proposed ap of 2 mm and ap of 1 mm for the two polyamides (PA66-GF30%) and (PA66), respectively. The results, on the other hand, revealed that the PSI method favored the maximization of MRR for both polyamides, whereas MABAC and MAIRCA methods favored the minimization of Ra, Fz, and Pc.