# ANN Surface Roughness Optimization of AZ61 Magnesium Alloy Finish Turning: Minimum Machining Times at Prime Machining Costs

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## Abstract

**:**

_{m}) and at prime machining costs (C). An ANN is built in the Matlab programming environment, based on a 4-12-3 multi-layer perceptron (MLP), to predict Ra, T

_{m}, and C, in relation to cutting speed, v

_{c}, depth of cut, a

_{p}, and feed per revolution, f

_{r}. For the first time, a profile of an AZ61 alloy workpiece after finish turning is constructed using an ANN for the range of experimental values v

_{c}, a

_{p}, and f

_{r}. The global minimum length of a three-dimensional estimation vector was defined with the following coordinates: Ra = 0.087 μm, T

_{m}= 0.358 min/cm

^{3}, C = $8.2973. Likewise, the corresponding finish-turning parameters were also estimated: cutting speed v

_{c}= 250 m/min, cutting depth a

_{p}= 1.0 mm, and feed per revolution f

_{r}= 0.08 mm/rev. The ANN model achieved a reliable prediction accuracy of ±1.35% for surface roughness.

## 1. Introduction

_{2}O

_{3}/TiC ceramic cutting tool using the response surface methodology (RSM). Krolczyk et al. [30] identified surface integrity of the turned workpieces using fused deposition modeling (FDM). Nieslony et al. [31] presented the problem of precise turning of 55NiCrMoV6 hardened steel mould parts and demonstrated a topographic inspection of the machined surface quality. Acayaba and Escalona [32] developed a model for predicting surface roughness in the low-speed turning of AISI316 stainless steel using multiple linear regression and ANN methodologies. D’Addona and Raykar [33] studied the influence of hard turning parameters—speed, feed rate, depth of cut, and nose radius (for wipers and regular inserts)—on surface roughness. Mia and Dhar [34] obtained an ANN model for predicting average surface roughness when turning EN 24T hardened steel. Jurkovic et al. [35] compared three machine-learning methods in predicting the observed parameters of high-speed turning (surface roughness (Ra), cutting force (Fc), and tool life (T)). Tootooni et al. [36] used a non-contact, vision-based online measurement method for measuring surface roughness while turning the external diameter of the workpiece. Mia et al. [37] focused on developing predictive models of average surface roughness, chip-tool interface temperature, chip reduction coefficient, and average tool flank wear when turning a Ti-6Al-4V alloy. Mia et al. [38] investigated the plain turning of hardened AISI 1060 steel and examined the effect of three sustainable techniques and the traditional flood cooling system on the following machining indices: cutting temperature, surface roughness, chip characteristics, and tool wear.

_{m}of the finished workpiece from high-strength steel using the ANN model that was later used to determine the optimum finishing cutting conditions. There is, therefore, little research dedicated to multi-objective optimization in turning. The most efficient approach to solving such problems is Pareto optimization. However, studies [48,49,50,51,52] are not concerned with multiobjective optimization of machining AZ61 magnesium alloys. Taking into account the high cost of this material, it is necessary to ensure the design roughness value of the machined surface and the minimum processing time of the material volume at minimal processing costs.

_{m}, the minimum surface roughness, Ra, and the minimum cost of machining one part, C.

## 2. Experiment

_{r}), back angle (α), and nose radius (r

_{o}) were set at 35°, 5°, and 0.4 mm, respectively. Workpiece length (L), workpiece diameter (D) and tolerance (l

_{1}) were set at 20, 40, and 2 mm, respectively. Experiments were carried out under wet cutting conditions. A TESA Rugosurf 90-G surface roughness tester (TESA, Bugnon, Switzerland) was used. Figure 1 illustrates the test rig used to measure surface roughness.

_{c}, was used (100, 150, 200, and 250 m/min). Each set of experiments was machined using four different depths of cut a

_{p}(0.25, 0.50, 0.750, and 1.00 mm). Each depth of cut was processed using four levels of feed rate f

_{r}(0.04, 0.08, 0.12, and 0.16 mm/rev). The surface roughness values for the different cutting conditions are presented in Table 2.

## 3. System Adaptation Procedure

_{1}, f

_{2}, …, f

_{m})—vector-valued criterion; Y = f(X)—a set of possible vectors (estimates); R

^{m}—Euclidean space of m-dimensional vectors with real components; >

_{X}—preference relation of DM specified in the set X; >

_{Y}—preference relation of DM, induced on the set with >

_{X}and specified in the set Y; >—relation >

_{Y}continued in the entire space R

^{m}; Sel X—a set of selected decisions; Sel Y—a set of selected vectors (estimates); Ndom X—a set of non-dominated decisions; Ndom Y—a set of non-dominated vectors (estimates); P

_{f}(X)—a set of Pareto optimal decisions; P(Y)—a set of Pareto optimal vectors (Pareto optimal estimates).

## 4. Formulation of an Optimization Problem

_{1}—surface roughness (Ra, μm) and f

_{2}—unit-volume machining time volume in one cutting tool pass (T

_{m}, min/cm

^{3}); and, f

_{3}—the cost price of processing one component part (C, $), i.e., m = 3. Relatively, a set of possible Y estimates in the two-dimensional space, R

^{3}, is formed with vectors f = (f

_{1}, f

_{2}, f

_{3}). The search is performed for a set of estimates with the minimum length of vector f, which is a vector from the coordinate origin to a point on the estimate surface. The criteria are presented in a normalized dimensionless form with index 1 assigned to the maximum actual numbers.

_{1}= [100 ÷ 250]; cutting speed, v

_{c}, m/min; x

_{2}= [0.25 ÷ 1.0]; depth of cut, a

_{p}, mm; x

_{3}= [0.04 ÷ 0.16]; and, feed rate, f

_{r}, mm/rev.

_{1}). The second criterion is unit-volume machining time, T

_{m}(min/cm

^{3}), and unit-volume dimensionless machining time T

_{m}*(f

_{2}). The third criterion is the cost price of processing one component part, C ($), or the dimensionless cost price of processing one component part, C* (f

_{3}). The fourth criterion is the dimensionless vector of estimates in a three-dimensional normalized space, f.

_{m}= 1/(1000 × v

_{c}× a

_{p}× f

_{r});

_{i}= (C

_{Mh}× T

^{/}) + (C

_{Toolmin}× T

^{/}) + C

_{w}, where is Machining Time in Turning T

^{/}= (L + l

_{1})/(n × f

_{r}), where

_{c})/(3.141·D);

_{i}/Ra

_{max};

_{m}* = T

_{m i}/T

_{m}

_{max};

_{i}/C

_{i}

_{max};

_{i}is surface roughness for the current combination of X... and f

_{r}; Ra

_{max}is the maximum surface roughness value of all the v

_{c}, a

_{p}, and f

_{r}combinations; T

_{m i}is the unit-volume machining time for the current values of v

_{c}, a

_{p}, and f

_{r}; T

_{m max}is the maximum unit-volume machining time of all the v

_{c}, a

_{p}, and f

_{r}combinations; С

_{i}is the cost price of processing one part for a given combination of v

_{c}, a

_{p}, and f

_{r}; C

_{i max}—the maximum value.

_{1}, f

_{2}, f

_{3}) and f(f

_{1}, f

_{2}, f

_{3}), as well as the functional Q to the plane f(f

_{1}, f

_{2}, f

_{3}). The ANN complex was constructed using the Skif AURORA-SUSU supercomputer cluster (South Ural State University, Chelyabinsk, Russia) [53].

## 5. Building a Neural Network Model

## 6. Graphical Representation of the Surface of Vector Estimates (D)

_{1}, x

_{2}, and x

_{3}were used to calculate f

_{1}, f

_{2}and f

_{3}.

_{2}f

_{3}(Tm* С*) and is shown Figure 6.

_{m}* and the quasi-horizontal plane with a dent U in the opposite area of C* and T

_{m}* values. We can also clearly see apexes B1, B2, and C with slopes towards the dent.

## 7. Establishment of a Pareto Frontier

_{m}* (see Figure 8). For this purpose, we shall consider surface projection estimates at fixed depths of cut—a

_{p}= 1 mm, a

_{p}= 0.75 mm, a

_{p}= 0.5 mm, and a

_{p}= 0.25 mm (Figure 8, Figure 9, Figure 10 and Figure 11).

_{p}= 1.0 to a

_{p}= 0.25 mm. It changes from a merged apex A and A’ to a maximum with two peaks in A1″, A″ with saddle A and B1′, and B2′ with saddle B. The largest displacement of the peaks occurs at the T

_{m}* coordinate with the distance between them increasing. The ridge (see Figure 10) is formed by peaks А, А′, А1″ and А2″, the mountain system, by peaks В1′and В2′, and the hill, with the long slope of peak В2′ and the dent, U.

^{2}= 0.986 (accuracy ±1.35%), a pattern was revealed for the AZ61 alloy. When the other two parameters were fixed and the cutting speed, v

_{c}*, was increased by 0.1 units, the values of Ra*, T

_{m}*, C*, and f* decreased by 0.001 units, 0.007 units, 0.003 units, and 0.005 units, respectively. When the other two parameters were fixed and the depth of cut, a

_{p}*, was increased by 0.1 units, the values of Ra*, T

_{m}*, C*, and f* decreased by 0.007 units, 0.010 units, 0.0001 units, and 0.005 units, respectively. When the other two parameters were fixed and the feed rate, f

_{r}*, was increased by 0.1 units, T

_{m}*, and C* decreased by 0.004 and 0.006 units, respectively, while Ra* and f* increased by 0.103 and 0.015 units, respectively. Compared to v

_{c}, surface roughness (Ra) was 7.3 times more affected by a

_{p}and 102.9 times more affected by f

_{r}; machining time (T

_{m}) was 1.5 times less affected by a

_{p}and 1.5 times more affected by fr; cost of production (C) was 26 times less affected by a

_{p}and 1.9 times more affected by f

_{r}; the integrated optimization criterion (f) was 1.04 times more affected by a

_{p}and 2.9 times more affected by f

_{r}. Hence, we should look for the optimal cutting conditions at the maximum cutting speed and depth of cut and the minimum feed rates.

_{p}= 1 mm, as in this case all values of Ra* are located near the minimum vector estimation F (0.0449; 0.8948; 0.1253), which is marked by point T.

_{p}= 1 mm, four graphic dependencies, Ra* = f(C*, T

_{m}*), were constructed, corresponding to the fixed v

_{c}= 250 m/min, v

_{c}= 200 m/min, v

_{c}= 150 m/min, v

_{c}= 100 m/min, and variable, f

_{r}, value. After matching the obtained curves with the projection (see Figure 8), we obtained the seven reference points of the Pareto frontier. They are shown in Figure 12: p

_{1}(0.0281; 0.8782; 0.8135); p

_{2}(0.0343; 0.8824; 0.8273); p

_{3}(0.0449; 0.8948; 0.1253); p

_{4}(0.0561; 0.9012; 0.1072); p

_{5}(0.0860; 0.9123; 0.0982); p

_{6}(0.1710; 0.9547; 0.0514); p

_{7}(0.2500; 1.0000; 0.7903).

_{1}and point p

_{2}, corresponds to v

_{c}= 250–200 m/min, a

_{p}= 1.00 mm, f

_{r}= 0.16 mm/rev. Section 2 between point p

_{2}and point p

_{3}corresponds to v

_{c}= 250 m/min, a

_{p}= 1.00 mm, f

_{r}= 0.16–0.08 mm/rev. Section 3 between point p

_{3}and point p

_{4}corresponds to v

_{c}= 250–200 m/min, a

_{p}= 1.00 mm, f

_{r}= 0.08 mm/rev. Section 4 between points p

_{4}and p

_{5}corresponds to v

_{c}= 200–150 m/min, a

_{p}= 1.00 mm, f

_{r}= 0.08 mm/rev. Section 5 between point p

_{5}and point p

_{6}corresponds to v

_{c}= 250 m/min, a

_{p}= 1.00 mm, f

_{r}= 0.12–0.16 mm/rev. Section 6 between point p

_{6}and point p

_{7}corresponds to v

_{c}= 150–100 m/min, a

_{p}= 1.00 mm, f

_{r}= 0.16 mm/rev. p

_{3}and p

_{6}are special points on the Pareto curve. These points correspond to absolute minimums; p

_{3}is the absolute minimum of the length of vector f; and p

_{6}is the absolute minimum of surface roughness.

## 8. The Optimum Settings

_{i}. In consequence, all vectors located above the blue vector that has the lowest f vector, plotted on the f

_{3}f

_{2}plane, at an angle of 7.89°, represent the Pareto non-dominated estimates (Figure 13). Point 3 on the Pareto curve coincides with the end point of this vector that was the global minimum in the case of unconditional optimization with the ranking f

_{1}:f

_{2}:f

_{3}= 1.0:19.9:2.7. Using the real coordinates, the global minimum corresponds to T

_{m}= 0.358 min/cm

^{3}, С = $8.2973, R

_{a}= 0.087 μm, v

_{c}= 250 rpm, a

_{p}= 1.0 mm, and f

_{r}= 0.08 m/min.

_{8}on the Pareto frontier curve, having the coordinates (0.0372, 0.8851, 0.300) in Figure 13. In this case (see Figure 13, blue estimates vector), the optimization criteria has a valid relation of importance that is–Ra*/T

_{m}*/С* = 1.0/27.6/9.3, and the valid preference for points eight and three becomes y

_{8}>

_{Y}y

_{3}with an induced preference of x

_{8}>

_{X}x

_{3}.

_{c}= 248 rpm, a

_{p}= 1.0 mm, f

_{r}= 0.10 mm/min).

_{c}= 250 m/min, depth of cut a

_{p}= 1.0 mm, and feed rate f

_{r}= 0.08 mm/rev) are presented in Figure 14 and Figure 15.

## 9. Conclusions

- (1)
- For the first time in the turning of magnesium alloy, the Edgeworth–Pareto methodology has been used for adapting the cutting tool–workpiece system to the state of the minimal value of the three-dimensional estimates of vector f in a normalized space: f
_{1}f_{2}f_{3}using an artificial intelligence-based model. - (2)
- An artificial neural network has been created in the Matlab programming environment based on an MLP 4-12-3 multi-layer perceptron that predicts the values of f
_{1}, f_{2}, f_{3}, f in the finishing turning of the AZ61 magnesium alloy workpiece with a width of X mm, a length of X mm, and a height of X mm, at a cutting speed of 100–250 m/min, a depth of cut from 0.25 to 1.0 mm, and a feed rate of 50–150 mm/rev with an accuracy of ±1.35%. - (3)
- According to the neural network model for the AZ61 alloy in finish turning, the value of the integrated optimization criterion, f, has mainly been influenced by feed rate, f
_{r}. Vector f is 2.9 times more influenced by feed rate than by cutting speed and depth of cut. Increasing the feed rate led to an increase in f, and increasing v_{c}and a_{p}led to a decrease in f. - (4)
- For the first time, an AZ61 magnesium alloy workpiece wafer plot of surface roughness after finishing turning has been generated at cutting speeds of 100–250 m/min, at a depth of cut from 0.25–1.0 mm, and at a feed rate of 50–150 mm/rev.
- (5)
- The global optimum in the finish turning of the alloy workpiece has been set as follows: the minimum length of 3D vector estimates with the coordinates Ra = 0.087 μm, T
_{m}= 0.358 min/cm^{3}, and С = $8.2973 corresponded to the following optimum conditions of finishing turning: cutting speed v_{c}= 250 m/min, depth of cut a_{p}=1.0 mm, and feed rate f_{r}= 0.08 mm/rev. - (6)
- Automated calculation with the Industry 4.0 Framework has been performed in the Matlab environment, to define the optimal turning conditions for magnesium alloy workpieces as products of intelligent computer-aided manufacturing systems.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The lowest mean squared error for the validationset in the multi-layer perceptron (MLP) 3-11-4 configuration (calculated in Matlab).

**Figure 3.**The lowest mean squared error for the validationset in the MLP 3-12-4 configuration (calculated in Matlab).

**Figure 4.**The lowest mean squared error for the validation set in the MLP 3-13-4 configuration (calculated in Matlab).

**Figure 5.**The lowest mean squared error in generalizing experimental data in MLP 3-12-4 (

**a**) with various validation sets: (

**b**) 5%; (

**c**) 15% (calculated in Matlab).

**Figure 6.**Wafer map of workpiece Ra* after machining with respect to changes in values of T

_{m}* and C*.

**Figure 7.**Apexes of the wafer map of Ra* values of the machined workpiece depending on the change in values of T

_{m}* and C*.

**Figure 8.**Surface projection of Ra* values depending on the change in the values of T

_{m}* and C* at fixed depth of cut, a

_{p}= 1 mm.

**Figure 9.**Surface projection of Ra* values dependent on the change in the values of T

_{m}* and C* at a fixed depth of cut, a

_{p}= 0.75 mm.

**Figure 10.**Surface projection of Ra* values depending on the change in the values of T

_{m}* and C* at a fixed depth of cut, a

_{p}= 0.5 mm.

**Figure 11.**Surface projection of Ra* values depending on the change in the values of T

_{m}* and C at a fixed depth of cut, a

_{p}= 0.25 mm.

**Figure 12.**Pareto frontier and seven reference points: p

_{1}(0.0281; 0.8782; 0.8135); p

_{2}(0.0343; 0.8824; 0.8273); p

_{3}(0.0449; 0.8948; 0.1253); p

_{4}(0.0561; 0.9012; 0.1072); p

_{5}(0.0860; 0.9123; 0.0982); p

_{6}(0.1710; 0.9547; 0.0514); p

_{7}(0.2500; 1.0000; 0.7903).

**Figure 15.**Profile of surface roughness graph from the surface roughness tester for the optimal machining parameters.

Element | Aluminum | Zinc | Copper | Silicon | Iron | Nickel | Magnesium |
---|---|---|---|---|---|---|---|

Mass % | 6 | 0.90 | 0.02 | 0.008 | 0.007 | 0.003 | Balance |

Cutting Speed: v_{c}, (m/min) | Feed: f_{r}, (mm/rev) | Surface Roughness: Ra (µm) | |||
---|---|---|---|---|---|

Depth of Cut: a_{p}, (mm) | |||||

0.25 | 0.5 | 0.75 | 1.0 | ||

100 | 0.0400 | 0.1730 | 0.1660 | 0.1500 | 0.1290 |

100 | 0.0800 | 0.3880 | 0.3610 | 0.3530 | 0.4400 |

100 | 0.1200 | 0.8720 | 0.9520 | 1.0470 | 1.0200 |

100 | 0.1600 | 1.6780 | 2.1040 | 2.1790 | 2.6290 |

150 | 0.0400 | 0.1460 | 0.1320 | 0.1160 | 0.1890 |

150 | 0.0800 | 0.3440 | 0.3480 | 0.3150 | 0.4130 |

150 | 0.1200 | 0.9310 | 1.0540 | 0.9840 | 0.9990 |

150 | 0.1600 | 1.6370 | 1.7640 | 1.7020 | 1.8840 |

200 | 0.0400 | 0.1820 | 0.1800 | 0.2040 | 0.1500 |

200 | 0.0800 | 0.3670 | 0.3860 | 0.3970 | 0.3550 |

200 | 0.1200 | 0.8450 | 1.0240 | 1.0340 | 1.2140 |

200 | 0.1600 | 1.9760 | 1.9220 | 1.9350 | 2.0140 |

250 | 0.0400 | 0.1230 | 0.1830 | 0.1370 | 0.2240 |

250 | 0.0800 | 0.3590 | 0.3890 | 0.3580 | 0.3250 |

250 | 0.1200 | 0.9370 | 0.9680 | 0.9500 | 1.0000 |

250 | 0.1600 | 2.0880 | 1.9540 | 2.0170 | 1.8930 |

Mater. | Cost of Machining/Hour (SR 400), CMh: $ | Cost of Tool Holder, CToolh: $ | Tool Holder Life: LTToolh min | Cost of Insert, CIn: $ | Setup Insert: k | Unit Cost of Work-Piece: Cw: $ | Tool Life: T Min | Cost of Tool Minute: CToolmin, $ CToolmin = (CIn/(T×k)) + (CToolhLTToolh) |
---|---|---|---|---|---|---|---|---|

AZ61 | 106 | 85 | 5 Year × 365 Day × 24 h × 60 min = 2,628,000 | 10 | 2 | 8 | 60 | 0.083 |

**Table 4.**Optimization criteria for the variable machining parameters at a fixed depth of cut—a

_{p}= 0.25 mm.

Variable Parameters | Optimization Criteria | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

x_{1}Cutting Speed: v _{c}, (m/min) | x_{2}Depth of Cut: a _{p}, (mm) | x_{3}Feed: f _{r}, (mm/rev) | Surface Roughness: Ra (µm) | Dimensionless Surface Roughness: f_{1} (Ra*), u | Unit Volume Machining Time: T_{m} (min/cm^{3}) | Dimensionless Volume Machining Time: f_{2} (T_{m}*), u | Unit cost Price of Processing One Part: C, ($) | Dimension-less Cost Price of Processing One Part: f_{3} (C*), u | Length of Estimates Vector: f, u | Length of Estimates Vector: f*, u |

0.4 | 0.25 | 0.25 | 0.1730 | 0.0660 | 1.0000 | 1.0000 | 9.2729 | 1.0000 | 1.4160 | 1.0000 |

0.4 | 0.25 | 0.5 | 0.3880 | 0.1480 | 0.5000 | 0.5000 | 8.6374 | 0.9310 | 1.1780 | 0.8319 |

0.4 | 0.25 | 0.75 | 0.8720 | 0.3320 | 0.3333 | 0.3330 | 8.4237 | 0.9080 | 1.1260 | 0.7952 |

0.4 | 0.25 | 1.0 | 1.6780 | 0.6380 | 0.2500 | 0.2500 | 8.3187 | 0.8970 | 1.2090 | 0.8538 |

0.6 | 0.25 | 0.25 | 0.1460 | 0.0560 | 0.6667 | 0.6670 | 8.8492 | 0.9540 | 1.2570 | 0.8877 |

0.6 | 0.25 | 0.5 | 0.3440 | 0.1310 | 0.3333 | 0.3330 | 8.4237 | 0.9080 | 1.0840 | 0.7655 |

0.6 | 0.25 | 0.75 | 0.9310 | 0.3540 | 0.2222 | 0.2220 | 8.2837 | 0.8930 | 1.0700 | 0.7556 |

0.6 | 0.25 | 1.0 | 1.6370 | 0.6230 | 0.1667 | 0.1670 | 8.2118 | 0.8860 | 1.1580 | 0.8178 |

0.8 | 0.25 | 0.25 | 0.1820 | 0.0690 | 0.5000 | 0.5000 | 8.6374 | 0.9310 | 1.1710 | 0.8270 |

0.8 | 0.25 | 0.5 | 0.3670 | 0.1400 | 0.2500 | 0.2500 | 8.3187 | 0.8970 | 1.0360 | 0.7316 |

0.8 | 0.25 | 0.75 | 0.8450 | 0.3210 | 0.1667 | 0.1670 | 8.2118 | 0.8860 | 1.0270 | 0.7253 |

0.8 | 0.25 | 1.0 | 1.9760 | 0.7520 | 0.1250 | 0.1250 | 8.1584 | 0.8800 | 1.2100 | 0.8545 |

1.0 | 0.25 | 0.25 | 0.1230 | 0.0470 | 0.4000 | 0.4000 | 8.5084 | 0.9180 | 1.1160 | 0.7881 |

1.0 | 0.25 | 0.5 | 0.3590 | 0.1370 | 0.2000 | 0.2000 | 8.2542 | 0.8900 | 1.0050 | 0.7097 |

1.0 | 0.25 | 0.75 | 0.9370 | 0.3560 | 0.1333 | 0.1330 | 8.1695 | 0.8810 | 1.0180 | 0.7189 |

1.0 | 0.25 | 1.0 | 2.0880 | 0.7940 | 0.1000 | 0.1000 | 8.1271 | 0.8760 | 1.2240 | 0.8644 |

**Table 5.**Optimization criteria values for the variable parameters of machining at a fixed depth of cut—a

_{p}= 0.5 mm.

Variable Parameters | Optimization Criteria | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

x_{1}Cutting Speed: v _{c}, (m/min) | x_{2}Depth of Cut: a _{p}, (mm) | x_{3}Feed: f _{r}, (mm/rev) | Surface Roughness: Ra (µm) | Dimensionless Surface Roughness: f_{1} (Ra*), u | Unit Volume Machining Time: T_{m} (min/cm^{3}) | Dimensionless Volume Machining Time: f_{2} (T_{m}*), u | Unit Cost Price of Processing One Part: C, ($) | Dimensionless Cost Price of Processing One Part: f_{3} (C*), u | Length of Estimates Vector: f, u | Length of Estimates Vector: f*, u |

0.4 | 0.5 | 0.25 | 0.1660 | 0.0630 | 0.5000 | 0.5000 | 9.2729 | 1.0000 | 1.2260 | 0.8658 |

0.4 | 0.5 | 0.5 | 0.3610 | 0.1370 | 0.2500 | 0.2500 | 8.6374 | 0.9310 | 1.0660 | 0.7528 |

0.4 | 0.5 | 0.75 | 0.9520 | 0.3620 | 0.1667 | 0.1670 | 8.4237 | 0.9080 | 1.0590 | 0.7479 |

0.4 | 0.5 | 1.0 | 2.1040 | 0.8000 | 0.1250 | 0.1250 | 8.3187 | 0.8970 | 1.2530 | 0.8849 |

0.6 | 0.5 | 0.25 | 0.1320 | 0.0500 | 0.3333 | 0.3330 | 8.8492 | 0.9540 | 1.1160 | 0.7881 |

0.6 | 0.5 | 0.5 | 0.3480 | 0.1320 | 0.1667 | 0.1670 | 8.4237 | 0.9080 | 1.0040 | 0.7090 |

0.6 | 0.5 | 0.75 | 1.0540 | 0.4010 | 0.1111 | 0.1110 | 8.2837 | 0.8930 | 1.0340 | 0.7302 |

0.6 | 0.5 | 1.0 | 1.7640 | 0.6710 | 0.0833 | 0.0830 | 8.2118 | 0.8860 | 1.1480 | 0.8107 |

0.8 | 0.5 | 0.25 | 0.1800 | 0.0680 | 0.2500 | 0.2500 | 8.6374 | 0.9310 | 1.0590 | 0.7479 |

0.8 | 0.5 | 0.5 | 0.3860 | 0.1470 | 0.1250 | 0.1250 | 8.3187 | 0.8970 | 0.9750 | 0.6886 |

0.8 | 0.5 | 0.75 | 1.0240 | 0.3900 | 0.0833 | 0.0830 | 8.2118 | 0.8860 | 1.0100 | 0.7133 |

0.8 | 0.5 | 1.0 | 1.9220 | 0.7310 | 0.0625 | 0.0630 | 8.1584 | 0.8800 | 1.1710 | 0.8270 |

1.0 | 0.5 | 0.25 | 0.1830 | 0.0700 | 0.2000 | 0.2000 | 8.5084 | 0.9180 | 1.0240 | 0.7232 |

1.0 | 0.5 | 0.5 | 0.3890 | 0.1480 | 0.1000 | 0.1000 | 8.2542 | 0.8900 | 0.9560 | 0.6751 |

1.0 | 0.5 | 0.75 | 0.9680 | 0.3680 | 0.0667 | 0.0670 | 8.1695 | 0.8810 | 0.9890 | 0.6984 |

1.0 | 0.5 | 1.0 | 1.9540 | 0.7430 | 0.0500 | 0.0500 | 8.1271 | 0.8760 | 1.1700 | 0.8263 |

**Table 6.**The values of optimization criteria for the variable parameters of machining at fixed depth of cut—a

_{p}= 0.75 mm.

Variable Parameters | Optimization Criteria | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

x_{1}Cutting Speed: v _{c}, (m/min) | x_{2}Depth of Cut: a _{p}, (mm) | x_{3}Feed: f _{r}, (mm/rev) | Surface Roughness: Ra (µm) | Dimensionless Surface Roughness: f_{1} (Ra*), u | Unit Volume Machining Time: T_{m} (min/cm^{3}) | Dimensionless Volume Machining Time: f_{2} (T_{m}*), u | Unit Cost Price of Processing One Part: C, ($) | Dimensionless Cost Price of Processing One Part: f_{3} (C*), u | Length of Estimates Vector: f, u | Length of Estimates Vector: f*, u |

0.4 | 0.75 | 0.25 | 0.1500 | 0.0570 | 0.3333 | 0.3330 | 9.2729 | 1.0000 | 1.1560 | 0.8164 |

0.4 | 0.75 | 0.5 | 0.3530 | 0.1340 | 0.1667 | 0.1670 | 8.6374 | 0.9310 | 1.0260 | 0.7246 |

0.4 | 0.75 | 0.75 | 1.0470 | 0.3980 | 0.1111 | 0.1110 | 8.4237 | 0.9080 | 1.0460 | 0.7387 |

0.4 | 0.75 | 1.0 | 2.1790 | 0.8290 | 0.0833 | 0.0830 | 8.3187 | 0.8970 | 1.2550 | 0.8863 |

0.6 | 0.75 | 0.25 | 0.1160 | 0.0440 | 0.2222 | 0.2220 | 8.8492 | 0.9540 | 1.0650 | 0.7521 |

0.6 | 0.75 | 0.5 | 0.3150 | 0.1200 | 0.1111 | 0.1110 | 8.4237 | 0.9080 | 0.9750 | 0.6886 |

0.6 | 0.75 | 0.75 | 0.9840 | 0.3740 | 0.0741 | 0.0740 | 8.2837 | 0.8930 | 1.0060 | 0.7105 |

0.6 | 0.75 | 1.0 | 1.7020 | 0.6470 | 0.0556 | 0.0560 | 8.2118 | 0.8860 | 1.1220 | 0.7924 |

0.8 | 0.75 | 0.25 | 0.2040 | 0.0780 | 0.1667 | 0.1670 | 8.6374 | 0.9310 | 1.0200 | 0.7203 |

0.8 | 0.75 | 0.5 | 0.3970 | 0.1510 | 0.0833 | 0.0830 | 8.3187 | 0.8970 | 0.9540 | 0.6737 |

0.8 | 0.75 | 0.75 | 1.0340 | 0.3930 | 0.0556 | 0.0560 | 8.2118 | 0.8860 | 0.9980 | 0.7048 |

0.8 | 0.75 | 1.0 | 1.9350 | 0.7360 | 0.0417 | 0.0420 | 8.1584 | 0.8800 | 1.1650 | 0.8227 |

1.0 | 0.75 | 0.25 | 0.1370 | 0.0520 | 0.1333 | 0.1330 | 8.5105 | 0.9180 | 0.9890 | 0.6984 |

1.0 | 0.75 | 0.5 | 0.3580 | 0.1360 | 0.0667 | 0.0670 | 8.2553 | 0.8900 | 0.9370 | 0.6617 |

1.0 | 0.75 | 0.75 | 0.9500 | 0.3610 | 0.0444 | 0.0440 | 8.1702 | 0.8810 | 0.9750 | 0.6886 |

1.0 | 0.75 | 1.0 | 2.0170 | 0.7670 | 0.0333 | 0.0330 | 8.1276 | 0.8760 | 1.1780 | 0.8319 |

**Table 7.**The values of optimization criteria for the variable parameters of machining at fixed depth of cut—a

_{p}= 1.0 mm.

Variable Parameters | Optimization Criteria | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

x_{1}Cutting Speed: v _{c}, (m/min) | x_{2}Depth of Cut: a _{p}, (mm) | x_{3}Feed: f _{r}, (mm/rev) | Surface Roughness: Ra (µm) | Dimensionless Surface Roughness: f_{1} (Ra*), u | Unit Volume Machining Time: T_{m} (min/cm^{3}) | Dimensionless Volume Machining Time: f_{2} (T_{m}*), u | Unit Cost Price of Processing One Part: C, ($) | Dimensionless Cost Price of Processing One Part: f_{3} (C*), u | Length of Estimates Vector: f, u | Length of Estimates Vector: f*, u |

0.4 | 1.0 | 0.25 | 0.1290 | 0.0490 | 0.2500 | 0.2500 | 9.2729 | 1.0000 | 1.1190 | 0.7903 |

0.4 | 1.0 | 0.5 | 0.4400 | 0.1670 | 0.1250 | 0.1250 | 8.6374 | 0.9310 | 1.0100 | 0.7133 |

0.4 | 1.0 | 0.75 | 1.0200 | 0.3880 | 0.0833 | 0.0830 | 8.4237 | 0.9080 | 1.0290 | 0.7267 |

0.4 | 1.0 | 1.0 | 2.6290 | 1.0000 | 0.0625 | 0.0630 | 8.3187 | 0.8970 | 1.3670 | 0.9654 |

0.6 | 1.0 | 0.25 | 0.1890 | 0.0720 | 0.1667 | 0.1670 | 8.8492 | 0.9540 | 1.0400 | 0.7345 |

0.6 | 1.0 | 0.5 | 0.4130 | 0.1570 | 0.0833 | 0.0830 | 8.4237 | 0.9080 | 0.9650 | 0.6815 |

0.6 | 1.0 | 0.75 | 0.9990 | 0.3800 | 0.0556 | 0.0560 | 8.2837 | 0.8930 | 0.9990 | 0.7055 |

0.6 | 1.0 | 1.0 | 1.8840 | 0.7170 | 0.0417 | 0.0420 | 8.2118 | 0.8860 | 1.1580 | 0.8178 |

0.8 | 1.0 | 0.25 | 0.1500 | 0.0570 | 0.1250 | 0.1250 | 8.6374 | 0.9310 | 0.9980 | 0.7048 |

0.8 | 1.0 | 0.5 | 0.3550 | 0.1350 | 0.0625 | 0.0630 | 8.3187 | 0.8970 | 0.9410 | 0.6645 |

0.8 | 1.0 | 0.75 | 1.2140 | 0.4620 | 0.0417 | 0.0420 | 8.2118 | 0.8860 | 1.0200 | 0.7203 |

0.8 | 1.0 | 1.0 | 2.0140 | 0.7660 | 0.0313 | 0.0310 | 8.1584 | 0.8800 | 1.1800 | 0.8333 |

1.0 | 1.0 | 0.25 | 0.2240 | 0.0850 | 0.1000 | 0.1000 | 8.5084 | 0.9180 | 0.9750 | 0.6886 |

1.0 | 1.0 | 0.5 | 0.3250 | 0.1240 | 0.0500 | 0.0500 | 8.2542 | 0.8900 | 0.9260 | 0.6540 |

1.0 | 1.0 | 0.75 | 1.0000 | 0.3800 | 0.0333 | 0.0330 | 8.1695 | 0.8810 | 0.9770 | 0.6900 |

1.0 | 1.0 | 1.0 | 1.8930 | 0.7200 | 0.0250 | 0.0250 | 8.1271 | 0.8760 | 1.1450 | 0.8086 |

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## Share and Cite

**MDPI and ACS Style**

Abbas, A.T.; Pimenov, D.Y.; Erdakov, I.N.; Taha, M.A.; Soliman, M.S.; El Rayes, M.M.
ANN Surface Roughness Optimization of AZ61 Magnesium Alloy Finish Turning: Minimum Machining Times at Prime Machining Costs. *Materials* **2018**, *11*, 808.
https://doi.org/10.3390/ma11050808

**AMA Style**

Abbas AT, Pimenov DY, Erdakov IN, Taha MA, Soliman MS, El Rayes MM.
ANN Surface Roughness Optimization of AZ61 Magnesium Alloy Finish Turning: Minimum Machining Times at Prime Machining Costs. *Materials*. 2018; 11(5):808.
https://doi.org/10.3390/ma11050808

**Chicago/Turabian Style**

Abbas, Adel Taha, Danil Yurievich Pimenov, Ivan Nikolaevich Erdakov, Mohamed Adel Taha, Mahmoud Sayed Soliman, and Magdy Mostafa El Rayes.
2018. "ANN Surface Roughness Optimization of AZ61 Magnesium Alloy Finish Turning: Minimum Machining Times at Prime Machining Costs" *Materials* 11, no. 5: 808.
https://doi.org/10.3390/ma11050808