1 Introduction

Nowadays, face milling is widely used in many industries such as machine and machine tool building, the automobile industry, etc. [1]. One of the major control parameters in face milling is surface roughness [2,3,4,5,6,7]. Grade-H steel materials have various uses and many industrial applications, such as cold-formed components in the automobile industry, among others, on account of their high tensile strength. It is specifically used in safety-related vehicle components, because its high strength means cost-efficient weight reduction of products, without affecting the integrity of the designed components. Increased rigidity of mechanical components can be achieved by using high-strength grade-H steel with no need for further reinforcements. Seat chassis, bumper reinforcement, and door impact beams are typical examples of its use in the automobile industry. Industrial and construction applications of high-strength grade-H steel include tubes for structural applications and hydraulic cylinders and heat- and wear-resistant sheet plates [8]. Some examples of precision face milling of high-pressure products from grade-H steel are breech rings and breech blocks for the manufacturing of basic parts of heavy cannons. The limited resources of the modern world mean that maximum efficiency in their use is an overarching objective. The development of resource-saving technologies including those for face milling is therefore becoming crucial. Optimal employment of resources is an important task when machining costly essential components from grade-H steel, as well as surface quality. The minimum machining time per unit volume and the minimum surface roughness, Ra, are both simultaneously essential. With the fast advance in technological and computational fields, analyzing large bodies of data using AI becomes increasingly relevant.

Research work studying regression models and changes of surface roughness in face milling has been reported in the literature [9,10,11,12,13,14]. Bruni et al. [9] studied the effect of lubrication-cooling condition and various cutting speeds on surface roughness in the face milling of AISI 420 B. Lela et al. [10] investigated the changes in surface roughness in face milling, corresponding to the changes in cutting speed, feed rate, and depth of cut of St 52-3 steel. They applied three different modeling methodologies to experimental data: Bayesian neural network, regression analysis, and support vector machines. Kovac et al. [11] performed the same study as Lela et al. on AISI 1060 steel. The paper demonstrated the advantage of empirical models using the fuzzy logic modeling technique over traditional regression analysis. Simunovic et al. [12] presented research on machined surface roughness in the face milling of aluminum alloys at various machining parameters. Pimenov [13] presented a geometric model of micro-roughness height in face milling on a machined flat surface, taking into consideration the wear of the cutting tool. Werda et al. [14] performed a similar study on X100CrMoV5 to investigate tool life and surface roughness under different cutting conditions: dry machining; minimum quantity lubrication, through inner channels oriented towards the insert rake face; and minimum quality lubrication through inner channels oriented towards the insert flank face.

Other research work studied roughness prediction models for face-milling applications [15,16,17,18,19,20,21,22,23,24]. Baek et al. [15] developed a newly developed model for surface roughness prediction in the face milling of AISI 1045 steel, taking into account both static and dynamic components of the cutting process. Benardos and Vosniakos [16] presented a neural network modeling approach for predicting surface roughness (Ra) in the CNC face milling of a series 2 aluminum alloy. Yazdi and Chavoshi [17] studied the influence of cutting parameters and machining force on material removal rate and surface roughness on AL6061 in CNC face-milling operations. The study used two different modeling techniques and showed that the MLP neural network was more powerful than regression analysis, and they simultaneously performed the estimation of Ra and MRR. Rosales et al. [18] offered a method for estimating surface roughness depending on spindle speed, feed, cutting depth, tool geometry, and run-out, starting from the register of cutting forces in the process. Bajić et al. [19] investigated the effect of various cutting conditions on surface roughness, tool wear, and cutting forces in face milling for the implementation of off-line process control. Two modeling methodologies, namely regression analysis and neural networks, were used in the study. Chavoshi [20] suggested a model for predicting surface roughness depending on various cutting speeds, feed rates, and cutting depth options in the CNC face milling of Stellite 6 alloy and various soft-computing techniques including multi-layer perceptron (MLP), generalized feed forward (GFF), modular neural network (MNN), and a co-active neuro-fuzzy inference system (CANFIS). Saric et al. [21] modeled surface roughness in the face milling of structural steel S235JRG2. Samtas [22] established surface roughness values in relation to cutting parameters in the face milling of an AISI 1040 steel and aluminum alloy 5083 using optical tools. A program was developed from the study to predict optical surface roughness values using a MATLAB m-file and GUI programming. Sheth and George [23] proposed a model for predicting surface roughness and flatness at different spindle speeds, feed rates, and cutting depths in the face milling of grade B Wrought Cast Steel (WCB). Simunovic et al. [24] offered an analysis on surface roughness in the face milling of Al6060 aluminum alloy based on digital imaging technology of the surfaces that were machined with various spindle speeds, feeds, and cutting depths. Other studies [15,16,17,18,19,20,21,22,23,24] proposed models for predicting surface roughness, with no successful optimizations of cutting conditions for better surface roughness.

The following studies may be found on the optimal surface roughness parameters in turning operations [25,26,27,28,29,30]. Bajić et al. [25] studied the influence of various cutting parameters on surface roughness in the face milling of St 52-3 carbon steel. The surface roughness prediction models were obtained using two different modeling approaches: regression analysis and neural networks. The optimal cutting parameters were established using the simplex optimization algorithm. Aykut et al. [26] proposed a hybrid NN with genetic algorithm (GONNS) for the optimization of cutting conditions in the face milling of workpieces from Stellite 6 with 44 Rockwell C hardness, based on experimental data with minimal operator involvement. End surface roughness and cutting force were measured for different cutting speeds, feed rates, and cutting depths without using coolant. Sukumar et al. [27] used the Taguchi method to identify the optimal combination of influential factors in the milling of an AL 6061 alloy. The input parameters taken into account included speed, feed rate, and depth of cut, and the output parameter was surface roughness. Simunovic et al. [28] investigated the influence of optimal machining parameters on the surface roughness of face-milled structural steel. Moghaddam [29] conducted an experimental and numerical study of the face milling of AISI1045 workpieces. He studied the influence of cutting speed, feed rate, and cutting depths on surface quality. In the last section of his research, a mathematical model was developed for surface roughness prediction using particle swarm optimization (PSO) on the basis of experimental results. Rodríguez et al. [30] used artificial intelligence to optimize choosing the right cutting tools in the design of face-milling operations, where end surface roughness is a key criterion. However, the solutions presented in studies [25,26,27,28,29,30] for establishing the optimum cutting conditions only took account of final surface quality, ignoring its relation to machining performance and machining time per unit volume, which is unacceptable when machining such expensive materials as grade-H steel.

Considering the above, let us now look at the studies that establish the optimal parameters of face milling using multi-objective optimization [31,32,33]. Fratila and Caizar [31] outlined the Taguchi optimization methodology, which is applied to optimize the cutting parameters in face milling when machining AlMg3 with a high-speed steel (HSS) tool, in order to obtain the best surface roughness with minimum power consumption. Yang et al. [32] solved the problem of multi-objective optimization of multi-pass face-milling parameters using Pareto optimal solutions. The optimization consists in simultaneously minimizing production time and cost and maximizing profit rate, conditional upon satisfying the constraints on machine power, cutting force, machining speed, feed rate, and surface roughness. Abbas et al. [33] investigated the effect of changing cutting parameters using a full-factorial technique; the studied parameters were the speed (n), depth of cut (ap), and feed (vf), and the measured parameters were the surface roughness parameters (Ra and Rt). Multi-objective optimization was used for minimizing Ra and for maximizing the metal removal rate, Q, and then the results were presented. There are therefore few papers discussing multi-objective optimization of face milling that have stated that the most prominent method to solve this kind of problems is considered to be the Pareto method. However, in studies [31,32,33], no multi-objective optimization of grade-H steel machining is featured. This kind of material is widely used when materials have to be of high operational strength. Considering the high cost of this material, the required surface roughness has to be precisely set and the unit-volume machining time has to be minimized.

The objective of this paper is therefore to establish the face-milling conditions of grade H steel that provide for either minimum cutting power or maximum machining time per unit volume, Tm, while maintaining the design Ra, and taking into account the cost of machining based on an ANN model for predicting surface roughness.

2 Materials and method

2.1 Experimental conditions

Grade-H high-strength steel is used for testing specimens. Table 1 shows its chemical composition following the DEFSTAN 10-13/2005 standard. The heat treatment consisted of heating samples to 870 °C for 4 h followed by oil quenching. Subsequently, heat was applied in a tempering process for 1.5 h at 650 °C and was maintained over a 10-h period, followed by air cooling. Table 2 shows the mechanical properties.

Table 1 Chemical composition for H-steel material according to DEFSTAN 10-13/2005
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Table 2 Mechanical properties for H-steel according to DEFSTAN 10-13/2005
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The machining of the test specimens was performed on a vertical mill as shown in Fig. 1. The surface area of the sample had the following dimensions b = 40 mm, l = 100 mm, and h = 60 mm. The width of cut for all the runs was 40 mm. The cutting tools with carbide inserts were used in a face mill with a Sandvik R245-063Q22-12M tool holder and Sandvik carbide-coated inserts R245-12 T3M-PM4240. The cutter diameter was d = 63 mm with z = 5 edges. This tool is known for providing high-quality surface finishes with efficient material removal rates. It is commonly used for all types of materials from stainless steel to titanium alloys. The investigation was divided into 25 groups of five runs. Clusters of five groups were subject to one common spindle speed n starting from 400 rpm with a 100 increment till 800 rpm. Depth of cut ap varied (0.50, 0.75, 1.00, 1.25, 1.50 mm). Feed rate, vf, varied (50, 75, 100, 125, 150 mm/min). A TESA Rugosurf 90-G model was used for surface roughness measurement.

Fig. 1
figure 1

Machined workpiece setup

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2.2 Strategy for determining the optimal conditions

ANN milling model based on experimental data [34,35,36] was built to achieve our objective of solving the optimization problem in a multi-objective setting using both Edgeworth and the Pareto frontiers [31,32,33, 37,38,39].

The strategy for determining the optimum conditions was as follows:

  1. Step 1.

    Define both the limitations and the boundary conditions to set the optimization criteria. Define the working domain.

  2. Step 2.

    Perform an approximation of three variable functions using Data Mining technique [40,41,42,43], based on experimental data and NN.

  3. Step 3.

    Identify the Pareto curve: optimal decisions and estimates.

  4. Step 4.

    Determine Pareto non-dominated estimates.

  5. Step 5.

    Establish a set of Pareto points that may contain an optimum decision.

As shown in the diagram in Fig. 2, the experimental machining parameter settings were agreed upon between the researchers and the experts. The experts then went on to develop the set of optimal decisions, and the decision-maker selected the most appropriate one. The symbols in use were the same ones used in [44]: DM—decision; FZ—valid estimates; f = (f1, f2, …, fm)—objective function; Y = f(X)—the set of estimates; Ndom X—set of non-dominated estimates; Ndom Y—set of non-dominated estimates; Pf (X)—the set of Pareto optimal frontiers; P (set of Pareto optimal vectors-Pareto optimal estimates).

Fig. 2
figure 2

Flow chart for the multi-objective optimization problem

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3 Results and discussion

The first three steps followed in the research are described below.

3.1 Optimization problem statement (the first step of the strategy)

Based on the objective, the optimization criteria in the milling of the cubic workpiece were established as f1 surface roughness (Ra, μm) and f2 machining time per unit volume (Tm, min/cm3), i.e., m = 2. Consequently, a set of possible Y estimates in the two-dimensional space, R2, formed the vectors f = (f1, f2). A search was then performed for a set of estimates that would have the minimum f vector lengths. The criteria were normalized and placed in a dimensionless form.

Table 3, 4, 5, 6, and 7 shows the results of experiments and also includes the parameters and limitations to the optimization problem: х1 = [400 ÷ 800]—spindle speed, n, rpm; х2 = [0.5 ÷ 1.5]—depth of cut, ap, mm; х3 = [50 ÷ 150]—feed rate, vf, mm/min.

Table 3 Results of experiment and optimization parameter values for variable workpiece milling at ap = 0.5 mm
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Table 4 Results of experiment and optimization parameter values for variable workpiece milling at ap = 0.75 mm
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Table 5 Results of experiment and optimization parameter values for variable workpiece milling at ap = 1.0 mm
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Table 6 Results of experiment and optimization parameter values for variable workpiece milling at ap = 1.25 mm
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Table 7 Results of experiment and optimization parameter values for variable workpiece milling at ap = 1.5 mm
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Surface roughness (Ra, μm) was measured, and the machining time was calculated as follows:

$$ {T}_m=1000/\left({v}_c\times {a}_p\times {v}_f\right) $$
(1)

The dimensionless surface roughness value, f1 (Ra*), and the machining time per unit volume, f2 (Tm*), were calculated on the basis of the following formulas:

$$ {Ra}^{\ast }={Ra}_i/{Ra}_{\mathrm{max}} $$
(2)
$$ {T_m}^{\ast }={T}_{mi}/{T}_{m\ \max }, $$
(3)

where, Rai—surface roughness for current n, ap; Ramax—maximal value of surface roughness, Ra, in the experiment; Tmi—machining time per unit volume for current n, ap, and vf, parameter combinations; Tm max—maximal value of machining time per unit volume.

Tables 3, 4, 5, 6, and 7 reflect the pattern of the optimization criteria changes depending on the machining conditions. These values are also going to be used as a training set for the neural network.

The SKIF Aurora-SUSU supercomputer cluster was used to approximate the surface roughness function, Ra* = f(x1, x2, x3).

3.2 Creating an ANN-based surface roughness prediction model (the second step of the strategy)

A licensed version of Matlab R2010b served as a tool for building the neural network employed in this paper. The Levenberg–Marquardt algorithm was employed to train the NN. Two layers were used to structure the NN: a layer of sigmoid neurons and a linear layer of output neurons. A normalization process was used for all values. It ensured that all inputs complied with the {0,1} domain to enhance the NN training process. The overfitting was dealt with by enhancing the generalization process of the network. A training set was used to update the weighted values of the neurons and a validation set for stopping the training in case of errors. The least mean squared error was used to establish the number of neurons in the hidden layer.

First of all, the multi-layer perceptrons were trained with 9–11 neurons in the hidden layer, with 15% of the records used to validate the model. The lowest error values are presented in Fig. 3.

Fig. 3
figure 3

The lowest mean squared error for the validation set in the … configuration (calculated in Matlab). a MLP 3-9-1. b MLP 3-10-1. c MLP 3-11-1

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An analysis of the graphical functions in Fig. 3a–c led to the conclusion that the lowest error of 0.23% in the validation set was provided by MLP 3-10-1. The determination coefficient of the model was 0.942, which reflects its highly accurate predictions of surface roughness (± 5.78%).

A similar performance appeared to be the best at generalization performance in the cases of allocating 10 and 20% in the validation set of tabular data (see Fig. 4). The errors of both the first and the second training variants were 0.41% (see Fig. 4b) and 0.61% (see Fig. 4c), respectively.

Fig. 4
figure 4

The lowest mean squared error in generalizing the experimental data in MLP 3-10-1 (a) with various validation sets: b 10% and c 20% (calculated in Matlab)

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The following tendency was revealed when using an MLP 3-10-1. Increases of 0.1 items in both n and vf increased the Ra* values by 0.3028 (0.073 μm) and 0.291 items (0.071 μm), respectively. An inverse effect of ар on surface roughness was observed: an increase of 0.1 items in ар led to a decrease in Ra* of 0.009 (0.002 μm). The weight ratio of the effects of technological parameters on Ra* was ар: n: vf = 1:25:25.

3.3 3.3. Determining the Pareto curves (the third step of the strategy).

Ra* values, calculated for the experimental values x1, x2, x3 (see Tables 3, 4, 5, 6, and 7) using the neural network, are used to show the Pareto frontiers in Fig. 5.

Fig. 5
figure 5

Pareto frontiers for machining time of unit volume Tm* with surface roughness, Ra*, at a fixed ap, and n, with varying feed rates vf (at higher feed rates, Ra* increases, and Tm* decreases)

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Intercepts АВ, BC, and DE are plotted at the end points of the curves closest to the reference points Ra* and Tm* (Fig. 6).

Fig. 6
figure 6

Intercepts demonstrating the relation of Ra* and Tm*: АВ corresponds to n = 600 rpm, vf = 50 mm/min, ap = 0.5…0.75 mm; BC corresponds to n = 600 rpm, vf = 50 mm/min, ap = 0.75…1.0 mm; DЕ corresponds to n = 600 rpm, vf = 50 mm/min, ap = 1.25…1.5 mm. Point coordinates: А (1.000; 0.293), В (0.280; 0.666), C (0.262; 0.500), D (0.340; 0.400), E (0.361; 0.325)

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Intercept AB corresponds to n = 600 rpm, vf = 50 mm/min, ap = 0.5…0.75 mm. Intercept BC corresponds to n = 600 rpm, vf = 50 mm/min, ap = 0.75…1.0 mm. Intercept DЕ corresponds to n = 600 rpm, vf = 50 mm/min, ap = 1.25…1.5 mm. The coordinates of the intercept ends are А (1.000; 0.293), В (0.280; 0.666), C (0.262; 0.500), D (0.340; 0.400), E (0.361; 0.325). Following the trend of the target function f, 11 points on the Pareto front were established with the following new numbers of the points: А → number 1, В → number 2, C → number 3, G → number 4, H → number 5, E → number 6, I → number 7, J → number 8, K → number 9, L → number 10, F → number 11 (Fig. 7).

Fig. 7
figure 7

The intersection coordinates of the plotted intercepts, the Ra* and Tm* curves closest to the reference points, and the coordinates of their intersections with each other. It includes the lowest point of those functions, crucial for building the Pareto frontier, with the following values: G (0.283; 0.471), H (0.360; 0.352), I (0.400; 0.310), J (0.450; 0.252), K (0.500; 0.215), L (0.525; 0.197), and F (0.582; 0.111)

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Ten sections were established from the Pareto frontier analysis. Section I, between point 1 and point 2, relates to ap = 0.5…0.75 mm, vf = 50 mm/min, and n = 600 rpm; section II between point 2 and point 3, relates to ap = 0.75…1.0 mm, vf = 50 mm/min, and n = 600 rpm; section III, between point 3 and point 4, relates to ap = 1.0…1.12 mm, n = 600 rpm, and vf = 50 mm/min; section IV, between point 4 and point 5, relates to ap = 1.12…1.25 mm, n = 600 rpm, and vf = 50 mm/min; section V, between point 5 and point 6, relates to ap = 1.25…1.5 mm, vf = 50 mm/min, and n = 600 rpm; section VI, between point 6 and point 7, relates to ap = 1.5 mm, n = 600 rpm, and vf = 50…100 mm/min; section VII, between point 7 and point 8, relates to ap = 1.5…0.75 mm, n = 600 rpm, vf = 100 mm/min; section VIII, between point 8 and point 9, relates to ap = 0.75…1.25 mm, n = 600 rpm and vf = 100…200 mm/min; section IX, between point 9 and point 10, relates to ap = 1.25…1.0 mm, n = 600 rpm, and vf = 120…150 mm/min; section X, between point 10 and point 11, relates to ap = 1.0…1.5 mm, n = 600 rpm, and vf = 150 mm/min. Point 10, at a special point on the Pareto frontier, was at an intersection of three curves for ap = 1.0 mm, n = 600 rpm, and vf = 150 mm/min; ap = 1.25 mm, n = 600 rpm, and vf = 150 mm/min; ap = 1.5 mm, n = 600 rpm, and vf = 150 mm/min.

3.4 3.4. Determining the Pareto frontier (the fourth step of the strategy).

Expert assessments were employed, so as to narrow the set of Pareto points, in order to determine the higher weighing factor of the machining time criteria, Tm*, over the surface roughness, Ra*. As we can see in Fig. 8a, blue line indicates the Pareto non-dominated estimates. The end point of this vector with the coordinates (0.358; 0.358). As seen in the figure, point no. 12 can be considered as the non-dominated Pareto point for an unconstrained optimization with equivalent criteria f1 and f2. The global optima in this case corresponds to the following values: Tm = 0.358 min/cm3, Ra = 0.087 μm, n = 600 rpm, ap = 0.75 mm, and vf = 82 mm/min.

Fig. 8
figure 8

Global optima Pareto frontier for the dimensionless parameters: Tm = 0.358 min/cm3, Ra = 0.087 μm, n = 600 rpm, ap = 0.75 mm, vf = 82 mm/min, with the corresponding real parameters: Tm = 0.111 min/cm3, Ra = 0.142 μm, n = 600 rpm, ap = 1.5 mm, vf = 150 mm/min

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3.5 Determining the optimal cutting parameters (the fifth step of the strategy)

In the final step of the algorithm, information gained in the DM was used to set the maximum degradation in the surface roughness value. It was equal to Ra = 0.142 μm corresponding to the 11th point of the Pareto frontier (0.582, 0.111)—Fig. 8b. In this case, the optimization criteria Tm* was five times smaller than Ra and for points 12 and 11, the valid preference was y11 ⊱ Y y12 and the induced preference was х11 ⊱ X х12. As a result, the selected points are presented above, the green vector pointing to values of (0.142, 0.111), of the optimum cutting parameters (n = 600 rpm, ap = 1.5 mm, vf = 150 mm/min).

Figure 9a and b show the profile of the machined surface corresponding to the global optimum conditions (Tm = 0.358 min/cm3, Ra = 0.087 μm, n = 600 rpm, ap = 0.75 mm, vf = 82 mm/min), and Fig. 9c, d shows the same results for the local optimum (Tm = 0.111 min/cm3, Ra = 0.142 μm, n = 600 rpm, ap = 1.5 mm, vf = 150 mm/min). The stylus of surface roughness tester moved in a parallel direction to the cutting direction.

Fig. 9
figure 9

Optical microscopy results and surface profile for the optimized machining conditions. a, b Global optimum (Tm = 0.358 min/cm3, Ra = 0.087 μm, n = 600 rpm, ap = 0.75 mm, vf = 82 mm/min). c, d Local optimum (Tm = 0.111 min/cm3, Ra = 0.142 μm, n = 600 rpm, ap = 1.5 mm, vf = 150 mm/min)

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In summary, the accurate values of the Pareto curve and the vector coordinates were automatically obtained in Matlab using a customized NN algorithm. The vector of estimates, f, was selected as the optimization criterion, with a boundary limit of Tm*/Ra* = 1/5. The Newton method with quadratic convergence was employed to solve the non-linear constrained optimization problem. The algorithm implemented in Matlab permitted rapid convergence towards the local optimal cutting parameters in CNC robot-aided machining depending on spindle speed and depth and their limitations.

The Edgeworth-Pareto method in the analyses generated the optimal surface treatment conditions for high-strength grade-H steel as part of the experimental conditions. In the analyses that were performed, as well as the variable cutting depth values, the machine tool settings were used as input parameters: n—spindle speed, vf—feed table.

Experiments were performed using a 5-tooth cutter with a diameter of 63 mm. The results could not therefore be used for other parameters of the tool and machine tool settings. Further analysis was performed to apply the other analytical results under other conditions (tool diameter, number of teeth) relating to high-strength grade-H steel processing.

The cutting speeds corresponding to the test conditions were determined (Table 8).

Table 8 Cutting speed for different spindle speed values (d = 63 mm)
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The milling process was significantly influenced by feed rate per tooth. The analyses were performed at the table feed, vf.

Based on the known dependence, the feed rate per tooth values were determined (Table 9) for the different machining process conditions used in the tests (n and vf), as presented in Table 9 and Fig. 10.

Table 9 Values of feed per tooth, fz, for different relations of parameters n and vf used in the experiment
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Fig. 10
figure 10

Results of theoretical fz values used in the experiments presented in the paper

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There was no reduction in surface roughness at low feed rate values per tooth, fz. One reason was the minimal thickness of the uncut chip [45,46,47,48,49]. The influence of the feed rate on surface condition was not observable in the curves of the graphs (Figs. 5, 6, and 7).

The cutting speed in the tests and the feed-rate values per tooth can be adapted to incorporate the test results and analyses from other steel processing conditions.

On the basis of the analysis of the optimal conditions for the tool (Ra = 0.142 μm at a minimal unit-volume machining time of Tm = 0.111 min/cm3) that has been presented, the following machining values were obtained: vc = 118.69 m/min and fz = 0.05 mm/tooth at ap = 1.5 mm. These values can be used for other cutters and machine tools in the milling of high-strength grade-H steel.

3.6 Estimating the cost and power of milling operations for the Pareto curve

The cost price of processing one part, Сi, is determined by the formula:

$$ {\mathrm{C}}_i=\left({C}_{Mh}\times {T}^{'}\right)+\left({C}_{Toolmin}\times {T}^{'}\right)+{C}_w, $$
(4)

where turning time is T/ = (L + l1)/(n × vf), where n = (1000 × vc)/(π × dt); (4) where CMh—cost of machining per hour (SR 400) (CMh = $140); CToolh—cost of tool holder (CToolh = $100); LTToolh—tool holder life (LTToolh = 5 Year × 365 Day × 24 h = 31,536,000 min); CIn—cost of insert (CIn = $14); k/—insert setup (k/ = 4); z—number of cutting teeth (edges) (z = 5); Cw—unit Cost of workpiece (Cw = $23); T—tool life (T = 60 min); CToolmin—tool cost per minute (CToolmin = [(CIn × z)/(T × k/)] + (CToolh × LTToolh) = $0.33).

The tool manufacturer, Sandvik, offers a formula for the calculation of cutting power (cutting power) (kW):

$$ {P}_c=\left({a}_p\times {a}_e\times {v}_f\times {K}_c\right)/\left(60\times 106\right) $$
(5)

where ae—work engagement (milling width) (ae = 40 mm in our case); Kc—specific power (specific energy) N/mm2 (Kc = 3750 N/mm2 for high-strength steel grade-H).

If we calculate the cost of the milling operation and the input cutting power along the Pareto curve from point 1 to point 11 (see Fig. 8) in accordance with formulas (4) and (5), then C appears to decrease from $27.08 to $24.36, and Pc increases from 0.0625 kW to 0.5625 kW. The values of these parameters for the optimums are global optimum C = $25.48, Pc = 0.359 kW, local optimum C = $24.36, Pc = 0.5625 kW. The milling conditions change from the blue vector to the green one at the same time as Tm increases 3.2 times and Pc 1.5 times, while the manufacturing cost decreases by 5%.

4 Conclusion

The optimization of cutting conditions for CNC face-milling operations on high-strength grade-H steel has been presented in this article based on artificial neural networks and the Edgeworth-Pareto method.

Surface roughness, Ra, has been predicted using an ANN MLP 3-10-1 multi-layer perceptron after finishing the face-milling process with the following ranges of parameters: cutting speed from vc = 78 to 158 m/min, cutting depth ap = 0.5–1.5 mm, and the feed per tooth fz = 0.013–0.075 mm/tooth with a precision of ± 5.78%.

Using neural network models in the study of the milling of grade-H steel: a positive effect of n and vf and a negative effect of depth of cut, ap, on surface roughness have been established. The emphasis of n and vf was 25 times higher than that of ap.

The global optimum for the milling of a grade-H steel workpiece has been established: a surface roughness of Ra = 0.087 μm and a unit-volume machining time of Tm = 0.358 min/cm3 corresponded to the optimum conditions at a face milling—cutting speed vc = 118.7 m/min, a depth of cut ap = 0.75 mm, and feed per tooth fz = 0.027 mm. Under these conditions, given a small (5%) increase in manufacturing cost, Pc decreased 1.5 times as compared to the local optimum machining conditions.

The local optimum for the milling of a grade-H steel workpiece has been established: a surface roughness of Ra = 0.142 μm and a minimum unit-volume machining time of Tm = 0.111 min/cm3 corresponded to the optimum conditions at a face milling—cutting speed vc = 118.7 m/min and feed per tooth fz = 0.05 mm/tooth at a depth of cut ap = 1.5 mm. Under those conditions, the maximum decline in surface roughness, Ra, to 0.142 μm yielded a 3.2-times increase in Tm as compared to the global optimum.

The tests in the form of the global and local optima, obtained by analyzing the test results with the methodology based on the use of an ANN together with Edgeworth-Pareto methods, can be applied to other technological conditions for Grade-H steel face-milling machining, by adjusting the cutting speed, vc; the feed per tooth, fz; and the cutting inserts.