Development of Machine Learning Model for Analysis of Total Manufacturing Cost in Medium Turning of C45E Steel

Abstract

The primary goal of manufacturing technologies in the metalworking industry is to provide products with specified quality characteristics, while maximizing time and cost efficiency. The total manufacturing cost in turning depends on a number of factors. The analysis of their effects and the estimation of the total manufacturing cost are of practical importance in process planning. Therefore, in the present study, the relationship between four inputs (depth of cut, feed rate, cutting speed and volume of material to be removed) and the total manufacturing cost in medium turning of C45E steel was modeled by using an artificial neural network (ANN). The developed ANN model was used for the analysis of the main and interaction effects of the aforementioned inputs on the total manufacturing cost. Verification of the observed effects was also carried out by applying the connection weight approach. The total manufacturing cost was mostly affected by depth of cut, while the effect of cutting speed was least pronounced. In addition, the results also revealed the presence of two-way interactions associated with cutting speed. For the given case study (with defined volume of material to be removed and specified machine tool), an optimized cutting regime was determined by developing and solving a single-objective turning optimization problem with three constraints related to chip slenderness, cutting power and depth of cut. Cutting force, needed for the estimation of cutting power, was estimated by using the dimensional analysis-based prediction model.

1. Introduction

Efficient and effective application of manufacturing technologies is the most significant factor of success on the market for production companies [1,2]. Actually, the primary goal of manufacturing technologies in the metalworking industry is to provide products with specified quality characteristics, while maximizing time and cost efficiency [3,4,5], i.e., to achieve an optimal balance between productivity, flexibility, quality and cost [6,7]. In this regard, analysis of production costs is an important and significant issue, particularly in the context of unpredictable markets, the price of raw materials, energy costs, workforce availability, the changing demands of customers, etc. In response to ecological challenges and economic pressures, manufacturing industries are prioritizing the reduction of carbon emissions in manufacturing processes with the ultimate aim of reducing costs and attracting more customers [8].

In parallel with the development and automation of lathes, cutting tools and cutting fluids, there is a constant need to determine optimal process parameters in order to minimize the cost. Cost optimization of turning operations is an active field of research where different methods and approaches are used for determining the most suitable cutting conditions for a given machining system [4]. A number of recent research studies have addressed issues regarding analysis, modeling and optimization of costs in turning from different aspects. Recent relevant studies in the field include the following studies.

Mir and Wani [9] made an economic analysis and total cost per part comparison when using PCBN, coated carbide and mixed ceramic inserts in finish turning of AISI D2 steel. The results obtained showed that the use of mixed ceramic inserts significantly reduces costs. Tebaldo et al. [10] made an analysis of turning costs under different lubrication conditions (wet, dry, MQL, MQC and cryo) in turning Inconel 718. It turned out that the use of cutting fluid is the most economically viable option if one considers production cost per unit volume of removed material. Bagaber and Yusoff [11] developed a Pareto turning optimization problem by considering machining cost and energy consumption in dry turning of stainless steel. With the use of the non-dominated sorting genetic algorithm II (NSGA-II), optimized cutting conditions were determined at the same time, ensuring energy and cost savings. Ranjan Das et al. [12] performed a cost analysis based on Gilbert’s approach for estimating total machining cost per part in hard turning of AISI 4340 steel using a coated carbide insert. Likewise, Kumar et al. [13] made a cost analysis in hard turning of AISI D2 steel using a mixed ceramic insert. Gilbert’s approach was implemented for the optimal combination of cutting parameters with respect to surface roughness, tool life and flank wear, which was determined using Taguchi-based grey relational analysis. Abbas et al. [14] developed a multi-layer perceptron (MLP) model for the prediction of surface roughness, machining time and prime machining cost, in terms of depth of cut, feed rate and cutting speed, in the finish turning of the AZ61 magnesium alloy. Furthermore, Edgeworth–Pareto optimization was performed so as to determine the optimal parameter settings. Balogun et al. [15] determined optimal cutting parameter settings for the minimization of production cost (machine, tool and electricity costs) in the turning of AISI 1040 steel. It was observed that, compared to the cutting tool manufacturer’s recommended conditions, up to approximately 50% in cost per volume of material removed can be saved. In addition, since energy cost is significantly lower than cutting tool cost, it was concluded that cost can be optimized fairly accurately without the need to model energy demand. Tian et al. [8] proposed an approach for integrated optimization of machining parameters and cutting tools with respect to carbon emissions, cost and time for machining features of parts. A case study of dry turning of alloy steel 38CrMoAl verified the adequacy of the proposed multi-objective optimization model and applied the NSGA-II algorithm for solving the model. Comparative analysis of total machining cost in hard turning under dry, MQL and cryogenic conditions was performed by Jamil et al. [16]. It was concluded that the cryogenic-assisted machining yields the lowest overall machining cost. Zubair and Abu Mansor [17] developed a CAPP system aimed at selecting the most appropriate cutting insert based on the specific volume of material to be removed, while simultaneously optimizing cutting speed, feed rate and depth of cut for minimal unit production cost. Abbas et al. [18] analyzed total machining costs in turning AISI 1045 steel under dry, flood and MQL-nanofluid lubrication conditions. In addition, Pareto-optimized cutting regimes were determined for simultaneous minimization of total machining costs and CO₂ emissions, as well as total machining costs, surface roughness and power consumption. With the application of the genetic algorithm, Radovanović [4] determined a set of Pareto optimization solutions, for both roughing and finishing, with respect to machining cost and machining time needed to remove a unit volume in turning AISI 1064 steel. Torres et al. [19] used stochastic programming coupled with the response surface methodology (RSM) to formulate and solve an optimization problem aimed at minimizing the total process cost per piece in hard turning of AISI 52100 steel. In contrast to most research studies, the authors, in addition to the main cutting parameters, also considered other stochastic variables, such as setup time, insert changing time, batch size, machine and labour costs, etc. Trifunović et al. [20] formulated a Pareto optimization problem of multi-pass turning of grey cast iron with part production cost and consumed energy as objective functions; they applied a deterministic approach, i.e., brute force optimization algorithm, for its solving. Cutting speed was identified as the decision variable of the highest significance. Uysal et al. [21] analyzed the effects of cutting speed, undeformed chip thickness and cutting conditions (dry, MQL and cryogenic) on machining cost in turning of AISI 304 austenitic stainless steel. In addition, the authors developed a multi-objective optimization model with carbon emissions, machining costs and human health factor as objectives. The NSGA-II algorithm was implemented to determine a set of Pareto optimal solutions. Khan et al. [22] developed a process performance simulator for the sustainability assessment of the Al₂O₃ nanofluid-assisted turning of cobalt-based superalloy (Haynes 25). It was observed that feed rate and cutting speed had the greatest effect on production cost, while the effect of nanoparticle concentration was the least pronounced. Khan et al. [23] analyzed tool wear and its morphology of ceramic cutting tools in the turning of lightweight composites. By the application of the Adam gene algorithm, the authors determined the minimal production cost and corresponding cutting parameters for two cutting tools. Machinability and economic aspects of Haynes 25 turning under novel cryogenic-LN oils-on-water (LNOoW) conditions were investigated by Khan et al. [24]. The authors proposed a new mathematical model to estimate the part production cost. It was observed that the combination of the highest feed rate and intermediate cutting speed values yielded the lowest production cost. Chung et al. [25] proposed a novel concept for cutting parameter optimization, while considering the effect of part geometry on tool wear. The goal was to determine optimized cutting conditions using the conjugate gradient descent method so that the resulting tool life would coincide with the completion of the machining operation. Cesén et al. [26] proposed a general cost model for estimating manufacturing costs of a turned part, taking into account the input–output turning system. The validity of the developed model was experimentally verified for the turning of AISI 1018 steel. Saatçi et al. [27] analyzed the estimated total machining cost for different cutting speeds, feed rate values and cutting conditions in the turning of AISI 310S austenitic stainless steel. Statistical and experimental results showed that total machining cost was mostly affected by cutting speed, followed by feed rate, while the effect of cutting conditions was least pronounced. The authors also performed sustainability-oriented optimization, with the application of the NSGA-II algorithm and TOPSIS, so as to determine optimal cutting conditions and machining regime, while considering at the same time resultant force, surface roughness, carbon emission, and total machining cost. Xie et al. [5] proposed the use of a Gaussian quantum-behaved bat algorithm (GQBA) for minimizing unit production cost in turning. By considering the well-known benchmark multi-pass turning optimization problem with multiple constraints [28], the authors concluded that GQBA outperformed other metaheuristic algorithms in terms of cost reduction. By solving the same multi-pass turning optimization problem, the same group of authors outlined the strong optimization capability of an optimization approach combining GQBA and the divide-and-conquer strategy [29]. Doan et al. [30] investigated environmental and quality indicators in rotary turning of Ti6Al4V. In addition, by formulating and optimizing the machining cost model, optimal values of machining parameters (inclination angle, turning depth, feed rate and turning speed) were determined. Buchmeister et al. [31] investigated turning optimization by considering two objective functions (machining time and machining costs) as a function of multiple influencing independent variables, such as machine tool, workpiece material, cutting tool, cutting parameters, etc. To this aim, the model was designed in an Excel spreadsheet and solved using the approximation method while considering two case studies, a special low-carbon structural steel EN 6CrMo15-5 turning on a conventional universal lathe and gray cast iron EN-GJL-250 turning on a CNC lathe. Agarwal and Khare [32] applied the desirability function approach for multi-objective optimization of energy consumption, processing time and total machining cost in turning of AISI 1040 steel using a carbide cutting tool. The analysis of optimization results revealed a trade-off between these performances. Pangestu et al. [33] proposed a multi-objective model for multi-pass CNC turning optimization for sustainable manufacturing. It considered spindle speed, feed rate, depth of cut and number of roughing passes as independent variables and energy consumption, carbon emissions, production time and production cost as objective functions and key sustainable manufacturing metrics. In addition, a sensitivity analysis was performed to analyze the effects of independent variables on objective functions. In multi-pass turning optimization, Pujiyanto et al. [34] considered five objective functions, i.e., total energy, surface roughness, total noise, total cost and total carbon emission. The optimal values of cutting speed, depth of cut, feed rate and number of roughing passes were determined using NSGA-II, while the final solution was selected using the TOPSIS method.

Most of the reviewed studies dealt with optimization with case study and comparative analysis of alternative cutting fluid types and applications (techniques), as well as application of metaheuristics to benchmark problems. Analysis of the main and particularly the interaction effects of parameters on total costs was limited.

As noted by Bagaber and Yusoff [11], a systematic analysis of the effects of process parameters and their appropriate selection not only ensures quality but also reduces cost. Moreover, by optimizing feed rate, depth of cut and cutting speed, costs can be reduced by as much as 47% when compared to using the cutting tool manufacturer’s recommended values [15]. Ultimately, in order to consider different machining constraints, determining optimized machining parameter values for the minimization of machining costs should be based on the development of turning optimization models [29]. By optimizing manufacturing costs in turning, resources can be used appropriately, waste can be avoided, and an economic benefit can be generated, in line with the concept of sustainable manufacturing [26,35].

As can be observed from the literature review, analyses of machining and manufacturing costs were mostly related to specific case studies, comparative evaluations of alternative cutting and lubrication conditions, and the application of different metaheuristics while solving well-known benchmark problems, with limited analysis of the quantitative and qualitative effects of process parameters, especially interaction effects. In this regard, the aim of the present study is to analyze main and interaction effects of depth of cut, feed rate, cutting speed and volume of material to be removed on the total manufacturing cost in medium turning of C45E steel, and determine the optimized cutting regime for the given case study by developing and solving a single-objective turning optimization problem with three constraints related to chip slenderness, cutting power and depth of cut. It is believed that the implementation of such a study opens up the possibility of better technological process planning for a specific case study, by focusing on parameters that have a decisive influence on the total manufacturing cost.

2. Overview of the Applied Approach

In the present study, the relationship between four inputs (depth of cut, feed rate, cutting speed and volume of material to be removed) and the total manufacturing cost in medium turning of C45E steel was modeled by using an ANN. The total manufacturing cost model combines two objective functions proposed in previous research and is based on models for the tool life consumed during machining, volumetric material removal rate, tool life and production rate. The developed ANN model was used for the analysis of the main and interaction effects of the aforementioned inputs on the total manufacturing cost. A case study is presented for the purpose of determining optimal cutting parameter values with regard to minimizing the total manufacturing cost in multi-pass medium turning of steel C45E for a specific volume of material to be removed, considering three constraints related to chip slenderness, cutting power and depth of cut. Cutting force, needed for the estimation of cutting power, was estimated by using the dimensional analysis-based prediction model. Figure 1 gives an overview of the applied approach.

3. Total Manufacturing Cost Model

Although analytical formulations exist for estimating cutting tool, labour and overhead costs, multiple nested equations with iterative calculations are needed for estimating the total manufacturing cost under different cutting conditions. Therefore, a machine learning-based surrogate model is proposed to provide a unified, computationally efficient and practically applicable manufacturing cost prediction framework. This surrogate model not only enables systematic exploration of the main and interaction effects of process parameters on manufacturing cost but also supports sensitivity and perturbation analyses and can be employed as an optimization function. In recent years, numerous machine learning models have been proposed for the analysis, modeling, and optimization of production processes [36]. Among various machine learning models, the ANN model was selected in this research for modeling the input–output relationships, due to the powerful approximation and generalization capabilities of ANNs [37]. Based on the applied experimental design, applied models for estimating tool life, material removal rate, production rate, used tool life, cutting insert data, and labour and overhead costs, an attempt has been made to model the relationship between the inputs (depth of cut, feed rate, cutting speed and volume of material to be removed) and the output (total manufacturing cost) by using an ANN. This model would allow for detailed analysis of the effects of the aforementioned inputs on the change in the resulting total manufacturing cost. It should be noted that this preliminary analysis does not take into account the resulting cutting force, assuming that all combinations of cutting parameters are feasible, according to the recommended ranges.

The total manufacturing cost model combines two objective functions proposed in previous research [38], and has the following form:

$$C_{tot}=C_t+C_{lo}=\epsilon \cdot {\displaystyle \frac{C_{ins}}{n_{ce}}}+{\displaystyle \frac{\tau }{60}}\cdot C_{slo}$$

(1)

where C_tot (EUR) is the total manufacturing cost, C_t (EUR) is the tool cost, C_lo (EUR) is the labour and overhead cost, ε is the tool life consumed during machining, C_ins (EUR) is the cutting insert cost, n_ce is the number of cutting edges, τ (min) is the production rate, and C_slo (EUR/h) is the specific labour and overhead cost.

The tool life consumed during machining model has the following form:

$$\epsilon ={\displaystyle \frac{V}{Q_W\cdot T}}={\displaystyle \frac{V}{\left(a_p\cdot f\cdot v\right)\cdot \left(C\cdot {a_p}^x\cdot f^y\cdot v^z\right)}}$$

(2)

where ε is the tool life consumed during machining, V (cm³) is the volume of material to be removed, Q_W (cm³/min) is the volumetric material removal rate, T (min) is the tool life, a_p (mm) is the depth of cut, f (mm/rev) is the feed rate, v (m/min) is the cutting speed, and C, x, y and z are the empirical constants determined using the available experimental data [39].

The production rate model has the following form:

$$\tau ={\tau }_s+{\displaystyle \frac{V}{Q_W}}\cdot \left(1+{\displaystyle \frac{{\tau }_{tc}}{T}}\right)+{\tau }_0$$

(3)

where τ (min) is the production rate, τ_s (min) is the setup time, V (cm³) is the volume of material to be removed, Q_W (cm³/min) is the volumetric material removal rate, τ_tc (min) is the cutting tool change time, T (min) is the tool life, and τ₀ (min) is the cutting tool idle time. The setup time, cutting tool change time and cutting tool idle time are assumed as 0.15, 0.5 and 0.05 min, respectively.

3.1. ANN Model Development

For the establishment of a mathematical relationship between the inputs (depth of cut, feed rate, cutting speed and volume of material to be removed) and the total manufacturing cost in medium turning of C45E steel, a single hidden layer ANN model was proposed. It has been previously proven that a feed-forward single hidden layer ANN can approximate any continuous function to a given degree of accuracy, provided that a sufficient number of hidden neurons are used along with appropriate activation functions [40,41]. There are various methodologies to estimate the number of neurons in the hidden layer. It can be determined based on the number of input neurons [42]. The number of neurons in the hidden layer in this study was determined based on the following logic: For an ANN to be mathematically defined it is necessary that the number of free parameters, i.e., synaptic weights and biases, be less than or equal to the number of available data for ANN training [43]. The experimental data from six stacked nearly orthogonal Latin hypercube designs were used, wherein training, testing and validation sets were selected by a random method. A stacked nearly orthogonal Latin hypercube design was chosen because it provides good coverage of the entire multidimensional parameter range by data points needed for obtaining realistic and accurate results [44]. The Levenberg–Marquardt algorithm was used for ANN training due to its stability and fast convergence [45,46]. The training process was monitored via mean squared error. In order to develop an accurate ANN model with good generalization capability, the training process was repeated several times using different initial weights and architectures with a varying number of hidden neurons. The training and architectural parameters of the developed ANN model are given in

Table 1. Total manufacturing cost ANN model parameters.

Number of input/hidden/output neurons	4/6/1
Training algorithm	Levenberg–Marquardt
Maximum number of training epochs	500
Transfer function in hidden layer	Hyperbolic tangent sigmoid
Transfer function in output layer	Linear
Data scaling range	[−1, 1]

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3.2. ANN Model Assessment

To assess the validity and prediction accuracy of the developed ANN model for predicting the total manufacturing cost, statistical indices, such as correlation coefficient (R) and mean absolute percentage error (MAPE), were used. The results obtained are given in

Table 2. Prediction performance of the developed prediction model.

Prediction Model	R	MAPE (%)
ANN	0.995	3.59

.

In addition, residual analysis is given in Figure 2. As can be observed, there is a random pattern for residual distribution. Therefore, the developed ANN model for predicting the total manufacturing cost is appropriate. Considering the above, one can argue that the developed ANN model proved to be able to model the underlying relationships between the process input parameters and the resulting total manufacturing cost in medium turning of C45E steel. Thus, it can be used for more detailed analysis of the effects of depth of cut, feed rate, cutting speed and volume of material to be removed on the total manufacturing cost.

The weights and biases of the trained ANN model for the prediction of the total manufacturing cost are given in

Table 3. The weights and biases of the trained ANN model.

IW1				IW2	b1	b2
1.503	1.388	−0.178	2.377	−0.113	−2.085	1.366
0.037	1.759	−2.389	2.924	0.008	2.296
2.146	2.974	−1.463	−0.559	−0.009	0.038
0.876	0.349	0.139	−0.289	−2.213	1.588
0.932	−0.447	−1.377	0.068	0.074	−0.769
−0.199	−0.168	0.191	0.616	0.477	0.201

IW₁—weights between input and hidden layer; IW₂—weights between hidden and output layer; b₁—hidden neuron’s biases; b₂—hidden neuron bias.

.

4. Results and Discussion

The effects of the process input parameters on the resulting total manufacturing cost were examined for different two-way interaction combinations, while keeping the third and fourth parameters constant at the central level. Six 3D surface plots were generated using the developed ANN model and are given in Figure 3.

As can be observed from Figure 3a–c, for different volumes of material to be removed, an increase in depth of cut decreases the total manufacturing cost consistently and non-linearly, regardless of the values of feed rate and cutting speed. Likewise, Figure 3a,d,e indicate that there exists an approximately linear consistent decrease in the total manufacturing cost with an increase in feed rate, for different combinations of inputs. By increasing either feed rate or depth of cut, material removal rate is increased, which is significantly reflected in the reduction in machine tool usage and labour costs. As expected, due to the prolonged cutting time required to carry out the process, the increased volume of material to be removed results in a higher total manufacturing cost, wherein the magnitude of the total manufacturing cost rise is particularly expressed at the lowest depth of cut and feed rate values (Figure 3c,e), while almost being unaffected regarding the different values of cutting speed (Figure 3f). The most interesting effect on the total manufacturing cost comes from cutting speed (Figure 3b,d,f). Namely, for low to medium depth cut values, an increase in cutting speed decreases the total manufacturing cost, while for medium to high depth cut values, an increase in cutting speed increases the total manufacturing cost. Also, the effect of cutting speed on the total manufacturing cost is variable and depends on feed rate. As can be observed, for low to medium feed rate values, an increase in cutting speed slightly decreases the total manufacturing cost, while for medium to high feed rate values, an increase in cutting speed slightly increases the total manufacturing cost. Likewise, for volumes of material to be removed up to 162.5 cm³, an increase in cutting speed up to 243 m/min decreases the total manufacturing cost, after which the total manufacturing cost starts to increase with an increase in cutting speed. For larger volumes of material to be removed, this change in influence on the total manufacturing cost occurs at higher values of cutting speed. The variable effect of cutting speed could be explained considering that it simultaneously influences both material removal rate and particularly tool life, with a more pronounced effect in comparison to feed rate and depth of cut. Thus, although increasing cutting speed reduces machining time and associated machine tool usage and labour costs, higher cutting speeds result in increased cutting tool wear progression, i.e., decreased tool life, which results in a higher cutting tool cost. The study by Khan et al. [24] also revealed that cutting speed for different cutting environments may have a variable effect on the resulting costs. Also, as noted by Torres et al. [19], extending cutting tool life by the appropriate selection of cutting conditions may not necessarily reduce the process cost, and actually can even increase it. A previous empirical study by Saatçi et al. [27] argued that the use of higher cutting speeds may be beneficial for cost reduction due to the observed decrease in the built-up edge formation.

From the analysis of Figure 3, one can argue that depth of cut, followed by feed rate and volume of material to be removed, has maximum influence on the total manufacturing cost. The effect of cutting speed on the total manufacturing cost is least pronounced but is characterized by significant interactions. The pronounced effect of feed rate observed here agrees well with observations reported in prior studies [11,21,22,24].

It has to be noted that, given that machine tool usage and labour costs have risen significantly, in contrast to the relatively modest increase in cutting tool and tool change costs, the use of cutting conditions with higher material removal rates and corresponding reduced cutting tool life ultimately leads to lower manufacturing costs [3,25,27].

It should also be noted that the conducted analysis and results are valid for labour-, cutting tool- and machine tool-related costs, and are specific for the present case study. Likewise, the effects of other related variables, such as setup time, cutting insert change time, batch size, etc., were not included in the optimization problem, although these stochastic variables can have a certain impact on the resulting manufacturing cost [19].

The observed effects of the considered inputs on the manufacturing cost were also verified by applying the connection weight approach (CWA). This approach was selected as one of the best methodologies for quantifying variable importance in applications of ANN models in terms of accuracy and precision [47,48]. It provides the relative importance of ANN inputs in terms of magnitude and the direction of the relationship with respect to the output [49]. The relative importance of inputs, i.e., depth of cut, feed rate, cutting speed and volume of material to be removed, and their ranking are given in

Table 4. Relative importance of input process parameters.

Process Input Parameter	Relative Importance	Rank
Depth of cut, ap	−2.155	1
Feed rate, f	−1.056	2
Cutting speed, v	−0.304	4
Volume of material to be removed, V	0.700	3

.

As can be observed from Table 4, depth of cut is the most important predictor for the total manufacturing cost in medium turning of C45E steel, while the effect of cutting speed is the least pronounced. The negative signs indicate that with an increase in depth of cut, feed rate or cutting speed, the total manufacturing cost decreases. On the other hand, the positive sign in the relative importance of volume of material to be removed clearly shows that removing a higher volume of the material incurs higher costs. The results obtained are logical from a machining perspective and also support the previous discussion and reported results.

Furthermore, the order of cutting parameter importance with respect to the manufacturing cost corresponds to the recommended order of cutting parameter adoption [3] (depth of cut followed by feed rate, with cutting speed selected last). However, although cutting speed may be freely selectable across a wide range utilizable for optimization [3], the obtained results indicate that for additional gains in manufacturing cost minimization, one must consider cutting speed interaction effects for its adequate determination.

It has to be noted that CWA allowed the assessment of individual input importances. However, due to the existence of two-way interactions, and perhaps higher-order interactions, the determination of the optimized cutting regime for a given case study (with a defined volume of material to be removed and a specified machine tool) requires developing and solving a turning optimization problem with constraints.

5. Case Study

A case study is also presented for the purpose of determining optimal cutting parameter values with regard to minimizing the total manufacturing cost in multi-pass medium turning of steel C45E. The analyzed workpiece material is an unalloyed, medium carbon steel, widely used in the industry for the manufacture of shafts, axles, gears, bolts, studs and other mechanical components [50], due to its good strength, toughness, moderate wear resistance and excellent machinability [51], as well as its suitability for heat treatment. The chemical composition of C45E steel mainly consists of C (0.465%), Mn (0.695%), Si (0.118%), P (0.015%), and S (0.013%), while Fe constitutes the balance. The tensile strength of R_m = 680 N/mm² was experimentally determined under laboratory conditions. The machine tool is the CNC lathe OKUMA GENOS L250II-e, with the motor power of P_m = 7.5 kW, maximum spindle speed of n_max = 3000 rpm, maximum torque of M_max = 71 Nm, and the efficiency of η = 0.9. Dry multi-pass medium longitudinal turning is realized using the Sandvik Coromant PCLNR 2525M 12 tool holder (cutting edge angle of κ = 95°, rake angle of γ_oh = −6°) with the Pramet CNMG 120408E-SF cutting insert (nose radius of r_ε = 0.8 mm, γ_oi = 14.5°) of T8430 grade (coated carbide). The recommended cutting parameter ranges are a depth of cut of a_p = 0.8–3.0 mm, feed rate of f = 0.12–0.30 mm/rev, and cutting speed of v = 149–337 m/min [52]. The volume of material to be removed is V = 116.4 cm³, which corresponds to multi-pass medium longitudinal turning with a starting diameter of 70 mm, end diameter of 44 mm, and machining length of 50 mm (Figure 1).

Ensuring minimum cost by itself would not yield adequate results if the corresponding cutting regimes did not ensure stable cutting conditions and favourable chip form [3], nor consider the technical capabilities of the available machine tool. For these reasons, the following optimization problem with constraints is formulated:

$$\begin{gathered} Minimize Total manufacturing costs \\ Subject to: P_c={\displaystyle \frac{F_c\cdot v}{60\cdot {10}^3}}\le P_m\cdot \eta (\mathrm{c}\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{t}) \\ {\xi }_{min}\le {\displaystyle \frac{a_p}{f}}=\xi \le {\xi }_{max} (\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{p} \mathrm{s}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{t}) \\ {a}_p·i={\displaystyle \frac{D_0-D_1}{2}} (\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e} \mathrm{s}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{y} \mathrm{v}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{d} (\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}) \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{t}) \\ {a}_{pmin}\le a_p\le a_{pmax} (\mathrm{m}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}) \\ {f}_{min}\le f\le f_{max} (\mathrm{m}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}) \\ {v}_{min}\le v\le v_{max} (\mathrm{m}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}) \\ n\le n_{max} (\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{l}\mathrm{e} \mathrm{s}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d} \mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}) \end{gathered}$$

(4)

where D₀ (mm) is the workpiece initial diameter, D₁ (mm) is the workpiece diameter after multi-pass turning, i is the number of passes, ξ is the chip slenderness ratio, η is the CNC lathe mechanical efficiency, F_c (N) is the cutting force, P_c (kW) is the cutting power and P_m (kW) is the CNC lathe motor power.

For the given volume of material to be removed (Figure 1), the total manufacturing cost was estimated using the developed and validated ANN model as a function of the depth of cut (a_p), feed rate (f) and cutting speed (v).

A cutting power constraint was included, given that within the recommended interval ranges, certain cutting parameter combinations, particularly those corresponding to higher material removal rates, may result in cutting power that exceeds the maximum available power of the machine tool used. This not only prevents machine tool overload, but also ensures that the available machine and cutting tools are used to their full capacity. It should be noted that the cutting force model, which figures in the cutting power constraint, was developed by using the dimensional analysis-based methodology described in the literature [53]. The dimensional analysis-based cutting force model predicts the cutting force based on the tensile strength of workpiece material, and six independent parameters (cutting speed, feed velocity, depth of cut, feed rate, cutting edge angle and rake angle), arranged in three dimensionless groups. It is a multiplicative power model able to adapt to specific manufacturing conditions by applying correction coefficients so as to accommodate for different cutting tools (tool nose radii, tool geometry), tool holders (cutting edge angle, rake angle), workpiece conditions and cutting and lubrication conditions. In order to avoid unfavourable chip forms (saw-tooth, wrinkled, snarled), which are related to multiple issues, such as problems with chip evacuation and control, damage to machined surface, safety hazards, etc., the chip slenderness ratio constraint was considered. The minimum and maximum values of the chip slenderness constraint were determined considering favourable chip slenderness intervals for the used workpiece material [54], based on which the experimental hyper-space for the conducted empirical study was defined. Finally, the third constraint of equality type was formulated, based on stock and part geometry, and was considered to ensure required machining dimensions, as well as consistent cutting conditions (cutting tool load (force), wear, chip formation) in each pass.

The values of machining parameter bounds and constants in the proposed optimization problem are given in

Table 5. Multi-pass turning optimization parameter data.

Parameter	Value	Parameter	Value	Parameter	Value
apmin	0.8 mm	fmin	0.12 mm/rev	vmin	149 m/min
apmax	3 mm	fmax	0.3 mm/rev	vmax	337 m/min
ξmin	6	D0	70 mm	Pm	7.5 kW
ξmax	9	D1	44 mm	η	0.9
Cins	11.16 EUR	nce	4	Cslo	30 EUR/h

Machine tool: CNC lathe OKUMA GENOS L250II-e (OKUMA, Oguchi-cho, Niwa-gun, Japan). Cutting tool: Sandvik Coromant PCLNR 2525M 12 (Sandvik Coromant, Sandviken, Sweden), Pramet CNMG 120408E-SF T8430 (Dormer Pramet, Sumperk, Czech Republic). Workpiece material: steel C45E.

.

For solving the developed optimization model (Equation (4)), which represents a single-objective problem with three functional constraints, of which two are of the inequality type and one is of the equality type, a genetic algorithm (GA) was applied. The GA can be considered as a directional stochastic, population-based optimization algorithm that iteratively explores the solution space while gradually converging toward an optimal or near-optimal solution [55]. This algorithm was selected due to its robustness [56]. It was implemented with a population size of 20 individuals, stochastic uniform selection for parent selection, elitism-based reproduction with an elite count to preserve the best individuals, constraint-dependent mutation to maintain feasibility, scattered crossover with a crossover probability of 0.8 and a stopping criterion of 100 generations.

Given that meta-heuristic algorithms do not guarantee convergence to the global optimum, even after multiple optimization runs [20,57], the obtained solutions were validated using the Brutomizer software tool (version 1.1) [58]. This software tool incorporates the brute force algorithm, a deterministic approach to guarantee the optimality and feasibility (constraints satisfaction) of the determined optimization solutions.

As a result of the optimization process, the identified optimization solution is the cutting regime with a depth of cut of a_p = 2.6 mm (five passes), feed rate of f = 0.3 mm/rev, and cutting speed of v = 244 m/min, ensuring a minimum total manufacturing cost of 0.647 EUR. Under this cutting regime, the production rate and the used tool life are equal to 0.85 min and 7.66%, respectively. By checking the constraints, it can be shown that all three imposed constraints are satisfied (cutting force F_c = 1579 N, cutting power P_c = 7.13 kW, chip slenderness ratio of 8.67, and multi-pass turning in five passes with a_p = 2.6 mm equals 13 mm, corresponding to a 26 mm diameter reduction after machining). If one considers the recommended cutting regime for the cutting insert, i.e., a depth of cut of a_p = 1 mm, feed rate of f = 0.2 mm/rev, and cutting speed of v = 272 m/min (the corrected starting cutting speed was determined by multiplying the starting cutting speed with the correction factors and the coefficient of the used workpiece material) [59], even assuming that all imposed constraints are met, the resulting total manufacturing cost is 1.74 EUR.

It should be noted that the determined optimal cutting regime is valid for this specific problem setting. This cutting regime can be considered conditionally optimal if the machining would be done under different conditions, such as different lubrication conditions [60].

6. Conclusions

The present study focused on the analysis of main and interaction effects of depth of cut, feed rate, cutting speed and volume of material to be removed on the total manufacturing cost in medium turning of C45E steel. For the given case study, the optimized cutting regime was determined by developing and solving a single-objective turning optimization problem with three constraints. Based on the conducted analyses and optimization results obtained in the case study, the following conclusions can be drawn:

• Depth of cut, followed by feed rate and volume of material to be removed, has the maximum influence on the total manufacturing cost. The effect of cutting speed is least pronounced.
• An increase in depth of cut or feed rate decreases the total manufacturing cost for different combinations of inputs. The effect of cutting speed on the total manufacturing cost is variable and should be considered with the analysis of the interaction effects with other parameters.
• The derived conclusions may change depending on the chosen cutting insert, which has defined ranges of cutting parameter values.
• Due to the existence of interactions between cutting speed and other parameters, it is necessary to determine the optimal cutting regime for each specific operation. In other words, cutting regimes that minimize the total manufacturing cost in turning different volumes of C45E steel will differ. For practical implementation, it is also necessary to consider additional constraints, such as the one related to chip slenderness.
• The results showed that maximizing tool life increased the total manufacturing cost, because it required a decrease in the cutting parameter values.
• By optimizing cutting parameter values, the total manufacturing cost can be reduced by 62.82% compared to the cost for the recommended cutting regime for the cutting insert, even assuming that all imposed constraints are met. The results obtained are valid for cutting insert costs and specific labour and overhead costs for a given market.

The limitation of the conducted study was that different workpiece shapes, cutting insert types and cutting fluid types were not analyzed. In future research, the ANN model can be adapted for different setup times, cutting tool change times, cutting tool idle times and batch sizes.

Future work will also focus on the analysis and comparison of manufacturing costs of different workpiece materials that can be processed using the same cutting insert. Additionally, the analysis will consider cutting economics under machine tool constraints (such as power and spindle speed limitations) and will be applied to parts with more complex geometries containing multiple features.