1 Introduction

Up-to-date production is faced with numerous requirements such as to reduce costs, minimize time, reduce power, reduce energy consumption, maximize flexibility, increase productivity, increase quality of products and minimize negative effect on environment. All of these requirements need to be addressed in the optimal way. This is a long, often iterative process with an uncertain ending, in particular if stochastics and a great number of mutually conflicting and contradictory requirements are taken into consideration. On the other hand, a great number of experiments are necessary for a complete study of the process, which considerably increases the costs. Therefore, a balance between the number of experiments and the necessary pieces of information has to be made in order to minimize the required resources, costs and time. This is why scientific researches lay great stress on modelling, prediction and optimization of all requirements facing contemporary production. In this way, costs and time are reduced and productivity is increased. To solve the mentioned problems, various optimization techniques and methods are applied [1,2,3,4]. On the other hand, it is necessary to perform full discretization of the process in order to analyse the full region of feasible solutions and to obtain the region of optimal solutions for the given constraints. In this way, it is possible to obtain a greater number of optimal solutions of the input process parameters depending on the required output process parameters, i.e. requirements of production.

Numerous factors affect efficiency of output process parameters of machining. Their realization requires adequate technological equipment, processes, machining parameters, etc. Various machining operations are applied in mechanical engineering industry. One of the most frequently used machining operations is turning. It is mostly applied in machining cylindrical workpieces at all series levels. Using this operation, workpieces can be machined roughly, as semi-finished or finished to final measure. Workpieces from different materials can be machined by turning. Steels are a group of materials that are most often used in manufacturing engineering products. Their groups and number are various. Carbon steels belong to a special group of steels whose demand has lately been in constant increase. Significant is the application of AISI 1040 steel in different industrial sectors. This steel is used in couplings and crankshafts, as well as in production of different kinds of bolts, rods, springs, studs, etc.

Turning of workpieces made of AISI 1040 carbon steel has been researched from different perspectives and on various bases through numerous theoretical, simulation and experimental studies. Tuffy et al. [5] evaluated the influence of TiN coating thickness on the performance of turning inserts made of WC. Gunay et al. [6] evaluated the effect of rake angle on the cutting forces during turning. Yaldiz et al. [7] presented a fuzzy rule-based model for estimating the cutting forces depending on the cutting speed, feed and depth of cut. Saglam et al. [8] investigated the influence of machining parameters, rake angle and approach angle on cutting force component and temperature generated on the turning insert. Salgado and Alonso [9] presented a system for online tool wear monitoring applying the feed motor current and the sound signal. Asilturk and Cunkas [10] used multiple regression and artificial neural network (ANN) approach to estimate the surface roughness varying the cutting speeds, feeds and depths of cut. Neseli et al. [11] researched the influence of corner radius, approach angle and rake angle on the surface roughness by the use of response surface methodology (RSM). Topal and Cogun [12] proposed an ANN model for error estimation depending on cutting speed, depth of cut, feed, workpiece diameter and length. Cohen et al. [13] predicted cutting temperature by the use of the analytical model of thermal distribution on the workpiece surface by the use of ANN and multiple regression, Asiltur [14] optimized cutting speed, feed, depth of cut, and insert radius affecting the surface roughness using the Taguchi method and analysis of variance (ANOVA). Venkata Rao et al. [15] estimated the influence of turning parameters on workpiece vibration, surface roughness and volume of metal removed. Venkata Rao et al. [16] evaluated the influence of cutting speed, insert radius and feed on tool life analysing the surface roughness and workpiece vibration. Prasad et al. [17] demonstrated the 3D finite element analysis to predict the workpiece displacements in feed direction and corresponding tool wear. Venkata Rao et al. [18] applied ANN to estimate surface roughness and vibration of workpiece for different cutting speeds, feeds and nose radii. Yadav [19] analysed the influence of cutting speed, feed, primary and secondary depth of cut on average surface roughness. Haque et al. [20] optimized roughness characteristics under different cutting conditions using grey relational analysis. Akkus [21] researched the influence of cutting speed, feed and depth of cut on surface roughness by the use of Taguchi method. Jhodkar et al. [22, 23] described the comparative study of untreated and microwave treated carbide inserts at different spindle speeds, feeds and depths of cut.

Moreover, researches in the field of turning of steel have also considered and analysed the application of alternative cooling and lubricating techniques in machining aimed at minimizing the negative effect on environment. Conventional coolants and lubricants are replaced by other methods and techniques such as minimum quantity cooling lubrication (MQCL), minimum quantity lubrication (MQL), cryogenic cooling and lubrication and minimum quantity solid lubrication. Dhar et al. [24] researched the influence of cryogenic cooling on tool wear, dimensional deviation and surface quality. Dhar et al. [25] compared the performance of MQL to dry turning based on cutting temperature, chip reduction coefficient, cutting forces, tool wear, surface quality and dimensional deviation. Vamsi Krishna et al. [26] developed an empirical model to estimate the surface roughness and tool wear in turning performed by the use of solid lubricant. Ramana et al. [27] studied the influence of variation of nano level boric acid particle size on the cutting forces, tool temperatures and surface roughness. Vamsi Krishna et al. [28] studied the effect of cutting speed and feed on temperature, flank wear and surface roughness varying the type of lubricants. Amrita et al. [29] evaluated the temperature, tool wear and cutting forces at different lubrication conditions. Srikiran et al. [30] investigated the application of nano-level particulate graphite powder as a solid lubricant during turning. Gupta et al. [31] investigated the effect of cryogenic cooling on tool wear, surface roughness, cutting forces and cutting temperature at various feeds and constant cutting speed and depth of cut. Padmini et al. [32] evaluated the main cutting force, temperature, surface roughness, and flank wear to estimate the performance of vegetable oil based nanofluids. Ajay Vardhaman et al. [33] evaluated the influence of lubricants on tool wear, friction coefficient, surface quality and chip morphology under dry, wet, coconut oil and MQL cutting conditions. Mia and Dhar [34] optimized the cutting speed, feed rate and workpiece hardness to determine minimal surface roughness using composite desirability function and genetic algorithm. Usha and Rao [35] investigated the effect of MQL flow rate, cutting speed, feed and depth of cut on cutting force, surface roughness and temperature. Sahinoglu and Rafighi [36] examined the influence of machining parameters and workpieces hardness on surface roughness and power consumption by the use of RSM and ANOVA. Gugulothu and Pasam [37] examined machining performance in turning at constant cutting conditions by the use of hybrid nanofluids under MQL.

Most of the previously mentioned researches dealt with the effects of machining parameters (cutting speed, feed, depth of cut) on the AISI 1040 carbon steel turning output process parameters. Some of the researches took into consideration cutting tool geometry, energy consumption, chip quantity, effects of coolants and lubricants, etc. The most frequently analysed output process parameters of turning were surface roughness of workpiece, cutting tool life, machining time, etc. The previously mentioned researches were performed as wet or dry turning, and some of the researches were based on the minimization of the coolants and lubricants’ negative effect on environment by considering alternative techniques of cooling and lubricating. The alternative techniques of cooling and lubricating have a smaller negative effect on environment. However, negative effects are produced in these cases too, only to a lesser extent. Costs are also increased due to the additional equipment and the coolants and lubricants, the fact that can have a negative effect on expenses, particularly in the cases of small batch. In addition to experimental investigations, a small number of investigators reported on modelling and optimizing of machining performances by the application of statistical methods and artificial intelligence.

On the other hand, there are numerous studies in turning operations, but not for AISI 1040 carbon steel, in which single-objective and multi-objective optimization techniques were applied [38, 39]. The optimization of the turning process consists of two steps. In the first step, the turning process is modeled. The model describes the functional relationship between the input and output turning parameters. The methods that had the greatest application for the modeling of the turning process are: regression analysis (RA), artificial neural network (ANN), fuzzy logic (FL), support vector machine (SVM), etc. [40, 41]. When applying these methods, problems may arise in collecting data in an economical and efficient way, filtering outliers, extracting statistical characteristics of data, etc. If these problems are solved successfully, industrial application becomes possible. In the second step, after modeling the turning process, the processing parameters are optimized. Optimization determines the values of processing parameters that provide the optimal value of the appropriate optimality criterion. The optimality criterion is used to quantitatively describe the performance of the turning process. Various techniques such as ant colony optimization (ACO), simulated annealing (SA), particle swarm optimization (PSO), genetic algorithm (GA), artificial bee colony (ABC), etc., were used for optimization [42,43,44,45,46,47]. These techniques, due to the tolerance of imprecision, uncertainty and the possibility of approximation, have great application for solving highly nonlinear, multidimensional, and complex problems that occur in the turning process [48]. All modeling and optimization techniques, and as some of their combinations, have certain advantages but also disadvantages [49,50,51,52,53], and it is practically very difficult to define unambiguous and generally accepted criteria for the selection of modeling and optimization techniques, i.e. when and how to apply any of these techniques. Different techniques require setting the appropriate parameters, which vary depending on the selected method. In studies that determined the results obtained using different techniques, different results were obtained. No technique has shown absolute dominance over others. In fact, whether an adequate technique was chosen and whether good parameters of that same technique were chosen, can be proven only with additional-confirmation experiments and by comparing the obtained results with their prediction, i.e. with the calculation of the errors that occur. A review of the literature found that GA was very often used in turning process optimization, and that in most cases gave good results [54,55,56,57,58,59,60]. Furthermore, in all previous research, one optimal solution was always obtained, i.e. one optimal combination of input process parameters and one optimal output process parameter. This is not realistic, especially not from the point of view of industrial application. One optimal solution is possible only for one combination of input process parameters. Output parameters and thus input parameters, as well as the manufacturing process, are a dynamic category [61]. It is often necessary, due to the adaptation of the turning process to the current manufacturing conditions, to change the values of turning parameters in a certain range, so it is very useful to provide more sets of optimal turning parameters in different value ranges [62]. Furthermore, in previous studies, the assessment of the validity of the developed methodology has been performed typically through one confirmation experiment, since there was only one optimal solution to be validated. In doing so, small errors in the confirmation experiment may be accidental. Therefore, it is necessary to perform a larger number of confirmation experiments to assess the validity of the applied method.

It seems that the study of the turning process based on experimental data in combination with the methods of optimization is the most promising for future researches. By combining these two approaches, one or several output process parameters can be optimized. The turning process performances can be evaluated and improved in this way. Besides, in the previous researches the variant multi-objective optimization of turning of AISI 1040 steel workpieces was not performed considering simultaneous optimization of surface roughness, tool wear and productivity. Optimal combinations of machining parameters and the cutting tool geometric and technologic characteristics depend on numerous factors in view of the fact that the turning process is characterized by a large number of the process dynamics variables. For different conditions of machining and different requirements of production, optimal parameters are also different.

Different from previous, the present research provides a multi-objective optimization of input process parameters, but for various impacts of output process parameters. In contrast to previous studies in which the main objective of optimization was to obtain one optimal combination of input process parameters, the basic idea of this research was to obtain a range, i.e. more potential Pareto optimal combinations. The main motivation was to generate optimal combinations of input process parameters (cutting speeds, feeds, depths of cut and corner radii) for various requirements of output process parameters (arithmetical surface roughness, average width of flank wear and material removal rate). In fact, it means that different combinations of input process parameters were obtained for the different weights or importance of surface quality (arithmetical mean roughness), tool life (average width of flank wear) and productivity (material removal rate). In addition, the novelty of the present paper is that a higher number of turning inserts have been used with different corner radii during the experimental research of AISI 1040 carbon steel.

2 Materials and methods

The research methodology is presented in Fig. 1. Input process parameters are the workpiece material, machining parameters which are defined on the specific machine tool and turning inserts which are located and clamped into a corresponding toolholder.

Fig. 1
figure 1

Research methodology

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The research was conducted on workpieces made of carbon steel AISI 1040 whose chemical composition is: (98.6–99) % Fe, (0.60–0.90) % Mn, (0.37–0.44) % C, ≤ 0.05% S and ≤ 0.040% P. Furthermore, the mechanical, physical and thermal characteristics of carbon steel AISI 1040 are: density = 7.845 g/cm3, hardness = 201 HB, tensile strength = 620 MPa, yield strength = 415 MPa, modulus of elasticity = 200 GPa, Poisson ratio = 0.29, thermal conductivity (at 0 °C) = 51.9 W/mK and thermal expansion co-efficient = 11.3 µm/m °C.

Considering the environment protection principles, the present research was conducted in dry environment. In dry machining conditions, special attention must be given to cutting tool selection and machining parameters optimization. The values of machining parameters as well as the geometric and technologic characteristics of cutting tool have to be selected in the way that they do not speed up the processes of the cutting tool wear.

Machining was performed with four turning inserts (designation: CNMG120404, CNMG120408, CNMG120412, CNMG120416) set into the same toolholder (designation: DCLBR3232M12). The turning inserts were selected in accordance with recommendations of the manufacturer, for the operation of longitudinal turning and the workpiece material AISI 1040 steel. Selected inserts are suitable for dry turning in all types of production. This is due to their geometry, material, coating process and specially designed chipbreaker.

The turning inserts differ in the size of corner radius (4 levels; r = 0.4–0.8–1.2–1.6 mm), the other geometric and technological parameters being identical:

  • Grade: Duratomic grade TP2501-CVD coated,

  • Coating: Ti(C,N) + Al2O3,

  • Clearance angle major: 0°,

  • Insert side clearance angle: 5°,

  • Rake angle: − 6°,

  • Inclination angle: − 6°,

  • Lead angle: 95°,

  • Insert included angle: 80°,

  • Fixing hole diameter: 5.2 mm,

  • Inscribed circle diameter: 12.70 mm,

  • Theoretical cutting edge length: 12.90 mm.

Moreover, based on the manufacturer of the turning inserts, the following machining parameters were selected:

  • 4 levels for feed, f = 0.1–0.2–0.3–0.4 (mm/rev),

  • 3 levels for cutting speed, vc = 300–350–400 (mm/min),

  • 3 levels for depth of cut, ap = 2–2.5–3 (mm).

The machine tool used for machining was CNC lathe DMG Mori Seiki CTX. Locating and clamping of workpiece was performed by means of the chuck and rotating centre. Machine tool and fixtures were selected so that the machining could be performed reliably, with high accuracy and stability of the machining process, aimed at minimizing vibrations of the workpiece and compliance of the workpiece-fixtures system. The original workpiece diameter was Ø60 mm and the length was 450 mm. Turning was carried out in 5 passes. For each experiment, a new turning insert was used.

Experimental research was conducted in accordance with randomized full factorial design, which enabled the research of all combinations of levels of input variables. Considering the fact that 4 levels were adopted for corner radius, 4 levels for feed, 3 levels for cutting speed and 3 levels for depth of cut, total number of experiments was 4 × 4 × 3 × 3 = 144.

After the experimental researches, the turning insert flank wear and the machined surface roughness were measured. The turning insert flank wear VB measuring was carried out on MT8500 microscope. Surface roughness measurement (of parameter Ra) was conducted on measuring instrument Talysurf with probe tip radius rt = 2 μm. Measurement was performed with a cut-off length of 0.8 mm and sampling length of 4 mm. Roughness was measured in 3 directions moved by 120 degrees in relation to the workpiece axis. The surface roughness Ra values were calculated as average values from three different directions of measuring.

Material removal rate (chip quantity in time) was calculated by equation [63]:

$$ Q = \frac{V}{t} \left( {{\text{mm}}^{3}/{\text{s}}} \right) $$
(1)

where V is total chip volume, t is turning time.

Turning time for the ith pass at external longitudinal turning equals [63]:

$$ t_{i} = \frac{{60 \cdot l_{i} }}{{f_{i} \cdot n_{i} }}\,({\text{s}}) $$
(2)

where li (mm) is turning length for the ith pass, fi (mm/rev) is feed for the ith pass, and ni (rev/min) is number of revolutions for the ith pass.

Number of revolutions for the ith pass equals [63]:

$$ n_{i} = \frac{{1000 \cdot v_{ci} }}{{\pi \cdot d_{i} }}\,\left( {\text{rev/min}} \right) $$
(3)

where vci (m/min) is cutting speed for the ith pass, di (mm) is diameter after the ith pass.

Diameter after the ith pass equals [63]:

$$ d_{i} = d_{k} - 2 \cdot a_{pi} \,({\text{mm}}) $$
(4)

where dk (mm) is diameter before the ith pass, api (mm) is depth of cut for the ith pass.

Substitution of Eqs. (3) and (4) into Eq. (2) gives the turning time for the ith pass:

$$ t_{i} = \frac{{60 \cdot l_{i} }}{{f_{i} \cdot \frac{{1000 \cdot v_{ci} }}{{\pi \cdot d_{i} }}}} = \frac{{60 \cdot \pi \cdot l_{i} \cdot d_{i} }}{{f_{i} \cdot 1000 \cdot v_{ci} }} = \frac{{60 \cdot \pi \cdot l_{i} \cdot \left( {d_{k} - 2 \cdot a_{pi} } \right)}}{{f_{i} \cdot 1000 \cdot v_{ci} }}\,({\text{s}}) $$
(5)

If machining is performed with constant cutting speed (vc = vci = const), constant feed (f = fi = const), constant depth of cut (ap = api = const) and at constant length of cut (l = li = const), machining time for the ith pass is:

$$ t_{i} = \frac{{60 \cdot \pi \cdot l \cdot d_{i} }}{{f \cdot 1000 \cdot v_{c} }} = \frac{{60 \cdot \pi \cdot l \cdot \left( {d_{k} - 2 \cdot a_{p} } \right)}}{{f \cdot 1000 \cdot v_{c} }}\,({\text{s}}) $$
(6)

For the machining in n passes, total turning time is:

$$ t = \mathop \sum \limits_{i = 1}^{n} t_{i} \,({\text{s}}) $$
(7)

where n is total number of passes.

Chip volume for the ith pass is equal to the volume before and after the ith pass:

$$ V_{i} = \left( {\left( {\frac{{d_{k} }}{2}} \right)^{2} - \left( {\frac{{d_{i} }}{2}} \right)^{2} } \right) \cdot \pi \cdot l_{i} \,\left( {{\text{mm}}^{3} } \right) $$
(8)

Substitution of Eq. (4) to (8) gives:

$$ V_{i} = \left( {\left( {\frac{{d_{k} }}{2}} \right)^{2} - \left( {\frac{{d_{k} - 2 \cdot a_{pi} }}{2}} \right)^{2} } \right) \cdot \pi \cdot l_{i} \,\left( {{\text{mm}}^{3} } \right) $$
(9)

For turning in n passes, total chip volume equals:

$$ V = \mathop \sum \limits_{i = 1}^{n} V_{i} \,({\text{mm}}^{3} ) $$
(10)

If several passes machining is performed with constant depth of cut (ap = api = const) and at constant length of cut (l = li = const) total chip volume equals:

$$ V = \mathop \sum \limits_{i = 1}^{n} V_{i} = \left( {\left( {\frac{{d_{\max } }}{2}} \right)^{2} - \left( {\frac{{d_{\min } }}{2}} \right)^{2} } \right) \cdot \pi \cdot l\,({\text{mm}}^{3} ) $$
(11)

where dmax is original diameter, dmin is final diameter.

As final diameter of machining equals:

$$ d_{{{\text{min}}}} = d_{{{\text{max}}}} - 2 \cdot a_{p} \cdot i\,({\text{mm}}) $$
(12)

Substitution of Eq. (12) to (11) gives total chip volume after i passes, at constant depth of cut (ap = api = const) and constant length of cut (l = li = const), being equal to:

$$ V = \left( {\left( {\frac{{d_{\max } }}{2}} \right)^{2} - \left( {\frac{{d_{\max } - 2 \cdot a_{p} \cdot i}}{2}} \right)^{2} } \right) \cdot \pi \cdot l\,({\text{mm}}^{3} ) $$
(13)

In case the machining is carried out with constant cutting speed (vc = vci = const), constant feed (f = fi = const), constant depth of cut (ap = api = const) and at constant length of cut (l = li = const), chip quantity in time (material removal rate) equals:

$$ Q = \frac{V}{t} = \frac{{\left( {\left( {\frac{{d_{\max } }}{2}} \right)^{2} - \left( {\frac{{d_{\max } - 2 \cdot a_{p} \cdot i}}{2}} \right)^{2} } \right) \cdot \pi \cdot l}}{{\mathop \sum \nolimits_{i = 1}^{n} \frac{{60 \cdot \pi \cdot l \cdot d_{i} }}{{f \cdot 1000 \cdot v_{c} }}}}\,\left( {{\text{mm}}^{3} /{\text{s}}} \right) $$
(14)

Following the performed experimental investigations, statistical processing, i.e. regression modelling, was conducted for obtained results. The regression modelling was conducted for different levels of four factors, i.e. four input parameters—corner radius (r), cutting speed (vc), feed (f) and depth of cut (ap). After obtaining regression equations, transformation and reduction were performed in order to obtain models with a high coefficient of determination to predict the measured responses (output process parameters)—flank wear of turning insert (VB) and surface roughness (Ra). Analysis of variance (ANOVA) was performed, as well as diagnostics and adequacy checking of regression models.

Multi-objective optimization is suitable for solving complex optimization problems whose individual objective functions are conflicting, i.e. one is opposed to the other. In this paper, GA is applied to multi-objective optimization. The main idea is to determine the optimal values of input process parameters (cutting speeds, feeds, depths of cut and corner radii) in order to minimize the flank wear (VB) and the surface roughness (Ra) and at the same time to maximize the material removal rate (Q). These quantities are expressed using regression model, described in the next chapter.

To conduct the multi-objective optimization, some modification of standard GA must be made. In this research, a version of elitist genetic algorithm [64] is used, which favours individuals with better fitness value. Also, some boundary constraints to input parameters, described in detail in the next chapter, must be obeyed during the optimization.

The input process parameters (cutting speeds, feeds, depths of cut) are real numbers in different, given ranges, and corner radii have discrete values, as mentioned before, which represent boundary-type constraints (the solution must be within these given constraints). Total number of parameters, i.e. variables, is four. Therefore, GA individuals are defined as vectors of four values representing these parameters.

The algorithm starts by creating the initial population, consisting of known number of individuals, defined as population size. The individuals are created randomly, using uniform distribution, and considering the feasible space defined by boundary constraints. The fitness function is then evaluated for each individual. According to the fitness value, all individuals are ranked in the following way:

  • All undominated individuals [64] have the rank 1.

  • Rank 2 individuals are those dominated only by rank 1 individuals. In general, rank k is given to individuals dominated by individuals of rank k − 1 and lower.

  • Having in mind that during the optimization process some infeasible individuals can occur, their rank value is always worse than of any feasible individual. Among the infeasible part of population, the rank is sorted by the infeasibility measure (individuals with less feasibility discrepancy have lower rank).

After the ranking procedure is conducted, the next phase of the algorithm is selection. The individuals are selected using binary tournament selection function. Two pairs of individuals are chosen randomly, and the ranks of individuals in each pair are compared. The individual with lower rank is selected. If the individuals have the same rank, the one with higher distance measure is selected. Distance measure shows how far the individual is from the other individuals of the same rank [64]. In this way, the population diversity is kept during the optimization process. The “wining” individuals from both pairs are chosen as parents.

The next step is the crossover phase. As the variables are dominantly real numbers, heuristic crossover has been used, meaning that the new individual (child) is created on the line containing both parents, in the vicinity of the better ranked parent. The parameter named crossover ratio determines how far from the better parent will a child be positioned. This parameter is usually taken in the range [0.8, 1.6]. The value 1.2, for example, means that the child is positioned away from the better parent at the distance of 20% of total distance between parents, in the direction opposite to the worse parent.

After the crossover, the mutation operator will be applied. Mutation means that some fractions of individuals will be changed. The probability of mutation for each fraction of individual (i.e. the variable) is determined by parameter named mutation rate, usually taken in the range [0.005, 0.02]. The fractions chosen for mutation are then replaced by random number selected uniformly from the range for that variable.

Final step consists of creation of new generation of individuals. The new generation consists of two parts. First part represents elite individuals, i.e. a part of the Pareto set of the current generation, determined by the parameter named Pareto front population fraction. The value of this parameter is usually in the range [0.2, 0.5], meaning that 20–50% of Pareto set will be kept for the next generation, as elite individuals. The second part of new generation is created by crossover and mutation. The number of individuals created by crossover is determined by parameter named crossover fraction, which is usually in the range [0.7, 0.9], meaning that 70–90% of this part is created by crossover, while the rest of the individuals are the result of mutation.

The described procedure is repeated for a given number of iterations. After the final iteration is completed, the undominated individuals, creating the Pareto front for the considered multi-objective optimization problem, are taken as a solution.

After the multi-objective optimization, optimal combinations of output process parameters (VBopt, Raopt, Qopt) are obtained with the corresponding optimal combinations of input process parameters (fopt, vcopt, apopt, ropt). After generating optimal values of input and output process parameters confirmation experiments were conducted to validate the use of genetic algorithm for multi-objective optimization. Finally, the obtained results were analysed and discussed.

3 Results and discussion

3.1 Experimental research

Results of measuring and calculating output process parameters for different combinations of input process parameters are displayed in Table 1.

Table 1 Results of measuring and calculating
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Obtained results point to the significant dispersion of results. Appropriate selection and various combinations of cutting speed, feed, depth of cut and corner radius can result in different values of surface roughness, flank wear and material removal rate. In other words, appropriate selection i.e. appropriate combination of input process parameters, can significantly affect output process parameters. In the conducted experimental researches, depending on the input process parameters, flank wear appears in the wide range from 0.147 to 0.293 mm. This means that lower but also higher flank wear can be obtained by the appropriate choice of the input process parameters. The lowest flank wear (VB = VBmin = 0.147 mm) is achieved at machining with the lowest values of machining parameters (vc = vcmin = 300 m/min, f = fmin = 0.1 mm/rev, ap = apmin = 2 mm) and the highest corner radius (r = rmax = 1.6 mm), but then the chip quantity in time is also the lowest (Q = Qmin = 1041.67 mm3/s). The highest flank wear (VB = VBmax = 0.293 mm) is obtained at machining with the highest values of machining parameters (vc = vcmax = 400 m/min, f = fmax = 0.4 mm/rev, ap = apmax = 3 mm) and the lowest corner radius (r = rmin = 0.4 mm), but then the chip quantity in time is also the highest (Q = Qmax = 8571.43 mm3/s).

Results of experimental research reveal that smaller roughness of machined surface and smaller flank wear are obtained at higher corner radii of turning insert, smaller feeds, smaller cutting speeds and smaller depths of cut. Different from this, shorter turning time and higher chip quantity are obtained at higher cutting speeds, higher feeds and higher depths of cut. It actually means that better quality of machined surface and longer tool life are obtained if corner radius is at maximal level and machining parameters at minimal level. Different from this, the highest material removal rate is reached if machining parameters are at maximal level. It is evident that there are conflicting requirements if output process parameters are considered separately, i.e. independently ones from the others (Table 2).

Table 2 Optimal values of input process parameters if output process parameters are considered independently one from the other
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3.2 Regression modelling

Statistical processing of measured data for VB and Ra is carried out by Design Expert software. Significant quadratic, transformed and reduced regression models are obtained (non-significant terms with the probability of type I error, p value, greater than 0.05 are removed).

Final equations in terms of actual factors are shown in Tables 3 (for VB) and 4 (for Ra) for each level of turning insert type (corner radius r) as a categorical factor.

Table 3 Regression equations for flank wear VB for each corner radius r
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Table 4 Regression equations for surface roughness Ra for each corner radius r
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Tables 5 and 6 present the analysis of variance (ANOVA) for the regression model for VB and for Ra. These tables also present the coefficients of determination: R2, R2 adjusted and R2 for prediction, as well as the other ANOVA output (adequate precision, standard deviation, mean, coefficient of variation, predicted residual sum of squares).

Table 5 ANOVA table—flank wear VB
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Table 6 ANOVA table—surface roughness Ra
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From Tables 5 and 6, it is visible that the regression models for VB and Ra are statistically significant because the probabilities for the F variable, i.e. p values are smaller than 0.05 (probability of type I error, i.e. significance level). It means that the null hypothesis should be rejected and alternative one accepted (at least one of the regression variables significantly contributes to the model, i.e. has a regression coefficient different from zero).

Normal probability plots of the internally studentized residuals to check for normality are shown in Fig. 2a (for VB) and b (for Ra).

Fig. 2
figure 2

Normal probability plot. a Flank wear VB, b surface roughness Ra

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Graphical presentations of internally studentized residuals versus predicted values (obtained by regression models) to check for constant error are given in Fig. 3a (for VB) and b (for Ra). It is visible from Figs. 2a and 3a that out of 144 internally studentized residuals, there is deviation of only 3 data for VB (− 5.145, 4.686 and 4.682 for runs 40, 90 and 120, Table 1), while in Figs. 2b and 3b the question is about the deviation of only 2 data for Ra (− 3.453 and − 3.327 for runs 15 and 105, Table 1).

Fig. 3
figure 3

Internally studentized residuals versus predicted values. a Flank wear VB, b surface roughness Ra

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Externally studentized residuals to look for outliers, i.e. influential values (Fig. 4a, b), show that for runs 40, 90 and 120 (Table 1) for VB there are deviations out of limits (Fig. 4a), while it is not the case for Ra (Fig. 4b) for runs 15 and 105, Table 1. However, for runs 90 and 120 (Table 1) for VB, for measures DFFITS (Difference in fits) and Cook's distance there is no deviation in relation to other runs (they are 1.8204, 1.8183 and 0.23197, 0.23152, respectively). These measures do not deviate from other runs even for runs 15 and 105, Table 1 for Ra (they are − 1.6593, − 1.4289 and 0.15728, 0.11748, respectively). Out of 5 analysed runs (3 for VB and 2 for Ra), only for run 40, Table 1, for VB, there is measure DFFITS which has a value different from other runs (it is − 2.04), however with a good indicator of Cook's distance (it is 0.2796).

Fig. 4
figure 4

Externally studentized residuals versus run number. a Flank wear VB, b surface roughness Ra

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After the conducted diagnostics of regression models, it can be concluded that the models can be applied as inputs for multi-objective optimization as, in addition to the above-mentioned analysis connected with outliers and influential values (Figs. 2, 3, 4), the other indicators are also very good (high and equivalent values of Adj R-Squared and Pred R-Squared, small MSE, that is, St. Dev., small values of indicator PRESS and high values of indicator Adeq Precision), which can be seen from Tables 5 and 6. Furthermore, the correlation between measured and predicted regression values for VB and Ra is given in Fig. 5. Evaluation of obtained results was performed based on the following errors and coefficient [64]:

$$ MAE_{x} = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left| {x_{ipv} - x_{imv} } \right|\quad i = 1,2, \, \ldots , \, 144\quad x = VB,Ra $$
(15)
$$ RMSE_{x} = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {x_{ipv} - x_{imv} } \right)^{2} }}{n}} \quad i = 1,2, \, \ldots , \, 144\quad x = VB,Ra $$
(16)
$$ MRE_{x} = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left| {\frac{{x_{ipv} - x_{imv} }}{{x_{imv} }}} \right|\quad i = 1,2, \, \ldots , \, 144\quad x = VB,Ra $$
(17)
$$ R_{x} = \sqrt {\mathop \sum \limits_{i = 1}^{n} x_{imv}^{2} - \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {x_{ipv} - x_{imv} } \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{n} x_{imv}^{2} }}} \quad i = 1,2, \, \ldots , \, 144\quad x = VB,Ra $$
(18)

where MAE—mean absolute error, RMSE—root mean square error, MRE—mean relative error, R—coefficient of correlation, xipv—predicted value, ximv—measured value.

Fig. 5
figure 5

Predicted versus measured values. a Flank wear VB, b surface roughness Ra

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The statistical measures—mean absolute error (MAE), root mean square error (RMSE), mean relative error (MRE) and coefficient of correlation (R) given in Table 7 prove that there is a very strong correlation between measured and predicted values obtained by the regression models. Finally, the calculated p values for the correlation are much smaller than 0.05 which means that the correlation is statistically significant (i.e. alternative hypothesis at significance level of 0.05 should be accepted).

Table 7 Statistical measures for comparison of predicted and measured values
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The results of regression analysis show that smaller flank wear, i.e. higher turning insert life is obtained at smaller depths of cut, smaller feeds, smaller cutting speeds and higher corner radii of turning inserts (Fig. 6). In this process, the highest influence on flank wear is exerted by the corner radius of turning insert (correlation coefficient, r = − 0.655), somewhat smaller is the influence of cutting speed (r = 0.614), even smaller is the influence of feed (r = 0.400), and the smallest is the influence of depth of cut (r = 0.075).

Fig. 6
figure 6

Influence of machining parameters and corner radius of turning insert on flank wear. a Flank wear versus turning insert type, b flank wear versus cutting speed, c flank wear versus feed, d flank wear versus depth of cut

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In this research, only fine tracks of abrasive wear of the flank and rake surface were visible on microscope, which means that the cutting insert had good toughness and resistance to wear even when machining at higher cutting speeds. Increased feed results in increased flank wear. The smaller the feed, the less the wear, and as the feed increases, so does the wear. Increased feed results in increased amount of work of cutting insert. Increasing the feed affects the increase in cutting forces, and thus the increase in temperature. Increased feed results in increased chip thickness and accordingly increased cutting forces and temperature in the machining zone. In the present research the feed did not affect flank wear as much as corner radius and cutting speed, due to the ability of cutting insert to resist heat and fracture. Increased depth of cut can be of influence to the increase of dynamic loads, therefore to the increase of temperature. In the present research, the effect of depth of cut to the increase of flank wear is very small. The effect of depth of cut on flank wear is minimized by the hardness of CVD coating. Appropriate combination of feed and depth of cut can result in the mimimization of their common negative effect.

Flank wear appears as a result of abrasive wear. It is the result of friction between the cutting tool flank surface and the workpiece. The increased friction generates higher heat, i.e. temperature in the machining zone. In conditions of high temperatures, flank wear is intensified. The increase of cutting speed, feed and depth of cut results in the increase of friction between the turning insert and the workpiece. The increased friction generates higher temperature in the machining zone. The higher temperature, at the same value of the corner radius of turning insert, causes higher wear. At this, the cutting speed has a higher influence on temperature increase, feed has a smaller influence and depth of cut the smallest. By increasing corner radius of turning insert, the contact surface between the turning insert and the workpiece is increased while at smaller values of the turning insert radius the contact surface between the turning insert and the workpiece is reduced. For the same values of cutting speed, feed and depth of cut, if the contact surface is bigger (higher corner radius) the generated heat is disposed at the turning insert bigger surface so the turning insert wear is also smaller. Generally, at higher corner radii of turning inserts with smaller speed, feed and depth of cut, lower temperature is generated in the machining zone and at this, it is simultaneously disposed at the turning insert bigger surface. This directly causes reduction of the turning insert flank wear. At smaller corner radii of turning inserts, higher cutting speeds, feeds and depths of cut, higher temperature is generated in machining zone and is concentrated at the smaller contact surface of the turning insert. As a result, the wear of the turning insert is increased.

Duratomic CVD coating technique provides a balance between two opposite requirements—good hardness and good toughness of cutting insert. Besides, these cutting inserts are characterized by sharp cutting edges and zones enriched by cobalt and aluminium oxide. This helps to improve tribological effects at the point of contact of the workpiece and the cutting insert, i.e. there is less friction, which results in lower temperatures and contact pressures. In this way, mechanical properties of the cutting insert are improved as well as the thermal and chemical inertia in a wide range of possible input process parameters. Temperature reduction resulted in reduction of abrasive, adhesion and diffusion wear. Higher temperatures can reduce yield strength, therefore the workpiece hardness as well. High stresses and temperatures can cause flaking and chipping of material, thus causing formation of micro-welds on rake and flank surface of the cutting insert. Besides, specially designed geometry of chipbreaker is incorporated in the cutting insert which can be successfully used for the final as well as medium and rough machining. For selected levels of the input process parameters, this geometry provided reliable chip control and its breaking apart. During experimental researches, the chip short form was prevailing. This is the most desirable form of the chip considering the fact that it causes less damage.

The results of the regression analysis indicate that smaller surface roughness, i.e. better quality of machined surface, is achieved at smaller depths of cut, smaller feeds, smaller cutting speeds and higher radii of turning inserts (Fig. 7). Feed (correlation coefficient r = 0.676) and corner radius of turning insert (r = − 0.521) exert the highest effect on surface roughness while the effect of depth of cut and cutting speed is small (correlation coefficients are r = 0.058 and r = 0.014).

Fig. 7
figure 7

Influence of machining parameters and corner radius of turning insert on surface roughness. a Surface roughness versus feed, b surface roughness versus turning insert type, c surface roughness versus depth of cut, d surface roughness versus cutting speed

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There is interaction between feed and turning insert type (corner radius) (Fig. 8). At higher values of corner radius of turning insert (1.2 and 1.6 mm), by increasing feed, surface roughness increase is less expressed. At smaller corner radii of turning insert (0.8 mm) and particularly at the smallest corner radius of turning insert (0.4 mm) the effect of feed on surface roughness is more expressed, i.e. by higher feed, surface roughness increases fastest for the smallest corner radius of turning insert. At smaller feeds the effect of corner radius of turning insert on surface roughness is less expressed. At higher feeds the turning insert corner radius effect on surface roughness is more and more expressed and it increases as the feed increases. It can also be seen from Fig. 8 that higher corner radius of turning insert, at the same feed, generates smaller values of surface roughness i.e. better quality of machined surface. Higher values of feed at the same value of corner radius of turning insert generate higher values of surface roughness, i.e. worse quality of the machined surface. Increased surface roughness at small corner radii of turning inserts and higher feeds is the result of obtaining wider and higher peaks as well as of wider and higher valleys in the machined surface. Therefore, the highest surface roughness is achieved at the smallest corner radius of turning insert and the highest feed.

Fig. 8
figure 8

Feed-turning insert type interaction

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It was found that during the turning process, the surface roughness increases, in different amount, however, under different combinations of input parameters of machining. The surface roughness increase is similar to the trend of flank wear increase. It is more or less identical for all combinations of the input process parameters. These observations point to the strong interdependence of flank wear and surface roughness. It has to be pointed out that the technology of coating forming of used cutting inserts also provides for obtaining sharp cutting edges and smooth surface of a cutting insert which helps in obtaining smoother surface of the workpiece, and this again adds to creating more suitable tribological conditions i.e. smaller generation of friction and less generated heat in the zone of machining.

3.3 Multi-objective optimization

As introduced earlier, individual objective functions are minimization of flank wear (VB), minimization of surface roughness (Ra) and maximization of material removal rate (Q). Based on this, the following multi-objective function is defined:

$$ F = {\text{MIN}}(w_{VB} \cdot VB_{x} ) + {\text{MIN}}\left( {w_{Ra} \cdot Ra_{x} } \right) + {\text{MAX}}(w_{Q} \cdot Q_{x} ) $$
(19)

where \(w_{VB}\), \(w_{Ra}\), \(w_{Q}\) are weight coefficients, \(VB_{x}\), \(Ra_{x}\), \(Q_{x}\) are normalized—transformed values of output process parameters.

Normalized values of output process parameters are calculated using the following equations:

$$ VB_{x} = \frac{{VB - VB_{\min } }}{{VB_{\max } - VB_{\min } }},Ra_{x} = \frac{{Ra - Ra_{\min } }}{{Ra_{\max } - Ra_{\min } }},Q_{x} = \frac{{Q - Q_{\min } }}{{Q_{\max } - Q_{\min } }} $$
(20)

It is however necessary to define the constraints, which limit the multi-objective optimization process to the desired feasible region. These constraints are defined by technological equipment performance (machine tool and measuring instrumentations), experimental research conditions and the obtained experimental results (Table 8).

Table 8 Multi-objective optimization process constraints
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After defining the objective function and boundary constraints for input parameters, the characteristic GA parameters must be set. When performing some initial test, it has been noticed that the objective function converges after approximately 200 iterations and stays within a narrow range for the rest of the optimization process, with the population size set to 100. In the current research, the problem is solved offline, and the results are stored for the future use, so duration of the optimization process is not essential. Therefore, the maximum number of generations and the population size are set to 400 and 100, respectively, to ensure the quality of the optimization results. To the authors’ best knowledge, the values of other GA parameters, crossover ratio, mutation rate, crossover fraction and Pareto front population fraction have more crucial influence to the optimization quality. Setting of these parameters is a very complex task, since there is no exact recommendation for values of these parameters; it is strictly problem-dependent. Therefore, it is common to apply trial-and-error procedure to determine the appropriate parameter values. However, in this research, a more systematic approach has been applied. All parameter values have been varied in ranges recommended in the literature, listed in the previous chapter, and for each combination, the calculation of objective function value was performed. Based on the obtained results, the parameters providing the best value of the objective function have been chosen: crossover ratio is set to 1.2, mutation rate is set to 0.01, crossover fraction is set to 0.8 and Pareto front population fraction is set to 0.35. In order to determine the influence of GA parameter values to the optimization results and to confirm the parameter tuning procedure, a sensitivity analysis has been performed.

In order to conduct sensitivity analysis, the authors investigated how a change in GA parameters will affect a change in multi-objective function (expression 19). Four GA parameters, Pareto front fraction, crossover ratio, mutation rate and crossover fraction, were varied at three levels. The level adopted as optimal after the initial analysis was the reference, as shown in Table 9.

Table 9 The levels of GA parameters
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By statistical processing of all 81 (34) combinations of parameter levels, a significant quadratic reduced model was obtained with p value less than 0.05, which can be seen in Table 10 showing the analysis of variance. Some non-significant terms (A—Pareto front fraction, C—mutation rate and D—crossover fraction) are retained in the model because of the hierarchy. The terms which significantly contribute to the model are B—crossover ratio, interaction CD between mutation rate and crossover fraction as well as quadratic terms of all GA parameters.

Table 10 ANOVA table—multi-objective function
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Based on the response surface of the model, shown in Fig. 9, it was proved that the best value (minimum) of multi-objective function is achieved at the initially adopted level of GA parameters.

Fig. 9
figure 9

3D response surface of multi-objective function

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Thus, by the sensitivity analysis of GA parameters, it was proved that decreasing or increasing the value of the parameters relative to the optimal level would affect the increase of the multi-objective function value. It means that the initially adopted values (optimal ones) gave the best value (minimum) of multi-objective function.

Using the GA settings explained above, multi-objective optimization has been performed, resulting in optimal Pareto set consisting of totally 60 points. These points were used to generate the Pareto diagram shown in (Fig. 9) for the defined objective function (Eq. 19). The surface in the diagram (Fig. 9) represents an optimal surface, meaning that the points that are on the surface give one of the combinations of optimal solution, i.e. to every point of this surface corresponds one Pareto optimal solution of output process parameters—optimal surface roughness (Raopt), optimal flank wear (VBopt) and optimal material removal rate (Qopt). To every optimal combination of output process parameters corresponds an optimal combination of input process parameters—optimal feed (fopt), optimal cutting speed (vcopt), optimal depth of cut (apopt) and optimal corner radius (ropt). It has to be mentioned that the surface shown in the diagram (Fig. 9) is obtained by cubic interpolation among 60 points which represent individual solutions obtained by genetic algorithm in a feasible region of possible optimal solutions (Fig. 10). The values of the output and input process parameters in the mentioned points are presented in Table 15 (Appendix).

Fig. 10
figure 10

Optimal surface and points for different combinations of output process parameters

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Based on the results of multi-objective optimization, it can be concluded that the optimal surface roughness is in the range Raopt = 0.315–5.808 μm, optimal flank wear is in the range VBopt = 0.150–0.218 mm, and optimal material removal rate is in the range Qopt = 1088.384–8228.571 mm3/s. Also, optimal value of corner radius of turning insert is always at maximal level ropt = 1.6 mm. Optimal values of machining parameters cover the nearly complete feasible region i.e. ranges of input process parameters. Optimal value of feed is in the range fopt = 0.10–0.40 mm/rev. Optimal value of cutting speed is in the range vcopt = 307–384 mm/min. Optimal value of depth of cut is in the range apopt = 2.04–3 mm. In fact, it means that, for the given conditions of experimental research, turning has always to be performed with the turning insert of the highest corner radius, and that optimal combination of machining parameters can vary depending on production demands, i.e. on surface quality, tool life and productivity.

In order to substantially comprehend the influence of obtained results of multi-objective optimization, it is possible to divide the region of optimal solutions in Fig. 10 into 27 equal areas (Fig. 11, 2D displays, gray arrows). In this way, the individual, but also common influences of input process parameters on output process parameters can be easily observed. In accordance with this, a decomposition of each of the output process parameters is performed into three equal ranges (three levels)—low, mid and high. This is done by dividing the ranges of obtained optimal values (Raopt = 0.315–5.808 μm, VBopt = 0.150–0.218 mm, Qopt = 1088.384–8228.571 mm3/s) into three equal parts (Table 11).

Fig. 11
figure 11

Discretized areas—regions with optimal points in which there are combinations of output process parameters

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Table 11 Ranges of optimal output process parameters
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In an ideal case, for the three levels (three ranges in Table 11) and three variables (optimal levels of the output process parameters), all possible combinations, i.e. all together 3 × 3 × 3 = 27 combinations, can be generated. However, in this concrete case it is not possible to generate all the combinations in view of the fact that for some combinations (e.g. low Raopt, low VBopt, high Qopt; etc.) the areas do not exist in the region of feasible optimal values of output process parameters (Fig. 11). This is a consequence of the conflicting individual objective functions for Ra, VB, on the one hand and Q, on the other. In accordance with the foregoing, Table 12 displays possible combinations (for the regions for which solutions exist) of the optimal output process parameters along with the corresponding optimal combinations of the input process parameters.

Table 12 Possible combinations of optimal input and output process parameters
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The obtained results (Table 12) point to the following:

  • high quality of machined surface (low level Raopt) can be realized in correspondence with small or medium flank wear (low level VBopt and mid-level VBopt) and small or mid-level productivity (low level Qopt or mid-level Qopt).

  • mid-quality of machined surface (mid-level Raopt) can be realized only in correspondence with the simultaneous low flank wear and low productivity (low level VBopt and low level Qopt) or with the simultaneous mid-flank wear and mid-level productivities (mid-level VBopt or mid-level Qopt).

  • high level of cutting tool life (low level VBopt) can be realized only in correspondence with the simultaneous better quality of machined surface and low productivities (low level Raopt and low level Qopt), or with better quality of machined surface and mid-level productivity (low level Raopt and mid-level Qopt), or with mid-quality of machined surface and low level productivity (mid-level Raopt and low level Qopt).

  • mid-level of cutting tool life (mid-level VBopt) can be realized in correspondence with the simultaneous low quality of machined surface and low productivity (low level Raopt and low level Qopt), mid-quality of machined surface and mid-level of productivity (mid-level Raopt and mid-level Qopt), low quality of machined surface and higher productivity (high Raopt and high Qopt), as well as in combination of higher quality of machined surface and mid-level productivity (low level Raopt and mid-level Qopt) and low quality of machined surface (high level Raopt) and mid-level productivity (mid-level Qopt).

  • mid-level productivity (mid-level Qopt) can be realized in correspondence with the simultaneous higher quality of machined surface and/or low and mid-level flank wear (low level Raopt and/or low level VBopt), mid-quality of machined surface and mid-level flank wear (mid-level Raopt and mid-level VBopt), mid- or worse quality of machined surface and mid- or higher flank wear (mid- or high level Raopt and mid- and high level VBopt).

  • high productivity (high level Qopt) can be realized in correspondence with mid- or low quality of machined surface (mid-level Raopt or high level Raopt) and/or mid- or low level of cutting tool life (mid-level VBopt or high level VBopt), but not with the simultaneous mid-quality of machined surface and mid-level of cutting tool life (mid-level Raopt and mid-level VBopt).

For the purpose of validation of optimal points obtained by multi-objective optimization, confirmation experiments were conducted. The confirmation experiments were carried out in accordance with the plan presented in Table 13. The experiments were performed for 12 optimal points, i.e. 12 optimal combinations of input process parameters (fopt, vcopt, apopt, ropt). Following the conducted additional experiments, measurement of flank wear (VB) and arithmetical mean roughness (Ra) was carried out. Ordinal numbers of optimal points shown in Table 13 correspond with ordinal numbers of optimal points presented in Figs. 10 and 11. Optimal combinations of input process parameters were selected to include the complete optimal surface shown in Figs. 10 and 11. This was done so that 10 optimal points were selected from the limits of optimal surface and 2 points from the mid-part of optimal surface. In this way, objectivity in the selection of points was achieved owing to the fact that all of the points were not concentrated in a single part of the optimal surface; this might have happened in the case of random selection of the points for confirmation.

Table 13 Results of confirmation experiments
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Quantitative evaluation of obtained results was performed based on the following equations:

$$ PE_{{x_{i} }} = \frac{{\left| {x_{ipov} - x_{imv} } \right|}}{{x_{imv} }} \cdot 100\% \quad i = 1,2, \, \ldots , \, 12\quad x = VB,Ra $$
(21)
$$ MPE_{x} = \frac{100\% }{n}\mathop \sum \limits_{i = 1}^{n} \left| {\frac{{x_{ipov} - x_{imv} }}{{x_{imv} }}} \right|\quad i = 1,2, \, \ldots , \, 12\quad x = VB,Ra $$
(22)

where PE—percentage error, MPE—mean percentage error, xipv—predicted optimal value, ximv—measured value.

Errors were calculated only for the output process parameters that were measured i.e. for flank wear (VB) and arithmetical mean roughness (Ra). Error was not calculated for material removal rate (Q), this being the output process parameter that was being calculated. Very low values of PE and MPE are to be noticed. PE for flank wear is in the range (1.130–1.878) %, for arithmetical mean roughness it is in the range (1.025–1.286) %. MPE are 1.478% and 1.146%, for flank wear and arithmetical mean roughness, respectively.

The same as for the regression models, the correlation between confirmed measured and obtained optimal GA values for VB and Ra is given in Fig. 12. The statistical measures—mean absolute error MAE, root mean square error RMSE, mean relative error MRE and coefficient of correlation R given in Table 14 prove that there is a very strong correlation between confirmed measured and obtained optimal GA values.

Fig. 12
figure 12

Optimal GA versus measured values. a Flank wear VB, b surface roughness Ra

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Table 14 Statistical measures for comparison of predicted GA optimal and measured values
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Finally, the calculated p values for the correlation between measured and obtained GA optimal values are much smaller than 0.05 which means that the correlation is statistically significant.

The results of multi-objective optimization support the hypothesis that it is possible to define a great number of optimal combinations of input parameters. At this the combination of optimal values of feed, cutting speed and depth of cut is interdependent. It depends on the requirements of output process parameters, i.e. on the importance and interrelation of surface quality, tool life and productivity.

With the increase of requirements for higher quality of machined surface, the importance of the machined surface roughness also increases. In these cases, if there are no high requirements for productivity, it is necessary to adopt feeds ranging from minimal level to somewhat less than mid-level (fopt = 0.1–0.19 mm/rev), cutting speeds ranging from close to minimal level to close to mid-level (vcopt = 307–342 m/min) and depths of cut in a wide range from minimal to maximal level (apopt = 2.04–2.9 mm). As the significance of productivity increases, feed also increases (fopt = 0.23–0.25 mm/rev) as well as cutting speed (vcopt = 329–381 mm/rev) and depth of cut (apopt = 2.53–2.82 mm). Higher corner radii of turning inserts ensure better quality of machined surface. In doing so, the higher radius compensates for the negative effect of increasing feed on the quality of machined surface. Therefore, the higher corner radius of turning insert enables the machining even with higher feeds and obtaining the corresponding quality of machined surface.

With increased requirements for dimensional accuracy and precision, higher requirements for geometric specifications of products and higher requirements for reducing costs, the importance of the turning insert wear also increases. In these cases, the feed should be adopted ranging from the minimal to the mid-level (fopt = 0.1–0.24 mm/rev), depth of cut in a wide range from the minimal to the maximal level (apopt = 2.04–2.9 mm), and cutting speed ranging from close to minimal level to close to mid-level (vcopt = 307–342 m/min). As the significance of productivity increases, feed also increases (fopt = 0.23–0.25 mm/rev) as well as cutting speed (vcopt = 324–374 mm/rev) and depth of cut (apopt = 2.53–2.82 mm). The level of the cutting speed, feed and depth of cut rises gently with the increase of requirements for higher productivity and decreases gently with the increase of requirements for the quality of machined surface.

By increasing the batch size and/or reducing the times of production, the importance of productivity also increases, i.e. the importance of material removal rate (chip quantity in time). In such cases, feeds should be adopted ranging from close to mid-level to maximal level (fopt = 0.32–0.4 mm/rev), cutting speeds in the range from close to mid-level to close to maximal level (vcopt = 340–384 mm/rev) and depth of cut is either close to maximal level or is at the maximal level (apopt = 2.69–3 mm). As the importance of the machined surface quality and the cutting insert wear increases, lower values of feed (close to the mid-level, fopt = 0.32 mm/rev) and lower values of cutting speed (close to the mid-level vcopt = 334–340 mm/rev) should be adopted as well as higher values of the depth of cut (close to maximal level, apopt = 2.79–2.87 mm). This is the result of the fact that depth of cut has no essential impact on surface roughness and flank wear. In other words, the higher the importance of productivity for machining process, the higher depths of cut are to be used.

Based on the foregoing the following statements in connection with the common action of two of the three considered output process parameters can be presented. Better surface quality and higher tool life are achieved at smaller feeds and smaller cutting speeds. In case of smaller requirements for productivity, depth of cut is to be smaller. With increased requirements for productivity, it is desirable to increase depth of cut. The increased depth of cut has the least negative effect on surface roughness increase and flank wear. Better surface quality and higher productivity are achieved at smaller feed and higher depths of cut. In case of less requirements for tool wear, cutting speed is to be higher. With increasing requirements for smaller wear, it is desirable to keep cutting speed reducing. Increasing the depth of cut has the least negative effect on declining the surface quality and productivity. Longer tool life and higher productivity are achieved at smaller cutting speeds and higher depths of cut. In case of smaller requirements for surface quality, feed is to be higher. With increased requirements for better surface quality, it is desirable to reduce feed.

Lower levels of optimal combinations of values of surface roughness, flank wear and productivity (material removal rate) are achieved at maximal values of corner radius of turning insert (ropt = 1.6 mm), feeds in the range from minimal values to those close to mid-level (fopt = 0.1–0.19 mm/rev), cutting speeds in the range from minimal values to close to mid-values (vcopt = 307–342 m/min) and depths of cut in the wide range from minimal values to maximal values (apopt = 2.04–2.9 mm). Productivity increase is easiest to achieve on account of depth of cut.

Mid-levels of optimal combinations of values of surface roughness, flank wear and productivity (material removal rate) are achieved at maximal values of corner radius of turning insert (ropt = 1.6 mm), feeds that correspond to mid-values (fopt = 0.26–0.233 mm/rev), cutting speeds in the range from those close to minimal value to those close to mid-value (vcopt = 319–357 m/min) and depths of cut in the range from mid-values to maximal values (apopt = 2.52–2.96 mm). In this case, smaller values of cutting speed correspond to higher values of feed and vice versa.

Higher levels of optimal value combinations of surface roughness, flank wear and productivity (material removal rate) are achieved at maximal values of corner radius of turning insert (ropt = 1.6 mm), feeds in the range from the values close to mid-values to maximal values (fopt = 0.34–0.4 mm/rev), cutting speeds in the range from close to mid-values to close to maximal values (vcopt = 340–384 m/min) and depths of cut in the range from values close to mid-values to maximal values (apopt = 2.73–3 mm).

Roughness of machined surface is a parameter of quality, which depends on geometric and technologic requirements of production. It may be an essential matter but a less essential as well in particular if some other finishing machining operation is performed after the operation of turning. As a rule, the flank wear of cutting insert directly affects the dimensional and geometric accuracy of machining. Apart from this it also affects machining costs considering the fact that the cutting insert which wears quicker becomes unusable and has to be replaced. The chip quantity is an essential output characteristic in case of larger batches or when production times are very short. In these cases, it is necessary to remove chip as soon as possible. That is why it is not always necessary to insist on general combination of optimal output process parameters. The essential point is to achieve the necessary values, i.e. optimal combinations in the complete feasible region. The achieved results of multi-objective optimization show that without regard to technological requirements of production, it is always suitable to perform turning with higher corner radii of turning inserts. This is the consequence of the fact that the turning inserts with bigger corner radius wear slower and at the same time generate a better quality surface i.e. surface with smaller roughness. Moreover, by increasing the importance of machined surface quality and/or tool life, machining should always be performed with higher corner radii of turning inserts, smaller cutting speeds, smaller feeds and higher depths of cut. With increased depth of cut, productivity (material removal rate) is also increased without notably affecting the surface quality and tool life.

4 Conclusion

In the present research, the authors conducted multi-objective optimization of cutting speed, feed, depth of cut and corner radius of turning insert from the point of their effect on the surface roughness, flank wear and material removal rate. Instead of the usual approach of obtaining only one combination of optimal parameters, a large number of combinations of optimal input process parameters were obtained, depending on the importance of each output process parameter, i.e. requirements of production.

Based on experimental researches, the arithmetical mean roughness of the surface is in the range Ra = 0.313–27.851 μm. This means that by an appropriate choice of the input process parameters exceptionally high quality of a machined surface can be obtained, but worse quality as well. In other words, by an appropriate combination of the input process parameters the turning operation can be conducted as finish machining, medium machining and rough machining. Further on, a cutting insert flank wear covers the range of VB = 0.147–0.293 mm. The ratio maximum to minimum is approximately 2. This means that an appropriate combination of the input process parameters can result in a considerable influence on the intensity of cutting insert wear. Consequently, there is a possibility of considerable influence on the necessary dimensional accuracy and precision, geometric specifications and costs. To be noticed is also the possibility of considerable influence on productivity taking into consideration its wide range of Q = 1041.67–8571.43 mm3/s.

Optimal combinations of input process parameters (Raopt = 0.315–5.808 μm, VBopt = 0.150–0.218 mm, Qopt = 1088.384–8228.571 mm3/s) vary on a large scale. Optimal combinations of input process parameters depend on requirements defined in production plan, i.e. on importance of output process parameters. For different relations in importance of surface quality, tool life and productivity, different optimal combinations of input process parameters are obtained (Raopt = 0.315–5.808 μm, VBopt = 0.150–0.218 mm, Qopt = 1088.384–8228.571 mm3/s). The trend of the optimal input parameters’ influence on the optimal output process parameters is similar to that for experimental researches. It is of the essence to emphasize that there are very small differences in the ranges of experimental results and optimal results. This means that the possibilities will not be limited largely by optimization, and on the other hand, there is a possibility to have a considerable impact on obtaining better output parameters of the turning process.

The obtained results indicate that AISI 1040 steel workpiece turning with CVD coated turning inserts is convenient to perform with larger corner radii of turning inserts (r = 1.6 mm) regardless of production requirements. For the same machining parameters, turning inserts with larger corner radii always generate better quality of the machined surface and less flank wear, i.e. greater tool life. Depth of cut shows the least negative effect on flank wear and machined surface quality in terms of their getting worse. At the same time, higher depth of cut affects the increase of material removal rate. Therefore, optimal values of depth of cut increase with increase of the importance (weight) of productivity. It is appropriate to conduct machining with higher depths of cut for almost the complete feasible region of solutions. High depths of cut should not be applied only in case of small batches and in case the times of production are not short.

Earlier application of GA in turning operations indicated some disadvantages. Convergence of the GA is not always guaranteed. There exists no universal rule for selection of appropriate GA parameters. Also, GA may need substantial time to arrive at optimal solutions, accompanied by low speed of convergence and repeatability of results. Therefore, as a rule, a confirmation experiment is conducted to confirm or refute the validity of the model. The model is considered applicable in practice, if the MPE occurred is less than 5%, since these errors ensure that the turning operation is performed within the machining tolerances and the geometric product specifications. In this study, the results of confirmation experiments confirmed the validity of results obtained by optimization, proving that in spite of possible disadvantages, the solutions close to global optimum could be found by the use of genetic algorithm. The calculated PE and MPE, between the results obtained by multi-objective optimization and the results obtained by measuring after the confirmation experiments, are low. MPE of 1.478% for flank wear and 1.146% for arithmetical mean roughness are considered highly acceptable for practical application of multi-objective optimization based on genetic algorithm, in industry.

The applied methodology is limited to conditions, equipment and parameters used in performing this experimental research and to the workpiece material AISI 1040 steel. This presents its basic limitation. However, the methodology is quite generalized and universal so that with additional experimental researches it can also be applied to other machining operations, workpiece materials, conditions of machining, etc. In addition, all types of wear mechanisms and chip morphology will be analysed in detail in future research. Furthermore, future researches will be directed to considering higher number of input process parameters. This will contribute to the optimization of the process considering a higher number of criteria. In addition, in the future it is planned to develop, apply and compare the performance of different modeling and optimization techniques, as well as the development of hybrid approaches, in order to improve the turning process.