Modelling and Simulation in Engineering
Volume 2017 (2017), Article ID 1940635, 10 pages
https://doi.org/10.1155/2017/1940635

Research Article

Multipass Turning Operation Process Optimization Using Hybrid Genetic Simulated Annealing Algorithm

Abdelouahhab Jabri, Abdellah El Barkany, and Ahmed El Khalfi

Laboratory of Mechanical Engineering, Faculty of Sciences and Techniques, University of Sidi Mohamed Ben Abdellah, Fes, Morocco

Correspondence should be addressed to Abdelouahhab Jabri; abdelouahhab.jabri@gmail.com

Received 4 March 2017; Accepted 8 May 2017; Published 21 June 2017

Academic Editor: Enmin Feng

Copyright © 2017 Abdelouahhab Jabri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

For years, there has been increasing attention placed on the metal removal processes such as turning and milling operations; researchers from different areas focused on cutting conditions optimization. Cutting conditions optimization is a crucial step in Computer Aided Process Planning (CAPP); it aims to select optimal cutting parameters (such as cutting speed, feed rate, depth of cut, and number of passes) since these parameters affect production cost as well as production deadline. This paper deals with multipass turning operation optimization using a proposed Hybrid Genetic Simulated Annealing Algorithm (HSAGA). The SA-based local search is properly embedded into a GA search mechanism in order to move the GA away from being closed within local optima. The unit production cost is considered in this work as objective function to minimize under different practical and operational constraints. Taguchi method is then used to calibrate the parameters of proposed optimization approach. Finally, different results obtained by various optimization algorithms are compared to the obtained solution and the proposed hybrid evolutionary technique optimization has proved its effectiveness over other algorithms.

1. Introduction

Metal removal processes such as turning operations involve different variables ranging from input variables (cutting speed, feed rate, depth of cut, and number of passes) to output variables (production cost, production time, tool life, dimensional accuracy, surface roughness, cutting forces, cutting temperature, and power consumption, etc.). Selecting appropriate cutting conditions has a significant impact on machining cost, product quality, and manufacturing cycle-time. These parameters are feed rate (), speed of cut (), depth of cut (), and number of passes (). Turning operation can be illustrated in Figure 1.

Figure 1: Example of turning operation.

The first paper related to parameter selection problem of multipass turning presented by Shin and Joo [1] initiated a series of studies. Authors from different backgrounds attempted optimization of the same problem using various optimization approaches [2, 3].

Chen and Tsai [4] proposed a hybrid technique based on Simulated Annealing Algorithm and the Hooke-Jeeves pattern search (SA-PS) to minimize the unit production cost; Onwubolu and Kumalo [5] presented an optimization based on Genetic Algorithm approach and have shown that GA optimization technique gives better results than [4]. However, M.-C. Chen and K.-Y. Chen [6] demonstrated that the result obtained by [5] was invalid, due to incorrect handling of the equality constraint of cutting depth. Vijayakumar et al. [7] have attempted the same problem using the Ant Colony Optimization (ACO) Algorithm. Subsequently, Wang [8] proved that the optimal solution, as found by [7], was also not valid. Srinivas et al. [9] used Particle Swarm Optimization (PSO) to minimize the unit production cost similarly to Chen and Tsai [4].

In the last decade, hybrid optimization techniques continue to attract more researchers by their ability to solve complex combinatorial optimization problems and various combinations approaches have been proposed by several authors. Costa et al. [10] proposed a Hybrid Particle Swarm Optimization (HPSO) technique for solving multipass turning optimization problem as presented by [4]. Lee and Ponnambalam [11] used a hybrid algorithm by combining the GA and the Artificial Immune Algorithm and attempted the same model as [7]. Xie and Guo [12] combined Ant Colony Optimizations (ACO) with a Pass Enumerating method to minimize unit production cost as well as [4].

Yildiz implemented several optimization techniques for solving the multipass turning operations problem, such as hybrid approach based on the Differential Evolution algorithm [13], Hybrid Robust Differential Evolution (HRDE) algorithm [14], Hybrid Artificial Bee Colony algorithm [15], and Hybrid Robust Teaching-Learning-Based Optimization (HRTLBO) algorithm [16] and hybrid PSO combined with Receptor Editing property of immune system [17]. However, any information about the optimal values of the machining parameters was provided.

Aryanfar and Solimanpur [18] presented an optimization of multipass turning operation based on GA technique which aims to minimize the unit production cost. Jabri et al. [19] proposed GA approach to solve a multiobjective problem where machining cost and tool life cost are simultaneously optimized. Rao and Kalyankar [20] used the Teaching-Learning-Based Optimization algorithm to solve the same problem (multipass turning operation). They concluded that the TLBO can converge to the optimal solution within a lower number of iterations. Belloufi et al. [21] proposed a Firefly Algorithm (FA), but constraint limitation such as cutting force was incorrectly handled.

Mellal and Williams [22] utilized Cuckoo optimization algorithm seeking to minimize production cost and comparing the obtained results with previously published results. In this study, the algorithm obtained the known optimal parameters in one case but failed in another. Chauhan et al. [23] proposed a Totally Disturbed Particle Swarm Optimization (TDSPO) for this problem where the phenomenon of chaos is embedded at PSO in order to disturb the particles at later iterations to prevent stagnation. Compared with the basic PSO the proposed method has given best results. Gayatri and Baskar [24] developed a sequential hybrid approach which is a fusion of GA, SA, and PSO optimization techniques to solve the multipass turning problem. In their work, three different selection methods of GA are studied which are linear ranking, tournament selection, and roulette wheel selection; the former method is the best selection method of GA technique and the hybrid approach has proven effectiveness when compared to each one. Another work related to the multipass turning problem is recently published [25]; it is based on a novel optimization technique called Flower Pollination Algorithm (FPA). The global and local search operators made of this optimization technique a powerful approach and it has given satisfactory results.

As presented in the above literature review, turning process is a complex problem to solve since it involves numerous elements. In fact, various optimization approaches are proposed by authors from different areas hoping to improve quality results in terms of precision and computational time. Unfortunately, it appears that more efforts are needed to be made to assist the process planner with reliable tool while selecting best machining parameters. In this work, a Hybrid Genetic Simulated Annealing Algorithm is developed to minimize the unit production cost of multipass turning process; the performance of the proposed optimization approach is highlighted by comparing the obtained results with different optimal costs presented by various optimization techniques.

The rest of the paper is organized as follows. In the next section, we present the problem concerned which is an objective function to minimize subject to various operational and practical constraints. In Section 3 the proposed hybrid algorithm optimization for multipass turning operation is presented. Parameters calibration of the proposed algorithm is then studied and obtained results are presented and compared to previously published works in Section 4. Finally, concluding remarks and some possible future works are given.

2. Multipass Turning Mathematical Model

The mathematical model of multipass turning operation proposed by [1] has remained a topic of interest for researchers; it is adopted in the present paper and the unit production cost subject to constraints will be minimized. Constraints including parameter bounds and cutting condition restraints on the tool and the workpiece are taken into account in this study. A detailed presentation of the objective and the constraints is given below.

2.1. Objective Function

The unit production cost, UC, for multipass turning operations can be divided into four basic cost elements.

(i)Cutting cost by actual time:

(ii)Machine idle cost due to loading and unloading operations and idling tool motion:

(iii)Tool replacement cost:

(iv)Tool cost:

where

Finally, objective function which is the sum of these costs can be expressed as follows:

2.2. Constraints

Constraints are technical and operational limitations that should be taken into consideration during the roughing and finishing operations. These constraints involve parameter bounds, operating constraints, and cutting condition relations.

(i) Parameter Bounds

(ii) Operating Constraints. The cutting force and the power constraints for both roughing and finishing operations are expressed as follows:

The chip-tool interface constraint of (10) follows the formulation of Hati and Rao (1976) [26]:

The stable cutting region constraint of (11) and the surface finish constraint of (17) follow the formulation of Narang and Fischer (1993) [27]:

(iii) Cutting Parameter Relations

3. Proposed Solution Algorithm

The proposed HSAGA is based on the combination between the GA algorithm and SA-based local search. In this section we present a brief overview on SA and GA techniques. GA is a randomized global search technique that solves problems by imitating processes observed from natural evolution; it has been introduced by Holland in 1960s and further described by Goldberg [28]. So far, GA has been successfully adopted in many complex optimization problems and shows its merits over traditional optimization methods. GA starts with a population of candidate solutions that evolves to reach near optimal solution. Each candidate is usually coded as a binary string “chromosome” in order to undergo reproduction; the fitness of each individual is then evaluated after the operation of decoding chromosome.

On the other hand, The SA algorithm mimics the behaviour of a physical system that is heated and then cooled slowly, such as growing crystals or annealing metals. Presented for the first time by Kirkpatrick et al. [29], it can be used to solve many combinatorial process optimization problems. This technique starts with selection of an initial random process decision vector and moves to new neighbourhood decision vector that improves objective function value. SA technique may accept inferior decision vector based on certain probabilistic measure “” called the Boltzmann statistics (see (18)), which helps the algorithm to escape entrapment in local optima and eventually reach the global optima:

where represents the difference in the objective value between the generated solution and the current solution (see (19)). “” is the annealing initial temperature

GA optimization technique has strong global search ability, but it also has limitations such as a premature and slow convergence rate, as well as weak local search ability. On the other hand, SA algorithm has strong local search ability and no premature problems. In this paper, a Hybrid Simulated Annealing Genetic Algorithm (HSAGA) is proposed to overcome the limits of each of the two techniques, bringing into play their respective advantages, and improve the solving efficiency.

3.1. Initialization

In this step of Genetic Algorithm a population of individuals (for each variable) is generated using the random uniform distribution in the defined interval, for both roughing and finishing operations. Feed rate () is a real number generated between ; cutting speed () and depth of cut () are integer numbers generated in the ranges , , respectively. In order to take into account the relation between and , the depth of cut of roughing is derived, subject to the fact that they lie between the accepted limits of . It should be noted that the population size “Popsize” has a considerable impact on quality solution and computational duration; thus it needs to be fixed at the appropriate value. Each element of the solution space “” has a cost value obtained using the objective function: .

3.2. Evaluation and Constraints Handling

The purpose of this optimization is to compute the minimal cost of a turning operation. A score function value is associated with each element of solution space (competitiveness). The penalty function is used to penalize the individuals who violate the constraints. The more the constraints are violated, the heavier the penalty will be done. As a result, the fitness of them will be small. In this way, unfeasible solutions have more chance to be eliminated from the solution space. Using the objective function the probability of selection is calculated as follows:

where

is the value of objective function of the current element “”. is the worst value of the current solution space (the maximum value of objective function).

The probability gives to the element the chance to be reproduced based on its competitiveness.

3.3. Solution Encoding

Individuals which represent machining parameters have to undergo some genetic operations such as crossover and mutation. To do so, they should be encoded appropriately. A proper encoding scheme, which is indicative of the characteristics of a solution, has considerable influence on the performance of quality solution. In this step we adopt the same encoding scheme of [4] where individuals represented as string-bit block called (chromosomes) are converted to a binary string and allocated to a 22-bit block.

The operation of converting each individual to a binary string from a real number “” to a binary string is calculated by using the following equation:

is then transformed to a binary string. For example, a feed rate value of “0.729” has a string bock as follows:

With this representation genetic operators can be applied to individuals and they can be reconverted into integer or real number again, using the binary mapping technique.

3.4. Genetic Simulated Annealing Operators Perturbation

3.4.1. Two Cut-Points Crossover

Crossover is a mechanism for diversification that encourages GA to examine unvisited regions. Two-point crossover calls for two randomly generated points to be selected on the parent strings and all bits between the two points are swapped between the parent strings, rendering two child strings. For example, two strings of the first block, having cutting speed values of 78 and 108 m/min, respectively, after crossover operation, become 110 and 76 m/min, respectively:

3.4.2. Mutation and Neighbourhood Principle

Mutation is a random modification of randomly selected bits. This operator performs modification into chromosomes with certain probability “” and is considered as a perturbation mechanism to explore neighbours of possible solutions. For example, the previous new strings 1 (mutated positions are underlined) become

Applying the crossover and mutation operations, new different individuals from parent strings are generated. The metropolis acceptance criterion expressed by (19) is then applied to select which of individuals go into the next generation. In fact, the Old_pop and New_pop go to the next generation through competition; throughout this competition, two elements are extracted from the new solution: f_newbest and f_worst (the minimum and the maximum solutions of the offspring individuals, resp.). The algorithm accepts replacing the old solution in two cases:

(i)1st case: if _newbest is inferior to _best (the best solution of the parents population)(ii)2nd case: when _newbest is superior to _best and the probability expressed by (19) is superior to random value generated between

3.5. Cooling Schedule and Termination Condition

In this paper, the linear pattern of temperature reduction has been used, where the temperature is obtained from the following equation:

The termination condition is when the program makes iterations without improvement of the best solution. The HSAGA algorithm is shown in Algorithm 1.

Begin

Set , , , and

Generate feasible solution Pop⁰

For to pop_size

Calculate fitness values (Pop⁰)

end

while :

While :

Select two individuals from

If :

Apply crossover

If :

Apply mutation

end

end

new-pop

for to pop_size:

Calculate new fitness values (new-pop)

end

if :

Temp-pop = new-pop,: ]

else if :

Temp-pop = new-pop

if :

if :

Temp-pop = new-pop[,: ]

end

new-pop = Temp-pop

end

Return optimal value

end

Algorithm 1: HSAGA proposed algorithm.

4. Results and Discussion

In this section, minimization of unit production cost as reported in the literature is considered. The hybrid SAGA optimization technique is conducted in order to compute the optimal machining parameters of the metal cutting problem. The example considered is a cylindrical workpiece with 500 mm length and 50 mm diameter and the desired depth to cut is 6mm. Machining data is given in Table 1.

Table 1: Machining data.

Parameter
Value

50 mm

500 m/min

0.9 mm/rev

1 mm

3 mm

200 Kgf

1000°C

108

2.5

1.75

0.95

1.2 mm

2.5 $/min

0.75 min/piece

300 mm

500 m/min

50 m/min

0.1 mm/rev

3 mm

25 min

5 KW

0.5

132

1.0

0.75

0.85

0.4

0 .105

6 mm

50 m/min

0.1 mm/rev

0.9 mm/rev

1 mm

45 min

SC
140

10 µm

1.0

5

0.75

2

0.2

0.5 $/min

0.3

1.5 min/edge

The hybrid SAGA is written in python 3.5 and runs on Intel(R) Core(TM) i5-2430M CPU @2.4 GHz, with 4 GB of RAM. A set of parameters are involved in the proposed algorithm; Table 2 shows this optimization algorithm parameters. As shown in the second column, a range is given for each parameter. Thus, a tuning parameter is needed to select the best values that ensure the program convergence and stability.

Table 2: HSAGA parameters.

Parameters
Range/values

Population size (Popsize)
20–100

Maximum iteration number
50–200

Crossover probability ()
70%–80%

Mutation probability ()
10%–30%

Initial temperature
500–900

Cooling rate (α)
0.70–0.90

Penalty ($)
1–10

4.1. Parameters Tuning

In this stage, Taguchi method is used to select suitable combination of the above-mentioned factors levels. Widely used in robust design, Taguchi technique is a universal approach [30]. The robust parameter design is based on the minimization of the sensitivity of noise by determination of factors levels. In such optimization, the response at each setting of parameters is treated as a measure to indicate the mean and the variance of some quality characteristics. These two indicators are combined into a single performance measure so called the signal-to-noise () ration. Taguchi proposed different categories of robust parameter design, smaller better, larger better, and so on [31]. The category smaller better is the suitable function for our problem, and it is expressed by

where , , and denote the trial number, response variable, and the number of replications, respectively. In the proposed algorithm, seven factors should be calibrated, each with three levels.

The selected L27 orthogonal array has 27 different combinations of parameter levels, which means that a total of 27 tests are conducted, and each test is replicated five times in order to obtain more reliable results. Table 3 shows the obtained results of each combination of factors proposed by Taguchi method.

Table 3: L27 orthogonal array.

Popsize
Temp

Cool-Rate

Max-It
Pen.

20
500
0,70
0,7
0,10
50
1
2,34
2,39
2,79
3,06
3,05

20
500
0,70
0,7
0,15
100
5
3,15
2,39
2,50
2,15
2,70

20
500
0,70
0,7
0,20
200
10
2,61
2,41
2,63
2,70
2,16

20
750
0,75
0,8
0,10
50
1
2,46
2,76
2,26
2,51
2,45

20
750
0,75
0,8
0,15
100
5
4,74
2,34
3,16
2,87
2,08

20
750
0,75
0,8
0,20
200
10
3,11
2,21
2,18
2,50
2,48

20
900
0,80
0,9
0,10
50
1
2,68
2,55
3,11
2,08
2,40

20
900
0,80
0,9
0,15
100
5
3,59
2,07
2,37
2,56
2,49

20
900
0,80
0,9
0,20
200
10
3,14
3,11
2,55
2,87
2,32

50
500
0,75
0,9
0,10
100
10
2,18
2,21
2,07
2,07
2,44

50
500
0,75
0,9
0,15
200
1
2,07
2,08
2,07
2,08
2,08

50
500
0,75
0,9
0,20
50
5
2,44
2,21
2,07
2,48
2,09

50
750
0,80
0,7
0,10
100
10
2,18
2,08
2,21
2,07
2,19

50
750
0,80
0,7
0,15
200
1
2,18
2,08
2,08
2,18
2,07

50
750
0,80
0,7
0,20
50
5
2,21
2,08
2,39
2,08
2,07

50
900
0,70
0,8
0,10
100
10
2,07
2,07
2,07
2,19
2,08

50
900
0,70
0,8
0,15
200
1
2,21
2,07
2,07
2,19
2,07

50
900
0,70
0,8
0,20
50
5
2,19
2,07
2,17
2,08
2,07

100
500
0,80
0,8
0,10
200
5
2,20
2,08
2,07
2,07
2,21

100
500
0,80
0,8
0,15
50
10
2,07
2,08
2,07
2,07
2,12

100
500
0,80
0,8
0,20
100
1
2,07
2,07
2,07
2,07
2,12

100
750
0,70
0,9
0,10
200
5
2,08
2,07
2,07
2,08
2,08

100
750
0,70
0,9
0,15
50
10
2,08
2,12
2,10
2,07
2,08

100
750
0,70
0,9
0,20
100
1
2,06
2,07
2,08
2,07
2,17

100
900
0,75
0,7
0,10
200
5
2,08
2,07
2,07
2,06
2,21

100
900
0,75
0,7
0,15
50
10
2,07
2,07
2,21
2,39
2,16

100
900
0,75
0,7
0,20
100
1
2,07
2,08
2,09
2,07
2,14

To illustrate the performance of Taguchi method, Figure 2 presents evolution of mean S/N ratios of each factor. The levels with the highest values of S/N ratio are selected as the optimal value for each of them; these parameters values are gathered in Table 4.

Table 4: HSAGA optimal parameters values.

Algorithm parameter
Optimal value

Population size (Popsize)
100

Maximum iteration number
200

Crossover probability ()
70%

Mutation probability ()
10%

Initial temperature ()
500

Cooling rate ()
0.70

Penalty (Penal)
1

Figure 2: The mean plot for each level of the HSAGA factors.

4.2. Comparisons with Other Results

Using the above-mentioned algorithm factors, optimal cutting parameters are obtained and related minimal cost is 2.06$. Table 5 summarizes the optimal machining parameters, constraints values, and limitation; meanwhile it can be seen clearly that all constraints are respected. Moreover, convergence curves of the unit cost using HSAGA are given in Figure 3. It can be observed from this figure that obtained costs solutions are the same for each test, which indicates that the proposed algorithm with selected parameter values based on Taguchi method has good stability.

Table 5: Obtained optimal unit cost using HSAGA approach.

Variables
Range/limit
HSAGA result

Process parameters
(m/min)
50–500
111

(mm/rev)
0.1–0.9
0.565

(mm)
1.0–3.0
3

(m/min)
50–500
171

(mm/rev)
0.1–0.9
0.225

(mm)
1.0–3.0
3

Constraints for rough cut
(min)
25–45
44.26

(Kgf)
200
199.16

(Kw)
5
4.22

SCr
140
2278

(°C)
1000
866

Constraints for finish cut

25–45
24.87

(Kgf)
200
98.5

(Kw)
5
3.25

SCs
140
2164

(°C)
1000
857

(µm)
10
5.3

Constraints on variable relations

1.0
1.54

2.5
2.51

1.0
1

Minimum cost ($)

2.06

Figure 3: Trend of unit cost by HSAGA.

As presented in the Introduction, there are abundant results in the literature which make it important to compare our results with the others. The hybrid approach, used for the work reported here, is compared with obtained results of some research papers cited previously in the literature. Table 6 shows that our optimization technique gives better solution compared to other optimization approaches.

Table 6: Comparison of obtained results of various optimization techniques.

References
Opti. method
Cutting parameters
Unit cost ($)
Viol. const.

Present
HSAGA
111
0.565
3
171
0.225
3
2.06
—

[25]
FPA
109.66
0.5655
3
169.97
0.2262
3
2.03
—

[24]
HGSS
499.99
0.8939
2.5
93.89
0.8836
1
2.22
Force

[23]
Chaotic PSO
123.34
0.5655
3
169.97
0.2262
3
1.95
Tool life

[22]
COA
123.34
0.5655
3
169.978
0.2262
3
1.959
Tool life

[21]
Firefly
98.4
0.82
3
162.28
0.25
3
1.93
Force

[20]
TLBO
110
0.565
3
170
0.226
3
2.28
Tool life

[18]
GA
109.663
0.566
3
169.986
0.226
3
1.12

[17]
Hybrid PSO
—
—
—
—
—
—
2.04

[16]
HDRE
AIA
—
—
—
—
—
—
—
—
—
—
—
—
2.04
2.12

[15]
HABC
—
—
—
—
—
—
2.04

[14]
HRTLBO
—
—
—
—
—
—
2.04

[13]
DERE
ABC
DE
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
2.04
2.11
2.13

[12]
ACO
109.66
0.565
3
169.97
0.226
3
2.07
—

[11]
GA-AIS
PSO
133.28
141.7
0.8
0.71
2.9
2.8
333.23
330.74
0.8
0.71
2.5
2.3
1.5
1.6
Force
Force

[10]
HPSO
109.66
0.5655
3
169.97
0.2262
3
2.03
Tool life

[9]
PSO
106.69
0.897
2
155.89
0.28
2
2.27
—

[8]
Analytic method
103.05
0.9
3
162.02
0.24
3
1.96
Force

[7]
ACO
103.05
0.9
—
162.02
0.24
—
1.62
Force

[6]
GA
—
—
—
—
—
—
2.3

[5]
GA
—
—
—
—
—
—
1.76

[4]
SA-PS
—
—
—
—
—
—
2.29

5. Conclusion

This work deals with metal removal process and particularly multipass turning operation. A new proposed optimization approach is applied to find minimum unit production cost of this turning operation. This approach is based on combination of SA local search technique with GA global optimization providing a strong possibility to GA of avoiding local minima and having a marked improvement of its performance. Compared to various optimization techniques, obtained results show that HSAGA is an efficient optimization technique and it can be interesting to be implemented in CAPP.

Nomenclature

: | Unit production cost, excluding material cost ($/piece)
: | Cutting cost by actual time in cutting ($/piece)
: | Machine idle cost due to loading and unloading operations and tool idle motion time ($/piece)
: | Tool replacement cost ($/piece)
: | Tool cost ($/piece)
and : | Cutting speeds in rough and finish machining, respectively (m/min)
and : | Lower and upper bounds of cutting speed in rough machining, respectively (m/min)
and : | Lower and upper bounds of cutting speed in finish machining, respectively (m/min)
and : | Depths of cut for each pass of rough and finish machining, respectively (mm)
and : | Lower and upper bounds of depth of roughing cut, respectively (mm)
and : | Lower and upper bounds of depth of finishing cut, respectively (mm)
and : | Feed rates in rough and finish machining, respectively (mm/rev)
and : | Lower and upper bounds of feed rate in rough machining, respectively (mm/rev)
and : | Lower and upper bounds of feed rate in finish machining, respectively (mm/rev)
: | Number of rough cuts, an integer
: | Depth of metal to be removed (mm)
, : | Diameter and length of workpiece, respectively (mm)
: | Direct labor cost, including overheads ($/min)
: | Cutting edge cost ($/edge)
and : | Time required to exchange a tool and tool replacement time, respectively (min)
and : | Constants pertaining to tool travel and approach/departing time, respectively (min)
and : | Expected tool life for rough and finish machining, respectively (min)
: | Tool life of weighted combination of and (min)
: | Weight for ,
, : | Upper and lower bounds for tool life, respectively (min)
, , , and : | , , , and , respectively
SC: | Limit of stable cutting region
: | Nose radius of the cutting tool (mm)
and : | Cutting forces during rough and finish machining, respectively (kgf)
: | Maximum allowable cutting force (kgf)
, , and : | Constants of the cutting force equation
and : | Cutting powers during rough and finish machining, respectively (kW)
: | Maximum allowable cutting power (kW)
: | Power efficiency
, : | Constants related to expression of the stable cutting region
and : | Constraint chip-tool interface temperatures during rough and finish machining, respectively (°C)
: | Maximum allowable chip-tool interface temperature (°C)
, , , and : | Constants related to the equation of chip-tool interface temperature
, , and : | Constants for roughing and finishing parameter relations
: | Maximum allowable surface roughness (mm).

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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