Multi-pass turning process parameter optimization using teaching–learning-based optimization algorithm

Abstract

The primary objective in multi-pass turning operations is to produce products with low cost and high quality, with a lower number of cuts. Parameter optimization plays an important role in achieving this goal. Process parameter optimization in a multi-pass turning operation usually involves the optimal selection of cutting speed, feed rate, depth of cut and number of passes. In this work, the parameter optimization of a multi-pass turning operation is carried out using a recently developed advanced optimization algorithm, named, the teaching–learning-based optimization algorithm. Two different examples are considered that have been attempted previously by various researchers using different optimization techniques, such as simulated annealing, the genetic algorithm, the ant colony algorithm, and particle swarm optimization, etc. The first example is a multi-objective problem and the second example is a single objective multi-constrained problem with 20 constraints. The teaching–learning-based optimization algorithm has proved its effectiveness over other algorithms.

Keywords

Multi-pass turning process ; Parameter optimization ; Teaching–learning-based optimization algorithm

1. Introduction

Turning is an important and widely used manufacturing process in engineering industries. The study of metal removal focuses on the features of tools, input work materials, and machine parameter settings. The technology of metal removal using turning operations has grown substantially over the past decades and several branches of engineering have contributed to this to achieve the various objectives of the process. Selection of optimal machining conditions is a key factor in achieving these objectives.

There are large numbers of variables involved in the turning process. These can be categorised as input variables and output variables. Various input variables involved in the turning process are: cutting speed, feed rate, depth of cut, number of passes, work material and its properties, tool material and tool geometry, cutting fluid properties and characteristics, etc. Similarly, the output variables associated with the turning process are: production cost, production time, tool life, dimensional accuracy, surface roughness, cutting forces, cutting temperature, and power consumption, etc. For optimization purposes, each output variable is taken as a function of a set of input variables. To achieve several conflicting objectives of the process, optimum setting of the input variables is very essential, and should not be decided randomly on a trial basis or by using the skill of the operator. Use of appropriate optimization techniques is needed to obtain the optimum parameter settings for the process.

In the past, researchers have used some optimization techniques such as fuzzy logic, neural networks, simulated annealing, genetic algorithms, ant colony optimization, and particle swarm optimization, etc. to optimize both single and multi-pass turning operation problems. Chua et al.  [1] used a sequential quadratic programming technique for optimising the cutting conditions for multi-pass turning operations. Gopalakrishnan and Faiz  [2] used the analytical approach of geometric programming to handle the constrained problem of the turning process. Shin and Joo  [3] proposed a mathematical model for the multi-pass turning process, which was subsequently used by many researchers.

A feed-forward neural network was used by Wang  [4] for solving the multi-objective problem, which involved productivity, operation cost and cutting quality. Gupta et al.  [5] worked on the optimality of depth of cut of the multi-pass turning operation using an integer programming model. Chen and Tsai  [6] applied the simulated annealing approach to solve the optimization problem for minimum unit production costs of the multi-pass turning process.

Kee  [7] outlined the optimisation strategies for multi-pass rough turning on conventional and CNC lathes with practical constraints, such as force and power. Prasad et al.  [8] combined geometric and linear programming techniques for determining the machining parameters by involving the tolerance and workpiece rigidity constraints. The Taguchi method was used by Yang and Tarng  [9] , whereas Nian et al.  [10] carried out the optimization of turning operations based on the Taguchi method and considered various multiple performance characteristics, such as tool life, cutting force, and surface finish. Alberti and Perrone  [11] used the genetic algorithm to solve a fuzzy probabilistic optimization model for determining the cutting parameters. Arezoo et al.  [12] developed an expert system to select cutting tools and conditions of turning operations using Prolog. The system can select the tool holder, and the insert and cutting conditions, such as cutting speed, feed rate and depth of cut. Dynamic programming was used to optimize the cutting conditions. Dereli et al.  [13] developed an optimization system for cutting parameters of prismatic parts based on genetic algorithms.

Onwubolu and Kumalo  [14] used the mathematical model of Chen and Tsai  [6] and applied the genetic algorithm to minimize the unit production cost. They showed that the genetic algorithm had given better results than simulated annealing. However, subsequently Chen and Chen  [15] proved that the result of Onwubolu and Kumalo  [14] was invalid, due to incorrect handling of the mathematical model. Al-Ahmari  [16] presented a nonlinear programming model for the optimization of machining parameters and subdivisions of the depth of cut in multi-pass turning operations. Wang et al.  [17] used the genetic algorithm to select optimal cutting parameters and cutting tools in multi-pass turning operations with more focus on the tool wear and chip breakability aspects of the process.

Vijayakumar et al.  [18] used the ant colony optimization algorithm and attempted the same mathematical model as Chen and Tsai  [6] and Onwubolu and Kumalo  [14] . However, Wang  [19] demonstrated that the optimal solution, as found by Vijayakumar et al.  [18] , was not valid. Wassila  [20] established a methodology for the prediction of cycle time during high speed turning operations. Wang and Jawahir  [21] proposed a new GA-based methodology, whose research was focused on the selection of different cutting tools for different passes of turning operations and allocation of the depth of cut.

Sardinas et al.  [22] used the micro-genetic algorithm for attempting the multi-objective optimization model and obtained the Pareto front result. Abburi and Dixit  [23] developed an optimization methodology, which was a combination of a real-coded genetic algorithm and sequential quadratic programming, to obtain Pareto optimal solutions for minimizing the production cost. Yildiz  [24] attempted the same mathematical model as Vijayakumar et al.  [18] using the hybrid Taguchi-harmony search algorithm. Ojha et al.  [25] used a neural network fuzzy set and genetic algorithm-based soft computing methodology to optimize process parameters in multi-pass turning operations. Srinivas et al.  [26] used particle swarm intelligence for selecting the optimum machining parameters in multi-pass turning operations. Kim et al.  [27] explored the applicability of real coded genetic algorithm in machining optimization and compared their results with simulated annealing, genetic algorithms and the generalized reduced gradient method. Tzeng et al.  [28] obtained the optimal parameter combination of the turning operation using grey relational analysis.

Yildiz  [29] presented a hybrid optimization approach, based on the particle swarm optimization algorithm and the receptor editing property of the immune system, and applied it to several different examples, including the same mathematical model previously used by Vijayakumar et al.  [18] . Zheng and Ponnambalam  [30] used the hybrid algorithm by combining the genetic algorithm and the artificial immune system, and attempted the same model as that of Vijayakumar et al.  [18] . Ilhan and Mehmet  [31] used artificial neural networks and a multiple regression approach to model the surface roughness during turning at different cutting parameters. Suleyman et al.  [32] showed the influence of tool geometry on the surface finish by developing a prediction model using response surface methodology.

It is observed from the literature that various conventional methods, like quadratic programming  [1] , geometric programming  [2]  and [8] , dynamic programming  [12] , and linear and nonlinear programming  [8]  and [16] etc., were used by some researchers. However, the results obtained by these conventional methods were not optimum and this was proved in subsequent research work. Application of some advanced optimization techniques was also observed in the literature, such as simulated annealing  [6] , genetic algorithms  [13] , [14] , [17] , [22] , [23]  and [25] , the ant colony algorithm  [18] , and particle swarm optimization  [26]  and [29] , etc., but subsequently, it was proved that many of those techniques were not handled properly and their results were not accurate. Moreover, all the evolutionary and swarm intelligence-based algorithms are probabilistic algorithms and require algorithm-specific control parameters in addition to the common controlling parameters like population size and number of generations. For example, the genetic algorithm uses mutation and crossover rates. Similarly, particle swarm optimization uses inertia weight, and social and cognitive parameters. Simulated annealing requires the fine setting of initial temperature, which affects its effectiveness in giving an optimum solution. Ant colony optimization also involves the setting of random walk, mutation, trail diffusion and the evaporation rate of pheromones. The proper tuning of the specific parameters of the algorithm is a very crucial factor, which affects the performance of the above mentioned algorithms. The improper tuning of algorithm-specific parameters either increases the computational efforts or yields the local optimum solution. Considering this fact, recently, Rao et al.  [33]  and [34] introduced a Teaching–Learning-Based Optimization (TLBO) algorithm which does not require any algorithm-specific parameters. TLBO requires only common controlling parameters, like population size and number of generations, for it to work. In this way, TLBO can be said to be an algorithm-specific, parameter-less algorithm.

Hence, an attempt is made here to use the TLBO algorithm to provide more accurate and global optimum solution in less time compared to other optimization techniques. Efforts are carried out to use the algorithm for the process parameter optimization of the multi-pass turning process.

2. Teaching–learning-based optimization algorithm

The Teaching–Learning-Based Optimization algorithm (TLBO) is a teaching–learning process inspired algorithm recently proposed by Rao et al.  [33]  and [34] , based on the effects of the influence of a teacher on the output of learners in a class. The algorithm mimics the teaching–learning ability of teacher and learners in a classroom. Teachers and learners are the two vital components of the algorithm, which describe two basic modes of learning, through the teacher (known as the teacher phase) and interaction with other learners (known as the learner phase). A high quality teacher is usually considered a highly educated person who trains learners so that they can have better results in terms of their marks or grades. Moreover, learners also learn from interaction among themselves, which also helps in improving their results.

In this algorithm, a group of learners is considered the population and different subjects offered to the learners are considered to be different design parameters; a learner’s result is analogous to the ‘fitness’ value of the optimization problem. The best solution in the entire population is considered to be the teacher. The design parameters are actually the parameters involved in the objective function of the given optimization problem, and the best solution is the best value of the objective function. The teacher phase is the first part of the algorithm, where learners learn through the teacher. During this phase, a teacher tries to increase the mean result of the class room to his or her level. The learner phase is the second part of the algorithm, where learners increase their knowledge by interaction among themselves. A learner interacts randomly with other learners for enhancing his or her knowledge. A learner learns new things if the other learner has more knowledge than him or her. The working of the TLBO algorithm is described in detail by Rao et al.  [33]  and [34] . The same explanation of the teacher and learner phases, along with mathematical and implementation steps, are referred to here for the working of the TLBO algorithm. The flowchart of the TLBO algorithm  [35] is shown in Figure 1 .

Figure 1. Flowchart of TLBO algorithm.

The concept of elitism and duplicate solution removal was also implemented in the TLBO algorithm by Rao and Patel [36] in order to see their effect on the performance of the algorithm. Elitism is a mechanism by which to preserve the best individuals from generation to generation. Thus, the system never loses the best individuals found during the optimization process. Elitism can be created by placing one or more of the best individuals directly into the population for the next generation. The algorithm had already been experimented with using different elite sizes by Rao and Patel  [36] , and it was observed that better results were obtained when considering elitism than when no elitism was considered, in some cases. Hence, provision of the elitism concept may enhance the performance of the TLBO algorithm. However, duplicate solutions may exist in the algorithm as the worst solutions get replaced by elite solutions, and it becomes necessary to modify the duplicate solutions in order to avoid trapping in the local optima. These duplicate solutions can be modified or removed by mutation on randomly selected dimensions of the duplicate solutions before executing the next generation.

The TLBO algorithm has been already tested on a large number of constrained and unconstrained benchmark functions and proved to be better than other advanced optimization techniques, like PSO, DE and ABC, etc.  [36] . It also proved better in other fields of engineering, having been used successfully by various researchers, such as those reported by Niknam et al.  [37] , [38] , [39]  and [40] , Krishnanand et al.  [41] and Satapathy et al.  [42] in the field of electrical engineering, Togan  [43] in civil engineering, Satapathy and Naik  [44] in computer engineering, Rao and Kalyankar  [45] , [46]  and [47] in manufacturing engineering, and Rao and Patel  [48]  and [49] in thermal engineering, etc. Crepinsek et al.  [50] raised some doubts about the algorithm-specific parameter-less concept of the TLBO algorithm and some other issues. However, Rao and Patel  [36] had already cleared all those issues and justified that the TLBO algorithm is an algorithm-specific parameter-less algorithm, which requires only common control parameters, such as population size, number of generations and elite size. In the next section, the TLBO algorithm is used for the parameter optimization of a multi-pass turning process.

3. Application examples

Two different examples are considered here for the parameter optimization of a multi-pass turning process, and the results obtained by the TLBO algorithm are discussed and compared with results obtained by previous researchers.

3.1. Example 1: multi-objective optimization of multi-pass turning

In this example, the work attempted by Sardinas et al.  [22] is considered, in which the genetic algorithm was used for multi-objective optimization of cutting parameters in a multi-pass turning process. Two conflicting objectives were simultaneously optimised by Sardinas et al.  [22] in which “production time” was to be minimised, whereas “used tool life” was to be maximised. The cutting parameters involved in the model were cutting speed ‘${\textstyle \upsilon }$ ’, feed ‘${\textstyle f}$ ’ and depth of cut ‘${\textstyle a}$ ’.

The first objective considered by Sardinas et al.  [22] was related to minimization of “production time” and the same is given by Eq. (1) :

${\mbox{Production time}},\tau =\tau _{s}+(V/M)(1+(\tau _{TC}/T))+\tau _{0}{\mbox{.}}$

( 1)

In this model, same notations are used as given by Sardinas et al.  [22] in which ${\textstyle \tau _{s}}$ is tool set-up time, ${\textstyle \tau _{TC}}$ is tool change time, ${\textstyle \tau _{0}}$ is tool idle time, ${\textstyle V}$ is the volume of the removed metal, ${\textstyle T}$ is the tool life and ${\textstyle M}$ is the material removal rate.

The tool life ‘${\textstyle T}$ ’ is given by Eq. (2) and the material removal rate is given by Eq. (3) :

$T=5.48\times 10^{9}/(\upsilon ^{3.46}\times f^{0.696}\times a^{0.460}){\mbox{,}}$

( 2)

The second objective considered by Sardinas et al.  [22] was associated with maximization of “used tool life”, considered as the part of the whole tool life that was consumed in the process. Eq. (4) represents the model of used tool life in %, as given by Sardinas et al.  [22] :

${\mbox{Used tool life}},\xi =(V/MT)\times 100\%{\mbox{.}}$

( 4)

The multi-objective optimization problem considered by Sardinas et al.  [22] involves following two important constraints related to a machine, i.e. cutting force and cutting power constraints. These constraints are given below in Eqs. (5) –(6) :

${\mbox{Cutting force}},F_{C}=(6.56\times 10^{3}\times f^{0.917}\times a^{1.10})/(\upsilon ^{0.286}){\mbox{,}}$

( 5)

The maximum allowable cutting force in the process was 5000 N and the cutting power should not exceed 7.5 kW. One more constraint considered by Sardinas et al.  [22] was related to surface roughness, which should be less than the specified value of surface roughness. Eq. (7) shows the mathematical model of surface roughness:

${\mbox{Surface roughness}},R=(125f^{2})/r_{E}\leq R_{MAX}{\mbox{,}}$

( 7)

where ${\textstyle r_{E}}$ is the tool nose radius. Even though the surface roughness constraint was mentioned by Sardinas et al.  [22] , it is observed that the same was not used for computing purposes, and was also not reported. The value of the tool nose radius was also not reported by Sardinas et al.  [22] . Hence, in the present work, also, the surface roughness constraint is not described.

Following limits were allowed on the cutting parameters for the application examples shown by Sardinas et al.  [22] , and the same are used in the present work.

${\mbox{Cutting speed}},\upsilon _{MIN}=250m/min\quad {\mbox{and}}\quad \upsilon _{MAX}=400m/min{\mbox{.}}$

A genetic algorithm was used by Sardinas et al.  [22] to obtain the optimum cutting parameters, in order to meet these two conflicting objectives. A result in the form of a Pareto front was given by Sardinas et al.  [22] and a minimum production time of 0.85 min and a maximum used tool life of 9.22% were reported as the best result. However, it is observed that the constraint value of the cutting power, reported by Sardinas et al.  [22] as 7.5 kW, is not correct, and is exceeded in most cases given in the Pareto front result presented by Sardinas et al.  [22] . The results obtained by Sardinas et al.  [22] are given in Table 1 , along with the corrected values of the power constraint. Table 1 clearly shows that the power constraint is violated in most cases by Sardinas et al.  [22] . Also, the results in the form of a Pareto front reported by Sardinas et al.  [22] are not correct. The Pareto front result given by Sardinas et al.  [22] shows that the production time and used tool life are both worsened from one point to another point. Hence, they should not be treated as Pareto front points and should only be referred to as deteriorated results from one point to another point. Thus, out of 14 results reported by Sardinas et al.  [22] , only six results satisfy the power constraint. Hence, in the present work, the better result from among those six results, i.e., minimum production time of 0.91 min and maximum used tool life of 3.90%, is considered for the result comparison.

Table 1. Result given by Sardinas et al.  [22] and the corrected values of power constraint.

No.	${\textstyle \tau }$ (min)	${\textstyle \xi }$ (%)	${\textstyle a}$  (mm)	${\textstyle f}$  (mm/rev)	${\textstyle v}$  (m/min)	${\textstyle M(mm^{3}/min)}$	${\textstyle T}$  (min)	${\textstyle F}$  (N)	${\textstyle P}$  (kW)	Corrected ${\textstyle P}$  (kW)
1	0.85	9.22	1.6	0.55	400	346,520	6.9	1125	7.5a	7.63
2	0.87	6.75	1.7	0.55	358	333,372	9.8	1257	7.5a	7.54
3	0.91	3.90	1.9	0.55	294	311,495	18.1	1528	7.5	7.40
4	0.94	2.77	2.1	0.55	261	298,575	26.6	1727	7.5a	7.59
5	0.95	2.46	2.1	0.55	250	294,281	30.3	1800	7.5	7.36
6	0.96	2.45	2.2	0.53	250	292,619	30.6	1800	7.5a	7.51
7	0.98	2.39	2.5	0.45	250	284,831	32.3	1800	7.5	7.42
8	1.00	2.33	2.9	0.38	250	277,270	34.0	1800	7.5	7.48
9	1.02	2.28	3.3	0.32	250	270,135	35.7	1800	7.5	7.37
10	1.03	2.24	3.7	0.29	250	265,117	37.0	1800	7.5a	7.63
11	1.06	2.18	4.3	0.24	250	256,782	39.4	1800	7.5a	7.57
12	1.08	2.13	4.9	0.20	250	250,600	41.2	1800	7.5	7.39
13	1.10	2.08	5.5	0.18	250	244,608	43.2	1800	7.5a	7.62
14	1.12	2.05	6.0	0.16	250	240,880	44.5	1800	7.5a	7.53

a. Power constraint violated by Sardinas et al.  [22] .

In the present work, the same example as considered by Sardinas et al.  [22] is attempted using the TLBO algorithm. The population size of 50 is used for the TLBO algorithm, whereas Sardinas et al.  [22] had reported the population size of 500 using the genetic algorithm. Due to the conflicting nature of both objectives, along with the involvement of constraints, various trials were performed using different values of population size and number of iterations to run the TLBO algorithm. Finally, consistent results are obtained with a population size of 50, with 50 iterations. The result obtained by the TLBO algorithm and its comparison with the correct result of Sardinas et al.  [22] using the genetic algorithm is given in Table 2 . The convergence of results for production time and used tool life is shown in Figure 2  and Figure 3 , respectively.

Table 2. Comparison of results of example 1.

Parameter	GA	TLBO
Objectives:
${\textstyle \tau }$  (min)	0.91	0 .85
${\textstyle \xi }$  (%)	3.90	9 .50
Process variables:
${\textstyle a}$  (mm)	1.9	1.57
${\textstyle f}$  (mm/rev)	0.55	0.55
${\textstyle \upsilon }$  (m/min)	294	400
Constraints:
${\textstyle F_{C}}$  (N)	1125	1122
${\textstyle P}$  (kW)	7.40	7.48

Figure 2. Convergence of production time of example 1.

Figure 3. Convergence of used tool life of example 1.

The results obtained by the TLBO algorithm are compared with the correct result shown by Sardinas et al.  [22] , and it is observed that the TLBO algorithm has made a significant improvement in the result. The production time is reduced from 0.91 min to 0.85 min, whereas the used tool life is increased from 3.90% to 9.50%. This result is also better, compared to the best result claimed by Sardinas et al.  [22] , i.e., production time of 0.85 min and used tool life of 9.22%. Thus, the newly developed TLBO algorithm has proved its effectiveness for parameter optimization of the multi-pass turning process, and has effectively handled the multi-objective problem in less time.

3.2. Example 2: multi-constrained single objective multi-pass turning

The example considered in this section for the process parameter optimization of a multi-pass turning process shows the ability of the TLBO algorithm to effectively handle a large number of constraints. This example was attempted by various researchers using different optimization techniques. Shin and Joo  [3] attempted this problem using a dynamic programming approach, Chen and Tsai  [6] proposed an approach that combined the simulated annealing algorithm and a pattern search technique, Onwubolu and Kumalo  [14] used a genetic algorithm, whereas Vijayakumar et al.  [18] used ant colony optimization methodology. Subsequently, Chen and Chen  [15] proved that the work of Onwubolu and Kumalo  [14] is impractical, whereas the work carried out by Vijayakumar et al.  [18] proved to be invalid by Wang  [19] . Later, Yildiz  [24]  and [29] and Zheng and Ponnambalam  [30] also attempted the same model using different optimization techniques.

In this work, the values of all 20 constraints are analyzed critically, which was not done by previous researchers. In some previous works, the optimised parameters values were also not mentioned and, because of this, it is difficult to justify the optimum result and the constraints values. For result comparison purposes, same mathematical models, as used by previous researchers  [3] , [6] , [14] , [15] , [18] , [19] , [24] , [29]  and [30] , are considered in the present work.

The problem is associated with minimisation of the unit production cost ‘${\textstyle UC}$ ’ for the multi-pass turning operation. This production cost was divided into four basic cost elements.

• Cutting cost by actual time in cut (${\textstyle C_{M}}$ );
• Machine idle cost due to loading and unloading operations and idling tool motion (${\textstyle C_{I}}$ );
• Tool replacement cost (${\textstyle C_{R}}$ ) and;
• Tool cost (${\textstyle C_{T}}$ ). The unit production cost was defined as the sum of the cutting cost, ${\textstyle C_{M}}$ , the machine idling cost, ${\textstyle C_{I}}$ , the tool replacement cost, ${\textstyle C_{R}}$ , and the tool cost, ${\textstyle C_{T}}$ :

$UC=C_{M}+C_{I}+C_{R}+C_{T}{\mbox{.}}$

( 8)

The cutting cost, ${\textstyle C_{M}}$ , was expressed as:

${\begin{array}{l}\displaystyle C_{M}=k_{0}[(\Pi DL/(1000V_{r}f_{r}))(d_{t}-d_{s}/d_{r})+\\\displaystyle +(\Pi DL/(1000V_{s}f_{s}))]{\mbox{.}}\end{array}}$

( 9)

The machine idle cost, ${\textstyle C_{I}}$ , is expressed as:

$C_{I}=k_{0}[t_{c}+(h_{1}L+h_{2})((d_{t}-d_{s}/d_{r})+1)]{\mbox{.}}$

( 10)

The tool replacement cost, ${\textstyle C_{R}}$ , is expressed as:

${\begin{array}{l}\displaystyle C_{R}=k_{0}(t_{e}/T_{P})[(\Pi DL/(1000V_{r}f_{r}))(d_{t}-d_{s}/d_{r})+\\\displaystyle +(\Pi DL/(1000V_{s}f_{s}))]{\mbox{.}}\end{array}}$

( 11)

The tool cost, ${\textstyle C_{T}}$ , is expressed as:

${\begin{array}{l}\displaystyle C_{T}=(k_{t}/T_{P})[(\Pi DL/(1000V_{r}f_{r}))(d_{t}-d_{s}/d_{r})+\\\displaystyle +(\Pi DL/(1000V_{s}f_{s}))]{\mbox{.}}\end{array}}$

( 12)

The process model under consideration consists of 5 constraints for a rough cutting operation and 6 constraints for the finish cut operation. Six parameter bounds are due to the three process variables, and three more constraints are imposed for the relationship of the process variables of rough and finish cuts. All these constraints are given below in Eqs. , , , , , , , , , , , , , , , , , ,  and  .

Constraint for the rough cutting operation:

$1.{\mbox{Tool-life constraint}}:T_{L}\leq T_{r}\leq T_{U}{\mbox{.}}$

( 13)

Constraint for the finish cutting operation:

$1.{\mbox{Tool-life constraint}}:T_{L}\leq T_{s}\leq T_{U}{\mbox{.}}$

( 18)

Limits for the process variables during rough cut:

$1.{\mbox{Cutting speed}}:V_{rL}\leq V_{r}\leq V_{rU}{\mbox{.}}$

( 24)

Limits for the process variables during finish cut:

$1.{\mbox{Cutting speed}}:V_{sL}\leq V_{s}\leq V_{sU}{\mbox{.}}$

( 27)

Constraints for process variables relation of rough and finish cut operation.

$1.V_{s}\geq k_{3}V_{r}{\mbox{.}}$

( 30)

The mathematical modelling of this problem was attempted by various researchers. The minimum production cost given by Shin and Joo  [3] was $2.385/unit, whereas Chen and Tsai  [6] had reported a minimum production cost of $2.2974/unit. Onwubolu and Kumalo  [14] attempted the same model and produced a minimum production cost of $1.761/unit. However, subsequently, this result was proved to be impractical by Chen and Chen  [15] who highlighted that the machining model was handled incorrectly, since the number of rough cuts in their method was not limited to an integer value. Vijayakumar et al.  [18] had solved the same optimization problem and claimed a minimum production cost of $1.6262/unit, but the work carried out by Vijayakumar et al.  [18] was also proven invalid by Wang  [19] .

Subsequently, a few more researchers also attempted the same model. Zheng and Ponnambalam  [30] used a hybrid algorithm by combining the genetic algorithm and the artificial immune system and produced several results for minimum production cost, but optimum variable values were not given in their work and, also, nothing was mentioned about constraints. Al-Ahmari  [16] used a nonlinear programming solver to solve this problem and obtained a minimum production cost of $1.939/unit. However, in this case also, the constraint part was not described and some of the optimum variables were missing. Yildiz  [24]  and [29] used a hybrid Taguchi-harmony search algorithm and a particle swarm optimization approach independently to attempt this model, and showed the minimum production cost. Again, in this case also, Yildiz  [24]  and [29] had not described the optimum variables result and nothing was mentioned about the constraints involved in the process model.

After critically analysing all previous research work related to the same mathematical model, it is observed that in most cases, details on the constraints were not mentioned, even though the model involved a large number of constraints. Also, in many cases, only the minimum production cost obtained was given, without mentioning the optimum process variables, due to which, the results claimed by those researchers cannot be verified. Hence, keeping all these points in mind, efforts have been made in this work to obtain optimum process variables that satisfy all constraints. The model explained above in Eqs. , , , , , , , , , , , , , , , , , ,  and  is now attempted by the TLBO algorithm. However, it is observed that in the case of a cutting force constraint, given by Eqs.  and  , the maximum allowable cutting force, ${\textstyle F_{U}}$ , considered by some researchers, was 5.0 kgf; some researchers had taken it to be 200 kgf. By closely observing the equation of the cutting force constraint and the range of process variables, it is concluded here that the maximum allowable cutting force of 5.0 kgf is not feasible. Hence, in the present work, the maximum allowable cutting force is considered as 200 kgf. The model is associated with a different number of rough cuts, with the necessary depth of cut in each pass. However, in most cases, the comparison was made with the result obtained for a single rough cut, along with a final finish cut. Hence, in this work, an attempt is made with one rough cut and, accordingly, the depth of the rough and finish cuts is obtained, keeping in mind the total material to be removed from the workpiece and the equation for obtaining the number of rough cuts.

Initially, various trials were undertaken to decide the population size and number of iterations, and consistent results are obtained with a population size of 200 and 100 iterations; the results obtained by the TLBO algorithm are given in Table 3 . The TLBO algorithm is run several times to confirm the values of the optimum input parameters. The optimum process variables obtained using the TLBO algorithm have given a minimum production cost of $2.2885/unit and, at the same time, it is shown that all constraints have been satisfied simultaneously.

Table 3. Result obtained by the TLBO algorithm for example 2.

	Variables	Range/limit	TLBO result
Process parameters	${\textstyle V_{r}}$	50–500	110 (m/min)
	${\textstyle f_{r}}$	0.1–0.9	0.565 (mm/rev)
	${\textstyle d_{r}}$	1.0–3.0	3.0 (mm)
	${\textstyle V_{s}}$	50–500	170 (m/min)
	${\textstyle f_{s}}$	0.1–0.9	0.225 (mm/rev)
	${\textstyle d_{s}}$	1.0–3.0	3.0 (mm)
Constraints for rough cut	${\textstyle T_{r}}$	25–45	44.38 (min)
	${\textstyle F_{r}}$	≤ 200	199.85 (kgf)
	${\textstyle P_{r}}$	≤ 200	4.22 (kW)
	${\textstyle SC}$	≥ 140	19 413
	${\textstyle Q_{r}}$	≤ 1000	866 (°C )
Constraints for finish cut	${\textstyle T_{s}}$	25–45	25.22 (min)
	${\textstyle F_{s}}$	≤ 200	100.19 (kgf)
	${\textstyle P_{s}}$	≤ 200	3.27 (kW)
	${\textstyle SC}$	≥ 140	18 465
	${\textstyle Q_{s}}$	≤ 1000	857 (°C )
	${\textstyle SR_{U}}$	≤ 0.01	0.0053 (${\textstyle \mu m}$ )
Constraint on variable relations	${\textstyle k_{3}}$	≥ 1.0	1.54
	${\textstyle k_{4}}$	≥ 2.5	2.51
	${\textstyle k_{5}}$	≥ 1.0	1.0
Unit production cost			2.2885 ($)

Figure 4 shows the convergence of the result obtained by the TLBO algorithm. Even though previous researchers  [14] , [16] , [18] , [19] , [24] , [29]  and [30] had shown some better results than the TLBO algorithm, comparison with these results cannot be made, because, in all those cases, only the result of minimum production costs was produced, without giving the values of all necessary process variables and validation aspects of the constraints. Also, some of those results were proven incorrect. Chen and Tsai  [6] had reported a minimum production cost of $2.2974/unit, and it was considered by many researchers for comparison purposes. The minimum production cost of $2.2885/unit given by the TLBO algorithm shows an improvement over the result given by Chen and Tsai  [6] . Thus, the TLBO algorithm has proved its ability of handling the multi-constrained problem effectively.

Figure 4. Convergence of production cost of example 2.

It has been observed that the optimum values of input parameters were not reported by many researchers for the examples considered in the present work. Only the optimum values of output parameters (i.e. values of objective functions) were reported, without giving any idea about the values of the corresponding optimum values of input parameters. The disadvantage of such an approach is that those results could not be verified by the readers/users subsequently. To avoid this problem, the complete set of values of optimum input parameters obtained in the present work is presented for each example and, based on that, a confirmation experiment is undertaken in the case of objective functions, as well as constraints. The readers of this paper will also be in a position to confirm these results by substituting the set of optimum values of the input parameters of each example in the corresponding objective functions and constraints.

4. Conclusion

In the present work, optimization aspects of process parameters of a multi-pass turning operation is considered using a recently developed advanced algorithm, known as the teaching–learning-based optimization algorithm. Two different examples are considered, out of which the first example is a multi-objective problem and the second example is a single objective multi-constrained problem with 20 constraints. The performance of the TLBO algorithm is studied in terms of the convergence rate and accuracy of the solution. Compared to other advanced optimization methods, the TLBO algorithm does not require selection of algorithm-specific parameters. It makes this algorithm applicable to real life optimization problems, easily and effectively. The TLBO algorithm required a lower number of iterations for convergence to the optimal solution. The algorithm has shown its ability in handling multi-constrained problems. The algorithm can also be easily modified to suit optimization of the process parameters of other manufacturing processes, such as casting, forming, and welding, etc.

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