Non-dominated sorting modified teaching–learning-based optimization for multi-objective machining of polytetrafluoroethylene (PTFE)

A non-dominated sorting modified teaching–learning-based optimization (NSMTLBO) is proposed to obtain the optimum solution for a multi-objective problem related to machining Polytetrafluoroethylene. Firstly, an experimental design is done and the L27 orthogonal array with three-level of cutting speed \( \left( {V_{c} } \right) \), feed rate (f), depth of cut (ap) and nose radius \( \left( {N_{r} } \right) \) is formulated. A CNC turning machine is used to perform experiments with cemented carbide tool at an insert angle of 80° and the response variables known as surface finish and material removal rate are measured. A response surface model is rendered from the experimental results to derive the minimization function of surface roughness \( \left( {R_{a} } \right) \) and maximization function of material removal rate (MRR). Both optimization functions are solved simultaneously using NSMTLBO. A fuzzy decision maker is also integrated with NSMTLBO to determine the preferred optimum machining parameters from Pareto-front based on the relative importance level of each objective function. The best responses R_a = 2.2347 µm and MRR = 96.835 cm³/min are predicted at the optimum machining parameters of V_c = 160 mm/min, f = 0.5 mm/rev, ap = 0.98 mm and N_r = 0.8 mm. The proposed NSMTLBO is reported to outperform other six peer algorithms due to its excellent capability in generating the Pareto-fronts which are more uniformly distributed and resulted higher percentage of non-dominated solutions. Furthermore, the prediction results of NSMTLBO are validated experimentally and it is reported that the performance deviations between the predicted and actual results are lower than 3.7%, implying the applicability of proposed work in real-world machining applications.

CFRP:

Carbon fibre reinforced polymer

DOE:

Design of experiment

ECM:

Electrochemical machining

EDM:

Electric discharge machining

EMOTLBO:

Enhanced multi-objective teaching–learning-based optimization

FIB:

Focused ion beam

ICA:

Imperialist competitive algorithm

microEDM:

Micro-electric discharge machining

MOEA:

Multi-objective evolutionary algorithms

MOGWO:

Multi-objective grey wolf optimizer

MO-ITLBO:

Multi-objective improved teaching–learning based optimization

MOP:

Multi-objective optimization problem

MOPSO:

Multi-objective particle swarm optimization

MOTLBO:

Multi-objective teaching–learning-based optimization

NSGA-II:

Non-dominated sorting genetic algorithm II

NSMTLBO:

Non-dominated sorting modified teaching–learning-based optimization

NSTLBO:

Non-dominated sorting teaching–learning-based optimization

PSO:

Particle swarm optimization

PCD:

Polycrystalline diamond

PTFE:

Polytetrafluoroethylene

RSM:

Response surface model

TLBO:

Teaching-learning-based optimization

WEDM:

Wire-electric discharge machining

d :

Index of each dimension component of learner

m :

Index of objective function

n :

Index of learner

r :

Index to indicate the rank of a given front

s :

Index of learner that is randomly selected for comparison

\( F_{r} \) :

Set containing all learners with the r-th rank value

\( L_{m,r} \) :

Set containing all sorted members of the r-th front for the m-th objective

Q :

Set containing all learners to create the next front

\( S_{n} \) :

Set containing all solutions dominated by the n-th learner

\( {\mathbf{Rank}} \) :

Set containing all non-domination rank values of N learners

\( {\mathbf{F}} \) :

Set containing all members from R fronts

\( {\varvec{\Delta}} \) :

Set containing the crowding distances of all N learners

\( {\mathbf{P}} \) :

Set containing all population members

\( {\mathbf{P}}^{{{\mathbf{off}}}} \) :

Set containing all offspring members

\( {\mathbf{P}}^{{{\mathbf{comb}}}} \) :

Set containing the combination of both population and offspring members

\( {\varvec{\Psi}}^{{\mathbf{U}}} \) :

Set containing the utopia point of a multi-objective optimization problem with M objective functions

\( {\varvec{\Psi}}^{{{\mathbf{SN}}}} \) :

Set containing the pseudo nadir point of a multi-objective optimization problem with M objective functions

R _P :

Set containing the Pareto non-dominated solution set of NSMTLBO

S _P :

Set containing the Pareto non-dominated solution set of MOPSO

T _P :

Set containing the Pareto non-dominated solution set of NSGA-II

U _P :

Set containing the Pareto non-dominated solution set of MOGWO

V _P :

Set containing the Pareto non-dominated solution set of MOTLBO

W _P :

Set containing the Pareto non-dominated solution set of MO-ITLBO

X _P :

Set containing the Pareto non-dominated solution set of NSTLBO

\( \psi \left( { \cdot , \cdot , \cdot , \cdot } \right) \) :

An operator that returns the response variable value based on the given control variables

\( \varPsi_{m} \left( \cdot \right) \) :

An operator that returns the value of the m-th objective function based on the given individual solution

\( Trunc\left( { \cdot , \cdot } \right) \) :

An operator that returns the best N members with the lowest ranking and highest crowding distance values

\( C\left( { \cdot , \cdot } \right) \) :

An operator that returns the percentage of solution set from one Pareto front that is dominated by solution set from another Pareto front

\( \prec_{cco} \) :

Crowding-comparison operator to compare the superiority of two solutions

\( V_{c} \) :

Cutting speed

f :

Feed rate

ap :

Depth of cut

\( N_{r} \) :

Nose radius

\( R_{a} \) :

Surface roughness

\( \Delta R_{a} \) :

Error rate of surface roughness

MRR :

Material removal rate

\( \Delta MRR \) :

Error rate of material removal rate

\( D_{initial} \) :

Initial diameter of PTFE sample before machining process

\( D_{final} \) :

Final diameter of PTFE sample after machining process

L :

Length of cut of PTFE sample

T :

Time taken to cut PTFE sample

Y :

Response term of regression equation

\( \alpha_{0} \) :

Free term of regression equation

\( X_{i} \) :

Control variable term of regression equation

\( \beta_{i} \) :

Linear coefficient term of regression equation

\( \beta_{ii} \) :

Quadratic coefficient term of regression equation

\( \beta_{ij} \) :

Interacting coefficient term of regression equation

D :

Number of decision variables to be optimized

N :

Population size

\( T_{f} \) :

Teaching factor that can be set as either 1 or 2

\( T_{f1} ,T_{f2} \) :

Teaching factors with the range of 1 to 2 generated from uniform distribution

\( C_{n} \) :

Domination count to indicate the number of solutions dominate the n-th learner

\( Rank_{n} \) :

Non-domination rank value of the n-th learner

\( \Delta_{a,r} \) :

Crowding distance of the a-th member in the r-th front

R :

Upper limit of front counter

\( \left| {F_{r} } \right| \) :

Number of members in the r-th front

\( X_{n,d} \) :

The d-th component of n-th candidate solution

\( X^{teacher} \) :

Solution vector that represents the best solution known as teacher

\( X^{mean} \) :

Solution vector that represents the average knowledge level of population

\( X_{n}^{new} \) :

New solution vector produced by the n-th learner during the teacher or learner phases

\( X_{d}^{U} \) :

Upper limit of the d-th dimensional component

\( X_{d}^{L} \) :

Lower limit of the d-th dimensional component

\( X_{a,d}^{Cand} \) :

Solution vector of the d-th dimensional component of a-th candidate teacher

\( X^{preferred} \) :

Solution vector of most preferred Pareto optimal solution

\( X_{n}^{teacher} \) :

Solution vector of teacher assigned to the n-th learner

\( \tilde{X}_{n}^{mean} \) :

Weighted mean position vector assigned to the n-th learner

\( E_{n,a} \) :

Normalized Euclidean distance between the n-th learner and the a-th candidate teacher

\( r_{1} ,r_{2} ,r_{3} ,r_{4} \) :

Random numbers with the range of 0 to 1 generated from uniform distribution

\( r_{5} \) :

Random numbers with the range of − 1 to 1 generated from uniform distribution

\( P_{cr} \) :

Crossover rate

\( P_{mut} \) :

Mutation probability

\( d_{r} \) :

Randomly selected dimensional component for mutation

\( \varPsi_{m}^{U} \) :

Utopia point of a multi-objective optimization problem in the m-th objective function

\( \varPsi_{m}^{SN} \) :

Pseudo nadir point of a multi-objective optimization problem in the m-th objective function

\( \mu_{a}^{m} \) :

Membership value of the a-th Pareto optimal solution in the m-th objective function

\( \mu_{a} \) :

Total degree of optimality of each a-th Pareto optimal solution

\( w_{m} \) :

Relative importance of each m-th objective function

\( w_{1} \) :

Relative importance level of minimizing surface finish

\( w_{2} \) :

Relative importance level of maximizing material removal rate

\( \gamma \) :

Counter of function evaluations

\( \varGamma \) :

Maximum fitness evaluation numbers

\( d_{a} \) :

Smallest Euclidean distance between the a-th and b-th Pareto optimal solutions

\( \bar{d} \) :

Average value of all smallest Euclidean distance

S :

Spacing measure

SD :

Standard deviation

R ² :

Percentage of variation of data

P :

Significance of control variables

\( c_{1} ,c_{2} \) :

Acceleration coefficients

\( \alpha \) :

Grid inflation rate

\( nGrid \) :

Number of grid per dimension

\( nGroup \) :

Number of group created for multiple group learning

\( \varepsilon \) :

Parameter used for epsilon dominance method

\( \left| A \right| \) :

Archive size

This work is partially supported by UCSI University Pioneer Scientist Incentive Fund (PSIF) with Project Code of Proj-In-FETBE-34 and Proj-In-FETBE-50.

Authors and Affiliations

1. Faculty of Engineering, Technology and Built Environment, UCSI University, 56000, Kuala Lumpur, Malaysia

   Elango Natarajan, Wei Hong Lim & Sew Sun Tiang

2. Department of Mechanical Engineering, Sona College of Technology, Salem, Tamil Nadu, 636005, India

   Varadaraju Kaviarasan

3. School of Engineering (Mechatronics), Monash University, Subang Jaya, Malaysia

   S. Parasuraman

4. FTMS College, Cyber Jaya, Malaysia

   Sangeetha Elango

Corresponding author

Correspondence to Elango Natarajan.

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