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

Abstract

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 Ra = 2.2347 µm and MRR = 96.835 cm3/min are predicted at the optimum machining parameters of Vc = 160 mm/min, f = 0.5 mm/rev, ap = 0.98 mm and Nr = 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.

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Abbreviations

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

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Acknowledgements

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.

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Correspondence to Elango Natarajan.

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Natarajan, E., Kaviarasan, V., Lim, W.H. et al. Non-dominated sorting modified teaching–learning-based optimization for multi-objective machining of polytetrafluoroethylene (PTFE). J Intell Manuf 31, 911–935 (2020). https://doi.org/10.1007/s10845-019-01486-9

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Keywords

  • Design of experiments
  • Multi-response
  • Non-dominated sorting modified teaching–learning-based optimization
  • Response surface model
  • Surface roughness