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An integrated evolutionary approach for modelling andoptimisation of CNC end milling processD. Kondayya a & A. Gopala Krishna ba Department of Mechanical Engineering, Sreenidhi Institute of Science and Technology,Hyderabad, 501301, Andhra Pradesh, Indiab Department of Mechanical Engineering, University College of Engineering, JawaharlalNehru Technological University, Kakinada, 533003, Andhra Pradesh, IndiaVersion of record first published: 14 May 2012.

To cite this article: D. Kondayya & A. Gopala Krishna (2012): An integrated evolutionary approach for modelling andoptimisation of CNC end milling process, International Journal of Computer Integrated Manufacturing, 25:11, 1069-1084

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An integrated evolutionary approach for modelling and optimisation of CNC end milling process

D. Kondayyaa* and A. Gopala Krishnab

aDepartment of Mechanical Engineering, Sreenidhi Institute of Science and Technology, Hyderabad 501301, Andhra Pradesh, India;bDepartment of Mechanical Engineering, University College of Engineering, Jawaharlal Nehru Technological University, Kakinada

533003, Andhra Pradesh, India

(Received 8 August 2011; final version received 1 April 2012)

In this research, a novel integrated evolutionary based approach is presented for the modelling and multi-objectiveoptimisation of a machining process. Computer numerical control end milling process has been considered in thepresent work as it finds significant applications in diversified engineering industries. Firstly, genetic programming(GP) has been proposed for explicit formulation of non-linear relations between the machining parameters (spindlespeed, feed and depth of cut) and the performance measures of interest (material removal rate and tool wear) usingexperimental data. Genetic programming approach optimises the complexity and size of the model during theevolutionary process itself and hence this technique has the potential to identify the true models avoiding theproblems of conventional methods. Central composite second-order rotatable design had been utilised to plan theexperiments and the effect of machining parameters on the performance measures is also reported. In the secondpart, as the chosen responses are conflicting in nature, a multi-objective optimisation problem has been formulated.A non-dominated sorting genetic algorithm-II (NSGA-II) has been used to simultaneously optimise the objectivefunctions. The Pareto-optimal set generated is useful for process planning which is a critical link in computer-integrated manufacturing (CIM).

Keywords: end milling; modelling; genetic programming; multi-objective optimisation; NSGA-II

1. Introduction

Machining processes find extensive applications inmost of the engineering industries such as aerospace,automotive, defence and tool and die industries. In thiswork, one of the most widely used machiningprocesses, computer numerical control (CNC) endmilling is considered for investigation. End milling is acomplex and costly shape machining process used forobtaining profiles, slots, engraving, surface contouringand pockets on finished components. This machiningprocess has become indispensable in industry and iscontinuously finding further applications with thedevelopments of new cutting tool materials.

Computer numerical control machining being anexpensive process (Tolouei-rad et al. 1997), there canbe a big payoff from even small increase in perfor-mance. Hence, the selection of optimum processparameters is highly essential to reduce the machiningcost and increase the production rate. Optimisationthrough full enumerations is not possible owing to thecomplex nature of the machining process and compli-cated interaction between the variables of machining.Moreover, the said procedures neither lead to optimaluse of the machines nor the quality of surfacegenerated. Though considerable number of researchers

carried out various investigations for improving theprocess performance, proper selection of machiningparameters for best process performance is never-theless a challenging job. Owing to the highlycomplicated interactions between process parameters,current analytical models and analyses cannot provideaccurate prediction for better quality control andhigher throughput. Therefore, an efficient method isneeded to determine the optimal machiningparameters.

In case of CNC milling, material removal rate(MRR) and tool wear (TW) are the most importantresponse parameters. Specifically in rough machiningoperations, these responses directly affect the economyof the process. While material removal rate indicatesthe productivity, tool wear is a measure of the qualityof machined components. As CNC milling, is acomplex process (Wang et al. 2006), it is very difficultto determine the optimal machining parameters for thebest machining performance. Moreover, the perfor-mance measures, namely MRR and TW, are conflict-ing in nature as it is desirable to have higher MRRwith lower value of TW.

The overall objective of this research is to apply anew methodology for modelling and optimisation of

*Corresponding author. Email: d_kondayya@yahoo.co.in

International Journal of Computer Integrated Manufacturing

Vol. 25, No. 11, November 2012, 1069–1084

ISSN 0951-192X print/ISSN 1362-3052 online

� 2012 Taylor & Francis

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the non-linear and complex CNC end milling process.With this aim, accurate prediction models to estimateMRR and TW were developed from the experimentaldata using a potential evolutionary algorithm calledgenetic programming (GP). Subsequently, the devel-oped models were used for optimisation of the process.As the chosen objectives, MRR and TW, are conflict-ing in nature, the problem was formulated as a multi-objective optimisation problem. A popular evolution-ary algorithm, non-dominated sorting genetic algo-rithm II (NSGA-II), was then used to solve andthereby retrieve the multiple optimal sets of inputvariables.

2. Literature survey

Many approaches have been proposed for end millingto select the appropriate process parameters. A reviewof literature is presented here:

Machinability database systems were in use forlong time for the selection of appropriate cuttingconditions. Model building in machinability databasesystems was studied using regression based andartificial neural networks (ANNs) techniques(Choudhury and El-baradie 1996, Wong and Hamou-da 2003).

Optimum cutting parameters for multi-pass millingutilising the two mathematical techniques dynamicprogramming and geometric programming were de-termined by I.hsan Sonmeza et al. (1999). Shunmugamet al. (2000) used genetic algorithm to optimise themachining parameters for multi-pass milling opera-tions. El-Mounayri et al. (2002) performed modellingand optimisation of flat end milling process usingartificial neural networks (ANNs) and particle swarmoptimisation (PSO) respectively. Linear regressionmodels were developed for cutting force using ANNs,while a production cost objective function was used tooptimise the variables, spindle speed and feed rate.Tandon et al. (2002) applied the evolutionary compu-tation technique, PSO to optimise multiple machiningparameters simultaneously for the case of pocket-milling on CNC milling machine. A hybrid approachbased on genetic algorithm (GA) and simulatedannealing (SA) was developed by Wang et al. (2004)to select the optimal machining parameters for plainmilling process, the obtained results demonstrate thatthe hybrid method is superior to those using GA alone.Baskar et al. (2005) demonstrated the efficiency ofseveral optimisation procedures such as GA, tabusearch, ant colony algorithm and particle swarmoptimisation for the optimisation of machining para-meters for milling operation. Oktem et al. (2006)optimised the cutting parameters for minimum surfaceroughness in end milling mould surfaces of a

component used in biomedical applications by cou-pling ANN and GA. Yih-fong (2007) applied Taguchimethods coupled with principal component analysis inthe process optimisation of the high-speed CNCmilling. Krain et al. (2007) experimentally optimisedthe tool life, tool wear and productivity when endmilling Inconel 718. For optimisation they consideredthe effects of fed rate, radial depth of cut, tool materialand geometry. They concluded that no single toolmaterial or geometry gave the best overall perfor-mance. Palanisamy et al. (2008) used regression andANN techniques for developing models to predict toolwear on AISI 1020 steel using a carbide cutter in auniversal milling machine. In their work, flank wearwas taken as the response variable measured duringmilling, while cutting speed, feed and depth of cut asinput parameters. Bala Murugan et al. (2009) carriedout experimental investigations for machinabilitystudy of hardened steel and obtained optimum processparameters by grey relational analysis. The machiningprocess parameters considered were cutting speed,feed, depth of cut and width of cut and the multipleresponses that were optimised were volume of materialremoved, surface finish, tool wear and tool life. Oktem(2009) used ANNs to model surface roughness andapplied GA to optimise the cutting parameters whenend milling of AISI 1040 steel material with TiAlNsolid carbide tools under wet condition. Lu et al.(2009) applied grey relational analysis coupled withprincipal component analysis for optimisation ofcutting parameters for rough cutting in high speedend milling on SKD61 tool steel. Modelling results forsurface roughness in end milling of 6061 aluminiumusing combined adaptive neuro-fuzzy inference systemand genetic algorithms were presented by Samanta(2009). Ding et al. (2011) investigated the main effectsand interaction effects and optimised the cuttingparameters for desirable surface roughness in endmilling of hardened AISI H13 steel with PVD coatedcarbide insert. The best surface roughness wasachieved with high cutting speed, small axial depth ofcut, high feed and small radial depth of cut. Azlanet al. (2011) applied an integrated approach to searchfor optimal cutting conditions leading to the minimumvalue of surface roughness in end milling of Ti-6AL-4Valloy. Regression equations were developed for surfaceroughness which were then optimised by conventionalGA and conventional simulated annealing (SA) algo-rithms separately and then by an integrated GA andSA technique.

3. Critique on literature survey

The literature survey reveals that specific efforts weredevoted to determine the most accurate empirical

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models for process performances such as surfaceroughness, material removal rate and tool wear. Thesemodels were utilised as objective functions andoptimised to obtain the machining conditions.

The other main findings of literature survey are:

(1) The predominant modelling technique forestablishing the process models were regressionbased techniques and ANNs.

(2) Multi-objective optimisation for multi passmilling has been performed considering pro-duction time and production cost as objectivefunctions.

(3) Efforts were mainly focused on optimisation ofonly single performance characteristic in CNCend milling.

(4) No published work available on multi-objectiveoptimisation of the conflicting objectives, MRRand TW.

In statistical-based techniques like regression ana-lysis, a prediction model has to be determined inadvance and a set of coefficients have to be found. Thepre-specified size and shape of the model means thatthese are not adequate to capture the complex relationbetween influencing variables and output parameters.Although ANNs have also been used in the literaturefor modelling, they have the drawback of not beingable to quantify explicitly the relationships betweeninputs and outputs (Baykaso�glu 2008). Consequently,great care must be taken with the ANN approach toprevent over-fitting (Benyamin and Daniel 2002).However, the accuracy and possibility of determiningthe global optimum solution depends on the type ofmodelling technique used to express the objectivefunction and constraints as functions of the decisionvariables (Jain et al. 2007). Therefore, effective,efficient and economic utilisation of the CNC millingprocess necessitates an accurate modelling and opti-misation procedure.

4. Proposed methodology

In this paper, a novel approach using GP is presentedfor modelling of material removal rate and tool wear.Genetic programming is a technique pioneered byKoza (1992) and belongs to the class of genetic orevolutionary algorithms. Since its introduction in theearly 1990’s GP has a number of significant achieve-ments in tackling industrial scale modelling, dataanalysis, search and optimisation problems (Rioloet al. 2007). The capabilities of GP in capturing theempirical relationship between input variables andoutput features are of primary significance for model-ling machining processes. The distinctive aspect of GP

as compared to traditional approaches is in the modelstructure definition which is not apriori. Therefore, anadvantage of this approach is that fewer assumptionshave to be made regarding the final form of the modelas the algorithm can evolve the model structure fromelementary building blocks. The generated model helpsdirectly to obtain an interpretation of the parametersaffecting the process. More details of this methodologyare discussed in section 5. The models developed byGP were subsequently used for optimisation.

In the current study, the optimisation problem ofCNC milling was explicitly formulated as a multi-objective optimisation problem, as the determinationof the optimal machining conditions involves a conflictbetween maximising MRR and minimising the TW. Itcan be noted that the classical optimisation methods(gradient descent, weighted sum methods, goal pro-gramming, min–max methods, etc.) are not efficient forhandling multi-objective optimisation problems. Theaforesaid methods do not find multiple solutions in asingle run, and therefore it is necessary for them to beapplied as many times as the number of desired Pareto-optimal solutions (Sardinas et al. 2006). In addition,classical methods fail when the objective functionbecomes discontinuous. The above-mentioned difficul-ties of classical optimisation methods are eliminated inevolutionary algorithms, as they can find the multiplesolutions in a single run. As a result, one of the bestevolutionary approaches, the NSGA-II is proposed inthis paper for multi-objective optimisation of CNCmilling. Genetic algorithm (GA) based multi-objectiveoptimisation methodologies have been widely used inthe literature to find Pareto-optimal solutions (Fonsecaand Fleming 1993, Horn et al. 1994, Zitzler and Thiele1998). In particular, NSGA-II has proven its effective-ness and efficiency in finding well-distributed and well-converged sets of near Pareto-optimal solutions (Deb2002). The proposed methodology of integrating GPand NSGA-II is depicted in Figure 1.

5. Modelling using GP

The Evolutionary Algorithms (EAs) are stochasticsearch methods which combine important character-istics like robustness, versatility and simplicity. Theability of EAs to search for complete and globalsolutions for a given problem makes them powerfulproblem solving tools. EAs exploit not only theinformation contained in each individual, but alsothe information of the population as a whole. Geneticprogramming is one such EA to generate an optimummodel structure with interaction terms or higher orderterms. Genetic programming is a fairly recent EAmethod compared to other three variants of evolu-tionary computation viz., genetic algorithms,

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evolutionary strategies and evolutionary programming(Angeline 1995).

The cardinal principles of GP were originallyproposed by Koza (1992). Since then this techniquehas found widespread applications in diversified fieldssuch as industrial robotics (Dolinsky et al. 2007),chemical process modelling (Greeff and Aldrich 1998),prediction of shear strength of beams (Ashour et al.2003), etc. However, it is noteworthy that the potentialof this tool has not been exploited in the field ofmanufacturing and few applications in the field ofmanufacturing can be found in the literature (Nastranand Balic 2002, Brezocnik et al. 2004, Kovacic et al.2007). Therefore, in the present work, the proposedmethod is used to model the machining process.

Genetic Programming principles are primarilyderived from GAs, and hence share most of GAs’properties. The main difference between GP and GAslies in the individual representations. In GAs popula-tion of individuals is represented as strings whereas inGP individuals are represented as tree structures. Thetree like structures of GP, which are in fact solutions toproblems, have hierarchical compositions of terminalsand functions. Terminals are input variables appro-priate to a particular problem domain and userspecifies a number of functions that manipulateterminals. The primitive functions typically include:

arithmetic operations (þ , –, *, /), Boolean operations(AND, OR, NOT), logical operations (IF–THEN–ELSE), and non-linear functions (sin, cos, tan, exp,log). Compatibility between the functions and term-inals must be ensured in order to pass informationimpeccably between each other (Sette and Boullart2001). Figure 2 shows a typical example of GPprocedure. Representation of an individual in GP isshown in Figure 3 which corresponds to the expression

Figure 1. Proposed methodology.

Figure 2. Example of GP procedure.

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of (x72)+y/(z2). The set of functions in the represen-tation are {þ , –, *, /} whereas the set of terminals are{x, y, z}.

5.1. Initial population

Genetic programming algorithm begins with a set ofrandomly created individuals called initial population.Each individual is a potential solution represented as atree. Each tree is constructed by random compositionsof the sets of functions and terminals. The shape of atree has an influence on its evolution and both sparseand bushy trees should be present in the initialpopulation. To ensure this, a ramped half-and-halfmethod, suggested by Koza (1992) is usually used inGP. In this method, half of the trees in the populationare generated as full trees and other half as randomtrees. Once the population is generated, a suitablefitness function should be given for evaluating thefitness value of each individual. Then, a set ofindividuals with better fitness value will be selectedand used to generate new population of next genera-tion by the predefined genetic operators while thepopulation size is kept constant.

5.2. Genetic operators

The purpose of genetic operators is to evolveindividual trees. The main operators generally includereproduction, crossover and mutation. Reproductioninvolves copying of the selected individual into the nextgeneration population. In general, about 10% of thepopulation is selected for simple reproduction and90% is selected through crossover. Crossover is usuallythe most important genetic operator in GP. Itsapplication produces two children trees from twoparent trees by exchanging randomly selected sub-treesof each parent. Both parents are selected using one of

the stochastic selection methods such as fitnessproportional selection or tournament selection.

The cross-over operator is illustrated in Figure 4.To avoid excessive growth of the branches of theprogram trees, a maximum depth value is usuallyestablished. If the crossover operation produces anoffspring of impermissible depth, this offspring isdisregarded and its parent is reproduced as it is.Koza (1992) uses the default maximum depth of 17.Similar to GAs, GP uses the mutation operator inorder to avoid falling into the local optimal solution.Mutation is typically a replacement of a randomlyselected single terminal of an individual with a newrandomly generated terminal. While mutation plays animportant role in maintaining genetic diversity in thepopulation, most new individuals in a particulargeneration result from crossover. The mutation op-erator is illustrated in Figure 5.

5.3. Result designation and termination criteria

Execution of the aforementioned three genetic opera-tors constitutes one generation and the procedure isrepeated until a termination criterion is met. The singleindividual with the best value of fitness over all thegenerations is designated as the result of a run. Thetermination criterion can be either a fixed number ofgenerations or specified quality of the solution. Thenumber of runs required for a satisfactory solutiondepends on the complexity of the problem underconsideration.

Figure 3. Representation of an individual in GP. Figure 4. Illustration of cross-over operator.

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6. Optimisation using NSGA-II

Multi-objective evolutionary algorithms (MOEAs) areoptimisation methods that make the best use of thefeatures of multipoint search of EAs, and can obtainPareto approximation set at the same time. ElitistNon-dominated Sorting Genetic Algorithm (NSGA-II) proposed by Deb et al. (2002) is a well known andan excellent implementation of MOEA.

NSGA-II improves upon the original version byincorporating the following main features:

(1) At each generation, the best solutions found arepreserved and included in the following gen-eration using an elite-preserving operator.

(2) A fast algorithm is used to sort the non-dominated fronts.

(3) A two level ranking method is used to assignthe effective fitness of solutions during theselection process.

6.1. Principle of NSGA-II

Initially, a random parent population Pgt (the sub-script t indicates the generation) of size N is created.The population is sorted based on the non-domination

principle. Each solution is assigned a fitness (i.e. rank)equal to its non domination level (1 is the best level, 2is the next-best level, and so on). Thus, maximisationof fitness is assumed. At first, the usual binarytournament selection, recombination, and mutationoperators are used to create an offspring populationQgt of size N. Since elitism is introduced by comparingcurrent population with the previously best found non-dominated solutions, the procedure is different afterthe initial generation. The procedural steps areillustrated in Figure 6. First, a combined populationRgt¼Pgt U Qgt is formed. The population Rgt is of size2N. Then, a fast non-domination sorting algorithm isused to rank the solutions according to their dom-inance rank and organise fronts of equal rank.

In this ranking method, an individual, k, israndomly chosen from the population Rgt and insertedin an intermediate set named F1. Then, anothersolution k0 is drawn from Rgt and compared to allindividuals from F1. If k0 dominates k, k0 enters F1 andk is deleted, but if k dominates k0, then k0 is deleted andk stays in F1. Continuing this comparison for allindividuals, F1 will consist of all non-dominatedindividuals of Rgt, the Pareto front by definition.Now the first Pareto front is removed from the originalpopulation and the same procedure is iterativelycontinued to identify other layers of Pareto fronts {Fi

i¼ 1,2,3 . . . }. Subsequently, individual solutions withineach front are ranked according to a density measureusing the crowding operator. This operator, as picturedin Figure 7, measures the diversity of each individualby measuring half of the perimeter of the rectangle thatencloses a solution in the objective function space andassigning infinite distance to the extreme points of thePareto-front. The next generation, Pgtþ 1, which hasthe same size as the first generation is filled withconsecutive Pareto fronts {Fi i¼ 1,2,3 . . . } until no fullPareto front can be fully accommodated anymore.Figure 5. Illustration of mutation operator.

Figure 6. Procedural steps of NSGA-II.

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Then, the solutions in the next Pareto layer are sortedin descending order according to their distance assign-ment and the empty spaces in the proceeding genera-tion are filled with higher ranked solutions.

The next offspring population, Qgtþ 1, is created byusing the crowded tournament selection operator. Twoattributes can be considered for each individualsolution: First, a non-domination rank (equal to thePareto layer rank) and second, a crowding distance. Inthe tournament selection, competitions are set upbetween individuals. The tournament is ‘won’ by thatindividual having a better non-dominated rank (lies onan outer Pareto front). If both individuals are on thesame Pareto front, ties are broken by the crowdeddistance and the tournament is ‘won’ by the one whichis least crowded. The procedure outlined in Figure 7 isrepeated until the termination criterion is met and thebest-known Pareto front is saved in an archive. Thesolutions in this archive are the Pareto optimalsolutions of the problem under consideration.

7. Experimental details

The machining experiments were conducted on apowerful and precise 3-axis CNC vertical machiningcentre (model: AGNI BMV45 made by Bharat FritzWerner Ltd.) employing a continuously variablespindle speed up to a maximum of 6000 rpm andwith a maximum spindle power of 5.5 kW. The feedrates can be set up to a maximum of 10m/min.Workpieces of AISI 52100 steel of size 200 mm6 300mm6 25 mm with hardness 50 HRC after heattreatment were prepared and utilised. All of theexperiments were performed under dry conditionsalong the 300 mm length in conventional millingmode, as recommended by the tool supplier for the

specific work material. Coated carbide inserts (ISO:APKT11T308-PM) were used to machine the toolsteel. The inserts have rake angle of 08 and clearanceangle of 118. End milling inserts with this geometry aresuited for semi-dry and dry machining operations ofsteels. A commercially available double end mill insertholder of type EMP01-020-G20-AP11–02 of 20 mmdiameter and overall length 100 mm was utilised forexperiments. Figures 8 and 9 show the CNC set-up ofthe experiment and view of the machining zone.

CNC end milling involves several control variablessuch as surface cutting speed, spindle speed, axialdepth of cut, radial depth of cut, feed and radialengagement of tool. The workpiece and cutting toolmaterial also influence the performance of the process.However, based on the literature survey and the trialexperiments, the variables, namely, spindle speed (x1),feed (x2), and axial depth of cut (x3) were consideredas the decision (control) variables and the MRR andthe tool wear were considered as the output responses.Characterisation of wear on cutting tool insert isspecified primarily by flank wear and its progressivegrowth (Urbanski et al. 2000). Flank wear evaluationof insert was made off line by a tool maker’smicroscope (make: Mitutoyo TM500) with 30Xmagnification and 1 mm resolution. The admissiblewear was ascertained in accordance with ISO8688:1989 standard and measured by taking theaverage flank wear for two inserts for each experiment.Each experiment was repeated twice and a new insertwith same specification was used every time andmachining was stopped after five passes for tool wearmeasurement. Machining time is noted at the end ofeach pass and the material removal rate is calculatedby loss of weight method. The weight of work piece ismeasured accurately by digital balance with 0.001 mmaccuracy and the material removal rate is expressed ing/min.

Experimental designs constitute powerful metho-dology for accumulating and analysing informationabout a given process rapidly and efficiently from asmall number of experiments, thereby minimising theexperimental costs. The GP models for materialremoval rate and tool wear were trained by a datasetconstructed using design of experiment based rotatablecentral composite design (CCD). The 27 runs CCDmethod is chosen since the method provides a widercovering region of parameter space and good con-sideration of variables interaction in the model. Theranges of machining parameters used were as follows:spindle speed (x1) of 900, 1200 and 1500 RPM, feedrates (x2) of 30, 45, 60 mm/min and axial depth of cut(x3) of 0.4, 0.5, 0.6mm. The radial depth of cut wasmaintained constant at half of the cutter diameter. Themeasured values of tool wear and material removalFigure 7. Illustration of Crowded comparision operator.

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rate for 27 experiments conducted as per CCD areshown in Table 1. This constitutes the training datasetfor the GP algorithm. In addition, the developedmodel is validated to ensure the generalisationcapability of the predicted model for unseen casesand the data set used for validation is presented inTable 2.

8. Implementation issues

GP, being a stochastic search technique, makes noprior assumptions about the actual model form. Thestructure and complexity of the model evolve auto-matically. The populations of models of initial treegeneration for MRR and tool wear are first initialisedusing the ‘ramped half-and-half’ method. The terminalset T and the function set F were defined as: T¼ (x1,x2, x3, <) and F¼ (þ,7, *, /). The terminal set consistsof all input variables of the CNC end mill process anda random ephemeral constant <2 (7200, 200). Thefunction set contains addition, subtraction, multi-plication and protected division. The genetic program-ming run is controlled by many parameters of whichthe two major numerical parameters are the popula-tion size and the maximum number of evolutionarygenerations. These two parameters depend on thedifficulty of the problem involved. Other minornumerical parameters include the probability of cross-over, reproduction and mutation (Koza 1992). There-fore, some preliminary experiments were carried out to

obtain some estimates of workable parameters. Thesepreliminary test runs in the GP system were executedfor the output parameters independently. Based onthese experiments, the final parameters used togenerate the models are given in Table 3.

Evolutionary algorithms are generally robust tovariations of control parameters and some guidelinesare provided (Koza 1992) for choosing the controlparameters of standard GP. Population size andnumber of generations were optimally set to 500 and40 respectively as low population size did not showbetter results while still large population size appearedto yield little advantage in terms of the improvement offitness. Also there was increase in computation timeswith large population size and higher number ofgenerations. GP being probabilistic in application, itis important to study the problem by carrying outmultiple runs of the algorithm, and analyse the resultsbefore arriving at conclusions, thus, multiple runs ofthe algorithm were conducted. After the investigationof various alternative models, the following expres-sions for MRR and TW were found to have the bestfitness value.

MRR ¼ 0:9896x3����

x2x32�þ�22:46=x1

����0:24x2x3

��

þ�x2x3

�1� x3

���ð1Þ

TW ¼�x2x3

�����x2���24=��x3

2���0:997x2

�þ 64:85

��160�x1 þ 36

�=x3���þ 89

��: ð2Þ

Figure 8. CNC set-up of the experiment.

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The fitness measure while developing the abovemodels is considered as the correlation coefficient, R2

as this produces an accurate solutions in fewergenerations. The algorithm run was continued until avalue as close as possible to the maximum value of R2

was obtained. The correlation coefficient value lies inthe range [0, 1] and measures the way in which thepredicted values and actual values vary together.Therefore credit is given to the solutions that are closeto the correct form. Figures 10 and 11 show theconvergence graphs of the models predicted by theproposed algorithm. The fitness measures for MRRand TW have gradually improved with the number ofgenerations and finally converged to 0.9980 and0.8950, respectively. On the basis of the high valuesof R2 it can be said that the models are adequate inrepresenting the process. The normal probability plotsof the residuals for the output responses are alsoshown in Figures 12 and 13. A check on these plotsreveals that the residuals are located on a straight line,which means that the errors are distributed normallyand the obtained models are fairly well fitted with theobserved values.

Table 1. Training dataset.

S.no.

Speed (x1)(RPM)

Feed (x2)(mm/min)

Depth ofcut (x3)(mm)

MRR(g/min)

TW(mm)

1 1500 30 0.6 5.587 0.1632 900 30 0.4 3.487 0.093 1500 60 0.5 8.494 0.2094 1200 45 0.6 7.87 0.1995 1200 30 0.5 4.888 0.136 900 30 0.6 5.587 0.1397 1500 30 0.4 3.492 0.1158 1200 60 0.4 5.924 0.1459 900 45 0.4 4.744 0.15110 1200 45 0.5 6.641 0.17411 900 45 0.5 6.741 0.14912 1200 30 0.4 3.497 0.11513 1200 45 0.4 4.744 0.14814 1500 45 0.6 7.645 0.20715 1500 60 0.6 9.479 0.22516 900 45 0.6 7.59 0.17517 1500 45 0.4 4.844 0.1118 900 60 0.6 9.479 0.22519 900 30 0.5 4.888 0.1320 1500 30 0.5 4.988 0.11621 900 60 0.5 8.376 0.18922 1500 60 0.4 5.925 0.17523 1200 60 0.5 8.294 0.20224 1500 45 0.5 6.654 0.18825 1200 60 0.6 9.579 0.23526 900 60 0.4 5.624 0.17327 1200 30 0.6 5.594 0.143

Table 2. Validation dataset.

S.no.

Speed(x1)

(RPM)Feed (x2)(mm/min)

Depth of cut(x3) (mm)

MRR(g/min)

TW(mm)

1 1300 35 0.55 6.1 0.1592 1000 35 0.42 4.429 0.1233 1100 50 0.55 7.988 0.194 1300 40 0.45 5.68 0.1465 1100 35 0.45 4.89 0.1236 1300 35 0.32 2.832 0.0977 1000 35 0.55 5.99 0.1588 1000 50 0.45 6.461 0.1679 1000 40 0.32 3.221 0.09810 1100 40 0.45 5.34 0.143

Table 3. GP control parameters.

Terminal set: {x1, x2, x3}Function set: {þ ,7 ,*, /}Population size: 500Number of generations (max.): 55Number of independent runs: 10Crossover probability (%): 85Mutation probability (%): 5Reproduction probability (%): 10Selection method: TournamentFitness measure: R2

Figure 9. View of machining zone.

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A comparison of the predicted models and theexperimental values for the validation datasets ofMRR and TW are shown in Figures 14 and 15respectively. Very high values of R2 for MRR and TWof validation sets are obtained and found to be 0.9989and 0.9988 respectively. These indicate that thedeveloped models satisfactorily represent the outputs.

8.1. Effects of cutting parmeters on the responses

Surface plots have been drawn using MINITAB forthe convenience of understanding the surface effectsand selecting the best combinations of cutting para-meters. The MRR and tool wear variation for differentcombinations of cutting parameters are shown inFigures 16 and 17.

8.1.1. MRR

Figure 16a shows how the material removal ratedepends on the feed (x2) and depth of cut (x3) in thecase when the spindle speed (x1) of 1200 RPM is keptconstant. It can be seen that even though both factorshave influence on MRR the feed is a more dominantfactor. The combination of high value ofwork piece feedand the depth of cut provide the highest materialremoval rate. The influence of the spindle speed and

Figure 10. Convergence graph of MRR model.

Figure 11. Convergence graph of Tool wear model.

Figure 12. Probability plot of the residuals for the outputresponse MRR.

Figure 13. Probability plot of the residuals for the outputresponse Tool wear.

Figure 14. Comparison of the predicted model and theexperimental value for the validation datasets of MRR.

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Figure 15. Comparison of the predicted model and the experimental value for the validation datasets of TW.

Figure 16. Surface plots of speed, feed and depth of cut on MRR.

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depth of cut on the MRR for the constant feed of 45(mm/min) is shown in Figure 16b. Both the factors showsimilar intensity of influence on MRR. The MRRdecreases if the cutting speed increases. The depth of cuthas the opposite effect, that is, MRR decreases whendepth of cut decreases. Figure 16c shows the influence ofthe spindle speed and feed on theMRR for the constantdepth of cut of 0.5mm. Inherent to milling machines isthe cutting speed which is independent of table feed.Thus the MRR cannot be increased by increasing thecutting speed (Kopac and Krajnik 2007). It is obviousfrom the figure that MRR increases only marginallywhen the speed increases. The feed is again, by far, amore influential factor.

8.1.2. Tool wear

Figure 17a shows the distribution of tool wear withinput parameters feed and depth of cut. It is evidentfrom the contour surface that tool wear is maximum(about 0.24 mm) when x2 and x3 are at their higherlimits and is minimum (about 0.10 mm) when x2 and x3are at their lower limits. At lower values of depth ofcut, fewer work piece material holds to the flank thanat larger depth of cut. As the forces and heat generatedduring the machining process are higher at larger depthof cut, it is inferred that the higher temperature and thehigher force are the major reasons that cause the

adhesion of work piece material onto the tool flankface, thus hastening the tool wear. From Figure 17b itis clear from the contour surface that, wear ismaximum (about 0.20 mm) when x1 and x2 are attheir higher limits and is minimum (about 0.136 mm)when x1 and x2 are at their lower limits. The Figure 17cshows the disposition of tool wear with input para-meters. In this figure, it could be generally inferred thatas the spindle speed, feed, and depth-of-cut increase,the tool wear increases. These parameters clearlyaffect the tool wear. It is evident from the contoursurface that wear is maximum (about 0.20 mm) whenx2 and x1 are at their higher limits and is minimum(about 0.14 mm) when x2 and x1 are at the lower limit.

9. Formulation of multi-objective optimisation

The two objective functions considered in this studyare:

(1) maximisation of material removal rate and(2) minimisation of tool wear

which are given by Equations (1) and (2),respectively.

The two objective functions are optimised subjectto the feasible bounds of input variables. The opti-misation problem is defined as follows:

Figure 17. Surface plots of speed, feed and depth of cut on Tool wear.

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Maximise

MRR ¼ 0:9896x3����

x2x32�þ�22:46=x1

��=�0:24x2x3

��

þ�x2x3

�1� x3

���ð3Þ

Minimise

TW ¼�x2x3

�=���

x2���24=��x3

2���0:997x2

�

þ 64:85��160�x1 þ 36

�=x3���þ 89

��: ð4Þ

Subject to

900 � x1 � 1500

30 � x2 � 60

0:4 � x3 � 0:6

10. Results and discussions

The source code of the proposed optimisation algo-rithm is based on the description provided by Debet al. (2002). For the current work, the code was run ona Pentium IV processor using Microsoft VCþþ pro-gramming language on a Windows XP platform.Tournament selection, Simulated Binary Crossover(SBX) and polynomial mutation operators wereselected as the genetic operators of the real-codedNSGA-II algorithm. NSGA-II starts with a randomgenerated population. To limit the effect of random-ness on the results, the algorithm had to be run anumber of times. Analysing the results of varioussimulations on number of test cases revealed that, afterrunning the algorithm for about 15 times, nosignificant improvement could be obtained for thePareto solutions. The control parameters required forimplementation of the algorithm are listed in Table 4.For achieving better convergence, 350 generationswere used in the study. The algorithm found thePareto optimal front of conflicting objective functionswith good diversity of solutions, as shown in Figure 18.

The optimal input variables and their correspondingobjective function values are presented in Table 5.Since none of the solutions in the non-dominated set isabsolutely better than any other, any one of them is anacceptable solution. The choice of one solution overthe other depends on the requirement of the processengineer. By analysing the Pareto front, some decisionscould be taken, depending upon specific requirementsof the process. Analysing point 1 which is at theextreme of the front, the highest value of materialremoval rate (maximum productivity) is achieved butthe highest value of tool wear (worst surface quality) isachieved at this point too. On the lower side of thefront at point 1 a component can be machined withlow tool wear (best surface quality) but with minimumMRR (minimum productivity). All the other points areintermediate cases. They must be employed when acertain response factor is established beforehand. Forexample, to maintain accuracy of product, if themachining condition can allow a tool wear of1.6 mm, the process engineer can choose the parametersetting to obtain maximum MRR at the specified valueof tool wear. As can be observed from the graph, nosolution in the front is better than any other as all ofthem are non-dominated solutions. The choice of asolution has to be made purely based on productionrequirements. To further illustrate the advantage of theproposed methodology, some examples are explainedas follows.

From the experimental results of Table 1, theparameters listed in the ninth experiment lead to theTW value of 0.151 mm and the MRR of 4.744 g/min.After optimisation, it can be noted that the MRR isincreased to 5.831 g/min for the same surface finish (S.no. 42, Table 5), with a 23% increase in MRR. Inanother instance, from Table 1, first experiment, the setof input variables leads to the MRR of 5.587 g/minand the TW value of 0.163 mm. After optimisation,the TW value is reduced to 0.1487 (Sl No. 13, Table 5)

Table 4. NSGA-II Control parameters.

Population size (N): 100Number of generations (ngen): 350Selection strategy: TournamentCrossover probability (pc): 0.85Mutation probability (pm): 0.15Distribution index for crossover

operator: (nc)20

Distribution index for mutationoperator: (nm)

20

Population size (N): 50Figure 18. Pareto optimal front.

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with almost the same value of MRR. Experimentnumber 11 of Table 1 leads to MRR of 6.741 g/minand TW of 0.149 mm for speed of 900 RPM, feed of 45mm/min of and depth cut of 0.5 mm. In the optimal setfor S. no. 7, Table 5 for speed of 900.798 RPM, feed of45.5389 mm/min and depth of cut of 0.4854 mm theMRR is 6.438 g/min and TW is 0.1643mm. The abovediscussion signifies that there is conformity between

values obtained from the optimisation technique withthe experimental values for relatively same parametersettings.

The scanning electron microscopy (SEM) photo-graphs of the tool inserts that correspond to the bestvalues of MRR and TW are shown in Figures 19 and20. Those values of MRR and TW correspond to theextreme positions (point 1 and point 2) of the Paretooptimal set shown in Figure 18. As can be observedfrom the photographs, the variation of tool wear andresultant effect on tool edge is apparent with respect tothe different optimal sets of input variables.

11. Conclusions and future work

Material removal rate and tool wear are importantmachining performance measures which directly influ-ence the productivity and accuracy of the process. Inthis paper, for the first time evolutionary basedapproaches were used to both model and optimisethe CNC end milling process. Experimental data basedon CCD were used to develop empirical models forMRR and TW in terms of prominent machiningparameters such as spindle speed, feed and depth ofcut, with an efficient evolutionary algorithm namely,GP. Genetic programming is a domain independentmethodology which does not assume any a priorifunctional form of the solution and hence it canaccurately model the complex relationships of theprocess. The high R2 value of the models and thenormal probability plots prove the effectiveness of theGP approach to establish substantially valid models.NSGA-II has been used to simultaneously optimise theconflicting objectives of material removal rate and toolwear. Accordingly the Pareto-optimal solution set isgenerated and presented. This enables the manufactur-ing engineer to select a particular optimal set of inputvariables according to production requirements. Theoptimum values are essential for the automation of theprocess and implementation of a computer integratedmanufacturing system.

While the present study is sufficient to demonstratethe potential of the approach, for extending itscapability towards real industrial implementation,future studies may also concentrate to include otherresponses such as surface finish, cutting forces andchatter for multi-objective optimisation. Furthermore,the models developed may include additional inputparameters as encountered in industrial processes suchas radial depth cut, geometry of cutting edge, diameterof tool etc. In such a case, the major problem is oftenone of determining the relevant input variables from ahighly correlated data set. Also, it would be interestingto extend the proposed approach to other cuttingconditions and workpiece/tool combinations as the

Table 5. Final optimal solutions.

S.no

SpindleSpeed(RPM)

Feed(mm/min)

Depthof cut(mm)

MRR(g/min)

Toolwear(mm)

1 931.433 50.734 0.5012 7.629 0.18872 906.919 50.541 0.5146 7.524 0.18643 933.998 36.642 0.5224 5.855 0.15244 922.752 41.747 0.5008 6.209 0.15995 903.403 50.566 0.5159 7.547 0.18696 905.414 45.905 0.4892 6.543 0.16657 900.798 45.538 0.4854 6.438 0.16438 928.707 41.783 0.5025 6.238 0.16069 906.845 37.079 0.4898 5.483 0.144110 929.41 36.783 0.5141 5.768 0.150411 904.941 41.357 0.4989 6.133 0.158312 910.858 41.663 0.5104 6.336 0.162813 906.549 36.899 0.5072 5.593 0.148714 933.277 37.107 0.5218 5.907 0.153615 907.317 50.912 0.4897 7.157 0.179716 905.422 45.630 0.5099 6.829 0.172917 935.146 51.310 0.5025 7.422 0.184318 901.12 45.752 0.4893 6.526 0.166219 933.709 50.277 0.5226 7.616 0.188720 929.195 45.836 0.5043 6.769 0.171421 938.027 46.394 0.5225 7.113 0.179122 935.205 51.082 0.5025 7.394 0.183823 904.108 37.362 0.4968 5.613 0.146924 900.148 50.660 0.5096 7.458 0.184925 901.683 51.343 0.4884 7.187 0.180226 931.34 41.747 0.4994 6.189 0.159527 911.594 50.480 0.5096 7.435 0.184528 907.317 51.122 0.4892 7.173 0.180029 932.903 37.572 0.5219 5.968 0.154930 904.71 37.189 0.5147 5.826 0.151731 900.261 45.735 0.5029 6.735 0.170732 928.964 45.978 0.5149 6.948 0.175433 933.998 41.293 0.5216 6.443 0.165334 920.798 50.221 0.5157 7.501 0.186135 935.205 50.541 0.4976 7.246 0.181336 907.586 37.420 0.5105 5.801 0.151137 936.665 45.741 0.5212 7.010 0.177038 904.03 43.960 0.4984 6.447 0.164839 905.456 42.002 0.4889 6.066 0.156840 933.709 50.177 0.5226 7.602 0.188441 901.143 50.386 0.5129 7.477 0.185442 900.309 37.190 0.5150 5.831 0.151843 905.451 42.131 0.5096 6.383 0.163744 900.014 41.233 0.4893 5.979 0.154945 903.628 41.266 0.4972 6.099 0.157546 934.749 50.605 0.5178 7.582 0.187747 902.692 45.769 0.5006 6.705 0.170048 930.785 50.846 0.5026 7.366 0.183349 931.433 50.734 0.5012 7.328 0.182450 935.714 36.642 0.5223 5.485 0.1423

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modelling and optimisation algorithms are of generictype and the approach is systematic.

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Figure 19. SEM photograph of cutting tool insert corresponding to point 1 of Pareto-optimal front. Machining conditions:x1¼ 931.433 RPM, x2¼ 50.734 mm/min, x3¼ 0.5012 mm, MRR¼ 7.629 g/min, TW¼ 0.1887 mm.

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