(2007) 153–162www.elsevier.com/locate/powtec

Powder Technology 173

Parameters optimization of a nano-particle wet milling process using theTaguchi method, response surface method and genetic algorithm

Tung-Hsu Hou a,⁎, Chi-Hung Su a, Wang-Lin Liu b,c

a Department of Industrial Engineering and Management, National Yunlin University of Science and Technology, Taiwan, ROCb Department of Electrical Engineering, National Changhua University of Education, Changhua, Taiwan, ROC

c Precision Machinery Research and Development Center, Taichung, Taiwan, ROC

Received 3 June 2006; received in revised form 15 November 2006; accepted 23 November 2006Available online 5 December 2006

Abstract

Nano-particles have been successfully and widely applied in many industrial applications. The wet-type mechanical milling process is a popularmethod used to produce nano-particles. Therefore, it is very important to improve milling process capability and quality by setting the optimal millingparameters. In this research, the parameter design of the Taguchi method, response surface method (RSM) and genetic algorithm (GA) are integrated and

applied to set the optimal parameters for a nano-particlemilling process. The orthogonal array experiment is conducted to economically obtain the responsemeasurements. Analysis of variance (ANOVA) and main effect plot are used to determine the significant parameters and set the optimal level for eachparameter. The RSM is then used to build the relationship between the input parameters and output responses, and used as the fitness function to measurethe fitness value of the GA approach. Finally, GA is applied to find the optimal parameters for a nano-particle milling process. The experimental resultsshow that the integrated approach does indeed find the optimal parameters that result in very good output responses in the nano-particlewetmilling process.© 2006 Elsevier B.V. All rights reserved.

Keywords: Nano-particle; Wet-type milling process; Taguchi method; Response surface method (RSM); Genetic algorithm (GA)

1. Introduction

Nano-particles are advanced materials with 1–100 nm grainsize. They have been widely applied in photo-catalyst, carbonnano-tube, nano-ceramics, fabric fiber, and compound materialindustries. The techniques for manufacturing the nano-particlescan be classified into “top-down” and “bottom-up”methods [1].The top-down method transforms the material with an initialsize of a few micrometers into nano-particles with a size of only40–200 nm. Mechanical ball milling, sputtering, chemicaletching and laser ablation are popular top-down methods. Thebottom-up method is to generate the nano-particles by heapingup atoms, or assemble the nano-particles from nano-buildingblocks. It can produce nano-particles in any desired sizes.Examples of bottom-up methods include aerosol compaction,chemical synthesis, chemical vapor deposition and gas-atomization methods. No matter what kind of methods areused to fabricate the nano-particles, the nano-particles will

⁎ Corresponding author. Tel.: +886 5 5342601x5115; fax: +886 5 5312073.E-mail address: houth@yuntech.edu.tw (T.-H. Hou).

0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.powtec.2006.11.019

aggregate again because of the effects of Coulomb electrostaticforce and Van der Waal force as soon as the grain size of thenano-particles is smaller than 100 nm. In this situation, thenano-particles don't remain at the nanometer size any more.

Recently, the wet-type milling machine has been developed toproduce the nano-particles and avoid the aggregation effect. It is apopular and suitable method for producing the nano-particlesbecause of its simplicity and applicability to all classes ofmaterials [2–4]. One of the most popular mechanical millingmethods is stirred ball milling which consists of a rotating agitatorwith grinding ball media. The wet-type mechanical millingprocess uses the stirring and colliding motions in the stirred ballmill that generates the impact of collisions between milling ballsand particles to break the materials into a nanometer size [6,5,7].

Many advanced mechanical milling machines for makingnano-particles have been rapidly developed. The verticalchamber milling machine and the horizontal chamber millingmachine are the examples. In addition, in order to increase theproduction capacity of the nano-materials, a lot of techniques ormodules are integrated into the mechanical milling machine,such as the design of rotors, geometrical shape of the milling

154 T.-H. Hou et al. / Powder Technology 173 (2007) 153–162

mill, the design of the flowing field within a milling mill, theseparating system, the circulation and cooling system, thecontrol system, and the user interface… etc. The integration ofthese modules makes the mechanical milling process a complexprocess. Therefore it is critical to control the multiple variablesto achieve the desired product quality.

In this research, a vertical chamber milling machinedeveloped by the Precision Machinery Research and Develop-ment Center (PMC) is applied to make the titanium oxide (TiO2)nano-particles. The vertical milling machine consists of a millwith 1000 CC. capacity, a vane agitator, a circulation system, acooling system with 24,000 BTU/h, a filter separating system, astorage tank with 3000 CC. capacity, a power motor with2400 rpm and 5500 W. Fig. 1 shows the structural diagram ofthe vertical chamber milling machine. The operation of themilling process is a wet-type and circulated mode. First, theprocessed material, the titanium dioxide (TiO2), has to bedissolved with a corresponding solvent, glycol, to make acolloidal solution in a suitable weight ratio. Second, thiscolloidal solution and a dispersant (phosphate) are poured into astirring tank to mix up together, and sampled to measure thegrain size before the milling process is started. Then thecolloidal solution with dispersant, and the grinding ball media,

Fig. 1. The structural drawing of the vertical cha

Zircon dioxide (ZrO2), are put into the mill. Forth, the powermotor is started to drive the rotating vane agitator. In the mill,the impact of collisions between grinding ball media andparticles is produced to break the TiO2 material to a nanometersize. Fifth, samples of colloidal solution are randomly sampledafter operating the milling process some hours later to measurethe quality responses, mean of grain size and variance of grainsize, by using the Coulter Multisizer machine. Finally, the TiO2

nano-particles with nanometer size are produced by separatingthe solute (TiO2) and solvent (glycol) in a centrifuge machine.

The required qualities of the wet-type milling process arethat the mean of grain size and the variance of grain size forthe nano-particles must be kept small. The parameters thatmay affect these qualities are the milling time, the flow velocityof circulating system, rotation velocity of agitator, solute-to-solvent weight ratio, filling ratio of grinding media, size ofgrinding media, grinding media material, mill geometry, millsize, temperature of the milling, type of dispersants and materialtypes of milling mill…etc. However, the effects of these para-meters on the output quality responses in the nano-particlemilling process remain unknown. Moreover, it is important tofind the optimal milling parameters in order to produce highquality nano-particles.

mber milling machine developed by PMC.

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Recently, soft computing intelligence techniques, such asartificial neural networks (ANN) and genetic algorithms (GA),and statistical experiment design techniques, such as the Taguchimethod and response surface method (RSM), have been widelyapplied in many engineering optimization problems. Sunadaand Bi [8] used polynomial regression and an artificial neuralnetwork (ANN) to optimize the formulation and preparation of arapidly disintegrating tablets process. Reh and Ye [9] applied theneural networks, Taguchi method and multiplier and Lagrangianmethods to form the on-line prediction and optimization of acirculating fluidized bed process. They used the orthogonalexperiments to gain experimental data and applied the back-propagation neural networks (BPN) to provide on-line predic-tions. Finally, the multiplier and Lagrangian methods are used tooptimize the operation of the process using these BPN trainedmodels. Fan [10] combined a specialized response surfacemethod with state-of-the-art mathematical programming tech-niques to improve the quality of the chemical-mechanical pla-narization (CMP) in a semiconductor manufacturing process. Heused the RSM as an external modeling technique and appliednon-linear programming (NLP) approaches as an optimum-seeking procedure to set the optimal parameters. This methodcould obtain a better polishing quality for the CMP process.

Doniavi et al. [11] applied the integrated computer aidedmanufacturing definition (IDEF), experimental design and RSMto analyze, control and optimize the photolithography process inan electronic manufacturing environment. They used the IDEFtechnique to describe a manufacturing system with a modelingstructure, and then used the RSM approach to suggest theoperations with a significant statistical inference to find anoptimal response for the given objectives. The optimal settings ofthe processing equipment resulted in an increase in the processyield. Kwak [12] proposed an application of the Taguchi methodand RSM to minimize the geometric error and find the optimalgrinding conditions in the surface grinding process. He evaluatedthe grinding parameters' effect on a geometric error by theTaguchi method and used the RSM to predict the geometric error.Zhou et al. [13] applied the Taguchi method and RSM to optimizethe rapid prototyped stereo-lithography parts process parametersand improve the accuracy of the process. Kurtarn et al. [14] usedthe RSM and GA approaches to optimize the efficient warpage ofthin shell plastic parts. TheRSM is used to predict the response forwarpage data and is then combined with an effective GA to findthe optimum process parameter values. Moreover, He et al. [15]pointed out that the process optimization of the flow behavior in agrinding chamber can enhance energy efficiency and throughputin wet ultra-fine grinding operation.

The Taguchi method combined with the RSM is a powerfuloptimum design tool in many engineering applications. It notonly can save a lot of time but also can build models quickly andaccurately in an optimization design. GA is one of the mostpowerful and broadly applicable optimization techniques inengineering design problems. Therefore, the objectives of thisresearch are to integrate the parameter design of the Taguchimethod, RSM and GA approach to find the critical parametersfor a nano-particle milling process and to optimize the processparameters. In the following sections, the research methods

of Taguchi method, RSM and GA are briefly described inSection 2. The proposed integrated approach is then presentedin Section 3. The implementation results of the proposedmethod are then illustrated in Section 4. Finally, concludingremarks are made in Section 5.

2. Research method

2.1. Taguchi method

The Taguchi method, proposed by Genichi Taguchi, containssystem design, parameter design, and tolerance design proceduresto achieve a robust process and result for the best product quality.The purpose of system design procedure is to determine thesuitable working levels of the design factors. The parameterdesign procedure determines the factor levels that can generate thebest performance of the product or process under study. Thetolerance design procedure is used to fine-tune the results ofparameter design by tightening the tolerance levels of factors thathave significant effects on the product or process. Regardless ofthe on-line or off-line engineering, the Taguchi method canefficiently improve the effectiveness of the product or process byusing a loss function and achieve the robust product quality interms of the parameter design. In general, the parameter design ofthe Taguchi method utilizes orthogonal arrays (OAs) to minimizethe time and cost of experiments in analyzing all the factors anduses the signal-to-noise (S/N) ratio to analyze the experimentaldata and find the optimal parameter combination. Moreover, ananalysis of variance (ANOVA) is employed to estimate the errorvariance and determine the significant parameters. Procedures forconducting a parameter design include the following steps:

1. Planning experiment(1) Determine the control factors, noise factors and quality

responses of the product or process.(2) Determine the levels of each factor.(3) Select an appropriate orthogonal array (OA) table.

The selection of the most appropriate OA depends on thenumber of factors and interactions, and the number oflevels for the factors. For example, an L27(3

13) OA canlay out 27 trials, up to 13 factors in columns, and 3 factorlevels.

(4) Transform the data from the experiments into a properS/N ratio.

2. Implementing experiment3. Analyzing and examining result

(1) Execute an ANOVA analysis to determine the significantparameters.

(2) Conduct a main effect plot analysis to determine theoptimal level of the control factors.

(3) Execute a factor contribution rate analysis.(4) Confirm experiment and plan future application.

2.2. Response surface method

The response surface method (RSM), developed by Box andWilson in the early 1950s, is a collection of mathematical and

156 T.-H. Hou et al. / Powder Technology 173 (2007) 153–162

statistical techniques that are used to model and analyze engi-neering applications. In these engineering applications, a responseof interest is usually influenced by several variables and theobjective of the engineering applications is to find the variablesthat can optimize the response. The RSM has been applied in awide variety of industrial setting and parameter optimizations suchas, chemical, semiconductor and electronic manufacturing,machining, and metal cutting processes. In general, the procedureof RSM consists of the following steps [16]:

Step 1. Designing and conducting a series of experiments to getadequate and reliable measurements of the interestingresponse (e.g. orthogonal array experiment).

Step 2. Developing mathematical models of the first and secondorder response surface with the best fittings.

Step 3. Finding the optimal set of process parameters thatproduce a maximum or minimum value of the response.

Step 4. Representing the direct and interactive effects of theprocess parameters through two and three dimensionalplots.

If all variables are assumed to be measurable, the responsesurface can be expressed as follows:

Y ¼ f ðx1; x2; N xnÞ ð1Þwhere n is the number of variables.

The goal is to optimize the response variable Y. It is assumedthat the independent variables are continuous and controllableby experiments with negligible errors. It is necessary to find asuitable approximation for the true functional relationshipbetween independent variables and the response surface.

2.3. Genetic algorithm

The genetic algorithm (GA), introduced by John Holland(1971), is a stochastic search technique based on the mechanismof natural selection and natural genetics to imitate living beingsfor solving difficult optimization problems with high complex-ity and an undesirable structure. The GA approach represents apowerful, general-purpose optimization paradigm in which thecomputational process mimics the theory of biological evolu-tion [17,18]. It has been successfully used in job-shopscheduling, production planning, line balancing, lumber cuttingoptimization, and process optimization. Goldberg [19] proposedthe most common and useful form of GA. Different fromtraditional point-to-point descending and ascending searchtechniques, a GA starts from one set of random solutions calleda population. Each individual solution in the population iscalled a chromosome. At each generation, the GA performsgenetic operations such as crossover and mutation on theselected chromosomes to yield offspring to produce the nextgeneration. During each generation, these chromosomes evolveinto better fitness by applying an evolution operation, called theselection. From generation to generation, eventually, thechromosomes in the population will converge. In this case,the best chromosome is found. Generally, the basic steps of a

GA approach in solving an optimization problem can besummarized as follows:

1. Represent the problem variable as a chromosome of a fixedlength, and choose the size of a chromosome population,the crossover probability, and the mutation probability.

2. Define a fitness function to measure the fitness of anindividual chromosome in the problem domain.

3. Randomly generate an initial population of chromosomes.4. Calculate the fitness of each individual chromosome.5. Select a pair of chromosomes for mating from the current

population. Parent chromosomes are selected with a proba-bility related to their fitness. Highly fit chromosomes have ahigher probability of being selected for mating. The roulettewheel method is usually applied in chromosome selection.

6. Create a pair of offspring chromosomes by applying thegenetic operators: crossover and mutation.

7. Place the created offspring chromosomes in the newpopulation.

8. Repeat step 5 until the size of the new chromosomepopulation is equal to the size of the initial population.

9. Replace the initial (parent) chromosome population withthe new (offspring) population.

10. Repeat steps 4–9 until the termination criterion is satisfied.

3. The proposed integrated approach

In this research, an integrated method is proposed toinvestigate the effects of the process parameters for a nano-particle milling process with multiple output responses and tofind the optimal process parameters for the process. First of all,the parameter design of Taguchi method is used to determine thesignificant parameters and set the optimal level for eachparameter based on the results of analysis of variance (ANOVA)and the main effect plot. The response surface method (RSM) isthen applied to establish the linear and nonlinear multivariaterelationships between the nano-particle milling process para-meters and the quality responses. Finally, a GA approach isapplied to find the optimal parameters by using the responsefunction of the RSM model as the fitness function to measurethe fitness value. The flow diagram of the proposed integratedapproach is shown in Fig. 2. The detailed procedures of theproposed approach are stated in the following subsections.

3.1. Parameter design and response surface method (RSM)

As stated in the Introduction section, a lot of parameters mayaffect the output quality responses in the nano-particle millingprocess. Therefore, the optimization of the mechanical millingprocess is a complex process. But the design experience ofengineers show that some parameters have to be consideredsimultaneously when the milling machine is used to reduce aspecific material into the nanometer size particles. For example, theparameters of mill size, mill material and mill geometry are alwaysconsidered simultaneously in designing a mill. In order to reducethe complexity of themilling parameters, themill size,millmaterialand mill geometry are fixed and not considered as the controllable

Table 1Factors and levels (%: the percentage of mill volume)

Factors Levels

Level 1 Level 2 Level 3

A Milling time 2(H) 5(H) 8(H)B Flow velocity of circulation system 1 (L/min) 2 (L/min) 3 (L/min)C Rotation velocity of agitator shaft 1200 (rpm) 1800(rpm) 2400(rpm)D Solute-to-solvent weight ratio 1 wt.% 4 wt.% 7 wt.%E Filling ratio of grinding

media (ZrO2)5% 15% 25%

Fig. 2. The flow diagram of the proposed integrated approach.

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process parameters in this research. Themill size is set at 1000CC.,themill material is stainless steelmaterial and themill geometry is avertical cylinder. Similarly, the zircon dioxide (ZrO2) is used as thegrinding media material and the size of grinding media is set at500 μm. These two parameters are fixed and not considered as thecontrollable process parameters either. In addition, in order to keepthe nano-particles in the nanometer at room temperature, themilling temperature is set at 20–25 °C and not considered as acontrol parameter, and the type of dispersants is set to thephosphate. In short, the five controllable process parametersconsidered in this research include the milling time, flow velocityof circulation system, rotation velocity of agitator shaft, solute-to-solvent weight ratio and filling ratio of grinding media.

The choices of the levels on these five parameters are based onthe property of the milling process and the limit of the millingmachine. For the parameter of the milling time, the milling timeshould be limited at a reasonable range in order to achieve themilling efficiency and to decrease the degree of contamination.According to the engineers' experience, the reasonable range ofmilling time is between 2 and 8 hours. Therefore, the levels of themilling time are set at 2, 5 and 8 hours. For the parameter of theflow velocity of circulation system, the factor levels are set basedon themill size and storage tank size. In this research, the mill size

is 1000 CC and the storage tank size is 3000 CC. The maximumvolume for the colloidal solution is 4000CC. Therefore, the levelsof the flow velocity of circulation system are set at 1, 2 and 3 (L/min). For the parameter of rotation velocity of agitator shaft, thefactor levels are set based on the limit of the power motor andthe abrasion of the vane agitator. In this research, the limit of therotation velocity of the power motor is 2400 rpm. Therefore, thelevels of the rotation velocity of agitator shaft are set at 1200, 1800and 2400 rpm. For the parameter of the solute-to-solvent weightratio, the factor levels are set based on the processed material andits corresponding solvent (e.g. TiO2 and glycol). According to theengineers' design experience, if the consistency of the colloidalsolution is too high, it will cause the milling equipment to shutdown. Therefore, the levels of the solute-to-solvent weight ratioare set at 1, 4 and 7 (wt.%). For the parameter of the filling ratio ofgrinding media, the factor levels are set based on the mill size.Since themill volume size is 1000CC., the filling ratio of grindingmedia should be set at a reasonable range in order to achieve themilling efficiency. According the engineers' design experience,the levels of the filling ratio of grinding media are set at 5, 15 and25 percentage of the mill volume.

The five process parameters, each with three levels, andsome interactions between process parameters are investigatedin this research. The five process parameters and their factorlevels are summarized in Table 1. The interest output responsesof the nano-particle milling process are the mean grain size andthe variance of grain size. Both of the output responses need tobe kept to a minimum.

In order to save on experimental costs and time, the orthogonalarray (OA) experiment rather than a full factorial experimentdesign is applied to obtain the response measurements of themilling process. The selection of an OA to be used depends on thedegree of freedom of the factors and interactions. In this research,five process parameters, eachwith three levels, and the interactionsbetween milling time (A) and rotation velocity of agitator (C),milling time (A) and number of grinding balls (E) are investigated.The calculations of the degrees of freedom are shown as follows:

The degrees of freedom of the main factors= (3−1)⁎5=10;The degrees of freedom of the interactions=(3−1)⁎ (3−1)⁎2=8.

The summation of the degrees of freedom from the mainfactors and interactions is 18. This means we have to choose anL18 OA at least. However, all of the parameters have threelevels, the OA must be a cube of 3 (e.g. L9, L27, L81). Therefore

Table 2The L27 (3

13) orthogonal array and output responses

No Process parameters Outputresponses

A C A⁎C A⁎C E A⁎E A⁎E B e e D e e Mean(grainsize)

Variance(grainsize)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 232.7 8.52 1 1 1 1 2 2 2 2 2 2 2 2 2 182.5 21.13 1 1 1 1 3 3 3 3 3 3 3 3 3 154.0 5.74 1 2 2 2 1 1 1 2 2 2 3 3 3 181.0 0.85 1 2 2 2 2 2 2 3 3 3 1 1 1 159.2 1.26 1 2 2 2 3 3 3 1 1 1 2 2 2 224.8 8.47 1 3 3 3 1 1 1 3 3 3 2 2 2 169.1 5.18 1 3 3 3 2 2 2 1 1 1 3 3 3 171.8 5.99 1 3 3 3 3 3 3 2 2 2 1 1 1 207.2 6.610 2 1 2 3 1 2 3 1 2 3 1 2 3 217.9 15.911 2 1 2 3 2 3 1 2 3 1 2 3 1 174.2 16.612 2 1 2 3 3 1 2 3 1 2 3 1 2 133.0 1.813 2 2 3 1 1 2 3 2 3 1 3 1 2 164.5 6.114 2 2 3 1 2 3 1 3 1 2 1 2 3 147.6 5.715 2 2 3 1 3 1 2 1 2 3 2 3 1 158.8 5.216 2 3 1 2 1 2 3 3 1 2 2 3 1 156.7 4.217 2 3 1 2 2 3 1 1 2 3 3 1 2 169.1 6.518 2 3 1 2 3 1 2 2 3 1 1 2 3 142.6 3.819 3 1 3 2 1 3 2 1 3 2 1 3 2 213.2 8.320 3 1 3 2 2 1 3 2 1 3 2 1 3 174.4 10.621 3 1 3 2 3 2 1 3 2 1 3 2 1 130.3 2.6

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an L27(313) OA is used to conduct the experiment in this

research. The L27(313) can lay out 27 trials, up to 13 factors in

columns, and 3 factor levels. The layout of the L27(313) OA is

shown in Table 2. It also shows the two output responses foreach experimental trial.

The response surface method (RSM) combined with theparameter design is a powerful optimization design tool in manyengineering applications. The RSM can be applied to obtain anapproximation for a response function in terms of predictorvariables. The linear and quadratic models are the mostcommon response models used to build the relationshipbetween the input parameters and output responses. They aregenerally stated as follows:

For the linear (1st order) model:

y ¼ b0 þ b1X1 þ b2X2 þ N þ bKXK þ e ð2Þwhere K is the number of variables, ε is the error item.

For the quadratic (2nd order) model:

(3)

y ¼ b0 þXK

i¼1

biXi þXK

i¼1

biiX2i

þXX

ibj

bijXiXj þ e

22 3 2 1 3 1 3 2 2 1 3 3 2 1 151.0 2.923 3 2 1 3 2 1 3 3 2 1 1 3 2 138.7 5.724 3 2 1 3 3 2 1 1 3 2 2 1 3 152.3 4.725 3 3 2 1 1 3 2 3 2 1 2 1 3 148.1 2.526 3 3 2 1 2 1 3 1 3 2 3 2 1 165.7 5.627 3 3 2 1 3 2 1 2 1 3 1 3 2 138.6 3.5

3.2. The integrated GA and RSM approach

In this research, an integration of GA and RSM is applied tofind the optimal process parameters in a nano-particle millingprocess. The RSM is used to establish the linear and nonlinearrelationships between the milling process parameters and theresponses. The GA approach is then applied to find the optimalprocess parameters using the RSM response models as thefitness function to measure the fitness value for the processparameters. The detailed procedures of the integrated GA andRSM approach are stated as follows:

Step 1. Encoding and generating the initial populationsThe process parameters are used to create the solutionspace for the GA approach. The searching ranges for thefive parameters are shown in Table 1. Followingnormalization, each searching range is divided into1000 equal intervals. The binary string representationfor coding space (chromosome) is adopted for this GAapproach and each input parameter is encoded into 10binary digits. For example, if X1=0.010, X2=0.036,X3=0.064, X4=0.256 and X5=0.065, then the chromo-some will be 0000001010,0000100100,0001000000,0100000000,0001000001. Following the encoding pro-cess, we randomly generate Npop chromosomes to findthe optimal process parameters (whereNpop is the numberof populations).

Step 2. Calculate the fitness valueThe RSM response functions are used as the fitnessfunction in the GA approach to calculate the fitnessvalues. In this research, the required qualities of the nano-particle milling process are that the mean grain size and

the variance of grain size of the nano-particle must bekept small. Therefore, these two output responses resultin a two-objective optimization problem, i.e., minimizingboth the mean grain size and the diameter variation.Because these two output responses have to be mini-mized, we can convert the multi-objective functions intoa single objective using two strategies. One strategy is tomerge these two output responses into one response byaggregating operator, and the other strategy is to combinethem into one response by using the statistical operator,coefficient of variance. For example, if the RSMmodel ofresponse 1 Y1 is and the RSM model of response 2 is, Y2then the final fitness function used to calculate the fitnessvalues by using the two strategies will be as follows:

Fitness function of the first strategy ¼ Y1 þ Y2

ð4Þ

Fitness function of the second strategy

¼ variancemean

¼ Y2Y1

ð5Þ

Step 3. SelectionSelect a pair of chromosome for mating, e.g. crossoverand mutation, from the current population. Parentchromosomes are selected with a probability related to

Table 3The ANOVA table of mean grain size

Source Sum of square Degrees of freedom Mean square F0

A 1670.8 2 835.4 117.00⁎

B 5613.2 2 2806.6 393.10⁎

C 2650.8 2 1325.4 185.70⁎

D 936.43 2 468.2 65.59⁎

E 4780.6 2 2390.3 334.80⁎

A⁎C 65.21 4 16.3 2.28A⁎E 143.97 4 35.99 5.04⁎

Error 57.11 8 7.14Total 15918 26

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their fitness. Highly fit chromosomes have a higherprobability of being selected for mating. The roulettewheel method is applied to the chromosome selection inthis research.

Step 4. CrossoverThe crossover operator is used to create a pair ofoffspring chromosomes. For each selected pair, a two-cut-point crossover operation is applied to generate anoffspring with the crossover probability Pc.

Step 5. MutationIn this study, the one-gene mutation operation with apreset mutation probability Pm is applied to generatenew chromosomes. By using the crossover andmutationoperations, new Npop(offspring) populations are created.

Replace the initial chromosome populations with the newNpop(offspring) populations and repeat Steps 2 3 4 5 until thetermination criterion is satisfied.

4. Results and discussions

4.1. Results of the orthogonal array experiment and RSM

Results of the orthogonal array experiment are shown inTable 2. The mean grain size and the variance of grain size are the

Fig. 3. The measured results of mean grain size and variance of grain size for the N

measurements from the Coulter Multisizer equipment. In eachexperiment, a sample of colloidal solution is randomly sampledand put into the Coulter Multisizer equipment to measure themean grain size and the variance of grain size. For the sample ofcolloidal solution in the Coulter Multisizer equipment, themeasurement of grain size is repeated five times and then themean and variance are calculated from these 5 measurements. Forexample, the No. 21 output response of the orthogonal arrayexperiment (shown in Table 2) is the measured result using theCoulter Multisizer equipment which is shown in Fig. 3.

Each experimental trial results in one measurement. Due to thelimitation of experimental cost, one trial was conducted for each

o. 21 orthogonal array experiment by using the Coulter Multisizer equipment.

Table 4The ANOVA table of variance of grain size

Source Sum of square Degrees of freedom Mean square F0

A 21.49 2 10.745 1.64B 85.06 2 42.53 6.51⁎

C 193.86 2 96.93 14.84⁎

D 81.92 2 40.96 6.27⁎

E 94.63 2 47.315 7.24⁎

A⁎C 33.22 4 8.305 1.27A⁎E 27.86 4 6.965 1.07Error 52.26 8 6.532Total 590.3 26

Fig. 4. Main effect plot of response 1.

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experiment. Analysis of variance (ANOVA) is then applied todetermine the significant parameters and the main effect plot isused to set the optimal level for each parameter in the nano-particle milling process. The MINITAB 13.0 software is used toanalyze the experimental data. Table 3 shows the ANOVA resultswith respect to the “mean grain size”. Because the F0.05(2,8)=4.46 and F0.05(4,8)=3.84, it is obvious to conclude that millingtime (A), flow velocity of circulation system (B), rotation velocityof agitator (C), solute-to-solvent weight ratio (D), number ofgrinding balls (E) and interaction between milling time andnumber of grinding balls (A⁎E) significantly affect the “meangrain size” with 95% confidence intervals. Based on the maineffect plot, shown in Fig. 4, the optimal level of each parameter isset at A3B3C3D3E3 for the “mean grain size” response.

Similarly, Table 4 shows the ANOVA results with respect to“variance of grain size”. It can be clearly seen that flow velocityof circulation system (B), rotation velocity of agitator (C ),solute-to-solvent weight ratio (D) and number of grinding balls(E) have significant effects on the “variance of grain size”response. Based on the main effect plot, shown in Fig. 5, theoptimal level of each parameter is set at A3B3C2D3E3.

Moreover, RSM is used to build the relationship between theprocess parameters and output responses. Both the 1st ordermodel and the 2nd order model of RSM are conducted using theDesign Expert version 6.0 software. The results are stated asfollows:

(1) Mean grain size (Response 1):For the 1st order model (Linear, R2 =0.929, AdjustedR2 =0.877):

Y1 ¼ 279:23−3:14X1−17:58X2−0:018X3−2:11X4−1:63X5

ð6ÞFor the 2nd order model (Quadratic, R2 =0.986, AdjustedR2 =0.907):

Y1 ¼ 380:99−7:15X1−17:58X2−0:12X3−9:67X4−1:63X5

þ 0:4X 21 þ 0:0000284X 2

3 þ 0:91X 24

ð7Þ

(2) Variance of grain size (Response 2):For the 1st order model (Linear, R2 =0.454, AdjustedR2 =0.324):

Y2 ¼ 14:26−0:0046X3 ð8Þ

For the 2nd order model (Quadratic, R2 =0.826, AdjustedR2 =0.624):

Y2 ¼ 36:25−1:8X2 þ 0:035X3 þ 3:66X4 þ 1:0X5

þ 0:0000085X 23 −0:5X

24 −0:036X

25 ð9Þ

It is found that the 2nd order models yield a larger R2 valuethan the 1st order models in both responses. Therefore, we caninfer that the relationships between the process parametersand output quality responses are nonlinear for both the meangrain size and the variance of grain size. The 2nd order modelresponses, Eqs. (7) and (9), are used as the fitness function tomeasure the fitness value in the GA approach. In addition, the2nd order model responses, Eqs. (7) and (9), can be used totest the sensibility of the process parameter for the mean ofgrain size and the variance of grain size respectively. Let ustake the Eq. (7) as an example, the coefficient of X2 (flowvelocity of circulation system) is −17.58, it means if weincrease one unit flow velocity of circulation system, thecorresponding response Y1, the mean grain size of nano-

Fig. 5. Main effect plot of response 2.

Table 5The best solutions found by the integrated RSM and GA approach

Parameter Response

X1 (A) X2 (B) X3 (C) X4 (D) X5 (E) Y1 Y2

Strategy 1 6.42 2.97 1793 6.96 24.86 109.55 0.24Strategy 2 6.27 2.96 1968 5.80 24.65 108.13 0.54

161T.-H. Hou et al. / Powder Technology 173 (2007) 153–162

particles, will reduce 17.58 nm. Similarly, the coefficients ofX1, X3, X4 and X5 can be used to test the sensitivity for eachprocess parameter for the mean of grain size using Eq. (7).Similarly, the Eq. (9) can be used to test the sensibility of theprocess parameters, X2, X3, X4 and X5, for the variance ofgrain size.

4.2. Optimal parameter setting using integrated GA and RSMapproach

The MATLAB 7.0 software is applied to develop theintegrated GA and RSM approach and determine the optimalparameters in the nano-particle milling process. The 2nd ordermodels of responses are used as a fitness function to measurethe fitness value in the GA approach. In this research, the tworesponse models are transformed into one single model. Onetransformation only adds the two models together and the othertransformation uses the coefficient of variance. By using thesetwo transformations, the new fitness functions are defined asfollows:

For the fitness function 1 which uses the “add together”transformation:

F1 ¼ 417:24−7:15X1−19:38X2−0:085X3−6:01X4−0:63X5

þ 0:4X 21 þ 0:0000396X 2

3 −0:41X24 −0:036X

25

ð10ÞFor the fitness function 2 which uses the “coefficient of

variance” transformation

F2 ¼ Y2Y1

¼ Eq:ð9ÞEq: ð7Þ ð11Þ

These two new fitness functions are then used to measure thefitness value for each chromosome in the GA procedure. InGA's procedure, the population size is set at Npop=30. Two-cut-point crossover is employed with the crossover probability0.98. The one-gene mutation operation probability is specifiedas Pm=0.015 and the iteration number is 2000. Under theseconditions, the best solutions are summarized in Table 5.

Table 5 shows the five process parameter values and theircorresponding output responses for the two fitness functiontransformations. The mean grain size and variances of grain sizeof the optimal parameters are clearly smaller than the values inTable 2. In addition, the obtained mean grain size is close to100 nm. Therefore, we can conclude that the proposed approachdoes find the optimal process parameters that result insatisfactory output responses. Moreover, the obtained parametervalues are continuous values. In the comparison of the optimalprocess parameters derived from the optimization procedurewith the previously obtained optimal level form the main effectplots, the obtained A parameter value falls between level 2 andlevel 3. The B parameter value is closed to level 3. The Cparameter is closed to level 2 for the fitness transformationstrategy 1 and falls between level 2 and level 3 for the fitnesstransformation strategy 2. The D and E parameter values areclosed to level 3. The results are similar to the result of the maineffect plot. However, the proposed approach obtains moreprecise parameter values than the main effect plot. It is alsoclearly found from Table 5 that different transformationstrategies of GA's fitness function result in different optimalsolutions. Using the coefficient of variance fitness functionresults in a better optimal solution in mean grain size than theaggregating strategy, but it results in a worse solution invariance of grain size than the aggregating strategy. Althoughthese two output responses have to be minimized, they areconflict. According to the engineering practices in producingnano-particles, the response of mean grain size is moreimportant than the response of variance of grain size. Therefore,the optimal process parameters of the wet-type nano-particlemilling process are set as follows: milling time (A)=6.27, flowvelocity of circulation system (B)=2.96, rotation velocity of

162 T.-H. Hou et al. / Powder Technology 173 (2007) 153–162

agitator (C)=1968, solute-to-solvent weight ratio (D)=5.8, andnumber of grinding balls (E)=24.65.

5. Conclusions

Using a wet-type mechanical milling process to producenano-particles is a popular method because of the applicabilityto all classes of materials. The required qualities of the millingprocess are that the grain size and the variance of grain size ofthe nano-particle must be kept small. There are severalparameters that may affect these qualities. However, the effectsof these parameters on the output quality responses in the nano-particle milling process remain unknown. Moreover, it is alsovery important to improve the milling process capability andquality by setting optimal milling parameters in the nano-particle milling process.

In this research, the parameter design of the Taguchi method,response surface method (RSM) and genetic algorithm (GA) areintegrated and applied to set the optimal parameters for a nano-particle milling process. The orthogonal array experiment isconducted to economically obtain the response measurementsand analysis of variance (ANOVA) and main effect plot are usedto determine the significant parameters and set the optimal levelfor each parameter. The RSM is then used to build therelationship between the input parameters and output responses,and used as the fitness function to measure the fitness value inthe GA approach. Finally, GA is applied to find the optimalparameters for the nano-particle milling process. In thisresearch, there are two output responses and therefore, thereare two response models. The two response models areconverted into a single fitness function. The experimentalresults show that these two transformations can obtain similaroptimal parameters in a nano-particle milling process. However,this research also finds that there are conflicts between the twooutput responses. The optimal process parameters that result ina smaller mean grain size may unfortunately result in a largervariance of grain size. Therefore, multi-objective optimizationtechniques may be applied to solve this problem in futureresearch.

Acknowledgements

This research was partially supported by the PrecisionMachinery Research and Development Center (PMC) and theNational Science Council of the Republic of China undergrant number NSC 93-2218-E-224-002 and NSC 94-2218-E-224-001. The authors also like to show their appreciation tothe anonymous reviewers very much for their valuable

comments that are very helpful for the improvement of thispaper.

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