optimization of edm by sa_ann method

7/28/2019 Optimization of EDM by SA_ANN Method

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Optimization of the EDM process parameters using a multi-objective

simulated annealing - based back propagation neuralnetwork

Journal: International Journal of Production Research

Manuscript ID: TPRS-2008-IJPR-1002

Manuscript Type: Original Manuscript

Date Submitted by theAuthor: 11-Dec-2008

Complete List of Authors: Laha, Dipak; Mechnical Engineering DepartmentBanerjee, Simul; Mechanical EngineeringPal, Amit; Mechanical Engineering

Keywords:META-HEURISTICS, MULTI-CRITERIA DECISION MAKING, NEURALNETWORK APPLICATIONS, SIMULATED ANNEALING

Keywords (user):electrical discharge machining (EDM), optimization, multi-objectivesimulated annealing (MOSA), back propagation neural network

http://mc.manuscriptcentral.com/tprs Email: [email protected]

International Journal of Production Research


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simulated annealing - based back propagation neural network

Amit Kumar Pal, Dipak Laha*, Simul Banerjee

Department of Mechanical Engineering

Jadavpur University, Kolkata 700032, India

*Communicating author; e-mail: [email protected]; Phone / Fax: + 91 33

2414 6890

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simulated annealing - based back propagation neural network

Abstract

This paper attempts to optimize the response parameters, namely, material removal rate

(MRR) and centre line average value of surface roughness (Ra), of the electrical

discharge machining (EDM) process using a multi-objective simulated annealing - based

neural network. In this study, the back propagation neural network is selected for the

purpose of modeling due to its good learning ability and to predict the relationships

between the input and the output variables of the highly complex and stochastic EDM

process. A multi-objective simulated annealing algorithm is subsequently applied to the

developed model for searching of a Pareto optimal solution set. Based on this set a

process engineer can take decision regarding the optimal setting of the process

parameters for a specific need-based requirement.

Keywords: electrical discharge machining (EDM), optimization, multi-objective

simulated annealing (MOSA), back propagation neural network (BPNN).

*Communicating author; e-mail: [email protected]; Phone / Fax: + 91 33 2414 6890

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mailto:[email protected]:[email protected]


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Introduction

Electrical discharge machining (EDM) process is extensively used in the machining

industry for the manufacture of mould, die, automotive, aerospace and surgical

component. The material is removed in this process by the erosive action of spatially

discrete and chaotic [1] high-frequency electrical discharges (sparks) of high power

density between a tool electrode and the work-piece electrode with a dielectric fluid in

the gap between them. The application of dielectric fluid makes it possible to flush away

eroded particles from the gap and cool it. The process has the capacity of producing

complex three-dimensional shapes on any material regardless of its hardness, strength

and toughness provided its electrical resistivity is not more than 100 ohm-cm [2]. This is

a costly process and optimal selection of process parameters is very much essential to

meet the specific requirement of process engineer for removal rate and the desired

surface integrity of work-piece. These performance parameters, however, are conflicting

in nature.

In the EDM process, material removal rate (MRR), centre line average value of

surface roughness (Ra) and fractal dimension (Rf) are the important response parameters

to evaluate the machining performance. Since the EDM process is complex and

stochastic in nature, modeling and determining the optimal machining performance is

reasonably difficult.

However, based on modeling and optimization, several researchers have proposed

various methodologies for improving the performance of the EDM process. Back

propagation neural network (BPNN) [3] and response surface methodology have been

successfully used to model and predict the complex relationships between the EDM

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process parameters and its machining performances [4]. It is revealed from the literature

[5-8] that among the optimization techniques so far applied to the EDM process, the non-

dominating sorting genetic algorithm (NSGA-II) [9] is one of the most widely used

methods for generating a Pareto optimal solution set to resolve the multi-objective

optimization problem in EDM.

Mandal et al. [5] used artificial neural network (ANN) and non-dominating sorting

genetic algorithm II (NSGA-II) to model and optimize machining parameters in EDM

process considering material removal rate and absolute tool wear rate as cutting

parameters. The ANN model has been trained based on experimental data. The NSGA-II

with multi-objective functions was adopted to neural network to obtain a Pareto optimal

solution set of response parameters.

Yuan et al. [6] proposed a predictive reliability multi-objective optimization

procedure based on Gaussian process regression (GPR), in an attempt to optimize the

high-speed wire-cut EDM process (WEDM-HS). Material removal rate and surface

roughness were taken as response parameters. In order to obtain an accurate estimation,

the non-liner electrical discharging and thermal erosion process of the WEDM-HS with

measurement noise is identified by the multiple GPR models. The experimental results

reveal that GPR model is better than other regressive models with respect to the model

accuracy, feature scaling and probabilistic variance.

Optimization of machining parameters in WEDM using NSGA has been studied by

Kuriakose and Shunmugam [7]. They considered surface roughness and cutting speed as

the output parameters. A multiple regression model is used to represent the relationship

between input and output parameters. NSGA is then applied to optimize the WEDM

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process. Thirty-six non-dominated solutions, thus obtained, have been reported in this

study.

Kanagarajan et al. [8] used NSGA-II to optimize the machining parameters of the

EDM process with WC/Co composites considering the input parameters as pulse current,

pulse on time, electrode rotation and flushing pressure. Based on experimental data, a

second order polynomial regression equation is developed for predicting different process

characteristics. The NSGA-II was applied to the regression equation to optimize the

processing conditions. Finally, a non-dominated solution set has been proposed.

However, to the best of the authors knowledge, no literature is available where

multi-objective simulated annealing (MOSA) was applied to generate a pareto-optimal

solution set for optimizing response parameters in electrical discharge machining process.

It is to be noted here that a pareto-optimal solution set assists a process engineer to select

the optimal parameter setting for a specific need-based requirement.

In the present study, therefore, the electrical discharge machining process is modeled

using back propagation neural network (BPNN) procedure and optimized by multi-

objective simulated annealing (MOSA). The back propagation neural network (BPNN)

model is developed with current setting, pulse-on time and pulse-off time as the input

process parameters to predict material removal rate (MRR), centre line average value of

surface roughness (Ra) and fractal dimension (Rf) of the work-piece. MOSA is then

applied to the trained back propagation neural network model to search for a set of Pareto

optimal solutions for two conflicting responses; material removal rate and surface finish

of the work-piece machined by EDM process.

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2. The BPNN for modeling the EDM process

The feed forward BPNN is used in the present work to predict the EDM process outputs,

namely, MRR, Ra, and Rf taking a set of input process parameters consisting of current

setting (C), pulse-on time (Ton) and pulse-off time (Toff). The neural network, selected in

the present study, is comprised with a chosen input layer, one or more hidden layers (to

be decided with the number of neurons during the process of modeling) and a chosen

output layer as shown in Figure 1.

[Insert Figure 1 here]

Each neuron in the input layer contains specific information (e.g. current setting) to

receive the input data. Lines joining each of the neurons from the input layer to the

hidden layer and to the output layer denote the connection weights that are used to

associate the synaptic strength (activation level) of the neurons connecting one layer to

the next. A neuron may or may not be active depending upon the activation levels of the

neurons in the previous layer. The neuron-bias at each input and hidden layers allow for a

more rapid convergence of the learning process. The hidden layer consists of neurons and

the weight values are modified as the network is getting trained. The output layer consists

of three neurons as three of the output parameters of the EDM process are considered.

The neural network predicts these output parameters using back propagation algorithm

based on gradient descent technique with backward error correction during learning.

2.1. The back propagation learning method

The feed forward back propagation algorithm is used to adjust the connection weights in

the network so that completely new information in the input layer will minimize the

objective function (sum-of -squares error). This error is simply the difference between the

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actual output and the desired output (e.g., a known MRR) value. The error is calculated

by training the network in the forward pass from the input layer, to the hidden layer, and

finally, to the output layer and this error is propagated backward from the output layer to

the hidden layer and then the input layer for adjusting the weights by means of the

gradient descent algorithm. In the training process, the weights are initially randomized

and then adjusted using the delta rule. As the number of iterations for training the

network increases this error is gradually decreasing and finally, the network converges to

a steady state of weights between any two consecutive layers with a minimum

corresponding error.

During the forward pass, the EDM process information enters the input layer through

neuron in where n = 1, 2, , k, k being the number of neurons in the input layer. Each of

the input process parameters is applied to each neuron in the input layer. A variable, Y, is

calculated for the input to each neuron in the hidden layer by summing up the product of

input vectors supplied to the input layer neurons and the corresponding weights

associated with the hidden neurons.

For instance, the variable Y for the first neuron in the hidden layer is calculated

as h1biasnnh1k

1nh1 wxwY

=+= .(1)

where, xn = the input vector supplied to nth input neuron, wnh1 = the weight connected

between nth neuron in the input layer and h1th neuron in the hidden layer, wbias-h1 = bias

at the h1th neuron in the hidden layer.

A transfer function (sigmoid) is used to transfer the combined effects into an output

signal represented by variable OUTh1 for each neuron in the hidden layer. The output

value of OUTh1 of the activation function is the input of the neuron in the consecutive

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layer. OUTh1 for each neuron in the hidden layer is then multiplied by its weight

connecting to the output layer, and these products are then summed to produce in Yo for

the lone neuron in the output layer. Applying the same logic, an OUT o for this neuron is

obtained.

.(2)....................)exp(-Y)exp(Y

)exp(-Y-)exp(Y)Ytanh()(YfOUT

h1h1

1hh11hh1h1 +===

The activation function is used to check whether the neuron is activated or non-

activated to show its contributory effects on the consecutive layers neurons. Usually, the

activation function of a neuron is represented by a sigmoid function. Since it is

continuous, nonlinear, and easily differentiable, it can be suitably applied to the modeling

of a complex system such as the EDM process. However, the activation function,

hyperbolic tangent, is considered here to obtain the output of the neuron lying between

two extreme saturated portions of the hyperbolic tangent curve; namely, -1 and 1,

representing the state of its activation level and contributes to the next layers neurons.

The backpropagation algorithm uses the gradient descent search method to minimize

an objective function (here, the error function). The most popular error function used, i.e.,

mean squared error (MSE) is calculated using the following formula:

( ) )3(OUTT2

1

m

1

N

1MSE

2

pjpj

m

1j

N

1p

===

where, N is the number of training patterns, Tpj is the desired output value of the jth

neuron for pattern p, OUTpj is the actual output of the jth neuron, and m is the number of

neurons in the output layer. Batch mode of training has been used for the training of the

network. The back propagation algorithm uses the following delta rule to minimize the

error function E:

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[ ] [ ] ....(4)....................wwE

w kjiji

1kji +

=+

where, [wji]k+1 is the adjustment of weight for each weight between neurons i and j

for the (k+1)th iteration, wji is the weight connected between neurons i and j, is the

learning rate, and is the momentum coefficient.

The constants and are chosen between 0 and 1 in a manner that optimizes

learning speed and accuracy. The learning rate, , dictates how greatly the weights are

varied, whereas the momentum coefficient, , filters out high frequency changes in w

[10].

3. Experimentation and the neural network modeling

The first objective of the present investigation is to develop a model of the electric

discharge machining process for predicting material removal rate, surface roughness and

fractal dimension using artificial neural network. The input variables chosen in this study

are the current setting, the pulse on time and the pulse off time in the roughing and semi-

finishing region. A full factorial (4x4x4) design is selected for conducting the

experiments for this purpose.

The equipment used to perform the experiments is an EDM machine (Tool Craft A25

EDM Machine). The machine operates with commercially available kerosene oil as

dielectric medium and an open circuit voltage of 70 volts. High speed steel of

specification C-0.80%, W-6%, Mo-5%, Cr-4%, V-2% equivalent to grade M2 was

chosen as the work-piece material. The density of material is 8144 kg/m3. The tool

material selected is electrolytic copper [11] with density 8904 kg/m3

and has circular

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cross section of 14 mm. The polarity of the tool electrode is set as positive while that of

work-piece as negative.

For the purpose of determining the material removal rate the mass is measured by

DHONA Instrument Calcutta, Model No. DND 200. It is then divided by the density of

work-piece material in order to convert it into volumetric term and is further divided by

the actual machining time to obtain the material removal rate in terms of mm3/min. The

surface roughness of the machined work piece and the fractal dimension are then

evaluated by the Taylor Hobson Precision Surtronis 3+

Roughness Checker. Here a

sample length of 4 mm was taken and stylus tip radius of 5 m was used. The value of

surface roughness parameter Ra in micron and fractal dimension for each experiment was

obtained directly from the Taly-profile software integrated with the machine. The

arithmetic mean of the values of the measurements taken along three mutually 1200 apart

directions over the area subjected to the EDM process was taken as the representative

value of Ra and fractal dimension.

Data set (60 of 68 out of which 64 data is based on the full factorial design) as shown

in Table 1 containing input and output parameters of the EDM process, is presented for

the training of the network. The rest data set shown in Table 2 is used for testing the

accuracy of training of the network model. Usually, it is desirable to consider the input

parameters small in magnitude for the network otherwise they will swamp the activation

function and make the learning very difficult [12]. It is, therefore, necessary to scale the

input values so that they lie approximately between 1.0 and 1.0 as referred in Table 3.

The initial weight values connected between two neurons in two consecutive layers have

been chosen randomly between 1.0 and 1.0.

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[Insert Tables 1, 2, 3 and 4 here]

Neural networks varying the learning rate, the momentum coefficient, number of

hidden layers and number of hidden neurons in each hidden layer were trained to

investigate the effect of these parameters on the overall performance of the EDM model.

The performance of different network architectures are studied with the learning rate and

the momentum coefficient both varying in the range 0.5 to 0.9. The number of iterations

for training each of the networks is chosen as 5000 for the purpose of drawing a

comparison among the different architectures. A program was written in the MATLAB

environment to simulate the BPNN and was run on Pentium 4, 2.2 GHz Computer. Some

typical results are shown is Table 4. Figures 2 through 4 show the performance of the

single layer and the double layer architectures with respect to the error function (equation

3). It is observed from the results of Table 4 and Figures 2 through 4 that the network

architecture with 3 9 9 3 yields the least sum-of-squares error for the learning rate

of 0.9 and the momentum coefficient of 0.5 and therefore, is selected for modeling.

[Insert Figures 2 - 4 here]

Table 5 summarizes results of the performance of the 3 9 9 3 back propagation

neural network to compare with the experimental results as referred in Table 2. For the

purpose of testing the network, the absolute percent error (APE) is considered, as it is

perhaps the easiest measure to interpret and does not depend on the magnitude of the item

(e.g., MRR) being predicted. It is calculated as

APE = 100*valueactual

valuepredictedvalueactual ..(5)

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[Insert Table 5 here]

Comparing with the experimental results (Table 2) as referred in Table 5 it is

observed that the 3 9 9 3 back propagation neural network performs quite well with

the mean absolute percent error of 4.13 for the MRR, 4.12 for the Ra and 1.63 for the Rf.

Figures 5 through 7 show the comparison of the absolute actual values of the

experimental results and the absolute predicted values using the neural network for the

outputs MRR, Ra, and the Rf respectively. The proposed BPNN model, therefore,

performs comparable to the experimental results by learning the unknown correlation

between the input and the output parameters of the highly complex and stochastic EDM

process and selected further for multi-objective optimization by simulated annealing and

prediction of pareto optimal solution set to fulfill different specific need-based

requirements.

[Insert Figures 5 - 7 here]

4. Multi-objective optimization for the EDM process

The EDM process can be considered as a system with multiple inputs and multiple

outputs. Since EDM is a complex process and the outputs are conflicting in nature, it is

desirable to optimize multiple outputs simultaneously rather than single output to obtain

the best machining performance. Usually, it is observed that as the MRR increases the R a

also increases. However, the optimum machining performance depends on the higher

value of MRR and lower value of Ra. In the present study, these two conflicting process

outputs are considered, as because higher MRR results in higher productivity, whereas

lower Ra leads to the better surface characteristics. So, the multi-objective optimization

problem takes the following form:

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( ){ }xx 21 f),(fMinimize

where, MRR

1)(f1 =x

aR)(

2f =x where, x is a vector of input parameters.

Let x0, x1, x2S and S be a feasible solution space. x0 is called optimal solution [13] in

the minimization problem if the following two conditions hold.

[1]If fi (x2) < fi (x2), for all i {1, 2,, k}

and fi (x2) < fi (x1) for at least one i {1, 2,., k} then x2 is said to dominate x1 .

2. If there is no x S so that x dominates xo, then xo is the Pareto optimal solution.

So, if a set of input parameters or vector x/ is not dominated by other sets of input

vectors (x) of the multi-objective optimization problem, then the solution x/ is called a

non-dominated solution or Pareto optimal solution. The set of non-dominated solutions of

the multi-objective problem is also referred to as Pareto optimal solutions. The proposed

MOSA based neural network tries to determine a set of non-dominated or Pareto optimal

solutions for the EDM process that considers three input parameters (within working

conditions) and two output functions.

5. SA algorithm

Simulated annealing (SA) of Metropolis et al.[14] and Kirkpatrick et al.[15] is a

metaheuristic based computational search process that has been found effective for

solving various optimization problems. It is based on the work of Metropolis et al.[14] in

statistical mechanics. It simulates the annealing of physical system by controlling a

temperature parameter following the Boltzmann probability distribution. Kirkpatrick et

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al. [15] pointed out the relevance of simulated annealing in optimization problems. Van

Laarhoven and Aarts [16] have reviewed a wide variety of applications of the SA. It

attempts to overcome the disadvantage of the gradient descent method. The unique

characteristic of the SA is that it considers the probability of accepting a worse point by

means of uphill move to approach global optimization instead of getting trapped in a

local optimum. At the beginning of the SA run, the probability of accepting an inferior

solution is high. In order to have high probability, a control parameter, called the

temperature (analogous to the temperature of the physical process) is kept high initially.

As the number of SA run increases, the probability is reduced due to the gradual

reduction of temperature according to a cooling (annealing) schedule.

5.1 Marching procedure

For the multi-objective optimization of the EDM process outputs, the SA algorithm

begins with the randomly generated initial point (solution) x1 (C1, Ton1, Tof1), known as

the current solution and a high temperature T1. An adjacent solution x

2(C

2, T

on2, T

of2) in

the neighborhood of the current solution is then generated using the normal distribution.

The response f(x1) and f(x2) are evaluated using a well-trained back propagation neural

network from the control points x1 and x2 respectively. The difference in the function

values at these two solution points (E = f(x2) f(x1)) is calculated. IfE < 0, i. e., the

neighbor is found to be better than the current point and it is unconditionally accepted as

the new current point. Otherwise, it is rejected outright, but accepted with a probability

exp(-E/T). The neighborhood point x2 is generated randomly using the normal

distribution [17] as follows.

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,2

nr i

n

1i

+==

11112222xxxxxxxx

where, n = number of random numbers, ri = ith random number, and

6

valueimumminvalueimummaxparameteraofiancevar2 == .

The SA algorithm is described as follows:

Choose an initial solution x1 and a high initial temperature T1.

while not yet frozen perform loop k times

perform the following loop m times

generate randomly a neighboring point (x2) ofx1 using the normal distribution

and E = E (x2) E (x1).

if E < 0 (downhill move)

then set x1 = x2

if E 0 (uphill move)

then set x1 = x2 with probability e-E/T

set T = c * T (reduction in temperature) /* c is the cooling factor */

return the best point as the final solution (x1).

The procedure for implementing the SA algorithm is shown in Figure 8. The multi-

objective SA algorithm in the MATLAB environment was run on Pentium 4, 2.2 GHz

Computer. The parameters used in the SA algorithm are given as: T1 = 2C, k = 60, m =

20, and c = 0.95. In the present study, the geometric cooling schedule considered is T = c

* T as suggested by Collins et al. [18]. It is selected because of its simplicity, rapid

cooling and good performance in solving various optimization problems. The SA

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algorithm is then applied to the BPNN for searching for the Pareto optimal process

parameters with the optimal EDM process outputs.

[Insert Figure 8 here]

6. Pareto optimal solution set for the EDM process

Since the present study considers multi-objective optimization problems, in most cases,

there could be a number of optimal solutions as compared to obtaining an optimal

solution for the single-objective problems. Single optimal solution sometimes restricts the

choice of the users for setting the input control parameters. However, multiple optimal

solutions are desirable for setting the control parameters depending on the requirement of

the process engineers. Table 6 shows the initial optimal solution set of different process

parameters with the process outputs. Also, the optimal solution points for the EDM

process are given in Figure 9. Using MOSA-based back-propagation network technique,

46 initial optimal solution sets are obtained. Among the 46 solution sets over the feasible

region, only 19 solution sets are Pareto optimal solution, and none of these solutions is

better than any other solutions in the set and hence these solutions are called non-

dominated or Pareto optimal solutions. Figure 10 displays the Pareto optimal set for the

EDM process. Therefore, any of them can be considered as an alternative accepted

solution to meet the requirement of the desired EDM process performance.

[Insert Figures 9 and 10 here]

From the experimental results presented in Table 1, the input parameters of the

experiment number 8 gives the MRR of 9.0118 mm3/min and Ra of 8.26 m for the

EDM surface. Using the MOSA-based BPNN models, it can be observed from the serial

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number 1, Table 7 that the MRR could be increased to 13.553mm3/min even with the

lower value of Ra as 8.19 m. Also, from Pareto optimal solution set in Table 7, it can be

seen that there exists two more solution sets (serial no. 17 and 18) when values of MRR

are more than 9.0118mm3/min yet the resultant Ra values are considerably low.

[Insert Tables 6 and 7 here]

7. Conclusion

In this paper, an intelligent modeling and optimization for the EDM process using a

multi-objective simulated annealing-based back propagation neural network have been

proposed. The back propagation network have been trained to model and found effective

with the network architecture of 3-9-9-3 with the learning rate of 0.9 and the momentum

coefficient of 0.5. Finally, a set of Pareto optimal solutions, using multi-objective

simulated annealing based on the developed network, is obtained by optimizing both the

output measures of the EDM process, namely, MRR and Ra simultaneously.

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1

Figure 1. The backpropagation neural network for single hidden layer

Figure 2. Change in SSE with the number of hidden neurons for single layer

MRR

Ra

Rf

current

on-time

off-time

input layer hidden layer output layer

neuron : bias

neuron : bias

neuron h1

8 9 10 11 12 13 14 15 16 175

6

7

8

9

10

11

12

13x 10

-3

Hidden Neurons

SSE

Sl. No

1 0.7 0.9

2 0.6 0.93 0.9 0.9

4 0.5 0.95 0.7 0.96 0.8 0.9

7 0.9 0.9

8 0.7 0.99 0.6 0.9

1

2 34

5

67 8 9

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2

Figure 3. Change in SSE with the hidden neurons for double layer

Figure 4. Minimum value of SSE with respect to the single and double hidden layers

SSE

SSE

3 14 3

= 0.9

= 0.8

3 9 9 - 3

= 0.9

= 0.5

Sl. No

1 0.9 0.9

2 0.7 0.93 0.5 0.9

4 0.6 0.95 0.8 0.9

6 0.7 0.97 0.7 0.9

8 0.8 0.9

9 0.8 0.9

10 0.8 0.911 0.7 0.9

2

1

3 4

5

6 7

8 910

11

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0

2

4

6

8

10

12

14

1 2 3 4 5 6 7 8

Test pattern

MRR

(mm

3/min)

Experimental

Predicted

Figure 5. Comparison between the experimental and the predicted MRR values

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8Test pattern

Ra

(micron)

Experimetal

Predicted

Figure 6. Comparison between the experimental and the predicted Ra values

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1.18

1.2

1.22

1.24

1.26

1.28

1.3

1.32

1.34

1 2 3 4 5 6 7 8

Test Pattern

FractalDimension

Experimental

Predicted

Figure 7. Comparison between the experimental and the predicted Rf values

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Initialization: Set an initial solution x1S

(S is the feasible solution region), initial

temperature T1>0, number of iterations k,

epoch length m, cooling factor

Set=x1, T = T1

Randomly generate a neighborhood

solution x2 using normal distribution.

Evaluate f(x2) and f(x1) (Evaluate

these response values by the well-

trained back-propagation network).

Compute E = f(x2) f(x1)

E


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Pareto-optimal set for ANN

0

1

2

3

4

56

7

8

9

0 2 4 6 8 10 12 14

Material Removal Rate

SurfaceRough

nes

Figure 9. Optimal solution for the MOSA based BPN model

0 2 4 6 8 10 12 14

2

3

4

5

6

7

8

9

SurfaceRoughnes

s

MRR (mm3/min)

Figure 10: Pareto-optimal set

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Table 1. Experimental data for the training of the BPNN model

ExperimentNumber

C (A) Ton (s) Toff(s) MRR (mm3/min) Ra (m) Rf

1 3 50 50 0.3427 2.31 1.353

2 6 100 100 4.886 4.36 1.3033 12 200 200 11.1278 8.78 1.243

4 9 200 200 6.1896 6.12 1.2435 3 100 100 0.6497 2.41 1.35

6 9 100 100 8.1302 6.15 1.283

7 3 150 200 0.4738 2.77 1.323

8 12 150 200 9.0118 8.26 1.259 6 200 200 2.7525 5.25 1.247

10 9 50 200 3.0221 5.63 1.29

11 9 200 50 9.1232 5.34 1.24712 6 50 200 1.9181 4.87 1.317

13 3 200 100 0.5306 1.92 1.314 12 200 100 15.3242 9.07 1.2315 6 100 50 5.7718 5.02 1.3

16 12 50 200 4.8901 5.97 1.277

17 12 150 50 18.8113 7.45 1.24

18 3 200 50 0.3924 2.40 1.22719 9 200 100 9.4129 6.77 1.247

20 6 200 100 4.6276 5.21 1.277

21 12 200 50 17.2259 8.35 1.24322 12 200 150 15.3902 7.79 1.223

23 3 50 200 0.4229 2.78 1.263

24 9 100 50 11.4031 6.65 1.26325 6 150 50 5.1503 4.33 1.283

26 12 100 150 10.8249 7.28 1.263

27 12 100 50 14.1423 6.53 1.2828 9 150 200 5.0682 5.89 1.263

29 3 150 50 0.7899 2.48 1.33

30 6 200 50 4.1619 4.33 1.267

31 3 200 200 0.5608 2.15 1.29732 9 100 150 7.5987 6.24 1.27

33 6 50 150 2.6178 4.59 1.307

34 9 150 100 8.4306 6.46 1.24735 3 50 100 0.7731 2.57 1.357

36 9 100 200 5.0313 6.16 1.277

37 12 50 100 7.8381 6.62 1.28338 6 100 150 3.5916 4.81 1.297

39 3 100 150 0.7061 2.48 1.33

40 3 100 200 0.4599 2.58 1.333

41 12 100 200 7.1771 6.56 1.25742 9 50 150 5.108 5.83 1.297

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Table 1: Continued

ExperimentNumber


43 6 100 200 2.7543 5.03 1.283

44 3 150 100 0.6099 2.22 1.29745 3 200 150 0.6134 2.24 1.293

46 6 200 150 3.3422 4.75 1.2747 12 150 100 12.6351 6.51 1.26

48 9 50 50 6.5304 5.34 1.31

49 12 50 50 11.7172 5.65 1.3

50 9 200 150 7.8196 6.46 1.2551 12 150 150 12.3972 7.18 1.26

52 9 50 100 5.3188 5.88 1.293

53 3 100 50 0.9071 2.57 1.3654 6 50 50 3.7873 4.54 1.307

55 6 150 200 2.7029 4.99 1.27356 12 50 150 8.2535 5.77 1.357 3 50 150 0.8559 2.90 1.36

58 6 50 100 2.4466 4.54 1.303

59 6 150 100 3.7811 4.77 1.29

60 9 150 50 10.1742 6.29 1.257

Table 2. Experimental data for testing the BPNN model

Experiment

Number


1 9 150 150 8.3452 7.25 1.25

2 6 150 150 1.9007 4.88 1.27

3 12 100 100 12.7374 7.33 1.274 3 150 150 0.4239 2.22 1.323

5 3 200 75 0.7454 2.29 1.306 4.5 75 75 1.6699 3.40 1.32

7 4.5 200 50 1.594 3.11 1.308 10.5 50 50 11.4686 6.21 1.30

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Table 3. Factors and levels of the input parameters for the BPNN model

LevelsFactors

-1 -0.3333 +0.3333 +1

C (A) 3 6 9 12

Ton (s) 50 100 150 200

Toff (s) 50 100 150 200

Table 4. Results of different architectures of the BPNN model

Serial no. Network architecture SSE after 5000 iterations

1 3-8-3 0.9 0.7 0.01242 3-10-3 0.9 0.6 0.0088

3 3-11-3 0.9 0.9 0.00904 3-12-3 0.9 0.5 0.0087

5 3-13-3 0.9 0.7 0.0072

6 3-14-3 0.9 0.8 0.0053

7 3-15-3 0.9 0.9 0.00588 3-16-3 0.9 0.7 0.0057

9 3-17-3 0.9 0.6 0.0061

10 3-4-4-3 0.9 0.9 0.013111 3-7-7-3 0.9 0.7 0.0058

12 3-9-9-3 0.6 0.9 0.003313 3-9-9-3 0.7 0.6 0.003314 3-9-9-3 0.9 0.5 0.0030

15 3-9-9-3 0.8 0.5 0.0032

16 3-9-93 0.5 0.9 0.003617 3-10-10-3 0.9 0.6 0.0032

18 3-11-10-3 0.9 0.8 0.0034

19 3-10-9-3 0.9 0.7 0.0032

20 3-9-10-3 0.9 0.7 0.003221 3-8-8-3 0.9 0.8 0.0049

22 3-9-8-3 0.9 0.8 0.0050

23 3-6-7-3 0.9 0.8 0.005924 3-5-6-3 0.9 0.7 0.0079

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Table 5. Comparison between the experimental test data and the BPNN based predicted

data

Sl.

No

C

(A)

Ton

(s)

Toff

(s)

Exp.

MRR

(mm3/min)

Predicted

MRR by

BPN

(mm3/min)

%

APE

Exp.

Ra(m)

Predicted

Raby

BPN

(m)

%

APE

Exp.

Rf

Predicted

Rfby

BPN

%

APE

1. 9 150 150 8.3452 8.0635 3.37 7.25 7.48 3.17 1.25 1.24 0.8

2. 6 150 150 1.9007 1.9481 2.49 4.88 4.94 1.23 1.27 1.268 0.157

3. 12 100 100 12.7374 11.1985 12.1 7.33 7.02 4.23 1.27 1.26 0.78

4. 3 150 150 0.4239 0.4055 4.34 2.22 2.31 4.05 1.32 1.29 2.27

5. 3 200 75 0.7454 0.7185 3.61 2.29 2.37 3.49 1.30 1.26 3.07

6. 4.5 75 75 1.6699 1.7315 3.69 3.40 3.57 5.0 1.32 1.33 0.75

7. 4.5 200 50 1.5594 1.5515 0.51 3.43 3.11 9.33 1.30 1.24 4.62

8. 10.5 50 50 11.4686 11.8015 2.90 6.21 6.06 2.42 1.30 1.29 0.77

Average APE of predicted

MRR = 4.13 %

Average APE of

predicted Ra = 4.12 %

Average APE of

predicted fractal

dimension = 1.63 %

Overall average prediction error of the ANN model = 3.29 %

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Table 6. Non-dominated solution set for the MOSA - based BPNN model

Sl. No. C (A) Ton (s) Toff(s) MRR (mm3/min) Ra (m)

1. 11.91 198.76 199.48 11.2576 8.41

2. 10.79 189.24 145.97 13.1553 8.193. 7.75 192.53 75.87 8.2703 5.88

4. 4.82 169.87 82.18 1.325 3.07

5. 7.06 52.22 67.64 4.7733 5.55

6. 5.32 151.28 50.79 4.3255 3.87

7. 4 166.54 129.93 0.4683 2.44

8. 4.43 151.21 119.50 0.5588 2.78

9. 4.89 106.93 191.17 0.7943 3.34

10. 5.33 106.63 106.88 2.3576 3.96

11. 3.42 86.46 69.95 0.8460 2.72

12. 3.40 161.52 134.84 0.4203 2.32

13. 4.53 175.27 85.55 0.9466 2.73

14. 3.94 138.05 198.94 0.6013 2.83

15. 3.82 95.34 167.83 0.5034 2.7816. 4.99 118.45 157.76 0.7176 3.18

17. 5.21 136.68 98.97 1.3889 3.39

18. 4.40 155.26 71.98 1.0759 2.66

19. 4.01 56.02 101.86 1.0805 2.88

20. 3.92 184.96 188.77 0.5468 2.61

21. 5.92 82.9 100.87 3.6366 4.52

22. 4.81 74.20 51.17 2.0954 3.73

23. 3.23 72.04 138.67 0.5329 2.75

24. 7.59 186.52 181.01 4.6293 5.92

25. 6.99 129.23 100.20 6.8787 5.68

26. 7.86 117.25 82.24 8.3682 6.16

27. 10.11 69.97 55.81 13.0602 6.83

28. 9.97 112.79 67.16 11.2459 6.8729. 8.71 137.35 52.72 10.2723 6.22

30. 8.83 145.75 111.42 9.1282 6.91

31. 6.95 126.10 133.08 6.9803 6.99

32. 6.42 136.36 192.57 3.5165 5.22

33. 6.38 106.10 188.94 2.9994 5.21

34. 5.35 199.91 158.61 1.3788 3.52

35. 5.44 181.69 154.17 1.0787 3.47

36. 5.62 186.14 139.20 1.3225 3.57

37. 3.42 177.69 150.76 0.4221 2.31

38. 3.73 184.40 128.46 0.4664 2.29

39. 3.66 153.53 154.59 0.4517 2.51

40. 3.67 147.16 131.26 0.4489 2.49

41. 3.39 151.83 128.11 0.4230 2.35

42. 3.12 150.19 150.51 0.4119 2.34

43. 3.53 196.74 130.72 0.4840 2.26

44. 3.18 190.05 82.03 0.6253 2.30

45. 3.23 185.57 61.86 0.6253 2.31

46. 3.07 199.40 61.10 0.6428 2.31

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Table 7: Pareto optimal solution set for the MOSA-based BPNN model

Sl. No. C(A) Ton(s) Toff(s) MRR(mm3/min) Ra (m)

1 10.79 189.24 145.97 13.1553 8.19

2 7.75 192.53 75.87 8.2703 5.88

3 4.82 169.87 82.18 1.325 3.07

4 7.06 52.22 67.64 4.7733 5.55

5 4 166.54 129.93 0.4683 2.44

6 5.33 106.63 106.88 2.3576 3.96

7 3.42 86.46 69.95 0.8460 2.72

8 3.40 161.52 134.84 0.4203 2.32

9 4.53 175.27 85.55 0.9466 2.73

10 5.21 136.68 98.97 1.3889 3.39

11 4.01 56.02 101.86 1.0805 2.88

12 3.92 184.96 188.77 0.5468 2.61

13 5.92 82.9 100.87 3.6366 4.52

14 4.81 74.20 51.17 2.0954 3.73

15 6.99 129.23 100.20 6.8787 5.68

16 7.86 117.25 82.24 8.3682 6.16

17 9.97 112.79 67.16 11.2459 6.87

18 8.71 137.35 52.72 10.2723 6.22

19 3.39 151.83 128.11 0.4230 2.35

ge 31 of 31 International Journal of Production Research

optimization of edm by sa_ann method

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