Electronic discharge machining

8/13/2019 Electronic discharge machining

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CHAPTER-1

INTRODUCTION

1.1 Introduction

A number of computationally efficient optimization tools are being developed now a day, but the numerical tools for multi-criterion optimization problems have their own

significance. The present work aims beyond the mini-max approaches use for such

optimization problems. In order to ustify the related work, a machining method, known

as !lectric "ischarge #achining $!"#% process has been chosen. The work contained in

this minor proect report covers the relationships establishment between the various input

variables and parameters for the !"# machine. &egression analysis techni'ues have

been adopted to obtain the relationships. (nce the problem of !"# process has been

modeled, it has been tried to ustify the importance of bicriterion approaches in the

manufacturing environment. The results have been obtained for the given set of

constraints and the individual obective functions and are compared to those obtained by

using bicriterion approaches. The generalized genetic algorithms have been used for the

nonlinear obective functions for a given constraints in the form of crisp value bounded

domains. A brief overview of the !lectric discharge machining process is as under.

1.2 Basics of EDM

The use of thermoelectric source of energy in developing the non-traditional techni'ues

has greatly helped in achieving an economic machining of the extremely low

machinability materials and difficult obs. The process of material removal by a

controlled erosion through a series of electric sparks, commonly known as !"#, was

first started in )*+ in &. /hen a discharge takes place between two points of the

anode and cathode, the intense heat generated near the zone melts and evaporates the

materials in the sparking zone. 0or improving the effectiveness, the workpiece and the

tool are submerged in the "ielectric fluid. The basic !"# process has been shown in

0ig. ).). it has been observed that if both the electrodes are made of the same material,

the electrode connected to the positive terminal generally erodes at faster rate. 0or this

reason, the work piece is generally made the anode. A suitable gap, known as the spark

gap, is maintained between the tool and the workpiece surfaces. The sparks are made to

discharge at a high fre'uency with a suitable source. ince the spark occurs at the spot



where the tool and the work piece surfaces are the closest and, since, the spot changes

after each spark, the spark ravel all over the surface. This results in a uniform material

removal all over the surface and finally the work face confirms to the tool surface. Thus

the tool produces the re'uired impression on the workpiece.

0or maintaining the predetermined spark gap, a servo control unit is generally used. The

gap is sensed through the average voltage across it and this voltage is compared with a

preset value. The difference is used to control the servomotor. The efficiency and

accuracy of performance have been found to improve when a forced circulation of the

dielectric fluid is provided. In the present work, the tool material is copper for conducting

the experiment or investigations.

!lectric discharge machining is a chipless metal removal process that uses the principleof metal erosion by an interrupted electric spark discharge between the tool and the

workpiece. This process is also called spark erosion process.



Princip! of op!ration

/e know that whenever an arc is caused by an accidental short circuit, pitting erosion

occurs on the surface of the shorted material. !"# also works on the same principle of

erosion by arcing. It involves the controlled erosion of electrically conducting materials

by rapid and repetitive discharge of spark between the electrode tool and workpiece

$hence the name spark erosion% the tool is usually made the cathode and the workpiece

made the anode. The workpiece and tool separated by a small gap and termed as the

sparkgap. The spark gap ranges from 1.112 mm to 1.12mmdepending upon the cutting

action re'uired and the current density, this spark gap is either flooded or immersed in a

dielectric fluid, the spark discharge is produced by the controlled pulsing and direct

current. The fre'uency ranges from a few hundred to several thousand kilohertz with the

application of a suitable voltage across the anode and cathode, electrons are the emitted

from the cathode and cause the ionization of the fluid in the sparkgap, when more



electrons are collected in the gap, the resistance drops causing an electric spark to ump

between the work and the tool gap. The spark causes a focused stream of electrons to

move with a high velocity and acceleration from the cathode toward the anode ,thus

creating high compression shock waves .such shock waves result in local rise in

temperature to the order of about )1,111 c and cause melting of the metal. The forces of

electric and magnetic fields caused by the spark produced a tensile force and tear off

particles of molten and soften metal from the workpiece, Thereby resulting in the metal

and carried away by the flowing dielectric fluid.

/orkpiece and the tool are separated by the dielectric fluid in a container. The dielectric

breaks down when a proper "3 voltage $21-+21% 4 is applied across the anode and the

cathode, and electrons are emitted from the cathode and the gap is ionized, there by

causing electrical discharge and machining operation. The electro-magnetic field cause

compressive forces to act on the cathode thus metal removal from the tool is much slower

than the workpiece .the duration of the electric pulse is about 1.11) seconds, hence the

whole cycle of sparking and metal removal take place in a few microseconds. The

particles of the metal so removed are driven away by the flowing dielectric fluid .the

current density and the power density used is the order of )1,111a5cm6 and 211mw5cm6

respectively.

1."# $actor aff!ctin% EDM proc!ss

A number of controlling variables play an important role in the entire !lectric "ischarge

machining process. A few of them are7

$i%. 8ulse duration

$ii%. 8ulse interval time

$iii%. "ischarge current

$iv%. !rosion diameter

$v%. !rosion depth

The few variables 5 parameters which are useful in analyzing the !"# process accuracy

and efficiency are as below7

$i%. #etal removal rate

$ii%. !lectrode wear

$iii%. urface &oughness



$iv%. 8ower consumption by the #achine

M!ta r!&o'a rat!

#etal removal rate it is direct proportional to the current density used. it is defined as the

volume of metal removed per unit time per ampere. The metal removal rate in

roughening operations of steel with a graphite electrode 21 A current is about

+11mm5min and with +11A current it is about +911mm5min . :ut for high precision

works with use of high fre'uency $211-)111% k;z and low current $)-6A%, metal removal

rate is as low as 6mm5min.

Accurac(

The accuracy of the process mainly depends on the spark gap. The smaller the gap the

higher is the accuracy, but a smaller gaps leads to a lower working voltage and hence a

slow metal removal rate. Thus an optimum gap is necessary for higher accuracies

tolerances of <-1.12 mm can be obtained in normal !"# operations. In precision

operations, with close control of process variables, tolerance up to <-1.11 mm can be

achieved. !"# also produces taper, overcut and corner radii, which are not desirable.

The taper is of the order of 1.112 to 1.12mm per )1 mm depth. The taper effect reduces

gradually to zero after about =2mm penetration .taper effect can be eliminated by the use

of vacuum flushing of dielectric fluid.

The range of overcut is 2 to )11 microns and depth on the roughening operations .the

effect of corner radii is e'ual to the spark gap. Its value is lower in finishing operations,

where low park gaps are used.

)urfac! finis*

In !"# operations, each electrical spark discharge develops a spherical crater in the

workpiece, as well as in the electrode. The volume of crater is proportional to the energy

in the spark. Thus the depth of the crater is proportional to the energy in the spark. Thus

the depth of the crater on work piece defines the surface finish and it depends upon the

current density, fre'uency and the electrode material. usually high fre'uency and low

current density give better surface finish, the best surface finish on steel is of the order of

1.+ micron $at )111khz >)A%.in a typical no-wear !"# ,the surface finish is about

.6micron $generally roughing operations%



H!at !ff!ct!d +on! ,HA

The instant heating and vaporization of metal due to spark, leaves behind a small amount

of molten metal on the machined surface which re-solidifies and due to fast cooling

action of the dielectric fluid forms a hard surface. This becomes the heat affected zone. In

!"# operations .the ;A? is about 6 to )1micron )1 micron deep on the work surface

and its hardness is about @1;&3.the hard surface is a source for thermal stresses ,plastic

deformation and fine cracks at the grain boundaries .the depth of ;A? is small in

finishing operations which can be removed by producing after !"# operations.

Too !!ctrod! /!ar

/hile performing the operations ,the tool gets eroded due to sparking action .so the tool

material should be a material should which is difficult to machine such as a graphite

which goes to vaporization without melting.

/ear ratiomaterial removal from work5material removal from the toolB

Typical electrode materials are,

). 3opper,

6. Tungsten

. Craphite

!lectrode material is selected on the basis of wear ratio, metal removal rate ,cost and ease

of grinding the electrode most commonly used materials may include brass copper

graphite ,al-alloys ,cu-alloys etc. flow hole should be made while making for the

circulation of dielectric flow so as to attain large flow rates at low pressure.



+. It should be able to deionise immediately after the spark discharge. It should have a

high flash point. It should not emit toxic vapours and should not have unpleasant odours.

2. It should not alter its basic properties under operating conditions of temperature

variations, contamination by metal particles and products of decomposition. It should be

economical for use.

)!!ction of di!!ctric fuid

The main consideration in the selection of a dielectric fluid is the operating conditionsE

the choice of dielectric fluids depends on the size of workpiece, complexity of the shape,

surface finish re'uired and material removal rate. ome of the dielectric fluids a, their

machining rate and wear ratios.

The choice of a particular dielectric fluid depends on the tolerance re'uired, size and

shape of the work material removal rate and the type of electrode. 0or example, white

spirit is used to machine tungsten carbide and metals for with intricate shapes and

re'uiring high surface finish. Cenerally, low viscosity fluids are used for very high

surface finishesE the used dielectric fluid is recirculated and after proper filtering action

remove the metal particles. 0iltering medium like wound cotton yarn cartridge

diatomaceous earth filter are used to remove wastes from the medium for normal

precision works filters better than 6 microns sizes are used.

The commonly used dielectrics are kerosene, paraffin transformer oil or mixture of

various oils.

)par %ap

The spark gap between the tool and the work is in the range of 1.112 to 1.12mm.this gap

depends on the type of tool, work material. "ielectric fluid and the current density used.

:efore solving the optimization problem for this process, some parameters or variables

are generally specified such as erosion diameter, erosion depth, applied working voltage,

dielectric pressure etc. once the operating conditions and workpiece-tasking is defined,

the same machine is to be run for various investigations with in the controlled parameter

variation. The parameter variation ranges are specified in upcoming chapter on problem

formulation and solutions.



Ad'anta%!s

!lectrical discharge machining has advantages over other machining techni'ues due to its

ability to create complex and intricate parts with a high degree of accuracy. This process

is able to machine hard materials that other machining processes would have difficulties

with. Another advantage of !"# is its ability to machine parts on an extremely small

scale. /hile using this process, the workpiece is not deformed from impact because there

is no direct contact between the electrode and the material. The workpiece is burr-free

after completion and saved from heat damage because very little is actually generated

during the procedure that would harm the material. As well as dimensional factors of size

and shape, an important consideration before undergoing this procedure is the material of

the workpiece, since the material of the electrode has to be specially matched. ome of

the common applications for !"# include producing plastic molds, die casting dies from

hardened steel and forging dies. (ther purposes include the manufacturing of engine

parts like compressor blades of titanium alloys and nickel based super alloys. Industries

that benefit from the use of the electrical discharge machining process include food and

beverage, automobile, stamping, extruding, defense, electronics, aerospace and medical.



CHAPTER 2

ITERATURE RE3IE4

!fforts are being made now a day to solve complex engineering problems through

mathematical algorithms with high computations accuracy. A number of research papers

exist in the literature related to these newly developed optimization techni'ues. I have

tried to use the concept of genetic algorithms for solving the complex formulated

problems of the !lectro "ischarge #achining process. The stress has been given on

understanding the problem from the core and some of the research papers used for

formulation and understanding the problem are outlined as below7

Foopelli, 4 G)H has tried to model and formulate the !lectric "ischarge #achining

process with the optimization of obective functions related to moving traectories of

machine tool electrode. Cradient based methods have been used to optimize the single

obective function variable. A moving frame reference has also been used to locate the

tool electrode at any instant along its traversed traectories.

Fain 4. . G6H has formulated the generalized !lectro "ischarge #achining method with

the limited constraints related to 8ulse interval time and pulse duration. #any operating

variables are considered as parameters with fixed working values over a given erosion

depth and erosion rate on the work piece. The formulated problem has been analysed by

using a simple optimization algorithm by keeping other obective functions unaffected

and the results are concluded to give the suitable operating variable value selection on the

basis of output obtained.

ahng, 3. ; GH states that operating working voltage and the pulse interval plays an

important role in obtaining the re'uired surface finish. The flow movement of the

dielectric fluid controls the homogeneous surface characteristics in the entire !"#

controlled region. The spark gap control has also been explained to obtain the desired

level of surface roughness for a given set of operating variables. 0luttering of the edges in

the !"# region has been investigated for the variation in the controlling parameters.



J.3. Jim, ;.;. Ju G2H has specified the basic thumb rules for the analysis of surface

features of !lectro discharge machining process. This paper is mainly meant for skilled

workmanship towards achieving the desired surface characteristics in minimum time and

with safety. The saving of the production cost is ustified for the !"# process carried

out. 3ommon measures and precautions which are helpful in carrying out the !"#

process for efficient operation are also been suggested. This paper is recommended for

peer mainly.

#adhu, 8., Fain, 4. G @ H has developed the governing e'uations for the analysis of

!lectro "ischarge machining process under a controlled environment. A computer

program has also been developed in the form of subroutines for the calculation of

electrode wear rate. #etal removal rate and dielectric material effect on the !"#

process. The results obtained by the formulation used with the help of 'uadratic elements

have shown a good convergence with those obtained by the commercial packages.

8andit, . # G9H ahs stated the critical factors affecting the performance of the !lectro

"ischarge machining process when the workpiece material is 3emented carbide. A

suitable hard alloy material is selected as electrode tool material. And the dielectric fluid

is given turbulent flow in and around the !"# region. The operating variables like 8ulse

duration, "ischarge 3urrent, (perating voltage, 8ulse Interval time and heat dissipation

rate differ in operating ranges considerably as compared to electro discharge machining

of teel alloys. ;owever, it has been claimed that consistency and repeatability of the

machine towards maintaining the minimum deviation in the operating conditions helps a

lot in the #achining accuracy in the process.

8andit, . # G*H has considered the theoretical aspects of the !lectro "ischarge

machining 8rocess. The energy parameters and the #etal removal rate relationships have

been developed for the given set of operating voltages and the dielectric pressure. The

relationships obtained have been used to plot graphs for the variation in the operating

controlling parameters and their effects on the conse'uents such as metal removal rate,

surface roughness and the power consumption by the machine etc. These graphs can be

directly used for the selection of given constrained condition of operating variables for

the desired obectives.



&aurkar, .8 ., and ?hu, "G)1H have elaborated a number of alternatives for the

improvement in the electrochemical machining process by changing the sensitivity

parameters and have stated that in comparison to other operating variables, the traectory

of the electrode movement plays a vital role in improving the surface characteristics of

the !3# process.

&aurkar, . 8. has developed a network based flow diagram for the !lectric "ischarge

machining processes. The intermediate processes are shown in the diagram along with

their precedence events and successors. The dependencies have also been evaluated at a

particular instant by assuming some of the operating conditions fixed. The flow diagram

had been used to optimize the !"# process by selecting the independent paths for a

given set of parameter values.

pedding, T.A G))H have used the concept of conformal transformation of the operating

characteristic variables. The variables are parameterized and the parametric

representation of the metal removal rate, surface roughness has been mapped onto

parametric surface. The surface characteristics of the wire3ut !"# process have been

analysed for the sensitivity of the operating variables. The dependencies of the decision

variable on each other are represented and a computation algorithm has been proposed to

evaluate the mapped point for specified surface characteristics onto parametric plane.

pedding, T.A. and /ang, ?.K G)6H has considered the theoretical aspects of the

modelling of the wire3ut !"# process. A user friendly approach has been adopted for

the definition of process parameters and these parameters are compared to other various

ranges of operating variables. The interpretation of output variables variations has been

carried out for the wire 3ut !"# process and the suggested ranges of the input variables

are given for a desired set of output variables in terms of metal removal rate and power

consumption etc.

myers, ., Cuha, A.G) H has stated a practical approach for #achining the :eryllium

3opper alloys as workpiece by !lectro "ischarge #achining process. #ethods have

been suggested for obtaining the desired level of surface characteristics by using this

!"# method. afety precautions and the indicative measures are suggested for the

fruitful implementation of the process. A brief note is also given for specifying the

operating characteristics and safety precautions for the !"# process to be carried out.



/ang /.#.G)+H states that spark gap and the controlling parameters for a sensitive !"#

process layout can be controlled in number of ways. A feedback system with real time

stability analysis and process monitoring through digital modern sensors and transducers

can give an efficient responding mechanism for the !"# process control. The author

also states that transducers, circuitry, encoders etc. can be selected to give influence of

simultaneous variation of operating variables and their response data storage facility.

Indurkhya, C has developed an artificial neural network for the entire !lectro "ischarge

#achining process. The relationships between the operating intermediate processes along

with decision variables have been framed. The performance index evaluation for the

!"# process for a given specified crisp sets of the operating variables helps in

understanding the efficiency of the process to be carried out. The performance Index

evaluated by using this method helps in analysing the adverse and positive gradient

effects of the variation in the #etal removal rate, the surface roughness and the power

consumption by the machine.

?hang, : G)2H the report submitted by the author and his team members have calculated

the effect of motion and turbulence level in the dielectric material during various stages

of the electo discharge machining process. The results are tabulated and graphs have

been recommended for use for the machining of steel materials. The effect of the

selection of dielectric fluid has also been analysed for a given set of electrode tool and

material combination.

?hang 3 G)@H and his team members have developed tolerances for the different

machining parameters,the results have recommended for the machining parameters.

?immermann ;. F G)=H. has developed 0uzzy 8rogramming and Jinear 8rogramming

with everal (bective 0unctions.



CHAPTER "

METHODO56 ADOPTED

everal multiobective optimization methods are available in the literature including the

weighted multiobective method, Archimedean goal programming, non-Archimedeangoal programming, and the original 3hebyshev goal programming methods. The

weighted multiobective method involves subectivity and bias in specifying the weights

in order to aggregate several non commensurable and conflicting obectives into a single

e'uivalent function. Archimedean and non-Archimedean goal programming methods

suffer from the difficulties in determining the weights for deviation variables within a

lower priorityE and the ranking of the goals in a preemptive preference order. /ith the

original 3hebyshev approach, the final solution could be dictated by a single goal. The

modified 3hebyshev goal programming #3C8 approach helps to avoid the above

difficulties and will be used in this study. It should also be noted that the #3C8 is in fact

a fuzzy programming approach and hence can be used to effectively deal with obectives

that are imprecise or fuzzy in nature, such as the intangible cost mentioned above. The

#3C8 can be implemented using the Jingo software. The procedure is detailed as

follows7

!ach obective function can be represented as an function in terms of minimization or of

maximization type as under.

tep )7 0ormulate the bi-criterion model

tep 67 olve the problem with one obective at a time olving #in sumLcost with all the

constraints, we obtain

sumLcost)

M valueLcalculated and sumLtol)



MvalueLsumLcalculated . (ther outputs are omitted due to space limit. olving #ax

sumLtol with all the constraints,

/e obtain sumLcost6

MvalueLsumLcalculatedM and sumLtol6

M value calculatedM the obective values of the above solutions are summarized below7

As we can see, the two obectives conflict with each other. #inimizing sumLcost leads to

a lower sumLtol and maximizing sumLtol causes an increase in sumLcost. Deither

situation is desirable particularly when the differences are as substantial as shown in the

above table. This calls for a compromised solution by solving the modified 3hebyshev

goal programming problem in the next step.

tep 7 0ormulate and solve the modified 3hebyshev goal programming

8roblem

The notion is that the best deviation from the two worst values of two obectives is

obtained by maximizing L. olving this problem, we obtain

1.*2), sum-costM )9.+*@, and sum-tolM 1.96@.



CHAPTER 7

$ORMUATION O$ THE EDM PROCE)) PROBEM AND IT) )OUTION

In many engineering activities, it is very difficult to obtain an explicitly or implicitlyformalized description of the system which could them be optimized. &ecently most of

the research work is devoted to the methods for finding a statistically- experimental

model of such systems. These methods are based on the experiments and their aim is to

determine an investigation program which s compromise between a re'uired number of

investigations in real life condition and their informativeness. The data thus obtained are

analyzed by means of regression methods and then the mathematical model of the system

is then obtained. This model describes the functional dependence between input variables

and out variables. The forms of approximation functions can be considered different. 0or

the !"# process formulation same techni'ues have been applied and the relationships

have been developed as specified obective functions in the problem.

In !lectric "ischarge machining process, main input 'uantities are7

a%. 8ulse "uration

b%. 8ulse Interval

c%. Amplitude of "ischarge current

d%. !rosion surface

e%. !rosion depth etc.

/hereas the main output 'uantities are

i%. #etal removal rate

ii%. !lectrode wear

iii%. 8ower consumption

iv%. urface roughness

v%. "imensional shape accuracy of the work piece

A theoretically recommended approach to the problem of finding a mathematical

description of the !"# process would be to carry out investigations in the whole region



of the space of input variables. The space of variables can be restricted to the region

which is physically sensible and sets of e'uations are obtained.

These sets of e'uations give a mathematical description of the !"# process which is the

basis for building the optimization model. In the present model, the decision variables are

those input 'uantities whose values are set on the machine i.e. 8ulse duration, 8ulse

interval time and discharge current. If the erosion surface does not change in the

machining process and depth of erosion is also known in advance, these two input

'uantities are treated as parameters.

The choice of the obective and constraint functions depends on the user re'uirement.

The output 'uantities usually chosen as obective functions are7 maximizing metal

removal rate, minimizing tool electrode wear rate.

The surface roughness and the dimension and shape accuracy of the workpiece can be

taken as the third and fourth obective functions in case of the accurate machining. In this

process rough machining has been considered, so these two 'uantities may be considered

as constraints or they may be omitted. imilarly, the power consumption by the machine

may be either a new obective function or the constraint or it may be discarded.

In the problem formulation a particular case of !"# process has been analyzed. It is

granted that cylindrical copper electrode is to be used as a tool and tool steel as the

machining material. It is assumed to have constant dielectric pressure and average

working voltage while taking the experimental readings which are to be used further for

regression analysis.

The (bective 0unctions of the problem formulated after regression analysis of the

investigations are as below

K e)).=++I)).=++<1.16lnTi<1.166lnTo<1.1612lnN<1.16@lngTi-).222<1.1+=lnTo<6.=@lnN<1.12)lngTo-1.)1=-

1.)=+lnN<1.)22lngN-).1@=-1.)6+lngg-1.=+6

O e-9).21*I2.@+-1.+*lnTi-1.2lnTo<1.))*lnN<1.)[email protected])nTo-1.++lnN<1.62lngTo).@1*-

6.1+2lnN<1.61=lngN)6.6)*-1.)=)lngg-.)16

D e-1.@@I).+)-1.1@@lnTi-1.))*lnTo<1.)+lnN<1.12lngTi1.6<1.1=))nTo-1.1+9lnN<1.1.1)@lngTo1.9+2-

1.)*=lnN- 1.129lngN1.22=-1.11lngg1.112

ubected to 3onstraints

Ti P 211 x )1-@ The pulse duration in econds does not exceed



Ti Q 6111 x )1-@ These minimum and #aximum Jimits

To Q 621 x )1-@ The pulse Interval time in econds does not exceed

To P )62 x )1-@ These minimum and #aximum Jimits

I Q )69 The "ischarge 3urrent in Amperes does not exceed

I P @+ These minimum and #aximum Jimits

N Q =1 The !rosion "iameter in mm does not exceed

NP 21 These minimum and #aximum Jimits

g Q )1 The !rosion "epth in mm does not exceed

g P 2 These minimum and #aximum Jimits

4 2 R The applied working 4oltage is kept

almost constant

8 @1.9 h8a The "ielectric 8ressure is kept almost constant

and e'ual to this value for a case when cylindrical

3opper electrode is the tool material and Tool

steel is the machined material

This problem is a case of #ulti- 3riteria (ptimization type with different #ini-#ax

constraints and obective 0unctions 3onditions. 0or a given set of constraint each

obective function can be evaluated by using any nonlinear programming techni'ue or by

using Cenetic algorithms. ;owever, in each case there is no guarantee to get optimized

results of the parameters when other obective functions are not taken into account. A

number of some other approaches exist to solve a #ulti-obective function problem but

all of them are either derivative function based or assumes suitable penalty functions for

the account of other obective function.

/hile solving such cases, the approximate solution is achieved but is considerably away

from the exact solutions. The integrated problem is formulated as a bicriterion model to

handle both tangible and intangible costs. The model is solved using a modified

3hebyshev goal programming method to achieve a preferred compromise between the

two conflicting and noncommensurable criteria.



In the above problem formulation, it is clear that for a given set of entry constraints

variables, the obective functions are of following types7

$i%. K, #aterial removal rate 7 To be maximized

$ii%. O, !lectrode /ear in percentage 7 To be #inimized

$iii%. D, power consumption in /atts 7 To be #inimized

/hen multi- criteria model is developed, the nature of the obective functions are

conflicting type and the combination of obective 0unction for #etal removal rate with

other two obective functions in terms of electrode wear and power consumption makes

obective function mini-max type.

According to the :i-3riteria #odel formulation to the above problem, such combination

of #ini-#ax type obective functions along with constraints is considered and the

optimal values of the parameters and obective functions are evaluated at particular

instances.

)t!p 17 0ormulation as per bicriterion model

Cas! I7 considering obective functions $i% and $i%

#aximize K e)).=++ I)).=++<1.16lnTi<1.166lnTo<1.1612lnN<1.16@lngTi-).222<1.1+=lnTo<6.=@lnN<1.12)lng

To-1.)1=-1.)=+lnN<1.)22lngN-).1@=-1.)6+lngg-1.=+6

#inimize O e-9).21*I2.@+-1.+*lnTi-1.2lnTo<1.))*lnN<1.)[email protected])nTo-1.++lnN<1.62lng

To).@1*-6.1+2lnN<1.61=lngN)6.6)*-1.)=)lngg-.)16


Ti P 211 x )1-@ Ti Q 6111 x )1-@

To Q 621 x )1-@ To P )62 x )1-@

I Q )69 I P @+

N Q =1 NP 21

g Q )1 g P 2

Cas! II# considering obective functions $i% and $iii%

#aximize K e)).=++ I)).=++<1.16lnTi<1.166lnTo<1.1612lnN<[email protected]<1.1+=lnTo<6.=@lnN<1.12)lng

To-1.)1=-1.)=+lnN<1.)22lngN-).1@=-1.)6+lngg-1.=+6

#inimize D e-1.@@I).+)-1.1@@lnTi-1.))*lnTo<1.)+lnN<1.12lngTi1.6<1.1=))nTo-1.1+9lnN<1.1.1)@lng

To1.9+2-1.)*=lnN-1.129lngN1.22=-1.11lngg1.112




Ti P 211 x )1-@ Ti Q 6111 x )1-@

To Q 621 x )1-@ To P )62 x )1-@

I Q )69 I P @+

N Q =1 NP 21

g Q )1 g P 2

tep 67 olve the problem with one obective at a time

The problem is solved by using generalized genetic algorithm for the given set of

constraints and only one obective function at a time. The values obtained for each case

are tabulated as below7

3ase I

#inimize O, mm #aximize K, mm5min "ifference4alue L#in O 1.)96+*9 1.*9@@=9* 1.91+6*)

4alueLmax K +)9.*=9@+2 *2).2+2@= 26.2@+*663ase II

#inimize D, /att #aximize K, mm5min "ifference

4alueLmin D [email protected]=@2+2 =112.2@=+ +6+).@*19924alueLmax K =6.9*=@2+2 *2).2+2@= 2=9.@+2*)62

As we see the two obectives conflict to each other in each case. #inimize O leads to a

lower value of K while maximizing K leads to an considerable increase in the value of O

in case I. in case II, #inimize D leads to a lower value of K while maximizing K leads to

an considerable increase in the value of D.

Deither solution can be considered desirable particularly when the differences are

substantial as shown in the table above.

This calls for a compromise solution by solving modified 3hebyshev Coal programming

problem in the next step.

tep III 7 0ormulation and solution by modified 3hebyshev Coal programming problem.

The problem can be formulated by using this bicriterion approach for each case as below7

3ase I 7

#aximize S

ubected to

$valueLmaxK-valueLmaxKM%

S Q ---------------------------------- $valueLmaxK-+)9.*=9@+2%5 26.2@+*66



$valueLmaxK6M-valueLmaxK)M%

$valueLminO6M-valueLminO%

S Q ---------------------------------- $1.*9@@=9*- valueLminO%5 1.91+6*)

$valueLminO6M-valueLminO)M%

Ti P 211 x )1-@ Ti Q 6111 x )1-@

To Q 621 x )1-@ To P )62 x )1-@

I Q )69 I P @+

N Q =1 NP 21

g Q )1 g P 2

imilarly,

for case II7

#inimize

ubected to

$valueLmaxK-valueLmaxKM%

Q ---------------------------------- $valueLmaxK-=6.9*=@2+2%5 2=9.@+2*)62

$valueLmaxK6M-valueLmaxK)M%

$valueLminD6M-valueLminD%

Q ---------------------------------- $=112.2@=+- valueLminD%5 +6+).@*1992

$valueLminD6M-valueLminD)M%

Ti P 211 x )1-@ Ti Q 6111 x )1-@

To Q 621 x )1-@ To P )62 x )1-@

I Q )69 I P @+

N Q =1 NP 21

g Q )1 g P 2

The notion is that the best deviation from the two worst values of two obectives is

obtained by maximizing S and minimizing D.

olving this problem,

/e obtain the following results



3ase I 7 maximizing S

8revious calculated value 0inal calculated values "ifference

4alue L#in O 1.)96+*9 1.)*1+62 1.119196=4alueLmax K +)9.*=9@+2 +6+.@2+=9 2.@=@)2

3ase II 7 minimizing

8revious calculated value 0inal calculated values "ifference4alueLmin D [email protected]=@2+2 6)+.9@+* +21.*9@*+2

4alueLmax K =6.9*=@2+2 9+.*+*= )6.1=)22

0rom the above result computation it is clear that the results are close enough to take a

decision about the behavioural response of the obective functions and the constraints.

The developed bicriterion method for the !lectric "ischarge machining process

optimization is also 'uite useful for observing the sensitivity of the obective functions.

Any set of the obective functions with respect to any set of constraint can be analyzed

for their significant behavioural response. This techni'ue seems to be 'uite useful for

solving such constraint based machining parameter optimization problems.

CHAPTER 8

RE)UT) AND CONCU)ION

The present method adopted to solve the optimization problem of !"# process is simple

enough and is flexible in selection of obective functions and the constraints for such

machining processes. At any stage, the dominance factor of the input variables and output



variables contained in the constraints and obective functions can be computed. This

techni'ue helps in getting the reliable multiobective decisions under constrained

penalties for the constrained optimization of such processes.

"uring the solution of the problem, it has been found that the results obtained by the

bicriterion approach show their convergence towards the exact solutions obtained by

optimization of obective functions under min-max condition. ;owever, the absolute

values of the obective function differ significantly for their absolute values under max-

max or min-min condition.

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