Multi-Objective Optimization of Grinding Processes with ... - CiteSeerX

Multi-Objective Optimization of Grinding Processes with Two Approaches: Optimal Pareto Set with Genetic Algorithm and Multi-attribute Utility Theory Angela Adamyan, David He, Ioan Marinescu, and Radu Coman Department of Mechanical, Industrial, and Manufacturing Engineering College of Engineering The University of Toledo Toledo, OH 43606

Abstract

and an example is presented. A new

Optimization of grinding process is a multi-

formulation for multi-objective optimization

objective optimization problem. However, it

of grinding process is developed.

traditionally has been solved as a single

results of the formulation represent the

objective problem. In this paper, general,

tradeoffs the designers are willing to make

useful

multidisciplinary

between work piece surface roughness, tool

optimization methods are discussed for

life, grinding ratio, and material removal

optimal

grinding

rate. Four examples are used to illustrate the

processes. The methods allow designer to

application of the formulation for multi-

explicitly consider and control multiple

objective

design objectives as an integrated part of the

grinding processes. An example for defining

optimization process, and easily choose and

the

set up preferences for the objectives in order

optimization process is also presented.

and

practical

design

of

surface

optimization

preferences

in

the

of

the

The

surface

multi-objective

to increase productivity and quality of the

1. Introduction

workpiece surface.

More and more brittle materials such as The methods discussed in this paper are

ceramics, glasses, sapphires, etc are now

Pareto efficient set and multi-attribute utility

used to produce different parts in many

theory based optimization approaches. The

industries such as aerospace industry,

algorithm for finding Pareto set is proposed

automotive industry, etc. In order to obtain a 1

high accuracy, these parts are necessary to

these works is that the grinding process was

be machined, then finishing processes like

considered as a single objective problem.

grinding, lapping, polishing, etc must be applied.

Optimization

Grinding is one of the most

of

the

grinding

process

important processes for finishing brittle

includes the determination of suitable

materials. The random distribution, random

abrasive

orientation,

and

conditions. Being therefore a multi-objective

geometry of abrasive grains on the wheel

optimization problem (MOP). The solution

surface create the difficulties of creating

to this problem requires a good knowledge

mathematical models.

of the relationship between quantitative

and

random

position

wheel

and

the

machining

parameters of the grinding process and of Researches have been carried out to

the wheel characteristics, the grinding

investigate the grinding mechanism. Werner

parameters and the dynamic behavior of the

(1983) has derived a linear relationship

grinding

between the normal and tangential force by

Sathyanarayanan (1992), the conventional

making assumptions about the wheel-

approach presented in the literature for

workpiece

optimization of grinding suffers from few

contact,

grit,

normal

and

machine.

As

discussed

by

tangential force equation, hardness of the

limitations.

workpiece and the ploughing pressure. The

experimental data has to be generated in

stages from friction to ploughing and cutting

order

have been described by utilizing a slip-line

accurately in the traditional way. Second,

field

existing

due to the conventional serial computing

boundary conditions (Lortz, 1979). The slip-

process, the derived model can not adjust

line field has been developed by starting

itself to simultaneous changes of the

from the velocity relation at the level of

parameters’ values. Third, the problem was

penetration cutting edge and characterizing

considered as a single objective problem.

which

satisfies

all

the

to

First, a large amount of

model

the

system

response

the frictional condition at the interface between the cutting edge and the work

The goal of this paper is to develop general,

material. Liao et al. (1989) utilized factorial

useful

experimental design to derive an empirical

optimization tools for robust design of

model for creep feed.

products, systems and processes. The tools

The drawback of

and

practical

multidisciplinary

allow designer to explicitly consider and 2

control, as an integrated part of the

illustrated with an example for grinding

optimization process, the multiple design

Basalt (II) with diamond wheel. Then the

objectives.

Easily choose and set up

models based on multi-attribute utility

preferences for the objectives in order to

theory for Basalt (I), Basalt (II), Ceramic

increase productivity and quality of the

(I), and Ceramic (II) are developed and

workpiece surface.

solved. Finally, the conclusions based on the results

obtained

from

the

mentioned

examples are drawn in Section 4.

The paper addresses several key questions in developing multi-objective methods for the design of grinding processes. •

“How

1.1. Brief Description of Surface Grinding

should

Process

different

objectives be incorporated into one

The optimization problems we are trying to

model?”

solve are concerned with the surface

•

“How should multi-objective

grinding. Surface grinding is performed on

optimization of grinding problem be

either a horizontal spindle or vertical a

formulated as a multidisciplinary

spindle grinding machine. The work piece

design optimization problem?”

is secured, unusual for ceramics, on a

•

magnetic chuck, which is attached to the

“How should the formulated

worktable. The main grinding parameters

models be solved?”

are shown in Figure 1 and listed below. The rest of the paper is organized as follows.

Geometric

description

some

of

First, a brief description of the surface

parameters is shown in Figure 1 as well.

grinding process and the experiment design

Vs – wheel wear, mm3

used in supporting this study are provided.

Vw – total stock removal, mm3

Then, a literature review of multi-objective

vs – wheel speed, m/sec

optimization in finding Pareto set with

vw – feed rate, m/min

Genetic

a

Algorithm,

and

multi-attribute

these

– depth of cut, µm

utility theory approach are presented. The

G – grinding ratio (Vw/Vs)

research approach developed for solving the

Qw – material removal rate, mm3/min

multi-objective optimization problem is

heq – equivalent chip thickens, µm

discussed in Section 3. The algorithm for

T

finding the Pareso set is presented and

life), min 3

– life of grinding wheel (tool

Ra – roughness, µm

Infeed

0.01 to 0.1

Coolant

water based synthetic water based synthetic

Flow rate

0.005 to 0.05

emulsion 2%,

emulsion 2%,

5

5 l/min

l/min 2

193 cm2

Workpiece area 193 cm heq range

µm

0.05 – 1

0.05 – 0.5 µm

vs

Table 2. Properties of the Materials used

bs vw

Properties

Ceramic

Basalt

Density

3.90 g/cm3

2.75

Thermal conductivity 39

a

Bending strength

9.1

W/cm K 5 2

N/mm

Max utilization temp. 1800 °C

g/cm3 W/cm K

7.5-8 N/mm2 400

°C

Table 3. Wheel Specifications Figure 1. Illustration of the Surface Grinding Process

Wheel Specifications

Showing Various Process Variables

Characteristics Metal Bond

Resin Bond

Type

1A1

1A1

1.2. Experimental Procedure

Dimensions

300-6-3 mm

250-10-3 mm

A surface grinder type Rubin 026 VA II

Abrasives

Medium friable

Friable Ni-coated 56%

(ELB

Schliff)

instrumentation

is

equipped

for measuring

synthetic diamond synthetic diamond

with grinding

forces, vibration, velocities, diamond wheel

Grit size

D126

D 107

Concentration

100

100

Hardness

HB 128

R

wear, and energy consumed during wet Details of the wheel

Throughout the rest of the paper, the

specifications and the test condition are

following parameters have been used. For

given in the Tables 1, 2, and 3.

grinding Basalt (I) metal bond has been used

grinding process.

and the following characteristics were applied: wheel 1A1 300-6-3 D 126 C100

Table 1. Wheel and bond characteristics Wheel bond

Metal Bond

Resin Bond

HB 128, vs = 30 m/sec, vw =16 m/min, b = 2

Wheel speed

25

m/sec

25

m/sec

mm. Basalt (II) means that the resin bond

Workpiece speed 16

m/min

6

m/min

Crossfeed

mm

5

mm

3

has been used and wheel 1A1 250-10-3 4

D107 C100 R, vs = 25m/sec, vw =4 m/min, b

Pareto front were found with VEGA,

= 6 mm. To process Ceramic (I) metal bond

whereas the niched Pareto GA gave better

wheel has been used, wheel 1A1 250 –10-3

results.

D 126 M100 vs = 25 m/sec, vw =16m/min, b = 3 mm. To grind Ceramic (II) resin bond

Viennet et al. (1996) has developed the

has been used and wheel 1A1 300-15-2 D76

multi-objective optimization algorithm that

R75, vs = 25 m/sec, vw =6 m/min, b = 5 mm.

uses GA properties and a non-dominated solution selection algorithm to generate a set

2. Literature Review

of Pareto efficient points. This technique

In this section, literature that related to two

permits one to determine the Pareto set with

methods

multi-objective

its contour line. In this method each

optimization problems is viewed. These two

objective is weighted and summed to form a

methods are Pareto set generation and multi-

scalar objective that is minimized or

attribute utility theory.

Recently several

maximized. Another technique in generating

multi-objective

Pareto points is to transform all objectives

optimization problems have been developed.

except one into constraints. With this

For example, Horn et al. (1994) modified a

method it is very difficult to take into the

genetic algorithm (GA) by introducing the

account variability of raw materials and

concept of Pareto’s domination in the

working conditions to get good results.

selection operator. In order to improve their

These techniques are neither accurate nor

results, they applied a niching pressure to

general. The advantage of this algorithm is

spread population out along the Pareto

that any function can be simultaneously

optimal surface, by alerting tournament

optimized. And the result consists of an

selection in the two ways. They added

optimal oneness zone in which the decision-

Pareto dominant tournaments, and when

maker is able to choose the working

they obtained a tie the winner was

conditions. This method is based on diploid

determined by implementing a sharing.

genetic

Consequently, they called this algorithm a

procedure using the pressure of Pareto.

niched Pareto GA. These authors compared

Compared with the algorithm of Horn et al.

their approach with vector evaluated GA

(1994), this pressure is not applied in the

(VEGA) which has been developed by

selection stage of this algorithm. The way

Schaffer (1985). Only extreme points on the

they applied their domination criterion

methods

in

for

solving

solving

5

algorithm

and

on

selection

their

(2) Representation of single attribute

algorithm needs more generations to define

utility function in terms of decision

optimal

variables and evaluation of single

would

seem

less

efficient,

Pareto

set.

and

Multicriteria

optimization algorithm uses a GA only to

attribute

utility

functions

find intermediate population. No Pareto–

preferences based on tradeoffs

and

(3) Aggregation of single attribute utility

efficient points are eliminated from these

functions into an overall multiple

populations with selection procedure.

attribute utility function Another technique to solve MOP is Multi-

(4) Optimization of the multiple attribute

attribute Utility Theory. Utility theory was

utility function for selection of

originally devised by von Neumann and

optimal design

Morgenstern

(1947)

for

economic

application and later was developed for

A

methodology

for

determining

and

multiple objective decision making purpose

evaluating single and multiple attribute

by Keeney and Raiffa (1976). Utility theory

utility functions in engineering design is

provides a formal approach in constructing a

described in Thurston (1991) and Thurston

design evaluation function that represents

et al. (1991). Thurston (1991) concluded

the designer's preferences and willingness to

that utility theory based approach measures

make tradeoff between conflicting design

more accurately designer's preferences and

objectives.

allows for a nonlinear relation between preference and attribute level, and nonconstant tradeoffs.

Similar to the application of the utility theory to decision-based design described by Gold and Krishnamurty, (1997); Olson and

3. Research approach

Moshkovich, (1995). The utility theory

The research in this paper consists of three

based multidisciplinary design optimization

tasks. The first task is to incorporate the

technique should include the following four

different

steps:

objective ginning optimization problem.

(1) Determination of design attributes, design

constraints,

and

objectives

in

solving

multi-

The next task is to formulate the multi-

decision

objective grinding process problems. The

variables

final task is to develop a general method for solving 6

the

formulated

multi-objective

grinding process problem. To accomplish

Basalt (II)

Ceramic(II)

those tasks two approaches were considered:

Ra =0. 4369(heq) + 6.9158

Ra = 0.8841Ln(heq) +5.7158

Pareto efficient set with Genetic Algorithm

G = -99.52Ln(heq)+299.06

G = 438.17e-13.719(heq)

T = 7317.4e-0.3091(heq)

T =2.7298(heq)-2.0468

Qw = 572.59e0.138(heq)

Qw =17372(heq) + 98.98

approach and multi-objective utility theory approach.

Characteristically, a MOP has no unique 3.1. Pareto Set with Genetic Algorithm

solution that can simultaneously optimize all

Approach The

optimization

objectives. of

grinding

In

multi-objective

decision

process

making situation we can find Pareto optimal

includes both objectives that should be

solution. A Pareto optimal solution, is

minimized and objectives that should be

defined as following:

maximized. This makes the optimization of

A solution x to a multiple- objective

the grinding process a very complex

problem is a Pareto optimal if no other

problem. The objectives that should be

feasible solution is at least as good as x with

maximized include G-ratio (G), Tool life

respect to every objective and strictly better

(T), and Material Removal Rate (Qw). The

than x with respect to at least one objective

objectiv that should be minimized is

(Winston 1997).

Roughness (Ra). Decision variable is the Equivalent Chip Thickness (heq) that in tern

Any point in a Pareto optimal set can

3

depends on wheel wear, (Vs) mm , total

become an “optimal solution” depending on

3

stock removal, (Vw) mm , and depth of cut

decision-maker’s preferences. There is no

(a), mm. According to Marinescu (1984),

universal standard to judge merits of the

heq is defined as following:

solution. Designers may choose on the basis

heq =

av w 60v s

of tradeoffs among design objectives by determining

interrelationship.

A

Pareto

optimal set carries information on those Applying simple regression analysis to the

tradeoffs. Thus, an ideal way to solve a

data generated from the experiment, the

MOP is to obtain an evenly distributed

objectives Ra, G, T, and Qw can be

subset of a Pareto optimal set, to explore this

expressed as functions of heq for Basalt (II)

subset, and to select the preferred solution.

and Ceramic (II) as follows:

7

GAs are based on the biological evolution mechanism and Darwin’s survival-of-the

ì 5x + 3 ï1 + (5 x + 3) , where f 1 is to be minimized ï min f i = í 1 ï , where f 2 is to be maximized ïî1 + (3 x + 6 ) Here it is assumed that the fi ≥ 0.

fittest theory, are intelligent search methods for solving complex optimization problems. They are problem independent algorithms and can process information generated at previous stages of a search process. A search

To illustrate the concept, let’s consider an

process comprises of natural selection, quick

example. We have two objectives: maximize

exploration, and information collection in

f1=3x+6 and minimize f2=5x+ 3. Applying

design space. In contrast to more classical

the normalized equations, the two objectives

optimization methods, a GA requires no

are converted into a minimization objective

gradient information and produces multiple

as in the following:

optima rather than a single local optimum. These characteristics make GA a powerful

To minimize

tool for solving multi-objective optimization

1 1 + (3 x + 6 )

is the same as to

maximize f1=3x+6 and to minimize

problems.

5x + 3 is 1 + (5 x + 3)

same as to minimize f2=5x+ 3.

To

apply

multicriteria

optimization

algorithm for the grinding process some

If any constraint is used the problem should

modification

be

should

be

done.

The

converted

into

the

unconstrained

multicriteria optimization algorithm does not

optimization problem by applying Lagrange

consider

penalty function.

the

functions

with

partial

The Pareto points

generation algorithm is presented below:

maximization (max {f1(x), f2 (x), ….f3(x)}) and partial minimization (min{f1(x), f2 (x), ….f3(x)}) problems. To overcome this limitation,

the

following

Pareto Points Generation Algorithm:

normalized

equations are used to convert the problem

Step 1.

into a minimization one. ì fi ï(1 + f ï i min f i = í 1 ï ï î(1 + f i

) )

Each function is calculated for the minimum values of the other functions and the maximum value

, where f i is to be minimized

will be denoted as Fi. For , where f i is to be maximized

example, minimum values of material removal rate and tool life 8

Step 2.

Step 3.

will be computed, then G ratio

Multi-objective

will be calculated for those

points that are not Pareto-efficient. In the

minimum values.

first two steps, the population will be created

For each function, one zone Ωi is

and in the third step, individuals of those

defined by fi(x)≤Fi. The solutions

populations, which are dominated, will be

for every individual function

eliminated.

construct populations Pi .

multicriteria optimization algorithm.

Eliminate

eliminates

Figure 2 summarizes the

Pareto-inefficient Optimization of each function fi

points by applying multicriteria optimization. combined

optimization

to

Populations form

one

are P

Calculation of Fi

population. Then the solution x for the first function is taken as a

Definition of Ωi by fi(x)≤Fi

reference and compared with all other individuals x in P, Pareto

Elimination of Pareto-inefficient

inefficient points are eliminated by using the function hi defined by :

Optimal Pareto Figure 2. Block diagram of theset Multi criteria

hi = 0, if f i (x) > f i ( x ),

optimization algorithm.

hi = 1, if f i (x) = f i ( x ), hi = 2, if f i (x) < f i ( x ). If

åh

i

Next, the Pareto points generation algorithm is illustrated for surface grinding of Basalt

> 1 , then the reference is

(II).

i

better than the individual ( x ) so the latter will be eliminated. If it

Illustrative Example 1

is not eliminated point ( x ) is used

In this example the results are computed for

as reference, and so on. By

Basalt (II).

applying this procedure the Pareto

Step 1.

All functions are normalized in the

set could be obtained.

following

way:

The

roughness of the workpiece should be minimized and the 9

normalization

takes

the

Step 2.

following form:

Each function is calculated for the minimum of the other three functions

f1 =

0. 4369heq + 6.9158

(1 + 0. 4369h

eq

and

the

maximum

values are denoted Fi. Results are

+ 6.9158)

presented in the Table 5.

The other objectives should be maximized,

so

they

take

the

Table 5. Functions value for different decision variable values

following forms: f2 = f3 =

1 (1 + -99.52Ln(heq ) + 299.06) 1

(1 + 7317.4e

-30.798 heq

fi(h eq1) Ra

)

1

(1 + 572.59e

0.138 heq

fi(h eq3)

fi(h eq4)

0.003355366 0.519554975 0.519554975

0.00017703 0.000177049

0.059157469 0.059157469

Qw 0.001518684 0.001518635 0.001514713

)

Fi

0.88028973 0.88040714 0.939953644 0.939953644

G 0.003334437 T

f4 =

fi(h eq2)

Step 3.

0.001518684

Let’s consider heq1 as reference point and compare with heq2, then

Where f2 is the function of G- ratio, f3 is the

Ra(1.001602289)= 0.880288266

function of the tool life and f4 the function of

material

removal

rate.

To

< Ra(1.001835818) = 0.88028972,

create

G(1.001602289) = 0.003334437

intermediate population all functions are

< G(1.001835818) = 0.003334695,

minimized with GA. The mortality rate of

h2=2

T(1.001602289) = 0.000177036
Qw(1.001835818) = 0.001518635, h4=2

h1 +

Table 4. Results of function minimization

heqi

h2+

h3 +

h4>1,

so

the

reference heq1 is better than the

Solutions Functions

h1=2

fi(heqi)

individual heq2 hence heq1 is included to the Pareto set.

f1(heq1)

1.001602289

0.880288266

f2(heq2)

1.001835818

0.003334695

f3(heq3)

1.020606754

0.000178079

f4(heq4)

20

0.000110524

Pareto set points lie on the interval [1.00, 20]. Since every objective function is ether monotonically decreasing or increasing any point taken from that interval according to 10

the Pareto set definition will be a Pareto

means of lottery questions to identify

efficient point.

certainty equivalents.

These certainty

equivalents are different from expected values for lotteries and reflect uncertainties

3.2. Multi-attribute Utility Theory Based

and risks.

Approach

Although certainty equivalents

In this section, a multi-attribute utility

are generally unique to each individual, it is

theory based approach is developed to solve

possible to characterize decision makers into

the

grinding

risk averse or risk prone categories subject

In this approach,

to the certainty equivalents chosen for a

important design criteria are identified and

given lottery. The categorization of decision

weighting factors are assigned to each

makers in this manner essentially implies the

criterion to represent relative tradeoffs. This

confidence that the decision makers have in

approach applies utility theory to assemble

a lottery for achieving the best possible

the important attributes into a utility

outcome for the attribute levels under

function using rigorously determined scaling

analysis.

constants that represent the acceptable

risk averse, prone, and neutral attitudes for a

tradeoffs

monotonically decreasing single attribute

multi-objective

surface

optimization problem.

between

attributes.

When

compared utility theory based approach to other

approaches,

Thurston

Figure 3 shows graphically the

utility function.

(1991)

concluded that utility theory based approach measures

more

accurately

Risk Neutral

designer's 1.

preferences and allows for a nonlinear relation between preference and attribute

Utility

Risk Averse

0.

level on one hand, and non-constant 0

tradeoffs on the other hand.

Risk Prone

Attribute

Figure 3. Graphical representation of risk attitudes

Utility is determined as a function of performance level of attributes that a design alternative exhibits. utility

functions

In developing single for

attributes,

A methodology for determination and

the

evaluation of single and multiple attribute

preferences for the performance level of

utility functions in engineering design is

each attribute in a design are quantified by

described in Thurston (1991) and Thurston 11

et al. (1991). The relevant points in these

attribute scaling constants, ai, represent the

two papers are summarized below.

tradeoff between attributes the designer is willing to make. These scaling constants are

Given conditions of preferential, utility, and

determined by a "lottery" type question.

additive independence of attributes, the

The value of ai is equal to the multi-attribute

overall multi-attribute utility of a design

utility where attribute yi is at its best level,

alternative is calculated with:

yib, and all other attributes are at their worst

U(Y) =

n

levels, yjw (for all j ≠ i). The designer is

i =1

asked to compare a design with attribute

å ai U i ( y i )

where:

levels that are certain to a design with vs.

Lottery Design with uncertain attribute

Certainty

α

Design with certain attribute levels

y1b,

y1b, 1-α

y1w,

U(Y) = the overall utility of

uncertain attribute levels, as shown in Figure

an alternative characterized

4.

by the attribute vector Y = ( y1 , y 2 ,..., y n )

On the left side, the alternative with no uncertainty in the two attribute levels is

yi = the performance level of

attribute i, i = 1, …, n Figure 4. Lottery questions to assess scaling constant a1

Ui(yi) = the single attribute

utility function for attribute i, i = 1, …, n

compared with the "lottery" shown on the ai = the single attribute

right

scaling constant

of

which

has

attribute

levels

dependent on probability, α. The value of α at which the designer is indifferent between

A detailed instruction on testing the conditions

side

preferential,

utility,

the lottery and the certain attribute levels is

and

obtained by iterating between extreme

additive independence of attributes is given

values of α. When evaluated at the best and

in Keeney and Raiffa (1976). The single 12

worst values of all attributes, Yb and Yw, the Figure 5. The overall utility when reliability

multi-attribute utility function is unity and

decreases with decomposition of the system

zero, respectively. The value of ai is then found from

The attributes are: G-ratio (G), Tool life (T),

U ( y1w ,..., yib ,..., y nw ) = αU (Yb ) + (1 − α )U (Y w )

and Material removal rate (Qw). Decision

= αU (1) + (1 − α )U (0 )

variables are: wheel wear, (Vs) mm3, total

=α where

ai

=

U ( y1w ,..., yib ,..., y nw ) = ai .

α,

stock removal, (Vw) mm3, and depth of cut

since

(a), mm.

The single attribute utility

functions of attributes take a linear form that

In solving the grinding optimization process

corresponds to a neutral risk attitude. Note

problems, as the objective function is

that the single attribute utility can also take a

represented by a multiple attribute utility

non-linear form depending on the decision-

function, it represents all the design

maker’s attitude toward the risk.

attributes under consideration and reflects the designers' willingness to make tradeoff

Let Ramax and Ramin be the maximum and

between the attributes. For example, if only

minimum

two attributes, roughness and material

respectively.

removal rate, are included in the objective

values

of

the

roughness,

The linear relation between

single utility function of the roughness

function, then the optimal solution of the

U1(Ra) and the level of roughness is shown

problem reflects the design preference over

in Figure 6.

the tradeoff between the reliability and material handling cost (see Figure 5). 1 U1(Ra) Optimal point

Utility of Roughness Utility of Material Removal Rate Overall Utility

U2(y2)

Utility

U(Y) = a1U1(y1)+a2U2(y2)

U1(y1)

13 High degree of Roughness Low degree of Material Removal Rate

Low degree of Roughness High degree of Material Removal Rate

0

Ramax Ramin Total value of Roughness, Ra

U 3 (T ) =

T min − T T min − T max

U (Q ) = Q

Qw

4

Figure 6. Single utility function for Roughness, Ra

The

min w

− Qw

− Qw

max

utility

function

of

the

Given the definition of the multi-attribute

measure

U1(Ra)

can

be

utility function of the overall objective

single

dissimilarity

w

min

function, the optimization of grinding

computed as following:

U 1 (Ra ) =

processes problem can be formulated as

Ra max − Ra Ra max − Ra min

follows: Maximize

Let Gmax and Gmin be the maximum value and

minimum

respectively.

value

of

the

é Ra max − Ra ù é G min − G ù + U ( heq ) = a1 ê max a 2ê ú min max ú − Ra min û ë Ra ëG − G û é Q w min − Q w ù é T min − T ù + a 3 ê min + a 4ê max ú min max ú ëT − T û ëê Q w − Q w ûú

G-ratio,

The linear relation between

single utility function of the G-ratio U2(G) and the value of the G-ratio is shown in

Illustrative Example 2

Figure 7.

For this example we have used material 1

Basalt (I). The following equations were

U1(Qw

obtained form Marinescu et al. (1984): Gmin

0 Gmax

æ av ö Ra = 2.761çç w ÷÷ è vs ø

Total value of G-ratio, G Figure 7. Single utility function for G-ratio, G

æ av ö G = 382.79çç w ÷÷ è vs ø

The single utility function of the G-ratio

æ av T = 89.927çç w è vs

U2(G) can be computed as following:

U 2 (G ) =

G min − G G min − G max

ö ÷÷ ø

0.283

−0.8704

−1.7085

æ av ö Qw = 13.059çç w ÷÷ è vs ø

In a same way we can find single utility

0.84

For different values of the single attribute

functions for Material Removal Rate (Qw)

scaling constants ai optimal solutions were

and Tool life (T). 14

with different values of ai

obtained using Genetic Algorithm software.

Computational Results

The summary of results is presented in the U

table 6.

heq (µm)

a2

a1

0.997 0.02681 0.3333 0.74803 0.02681 Table 6. Overall Utility Function values for Basalt (I)

heq 1

0.78271

a1

a3

a4

0.045 0.3333 0.3333 0.3333 0.045187

0.523953

0.25

0.25

0.5485

0

0.25 0.25

0.89993

0.15

0.15

0.899925 0.13333 0.13333 0.13333 0.6

0.751314

0.899978 0.08333 0.08333 0.08333 0.75 0.9

0

0.25

0

0.25

0.25

0.16667 0.16667

0.5

0

0.1594

0.15

0.15

0.15

0.55

0.1594 0.13333

0.13333 0.13333

0.6

0.74726

0.1594 0.08333

0.08333 0.08333

0.75

1

0.15 0.55

0.602139

1

0.25

0

0.59825

0.0045 0.16667 0.16667 0.16667 0.5

0.552429

0.3333 0.3333

0.49906 0.02681 0.16667 a2

a4

1

with different values of ai U

a3

0.9988

0.1598

0

0

0

1

1

Table 8. Overall Utility Function values for Ceramic

The same experiment was performed for

(I) with different values of ai

Basalt (II), Ceramic (I) and Ceramic (II).

Computational Results U

The results are presented in the Tables 7, 8,9

heq

a2

a1

a3

a4

(µm)

respectively.

0.9717595 0.1109 0.3333 0.7415571 0.1084

The following equations for Basalt(II),

0.25

0.3333 0.3333 0.25

0

0.25 0.25

0.499801 1.0629 0.16667 0.16667 0.16667 0.5

Ceramic(I), and Ceramic (II) were obtained

0.549776

from Marinescu et al. (1984):

0.599816 1.0658 0.13333 0.13333 0.13333 0.6

1.065

0.15

0.15

0.15 0.55

0.749596 1.0658 0.08333 0.08333 0.08333 0.75 Basalt (II)

Ceramic (I)

0.99923 1.0658

Ceramic(II)

Ra =0. 4369heq + 6.9158

Ra = 8.6978heq2 - Ra = 0.8841Ln(heq) +

G = -99.52Ln(heq) +

3.3626heq+7.23717 5.7158

299.06

G = 1440.9heq-0.5573 G = 438.17e-13.719heq

T = 7317.4e-0.3091heq

T = 164.86heq-1.4156 T =2.7298heq-2.0468

Qw = 572.59e0.138heq

Qw =

Qw =17372heq +

45154.4heq0.8745

98.98

0

0

0

1

Table 9. Overall Utility Function values for Ceramic (II) with different values of ai Computational Results U

heq (µm)

A1

a2

a3

0.9857205 0.02031185 0.3333 0.3333 0.3333 0.7397235 0.02031185 Table 7. Overall Utility Function values for Basalt (II)

0.25

0.25

0

0.25 0.25

0.499659 0.19998444 0.16667 0.16667 0.16667

15

a4

0.5

0.549967 0.19998444

0.15

0.15

0.15 0.55

0.599955 0.19998444 0.13333 0.13333 0.13333

0.6

heq

0.749994 0.19998444 0.08333 0.08333 0.08333 0.75 0.9998637

0.19999

0

0

0

a1

a2

a3

a4

U

0.0267 0.69 0.12 0.08 0.11

1

0.04 0.96 0.03 0.055

3.3.Definition of Preferences

1

0.89

0 0.01 0.8882

0

0

0 0.7875

0.08 0.96 0.02

0 0.02 0.5917

0.1 0.04 0.02

0 0.94 0.5381

In this section we try to optimize ai

From the results we can conclude that the

coefficients for any given value heq, which

better the quality of surface the lesser the

will allow to find out whether the objective

utility of the tool life. The best value of the

function is dependable on ai coefficients.

utillity function is obtained when the heq is

This method has the following advantages:

the smallest, which means the smallest depth

1) completely eliminates the need for

of cut a since for this experement feed rate

iterative weight setting, i.e. once the

vw and wheel speed vs where fixed.

designer’s

preferences

are

articulated,

obtaining the corresponding optimum design

4. Conclusions

is a nonnutritive process, hence improve the

The surface grinding problems traditionally

computational efficiency; 2) provides the

have been solved as a single objective

means to reliable employ commercial

optimization problem.

optimization tools with minimal prior

designs

knowledge.

aproach may omit other import design

generated

As such process

with

such

solution

attributes of the process and thus result in The optimization model is defined as

process that performs poorly.

follows: This paper presents two methods for solving U=a1U1+a2U2+a3U3+a4U4

Max

the

Subject to:

optimization problems.

4

åa i =1

multi-objective

i

=1

surface

grinding

The first method

finds Pareto efficient set and second method applies multi-attribute utility theory in

0 ≤ ai ≤ 1

solving the problems.

The algorithm for

finding Pareto set was developed and an

The following results were obtained for

example was presented. A new formulation

Basalt (II): 16

for multi-objective optimization of grinding

ratio and material rate while cause

process was developed and represents the

simultaneous increase in the value of

tradeoffs the designers are willing to make

objective function for roughness.

between roughness, tool life, and material removal rate. Four examples were used to

(1) According to the Utility theory the

illustrate the application of the formulation

preferences should be given to the

for

grinding

material removal rate. For all kinds

optimization problems. The example for

of materials the Utility function

definition of the preferences was also

value is the highest when the single

presented.

attribute scaling constant for the

solving

multi-objective

material rate a4=1: U(Basalt (I)) =1, According to the computational results the

heq= 0.9µm; U(Basalt (II)) =0.9988,

following conclusions can be drawn:

heq=

0.159µm;

U(Ceramic

(I))

=0.9992, heq= 1.06µm; U(Ceramic

(1) The Pareto set for heq lies for Basalt (I) in [0.01µm:1µm], for Basalt

(II)) =0.9998, heq= 0.1µm.

[0.01µm:0.2µm], for Ceramic (I) in [0.1µm:2µm], for Ceramic (II) in

(3) For the fixed heq the result could

[0.01µm:0.25µm]. Our results show

vary. The best value of Utility

that the Pareto set lies between

function was achieved for Basalt (II)

extreme points. Any solution, which

when heq=0.267µm, and the single

lies in that interval, is good with

scaling constants for roughness, G-

respect at least to the one objective.

ratio, tool life, and material removal

1

For example if we will pick up heq =

rate

0.065µm and heq2 = 0.65µm we see

a2=0.12, a3=0.08, a4=0.11, U=0.86.

that heq1 increases the values of the

That

objective functions for tool life, G-

preferences should be give to the

ratio and material rate but cause

roughness. So the definition of the

simultaneous decrease in the value of

preferences method allow as setting

roughness objective function.

up preference toward objectives.

The

heq2 solution decreases the values of

objective functions for tool life, G17

respectively

means,

in

are

this

a1=0.69,

case

the

Multi-attribute utility theory is very flexible

Conference on Evolutionary Computation –

to allow adding as many objectives as

Proceedings, Vol. 1, p 82 - 87.

necessary for determining the best design model for grinding process. The tools allow

Keeney, R. L., and Raiffa, H., 1976,

designer to explicitly consider and control,

Decision

as an integrated part of the optimization

Preferences and Value Tradeoffs, Wiley

process, the multiple design objectives,

and Sons, New York.

with

Multiple

Objectives:

easily choose and set up preferences for the objectives in order to increase productivity

Liao, T. W., Sathyanarayanan, G., Storer, R.

and quality of the workpiece surface.

H., and Wilson, G. R., 1989, “Off-line

Besides it is possible to incorporate more

Optimization of Creep Feed Grinding of

decision variables.

Aluminum

Ceramics”

Grinding

Fundamentals and Applications, edited by

The future directions of the research will

S. Malkin and J. A. Kovach, ASME PED

include incorporating other important design

39.

variables such as Vs – wheel wear, mm3, Vw – total stock removal, mm3, vs – cutting

Lortz, W., 1979, “A model of cutting

speed, m/sec, vw – feed rate, m/min, a –

mechanism in grinding”, Wear, Vol. 53, pp.

depth of cut and other.

115-128

Marinescu I,.1984, “Tribo-Technological Aspects of Ceramic Diamond Grinding”,

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