1
ISCI 2012
Ordinal Optimization
Based
Framework
for Optimization of
Computationally Intensive Functions
N.
H. Kulkarni
Tata Research Development and
Design Centre, Tata Consultancy
Services, Pune.
nagesh
.kulkarni@tcs.com
P. R. Zagade
Tata Research Development and
Design Centre, Tata Consultancy
Services, Pune
.
pramod.zagade@tcs.com
Dr.
B. P. Gautham
Tata Research Development and
Design Centre, Tata Consultancy
Services, Pune.
bp.gautham@tcs.com
Abstract
Computational
simulations
are
increas
ingly
use
d
in design
and optimization
of complex systems.
These s
imulation
s
are computationally intensive
and
c
onventional optimization algorithms are
practically not
suitable,
as they may need
large number
of function evaluations of such function.
Different
from conventional optimization algorithms
,
Ordinal optimization (OO) works by
considering “order” in perform
ance rather than “value”
and provide “good enough” solution
.
OO
can be easily coupled with other optimization algorithm
s
such as Genetic Algorithm
s
(
hybrid
algorithm is called as Genetic Ordinal Optimization (GOO)
)
.
In this work, OO
and GOO are
applied to
an industrial problem involving cooling of asymmetric section long steel bars
, where
l
arge length

wise distortions are produced during cooling
which need to be reduced to
tolerable
level
.
Besides providing a possible solution, this work also shows the effi
cacy of OO
and GOO
for application to industrial problems.
Keywords:
Finite Element Modeling,
Genetic Ordinal Optimization,
Ordinal Optimization, Thermo

mechanical Analysis.
1.
Introduction
Computational s
imulations are increasingly used in design
and optimization
of complex
system such as
aircraft
s
,
automobile
s
or setting up/modif
ication
of manufacturing processes.
Thanks to the advancements in computing as well as the
understanding of
physics related to
the problem, it is possible to build high fi
delity simulation models of these complex systems.
High computational cost associated with these high fidelity models have offset the cost
reduction achieved by the advances in
computing
hardware. Design
and optimization
of
such
complex systems may involve
large number of design parameters, which in turn involves
exploring
a
large search space.
Most of the
conventional
optimization algorithms focus on
iteratively
exploring
the
search
space
for
the best solution
. In this process they
require
large
number
of
function and/or gradient
evaluations.
Intuitively, use of such conventional
optimization schema with computationally intensive function
s
is practically infeasible. To
overcome this, very often, approximate models are used that are derived from a small set
of
detailed simulations
(Qian et. al. (2006), Singh et. al. (2009)
)
.
However, this has limited
applicability over the range on which the data is sampled and
more importantly
these
optimization techniques generally do not
take into consideration
the accuracy level of
the
approximation
.
Ordinal Optimization (OO) developed by Ho et. al.
(2007)
is
a
technique
that
addresses these issues
. Different from conventional optimization technique, OO works by
considering “order” in performance rather than “v
alue” and provides set of “good enough”
solution
s
with a
probabilistic
guarantee instead of a unique best solution.
OO is based on two
basic ideas as given below:
“order is much more robust against noise than value”, and
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ISCI 2012
“don’t insist on getting the best b
ut be willing to settle for good enough
.
”
OO gives large reduction in
search space
with a probability guarantee for sufficient
ly
good
enough designs in the
reduced
search space
.
It
can be parallelized to take advantage of the
parallel computing to speed up
computation.
OO can be easily integrated with other
optimization algorithms like Genetic Algorithm (GA)
(
Zhang et. al. (2004)
)
and the hybrid
optimization algorithm is called as Genetic Ordinal optimization (GOO).
In t
his work OO
, GOO and GA
are
applied
to
an industrial problem of optimizing
cooling
conditions for asymmetric section steel long product (bar)
on the run

out table of a hot rolling
mill. Due to the asymmetric section geometry and phase transformation during cooling, the
section gets distorted
, sometimes, the bars get entangled; they also create post cooling
operational difficulties in straightening. Purpose of present work is to reduce this peak
distortion value
to
an
acceptable limit
by
designing
appropriate
cooling conditions.
2.
Ordinal
Optimization
OO was developed by Ho et al.
(2007)
, at Harvard University to deal with computationally
intensive simulation based optimization problems
. In practice engineers are looking for “good
enough” solution which gives sufficient improvement in the d
esired objective in
the
given
limited time rather than “best” fully converged solution, which may take large amount of
time.
OO
helps in this area by providing
a
set of “
good enough
”
solutions with high
probability (> 0.95) and large reduction in the
searc
h space
, instead of
finding
optimal
solution.
P
otential
set of “good enough”
solutions are determined in OO by their relative order
rather than performance value
, hence it is possible to use an approximate model to evaluate
and order designs in search spac
e
in a more robust way
.
2.
1
Terminologies and definitions used in OO
Let
Φ
be the entire search space and
Φ
N
, the search space sampled into
N
points.
Good
enough set (
G(Φ)
or simply subset
G
)
is set
which
contain
s
top

g designs of
Φ
.
Selected set
(
S(Φ
N
)
or simply subset
S
)
is set
which
contain
s
top

s designs of
Φ
N
.
Selection rule
is rule
used
determines how the designs will be selected to form set
S
.
Ordered Performance Curve (OPC)
:
it
is a curve obtained by ordering and plotting
performance of the approximate model (evaluated at the sampled
N
designs).
Lau et al.
(1997)
have carried out large number of experiments for different class of problems and they
observed that only five differen
t types of OPC viz. Flat, U shape, Neutral, Bell and Steep are
possible.
Noise/error level
(σ
2
)
: it
is estimated based on the domain knowledge, physics of the problem
and simplifying assumptions made while deriving the approximate model. It can also be
estimated by the maximum error between true function value and approximate function value.
Alignment level
(
k
):
it is
minimum number of common designs between
G
and
S
.
Alignment probability
(p)
: it
is defined as probability that
S
will contain at least
k
truly good
designs, it is expressed as below:
2.
2
Application procedure of OO
(Ho et. al. (2007))
Following is the application procedure of OO
1.
“Uniformly and randomly sample
N
designs from
Φ”
.
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ISCI 2012
2.
“Use an approximate and computationally fast model to estima
te the performance of
these
N
designs”.
3.
“Estimate OPC class of the problem and noise level or error level (referred as error
level henceforth) of the approximate model. Specify size of the good enough set
G
and
required alignment level
k”
.
4.
“Use Eq
.
1
, devi
sed by Ho et al.
(2007)
to calculate size
s
of selected set
S
as function
of
g
,
k
, OPC class, and error level”.
4
3
2
1
Z
g
k
e
s
Z
Z
Z
(1)
where
Z
1
,
Z
2
,
Z
3
and
Z
4
are the regressed values
taken from Ho et. al. (2007)
5.
“Select observed top
s
(which is the size of
S
) designs of the
Φ
N
as estimated by the
approximate model as the set
S”
.
6.
“Theory of OO ensures that
S
contains at least
k
truly good enough designs which will
be in
G
, with probability no less than 0.95”.
3.
Genetic Ordinal
Optimization (GOO)
Based on the
OO
philosophy
,
Zhang et al.
(2004) proposed a hybrid algorithm based on GA
and OO
called as Genetic Ordinal Optimization algorithm
(referred as GOO
algorithm
henceforth).
GOO combines the ability of OO to identify good enoug
h solutions with high
probability and ability of GA to efficiently explore the search space using genetic operators.
The difference between traditional GA and the hybrid algorithm GOO lies in selection of
appropriate fitness function, stopping criteria and
probability of copying (for crossover and
mutation) (Ho et. al. (2007), Zhang et. al. (2004)). GOO
uses
an approximate
model
as fitness
function.
F
itness function
is used
to evaluate designs in the current population, observed
“good
” designs
based on orde
r
are kept and
the
rest
are
rejected. These observed good
designs
are used in cross
over and mutation to produce next population.
GOO provides well

defined
stopping criteria as given in Eq
.
2
(Ho et. al.)
.
GOO provides upper bound on number of
designs,
N
, w
hich needs to be evaluated to obtain at least one design which belong to top
g
%
of the search space
Φ
.
%)
1
ln(
)
1
ln(
g
p
N
(2)
w
here
N
are designs
/population to be evaluated
in truly top
g
% with probability no less than
p
4
.
Problem Description
4
.1
Introduction
Cooling of hot rolled steel long
bars
with asymmetric sections (the typical section is
as
shown
in
Fig. 1
(a)
) on run

out table
of a hot rolling mill
is a complex process. The asymmetric
section long steel
bars are
hot rolled, cut to required
length and kept in
the
open on the run

out
table. They are supported at finite number of places along the length.
A mechanism moves the
bars slowly in discrete s
lots
; as a result they spend a fix
ed
amount of time
of approx.
30
seconds in each s
lot
before m
oving to
the
next s
lo
t.
Fig. 1 (b) shows the schematic of the
mechanism and placement of the sections.
4
ISCI 2012
Fig
1
:
Schematic of
(a)
rolled asymmetric section
, (b)
run

out table and placement of air
nozzles
The bars
get
cooled from
~
1000

950 °C to room temperature on run

out table as they are
moved slowly. Due to the asymmetric geometry
of the
section
,
differential cooling rates
develop
across the section
causing
significant
distort
ion
. This distortion is further enhanced
due to phase
transformation
s which take
place in the temperature range
8
00

65
0
°
C. As a
result of this large distortion, the long steel bars get entangled on run

out table. This creates
operational difficulties on
the
run

out table as well as post cooling straightenin
g operation
s
.
All these may result in stoppage of the
rolling
line resulting in significant loss of production
time.
Hence
it is desired to limit the distortion of the asymmetric section during cooling by
providing optimum set of cooling conditions
to a va
lue that does not cause operational
problems
.
Study of cooling process revealed that large temperature gradients develop in the
section, which causes different locations of the section undergo phase transformations at
different times and results in large d
istortion of bars. If we reduce the temperature gradient
across the section, then the resulting distortions can be reduced. Some earlier studies
done by
Pietzsch et. al. (2007) and Olden et. al. (1998),
provide external cooling in for
m
of air jets and
water
jets
respectively
to achieve the desired cooling
and reduce distortion
.
A similar concept
during initial numerical experiments
is
found
sufficient to reduce
the
distortion.
Based on it a
scheme of external cooling using air
nozzles
is proposed.
Tabl
e
1
. Air nozzles and corresponding time slots
Nozzle #
1
2
3
4
5
6
7

8
9

10
11

12
Time slot
(
s)
0

30
31
–
60
61
–
90
91
–
120
121
–
150
151
–
180
181
–
240
241
–
300
301
–
360
The schematic is as shown in
Fig
1 (b)
. Nine
sets of nozzles
are placed in nine discrete slots,
(details are provided in
Table 1
) which provide required cooling in
the
initial time span
of 360
s
,
during which phase transformation
s
are
taking place.
The arrangement of nozzles is as
shown in Fig.
1 (b)
.
An optimizati
on problem
is formulated
to obtain the cooling conditions
by adjusting the individual velocities of nin
e sets of nozzles
as follows:
4
.2
Problem Formulation
Objective
: Minimize the maximum effective in plane distortion.
Design Variables
: Individual
velocities of nine sets of nozzles.
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The section analyzed in current paper is as shown in Fig. 1
(a)
. It is made of C

Mn steel (0.1%
C, 1.3% Mn, 0.3% Si, 0.04% Al and 0.02% Nb). The temperature at exit of final roll is 950
°C. Bars are then cooled from 950
°C
to ambient temperature in natural or forced convection
and radiation mode of heat transfer as they move on the run

out table. The air nozzles are
assumed to supply air at maximum velocity of 15 m/s.
The ambient temperature is assumed to
be 40
°C
with
ra
diation
emissivity
of ~
0.8.
4
.2.1
H
igh

fidelity model
C
ooling
processes
of the asymmetric section long steel bar
on run

out table
is model
ed using
Finite Element Method (FEM)
. A thermo

mechanical analysis of cooling of the asymmetric
section is
carried o
ut.
Thermal analysis consists of cooling of hot
asymmetric section
profile
due to convection and radiation and also includes the effects of heat of phase transformation.
The effect of cooling rate on start and end of phase transformations (austenite to fer
rite) is
also considered.
Stress and deformation analysis
uses
the time and temperature histories along
cross section and length of the
asymmetric section profile from thermal analysis, computes
strain
due to thermal expansion as well as phase changes. Thi
s information is further used to
compute elastic

plastic
distortion due to thermal gradient in the cross section.
Fig
2
: Schematic of
the
finite element model descri
bing different surface segments
for
applying heat transfer conditions
Fig.
2
shows finite element
discretization
and schematic of air

cooling arrangements. The
section is inclined at 30
°
to horizontal as it is placed on the run

out table.
Air nozzle
is placed
below the section as shown in
Fig.
2
. T
he mode of heat transfer on
top s
urface is natural
convection and radiation, while it is forced
convection and radiation on
bottom
and left
surface
, except for the first nozzle set
, where air is supplied
from top and hence there is forced
convection and radiation is on top surf and natura
l convection and radiation on bottom
surface
. On the bottom surface
heat transfer coefficient is not constant and it is function of
d
istance from the edge where air nozzle
is placed. To facilitate application of different heat
tr
ansfer conditions on
surfac
e of the section, it is divided in
to
7 segments as shown in
Fig.
2
.
The heat transfer c
oefficient is calculated at mid

point of all
surfaces
using equation for
steady state forced convection heat transfer mode
.
The simulation is carried out using commerci
al FEA software package ABAQUS
®
using 8
noded solid elements with reduced integration.
The
effects of
phase transformations,
volumetric changes and heat transfer
are incorporated through appropriate user routines
.
T
hermal properties
required
for heat trans
fer analysis include density, heat capacity
, latent
heat of
phase transformations, thermal conductivity and temperatures at which austenite to
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ISCI 2012
ferrite transformations start and end. Density is assumed to be constant at 7850 kg/m
3
.
H
eat
capacity and
thermal conductivity as a function of temperature is taken from
Hot Rolled
Carbon
S
teel
T
hermal
P
roperties
(as seen Sept. 2011)
. The
information
for
temperature
at
which austenite to ferrite transformations start and end
depending on cooling rate,
is
calcu
lated
from CCT diagram for the corresponding
C

Mn
steel. The deformation analysis
requires coefficient of thermal expansion and volumetric changes due to phase
transformations,
which are
taken from
work of Mohapatra et. al. (2006)
. The data for
variation o
f yield strength
and
modulus of elasticity as function of temperature is taken from
Olden et. al. (1998)
and Basu et. al. (2004)
respectively.
4
.2.2
Approximate model development
It is observed that
simulation model of the cooling process is computationall
y intensive
hence
it
is not desirable to
use it
directly in OO
or GOO
. An approximate model is developed using
Support Vector Regression (SVR).
SVR is a machine

learning tool
used
for constructing data

driven non

linear models
.
It is based on Support Vecto
r Machine
(Vapnik (1995))
, which is
method of supervised learning used to analyze the data, classify it into groups and develop
regression models for them. In this paper, SVR model was implemented based on the work
done by Palancz et al.
(2005)
. SVR needs
training and testing datasets, these datasets were
generated by uniformly and randomly sampling designs in the search space.
Fig
3
shows
performance of SVR. It can be seen that most of the predictions are within 10% error
.
Fig
3
: Performance
of SVR model, (a) training set
(b) test set
5
. Results
OO
, GOO and GA
are
used to minimize the maximum distortion by optimizing individual
velocities of all nine sets o
f nozzles
. The value of
g
in Eq
.
1
is taken as 1
%. The
desired
alignment level is taken
as,
k
=
1 and desired alignment probability
p
is taken as 95%.
A
binary coded GA is applied to this problem so that the performance of GOO can be
benchmarked. For GA and GOO the probability for crossover is considered to be 1.0 and
probability for mutatio
n is considered to be 0.1.
Since the designs found in OO
, GOO and GA
are random in each run, we have conducted run of
them
five times
each
.
The detailed
simulation with natural convection/radiation mode of heat transfer in the existing setup is used
as
benchmark for comparison.
For the application of OO, a
total number of 300
design
s
were sampled
randomly
in the
search space
.
The
approximate
model developed using
SV
R
is used to evaluate the
performance of these
300
designs. “Horse Race” selection rule is
used and OPC is generated,
7
ISCI 2012
which
is
found to be
of Bell type.
The error
level
is estimated by evaluating the
approximate
model at the
train and test datasets of SVR
(for which the detailed simulation results are
available)
and
computing
the normalized err
or
.
The error level is found to be 0.
69. R
egression
parameters used to estimate size of
S
in Eq.
1
are based on
error
level of
1
.
0.
A linear
interpolation
suggested by Ho et al.
(2007)
is performed
to obtain the actual size of
S
,
which
is found out to be 85
. Top
85
designs from the OPC curve are selected and the detailed
FEM
based simulations are performed on them to find most acceptable solution
.
Subsequent to OO, we have applied GOO.
A
binary coded hybrid
GOO
algorithm gi
ven by
Zhang et al.
(2004)
is
used
in this study
.
T
he population
size required, for the specified
alignment probability and
G
,
is determined using Eq
.
2
, which comes out to be
N
= 298. A
n
initial population of 100
random design
s
is used.
GOO
is allowed to run for
3
generations
which results
in evaluation of
300
random design
s
.
The
approximate
model generated using
SV
R is used
as the fitness function
.
Top
l
(where
l
is the number of selected good designs, i.e.
taken as
60
as per Zhang et. al. (2004)
) designs out of the population of 100 desig
ns are
selected for crossover and mutation.
A selection probability, as given in Zhang et. al. (2004)
for
i
th
selected design proportional to
2(
l

i
+1)/(
l
(
l
+1))
is used.
O
n reaching the stopping
criteria
(which is evaluation of total
N
designs given by Eq
.
2
)
,
detailed simulations are
performed for the best
l
designs obtained through all generations and
the best design
among
them
is obtained.
Finally, we applied GA for the same.
A binary coded GA is applied for the optimization
problem. Initial population of
100 random designs is sampled from the search space.
The
approximate
model generated using the
SV
R is used
as the fitness function
.
All the designs
are considered for selection. GA is also allowed to run for 3 generations and the best design
found is cons
idered for detailed simulation. The total number of function evaluations are in
the optimization loop is same as that of GOO, but the detailed simulation is carried out for
only one final design point provided as optimum for GA as there is no specific guid
ance
exists.
Table 2 gives
shows the optimal values of distortions obtained using OO, GOO and GA in
five runs; average distortion of these five runs is calculated and reported in Table 2
.
The
average distortion obtained using OO, GOO and GA is compared ag
ainst the distortion
obtained in the existing setup with
no air cooling
in Table 3
.
Table
2
. Distortion (m) obtained using OO
, GOO and GA
Run #
OO
GOO
GA
1
0.499
0.569
0.522
2
0.495
0.560
0.579
3
0.483
0.504
0.612
4
0.543
0.484
0.559
5
0.571
0.522
0.525
Avg
0.518
0.528
0.559
Table
3
. Comparison of performance of OO
Method
Existing industrial
setup
OO
GOO
GA
Distortion, m
0.814
0.518
0.528
0.559
% Improvement
BENCHMARK
36.36
35.13
31.32
8
ISCI 2012
Table 3
shows that OO and GOO are able to achieve ~ 35% improvement in reduction of the
maximum effective distortion compared to GA which is able to achieve ~ 31% improvement.
The improvements obtained by OO & GOO as compared to GA are 7.3% and 5.5%
respectively.
An infinite continuous hyperspace in nine variables was reduced to testing of
300 random designs for OO and 100 random designs for GOO and these are evaluated using
approximate function. Based on the
OO output,
detailed evaluation of 85 and 17 designs for
OO and GOO respectively
are carried out using high fidelity model
.
The mutation
probability
and number of generations are found to
affect
the results of GOO.
Table 4 below shows
comparison
of
runs of two independent sets of GOO.
As stated earlier 5
runs of each set are carried out.
Table
4
: Effect of parameters on GOO
Distortion (m)
Set 1
(cross

over = 1.0
mutation = 0.1
no of gen. = 3)
Set 2
(cross

over = 1.0
mutation = 0.2
no of gen. = 10)
Average
0.528
0.488
min
0.484
0.481
max
0.469
0.499
std. deviation
0.036
0.008
From table 4, it can be seen that as the mutation and number of generations are increased the
results of GOO in terms of minimum, maximum and standard deviation are significantly
improved.
This can be attributed to the increased degree of randomness and wider exploration
of the search space.
6
.
Summary
The cooling process of asymmetric long steel bar made of C

Mn steel was modeled using
FEM. The simulations show that there is signi
ficant distortion of the section due to the
asymmetric cooling and phase transformation. A scheme of sequence of forced cooling using
air nozzles was proposed to minimize the distortion to acceptable value. OO
, GOO and GA
are
used to
obtain
the velocities
of the nine sets of nozzles to minimize the distortion.
It is
observed that OO and GOO perform better compared to GA when all the key parameters of
optimization are kept same.
A significant reduction is observed in the search space when OO
and GOO are used
.
OO when integrated with GA improves the performance of the later and
results in a better algorithm compared to individual GA. GOO require
lower number of
detailed evaluations as compared to OO
.
OO
and GOO are
observed to be promising tools for
obtaining
“good enough” solutions to many problems with reduced computational cost.
Acknowledgements
Authors greatly acknowledge the management of Tata Consultancy Services Ltd. for
providing support for carrying out this research.
9
ISCI 2012
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Biographies
N H Kulkarni
N H Kulkarni is a scientist at Tata Research Development and Design Centre, an innovation
Laboratory of Tata Consultancy Services
(TCS)
. He obtained his M Tech in Mechanical
Engineering from IIT Bom
bay.
He has been with TCS since 2003. His interests include
mathematical modeling, design and optimization.
10
ISCI 2012
P R Zagade
Pramod Zagade is a Scientist at Tata Research Development and Design Centre, an
Innovation Laboratory of Tata Consultancy Services. He o
btained his BE in Mechanical
Engineering and MTech in Design Engineering from Pune University. Pramod has been with
TCS since 2006. His interests include solving industrial problems using numerical simulation
based approach. He has worked in applying finit
e element models to hot rolling, sheet metal
stamping, simulation of tyre wear and low cycle fatigue using continum damage mechanics
approach. Pramod's current research focus is towards integrated computational materials
engineering.
B P Gautham
B P Gaut
ham is a Principal Scientist at Tata Research Development and Design Centre, an
Innovation Laboratory of Tata Consultancy Services. He obtained his BTech in Mechanical
Engineering and PhD in Applied Mechanics from Indian Institute of Technology

Madras,
Ch
ennai. Gautham has been with TCS since 1994. His interests include application of FEA
and other computational tools for industrial processes such as metal forming, solidification of
metals & ceramics and melt processing of polymers. The research focus is o
n developing
finite element based virtual models and simulation models for the above processes.
Gautham’s current research focus is towards integrated computational materials engineering.
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