Optimization of Simulated

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Dept. of Mathematical Information Technology

June 13
-
17, 2011

MCDM2011, Jyväskylä, Finland

On Metamodel
-
based
Multiobjective

Optimization of Simulated
Moving Bed Processes

Jussi Hakanen

Dept. of Mathematical Information Technology

University of Jyväskylä, Finland

jussi.hakanen@jyu.fi


Dept. of Mathematical Information Technology

June 13
-
17, 2011

Outline

Motivation

Simulated Moving Bed (SMB) process

Multiobjective optimization of SMBs

Metamodelling

Metamodelling
-
based global optimization of
SMBs

Conclusions and future research

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

Motivation

SMB processes are applied to many important separations in
sugar, petrochemical, and pharmaceutical industries

Dynamic

process operating on periodic cycles,
non
-
convex

(bilinear) functions

→ challenging optimization problem

Optimization of SMBs involves several conflicting objectives →
need for
multiobjective optimization

Efficient (gradient
-
based) local optimizers exist but using global
optimizers is time consuming (one simulation of an SMB takes
seconds)


Is there a need for
global optimization

of SMBs?

Can

metamodelling

techniques enable fast global
optimization of multiobjective SMBs?

June 13
-
17, 2011

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

June 13
-
17, 2011

Based on liquid chromatographic separation


Utilizes the difference in the migration speeds of different
chemical components in liquid


Simulated Moving Bed processes (SMB)

Periodic adsorption
processes for
separation

of
chemical products

* http://www.pharmaceutical
-
technology.com

*

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

June 13
-
17, 2011

5. Recover 2
nd

product

4. Recover 1
st

product

2. Feed

Desorbent

Feed (Mixture of
two components)

1.
Initial state


Column is filled with desorbent

3. Elution

Chromatography (single column)

Chromatographic Column

(Vessel packed with adsorbent particles)

Pump

Adapted from Y. Kawajiri, Carnegie Mellon University

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

June 13
-
17, 2011

Simulated Moving Bed

Cycle

Step

Liquid Flow
Feed
Desorbent
Extract
Raffinate
1
Liquid Flow
Feed
Desorbent
Extract
Raffinate
2
Liquid Flow
Feed
Desorbent
Extract
Raffinate
3
Liquid Flow
Feed
Desorbent
Extract
Raffinate
4
Liquid Flow
Feed
Desorbent
Extract
Raffinate
5
Liquid Flow
Feed
Desorbent
Extract
Raffinate
6
Liquid Flow
Feed
Desorbent
Extract
Raffinate
7
Liquid Flow
Feed
Desorbent
Extract
Raffinate
8
Liquid Flow
Feed
Desorbent
Extract
Raffinate
9
Liquid Flow
Feed
Desorbent
Extract
Raffinate
10
Liquid Flow
Feed
Desorbent
Extract
Raffinate
11
Liquid Flow
Feed
Desorbent
Extract
Raffinate
12
Liquid Flow
Feed
Desorbent
Extract
Raffinate
13
Liquid Flow
Feed
Desorbent
Extract
Raffinate
14
Liquid Flow
Feed
Desorbent
Extract
Raffinate
15
Liquid Flow
Feed
Desorbent
Extract
Raffinate
16
Liquid Flow
Feed
Desorbent
Extract
Raffinate
17
Adapted from Y. Kawajiri, Carnegie Mellon University

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

June 13
-
17, 2011

Cyclic Operation


Switching interval
(Step Time)


Liquid Velocities

Operating Parameters
:

Adapted from Y. Kawajiri, Carnegie Mellon University



Two inlet and two outlet
streams are
switched

in
the direction of the liquid
flow at a regular interval
(steptime)



Feed mixture and
desorbent are supplied
between columns
continuously



Raffinate and extract,
are withdrawn from the
loop also
continuously

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

June 13
-
17, 2011

Multiobjective SMB problem

MCDM2011, Jyväskylä, Finland

Hakanen et al.,
Control & Cybernetics
, 2007

Dept. of Mathematical Information Technology

June 13
-
17, 2011

Multiobjective SMB problem

Case study: separation of
glucose/fructose

(fructose
used in most soft drinks and candies, price varies
depending on purity)

4 objective functions



maximize T = Throughput [m/h]



minimize D = Desorbent consumption [m/h]



maximize P = Purity of the product [%]



maximize R = Recovery of the product [%]

Full discretization

of the SMB model (both spatial and
temporal discretization) → huge system of algebraic
equations

33 997 decision variables and 33 992 equality
constraints

5 degrees of freedom: 4 zone velocities and


steptime

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

June 13
-
17, 2011

Previous results (local optimizer)

4 objective SMB problem was solved by using an
interactive IND
-
NIMBUS software (
Hakanen et al.,
Control & Cybernetics
, 2007
)

IND
-
NIMBUS



an implementation of the NIMBUS
method for solving complex (industrial) problems
(Miettinen,
Multiple Criteria Decision Making '05
, 2006)

Scalarized single objective problems produced by
IND
-
NIMBUS were solved with
IPOPT

local
optimizer
(
Wächter & Biegler,
Math. Prog.
, 2006
)

13 PO solutions generated, single PO solution
took 16.4 IPOPT iterations (27.6 objective function
evaluations) and 65.8 CPU s on average


MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

Remarks of the results

Multiobjective SMB problem is
non
-
convex

(includes bilinear functions)

Can we obtain better results by using
global optimizers for scalarized problems?

One simulation of an SMB takes about 4
-
5
seconds → global optimization takes time

Can we use a faster model for simulation?

June 13
-
17, 2011

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

June 13
-
17, 2011

Metamodelling

Used for approximating computationally
costly functions

Training data
: a set of points in the decision
space and their function values evaluated
with the original model (or obtained from
measurements)

Idea: use training data to fit computationally
simple functions to mimic the behaviour of
the original model

Techniques e.g.
Radial Basis Functions
,
Kriging, Neural Networks, Support Vector
Regression, Polynomial Interpolation

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

Radial Basis Function (RBF)

Training data consists of pairs

Basis functions e.g.


Gaussian:


polyharmonic spline:

June 13
-
17, 2011

MCDM2011, Jyväskylä, Finland

k
i
y
x
i
i
,
,
1
),
,
(



,
5
,
3
,
1
,
)
(


j
r
r
j

0
,
)
(
2






r
e
r
Dept. of Mathematical Information Technology

June 13
-
17, 2011

Metamodelling
-
based
optimization of SMBs

Idea:
train metamodels for each objective function and
use a global optimizer to solve SMB problem

RBFs used in metamodelling with


2500 points in training data (
5
-
dimensional decision
space
); training took ≈ 5 s



for throughput and desorbent consumption



for purity and recovery


mean error
[%] for objectives in validation (50 points):


T:
0.05, D: 0.08, P: 2.6, R: 6.0

Filtered Differential Evolution (FDE)

used as a global
optimizer
(Aittokoski,
JYU Technical report
, 2008)


MCDM2011, Jyväskylä, Finland

2
8
)
(
r
e
r



3
)
(
r
r


Dept. of Mathematical Information Technology

Aim: study applicability of metamodelling
-
based
optimization in SMB problems

Comparison with existing results with IND
-
NIMBUS; PO
solutions produced by solving achievement scalarizing
problems (by Prof. Wierzbicki)




Global optimizer FDE gave
better

results than local
IPOPT:


88% better values (on the average) for the
achievement scalarizing function (from 27% to
121%) → solutions closer to the reference point


SMB optimization problem has local optima!

June 13
-
17, 2011

Results

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

June 13
-
17, 2011

Remarks

Solving an achievement scalarizing problem with
FDE (2000 function evals) took ≈ 15 s

Previously: single PO solution took 16.4 IPOPT
iterations (27.6 objective function evaluations) and
65.8 CPU s on average

Accuracy of metamodelling was excellent for the
first 2 objectives (error < 1%) and sufficient for the
other 2 (2% < error < 6%) → needs more studying

To summarize:
results obtained are promising but
more research is needed

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

June 13
-
17, 2011

Conclusions and future research

Metamodelling was succesfully applied to SMBs


accuracy

varied depending on the objectives

Metamodelling enabled
fast

global optimization for
SMBs

SMB problems seem to have
local

optima

Future research


study more metamodelling for Purity & Recovery
(try different metamodelling techniques)


adaptive

metamodel
-
based optimization


Evolutionary Multiobjective Optimization (EMO)

(or some hybrid) method with metamodelling

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

June 13
-
17, 2011

References

Aittokoski
,Efficient Evolutionary Optimization
Algorithm: Filtered Differential Evolution,
Reports of the
Dept. of Mathematical Information Technology, JYU
,
2008

Hakanen
,
Kawajiri
,
Miettinen

&
Biegler
,
Interactive
Multi
-
Objective Optimization for Simulated Moving Bed
Processes,

Control & Cybernetics
, 36, 2007

Miettinen
, IND
-
NIMBUS for Demanding Interactive
Multiobjective Optimization, In
Multiple Criteria Decision
Making '05
, 2006

Wächter

&
Biegler
,
On the Implementation of an
Interior
-
Point Filter Line
-
Search Algorithm for Large
-
Scale Nonlinear Programming,
Mathematical
Programming
, 106, 2006

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

June 13
-
17, 2011

Acknowledgements

Timo Aittokoski, Tomi Haanpää, Prof. Kaisa
Miettinen & Vesa Ojalehto, JYU

Prof. Lorenz T. Biegler and Yoshiaki Kawajiri,
Carnegie Mellon University, USA

Tekes, the Finnish Funding Agency for Technology
and Innovation (BioScen project in the Biorefine
Technology Program)

MCDM2011, Jyväskylä, Finland

Dept. of Mathematical Information Technology

June 13
-
17, 2011

Thank You!

Dr Jussi Hakanen


Industrial Optimization Group

http://www.mit.jyu.fi/optgroup/


Department of Mathematical Information Technology

P.O. Box 35 (Agora)

FI
-
40014 University of Jyväskylä

jussi.hakanen@jyu.fi

http://users.jyu.fi/~jhaka/en/

MCDM2011, Jyväskylä, Finland