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Computers chem. Engng Vol. 20, Su ppl.,pp. $389-$39 4, 1996Copyright © 1996 ElsevierScience Ltd

Pergamon S0098.13 54(96)00075 .0 Printed in Great Britain. All rights reserved0098-1354196 $15.0 0+0.0 0

R o b u s t D e s ig n U s i n g a n E f fic ie n t S a m p l i n g T e c h n iq u e

Urmi la M. D iwekar & Jayant R . K a lagnanam

D e p a r t m e n t o f E n g i n e e r i n g & P u b l i c P o l ic y

C a r n e g i e M e l l o n U n i v e r s i t y

P i t t s b u r g h , P A 1 5 2 1 3

Abstract

Th e c o n c e p t o f ro b u s t d e s ig n in v o lv e s i d e n t i f ic a t io n o f d e s ig n se t t i n g s t h a t ma k e th e p ro d u c t p e r fo rm a n c e l e ss se n s i t i v e

to th e e f fe c t s o f se a so n a l a n d e n v i ro n me n ta l v a r i a t io n s . Ba tc h p ro c e sse s a re m o re f l e x ib l e c o m p a re d to c o n t in u o u s

p ro c e sse s a n d a re f r e q u e n t ly u se d in th e se s i t u a t io n s . S to c h a s t i c o p t imiz a t io n me th o d s p ro v id e a g e n e ra l a p p ro a c h

to ro b u s t /p a r a m e te r d e s ig n a s c o m p a re d to c o n v e n t io n a l t e c h n iq u e s . H o w e v e r , t h e c o mp u ta t io n a l b u rd e n o f th e se

a p p ro a c h e s c a n b e e x t re me a n d d e p e n d s o n th e sa mp le s i z e u se d fo r c h a ra c t e r i z in g th e p a ra me t r i c v a r i a t i o n s a n d

u n c e r t a in t i e s . In t h i s p a p e r , w e p re se n t a n o v e l sa mp l in g t e c h n iq u e . Th e e x a m p le o f ro b u s t b a t c h d i s ti l l a t i o n c o lu mn

d e s ig n i l l u s t r a te s t h at t h e n e w sa mp l in g t e c h n iq u e o f fe r s s ig n i f i c a n t c o mp u ta t io n a l sa v in g s a n d b e t t e r a c c u ra c y .

K e y w o rd s : ro b u s t d e s ig n , O f f - l i n e q u a l i t y c o n t ro l , s t o c h a s t ic o p t imiz a t io n , sa mp l in g te c h n iq u e s , b a t c h d i s t i l l a -

t io n , M o n t e C a r l o M e t h o d s .

1 Introduction

Ro b u s t /Pa ra me te r d e s ig n i s a n o f f - l i n e q u a l i t y c o n t ro l me th o d p o p u la r i z e d b y th e Ja p a n e se q u a l i t y e x p e r t G . Ta g u c h i ,

fo r d e s ig n in g p ro d u c t s a n d ma n u fa c tu r in g p ro c e sse s t h a t a re ro b u s t i n t h e f a c e o f u n c o n t ro l l a b l e v a r i a t io n s (Be n d e l l

e t a l . , 1 9 9 0) . A t t h e d e s ig n s t a g e , t h e g o a l o f p a ra me te r d e s ig n i s t o i d e n t i fy d e s ig n se t ti n g s t h a t ma k e th e p ro d u c t

p e r fo rm a n c e l e ss s e n s i t i v e to t h e e f fe c t s o f ma n u fa c tu r in g a n d e n v i ro n me n ta l v a r i a t i o n s , a n d d e t e r io ra t io n .

Th e mo s t g e n e ra l a p p ro a c h to t h e ro b u s t o r p a ra m e te r d e s ig n p ro b le m i s t o c o u p le a n o p t im iz e r d i r e c t ly w i th t h e

c o m p u te r s imu la t io n mo d e l u s in g s to c h a s ti c d e sc r ip t io n s o f t h e n o i se f a c to r s (Bo u d r ig a , 1 9 9 0; D iw e k a r a n d R u b in ,

1 9 94 ) . Th e se g e n e ra l c l a ss o f p ro b le m s c a n b e . c h a ra c te r i z e d a s s to c h a s t i c o p t imiz a t io n p ro b le ms . Th e s to c h a s t i c

o p t imiz a t io n p ro b le m in v o lv e s t h e e v a lu a t io n o f a n a g g re g a te me a su re (u se d a s a p e r fo rm a n c e s t a t i s ti c s ) d e r iv e d f ro m

a m u l t iv a r i a t e p ro b a b i l i t y d i s t r i b u t io n . Fo r n o n l in e a r mo d e l s , t h i s i s d o n e n u m e r i c a l ly u s in g a r e p re se n ta t iv e sa m p le

f ro m th e mu l t iv a r i a t e sp a c e a n d h a s t o b e r e p e a t e d a t e a c h o p t imiz a t io n i te ra t io n . Th e re fo re , a n e f f i c i e n t sa mp l in g

sc h e me w h ic h r e d u c e s t h e n u m b e r o f sa m p le s r e q u i re d fo r e a c h i t e ra t io n c a n s ig n i f i c a n t ly imp ro v e th e c o mp u ta t io n a l

e f f i c a c y o f t h e s to c h a s t i c o p t imiz a t io n p ro c e d u re .

In t h i s p a p e r , w e p re se n t a n e w a n d e f f i c ie n t sa mp l in g t e c h n iq u e u s in g th e sh i f t e d H a m me rs l e y p o in t s fo r u n i fo rm ly

sa mp l in g a k -d ime n s io n a l u n i t h y p e rc u b e . Th i s n e w sa m p l in g te c h n iq u e s r e q u i re s f a r f e w e r sa mp le s a s c o mp a re d too th e r t e c h n iq u e s to a p p ro x ima te t h e me a n a n d v a r i a n c e o f d i s t r i b u t io n s d e r iv e d b y p ro p a g a t in g a r e p re se n ta t iv e sa mp le

( fo r t h e in p u t s ) o v e r n o n l in e a r fu n c t io n s . Fo r t h e ro b u s t d e s ig n p ro b le m p o se d in t e rms o f s to c h a s t i c o p t imiz a t io n , th e

u se o f t h i s e f f i c i e n t sa mp l in g t e c h n iq u e c a n s ig n i f i c a n t ly a l l e v i a t e t h e c o m p u ta t io n a l b u rd e n . We i l l u s t ra t e t h i s i n t h e

$389

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context of robust design of a batch distillation column which is subject to feedstock variations, modeling uncertainties

and measui'ement errors and report computational savings of upto a factor of 10.

2 M on te Carlo and Latin Hypercube Sampling Techniques

Perhaps one of the best known methods for sampling a probability distribution is the Monte-Carlo sampling technique

which based on the use of a pseudo random number generator to approximate a uniform distribution, U(0,1) with n

samples. The specific values for each input variables are selected by inverting the n samples over the cumulative

distribution function. A Monte Carlo sample has the property that successive points are independent. However, in

most applications, the actual relationship between successive points in a sample has no physical significance, hence

the independence/randomness of a sample for approximating a uniform distribution is not critical (Knuth, 1973).

Moreover, the error of approximating a distribution by a finite sample depends on the equidistr ibutionproperties of the

sample used for U(0,1) rather than it' s randomness. Once it is apparent that the uniformity properties are central to the

design of sampling techniques, constrained or stratified sampling becomes appealing (Morgan and Henrion, 1990).

Latin Hypercube Sampling (LHS) is one form of stratified sampling which can yield more precise estimates of the

distribut ion function (Iman and Shortencarier, 1984). The range of each u i is divided into non overlapping intervals

of equal probability. One value from each interval is selected randomly with respect to the probability density in the

interval. The n values thus obtained for ui are paired in a random manner with the n values of X2 and these n pairs are

combined with n values of X3 and so on to form n k-tuplets. The random pairing is based on a pseudo random number

generator. The main shortcoming with this stratification scheme is that it is one dimensional and does not provide

good uniformity properties on a k-dimensional unit hypercube. Until recently, the only known design for uniformity

on a d-dimensional hypercube was a uniform grid (Papageoriou and Wasilkowski, 1990). However, the uniform grid

requires exponential sample points in the number of variables for good equidistributionproperties. This paper presents

a new sampling technique which uses a quasi-random generator based on shifted Hammersley sequences. The shifted

Hammersley sequences were shown to be a minimumdiscrepancy design by Wozniakowski (1991 ). This new sampling

technique shows superior convergence properties.

3 The New Sam pling Technique

Since most of the stochastic optimization problems involve integrals o f some probabilistic functional, consider the

approximation of an integral o f a k-dimensional continuous function by sampling its values at a finite set of points.

Fo r sake of simplici ty let us assume that the integration is restricted to a k-dime.nsional uni t cube. One.straightforward

approach is to place the points along equally spaced intervals on a d-dimensiona l grid. Although this is a good

arrangement, the number of points needed to keep the average error less than e is roughly proportional to ~ . The

traditional alternative is to use a" Mot~te Carlo method where the points are chosen completely randomly using a

pseudo-random number generator. The approximation to the integral is then based on the funct ion evaluation at these

points. Although on the average, the number of points required to keep the error within ~ is bound by ~ there is no

methodical way for constructing the sample points to achieve the bound (Papageoriou and Wasilkowski, 1990).

In this section we describe a new sampling technique based on the use of the shifted Hammersley points. We

call this new technique - the Shifted-Hammersley Sampling (SHS) technique, The basic idea behind this technique

is to replace a Monte Carlo integration by a quasi-Monte Carlo scheme. The choice of an appropriate quasi-Monte

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Carlo sequence is based on the concept of discrepancy. Discrepancy is a quantita tive measure for the deviat ion of the

sequence from uniform distribution. Therefore it is typically desirable to choose a low discrepancy sequence. Some

examples o f low discrepancy sequences are the Hahon and Hammersley sequences. Without embarking on a detailed

discussion of these issues (the interested reader is referred to Niederreiter (1992), it is apparent that we are faced with

the issue of which sequence should one use for the design of a quasi-Monte Carlo sampling technique. The shifted

Hammersley sequence has been shown (Wozniakowski 1991) to have the minimum discrepancy for the class of real

continuous functions in the average case setting.

We provide a comparison of the performance of the Shifted Hammersley sampling (SHS) technique to that

of Latin Hypercube (LHS) and Monte Carlo (MCS) techniques. The comparison is performed by propagating

samples derived from each of the techniques for a set of n-input variables (X d, through various nonlinear functions

(Y = f (X l, X2, ..., Xn)) and measuring the number of samples required to converge to the mean and variance of

the derived distr ibution for Y. Since there are no analyt ic approaches (for stratified designs) to calculate the number

of samples required for convergence, we have conducted a large matrix of numerical tests. The design of the test

matrix included varying of the type of function, the number of input variables, Xi, type of input distribution, and

the correlation structures between them. The details of the test matrix are described below. A total of four sampling

techniques have been compared: Monte Carlo, random Latin Hypercube, median Latin Hypercube, and the shifted

Hammersley. The number of input variables used was varied between 2 and 10. Five different kinds of funct ions were

used including 1) linear additive function, 2) multiplicative, 3)quadratic, 4) exponential, and 5) logarithmic. Three

types of distributions have been used for the input variables X,. Two of them, uniform and normal are symmetric

and the the third is a skewed distribution, Lognormal. Three types of correlation structures have been used: the first

is a zero correlation, and the other two sets use a correlation of 0.5 and 0.9 between the input variables. This matrix

represents a total of 180 data sets (4 sampling techniques x 3 types of distributions x 3 correlation structures × 5

functions) for each set of input variables, Xi . As the number of input variables is varied from 2 to 10, it adds another

factor of 9, i.e. 1620 data sets. However, in the interests of space and clarity we will present only the results which

highlight the main findings of this numerical experiment. It has been established in the literature that LHS performs

significantly better than MCS. The numerical experiments supported this result. Furthermore, it has been found that

the SHS sampling technique has a much faster convergence rate as compared to LHS, anywhere from a factor of 1.5

to 100 and larger! For more discussion on the sampling technique and comparisons, please refer to Kalagnanam and

Diwekar (1995).

4 R o b u s t D e s i g n o f a B a t c h D i s t il la t i o n C o l u m n

Batch distillation is an important unit operation frequently used for small-scale production. Batch distillation is prefer-

able to continuous distillation when small quantit ies of high technology/high-value-addedchemicals and biochemicals

need to be separated. The most outstanding feature o f batch distillat ion is its flexibility. This flexibility allows one to

deal with uncertaint ies in feed stock or product specification (Diwekar, 1995). Therefore the concept of robust design

is particularly relevant to batch distillation.

The sudden increase in the product ion of high value-added, low-volume specialty chemicals and biochemicals in

recent years has generated renewed interest in batch dis tillation design. However, the current design procedures arestill based on deterministic framework. We first present a description of the robust design problem in the context of

batch dis tillation column design and then outline the sample size required to characterize the variance of the output

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T a b le 1 P A R A M E T E R S A N D T H E I R V A L U E S U S E D I N T H E S T U D Y

Pa ra me te r V a lu e s U n i t s E i%

c~ 2. 0 5

X F 0 .5 m o l e f r a c t i o n 10

R 2 .5 5

V 1 00 m o l / h r 5

F 1 0 0 m o l 10

x~9 0.95 m o l e f r a c t i o n

D spec 36 m o l e

f ro m th e p ro c e ss t h a t n e e d s to b e c o n t ro l l e d fo r q u a li t y . F in a l ly , w e p re se n t t h e c o mp u ta t io n a l b u rd e n o f so lv in g th e

ro b u s t d e s ig n p ro b le m fo r b o th th e sa m p l in g t e ch n iq u e s .

Fo r s imp l i c i t y , c o n s id e r a b in a ry m ix tu re w i th t h e f e e d c o mp o s i t i o n o f mo re v o la t i l e c o mp o n e n t t o b e X F ( m o l ef ra c t io n ) a n d to t a l f e e d F (mo le s ) t o b e se p a ra t e d in a b a t c h d i s t i ll a t i o n c o lu mn . Th e re a re v a r i a b i l i t i e s a n d f lu c tu a t io n s

in f e e d c o mp o s i t i o n a n d fe e d a mo u n t o v e r d i f f e re n t b a t c h e s w h ic h a mo u n t s t o sa y in g th a t t h e f e e d c o m p o s i t i o n (x F)

a n d fe e d ( F ) a re u n c e r t a in q u a n t i t ie s . Th e re a re m e a su re me n t e r ro r s i n q u a n t it i e s li k e v a p o r b o i lu p ra t e V ( fu n c t io n

o f h e a t i n p u t t o t h e r e b o i l e r ) , a n d r e f lu x r a t io R . Th e th e rmo d y n a mic e r ro r s l e a d to u n c e r t a in t i e s i n r e l a t i v e v o la t i l it y

(a ) p re d ic t io n s . G iv e n th e a b o v e v a r i a b i l i t i e s a n d u n c e r t a in t i es w e w a n t t o d e s ig n a b a t c h c o lu mn w h ic h w i l l ma in ta in

th e a mo u n t o f p ro d u c t w i th t h e g iv e n p u ri ty . Th e se v a r i a t io n s a re a ssu me d to b e a t E l % error leve ls in the inputs ,

E~ = a i /# ~ x 1 0 0 n o rma l ly d i s t ri b u te d . N u m e r i c a l v a lu e s o f t h e d e s ig n a n d n o i se v a r i a b le s (n o min a l v a lu e s , # i a n d %

e r ro r l e v e l s E i ) a re g iv e n in Ta b le 1 . So th e p ro b le m o f ro b u s t d e s ig n t ra n s l a t e s in to f i n d in g th e n u mb e r o f t h e o re t ic a l

p l a t e s a n d re f lu x r a t io r e q u i re d to m in imiz e th e v a r i a n c e in t h e d i s t i l l a te a mo u n t o f sp e c i f i c p u r i ty X D * .

Th e i t e ra t iv e n a tu re o f t h e ro b u s t d e s ig n c a l l s fo r u se o f s imp l i f i e d mo d e l s . Th e sh o r t c u t me th o d p re se n te d

b y D iw e k a r a n d Ma d h a v a n (1 9 9 1 ) p ro v id e s a n e ff i c ie n t a l t e rn a tiv e fo r ro b u s t d e s ig n . A p a r t fro m c o m p u ta t io n a l

e f f ic i e n c y , l o w e r m e mo ry re q u i re me n t s , a n d th e a lg e b ra i c e q u a t io n o r i e n t e d fo rm o f t h e sh o r t c u t me th o d , i t i s a l so

u se fu l i n i d e n t i fy in g f e a s ib l e re g io n o f o p e ra t io n c ru c i a l fo r d e s ig n p ro b le ms . Fu r th e rmo re , t h e d im e n s io n a l i t y o f

th e p ro b le m d o e s n o t i n c re a se w i th i n c re a s in g n u mb e r o f p l a t e s a n d th e d e s ig n v a r i a b l e n u mb e r o f t h e o re t i c a l p l a t e s

i s n o t a n in te g er . Th e se a t t r i b u te s ma k e s th e sh o r t c u t me th o d d e s i r a b l e fo r ro b u s t d e s ig n p ro c e d u re s . Th e ro b u s t

d e s ig n p ro b le m in v o lv e s so lu t io n o f a s to c h a s t i c o p t imiz a t io n p ro b le m w h e re th e o p t imiz e r i n t h e o u te r l o o p f in d s t h e

d e c i s io n v a r i a b l e s , n u mb e r o f p l a t e s N a n d re f lu x r a t io R a n d th e in n e r l o o p i s t he s to c h a s t i c mo d e l . Th e sh o r t c u t

me th o d p ro v id e s f e a s ib i l i t y c o n s id e ra t io n s w h ic h a re u se d in t h i s i t e ra t iv e p ro c e d u re . Th e o p t imiz e r min imiz e s t h e

e r ro r v a r i a n c e o f t h e e r ro r i n d i s t i l l a te a mo u n t c o l l e c t e d a n d th e sp e c i f i e d d i s t i ll a t e a mo u n t Dspec.

Sin c e th e o b je c t iv e f ro m a ro b u s t d e s ig n p e r sp e c t iv e is t o m in imiz e th e v a r i a n c e in t h e a m o u n t o f d i s t i l l a t e ,

w e c h a ra c t e r i z e t h e n u mb e r o f sa m p le s r e q u i re d to e s t ima te t h e v a r ia n c e to w i th in 1 % o f i t s v a lu e u s in g b o th La t in

H y p e rc u b e a n d Sh i f t e d H a m me rs l e y p o in t s . Th e S h i f t e d H a m me rs l e y p o in t s r e q u i re s a b o u t 6 5 0 p o in t s t o c o n v e rg e

a s c o mp a re d to 6 5 5 0 p o in t s r e q u i re d b y th e La t in H y p e rc u b e Sa mp le . F ig u re l a p lo t s th e v a r i a n c e a s a fu n c tio n o f

n u mb e r o f sa mp le s . F ig u re l b sh o w s th e v a r i a t i o n in t h e p ro d u c t a mo u n t b e fo re a n d a f t e r t h e ro b u s t d e s ig n , w h e re th e

v a r i a n c e i s r e d u c e d f ro m 3 6 4 to 1 0 8 b y c h a n g in g th e th e o re t i c a l n u mb e r o f p l a t e s f ro m 1 0 .9 5 to 2 0 .8 8 a n d re f lu x r a tio

f ro m 2 .5 to 3 .0 . Th i s v a r i a n c e r e d u c t io n a t th e e x p e n se o f o v e rd e s ig n in g is w o r th w h i l e w h e n th e p ro d u c t i s v a lu a b le

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A

1 9 5 0 . 4 1 I I I I

0 . 3 5 . - .J ~ " ~ ( \

i ~ - i! P : ! . . . . . . . . ~ . . . . . . . . . . . . L L

- : { l i l i ! i = -w1 8 5 .! . 1 . . . . . !7_i g o . 2 ,. _ ~ , aI I o . , 5 '• , ~

t' ~

1 8 0 ~ 0 . 1

f l " 0 . 0 5

1 7 5 ! 0 - 1 1 I I I

1 2 5 0 2 5 0 0 3 7 5 0 5 0 0 0 6 2 5 0 7 5 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0

n A m o u n t o f D i s t il la t e , m o l e s

Figu re 1: (a)Variance o f error in D using LH S (dotted l ine) and SHS (solid l ine). 1% error band is show n by broken

lines; (b) Variations in the am oun t o f dist il late, befo re (solid l ine) and a fter (broke n l ine) design for quali ty

(g iven that designs w hich can g ive very smal l am ount of product are avoided). The var iance can be fur ther reduced by

Taguch i ' s t o l e rance des i gn m et hod .

5 C o n c l u s i o n s

This paper presented a new sam pl ing technique based on shi fted Ham me rsley poin ts. This new sampl ing technique

is shown to ha ve bet ter uni formi ty proper ties which reduces the co mp utat ional intensi ty of s tochastic opt imizat ion

problem considerably . A robust design conc ept was in troduced in the context of batch d ist il lat ion colum n design

ope rating unde r internal and external uncertainties. Since the robust design con cep t essential ly involves solution o f

stochastic optimization problem, i t was found that this sampling technique is always preferable for robust/parameter

design problems. This i s because of i ts h igh precis ion and consis tent behavior coupled wi th great computat ional

efficiency.

R e f e r e n c e s

Bendeil A., J. Disney, and W. A. Pridm ore (1989), Tagu chi metho ds : applications in world industry, IF S

Publication, Springer-Verlag, UK.

Boud r iga S. (1990) , Evaluation of parameter design methodologies or the design of chem ical processing units,

Master 's thesis, Universi ty of Ottawa, Ottawa.

Diw eka r U. M. (1995), Ba tch Disti llat ion: Simulation, Optimal D esign and Control , Tay lor and Francis Inter-

national Publishers:W ashington D.C.

CAOE ZO:I3(A)-N

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$394 European Symposium on Computer Aided Process Engineering--6. Part A

Diwekar U. M. and K. P. Madhavan (1991), Multicomponent batch dist i l lat ion column design, Industrial and

Enginee ring Chemistry Research , 30, 713.

Diw eka r U. M. and E. S. Rub in (1994), Parameter D esign Metho d using Stochastic Optimization with AS PEN ,

I & E C R e s, 33, 292.

Iman , R.J . and Shor tencarier, M.J .(1984) A FO RTR AN 77 Program and U ser ' s Guide for Generat ion of Latin

Hypercube and Ran dom Sampl es fo r Use wi th C ompu t er Models , NU REG /CR- 3 6 2 4 , S AND 8 3 - 2 3 6 5 , Sandia

National Laboratories, Albuquerque, N.M.

Im an R. L. and W. J. Cono ver (1982), Small sam ple sensitivi ty analysis techniq ues for com pute r models, with

an application to risk assessment, Com munications in S ta tis t ics , A17, 1749.

Kalagnan am J . R. and U. M . D iwekar (1995) , A n eff icient sam pl ing technique for of f - l ine qual ity cont ro l , to

appear in Technometrics.

Knuth, D .E. (1973) , Th e Art of Com puter Program ming, Volume 1: Fundam ental Algor i thms, Addison-Wesley

Publi shing Co. , Reading MA .

Morgan G . and M. Henr i on (1990),Uncertainty: A guide to dealin g with uncertainty in quantitative risk an d

policy analysis, Cam bridge Univers i ty Press, N ew York.

Niederreiter, H. (1992), Ra n d o m Nu m b er G en era t io n a n d Q u a s i- Mo nte Ca r lo m e th o d s , SIAM, Philadelphia,

PA .

Papageo rgiou, A. and W asilkowski , G.W.(1990), On average case com plexi ty of mul t ivar iate problems, Jo u rn a l

o f Complexity , 6, pp: l -6 .

Wozniakowski, H.(1991), Ave rage Ca se Com plexity of Multivariate Integration, Bullet in o f the American

Ma thematica l Society, 24(1), pp: 185-194.

paper 69

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