Sch i en eihe CIplus, Band 4/2015
He ausgebe : T. Ba z-Beiels ein, W. Konen, B. Naujoks, H. S enzel
Sequen ial Pa ame e
Op imiza ion in Noisy
En i onmen s
Thomas Ba z-Beiels ein, Ch is ian Jung, Ma in Zae e e
Sequen ial Pa ame e Op imiza ion in Noisy
En i onmen s∗
Thomas Ba z-Beiels ein, Ch is ian Jung, Ma in Zae e e
SPOTSe en Lab
Cologne Uni e si y o Applied Sciences
S einm¨ulle allee 1
51643 Gumme sbach
www.spo se en.de
May 27, 2015
Abs ac
Sequen ial Pa ame e Op imiza ion is a model-based op imiza ion me hodology, which
includes se e al echniques o handling unce ain y. Simple app oaches such as sha p-
ening and mo e sophis ica ed app oaches such as op imal compu ing budge alloca ion
a e a ailable. Fo many eal wo ld enginee ing p oblems, he objec i e unc ion can be
e alua ed a di e en le els o ideli y. Fo ins ance, a CFD simula ion migh p o ide
a e y ime consuming bu accu a e way o es ima e he quali y o a solu ion.The
same solu ion could be e alua ed based on simpli ied ma hema ical equa ions, leading
o a cheape bu less accu a e es ima e. Combining hese di e en le els o ideli y
in a model-based op imiza ion p ocess is e e ed o as mul i- ideli y op imiza ion.
This chap e desc ibes unce ain y-handling echniques o me a-model based sea ch
heu is ics in combina ion wi h mul i- ideli y op imiza ion. Co-K iging is one powe -
ul me hod o co ela e mul iple se s o da a om di e en le els o ideli y. Fo he
i s ime, Sequen ial Pa ame e Op imiza ion wi h co-K iging is applied o noisy es
unc ions. This s udy will in oduce hese echniques and discuss how hey can be
applied o eal-wo ld examples.
1 In oduc ion
Sequen ial Pa ame e Op imiza ion (SPO) is a me a-model based sea ch heu is-
ic ha combines classical and mode n s a is ical echniques. I was o iginally
de eloped o he analysis o sea ch heu is ics such as simula ed annealing, pa -
icle swa m op imiza ion and e olu iona y algo i hms [6]. He e, SPO i sel will
∗This is a p ep in o he publica ion T. Ba z-Beiels ein, C. Jung, M. Zae e e . Unce -
ain y Managemen Using Sequen ial Pa ame e Op imiza ion. In G. Dellino and C. Meloni,
Ca lo, edi o s, Unce ain y Managemen in Simula ion-Op imiza ion o Complex Sys ems,
Sp inge , 2015 (in p in ). The o iginal publica ion is a ailable a www.sp inge link.com
1
be used as a sea ch heu is ic, i.e., SPO is applied o he objec i e unc ion di-
ec ly. An in oduc ion o he s a e-o - he-a Rimplemen a ion o SPO, he
so-called sequen ial pa ame e op imiza ion oolbox (SPOT), is p esen ed in [8].
Me a models, also called su oga e models, simpli y he simula ion op imiza-
ion, because he un imes a e gene ally much sho e han he o iginal unc ion
e alua ions (simula ion uns) [2, 25]. Cos -in ensi e op imiza ion p oblems in
enginee ing ha e o en less cos ly, less accu a e ep esen a ions which can be
e alua ed. Tha means, wo unc ions o di e en ideli y a e a ailable o he
op imiza ion p ocess, he ine unc ion (expensi e, ime-consuming, accu a e)
and he coa se unc ion. In e media e ideli y le els can be a ailable, oo. In
he emainde o his chap e , Medeno es he expensi e model, e.g., compu a-
ionally expensi e simula ions o eal-wo ld expe imen s such as c ash es . The
simpli ied (cheap) me a model will be deno ed as Mc. The combina ion o in o -
ma ion om Mcand Memodels will be e e ed o as mul i- ideli y analysis[24].
An in e es ing aspec is he compu a ional budge (numbe o unc ion e alua-
ions) ha is spen o selec ing new design poin s and he ela ionship be ween
e alua ions o he cheap and he expensi e model. A powe ul mul i- ideli y
echnique is co-K iging [17], which exploi s co ela ion be ween he di e en
ideli y le els o imp o e he me a model o he highes ideli y unc ion.
Unce ain y may a ise in many eal-wo ld op imiza ion se ings, e.g., om
noisy senso s, impe ec models, o he inhe en ly s ochas ic na u e o he simu-
la ed sys em. The e o e, unce ain y-handling echniques a e necessa y [1, 21].
An elemen a y app oach o cope wi h unce ain y is o inc ease he numbe
o unc ion e alua ions. SPOT in eg a es sha pening as a simple me hod,
which gua an ees a ai compa ison o he ob ain solu ions. Lasa czyk [27] and
Ba z-Beiels ein e al. [3, 4] analyzed he in eg a ion o a mo e sophis ica ed
con ol- heo e ic simula ion echnique called op imal compu ing budge alloca-
ion (OCBA) in o SPOT. The OCBA app oach can in elligen ly de e mine
he mos e icien eplica ion numbe s [12]. The goal is o ob ain he highes
decision quali y using a ixed compu ing budge o o a ain a desi ed simula-
ion decision quali y using a minimum compu ing budge . This SPOT-OCBA
a ian is compa ed o SPOT’s s anda d echnique o inc easing he numbe o
epea s.
Since Fo es e e al. [17] desc ibe co-K iging o de e minis ic se ings, i
is o g ea in e es o ex end his analysis o noisy en i onmen s. Cu en ly,
he e a e only a ew publica ions a ailable, which analyze co-K iging unde
unce ain y. Fo example, Wankhede e al. [40] compa e a co-K iging based
op imiza ion s a egy wi h a s anda d K iging based op imiza ion s a egy o
he design o a 2D combus o .
These conside a ions mo i a ed he cen al ques ion o his publica ion:
A e esul s om op imiza ion uns unde unce ain y, which a e
based on a la ge quan i y o cheap da a and a small quan i y o
expensi e da a, be e han esul s om uns which a e based on a
small quan i y o expensi e da a?
This ques ion mo i a ed he ollowing expe imen al se up. Two classes o me a
models, which ha e been p o en use ul in he SPOT amewo k, i.e., (i) ee-
based models such as andom o es [10, 9, 28] and (ii) s ochas ic p ocess models
(Gaussian p ocesses, K iging) [34, 29, 35], will be used. A compa ison o he
a he simple ee-based echniques wi h sophis ica ed K iging and co-K iging
2
echniques is o g ea in e es . To enable a ai compa ison, a sweeping me hod
based on La in hype cube sampling (LHS) is added o ou expe imen al po -
olio [30]. Summa izing, he ollowing po olio is used: (i) simple sweep o
he sea ch space by La in hype cube sampling, (ii) andom o es , (iii) K iging,
and (i ) Co-k iging models. This se up allows he in es iga ion o he ollowing
esea ch ques ions:
Ques ion 1 Does co-K iging pe o m well unde he p esence o noise, in com-
bina ion wi h unce ain y handling echniques like OCBA?
Ques ion 2 How do andom- o es based me a models pe o m in compa ison
o K iging-based me a models?
Resul s om his s udy a e applicable o o he me a-model sea ch heu is ics
such as sequen ial k iging op imiza ion [19].
This chap e , which desc ibes unce ain y-handling echniques o me a-
model based sea ch heu is ics in combina ion wi h mul i- ideli y analysis, is
s uc u ed as ollows. Sec ion 2 in oduces SPOT and he me a models such as
andom o es , K iging and co-K iging, used in his s udy. Unce ain y-handling
echniques a e desc ibed in Sec. 3. The expe imen al se up, e.g., objec i e unc-
ion and un leng h, numbe o epea s e c. and esul s a e p esen ed in Sec. 4.
A eal-wo ld example is desc ibed in Sec. 5. Finally, he chap e concludes wi h
a Summa y in Sec. 6.
2 SPO Va ian s
2.1 SPOT in a Nu shell
SPOT uses he a ailable budge (e.g., simula o uns, numbe o unc ion e al-
ua ions) sequen ially, i.e., i uses in o ma ion om he explo a ion o he sea ch
space o guide he sea ch by building one o se e al me a models. P edic ions
om me a models a e used o selec new design poin s. Me a models a e e ined
o imp o e knowledge abou he sea ch space. SPOT p o ides ools o cope
wi h noise, which ypically occu s when eal-wo ld applica ions, e.g., s ochas ic
simula ions, a e un. I gua an ees compa able con idence o sea ch poin s.
Use s can collec in o ma ion o lea n om his op imiza ion p ocess, e.g., by
applying explo a o y da a analysis (EDA) [39, 11]. Las , bu no leas , SPOT
p o ides mechanisms bo h o in e ac i e and au oma ed uning [7, 5]. An R
e sion o his oolbox o in e ac i e and au oma ic op imiza ion o algo i hms
can be downloaded om CRAN.1P og ams and iles om his s udy can be
eques ed om he au ho .
As can be seen om Algo i hm 1, SPOT equi es a mechanism o gene a e
an ini ial design. Addi ionally, SPOT gene a es new design poin s du ing he
sequen ial s ep. La in hype cube sampling was chosen as he gene a o o design
poin s du ing he ini ial and sequen ial SPOT s eps. LHS was chosen, because
i is easy o implemen and unde s and. Many design poin gene a o s a e
a ailable in R, see, e.g., he CRAN Task View: Design o Expe imen s (DoE)
& Analysis o Expe imen al Da a.2
1h p://c an. -p ojec .o g/web/packages/SPOT/index.h ml
2h p://c an. -p ojec .o g/web/ iews/Expe imen alDesign.h ml
3
Table 1: SPOT me a models used in his s udy
Type Name o he SPOT plugin Abb e .
K iging (Gaussian P ocesses) spo P edic Fo es e KR
Co-K iging (Mul i-Ou pu Gaus-
sian P ocesses)
spo P edic CoFo es e CK
Random o es spo P edic RandomFo es RF
The e is a s ong in e ac ion be ween design gene a o s and me a models,
because he op imali y o a design poin depends on he me a model [32, 35].
This pape modi ies SPOT’s me a models, while design gene a o s emain un-
changed. The impac o he a ia ion o he design gene a o s on he algo i hm’s
pe o mance will be subjec o a o hcoming pape .
2.2 Me a Models Used Du ing SPOT Runs
SPOT p ocesses da a sequen ially, i.e., s a ing om a small ini ial design,
u he design poin s a e gene a ed using a me a model. Many me a models
a e a ailable in R. Simila as o he design gene a o s he use has he op ion
o choosing be ween s a e-o - he-a me a models o uning his algo i hm o
w i ing his own me a model and use i as a plugin o SPOT. The de aul
SPOT ins alla ion con ains se e al me a models. The Rimplemen a ion o
andomFo es was chosen as SPOT’s de aul one. This is qui e obus and
can handle ca ego ical and nume ical alues needing only a compa ably small
amoun o compu a ional esou ces. Table 1 summa izes me a models used o
expe imen s desc ibed in his documen .
2.2.1 Random Fo es -based Pa ame e Tuning
The andom o es (RF) me hod om he Rpackage andomFo es implemen s
B eiman’s algo i hm, which is based on B eiman and Cu le ’s o iginal Fo an
code, o classi ica ion and eg ession [9]. I is implemen ed as a SPOT plugin,
which can be selec ed ia se ing he command seq.p edic ionModel. unc
acco ding o Table 1 in SPOT’s con igu a ion ile. A de ailed desc ip ion o he
SPOT con igu a ion is gi en in [8].
2.2.2 K iging-based SPO
K iging is one o he mos p omising su oga e models o op imiza ion p ob-
lems [26, 29]. I p o ides a e y lexible and e icien way o model con inuous
landscapes, p o iding a good p edic i e quali y o inding solu ions o inc eased
op imali y in he design space. K iging p o ides a way o es ima e he local un-
ce ain y o he model. Fo de e minis ic p oblems he unce ain y is ze o a
obse ed loca ions, and will inc ease wi h ising dis ance o such loca ions as
well as inc eased cu a u e o he model. This a iance es ima e allows o
an e icien way o balance be ween exploi a ion and explo a ion du ing he
op imiza ion p ocess. Jones e al. in oduced his me hod as e icien global
op imiza ion (EGO) [22]. Fo es e e al. [17] also u ilize a iance es ima es as
a penal y o impu a ion o ailed a ge unc ion e alua ions.
4

Se e al K iging implemen a ions a e a ailable in R, p o ided by packages
like mlegp,DiceK iging,ke nlab o ields [15, 33, 23, 18]. SPOT includes
examples o in e acing wi h se e al di e en implemen a ions. Mos no ably,
he SPOT package i sel p o ides wo implemen a ions, which a e a DACE (De-
sign and Analysis o Compu e Expe imen s) based implemen a ion [29] and an
implemen a ion based on Code by Fo es e e al. [17]. They we e chosen o
be eimplemen ed in he SPOT R-Ve sion, as hey we e also used in he ea -
lie SPOT ma lab e sion. Bo h a e nume ically obus and show good pe o -
mance. While he o me p o ides a lexible in e ace o choose di e en Ke nels
o polynomial unc ions, he la e includes a co-K iging implemen a ion. Co-
K iging will be in oduced below. In his a icle, he K iging implemen a ion
based on Fo es e e al. [17] is used.
2.2.3 Co-K iging
Fo many eal wo ld enginee ing p oblems, he a ge unc ion can be e alua ed
a di e en le els o ideli y o g anula i y. Fo ins ance, a CFD simula ion
migh p o ide a e y ime consuming bu accu a e way o es ima e he quali y
o a solu ion.The same solu ion could be e alua ed based on simpli ied analy ical
equa ions, leading o a cheape bu less accu a e es ima e. Combining hese
di e en le els o ideli y in a model-based op imiza ion p ocess is e e ed o as
mul i- ideli y op imiza ion. Kennedy and O’Hagan [24] explo e ways in which
models wi h di e en ideli ies can be used o make in e ence abou he ou pu
om he mos expensi e, complex o ine-g ained model.
One possible app oach o mul i- ideli y op imiza ion is co-K iging. Co-
K iging can be de ined as a a ian o k iging, which uses in o ma ion om
an addi ional, highly co ela ed a iable oge he wi h he p ima y a iable o
imp o e es ima es o he unc ion alues. Fo es e e al. [16] in oduce co-
K iging oge he wi h a simple es unc ion and a eal-wo ld example. They
show, how co-K iging can employ he lowe ideli y unc ion o imp o e he
model o he highe ideli y unc ion. The simple es - unc ion in oduced by
Fo es e e al. [16] will be used in a sligh ly changed way o he expe imen s
desc ibed in Sec. 4.
I has o be no ed, ha in his s udy, co-K iging equi es he design poin s
e alua ed on he ine a ge unc ion o be nes ed in o he la ge design o he
coa se a ge unc ion. In SPOT i is ensu ed ha he designs o he di e en
ideli y le els a e s ill space- illing. The c ea ion o he lowe le els design is
he e o e always based on he uppe le els design.
2.2.4 K iging/co-K iging and Noise
A s anda d K iging model would no be pe ec ly sui able o a noisy p oblem,
because K iging is a s ic ly in e pola ing app oach. Tha means, he p edic ed
mean alues exac ly ma ch wi h he known obse a ions. Howe e , a egula -
iza ion cons an can be in oduced (also called nugge e ec ) o ans o m he
model o a eg essing one, whe e p edic ion and obse a ion can de ia e om
each o he . I expec ed imp o emen (EI) [22] is used, his will lead o non-ze o
a iance es ima es a al eady e alua ed design poin s. This may de e io a e he
explo a i e p ope ies o EGO. Howe e , a ein e pola ing app oach can be used
o deal wi h his p oblem, bo h o K iging [17] and co-K iging [16].
5
Besides his, epea ed e alua ion o design poin s has o be conside ed o he
coa se unc ion. The unce ain y handling me hods in SPOT, namely OCBA
and sha pening, a e in oduced in Sec. 3. They a e me hods o selec design
poin s o e-e alua ion, which a e based on quali y and/o a iance. Sha pening
and OCBA a e no di ec ly applicable o he coa se unc ion design om he
Mcmodel. The coa se unc ion op imum can be comple ely meaningless o he
ue unc ion, which means ha he quali y alue becomes a he meaningless.
A sui able me hod should he e o e ei he ocus on a good global i o he
coa se unc ion (e.g. e en sp ead o epea s). This should be especially well
applicable when he unc ion is indeed e y cheap o e alua e. O he coa se
unc ion budge should ocus on he a ea o in e es , as iden i ied by ine unc ion
e alua ions. In his s udy, a la ge numbe o epea s is e enly sp ead o e he
whole design space. S ill, poin s o he ine unc ion design, which a e nes ed in
he coa se unc ion design and chosen o epe i ion, will also be e-e alua ed
on he coa se unc ion.
3 Unce ain y Handling Techniques
3.1 Sha pening
In he p esence o noise, a e aging o e se e al unc ion e alua ions may help o
manage unce ain y and o imp o e con idence. In he con ex o e olu iona y
algo i hms, S agge [36] demons a ed ha a educ ion o noise is no necessa y
o e e y single poin in he sea ch space bu only o he bes ones. The decision
which ones a e he bes is acili a ed by a e aging bu possibly a small numbe o
e alua ions is enough o ha decision. S agge [36] expe imen ally demons a ed
ha his idea can educe he numbe o unc ion e alua ions signi ican ly.
SPOT p o ides ools o managing unce ain y and imp o ing he con idence
du ing he sea ch. Fi s app oaches inc eased he numbe o epea s. An ea ly
SPOT implemen a ion p oceeded as ollows [6]:
A each s ep, wo new designs a e gene a ed and he bes is e-
e alua ed. This is simila o he selec ion p ocedu e in (1 + 2)-
E olu ion S a egies. The numbe o epea uns, k, o he algo i hm
designs is inc eased (doubled), i a design has pe o med bes wice
o mo e. A s a ing alue o k= 2 was chosen.
A sligh ly modi ied app oach, which will be e e ed o as sha pening (SHRP),
is implemen ed in mo e ecen SPOT e sions. Sha pening consis s o wo
phases, (i) he model cons uc ion and (ii) sequen ial imp o emen . Phase (i)
de e mines a popula ion o ini ial designs in algo i hm pa ame e space and
uns he algo i hm k imes o each design. Phase (ii) consis s o a loop wi h
he ollowing componen s: By means o he ob ained da a, he model is buil o
upda ed, espec i ely. Then, a possibly la ge se o design poin s is gene a ed
and hei p edic ed u ili y compu ed by sampling he model. A small se o he
seemingly bes design poin s is selec ed and he algo i hm is un k+ 1 imes o
each o hese. The algo i hm is also un once o he cu en bes design poin
and kis inc eased by one. No e, o he upda e ules o he numbe o epea s,
k, a e possible. The new design poin s a e added o he popula ion and he loop
s a s o e i he e mina ion c i e ion is no eached (usually a p ese budge is
6
g an ed o he p ocess). In consequence, his means ha he numbe o epea s
is always inc eased by one i he cu en bes design poin s ays a he op o
he lis o a newly gene a ed one ge s he e. Due o nonde e minis ic esponses
o he algo i hm, i may howe e happen ha nei he o hese is ound a he
op o he lis a e inishing he loop. In his case, kmay e ec i ely sh ink as
pe o mance compa isons ha e o be ai and hus shall be based on he same
numbe o epea s.
3.2 Op imal Compu ing Budge Alloca ion
The sha pening app oaches om Sec. 3.1 do no use any in o ma ion abou he
unce ain y ( a iance). He e come echniques such as op imal compu ing budge
alloca ion (OCBA) in o play [13, 20, 14]. OCBA was de eloped o ensu e a high
p obabili y o co ec selec ion (PCS). To maximize PCS, a la ge po ion o
he a ailable budge is alloca ed o hose designs ha a e c i ical o he p ocess
o iden i ying he bes candida es. OCBA uses sample means and a iances in
he budge alloca ion p ocedu e in o de o maximize PCS.
OCBA’s cen al idea can be explained as ollows. Conside a numbe o
simula ion eplica ions, say T, which can be alloca ed o mcompe ing design
poin s wi h means Y1, Y 2, . . . , Y mand ini e a iances σ2
1, σ2
2, . . . , σ2
m, espec-
i ely. The app oxima e p obabili y o co ec selec ion can be asymp o ically
maximized when
Ni
Nj
=σi/δb,i
σj/δb,j 2
, i, j ∈ {1,2, . . . , m},and i6=j6=b, (1)
Nb=σb
u
u
X
i=1,i6=b
N2
i
σ2
i
,
whe e Niis he numbe o eplica ions alloca ed o design i, and δb,j =Yb−Yi
deno es he di e ence o he i- h and b- h mean wi h Yb≤mini6=bYi. As can
be seen om (1), he alloca ed compu ing budge is p opo ional o a iance
and in e sely p opo ional o he di e ence om he bes design. Chen and Lee
p esen a comp ehensi e co e age o he OCBA me hodology [12].
Lasa czyk was he i s who combined SPOT and OCBA [27]. The OCBA
implemen a ion in his s udy is based on Lasa czyk’s wo k. SPOT wi h OCBA
is shown in Algo i hm 1. New design poin s which we e p oposed by he me a
model a e e alua ed se e al imes, e.g., wice. This alue can be modi ied using
he ini .design. epea s a iable in SPOT’s con ig ile. Du ing each SPOT
s ep, a ce ain budge (he e: spo .ocba = 3, as can be seen om Table 2) is
alloca ed o he candida e solu ions o ensu e a high PCS o he bes design
poin .
4 Expe imen s
4.1 Objec i e Func ion
To demons a e he e ec i eness o di e en app oaches he one- a iable es -
unc ion, ha Fo es e e al. [17] in oduced, is in es iga ed in he expe imen s.
Al hough his unc ion is a he simple, i allows a compa ison wi h p e ious
7
Algo i hm 1: SPOT-OCBA.
0=ini .design. epea s, =seq.ocba.budge ,
l=seq.design.size,d=seq.design.new.size
// phase 1, building he model:
le Fbe he uned algo i hm;
// design conside a ions necessa y:
gene a e an ini ial popula ion X={¯x1,...,¯xm}o mpa ame e ec o s;
le 0be he ini ial numbe o es s o de e mining es ima ed unc ion
alues;
o each ¯x∈Xdo
e alua e Fwi h ¯x 0 imes o de e mine he es ima ed unc ion alue
ˆyo ¯x;
end
// phase 2, using and imp o ing he model:
while e mina ion c i e ion no ue do
// OCBA:
le B⊆Xdeno e he subse o candida e solu ions wi h bes
es ima ed unc ion alue ˆy;
le deno e he OCBA budge ;
dis ibu e among B, i.e., gene a e OCBA dis ibu ion O;
// model conside a ions necessa y:
build me a model based on Xand {ˆy1,...,ˆy|X|};
// design conside a ions necessa y:
gene a e a se X0o lnew pa ame e ec o s by andom sampling;
o each ¯x∈X0do
calcula e (¯x) o de e mine he es ima ed unc ion alue (¯x) o ¯x;
end
selec se X00 o dpa ame e ec o s om X0wi h bes p edic ed
u ili y (dl);
e alua e Fwi h B ollowing he OCBA dis ibu ion O;// (imp o e
con idence)
e alua e F 0 imes wi h each ¯x∈X00 o de e mine he es ima ed
unc ion alues ˆy;
ex end he popula ion by X=X∪X00;
end
8
0.0 0.2 0.4 0.6 0.8 1.0
−5 0 5 10 15
RMSE: 0.722299747707585
x
(x)
●
●
_e(x)
T ue Op imum
_p edic ed(x) (KR + SHRP)
P edic ed Op imum
0.0 0.2 0.4 0.6 0.8 1.0
−5 0 5 10 15
RMSE: 0.743842140598047
x
(x)
●
●
_e(x)
T ue Op imum
_p edic ed(x) (KR + OCBA)
P edic ed Op imum
Figu e 4: The p edic ion o he inal model a e an op imiza ion un wi h 50
e alua ions wi h K iging and OCBA (uppe g aph) o sha pening (lowe g aph).
Table 6: S a is ical p ope ies o he esul s wi h co-K iging. (S) indica es sho
uns, (L) indica es long uns,
CK+SHRP (S) CK+OCBA (S) CK+SHRP (L) CK+OCBA (L)
Min -6.021 -6.021 -6.021 -6.021
1Q -6.020 -6.020 -6.019 -6.019
Med -6.017 -6.017 -6.011 -6.014
Mean -5.951 -5.936 -5.993 -6.012
3Q -6.014 -6.014 -5.976 -6.009
Max -2.807 -2.041 -5.874 -5.981
15

0.0 0.2 0.4 0.6 0.8 1.0
−5 0 5 10 15
RMSE: 0.639980337805978
x
(x)
●
●
_e(x)
T ue Op imum
_p edic ed(x) (CK + SHRP)
P edic ed Op imum
0.0 0.2 0.4 0.6 0.8 1.0
−5 0 5 10 15
RMSE: 0.625057542988798
x
(x)
●
●
_e(x)
T ue Op imum
_p edic ed(x) (CK + OCBA)
P edic ed Op imum
Figu e 5: The p edic ion o he inal model a e an op imiza ion un wi h 50
e alua ions wi h co-K iging and OCBA (uppe g aph) o Sha pening (lowe
g aph).
he RMSE is imp o ed o he co-K iging Models. The imp o ed RMSE can
be obse ed o mos expe imen s, bu does no lead o imp o ed op imiza ion
pe o mance.
In a eal-wo ld use case, one would o cou se ha e o conside ha CK needs
inc eased e o . This addi ional e o includes he e alua ions o he coa se
(supposedly cheap) a ge unc ion, as well as he mo e complex model building
and p edic ion. Thus, i ’s use ulness would depend on he di e ence in ime
consump ion o he coa se and ine unc ion, as well as he ime consump ion
o he model building o he gi en design space dimensionali y and numbe o
obse a ions.
4.6 Discussion o he Expe imen al Resul s
The expe imen s desc ibed a e o cou se only ela ed o a single one-dimensional
es - unc ion. This has se e al implica ions. Fi s ly, hings migh look di e en
16
o di e en unc ions o a ious dimensionali y. Secondly, eal-wo ld p oblems
p esen a la ge a ay o addi ional challenges no conside ed he e, o ins ance
he handling o ailed a ge unc ion e alua ions. S ill, he esul s do show ha
co-K iging can help o imp o e he op imiza ion pe o mance in he p esence o
noise. This gi es a p elimina y answe o Ques ion 1. Al hough his esul is
a he ague, i could be shown ha co-K iging is bene icial e en in op imiza ion
unde unce ain y.
Ano he impo an lesson o be lea ned om hese expe imen s is ha
he e can be no gene al ecommenda ion owa ds a single unce ain y handling
me hod. This clea ly depends on he a ailable budge as well as he choice o
op imiza ion p ocess pa ame e s, e.g., he chosen me a model. P oblem ea-
u es like he ype o noise will also ha e an e ec , bu a e no conside ed he e.
The e o e, no simple answe can be gi en o Ques ion 2.
OCBA and SHRP do ha e di e en in luences on he op imiza ion p o-
cess beha io , p omo ing ei he explo a ion o exploi a ion. A simila e ec o
OCBA could be assumed when expec ed imp o emen comes in o play because
i is a me hod o balance owa ds explo a ion as well. The di e ence he e is
o cou se, ha OCBA explo es he numbe o samples o each known loca ion
in he design space, while EI explo es egions no ye well ep esen ed by he
lea ned me a model.
5 Real-wo ld Example: Hea y Wid h Reduc-
ion o S eel Slabs
5.1 The Ho S eel Rolling P ocess
One impo an quali y pa ame e in he complex p ocess o ho s eel olling is
he p edic ion and op imiza ion o he wid h o pla es and s ips. Rec angula
s eel slabs, which a e used o he manu ac u e o all la s eel p oduc s such as
coils, a e ho olled. Wid h educ ion has become inc easingly impo an in he
p oduc ion o ho s eel s ips.
The olling p ocess is di ided in o se e al passes. Each pass can consis o
a hickness (ho izon al olling) and a wid h educ ion ( e ical olling). The
wid h educ ion is only pe o med in o wa d passes and he e ical olling
p ocess has no e ec in he backwa ds di ec ion. This si ua ion is depic ed in
Fig. 6.
In gene al he e ical olling p ocess is pe o med be o e he ho izon al
olling p ocess. Du ing his e ical olling p ocess a so called dogbone shape
is added o he p oduc which will hen again be la ened in he ho izon al
olling p ocess. The dogbone shape canno be measu ed because i only occu s
be ween he e ical and ho izon al s ands o he s eel mill and he e a e no
measu emen sys ems a ailable which a e wo king p ope ly in his en i onmen .
Con a y o he pla e and s ip hickness he wid h a e each pass canno be
se di ec ly and an accu a e model is needed o ob ain a p ope wid h shape
o he p oduc . Each de o ma ion s ep wi hou any wid h educ ion esul s in
an inc eased p oduc wid h. The numbe o passes in e e sing mills, say N,
has always o be odd because no mally he p oduc is ans e ed o u he
p ocesses away om he u nace. This leads o (N+1)/2 o wa ds passes and
(N-1)/2 backwa ds passes. Usually, he e e sing mills a e equipped only wi h
17
Fu nace Successi e
P ocesses
Ve ical olling
(wid h educ ion)
Ho izon al olling
( hickness educ ion)
Fu nace Successi e
P ocesses
Ve ical olling
deac i a ed
Ho izon al olling
( hickness educ ion)
Fo wa d pass
Backwa d pass
P eceding
P ocesses
P eceding
P ocesses
Me al P oduc
Me al P oduc
Figu e 6: Illus a ion o a olling p ocess s ep in se e al passes. The en y
side is on he le . Measu emen s a e a ailable on he igh a e each o wa d
pass, whe eas no measu emen s can be ob ained on he le . The olling p ocess
consis s o se e al, e.g., N= 7 o wa d and backwa d passes.
18
one wid h gauge a he exi side o he s and so he e a e only measu emen s
a e each o wa d pass. Due o he ac ha he wid h canno be measu ed
a he en y side, a hidden s a e p oblem occu s. The dogbone shape, which
esul s a e he wid h educ ion p ocess, is ha d o desc ibe analy ically. This
has only been done o a ew s anda d s eel g ades wi hin na ow geome ic
con ines. Sophis ica ed ime-consuming me hods ha e o be applied o cope
wi h he di e en wo king poin s. The occu ence o he dogbone will esul
in an addi ional sp ead in he ollowing p ocess o ho izon al olling. Assuming
ha he incoming geome ies o he p oduc be o e he i s de o ma ion p ocess
a e known hen he e a e wo successional p ocesses which modi y he p oduc
wid h.
5.2 Modeling
Va ious models can be cons uc ed o ep esen he p ocess desc ibed abo e.
They a e based on he ollowing inpu pa ame e s:
•p oduc a ibu es such as geome y ( hickness, wid h), ma e ial compo-
nen s (chemical decomposi ion), and he mo-mechanical p ope ies
•p ocess pa ame e s such as oll gap se ings, eloci y, and cooling.
The ou pu pa ame e is he wid h o he p oduc .
To model he comple e physical p ocess e e y de o ma ion s ep should be
modeled sepa a ely, including a model o he dogbone shape. Howe e , his is
no possible, because measu emen s a e no a ailable be ween he e ical and
ho izon al olling s ep. The e o e, he ollowing wo models will be conside ed
u he :
1. a model, which desc ibes each pass wi h i s inpu and ou pu pa ame e s,
igno ing he dogbone shape
2. a model, which neglec s he hidden s a e a e he backwa ds pass.
These wo models can be buil based on di e en app oaches:
1. using a da a-d i en app oach, which p ocesses eal-wo ld da a, o al e -
na i ely
2. using an analy ical model, o example as p esen ed in [31, 37, 38].
This classi ica ion allows he gene a ion o ou di e en models. Subjec o
ou cu en esea ch is he implemen a ion o models using di e en le els o
ideli y. Two models will be conside ed u he . The i s , high- ideli y model
Mewill be called he da a-d i en model. I uses da a om he eal-wo ld p ocess
o gene a e a K iging model. The second, coa se o lowe ideli y model, say
Mc, desc ibes he inpu -ou pu ela ionship using he simple analy ical o mula.
The second model will be e e ed o as he analy ical model. Co-K iging could
addi ionally exploi in o ma ion om he lowe ideli y analy ical model. No e,
ha o all da a-d i en models, expensi e da a p e-p ocessing is necessa y.
19
6 Summa y
This a icle illus a es ha co-K iging can wo k unde he p esence o noise in
he coa se and ine a ge unc ion, and can be combined wi h he unce ain y
handling echniques included in SPOT. S a ing poin o ou expe imen al anal-
ysis was he co-K iging es unc ion, which was in oduced by Fo es e e
al. [17]. We demons a ed ha co-K iging can be bene icial in unce ain en i-
onmen s. Unsu p isingly, no gene al ecommenda ions o unce ain y handling
echniques can be gi en. Each expe imen al se up has di e en equi emen s.
Modi ica ions o he compu a ional budge , e.g., inc easing he numbe o unc-
ion e alua ions om n= 20 o n= 50 leads o di e en esul s. As a ule
o humb, we can s a e ha complex models such as K iging equi e la ge
compu a ional budge s han simple models such as andom o es . Howe e ,
his di e ence anishes i in o ma ion om cheap and expensi e models can be
combined. Co-K iging seems o be a p omising app oach o cos ly eal-wo ld
op imiza ion p oblems.
The ho s eel olling p ocess was in oduced as an impo an eal-wo ld op i-
miza ion p oblem. This p oblem is subjec o ou cu en esea ch. A modeling
app oach, which combines in o ma ion om a simple ma hema ical model wi h
in o ma ion om an expensi e da a-d i en K iging model was p esen ed. This is
only one signi ican eal-wo ld p oblem whe e mul i- ideli y models a e o g ea
impo ance, which can be adap ed o o he a eas.
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23
Kon ak /Imp essum
Diese Ve ¨o en lichungen e scheinen im Rahmen de Sch i en eihe ”CIplus”. Alle
Ve ¨o en lichungen diese Reihe k¨onnen un e
h p://www.ciplus- esea ch.de
ode un e
h p://opus.bsz-bw.de/ hk/index.php?la=de
abge u en we den.
K¨oln, Janua 2012
He ausgebe / Edi o ship
P o . D . Thomas Ba z-Beiels ein,
P o . D . Wol gang Konen,
P o . D . Bo is Naujoks,
P o . D . Ho s S enzel
Ins i u e o Compu e Science,
Facul y o Compu e Science and Enginee ing Science,
Cologne Uni e si y o Applied Sciences,
S einm¨ulle allee 1,
51643 Gumme sbach
u l: www.ciplus- esea ch.de
Sch i lei ung und Ansp echpa ne / Con ac edi o s o ice
P o . D . Thomas Ba z-Beiels ein,
Ins i u e o Compu e Science,
Facul y o Compu e Science and Enginee ing Science,
Cologne Uni e si y o Applied Sciences,
S einm¨ulle allee 1, 51643 Gumme sbach
phone: +49 2261 8196 6391
u l: h p://www.spo se en.de
eMail: homas.ba z-beiels ein@ h-koeln.de
ISSN (online) 2194-2870