Robus In eg a ed P oduc ion-Main enance Scheduling
o an E apo a ion Ne wo k
C.G. Palac´ına,∗, J.L. Pi a cha, C. Jaschb, C.A. M´endezc, C. de P adaa
aSys ems Enginee ing and Au oma ic Con ol DPT, Uni e sidad de Valladolid.
C/ Real de Bu gos, s/n. EII Sede Me gelina. Valladolid, 47011, Spain
bLenzing Ak iengesellscha , We ks aße 2, 4860 Lenzing, Aus ia
cCen e o Ad anced P ocess Sys ems Enginee ing (CAPSE). INTEC (UNL - CONICET).
Indus ial Enginee ing DPT (FIQ-UNL). G¨uemes, 3450 - San a Fe (3000), A gen ina
Abs ac
This wo k aims o educe he global esou ce consump ion in an indus ial e ap-
o a ion ne wo k by be e asks managemen and coo dina ion. The ne wo k
wo ks in con inuous, p ocessing some p oduc s in se e al e apo a ion plan s, so
op imal load alloca ion and p oduc -plan assignmen p oblems appea . The
plan s ha e di e en ea u es (capaci y, equipmen , e c.) and hei pe o mance
is a ec ed by ouling inside he hea exchange s and ex e nal ac o s. He eby,
he op imize has o decide when main enance ope a ions ha e o be igge ed.
The e o e, a mixed p oduc ion/main enance scheduling p oblem a ises. The
plan beha io is app oxima ed by su oga e linea models ob ained expe i-
men ally, allowing hus he use o mixed-in ege linea op imiza ion ou ines o
ob ain solu ions in accep able ime. Fu he mo e, unce ain y in he wea he
o ecas and in he p oduc ion plan is also conside ed ia a wo-s age s ochas ic
p og amming app oach. Finally, a ade-o analysis be ween o he objec i es
o in e es is gi en o suppo he decision make .
Keywo ds: P oduc ion scheduling, S ochas ic op imiza ion, In eg a ion,
Fouling, Main enance p edic ion, Simila i y index
1. In oduc ion
Imp o ed coo dina ion o he ope a ion in indus ial si es can lead o eno -
mous sa ings in he ene gy and esou ce consump ion, and consequen ly o he
∗Co esponding au ho
Email add ess: [email p o ec ed] (C.G. Palac´ın)
P ep in submi ed o Compu e s & Chemical Enginee ing Janua y 6, 2018
educ ion o p oduc ion cos s, h ough he de elopmen o be e decision sup-
po sys ems. This is he eason which mo i a es he de elopmen o me hods
and so wa e o e iciency moni o ing, coo dina ed p ocess con ol and op imal
planning and p oduc ion scheduling o ac o ies, indus ial plan s and pa ks
unde dynamically changing ma ke condi ions (K ¨ame & Engell, 2017). In
pa icula , linking he eal- ime ope a ional laye s wi h he si e-wide op imi-
sa ion h ough scheduling app oaches is ecei ing inc eased in e es (Engell &
Ha junkoski, 2012; Adamson e al., 2017).
These ype o p oblems a e posi ioned on he uppe le els o he con ol hie -
a chy (see Figu e 1) and usually in ol e bo h eal- alued quan i ies (e iciency
indica o s, load assignmen , e c.) and choosing be ween di e en disc e e op-
ions ( ask execu ion, pa h decisions o a ailable equipmen ). A scheduling
p oblem has o conside h ee main poin s: which asks ha e o be pe o med
and when hey will be execu ed, he equipmen ha will pe o m hese asks
and he esou ces/ ime ha will be equi ed (and hen alloca ed) o each ask.
I is no ewo hy o say ha he p oblem complexi y is usually high and compu-
a ional demands inc ease exponen ially wi h he numbe o asks.
Figu e 1: Au oma ion py amid
Scheduling is a complex and “exo ic” ask in indus y, which is s ill o en
based on expe ules. Howe e , hese p oblems can be ma hema ically o mu-
la ed, ansla ed o op imiza ion p oblems which in ol e bo h bina y/in ege
and eal decision a iables, and sol ed e icien ly using mixed-in ege p og am-
ming (MIP) (M´endez e al., 2006). Acco ding o he na u e o he in ol ed
ma hema ical models (hyb id linea o nonlinea ), hese op imiza ions a e o -
mula ed ia cons ain p og amming, mixed-in ege linea (MILP) o nonlinea
(MINLP) p og amming (Floudas, 1995). Hence, nowadays he compu e -aided
decision suppo ools a e ge ing mo e and mo e signi icance in o de o help
ope a o s o ake be e decisions (Ha junkoski e al., 2014). Ne e heless, he e
s ill exis many challenges o ace wi h o he success ul deploymen o in e-
g a ed scheduling ools in indus ial en i onmen s(Ha junkoski, 2016).
This pape ocuses on he in eg a ion be ween eal- ime op imiza ion (RTO) and
scheduling in a con inuous indus ial e apo a ion ne wo k wi h se e al plan s
and se e al p oduc s o concen a e. RTO is o en used in la ge-scale sys ems
2
o seek online o he “op imal” con ol se poin s and design con igu a ions o
bes ace possible changes (p oduc ion, wea he dis u bances, equipmen ail-
u es, changeo e s, e c.) (Adamson e al., 2017), bu i usually bases on he
cu en plan s a e and does no conside wha can happen in he u u e. In
he e apo a ion ne wo k, each plan uns in con inuous wi h i s own nominal
e iciency, bu his e iciency also dec eases wi h ime due o ouling e ec s inside
he hea exchange s. Indeed, long- e m e ec s which educe pe o mance, such
as ouling o ca alys deac i a ion, a e a common issue in indus y. The e o e,
ins an aneous decisions may no be he bes ones looking some ime ahead.
This d awback can be pa ially add essed by including models o such long-
e m e ec s in each plan RTO, e.g., he p e ious wo k om he au ho s o an
e apo a ion plan in Pi a ch e al. (2017).
Fu he mo e, hese ouling e ec s o ce o pe iodically pe o m cleaning asks
in o de o eco e plan e iciencies, bu no all plan s can be s opped a he
same ime, so a mixed main enance/p oduc ion scheduling p oblem a ises. Se -
e al people ha e de o ed e o s along he las decades o deal wi h hese kind
o p oblems: Sma¨ıli e al. (1999); Pogia zis e al. (2012) p oposed a MINLP
app oach while La aja & Bagajewicz (2004) and Casas-Liza e al. (2005) also
p oposed a ime disc e iza ion o ecas he p oblem as a MILP one. Recen ly,
Biondi e al. (2017) p esen ed a amewo k o indus ial si es o in eg a e hese
wo in e ac i e decision p ocesses in a mul i-scale scheduling p oblem. In such
wo k, he decay in equipmen pe o mance is modeled using he concep o
“ esidual use ul li e” (RUL), a capaci y esou ce which dec eases o e ime. The
e apo a ion plan s in ou case do no p esen a loss in capaci y due o ouling (a
leas du ing he la ges ime hey ha e been in ope a ion wi hou cleaning), bu
an inc emen in he speci ic s eam consump ion o achie e he desi ed e apo a-
ion se poin . So he RUL app oach is no di ec ly applicable he e. We p esen
a p edic i e scheduling app oach which uses a su oga e model including oul-
ing o ep esen he plan beha io and we p opose an adap a ion o he gene al
p ecedence app oach o e icien ly ackle he mixed main enance/p oduc ion
p oblem, whe e h ee main ypes o asks coexis s: no mal ope a ion, s andby
and cleaning.
Mo eo e , unce ain y is always p esen when acing eal p oblems (mismodel-
ing, unplanned changes, dis u bances, e c.). The e o e, conside ing unce ain y
since he design phase in o de o sea ch o obus solu ions is key. Robus -
ness can be p o ided by o cing a single schedule o ul ill a bunch o scena ios,
sampled acco ding o expec ed ealiza ions o he unce ain y (Kou elis e al.,
2000). Howe e , his is usually a e y conse a i e solu ion, i.e., no mally he
wo s scena io does no ealize. Thus, a less conse a i e op ion is using a mul i-
s age s ochas ic op imiza ion (G ossmann e al., 2016), which bene i s om he
assump ion ha he unce ain y can be mo e p ecisely known in he u u e, so
he schedule can p o ide di e en decisions o each scena io beyond he ini-
ial s age. He e we p opose a wo-s age op imiza ion app oach by conside ing
unce ain y in he ou doo wea he and in he p oduc ion plan.
3
In addi ion, se e al con lic ing op imiza ion c i e ia such as e iciency, obus -
ness o p oduc i i y appea (Yenisey & Yagmahan, 2014; Ruiz-Femenia e al.,
2013). The e o e, based on he abo e app oach and he concep o simila i y
be ween schedules, we p opose a mul i-c i e ia scheduling p oblem. Then, he
o line analysis and isualiza ion o he Pa e o on (Reynoso-Meza e al., 2013)
gi es he plan enginee s a guide abou he igh ade-o s o accomplish.
The es o he pape is s uc u ed as ollows: nex sec ion desc ibes he e ap-
o a ion p ocess and discusses a sui able plan model; Sec ion 3 desc ibes he
e apo a ion ne wo k oge he wi h he p oblem cons ain s, p esen s he p o-
posed scheduling model and o mula es he o e all p oblem ia disjunc i e p o-
g amming; hen he wo-s age s ochas ic and mul iobjec i e op imiza ions a e
p esen ed in sec ions 4 and 5 espec i ely; Sec ion 6 shows some ob ained solu-
ions o no mal ope a ing condi ions and analyzes he ob ained Pa e o on ;
and inally, a las sec ion summa izes he main conclusions.
2. The e apo a ion p ocess
This wo k is pa o a se ies o ac ions o imp o e esou ce e iciency in Lenzing
AG, a la ge indus ial si e which p oduces man-made cellulose ibe s using wood
as aw ma e ial. The main sou ce o ene gy consump ion in his si e is loca ed
in he so-called spinning p ocess, whe e he cellulose pulp ex ac ed is ex uded
join ly wi h a solu ion o chemicals and wa e , in o de o p o ide he ibe s
wi h he desi ed mechanical p ope ies. These chemicals ha e a sensible eco-
nomic alue, so hey mus be eused. Thus, hese aqueous solu ions (hence o h
p oduc s) a e sen o an e apo a ion sys em in o de o emo e he main pa
o wa e . Finally, he ou pu concen a e en e s in o a c ys alliza ion p ocess.
2.1. P ocess desc ip ion
Each e apo a ion plan is o med by a se ies o hea exchange s, e apo a ion
chambe s, condense s and cooling sys ems (see Figu e 2). The e apo a ion is
achie ed by a mul iple-e ec p ocess, as ollows. The p oduc en e s he plan
by he inle loca ed in he ex eme o he e apo a ion chambe s 2. Then i
mixes wi h some concen a e p oduc and lows h ough he hea exchange s
in o de o each an adequa e empe a u e (con ol se poin ). The i s se
o hea exchange s euse s eam coming om e apo a ion chambe s 1, whe eas
he las se use esh s eam p o ided by boile s ( he main sou ce o ene gy
consump ion). A e wa ds, he hea ed p oduc en e s sequen ially in o he low
p essu e e apo a ion chambe s 1. Finally, a second e apo a ion is achie ed in
chambe s 2, whe e he p essu e is dec eased using a condense connec ed o a
cooling sys em, ypically a cooling owe . Las , pa o he concen a e lea es
he sys em by o e low and he emaining is mixed wi h he p oduc inle , being
eci cula ed o he p ocess (con ol se poin ).
4
Figu e 2: Scheme o a single mul i-e ec e apo a ion plan .
2.2. Plan su oga e modeling
The de ailed modeling and op imiza ion o a single e apo a ion plan was al-
eady add essed in p e ious wo ks om he au ho s (Palac´ın e al., 2015; Pi a ch
e al., 2017). In hese wo ks, some Resou ce E iciency Indica o s (REI) we e
i s de ined in o de o measu e p ocess e iciency in eal ime. The main REIs
we e he speci ic s eam consump ion and he no malized cos pe ime uni o
ope a ion. A nonlinea g ey-box model, whose co e is based on i s p inciples,
was de eloped o op imiza ion pu poses. Then, he p oposed Real-Time Op i-
miza ion (RTO) showed up some pa e ns o he op imal ope a ion o a single
plan and hese we e implemen ed ollowing he concep s o sel -op imizing con-
ol.
In addi ion, inside he hea exchange s a di laye g ows du ing no mal op-
e a ion, caused by he deposi ion o o ganic ma e ial p esen in he p oduc s.
This ouling e ec slowly dec eases he hea - ansmission coe icien o e ime.
Hence, a complex model o p edic he e olu ion o hese coe icien s was iden i-
ied expe imen ally and in oduced in o an economic op imiza ion. In ha way,
an op imal cleaning policy o a single plan ope a ing in isola ion was al eady
p oposed in Pi a ch e al. (2017). Howe e , when conside ing se e al plan s and
p oduc s in a ne wo k, o he aspec s such as op imal assignmen o p oduc s
o plan s o coo dina ion o he main enance ope a ions ha e o be conside ed.
Fo mula ing such p oblem ia MINLP becomes compu a ionally challenging i
he g ey models de eloped in he abo e e e ence a e used o ep esen he
plan s.
The e o e, hanks o he ad an age ha nea op imal ope a ion is cu en ly
achie ed in each plan by he sel -op imizing con olle , we can build local su -
oga e plan models compu ed in di e en ope a ing condi ions (Ba z-Beiels ein
& Zae e e , 2017), ei he om simula ion wi h he nonlinea model o di-
ec ly om p ocess measu emen s. So, gi en a ouling s a e, hese simula-
5

ions/measu emen s allow o eco d a s a ic map o he cos unc ion ( esh
s eam consump ion) as a unc ion o he ou doo empe a u e Tou and he p od-
uc inle P. This mapping u ned ou o be qui e linea wi h hese a iables,
so a linea app oxima ion has been compu ed by leas -squa es iden i ica ion.
In addi ion, using expe imen al da a eco ded om he plan ope a ing se e al
mon hs1, an a e age inc ease o he speci ic s eam consump ion a ound 16% can
be iden i ied be ween consecu i e cleaning ope a ions (see Figu e 3). The e o e,
a linea e olu ion o he ouling e ec can also be assumed, mainly depending
on he ime he e apo a o has been in ope a ion.
Figu e 3: E olu ion o he pe cen age inc ease o speci ic s eam consump ion due o ouling
o se e al ope a ion cycles.
In he end, he su oga e model ep esen ing he cos unc ion o an e apo a ion
plan , p ocessing a p oduc pa ime ins an , eads as ollows:
Cos ( , , p)=(KT( )·Tou ( ) + KE( )) ·P( , , p) + KF( , ) (1)
Whe e KT( ) depends on he e iciency o each cooling sys em, KE( ) ep esen s
he nominal e iciency o he e apo a ion plan , and KF( , ) is he inc ease
o cos due o he cu en s a e o ouling. This plan model (1) is suppo ed by
ex ensi e expe imen al wo k (as can be seen in Fig.3) and i is he basis o he
p oposed scheduling app oach in he ollowing sec ions. As illus a i e example,
Figu e 4 shows h ee su aces co esponding o h ee di e en ouling s a es.
Rema k 1. No e ha his app oach has no only he ad an age o sensibly
educing he compu a ional cos equi ed o sol e he op imiza ion, bu also
allows an easie main enance o plan models by he p ocess enginee s.
3. The e apo a ion ne wo k
The ne wo k consis s o se e al e apo a ion plan s and some p oduc s o be
concen a ed. On he one hand, each p oduc may be p ocessed in se e al e ap-
o a ion plan s a he same ime, bu a plan can only p ocess a single p oduc
1In o de o isola e he inc ease o s eam consump ion due o ouling, he plan is momen-
a ily d i en o e e ence condi ions be o e aking a measu emen .
6
Figu e 4: Su oga e model o he s eam consump ion in a single plan .
a a ime. The e o e, gi en se s o pp oduc s and e apo a ion plan s, p ob-
lems o plan assignmen o p oduc s and load alloca ion appea (see Figu e 5),
whe e he ope a ion cos di e s om one plan o ano he due o he pa icula
equipmen e iciencies.
Figu e 5: Plan assignmen o p oduc s and load dis ibu ion.
On he o he hand, he ouling, which educes plan e iciencies, o ces pe iodic
s ops o cleaning in o de o eco e he nominal alues. The e exis se e al
aspec s ela ed o cleaning o be conside ed: (A) he e a e di e en cleaning
ypes, each one wi h an associa ed cos KCo manpowe and cleaning p oduc s,
and achie ing di e en eco e ies; (B) because o limi a ions on he a ailable
pe sonnel, only one ask can be pe o med a a ime. The e o e, a main enance
scheduling p oblem appea s, whe e we need o coo dina e he plan cleaning
s ops ( igh ime and ype o ask) while keeping he o e all p oduc ion and
he use o esou ces in an op imal way. No e ha , i one e apo a o s ops o
cleaning, i s load mus be eassigned o he es o ope a i e plan s.
Con en ionally, p oduc ion scheduling p oblems ha e been ep esen ed ia ne -
wo k s uc u es using asks (any abs ac ed p ocess ope a ion) wi h se e al
ba ches o schedule as basic elemen s. Howe e , as he e we a e no deciding
a he long- e m planning, in his pape he meaning o ask is closely linked
o he s a e in which a plan can be (p ocessing a p oduc p, s andby o un-
7
de a cleaning ope a ion). No e ha he numbe o equipmen ha can be
used o ope a ion is known, bu he amoun o asks o any ype ha ha e
o be pe o med wi hin a ime ho izon is no known in ad ance, because he
p ocess is con inuous, he u u e p oduc ion may be unce ain and he bes mo-
men o cleaning he plan s is no known. Handling common esou ces ( o al
e apo a ion pe p oduc in his case) wi h a con inuous- ime app oach equi es
synch oniza ion cons ain s o be ul illed a all ime , which makes he p ob-
lem compu a ionally e y demanding, hence usually no sui able o eal- ime
implemen a ions.
3.1. P oposed modeling
To e icien ly handle he synch oniza ion cons ain s in he p edic i e schedul-
ing p oblem o he e apo a ion ne wo k, he p edic ion ho izon Hhas been dis-
c e ized using one-day leng h as he sho es ask uni . This choice is mo i a ed
by ee ac s: 1) one day is he ypical du a ion needed o comple e a cleaning
ask and he ouling will no change signi ican ly in one day, 2) esou ce-sha ed
cons ain s a e na u ally handled in disc e e ime and 3) compu a ional s udies
(Sunda amoo hy & Ma a elias, 2011) showed ha disc e e- ime o mula ions
usually pe o m be e in complex p oblems. Then, he unde lying ideas o
gene al p ecedence alloca ion (M´endez e al., 2006) a e used he e o o ce
he ope a ion acco dingly o he known ime e olu ion o he ouling e ec s,
which mus be ollowed by a limi ed numbe o cleaning al e na i es.
Th ee di e en classes o s ages a e de ined o he p oposed au oma on: wo k-
ing, cleaning and s andby. The no mal ope a ion wo k low o one e apo a o is
depic ed in Figu e 6, whe e he wo king s ages a e displayed as blue che ons.
These s ages a e ela ed o he ime ha an e apo a o has been in ope a ion
( .g ., one e apo a o ha has s a ed ope a ion oday will be in s age s0, and
one e apo a o ha has been wo king o wo weeks will be in s age s14). Using
hese s ages, we will be able o indica e he plan pe o mance deg ada ion wi h
ime due o he ouling, i.e., i a plan is in s age s, i will ge an associa ed
alue KF( , s) o he cos unc ion (1). In his way, he s a e o an e apo a o
will ad ance wi h ime h ough he cha s ages: i will s a wo king a s age s0
i i is ully clean o om a mo e ad anced one, le ’s name i sc, i he cleaning
was less deep (see Fig.6). In addi ion o his, in Pi a ch e al. (2017) i was
ound by he au ho s ha s opping a plan o cleaning du ing he i s days o
ope a ion a e a p e ious cleaning is no wo hwhile, because in such case he
no malized cos pe ime is huge ( he ixed cos KCassocia ed o a cleaning
ask is no amo ized ye ). Hence, he e will be a se o ini ial s ages whe e no
decision abou cleaning needs o be checked.
Then, a e lea ing his ini ial se , he subsequen se s o s ages sAand sB
include he possibili y o ei he con inue ope a ion, go di ec ly o cleaning ( o
he small ask A om sAand o he big ask B om sB) o s op in s andby
awai ing o he cleaning esou ces o be a ailable. Two a ailable ypes o
8
Figu e 6: Simpli ied scheme o he au oma on.
cleaning ope a ions a e ep esen ed in Figu e 6 by he g een hexagons, whe eas
he s andby s ages a e ep esen ed by he g ey ones. Also, an e apo a ion plan
can be in s andby a e cleaning because i is no needed, o because i is no
p o i able o s a wo king wi h his plan .
The op ion o s opping he plan in he middle o an ope a ion cycle o con inue
ope a ing wi hou cleaning a e wa ds is no conside ed, as i is clea ly subop-
imal. Hence, once a plan has s a ed ope a ion, i mus con inue in ope a ion
as many days as s ages a e de ined in he se o ini ial ones. This concep is
analogous o he minimum unning ime in Velez e al. (2015) equi ed once a
ask has s a ed.
3.2. Logic o mula ion
The e exis se e al al e na i es in o de o o mula e scheduling p oblems ia
mixed-in ege and disjunc i e p og amming (G ossmann, 2002), each o hem
in luencing he model s uc u e, kind o so wa e o be used and e iciency in
ob aining a solu ion. The e o e, unde s anding he associa ed ad an ages and
d awbacks o each op ion is key.
In his case, ollowing he au oma on p oposed in he p e ious sec ion, i e
di e en se s o en i ies a e es ablished:

Vdeno es he se o all he e apo a ion plan s.

Swill be he se o possible s ages o an e apo a ion plan . As subse s i
includes:
–SIas ini ial s ages, de ined as he ones whe e a s op o cleaning is no
wo hwhile. In pa icula , s0will be he i s s age and scdeno es a
p ede ined s age o e u n ope a ion a e a less deep cleaning, e.g. o
ype A(Fig. 6).
9
5. Mul i-c i e ia decision making
In he abo e sec ion, only an economic op imiza ion is conside ed as an aim.
Howe e , an e en ual decision make mus also conside addi ional aspec s han
economic ones in p ac ice. Hence, o he goals such as p oduc i i y o obus ness,
possibly con lic ing wi h J1in (16), may be o in e es o op imiza ion oo.
The wo-s age s ochas ic app oach al eady p o ides a adeo be ween obus -
ness agains he dis u bances conside ed in he scena io ee and economic pe -
o mance. Howe e , he e is always a small possibili y ha he sugges ed op-
imal schedule canno be ully applied when he unce ain y ealiza ion is no
explici ly conside ed in he scena io ee, e en i he o mula ion is linea and
he ealiza ion belongs o he con ex egion discussed in he p e ious sec ion5.
This d awback is inhe en o p oblems in ol ing disc e e decisions. Theo e i-
cally, wi h a linea o mula ion, he op imal schedule which co e s a ealiza ion
belonging o he conside ed unce ain y se is a con ex combina ion be ween he
solu ions co esponding o he e ex scena ios. Howe e his op imal schedule
canno be compu ed excep in, pe haps, a e y ew cases by shee luck, because
a con ex combina ion o disc e e alues does no usually e u n ano he disc e e
alue. The e o e, some alues mus be “ ounded” o he nea es o he mos
p obable disc e e one in o de o apply a schedule in p ac ice. This uns he
isk o being in he ac ual pe o mance a om he p edic ed one o , in he
wo s case, i may lead o in easibili y.
A s aigh o wa d way o minimize such isk is ei he conside ing mo e scena -
ios o enla ging HR(Rema k 2). The i s op ion is disca ded o p ac ical
implemen a ions (i inc eases he compu a ional bu den conside ably). Thus,
how o choose he leng h o HR o achie e a desi ed isk educ ion wi hou
being oo conse a i e would be he ques ion o answe . In his las case, an
index o measu e obus ness could be de ined as J2:= HR/H. Howe e , his
way may become conse a i e, as he wo s -case combina ion o unce ain y e-
aliza ions may happen a he beginning o he p edic ion ho izon Hso, when
HRincludes such day, he conse a ism- educ ion ad an ages o he wo-s age
app oach anish.
To o e come his d awback and o o ce a desi ed obus ness le el wi hou ex-
plici ly a ying HR, we in oduce he concep o simila i y be ween schedules
(Palac´ın e al., 2017). A simila i y index (SI) will be de ined in o de o indica e
whe he a sugges ed schedule is close o he mo e isky one ob ained by he
s anda d wo-s age app oach (Sec ion 4), o o he isk a e se single schedule
(HR=H). The idea is inspi ed in he concep o minimum ag eemen index
5I he o mula ion is nonlinea in decision a iables o he he ac ual ealiza ion does no
belong o he con ex hull o med by he conside ed egion o unce ain y, no gua an ees can be
ensu ed. The only op ion in such cases is eschedule including ha ealiza ion as an addi ional
scena io in he ee, al hough no easibili y gua an ees can be p o ided ei he .
16

o uzzy dueda e o uzzy comple ion ime (Sakawa & Kubo a, 2000) and is as
ollows. Fi s , disc e e bina y decisions aken each day o a pa icula e ap-
o a o and scena io a e uzzi ied along he su ounding days, e.g., a decision
akes a alue o 100 a he sugges ed day bu i also in luences he be o e and
ollowing days wi h a dec easing alue, p opo ional o he dis ance om he
cu en day. Then, he SI is de ined as he in e sec ion be ween he uzzi ied
schedules compu ed o all he scena ios, see Figu e 8.
(a) Fuzzi ica ion o a disc e e decision. (b) SI alues pe day o one plan .
Figu e 8: Illus a i e SI calcula ion.
Then, he alues compu ed o all days, s ages and plan s mus be agg ega ed
oge he in a single SI indica o . Thus, using jus he wo closes days o he
cu en one ( −1, + 1) o uzzi ica ion, and abusing no a ion se ing T ue=1
and False=0 o E se, he SI index eads
SI:= X
∈V X
s∈S X
∈MU F
min 100E se,50E ( +1)se,50E ( −1)se
n (200(nu−1) + 150) (19)
whe e n is he numbe o e apo a ion plan s and nuis he numbe o days in
MU. Hence, a SI= 100% means ha he schedules coincide o all scena ios,
so he e is jus a single schedule: he isk-a e se solu ion.
The eade may ealize ha he SI as de ined in (19) is nonlinea in decision
a iables. Howe e , a lowe bound o i can be se in oducing slack a iables
S p ∈R+join ly wi h he ollowing addi ional linea disjunc ions:
E se
S s ≤100 ∨E ( +1)se
S s ≤50 ∨E ( −1)se
S s ≤50 ∨
¬(E se ∨E ( +1)se ∨E ( −1) se)
S s = 0 ∀ ∈ MU F,
∀ ∈ V,∀s∈ S (20)
In his way, he SI can be bounded by:
J3:= X
∈V X
s∈S X
∈MU F
S s
n (200(nu−1) + 150) ≤SI(21)
Las , he plan manage migh e en ually be in e es ed in analyzing how a ying
he o e all p oduc i i y a ec s he o he objec i es, e.g., o i ma ke condi ions,
17
o maximize he p o i (i.e., minimize he cos pe on o p oduced ib e) o o
decide o e u u e equipmen in es men s. In o de o app oach his, no e ha
he e apo a ion demands o each p oduc a e se by pa ame e s SPp e in (15).
So, he lowes demand δ o all scena ios, p oduc s and days is:
J4:= δ= min SPp e∀e∈ E,∀ ∈ M,∀p∈ P (22)
Hence, he gaps ∆Pp e := SPp e −δcan be compu ed oo. Now, i δbecomes
decision a iable, a way o uni o mly decide o e he e apo a ion demands ia
(15) is a ying δ, adding cons ain s (23) o compu e new se poin s wi h he
al eady ixed ∆Pp e.
SPp e = ∆Pp e +δ∀e∈ E,∀ ∈ M,∀p∈ P (23)
In o de o simul aneously analyze he in e ac ions be ween hese opposi e ob-
jec i es (cos , obus ness and p oduc i i y) and o p o ide he manage wi h
he signi ican in o ma ion a a glance, we come up wi h a mul i-objec i e op i-
miza ion p oblem (MOOP) o mula ed as ollows:
minimize J= [J1,−J3,−J4]∈R3subjec o: (2) −(17); (20); (23);
A pe ∈ A ∀ ∈ M,∀e∈ E;E 0se ∈ S0∀e∈ E
C se, P pe, S s, δ ∈R+;E se, A pe ∈ {T ue,False}
(24)
In o de o sol e (24) using e icien MILP so wa e, we se addi ional cons ain s
wi h bounds in J3and J4, deno ed by J3and J4, de ining a g id wi hin he
pe inency egion6(Reynoso-Meza e al., 2014) so ha only J1is in he objec i e
unc ion. In his way, he o iginal MOOP is cas as a se o single-objec i e
op imiza ions:
minimize J1∈Rsubjec o: (2) −(17); (20); (23); J3≥J3;δ≥J4;
A pe ∈ A ∀ ∈ M,∀e∈ E;E 0se ∈ S0∀e∈ E
C se, P pe, S s, δ ∈R+;E se, A pe ∈ {T ue,False}
(25)
Finally, a Pa e o F on can be compu ed o line and i s analysis will esul in
aluable in o ma ion o he decision make o choose which le el o isk o
assume depending on he pe missible slack o a y p oduc ion demands.
6. Illus a i e esul s
In o de o check he e ec i eness o he p oposed app oach, se e al schedules
ha e been compu ed om simula ed da a. As a i s ial, we do no conside
unce ain y, so we look o an e icien de e minis ic solu ion o he ac ual
ne wo k. Then, unce ain y is in oduced in wo smalle ins ances o he p oblem
and he ob ained s ochas ic solu ions a e discussed.
6Range whe e p oduc i i y and obus ness is conside ed accep able.
18
6.1. De e minis ic solu ion
In his case, he whole ne wo k o 23 plan s o p ocess 5 p oduc s (A,B,C,D,E)
is conside ed. The allowed physical connec ions be ween p oduc s and plan s
as well as nominal e iciencies7a e lis ed in Table 1.
p1p2p3p4p5KE
13 7 7 3 3 0.6
23 3 7 3 3 0.7
33 7 7 3 3 0.8
43 7 7 3 3 0.9
53 3 7 3 7 1
63 3 7 3 3 1.1
73 7 7 3 7 1.2
83 7 7 3 7 1.3
93 7 7 3 3 1.4
10 33337 1.1
11 3 3 7 3 3 1.11
12 3 7 3 3 7 1.12
13 7 7 7 3 3 1.13
14 7 3 7 3 7 1.14
15 333331.15
16 733330.9
17 3 3 3 7 3 1
18 7 3 3 7 3 1.1
19 733331.2
20 333331.3
21 7 3 7 3 3 1.4
22 3 3 7 7 3 0.7
23 333330.8
Table 1: Connec ions p oduc -plan and nominal e iciencies.
Each plan canno ope a e unde a load o L = 15 T/h, bu hei maximum
capaci y a ies wi h he wea he condi ion, i.e. U = 30 + (Tou ), whe e (·)
is such ha U ≤35 T/h. Two ypes o cleaning asks ha e been conside ed,
small (A) and big (B), wi h hei co esponding associa ed cos s KC( , sLA) and
KC( , sLB) o manpowe and chemical p oduc s. Ma ginal cos s KS( , sP A) and
KS( , sP B) ha e been assigned o he wai ing s ages be o e cleaning o a oid
pe sis en si ua ions in ime whe e e apo a o s a e no used bu emain di y,
which may lead o an o e all loss o e iciency when hey will be needed ( o
ins ance agains unexpec ed p oduc ion inc emen s). Also, analyzing he plan
his o ian, i has been obse ed ha an e apo a o canno ope a e du ing mo e
han 40 days wi hou cleaning because i is clea ly subop imal. So, he se Sis
o med by {s0, s1, . . . , s40, sLA, sLB, sP A, sP B , sP LA, sP LB}.
Fo his es , he desi ed se poin s o e apo a ed wa e pe p oduc a e se o
SP1= 120, SP2= 76, SP3= 64, SP4= 146 and SP5= 68 T/h. Hence, gi en a
andomly ixed ini ial s a e o he ne wo k oge he wi h he abo e cons ain s,
we un he economic op imiza ion (16) o p o ide he op imal load alloca ion
as well as he ask schedule wi hin a p edic ion ho izon o H= 30 days.
7Scaled alues. Real ones a e no included due o con iden iali y ag eemen s wi h Lenzing
AG.
19
An op imal solu ion wi h ela i e gap less han 1% has been ound o his p ob-
lem (35150 bina y a iables, 3452 eal ones and 42613 cons ain s) in abou 11
min using up o 4 h eads o concu en op imiza ion in GAMS wi h GUROBI
7.0.2 o e an In el
®
i7-4510U CPU machine wi h 16 Gb o RAM memo y8. This
solu ion is depic ed in he Gan diag am o Figu e 9, whe e he s a e e olu ion
o each e apo a o is shown o e he ho izon al axis. Columns ep esen he
days, and each cell shows he p oduc load which has o be p ocessed in each
plan , i.e., he alue o Pe p. The p oduc ype is ep esen ed by he backg ound
colo , whe eas da kness indica es he plan ouling s a e.
Figu e 9: Op imal de e minis ic schedule.
The compu ed schedule shows how he op imize ies o a oid using he less
e icien plan s when possible, ei he because hey ge highe KEo hey a e
di ie han o he s. The cleaning asks a e scheduled in he be e way, in ol ing
swi ching o a di e en p oduc as long as o e all e iciency is achie ed. Finally,
only 4 plan s a e in a ela i ely di y s a e a he end o he p edic ion ho izon,
so he inal ne wo k s a e gua an ees easibili y in u u e uns.
6.2. Two-s age s ochas ic solu ion
Now we in oduce unce ain y as explained in Sec ion 4. Fi s , a handy example
wi h 3 plan s and 2 p oduc s is p o ided o a be e unde s anding o he u he
esul s. In his example, plan 1can wo k wi h bo h p oduc s whe eas plan
2is assigned o p1and 3 o p2. Plan e iciencies a e se o KE 1= 0.6,
KE 2= 0.7 and KE 3= 0.8. The se poin s o e apo a ed wa e a e ini ially
se o SP1= 32 and SP2= 25 T/h.
8Reducing he gap o 0.5% elapses 20 min and he p o en op imal solu ion (ze o gap) is
go in abou one hou , bu his ex a compu a ional e o is no wo hwhile in p ac ice.
20
Then, o simplici y only unce ain y in he p oduc ion is in oduced, se ing
HR= 7. The conside ed la ges de ia ions om he se poin s a e σp1=
6 and σp2= 4 T/h. Hence, jus conside ing he max/min e ex alues o
he unce ain y ealiza ions, a 4-scena io ee a ises. The ob ained wo-s age
s ochas ic schedule compu ed by sol ing (18) wi h a gi en ini ial plan s a es is
depic ed in Figu e (10).
Figu e 10: Two-s age s ochas ic schedule o 3 plan s and 2 p oduc s.
Now, a mo e complex ne wo k subse conside ing 3 p oduc s and 9 plan s is
se up. The se poin o e apo a ed wa e is se o 40 T/h pe p oduc and
he physical connec ions be ween p oduc s and plan s o his case a e shown
in Table 2.
V1V2V3V4V5V6V7V8V9
P17 3 3 3 3 7 3 3 7
P23 3 7 7 3 3 3 3 3
P33 3 3 3 3 3 3 7 7
KE1 0.88 1.1 1.01 0.77 0.95 1.2 1 1.05
Table 2: Connec ions p oduc -plan and nominal e iciencies.
He e we in oduce unce ain y in bo h he wea he and in he p oduc ion plan
o each p oduc . The expec ed la ges de ia ions o hese sou ces o unce -
ain y a e σTou = 7
°
C and σp= 6 T/h espec i ely. Hence, wi h 3 p oduc s,
a 16-scena io ee a ises. In his case, in o de o ge p o en op imal solu ions
in accep able imes, he p edic ion ho izon has been educed o H= 25 days.
P oblem size is 130536 bina y a iables, 7994 eal ones and 152863 cons ain s.
Sol ing (18) wi h CPLEX 24.8.5 in he same machine e u ns he 1%-gap op i-
mal solu ion in 5 minu es. Figu e (11) depic s he compu ed wo-s age s ochas ic
schedule o plan s 2 and 8 ( he es a e omi ed due o space cons ain s).
Fo comple eness, i (4) is elaxed o allow cleaning se e al plan s a he same
day, only a ela i e cos imp o emen a ound 0.02% is achie ed. So, he op ion
o hi ing mo e pe sonnel o cleaning does no seem po en ially wo hwhile.
21

Figu e 11: Two-s age s ochas ic schedule.
6.3. Mul i-objec i e analysis
Now, de ining a well dis ibu ed g id o poin s wi hin he pe inency ange o
J3and J4, op imal solu ions in he Pa e o sense can be compu ed by sol ing
(25) o line. A se o poin s app oxima ing he Pa e o on is depic ed in Figu e
12. Some in e es ing conclusions can be ex ac ed om i s shape:

E iden ly, he absolu e cos inc eases wi h he p oduc ion. Howe e , he
sensi i i y is highe o low p oduc ions, see Figu e 12b.

Sensi i i y om cos o obus ness is also highe a low p oduc ions. In-
deed, he amoun o di e en solu ions educes as p oduc ion inc eases
(see again Fig.12), ending o he single one wi h SI=100%, which sug-
ges s ha he wo-s age s ochas ic app oach is a was e o compu a ional
esou ces when p oduc ion is cons ained o be high.

Finally, i we look a he speci ic cos pe amoun o p oduc ion ( ep e-
sen ed by he colo map) ins ead o he absolu e cos , he lowes o e all
e iciency is achie ed o low p oduc ions. Howe e , his esul is due o
22
he ac ha all plan s in ope a ion mus be cleaned a e some ime, de-
spi e o whe he hey a e wo king a low load, so he ixed cos s o cleaning
asks make he o e all speci ic cos inc ease.
(a) 3D iew. Colo indica es speci ic cos . (b) 2D iew om objec i es J1and J3.
Figu e 12: App oxima ion o he Pa e o on .
7. Conclusion
This pape add esses a scheduling p oblem o an indus ial e apo a ion ne -
wo k a ec ed by long- e m ouling e ec s and unce ain y in ex e nal ac o s.
Consequen ly, he equipmen canno ope a e o e e wi hou s opping o pe -
o m main enance asks in o de o eco e e iciency. The main ea u e which
makes he p oblem singula om he o mula ion side is ha he RTO needs
o be ex ended o he ne wo k scheduling in a compu a ionally ac able way,
also conside ing unce ain y. A disc e iza ion in days and a modi ica ion o he
gene al p ecedence alloca ion me hod ha e been p oposed o e icien ly ackle
his p oblem.
Unce ain y has been in oduced in he wea he p edic ion and in he p oduc-
ion plan ia a wo-s age s ochas ic op imiza ion app oach. In his way, less
conse a i e obus solu ions a e ob ained by compu ing di e en schedules o
some expec ed unce ain y ealiza ions in he u u e. Mo eo e , he p oposed
app oach gi es solu ions in accep able ime, so we could also ake ad an age o
pe iodically measu ing he ac ual ex e nal ac o s and eschedule acco dingly i
needed.
Fu he mo e, a mul i-c i e ia op imiza ion is p oposed by adding o he objec-
i es o in e es o he economic one, in o de o p o ide plan manage s wi h
signi ican in o ma ion abou he possibili y a ying he p oduc ion o he as-
sumed isk agains unconside ed scena ios. A simila i y index be ween scena io-
based solu ions has been p oposed as a measu e o obus ness in o de o gi e he
schedule he possibili y o educing such isk a he p ice o inc eased conse -
a ism. The shape analysis o he ob ained Pa e o on , al hough dependen on
23
he cu en s a e o he ne wo k, p o ides in e es ing conclusions which p o ide
plan enginee s wi h aluable in o ma ion o design decision-suppo sys ems.
Finally, he app oach is es ed in simula ion wi h se e al ins ances o he e ap-
o a ion ne wo k. The esul s we e p omising so ha he wo-s age s ochas ic
app oach can be p og essi ely ex ended o he whole ne wo k. Ne e heless,
e en ually we will ace la ge p oblems when including mo e acili ies so, in
o de o keep he esolu ion imes wi hin easible anges, ou u u e wo k will
explo e decomposi ion me hods o he o e all p oblem.
Acknowledgmen s
These esul s a e pa o he CoP o p ojec which has ecei ed unding om
he Eu opean Union’s Ho izon 2020 esea ch and inno a ion p og amme unde
g an ag eemen No 723575 and om he Spanish Go e nmen wi h p ojec
INOPTCON (MINECO/FEDER DPI2015-70975-P).
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