

Ci a ion: Sei inge , W.; Cas aneda, J.;
Al endo e , K.; Panade o J.; Juan,
A.A. Applying Simheu is ics o
Minimize O e all Cos s o an MRP
Planned P oduc ion Sys em.
Algo i hms 2022,15, 40. h ps://
doi.o g/10.3390/a15020040
Academic Edi o : Meng Liu
Recei ed: 13 Decembe 2021
Accep ed: 25 Janua y 2022
Published: 27 Janua y 2022
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2022 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
algo i hms
A icle
Applying Simheu is ics o Minimize O e all Cos s o an MRP
Planned P oduc ion Sys em
Wol gang Sei inge 1,* , Juliana Cas aneda 2, Klaus Al endo e 1, Ja ie Panade o 2and Angel A. Juan 3
1School o Business and Managemen , Uni e si y o Applied Sciences Uppe Aus ia, Weh g abengasse 1-3,
4400 S ey , Aus ia; klaus.al endo e @ h-s ey .a
2IN3—Compu e Science Depa men , Uni e si a Obe a de Ca alunya, Rambla Poblenou 156,
08018 Ba celona, Spain; [email p o ec ed] (J.C.); jpanade [email p o ec ed] (J.P.)
3Depa men o S a is ics and OR, Uni e si a Poli ècnica de València, Plaza Fe andiz y Ca bonell,
03801 Alcoy, Spain; [email p o ec ed]
*Co espondence: wol gang.sei inge @ h-s ey .a
Abs ac :
Looking a cu en en e p ise esou ce planning sys ems shows ha ma e ial equi emen s
planning (MRP) is one o he main p oduc ion planning app oaches implemen ed he e. The MRP
planning pa ame e s lo size, sa e y s ock, and planned lead ime, ha e o be iden i ied o each MRP
planned ma e ial. Wi h inc easing p oduc ion sys em complexi y, mo e planning pa ame e s ha e o
be de ined. Simula ion-based op imiza ion is known as a aluable ool o op imizing hese MRP
planning pa ame e s o he unde lying p oduc ion sys em. In his a icle, a as and easy- o-apply
simheu is ic was de eloped wi h he objec i e o minimize o e all cos s. The simheu is ic se s
he planning pa ame e s lo size, sa e y s ock, and planned lead ime o he simula ed s ochas ic
p oduc ion sys ems. The de eloped simheu is ic applies aspec s o simula ion annealing (SA) o an
e icien me aheu is ic-based solu ion pa ame e sampling. Addi ionally, an in elligen simula ion
budge managemen (SBM) concep is in oduced, which skips eplica ions o no p omising i e a ions.
A comp ehensi e simula ion s udy o a mul i-i em and mul i-s aged p oduc ion sys em s uc u e
is conduc ed o e alua e i s pe o mance. Di e en simheu is ic combina ions and pa ame e s a e
es ed, wi h he esul ha he combina ion o SA and SBM led o he lowes o e all cos s. The
con ibu ions o his a icle a e an easy implemen able simheu is ic o MRP pa ame e op imiza ion
and a p omising concep o in elligen ly manage simula ion budge .
Keywo ds: MRP; planning pa ame e ; op imiza ion; simula ion budge ; heu is ic
1. In oduc ion
Fo manu ac u ing companies, an en e p ise esou ce planning sys em (ERP) is he
cen al sys em o plan and con ol p oduc ion- ela ed esou ces. Compa ed o he ech-
nological de elopmen o ERP sys ems, he used planning algo i hms ha e no changed
ha much du ing he las decade. Mos comme cial ERP sys ems s ill use he hie a chical
p oduc ion planning app oach o ma e ial equi emen s planning (MRP) o gene a e p o-
duc ion o de s [
1
]. The applicabili y in di e en indus ies and he s aigh o wa d and
scalable logic—independen om he p oduc complexi y— os e ed MRP’s impo ance
in indus y and also science. The h ee planning pa ame e s o con ol MRP a e lo size,
planned lead ime, and sa e y s ock. Many o he pa ame e s mus be se up and de ined
o ge a p oduc ion sys em on which an MRP can be applied, such as a bill o ma e ial
(BOM), p ocessing ime, se up ime, planning pe iod, machine a ailabili y, and many mo e.
The mas e p oduc ion schedule (MPS), which is based on he cus ome demands, de ines
he quan i ies and ime pe iods o he o de ed ma e ials and speci ies he g oss equi e-
men s o he MRP algo i hm. E en hough he MRP logic is no complica ed, keeping
an MRP sys em up o da e is a challenging ask. Suppose aw ma e ial is a ailable in
he ERP sys em, bu canno be handed o p oduc ion. This ype o misin o ma ion can
Algo i hms 2022,15, 40. h ps://doi.o g/10.3390/a15020040 h ps://www.mdpi.com/jou nal/algo i hms
Algo i hms 2022,15, 40 2 o 18
equi e upda ing he comple e MRP sys em wi h e-planning, including scheduled eceip s
and planned o de eleases [
1
]. Howe e , hese a e no sophis ica ed asks compa ed o
selec ing op imal MRP planning pa ame e s. To o e come a wo sening se ice le el, i is,
o example, possible o inc ease planned lead ime wi h he consequence o inc easing
s ocking le el [
2
]. Conside ing he e ec s o changing he alue o one MRP planning
pa ame e is manageable, bu doing his in a sys ema ic way o lo size, planned lead ime,
and sa e y s ock need o be suppo ed by me hods om he ield o p oduc ion sys em
simula ion and heu is ics. Using p oduc ion sys em simula ion o e alua e he pe o -
mance o a p oduc ion sys em elying on an MRP p o ides in e es ing insigh s in o he
beha io o he p oduc ion sys em pe o mance. In pa icula , he sys ema ic explo a ion
o sui able MRP pa ame e combina ions will help ind op imal alues wi h espec o
gi en pe o mance indica o s, such as in en o y and a diness cos s. P oduc ion sys ems
a e cha ac e ized by a high unce ain y le el, which is associa ed wi h hei planning.
This may be due o in e nal ac o s, such as s ochas ic p ocessing imes, machine ailu es,
limi ed a ailabili y o esou ces—ei he wo ke s o aw ma e ials—e c. I can also be
due o ex e nal ac o s associa ed wi h cus ome s’ demand, such as equi ed lead imes,
changes in o de s eques ed, o cancella ions [
3
]. Gi en his complexi y, he planning o
he MRP pa ame e s in a p oduc ion sys em equi es he de elopmen o me hodologies
capable o dealing wi h sys ems unde unce ain y. The implemen a ion o op imiza ion
echniques based on hyb id me hods be ween me aheu is ic and simula ion echniques
has p o en o be a me hodology wi h he po en ial o es ablish he s ochas ic MRP pa-
ame e s a nea op imal le els [
4
]. The e o e, his pape p oposes a simula ion heu is ic
(simheu is ic) o es ablish he alues o he MRP pa ame e s in a mul i-s age, mul i-i em
p oduc ion sys em. A disc e e e en simula ion model simula ing a s ochas ic MRP sys em
is combined wi h a me aheu is ic using a daemon-like p ocedu al c i e ion and simula ion
budge managemen (SBM). The s ochas ic MRP mimics a p oduc ion plan exposed o
he unce ain ies associa ed wi h s ochas ic o de amoun s, he cus ome - equi ed lead
imes, and machine se up imes. He eby, he ocus o his a icle is o in es iga e how o
apply di e en simheu is ic app oaches o sys ema ically ind he bes MRP pa ame e
se ings in o de o minimize o e all cos s. F om an algo i hmic pe spec i e, he challenge
is o in eg a e he simheu is ics in o he used simula ion amewo k. Da a om he ac ual
i e a ion mus be s o ed and p o ide he ounda ion o subsequen i e a ions. Fo SBM, i
is necessa y o keep ack o p e ious solu ion quali y. This in o ma ion is used o decide
i he emaining eplica ions o an i e a ion a e consumed. P oduc ion sys em modeling
associa ed challenges a e he selec ion o app op ia e s a ing pa ame e s including he
anges o he MRP pa ame e s. The ackled p oblem is o iden i y he combina ion o MRP
pa ame e s p o iding he lowes o e all cos s by applying he de eloped simheu is ics.
Hence, he pe o med simula ion s udy is used o answe he ollowing h ee esea ch
ques ions. To answe hese ques ions he de eloped simheu is ics a e e alua ed using an
MRP-based p oduc ion sys em:
Q1: Does he applica ion o an in elligen simula ion budge managemen (SBM)
p o ide lowe o e all cos s compa ed o a ixed numbe o eplica ions?
Q2: In he con ex o MRP pa ame e s se ing, can he use o simula ed annealing (SA)
help a oid ge ing apped in a local minimum ela ed o he o e all cos s?
Q3: Is he combina ion o SBM and SA leading o lowe minimum o e all cos s and, i
so, wha is he e ec o inc easing simula ion budge ?
The emainde o he a icle is s uc u ed as ollows: Sec ion 2p esen s a li e a u e
e iew on he use o simula ion and simula ion-based op imiza ion me hods o deal wi h
s ochas ic MRP sys ems. Sec ion 3in oduces he simula ion-based op imiza ion heu is ic
used o sol e he p oblem. Sec ion 4explains he cha ac e is ics o he s ochas ic MRP
sys em used in his wo k o es ou me hodology. Sec ion 5desc ibes he simula ion-
op imiza ion expe imen s pe o med. Sec ion 6discusses he esul s ob ained. Finally,
Sec ion 7concludes on he indings o his wo k and discusses u u e lines o esea ch.
Algo i hms 2022,15, 40 3 o 18
2. Rela ed Wo k
P oduc ion sys ems a e cha ac e ized by a high complexi y gi en by he la ge numbe
o pa ame e s in ol ed in hei wo king p ocesses. In gene al, MRP sys ems calcula e
he lo size o be p oduced o o de ed o each componen in each pe iod based on he
i ems’ demand. The e is a BOM in which hey a e o ganized hie a chically as componen s,
and i also includes he numbe o each pa pe inal p oduc [
3
]. This sec ion p esen s a
e iew o some o he wo ks ca ied ou abou he op imiza ion o he pa ame e s s udied
in his wo k, which a e associa ed wi h MRP. The objec i e is o iden i y he mos s udied
pa ame e s, hei combina ion, and he analysis o solu ion me hods. Please no e, ha he
objec i e unc ion o he op imiza ion p oblem discussed in his pape includes in en o y
and backo de cos s. This includes se ice and ime pe o mance e ec s, i.e., low se ice
le el leads o high backo de cos s and high p oduc ion lead imes lead o high in en o y
cos s (and p obably o highe backo de cos s). Consequen ly, he objec i e unc ion in his
a icle is in line wi h a lo o MRP pa ame e op imiza ion p oblems om he p esen ed
li e a u e.
The MRP in a p oduc ion line is modeled o simpli y he sys em wi h a se o as-
sump ions. They include all hose pa ame e s ha allow us o model he unce ain y
and complexi y o he sys em i sel . The op imiza ion o hese models—ei he wi h exac ,
heu is ic, me aheu is ic, o e en hyb id me hods—enables us o ind nea -op imal and
e icien solu ions o he sys em. Among he mos s udied pa ame e s unde unce ain y
in he MRP li e a u e a e he sa e y s ock and he planned lead ime [
5
]. The app oach
employed o manage an MRP unde unce ain y depends on he ype o unce ain y, he
signi icance o i s e ec , and he manage s’ p e e ences [
6
]. Table 1p esen s some o he
op imiza ion wo ks on one o mo e o he MRP pa ame e s s udied in his pape , wi h
hei espec i e solu ion echniques. F om he li e a u e e iew, i can be concluded ha
his a icle ackles he MRP pa ame e op imiza ion p oblem, which is al eady ea ed by
o he au ho s.
Table 1. Rela ed wo k e iewed.
Au ho App oach Pa ame e s Me hod o Analysis
Lead Time Sa e y S ock Lo Size Demand
Whyba k and Williams [7], Buzaco
and Shan hikuma [8], Enns [9]X X X Simula ion
Molinde [10] X X X X Hyb id: simula ed
annealing & simula ion
Al endo e [2] X X Heu is ic
Al endo e e al. [11] X X Simula ion
Teo e al. [12] X X Non-linea op imiza ion
Libe opoulos and Koukoumialos
[13] X X Simula ion
Al endo e and Minne [14] X X Gene al op imiza ion
model
Al endo e [15] X X X Heu is ic
Ba ios e al. [4] X Hyb id: heu is ic &
simula ion
Gans e e e al. [5] X X X
Hyb id: Va iable neighbo
sea ch & simula ion
Ka de e al. [16] X X X Hyb id: Gene ic
Algo i hm & simula ion
Among he ela ed wo k analyzed, i is ound ha simula ion echniques ha e been
widely applied in MRP models since hey allow us o obse e he beha io o he supply
Algo i hms 2022,15, 40 4 o 18
chain h ough he c ea ion o expe imen s based on he model o he sys em unde s udy.
A he same ime, i allows us o obse e he e ec s o changes in MRP pa ame e s. I
o e s he possibili y o planning and con olling he p oduc ion sys ems and imp o ing
s a egies a di e en o ganiza ional le els. Likewise, disc e e-e en simula ion mimics
he dynamics o sys ems in he eal wo ld, which has made i a e y popula me hod o
MRP modeling [
17
]. Whyba k and Williams
[7]
ha e been among he i s o p opose a
amewo k o cha ac e ize and s udy he unce ain y ha can a ec in en o y in es men
and se ice le el pe o mance in an MRP sys em. They used simula ion o compa e wo
pa ame e s, sa e y s ock and sa e y lead ime. They also es ed whe he he e is be e
in en o y con ol ha mee s ma ke unce ain y. In he same di ec ion, Buzaco and
Shan hikuma
[8]
, Enns
[9]
s udied he in luence o sa e y s ock size and planned lead ime.
Likewise, hese au ho s de eloped models ha simul aneously decide on he lo size and
planned lead ime in an en i onmen wi h cons an cus ome demand and s ochas ic lead
imes [
18
]. O he models a e de eloped o add ess he isk gene a ed by unce ain y in
lead imes and demand, while ocusing on deciding which o hese wo pa ame e s should
be used o make decisions [
10
]. The design o hese simula ion models o explo e he e ec
o demand unce ain y on he sys em pe o mance also a ies depending on whe he we
a e conside ing a single-s age manu ac u ing sys em [
2
] o a mul i-i em, mul i-s age MRP
p oduc ion sys em [11].
Typically, MRP simula ion models a e s udied o know he esponses (o ou pu s o
he sys em) o he ini ially de ined pa ame e s [
19
,
20
]. Howe e , his echnique does no
allow de e mining he op imal alues o he pa ame e s o achie e a gi en objec i e [
21
].
Pa ame e op imiza ion o con igu e he sys ems and ope a e as e icien ly as possible is
ecommended [
10
]. In he li e a u e, some models es ablish planned lead imes based on
capaci y equi emen s planning, using nonlinea op imiza ion o ind he op imal alues
o he planning pa ame e s minimizing p oduc ion- ela ed cos s [
12
]. O he models ely
on simula ion-based op imiza ion o pe o m nume ical explo a ions. The in eg a ion o
op imiza ion me hods and simula ion allows us o calib a e he pa ame e s o he s ochas ic
model. In his op imiza ion p ocess, he objec i e unc ion has an associa ed measu e o he
expe imen al simula ion ha is op imized [
22
]. Fo ins ance, i includes he op imiza ion o
planned s ock le els and lead ime in p oduc ion, as well as he op imiza ion o in en o y
sys ems wi h ad anced demand in o ma ion and some se ice le el equi emen s [
13
].
The e a e also analy ical models o he simul aneous op imiza ion o capaci y and planned
deli e y ime in a wo-s age p oduc ion sys em wi h di e en cus ome due da es [
14
].
Ano he wo k in es iga ing he applica ion o analy ical me hods and heu is ics was
p esen ed in [
15
]. The in en o y and he a diness cos s we e analyzed. In addi ion,
a heu is ic-based app oach o sol e he associa ed op imiza ion p oblem was used. All
h ee MRP pa ame e s wi h he limi a ion o only one p oduc ion s age and capaci y, we e
in es iga ed.
One o he me hods o simula ion-based op imiza ion is simheu is ics. I is a solu ion
me hod based on algo i hms combining simula ion me hods wi hin a heu is ic/me aheu is ic
op imiza ion amewo k. I allows o deal wi h s ochas ic combina o ial op imiza ion p ob-
lems [
23
,
24
]. Ba ios e al.
[4]
demons a ed ha simheu is ics a e a p omising app oach
o sol e a wo-s age s ochas ic MRP. They iden i ied he sa e y s ock le el o each i em
o minimize he o al expec ed manu ac u ing cos . Thei esul s demons a e ha he
use o simheu is ics p o ides ad an ages o e he use o simula ion app oaches alone.
Some app oaches a y be ween hem because o he p oposed op imiza ion heu is ics and
me aheu is ics. Fo example, Molinde
[10]
p oposed o de e mine he sa e y lead ime
wi h a hyb id app oach combining simula ion wi h op imiza ion based on he simula ed an-
nealing algo i hm. Gans e e e al.
[5]
p esen ed a simula ion-based op imiza ion app oach
wi h a sea ch p ocedu e based on a iable neighbo hood sea ch wi h a obus p oduc ion
planning as a esul . They ocused on he simul aneous op imiza ion o h ee pa ame e s,
planned lead ime, sa e y s ock, and lo sizes. In addi ion, Ka de e al.
[16]
op imized
he MRP pa ame e s, applied, and compa ed wo di e en e sions o e icien global
Algo i hms 2022,15, 40 5 o 18
op imiza ion o single-objec i e and mul i-objec i e unc ions. Thei esul s demons a e
how bo h app oaches a e compe i i e wi h each o he .
Thus, some o he wo ks ela ed o he op imiza ion o he MRP sys em ha e ocused
mainly on pa ame e s such as lead ime, ollowed by sa e y s ock and lo size. The li e -
a u e ha has analyzed he h ee pa ame e s has always used hyb id simula ion-based
op imiza ion app oaches o analyze and sol e he model. I is because inc easing he
numbe o pa ame e s inc eases he complexi y. The simples analysis me hods usually
in ol e only one o a mos wo pa ame e s. Simula ion is he mos widely used analysis
me hod, ollowed by hyb id simula ion-based op imiza ion me hods. Mo eo e , he la e
is also one o he mos ecen solu ion me hods in he li e a u e. Conside ing he end o
ela ed wo ks, his pape p esen s a simula ion-based op imiza ion me hodology o ind
he alues o he s udied pa ame e s o MRP ha gua an ee an e icien pe o mance o
he p oduc ion sys em, wi h he op imiza ion a ge o minimum o e all cos s.
3. Simula ion Heu is ic
In he con ex o MRP pa ame e se ing, simheu is ic algo i hms [
25
] a e a meaning ul
app oach o e alua e a b oad ange o pa ame e combina ions wi h a de ined objec i e
(e.g., minimizing o e all cos ). A i s simple simheu is ic wi h wo ex ensions o minimize
o e all cos s was p esen ed in [
26
]. Howe e , hei app oach is limi ed in he sense ha i
excludes lo sizing. Thei simheu is ic is signi ican ly ex ended in his pape and applied
also o he lo sizing decision which has a majo impac on he o e all cos . In addi ion,
he new simheu is ic algo i hm is u he in eg a ed in o a disc e e-e en simula ion model,
which implemen s a p oduc ion sys em elying on MRP. Du ing a simula ion expe imen ,
he simheu is ic is applied o compu e new planning pa ame e s o each planned ma e ial.
The nex simula ion i e a ion is hen pe o med wi h he changed planning pa ame e s.
In Sec ion 3.1, h ee e sions o he p oposed simheu is ic a e desc ibed. They all ely on
Algo i hm 1.
3.1. Ini ial Simheu is ic Algo i hm
The ini ial simheu is ic depic ed in Algo i hm 1allows us o de elop di e en a ian s,
named: s a ic ange (STR), exponen ial ange educ ion (ERR), and bes solu ion-se base
ange (BSBR). The ini ial simheu is ic p o ides a desc ip ion o he algo i hmic low, and
his gene ic s uc u e acili a es o subsequen ly in eg a e he a o emen ioned a ian s. The
algo i hm s a s by selec ing he bes solu ion o all p e iously inished i e a ions. The bes
solu ion ep esen s he minimum o e all cos o all inished i e a ions and p o ides a lis
wi h he used mode, lowe bound (LB), and uppe bound (UB).
Fo he STR e sion, a iangula dis ibu ion is used o compu e new alues using
he LB, he UB, and he mode gi en by he midpoin o he ange. Fo all h ee a ian s,
he possible alues o he MRP pa ame e s (planned lead ime and sa e y s ock) lie wi hin
he LB and he UB. While in [
26
] he lo size is no included, in his pape , we conside i s
pa ame e sampling o op imiza ion in he same way as o he pa ame e s planned lead
ime and sa e y s ock.
The second simheu is ic e sion, named ERR, upda es LB and UB using he bes
solu ion. This makes i possible o each a sea ch space ou side he o iginal bounds. The
solu ion ange is de ined as
0=
mode
−
UB o bo h MRP pa ame e s. The new bounds
o he MRP pa ame e s and o each i em a e compu ed as
i= (
1
−α)∗ i−1
, whe e, iis
he simula ion i e a ion numbe and
α
is he ange educ ion le el. The goal o he ERR
a ian is o educe he LB and UB when he numbe o i e a ions inc ease, so he possible
solu ion ange is i e a i ely sh inking.
In he hi d e sion, named BSBR, he LB and he UB a e compu ed using he minimum
and maximum o he bes nsolu ions.

Algo i hms 2022,15, 40 6 o 18
Algo i hm 1 Base Simheu is ic Algo i hm.
1: i Ini ializa ion hen
2: α←e.g., 0.025
3: opN ←e.g., 7
4: ini I e a ions ←e.g., 50
5: ini Pa ame e Range(lb,ub,mode)
6: n←0
7: endIni Phase ← alse
8: cu en Solu ion ←T iangula Dis ibu ion(pa ame e Range)
9: baseSolu ion ←cu en Solu ion
10: end i
11: while n≤maxReplica ions do
12: cos (cu en Solu ion)←Simula eMRP(cu en Solu ion)
13: i n>ini I e a ions hen
14: endIni Phase ← ue
15: end i
16: i cos (baseSolu ion)<cos (cu en Solu ion) hen
17: baseSolu ion ←cu en Solu ion
18: end i
19: baseSolu ion ←Ge Bes Solu ion()
20: i simheu is ic =STR ∨endIni Phase = alse hen
21: pa ame e Range.mode ←baseSolu ion
22: end i
23: i simheu is ic =ERR ∧endIni Phase = ue hen
24: oldRange ←(pa ame e Range.ub −pa ame e Range.lb)/2
25: newRange ←(1−α)∗oldRange
26: pa ame e Range.mode ←baseSolu ion
27: pa ame e Range.lb ←pa ame e Range.mode −newRange
28: pa ame e Range.ub ←pa ame e Range.mode +newRange
29: end i
30: i simheu is ic =BSBR ∧endIni Phase = ue hen
31: bes NSolu ions ←Ge Bes NSolu ions( opN)
32: pa ame e Range.mode ←baseSolu ion
33: pa ame e Range.lb ←min(bes NSolu ions)
34: pa ame e Range.ub ←max(bes NSolu ions)
35: end i
36: baseSolu ion ←T iangula Dis ibu ion(pa ame e Range)
37: n←n+ eplica ionsPe I e a ion
38: end while
The cons an
α
is se o he ERR ange educ ion and opN is equi ed only o he BSBR
e sion. An ini ializa ion phase can be applied o bo h he ERR and BSBR e sions. The
numbe o i e a ions is ep esen ed by he cons an ini I e a ions. The a iable endIni Phase
is used o con ol he end o he ini ializa ion phase. No ice ha , by design, ERR and BSBR
a e ex ensions o STR. The e o e, STR is applied du ing he ini I e a ions. Thus, o example,
50 i e a ions ou o a maximum o 300 i e a ions can be used o ind s a ing alues o
ange educ ion and he bes
n
solu ions. A key componen in he algo i hmic low is
he a ay pa ame e Range, as i ep esen s he s a ing LB and UB alues, as well as he
mode o each MRP pa ame e and planned i em. In addi ion, his a ay is con inuously
upda ed du ing he simula ion. The inal s ep o each i e a ion in each e sion is o pass
he adap ed pa ame e Range o he iangula dis ibu ion. This allows us o gene a e new
MRP pa ame e alues and apply hem du ing he MRP simula ion. A he beginning o
each i e a ion, he base solu ion is upda ed o he cu en Solu ion in case his is he new
bes - ound solu ion. The bes solu ion is ep esen ed by he minimum o e all cos o he
p e ious i e a ions, compu ed using he a e age o e all cos o he simula ion uns. The
objec i e is o minimize o e all cos s. A limi a ion is ha logis ic objec i es such as se ice
Algo i hms 2022,15, 40 7 o 18
le el and lead ime a e no conside ed. E en hough he o e all cos c i e ia a e common
in a lo o s udies (see Sec ion 2, e.g., [
5
,
13
,
15
]), his limi a ion implies ha he e ec o
he op imized planning pa ame e s on o he impo an key pe o mance indica o s is no
analyzed which es ic s he discussion o one dimension.
The algo i hm is epea ed un il a maximum numbe o uns (maxReplica ions) is
eached. Hence, o ins ance, o 100 i e a ions wi h 20 uns, he alue o maxReplica-
ions is 2000. The s a emen Simula eMRP is hen pe o med 20 imes pe i e a ion.
As shown in [
26
], e en basic e sions o hese concep s can be use ul o iden i y
pa ame e se ings ha minimize he o e all cos . In his pape , we ex end he ini ial
concep s o de elop a holis ic app oach o MRP pa ame e op imiza ion. Thus, he
ollowing no el aspec s ha e been conside ed in ou s udy: (i) he MRP pa ame e lo size
has been added; (ii) he o iginal heu is ic has been ex ended in o a ull simheu is ic; and
(iii) an in elligen simula ion budge managemen is included. The a ge o he simula ion
budge managemen is o es solu ion quali y a e each un wi hin an i e a ion. In o he
wo ds, o each pa ame e se , he o e all cos is compa ed o he pas esul s, and only i
he solu ion is su icien ly good, u he uns o he cu en i e a ion a e conduc ed. The
a ge is o pe o m mo e i e a ions wi h he same o e all numbe o uns—i.e., he same
simula ion budge —by a oiding unnecessa y uns.
3.2. Simula ed Annealing
Escaping a local minimum and explo ing a new solu ion ange equi es a conc e e
s a egy. A biased- andomiza ion algo i hm [
27
] o iangula dis ibu ion mode is de-
sc ibed in Algo i hm 2. This algo i hm applies aspec s o he SA amewo k [
28
]. In
addi ion, he eedom aspec o a demon algo i hm is used. Such a demon-based beha io
is explained in [
29
]. A e each i e a ion, he cos s o he base solu ion a e compa ed o he
cos s o he bes solu ion. Each ime he cos s o he base solu ion a e g ea e o equal o
he bes solu ion, a ese coun e is dec eased by one. When he ese coun e eaches 0,
he base solu ion is ese o he bes solu ion. The ese o he bes solu ion a oids was ing
simula ion budge in a no p omising solu ion space.
Algo i hm 2 Reac i e biased- andomiza ion o iangula dis ibu ion mode.
1: i Ini ializa ion hen
2: ese Coun e ←ini ialValue e.g., 5
3: bes Solu ion ←baseSolu ion
4: end i
5: δ=cos (baseSolu ion)−cos (cu en Solu ion)
6: i δ>0 hen
7: c edi ←δ
8: baseSolu ion ←cu en Solu ion
9: i cos (baseSolu ion)<cos (bes Solu ion) hen
10: bes Solu ion ←baseSolu ion
11: end i
12: else i −δ≤c edi hen
13: baseSolu ion ←cu en Solu ion
14: c edi ←0
15: end i
16: i cos (cu en Solu ion)≥cos (bes Solu ion) hen
17: ese Coun e
−−
18: end i
19: i ese Coun e =0 hen
20: baseSolu ion ←bes Solu ion
21: ese Coun e ←ini alValue
22: end i
Algo i hms 2022,15, 40 8 o 18
3.3. Simula ion Budge Managemen (Sbm)
Limi ing he simula ion expe imen by a maximum numbe o i e a ions and uns pe
i e a ion inc eases he p obabili y o exclude po en ial good solu ions, as hey a e excluded
due o he un ou o simula ion i e a ions. Always pe o ming a cons an numbe o
uns pe i e a ion will no necessa ily inc ease he po en ial o ind a new bes solu ion.
Consequen ly, i would be mo e meaning ul o in oduce a cons ain o skipping he
ac ual i e a ion and in es he emaining simula ion budge o explo e new solu ions wi h
addi ional i e a ions and a changed se o pa ame e alues. A simple simula ion budge
managemen pseudocode, which acili a es o un he simula ion expe imen , is desc ibed in
Algo i hm 3. This algo i hm allows as many i e a ions as budge is a ailable, and will s op
he las i e a ion when he maximum numbe o uns is eached. The bes solu ion always
has he maximum numbe o he gi en eplica ions due o he s opping c i e ia. The a e age
cos o he p e ious i e a ions (a gI e a ionO e allCos s) is equal o
1
n∑n
io e allCos si
, wi h
n=
maximum uns pe i e a ion (maxReplica ionsPe I e a ion). The a gI e a ionO e allCos s
is compa ed o he pe cen ile alue o all pas i e a ions, which depends on he eplica ion
numbe . The pe cen ile alue is compu ed using Ge Pe cen ilValue(se O AllSolu ions,
β)
,
which equi es he p e ious o e all cos o be equal o se O AllSolu ions, and a alue
educed a e each un om a UB, e u ning he alue associa ed wi h he posi ion in
he passed solu ions. These alues a e ep esen ed by
β
and pe cen ileS ep. A e each
eplica ion, he o e all cos is added o he alues o he cu en i e a ion and he new
pe cen ile alue is compu ed. Compa ed o he ini ial Algo i hm 1, we do no always
pe o m a ixed numbe o uns. The SBM s ops a e he simula ionBudge is exhaus ed.
Algo i hm 3 Simula ion budge managemen .
1: i Ini ializa ion hen
2: o alReplica ionCoun ←0
3: simula ionBudge ←maxI e a ions ∗maxReplica ionsPe I e a ion
4: s opCu en I e a ion ← alse
5: cu en Replica ionCoun ←1
6: pe cen ileS ep ←0.0175
7: end i
8: β←0.4
9: while cu en Replica ionCoun ≤maxReplica ionsPe I e a ion ∧
10: s opCu en I e a ion = alse do
11: o alReplica ionCoun
++
12: cu en Replica ionCoun
++
13: i a gI e a ionO e allCos s >Ge Pe cen ilValue(se O AllSolu ions,β)∧
cu en Replica ionCoun >3 hen
14: s opCu en I e a ion ← ue
15: end i
16: β=β−pe cen ilS ep
17: end while
18: s opCu en I e a ion ← alse
19: eplica ionsPe I e a ion ←cu en Replica ionCoun
4. Modeling a S ochas ic M p-Based P oduc ion Sys em
To model a s ochas ic MRP sys em, disc e e e en simula ion can be used [
30
]. F om
a echnical pe spec i e, his equi es agen s o simula e he cus ome s’ o de s beha io
and a simula ion amewo k capable o eading he gi en expe imen planning pa ame e s
and change hem a e each i e a ion. The expe imen planning pa ame e s include s a
alues and bounda ies o he MRP- ela ed planning pa ame e s o lo size, planned lead
ime, and sa e y s ock. In addi ion, MRP ela ed se ings, such as planning ho izon and
addi ionally echnical pa ame e s a e equi ed o he simula ion expe imen s, such as
he numbe o simula ion uns and algo i hm i e a ions [
31
]. Cus ome ’s o de agen s
Algo i hms 2022,15, 40 9 o 18
a e passed o a queue. Whene e a cus ome ’s o de canno be comple ed be o e he
deadline, i is classi ied as ‘delayed’. The same logic is applied o he p oduc ion o de s.
The p oduc ion o de s a e gene a ed based on cus ome ’s o de s, which a e he g oss
equi emen s o he applied MRP un. The ou pu o each MRP un is he p oduc ion
o de s, which p o ide he in o ma ion on quan i y, i em, s a and end da e. A e each
i e a ion he solu ion quali y is e alua ed, i.e., he a e age o all eplica ions’ o e all cos s is
calcula ed, and he simheu is ic is applied o se he new MRP planning pa ame e alues
o he nex i e a ion. The simula ion esul s o each i e a ion (MRP pa ame e s, backo de
cos s, in en o y cos s, i e a ion coun , eplica ion coun ,
. . .
) a e s o ed a e each i e a ion
in he linked in-memo y da abase. These alues a e hen a ailable o he subsequen
i e a ions. The esul s o he cu en i e a ion including each eplica ion a e hold in he
wo king memo y o he simula ion compu e , un il he las eplica ion is pe o med. The
esul s o he in-memo y da abase a e s o ed as da abase ile and a e he basis o he
pe o mance analysis o he unde aken simula ion s udy.
4.1. S ochas ic M p Se ing
Ou MRP simula ion model can handle s ochas ic demands as well as andom p o-
cessing imes, and p o ides he desc ibed s anda d MRP logic o ea cus ome s’ o de s.
S ochas ic beha io is in oduced and con olled using log-no mal p obabili y dis ibu ions
wi h expec ed alues (
µi
) and a iances (
σ2
i
) o machine se up ime, cus ome equi ed
lead ime and cus ome ’s expec ed o de amoun . The andom a iables can be ound in
Table 2, which also includes he espec i e coe icien s o a ia ion (CVi).
Table 2. Random a iables wi h log-no mal beha io .
Pa ame e I em µiσ2
iCVi
o de amoun 10 10 2 0.1414
o de amoun 11 15 6 0.1633
cus ome equi ed lead ime
10 6 9 0.5000
cus ome equi ed lead ime
11 6 10 0.5270
machine se up ime all 12 36 0.5000
The simula ed p oduc ion sys em is illus a ed in Figu e 1. The ole o he de eloped
simheu is ic is o se lo size, planned lead ime and sa e y s ock pa ame e s in o de o
minimize he o e all cos , which is he sum o in en o y and a diness cos s. Despi e being
a simple BOM ha conside s only h ee le els, he example allows o demons a e how
simheu is ics can be meaning ully in eg a ed in o an MRP sys em o op imize o e all cos s
o all h ee planning pa ame e s. Two inal p oduc s, p oduc s 10 and 11, a low-le el-code
(LLC) 0 a e p oduced on machine
M
2. The wo semi- inished p oduc s, ma e ials 20 and
21, a e p oduced on machine
M
1. Fo one uni o inal p oduc 10, 1 semi- inished p oduc
20 is needed. One piece o semi- inished p oduc 21 is needed o inal p oduc 11. The aw
ma e ial 100, which is a pu chased p oduc , is needed o he semi- inished p oduc s 20
and 21. This aw ma e ial is assumed o be always a ailable. Fo his simple p oduc ion
sys em, 12 di e en pa ame e s ha e o be op imized because du ing simula ion o each
o he ma e ials (10, 11, 20 and 21), all h ee MRP planning pa ame e s ha e o be es ed
o op imali y. Table 3shows he s a -up alues and espec i e anges o op imiza ion
pa ame e s. Please no e ha he MRP planning pa ame e alues a e ea ed and e alua ed
as in ege uni s. This means ha in he simula ion esul s, he inc ease o dec ease o
o e all cos s ela ed o each indi idual MRP pa ame e can be acked. Fu he mo e, he
in e ela ion be ween he single planning pa ame e s is in eg a ed in he simula ion model,
e.g., bo h a highe planned lead ime and a highe sa e y s ock lead o highe in en o y cos s
and lowe a diness cos s, also compa e o Buzaco and Shan hikuma
[8]
. A dec ease in
lo size, o example, leads o highe se up e o s and, he e o e, also a ec s he p oduc ion
Algo i hms 2022,15, 40 16 o 18
mo e MRP planning pa ame e s, i.e., mo e decision a iables. This equi es o upscale
he sys em o pa ame e handling and in e media e esul compu a ion. Fu he mo e, an
e icien MRP pa ame e upda e s a egy has o be de eloped. Resul s o such an implemen-
a ion wi h eal da a can p o ide aluable esul s abou op imal pa ame e combina ions
and he pe o mance issues o he espec i e p oduc ion sys em. The au oma ic compu-
a ion o op imal MRP planning pa ame e based on op imized in en o y and backo de
cos s p o ides use ul in o ma ion o p oduc ion planning ela ed decision making. F om
a scien i ic poin o iew, especially he gene al app oach o SBM is eusable o o he
simula ion s udies and can help consume he a ailable simula ion budge o es la ge
solu ion spaces.
Figu e 3. Compa ison be ween he bes esul ob ained in each simheu is ic.
The p esen ed simheu is ic does no explici ly conside he in e ela ion be ween
op imized planning pa ame e s. Igno ing he pa ame e dependency can ha e nega i e
e ec s on he op imized pa ame e s and he inal esul . Howe e , pa ame e in e ela ion
is also igno ed by o he heu is ics, compa e wi h [
16
]. In es iga ing MRP pa ame e
in e ela ion may p o ide addi ional insigh in o op imiza ion-based p oduc ion sys em
simula ions, bu is ou o con ex o his a icle.
7. Conclusions and Fu u e Wo k
This a icle p o ides insigh s in o he applica ion o simheu is ics in iden i ying op i-
mal MRP planning pa ame e alues o lo size, planned lead ime and sa e y s ock. The
aim is o ind he minimal o e all cos (in en o y cos s plus backo de cos s) compu ed
du ing he applica ion o he MRP planning algo i hm in a simula ion en i onmen . This
a icle is a u he de elopmen o he esul s based on h ee di e en simheu is ic e sions
o iden i y he minimal o e all cos only o he MRP planning pa ame e sa e y s ock and
planned lead ime. The i s ex ension is ha op imal alues o all h ee MRP planning
pa ame e s a e sea ched. A second ex ension is he de elopmen o a SA-based simheu is ic
ex ension. The hi d ex ension ocuses on inding op imal pa ame e s wi hou was ing
simula ion budge o non-p omising i e a ions. Hence, an in elligen simula ion budge
managemen (SBM) is in oduced o consume he a ailable eplica ions only o po en ial
bes solu ions, hus skipping solu ions ou side he de ined pe cen ile ange. SBM was
combined wi h he SA ex ension o addi ionally explo e new a eas o he solu ion space
and o escape local minima. The applica ion o he de eloped SBM and SA ex ensions,
combined wi h all h ee planning pa ame e s, allows o a sys ema ic iden i ica ion o
op imal planning pa ame e s wi h a clea op imiza ion a ge . The nume ical esul s show
ha he bes pe o ming e sion o his applica ion is ERR. F om a p oduc ion planning

Algo i hms 2022,15, 40 17 o 18
pe spec i e, he de eloped simheu is ic e sions, speci ically ERR, p o ides a as and well
pe o ming simula ion-based op imiza ion app oach which is simple o apply in p ac ical
en i onmen s. The pe o med simula ion s udy shows he imp o emen o he exis ing
simheu is ic e sions ega ding he iden i ica ion o he minimal o e all cos wi h he join
applica ion o SBM and SA. This inding is suppo ed by he lowes o e all cos ob ained
wi h he highes simula ion budge and he join applica ion o SBM and SA o all h ee
simheu is ic e sions. Limi a ions o his s udy a e ela ed o he applica ion ield and he
p oduc ion sys em wi hin which he simheu is ics a e es ed, i.e., only one applica ion ield
is e alua ed in his s udy and he espec i e p oduc ion sys em is s ill a he simple. How-
e e , his a icle p o ides he ounda ion o u u e in es iga ions, o sys ema ically apply
he simheu is ics concep o mo e complex p oduc ion sys em s uc u es and o o he ap-
plica ion ields. The de eloped simheu is ics ocus on a single objec i e unc ion, howe e ,
o eal p oduc ion sys ems, i is no always possible o ans o m all objec i e dimensions
in o cos s. The e o e, u u e wo k could in es iga e how o ex end he simheu is ic o a
mul i-objec i e app oach. A b oad sensi i i y analysis o p o e he p omising esul s o
SBM is also planned in u u e wo k.
Au ho Con ibu ions:
Concep ualiza ion, K.A. and A.A.J.; me hodology, K.A. and J.P.; so wa e,
W.S.; alida ion, K.A., W.S.; o mal analysis, K.A., W.S. and J.C.; in es iga ion, W.S. and J.C.; esou ces,
W.S.; da a cu a ion, W.S.; w i ing—o iginal d a p epa a ion, W.S. and J.C.; w i ing— e iew and
edi ing, K.A., W.S., J.C. and A.A.J.; isualiza ion, W.S. and J.C.; supe ision, K.A. and J.P.; p ojec
adminis a ion, K.A. and J.P. All au ho s ha e ead and ag eed o he published e sion o he
manusc ip .
Funding:
This wo k has been pa ially unded by he Spanish Minis y o Science (PID2019-111100RB-
C21/AEI/10.13039/501100011033 and RED2018-102642-T), he E asmus+ P og am (2019-I-ES01-
KA103-062602), and unded by he Aus ian Science Fund (FWF): P32954-G.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : No applicable.
Con lic s o In e es :
The au ho s decla e no con lic o in e es . The unde s had no ole in he design
o he s udy; in he collec ion, analyses, o in e p e a ion o da a; in he w i ing o he manusc ip , o
in he decision o publish he esul s.
Abb e ia ions
The ollowing abb e ia ions a e used in his manusc ip :
BOM Bill o Ma e ial
CU Cos Uni
CV Coe icien o a ia ion
ERP En e p ise Resou ce Planning
FOP Fixed O de Pe iod
FOQ Fixed O de Quan i y
LB Lowe Bound
MRP Ma e ial Requi emen s Planning
UB Uppe Bound
SA Simula ed Annealing
SBM Simula ion Budge Managemen
Re e ences
1. Hopp, W.J.; Spea man, M.L. Fac o y Physics, 3 d ed.; Wa eland P ess: Long G o e, IL, USA, 2011.
2.
Al endo e , K. In luence o lo size and planned lead ime on se ice le el and in en o y o a single-s age p oduc ion sys em
wi h ad ance demand in o ma ion and andom equi ed lead imes. In . J. P od. Econ. 2015,170, 478–488. [C ossRe ]
3.
The enin, S.; Adulyasak, Y.; Co deau, J.F. Ma e ial equi emen s planning unde demand unce ain y using s ochas ic op imiza-
ion. P od. Ope . Manag. 2021,30, 475–493. [C ossRe ]
Algo i hms 2022,15, 40 18 o 18
4.
Ba ios, B.B.; Juan, A.A.; Panade o, J.; Al endo e , K.; Pei lei ne , A.J.; Es ada-Mo eno, A. On he use o simheu is ics o op imize
sa e y-s ock le els in ma e ial equi emen s planning wi h andom demands. In P oceedings o he 2020 Win e Simula ion
Con e ence (WSC), O lando, FL, USA, 14–18 Decembe 2020; pp. 1539–1550.
5.
Gans e e , M.; Almede , C.; Ha l, R.F. Simula ion-based op imiza ion me hods o se ing p oduc ion planning pa ame e s. In . J.
P od. Econ. 2014,151, 206–213. [C ossRe ]
6.
Koh, S.C.L.; Saad, S.M.; Jones, M. Unce ain y unde MRP-planned manu ac u e: Re iew and ca ego iza ion. In . J. P od. Res.
2002,40, 2399–2421. [C ossRe ]
7. Whyba k, D.C.; Williams, J.G. Ma e ial equi emen s planning unde unce ain y. Decis. Sci. 1976,7, 595–606. [C ossRe ]
8.
Buzaco , J.A.; Shan hikuma , J. Sa e y s ock e sus sa e y ime in MRP con olled p oduc ion sys ems. Manag. Sci.
1994
,
40, 1678–1689. [C ossRe ]
9. Enns, S. MRP pe o mance e ec s due o lo size and planned lead ime se ings. In . J. P od. Res. 2001,39, 461–480. [C ossRe ]
10.
Molinde , A. Join op imiza ion o lo -sizes, sa e y s ocks and sa e y lead imes in an MRP sys em. In . J. P od. Res.
1997
,
35, 983–994. [C ossRe ]
11.
Al endo e , K.; Felbe baue , T.; Jodlbaue , H. E ec s o o ecas e o s on op imal u ilisa ion in agg ega e p oduc ion planning
wi h s ochas ic cus ome demand. In . J. P od. Res. 2016,54, 3718–3735. [C ossRe ]
12.
Teo, C.C.; Bha naga , R.; G a es, S.C. Se ing planned lead imes o a make- o-o de p oduc ion sys em wi h mas e schedule
smoo hing. Iie T ans. 2011,43, 399–414. [C ossRe ]
13.
Libe opoulos, G.; Koukoumialos, S. T adeo s be ween base s ock le els, numbe s o kanbans, and planned supply lead imes in
p oduc ion/in en o y sys ems wi h ad ance demand in o ma ion. In . J. P od. Econ. 2005,96, 213–232. [C ossRe ]
14.
Al endo e , K.; Minne , S. Simul aneous op imiza ion o capaci y and planned lead ime in a wo-s age p oduc ion sys em wi h
di e en cus ome due da es. Eu . J. Ope . Res. 2011,213, 134–146. [C ossRe ]
15.
Al endo e , K. E ec o limi ed capaci y on op imal planning pa ame e s o a mul i-i em p oduc ion sys em wi h se up imes
and ad ance demand in o ma ion. In . J. P od. Res. 2019,57, 1892–1913. [C ossRe ]
16.
Ka de , J.; Beham, A.; Pei lei ne , A.; Al endo e , K. Su oga e-Assis ed Mul i-Objec i e Pa ame e Op imiza ion o P oduc ion
Planning Sys ems. In In e na ional Con e ence on Compu e Aided Sys ems Theo y; Sp inge : Be lin/Heidelbe g, Ge many, 2019;
pp. 239–246.
17.
Ingalls, R.G. In oduc ion o simula ion. In P oceedings o he 2011 Win e Simula ion Con e ence (WSC), Phoenix, AZ, USA,
11–14 Decembe 2011; pp. 1374–1388.
18.
Louly, M.A.; Dolgui, A. Op imal MRP pa ame e s o a single i em in en o y wi h andom eplenishmen lead ime, POQ policy
and se ice le el cons ain . In . J. P od. Econ. 2013,143, 35–40. [C ossRe ]
19.
And adó i , S. Simula ion op imiza ion. In P oceedings o he Win e Simula ion Con e ence, San Diego, CA, USA, 8–11
Decembe 2002; pp. 307–334.
20. Fu, M.C. Op imiza ion ia simula ion: A e iew. Ann. Ope . Res. 1994,53, 199–247. [C ossRe ]
21.
Gosa i, A. Simula ion-based op imiza ion: An o e iew. In Simula ion-Based Op imiza ion; Sp inge : Be lin/Heidelbe g, Ge many,
2003; pp. 47–55.
22. Fu, M.C.; Glo e , F.W.; Ap il, J. Simula ion op imiza ion: A e iew, new de elopmen s, and applica ions. In P oceedings o he
Win e Simula ion Con e ence, O lando, FL, USA, 4 Decembe 2005; p. 13.
23.
Rabe, M.; Deininge , M.; Juan, A.A. Speeding up compu a ional imes in simheu is ics combining gene ic algo i hms wi h
disc e e-e en simula ion. Simul. Model. P ac . Theo y 2020,103, 102089. [C ossRe ]
24.
G ule , A.; Panade o, J.; de A mas, J.; Mo eno, J.A.M.; Juan, A.A. Combining a iable neighbo hood sea ch wi h simula ion o
he in en o y ou ing p oblem wi h s ochas ic demands and s ock-ou s. Compu . Ind. Eng. 2018,123, 278–288. [C ossRe ]
25.
Chica, M.; Juan, A.A.; Bayliss, C.; Co dón, O.; Kel on, W.D. Why simheu is ics? Bene i s, limi a ions, and bes p ac ices when
combining me aheu is ics wi h simula ion. SORT-S a . Ope . Res. T ans. 2020,44, 311–334. [C ossRe ]
26.
Sei inge , W.; Al endo e , K.; Cas aneda, J.; Panade o, J.; Juan, A. Applying Simheu is ics o Sa e y S ock and Planned Lead
Timeop imiza ion in a Rolling Ho izon M p Sys em unde Unce ain y. In P oceedings o he 2021 Win e Simula ion Con e ence,
Phoenix, AZ, USA, 13–15 Decembe 2021; Kim, S., Feng, B., Smi h, K., Masoud, S., Zheng, Z., Szabo, C., Lope , M., Eds.; Ins i u e
o Elec ical and Elec onics Enginee s, Inc.: Pisca away, NJ, USA, 2021; p. 999.
27.
Fe one, D.; G ule , A.; Fes a, P.; Juan, A.A. Enhancing and ex ending he classical GRASP amewo k wi h biased andomisa ion
and simula ion. J. Ope . Res. Soc. 2019,70, 1362–1375. [C ossRe ]
28.
Panade o, J.; Juan, A.A.; Bayliss, C.; Cu ie, C. Maximising ewa d om a eam o su eillance d ones: A simheu is ic app oach
o he s ochas ic eam o ien ee ing p oblem. Eu . J. Ind. Eng. 2020,14, 485–516. [C ossRe ]
29.
Juan, A.A.; Lou enço, H.R.; Ma eo, M.; Luo, R.; Cas ella, Q. Using i e a ed local sea ch o sol ing he low-shop p oblem:
Pa alleliza ion, pa ame iza ion, and andomiza ion issues. In . T ans. Ope . Res. 2014,21, 103–126. [C ossRe ]
30.
Law, A.M. Simula ion Modeling and Analysis, 5 h ed.; McG aw-Hill Se ies in Indus ial Enginee ing and Managemen Science;
McG aw-Hill Educa ion: New Yo k, NY, USA, 2014.
31.
Bo shche , A. The Big Book o Simula ion Modeling: Mul ime hod Modeling wi h Anylogic; AnyLogic No h Ame ica: Oakb ook
Te ace, IL, USA, 2013.