Eu opean Jou nal o Ope a ional Resea ch 303 (2022) 1137–1150
Con en s lis s a ailable a ScienceDi ec
Eu opean Jou nal o Ope a ional Resea ch
jou nal homepage: www.else ie .com/loca e/ejo
P oduc ion, Manu ac u ing, T anspo a ion and Logis ics
Pe o mance measu es o nons a iona y in en o y models o
pe ishable p oduc s unde he EWA policy
Ca los Go ia
a
, Mikel Lezaun
a
, F. Ja ie López
b , ∗
a
Dp o. Ma emá icas, Uni e sidad del País Vasco - UPV/EHU, Spain
b
Dp o. Mé odos Es adís icos and BIFI, Uni e sidad de Za agoza, Spain
a i c l e i n o
A icle his o y:
Recei ed 11 Feb ua y 2021
Accep ed 9 Ma ch 2022
A ailable online 13 Ma ch 2022
Keywo ds:
In en o y
Pe ishable
Pe iodic e iew
S ochas ic demand
Pla ele s
a b s a c
Accu a ely es ima ing key pe o mance indica o s in in en o y models o pe ishable i ems is essen ial
in o de o assess and imp o e he managemen s a egy o hese sys ems. We analyse he p oduc ion
o pla ele concen a es a blood banks unde he EWA eplenishmen policy. We gi e analy ical app oxi-
ma ions o he mos impo an pe o mance measu es, such as he size o o de s, he size o s ocks, he
pe cen age o ou da ing, he age dis ibu ion o s ocks and he eshness o uni s issued, among o he s.
The p oduc ion o pla ele concen a es is a p o o ypical example o in en o y models o sho li e i ems
wi h andom demand and a weekly pa e n, whe e a high se ice le el is equi ed. The me hodology and
he app oxima ions p esen ed he e can be easily adap ed o o he in en o y sys ems wi h simila cha -
ac e is ics. Mos o he o mulae in his a icle a e new o nons a iona y models unde he EWA policy;
indeed, o mulae o he age dis ibu ion o uni s in s ock and o uni s issued ha e no appea ed in he
li e a u e e en o he simple base-s ock eplenishmen policy. We apply ou esul s o a eal blood bank
and find e y close ag eemen be ween he o mulae and he esul s o Mon e Ca lo simula ions. The ac-
cu acy o ou app oxima ions is also es ed in se e al scena ios, depending on he li e ime o uni s, sa e y
s ock le els and he p obabilis ic dis ibu ion o demand.
© 2022 The Au ho (s). Published by Else ie B.V.
This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ )
1. In oduc ion
1.1. Backg ound and mo i a ion
In en o y managemen o pe ishable i ems is o g ea impo -
ance in many sec o s o he economy. Food and blood p oduc s
a e jus a couple o examples o pe ishable goods. The ma hema -
ical analysis o in en o y models o hese p oduc s is much mo e
difficul han o nonpe ishable i ems. Also, al hough he e a e a
g ea many esea ch pape s on models o pe ishable goods hey
a e s ill a ewe in numbe han he pape s and books de o ed o
nonpe ishable p oduc s; see, e.g., Sil e , Pyke, & Pe e son (1998) .
Much o he esea ch on in en o y managemen o pe ish-
able p oduc s has ocused on blood p oduc s, and has been pub-
lished bo h in medical and ma hema ical jou nals; see A kinson,
Fon aine, Goodnough, & Wein (2012) , Beliën & Fo cé (2012) ,
Ci elek, Ka aesmen, & Schelle -Wol (2015) , Ensafian & Yaghoubi
(2017) , Rajend an & Ra ind an (2019) . Blood p oduc s a e used o
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (C. Go ia), [email p o ec ed]
(M. Lezaun), ja ie .lopez@uniza .es (F.J. López).
ans usion in mos hospi als and a e seen as a sca ce, p ecious
esou ce. They a e o i al impo ance o pa ien s, so a sufficien
s ock mus be kep in o de o a oid s ockou s. Di e en compo-
nen s o blood, such as ed blood cells, plasma and pla ele s, a e
used o ans usion. Among hem, pla ele concen a es a e con-
side ed as a c i ical p oduc since hey ha e a sho li e ime (usu-
ally 5 o 7 days). They a e also expensi e ( o ins ance, Haijema,
an de Wal, & an Dijk (2007) assume a cos o mo e han 450 Eu-
os pe pa ien pe ea men ), and o e cau ious policies in keeping
big s ocks esul in a la ge numbe o ou da ed uni s, leading o an
unnecessa y was e o money and e hical conce ns.
In his pape we ocus on a pe iodic e iew model o fixed li e-
ime pe ishable goods such ha s ockou s mus be kep o a min-
imum. Pla ele s a e a clea example o such p oduc s, and we use
he in en o y model o he Basque Cen e o T ans usion and Hu-
man Tissues (CVTTH) in Galdakao, Bizkaia, Spain, o he de i a ion
o ou o mulae. This esea ch o igina ed in collabo a ion be ween
he Uni e si y o he Basque Coun y and he CVTTH o he imple-
men a ion o a ma hema ical model o he managemen o blood
p oduc s; see Pé ez Vaque o, Go ia, Lezaun, López, Monge, Eguiz-
abal, & Vesga (2016) , Go ia, Laba a, Lezaun, López, Pé ez Aliaga,
& Pé ez Vaque o (2020) . The pape ocuses on his model o he
h ps://doi.o g/10.1016/j.ejo .2022.03.018
0377-2217/© 2022 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ )
C. Go ia, M. Lezaun and F.J. López Eu opean Jou nal o Ope a ional Resea ch 303 (2022) 1137–1150
p oduc ion o pla ele concen a es, bu ou analysis can easily be
adap ed o o he pe ishable goods whe e he se ice le el needs o
be high.
1.2. Objec i e o he pape
In p ac ice, i is e y use ul o ha e app oxima ions o he pe -
o mance measu es o an in en o y sys em, which can be used o
alida e policies and op imise he pa ame e s o he model. When
such app oxima ions a e no a ailable, alida ion and op imisa ion
mus ely on simula ions. This pape se s ou o de i e analy ical
app oxima ions o he main pe o mance measu es o nons a ion-
a y models when he Es ima ed Wi hd awal and Aging (EWA) e-
plenishmen policy is used and a high se ice le el is needed. EWA
is a modifica ion o he base-s ock eplenishmen policy whe e he
in en o y posi ion is modified by sub ac ing an es ima ion o he
amoun o ou da ing, o placing a new o de .
We do no assume any specific o m o he p obabilis ic dis-
ibu ion o he daily demand. Among o he s, we ob ain o mulae
o he expec ed on-hand in en o y, he p obabili y o s ockou , ex-
pec ed ou da ing, he age dis ibu ion o s ocks and he eshness
o uni s issued. The models ha we conside a e no s a iona y,
bu weekly s a iona y: he weekly pa e n exis s in he dis ibu ion
o he demand and in he ope a ional assump ions o he sys em
( o ins ance, o de s a e placed e e y weekday bu no on week-
ends). Since ou model has a weekly pa e n, all he o mulae a e
ob ained o each day o he week.
We de i e he app oxima ions o a pa icula model o be de-
sc ibed in de ail in Sec ion 2 . This may be seen as somewha e-
s ic i e, bu we see wo ad an ages in i . Fi s , he model is eal-
is ic, since i is e y close o he ope a ion o a eal blood bank.
Second, i includes a a ie y o si ua ions (e.g. no all days ha e
he same lead o e iew imes and uni s a i ing on Mondays ha e
a di e en emaining li e ime han uni s a i ing on o he days);
hus, he easoning and de i a ion o he o mulae o he model
can be adap ed wi hou difficul y o o he sys ems.
To he bes o ou knowledge, his is he fi s pape whe e ap-
p oxima ions o pe o mance measu es a e gi en when he model
is nons a iona y and he EWA policy is used (excep o he fill
a e, which has al eady been app oxima ed in an Donselaa &
B oekmeulen, 2011 ). A key poin in ou analysis conce ns he o -
mulae o νi
, he expec ed numbe o uni s o de ed on day which
a e in s ock a he end o day + i , de i ed in Sec ion 3.3 . These
o mulae enable us o gi e app oxima ions o many pe o mance
measu es o he model, such as expec ed ou da ing. F eshness, de-
fined as he expec ed emaining li e ime o uni s issued, can also
be easily compu ed om νi
. In ac , we gi e mo e comp ehensi e
in o ma ion on uni s issued by de i ing app oxima ions o he al-
ues w
, he expec ed numbe o uni s issued on day wi h days
o emaining li e ime.
We also use νi
o compu e he age dis ibu ion o he s ock,
i.e. he expec ed numbe o uni s in s ock wi h emaining li e ime
1 , . . . , m a he beginning o a day in he long un. This dis ibu ion
gi es ull in o ma ion on he beha iou o he sys em. We poin
ou ha age dis ibu ion o s ock in nons a iona y models o pe -
ishable i ems has no appea ed in he li e a u e e en unde he
simple base-s ock policy.
While in mos ins ances o in en o y sys ems demand is dis-
c e e in na u e, con inuous dis ibu ions a e o en used o model
i . Th oughou he pape we assume ha demand is well modelled
by a con inuous dis ibu ion, so ou o mulae a e exp essed using
in eg als and p obabili y densi y unc ions (PDF); i a disc e e dis-
ibu ion o demand is o be used, hen, PDFs mus be eplaced
by p obabili y mass unc ions and in eg als by sums.
1.3. Li e a u e e iew
The e is a g ea a ie y o ma hema ical models o pe ishable
i ems. They di e in many cha ac e is ics, such as de e minis ic o
andom demand, fixed o andom li e ime, ze o o posi i e lead
ime, s a iona y o ime- a ying demand, among o he s. We wo k
wi h a andom demand, fixed li e ime model he e, so we es ic
ou sel es o ha se ing o he es o he pape .
An excellen e iew o he esea ch published on in en o y
models o pe ishable i ems can be ound in Nahmias (1982) o
ea ly pape s on he subjec ; in Raa a (1991) o pape s up o 1991;
in Goyal & Gi i (2001) o publica ions om he ea ly 90s o 20 0 0;
and in Bakke , Riezebos, & Teun e (2012) , Janssen, Claus, & Saue
(2016) and Chaudha y, Kulsh es ha, & Rou oy (2018) o mo e e-
cen wo k. The fi s ma hema ical s udies on in en o y sys ems
o pe ishable i ems se ou o find op imal solu ions in e ms
o minimising cos unc ions. Howe e , in con as o wha hap-
pens o nonpe ishable i ems, whe e op imal solu ions a e known
in a wide a ie y o se ings, esea che s ound ha models o
pe ishable i ems we e much ha de o analyse, a leas when de-
mand was andom. Thus, op imal solu ions we e ob ained only in a
limi ed numbe o si ua ions such as e y sho p oduc li e imes
( m = 1 , 2 ) o ze o lead ime; see Nahmias (1982) and e e ences
he ein.
One way o finding an op imal solu ion is by using dynamic
p og amming, which is a sui able ool o hese models, aking
a s a e space defined by he age dis ibu ion o he s ock and a
s ochas ic ans e unc ion (see Nahmias, 1975 ). This echnique
sol es, a leas heo e ically, he p oblem o finding a policy which
minimises he cos unc ion subjec o a se ice cons ain . How-
e e , due o he “cu se o dimensionali y” o dynamic p og am-
ming, he s a e space o he p oblems becomes huge e en o mod-
e a e alues o m (li e ime) and maximum s o age capaci y and he
p oblems become unsol able in p ac ice. In he las ew decades
he inc easing speed and capaci y o compu e s ha e led o a e-
eme gence o his echnique, al hough i s ill needs o be combined
wi h agg ega ion o s a es o simula ion o find solu ions in a ea-
sonable ime. Haijema e al. (2007) , who combine dynamic p o-
g amming wi h simula ion, wo k wi h a model whose s a e space
is la ge han 10
8
, which implies a complexi y o he o de o 10
13
o one week i e a ion, so a downscaling o ou o one uni s is
ca ied ou . Algo i hms based on agg ega ion o s a es in mul iple
le els a e p oposed in Voelkel, Sachs, & Thonemann (2020) .
Ano he app oach o finding good policies in in en o y mod-
els is disc e e e en simula ion. I consis s o modelling he sys-
em and implemen ing i in simula ion so wa e. By unning he
simula ion wi h di e en policies and in a ious se ings, he pe -
o mance o he policies can be compa ed wi h a iew o choos-
ing he bes . Simula ion has been widely used o model eal blood
banks. Fo ins ance, Ry ilä & Spens (2006) compa e di e en sce-
na ios o p oduc ion and dis ibu ion o blood componen s in Fin-
land. Asllani, Culle , & E kin (2014) build a model o a blood bank
cen e supplying 50 heal h ca e acili ies in he US o sea ch o he
bes pla ele p oduc ion policy in he week, when pla ele s a e di -
e en ia ed by blood ype. Dalalah, Ba aineh, & Alkhaledi (2019) use
a simula ion-op imisa ion app oach o find an op imal policy when
demand is di e en ia ed by he age o pla ele s, and apply i o
Kuwai public hospi als. Go ia e al. (2020) use da a om wo
blood banks in Spain o s udy he dec ease in ou da ing when
he li e ime o pla ele concen a es is ex ended om 5 o 7 days
ia pa hogen educ ion echnologies and analyse wha days o he
week a e mos app op ia e o applying hese echnologies. An ad-
an age o he simula ion app oach is ha he model can be as
ealis ic as desi ed. Howe e , no analy ical exp essions o he op-
imal solu ion o he pe o mance measu es o he model a e ob-
ained, which p e en s he pa ame e s o he model om being in-
1138
C. Go ia, M. Lezaun and F.J. López Eu opean Jou nal o Ope a ional Resea ch 303 (2022) 1137–1150
e p e ed di ec ly; mo eo e , simula ions mus be un e e y ime a
change in he pa ame e s is obse ed.
Ye ano he app oach is o use heu is ics o find a good so-
lu ion. This app oach does no seek o find he bes o all easi-
ble solu ions, bu a he o p opose easonable, easy- o-implemen
policies which pe o m well in p ac ice. Many o hese policies a e
myopic, in he sense ha hey make pe iod-by-pe iod decisions,
and/o in ol e a simplifica ion o he s a e space ( o ins ance, us-
ing a simple unc ion o he composi ion o he s ock ins ead o i s
comple e age dis ibu ion). Nandakuma & Mo on (1993) desc ibe
heu is ic solu ions in he case o ze o lead ime which a e close
o op imal. Fo posi i e lead ime, Chiu (1995) de elops a solu ion
based on a pe iod-by-pe iod op imisa ion o an app oxima ion o
he cos unc ion which only akes in o accoun he size o he
s ock bu no i s age dis ibu ion. Haijema & Minne (2019) gi e
an o e iew o some o he mos impo an s ock-age dependen
o de policies, p opose new ones and compa e hem in a b oad se
o scena ios.
One heu is ic de eloped o pe iodic e iew and fixed li e-
ime models, which yields good esul s, is he EWA policy, in o-
duced in B oekmeulen & an Donselaa (2009) . I is known (see
B oekmeulen & an Donselaa , 2009; Haijema & Minne , 2019 )
ha he EWA policy significan ly ou pe o ms he base-s ock pol-
icy. The EWA policy has been analysed by an Donselaa & B oek-
meulen (2011) , an Donselaa & B oekmeulen (2012) , B oekmeulen
& an Donselaa (2019) . an Donselaa & B oekmeulen (2011) gi e
app oxima ions o he fill a e bo h when demand is s a iona y
and when i has a weekly pa e n. In he case o s a iona y mod-
els, an Donselaa & B oekmeulen (2012) gi e analy ical exp es-
sions o app oxima e he expec ed ou da ing; hese app oxima-
ions a e hen imp o ed by simula ing a la ge numbe o sce-
na ios and fi ing a eg ession model. Also o s a iona y mod-
els, B oekmeulen & an Donselaa (2019) p opose se e al poli-
cies o educing was e and inc easing eshness. F eshness is a
e y impo an pe o mance measu e when dealing wi h pe ish-
able i ems, since was e due o ou da ing occu s e y equen ly
a cus ome le el: e.g. households o ood ( Secondi, P incipa o, &
Lau e i, 2015; an Ge en, an He pen, & an T ijp, 2020 ) o hos-
pi als o pla ele s ( Flin , McQuil en, I win, Rush o d, Haysom, &
Wood, 2020; Pé ez Vaque o e al., 2016 ). Mo eo e , eshe uni s
a e usually p e e ed by cus ome s in he case o bo h ood p od-
uc s ( Li & Teng, 2018 ) and pla ele s, since hey ha e be e p ope -
ies han olde ones, Ca am-Deelde , K euge , Jacobse, an de Bom,
& Middelbu g (2016) , Aub on, Flin , Ozie , & McQuil en (2018) .
B oekmeulen & an Donselaa (2019) we e he fi s o ob ain an
app oxima ion o eshness in an in en o y model o pe ishable
i ems. To es ima e eshness, hey p opose an exp ession based on
Li le’s o mula o queuing heo y; he exp ession uses an es ima-
ion o expec ed ou da ing, so simula ions mus be un and a e-
g ession model fi ed as in an Donselaa & B oekmeulen (2012) in
o de o compu e he app oxima ion o eshness.
A mo e comp ehensi e desc ip ion o he pe o mance o an
in en o y sys em o pe ishable i ems is achie ed by compu ing
he age dis ibu ion o he s ock. Fo mulae o he age dis ibu-
ion o s ock a e challenging o ob ain e en in he s a iona y case
and unde he base-s ock policy. They ha e been compu ed only
o con inuous- e iew models assuming ha demand ollows a
Poisson p ocess, using he heo y o queuing ne wo ks. See Kouki,
Leg os, Babai, & Jouini (2020) and e e ences he ein.
The es o he pape is o ganised as ollows. Sec ion 2 desc ibes
he model used o de eloping ou app oxima ions and how he
EWA policy applies o i . The o mulae o he app oxima ions a e
gi en in Sec ion 3 . The accu acy o he app oxima ions is assessed
ia compa ison wi h Mon e Ca lo simula ions in a eal example in
Sec ion 4 . Sec ion 5 shows he esul s o 72 scena ios and analy-
ses he ex en o which ou app oxima ions can be ega ded as e-
liable. Conclusions and ideas o u u e wo k a e gi en in Sec ion 6 .
The pape has ou appendices wi h some addi ional o mulae and
in o ma ion and a Supplemen a y Ma e ial file wi h ables ela ed
o Sec ions 4 and 5 .
2. The model
Fo ease o exposi ion, we de elop ou esul s o a pa icula
model, i.e. he p oduc ion o pla ele concen a es in he CVTTH.
The cha ac e is ics o he model p esen ed a e common o many
blood banks. Fo ins ance, a simila model is analysed by Haijema
(2013) using dynamic p og amming. The o mulae de i ed he e can
be easily adap ed o any model wi h pe iodic e iew, s ochas ic de-
mand, fixed li e ime and a weekly pa e n.
We conside a FIFO issuing policy (olde i ems a e issued fi s ).
FIFO is he mos common issuing policy in he li e a u e on pe -
ishable i ems, especially when dealing wi h blood p oduc s; i is
known ha he eshness o uni s issued is lowe using a FIFO pol-
icy han wi h o he issuing policies such as LIFO (newe i ems a e
issued fi s ), bu ou da ing is lowe wi h FIFO han wi h LIFO; see,
o ins ance, Cohen & P as acos (1981) , S ange , Ya es, Wilding, &
Co on (2012) .
The e is daily demand om hospi als om Monday o Sunday
whose dis ibu ion depends on he day o he week. Le D
be he
andom a iable ep esen ing he demand on day ≥1 . We ake
day = 1 o be a Monday. As usual in hese models, we assume
ha demands on di e en days a e independen o each o he .
Simila ly, we assume ha he demand is weekly s a iona y, i.e. he
p obabili y dis ibu ion o D
+7 k
is iden ical o D
o all 1 ≤ ≤7 ,
k ≥1 . We deno e by F
he cumula i e dis ibu ion unc ion (CDF)
associa ed wi h D
and i s mean and a iance by μ
= E[ D
] and
σ2
= V a [ D
] , espec i ely.
Fo , i ≥1 , he agg ega ed demand du ing he in e al [ , + i ]
is deno ed by
D
, + i
=
i
j=0
D
+ j
.
Fo ≤s , le F
,s be he CDF o D
,s and μ ,s
= E[ D
,s
] and σ2
,s
=
V a [ D
,s
] he co esponding mean and a iance. We also w i e F
,s
wi h 1 ≤s < ≤7 o deno e he CDF o D
,s +7
, which has he same
dis ibu ion as D
, 7
+ D
1 ,s
; o ins ance, F
5 , 2 ep esen s he dis i-
bu ion o D
5 , 9
, he demand om a F iday o he ollowing Tues-
day. Acco dingly, o 1 ≤s < ≤7 , le μ ,s
= E[ D
, 7
] + E[ D
1 ,s
] and
σ2
,s
= V a [ D
, 7
] + V a [ D
1 ,s
] , he mean and a iance o F
,s
.
In p ac ice, his o ical da a is used o fi a dis ibu ion o F
and
o es ima e μ and σ
, = 1 , . . . , 7 . The es ima ions o F
,s
, μ ,s and
σ ,s can be compu ed om he es ima ions o F
, μ and σ
, =
1 , . . . , 7 since we assume independence o he andom a iables D
.
Pla ele concen a es ha e a fixed li e ime o m days. We de i e
ou o mulae o gene al m ; when a specific alue o m is needed
we ake m = 5 since his is he mos common li e ime o pla ele
concen a es and he one used in Pé ez Vaque o e al. (2016) . P o-
duc ion o de s can be placed e e y day om Monday o F iday.
This is equi alen o saying ha he e is a e iew in e al o one
day om Monday o Thu sday, R = 1 , and o h ee days on F iday,
R = 3 . An o de means ha blood is p ocessed immedia ely a e
he o de and pla ele concen a es a e p oduced du ing he day.
I he o de is placed on any day be ween Monday and Thu sday,
he concen a es a e eady o use in he mo ning o he ollowing
day, wi h a emaining li e ime o m days; o de s placed on F iday
a e eady o use on Monday mo ning, wi h a emaining li e ime
o m −2 days. Tha means ha he lead ime is L = 1 o o de s
placed om Monday o Thu sday and L = 3 o o de s placed on
F iday. No e ha his model is no daily s a iona y, bu weekly s a-
iona y: he dis ibu ion o demand, he e iew in e al and he
1139
C. Go ia, M. Lezaun and F.J. López Eu opean Jou nal o Ope a ional Resea ch 303 (2022) 1137–1150
lead ime depend on he day o he week. We assume ha he
se ice le el is high, which is cus oma y in blood banks, and ha
unsa isfied demand is no backlogged (in p ac ice, s ockou s a e
co e ed by an u gen eques o a neighbou ing bank). O de s a e
placed a he beginning o he day, aking in o accoun he on-hand
in en o y and he a i ing uni s bu be o e he demand o he day
is known. I a uni has no been used by he m h day o li e i is
disposed o ; o ins ance, i m = 5 , concen a es o de ed on Mon-
day and no used by Sa u day a e disca ded. The o de o e en s
each day is: (1) ecei e he incoming o de (only o weekdays);
(2) place he new o de (only o weekdays); (3) obse e and mee
he demand; (4) dispose o ou da ed uni s.
2.1. The EWA policy
Ou s is a ypical model o pe ishable i ems wi h fixed li e imes.
As commen ed in he In oduc ion, he e is no known op imal pol-
icy o such a model (weekly pa e n, nonze o lead ime, s ochas ic
demand and li e ime g ea e han 2). A heu is ic app oach o find
a easonable solu ion is he base-s ock policy, which is a simple
o de -up- o policy. The EWA policy is an imp o emen o ha pol-
icy. We desc ibe bo h hese policies he e.
The base-s ock policy has been widely used as a heu is ic o
s ochas ic demand, fixed li e ime in en o y sys ems; see o in-
s ance Nahmias (1982) , Coope (2001) and e e ences he ein. I
has been also used as a benchma k o compa ison wi h mo e
complex policies: Tekin, Gü le , & Be k (2001) , B oekmeulen & an
Donselaa (2009) , Duan & Liao (2013) , Haijema & Minne (2019) .
The base-s ock policy wo ks like an o de -up- o policy o non-
pe ishable i ems. I s a ionale is o ha e sufficien on-hand in-
en o y o mee demand un il he in en o y eplenishmen co e-
sponding o he nex o de . A each e iew poin an o de o size
Q
B
= max { SS
+ μ , + L + R −1
−IP
, 0 } (1)
is placed. He e SS
is he sa e y s ock o day , μ , + L + R −1 is he
expec ed demand om he placemen o he o de un il he a -
i al o he nex o de and IP
is he in en o y posi ion. No e ha
bo h L and R may depend on he day o he week. When al-
ues a e assigned o he subsc ip + L + R −1 , he R co espond-
ing o day and he L co esponding o day + R a e aken. Le
S
= SS
+ μ , + L + R −1
be he o de -up- o quan i y o day .
This policy akes in o accoun he in en o y posi ion o plac-
ing an o de , bu no i s age dis ibu ion. The EWA policy, p oposed
by B oekmeulen & an Donselaa (2009) , wo ks like he base-s ock
policy, bu he in en o y posi ion is dec eased by an es ima ion o
he expec ed ou da ing be ween he placemen o an o de and
day L + R −1 he ea e . The o de quan i y unde his mo e so-
phis ica ed policy o day is
Q
= max { SS
+ μ , + L + R −1
−IP
+ ˆ
E
, 0 } ,
whe e ˆ
E
is an es ima ion o E
, he expec ed ou da ing du ing
he in e al [ , + L + R −2] . No e ha he ou da ed quan i y on
day + L + R −1 is no included because i does no a ec he
abili y o mee demand ha day. In he EWA policy, he es ima-
ion ˆ
E
is compu ed assuming ha demands du ing he in e al
[ , + L + R −2] a e equal o hei mean alues. Mo e accu a e es-
ima ions o was e can be compu ed by using he exac dis i-
bu ion o demand ins ead o i s mean, bu hey show li le im-
p o emen and a e e y ime-consuming (see Haijema & Minne ,
2019 ). The e is an i ele an shi o he index o he days o be
conside ed o ou da ing in he compu a ion o ˆ
E
, when com-
pa ed o B oekmeulen & an Donselaa (2009) . In hei pape he
days o be conside ed a e + 1 , . . . , + L + R −1 while we ake
, . . . , + L + R −2 . This is because in hei s udy o de s a e placed
a he end o he day bu in ou s hey a e placed a he beginning
o he ollowing day.
To es ima e E
, we need some no a ion. Le B
, = 1 , . . . , m be
he numbe o uni s in s ock (on-hand) a he beginning o day
wi h days o emaining li e ime a e he a i al o new i ems,
and B
= B
1
+ ···+ B
m
. No e ha B
is equal o IP
excep o Sa u -
days and Sundays, whe e F iday’s o de is included in IP
bu no
in B
. Le W
be he numbe o uni s wi h days o emaining li e-
ime which a e issued on day . The ollowing ecu si e o mulae
ela e he ou da ed quan i y O
on day o he on-hand uni s B
,
he uni s issued W
and he demand D
(see B oekmeulen & an
Donselaa , 2009 ):
O
= B
1
−W
1
= max
B
1
−D
, 0
,
W
= min
B
, D
−
−1
k =1
W
k
, = 1 , . . . , m,
B
−1
+1 = B
−W
+ A
−1
+1
, = 2 , . . . , m, B
m
+1
= A
m
+1
, (2)
whe e A
is he numbe o uni s a i ing on day wi h days o
emaining li e ime. The e m A
−1
+1
is no included in o mula (6)
o B oekmeulen & an Donselaa (2009) because in hei case all
uni s en e he sys em wi h m days o emaining li e ime. In ou
case, uni s a i ing om Tuesday o F iday ha e m days o emain-
ing li e ime while uni s a i ing on Monday ha e m −2 . Due o he
ecu si e na u e o (2) , he e is no simple way o exp ess O
+ i
as a
unc ion o B
1
, . . . , B
m
and D
, . . . , D
+ i
. Mo eo e , since D
, ... , D
+ i
a e andom, so a e O
, . . . , O
+ i
, bu a de e minis ic alue is needed
o he la e in o de o app oxima e he expec ed ou da ing E
.
The EWA policy assumes ha D
, . . . , D
+ L + R −2 a e equal o hei
expec ed alues, uses (2) o ge he es ima es ˆ
O
, . . . , ˆ
O
+ L + R −2 o
O
, . . . , O
+ L + R −2 and hen akes ˆ
E
= ˆ
O
+ ···+ ˆ
O
+ L + R −2
.
2.2. Applica ion o he EWA policy o he model
We fi s show he applica ion o he base-s ock policy o ou
model, which is needed o he EWA policy. Recalling (1) , and
due o weekly s a iona i y, we need o define SS
, = 1 , . . . , 5 , he
sa e y s ock o Mondays, Tuesdays, Wednesdays, Thu sdays and
F idays, espec i ely (no o de s a e placed on Sa u days o Sun-
days). No e ha he alues o R and L depend on he day o he
week. The alues o R and L o be used o he o de placed on
day a e he ime un il he nex o de is placed ( R ) and he ime
be ween he placemen o ha o de and i s a i al ( L ). Fo Mon-
day, = 1 , he e iew in e al is R = 1 , since a new o de will
be placed on Tuesday, and he lead ime is L = 1 , since he o -
de placed on Tuesday will a i e on Wednesday. Also, L = R = 1
o Tuesday and Wednesday ( = 2 , 3 ). Fo Thu sday, = 4 , he e-
iew in e al is R = 1 , since a new o de will be placed on F iday,
and L = 3 , since he o de placed on F iday will a i e on Monday;
o F iday, = 5 , we ha e R = 3 , L = 1 . In o he wo ds, he pe iod
om o + L + R −1 co esponds o Mon-Tue o o de s placed
on Monday, Tue-Wed o o de s placed on Tuesday, Wed-Thu o
o de s placed on Wednesday, Thu-Sun o o de s placed on Thu s-
day and F i-Mon o o de s placed on F iday. The e a e di e en
ways o de e mine he sa e y s ock o day . A common op ion, o
bo h nonpe ishable and pe ishable i ems, is o ake i as a ac o o
he s anda d de ia ion o he demand; see Chap e 7 in Sil e e al.
(1998) . We ake i as
SS
=
kσ , + L + R −1
+ k
1 o = 1 , 2 , 3 ,
kσ , + L + R −1
+ k
2 o = 4 , 5 ,
wi h k, k
1
, k
2
≥0 , whe e he alues o L, R depend on he day o
he week as explained abo e. Tha is, he sa e y s ock is p opo -
ional o he s anda d de ia ion o he demand o be co e ed, plus
a fixed alue ( k
1 o k
2
); we allow di e en alues o k
j
o Mon-
days, Tuesdays and Wednesdays, which co e only 2 day demand,
1140
C. Go ia, M. Lezaun and F.J. López Eu opean Jou nal o Ope a ional Resea ch 303 (2022) 1137–1150
Table 1
Random a iables ela ed o he in en o y sys em.
B
≥1 ; = 1 , . . . , m numbe o uni s in s ock (on-hand) a he beginning o day wi h days o
emaining li e ime once he incoming o de has a i ed
B
≥1 o al numbe o uni s in s ock (on-hand) a he beginning o day once he
incoming o de has a i ed
Q
≥1 o de quan i y o day
S
≥1 o de -up- o quan i y o day
O
≥1 numbe o uni s ou da ed on day
E
≥1 expec ed ou da ing in he in e al [ , + L + R −2]
W
≥1 ; = 1 , . . . , m numbe o uni s issued on day wi h days o emaining li e ime
V
i
≥1 , i = 1 , . . . , m numbe o uni s o de ed on day which a e in s ock (on-hand) a he end o
day + i be o e ou da ed uni s a e disca ded
H
≥1 numbe o uni s in s ock (on-hand) a he end o day once ou da ed uni s
a e disca ded
U
≥1 unsa isfied demand on day
and o Thu sdays and F idays, which co e 4 day demand. We de-
cided o use only wo pa ame e s, k
1 and k
2
, ins ead o fi e di e -
en pa ame e s k
, = 1 , . . . , 5 , one o each weekday, o easons
o simplici y and efficiency. This asse ion is based on he conclu-
sions o a simula ion-op imisa ion model o he managemen o
pla ele s used in Pé ez Vaque o e al. (2016) , whe e he op imal
solu ions in e ms o ew ou da ings and eshness o uni s issued
we e ound o he empi ical da a o 52 weeks in he CVTTH. In
any e en , he choice o his o m o sa e y s ocks does no a ec
he de i a ion o ou o mulae, and hey can be s aigh o wa dly
adap ed o any o he o m o sa e y s ocks, such as aking a di e -
en pa ame e k
o = 1 , . . . , 5 .
We now u n o he applica ion o he EWA policy. To compu e
ˆ
E
, we conside se e al cases, depending on he day o he week.
When day is a Monday, om (2) , O
= (B
1
−D
)
+
, whe e a
+
=
max { a, 0 } . Since he EWA policy assumes D
= μ1
, i ollows ha
ˆ
O
= (B
1
−μ1
)
+
. As L + R −2 = 0 in his case, ˆ
E
= (B
1
−μ1
)
+
. The
same exp ession is alid when day is a Tuesday o a Wednesday
( eplacing μ1 by μ2
, μ3
, espec i ely).
When day is a Thu sday o a F iday, L + R −2 = 2 . Thus, O
=
B
1
−D
+ and ˆ
O
=
B
1
−μ
+
. Also, O
+1
=
B
1
+1
−D
+1
+
, wi h
W
1
= min
B
1
, D
, W
2
= min
B
2
, D
−W
1
, which yields B
1
+1
=
B
2
−D
−B
1
+
+ and O
+1
=
B
2
−D
−B
1
+
+
−D
+1
+
. The
de i a ion o O
+2
=
B
1
+2
−D
+2
+ is mo e in ol ed. No e ha
B
1
+2
= B
2
+1
−W
2
+1
, wi h B
2
+1
=
B
3
−D
−W
1
−W
2
+
and W
2
=
min
B
2
,
D
−B
1
+
. We ha e
B
2
+1 =
B
3
−D
−B
1
+
−B
2
+
+
,
W
1
+1 = min
B
2
−D
−B
1
+
+
, D
+1
,
W
2
+1 = min
B
2
+1
,
D
+1
−B
2
−D
−B
1
+
+
+
.
The e o e, B
1
+2
=
B
3
−D
+1
+
D
−B
1
+
−B
2
+
+
. Collec ing
all he e ms abo e gi es O
+2
= (
B
3
−D
+1
+
D
−B
1
+
−B
2
+
+
−D
+2
)
+
. The alue o ˆ
E
is ob ained by summing up O
, O
+1
and O
+2 and eplacing D
, D
+1 and D
+2 by μ
, μ +1 and μ +2
,
espec i ely.
The exp essions a e alid o gene al m . They a e simple o
Monday, Tuesday and Wednesday, bu a e a he complica ed o
Thu sday and F iday. Howe e , hey ge simple when a conc e e
alue o m is aken since some B
a e equal o 0. Fo ins ance, m =
5 gi es ˆ
E
= (B
3
−μ4 , 6
)
+ o Thu sdays and ˆ
E
= (B
2
−μ5 , 6
)
+
+
(B
3
−(μ5 , 6
−B
2
)
+
−μ7
)
+ o F idays.
3. App oxima ions o pe o mance measu es unde he EWA
policy
We now de i e analy ical app oxima ions o he main pe o -
mance measu es in he model. Table 1 summa ises he andom
a iables ela ed o he in en o y sys em. We use he same no a-
ion, wi h small ins ead o capi al le e s, o hei expec ed alues
in he s eady s a e. The model is weekly s a iona y, so hese ex-
pec ed alues depend on he day o he week, which means ha
he expec ed on-hand s ock h
, say, is he same o all + 7 k , k ≥0 .
See Appendix A o a heo e ical jus ifica ion o he exis ence o
he long- un dis ibu ion and i s pe iodici y. In he es o he Sec-
ion we w i e he o mulae o = 1 , . . . , 7 only. On some occa-
sions he subsc ip s in he o mulae become nega i e o ze o: o
ins ance when day is a Monday ( = 1 ) and we w i e D
−3 , −1
; in
hose cases he alue in he o mula mus be unde s ood as + 7 .
In his sec ion we keep m gene al as long as we can. When he
app oxima ions need a specific alue o m , we ake m = 5 .
3.1. App oxima ion o he o de quan i ies
No e ha he o de quan i y o day , Q
, is (S
−B
)
+
, whe e
S
= SS
+ μ , + L + R −1
+ ˆ
O
+ ···+ ˆ
O
+ L + R −2
, and ˆ
O
+ j
is he es ima-
ion o he ou da ed quan i y on day + jin Sec ion 2.2 . App oxi-
ma ing ˆ
O
by o
gi es an app oxima ion o he o de -up- o quan i-
ies:
s
∼μ , +1
+ kσ , +1
+ k
1
+ o
, = 1 , 2 , 3
s
4 ∼μ4 , 7
+ kσ4 , 7
+ k
2
+ o
4
+ o
5
+ o
6
,
s
5 ∼μ5 , 1
+ kσ5 , 1
+ k
2
+ o
5
+ o
6
+ o
7
. (3)
No e ha , depending on he pa icula alue o m , some o he
o
a e 0. Fo ins ance, i m = 5 , hen o
4
= o
5
= 0 since he e is no
p oduc ion on Sa u days o Sundays, so he e is no ou da ing on
Thu sdays o F idays.
We now make wo assump ions. The fi s is (B
−D
)
+
∼B
−
D
, which is qui e easonable since he se ice le el is high, so
mos days we ha e B
≥D
and, e en i B
< D
, he di e ence
D
−B
= U
(unsa isfied demand on day ) is small. In ac , his
assump ion is common when analysing in en o y sys ems: o in-
s ance, Sil e e al. (1998) asse (p. 253) ha a usual assump ion
o he in en o y managemen o i ems wi h andom demand is
“Uni sho age cos s (explici o implici ) a e so high ha a p ac-
ical ope a ing p ocedu e will always esul in he a e age le el o
backo de s being negligibly small when compa ed wi h he a e -
age le el o he on-hand s ock”. The second assump ion is S
≥B
o e e y weekday , as o he wise he s ock a he beginning o he
day is e y la ge and Q
= 0 , which is in equen in many in en o y
models, such as models o he p oduc ion o pla ele concen a es
in blood banks, whe e he size o de s a e posi i e a e e y e iew
poin .
Now we app oxima e b
. Fo = 2 , 3 , 4 , 5 , we ha e
B
=
(
B
−1
−D
−1
)
+
+ Q
−1
−O
−1
=
(
B
−1
−D
−1
)
+
+
(
S
−1
−B
−1
)
+
−O
−1
∼S
−1
−D
−1
−O
−1
.
1141
C. Go ia, M. Lezaun and F.J. López Eu opean Jou nal o Ope a ional Resea ch 303 (2022) 1137–1150
By (3) , b
∼μ
+ kσ −1 ,
+ k
1 o = 2 , 3 , 4 and b
5
= μ5 , 7
+ kσ4 , 7
+
k
2
+ o
5
+ o
6
. Fo Sa u day, B
= (B
−1
−D
−1
−O
−1
)
+
∼S
−2
−
D
−2 , −1
−O
−2
−O
−1 which yields b
6
∼μ6 , 7
+ kσ4 , 7
+ k
2
+ o
6
.
Fo Sunday, B
= (B
−1
−D
−1
−O
−1
)
+
∼S
−3
−D
−3 , −1
−
O
−3
−O
−2
−O
−1 and b
7
∼μ7
+ kσ4 , 7
+ k
2
. Las , o Monday,
B
∼S
−3
−D
−3 , −1
−O
−3
−O
−2
−O
−1
, so b
1
∼μ1
+ kσ5 , 1
+ k
2
.
Since Q
= (S
−B
)
+
∼S
−B
, he app oxima ions o he ex-
pec ed o de quan i ies a e
q
1 ∼μ2
+ k (σ1 , 2
−σ5 , 1
) + k
1
−k
2
+ o
1
q
2 ∼μ3
+ k (σ2 , 3
−σ1 , 2
) + o
2
q
3 ∼μ4
+ k (σ3 , 4
−σ2 , 3
) + o
3
q
4 ∼μ5 , 7
+ k (σ4 , 7
−σ3 , 4
) + k
2
−k
1
+ o
4
+ o
5
+ o
6
q
5 ∼μ1
+ k (σ5 , 1
−σ4 , 7
) + o
7
No e ha he o mulae abo e depend on o
, = 1 , . . . , 7 ,
which a e unknown. We gi e app oxima ions o hei alues in
Sec ion 3.4 .
3.2. Expec ed on-hand in en o y
Since H
= (B
−D
)
+
−O
and s ockou s a e assumed o be un-
common, we can app oxima e H
∼B
−D
−O
and he o mulae
in Sec ion 3.1 gi e
h
1 ∼kσ5 , 1
+ k
2
−o
1
h
∼kσ −1 ,
+ k
1
−o
, = 2 , 3 , 4
h
5 ∼μ6 , 7
+ kσ4 , 7
+ k
2
+ o
6
h
6 ∼μ7
+ kσ4 , 7
+ k
2
h
7 ∼kσ4 , 7
+ k
2
−o
7
3.3. A o mula o
i
and he age dis ibu ion o he s ock
In his sec ion we de i e a o mula o E[ V
i
] , he expec ed num-
be o uni s o de ed on day which a e in s ock a he end o
day + i be o e ou da ed uni s a e disca ded. Fi s no e ha V
i
= 0
o i = 1 , . . . , m when = 6 is a Sa u day o = 7 is a Sunday;
also, V
1
= V
2
= 0 i = 5 is a F iday, since uni s a i e on Mon-
day. Fo he es o he alues V
i
, ecall ha Q
= (S
−B
)
+
. Now,
since he o de placed on day se s he in en o y posi ion o
S
and we use a FIFO issuing policy, V
i
can be app oxima ed by
(S
−D
, + i
−(O
+ ···+ O
+ i −1
))
+
, wi h a limi o (S
−B
)
+
. Tha
is,
V
i
∼min
(
S
−D
, + i
−(
O
+ ···+ O
+ i −1
) )
+
, (S
−B
)
+
.
In o de o compu e E[ V
i
] , we condi ion on B
. In wha ollows, we
ake S
as i i was de e minis ic, which is no ue because unde
he EWA policy i depends on he age dis ibu ion o he s ock. Le
˜
D
, + i
= D
, + i
+ O
+ ···+ O
+ i −1
, o ≥1 , i = 1 , . . . , m . No e ha i
B
≥S
, hen V
i
= 0 o e e y i = 1 , . . . , m ; i B
< S
, hen
E[ V
i
| B
] ∼(S
−B
) P (
˜
D
, + i
≤B
) +
S
B
(S
−x )
˜
D
, + i
(x ) dx
=
{ (x,y ): B
<y<S
, 0 <x<y }
˜
D
, + i
(x ) d xd y
=
S
B
F
˜
D
, + i
(x ) dx,
whe e
˜
D
, + i
and F
˜
D
, + i
a e he PDF and CDF, espec i ely, o he
demand in he in e al [ , + i ] plus he ou da ed uni s in he in-
e al [ , + i −1] . Thus,
E[ V
i
| B
] ∼
S
B
F
˜
D
, + i
(x ) dx i B
< S
,
0 i B
≥S
.
By he p ope ies o condi ional expec a ion,
E
V
i
∼ S
0 S
y
F
˜
D
, + i
(x ) dx
B
(y ) dy
=
{ (x,y ):0 <y<x<S
}
F
˜
D
, + i
(x )
B
(y ) d xd y
=
S
0 x
0
B
(y ) dy
F
˜
D
, + i
(x ) dx
=
S
0
F
B
(x ) F
˜
D
, + i
( x ) dx. (4)
To apply o mula (4) , we need app oxima ions o F
˜
D
, + i
(x ) and
F
B
(x ) . Since we app oxima e O
by i s expec ed alue o
, we ha e
F
˜
D
, + i
(x ) = P
(
D
, + i
+ O
+ ···+ O
+ i −1
≤x
)
∼F
, + i
(x −(o
+ ···+ o
+ i −1
)) .
The app oxima ion o B
depends on he day o he week. When
day is a Monday, using he app oxima ions in Sec ion 3.1 ,
F
B
(x ) ∼P (S
−3
−D
−3 , −1
−O
−3
−O
−2
−O
−1
≤x )
∼P (μ5 , 1
+ kσ5 , 1
+ k
2
−D
−3 , −1
≤x )
= F
5 , 7
(μ5 , 1
+ kσ5 , 1
+ k
2
−x ) ,
whe e F ( ) = 1 −F ( ) .
When day is Tuesday, Wednesday o Thu sday,
F
B
(x ) ∼P (S
−1
−D
−1
−O
−1
≤x )
∼P (D
−1
≥μ −1 ,
+ kσ −1 ,
+ k
1
−x )
= F
−1
(μ −1 ,
+ kσ −1 ,
+ k
1
−x ) .
wi h = 2 , 3 , 4 , espec i ely.
Analogously, o F iday:
F
B
(x ) = P (S
−1
−D
−1
−O
−1
≤x )
∼F
4
(μ4 , 7
+ kσ4 , 7
+ k
2
+ o
5
+ o
6
−x ) .
The abo e exp essions, oge he wi h (4) , yield he equi ed ap-
p oxima ions o
i
, = 1 , . . . , 5 , i = 1 , . . . , m . Fo ins ance, he ex-
pec ed numbe o uni s ha a e o de ed on Thu sday and a e in
s ock a he end o Sunday,
3
4
, can be app oxima ed by
μ4 , 7
+ kσ4 , 7
+ k
2
+ o
4
+ o
5
+ o
6
0
F
3
(μ3 , 4
+ kσ3 , 4
+ k
1
−x )
×F
4 , 7
(x −(o
4
+ o
5
+ o
6
)) dx.
These o mulae a e explici ; howe e , hey depend on o
1
, . . . , o
7
,
he expec ed numbe o ou da ed uni s each day. In he nex sec-
ion we show how o es ima e hese quan i ies. Once hey ha e
been es ima ed, hei alues can be plugged in o he abo e o mu-
lae o compu e he app oxima ions o
i
, since he dis ibu ions F
,s
a e known.
The o mulae abo e enable us o app oxima e he age dis i-
bu ion o he s ock b
. In ac , when day is no a Monday, he
numbe o uni s wi h days o emaining li e ime a he begin-
ning o day , once he incoming o de has a i ed, B
, is V
m −
+ −m −1
,
o = 1 , . . . , m −1 , and B
m
= Q
−1
. When day is a Monday, B
=
V
m −
+ −m −1
, o = 1 , . . . , m −3 , B
m −2
= Q
−3 and B
m −1
= B
m
= 0 . The
app oxima ion o b
is ob ained by subs i u ing he alues o Qand
V by he co esponding app oxima ions o q and .
3.4. Expec ed ou da ing
In his sec ion we se m = 5 . De i a ion o he o mulae when
m is 4 and 6 can be ound in Appendix B . O he alues o m can
be wo ked ou in a simila way.
1142
C. Go ia, M. Lezaun and F.J. López Eu opean Jou nal o Ope a ional Resea ch 303 (2022) 1137–1150
Le m = 5 . Recall fi s ha o
4
= o
5
= 0 since he e is no ou da -
ing on Thu sdays o F idays. Now, since O
= V
5
−5
, we use o mula
(4) wi h i = 5 , and ge
o
1 ∼ μ3 , 4
+ kσ3 , 4
+ k
1
+ o
3
0
F
2
(μ2 , 3
+ kσ2 , 3
+ k
1
−x ) F
3 , 1
(x −o
3
−o
6
−o
7
) dx,
o
2 ∼ μ4 , 7
+ kσ4 , 7
+ k
2
+ o
6
0
F
3
(μ3 , 4
+ kσ3 , 4
+ k
1
−x ) F
4 , 2
(x −o
6
−o
7
−o
1
) dx,
o
3 ∼ μ5 , 1
+ kσ5 , 1
+ k
2
+ o
6
+ o
7
0
F
4
(μ4 , 7
+ kσ4 , 7
+ k
2
+ o
6
−x )
×F
5 , 3
(x −o
6
−o
7
−o
1
−o
2
) dx,
o
6 ∼ μ1 , 2
+ kσ1 , 2
+ k
1
+ o
1
0
F
5 , 7
(μ5 , 1
+ kσ5 , 1
+ k
2
−x ) F
1 , 6
(x −o
1
−o
2
−o
3
) dx,
o
7 ∼ μ2 , 3
+ kσ2 , 3
+ k
1
+ o
2
0
F
1
(μ1 , 2
+ kσ1 , 2
+ k
1
−x ) F
2 , 7
(x −o
2
−o
3
−o
6
) dx.
(5)
The o mulae abo e a e cyclical, since o
s is needed o compu e
o
. We sol e his ia an i e a i e p ocedu e: we se all o
= 0 , com-
pu e he o mulae in (5) o ge an app oxima ion o o
and plug
he new alues in o he o mulae o ge ano he app oxima ion.
This p ocedu e is i e a ed un il he changes in he o
a e smalle
han a ole ance alue. While we ha e no p o ed analy ically ha
his p ocedu e con e ges, in all ou se ings below a small numbe
o i e a ions (less han 8) we e needed o a ole ance o 10
−3
.
3.5. Remaining li e ime o uni s issued and eshness
Fo a gi en day , he numbe o uni s issued wi h a emain-
ing li e ime o days, W
can be exp essed as W
= V
m −
+ −m −1
−
V
m +1 −
+ −m −1
, o = 1 , . . . , m −1 and W
m
= Q
−1
−V
1
−1
. This o mula
has wo excep ions ela ed o he weekend: when is a Monday
and = m −2 , W
m −2
= Q
−3
−V
3
−3
, and when is a Sa u day and
= m , W
m
= 0 . Thus, he alues w
can be app oxima ed by sub-
s i u ing Qand V by hei app oxima ions in Sec ions 3.1 and 3.3 ,
espec i ely. Also, eshness o uni s issued on day , = 1 , . . . , 7
can be app oxima ed by
m
=1
w
m
=1
w
.
No e ha ex ending he o mula de i ed by B oekmeulen & an
Donselaa (2019) o eshness in s a iona y models o he p esen
si ua ion is difficul , since he dis ibu ion o he demand, he al-
ues o he lead ime L and he e iew in e al R , and he emaining
li e ime o uni s when hey en e he in en o y depend on he day
o he week. Thus, he e seems o be no easy way o using Li -
le’s o mula in ou con ex o find he eshness o uni s deli e ed
each day o he week. Appendix C shows how ha app oach can
be used o ge a o mula o he eshness o uni s wi hou di e -
en ia ing by he day o issue, al hough i s ill equi es he app ox-
ima ions in Sec ion 3.4 .
3.6. Expec ed sho age
We a e assuming ha s ockou s a e a e, bu hey may s ill oc-
cu , so i is impo an o ha e an app oxima ion o he expec ed
sho age, u
, whe e U
= (D
−B
)
+
. We begin when day is a
Tuesday, Wednesday o Thu sday. Using he app oxima ions o B
,
S
and O
in he p e ious sec ions, U
∼(D
−S
−1
+ D
−1
+ O
−1
)
+
,
so
u
∼E
(D
−1 ,
−(μ −1 ,
+ kσ −1 ,
+ k
1
))
+
=
∞
μ −1 ,
+ kσ −1 ,
+ k
1
(x −(μ −1 ,
+ kσ −1 ,
+ k
1
))
−1 ,
(x ) dx
=
∞
μ −1 ,
+ kσ −1 ,
+ k
1
F
−1 ,
(x ) dx,
o = 2 , 3 , 4 .
When is a F iday, U
∼(D
−S
−1
+ D
−1
+ O
−1
)
+ and
u
5
∼ ∞
μ4 , 7
+ kσ4 , 7
+ k
2
+ o
5
+ o
6
F
4 , 5
(x ) dx.
When is a Sa u day, U
∼(D
−S
−2
+ D
−2 , −1
+ O
−2
+ O
−1
)
+
,
so
u
6
∼ ∞
μ4 , 7
+ kσ4 , 7
+ k
2
+ o
6
F
4 , 6
(x ) dx.
Fo Sunday, U
∼(D
−S
−3
−D
−3 , −1
+ O
−3
+ O
−2
+ O
−1
)
+
and
u
7
∼ ∞
μ4 , 7
+ kσ4 , 7
+ k
2
F
4 , 7
(x ) dx.
Las ly, when is a Monday, U
∼(D
−S
−3
−D
−3 , −1
+ O
−3
+
O
−2
+ O
−1
)
+
, so
u
1
∼ ∞
μ5 , 1
+ kσ5 , 1
+ k
2
F
5 , 1
(x ) dx.
The fill a e o day can be app oxima ed by 100(1 −u
/μ
) .
3.7. P obabili y o on-hand in en o y being lowe han a h eshold
The p obabili y o he on-hand in en o y a he end o day
(be o e ou da ed uni s a e disca ded) being less han a is P [ D
>
B
−a ] , which can be app oxima ed in a simila way o he p e i-
ous sec ion, ob aining:
P (D
> B
−a )
=
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
F
5 , 1
(μ5 , 1
+ kσ5 , 1
+ k
2
−a ) i day is a Monday ,
F
−1 ,
(μ −1 ,
+ kσ −1 ,
+ k
1
−a ) i day is a Tuesday ,
Wednesday o Thu sday ,
F
4 , 5
(μ4 , 7
+ kσ4 , 7
+ k
2
+ o
5
+ o
6
−a ) i day is a F iday ,
F
4 , 6
(μ4 , 7
+ kσ4 , 7
+ k
2
+ o
6
−a ) i day is a Sa u day ,
F
4 , 7
(μ4 , 7
+ kσ4 , 7
+ k
2
−a ) i day is a Sunday .
In pa icula , aking a = 0 gi es an app oxima ion o he p obabili y
o a s ockou , i.e. 1 −se ice le el.
4. Applica ion o he CVTTH da a
We assess he accu acy o ou app oxima ions by compa ing
hem wi h he alues ob ained by Mon e Ca lo simula ions in a
eal example. Fo he dis ibu ions o he daily demand we choose
hose fi ed o he da a o he CVTTH o 2012 (which we e consid-
e ed by Pé ez Vaque o e al. (2016) ), i.e. (disc e ised) no mal dis i-
bu ions wi h means and s anda d de ia ions as shown in Table 2 .
The li e ime o pla ele concen a es is m = 5 .
The no mali y assump ion was checked using he Shapi o es
o no mali y. Significan p− alues we e ound o Sa u day and
Sunday. No e howe e ha he dis ibu ions o he demand on
hose days a e ne e used on hei own in ou o mulae; ins ead
hey a e used a leas oge he wi h F iday ( o Sa u day) and wi h
F iday and Sa u day ( o Sunday). The e o e, wha needs o be
checked is no he no mali y o he demand on Sa u days and on
Sundays bu he no mali y on F iday + Sa u day and F iday + Sa -
u day + Sunday. Table 3 shows ha he no mali y assump ion is
easonable.
We ema k ha he hypo hesis o no mali y is no necessa y o
he de i a ion o ou o mulae: hey can be applied wi h any o he
dis ibu ion, including he empi ical dis ibu ion o his o ical da a
i a ailable.
Rega ding independence o he daily demand, we pe o med
he (Pea son) co ela ion es o each pai o consecu i e days.
Two o he pai s we e ound o be significan , namely Tue-Wed
( p− alue 0,045) and Thu-F i ( p− alue 0,004). An analysis o he
1143
C. Go ia, M. Lezaun and F.J. López Eu opean Jou nal o Ope a ional Resea ch 303 (2022) 1137–1150
Table 2
Means and s anda d de ia ion o daily demand o pla ele concen a es in CVTTH in 2012.
Monday Tuesday Wednesday Thu sday F iday Sa u day Sunday
μ 27,75 23,71 24,57 22,16 29,39 13,29 11,82
σ 6,85 5,65 7,86 6,90 7,81 4,89 4,38
Table 3
p− alues o he Shapi o es o no mali y o demand o pla ele concen-
a es in CVTTH in 2012.
Mon Tue Wed Thu F i F i + Sa F i + Sa + Sun
0,05 0,32 0,95 0,22 0,50 0,59 0,41
Table 4
p− alues o he Pea son co ela ion es o independence o demand on consecu-
i e days in CVTTH in 2012.
Mon-Tue Tue-Wed Wed-Thu Thu-F i F -Sa Sa -Sun Sun-Mon
0,10 0,06 0,38 0,11 0,67 0,07 0,79
sca e plo s o hese wo pai s e eals he exis ence o influen ial
poin s which co espond o e y low demand (unde 10 uni s) on
public holidays o e he yea . Once hese poin s a e emo ed he
p− alues a e g ea e han 0,05, so independence can be assumed:
see Table 4 .
We compa e ou app oxima ions wi h he es ima ions ob-
ained om simula ions o di e en alues o pa ame e s k, k
1
, k
2
.
Namely, we ake all combina ions whe e k anges in { 1 . 5 , 2 , 2 . 5 , 3 }
and (k
1
, k
2
) is (0,0) o (10,5). Fo he Mon e Ca lo simula ions,
we se 10 0 0 uns and a un leng h o 520 weeks (10 yea s),
whe e he fi s 52 weeks o each simula ion un a e aken as he
wa m-up pe iod. The numbe o simula ions and hei leng h a e
chosen o gi e a small simula ion e o . This can be checked in
Table D.8 in Appendix D which shows, o he case k = 1 . 5 , k
1
=
k
2
= 0 , he s anda d de ia ion o he es ima ions using simula ion
oge he wi h hei ela i e e o s, measu ed as he a io o he
hal -wid h o he 95 % confidence in e al o e he sample mean.
In almos e e y ins ance he ela i e e o is smalle han 2% , wi h
he only excep ions being quan i ies wi h a e y small sample
mean. Inc easing he leng h o he simula ion uns o hei num-
be (and hus inc easing he simula ion ime) p oduces almos no
changes in he sample means. The compu a ion o ou o mulae in
Sec ion 3 o he se ing k = 1 . 5 , k
1
= 0 , k
2
= 0 akes 4 ms. on an
In el(R) Co e(TM) i5 (3.30 GHz), while simula ion akes 1130 ms.
4.1. Resul s o he CVTTH da a
Table 5 shows he esul s o ou o mulae and he simula ion
o k = 1 . 5 , k
1
= 0 , k
2
= 0 . The ables o he es o he cases a e
a ailable in Sec ion 1 o he Supplemen a y Ma e ial file. In he
ables he e a e wo ows o each quan i y: he uppe ow (plain
on ) is he esul o he applica ion o he o mulae in Sec ion 3 ;
he lowe ow (i alic on ) is he esul ob ained by a e aging he
esul s o e he simula ion uns. We explain Table 5 in de ail, and
he es o he ables ha e he same s uc u e.
The fi s pa o he able gi es he esul s o each day o he
week. The fi s column shows he alues o b
; o ins ance b
1
, he
expec ed numbe o uni s in s ock a he beginning o a Monday is
46,2 by ou o mula in Sec ion 3.1 and 46,8 by he simula ion. Be-
low we w i e fi s he esul o ou o mula and hen, in b acke s,
he esul o simula ion, i.e. b
1 is 46,2 (46,8). The second column
is q
, he expec ed o de size; hus, o ins ance, he expec ed o -
de size on Mondays is 18,6 (18,3). The nex column gi es o
, he
expec ed quan i y ou da ed on day : so he e a e, on a e age, 0,17
(0,14) uni s ou da ed on Wednesdays. The nex column is h
, he
Fig. 1. Efficien F on ie ela ing was e wi h he fill a e (solid cu e) o m = 4
(blue), m = 5 (g een) and m = 6 ( ed). Do ed cu es ep esen he eshness o
uni s issued in each op imal configu a ion. (Fo in e p e a ion o he e e ences o
colou in his figu e legend, he eade is e e ed o he web e sion o his a icle.)
expec ed numbe o uni s on-hand a he end o day . Column u
gi es he alue o u
, he expec ed sho age on day . The ollowing
wo columns gi e he se ice le el and he p obabili y o he on-
hand s ock being below a h eshold a he end o he day; we use
5 uni s as he h eshold in all se ings. Thus, o ins ance, he se -
ice le el on Tuesdays is 94,0 % (95,2 % ) and 14,7 % (14,1 % ) o Tues-
days end wi h an on-hand s ock lowe han 5 uni s. The ollowing
columns b(1) , ... , b(5) a e he alues o b
, he expec ed numbe
o uni s wi h a emaining li e ime o days a he beginning o
day ; o ins ance, he expec ed numbe o uni s wi h 2 days o
emaining li e ime a he beginning o a Monday is 18,8 (18,9). The
ollowing fi e columns w (1) , ... , w (5) a e w
, he numbe o uni s
issued on day wi h emaining li e ime equal o days; o in-
s ance, he expec ed numbe o uni s issued on Monday wi h 2
days o emaining li e ime is 16,7 (16,7). The las column is esh-
ness, he expec ed emaining li e ime o uni s issued on day ; o
ins ance, uni s issued on Wednesday ha e an expec ed emaining
li e ime o 4,14 (4,09).
The second pa o he able, below he “Week” line, sum-
ma ises he alues o e he whole week, compu ing sums, a e -
ages o pe cen ages o e he 7 days whe e app op ia e. The fi s
column gi es he a e age o he b
alues, i.e. he a e age num-
be o uni s on-hand a he beginning o a day: 43,9 (44,1). q is
he o al numbe o uni s o de ed o e he week: 152,9 (151,7),
which co espond o 100,2 % (99,4 % ) o he demand. The nex col-
umn shows ha 0,25 (0,22) uni s go ou o da e o e he week,
i.e. 0,16 % (0,15 % ) o he uni s o de ed. The a e age on-hand in en-
o y a he end o he day is 22,0 (22,4) uni s. The e a e, on a e -
age, 1,426 (1,243) uni s no se ed o e he whole week, i.e. 0,93 %
(0,81 % ) o he demand, so he fill a e is 99,07 % (99,19 % ). The ol-
lowing wo columns show ha he se ice le el is 95.6 % (96,2 % );
i.e. 4,4 % (3,8 % ) o he days ha e a s ockou , and 9,6 % (9,5 % ) o he
days end wi h less han 5 uni s in s ock. Columns b(1) , ... , b(5)
1144
C. Go ia, M. Lezaun and F.J. López Eu opean Jou nal o Ope a ional Resea ch 303 (2022) 1137–1150
Table 5
Pe o mance measu es o he CVTTH da a in Sec ion 4 : app oxima ions by o mulae (plain on ) and simula ion (i alic on ). k = 1 . 5 ; k
1
= 0 ; k
2
= 0 .
Day b q o h u s.l. P(u. .) b(1) b(2) b(3) b(4) b(5) w(1) w(2) w(3) w(4) w(5) eshness
Monday 46,2 18,6 0,00 18,4 0,328 0,938 0,120 0,0 18,8 27,7 0,0 0,0 0,0 16,7 11,1 0,0 0,0 2,40
46,8 18,3 0,00 19,3 0,226 0,954 0,110 0,0 18,9 27,9 0,0 0,0 0,0 16,7 10,8 0,0 0,0 2,39
Tuesday 37,0 25,9 0,08 13,2 0,229 0,940 0,147 2,2 16,6 0,0 0,0 18,6 2,1 13,6 0,0 0,0 8,0 2,93
37,6 25,4 0,08 14,0 0,197 0,952 0,141 2,2 17,1 0,0 0,0 18,3 2,1 13,9 0,0 0,0 7,5 2,87
Wednesday 39,1 23,5 0,17 14,4 0,252 0,940 0,139 3,1 0,0 0,0 10,5 25,9 2,9 0,0 0,0 9,6 12,1 4,14
39,5 22,6 0,14 15,0 0,238 0,947 0,138 3,2 0,0 0,0 10,8 25,4 3,1 0,0 0,0 9,8 11,5 4,09
Thu sday 37,8 57,3 0,00 15,7 0,275 0,939 0,132 0,0 0,0 0,9 13,7 23,5 0,0 0,0 0,9 11,9 9,3 4,38
37,5 57,5 0,00 15,6 0,254 0,944 0,141 0,0 0,0 1,0 14,0 22,6 0,0 0,0 1,0 12,2 8,8 4,36
F iday 73,0 27,7 0,00 43,6 0,000 1,000 0,000 0,0 0,0 1,8 14,2 57,3 0,0 0,0 1,8 13,2 14,7 4,43
73,1 27,9 0,00 43,7 0,000 1,000 0,000 0,0 0,0 1,8 13,8 57,5 0,0 0,0 1,8 12,9 14,6 4,44
Sa u day 43,6 0,0 0,00 30,3 0,013 0,996 0,011 0,0 0,0 1,0 42,6 0,0 0,0 0,0 0,8 12,4 0,0 3,93
43,7 0,0 0,00 30,4 0,011 0,997 0,011 0,0 0,0 0,9 42,8 0,0 0,0 0,0 0,8 12,5 0,0 3,94
Sunday 30,3 0,0 0,00 18,5 0,329 0,938 0,120 0,0 0,1 30,2 0,0 0,0 0,0 0,1 11,4 0,0 0,0 2,99
30,4 0,0 0,00 18,9 0,317 0,942 0,124 0,0 0,1 30,3 0,0 0,0 0,0 0,1 11,4 0,0 0,0 2,99
Week
A e age 43,9 22,0 0,956 0,096 0,8 5,1 8,8 11,6 17,9
44,1 22,4 0,962 0,095 0,8 5,2 8,8 11,6 17,7
Sum 152,9 0,25 1,426 5,0 30,4 25,9 47,2 44,2 3,62
151,7 0,22 1,243 5,2 30,7 25,8 47,4 42,4 3,60
Pe cen age 100,2% 0,16% 0,93% 3,3% 19,9% 17,0% 30,9% 28,9%
99,4% 0,15% 0,81% 3,4% 20,3% 17,0% 31,3% 28,0%
Table 6
Pe o mance measu es o he CVTTH da a in Sec ion 4 : app oxima ions by o mulae (plain on ) and simula ion (i alic on ). Agg ega ed weekly esul s.
Sa e y s ock b q o h u s.l. P(u. .) b(1) b(2) b(3) b(4) b(5) w(1) w(2) w(3) w(4) w(5) eshness
k = 1,5, k1 = 0, k2 = 0 43,9 100,2% 0,16% 22,0 0,93% 0,956 0,096 0,8 5,1 8,8 11,6 17,9 3,3% 19,9% 17,0% 30,9% 28,9% 3,62
44,1 99,4% 0,15% 22,4 0,81% 0,962 0,095 0,8 5,2 8,8 11,6 17,7 3,4% 20,3% 17,0% 31,3% 28,0% 3,60
k = 1,5, k1 = 10, k2 = 5 51,0 100,4% 0,36% 29,1 0,19% 0,990 0,024 1,3 6,3 10,7 14,8 17,9 5,5% 23,2% 20,0% 36,9% 14,4% 3,31
51,0 100,1% 0,31% 29,2 0,18% 0,991 0,025 1,3 6,4 10,7 14,8 17,8 5,6% 23,3% 20,1% 36,8% 14,2% 3,31
k = 2, k1 = 0, k2 = 0 49,4 100,4% 0,43% 27,5 0,26% 0,985 0,038 1,4 6,5 10,1 13,6 18,0 6,0% 23,4% 16,4% 34,1% 20,1% 3,39
49,5 100,1% 0,36% 27,6 0,25% 0,987 0,039 1,4 6,5 10,1 13,6 17,8 6,1% 23,5% 16,5% 34,4% 19,5% 3,38
k = 2, k1 = 10, k2 = 5 56,6 100,9% 0,86% 34,6 0,04% 0,997 0,007 2,2 7,8 12,4 16,2 18,0 9,1% 25,7% 21,2% 35,4% 8,6% 3,09
56,5 100,7% 0,72% 34,5 0,05% 0,998 0,008 2,2 7,8 12,5 16,1 17,9 9,2% 25,7% 21,4% 35,3% 8,3% 3,08
k = 2,5, k1 = 0, k2 = 0 55,0 101,0% 1,00% 33,0 0,06% 0,996 0,012 2,3 7,9 11,6 15,2 18,1 9,8% 25,4% 16,8% 35,1% 13,0% 3,16
54,8 100,7% 0,82% 32,8 0,06% 0,996 0,013 2,3 7,8 11,5 15,2 18,0 9,8% 25,4% 16,9% 35,3% 12,7% 3,16
k = 2,5, k1 = 10, k2 = 5 62,2 101,8% 1,80% 40,0 0,01% 0,999 0,002 3,4 9,2 14,2 17,2 18,3 13,6% 26,7% 22,8% 32,1% 4,7% 2,88
61,8 101,5% 1,47% 39,7 0,01% 1,000 0,002 3,3 9,1 14,2 17,1 18,1 13,6% 26,5% 23,5% 31,8% 4,5% 2,87
k = 3, k1 = 0, k2 = 0 60,6 102,1% 2,02% 38,4 0,01% 0,999 0,003 3,5 9,2 13,1 16,6 18,3 14,1% 25,8% 18,1% 34,1% 7,9% 2,96
60,6 101,7% 1,69% 38,4 0,01% 0,999 0,004 3,5 9,1 13,1 16,6 18,3 14,5% 25,6% 18,1% 34,0% 7,8% 2,95
k = 3, k1 = 10, k2 = 5 67,8 103,4% 3,28% 45,2 0,00% 1,000 0,000 4,7 10,5 15,9 18,1 18,6 18,3% 26,5% 24,5% 28,2% 2,5% 2,70
67,7 102,9% 2,78% 45,2 0,00% 1,000 0,000 4,7 10,4 15,9 18,1 18,6 18,7% 26,1% 25,5% 27,4% 2,4% 2,69
1145
