Hadas, Yu al; Figliozzi, Miguel A.
A icle
Modeling op imal d one lee size conside ing s ochas ic
demand
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Modeling op imal d one lee size conside ing s ochas ic demand
Yu al Hadas
a
, Miguel A. Figliozzi
b
,
*
a
Depa men o Managemen , Ba -Ilan Uni e si y, Rama Gan, 5290002, Is ael
b
T anspo a ion Technology and People (TTP) Lab, Depa men o Ci il and En i onmen al Enginee ing, Po land S a e Uni e si y, Po land, OR, 97201, USA
ARTICLE INFO
Keywo ds:
D one
Flee sizing
S ochas ic demand
Op imiza ion
ABSTRACT
The las mile deli e y is pa icula ly challenging o s ochas ic deli e ies wi h na ow ime windows. This opic is
imely due o he ise o e-comme ce and cou ie ype se ices and he impac s o lee size and ehicle ype on
deli e y cos s. A no el con ibu ion o his esea ch is o p o ide an op imiza ion app oach, ex ending he
news endo model, o p o ide an op imal d one lee sizing solu ion wi h s ochas ic demand in e ms o numbe
o deli e ies and deli e ies weigh o payload om one cen al depo . The solu ions ob ained a e obus , as
shown in a comp ehensi e sensi i i y analysis.
1. In oduc ion
In ecen yea s, he e ha e been a lo o exci ing announcemen s and
pilo s udies o d one, o Unmanned Ae ial Vehicles (UAVs), de-
ploymen s in eigh anspo a ion and logis ics. UAVs ha e been
ea u ed equen ly in he media ollowing announcemen s made by
la ge co po a ions such as Amazon (Vincen and Ga enbe g, 2019).
In he US e-comme ce g ew a a 30% a e in 2020 (eMa ke e , 2020)
and d one deli e ies a e expec ed o become a 7 billion US dolla ma ke
by 2027 (Insigh s, 2020). D ones a e inc easingly being u ilized o
deli e medical supplies and in cou ie ype se ices. The COVID-19
pandemic has accele a ed his end. D ones a i e quickly by aking
mo e di ec pa hs and a oiding g ound-based obs uc ions o conges ed
oads.
Mos o he d one li e a u e has ocused on ou ing, pa h op imiza-
ion, o scheduling o uck-d one eams. This esea ch s udies a no el
d one lee sizing p oblem wi h s ochas ic demand in e ms o demand
olume and size o he deli e y. Unlike mos s udies, in his esea ch, he
objec i e unc ion is p o i maximiza ion when he e a e cos s associa ed
wi h he ai c a size and unme cus ome demands. This pape is
o ganized as ollows: a li e a u e e iew is p esen ed in he nex sec ion,
ollowed by a o mula ion o he d one lee size op imiza ion p oblem
wi h a s ochas ic numbe o demands and demand weigh s o payload.
Payload is impo an because o d ones, unlike g ound ehicles,
payload is a majo cons ain ha se e ely educes d one ange and
impac s on i s cos s. A comp ehensi e simula ion and sensi i i y analysis
o a lee sizing scena io is la e p esen ed. The pape ends wi h
conclusions.
2. Li e a u e e iew
This sec ion p o ides a b ie backg ound ega ding d one- uck ap-
plica ions, lee sizing, and d one emissions. The idea o u ilizing bo h
UAV and ucks (Mu ay and Chu, 2015) o imp o e o e all deli e y
e iciency has been analyzed by many au ho s ocusing on he ac ual
design o ou es and logis ics sys ems in he las i e yea s. The e has
been an explosion in he numbe o pape s ela ed o d one ou ing
op imiza ion. Se e al e iews p esen an o e iew o modeling e o s.
Since he ocus o his esea ch is on d one lee sizing, no on ou ing o
scheduling, he eade is e e ed o he ollowing su eys o d one
applica ions, ehicle ou ing, and scheduling. A su ey o applica ions o
d ones in ci il p oblems is p esen ed by O o e al. (2018). A mo e ecen
su ey by Mac ina e al. (2020) ocuses mos ly on ou ing p oblems wi h
d ones and he e iew by o Chung e al. (2020) on d one- uck com-
bined ope a ions.
In he d one li e a u e, i is possible o ind d one a small numbe o
esea ch e o s and models ha inco po a e s ochas ic elemen s. Fo
example, Baloch and Gza a (2020) s udy pa cel deli e ies wi h d ones
unde compe i ion using mul inomial logi models. In his wo k, he e is
compe i ion and unce ain y ega ding d one ma ke sha e bu also
ega ding o e all ma ke size. The wo k o Chen e al. (2021) use a
Ma ko decision p ocess o de elop closed analy ical solu ions ha a e
use ul o ge insigh s ega ding d one ee s uc u es and deli e y ca-
paci y o d one deli e y ope a ions wi h andom demands, di e en
p oduc ca ego ies, and mul iple se ice zones. D one lee sizing has
ecei ed scan a en ion in ela ion o d one- uck ou ing. The wo k o
Lee (2017) u ilizes modula i y and simula ion in d one design o
* Co esponding au ho .
E-mail add esses: [email p o ec ed] (Y. Hadas), [email p o ec ed] (M.A. Figliozzi).
Con en s lis s a ailable a ScienceDi ec
EURO Jou nal on T anspo a ion and Logis ics
jou nal homepage: www.sciencedi ec .com/jou nal/eu o-jou nal-on- anspo a ion-and-logis ics
h ps://doi.o g/10.1016/j.ej l.2024.100127
Recei ed 13 July 2022; Recei ed in e ised o m 18 Augus 2023; Accep ed 18 Janua y 2024
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100127
2
es ima e lee size and op imize hei ope a ion. Rab a e al. (2018)
s udies a d one lee model o las mile deli e ies in humani a ian lo-
gis ics using a MIP model whe e he objec i e minimizes he o al
a eling dis ance o cos aking in o accoun d one payload and ene gy
cons ain s and he ins alla ion o echa ging s a ions. Chauhan e al.
(2019) p esen a MIP o mula ion and heu is ics o maximize d one lee
co e age, also aking in o accoun d one payload and ene gy cons ain s
and p o iding acili y loca ion, alloca ing d ones o acili ies and d ones
o cus ome s. This wo k is la e ex ended o p o ide obus solu ions
when accoun ing o unce ain y in ini ial ene gy ba e y a ailabili y
and ene gy consump ion (Chauhan e al., 2020). To abbeigi e al. (2020)
o mula e a se co e ing p oblem and a MIP model, conside ing payload
and ba e y consump ion, o loca e acili ies and minimize he numbe
o d ones in a pa cel deli e y sys em. Poin - o-poin ope a ions ha
ex end he ange o d one ha e been also s udied (Pin o and Lago io,
2022).
Flee sizing unde demand unce ain y has been s udied o speci ic
ime-sensi i e applica ions. Fo example, Bou ilie e al. (2017) akes
in o accoun spa ial demand unce ain y and u ilizes op imiza ion and
queuing o de e mine d one lee size o he eme gency deli e y o
au oma ed ex e nal de ib illa o s. This wo k was ex ended by Glick e al.
(2021) o include no only demand unce ain y, bu also he impac o
wea he condi ions on d one lee sizing. An in eg a ed loca ion-queuing
model was la e de eloped using ealis ic esponse imes and analyzing
he impac o conges ion (Bou ilie and Chan, 2022). O he p ac ical
applica ions include wo k on cold chain ne wo k design o accine
dis ibu ion wi h ime limi s (Enaya i e al., 2023).
The models and esul s p esen ed in his esea ch a e no el in se e al
ways: (a) he e is unce ain y ega ding he numbe o eques s pe
pe iod, (b) he e is unce ain y in e ms o demand cha ac e is ics such
as he pa cel weigh , and (c) i is a p o i maximiza ion p oblem whe e
companies o d one ope a o s ace adeo s in e ms o lee size, ype o
d one, e enue, ope a ing cos s, and los sales. The p o i maximiza ion
app oach is mo e app op ia e when p i a e ope a o s a e assumed and
gi en he e enue/cos adeo s no all cus ome s could be se ed. To
he bes o he au ho s’ knowledge, he e is no simila published
esea ch ela ed o d one lee size op imiza ion.
3. D one lee size p oblem o mula ion
The o mula ion o he d one lee size model u ilizes ideas ela ed o
he news endo model o lee sizing. The news endo model is a basic
p oblem in s ochas ic in en o y con ol and om he 1950s has been
widely s udied and applied o ope a ions esea ch and supply chain
managemen p oblems (Woold idge, 2015). The news endo app oach
has been used in he anspo a ion li e a u e, o example, o de e mine
op imal lee size in public anspo a ion se ings whe e agency cos s
a e comp ised by ehicle size, emp y sea s, and los sales and use cos s
a e ela ed o wai ing and o e c owding (He bon and Hadas, 2015) o
anspo a ion adap a ion o a supply chain model (Hadas and Shnai-
de man, 2012). Howe e , unlike he abo e-men ioned models, which
a e minimum cos models, he p oposed model is based on p o i
maximiza ion and applied o d ones, which equi es a di e en o mu-
la ion inco po a ing payload cons ain s.
3.1. Assump ions
The main assump ions in his esea ch a e he ollowing.
1. Deli e y imes a e sho like in a ypical cou ie se ice. Wi hou loss
o gene ali y in his esea ch, i is assumed ha he planning pe iod is
½ hou and ha d one deli e ies a e made in his ½ hou pe iod.
Hence, o lee size planning pu poses, a ½ peak demand pe iod is
assumed o de e mine d one lee size.
2. The dis ibu ion o demand is known, bu he ac ual demand o be
se ed in each pe iod is only known a he s a o he pe iod.
3. One ip o one cus ome pe d one can be comple ed in he ½ hou
demand pe iod. This is a easonable assump ion, aking in o accoun
lying ime, akeo and d op-o ime, ca go p epa a ion and ba e y
swapping ime.
4. D one pu chase cos is a unc ion o payload capabili y. D one pu -
chase cos and ope a ional cos is an inc easing unc ion o d one
size.
5. D one se ice a ea and ange a e cons an and independen o d one
size. D one cos s inc ease as a unc ion o payload and e lec la ge
ba e ies o se e he same se ice a ea.
6. A cus ome can be se ed by a d one o a uck, bu d ones a el
om he depo di ec ly o he cus ome , i.e. he d one is no using he
uck as a base o ake-o o landing.
The i s wo assump ions a e simila o assump ions ound in he
adi ional news endo p oblem, whe e he decision ( lee size in his
case) mus be made be o e ac ual demand ealiza ion is known. In
pa icula , he ½ hou ime window in he case o d one deli e ies can be
conside ed as planning o he peak hou demand whe e cus ome s a e
willing o pay an ex a ee o p emium o a as deli e y se ice as well
discussed in Baloch and Gza a (2020). The ollowing h ee assump ions
a e needed o d ones and no ound in he news endo li e a u e. As a
esul o hese h ee assump ions, i is necessa y o add a decision a -
iable: d one payload (size) ha g ea ly inc eases he complexi y o he
p oblem. These assump ions a e ealis ic and no e y es ic i e. The
main limi a ion o he news endo model is ha demands canno be
ca ied o e on o he nex pe iod, i.e. demands ha a e no se ed in a
gi en pe iod a e los . I is no possible o pos pone se icing a demand o
o allow o demand backlogs in in en o y con ol e minology. Ano he
limi a ion o he model is ha in he eal-wo ld, lee size is an in ege
a iable and he e is a ini e se o d ones (payloads) o choose om. The
esul s o he model p o ide he bes solu ion, assuming con inuous
a iables, bu because he objec i e unc ion is ai ly “ la ” a ound he
op imal, he ounded alues p o ide a e y good app oxima ion.
Al e na i ely, gi en ha he numbe o po en ial payloads is limi ed, he
payload alue can be se and he model can be un op imizing only lee
size. The model is mean o p o ide manage ial insigh s o guide d one
lee sizing decisions. In p ac ice, he e could be many o he applica ion
speci ic cons ain s, like inancing and cash lows ha a e ou side o he
scope o his pape .
The model conside s wo decision a iables, he lee size (N) and
d one’s capaci y o payload (V). Howe e , o simpli y he p esen a ion
o he model, and wi hou loss o gene ali y, N is also e e ed o as he
maximum numbe o planned ips pe pe iod because i is assumed ha
each d one can only se e a cus ome pe pe iod. Ha ing N as he
numbe o planned ips, can u he elax he non-in ege esul s, which
is accep able as N is he a e age numbe o ips pe ime window,
simila o he way public anspo headways a e being used (Hadas and
Shnaide man, 2012).
3.2. Random a iables and dis ibu ions
The decision a iables a e ela ed o wo dis ibu ions o andom
a iables: he demand dis ibu ion o numbe o ips pe pe iod and he
dis ibu ion o cus ome payloads. In he p oposed model, he wo
andom a iables should ollow h ee equi emen s: 1) non-nega i e
in e als o ini e leng h, 2) wice di e en iable, and 3) i o es ima e
common dis ibu ions. The i s equi emen s ems om he na u e o
he decision a iables, i.e. bo h a e non-nega i e, wi h he payload
bounded by minimal and maximal size ela ed o economical and
physical p ope ies, while he demand can be es ima ed wi hin a ce ain
con idence in e al. The second equi emen e lec s he need o a
closed- o m de i a ion o he op imal solu ion. The las equi emen
sa is ies he need o es ima e di e en dis ibu ions ep esen ing de-
mand and weigh pa e ns.
The ou pa ame e Be a dis ibu ion (McDonald and Xu, 1995), wi h
Y. Hadas and M.A. Figliozzi
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100127
3
he shape pa ame e s
α
and β in he posi i e in e al [l,u]sa is ies all
h ee equi emen s. I is bounded by pa ame e s l and u and di e en-
iable pe equa ion (1).
(x) = (x−l)
α
−1(u−x)β−1
(u−x)
α
+β−1Γ(
α
)Γ(β)
Γ(
α
+β)
(1)
Gi en ha he Gamma unc ion is de ined as Γ(n) = (n−1)!equa ion
(1) is also polynomial, which will be use ul la e o ob aining he global
op imal solu ion. The shape pa ame e s
α
and β can be modi ied o
de ine bo h symme ic (
α
=β=1)and unimodal skewed
(
α
>1,β>1,
α
∕= β)dis ibu ions, wi h ela i ely s aigh o wa d
calcula ion o he i s wo momen s (equa ion (2) and equa ion (3)).
E(x) = l+
α
α
+β(u−l)(2)
Va (x) =
α
β
(
α
+β)2(
α
+β+1)(u−l)2(3)
Fu he mo e, he mos common dis ibu ions (no mal, logis ic, e c.)
can be app oxima ed using a Be a dis ibu ion. Fu he mo e, bo h uni-
o m and exponen ial dis ibu ions a e special cases when he pa ame-
e s a e (
α
=β=1)and lim
n→∞nBe a(1,n) = Exp(1) espec i ely. Fig. 1
p o ides ou examples o he ou pa ame e s Be a dis ibu ion o
di e en pa ame e s
α
, β , l and u, in his esea ch he demand a ibu es
a e numbe o cus ome s pe pe iod and payload pe cus ome (bo h
s ochas ic a iables).
3.3. Model o mula ion
The model maximizes a p o i unc ion which is composed o ou
componen s: e enues (T ), deli e y cos s (TCe), ope a ion cos s (TCo),
and los sales when cus ome eques s canno be sa is ied due o lee
size and payload limi s (TCl).
Unlike he adi ional news endo model, which only has one deci-
sion a iable, he op imiza ion model u ilized in his esea ch has wo
decision a iables. In he adi ional news endo model, he e is a one-
dimensional p obabili y space wi h wo egions, sho age and su plus. In
he p oposed model, he e is a join p obabili y space wi h wo di-
mensions ha a e independen o each o he based on he ollowing
p ope ies.
1) The demand and weigh dis ibu ions a e independen . The o me
de ines he numbe o deli e ies, while he la e deno es he indi-
idual weigh o each deli e y.
2) The lee size and payload decision a iables a e each uniquely
associa ed wi h he demand and weigh , espec i ely.
3) Following 1) and by de ini ion, he decision a iables a e indepen-
den . Gi en he complexi y o he p oposed news endo model wi h
wo independen a iables, a case whe e demand numbe and
payload a e posi i ely o nega i ely co ela ed is le as a u u e
esea ch e o .
Based on he p e ious h ee assump ions he wo-dimensional
p obabili y space can be di ided in o ou egions, as illus a ed in
Fig. 2. The bounds o he demand in e al a e deno ed [ld,ud]and he
bounds o he payload in e al a e deno ed [lw,uw]. The wo axes, x and
y, co espond o he payload and lee size, espec i ely. Each axis is
bounded by he a iable’s minimum and maximum alues. The e ical
and ho izon al lines a e p ojec ing a gi en alues o he decision a i-
ables. Region 1 e e s o he so-called su plus egion, as hese ac ual
demand and weigh can be sa is ied. Region 4 e lec s he sho age
ela ed o bo h payload and lee size. Region 2 e e s o he si ua ion in
which unsa is ied demand is ela ed o payload only (payload sho age).
Region 3 e e s o he si ua ion in which unsa is ied demand is
ela ed o insu icien lee size ( lee size sho age). Unlike p e ious
egions, egion 3 can be di ided in o wo sub- egions, 3a and 3b, in
which he la e can be sa is ied by unused ips associa ed wi h egion
2. Hence, he o mula ion o he p oblem should include he no a p io i
ob ious case whe e he demand pe pe iod exceeds he numbe o
d ones a ailable bu some o he sho age is cap u ed al eady in egion 2
and should no be double coun ed.
Fu he mo e, each egion is associa ed wi h one o mo e o he
e enue o cos componen s. The ealized e enue and deli e y cos s a e
associa ed wi h egion 1, as hey a e a unc ion o he ac ual demand
bounded by he selec ed lee size and payload. Los sales due o payload
limi occu o all eques s wi h weigh o e he payload limi ( egion 2).
Los sales due o lee size limi occu o lee size limi s ( egion 3), and
los sales due o bo h a iables a e associa ed wi h egion 4.
The exac o mula ion o each componen is he ein de ined, wi h
d(x)and w(y)as he demand and weigh p obabili y dis ibu ion
unc ions, espec i ely. Since bo h p obabili y dis ibu ions a e inde-
penden he join p obabili y dw(x,y)can be w i en as d(x) w(y).
Following ha , he in eg als o e d(x)and w(y)can be eplaced by he
p obabili y unc ions:
Fig. 1. Examples o he ou pa ame e s Be a dis ibu ion. Fig. 2. p obabili y space’s zones.
Y. Hadas and M.A. Figliozzi
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100127
4
Fd(V) = ∫
N
ld
d(x)dx (4)
Fw(V) = ∫
V
lw
w(y)dy (5)
In u n, equa ions (6)–(9) de ine he a o emen ioned egions’
expec ancies.
Ed(N,V) = ∫
ud
ld
Min(x,N) d(x)dx Fw(V) = ⎛
⎝∫
N
ld
x d(x)dx +N(1−Fd(N))⎞
⎠
Fw(V)
(6)
Ew(N,V) = ∫
ud
ld
w(N,V,x) d(x)dx =∫
ud
ld
Min(x,N) d(x)dx (1−Fw(V))
=⎛
⎝∫
N
ld
x d(x)dx +N(1−Fd(N))⎞
⎠(1−Fw(V))
(7)
Es(N,V) = ∫
ud
N
s(N,V,x) d(x)dx =⎛
⎝∫
ud
N
x d(x)dx
−N(1−Fd(N))⎞
⎠Fw(V)(8)
Esw(N,V) = ⎛
⎝∫
ud
N
x d(x)dx −N(1−Fd(N))⎞
⎠(1−Fw(V)) (9)
The addi ional deli e ies based on he unused ips esul ing om he
payload limi s can be o mula ed as he expec ancy o he minimum
be ween egions 2 and 3. Equa ion (10) de ines he addi ional success ul
deli e ies esul ed wi h he unused ips due o payload limi s. i.e., as
each planned ip has a p obabili y Fw(V)o no being se ed i can
po en ially sa is y pa o he unme deli e ies.
Ed
′
(N,V) = ∫
ud
N
Min (s(N,V,x),w(N,V,x))dx
=∫
ud
N
Min ((x−N)Fw(V),N(1−Fw(V))) d(x)dx (10)
Equa ion (11) de ines he o al e enues’ expec ancy, whe e R is he
e enue o p ice paid pe deli e y. Equa ion (12) and equa ion (13)
de ine he ope a ion cos s’ expec ancies ela ed o he lee size and
d one size (maximal payload). The o me a e he cos s incu ed by he
ac ual deli e ies, wi h he coe icien Ce, which is he a e age cos o
comple e a ip ela ed o ene gy and ba e y consump ion. The la e is
he ixed cos equi ed o secu e he a ailabili y o he d ones ega dless
o he ac ual deli e ies (pu chasing, labo , acili y). Two coe icien s a e
associa ed wi h he ixed cos s, C and C , he o me wi h he d one size
and he la e wi h he lee size.
Finally, equa ions (14)–(16) de ine he penal ies’ expec ancies
associa ed wi h unse ed cus ome s due o lee size and payload limi s.
Bo h penal y componen s a e associa ed wi h he same penal y pe un-
se ed cus ome Cl.
T (N,V) = R(Ed(N,V))+Ed
′
(N,V)(11)
TCe(N,V) = CeV(Ed(N,V) + Ed
′
(N,V)) (12)
TCo(N,V) = N(C +C V)(13)
TCs(N,V) = ClEs(N,V)(14)
TCw(N,V) = ClEw(N,V)(15)
TCsw(N,V) = ClEsw(N,V)(16)
The objec i e unc ion is hen o maximize he p o i s ( o cla i y, he
decision a iables we e emo ed om he unc ions).
max Tp =T −TCo −TCe −TCs −TCw −TCsw
ld ≤N≤ud
lw ≤V≤uw
Howe e , pa o he penal ies should be elimina ed. This ac ollows
om equa ion (10) whe e addi ional deli e ies a e ca ied ou . Hence,
TC sw, which is he absolu e di e ence be ween w(N,V,x)and s(N,V,x),
eplaces TCs and TCw as ollows:
max Tp =T −TCo −TCe −TC sw −TCsw.
ld ≤N≤ud
lw ≤V≤uw
(17)
whe e
Esw (N,V) = ∫
N
ld
x(1−Fw(V)) d(x)dx +∫
ud
N
|N(1−Fw(V))
− (x−N)Fw(V)| d(x)dx (18)
TC sw (N,V) = ClE sw (N,V)(19)
The model can be ew i en as ollows. Equa ion (10) can be
simpli ied gi en ha egions 3a and 3b a e di ided a he line de ined by
equa ion (20).
N
′
(V) = N
Fw(V)(20)
as a esul , equa ion (10) can be ew i en as:
Ed
′
(N,V) =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
∫
N
Fw(V)
N
(x−N) d(x)dx
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
Fw(V)
+
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
∫
N
Fw(V)
ud N d(x)dx
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(1−Fw(V)) (21)
Wi h N
′
(V) ≤ ud, elimina ing he use o he minimum ope a o (equa ion
(10)) by di iding he in eg al om N o ud in o wo sepa a e in eg als
om N o N
Fw(V)and om N
Fw(V) o ud. Fo cla i y, he ull de elopmen o
equa ion (21) is p o ided in Appendix A. Based on equa ion (20),
equa ion (18) can be ew i en as:
Esw (N,V) = ∫
N
ld
x(1−Fw(V)) d(x)dx +∫
N
Fw(V)
N
(N−x−Fw(V)) d(x)dx
+∫
N
Fw(V)
ud (x Fw(V) − N) d(x)dx
(22)
Y. Hadas and M.A. Figliozzi
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100127
5
Once again, o cla i y, he ull de elopmen o equa ion (22) is
p o ided in Appendix A. As a esul , he e ised objec i e unc ion is
max Tp =T −TCo −TCe −TC sw −TCsw.
ld ≤N≤ud
lw ≤V≤uw
s. .N
′
(V) ≤ ud
(23)
4. Global op imal sea ch algo i hm
The objec i e unc ion (23) is non-con ex, as depic ed in Fig. 3,
hence adi ional op imiza ion algo i hms does no gua an ee inding
he global op ima. Howe e , he model has se e al p ope ies ha
enable he cons uc ion o an e icien global op imal sea ch algo i hm.
The objec i e unc ion componen s a e wice con inuously di e -
en iable unc ions, as such, exis ing algo i hms ha use con ex elaxa-
ion o non-linea wice con inuously di e en iable unc ions can ob ain
he global op ima. Hence he
α
BB algo i hm (And oulakis e al., 1995) o
he QBB algo i hm (Zhu and Kuno, 2005) a e possible algo i hms ha
can be used. Fu he mo e, as d(x)and w(y)a e polynomial unc ions i
ollows ha Tp is a polynomial unc ion as well, as Tp’s componen s a e
in eg als o e polynomial unc ions. This in u ns, simpli ied he sea ch
o he global op ima, e en hough Tp is non-con ex. Speci ically, he
sea ch p ocedu e can be cons uc ed as ollows.
4.1. S ep A: model elaxa ion
1) Objec i e unc ion (23) is sol ed uncons ained.
2) Find all oo s o he uncons ained unc ion (23):
∂
Tp
∂
N=0,
∂
Tp
∂
V=0(24)
3) I e a e o e all oo s and iden i y he global op ima.
4) I he global op imum is easible, gi en he cons ain , s op. O he -
wise con inue o s ep B.
Sol ing (24) is a ela i ely simple ask, as he p oblem o inding all
oo s o a mul i a ia e polynomial is a well-s udied p oblem (Geil,
2015), speci ically when sol ing o e a bounded in e al, such as he
p oblem a hand. Then, comme cial packages can be used ha imple-
men adi ional sea ch algo i hms such as he New on-Raphson.
4.2. S ep B: Lag ange mul iplie
I he global op imum is ou side he easible egion, hen sol e he
objec i e unc ion wi h equali y cons ain , as he solu ion esides along
he cons ain (equa ion (25)).
max Tp
′
=T −TCo −TCe −TC sw −TCsw −λ(N
′
(V) − ud)
ld ≤N≤ud
lw ≤V≤uw
(25)
1) Find all oo s o equa ion (25) based he Lag ange mul iplie s
me hod.
∂
Tp
′
∂
N=0,
∂
Tp
′
∂
V=0,
∂
Tp
′
∂
λ=0(26)
2) I e a e o e all oo s and iden i y he global op imum.
5. D one lee sizing esul s
The implemen a ion o he o mula ion and sea ch algo i hm is
applied in his sec ion o a base case s udy. To u he s udy he s abili y
o he solu ion, a sensi i i y analysis and noise in he pa ame e s a e also
s udied in his sec ion.
6. Case s udy
The model was implemen ed wi h Maple 2020 package (Mapleso ,
2020). Solu ions we e ob ained ins an ly, as he model is di e en iable
wi h only ange cons ain s o he decision a iables. Assuming a ½
hou pe iod, wi h 1 ip pe pe iod pe d one, he ollowing pa ame e s
we e selec ed o a case s udy. The e enue (R)pe deli e y is based on
ush cou ie deli e y a es (B eakaway, 2021) a e conside ing he
se ice a ea o he d one and he numbe o deli e ies pe hou . Fo he
d one cos s, wo key assump ions a e labo cos s pe ip and he use ul
li e o he ai c a (Figliozzi, 2018) and a base cos o unme deli e y is
he cos o an al e na i e g ound deli e y like Ube Ea s (G idwise,
2020).
The appeals o d ones a e bo h po en ial deli e y ime sa ings and
cos (Vincen and Ga enbe g, 2019). I is impo an o no e ha he
p o i o a d one deli e y assumes a p emium due o as and eliable
deli e y and he cos o unme demand is ela ed o using a g ound
se ice o co e he unme d one demand and/o he loss o a cus ome
o a i al se ice p o ide . D one se ices a e s ill in i s in ancy and i is
no i ial o es ima e he alue o los cus ome (Hogan e al., 2003),
due o he di icul ies o iden i y p ecise o na ow anges o mos o he
pa ame e s an ex ensi e sensi i i y analysis exe cise is pe o med in he
ollowing sec ion.
R =$12.5/deli e y
Cl =$ 5.0/unme deli e y
C =$ 1.5/deli e y
Ce =$ 0.2/deli e y-kg payload
C =$ 0.1/deli e y-kg payload
Fo his s udy, we assume ha a mul i-cop e can ca y up o 2.5 kg o
payload wi h an e ec i e ange o 10 km. As a e e ence, Amazon is
s udying d ones o deli e up o 5 pounds (2.27 kg) in 1/2 h o less,
which oughly ag ees wi h he assump ions made in his pape (Amazon,
2021). In addi ions, up o 5 pounds accoun s o 80–90% o Amazon
packages (Manjoo, 2016). As a e e ence, an e ec i e ange o 5 km
co e s down own Manha an in New Yo k Ci y. Fo se ice imes, i is
assumed ha an a e age d one deli e y equi es 30 min. No e ha o a
o al ound ip o 10 km, lying a an a e age c uise speed o 18 m s he
lying ime is app oxima ely 9–10 min, and he es o ime is consumed
by o he ac i i ies which include: ake-o p epa a ion, deli e y ime,
Fig. 3. Objec i e unc ion (Tp) as a unc ion o payload V and lee size N.
Y. Hadas and M.A. Figliozzi
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100127
6
landing, ba e y swap, and ca go loading and secu ing (Glick e al.,
2021).
The alues o Ce and C a e de e mined aking in o accoun ha o
deli e y d ones he payload is app oxima ely 1/5 o he d one o al
akeo weigh and ha d one ene gy consump ion and cos s a e a
unc ion o d one size (Figliozzi, 2023a). I is implici ly assumed ha
many o he d one dimensions and cos da a a e app oxima ely linea o
small size d ones wi h a a e (emp y weigh wi hou payload and ba -
e y) o up o 10 kg as shown when analyzing eal-wo ld d one speci-
ica ions (Figliozzi, 2018). The a e o he d one is assumed o be 8 kg
and he ba e y 2.5 kg. The es ima ed cos ela ed o ene gy and ba e y
eplacemen a e app oxima ely $0.5 pe deli e y assuming: a payload o
2.5 kg, a ba e y ene gy densi y o 200 wh/kg, a cos o $275/Kwh, a
ba e y li e o 300 cycles, a 50% ba e y u iliza ion o accoun o he
sa e y ac o and aging, a cos pe Kwh o app oxima ely $0.15/Kwh,
and inally a d one ene gy consump ion o 25 wh/km when lying. The
es ima ed cos o d one as a unc ion o size is es ima ed as $0.25 pe
deli e y assuming: a payload o 2.5 kg, 3000 ligh hou s pe d one, and
an ini ial d one cos o $7500. A comp ehensi e sensi i i y analysis is
p o ided because d ones and ba e y echnologies a e quickly e ol ing
(Figliozzi, 2023b) and he e o e i is ecommended ha assump ions
ega ding d one cos s and capabili ies in u u e esea ch e o s should
be adequa e o he speci ic applica ion and yea o analysis.
Demand and weigh p obabili y dis ibu ions pa ame e s we e se o
d(x) ∼ Be a(
α
=3,β=3,l=0,u=100)and w(y) ∼ Be a(
α
=3,β=3,
l=0,u=2.5) espec i ely. The weigh dis ibu ion is in kilog ams. Wi h
hese pa ame e s he expec ed demand and a iance a e 50 and 357
espec i ely and he expec ed weigh and a iance o 1.25 and 0.22
espec i ely. The op imal solu ion alue is $458 ob ained o N =75
d ones and V =2.38 kg (5.25 lbs) o maximum payload pe d one.
The solu ion space is p esen ed in Fig. 3, along wi h he b eak-e en
plane. The shape o each o he componen s and alues associa ed wi h
he op imal solu ion a e p esen ed in Appendix B. All he six igu es a e
aligned wi h he axes o Fig. 3.
6.1. Sensi i i y analysis
The sensi i i y analysis cons i u es o he ollowing assessmen s: 1)
dis ibu ions’ shape analysis, 2) coe icien s sensi i i y, 3) noise le el
analysis and 4) obus ness analysis.
6.1.1. Dis ibu ion shapes
Fi s , he e ec o he demand and weigh dis ibu ion shapes was
in es iga ed. Fo ha , he base solu ion, which has symme ical dis i-
bu ions was compa ed o posi i e and nega i e skewed dis ibu ions.
The esul s a e p esen ed in Table 1.
When compa ed o he symme ical dis ibu ion, he posi i e and
nega i e skewed dis ibu ions ha e a d as ic e ec on he lee size and
d one size. Posi i e skew dis ibu ions dec ease he op imal lee size
and d one size (demand and weigh a e ages a e smalle ), while nega-
i e skew dis ibu ions inc ease hem. Mo eo e , as d(x)and w(y) a e
independen , each one a ec s i s co esponding decision a iable. The
sligh change o he o he decision a iable can be a ibu ed o TCo
(equa ion (13)), which has a mul iplica ion o N and V. Please no e ha
N was ounded, he sligh change (0.1–0.2 o each column) is no
p esen ed.
6.1.2. Coe icien s sensi i i y
Addi ional sensi i i y analysis was pe o med o each o he co-
e icien s wi h selec ed coe icien alues, while holding all o he co-
e icien s o hei ini ial alues. This analysis examines he e ec on N
and V esul ing om inc easing o dec easing each coe icien . The e-
sul s a e summa ized in Table 2. The esul s p o ide insigh in o he
change o N and V wi h ega ds o he speci ic coe icien . The coe icien
Table 1
Sensi i i y analysis o he demand and weigh dis ibu ions.
Demand dis ibu ion
Posi i e skew
E=29, V=16
2
α
=2,β=5
Symme ical
E=50, V=19
2
α
=3,β=3
Nega i e Skew
E=71, V=16
2
α
=5,β=2
Weigh dis ibu ion Posi i e skew
E=0.71, V=0.4
2
α
=2,β=5
N 51 75 91
V 1.90 1.93 1.94
Tp 243 465 703
Symme ic
E=1.25, V=0.47
2
α
=3,β=3
N 51 75 91
V 2.37 2.38 2.39
Tp 239 548 685
Nega i e Skew
E¼1.79, V¼0.4
2
α
=5,β=2
N 51 75 91
V 2.49 2.50 2.50
Tp 238 457 693
Y. Hadas and M.A. Figliozzi
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100127
7
ha is being examined has a g ay backg ound. The objec i e is o e i y
ha inc easing o dec easing each coe icien is logical: 1) Inc easing R
( om 12.5 o 30, 100, and 550) inc eases bo h N and V, as highe e -
enue educes he e ec o he cos s. The same applies o dec easing R
( om 12.5 o 10, 5, and 1). 2) Simila beha io can be obse ed om Cl.
Highe sho age cos s mus educe hen sho age, which is ealized by
inc easing N and V. 3) C and C a e he coe icien s ela ed o he so
called ixed cos s (equa ion (13)), so highe ixed cos s (inc easing he
coe icien s) dec eases bo h N and V. Howe e , as C is only ela ed o N,
he e ec on V is minimal. 5) Ce is ela ed o he ope a ional cos s
(equa ion (12)), and ha e simila e ec o C and C . Howe e , C is
mul iplied by V (aside om he demand), hence i a ec s only V when
he change is close o he ini ial coe icien alue. Howe e , due o he
powe e ec , he la ge he coe icien , he highe he change o bo h N
and V.
The esul s e i y ha he beha io o all he coe icien s a e logical.
6.1.3. Noise le el analysis
Each o he objec i e unc ion coe icien s was injec ed wi h a
andom, uni o mly dis ibu ed noise, in which c’ is a andomized coe -
icien c wi h a noise le el in a gi en % ange p . The noise was injec ed
o each coe icien be o e ob aining he op imal solu ion. A o al o
1000 independen simula ions we e pe o med o each scena io wi h
he op imal decision a iables esul s eco ded.
c
′
=c+ and (1−p …1 +p (27)
Fig. 4 illus a es he dis ibu ion o he decision a iables op imal
alues in a ange de e mined by p = ±20% o each o he coe icien s
(Cl,Ce,C ,C ,R). The black do ep esen s he o iginal op imal solu ion.
All he coe icien s ha e linea e ec on N and V. Clea ly, based on he
slopes, C a ec mos ly N, while Ce,C has mo e e ec on V. The pa-
ame e s Cl,R ha e simila impac on N and V.
In o de o in es iga e ex eme alues, each o he coe icien s was
andomized wi hin he ollowing ange: Cl−0.1000, Ce−0.5.1, C –
0.10, C – 0.5.1, and R – 1.500.
Fig. 5 illus a es ha o a wide ange o noise le els, no all he
coe icien s a e linea , especially he e ec o R, C and Ce on N and V.
Table 2
Sensi i i y Analysis o he Model’s Coe icien s.
Fig. 4. Decision a iables dis ibu ion by coe icien wi h p ±20%.
Fig. 5. Decision a iables dis ibu ion by coe icien wi h ex eme alues.
Y. Hadas and M.A. Figliozzi
EURO Jou nal on T anspo a ion and Logis ics 13 (2024) 100127
8
6.1.4. Robus ness analysis
The obus ness o he solu ion was in es iga ed by in oducing noise
o all coe icien s, excluding R. This is due o he unce ain y wi h
d ones’ ope a ions, while R is associa ed wi h he deli e y cos and is
mo e p edic able.
Th ee scena ios we e cons uc ed o h ee noise le els: 5%, 10%,
and 20%. The decision a iables dis ibu ion is illus a ed in Fig. 6, as a
unc ion o op imal lee size (N) and payload (V). The black do is he
op imal solu ion wi hou noise. The obse a ions wi h less noise (5%)
a e close o he black do and mos ly co e ed by he obse a ions wi h
highe noise le els. To acili a e he obse a ion o he dis ibu ion,
he e a e h ee boxes bounded by he wo-dimensional 5 and 95 pe -
cen iles o N and V and by noise le el.
I is e iden ha he solu ion is obus , o he noisies le el (20%)
he solu ion ange is +/−~1.5 d ones and +/−~0.012 kg. when
conside ing he 5 and 95 pe cen iles bounds.
The impac o noise on p o i s is obse ed in Fig. 7 u ilizing his o-
g ams. As expec ed, he his og am ange is oughly p opo ional o he
noise le el. The black e ical ba is he op imal solu ion wi hou noise.
I is also obse ed ha he solu ion is also obus in e ms o p o i wi h a
ange ±5% wi h he noisies le el (20%) wi h espec o he p o i
wi hou noise.
7. Conclusions
The las mile deli e y is pa icula ly challenging o s ochas ic de-
li e ies wi h na ow ime windows. Due o i s cha ac e is ics, d ones a e
sui ed o si ua ions whe e as and eliable deli e ies a e needed. This
esea ch de eloped a no el op imiza ion app oach o d one lee sizing,
ex ending he news endo model. The model p o ides an op imal d one
lee sizing solu ion wi h s ochas ic demand in e ms o wo decision
a iables: 1) numbe o deli e ies and 2) deli e ies weigh o payload.
Unlike o he s udies, in his esea ch, he objec i e unc ion is p o i
maximiza ion and he e a e cos s associa ed wi h he ai c a size and
unme cus ome demands. An e icien algo i hm gua an ees ha he
op imal solu ion is ound, which can be used in la ge-scale deli e y
scena ios. The solu ions ob ained a e obus , as shown in he comp e-
hensi e sensi i i y analysis. The sensi i i y analysis showed he highe
impo ance o lee size in ela ion o d one size, hough as d one
echnology is apidly e ol ing i is impo an o conside ha his
inding may change in he u u e o o he speci ic d one applica ions.
This esea ch ocused only on d one lee sizing, howe e , ha ing a
lee o d ones and a lee o ucks is a mo e esilien app oach since
d ones may no be able o ope a e e ec i ely wi h ad e se wea he
condi ions and simila ly g ound ehicles may be hinde ed some imes by
conges ion o g ound ne wo k dis up ions. Howe e , ha ing wo
di e en ehicle ypes is also likely o inc ease cos s no only in e ms o
labo bu also in e ms o acili ies and capi al cos s, and modeling hese
cos s and adeo s equi es a majo u u e esea ch e o . Fu he
esea ch is also necessa y o s udy bo h d one and uck lee sizes when
conside ing p o i and/o sus ainabili y goals. O he esea ch di ec ion
can ex end he model wi h addi ional decision a iable, he ange (o
co e age). The la ge he ange, he highe he po en ial demand.
Howe e , he numbe ips pe d one will dec ease due o he la ge
ound ip.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inancial
in e es s o pe sonal ela ionships ha could ha e appea ed o in luence
he wo k epo ed in his pape .
Acknowledgemen s
This esea ch was suppo ed by a g an om he F eigh Moni o ing
Resea ch Ins i u e (FMRI), a USDOT T anspo a ion Cen e . Any e o s
o omissions a e he sole esponsibili y o he au ho s.
APPENDIX A
In his appendix, equa ion (21) and equa ion (22) a e ully de eloped
and p oo ed. Fo bo h equa ions wo p oposi ions a e in oduced, ol-
lowed by each equa ion de elopmen .
Fig. 6. Decision a iables o 5%, 10%, and 20% noise le els.
Fig. 7. P o i dis ibu ion o 5%, 10%, and 20% noise le els.
Y. Hadas and M.A. Figliozzi
