Meal Delivery Routing Problem with Stochastic Meal Preparation Times and Customer Locations

Kancha la, Su end a Reddy; Van Woensel, Tom; Walle , S. T a is; Ukkusu i, Sa ish V.
A icle — Published Ve sion
Meal Deli e y Rou ing P oblem wi h S ochas ic Meal
P epa a ion Times and Cus ome Loca ions
Ne wo ks and Spa ial Economics
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Kancha la, Su end a Reddy; Van Woensel, Tom; Walle , S. T a is; Ukkusu i, Sa ish
V. (2024) : Meal Deli e y Rou ing P oblem wi h S ochas ic Meal P epa a ion Times and Cus ome
Loca ions, Ne wo ks and Spa ial Economics, ISSN 1572-9427, Sp inge US, New Yo k, NY, Vol. 24,
Iss. 4, pp. 997-1020,
h ps://doi.o g/10.1007/s11067-024-09643-1
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RESEARCH
Meal Deli e y Rou ing P oblem wi hS ochas ic Meal
P epa a ion Times andCus ome Loca ions
Su end aReddyKancha la1· TomVanWoensel2· S.T a isWalle 1,3·
Sa ishV.Ukkusu i4
Accep ed: 25 July 2024 / Published online: 18 Sep embe 2024
© The Au ho (s) 2024
Abs ac
We in es iga e he Meal Deli e y Rou ing P oblem (MDRP), managing cou ie assign-
men s be ween es au an s and cus ome s. Ou p oposed a ian conside s unce ain ies
in meal p epa a ion imes and u u e o de numbe s wi h hei loca ions, mi o ing eal
challenges meal deli e y p o ide s ace. Employing a olling-ho izon amewo k in e-
g a ing Sample A e age App oxima ion (SAA) and he Adap i e La ge Neighbo hood
Sea ch (ALNS) algo i hm, we analyze modi ied G ubhub MDRP ins ances. Conside -
ing ou e planning unce ain ies, ou app oach iden i ies ou es a leas 25% mo e p o i -
able han de e minis ic me hods elian on expec ed alues. Ou s udy unde sco es he
pi o al ole o e icien meal p epa a ion ime managemen , impac ing o de ejec ions,
cus ome sa is ac ion, and ope a ional e iciency.
Keywo ds Meal deli e y ou ing· Unce ain y· Sample a e age app oxima ion·
Adap i e la ge neighbo hood sea ch
1 In oduc ion
In eg a ing he gig economy in o i s and las -mile deli e y se ices o eigh
and passenge sec o s has signi ican ly e olu ionized u ban anspo a ion. This
shi is in esponse o he escala ing demands o e icien las -mile deli e ies.
* S. T a is Walle
s e en_ a is.w[email p o ec ed]
1 “FRIEDRICH LIST” Facul y o T anspo andT a ic Sciences, D esden Uni e si y
o Technology, He ne s aße 1-3, D esden01069, Saxony, Ge many
2 School o Indus ial Enginee ing andInno a ion Sciences, Eindho en Uni e si y o Technology,
P.O. Box513, 5600 MB, Eindho en, TheNe he lands
3 College o Enginee ing, Compu ing andCybe ne ics, Aus alian Na ional Uni e si y, 108 No h
Rd, Canbe a2601, Aus alia
4 Lyles School o Ci il Enginee ing, Pu due Uni e si y, 550 W S adium A e., Wes La aye e,
IN47907, USA
998
S.R.Kancha la e al.
Nume ous s a ups specializing in es au an meal deli e y, such as G ubhub,
Doo dash, Deli e oo, Swiggy, and Ube ea s, and on-demand anspo se ices
like Ube , Ly , and Ola, ha e eme ged o mee his g owing need. Despi e ac-
ing unique challenges, hese se ices undamen ally add ess he same issue: hey
enable cus ome s o con enien ly eques pickup and d op-o se ices h ough
mobile apps o websi es, ypically ensu ing deli e y wi hin a se ime ame.
These companies cha ge a deli e y ee, a po ion o which is paid o he se ice
p o ide . Success in his domain hinges on sa is ying all pa ies in ol ed: cus-
ome s expec p omp , a o dable, and eliable se ice; d i e s aim o subs an ial
ea nings; and es au an s seek o expand hei each and cus ome base h ough
hese deli e y se ices.
Ou esea ch add esses a i al issue in he eigh indus y: he in icacies o ou -
ing o es au an meal deli e ies. This is an expanded o m o he classical Pickup
and Deli e y P oblem (PDP), known o i s compu a ional complexi y. Unlike a-
di ional PDP, which o en ocuses on he limi a ions o ehicle numbe s, ou s udy
concen a es on he a iable elemen s ha can signi ican ly a ec he e iciency o
ope a ions. In he con ex o he gig economy, he concep o a ixed lee size is
eplaced by a po en ially limi less numbe o independen cou ie s who wo k on a
lexible schedule. This lexibili y, howe e , b ings unp edic abili y in se e al aspec s
- such as he cou ie s’ wo king hou s, hei s a ing poin s o deli e ies, and e en
hei disc e ion o accep o decline o de s.
Addi ionally, he a iable na u e o meal p epa a ion imes adds complexi y o he
ood deli e y sec o . Mino a ia ions in hese imes can lead o signi ican delays,
ad e sely a ec ing cus ome sa is ac ion and he e iciency o deli e y ope a ions.
The unp edic abili y o u u e o de s in ensi ies his challenge, as he speci ic loca-
ions and quan i ies o hese o de s, c ucial o ou e op imiza ion, emain unce ain.
Exis ing app oaches in he meal deli e y sec o , such as hose by Reyes e al. (2018)
and Yildiz and Sa elsbe gh (2019), p ima ily ely on de e minis ic models wi h ixed
meal p epa a ion imes and deli e y windows, ailing o accoun o he a iabili y
and unp edic abili y in hese ac o s. In con as , Ulme e al. (2021) and Zheng e al.
(2023) a emp ed o inco po a e unce ain ies in meal p epa a ion and a el imes,
ye hei models s ill lacked he dynamic adap abili y equi ed o eal-li e, unce ain
scena ios. Relying on a e age es ima es o meal p epa a ion and u u e o de s is
insu icien , as his me hod o e looks he ex ensi e ange o a iabili y and i s in lu-
ence on decision-making p ocesses. These a iables’ complex and o en unknown
p obabili y dis ibu ions ende simple a e age-based app oaches ine ec i e. This
unp edic abili y d ama ically expands he ange o po en ial scena ios, making a-
di ional de e minis ic solu ions un easible. Such de e minis ic models, no accoun -
ing o his a iabili y, end o unde es ima e necessa y esou ces, leading o ope a-
ional ine iciencies and dec eased p o i abili y. Ou goal is o de elop a p obabilis ic
model ha accu a ely accoun s o hese unce ain ies, allowing o he c ea ion o
mo e eliable and e ec i e ou ing s a egies ha enhance he pe o mance o he
meal deli e y sec o .
Gi en he in ica e challenges o unce ain meal p epa a ion imes and he unp e-
dic able low o o de s, we’ e iden i ied a collec ion o me hods pa icula ly adep
a na iga ing he Meal Deli e y Rou ing P oblem (MDRP). To add ess he dynamic
999
Meal Deli e y Rou ing P oblem wi hS ochas ic Meal P epa a ion…
and unce ain aspec s o MDRP, we u ilize a olling ho izon amewo k ha in e-
g a es he Sample A e age App oxima ion (SAA) me hod wi h he Adap i e La ge
Neighbo hood Sea ch (ALNS), selec ed o i s p o en e ec i eness in complex sce-
na ios akin o MDRP. SAA, known o i s s eng h in handling s ochas ic disc e e
op imiza ion p oblems, uses a scena io-based app oach o es ima e he expec ed al-
ues o decision a iables, conside ing a ious po en ial u u e s a es. This aspec is
i al when elying on a e age es ima es o unce ain ac o s, which could esul in
less op imal decisions. The SAA’s inco po a ion acili a es he dynamic modi ica-
ion o ou es in esponse o new in o ma ion, aligning wi h he cons an ly chang-
ing deli e y en i onmen . This p ocess in ol es epea edly applying he SAA on
selec ed scena ios o e di e en ime ho izons, assessing he solu ions agains a
b oade ange o scena ios o app oxima e he expec ed alue unc ion be e , and
upda ing ou es based on ac ual de elopmen s. Such a s a egy ensu es ha he solu-
ions a e always ele an and adap able o he immedia e ope a ional demands.
The Adap i e La ge Neighbo hood Sea ch (ALNS) algo i hm, highly e ec i e
o la ge-scale ou ing p oblems (Li e  al. 2016; Ghilas e  al. 2016a, b; Zhu and
Sheu 2018a), is an in eg al complemen o he SAA me hod in ou app oach. ALNS
(Ropke and Pisinge 2006) is adep a explo ing and op imizing wi hin ex ensi e
solu ion spaces, making i pa icula ly well-sui ed o add essing he Meal Deli e y
Rou ing P oblem (MDRP). This algo i hm is ailo ed o quickly adjus o changes in
ou ing pa ame e s, aligning wi h he unp edic able wo k schedules o cou ie s and
he a ying pa e ns o o de accep ance ypical in MDRP. Combining ALNS wi h
he SAA me hod wi hin a olling ho izon amewo k me ges he SAA’s abili y o
app oxima e s ochas ic elemen s wi h ALNS’s capaci y o e icien ly na iga e and
ine- une solu ions in a b oad and in ica e solu ion en i onmen . This syne gis ic
app oach enables us o de elop esilien ou ing solu ions in he ace o unce ain ies
and adap and espond o he dynamic na u e o meal deli e y ope a ions.
The s udy p ima ily ocuses on he challenges in he eigh sec o , pa icula ly
he es au an meal deli e y ou ing p oblem, a complex a ian o he Pickup and
Deli e y P oblem (PDP). This p oblem is cha ac e ized by unp edic able ac o s like
a ying cou ie a ailabili y, luc ua ing wo king hou s, and dynamic meal p epa a-
ion imes, signi ican ly a ec ing ope a ional e iciency. T adi ional de e minis ic
me hods a e inadequa e due o he complexi ies and unce ain ies in ol ed, includ-
ing he unp edic able na u e o u u e o de s. The s udy explo es ad anced me h-
odologies like he Sample A e age App oxima ion (SAA) and he Adap i e La ge
Neighbo hood Sea ch (ALNS) wi hin a olling ho izon amewo k o add ess hese
challenges. These me hods e ec i ely handle he s ochas ic elemen s and com-
plex solu ion spaces o he Meal Deli e y Rou ing P oblem (MDRP). The esea ch
includes de eloping ailo ed algo i hms o he MDRP, conduc ing compu a ional
es s wi h eal-wo ld da a, and pe o ming sensi i i y analysis o e alua e algo i hm
pe o mance unde a ying le els o unce ain y.
The main con ibu ions a e as ollows:
1. We de eloped inno a i e solu ion algo i hms c a ed explici ly o he meal deli -
e y ou ing p oblem. These algo i hms a e s a egically designed o adep ly handle
1000
S.R.Kancha la e al.
unce ain ies, such as he a iabili y in meal p epa a ion imes, he luc ua ing
numbe o o de s, and hei loca ions. This ailo ed app oach ensu es ha he
ou ing solu ions a e e icien and highly esponsi e o he dynamic na u e o
meal deli e y ope a ions.
2. Fu he mo e, we unde ake comp ehensi e compu a ional es ing using da a
de i ed om eal-wo ld scena ios. These es s we e ex ensi e, co e ing a wide
ange o scena ios ha included a ious combina ions o unce ain ies. This
app oach allowed us o igo ously e alua e he algo i hms in condi ions ha
closely mimic ac ual ope a ional en i onmen s, he eby ensu ing he obus ness
and eliabili y o ou solu ions in p ac ical se ings.
3. We conduc a de ailed sensi i i y analysis o sc u inize how he algo i hms pe -
o med unde di e en le els o unce ain y. This sys ema ic and ho ough analy-
sis p o ided deep insigh s in o he beha io and pe o mance nuances o he
algo i hms when con on ed wi h a ying deg ees and ypes o unce ain ies.
Th ough his sensi i i y analysis, we iden i ied s eng hs and po en ial a eas o
imp o emen in ou algo i hms, ensu ing hey a e e ec i e and adap able o he
complex and e e -changing landscape o meal deli e y se ices.
The pape ’s o ganiza ion is as ollows: Sec ion 2 p esen s he li e a u e e iew
ela ed o he MDRP and ela ed p oblems. Sec ion3 o mally in oduces he MDRP
and unce ain ies conside ed. Sec ion4 desc ibes he p oposed solu ion algo i hm
o MDRP wi h unce ain ies. Sec ion5 p esen s he es ins ances and discusses he
compu a ional esul s, ollowed by conclusions in Sec ion6.
2 Li e a u e Re iew
We di ide he li e a u e in o wo pa s. Fi s , we e iew he wo k on Meal-deli e y
ou ing and ela ed p oblems, ollowed by li e a u e on s ochas ic Pickup and Deli -
e y p oblems (PDPs).
2.1 Meal‑deli e y P oblems
Reyes e al. (2018) p oposed a dynamic de e minis ic a ian o a pickup and deli -
e y p oblem called he Meal Deli e y Rou ing P oblem (MDRP), and hey sol ed
i using a olling-ho izon app oach. La e , Yildiz and Sa elsbe gh (2019) p oposed
a ma hema ical o mula ion o he s a ic a ian and sol ed i using a simul ane-
ous ow and column gene a ion-based algo i hm. Bo h a icles assumed unlim-
i ed capaci y o cou ie s, a ixed deli e y window o 90 minu es om he o de ’s
place, and ixed meal p epa a ion imes. S ee e e al. (2019) elaxed he i s wo
assump ions and allowed o de placemen om mul iple es au an s. They p o-
posed a heu is ic ha accoun s o u u e o de s using equi y and dispe sion me -
ics. Ulme e al. (2021) elaxed he la e wo assump ions and p oposed an an ici-
pa o y cus ome assignmen policy ha accoun s o he andom meal p epa a ion
imes. Howe e , he abo e models (excep S ee e e al. (2019)) only ma ch cou ie s

1001
Meal Deli e y Rou ing P oblem wi hS ochas ic Meal P epa a ion…
wi h o de s and assume he cou ie s ake he bes ou es. Mo eo e , hey Yildiz and
Sa elsbe gh (2019); S ee e e al. (2019); Ulme e al. (2021) also en o ce he con-
s ain o isi ing all he o de s. Liu (2019) p oposes a MILP model o a p oblem
simila o MDRP, whe e d ones a e used ins ead o egula cou ie s. Using d ones
helps emo e unce ain y ela ed o a el imes and adds addi ional complexi y, like
cha ging equi emen s and limi ed capaci y. Liao e al. (2020) p oposes a wo-s age
solu ion algo i hm o a s a ic and de e minis ic a ian o he p oblem o minimize
ca bon oo p in . Recen ly, Zheng e al. (2023) p oposed an i e a i e g eedy algo-
i hm o sol e a meal deli e y p oblem wi h unce ain ies in meal p epa a ion imes
and a el ime and also p oposed wo ime-sa ing s a egies o imp o e he compu-
a ional e o . Howe e , hei ins ances a e no o dynamic scena ios close o he
eal-li e applica ion.
2.2 S ochas ic Pickup andDeli e y P oblems
S ochas ic a ian s o VRP ha e been ex ensi ely s udied (Lapo e e al. 2002; Ve weij
e al. 2003; Secomandi and Ma go 2009; Chu e al. 2015; Ghilas e al. 2016b; Zhu and
Sheu 2018b; Shi e al. 2018; Gyö gyi and Kis 2019; Ka oonsoon awong e al. 2020;
Fachini e al. 2022). We can ca ego ize he solu ion app oaches o hese a ian s in o
wo g oups. The i s g oup uses s ochas ic p og amming wi h ecou se, a well-known
amewo k o modeling unce ain y op imiza ion p oblems. In his me hod, some da a
is unknown a he momen o planning. Fi s , a decision is made, and hen he ecou se
cos s o he consequences o he plan a e minimized. The second g oup uses a mul i-
scena io app oach, ollowed in his s udy. This me hod app oxima es expec ed cos s
by e alua ing a solu ion based on gene a ed scena ios. Me aheu is ic algo i hms a e
gene ally used in implemen ing he mul i-scena io s ochas ic op imiza ion app oach.
A good e iew o me aheu is ic algo i hms o s ochas ic combina o ial op imiza ion
can be ound in Bianchi e al. (2009) and Gu jah (2011).
Unlike he li e a u e on s ochas ic VRP, li e a u e on Pickup and Deli e y P oblems
(PDP) wi h s ochas ic demands is limi ed. Powell e al. (1988) is one o he i s s udies
ha conside ed he dynamic PDP wi h s ochas ic demands. They also showed ha con-
side ing unce ain y in he planning p ocess esul s in subs an ial p o i s and inc eases
he se ice le el compa ed o he de e minis ic planning app oach. In Ghilas e  al.
(2016b) in eg a ed he PDP wi h he public anspo sys em and conside ed s ochas ic
demands. Zhu and Sheu (2018b) p oposed a ailu e-speci ic coope a i e ecou se s a -
egy o he simul aneous PDP wi h s ochas ic demand. Unlike he p e ious s udies, Shi
e al. (2018) conside ed unce ain y in a el and se ice imes, Gyö gyi and Kis (2019)
conside ed unce ain y in ime windows. The ypical esul in all he abo e s udies is
ha unce ain y in he planning p ocess leads o signi ican imp o emen s in objec i e
o e he de e minis ic case. In Zhang e al. (2023) in oduced app oxima ions based on
he knapsack p oblem o es ima ing ewa d- o-go. These app oxima ions se e as he
ounda ion o c ea ing e icien online scheduling policies and o line planning algo-
i hms. In Wang e al. (2023), ocused on mee ing cus ome deli e y punc uali y expec-
a ions by es ima ing a i al imes and success p obabili ies in unce ain scena ios.
1002
S.R.Kancha la e al.
Thei es ima ed success p obabili ies ended o be conse a i e lowe bounds. They
also in oduced a solu ion app oach based on a b anch-p ice-and-cu amewo k.
3 P oblem Desc ip ion
Gi en a g aph
G(N,A)
, whe e
N
deno es he se o nodes ep esen ing loca ions
such as es au an s and cus ome s, and
A
ep esen s he a cs be ween hese nodes.
We ha e a se
T
, which ep esen s asks, di ided in o
Tp
o pickups and
Td
o
deli e ies. The se
V
enume a es cou ie s in ol ed in he deli e y p ocess. Fo
each node
i∈T
, he e a e associa ed se ice imes
si
and ime windows de ined
by ea lies
ei
and la es
li
a i al imes. The demand a each node is gi en by
di
,
whe e posi i e alues indica e pickups and nega i e alues ep esen deli e ies.
Cou ie s
k∈V
ha e speci ic on- imes
ek
, o - imes
lk
, and capaci y cons ain s
𝛽k
.
They expec a minimum paymen
mk
pe uni ime. The a el ime be ween nodes
i
and
j
is deno ed by
ij
.
𝛼
, he cos pe uni ime, se es a pi o al ole in ou
model by con e ing ime me ics, like delay and wai ing imes, in o cos me ics.
Unlike
cij
, which di ec ly ep esen s he a el cos be ween nodes
i
and
j
, and
pi
,
which deno es he paymen ecei ed o o de
i
. Cus ome s a node
i∈Td
ha e a
maximum willingness o pay
wi
.
𝜇i
ep esen s he de e minis ic ime associa ed
wi h wai ing imes and delays a each node
i
. x
k
ij
is a bina y a iable ha is equal
o 1 i cou ie
k
a els di ec ly om node
i
o node
j
.
ak
ij
is a con inuous a iable
ep esen ing he a i al ime o cou ie
k
a node
j
om node
i
.
yk
ij
is a con inuous
a iable ep esen ing he load ca ied by cou ie
k
when a i ing a node
j
om
node
i
.
𝜉
is a andom ec o .
Subjec o:
(1)
max ∑
i∈Td
pi−
∑
i,j∈T
cij −𝛼
(∑
i∈Td
𝜇i−E[Q(x,𝜉)]
)
(2)
pi≤wi
∑
j∈N
∑
k∈V
xk
ji ∀i∈T
d
(3)
c
ij ≥
∑
k∈V
mkxk
ij ij ∀i,j∈T,i≠
j
(4)
∑
k∈V(∑
j∈T�{i}
xk
ji
)
≤1∀i∈T
(5)
∑
k∈
V
(∑
j∈T�{i}
xk
ij
)
=
∑
k
∈
V
(∑
j∈T�{i}
xk
ji
)
∀i∈T
1003
Meal Deli e y Rou ing P oblem wi hS ochas ic Meal P epa a ion…
The objec i e unc ion Eq.1 is designed o maximize expec ed p o i while accom-
moda ing he s ochas ic na u e o demand and se ice imes. I explici ly includes pen-
al ies o missed deli e ies and se ice delays, add essed wi hin he unc ion’s hi d
e m. This e m employs a cos ac o , scaled by
𝛼
, ha inc eases p opo ionally wi h
he de ia ion om scheduled deli e y imes. By inco po a ing his penal y, he unc-
ion e ec i ely quan i ies he inancial and se ice quali y impac o delays, ensu ing
ha ope a ional s a egies seek o op imize p o i abili y and uphold eliabili y and cus-
ome sa is ac ion. Cons ain s Eq.2 limi he maximum paymen expec ed om he
cus ome . Cons ain s Eq.3 ensu e cou ie s ecei e a leas he minimum expec ed
paymen . Cons ain s Eq.4 ensu e each pickup and deli e y pai is isi ed a mos
once. Cons ain s Eqs.5-6 ensu e low conse a ion a each node. Cons ain s Eq.7
ack he a i al ime a each node. Cons ain s Eq.8 ensu e he ea lies a i al ime a
a node is espec ed. Cons ain s Eq.9 ensu e assignmen s o cou ie s a e wi hin hei
shi ime. Cons ain s Eq.10 ensu e ha he a i al ime a he deli e y node is la e
han he pickup node. Cons ain s Eq.11 ensu e demand sa is ac ion a each node.
Cons ain s Eq.12 ensu e cou ie capaci y is no iola ed. Cons ain s Eq.13 ensu e
(6)
∑
j∈T
∑
i∈T
p
xk
ij =
∑
j∈T
∑
i∈T
d
xk
ji ∀k∈V,i≠
j
(7)
∑
j∈N�{i}
∑
k∈V
ak
ji ≤
∑
j∈N�{i}
∑
k∈V
ak
ij −(si+ ij)xk
ij ∀i∈T
(8)
e
ix
k
ij
≤a
k
ij
∀k∈V,i∈T,j∈T�{i
}
(9)
e
kx
k
ij
≤a
k
ij
≤lkx
k
ij
∀k∈V,i∈N,j∈N�{i
}
(10)
∑
j∈N�{i}
ak
ji ≤
∑
j∈T�{i+n}
ak
ji+n∀k∈V,i∈
P
(11)
∑
k∈
V
∑
j∈N�{i}
yk
ij −dixk
ij =
∑
k
∈
V
∑
j∈N�{i}
yk
ji ∀i∈T
(12)
yk
ij
≤𝛽kx
k
ij
∀k∈V,i∈N,j∈N�{i
}
(13)
𝜇
i+li≥
∑
k∈V
∑
j∈N
ak
ji +sixk
ij ∀i∈T
d
(14)
xk
ij
∈{0, 1}∀k∈V,i∈N,j∈
N
(15)
yk
ij
≥0, a
k
ij
≥0, pi≥0, cij ≥0∀k∈V,i∈N,j∈N
1004
S.R.Kancha la e al.
ha he ac ual delay expe ienced by he cus ome is a leas as la ge as he se ice ime
plus he a el ime, ensu ing ha no penal y is applied o ea ly o on- ime deli e ies.
Cons ain s Eqs.14 and 15 de ine he decision a iables’ domains, ensu ing ha ou es
a e bina y decisions and all o he a iables a e non-nega i e.
Unlike he adi ional ehicle ou ing p oblems, we allow he d opping o o de s.
Cons ain s Eq.4 make his d opping o o de s possible. The
xk
ij
e ms in cons ain s
Eqs.3, 4, 8 and 11 a oid conside ing he d opped o de in he es ima ion o paymen
om he cus ome s, paymen s made o cou ie s, a i al ime acking, and cou ie
load acking, espec i ely.
3.1 Recou se Ac ion
The meal p epa a ion imes a e ealized when he cou ie a i es a he es au an ,
and u u e o de s om a es au an can be ealized only when he o de is placed. The
longe wai ing imes due o delays in meal p epa a ion can iola e he ime windows.
Suppose he delay is wi hin he gi en bu e . In ha case, a delay penal y will be added
o he objec i e, o when he delay is beyond he allowed bu e , he o de s a e d opped,
and a penal y is applied o he objec i e. Un ealized o de s also impac he a ailabili y
o cou ie s o u u e o de s because o he de ou s aken o mee he ealized o de s.
3.2 Modeling o Meal P epa a ion Times
We model he meal p epa a ion ime a each es au an node as a andom a iable
gi en by
+𝛿i
, whe e
𝛿i≥0
is he du a ion o he s ochas ic dis up ion a node i and
is a de e minis ic meal p epa a ion ime. Speci ically,
𝛿i
ollows a gamma dis ibu ion
wi h a gi en shape pa ame e (k) and scale (
𝜃
) pa ame e ha depends on he de e -
minis ic meal p epa a ion ime. The Gamma dis ibu ion is commonly used in he
li e a u e o desc ibe s ochas ic imes, as hey ollow con olu ion and non-nega i i y
p ope ies. The pa ame e s k and
𝜃
allow o he gene a ion o scena ios conside ing
he deg ee by which p epa a ion imes a y, adjus ed by he coe icien o a ia ion
(
c
). Fo ou analysis,
c2
=
0.25
ga e he bes i o he meal p epa a ion imes.
We de i e pa ame e s k and
𝜃
o a gi en alue
c
as ollows:
3.3 Modeling o Fu u e O de s andThei Loca ions
We assume ha he numbe o o de s wi hin he upcoming in e al ollows a Poisson
dis ibu ion wi h a mean a i al a e
𝜆
, ep esen ed by
O +1∼Poisson(𝜆)
. The Pois-
son dis ibu ion is commonly used in he li e a u e o ep esen andom occu ences.
A e de e mining he numbe o o de s,
O +1
, we iden i y hei p obable loca ions.
We di ide he cus ome base in o cen al and pe iphe al segmen s using he Isola-
ion Fo es s me hod, as desc ibed by Liu e al. (2008). The p opo ion o cus om-
e s wi hin each segmen deno ed as
𝛼cen al
o he cen al segmen and
𝛼pe iphe al
o
(16)

c
=
𝔼(𝛿
i
)
Va (𝛿i)=k𝜃
k𝜃2
i
⇒k=1
c2
;𝜃i= c
2
1011
Meal Deli e y Rou ing P oblem wi hS ochas ic Meal P epa a ion…
s udy: a empe a u e se ing o 50, a empe a u e educ ion ac o o 0.98, a maxi-
mum o 10 imes he numbe o o de s o empe a u e educ ion i e a ions, and 20
imes he numbe o o de s o maximum i e a ions a a empe a u e. Addi ionally,
we se he alue o
Ω�
o 1000.
I ’s well-known ha inc easing he maximum numbe o eplica ions (N) gen-
e ally leads o imp o ed esul s, albei wi h a subs an ial inc ease in compu a-
ional ime. To s ike a balance be ween solu ion quali y and un ime e iciency,
we es ed a ious alues o N, anging om 20 o 100 wi h inc emen s o 20, on
h ee ins ances, each ep esen ing a di e en se ( e e o Table2). Ou expe imen s
e ealed ha inc easing N beyond 60 esul ed in only ma ginal imp o emen s in he
objec i e, accompanied by a signi ican ise in un ime. Consequen ly, we op ed o
N=60 in his s udy, inding i o be he op imal comp omise be ween solu ion quali y
and compu a ional e iciency.
Table 1 Ins ances De ails
Ins ance O de s Res au an s Cou ie s Time Win-
dow
WTP Demand
Mean S d Mean S d Mean S d
0o50 100s1p100mc1 1 126 64 40 57.02 8.94 7.40 3.38 1.99 0.80
0o50 100s1p100mc1 2 126 64 40 57.02 8.94 7.40 3.38 1.99 0.80
0o50 100s1p100mc2 1 126 60 40 56.17 8.60 7.43 3.41 1.91 0.79
0o50 100s1p100mc2 2 126 60 40 56.17 8.60 7.43 3.41 1.91 0.79
0o50 100s2p100mc1 1 126 64 47 57.02 8.94 7.40 3.38 1.99 0.80
0o50 100s2p100mc1 2 126 64 47 57.02 8.94 7.40 3.38 1.99 0.80
0o50 100s2p100mc2 1 126 60 47 56.17 8.60 7.43 3.41 1.91 0.79
0o50 100s2p100mc2 2 126 60 47 56.17 8.60 7.43 3.41 1.91 0.79
1o50 100s1p100mc1 1 134 64 35 56.78 9.33 7.31 3.30 2.09 0.80
1o50 100s1p100mc1 2 134 64 35 56.78 9.33 7.31 3.30 2.09 0.80
1o50 100s1p100mc2 1 134 66 35 56.40 8.83 6.81 3.39 2.16 0.80
1o50 100s1p100mc2 2 134 66 35 56.40 8.83 6.81 3.39 2.16 0.80
1o50 100s2p100mc1 1 134 64 35 56.78 9.33 7.31 3.30 2.09 0.80
1o50 100s2p100mc1 2 134 64 35 56.78 9.33 7.31 3.30 2.09 0.80
1o50 100s2p100mc2 1 134 66 35 56.40 8.83 6.81 3.39 2.16 0.80
1o50 100s2p100mc2 2 134 66 35 56.40 8.83 6.81 3.39 2.16 0.80
2o50 100s1p100mc1 1 177 93 61 58.25 9.49 7.30 3.34 1.90 0.79
2o50 100s1p100mc1 2 177 93 61 58.25 9.49 7.30 3.34 1.90 0.79
2o50 100s1p100mc2 1 177 88 61 58.67 9.54 7.25 3.17 2.04 0.85
2o50 100s1p100mc2 2 177 88 61 58.67 9.54 7.25 3.17 2.04 0.85
2o50 100s2p100mc1 1 177 93 70 58.25 9.49 7.30 3.34 1.90 0.79
2o50 100s2p100mc1 2 177 93 70 58.25 9.49 7.30 3.34 1.90 0.79
2o50 100s2p100mc2 1 177 88 70 58.67 9.54 7.25 3.17 2.04 0.85
2o50 100s2p100mc2 2 177 88 70 58.67 9.54 7.25 3.17 2.04 0.85

1012
S.R.Kancha la e al.
5.2 S ochas ici y inBo h Meal P epa a ion Times andFu u e O de s
This p oblem lacks es ablished benchma k esul s. Consequen ly, we employ ou
S ochas ic App oxima ion Algo i hm (SAA) amewo k o gene a e s ochas ic
and de e minis ic solu ions whe e all andom a iables a e subs i u ed wi h hei
expec ed alues. Subsequen ly, we compa e bo h solu ions’ ou ing cos s ( i s s age)
and ecou se cos s (second s age). Algo i hm2 is applied o compu e he expec ed
ecou se cos , u ilizing a scena io sample size o
|Ω|
= 1000. This expe imen illus-
a es he impac o inco po a ing unce ain y on a s ochas ic solu ion’s i s -s age
decisions and cos s.
In Fig.1, we p esen a compa ison be ween he de e minis ic and s ochas ic solu-
ions o he ins ances de ailed in Table1 using he ecou se R. I is impo an o
no e ha in R, no co ec i e ac ions a e implemen ed; he second-s age cos s solely
accoun o penal ies acc ued due o missed o de s and delays. This expe imen gen-
e a es scena ios wi h a coe icien o a iance
c
= 0.25. The esul s a e a e aged
o e h ee i e a ions o he SAA amewo k.
Table 3 indica es ha employing p obabilis ic in o ma ion, a he han elying
solely on expec ed alues, subs an ially boos s he objec i e alue. This inc ease is
p ima ily a ibu ed o a signi ican ise in he o al numbe o o de s se ed, wi h
an a e age inc ease o mo e han 13% ac oss all da ase s. Howe e , i ’s impo an
o no e ha he compu a ional un ime expe iences a subs an ial su ge, exceeding
90% in all cases. This inc ease in un ime can be p ima ily a ibu ed o he need o
epea ed sol ing o N eplica ions o he SAA.
We an wo scena ios o see which s ochas ic a iable is causing he signi ican
inc ease in he numbe o o de s se ed and, he eby, he objec i e alue. In he
i s case, s ochas ic in o ma ion abou he meal p epa a ion ime alone is consid-
e ed, and u u e o de s a e no conside ed. In he second case, s ochas ic in o ma ion
abou he numbe o u u e o de s alone is conside ed, and meal p epa a ion imes
a e assumed o be known.
5.3 S ochas ici y Only o Meal P epa a ion Times
Like he p e ious scena io, we analyze he ou ing and ecou se cos s in s ochas ic
and de e minis ic con ex s. Figu e2 and Table4 depic he ou comes. In e es ingly,
inco po a ing s ochas ic da a solely o meal p epa a ion imes did no signi ican ly
Table 2 E ec o he alue o N
on objec i e and un ime N Objec i e Run ime (s) %
↑
Obj %
↑
un ime
20 1624.02 3052 − −
40 1642.56 3759 1.14 23.17
60 1773.14 5867 7.95 56.05
80 1777.79 7526 0.26 28.28
100 1780.33 11348 0.14 50.79
1013
Meal Deli e y Rou ing P oblem wi hS ochas ic Meal P epa a ion…
enhance he solu ion compa ed o he de e minis ic app oach. This was despi e a
conside able inc ease in un ime due o epea ed p oblem-sol ing. Fu he mo e,
no conside ing u u e o de s led o a no able d op in he objec i e alue, exceeding
20% compa ed o he scena io whe e bo h meal p epa a ion ime and u u e o de s
Objec i e Recou seCos Rou eCos
Se −0Se −1Se −2 Se −0 Se −1Se −2 Se −0 Se −1 Se −2
−1000
0
1000
2000
Ins ance se
alue
Solu ion
De e minis ic
S ochas ic
Fig. 1 Compa ison o s ochas ic and de e minis ic solu ions
Table 3 Pe cen age change in
Objec i e alue, O de s missed,
and un ime by including
p obabilis ic in o ma ion o bo h
meal p epa a ion ime and u u e
o de s
Ins ance se Pe cen age change
↑
Objec i e alue
↓
O de s missed
↑
Run ime
Se -0 36.79 15.67 90.22
Se -1 27.56 17.16 90.18
Se -2 34.43 13.56 90.24
Objec i e Recou seCos Rou eCos
Se −0Se −1 Se −2 Se −0 Se −1 Se −2 Se −0 Se −1 Se −2
−500
0
500
1000
1500
Ins ance se
alue
Solu ion
De e minis ic
S ochas ic
Fig. 2 Compa ison o s ochas ic and de e minis ic solu ions wi h meal p epa a ion ime as he only s o-
chas ic a iable
1014
S.R.Kancha la e al.
we e conside ed s ochas ically. Howe e , i is impo an o no e ha he e was s ill
an imp o emen o e he pu ely de e minis ic solu ion.
5.4 S ochas ici y Only o Fu u e O de s
He e, we assume ha meal p epa a ion imes a e p ede e mined while he num-
be o u u e o de s ollows a p obabili y dis ibu ion. Figu e3 and Table5 illus-
a e he impac o inco po a ing p obabilis ic in o ma ion ins ead o elying solely
on expec ed alues. No ably, we obse e a subs an ial imp o emen in he objec-
i e alue, exceeding 32%, and an inc ease in he numbe o o de s se ed by mo e
han 13%. As expec ed, his enhancemen comes a he cos o a signi ican un -
ime inc ease o o e 90%. This app oach yields supe io solu ions when conside ing
meal p epa a ion imes and u u e o de s as s ochas ic a iables. The p ima y d i e
o his imp o emen is he p ecise knowledge o meal p epa a ion imes.
5.5 Va ia ion in heLe el o S ochas ici y o Bo h Meal P epa a ion Times
andFu u e O de s
Ou s udy aimed o explo e how a ying deg ees o andomness in he sys em
a ec i s pe o mance, speci ically looking a meal p epa a ion imes and o de
Table 4 Pe cen age change in
Objec i e alue, O de s missed,
and un ime by including
p obabilis ic in o ma ion only
o meal p epa a ion imes
Ins ance se Pe cen age change
↑
Objec i e
↓
O de s missed
↑
Run ime
Se -0 0.73 0.40 66.37
Se -1 3.60 0.93 62.45
Se -2 6.45 0.42 64.99
Objec i e Recou seCos Rou eCos
Se −0Se −1Se −2 Se −0 Se −1Se −2 Se −0 Se −1 Se −2
−1000
0
1000
2000
Ins ance se
alue
Solu ion
De e minis ic
S ochas ic
Fig. 3 Compa ison o s ochas ic and de e minis ic solu ions wi h he numbe o u u e o de s as he only
s ochas ic a iable
1015
Meal Deli e y Rou ing P oblem wi hS ochas ic Meal P epa a ion…
equencies. We employed wo s a is ical dis ibu ions o achie e his: he gamma
dis ibu ion o meal p epa a ion imes and he Poisson dis ibu ion o he numbe
o o de s. We adjus ed a ange o pa ame e s wi hin hese dis ibu ions o simula e
di e en le els o a iabili y.
Fi s ly, we al e ed he gamma dis ibu ion’s shape pa ame e , which con ols he
a iabili y o meal p epa a ion imes. The idea was o mimic eal-wo ld scena ios
whe e some meals migh be p epa ed quickly while o he s ake longe . By inc eas-
ing he shape pa ame e , we in oduced g ea e unp edic abili y in p epa a ion imes,
e lec ing a mo e ealis ic and challenging en i onmen o he sys em.
Secondly, we modi ied he lambda pa ame e in he Poisson dis ibu ion, which
dic a es he a e age equency o o de s. This allowed us o simula e high and low-
o de pe iods, examining how he sys em copes wi h luc ua ing demand.
Th ough hese adjus men s, we sough o unde s and he sys em’s esilience o
unce ain y comp ehensi ely. As illus a ed in Fig.4, he esul s e ealed a di ec
co ela ion: highe a iabili y in bo h meal p epa a ion imes and o de equencies
led o an inc ease in missing o de s. This, in u n, had a ipple e ec , causing a ise
Table 5 Pe cen age change in
Objec i e alue, O de s missed,
and un ime by including
p obabilis ic in o ma ion only
o u u e o de s
Ins ance se Pe cen age change
↑
Objec i e
↓
O de s missed
↑
Run ime
Se -0 46.06 16.67 91.19
Se -1 32.96 13.18 90.38
Se -2 47.11 14.50 90.30
Fig. 4 Compa ison o solu ions wi h a ia ion in he le el o s ochas ici y o bo h meal p epa a ion
imes and u u e o de s
1016
S.R.Kancha la e al.
in bo h ou e and ecou se cos s as he sys em s uggled o adap o he heigh ened
unp edic abili y.
This inding highligh s he challenge o dealing wi h inc eased unce ain y in he
sys em. As he o de s and meal p epa a ion imes became mo e unp edic able, he
sys em had di icul y managing esou ces and planning ou es e ec i ely. Despi e
e o s o adap , he sys em aced mo e missed o de s, d i ing up cos s. These obse -
a ions, as illus a ed in Fig.4, emphasize he delica e balance needed o handle he
complexi ies o heigh ened unce ain y.
5.6 Va ia ion in heLe el o S ochas ici y Only o Meal P epa a ion Times
To s udy he impac o s ochas ici y in meal p epa a ion imes, we main ained a con-
sis en o de dis ibu ion while a ying he le els o unce ain y in meal p epa a-
ion. As depic ed in Fig.5, he e was a subs an ial ise in he pe cen age o missed
o de s, exceeding 40% in ce ain ins ances, especially when he shape pa ame e
was inc eased by 50%. This end was consis en ly obse ed ac oss all h ee se s o
ins ances.
As unce ain y inc eased and he dis ibu ion sp ead wide , meal p epa a ion
imes g ew longe . This ex ended wai ing pe iod a ec ed deli e y d i e s, causing
hem o expe ience delays. Consequen ly, many o de s had o be ejec ed due o he
sho age o a ailable d i e s. This si ua ion leads o delayed deli e ies and esul s in
los sales oppo uni ies and ope a ional ine iciencies. Managing meal p epa a ion
imes e ec i ely is i al o op imizing esou ces, minimizing o de ejec ions, and
enhancing ope a ional e iciency and cus ome sa is ac ion.
Fig. 5 Compa ison o solu ions wi h a ia ion in he le el o s ochas ici y only o meal p epa a ion imes

1017
Meal Deli e y Rou ing P oblem wi hS ochas ic Meal P epa a ion…
5.7 Va ia ion in heLe el o S ochas ici y Only o Fu u e O de s
To s udy he impac o s ochas ici y on he numbe o o de s, we main ained consis -
en meal p epa a ion imes while in oducing di e en le els o unce ain y in u u e
o de numbe s. As illus a ed in Fig.6, Su p isingly, he a ia ion in u u e o de s
did no signi ican ly in luence he numbe o missed o de s.
A close analysis e eals an in e es ing phenomenon: a sligh ly highe in lux o
o de s wi hin sho e in e als migh ha e esul ed in he occasional missing o a
ew o de s du ing peak ush pe iods. Howe e , because he o al numbe o o de s
emained cons an , his b ie ush was ollowed by ela i ely quie e pe iods. Du ing
quie e pe iods, all o de s we e success ully deli e ed because mo e ime was a ail-
able o handle hese o de s, gi en ha hey we e sp ead ou o e a longe ime ame.
6 Conclusions
We in oduced unce ain ies in meal p epa a ion imes and u u e o de loca ions as
s ochas ic a iables, emphasizing he c ucial need o inco po a e hese unce ain ies
in ou e planning. Ou app oach in ol ed employing a Sample A e age App oxima-
ion me hod wi hin a olling ho izon amewo k, u ilizing he Adap i e La ge Neigh-
bo hood Sea ch algo i hm o he i s -s age p oblem, and implemen ing a ecou se
ac ion in he second s age.
We made signi ican obse a ions Th ough ex ensi e expe imen s on ins ances
de i ed om G ubhub MDRP ins ances. The u iliza ion o a iables, a he han
Fig. 6 Compa ison o solu ions wi h a ia ion in he le el o s ochas ici y only o u u e o de s
1018
S.R.Kancha la e al.
expec ed alues, esul ed in no ably p o i able ou es, p ima ily due o accommo-
da ing a la ge numbe o o de s. We explo ed scena ios whe e indi idual unce -
ain ies we e elaxed. No ably, he pe o mance did no signi ican ly imp o e when
unce ain y was conside ed, only in meal p epa a ion imes and dis ega ding u u e
o de s. I was, in ac , 20% wo se due o he absence o u u e o de conside a ions.
Con e sely, when only u u e o de s we e ega ded as unce ain and meal p epa a-
ion was assumed known, pe o mance signi ican ly imp o ed, displaying an a e -
age enhancemen o o e 30%.
Fu he mo e, ou s udy del ed in o he in ica e dynamics o ope a ional unce -
ain ies. We disco e ed ha inc eased s ochas ici y could lead o mo e missed o de s
and escala ed ope a ional cos s. In e es ingly, occasional o de misses we e obse ed
du ing peak demand pe iods, bu hese we e compensa ed o du ing quie e imes.
The e ec i e managemen o meal p epa a ion imes eme ged as a pi o al ac-
o in luencing o de ejec ions, cus ome sa is ac ion, and o e all ope a ional e i-
ciency. These indings unde sco e he necessi y o adap i e s a egies in balancing
hese ade-o s e ec i ely, o e ing aluable insigh s o decision-make s in he
ope a ional managemen domain.
Au ho Con ibu ions The au ho s con i m con ibu ion o he pape as ollows: s udy concep ion and
design: S.K and S.U; analysis and in e p e a ion o esul s: S.K, T.W, S.U, and S.W; d a manusc ip
p epa a ion: S.K. All au ho s e iewed he manusc ip .
Funding Open Access unding enabled and o ganized by P ojek DEAL. This esea ch ecei ed no spe-
ci ic g an om unding agencies in he public, comme cial, o no - o -p o i sec o s.
Da a A ailabili y Publicly a ailable da a sou ces a e used.
Decla a ions
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Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps
and ins i u ional a ilia ions.